Development and Applications of Relativistic Correlation Consistent Basis Sets for Lanthanide Elements and Accurate Ab Initio T

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DEVELOPMENT AND APPLICATIONS OF RELATIVISTIC CORRELATION

CONSISTENT BASIS SETS FOR LANTHANIDE ELEMENTS

AND ACCURATE AB INITIO THERMOCHEMISTRY

AND SPECTROSCOPY

QING LU

A dissertation submitted in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

WASHINGTON STATE UNIVERSITY Department of Chemistry

MAY 2017

To the Faculty of Washington State University:

The members of the Committee appointed to examine the dissertation of QING LU find it satisfactory and recommend that it be accepted

Kirk A. Peterson, Ph.D., Chair

Aurora E. Clark, Ph.D.

Ursula Mazur, Ph.D.

James A. Brozik, Ph.D.

ACKNOWLEDGEMENT

Having been at this step, I feel difficult to hold my emotions and think logically. So for whoever read this part, please tolerate with my ramblings.

Chronologically, the first person I want to acknowledge is Prof. Ming Xian, as he offered me to the WSU grad school. Unfortunately, things didn’t work out well during my first year as I had no idea how to do research at that time. Nonetheless, I still have a good time there and I want to appreciate the help from Dr. Jia Pan and Dr. Nelmi Devarie, who were in Prof. Xian’s group and gave me lots of help. Oh, I should also thank the graduate director and search committee, whose names I forget, in the University of New Hampshire, as they gave me the first offer to cheer me up when I was applying for a US grad school back in China, and allowed me to turn down their offer after I had accepted their offer. Many thanks!

Next I want to express my greatest gratefulness to my advisor, Prof. Kirk Peterson, as he gave me 2 important chances, which largely change my life. The first time is he accepted me to join his group. I still remember that after the one-year’s fail in organic chemistry, I was quite frustrated, and didn’t sleep well. I thought many times to quit the Ph.D program, but that is also a big shame to myself. Then Kirk accepted me to his lab and gave me a second chance to obtain my degree. I am really relieved at that time. The second time Kirk helps me is after my failed proposal defense. I am really grateful that he didn’t give me up, but give me another chance to take the prelim exam. I am sure this is a quite unusual thing in the department, so I can imagine there must be lots of work behind this and Kirk didn’t let me worry about it, so are my committee members, Prof. Aurora Clark, Prof. Ursula Mazur and Prof. Jim Brozik. I am very grateful for your help and patience for this confused student.

Especially, I am very grateful for my committee members to fail me at my first prelim exam.

iii

That fail, although painful, provides me a great opportunity to think about life and universe. And after exchanging emails with Prof. Peterson during the following months, I began to know how to do research. That is my biggest harvest. Now I feel I am a totally different person from 5 years ago. A lot of things, not only in chemistry, can be seen unprecedentedly clearly. Again, I want to thank my committee members, Prof. Peterson, Prof. Brozik, Prof. Mazur and Prof. Clark for their help, care and patience.

Lastly, I want to thank my family and my friends for their support during my dark time, especially my parents. They are always so supporting and patient to me, while I often release my pressure on them by shouting at them. I am so sorry for that and thank you for loving me. I wish

I can be your son again in the next life

I own too much to too many people. I am really sorry for my mistakes and stupidity. May

Buddha and God bless you.

DEVELOPMENT AND APPLICATIONS OF RELATIVISTIC CORRELATION

CONSISTENT BASIS SETS FOR LANTHANIDE ELEMENTS

AND ACCURATE AB INITIO THERMOCHEMISTRY

AND SPECTROSCOPY

Abstract

by Qing Lu, Ph.D. Washington State University May 2017

Chair Kirk A. Peterson

The relativistic correlation consistent basis sets for Lanthanide elements are developed with the 3rd-order Douglas-Kroll-Hess Hamiltonian. Atomic and molecular benchmark calculations show robust reliability of the new basis sets. Applications based on the new sets were performed, which include correcting the dissociation energy of the LuF molecule, resolving the geometry controversy of the LnX3 molecules (Ln= La, Nd, Gd, Dy, Lu; X=F, Cl, Br) and predicting the new molecules of isoelectronic to the Gd2 molecule.

Spectroscopy studies of coinage metal nitroxyl molecules as well as CCX (X=P, As) radicals are also performed. The calculations should provide the most accurate theoretical results up to date.

The first chapter of this dissertation briefly introduces the motivation and background of studied projects. Chapter 2 offers a summarization of the computational methods used in the study. Chapter 3 to chapter 7 present details of the studied projects.

Table of Contents

Page

ACKNOWLEDGEMENT ...... iv

ABSTRACT ...... vi

LIST OF TABLES ...... ix

LIST OF FIGURES ...... xi

CHAPTER

1. INTRODUCTION ...... 1

I Background ...... 1

II Summary of previous work ...... 4

III Motivation and goals of current work ...... 10

2. REVIEW OF COMPUTATIONAL METHODS USED ...... 20

I General consideration ...... 20

II Feller-Peterson-Dixon methodology ...... 21

III Hartree-fock theory ...... 23

IV Correlation methods...... 25

V Relativistic hamiltonians ...... 32

VI Choice of basis sets ...... 38

VII Spectroscopy calculations ...... 40

3. CORRELATION CONSISTENT BASIS SETS FOR LANTHANIDES. THE ATOMS LA – LU ...... 46

I. Introduction ...... 48

II. Basis set development ...... 51

III. Results and discussion ...... 56

IV. Conclusions ...... 69

4. THE THERMODYNAMIC AND SPECTROSCOPIC PROPERTIES OF LNX3 (LN= LA, ND, GD, DY, LU;

X=F, CL, BR) ...... 92

I. Introduction ...... 92

II. Computational details ...... 96

III. Discussion ...... 98

IV. Conclusions ...... 104

5. A COUPLED CLUSTER STUDY OF THE LUF MOLECULE WITH A REEVALUATION OF ITS BOND

DISSOCIATION ENERGY ...... 118

I. Introduction ...... 118

II. Computational details ...... 121

III. Results and discussion ...... 123

IV. Conclusions ...... 126

6. THE HIGH-ACCURACY STUDY OF HIGHEST SPIN DIATOMIC MOLECULE (GD2) AND ITS

ISOELECTRONIC ANALOGS ...... 135

I. Introduction ...... 136

II. Computational details ...... 138

III. Results and discussion ...... 139

IV. Conclusion ...... 146

7. A COUPLED CLUSTER STUDY OF COINAGE METAL NITROSYLS, M-NO0/+ (M = CU, AG, AU) .. 160

I. Introduction ...... 161

II. Computational details ...... 164

III. Results and Discussion ...... 168

IV. Conclusions ...... 178

8. ACCURATE AB INITIO VIBRONIC SPECTROSCOPY OF THE CCP AND CCAS RADICALS ...... 192

I. Introduction ...... 192

II. Computational details ...... 195

vii

III. Discussion ...... 198

IV. Conclusion ...... 206

9. CONCLUSIONS ...... 217

APPENDIX A ...... 219

Supplemental information for A coupled cluster study of coinage metal nitrosyls, M-NO0/+ ...... 219

APPENDIX B ...... 228

Supplemental information for Accurate ab initio vibronic spectroscopy of the CCP and CCAs radicals ...... 228

viii

LIST OF TABLES

Page

Table 3-1 Electronic ground states and configurations for the lanthanides (Ln) through Ln3+...... 72

Table 3-2 CCSD(T) results for the first ionization potentials (kcal/mol)...... 73

Table 3-3 CCSD(T) results for the second ionization potentials (kcal/mol)...... 75

Table 3-4 CCSD(T) results for the third ionization potentials (kcal/mol)...... 77

Table 3-5 DKH3-CCSD(T) results for Gd2 with comparison to experiment and previous calculations...... 79

Table 3-6 CCSD(T) results for t GdF with comparison to experiment and previous calculations...... 82

8 Table 3-7 CCSD(T) results for the atomization energy De (kcal/mol) of X A" GdF3...... 84

Table 4-1 Molecular geometries for LnX3 molecules optimized at CCSD(T)/VTZDK3 level of theory...... 105

Table 4-2 Harmonic frequencies calculated at CCSD(T)/VTZDK3 level of theory for the LnX3 molecules.. . 106

Table 4-3. The calculated harmonic frequencies for GdF3 and LaCl3 molecules...... 107

Table 4-4. Comparison with the recommended values by Kovacs and Konings...... 108

Table 4-5. The natural population analysis of the LaX3 and LuX3 molecules...... 109

Table 4-6. The atomization energies for the LnX3 molecules...... 110

Table 5-1. The contributions of each FPD term for the molecular properties...... 128

Table 5-2. Comparison between experimental values and theoretical values for LuF...... 129

Table 6-1. The T1 diagnostics and electronic energies of valence only and core-valence calculations...... 150

Table 6-2. The thermodynamic and spectroscopic data for Gd2 isoelectronic molecules…………………….151

Table 6-3. The natural population analysis of isoelectronic molecules...... 152

Table 7-1. The Convergence trend of BCCD(T) for CuNO...... 188

Table 7-2. The Convergence trend of BCCD(T) for AgNO...... 189

Table 7-3. The Convergence trend of BCCD(T) for AuNO...... 190

Table 7-4. The Convergence trend of CCSD(T) for CuNO+...... 191

Table 7-5. The Convergence trend of CCSD(T) for AgNO+...... 192

Table 7-6. The Convergence trend of CCSD(T) for AuNO+...... 193

Table 7-7. The calculated frequencies for CuNO and AgNO molecules comparing to experimentals...... 194

Table 7-8. The calculated intensities of each vibrational normal mode for the MNO and MNO+ molecules...195

Table 8-1. The molecular constants of the CCP radical ...... 220

Table 8-2. The molecular constants for the CCAs radical ...... 221

Table 8-3. The molecular constants obtained from A’ PES, A” PES and the averaged PES without SO ...... 222

Table 8-4. The molecular constants obtained from A’ PES and A” PES and the averaged PES with SO ...... 223

Table 8-5. Selected bending vibronic levels for CCP and CCAs ..…………………..…………….……..…...224

LIST OF FIGURES

Page

Figure 3-1. Contribution of angular momentum functions to the MRCI correlation energy for Gd atom...... 85

Figure 3-2. Basis set convergence relative to the CBS limits for the 3rd ionization potentials ...... 86

Figure 3-3. Comparison of the CCSD(T) basis set convergence...... 87

Figure 4-1. The z-matrix used throughout the calculation...... 111

Figure 4-2. The plot of Ln-X bond lengths vs the Ln atomic number for the LnX3 molecules...... 112

Figure 5-1. The DKH3 CCSD(T) equilibrium properties as a function of correlation consistent basis sets .... 131

Figure 6-1. Potential energy scan of EuTb molecule at UCCSD(T)/vdz level of theory...... 153

Figure 6-2. The bond length convergence trends of the isoelectronic molecules...... 154

Figure 6-3. The binding energy convergence trends of the isoelectronic molecules ...... 155

Figure 6-4. The frequency convergence trends of the isoelectronic molecules ...... 156

Figure 6-5. Molecular orbital diagrams of valence shell for Gd2 and LaLu molecules...... 157

Figure 6-6. Molecular orbital diagrams of valence shell for CeYb and EuTb molecules...... 158

Figure 6-7. Molecular orbital diagrams of valence shell for PrTm, NdEr, PmHo and SmDy molecules...... 159

Figure 8-1. Potential energy curves along the bending mode of the CCP and CCAs radical without SOC ..... 225

Figure 8-2. Potential energy curves along the bending mode of the CCP and CCAs radical with SOC ...... 226

Dedication

This dissertation is dedicated to my mother and father.

xii

Chapter 1

Introduction

1 Background

During the past decades, the computational chemistry has evolved into an important and independent sub-discipline of chemistry. Computational studies help the experimentalists predict experimental results, optimize experimental conditions, and rationalize reactions.

Among the many fields in computational chemistry, ab initio quantum chemistry may be the most promising theory in the sense that it requires the least amount of empirical parameters and the calculations are mostly based on rigorously derived mathematical equations.

However, although the theories and mathematics are available for computational chemists, the computation power is far behind the demand even with today’s technology, thus various approximations are necessary to reduce the computational cost. As P.A.M. Dirac states in 19291,

“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.”

The implementation of quantum chemistry can be generally categorized into two sub-classes: density functional theory (DFT) and wave function theory (WFT). The DFT approach fames

1 itself by presenting a very good performance to computational time ratio2. However, in terms of high-accuracy (1 kcal/mol) demands, the DFT approach quickly becomes less competent than a

WFT method. And in fact WFT methods are often used as a calibration for various DFT functionals3,4.

The WFT method can present a clear hierarchy in terms of the methods and their accuracy.

Although in DFT fields the so called “Jacob’s Ladder”5 can be considered as a hierarchy, it cannot theoretically reach the full configuration interaction (CI) limit, due to the lack of knowledge of the exact exchange-correlation functionals6. On the other hand, various approximations have been proposed for the exchange-correlation functionals, and it becomes an art to some degree to choose the best functional for the studied systems. Lastly, the development of some functionals more or less includes certain empirical parameters, such as perhaps the most popular functional B3LYP7,8. Therefore it becomes kind of a circular argument in the most restricted philosophic sense. That is, the empirical parameters are obtained by fitting a group of experimental data, but in turn they are used to reproduce the experimental data.

It should be noted that the WFT methods also make use of experimental data, but it is in a more subtle fashion9. As a result, from a pure ab initio point, the WFT method should be preferred. With a careful selection of the Hamiltonians, methods, basis sets, schemes to calculate composite energies etc. high accuracy is achievable. And this makes the computation valuable in narrowing down experimental errors, rationalizing chemical processes and predicting new molecules.

Currently, the widely-used electronic Hamiltonians include: (1) the non-relativistic

Hamiltonian; (2) the Zeroth-Order-Regular-Approximation (ZORA) Hamiltonian10,11; (3) the

Douglas-Kroll-Hess (DKH) Hamiltonian12-14; (4) the eXact 2-component (X2c) Hamiltonian15,16;

(5) the Dirac-Coulomb (DC) Hamiltonian and its associated Hamiltonians17 et al. Except for the non-relativistic Hamiltonian, all the other Hamiltonians are relativistic Hamiltonians derived from different approximations. Among them, the DKH and X2c Hamiltonians should be emphasized as the former has no negative-energy artifacts and the accuracy can be systematically improved18, while the latter could exactly reproduce the positive-energy spectrum of the parent four-component Hamiltonians15. On the other hand, the ZORA Hamiltonian is electric gauge dependent, that is, the ZORA energy depends on where the zero of the potential is chosen19.

While for the DC Hamiltonians, they are expensive to use and may suffer from a variational collapse problem20.

In terms of the correlation methods, the widely-used methods include: (1) coupled cluster methods21; (2) many-body perturbation methods22; (3) configuration interaction (CI) methods23;

(4) the multi-reference configuration interaction (MRCI) method24,25; and (5) multi-reference perturbation methods26 etc. Except for the last two methods, the others are single-reference methods. Especially, the coupled cluster singles, doubles, and perturbative triples method,

CCSD(T)27, is referred to as the gold standard28 as it provides the best balance between accuracy and computational cost. While for the CI methods and perturbation methods, they suffer a size-consistency problem and possible energy-overestimation problems, respectively. Between the two types of multi-reference methods, the MRCI method is more expensive while the

MR-perturbation method is not variational, suffering the same arguments as their single-reference counterparts.

For basis sets, currently popular basis sets include: (1) correlation consistent basis sets29; (2)

Pople style basis sets30; (3) Ahlrichs type basis sets31,32; and (4) atomic natural orbital basis sets33,34 etc. Among these basis sets, only the correlation consistent basis sets are in practical use

3 for basis set extrapolation, which can eliminate the basis set incompleteness error and facilitate analysis of computed results. While for other sets, although some of them may be smaller in size, yet the computation accuracy is at the expense.

Lastly for the schemes to calculate composite energies, available methodologies include: (1)

Feller-Peterson-Dixon (FPD) methodology35; (2) HEAT scheme36; and (3) Wn schemes37, etc.

The major advantage of the FPD methodology is its flexibility that each correction term can be calculated without restriction while other schemes must follow a fixed prescription.

The detailed discussion of the Hamiltonians, correlation methods, basis sets and composite scheme used in this work can be found in Chapter 2.

2 Summary of previous work

2.1 Correlation consistent basis sets for Lanthanide elements

It is well recognized that one of the main errors of the electronic structure calculations is the incompleteness of the basis sets. From the viewpoint of the functional analysis, the basis sets are a set of functions that span the Hilbert space, which describes the distribution of the electrons. A complete set of functions can exactly span the function space, but that is impossible to practice.

Therefore, it has to utilize a finite basis set to perform the actual calculations, and the way to develop the finite basis sets is important.

The correlation consistent basis sets are well appreciated as they group the correlation functions systematically with respect to the correlation energies29. As a result, one can develop different size of basis sets in a systematic manner and extrapolate to the complete basis set limit.

Such an extrapolation is of great significance as it can completely remove the basis set

4 incompleteness error and help analyze the electronic structure of the molecules.

By the year of 2015, the available basis sets and their development for lanthanide elements are reviewed by Peterson38. If the discussion is limited to all-electron basis sets designed for correlated calculations, the choices are fairly limited. Pantazis and Neese39 used both 2nd-order

DKH and ZORA Hamiltonians to develop the segmented-contracted basis sets. Such basis sets are designed for DFT calculations and only available with one size. Of similar size, Dolg40 and

Hirao41 used 2nd-order DKH and 3rd-order DKH Hamiltonian, respectively to develop the all-electron lanthanide basis sets, which are between the double- and triple-zeta quality.

In addition to these one-size basis sets, there are also three sets systematically ranging in size from double- through at least quadruple-zeta. The basis sets of Gomes et al.42 were developed with the 4-component Dirac-Coulomb Hamiltonian. These sets are most often used in their completely uncontracted forms. The Sapporo DK-nZP sets (denoted SAP-nZP in this work) of

Sekiya et al.43 were optimized with the 3rd-order DKH scalar relativistic Hamiltonian and range from DZ to QZ. These sets were developed for correlation down through the 4s electrons of the lanthanide. The third type basis sets are the atomic natural orbital (ANO) basis sets of Roos et al.44 This type of basis sets were developed with the DKH2 Hamiltonian and range in size from

DZ to beyond QZ ("large") by adding additional ANO contractions from a common primitive set.

Yet unfortunately, the correlation consistent basis sets are not currently available for the lanthanide elements. Therefore it is important to develop their correlation consistent basis sets to fill the blank of the lanthanide chemistry. A detailed discussion of the basis set development is in

Chapter 3.

Lanthanide chemistry, comparing to its counterpart of actinide chemistry, is much less focused. However, it has many unique properties and controversies, which need to be explored

5 and resolved.

2.2 Controversy of lanthanide trihalide (LnX3)

The first example involves the lanthanide trihalide (LnX3) molecules. The gas-phase lanthanide trihalides, LnX3, have found themselves in many industrial applications such as high temperature extraction and separation45, nuclear waste purification46 and high pressure metal halide discharge lamp47. Yet almost starting from the very beginning of the studies, people have been obsessed by one simple and basic properties of the molecules, their molecular geometry. their molecular geometry. These molecules can be characterized by two variables, the bond length and the bond angle that dictates how pyramidal the molecule is. Although the molecule looks simple, it has not yet reached a consensus with respect to their bond length and bond angle.

For the bond length, some studies48-51 claim the Ln-X bond length decreases linearly with respect to the atomic number of the Ln atom, while others52,53 claim the change in bond length should follow a quadratic pattern. For the bond angle, some reports54 claim the molecules are planar

55 53,56-65 while others claim they are pyramidal. For a selection of the LnX3 molecules (LaX3 ,

56-58,63,64,66 53,56-58,63,64,67,68 56,57,63,64,69-72 44,53,56-58,60,62-64,73-76 NdX3 , GdX3 , DyX3 , LuX3 ), there have been many publications but they cannot reach a conclusive agreement. The controversy has to be resolved with a high-level correlation method and a high-quality basis set, since the improper basis sets will only give invalid conclusions, and the incorrect conclusions will lead to a wrong estimation of the entropy. In Chapter 4, a full discussion and conclusion is given regarding to this topic.

2.3 Controversy of Lutetium monofluoride

The second example of the lanthanide chemistry controversy involves the LuF molecule, whose dissociation energy (De) has a huge disagreement between the experimental and

77,78 theoretical values. Based on studies of pure rotational spectroscopy , the De value was estimated by the usual Morse potential approximation, and range from 97-105 kcal/mol. By using Knudsen cell mass spectrometry79, the dissociation energy was estimated to be 136±12 kcal/mol based on periodic trends and not from direct measurements.

On the other hand, various theoretical studies have been published. The DFT studies with different functionals or Hamiltonians80-82 give the values falling in the range from 143 kcal/mol to 286 kcal/mol. The more sophisticated methods such as coupled cluster method, CCSD(T)82, determines De as 175.9 kcal/mol. Another wave function theory calculation, the equation of

83 motion (EOM) CCSD(T) , shows the De is 171.3 kcal/mol. However, with the same method and basis sets, an all-electron correlation calculation shows the De is 139.6 kcal/mol, which is quite rare and extraordinary for core electrons to have such a large contribution to the dissociation energy. For multi-reference calculations, the multi-reference configuration interaction (MRCI)

84 calculation didn’t provide the De value while the multi-reference averaged coupled pair functional (MRACPF)85 calculation determines it to be 180.6 kcal/mol.

In Chapter 5, the LuF molecule is examined. By applying proper methods and basis sets, the dissociation energy is calculated and is believed to be the correct value.

2.4 Calculation of Gd2 and prediction of its isoelectronic analogs

For the last studied example of lanthanide chemistry, the Gd2 molecule was studied, which perhaps has the most unique electronic structure as it is the experimentally confirmed86,87 highest

7 spin diatomic molecule, which has 18 unpaired electrons. With such an extraordinary spin

88 property, Gd2 found itself in many different applications such as molecular imaging , CO or N2 activation89,90, non-linear optical response91,92, and comparable studies with Gd clusters, surface and crystal93,94.

Despite its industrial and academic interests, the study of the model structure of Gd2, is very limited. The first thermodynamic95 and spectroscopic96 studies were performed more than 30 years ago with relatively simple measurements or estimates. The study of the molecule became dormant until 1992, when Dolg97 successfully predicted the molecule had a super-configuration

7 7 2 1 1 2 86 (4f 4f ζg ζu ζ g π u ) with a spin multiplicity of 19. Two years later, an ESR experiment confirmed the assignment and suggested that the Gd2 molecule should be the highest spin diatomic molecule as the f orbitals do not participate in bonding. One drawback of Dolg’s study, however, is that the 4f orbitals were included in the pseudo-potential core but not in the valence

98 87 space due to technical reasons. In 2000 and 2003 , Dolg revisited the Gd2 molecule with explicit treatment of the 4f orbitals and conducted the first coupled cluster calculation of the molecule. Other theoretical studies include a GASSCF study99, several DFT studies93,98,100,89,90, while other experimental studies include Raman spectra101, Stern-Gerlach experiment94, and

FTIR89,90. Reviewing these experimental and theoretical studies, some uncertainty in the results still remains. For example, the binding energy of the Gd2 molecule was estimated by fitting spectroscopic constants of a Raman spectrum. The resulting value has a large error range of

2.1±0.7 eV101. The best-matching calculation of the molecule, on the other hand, comes from the

GASSCF calculation99 where the dynamic correlation was not fully recovered. Therefore, to better understand the molecule, a thorough and systematic improvement has to be adopted.

Moreover, it can be found that the Gd element sits in the center of the lanthanide row. If an

8 early lanthanide element is paired with a late lanthanide element to form an isoelectronic molecule to Gd2, then what properties do they have? Will they be stable? Do they also have high spin-multiplicity? The answers to these questions will be addressed in Chapter 6.

2.5 Potential energy surfaces and coinage metal nitroxyl molecules

Not unlike in lanthanide chemistry, the transition metal chemistry and main group chemistry also have questions that need to be answered.

For the transition metal chemistry, there are arguments about the ground state of the CuNO molecule102,103. Schwarz104,105, Uzunova103 and Marquardt106,107 used either coupled cluster methods, multi-reference methods or DFT methods103 to calculate the molecule but didn’t reach an agreement whether the molecule has a 1A’ ground state of a 3A” ground state.

For its analogous molecule of AgNO, there is no high-accuracy theoretical calculation. For the AuNO molecule, previous calculations show the scalar relativistic effect could cause a difference in the characteristic frequencies by hundreds of wave numbers, which is quite extraordinary. The details of these molecules will be discussed in Chapter 7. Moreover, it will also discuss the difference between the conventional coupled cluster method and the Brueckner coupled cluster (BCC) method in Chapter 7 as new evidences show that the BCC method has some advantages over the conventional coupled cluster method even in such “simple” closed-shell molecules.

2.6 The Renner-Teller effect and CCX (X=P, As) radicals

In Chapter 8, the 2 radicals of CCP and CCAs will be discussed and studied. The existence of the CCP radical was proposed more than 20 years ago, along with the search of other

9 phosphorus-containing molecules in the interstellar media108. It was assumed that the radical might exist in the molecular hot core of star-forming regions provided the absence of oxygen atoms109. In 2008, the radical was first detected in the circumstellar gas of IRC +10216, as the fifth phosphorus-containing molecule found in interstellar space110. The CCP radical is of another academic interest in the sense that the radical exhibits the Renner-Teller (RT) effect. From previous studies, the CCP radical lies in the ground 2 state. The vibrational angular momentum thus is able to couple with the orbital angular momentum, leading to a breakdown of the

Born-Oppenheim approximation.

A natural extension to the CCP radical is its isovalent counterpart, CCAs, which is a potentially interesting molecule in the formation of durable films111. Similar to the CCP radical, the linear configuration with a single electron in the degenerate π orbitals makes the orbital angular momentum possible to couple with vibrational angular momentum, which leads to strong changes in the resulting bending vibrational levels. However, previous theoretical studies112-115 mainly focus on the bonding and electronic transition. A B3LYP study111 shows the C-C stretching frequency as 1749 cm-1, while the wave function based methods, QCISD(T)112 and

MRCI113 give the frequency as 1712 and 1685 cm-1, respectively. As with the CCP studies, no anharmonic correction has been calculated yet, and no SOC calculation has been performed for the frequency calculations.

3 Motivation and goals of current work

Having summarized the problems, it hereby reiterates the motivation and goals of the current work. The correlation consistent basis sets of lanthanide elements need to be developed. The application studies using the new basis sets were carried out and the chemistry involving

10 transition metal and main group elements were studied:

1) In order to achieve high-accuracy results, correlation consistent basis sets will be

indispensable, since only this kind of basis sets is able to do the energy extrapolation and

thus remove the basis set incompleteness error. However, there is lack of correlation

consistent basis sets for the lanthanide elements. Thus the development of the new basis sets

is necessary.

2) For the LuF molecule, which is important in high-pressure plasma and astrophysics, the

theoretical results have a good agreement with experiments in terms of most molecular

properties. But for the dissociation energy, there is a large discrepancy between and within

the experimental and theoretical values. Therefore, it is necessary to revisit the molecule.

This high accuracy theoretical study would reveal the reason behind this discrepancy.

3) For the geometry of the lanthanide trihalides, there is a long-standing controversy about their

geometry between a planar structure versus a pyramidal structure. Early studies reveal that

there is a very shallow potential energy curve along the bending coordinate and the

correlation energy is vital to study the potential energy well. Therefore, it is aimed to recover

the correlation energy systematically to well characterize the potential energy well and

determine the molecular geometry conclusively.

4) With the ability to achieve high accuracy results, it is thus paramount to do the theoretical

prediction. For the highest spin diatomic molecule (Gd2), it has tremendous impact on

medicinal chemistry. The current work aims to well characterize the molecule and predict its

isoelectronic molecules for the first time.

5) For the transition metal chemistry, the coinage metal nitroxyl molecules and their cations are

studied. The spectroscopic constants are calculated through the potential energy surface and

compared to the available experimental values. The results are aimed to solve controversies

and predict the spectroscopic constants and thermodynamic constants where experiments are

not available.

6) For the chemistry involving main group elements, the CCX (X=P, As) radicals are examined.

The Renner-Teller effect as well as its interaction with spin-orbit coupling effect are studied

for the first time.

As introduced above, the work in this dissertation corrects the literature value, quantifies previous explanations, provides tools for future theoretical studies, and predicts new molecules based on the current knowledge.

In the following chapters, a review of the computational methods used in this work will be first summarized in chapter 2. The work in chapter 3 has been published (c.f. J. Chem. Phys. 145,

054111) while the work in other chapters are in preparation to submit. The detailed introduction, study, and conclusion for each project mentioned above will be elaborated.

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Chapter 2

Review of computational methods used

1 General consideration

Quantum computational chemistry is built on the work of the Schrödinger equation, which is a second-order partial derivative equation. When written in the matrix form, the Schrödinger equation can also be considered as an eigenvalue equation, where the eigenvalues correspond to the energies of the system. Under the Born-Oppenheim approximation, the nuclear motion is separated from the electron motion, and thus the electronic Schrödinger equation forms an n-body problem, where n is the number of electrons.

However, due to the limitation of the mathematics in solving any many-body problems, an analytical solution is not possible for the Schrödinger equation, except for the Hydrogen-like atoms. As a result, approximations have to be made to solve the Schrödinger equation.

Firstly, in term of the wave function, the true wave function ψ, is expanded by an N-electron basis, which are also known as Slater determinants. Thus the true wave function is expressed as a sum of many Slater determinants. But the method to generate these many Slater determinants or the N-electron basis is not unique, and different methods achieve different levels of accuracy and have different computational cost. Thus it is important to be well aware of the choice of the methods, and the methods used in this work are discussed in Section 3 and 4 in this chapter.

Secondly, each Slater determinant is composed of a 1-electron basis, or the orbitals. For molecular orbitals, they are approximated as a linear combination of atomic orbitals and for

20 atomic orbitals, they are expanded by a set of basis functions, known as the basis sets. The basis functions describe the radial part of the Schrödinger equation and they are typically in the form of Gaussian functions. The quality of the basis sets depend on its size, contraction scheme, as well as the exponent and coefficient optimizations. As important as the method, the choice of basis sets also determines the accuracy of the computed results and they will be discussed in

Section 6 in this chapter.

Thirdly, the Schrödinger equation was initially introduced at the non-relativistic limit. But when the heavy elements are involved, the relativistic effects cannot be ignored and the

Hamiltonian must be modified to include them. During the past years, various approximations have been proposed, resulting in different relativistic Hamiltonians. The derivation of the relativistic Hamiltonian used in this work is briefly summarized in Section 5 of this chapter.

Lastly, it is computationally impossible to choose the best method, best basis set and best

Hamiltonian simultaneously as a one-step calculation. Therefore, a sophiscated methodology is needed to divide the one-step calculation into several less-challenging steps and then conquer these steps individually to achieve the desired composite results. Such a methodology will be introduced first in Section 2 of this chapter.

2 Feller-Peterson-Dixon methodology

In past years, the Feller-Peterson-Dixon (FPD) methodology1 has been proved to be a robust methodology to calculate the composite energy of molecules containing heavy elements2-4.

Therefore, throughout this whole study, the FPD methodology is used to recover various contributions to the molecular properties.

In essence, the FPD methodology breaks down the total energy into individual component energies and assumes an additive relationship between the components:

E =E +DE +DE +DE +DE +DE +DE (1) total VQZ CBS CV SR SO Gaunt Misc. where each term on the right hand side of Equation (1) stands for valence electronic energy, core-valence correlation contribution, scalar relativistic effects, spin-orbit coupling effect, Gaunt interaction5 and miscellaneous contributions, respectively.

To be specific, the first term on the right hand side is the molecular energy calculated with the VQZ size of basis sets, while the second term is the valence correlation energy extrapolated to the complete basis set (CBS) limit at the coupled cluster singles, doubles and perturbative triples, CCSD(T), level of theory or the multi-reference configuration interaction (MRCI) level of theory. The energies calculated with different size of correlation consistent basis sets are extrapolated to the complete basis sets (CBS) limit. The Hartree-Fock energies were extrapolated by the Karton-Martin formula6:

−6.57√푛 퐸퐻퐹(푛) = 퐸퐻퐹/퐶퐵푆 + 퐴(푛 + 1)푒 (2) and the correlation energies were extrapolated by a two-parameter formula, as Equation (3):

1 4 En =ECBS+A/(n+ ) (3) 2

푐표푟푟 where the energies calculated with TZ (n=3) or QZ (n=4) basis sets and EHF/CBS and 퐸퐹퐶/퐶퐵푆 are the extrapolated CBS limit HF energy and the correlation energy, respectively. The second term in Equation (1) is similar to ECBS but recovers the correlation energy from outer core electrons.

The outer core correlation calculation recovers the correlation energy from outer-core outer-core electron interaction and outer-core valence electron correlation. This calculation is important, as it has been well demonstrated that for high-accuracy calculations, the outer-core electron correlation is necessary.

The third, fourth and fifth terms originate from relativistic effects, which will be elaborated

22 in later sections.

The beauty of the FPD methodology is that it provides a flexible and accurate methodology that can systematically approach the total energy with a controllable computational cost. In addition, it can exhibit the importance of each contributing term, which helps to elucidate the chemical phenomena.

3 Hartree-Fock Theory

The starting point of all calculations is the Hartree-Fock (HF) theory. In 1926, Max Born interpreted the wave function as a probability wave. Therefore it may help to borrow some terms from probability theory to review the HF theory as well as more sophisticated theories.

The HF theory starts from the Hartree product, where the mean-field approximation is used.

The mean-field approximation assumes the electrons in atoms or molecules are independent from each other and the interactions with other electrons are introduced by an averaged Columbic potential. Thus, each electron i in the spin orbital φ(i) can be considered as an independent event and φ(i) is the probability to find electron i in φ(i). From probability theory, the total probability is the product of probability for each independent event. Thus, the total wave function is simply a product of each spin orbital, giving the Hartree product:

Hartree 1  2  3  4  5..... (4)

However, the Pauli principle must be satisfied since electrons are fermions, and the Hartree product is not an antisymmetric wave function. To overcome this problem, the linear combination of Hartree products has to be made, leading to the Slater determinant.

Philosophically, electrons are indistinguishable particles. But when labeling the electrons, it is actually distinguishing electrons. The electron i should not be restricted in orbital φ1. Its

23 occurrence probability in φn≠1 should also be counted. Therefore it needs a linear combination to account for the exchanged Hartree product and the negative sign in front of even-term Hartree products is a signature of the Fermion particles. The resulting sum of the Hatree products can then be rewritten as a Slater determinant.

To derive the energy of the Slater determinant, the HF theory is applied. In the HF theory using the mean-field approximation, the Fock operator can be constructed as:

1 M Z f()() i  2 A  V i (5) 2 i r HF A1 iA where the VHF potential is the averaged potential out of the mean-field approximation and it is a sum of coulomb (J) and exchange integrals (K) of electron i with every other electron.

N 1 ˆ ˆ VHF(i)=- å(Jj-Kj) 2 j 1 Jˆ = j2(r) j2(r)drdr (6) j òò i 1 r j 2 1 2 12 1 Kˆ = j (r)j (r) j (r)j (r)drdr j òò i 1 j 2 i 2 j 1 1 2 r12

The objective now is to determine a set of molecular orbitals, φ, that makes the energy a minimum, with the constraint that the molecular orbitals are orthonormal. To meet this condition, the Lagrange multiplier is used and the Lagrange function is required to be stationary:

Nelec d L = å (< dji | Fi | ji > + ) i (7) Nelec - å lij (< dji | j j > + < ji |dj j >) = 0 ij

where lij is the Lagrange multiplier, interpreted as the orbital energy. The variation of the molecular orbitals is to optimize the coefficients in front of the atomic orbitals as in the linear combination of atomic of orbital approximation:

(8) ji =åca.ifa.i a

Rearranging Equation (7) by making use of complex conjugate properties, the final set of HF equations may be written as:

Fiji =eji (9) where ε is the orbital energy.

The Fock equation is a non-linear dependence equation. This requires iterative solution until a convergence threshold is met. The Fock equation is solved variationally and the coefficients of molecular orbitals in Equation (8) are optimized to minimize the energy. Since the variation method is used, the HF equation provides the best single-determinant result.

4 Correlation methods

The HF solution typically can recover 99% of the energy of a molecular system7,8. However, the remaining 1% energy is crucial since its absolute value is of the same order of magnitude with energy change of a chemical reaction. Therefore a more accurate method is needed to recover the remaining energy.

To go beyond the HF theory, a modification of the mean-field approximation has to be made.

In HF theory, the mean-field approximation assumes electrons are independent, but in reality, electrons are not independent of each other. Using the statistical language, the electrons are not independent events, but are correlated events. The HF energy based on this approximation deviates from the exact energy. The remaining energy, as the difference between HF energy and exact energy, is therefore called the correlation energy.

To recover the correlation energy, more determinants have to be included9. Depending on how the extra determinants are generated, the correlation methods can be divided into two

25 general classes: the static correlation methods and the dynamic correlation methods.

For the static correlation methods, such as Complete Active Space Self Consistent Field

(CASSCF) method, the extra determinants are not generated from the HF determinant; while for the dynamic correlation methods, extra determinants are derived from the HF determinants by exciting electrons from occupied orbitals to virtual orbitals.

4.1 Configuration Interaction and Coupled Cluster Theories

The Configuration Interaction (CI) method10 is one of the simplest correlation methods. To some degree, it is comparable to a Taylor expansion, that extra determinants are generated from the reference HF determinant with single excitations, double excitations, etc.

ˆ aˆ aˆ b CI c0  HF  c 1 i  HF  c 2  i  j  HF (10)

The Equation (10) shows the expression of the CI wave function with single and double

a ˆ b excitations (CISD), where ˆi and  j are excitation operators that excite electron from occupied orbitals i,j to virtual orbitals a,b, and ci’s are the intermediate normalization coefficients.

The CISD method recovers the correlation energy and improves the results upon the HF calculation. However, the CISD method lacks the property of the size consistency so that the correlation energy it recovers does not scale properly with the size of the system.

To remedy this size consistency problem, one solution is to use the Davidson correction11-14.

The Davidson correction estimates the correlation energy from even higher excitations based on the perturbation theory:

2 EQ (1  a0 )( E CISD  E HF ) (11)

26 where a0 is the coefficient of the HF wave function. However, the Davidson correction only alleviates the size consistency problem, not completely fixing the problem. Thus the coupled cluster (CC) method is mainly used in this work.

The coupled cluster method15,16 is similar to CI but is a more sophisticated method. It generates the excited determinants by an exponential operator:

ˆ 1 1 Y = eT Y = (1+ Tˆ + Tˆ 2 + Tˆ 3 +...)Y (12) CC HF 2 3! HF

ˆ ˆ ˆ ˆ where TTTT1  2  3  ...

ˆ ˆ In practice, the excitation operator T is truncated toT2 , giving a CCSD wave function:

ˆ ˆ TT12 CCSD e  HF (13)

With the CCSD wave function, a perturbative treatment of the triple excitation operator is further applied to obtain the CCSD(T) energy, where (T) denotes the perturbative treatment of triple excitation amplitudes.

The CCSD(T) method is mainly used in this work since it balances the computational efficiency and the accuracy, and this method is often called the golden standard in the ab initio calculations.

An important technical detail needed to be mentioned here is that the CCSD(T) is a single reference method in the sense that all the excited determinants are generated based on a single

HF determinants. Therefore, the accuracy of the CCSD(T) method depends on how dominant the

HF wave function is in the total wave function. The less dominant the HF wave function is, the less accurate the CCSD(T) is, and the system has more multi-reference character. The indicator

17 for measuring this multi-reference character is termed as the T1 diagnostics . The lower the T1 is, the better the single reference is.

4.2 Brueckner Coupled Cluster theory

A variation of the coupled cluster theory, the Brueckner coupled cluster (BCC)18-21 theory is another method used in this work to recover correlation energy. In the standard coupled cluster theory, the singles operator are formulated in terms of the exponential operator:

ˆ a † exp(T1) = exp(åti aaai ) (14) ai where the indices a and i stand for the occupied and virtual orbitals, respectively. The a† and a operators are the creation and annihilation operator, respectively. In BCC theory, the singles operator is obtained by employing the orthogonal orbital-rotation operator

ˆ † † exp(-k ) = exp[-åk ai (aAai - ai aA )] (15) ai which performs the rotations between the occupied and virtual orbitals. It can be seen that

Equation (14) and (15) generate the same electronic state to the first order providing the same single-determinant reference state is used:

ˆ a † 2 exp(T1) | Y0 >= [1+ åti aaai + o(t )]| Y0 > (16) ai

ˆ † † 2 exp(-k ) | Y0 >= [1- åk ai (aaai - ai aa ) + o(k )]| Y0 > ai (17) † 2 = [1- åk aiaaai + o(k )]| Y0 > ai

Therefore the conventional coupled cluster wave function is transformed as:

ˆ ˆ | YBCC >= exp(-kˆ)exp(T2 +T3 +...) | Y0 > (18)

The desired BCC energy is obtained as the conventional coupled cluster method by projecting Equation (18) against the reference state. The cluster amplitudes are determined by

ˆ projection against the excited states | Yexc > spanned by higher excitation operators Texc :

ˆ ˆ ˆ = 0 (19)

The orbital-rotation parameters are determined by extending the projection to the single excitations with the restriction of the orbital condition:

ˆ ˆ ˆ < Yexc | exp(-Texc )H exp(Texc )| Y0 >= 0 (20)

In the end, the orbitals in the canonical HF determinant are transformed by a unitary matrix

ˆ so that the T1 amplitudes are controlled to be 0 (T1  0 ). As a result, the Brueckner orbitals are relaxed. Another consequence of this transformation is that the reconstructed Brueckner determinant has the maximum overlap with the exact wave function.

=max (21) B exact

As Sherrill22 summarized in 1992, Hampel, Peterson and Werner15 made a breakthrough in the efficiency of projective Brueckner coupled cluster with doubles computation by solving for orbitals and doubles amplitudes simultaneously, rather than performing a complete CCD calculation each orbital iteration. By avoiding full integral transformations every iteration, the implemented BCCD method is actually slightly less expensive than the conventional CCSD method.

The BCC method has an outstanding record in dealing with the symmetry-breaking problem22-24. In this work, the calculations of coinage metal-NO complexes show that the BCC method also has an excellent performance for weak multi-reference systems.

4.3 CASSCF

The Complete Active Space Self Consistent Field (CASSCF) method12,25,26 is a different

29 type of correlation method, which recovers static correlation energy. Instead of generating one single determinant as the reference function, CASSCF generates multiple determinants or configuration state functions (CSFs) serving as the reference functions.

To generate the extra reference determinants, CASSCF first partitions the MOs into active and inactive spaces based on the chemical intuition and calculation expense. The active electrons, usually the valence electrons, then exhaust every possible occupation in the active space. The resulting CSFs (f ) span the full-ci space within the active space. Note that the CASSCF wave function implicitly depends on the molecular orbitals (denoted as c). The whole set of occupations corresponds to a set of CSFs, which are used as the reference functions12:

(22) y 0 (B,c) = åBI fI I

In CASSCF, the CI parameters (B) and the orbital coefficients (c) are determined variationally.

¶E(B,c) ¶E(B,c) =0 , = 0 (23) ¶B ¶c I i

Therefore, each extra reference determinant is generated individually and is responsible to recover a part of the static correlation energy. Because of this importance, the selection of the active space is vital in calculations as large active space usually cannot be performed due to the computational cost while small active space may not correctly describe the problem or generate biased solutions.

In this work, the CASSCF method is only used during the development of lanthanide basis sets. For each lanthanide atom, the active space typically consists of 6s6p5d4f orbitals, which consists of 16 orbtials in the active space.

4.4 MRCI and CASPT2

Static correlation and dynamic correlation can be calculated simultaneously and this results in the Multi-Reference-Configuration-Interaction (MRCI) method27. For MRCI calculations, the

CASSCF wave function is chosen as the reference function. Excitation operators then operate upon the reference function and generate excited determinants for the dynamic correlation.

As mentioned above, a full excitation operator is not affordable. In practice, the operator is truncated to the singles and doubles, giving the single and double excitation MRCI:

a a a ab ab | Y >= c0 |y 0 > +åci |y i > + å cij |y ij > (24) i i< j,a

a,b Nref ab ab ref ab ab åcij |y ij > = å å(cm cij ) |y (m)ij > (25) i, j i< j,a

Equation (25) is also defined as uncontracted MRCI since both of the reference coefficients and CI expansion coefficients are going to be optimized during the iteration process, which is very computationally expensive.

An approximated alternative is to fix the reference coefficients, cref, by the CASSCF calculations so the number of variational parameters is largely reduced. Such a method is termed as the internally contracted MRCI (ic-MRCI) and is variationally bounded from below by the uncontracted MRCI energies.

Another approximated alternative, termed as the externally contracted MRCI, is to fix CI expansion coefficients for configurations with the same internal parts. For example, the double

31 excitations are:

ab ab |m(ij);ci>=åc(m)ij |y (m)ij > (26) a>b with the contraction coefficients determined from first-order perturbation theory:

ab ab c(m)ij = ab ab (27) E0-

In this study, the ic-MRCI scheme is adopted as it has been demonstrated successful in many different cases.

As all truncated CI methods, the MRCI suffers from the size consistency problem. Thus the

Davidson correction is often used to alleviate this problem.

In this study, the MRCI method was used to obtain correlation atomic natural orbitals and thus generate contraction coefficients for a given basis set, especially, the density matrices generated by the MRCI calculations are symmetry-averaged in order to make the basis sets isotropic.

Another commonly used multi-reference correlation method is the CASPT2 method28-30, which applies the 2nd order perturbation theory to the reference functions.

5 Relativistic Hamiltonians

For the lanthanide elements, the atomic numbers range from 57 to 71. With such a large atomic number, relativistic effects cannot be ignored. Therefore, the regular non-relativistic

Hamiltonian is not accurate for the study of lanthanide chemistry, and a proper relativistic

Hamiltonian should be used. The Dirac Hamiltonian is the start point.

The standard time-independent Dirac equation is a four-component vector equation:

æ Y ö ç La ÷ ç Y ÷ ˆ ˆ ˆ 2 ˆ Lb H DY = [ca ×p + bmc +V ]ç ÷ (28) ç YSa ÷ ç Y ÷ è Sb ø where α and β are 4 by 4 matrices consisting of 2 by 2 Pauli matrices ζx,y,z as well as the unit matrix I,

0  x,, y z ˆ I 0  x,, y x  ()   () (29)  x,, y z 0 0 I

The c, m, V are light of speed, mass of the electron and potential energy operator, respectively.

The top 2 components, also termed as the large components, can be considered as the solution for electrons. While the bottom 2 components, the small components, can be considered as the solution for positrons. In chemistry, the small components are usually irrelevant and expensive for calculation and may cause variational collapse during the solution of the partial differential equation. Thus, it often applies approximations to reduce the dimensions of the Dirac equation.

5.1 DKH3

The third-order Douglas-Kroll-Hess (DKH3) Hamiltonian31 32-34is one of the most used approximations in this work to reduce the dimension of the Dirac equation and generate a scalar relativistic Dirac equation. The generation of the DKH3 Hamiltonian can be summarized as the following steps:

1) The first step of the DKH transformation is to apply the free-particle Foldy-Wouthuysen

35 ˆ (FW) transformation U FW to the Dirac operator. The free-particle FW transformation operator has the form:

ˆ ˆ ˆ U FW = A(I + bR) (30) where

Eˆ  mc2 Aˆ  ()p 1/2 Eˆ  m2 c 4 c 2 p 2 (31) ˆ p 2Ep

1 Rˆ  Dcˆ α p Dˆ (32) ˆ 2 Ep mc

2) Applying the operator to the Dirac Hamiltonian, the one-electron Dirac

Hamiltonian is then transformed to the first-order DKH Hamiltonian

Eˆ 0 Vˆ+RˆVˆRˆ [Rˆ ,Vˆ] Hˆ =Uˆ Hˆ Uˆ-1 =( p )+Aˆ( 2 2 2 )Aˆ (33) 1 FW D FW 0 -Eˆ [Vˆ,Rˆ ] Vˆ+RˆVˆRˆ p 2 2 2

The operator R2 in Eq. (33) is the 2-spinor version of operator R in Eq. (32), where the α matrices are replaced by the Pauli matrices ζ.

3) As it has been found that it is not possible to obtain the FW transformation in closed form, the best one can do is to decouple the large and small components to a certain order. It is

ˆ ˆˆ possible to stop at Eq. (33) and then split the R2VR2 term into scalar and spin-orbit operators, but the calculated energies are too low at Ο(c-4)35. Thus it has to apply more transformation to further reduce the off-diagonal terms, resulting in a higher-order of DKH Hamiltonian.

Instead of using an exponential ansatz for a unitary operator, Douglas and Kroll write the unitary operator as:

ˆ ˆ 2 1/2 ˆ U1 = (1+W1 ) +W1 (34) where W1 is an integral operator with the kernel:

ARVˆ ˆ (,')'p p Aˆ  Aˆ '(,')'' V p p Rˆ Aˆ Wˆ (pp , ')  (35) 1 ˆ ˆ ()EEpp '

ˆ ˆ A power series in W1 is used to expand U1 , so that the expansion is the same as the exponential ansatz up to the second-order term. The potential integral operator in Eq. (35) is in the momentum space and has the form of:

21 V(pp , ') Ze2 ( ) 1/2 (36)  pp ' 2

Applying this new unitary transformation operator to the first-order Hamiltonian in Eq. (33), the second-order DKH Hamiltonian is obtained:

1 HEWEWWEˆ ˆ  ˆ   ([,])ˆ ˆ ˆ  ˆ 2 ˆ (37) 2p 1 1 p 12 1 p  where the diagonal operator:

1 1 1 ˆ ˆ ˆ 24 1 V 2 4[ cα p ,[ cα p , V ]]  2  ( cα p )  3 6  ( cα p )  ... (38) 8m c 2 mc 8 m c

4) The 2nd-order DKH Hamiltonian is widely used in quantum chemistry calculations. But from previous experience, the 3rd-order DKH Hamiltonian is preferred for high-accuracy calculations and the extra computational cost by generating the 3rd-order DKH Hamiltonian is negligible. Thus the 3rd-order DKH Hamiltonian is used in this work and it is obtained by:

ˆ ˆ ˆ ˆ ˆ H3 = H2 +[W 1,[W 1,e p ]] (39)

ˆ ˆ where W 1 and e p are defined same as Eq.(35) and Eq. (38).

It should be noted at last in this section that, at this point no relativistic correction has been

35 made to the two-electron operators. It is possible to perform the free-particle FW transformation on the two-electron operators on both electron coordinates, but it has been found that this first transformation is as important in scalar relativistic calculations as the transformations of the one-electron operators up to fifth order31. Higher-order of transformations of the two-electron terms are thus expected to be even smaller. Therefore, in practice the relativistic correction to the two-electron terms is often ignored. But if their contribution is of interest, the Guant interaction31, as the leading term in the relativistic two-electron operators, can be calculated at the HF level of theory.

5.2 X2c Hamiltonian

In this work, another relativistic Hamiltonian used is the exact two-component (X2c)

Hamiltonian36,37. The X2c Hamiltonian can reproduce exactly the positive-energy spectrum of the parent four-component Dirac Hamiltonian and it results from two the confluence of two important realizations:

Firstly, the problems interesting to chemists are the electron solutions. And the computational cost is dominated by quantities associated with two electron terms. But on the other hand, the two-electrons terms are not important in relativistic corrections, as discussed in the previous section. Thus the focus is to initially solve the one-electron problem, which is meaningful and inexpensive.

Secondly, the analytic expression of the exact coupling is not available in closed form. But the decoupling transformation can be carried out using matrix algebra without programing integrals over additional complicated operator expressions.

Therefore, the development of the X2c Hamiltonian can be summarized as follows:

1) The first step to generate the X2c Hamiltonian Hx2c, is to expand the 4-component spinors in finite non-orthogonal atomic orbitals with the one-electron Dirac Hamiltonian:

h0 c 0 S 0 c 0 (40)

The Equation (40) is then subsequently transformed into an orthonormal basis using canonical orthonormalization

† † h1 c 1 c 1 h1 V 1 h 0 V 1 VSVI1 0 1 (41)

2) Secondly, to enforce the kinetic balance condition, which prevents the variational collapse of the 4-component Dirac equations due to the existence of the negative energy solutions and ensures the kinetic energy is properly represented in the nonrelativistic limit, a transformation matrix W is applied to modify the Dirac equation

† h2 c 2 S 2 c 2 SWW2  (42) where

1 W()()  p V  p (43) 4mc22

3) Thirdly, another transformation V2 is applied again to restore the orthonomality

† h3 c 3 c 3 VSVI2 2 2 (44)

4) The eigenvalue Equation (43) is solved and the coupling matrix R is extracted

SL SL SL C RC C RC C RC (45)

L(S) where C+(-) is the block of the eigenvector matrix corresponding to the large (small) components of the positive (negative) energy solutions.

5) The Jensen and Ilias implementation37,38 is adopted such that the system can be solved by a Cholesky decomposition9

AR B ACC SS† BCC S L? (46)  

6) From the coupling matrix R in Equation (32), the decoupling transformation matrix U as in Equation (34) can be constructed. The renormalization transformation W2 can be constructed using canonical orthonormalization:

 0 W2  [ ] (47) 0  where

1 1   (48)  †  † 1 RR 1 RR

7) The final matrix representation of hx2c can be constructed and subsequently transformed back to the initial AO basis.

In this study, all the spin-free, spin-orbit and Gaunt interaction calculations are carried out by DIRAC39.

6 Choice of basis sets

The one-electron Hydrogen-like orbital is expressed as a sum of basis functions:

n ji = åca.i ca.i (49) a where c’s are the coefficients of the basis functions χ.

The first generation of the basis functions is of Slater type:

n-1 -xr m c(r,q,j) = r e Yl (q,j) (50)

m where Yl (q,j) is the spherical harmonic wave function. Although the Slater type functions have the correct asymptotic behavior, they are seldom used in practice due to its difficulties in

38 integral calculations. To overcome this disadvantage, the Gaussian type functions are proposed:

n-1 -ar2 m c(r,q,j) = r e Yl (q,j) (51) and n is commonly fixed at 1 to generate the Hydrogen-like orbitals. This Gaussian type orbital

(GTO) has much more efficiency in integration and the linear combination of GTOs can at a large extent approximate the asymptotic behavior of Slater type orbitals.

Depending on the function form, contraction scheme, Hamiltonian and location of coordinate center, etc., there are many types of the basis sets. In this study, the correlation consistent basis sets are used throughout the whole work.

The correlation consistent basis sets are generally-contracted basis sets using Gaussian type functions. The name “correlation consistent” indicates that the basis sets are designed to group the functions that contribute similar amounts of correlation energy at the same stage, independent of the angular momenta. By doing so, different sizes of correlation consistent basis sets can be developed by systematically increasing the functional space as well as inclusion of higher angular-momentum functions. The most valuable feature of the correlation consistent basis sets is that the complete basis set limit can be reached by extrapolation of energies calculated by using correlation consistent basis sets. As a result, the basis set incompleteness error can be eliminated and assist analysis of the computation errors.

Currently, the correlation consistent basis set are one of the most used basis sets, and they cover most elements in the periodic table (cf. light main group element by T. Dunning and co-workers40, transition metals41, heavy main group42,43, and actinides44 by Peterson and co-workers).

It should be noted that by default the above-mentioned basis sets are designed to recover the

39 correlation energies of valence electrons. While for the outer core correlation calculations, additional correlation functions should be included in the valence basis sets. In this work, the weighted core-valence correlation consistent basis sets are used, which are designed by the weighted core-valence scheme45. The weighted core-valence scheme takes advantage of the fact that the core-valence electron interaction dominates the chemical interest than the core-core electron interaction. Therefore the optimization of the exponents is heavily biased on the core-valence correlation energy. The resulting weighted core-valence sets can better describe the behavior of the outer core electrons.

7 Spectroscopy calculations

In this study, the spectroscopic constants are calculated by derivatives of the near-equilibrium potential energy surfaces (PES) for triatomic molecules or potential energy curves (PEC) for diatomic molecules.

For diatomic molecules, the PEC is fitted by a 7-point grid near the equilibrium geometry to

th a 5 -order polynomial. The grid points are sampled as re-0.3, re-0.2, re-0.1, re, re+0.1, re+0.3, and re+0.5 bohr where re is the equilibrium bond length. For each grid point, the FPD methodology is employed and thus the PEC at different level of theories can be established. The Dunham analysis46 is then applied to the PEC to calculate the spectroscopic constants.

For triatomic molecule, the PES is fitted by a 50-point grid for linear molecules or an

84-point grid for non-linear molecules. These grid points are sampled by symmetry and lying in the range from re-0.3, to re+0.5 bohr along the bond length coordinates, and θe-30° to θe+30° along the bond angle coordinate, where re and θe are equilibrium bond length and bond angle, respectively.

Having calculated the FPD energies for each point, the grids are then fitted to a 5th order polynomial:

i j k V  cijk R1 R 2 R 3 (52) i, j , k  1 where R1, R2 and R3 corresponds to the internal displacement coordinates in term of r1, r2 and θ, respectively. The summation includes a full set of 5th-order terms. Additional 6th-order diagonal terms are included during the fitting process. The typical root-mean-square (RMS) errors in the fits are about 3 cm-1.

Especially, for the neutral bent molecules, the angle displacement was further expanded into a cubic polynomial to minimize the RMS error.

2 3 (53) R3 = A0Dq + A1Dq + A2Dq

47 The value of A1 and A2 was determined by the boundary conditions , while A0 was roughly optimized during the fitting process.

The PES fitting as well as calculation of spectroscopic constants via 2nd order perturbation theory48 were carried out by the Surfit program49.

To calculate the dipole moment and then the intensities of the vibration spectrum, the finite field perturbation method50 is applied. Setting the external finite field strength as ±0.002 a.u., the dipole moments are calculated as the derivatives of energies with respect to the external field strength:

¶E(F ) E(+F )- E(-F ) b = b b (54) ¶Fb 2Fb where E(Fβ) is the filed-dependent energy and Fβ (β=x,y,z) is the field strength along x, y and z directions.

The obtained dipole moment surfaces in Cartesian coordinates are then rotated into the

Eckart frame51 and fitted to the 4th-order polynomials. The new derivatives of dipole moments with respect to each normal mode coordinate are then calculated and used to calculate the intensities I for each mode by the formula52:

('')dd 2 I( cm21 atm at 300 K) 65.785   i,, x i y (55) ii2 where ωi is the harmonic frequency for each mode. The d’i,x and d’i,y are the first-order derivatives of dipole moments with respect to the dimensionless normal coordinates along the x and y principal axes, respectively.

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Press, 2007).

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wirtten by T. Saue, L. Visscher, H. J. Aa. Jensen, and R. Bast, with contributions from V.

Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faßhauer, T.

Fleig, O. Fossgaard, A. S. P. Gomes, T. Helgaker, J. Henriksson, M. Iliaš, Ch. R. Jacob, S.

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Chapter 3

Correlation consistent basis sets for lanthanides. The atoms

La – Lu

Qing Lu and Kirk A. Petersona) Department of Chemistry, Washington State University, Pullman, WA 99164-4630, USA

Abstract Using the 3rd-order Douglas-Kroll-Hess (DKH3) Hamiltonian, all-electron correlation consistent basis sets of double-, triple-, and quadruple-zeta quality have been developed for the lanthanide elements La through Lu. Basis sets designed for the recovery of valence correlation

(defined here as 4f5s5p5d6s), cc-pVnZ-DK3, and outer-core correlation (valence+4s4p4d), cc-pwCVnZ-DK3, are reported (n = D, T, Q). Systematic convergence of both Hartree-Fock and correlation energies towards their respective complete basis set (CBS) limits are observed.

Benchmark calculations of the first three ionization potentials (IPs) of La through Lu are reported at the DKH3 coupled cluster singles and doubles with perturbative triples, CCSD(T), level of theory, including effects of correlation down through the 4s electrons. Spin-orbit coupling is treated at the 2-component HF level. After extrapolation to the CBS limit, the average errors with respect to experiment were just 0.52, 1.14, and 4.24 kcal/mol for the 1st, 2nd, and 3rd IPs, respectively, compared to the average experimental uncertainties of 0.03, 1.78, and

a) Electronic mail: [email protected] Reproduced with permission from The Journal of Chemical Physics, 145, 054111. (2016); doi: 10.1063/1.4959280 Copyright American Institute of Physics 2016

2.65 kcal/mol, respectively. The new basis sets are also used in CCSD(T) benchmark calculations of the equilibrium geometries, atomization energies, and heats of formation for Gd2,

GdF, and GdF3. Except for the equilibrium geometry and harmonic frequency of GdF, which are accurately known from experiment, all other calculated quantities represent significant improvements compared to the existing experimental quantities. With estimated uncertainties of about ±3 kcal/mol, the 0 K atomization energies (298 K heats of formation) are calculated to be

(all in kcal/mol): 33.2 (160.1) for Gd2, 151.7 (–36.6) for GdF, and 447.1 (–295.2) for GdF3.

I. Introduction

It is now commonly recognized that one of the main sources of error in electronic structure calculations is the incompleteness of the 1-particle basis set. One method to address this is to just use very large basis sets, but a more efficient and reliable approach is to use a sequence of basis sets that systematically approach the complete basis set (CBS) limit. In particular, with an accurate estimate of the CBS limit, the intrinsic error of the underlying correlation method can be unambiguously assessed. By far the most common choice of basis sets for this purpose is the correlation consistent (cc) family of basis sets first introduced by Dunning.1 These sets are designed to yield regular convergence to both the Hartree-Fock (HF) and correlated CBS limits as they increase in size,2,3 leading to numerous efforts to exploit this behavior via extrapolation in the basis set index or the maximum angular momentum present in the basis set (see, e.g., Ref.

4). Combined with the use of highly correlated wavefunction methods, these sets are the cornerstone of nearly all composite ab initio thermochemistry and spectroscopy methodologies.5

While cc basis sets are available throughout most of the periodic table, they have only recently been developed for f-block elements, namely the early actinides.6,7 The current contribution extends the availability of cc basis sets to the entire lanthanide row. The extension to the remainder of the actinide elements will be the subject of a separate report.

Similar to the actinide elements, the lanthanides start (La) and end (Lu) like d-block transition metals with the ground states occupying the 5d orbital with the 4f either empty (La) or completely filled (Lu). Partial occupation of the 4f begins with Ce with a 4f5d configuration and the 5d is then unoccupied until the middle of the row where single occupation of the 5d in Gd is favored over adding an additional f electron beyond a half-filled 4f shell. Analogous to the actinides, the 6s orbital is doubly occupied in the ground state configurations of all the

48 lanthanides. As discussed in Ref. 8, the radial extent (root-mean-square and 95% density) of the lanthanide atomic spinors are in shell order, 4d < 4f << 5s < 5p < 5d << 6s. In addition and contrastingly to the actinides, the radii containing 95% of the 4f spinor density are always less than those of the 5s, implying that the 4f is not very accessible for bonding since it lies inside the semi-core 5s and 5p orbitals. The 4f orbital does, however, lie energetically above the 5s and 5p orbitals, and since for nearly all of the lanthanides the 4f is partially occupied, it should be treated as a valence orbital from the point of view of the basis set (as well as the 5s, 5p, 5d, and

6s orbitals). Last, even though the 6p orbital is not occupied in the ground states of any of the lanthanides, it is often important in molecular systems and should be considered as part of a balanced valence basis set. This then leaves the 4s, 4p, and 4d orbitals as the outer-core, which should be treated in the development of core-valence basis sets. As noted previously by Gomes et al.,9 correlation of the 4d may be particularly important in the lanthanides due to the compactness of the 4f.

The currently available (before this work) basis sets and their construction for the lanthanide elements was the subject of a recent review.8 If the discussion is limited to all-electron basis sets designed for correlated calculations, the choices are fairly limited. Several consist of just one size

– the compact SARC basis sets of Pantazis and Neese (using both the 2nd-order

Douglas-Kroll-Hess10-12 (DKH2) and zero-order regular approximation13 (ZORA) Hamiltonians) designed for density functional theory calculations,14 the comparably-sized DKH2 sets of Dolg,15 and the 3rd-order Douglas-Kroll (DK3) sets of Hirao and co-workers16 that are between double- and triple-zeta quality. Three other families of sets could in principle be used to obtain estimates of the CBS limit since they systematically range in size from double- through at least quadruple-zeta. The basis sets of Gomes et al.9 were developed in the style of correlation

49 consistent basis sets but explicitly for full 4-component Dirac-Coulomb relativistic calculations.

They range in size from valence double-zeta (DZ) to quadruple-zeta (QZ) with additional tight functions available for correlation of the outer-core 4s4p4d electrons as well as functions for dipole polarization. These sets are most often used in their completely uncontracted forms. The

Sapporo DK-nZP sets (denoted SAP-nZP in this work) of Sekiya et al.17 were optimized in conjunction with the 3rd-order DKH (DKH3) scalar relativistic Hamiltonian and range from DZ to QZ. These sets were developed for correlation down through the 4s electrons of the lanthanide elements. Extensions of these sets obtained by adding an additional diffuse function in each angular momentum are also available and are denoted in this work as SAP-nZP+diffuse. The last basis sets to mention here are the atomic natural orbital (ANO) basis sets of Roos et al.,18 denoted ANO-RCC (relativistic semi-core correlation), which range in size from DZ to beyond

QZ ("large") by adding additional ANO contractions from a common primitive set. These sets were optimized with the DKH2 Hamiltonian and are designed for correlation down through the

4f and 5s5p semi-core electrons of the lanthanide elements.

In the present work new correlation consistent basis sets ranging in size from double- to quadruple-zeta quality have been developed using the DKH3 scalar relativistic

Hamiltonian10,11,19,20 for both valence and outer-core electron correlation of the lanthanides La through Lu. The details of the basis set construction are given in Sec. II. To demonstrate the efficacy of the resulting sets, they are then used in both atomic and molecular benchmark calculations using the coupled cluster single and doubles with perturbative triples, CCSD(T), level of theory. These results are shown and discussed in Sec. III. Conclusions are presented in

Sec. IV.

II. Basis Set Development

The development of correlation consistent basis sets for lanthanide elements has been carried out in a very similar manner as recently reported for the first few actinides.6 All calculations pertaining to the basis set development utilized the MOLPRO suite of ab initio programs.21,22 The orbitals from the HF calculations below were fully symmetry equivalenced, which generally required a state-averaged multiconfigurational self-consistent field (MCSCF) treatment. The DKH3 scalar relativistic Hamiltonian has been used throughout. This order of the

DKH Hamiltonian was chosen in order to be consistent with earlier work on the 6p elements,23 as well as considering the recent results6 of CCSD(T) atomization energy calculations of the heavier ThO2 and UF6 molecules where an extension from DKH3 to DKH4 resulted in changes of only 0.1 kcal/mol.

A. Hartree-Fock primitive sets

Just as in the actinide correlation consistent basis sets,6 the HF spdf primitive exponents were bootstrapped from the primitive sets of Dyall and co-workers,9 which were optimized in

4-component Dirac-Hartree-Fock calculations using the Dirac-Coulomb Hamiltonian. Since they employed a finite nucleus model, these sets avoid the problem of heavy element basis sets optimized in a point charge nucleus framework, whereby particularly the s-type primitive exponents become large enough to cause numerical difficulties in the integral evaluation. In the present work the innermost spd functions from the valence double-zeta (VDZ), triple-zeta (VTZ), and quadruple-zeta (VQZ) sets of Gomes et al.9 were retained unchanged, and the exponents representing approximately the outer two radial maxima of the valence orbitals, as well as all the f exponents, were re-optimized at the HF level using the DKH3 scalar relativistic Hamiltonian.

Specifically, the total sizes of the HF primitives sets are (24s19p13d8f), (30s24p16d11f), and

(35s30p19d13f) for DZ, TZ, and QZ, respectively, from which the most diffuse (5s5p7d),

(7s7p8d), (9s9p10d) exponents (and all the f functions) were reoptimized in this work. For Ce through Yb the s-type exponents were optimized for the lowest electronic state with a 4fn 6s2 configuration (in this context n=1-14 for La through Yb). In the case of Lu, the lowest electronic state with configuration 4f14 5d 6s2 was used. For the p-type exponents, the average energy from the lowest states with configurations 4fn 6s 6p and 4fn-1 6s2 6p were used for Ce through Yb

(4f145d6s6p and 4f146s26p for Lu). Likewise the d-type exponents for Ce through Yb were optimized for the energy average of states associated with 4fn-1 5d 6s2 and 4fn-1 5d2 6s. Those for

Lu utilized states with configurations 4f14 5d 6s2 and 4f14 5d2 6s. Finally, for the f-type exponents, states with configurations 4fn 6s2 were used for Ce through Yb and 4f14 5d 6s2 for Lu.

As noted previously by Weigand et al.,24 optimization of f-type primitives for the Ac atom using the 5f 7s2 configuration leads to exponents that are too diffuse, hence they utilized optimizations on Ac2+. In the present work, the f-type exponents for La were optimized for the 3F state of La+ , which corresponds to a 4f 6s configuration. This yielded f exponents consistent with the trend of exponents from nearby elements of the row. In addition, the s-type primitives for La were optimized for the 5d 6s2 configuration of neutral La, while the d-type exponent optimizations involved both the 5d 6s2 and 5d2 6s configurations (La). Finally, the p-type exponents of La were optimized for the state-average of both the 6s2 6p of La and the 6s 6p of

La+.

In all cases in order to get a proper distribution of f-type primitives, i.e., a primitive set that well described the 4f orbital in a HF sense, as well as providing functions appropriate for correlation (see below), the optimizations of the f exponents were carried out in two steps. First

52 the most diffuse f exponent from the 4f-dipole polarization set of Gomes et al.9 was removed and the remaining exponents optimized at the HF level as described above. Then the diffuse f exponent of Gomes et al. was combined with the most diffuse f resulting from the first HF optimization and reoptimized with all tighter exponents now frozen.

The resulting HF spdf primitives were generally contracted to [6s5p3d1f] using atomic orbital (AO) coefficients from state-averaged MCSCF calculations. Of these contracted functions, all but the last p and d contractions were obtained for Ce – Yb by state averaging the 4fn 6s2 and

4fn-1 5d 6s2 states of the neutral atoms. An additional d contraction was added from the 5d AO of the 4fn-1 5d 6s2 state and a p-type contracted function was added from the 6p AO of a state-average of the 4fn 6s 6p and 4fn-1 6s2 6p states. For Lu all of the contraction coefficients except the 6p orbital were taken from the 4f14 5d 6s2 configuration. The Lu 6p contraction was taken from a state average of the lowest states corresponding to the 4f146s26p and 4f145d 6s 6p configurations. Finally, for La all contraction coefficients except those for the 6p and 5d AOs were taken from a state-average of states arising from the 5d 6s2 of La and 4f 6s of La+. The 5d contraction coefficients were taken from the 5d 6s2 (La) while the 6p utilized a state-average of the 6s2 6p of La and the 6s 6p of La+.

B. Correlating functions – valence correlation (5s5p4f5d6s)

Correlating functions for valence correlation, defined here to be the 4f, 5d, and 6s electrons together with the 5s and 5p semi-core electrons, were also optimized in the same manner as the actinide elements, assuming identical correlation consistent groupings.6 The latter choice is borne out by inspection of the incremental correlation recovery as a function of angular momentum functions at the frozen-core multireference configuration interaction (MRCI) level of theory25,26

53 for the Gd atom as shown in Figure 1. To construct this figure, the minimally-contracted QZ basis set described above was used with the 5 most diffuse s and p functions uncontracted.

Angular momentum functions of d, f, g, h, and i symmetries were then optimized for 5s5p4f

8 2 7 correlation on the lowest energy state corresponding to the 4f 6s configuration ( Fg) one at a time following an even-tempered prescription; first 1-4 d-type functions, then 1-4 f-type functions were added and optimized in the presence of the optimal 4d set, then 1-4 g-type functions were optimized in the 4d4f set, etc. In all cases the resulting optimized values of the exponents followed a very regular progression. As seen in Fig. 1, the incremental correlation energy lowerings within each angular momentum show nearly an exponential trend. It is observed, however, that the g-type functions contribute more correlation energy than the h or i angular momenta, making the correlation consistent groupings for DZ and TZ somewhat ambiguous. However, to remain consistent with previous work, only 1 g-type function is included for 5s5p4f correlation at the DZ level and 2g1h for TZ. The QZ level set should consist then of 3g2h1i functions for 5s5p4f correlation.

The higher angular momentum correlating functions were optimized at the MRCI level separately for 5s5p4f and 5d6s correlation due to the very different radial extents of these AOs.

As described above for Gd (see also Fig. 1), functions added for 5s5p4f correlation (utilizing states associated mainly with the 4fn 6s2 configurations, except for the 4f 6s configuration for La+ and 4f14 5d1 6s2 for Lu) corresponded to 1g for DZ, 2g1h for TZ, and 3g2h1i for QZ. Additional sets of 1g and 2g1h functions were added to the TZ and QZ sets, respectively, for 5d6s correlation (using states associated with the 5d 6s2 configurations). In the TZ and QZ cases, the two sets of exponents for these different correlation spaces were optimized together and iterated until consistency was reached. In all cases the ratios between consecutive exponents were

54 constrained to be greater than a factor of 1.6 in order to minimize linear dependency issues.

Correlating functions for the occupied angular momenta were represented as ANO contractions obtained using averaged density matrices from MRCI calculations on the lowest states associated with the 4fn 6s2 and 4fn-1 5d 6s2 configurations. Those for La involved the 4f 6s configuration of

La+ and the 5d 6s2 configuration of La, while Lu utilized only the 4f14 5d 6s2 configuration. The contractions of [1s1p1d1f] for DZ, [3s3p3d3f] for TZ, and [4s4p4d4f] for QZ were supplemented by uncontracting the most diffuse exponent of each angular momentum to provide additional flexibility. Note that the ANO sets for TZ and QZ are one function larger (in each angular momentum) than the previously reported sets for Th – U.6,7 The larger sets were found to be necessary to reduce basis set superposition error (BSSE) in initial benchmark calculations on Gd2.

The total contracted sizes of the resulting cc-pVnZ-DK3 basis sets were [8s7p5d3f1g],

[10s9p7d5f3g1h], and [11s10p8d6f5g3h1i] for n=D, T, and Q, respectively.

C. Correlating functions - core correlation (4s4p4d)

As noted previously by Gomes et al.,9 due to the compactness of the 4f orbital, correlation of the 4d could be very important for the lanthanide elements. In this work additional functions were optimized for 4s4p4d correlation using the weighted core-valence scheme,27 which emphasizes intershell core-valence correlation over intrashell core-core correlation. These additional groups of correlating functions corresponded to 1s1p1d1f for DZ, 2s2p2d2f1g for TZ, and 2s2p2d3f2g1h for QZ (the 3 f-type functions were constrained to an even-tempered sequence). The MRCI optimizations utilized states associated with the 4fn 6s2 configurations

(with the same modifications for La and Lu as described above). The resulting basis sets are denoted cc-pwCVnZ-DK3 with n=D, T, Q. To avoid linear dependency issues in the DKH

55 integral evaluation, many of the optimized correlating functions corresponding to occupied angular momenta were replaced by functions obtained by simply uncontracting underlying HF primitives based on their proximity to the optimal functions. As in the actinide elements, due to the possibility of linear dependency between the high angular momentum core correlating functions with the exponents of inner valence functions (from cc-pVnZ-DK3), the innermost valence correlating functions were reoptimized (for 5s5p4f correlation) along with the 4s4p4d correlating g- and h-type functions at the TZ and QZ basis set levels. Hence the final cc-pwCVnZ-DK3 sets (TZ and QZ) are not exactly equivalent to simply adding tight functions to the cc-pVnZ-DK3 valence basis sets.

III. Results and Discussion

A. Atomic ionization potentials

As an initial assessment of the new basis sets, the first three ionization potentials (IPs) of the lanthanide atoms were calculated at the CCSD(T) level of theory. Previously there have been several benchmark quality ab initio studies of these IPs, which have been fairly well characterized by experiment. In particular Cao and Dolg28,29 have carried out large basis set average coupled pair functional (ACPF) pseudopotential (PP) calculations on the first 4 IPs of the lanthanide atoms, as well as CCSD(T) calculations on a subset of these (all including correlation from the 6s through the 4s electrons). Extrapolation to the CBS limit, spin-orbit corrections, and PP corrections were applied, resulting in average errors of ~3 to ~8 kcal/mol.

Where CCSD(T) was applied, this typically resulted in lower errors by up to a factor of 6. In a latter study, Roos et al.18 calculated the 1st IPs of the lanthanide atoms using the complete active

56 space 2nd-order perturbation theory (CASPT2) method with the DKH2 scalar relativistic

Hamiltonian and their new ANO-RCC basis sets. Spin-orbit effects were not included but comparisons to J-averaged experimental data exhibited agreement to generally within 0.1 eV (2 kcal/mol) or better. Results for Ce and Gd, however, differed from experiment by about 4 kcal/mol. Recently Wilson and co-workers30 reported an investigation of the 3rd IPs of La through Eu using the DKH3 scalar relativistic Hamiltonian with CCSD(T) and the Sapporo sequence of basis sets. Somewhat similar to the earlier work of Cao and Dolg, they included spin-orbit (SO) corrections based on 4-component DHF and complete open-shell configuration interaction (COSCI) calculations. With an uncontracted basis set of quadruple-zeta quality, they reported an average deviation from experiment of just above 5 kcal/mol when the 4s through 6s electrons were correlated.

1. Computational details

In the present work the CCSD(T) calculations employed restricted open-shell HF (ROHF) orbitals (symmetry broken) and an open-shell variant of CCSD that allows for small amounts of spin contamination, i.e., ROHF-CCSD(T) or R/UCCSD(T),31-34 using the MOLPRO program.21,22 The electronic states corresponded to those shown in Table I, except for the case of neutral Ce atom where the low-spin open-shell 1G state is not amenable to single determinant

CCSD(T) calculations. In that case the low-lying 3F state (which like the 1G ground state level corresponds to the 4f5d6s2 configuration) was employed, and the final CCSD(T) ionization potential was corrected using the experimental excitation energy.35 The final CCSD(T) IPs were calculated in a Feller-Peterson-Dixon (FPD) composite scheme5,36,37 via

IP = IPCBS + CV/CBS + SO (1)

57 where IPCBS is the ionization potential calculated at the frozen-core (FC) CCSD(T) level of theory at the extrapolated CBS limit, CV/CBS is the effect of correlating the outer-core electrons

(4s4p4d) at the CCSD(T) level also extrapolated to the CBS limit, and SO is the contribution to the IP from spin-orbit coupling. In the FC calculations the CBS limits were obtained by separate extrapolation of the HF total energies and CCSD(T) correlation energies using cc-pVTZ-DK3 and cc-pVQZ-DK3 basis sets. The HF extrapolations used the Karton-Martin formula originally developed for molecules containing light elements,38

-6.57 n En = ECBS + A(n +1)e (2) with n=3 and 4, while the CCSD(T) correlation energy utilized39

B En = ECBS + 1 4 (n + 2) (3)

These choices have been previously used to good effect in similar calculations for actinide atoms and actinide-containing molecules.6,40-42 In particular Eq. (3) was chosen for correlation energies based on its robustness for yielding accurate CBS limits in calculations involving main group elements.4

The CCSD(T) calculations for CV used the cc-pwCVnZ-DK3 basis sets, both for frozen-core and outer-core (4s4p4d) correlated calculations. The difference in these two calculations for each atom yielded the CV correction to the IP, from which the TZ and QZ results were extrapolated to the CBS limit via Eq. (3). Of course by extrapolating CV to the

CBS limit a decomposition of FC and CV effects is not actually needed, but is very useful in order to assess both the importance of outer-core correlation and the impact of additional tight basis functions on the FC results (compared to the normal cc-pVnZ-DK3 basis sets).

Last, contributions from SO coupling were calculated at the 2-component HF level of theory

58 with the Exact 2-component (X2C) Hamiltonian43 using uncontracted cc-pVDZ-DK3 basis sets.

Two-electron spin-same-orbit and spin-other orbit (Gaunt) corrections were included using the atomic-mean-field integral (AMFI) approximation.44,45 Specifically 2-component average-of-configuration HF calculations were first carried out by distributing just the open-shell electrons among the 4f, 6s, or 5d spinors, the choice of which corresponded to the minimal set depending on the electronic term required. The energy of the ground state configuration (or 3F in the case of Ce atom) was then recovered from a COSCI calculation. The SO contribution was then derived from calculations both including and excluding SO terms in the Hamiltonian. All

SO calculations were carried out with the DIRAC program package.46

2. Results and discussion

The results for the first, second, and third IPs of the lanthanide atoms are given in Tables II,

III, and IV, respectively. In each case in addition to the FC and CV/CBS limits and SO contributions, the basis set convergence is represented by the difference of a FC IP or CV contribution calculated from a finite basis set from that at the extrapolated CBS limit, e.g., TZ is the difference between a TZ quantity from its CBS limit. In nearly all cases the FC CCSD(T)

IPs converge to their CBS limits from below, i.e., the IPs increase with increasing basis set cardinal number. The only two exceptions are the first IPs of both La and Ce, in which the ionization involves removal of a 6s electron accompanied by an excitation from the 6s to 5d orbital. All other first IPs, except for Lu where the 1st IP also involves ionization of a 5d electron, involve ionization of a 6s electron and the basis set convergence is very rapid. The different ionization processes between Yb and Lu (6s in Yb and 5d in Lu) is reflected by the much slower basis set convergence of IP1 in the case of Lu where the FC cc-pVQZ-DK3 value is still 1.2 kcal/mol below the CBS limit. At this same basis set level, all the first IPs involving 6s

59 ionization had QZ values of just –0.2 to –0.3 kcal/mol. In regards to the effect of correlating the outer-core 4s4p4d electrons on the 1st IPs, this is calculated to be nearly negligible in the cases involving 6s ionization, i.e., generally +0.3 kcal/mol or less. Contrastingly, the CV values for

La and Ce lie between +2 and +3 kcal/mol while that of Lu is the only IP1 where outer-core correlation decreases the IP, albeit by just -0.7 kcal/mol. In all cases the basis set convergence is rapid. Where ionization involves just the 6s electron, not surprisingly the SO contribution to IP1 is nearly negligible, on the order of 0.1 kcal/mol. The largest effect is for IP1 of Lu (5d ionization), which is calculated to be 2.4 kcal/mol at the present 2c-HF level of theory. The other two IP1 values that involve the 5d electrons, i.e., those of La and Ce, also have non-negligible

SO values, -0.5 and -1.2 kcal/mol, respectively. All together the calculated first IPs are in excellent agreement with experiment47 with a mean unsigned deviation (MUD) of just 0.52 kcal/mol. Not surprisingly the largest differences are observed for La, Ce, and Lu which range from 1 to about 3 kcal/mol. Even so, the present results represent an improvement over the previous extrapolated-CBS averaged coupled pair functional (ACPF) investigations of Cao and

Dolg28, as well as the complete active space 2nd-order perturbation theory (CASPT2) values of

Roos et al.18, where average deviations ranged from 2-3 kcal/mol. Cao and Dolg28 did note however that the use of CCSD(T) did improve upon their ACPF average errors by more than a factor of 2 in several cases.

Inspection of the calculated 2nd IPs of Table III reveals only a slightly stronger basis set dependence of the FC results, but the CV effects are still small and rapidly convergent. The only exception for the latter trend is the Ce atom, where IP2 involves the removal of two 5d electrons (see Table I), one by ionization and one by promotion to the 4f. This leads to a relatively large outer-core correlation contribution of just over 10 kcal/mol with a very slow

60 basis set convergence. The 2nd IP of La also involves ionization of a 5d electron, but this results in only a slightly stronger basis set dependence of the FC result compared to those involving 6s ionization and a small negative CV contribution. All atoms involving 6s ionization have small

(less than 1 kcal/mol) positive CV contributions. The final average errors with respect to experiment are still very good, just over 1.1 kcal/mol, with the largest errors (1-3 kcal/mol) observed for the early lanthanides La, Pr, Nd, and Pm. The average experimental uncertainty is

1.8 kcal/mol, however, and the present calculations yield 2nd IPs within 2 kcal/mol of the experimental values in all cases except La.

In the case of the 3rd IPs, except for La, Gd, and Lu, which involve ionization of a 5d (La and Gd) or a 6s (Lu) electron, the ionization process corresponds to removal of a 5f electron. As shown in Table IV, this is marked by a much stronger basis set dependence of the FC IPs compared to IP1 and IP2 and relatively large CV and SO contributions. As shown in Figure 2, the relatively slow, but regular, basis set convergence of the FC CCSD(T) IPs is dominated by the slow convergence of the correlation contribution. Even at the DZ level the HF contribution is within 2 kcal/mol of the HF limit and within about 0.2 kcal/mol with just cc-pVTZ-DK3. The convergence of the FC correlation contribution is much slower, with DZ values ranging from about -5 to -7 kcal/mol for La and Lu, respectively, which involve 5d ionization, to much larger values when a 5f electron is removed, i.e., up to -30 kcal/mol for Er. Obviously accurate calculation of IP3 strongly benefits from systematically convergent basis sets yielding the ability to extrapolate to the CBS limit. For example for Er the large cc-pVQZ-DK3 basis set still yielded a FC CCSD(T) IP3 value that was 4 kcal/mol below the extrapolated CBS limit. For the cases involving ionization of a 4f electron, outer-core correlation effects on IP3 were calculated to be relatively large (4 to 9 kcal/mol) for the early lanthanides, dropping to at most 1 to 2 kcal/mol

61 from Eu onwards. Not surprisingly SO effects were much larger when a 4f electron was ionized, ranging up to nearly 13 kcal/mol for Yb. Upon comparing the final composite values to experiment, the average error increases to 4.2 kcal/mol for these 3rd IPs, which is still less than 1% and suprisingly about half the average error as the ACPF treatment of Cao and Dolg.28 Some individual errors, however, reached values as large as nearly 11 kcal/mol (Tb) and this can be attributed to using CCSD(T) in situations where non-dynamical correlation effects are becoming important. Simply inspecting the sizes of the T1 diagnostic or the largest doubles amplitudes was not sufficient, however, to determine when the CCSD(T) results would lead to larger errors. For the most part the present results are similar to the recent CCSD(T) results of Wilson and co-workers30 (La-Eu) who used the Sapporo basis sets, specifically their FC1 (analogous to our

FC results) and FC3 (analogous to our FC+CV) values. The average errors, however, resulting from the present composite treatment were about 1 kcal/mol smaller than those reported in their work, presumably due to the use of CBS extrapolations in the present work that were enabled by the new cc-pVnZ-DK3 and cc-pwCVnZ-DK3 basis sets.

B. Molecular calculations

For representative molecular benchmark calculations, CCSD(T) calculations have been carried out on three gadolinium-containing molecules, Gd2, GdF, and GdF3. A composite scheme analogous to that described above for the atomic IPs has been utilized. Namely the DKH3

Hamiltonian is used in conjunction with ROHF-CCSD(T) and sequences of cc-pVnZ-DK3 and cc-pwCVnZ-DK3 basis sets (n=D-Q) for Gd. The standard diffuse-augmented aug-cc-pVnZ-DK and aug-cc-pwCVnZ-DK sets (n=D-Q) are used for F.1,27,48,49. Below the combinations cc-pVnZ-DK3/aug-cc-pVnZ-DK and cc-pwCVnZ-DK3/aug-cc-pwCVnZ-DK are denoted simply as VnZ-DK and wCVnZ-DK. In each case frozen-core calculations (correlating 4f and 5s

62 through 6s on Gd with 2s and 2p on F) were first carried out with the VnZ-DK sets while core correlation effects were determined using the wCVnZ-DK basis sets. In each case the final composite energies were determined as

EFinal = ECBS[TQ] + CVCBS[wTQ] + SO +QED (4) where ECBS[TQ] represents the frozen-core CCSD(T) energy at the CBS limit obtained by extrapolating the HF and CCSD(T) correlation contributions separately via Eq. (2) and (3), respectively, with VTZ-DK and VQZ-DK basis sets. The 2nd term, CVCBS[wTQ], is the contribution due to outer-core correlation (4s4p4d on Gd and 1s on F), obtained as the difference in two CCSD(T) calculations, one with FC and one with outer-core correlated, both in the same wCVnZ-DK basis sets. The contributions due to SO coupling were determined by a 2c-X2C-HF calculation together with COSCI in the case of Gd2 or from 2c-X2C-CCSD(T) calculations50 for

GdF and GdF3. Each of these used uncontracted VDZ-DK basis sets. The atomic spin orbit correction for F atom was obtained by J-averaging the experimental SO energy levels.51 Last, the final term in Eq. (3), QED (quantum electrodynamics), is an estimate of the Lamb shift on the

Gd atom and was calculated at the CCSD(T) level of theory with wCVDZ-DK basis sets with valence electrons correlated. These calculations used the model potential approach for both the vacuum polarization and self-energy contributions as first described by Pyykkö and Zhao.52 The current implementation has been previously described in detail.6

Spectroscopic properties of the two diatomic molecules were determined by fitting 7 calculated energies distributed about the equilibrium bond lengths (–0.3 ao ≤ r - re ≤ +0.5 ao) to

6th-order polynomials in internal displacement coordinates. The derivatives of these fits were then used in the usual 2nd-order perturbation theory expressions for e, e xe, etc. (see, e.g., Ref.

53).

1. Gd2

The gadolinium dimer is an interesting molecule in its own right since it has the honor of having the highest experimentally confirmed54 spin multiplicity ground state of any diatomic

19 S- molecule, 19 (18 unpaired electrons). The g ground state corresponds to the

(4 f 7)(4 f 7)s 2s 1s 1p 2 superconfiguration g u g u . As previously shown in the work of Cao and

55 Dolg, the doubly occupied ζg is a bonding orbital arising from the 6s atomic orbitals (AOs), while the singly occupied ζg and doubly occupied ζu can be attributed to the 5d AOs. The singly occupied ζu has predominately 6s character with strong contributions from the 6p AOs. The spectroscopic properties of the Gd dimer have been previously determined from Raman spectra in argon matrices by Lombardi and co-workers.56 In addition to deriving values for the harmonic frequency (138.7±0.4 cm-1) and anharmonicity constant (0.3±0.1 cm-1), a dissociation energy of

48±16 kcal/mol was derived based on the dimer force constant. Their dissociation energy can be compared to the value reported by Kant and Lin57 based on mass spectrometry of 41.1 ± 8.1 kcal/mol. The previous PP-based CCSD(T) calculations of Cao and Dolg yielded a dissociation energy of 31.8 kcal/mol.

CCSD(T) results using both VnZ-DK and wCVnZ-DK basis sets from the present work using the composite treatment of Eq. (4) are shown in Table V. Comparisons of identical calculations using the ANO-RCC basis sets of Roos et al.18 as well as the Sapporo basis sets

(SAP-nZP) of Sekiya et al.,17 are also given. The ANO-RCC sets range from double-zeta to quadruple-zeta ANO contractions with an additional ANO-RCC-Large set that is intermediate between QZ and 5Z. These sets are only designed for valence electron (4f and 5s through 6s) correlation and hence have not been employed in this work for outer-core correlation. The

Sapporo sets also range from DZ to QZ but were optimized for inclusion of outer-core correlation. Also shown in Table V are the SAP-nZP sets augmented with an additional diffuse function in each angular momentum, SAP-nZP+diffuse. The ANO-RCC sets are much more compact than either the VnZ-DK or SAP-nZP basis sets, while the SAP-nZP sets are only slightly larger than the VnZ-DK ones. The wCvnZ-DK sets of the present work are similar in size to the SAP-nZP+diffuse basis sets. In order to assist the interpretation of the resulting spectroscopic properties, particularly re and De, the basis set superposition error (BSSE) has also been calculated in each case using the standard function counterpoise method.58

Focusing first on the frozen-core CCSD(T) results, the VnZ-DK basis sets result in spectroscopic properties that exhibit a regular convergence towards their CBS limits. As also shown in Figure 3, particularly the ANO-RCC basis sets yield an irregular convergence pattern.

In the latter case, the dissociation energy initially converges from above and this can be attributed to the very large BSSE with the ANO-RCC-DZP set, which is nearly an order of magnitude larger than either VDZ-DK or SAP-DZP. While both the ANO-RCC-TZP and -QZP basis sets yield De values close to the estimated CBS limit, this is due to fortuitous BSSE contributions that are still 4.5 kcal/mol at the QZP level. In fact the ANO-RCC-Large result, where the BSSE is calculated to be below 1 kcal/mol, falls below the estimated CBS limit by about 2 kcal/mol. Without additional diffuse functions, the SAP-nZP basis sets are associated with small BSSE values similar to the VnZ-DK sets, but the convergence with increasing basis set size is very slow. This is remedied by adding additional diffuse functions, SAP-nZP+diffuse.

Presumably this occurs due to a bias in the standard SAP-nZP sets towards 4s4p4d correlation.

The FC spectroscopic constants obtained using the very large SAP-QZP+diffuse basis set are intermediate between the VQZ-DK and CBS[TQ] results.

The effects of correlating the 4s4p4d electrons are shown in Table V for the wCVnZ-DK and

SAP-nZP+diffuse basis set sequences. The convergence of re, e, and De are fairly regular for the correlation consistent basis sets but are somewhat irregular for the SAP sets. At the estimated

CBS limit, the effect of outer-core correlation is only –0.0012 Å, –0.13 cm-1, and -0.43 kcal/mol for re, e, and De, respectively. The effects of SO coupling is calculated to be nearly negligible for re and e, but predominately due to atomic SO it lowers the dissociation energy by 4.2 kcal/mol. The latter correction is about 0.9 kcal/mol smaller than what would be expected due to just the atomic fine structure (estimated from experiment, 5.1 kcal/mol). To the authors' knowledge this is the first ab initio calculation of the effects of SO coupling on the Gd dimer.

(The work of Cao and Dolg55 utilized the experimental atomic SO splitting.) The impact of the

Lamb shift is nearly negligible, increasing the dissociation energy by only ~0.2 kcal/mol.

The final composite results are in good agreement with both experiment and the previous ab initio results of Cao and Dolg. In particular the composite harmonic frequency differs from

(matrix) experiment result by less than 4 cm-1. The final D0 of 33.2 kcal/mol is consistent with the previous result of Cao and Dolg (31.8 kcal/mol) and at the lower end of the large experimental uncertainties. The current composite value, however, is estimated to be accurate to within about 3 kcal/mol, representing a significant improvement over the current experimental values. This error estimate is motivated by the ionization potential results discussed above in the cases where CCSD(T) is particularly valid, as well as previous CCSD(T) calculations on 3d transition metal molecules59 and those involving Th and U.6,60,61 Applying 298 K thermal corrections from standard harmonic oscillator-rigid rotor statistical mechanical partition functions yields a D298 value of 34.4 kcal/mol. Combining this with the experimental 298 K

62 formation enthalpy of Gd atom (97.25 ±0.48 kcal/mol) yields Hf(298) for Gd2 of 160.1

66 kcal/mol.

2. GdF

8 - Table VI displays the results of the current CCSD(T) calculations for the S7/2 ground state of the GdF molecule, which is well described as [Gd+][F–] where Gd+ is in a 4f7 6s2 configuration. Both the FC and CV results show good convergence towards their respective CBS limits, however some slight irregularities in e are observed at the DZ level presumably from

BSSE effects. In both cases the QZ quality basis set results are fairly close to the extrapolated

CBS limits. The effects of outer-core correlation are again calculated to be very small for re and

e, but nearly +0.9 kcal/mol for De. The contributions due to SO coupling are also small, except for De where it is lowered by 2.6 kcal/mol, which is nearly entirely due to atomic SO on Gd. The effects of the Lamb shift are completely negligible in this case. Compared to the accurate values

63 from spectroscopic measurements, the present composite results for re and e are in excellent

-1 agreement, being within about 0.002 Å and 2 cm , respectively. The values of D0 and D298 from the present work, 151.6 and 152.9 kcal/mol, respectively, are larger than the experimental value

(298 K) of 141.1 kcal/mol derived from mass spectrometric measurements,64 which has a stated uncertainty of ±4 kcal/mol. The present results, however, are consistent with the previous ab

65 initio calculations of Dolg et al. The uncertainty of the present result for D0 is also estimated to be about 3 kcal/mol. Combining the present D298 value with the experimental formation enthalpies at 298 K of both Gd(g) (97.25 ± 0.48 kcal/mol)62 and F(g) (18.97 ± 0.07 kcal/mol)66 leads to a Hf(298) value for GdF of –36.6 kcal/mol. The analogous experimental value from Ref.

64 is -40 ± 6 kcal/mol.

3. GdF3

The lanthanide trihalides have been the subject of many experimental and ab initio studies.

7 In these molecules the central Ln atom is in its +3 oxidation state and for GdF3 this leads to a 4f electron configuration on Gd+3. While there is still considerable uncertainty on whether their true equilibrium geometries are planar or pyramidal, it is clear that the out-of-plane bending mode is very soft with a concomitant small barrier to inversion.67 The current DKH3 CCSD(T) calculations are the most extensive to date, however the present structural results are qualitatively similar to the recent PP-based (predominately MP2) calculations of Lanza and

68 Minichino. The CCSD(T)/VTZ-DK level of theory yields a pyramidal C3v equilibrium structure with re = 2.049 Å and a F-Gd-F bond angle of 119.3º. The latter corresponds to the F atoms being just 4.9º below the D3h plane. An analogous optimization with the VDZ-DK basis set yielded a planar D3h structure with re = 2.061 Å. Experimentally the structure has been characterized by electron diffraction,69 giving a pyramidal structure with a thermally averaged bond length of

2.053 ±0.003 Å. An equilibrium distance has also been estimated in the same experimental work as 2.016 ±0.006 Å (see Ref. 70). Based on the present calculations, the experimentally-derived correction to the thermal average yielding the equilibrium bond length seems to be strongly overestimated.

Results pertinent to the CCSD(T) atomization energy of GdF3 are given in Table VII. Both in the frozen-core calculations with the VnZ-DK basis sets and those with the outer-core electrons correlated with the wCVnZ-DK sets, the calculated atomization energy smoothly converges from below their respective CBS limits. The CBS extrapolation is not negligible in this case, with the CBS limit being more than 3 kcal/mol larger than the QZ result. Correlation of the outer-core electrons (including 1s correlation on F) increases the atomization energy over the

68 frozen-core value by 0.88 kcal/mol, which is identical to the GdF case (see Table VI). After inclusion of SO and QED (the latter again negligible), the equilibrium atomization energy is calculated to be 449.5 kcal/mol. Employing the CCSD(T)/VDZ-DK harmonic frequencies, the 0

K and 298 K values are 447.1 and 449.4 kcal/mol, respectively (with estimated uncertainties of

±3 kcal/mol). The latter can be compared to the experimentally derived value of Myers,71 443 ±

5 kcal/mol or the experimental value of Zmbov and Margrave,64 443 ± 10 kcal/mol. The enthalpy of formation at 298 K is calculated in this work to be –295.2 kcal/mol.

IV. Conclusions

New correlation consistent basis sets have been developed in conjunction with the DKH3 scalar relativistic Hamiltonian for the lanthanide elements La through Lu. The basis sets range from double- to quadruple-zeta, encompassing those for valence correlation (4f5s5p5d6s, denoted cc-pVnZ-DK3) and outer-core correlation (valence + 4s4p4d, denoted cc-pwCVnZ-DK3). Both series of basis sets systematically converge the HF and correlation energies towards their respective CBS limits. Coupled cluster benchmark calculations were carried out for the 1st three ionization potentials of the lanthanide atoms, as well the Gd2, GdF, and GdF3 molecules. The ability to converge both the valence and outer-core correlation energies with respect to basis set was found to be essential for the accurate determination of the 3rd IPs, which involved the ionization of a 4f electron. The new correlation consistent basis sets of this work were compared to the ANO-RCC and SAP-nZP basis sets in CCSD(T) calculations on the

Gd dimer. Both the new correlation consistent and SAP sets were found to yield a more systematic convergence towards the CBS limit for the dissociation energy in comparison to the

ANO-RCC sets, the latter of which suffer from relatively large BSSE. The most comparable

69 results to the present cc-pVnZ-DK3 or cc-pwCVnZ-DK3 values were obtained by using the larger SAP-nZP+diffuse basis sets. The final CBS limit CCSD(T) results for atomization energies, which are estimated to be accurate to within 3 kcal/mol, are proposed to be an improvement over the current experimental values in each case. This is particularly true for Gd2 where the current experimental dissociation energies have uncertainties of 8 – 16 kcal/mol. The effect of correlating the outer-core 4s4p4d electrons was found to be negligible for the bond lengths and frequencies of Gd2 and GdF, but contributed up to 0.9 kcal/mol to the atomization energies. The final 0 K atomization energies (298 K heats of formation) are calculated to be (all in kcal/mol): 33.2 (160.1) for Gd2, 151.7 (–36.6) for GdF, and 447.1 (-295.2) for GdF3.

The new basis sets are provided in the Supplemental Material72 and will also be made available on the authors' website,73 the Basis Set Exchange,74 as well as the MOLPRO basis set library.21

Acknowledgments

The support of the U.S. DOE Office of Science, Office of Basic Energy Sciences, Heavy

Element Chemistry Program through Grant No. DE-FG02-12ER16329 is acknowledged.

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Table 1 Electronic ground states and configurations for the lanthanides (Ln) through Ln3+.

Ln Ln+ Ln2+ Ln3+

1 2 2 2 3 1 3 2 6 1 La 5d 6s D3/2 5d F2 5d D3/2 5s 5p S0 1 1 2 1 1 2 4 2 3 1 2 Ce 4f 5d 6s G4 4f 5d H7/2 4f H4 4f F5/2

3 2 4 3 1 5 3 4 2 3 Pr 4f 6s I9/2 4f 6s I4 4f I9/2 4f H4 4 2 5 4 1 6 4 5 3 4 Nd 4f 6s I4 4f 6s I7/2 4f I4 4f I9/2 5 2 6 5 1 7 5 6 4 5 Pm 4f 6s H5/2 4f 6s H2 4f H5/2 4f I4 6 2 7 6 1 8 6 7 5 6 Sm 4f 6s F0 4f 6s F1/2 4f F0 4f H5/2

7 2 8 7 1 9 7 8 6 7 7 Eu 4f 6s S7/2 4f 6s S4 4f S7/2 4f F0

7 2 9 7 1 10 7 1 9 7 8 Gd 4f 5d6s D2 4f 5d6s D5/2 4f 5d D2 4f S7/2 9 2 6 9 1 7 9 6 8 7 Tb 4f 6s H15/2 4f 6s H8 4f H15/2 4f F6

11 2 4 11 1 5 11 4 10 5 Ho 4f 6s I15/2 4f 6s I8 4f I15/2 4f I8 12 2 3 12 1 4 12 3 11 4 Er 4f 6s H6 4f 6s H13/2 4f H6 4f I15/2 13 2 2 13 1 3 13 2 12 3 Tm 4f 6s F7/2 4f 6s F4 4f F7/2 4f H6

14 2 1 14 1 2 14 1 13 2 Yb 4f 6s S0 4f 6s S1/2 4f S0 4f F7/2

14 1 2 2 14 2 1 14 1 2 14 1 Lu 4f 5d 6s D3/2 4f 6s S0 4f 6s S1/2 4f S0

Table 2 CCSD(T) results for the first ionization potentials (kcal/mol).

Frozen-core with cc-pVnZ-DK3 sets CV effects with cc-pwCVnZ-DK3 sets   d e f DZa TZb QZc CBS DZa TZb QZc CBS SO  Final Expt

La 1.38 0.79 0.28 128.76 -0.75 -0.05 -0.02 2.36 -0.53 130.58 128.61±0.01 g Ce 1.01 0.94 0.34 127.27 -1.10 -0.19 -0.07 2.85 -1.24 128.89 127.72±0.01 Pr -1.32 -0.49 -0.18 125.56 0.21 0.09 0.03 0.33 0.07 125.95 126.2±0.2

Nd -1.35 -0.51 -0.19 126.87 0.16 0.08 0.03 0.31 0.07 127.25 127.41±0.01

Pm -1.41 -0.54 -0.20 128.18 0.14 0.08 0.03 0.33 0.09 128.61 128.7±0.2 Sm -1.40 -0.56 -0.21 129.36 0.08 0.06 0.02 0.24 0.30 129.66 130.147

7 Eu -1.44 -0.57 -0.21 130.37 0.05 0.05 0.02 0.24 0.09 130.70 130.762

Gd -0.80 -0.46 -0.19 140.98 0.00 0.01 0.00 0.31 -0.08 141.21 141.818

Dy -1.63 -0.64 -0.24 136.65 0.01 0.03 0.01 0.22 0.13 137.00 136.958

Ho -1.69 -0.67 -0.25 138.56 0.00 0.03 0.01 0.20 0.12 138.87 138.86±0.01 Er -1.71 -0.67 -0.25 140.53 -0.01 0.02 0.01 0.17 0.13 140.83 140.85±0.02

Tm -1.78 -0.71 -0.26 142.33 -0.02 0.02 0.01 0.16 0.14 142.63 142.614

Yb -1.89 -0.76 -0.28 144.05 -0.04 0.00 0.00 0.18 0.16 144.37 144.224 h Lu -7.21 -3.42 -1.24 120.72 0.02 0.03 0.01 -0.68 2.40 122.44 125.124

MUD 0.52 0.03i

MAX 2.68

a Difference between DZ and the extrapolated CBS limit. b Difference between the TZ and the extrapolated CBS limit.

c Difference between the QZ and the extrapolated CBS limit. d Spin-orbit contribution calculated at the 2-component X2C-HF level of theory with uncontracted cc-pVDZ-DK3 basis sets. See the text. e Final composite result, CBS(FC+CV)+SO f Ref. 47. Values in italics were also taken from this reference but these were noted as being obtained by interpolation or extrapolation of experimental values or by semi-empirical calculation. The estimated error bars from experiment are given if greater than 0.001 kcal/mol. g The 3F excited state of neutral Ce atom was utilized since the low-spin 1G was not amenable to single reference CCSD(T). -1 1 3 Correspondingly the SO correction includes the experimental 228 cm excitation energy from G4 to F2. h A J-average of just the experimental 2D energy levels yields a SO correction of 3.42 kcal/mol. i Average experimental uncertainty.

Table 3 CCSD(T) results for the second ionization potentials (kcal/mol).

Frozen-core with cc-pVnZ-DK3 sets CV effects with cc-pwCVnZ-DK3 sets   d e f DZa TZb QZc CBS DZa TZb QZc CBS SO  Final Expt

La -5.81 -1.67 -0.60 255.19 0.26 -0.01 0.00 -0.84 0.11 254.45 257.930

Ce -1.93 -0.78 -0.30 259.74 14.05 5.33 1.95 -10.30 1.56 251.00 250.2±1.6 Pr -2.89 -0.74 -0.27 244.52 0.22 0.17 0.06 0.78 0.03 245.34 243.3±1.6

Nd -3.10 -0.82 -0.30 248.32 0.15 0.16 0.06 0.76 0.05 249.13 247.2±1.6

Pm -3.35 -0.91 -0.33 252.10 0.09 0.14 0.05 0.81 0.06 252.95 251.4±1.6 Sm -3.42 -0.94 -0.34 255.57 0.01 0.12 0.04 0.66 0.78 256.35 255.3±1.6

Eu -3.59 -1.03 -0.37 258.79 -0.03 0.11 0.04 0.67 0.10 259.56 259.2±0.1

Gd -4.57 -1.02 -0.35 279.27 -0.16 0.03 0.01 0.85 -0.44 279.69 278.8±1.6 Tb -3.85 -1.14 -0.41 266.04 -0.03 0.09 0.03 0.51 0.10 266.64 265.7±1.6

Dy -4.08 -1.22 -0.44 268.94 -0.04 0.08 0.03 0.50 0.12 269.56 269.1±1.6

Ho -4.19 -1.28 -0.46 271.97 -0.06 0.06 0.02 0.45 0.16 272.58 272.1±1.6 Er -4.25 -1.31 -0.47 275.14 -0.07 0.05 0.02 0.37 0.19 275.70 275.1±1.6

Tm -4.34 -1.36 -0.49 278.03 -0.09 0.03 0.01 0.33 0.21 278.57 277.9±1.6

Yb -4.49 -1.43 -0.52 280.75 -0.09 0.02 0.01 0.29 0.23 281.26 280.859 Lu -2.60 -0.99 -0.40 322.00 -0.07 0.00 0.00 0.28 0.21 322.49 321±9

MUD 1.14 1.78g

MAX 3.48

a Difference between DZ and the extrapolated CBS limit. b Difference between the TZ and the extrapolated CBS limit.

Table 4 CCSD(T) results for the third ionization potentials (kcal/mol).

Frozen-core with cc-pVnZ-DK3 sets CV effects with cc-pwCVnZ-DK3 sets   d e f DZa TZb QZc CBS DZa TZb QZc CBS SO  Final Expt

La -5.03 -1.54 -0.57 439.16 0.07 -0.13 -0.05 -0.56 2.22 440.80 442.24±0.01

Ce -9.15 -3.51 -1.26 448.93 -12.86 -5.13 -1.88 8.08 0.95 457.96 465.76±0.06 Pr -8.60 -3.86 -1.42 486.43 -10.54 -4.61 -1.69 5.98 -0.20 492.21 498.65±0.06

Nd -11.18 -4.65 -1.72 500.26 -7.44 -3.91 -1.43 6.93 -1.41 505.77 510.6±6.9

Pm -13.38 -5.55 -2.09 507.29 -5.60 -3.20 -1.17 9.18 -2.47 513.99 514±9 Sm -13.22 -5.51 -2.07 544.27 -3.88 -2.89 -1.06 4.44 1.55 545.82 540.3±6.9

8 Eu -13.03 -5.42 -2.02 578.97 -2.82 -2.81 -1.03 0.68 -9.60 570.06 574.7±2.3

Gd -8.62 -3.78 -1.38 470.18 0.17 0.14 0.05 -0.82 3.11 472.47 475.7±2.3 Tb -21.54 -9.16 -3.37 488.09 -2.18 -1.50 -0.55 3.33 3.17 494.57 505.3±2.3

Dy -21.33 -9.00 -3.33 519.51 -1.82 -1.42 -0.52 1.23 0.27 521.01 528.8±2.3

Ho -25.97 -10.28 -3.78 525.31 -1.15 -1.07 -0.39 1.92 -3.57 523.66 526.7±2.3 Er -28.97 -11.46 -4.20 527.92 -0.36 -0.79 -0.29 1.89 -5.87 523.93 524.4±2.3

Tm -28.70 -11.44 -4.29 559.47 1.06 1.03 0.38 0.15 -8.86 550.76 546.1±2.3

Yb -27.56 -10.89 -4.02 590.09 -0.55 -0.93 -0.34 -2.23 -12.67 575.19 577.74±0.58 Lu -7.35 -2.00 -0.70 483.01 -0.16 0.00 0.00 0.45 0.22 483.69 483.23±0.14

MUD 4.24 2.65g

MAX 10.73

a Difference between DZ and the extrapolated CBS limit. b Difference between the TZ and the extrapolated CBS limit.

19 - Table 5 DKH3-CCSD(T) results for the ground S9g state of Gd2 with comparison to experiment and previous calculations.

a # of basis functions re e De BSSE

Basis Set (Å) (cm-1) (kcal/mol) (kcal/mol)

Valence correlation

VDZ-DK 168 2.9722 133.7 32.71 2.91

VTZ-DK 290 2.9350 141.39 35.61 1.34

VQZ-DK 428 2.9160 142.0 37.41 0.89

CBS[TQ] 2.9053 142.1 38.42

ANO- RCC-DZP 126 2.8839 159.34 44.70 22.22

ANO- RCC-TZP 198 2.9081 145.7 37.89 7.07

ANO- RCC-QZP 270 2.9042 142.8 38.65 4.48

ANO- RCC-Large 384 2.9117 141.6 36.34 0.70

SAP-DZP 194 2.9059 141.6 28.65 1.98

SAP-TZP 292 2.9009 143.6 31.63 0.84

SAP-QZP 462 2.9109 143.8 35.94 0.35

SAP-DZP+diffuse 244 2.9139 138.1 35.35 1.60

SAP-TZP+diffuse 364 2.9158 141.5 36.00 0.52

SAP-QZP+diffuse 560 2.9098 142.7 37.48 0.37

Core correlation (4s4p4d)

wCVDZ-DK b 200 2.9595 136.1 33.30 3.41

(-0.0060) (+0.4) (+0.84)

wCVTZ-DK b 372 2.9319 141.5 35.03 0.85

(-0.0042) (+1.6) (-0.19)

wCVQZ-DK b 564 2.9166 142.0 36.58 0.48

(-0.0022) (+0.5) (-0.34)

CBS[wTQ] b 2.9081 142.2 37.45

(-0.0012) (-0.13) (-0.43)

SAP-DZP+diffuse 244 2.9094 138.5 35.42 2.03

SAP-TZP+diffuse 364 2.9128 142.3 35.73 0.72

SAP-QZP+diffuse 560 2.9074 143.0 37.09 0.52

CBS[wTQ] +SO 2.9077 142.6 33.26

CBS[wTQ] +SO + 2.9079 142.4 33.45 c QED [33.24]

Expt. 138.7±0.4d 41.1±8.1e

48±16d

CCSD(T)/QZ(PP)f 2.877±0.020 149±2 31.8±4.2

GASSCF/DZP(DK2)g 3.06 140 48

a Calculated at r = 2.9237 Å. b Core-valence correlation effects calculated with this basis set given in parentheses. c Includes anharmonic zero-point vibrational effects at the CCSD(T)/CBS[wTQ]+SO+QED level of theory. Standard thermal corrections yield a 298 K dissociation energy of 34.4 kcal/mol, which when combined with the experimental formation enthalpy of Gd(g),62

Hf(298) for Gd2 of 160.1 kcal/mol. d Spectroscopic value in Ar matrices from Ref. 56. e Obtained by 3rd law methods, Ref. 57. f Ref. 55; Their listed uncertainties are based on BSSE estimates from counterpoise calculations. g Ref. 75.

8 S- Table 6 CCSD(T) results for the X 7/2 ground state of GdF with comparison to experiment and previous calculations.

-1 Basis/Corr. Space re (Å) e (cm ) De (kcal/mol)

VDZ-DK/val 1.9875 600.1 150.15

VTZ-DK/val 1.9718 595.8 152.08

VQZ-DK/val 1.9653 603.6 153.65

CBS[TQ]/val 1.9621 607.1 154.44

a wCVDZ-DK/cv 1.9845 601.2 151.19

(-0.0032) (+2.9) (+1.46)

a wCVTZ-DK/cv 1.9685 598.4 152.66

(-0.0014) (+0.8) (+0.86)

a wCVQZ-DK/cv 1.9633 604.9 154.29

(-0.0013) (+0.7) (+0.87)

a CBS[wTQ]/cv 1.9606 607.9 155.18

(-0.0013) (+0.6) (+0.88)

CBS[wTQ]/cv+SO 1.9597 608.3 152.58

CBS[wTQ]/cv+SO+QED 1.9597 608.4 152.52 b [151.66]

d Expt. 1.962c 606.8c 141.1±4.0

e ACPF/sc-PP 1.939 642 156

a Core-valence correlation effects (from correlating the 4s4p4d electrons of Gd and the 1s of F) calculated with this basis set given in parentheses. b Includes anharmonic zero-point vibrational effects at the CCSD(T)/CBS[wTQ]/cv+SO+

QED level of theory. Standard thermal corrections yield a 298 K dissociation energy of 152.9 kcal/mol, which when combined with the experimental formation enthalpies of Gd(g) and F(g) 62 66 atoms, 97.25±0.48 and 18.97±0.07 kcal/mol, respectively, yields a Hf(298) for GdF of –36.6 kcal/mol. c Ref. 63. The equilibrium bond length was calculated using their B0 and α values. d D298 value, Ref. 64. e Ref. 65.

8 a Table 7 CCSD(T) results for the atomization energy De (kcal/mol) of X A" GdF3.

b Valence correlation Core correlation 

Basis Set De Basis Set De

VDZ-DK 437.12 wCVDZ-DK 439.31

VTZ-DK 443.84 wCVTZ-DK 444.08

VQZ-DK 449.58 wCVQZ-DK 449.86

CBS[TQ] 452.72 CBS[wTQ] 453.13

CBS[wTQ]/cv+SO 449.46

CBS[wTQ]/cv+SO+QED 449.49 c [447.07]

Expt. 443 ± 5d

e 443±10

a Calculated using the DKH3-CCSD(T)/VTZ-DK optimized (pyramidal C3v) geometry of r -Gd-F)=119.3º. (The F atoms are 4.9º below the D3h plane.) b Includes 4s4p4d correlation of Gd and 1s correlation on F. The CBS[wTQ] value for just valence correlation was 452.25 kcal/mol. c Includes harmonic zero-point vibrational effects at the CCSD(T)/VTZ-DK level of theory. Standard thermal corrections yield a 298 K dissociation energy of 449.4 kcal/mol, which when combined with the experimental formation enthalpies of Gd(g) and F(g) atoms (see footnote to Hf(298) for GdF3 of –295.2 kcal/mol. d Ref. 71 for T=298 K. e Ref. 64 for T=298 K.

Figure 1. Contribution of angular momentum functions to the MRCI correlation energy in the 5f86s2 7 ( Fg) state (5s5p4f electrons correlated) of the Gd atom. The absolute values of the incremental correlation energy lowerings, |∆Ecorr|, are plotted in mEh against the number of functions in the expansion of dfghi functions in a large contracted spdf set. See the text.

Figure 2. Basis set convergence relative to the estimated CBS limits for the (a) SCF and (b) frozen-core CCSD(T) correlation energy contributions for the 3rd ionization potentials of La, Eu, Tb, and Lu (in kcal/mol).

Figure 3. Comparison of the CCSD(T) basis set convergence of (a) the dissociation energy of Gd2 and (b) the corresponding basis set superposition errors (all in kcal/mol). The rightmost basis set, "large", corresponds to the ANO-RCC-Large set. The DKH3 scalar relativistic Hamiltonian was used throughout.

Chapter 4

The Thermodynamic and Spectroscopic Properties of

LnX3 (Ln= La, Nd, Gd, Dy, Lu; X=F, Cl, Br)

Abstract Coupled cluster theory has been employed with new correlation consistent basis sets to accurately calculate the thermochemical and structural properties of several LnX3 molecules

(Ln= La, Nd, Gd, Dy, Lu; X=F, Cl, Br). The calculations, which utilized the 3rd-order

Douglas-Kroll-Hess Hamiltonian (DKH3) and exact-2-component (X2c) scalar relativistic

Hamiltonian, show that to obtain reliable structure and vibrational frequencies the basis sets need to be at least of triple-zeta quality. Among the selected molecules, only LaF3, NdF3, and

GdF3 have a pyramidal C3v equilibrium geometry while the others possess a planar D3h structure. The bond lengths of the LnX3 molecules vary nearly linearly with atomic number, demonstrating the lanthanide contraction. Atomization energies (AE) are calculated under the

Feller-Peterson-Dixon composite coupled cluster framework and are generally in good agreement with experimental values where these are available. The disagreement of AE suggests a re-measurement for certain molecules.

I. Introduction

The gas-phase lanthanide trihalides, LnX3, are important in many industrial applications

92 such as high temperature extraction and separation1, nuclear waste purification2 and high

3 pressure metal halide discharge lamp . Yet despite rich publications regarding to LnX3 molecules due to the stable +3 oxidation state of the Ln, the molecules have obsessed people about one simple and basic property, their molecular geometry. For decades, people have argued about its controversial structure, through either experimental or theoretical approaches, and can hardly conclude whether the molecule is C3v or D3h, i.e. whether the molecule is planar or pyramidal.

The determination of a molecule’s geometry is important in theoretical studies since more sophisticated calculations would require an accurate geometry description. Moreover, the entropy from statistic mechanics depends on the symmetry number, which is different for

4 the two point groups: 3 for C3v and 6 for D3h . Therefore, if the point group is incorrectly assigned, the rotational partition function will either be a factor of 2 too large or too small.

In principle, it should be straightforward to determine the molecule’s point group by spectroscopy. From the IR irreducible representations, the C3v and D3h normal modes can be represented as:

C3v  2A1  2E

D3h  A1'A2"2E'

According to the selection rules, the ν1 (A1’) mode should be forbidden in the IR spectrum. However, previous experimental results are not consistent with respect to this point.

5 For the LaF3 molecule as an example, Hastie et al. observed the ν1 (A1) peak in different inert gas environment in 1970. But one year later DeKock6 published the opposite result that

5 no ν1 peak was observed in the same environment. For LuF3, Hastie et al. observed the ν1 fundamental in Ne and Ar gas, but no such an observation in the N2 gas. In regards to the out-of-plane bending mode (v2), previous studies found that the bending fundamental frequency is near the lower detection limit4. With such a low wave number, the matrix gas

93 interaction is able to shift the peak significantly7.

Despite the dispute of the planarity represented by the bond angle, as one of the two geometry parameters, there is also a controversy regarding to the other geometry parameter, the bond length. Early studies8-11 report or imply that the lanthanide-halide (Ln-X) bond lengths decrease linearly with respect to the Ln atomic number, representing the phenomenon of the lanthanide contraction. But another study12 claimed that the lanthanide-oxide (Ln-O) bond lengths decreases following a quadratic curve and the conclusion was applied to lanthanide trihalides13 to study the lanthanide contraction. This disagreement may indicate inaccurate calculations of the bond lengths and thus demands more accurate and reliable examinations.

In the past decades, many theoretical and experimental studies have been published concerning lanthanide trihalides. The work prior to 2002 has been thoroughly reviewed by

Kovacs and Konings4. It was concluded that (1) the Becke-Perdew86 Dirac-Slater (BP-DS) method with TZ, or TZd basis sets gave the best performance for LnX3 up to that date, implying the superiority of Slater type basis functions and relativistic effects. (2) If the effective core potential (ECP) is used, the importance of the extended basis set should be emphasized. (3) The joint experimental and theoretical information should be considered together with the given errors. For example of the electron diffraction (ED) technique, the uncertainty is at least 2 pm for bond lengths and 2° for bond angles. (4) For the assessment of the structural and vibrational data, the more accurate methods are required.

Since 2002, many papers have been published using new DFT functionals or other

13-23 14-16,21,22,24 methods such as MP2 in the hope to solve the problem, (see LaX3 , NdX3 ,

7,13-16,21,22,25 14-16,21,22,26-30 13-16,18,20-22,31-35 GdX3 , DyX3 , and LuX3 for selected examples as early, middle and late trihalides), and the basis sets may be customized by adding additional diffusion functions. Two important remarks must be mentioned about these studies. Firstly,

94 using the same MP2 method, different basis sets have led to contradictory conclusions36. In the study using basis sets without diffuse f and g functions, the LuF3 molecule was optimized

20 to be C3v while the diffuse fg-included basis set gives a D3h geometry . Secondly, using the same basis set, lower-level method gives inconsistent results with high-level method. It was reported that the MP2 method gives a C3v geometry while the coupled cluster with singles, doubles and perturbative triples, CCSD(T), gives the D3h geometry for the same LuF3 molecule20.

The nature behind these struggles is that the C3v configuration lies very closely in energy to the D3h configuration, with the energy difference being less than the zero-point energy

19 (ZPE) . As a result, the determination of LnX3 bond angles and bond lengths is very sensitive to the correlation energy. The overestimation or underestimation of the correlation energy will lead to an improper assignment of the molecular geometry as well as the fitting function of the bond lengths. Therefore, it is crucial to adopt a high-level theory and high-quality basis sets to recover the correlation energy accurately. Such a method should be size consistent and should contain high-order excitation operators36 in order to recover the correlation energy as accurate as possible, and such basis sets should include the correlation functions systematically to avoid any arbitrary inclusion of the correlation functions, such as the fg-included basis sets mentioned in the above LuF3 example.

On the other hand, the thermochemistry of LnX3 molecules received relatively less attention comparing to their spectroscopy studies. The probably most cited experimental thermodynamic values were recommended by Myers37 with an uncertainty up to 10 kcal/mol for the atomization energy. In recent years, various DFT functionals21,31,34,38-40 have been used to calculate the atomization energy. But the agreement is not very satisfactory for certain

LnX3 molecules, and it is not known whether the disagreement comes from experiment’s uncertainty or the computation methods. Taking the NdBr3 molecule as an example, the

95 all-electron DKH calculation with the PBE0 functional39 deviates from Myers’s experimental value by 10.9 kcal/mol, but the experiment’s uncertainty is also 10 kcal/mol. Thus it is hard to conclude which value is accurate and a calculation with higher level of theories is needed.

In this study, the CCSD(T) method and newly-developed correlation consistent basis

41 sets for the lanthanides are employed to study the LnX3 (Ln=La, Nd, Gd, Dy, Lu; X=F, Cl,

Br) molecules. The thermochemistry of these selected molecules are studied under the

Feller-Peterson-Dixon (FPD) framework42. For these representative lanthanide trihaldies, it would be the first time to use CCSD(T) and correlation consistent basis sets to calculate these molecules. The calculated results should provide the most accurate and conclusive results for these fundamental lanthanide halides.

II. Computational details

In this study, the spin-free, spin-orbital coupling and Gaunt calculations were carried out by the DIRAC43 code. The geometry optimization and frequency calculations were done with the MOLPRO package44. Especially, for the frequency calculations, unless explicitly mentioned, the CFOUR45 code was used to generate the grid point and fit to get the frequencies. The energies for each grid point were calculated by MOLPRO. The natural population analysis was carried out by NBO646.

Using MOLPRO, the geometry optimizations and frequency calculations of LnX3 molecules were carried out at the CCSD(T) level of theory with the 3rd-order

Douglas-Kroll-Hess (DKH) Hamiltonian47. The DKH correlation consistent basis sets for

Ln41 and equivalent basis sets with augmented functions for halogen48-50 were used at the triple zeta size. Specifically, the cc-pVTZ-DK3 and aug-cc-pVTZ-DK sets were used for Ln and halogen, respectively. Especially, the basis sets with additional d functions were used for the Cl atom51. For simplicity, they are abbreviated as VTZ sets.

To explore the frequency dependence on the basis sets, the CFOUR code was employed

96 to use its analytical gradient to eliminate numerical errors. The geometry and frequencies of

GdF3 and LaCl3 molecules were optimized and calculated with the exact 2-component (X2c)

Hamiltonian52. The DKH3 correlation consistent basis sets for Ln were recontracted with the

X2c Hamiltonian and gave the X2c basis sets with double zeta (VDZ), triple zeta (VTZ), and quadruple zeta (VQZ) sizes. The new X2c basis sets for Ln were used together with the aug-cc-pVnZ-DK (n=D, T, Q) sets for halides to calculate the molecules’ frequencies. Again, additional d functions were included for the Cl atom.

For the open-shell molecules of the NdX3 and DyX3, the state-averaged complete active space self-consistent field (SA-CASSCF) method was employed before the geometry optimization to determine the dominant configurations in the CASSCF wave function.

Subsequent CCSD(T) calculations then adopted the most dominant configuration as the reference configuration. The active space in the SA-CASSCF calculations consisted of the 7 f orbitals and the number of states to be averaged was chosen to be the same as the degeneracy of a neutral lanthanide atom at its lowest electronic state.

The thermochemistry of the selected molecules was calculated under the FPD framework42. The total composite energy is expressed as a sum of different FPD terms:

Ecomposite=Eval(CBS)+ΔEouter-core(CBS)+ ΔE(SO)+ ΔE(gaunt)+ ΔMisc. (1)

The first term on the right hand side in Eq. 1 stands for the energy at the complete basis set (CBS) limit. To obtain the energy at the CBS limit, the VTZ and VQZ basis sets were used to first calculate the energies at the CCSD(T)/VTZ optimized geometry (for the GdF3 and LaCl3 molecules, at the CCSD(T)/VQZ geometry). The HF energy was then extrapolated by Karton-Martin’s formula53:

-6.57 n En =EHF-Limit+A(n+1)e (2) where En are the HF energies calculated by VTZ and VQZ sets and n equals 3 and 4 as in the cardinal number in the basis set notation. While the correlation energy extrapolation used:

1 4 En E CBS  A/() n  2 (3) where En are TZ or QZ energies with n equals to 3 or 4, respectively.

Secondly, to recover the outer-core electron correlation, the weighted core-valence correlation consistent basis sets for Ln metals and equivalent weighted core-valence

Douglas-Kroll correlation consistent basis sets with augmented functions for halides were used together with the CCSD(T) method. The outer-core electrons consist of 4s4p4d for the

Ln metal, 1s for the F atom, 2s2p for the Cl atom and 3s3p3d for the Br atom. The same extrapolation formulas were used to obtain the CBS limit energies. The resulting energy contributions were added to the valence CBS energies following the FPD guideline.

The third and fourth terms in the FPD framework are the spin-orbit coupling and Gaunt contributions. The X2c Hamiltonian and the uncontracted DZ quality DK3 basis sets were used to calculate the spin-free (SF), spin-orbit coupling (SOC) and Gaunt energies at the average of configuration HF level of theory. The difference between SF and SOC, SOC and

Gaunt energies are added stepwise to the composite energies.

The last term in Eq.1 in this study includes the zero point energy (ZPE) as well as the thermal correction from 0K to 298K, which are derived from the frequency calculations and the usual statistical mechanics expressions, respectively.

III. Discussion A. Molecular geometry

For convenience of discussion, the coordinates used in the current study are depicted in

54 Fig. 1. The T1 diagnostic , maximum T2 amplitude, optimized Ln-X bond length (r), the angle between r and the C3v symmetry axis (θ), the deviation from planarity (Δθ) and the relative energy between D3h and C3v configurations (ΔE) are given in Table 1. As an

98 estimation of multi-reference character, it can be seen that for all molecules the T1 diagnostic and T2 amplitude are small except for NdX3 molecules relatively large T2 amplitude. This may cause some multi-reference concerns for the NdX3 molecules, but the details will be discussed later in the atomization energy section. But for other molecules, the small T1 diagnostic and T2 amplitude indicate good single reference character for the systems.

Among all the molecules, it can be seen that except for LaF3, NdF3, GdF3, and LaCl3, all of the other molecules have a planar equilibrium geometry. Among the four LaF3, NdF3,

GdF3 and the LaCl3 molecules that have pyramidal equilibrium structure, the LaF3 and NdF3 molecules are more strongly non-planar with Δθ values ranging up to ~13°. But still, the inversion barrier (ΔE) is no greater than 0.4 kcal/mol. It may be difficult to draw any conclusions regarding to the GdF3 and LaCl3 molecules, however, since they are only slightly non-planar with this basis set and have essentially the same energy as the D3h structure. This will be further quantified below by investigating the basis set dependence of the energies and harmonic frequencies to these two molecules. Yet overall, it can be concluded that as the Ln center or the halides become heavier and heavier, the molecules favor planar structure, in agreement with previous studies14,15,21.

Table 2 shows the harmonic frequencies of LnX3 molecules calculated at the

CCSD(T)/VTZDK3 level of theory, which are the most accurate theoretical results reported up to this date. It can be seen that when restricted in the D3h group, the LaF3, NdF3, GdF3 and the LaCl3 molecules have a distinct imaginary frequency at the A2” vibrational mode, which is the out-of-plane bend mode. But on the other hand, when the symmetry is relaxed to C3v, there are no imaginary frequencies. Hence, the D3h structure is a transition state connecting to the two umbrella-like C3v structures. Not unexpectedly, these four molecules at the D3h transition state are very similar to those calculated at the C3v minimum, with the exception of course of the imaginary mode. For all the other molecules, no imaginary frequencies appear,

99 thus they are confirmed as planar molecules.

Table 3 shows the dependence of the GdF3 and LaCl3 harmonic frequencies on increasing the basis set towards the CBS limit. It can be seen that as the basis set becomes more and more complete, the GdF3 molecule becomes more pyramidal, and the imaginary frequency of the A2” mode of the D3h transition state increases. With a lower-quality basis set, i.e. the DZ sets, the GdF3 molecule is actually predicted to be planar. This shows the incompleteness error of the DZ sets. For the LaCl3 molecule, on the other hand, the molecule is essentially flat and with no imaginary frequencies at the DZ and QZ quality of basis sets. The TZ geometry, however, is more bent and has an imaginary frequency. This conflict shows that the real minimum calculated by the DZ sets should fortuitously come from the error cancellation, while the imaginary frequency found in TZ calculation indicates the insufficiency of the TZ sets. Thus, the true minimum of the LaCl3 molecule should be planar as calculated the QZ sets. In term of the relative energies (C3v vs. D3h) for the 2 molecules, with the QZ basis set, the energy difference for the GdF3 molecule is about 0.1 kcal/mol while for the LaCl3 molecule the energy difference is still 0.

Based on these observations, it can be summarized that in order to obtain reliable results, the structures and frequencies of these molecules should be calculated with at least

TZ-quality basis sets. One may argue if other molecules will become pyramidal when QZ basis sets are used. Yet according to this and previous studies14,15,21, it has been found that the

LnX3 molecules become more planar as the lanthanide and halide atoms become heavier. It is thus highly unlikely for these molecules become pyramidal with basis sets larger than VTZ.

Therefore, it is concluded here that among the selected molecules, only the LaF3, NdF3, and the GdF3 molecules have the C3v minimum at the equilibrium while other molecules have the D3h minimum.

Having decided the point group of the LnX3 molecules, in Table 4 there shows the

100 calculated results in this work comparing to the experimental results from the IR measurements mainly in the Ar environment or the recommended theoretical values by

Kovacs and Konings4. It can be seen that the calculated results match experiments well. The mean absolute deviations (MAD) for each mode are 5.5, 11.7, 3.0, and 5.8 cm-1, respectively.

The difference between experimental and this theoretical work should mainly come from the anharmonicity, which has been shown at the MP2 level of theory20 that the anharmonicity can

-1 shift the ω2 harmonic frequencies by 10~40 cm for LnX3 (Ln=La, Lu; X=F, Cl) molecules.

But for other modes, the MP2 calculations show the anharmonicity has less influence. This is in consistence with this work that the ω2 mode has the largest MAD. Other factors include the basis set effect and the matrix effect. The basis set effect may be revealed in Table 3 that larger basis sets may shift the frequencies by at most ~20 cm-1. While the matrix effect can shift the frequencies 17 cm-1, as reviewed by Kovacs and Konings4. It should be noted at last that the recommended values of ω1 modes in Table 4 are actually theoretical results, not experimental values.

Focusing on the trends of the symmetric stretching mode (ω1) shown in table 4, it can be seen that the frequencies of LnF3 are larger than those of LnCl3, which in turn are larger than those of LnBr3. This correlates with the LnX3 bond lengths, whereby LnF3 bond lengths are shorter than LnCl3, which are shorter than LnBr3. Varying the Ln atom with a given halide, the symmetric stretching frequencies increase from La to Lu, also following the corresponding trend of LnX3 bond lengths, reflecting the lanthanide contraction. One outlier of this pattern is the DyBr3 molecule, whose ω1 frequency is lower than that of the GdBr3 molecule. This exception should come from the numerical error. For the out-of-plane bending mode (ω2), the frequencies are very small for all molecules. Upon comparison to the recommended values, the calculated results are lower in frequency and some are outside of the stated error bars of the recommended values. This may arise come from matrix effects as

101 well as basis set effect. For the experimentally well-characterized asymmetric stretching mode (ω3), similar trends can be observed as for the ω1 mode.

B. Lanthanide contraction and natural population analysis

The plot of LnX3 bond lengths versus the atomic number of the Ln atom is shown in

Figure 2. It can be seen that all the LnX3 bond lengths are well described by a linear function

2 with adjusted R as high as 0.984. As the Ln atomic number increases, the Ln-X bond lengths decrease. This is well known as the lanthanide contraction. However, previous study12 claimed a quadratic function for the lanthanide contraction, rather than a linear function. And it can be found that the confidence level for the calculated error bars is at 95% and the correlation factor R2 is up to 99%. As both fitting have reached high confidence level, it can only be concluded that the quadratic fitting has an overfitting problem. Thus the lanthanide contraction should simply follow a linear function.

On the other hand, it can be found from Fig. 2 that the three linear functions of lanthanide fluorides, chlorides and bromides have very similar slopes, indicating that the Ln center is in a similar environment with a common oxidation state. Especially, the LnCl3 and

LnBr3 slopes are more similar comparing to that of LnF3.

This similarity coincides with the natural population analysis (NPA) in Table 5. It can be seen that the Ln centers have more d occupations in the Cl and Br environment comparing to the F environment. In details, the Ln d occupations in trichlorides and tribromides are nearly twice of those in trifluoride and thus can be rounded up to 1 while the d occupations in trifluorides are close to 0. As a result, the Ln atoms in the Cl and Br environment are more like in the +2 oxidation state while in the F environment the Ln atoms are in +3 oxidation state. This finding matches the general chemical concept that Cl and Br are more similar than

102

C. Thermochemistry of LnX3 molecules

The atomization energies (AE) of the LnX3 molecules are shown in Table 6, which also shows the individual contributions for each FPD terms. They include CBS extrapolation, contributions from outer core electron correlations, spin-orbit coupling, zero point energy

(ZPE) and the thermal correction from 0K to 298K. It can be seen from this table that the

LnF3 molecules have larger AE than LnCl3 molecules than LnBr3 molecules, indicating stronger bonds in LnF3 than LnCl3 than LnBr3. Among each halide group, the general trend of the AE follows a “W” shape, in which the LaX3, GdX3 and LuX3 molecules have larger AE than the NdX3 and DyX3 molecules. This can be rationalized from the occupation of f electrons on the Ln atoms, where La, Gd and Lu have empty, half, and full occupation, respectively. The SOC effect is generally small for LnF3 and LnCl3 molecules, but more significant for the LnBr3 molecules. The Gaunt interaction, on the other hand, can be neglected for most molecules but for bromides especially with heavy Ln such as LuBr3, the

Gaunt contribution can be nearly 1 kcal/mol. The ZPE and the thermal correction generally have similar contribution to the AE, but with different signs, so they tend to cancel each other.

The final composite results are compared to the available experimental values. It can be seen that in general there is a good agreement between the calculated results and the experimental values. But for certain molecules there are still large discrepancies, such as LaF3,

DyF3, LuF3, LuBr3 molecules. Concerning the discrepancy between the experiments and calculations, it should first be noted that the experimental values were obtained by a 4-step thermochemical cycle. If all quantities were measured directly, the error bar is 5 kcal/mol, while if one or more quantities were estimated, the error bar is 10 kcal/mol. Thus it can be seen that the experimental error bars are too empirical and not well defined. On the

103 theoretical side, it may raise concerns about the multi-reference concerns about the selected molecules. But from Table 1, it can be seen that the molecules with large T2 amplitudes, such as NdX3 molecules, have a good agreement with both thermodynamic and spectroscopic experiments. Therefore it can conclude that the multi-reference character shouldn’t be the major source of the experimental-theoretical gap. This claim can be further supported by molecules such as LaX3, GdX3 or LuX3, which should have a very good single reference character in principle. But for molecules such as LaF3, LuF3, and LuBr3, the calculated AE are also outside the experimental range. Moreover, comparing homologous trihalides to these problematic molecules, such as comparing LaCl3 and LaBr3 vs. LaF3, it can be found that

LaCl3 and LaBr3 have a good agreement with experiments using the same computation prescription used for LaF3. Therefore it becomes questionable about the accuracy of the experimental values. Giving the facts of good agreement of frequencies as well as the experimental bars not well defined, it is thus suggesting re-measurements for some LnX3 atomization energies.

IV. Conclusions The CCSD(T) method combined with all-electron DKH3 and X2C scalar relativistic

Hamiltonian and new correlation consistent basis sets were used to optimize the geometries and calculate the harmonic frequencies of several representative LnX3 molecules to provide the most accurate theoretical work up to date.

The calculation shows that among the selected molecules, only the LaF3, NdF3, and the

GdF3 molecules have a C3v minimum while all the other molecules have a planar D3h geometry. The energy difference between the C3v and D3h equilibrium geometry is however very small.

The frequency dependence on the basis sets is explored. It is found that at least the TZ

104 quality basis sets should be used to accurately calculate harmonic frequencies. For the LaCl3 molecule, however, the QZ set should be used.

The Ln-X bond lengths are plot against the atomic number of the Ln metal. The decrease of the bond lengths along the lanthanide row reflects the well-known phenomenon of the lanthanide contraction and the contraction should be fitted in a linear function.

The atomization energies of the LnX3 molecules are calculated under the FPD framework and in general have a very good agreement with the experimental values. The experimental-theoretical gap should mainly come from the multi-reference character and not well-defined experimental error bars. For some of the LnX3 molecules, an experimental re-measurement is suggested.

105

Table 1. Molecular geometries for LnX3 molecules optimized at CCSD(T)/VTZDK3 level of theory. ΔE shows the energy difference of the D3h

structure from the C3v structure.

a a C3v T1 diag. Max T2 r(ang.) θ Δθ D3h T1 diag. Max T2 r(ang.) ΔE(kcal/mol) 1 1 A1 LaF3 0.02 2.1211 78.32 11.68 A1’ LaF3 2.1277 0.3 4 4 A2 NdF3 0.02 0.15 2.0784 102.80 12.80 A1” NdF3 0.02 0.16 2.0819 0.4 8 8 A2 GdF3 0.02 2.0459 94.70 4.70 A1” GdF3 0.02 2.0469 0.0 6 A1” DyF3 0.02 2.0238 0.0 1 A1’ LuF3 0.02 1.9763 0.0

1A LaCl 0.01 2.5956 86.36 3.64 1A ’ LaCl 0.01 2.5963 0.0 10 1 3 1 3

6 4 A1” NdCl3 0.01 0.18 2.5389 0.0 8 A1” GdCl3 0.01 2.4879 0.0 6 A1” DyCl3 0.02 2.4559 0.0 1 A1’ LuCl3 0.01 2.4029 0.0

1 A1’ LaBr3 0.01 2.7505 0.0 4 A1” NdBr3 0.02 0.15 2.6903 0.0 8 A1” GdBr3 0.02 2.6362 0.0 6 A1” DyBr3 0.02 2.6029 0.0 1 A1’ LuBr3 0.01 2.5505 0.0

a The maximum T2 amplitude is not listed if smaller than 0.05.

Table 2. Harmonic frequencies calculated at CCSD(T)/VTZDK3 level of theory for the LnX3 molecules. The unit is in cm-1.

C3v A1 A1 E E D3h A1’ A2” E’ E’ LaF3 531.2 54.7 501.0 119.0 LaF3 523.7 43.33i 498.5 122.4

NdF3 548.0 62.3 516.2 126.4 NdF3 540.5 14.77i 517.0 124.3 GdF3 557.0 28.3 538.3 133.6 GdF3 555.7 15.88i 537.8 131.7 DyF3 DyF3 564.7 32.8 545.4 140.4 LuF3 LuF3 584.0 39.9 573.9 140.2

LaCl3 306.9 8.4 323.5 66.7 LaCl3 306.5 7.7i 323.6 66.7 NdCl3 NdCl3 315.6 30.1 334.4 68.4 GdCl3 GdCl3 323.5 32.9 344.3 74.3 DyCl3 DyCl3 328.9 37.1 350.9 72.8 LuCl3 LuCl3 335.9 41.5 359.8 80.5

LaBr3 LaBr3 189.8 11.2 237.8 43.8 NdBr3 NdBr3 194.1 23.4 243.7 46.8 GdBr3 GdBr3 199.4 18.5 248.5 46.5 DyBr3 DyBr3 203.1 29.8 251.9 47.6 LuBr3 LuBr3 205.8 28.1 254.6 50.4

107

Table 3. The harmonic frequencies calculated at CCSD(T) level of theory with different sizes of basis sets for the GdF3 and LaCl3 molecules. The Δθ is in the unit of degree and the frequencies are in the unit of cm-1.

C3v Δθ A1 A1 E E D3h A1’ A2” E’ E’ GdF3 GdF3 vdz 0.00 558.3 12.7 540.9 128.8 vdz 558.2 12.8 540.9 128.8 vtz 4.63 556.9 19.4 537.9 131.9 vtz 555.8 14.4i 537.8 131.6 vqz 8.44 561.3 62.5 542.4 132.8 vqz 561.3 37.1i 543.2 131.0

LaCl3 LaCl3 vdz 0.00 302.6 10.6 318.7 66.9 vdz 302.6 10.6 318.7 66.9 vtz 3.33 309.1 9.1 325.1 67.1 vtz 308.5 6.4i 325.1 67.0 vqz 0.00 311.6 23.0 329.0 47.7 vqz 311.6 23.1 329.0 47.7

108

Table 4. Comparison with the recommended values by A. Kovacs and J. Konings. The harmonic frequencies are calculated at the CCSD(T)/VTZ level of theory except for the GdF3 and LaCl3 molecules which are calculated at the CCSD(T)/VQZ level of theory. The -1 estimated errors of recommend values are ±10 cm except for ν3 mode of LnBr3 molecules, whose errors are ±5 cm-1. The unit is in cm-1.

CCSD(T)/VTZ Exp. C3v C3v A1 A1 E E A1 A1 E E LaF3 531.2 54.7 501.0 119.0 LaF3 532 79 501 123 NdF3 548.0 62.3 516.2 126.4 NdF3 544 85 516 128 GdF3 565.5 49.8 544.4 133.8 GdF3 560 91 536 135 D3h D3h A1’ A2” E’ E’ A1’ A2” E’ E’ DyF3 564.7 32.8 545.4 140.4 DyF3 568 95 546 138 LuF3 584.0 39.9 573.9 140.2 LuF3 588 103 572 147

LaCl3 308.9 19.4 326.2 70.6 LaCl3 318 59 317 84 NdCl3 315.6 30.1 334.4 68.4 NdCl3 324 60 327 87 GdCl3 323.5 32.9 344.3 74.3 GdCl3 331 64 337 91 DyCl3 328.9 37.1 350.9 72.8 DyCl3 335 65 340 93 LuCl3 335.9 41.5 359.8 80.5 LuCl3 345 69 351 98

LaBr3 189.8 11.2 237.8 43.8 LaBr3 207 30 232 57 NdBr3 194.1 23.4 243.7 46.8 NdBr3 210 37 236 58 GdBr3 199.4 18.5 248.5 46.5 GdBr3 214 38 241 60 DyBr3 203.1 29.8 251.9 47.6 DyBr3 216 44 243 61 LuBr3 205.8 28.1 254.6 50.4 LuBr3 221 41 249 64

109

Table. 5 The natural electron configurations of Ln and X at the CCSD(T)/VTZ optimized geometry.

6s 4f 5d 2s 2p LaF3 La 0.01 0.12 0.39 F 1.96 5.85

NdF3 Nd 0.02 3.05 0.41 F 1.97 5.85 GdF3 Gd 0.02 7.01 0.39 F 1.97 5.87 DyF3 Dy 0.02 9.01 0.39 F 1.97 5.87 LuF3 Lu 0.04 14.0 0.38 F 1.97 5.87

LaCl3 La 0.07 0.18 0.70 Cl 1.94 5.71 NdCl3 Nd 0.08 3.07 0.71 Cl 1.94 5.71 GdCl3 Gd 0.11 7.01 0.71 Cl 1.95 5.73 DyCl3 Dy 0.12 9.01 0.71 Cl 1.95 5.73 LuCl3 Lu 0.16 14.0 0.71 Cl 1.94 5.72

LaBr3 La 0.10 0.19 0.79 Br 1.94 5.66 NdBr3 Nd 0.12 3.08 0.81 Br 1.94 5.66 GdBr3 Gd 0.15 7.01 0.80 Br 1.95 5.69 DyBr3 Dy 0.17 9.01 0.80 Br 1.94 5.68 LuBr3 Lu 0.21 14.0 0.79 Br 1.94 5.67

110

Table. 6 The atomization energies for the LnX3 molecules calculated at the CCSD(T)/VTZ optimized geometry except for the GdF3 and LaCl3 molecules which are optimized at CCSD(T)/VQZ level of theory. Experimental values are from ref37. The unit is in kcal/mol.

AE VQZ ΔCBS ΔCV(CBS) ΔSOa ΔZPE Δ H(0->298K) Composite results Exp.

LaF3 466.1 +3.2 +1.8 -2.7 -2.6 +2.0 467.8 458±5 NdF3 441.0 +2.6 -6.0 +0.0 -2.7 +1.9 436.8 439±5 GdF3 449.6 +3.1 +0.9 -3.3 -2.8 +2.2 449.8 443±5 DyF3 435.5 +1.2 -0.9 -1.5 -2.7 +2.0 433.6 400±10 LuF3 458.7 +2.8 +0.9 -3.3 -2.9 +2.1 458.3 425±10

LaCl3 363.6 +4.4 +1.6 -4.2 -1.6 +1.3 365.2 366±5

11 NdCl3 338.1 +3.5 -9.2 -1.2 -1.6 +1.3 330.7 334±5

1 GdCl3 349.8 +4.1 -1.3 -4.7 -1.7 +1.6 347.7 349±5 DyCl3 334.6 +2.2 -0.8 -2.8 -1.7 +1.3 332.8 327±5 LuCl3 358.6 +3.9 +1.0 -4.6 -1.8 +1.4 358.4 350±5

LaBr3 326.4 +4.3 +2.9 -11.8 -1.1 +0.9 321.7 316±10 NdBr3 300.6 +3.4 -7.9 -8.8 -1.1 +0.8 287.0 291±10 GdBr3 312.5 +3.9 +2.4 -12.3 -1.2 +1.2 306.5 301±10 DyBr3 296.7 +2.1 +0.6 -10.3 -1.2 +0.8 288.7 273±10 LuBr3 320.3 +3.9 +2.5 -12.2 -1.1 +1.0 314.3 295±10 a) including Gaunt contributions

Fig. 1 The z-matrix and molecular orientation used throughout the current work. Z is the C3v axis

112

Fig. 2 The plot of Ln-X bond lengths vs the Ln atomic number for LnX3 molecules.

113

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117

Chapter 5

A coupled cluster study of the LuF molecule with a reevaluation of its bond dissociation energy

Abstract

The spectroscopic and thermodynamic properties of the ground state of the LuF molecule were studied using the Feller-Peterson-Dixon (FPD) composite thermochemistry methodology using the coupled cluster method. The basis set incompleteness error was removed by basis set extrapolation using recently developed correlation consistent basis sets using the 3rd-order

Douglas-Kroll-Hess relativistic Hamiltonian. Outer-core electron correlation, spin-orbit coupling, and quantum electrodynamics contributions were calculated and found to play only a small role in the final molecular properties. The final composite results are believed to be the most accurate theoretical results to date and are in excellent agreement with the experimental values, except for the dissociation energy. The latter value, however, has not actually been directly measured by experiment. The 0K dissociation energy determined in this work, 172.4 kcal/mol, is estimated to be accurate to within 3 kcal/mol and should be the most reliable value currently available for this quantity.

I. Introduction

118

The thermodynamic and spectroscopic properties of lutetium monohalide (LuX) molecules are of great significance and interest in many different fields. For example in astrophysics, there has been detections of lutetium in the atmosphere of the extremely metal-poor galactic halo giant

CS22892-052,1 as well as in CS31062-0502. Considering the abundance of fluorine in the universe3, whether the LuF molecule exists in the interstellar medium will depend on the environment as well as its bond energy. Therefore, an accurately known dissociation energy of this molecule will be important in determining the physical/chemical environments of remote stars. For another example, high pressure lamps4 involve LuX molecules as important radiating species in the high pressure plasmas derived from doses containing LnX3. Such plasmas, with temperatures above 1000 K, play a vital role in the design and fabrication of these lamps, since the plasma directly influences the power loss, lamp voltage, ballast design and life of the device.

Other examples include the atomic Lu cation which is used in bond activation, such as S-F5 or

C-F6,7 bonds, where the Lu+ is in competition with the S or C atom to form LuF. The accurate thermodynamic data of these bond breaking and formation reactions involving LuF may help in the design of new chemical lasers5 and help understand reaction barriers in synthetic organic chemistry6,7.

Experimentally, the spectroscopic properties of the LuF molecule have been well characterized by both electronic and pure rotational spectroscopies. A total of 9 well-defined electronic band systems have been observed under high resolution by Effantin and co-workers.8-11 These studies led to accurate determinations of both the rotational and vibrational constants of the ζ ground state. From observed hyperfine structure,12 it was determined that the

LuF ground state is very well described by an ionic configuration with Lu+ being in its ground

6s2 electronic configuration. More recently, Cooke et al.13 carried out extensive pure rotational

119 spectroscopy experiments in the first three vibrational levels of LuF using Fourier Transform microwave spectroscopy. Their experiments yielded a very accurate equilibrium bond length for

LuF, as well as values for the harmonic frequency and anharmonicity constant that were in good agreement with the more accurate values obtained previously from analysis of the spectrum.10 In

11,13 regards to the dissociation energy (De) of LuF, the previous spectroscopy studies provided

2 estimates based on the usual Morse potential approximation, i.e., 퐷푒 = 휔푒 ⁄4휔푒푥푒 , which yielded values ranging from 97 – 105 kcal/mol. By analyzing the vapors effusing from a

Knudsen cell using mass spectrometry, Zmbov and Margrave14 estimated the dissociation energy of LuF to be 136±12 kcal/mol, but this value was based on periodic trends and not from direct measurements involving Lu.

Of course LuF has also been the subject of several theoretical studies as well. In an early coupled cluster (CC) study, Küchle et al.15 employed Stuttgart relativistic pseudopotentials (PPs) together with singles and doubles CC with perturbative triples, CCSD(T), as well as multireference averaged coupled pair functional, MRACPF, to compute the molecular properties of several lanthanide and actinide diatomic molecules, which included LuF. In addition to yielding high quality spectroscopic properties, the very close correspondence of CCSD(T) and

MRACPF indicated that the ground state of LuF is well described by single-determinant-based wavefunction methods. Similar PP-based CCSD(T) results but with a more extensive basis set were subsequently reported by Cao and Dolg.16,17 Similar results to the previous MRACPF properties were found latter by Hamada et al.18 in PP-based multireference configuration interaction (MRCI) calculations. This latter work has been followed by studies employing density functional methods (DFT),13,19,20 as well as 4-component Dirac-Fock-Roothaan calculations.21 Most recently, Schoendorff et al.22 carried out all-electron scalar relativistic

120 calculations on the ground state of LuF using Sapporo-style triple-zeta (TZ) basis sets and the completely-renormalized (CR-) CCSD(T) method within the framework of the 3rd-order

Douglas-Kroll-Hess Hamiltonian. Their resulting bond length for LuF was in excellent agreement with experiment while their frozen-core dissociation energy, like the previous coupled cluster and MRACPF results,15-17 was about 40 kcal/mol higher than experiment. Curiously, the latter study reported that subsequent CR-CCSD(T) calculations with all electrons correlated

(same basis set as before) resulted in a much lower dissociation energy that was in good agreement with experiment.

In the present paper the Feller-Peterson-Dixon composite methodology23 based on relativistic CCSD(T) calculations has been applied to the ground electronic state of the LuF molecule to accurately determine its spectroscopic properties and in particular its dissociation energy. As detailed below, these calculations include extrapolations to the complete basis set

(CBS) limit, effects of core correlation, spin-orbit coupling, and even estimates of quantum electrodynamic (QED, i.e., Lamb shift) effects. Excellent agreement with experiment is found for the spectroscopic properties, however the final predicted value for the dissociation energy is in line with the previous theoretical work and not values currently associated with experiment.

II. Computational details

Spectroscopic properties were derived from near-equilibrium potential energy curves defined by fitting a grid of 7 bond lengths distributed as –0.3 ao ≤ r - re ≤ +0.5 ao to 5th-order polynomials in displacement coordinates. For each bond length, the energy calculations adopted the Feller-Peterson-Dixon (FPD) composite methodology23 whereby a composite energy was calculated as the sum of several contributions:

121

퐸(푟) = 퐸퐹퐶/퐶퐵푆 + ∆퐶푉퐶퐵푆 + ∆푆푂 + ∆푄퐸퐷 (1) where E(r) is the grid point energy at a given bond length r. On the right hand side of Eq. 1, the first term EFC/CBS represents the frozen-core energy at the CCSD(T) level of theory extrapolated to the complete basis set (CBS) limit. The third-order Douglas-Kroll-Hess (DKH3)

Hamiltonian24,25 in conjunction with the recently developed relativistic correlation consistent basis sets26 were employed. Specifically, the cc-pVnZ-DK3 (n=D,T,Q) sets for Lu and the same quality DKH basis sets with diffuse functions, aug-cc-pVnZ-DK,27-29 were used for the F atom.

The CBS limit was obtained by separate extrapolation of the triple-zeta (TZ) and quadruple zeta

(QZ) Hartree-Fock (HF) energies and CCSD(T) correlation energies. For the HF energy extrapolation, the Karton-Martin formula30 was applied

−6.57√푛 퐸퐻퐹(푛) = 퐸퐻퐹/퐶퐵푆 + 퐴(푛 + 1)푒 (2) while for the correlation energies, the extrapolation formula used was31

4 푐표푟푟 푐표푟푟 ⁄ 1 퐸퐹퐶 (푛) = 퐸퐹퐶/퐶퐵푆 + 퐴 (푛 + 2) (3)

푐표푟푟 where the energies calculated with TZ (n=3) or QZ (n=4) basis sets and EHF/CBS and 퐸퐹퐶/퐶퐵푆 are the extrapolated CBS limit HF energy and CCSD(T) correlation energy, respectively. Note that in these calculations the present frozen-core definition leaves the 5s5p5d4f electrons of Lu and the 2s2p electrons of F in the valence space.

Going beyond the frozen-core approximation, the outer-core is calculated at the CCSD(T) level of theory by including correlation of the 4s4p4d electrons of

Lu and the 1s electrons of F. For these calculations the cc-pwCVnZ-DK326 and aug-cc-pwCVnZ-DK32 (n=D, T, Q) sets were used for Lu and F, respectively, for both frozen-core and outer-core correlated calculations. The difference in these latter two calculations with n=T and Q were extrapolated to the CBS.

122

The third term in Eq. (1) is the contribution from spin-orbit coupling. The calculation was carried out using the exact two-component (X2C) Hamiltonian33 with uncontracted double zeta

(DZ) correlation consistent basis sets as introduced above at the 2-component CCSD(T) level of theory.34 The frozen core approximation was adopted to be the same as the scalar relativistic calculations defined above and virtual orbitals with orbital energies higher than 10 a.u. were neglected in the correlation space. The atomic mean field integral approximation was also employed.35 In addition, the Gaunt interaction (spin-other-orbit) was also included using the X2C

Hamiltonian at the HF level of theory.

The last term in Eq. (1) is a contribution from quantum electrodynamics (QED). It was obtained by employing a local model potential as originally described by Pyykkö and Zhao36 to account for both the vacuum polarization and self-energy contributions to the Lamb shift. See also Ref. 37 for the present implementation. This contribution was calculated at the frozen-core

CCSD(T) level of theory with the DKH3 Hamiltonian and DZ correlation consistent basis sets.

After applying the above FPD scheme to each bond length on the near-equilibrium potential energy curve, spectroscopic constants were calculated from the fitted curves using second-order vibrational perturbation theory.38 All the scalar relativistic calculations of this work were carried out with the Molpro program package,39 while the 2-component SO calculations were performed with Dirac14.40

III. Results and Discussion

The accuracy of the present composite calculations hinge on the suitability of the CCSD(T) method to accurately describe the electronic structure of LuF. As mentioned above, previous

MRACPF calculations gave nearly identical results to CCSD(T) which implies that

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41 multireference effects are negligible for this molecule. In addition, the T1 diagnostic and the maximum T2 amplitude are often used as indices to estimate the validity of using the single reference CCSD(T) method. For all the 7 bond lengths on the potential curves of this work, the

T1 diagnostic was no more than 0.02 for any basis set and the maximum T2 amplitudes are just

0.10, 0.07, and 0.05 for DZ, TZ and QZ quality of basis sets, respectively. It can be seen that the

T2 amplitudes get smaller as the basis sets become more complete, indicating the single reference character becomes even more dominating. Therefore, multireference character is not expected to be important for the near-equilibrium potential energy curve of LuF and the CCSD(T) method should yield very reliable spectroscopic properties for this system.

The molecular constants re e, and De from valence electron correlation calculations are plotted as a function of basis set in Figure 1. As in the case of the total energies (not shown), the bond length smoothly converges with basis set towards the estimated CBS limit. As is typically the case for the harmonic frequency, the convergence is not nearly as monotonic presumably due to the basis set superposition error at the double-zeta level. This is also somewhat apparent for the dissociation energy but the convergence is nonetheless very rapid with a total change in De between the DZ result and the CBS limit of just 2.8 kcal/mol. Of course, some of this is due to basis set superposition error (BSSE) particularly in the valence DZ and TZ basis sets. Use of the larger cc-pwCVnZ-DK3 sets in frozen core calculations (which are needed to define the core correlation contributions) yield a CBS limit slightly smaller by 0.6 kcal/mol, i.e., 174.45 kcal/mol. This difference provides some estimate of the uncertainty in the present

CBS extrapolations.

All of the contributions to the final FPD spectroscopic constants and dissociation energy are given in Table 1. At the frozen-core CCSD(T) level, very small differences between the VQZ

124 and CBS limits are observed in almost all cases, e.g., +0.6 kcal/mol for De, –0.0037 Å for re, and

-1 +4.2 cm e. The effect of correlating the outer core electrons (4s4p4d for Lu and 1s for F) has only a relatively mi

shortening it by much less than 1 mÅ. The effects of spin-orbit coupling is largest for the dissociation energy, –2.35 kcal/mol, which is mostly attributable to the atomic spin-orbit fine structure in the Lu atom. Using perturbation theory with the experimental atomic term energies, this would be equal to about –3.8 kcal/mol (-3.42 kcal/mol from Lu and -0.39 kcal/mol from F),

-negligible molecular SO contribution. Last, as shown in Table 1 the effects of the Lamb shift are nearly negligible for all

26 properties studied in this work. This is consistent with previous work on Gd2 and GdF.

The final FPD composite results are compared to experiment and selected previous calculations in Table 2. In regards to the spectroscopic constants, the FPD results are in excellent agreement with the available experimental values with an equilibrium bond length in agreement to under a mÅ of the accurate microwave value13 and a harmonic frequency within a cm-1 of the experimental value determined from the electronic spectrum.10 In agreement with previous theoretical work, however, the dissociation energy is much larger than the currently accepted experimental values. As mentioned in the Introduction, none of the experimental values cited in

Table 2 are directly measured dissociation energies, so it perhaps not surprising they are much less accurate than expected. As in the earlier work of Cao and Dolg,16 the current work strongly recommends that the accepted experimental value for De should lie close to 173 kcal/mol. Based on previous work at a similar level of theory,26,37 it is expected that this is accurate to at least 3 kcal/mol.

125

One remaining discrepancy is to address the previous results of Schoendorff and Wilson where correlation of core electrons, i.e., the 1s through 4d of Lu, was calculated to decrease the dissociation energy by about 32 kcal/mol (relative to their frozen-core result). In the present work correlation of the outer-core, i.e., 4s, 4p, and 4d electrons of Lu (as well as the 1s of F), only decreased De by 0.65 kcal/mol. To determine if correlation of the inner core electrons of Lu

(1s through 3d) could so strongly lower the dissociation energy of LuF, additional CCSD(T) calculations were carried out in this work using a modified core correlation basis set, namely the tight spdf functions contained in the cc-pwCVTZ-DK3 set of Lu were replaced by uncontracting the most diffuse 11s,11p,10d, and 8f functions from the underlying HF set and then also adding 2 additional tight g-type functions and 1 additional tight h function using an even tempered extension (standard aug-cc-pwCVTZ-DK was used for F). It should be noted that in the work of

Ref. 22, the standard Sapporo TZP basis set was used, which is designed only for correlation down to the 4s electrons of Lu. The calculations of the present work yielded incremental core correlation effects on De of +0.37 kcal/mol for Lu 4s4p4d correlation, +0.14 for F 1s correlation, and just –0.003 kcal/mol for Lu 3s3p3d correlation. Correlation of even deeper core electrons of

Lu was not carried out due to the completely negligible contribution from 3s through 3d. The present calculations indicate that the previous core correlation effects of Ref. 22 were presumably due to very large BSSE effects.

IV. Conclusions The FPD composite methodology was applied to study the spectroscopic and thermodynamic properties of the LuF molecule. CBS limits at the CCSD(T) level of theory with the DKH3 Hamiltonian were obtained by basis set extrapolations using newly developed

126 correlation consistent basis sets. Contributions from outer-core electron correlation, spin-orbit coupling, and Lamb shift effects were considered. Compared to the experimental studies, the calculated spectroscopic constants showed excellent agreement. The final dissociation energy resulting from this work, D0 = 172.4 kcal/mol, which is expected to be accurate to within 3 kcal/mol, is in good agreement with most previous theoretical calculations but in strong disagreement with the currently accepted experimental values. The latter, however, are not based on direct observations and new experiments are strongly recommended for this property of LuF.

Acknowledgments The generous support of the U.S. Department of Energy Office of Science, Office of Basic Energy Sciences, Heavy Element Chemistry Program through Grant No. DE-FG02-12ER16329 is gratefully acknowledged.

127

Table 1. FPD contributions to the calculated CCSD(T) spectroscopic properties of X1+ 175LuF.

a De re ωe ωexe Be δe αe (kcal/mol) (ang.) (cm-1) (cm-1) (MHz) (kHz) (cm-1 x 1000)

VDZ 172.23 1.9533 593.1 2.50 7729.0 5.84 1.52 VTZb 173.24 1.9298 598.5 2.42 7918.9 6.17 1.55 VQZ 174.44 1.9223 607.2 2.43 7980.4 6.14 1.55 CBS 175.05c 1.9186 611.4 2.43 8011.1 6.12 1.55

CV 0.65 -0.0002 0.31 0.03 1.8 0.00 0.00 SO -2.35 -0.0012 0.58 0.00 10.1 0.01 0.01 QED -0.04 -0.0001 0.32 0.00 0.8 0.00 -0.01 Total FPD 173.32 1.9171 612.57 2.46 8023.8 6.13 1.55 a The centrifugal distortion constant is denoted here as e to avoid confusion with the dissociation energy De. b -1 -1 BCCD(T) results with this same basis set yielded re=1.9291 Å, e=598.8 cm , and exe=2.42 cm . c The frozen-core CBS limit De using the cc-pwCVnZ-DK3 sequence of basis sets is 174.45 kcal/mol.

128

Table 2. Comparison of the present composite results for LuF with previous theory and experiment.

De re ωe ωexe Be δe αe

(kcal/mol) (Å) (cm-1) (cm-1) (MHz) (kHz) (cm-1 x 1000)

This work 173.32 1.9171 612.57 2.46 8023.8 6.13 1.55

Expt Rotationala 96.8 1.9171 618±12 2.82±0.08 8023.74 1.56 Electronicb 105.3 1.9165 611.79±0.15 2.54±0.04 8023.7 6.12 1.56±0.04 Mass spec.c 136±12

Ref. 42d 124

Selected Theory CR-CCSD(T)e 171.3 1.917 CR-CCSD(T)-alle 139.6 PP-CCSD(T)f 173 1.923 604

PP-MRACPFg 175 1.940 582 DKH-DFTh 175 1.920 613 MRCIi 1.922 606.6 3.3

a Spectroscopic parameters correspond to Dunham parameters of 175LuF from Fourier transform microwave spectroscopy.13 The dissociation energy assumes a Morse potential, see the text. The equilibrium bond length was determined as 1.91711816 Å for 175LuF.

b From analysis of the electronic spectrum.10,11 The dissociation energy was determined assuming a Morse potential from the vibrational constants of the ground state. c Estimated value from mass spectrometry equilibrium measurements.14

d 42 Estimated D0 value obtained by using the experimental dissociation energies of LaF and GdF together with ligand field theory. e Using the DKH3 Hamiltonian with Sapporo TZP basis sets.22 f PP-based CCSD(T) with an extended basis set and counterpoise corrections.17 g PP-based MRACPF calculations with basis sets of intermediate TZ and QZ quality, including counterpoise corrections.15 Analogous CCSD(T) h Using the all-electron DKH2 Hamiltonian.19 i PP-based MRCI with DZ-quality basis sets.18

Figure 1. The DKH3 CCSD(T) equilibrium bond length (upper), harmonic frequency (middle) and dissociation energy (bottom) as a function of correlation consistent basis sets.

131

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Chapter 6

The high-accuracy study of highest spin diatomic molecule

(Gd2) and its isoelectronic analogs

Abstract

The highest spin diatomic molecule Gd2 and its isoelectronic analogs (LaLu, CeYb,

PrTm, NdEr, PmHo, SmDy and EuTb) were studied by the coupled cluster method with the new developed correlation consistent basis sets with the 3rd order Douglas-Kroll-Hess (DKH)

Hamiltonian. As the first systematic study, the series of molecules forms a unique pattern with respect to the spin multiplicity. The Feller-Peterson-Dixon (FPD) methodology was applied to study the spectroscopic and thermodynamic properties of molecules. The possible significant errors are ruled out by the FPD methodology and a good agreement is achieved for the experimentally confirmed Gd2 molecule. The prediction of the isoelectronic analogs is thus believed to be reliable. The results will be helpful to design new lanthanide complexes.

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I. Introduction

The lanthanide (Ln) dimer, either homonuclear or heteronuclear, is the model core structure of many complex compounds. Such compounds, with proper ligands on the lanthanide atoms, have tremendous impact in various fields due to their special thermodynamic, kinetic and magnetic properties derived from the Ln dimer1,2. In medical chemistry, various Ln dimer complexes have been used or proposed as the (next generation) molecular imaging agents. In the field of material fabrication, the

2 3 [Ln2(μ2-η -H2L)2(μ2-H2L)2(phen)2(NO3)2]2CH3OH (Ln=Eu, Sm, Tb, Dy) for example, is studied for its potential use in diode lasers and optical fibers. Among these studies, one of the key parts is to find the proper Ln dimer since they determine the structure and properties of the complex. Therefore, the study of lanthanide dimers is vital in helping in the prediction and design of these new types of complexes.

Among the various type of homonuclear or heteronuclear lanthanide dimers, the gadolinium dimer (Gd2) might be the most special since it has the highest known spin multiplicity of any diatomic molecule4,5 (2S+1=19) due to its unique electronic structure.

With such unusual spin and magnetic property, the Gd2 found itself in many different

6 7,8 applications such as molecular imaging , CO or N2 activation , non-linear optical response9,10, or comparison studies with Gd cluster, surface and crystal11,12.

Despite its industrial and academic interest, the study of the model structure, Gd2, is very limited. The first thermodynamic13 and spectroscopic14 studies were performed more than 30 years ago with relatively simple measurements or estimates. The molecule was then ignored by the world until 1992, when Dolg and co-authors4 successfully predicted the molecule has a super-configuration with spin multiplicity of 19 with the configuration interaction (CI) and density functional theory (DFT) methods. Two years later, the ESR experiments5 confirmed the assignment and suggested that the Gd2 molecule should be the highest spin diatomic

136

molecule since the f orbitals do not participate in bonding. One drawback of the previous coupled cluster study, however, was that the 4f orbitals were subsumed in the pseudo-potential core and were not included in the valence. In 200015 and 200316, Dolg and co-authors revisited the Gd2 molecule with explicit treatment of 4f orbitals and conduct the first coupled cluster calculation of the molecule. Other theoretical studies include generalized active space self-consistent field (GASSCF) study17, several DFT studies7,8,11,15,18 while other experimental work include Raman spectra19, Stern-Gerlach experiments12, and Fourier transform infrared (FTIR)7,8. Reviewing these experimental and theoretical studies, the results remain some uncertainty. For example, the binding energy of the Gd2 molecule was estimated with a large error range as 2.1±0.7 eV19. And this error bar (0.7 eV) is one third of the central value (2.1 eV), making this data barely useful. Therefore, a thorough and systematic improvement has to be adopted to better understand the molecule.

The first improvement upon the previous theoretical studies is to eliminate the incompleteness error from the finite basis sets. With the advance of the recently developed correlation consistent basis sets of lanthanide elements20, there comes an opportunity to eliminate this error by basis set extrapolation. In their benchmark calculations, the binding energy (De) was calculated as 38.42 kcal/mol at the complete basis set (CBS) limit. To give more accurate results, the 4s4p4d electrons were correlated explicitly under the frame of

21 Feller-Peterson-Dixon (FPD) methodology . The De value was then determined as 38.33 kcal/mol.

Going beyond the Gd2 molecule, the Gd element sits at the center of the 15 lanthanide elements. Therefore there are 7 isoelectronic molecules to Gd2 (LaLu, CeYb, PrTm, NdEr,

PmHo, SmDy and EuTb) and thus potentially there are 7 candidates available to serve as a core of a Gd2-like complex. But what are the properties of these isoelectronic molecules? Are they stable? Do they have similar electronic structures? To answers these questions, a

137

systematic and consistent study is required that accurately includes electron correlation and relativistic effects. The answers to these questions will provide a deeper understanding of lanthanide chemistry and physics, and will open a door to the design of new complex molecules.

In this study, the diatomic molecules isoelectronic to Gd2 (LaLu, CeYb, PrTm, NdEr,

PmHo, SmDy and EuTb) are studied under the framework of the Feller-Peterson-Dixon composite coupled cluster methodology. To the author’s knowledge, the studies of these molecules have not been reported. Their bond lengths, bond strengths, vibrational frequencies, and electronic states are predicted and compared with the Gd2 molecule, revealing the

“periodic” pattern among these molecules.

II. Computational details The most likely ground state of bounded molecule was determined by scanning the potential energy curves (PEC) of different configurations and states at given spin multiplicities. The near equilibrium PEC which was constructed by fitting a 7-point grid by a

5th order polynomial function.

The energy calculation for each grid point adopted the Feller-Peterson-Dixon methodology22, where the composite energy is expressed as the sum of the valence electron energy (VAL) and the outer core electron energy (CV) at their complete basis set (CBS) limits.

E(r)=VAL(CBS)+CV(CBS) (1)

The coupled cluster singles, doubles and perturbtive triples, CCSD(T), method was used with the 3rd order Douglas-Kroll-Hess (DKH) Hamiltonian. The correlation consistent basis sets, cc-pvXz (X=D,T,Q) were used and TZ and QZ energies were extrapolated to the CBS

138

limit. The HF energies were extrapolated by Karton-Martin’s formula23:

−6.57√푛 퐸퐻퐹(푛) = 퐸퐻퐹/퐶퐵푆 + 퐴(푛 + 1)푒 (2) while the correlation energies were extrapolated by:

1 4 En E CBS  A/() n  (3) 2 where En are the energies calculated with TZ or QZ basis sets and ECBS is the extrapolated energy at CBS limit.

For the outer core electrons correlation, it recovers the correlation energies from 4s, 4p, and 4d electrons. The cc-pwCVXz (X=D,T,Q) basis sets were used. The extrapolations were using the same formula as above. In all UCCSD(T) calculations, the T1 diagnostics were generally 0.02 to 0.03. In a very few cases, i.e. at the bond length furthest away from the equilibrium for certain molecule, the T1 diagnostics increased to 0.06. In term of the maximum T2 amplitude, the values generally range from 0.05 to 0.07 but with occasional circumstances the number jumps up to 0.1. However, if those terms with large T2 amplitude would make a very significant change to energy, the 7-point grid will hardly form a Morse potential-like PEC and the fitting error would catastrophically large. Referring the fitting errors of the PEC on the other hand, for all molecules the fitting error is at or below 10-6 in atomic unit. Thus the multi-reference shouldn’t be a main concern for the molecules.

The spectroscopic constants were calculated by the second-order perturbation theory.

The natural bond orbital (NBO) analysis was carried out using NBO624 interfaced to the

Molpro package. All the other calculations were carried out by Molpro25.

III. Results and discussion

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A. Electronic structure

The first challenge of calculations for the isoelectronic molecules is to determine their ground states. For the well-characterized Gd2 molecule, its electronic configuration is

4f74f7ζ2ζ1ζ1π2, giving a total spin multiplicity of 19. However, except for certain elements which have 4fn5d162 ground state configurations (La, Ce, Gd, Lu), most lanthanide elements have 4fn6s2 configuration. The first thought to determine the ground state might be a doubly-occupied ζ and ζ* orbitals out of the 6s2 valence electrons and the f electrons keep localized. The molecules’ chemistry nature, however, does not like this way. With this configuration, the two lanthanide centers will repel each other, and a bounded state cannot be found.

Instead, after attempts of different assignments of the electrons, the 6s electrons are found promoted to a higher orbital, therefore opening up the ζ orbitals and reducing the repulsion. In fact, similar phenomenon has been found in most homo-nuclear lanthanide dimers4, that the ζ2 configuration gives longer bond length than the molecular orbital configurations with singly-occupied ζ orbitals.

To elaborate the electronic structure, it may be helpful for discussion to categorize the molecules into 3 subgroups. The ground state atomic configurations for La, Ce, Gd and Lu are 4fn5d16s2 (n=0,1,7,14); while the configurations for other elements (Pr, Nd, Pm, Sm, Eu,

n 2 Tb, Er, Ho, Tm, Yb) are 4f 6s (n=2~13). Therefore, (1) the first group includes Gd2 and

LaLu molecules, by the criteria that both atoms have the f orbitals either empty (La), half occupied (Gd), or fully occupied (Lu), (2) the second group, by a similar criteria, includes

CeYb and EuTb, where only 1 atom is half occupied (Eu) or fully occupied (Yb), and (3) The remaining molecules, PrTm, NdEr, PmHo and SmDy form the third group.

1 Group of Gd2 and LaLu

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16 7 7 2 1 1 2 The ground state of Gd2 has been determined by Dolg as 4f 4f ζg ζg ζu πg , where the

6s electrons give 1 doubly occupied and 2 singly occupied ζ orbitals and 5d electrons give a doubly occupied π orbital. Since the 4f orbitals of lanthanide elements are buried far below the 6s5d orbitals26, it is therefore expected they will remain highly localized in the hetero-nuclear lanthanide dimers, too. As La and Lu have similar atomic configurations

(4fn5d16s2) with Gd, the ground state of LaLu is easily determined as 4f04f14ζ2ζ1ζ1π2, and only the f orbitals differ in electron occupation from Gd2.

2 Group of CeYb and EuTb

In this group, the ground state determination is more complicated. As mentioned above, the premature determination to the doubly-occupied ζ and ζ* orbitals gives a dissociative curve, as shown in Figure 1. Visiting Dolg’s study4,16 of the homonuclear lanthanide dimer, such as Tb2, it was found that the low-lying excited states of the Ln atom provide the electrons with orbitals easy to promote. Inspired by this, for the heteronuclear dimers, the 6s2 electrons are promoted to higher π orbitals, leading to the 4f14f14ζ1ζ1ζ1π2 configuration for

CeYb and 4f748ζ2ζ1ζ1π1 configuration for EuTb.

3 Group of PrTm, NdEr, PmHo and SmDy

To determine the ground state of the bounded molecules, similar strategy was adopted.

The energy scans of different configurations were carried out to compare the energies for each configuration and find the stable minimum on the PEC. It was found that although the

ζ2 configuration has a lower energy, it cannot lead to a bounded molecule. On the other hand, promoting 6s electrons can stabilize the molecule in the sense that the molecule will not fall part. As a result, the 4fm4nζ1ζ1π2 configuration is obtained, as the ground state configuration for the bounded molecule.

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B. Basis set convergence

Using the new-developed correlation consistent basis sets contracted for the 3rd order

DKH Hamiltonian, the molecules were systematically studied with different sizes of basis sets. It can be seen from Table 1 that the valence correlation energies are systematically converging to the basis set limit in the sense that the energies decrease monotonically and the energy spacing between neighboring sets shrinks as the basis set becomes more and more complete. Taking the LaLu molecule as an example, the TZ-DZ, QZ-TZ, and CBS-QZ differences are approximately 0.41, 0.13 and 0.08 a.u., respectively. On the other hand, it can be seen that using the largest basis set (VQZ), there is still about 0.08 a.u. basis set incompleteness error (BSIE) from the CBS limit. This indicates that the basis set convergence is relatively slow, which suggests explicit correlation methods to improve the results. But due to the unavailability of proper basis sets for explicit correlation, this improvement cannot be done at the moment.

To improve the CBS limit, the outer core (CV) electrons of 4s4p4d were correlated and added to the CBS values following the FPD methodology. It can be seen that similar to the valence calculations, the CBS+CV energies also have a monotonically convergence trend, and the spacing between neighboring sets also get smaller and smaller.

Comparing the energies between molecules, it can be found that the LaLu molecule has the most negative energy, followed by CeYb, PrTm, NdEr, SmDy, EuTb, and lastly the Gd2 molecule, with nearly equal energy spacing between the two molecules. It can been found in the later discussion that this order is not only for energy, but also for the spin multiplicity, which increases monotonically from LaLu to Gd2.

It should be lastly noted that for most calculations, the T1 diagnostics are relatively small, showing that the chosen configurations are well separated from the excited states and the

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UCCSD(T) calculations provide reliable results. For occasional cases where atoms are at either the mid-early or mid-late position of the lanthanide row, the multi-reference character makes the T1 diagnostic very large and the entry is thus leaving blank.

C. Thermodynamic and spectroscopic properties

The thermodynamic and spectroscopic properties of Gd2 isoelectronic molecules at

UCCSD(T)/CBS(valence and core) level of theory are summarized in Table 2. The diatomic molecules are ordered by the atomic number of the lighter atoms. It can be seen that the bounded molecules with the chosen configurations naturally forms an arithmetic progression with respect to the spin multiplicity. That is, the spin multiplicity increases synchronically with the atomic number of the lighter atom. And this pattern arises because there is a systematic change in the f electrons and the valence electron configurations are similar to form a bounded molecule. Such a character of spin multiplicities will help people to design and tune the Ln dimer complex property.

It can also be seen that only the LaLu and Gd2 molecule have a Σ ground state and they have a relatively larger vibrational frequencies. This is consistent with the above categorization of the molecules that these molecules have either empty, half, or fully occupied f orbitals. The third category molecules of PrTm, NdEr, PmHo and SmDy have partially filled f orbitals and they have similar bond length, binding energy as well as frequencies. For the remaining molecules (CeYb and EuTb) as the second category, their properties are at the boundaries between the first-type molecules and third-type of molecules.

To be specific, Firgure 2 shows the bond length (Re) convergence trend with respect to the valence basis sets for the isoelectronic molecules. It can be seen that the bond lengths are systematically converging to the CBS limit. This shows the signature property of the correlation consistent basis sets. For comparison of bond lengths among the molecules, the

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Gd2 molecule has the shortest bond length followed by its same-type molecule, LaLu. The

EuTb molecule, on the other hand, has the longest bond length, far away from its same type molecule CeYb. This is probably due to the fact that this molecule is the only one that has an f electron promoted to a higher molecular orbital among its isoelectronic species. The third type molecules have similar bond lengths and they lie closely on the top of Figure 2 with longer bond length. Comparing the bond length to the corresponding homonuclear molecules, for example, EuTb vs. Eu2 vs. Tb2, it can be found that the heteronuclear molecules are generally in-between the homonuclear molecules, although the method and basis sets are not exactly the same.

In Figure 3 there shows the convergence trend of binding energy with respect to the valence basis sets. The electronic configuration of the atoms at the diabatic limit is adopted as the natural electron configuration in Table 3. It can be noticed that the binding energies converge from different directions to the CBS limit. For CeYb, LaLu, and Gd2 molecules, the

DZ calculations underestimate the binding energy while for other molecules, the DZ calculations overestimate the binding energy. In addition, at the DZ level, basis set incompleteness error (BSIE) is very large for certain molecules, maximally about 7 kcal/mol.

This indicates that the good agreement between DZ and CBS results for some other elements results from fortuitous error cancellation. And the larger basis sets are necessary to remove the BSIE.

The Figure 4 shows the frequency convergence trends with respect to the valence basis sets and this figure may be the best one to expose BSIE. It can be seen that for the CeYb molecule the DZ frequency is relatively far away from the CBS frequency. Therefore in order to obtain accurate frequencies, it would be necessary to use the correlation consistent basis sets and do the extrapolation to eliminate the BSIE. Focusing on the CBS frequencies, it can be seen that the Gd2 and LaLu molecules stand out from other molecules with a relatively

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large frequency. This indicates that they have a larger force constant and thus a stronger bond.

This is consistent with the binding energy results. For the third-type molecules, they have similar frequencies, lying at the bottom of the figure. The second-type molecules, CeYb and

EuTb, have the frequencies in-between the other two types, supporting the classification as discussed above.

D. MO diagrams and NPA analysis

The molecular orbital (MO) diagrams for the valence electrons of the three types of molecules are shown in Fig. 5 to Fig. 7. The f electrons lie well below the valence shell and thus not shown in the diagrams. It may be surprising to find that the electronic structure of these molecules does not follow the Aufbau principle, i.e. that the electrons do not occupy the lower molecular orbitals first. This violation of the Aufbau principle is not uncommon in the high-spin open-shell calculations but its interpretation is still open to debate27-29. Based on results of the Figure 5-7, it may be helpful to adopt Schmidt’s conclusion29 that the orbital energies are arbitrary in high spin open-shell calculations and thus its ordering is arbitrary.

Having this in mind, it would be easier to accept the current MO diagrams and appreciate the similar pattern within each type of molecules. Between the Gd2 and LaLu molecules, the degenerate π orbitals are both doubly occupied. The ζ bonding orbitals consisting of dzz atomic orbitals are both singly occupied. For Gd2 dimer, another doubly occupied ζ orbital and singly occupied ζ* orbital are formed. While for the LaLu molecule, most likely due to their large difference of nuclei charge, the non-bonding orbitals are formed instead. Between the CeYb and EuTb molecules, the MO diagrams are relatively less comparable. The non-bonding orbital of CeYb consists of the d orbital from the Ce atom while the non-bonding orbital of EuTb mainly comes from Tb 6s orbital and it is doubly occupied. The

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singly occupied ζ and ζ* orbitals of CeYb dominantly consist of s orbitals from component atoms while the singly occupied ζ and ζ* orbitals of EuTb blend in some 5d(Tb) and 6p(Eu) character. The π orbitals in both cases are similar to the first type molecules. Note that due to the broken symmetry, the other component of π orbital in the EuTb molecule is not shown in the MO diagram. Lastly, among the third type of molecules, the molecular orbital, occupation number, and orbital composition are highly similar. Thus only one set of the labeling is shown in the Figure 7. The only difference between these MO diagrams lies in how much 5d or 6p character are mixed in the bonding or anti-bonding orbitals.

The natural population analysis (NPA) was done at the equilibrium for each molecule.

Table 3 shows the natural electron configuration of molecules comparing to the ground configuration of each component atom. It can be seen that except for Tb in the EuTb molecule, all other molecules have the same number of f electrons in both molecular environment and the atomic environment. This follows the general belief that the 4f electrons are less active. On the other hand, the loss of 1 f electron of Tb in this study is not the only

16 30 case in chemistry. In fact, in both Tb dimer molecule and TbF3 molecule , the Tb atom will lose the f electron, too.

For the other molecules, it can be generalized from NPA that the component atoms are prone to adopt the Gd-like configuration, i.e. the 4fn5d16s1, where one of the 6s electron is promoted to the d orbital. Therefore it concludes that the 5d orbitals play an important role in forming a bounded Lanthanide heteronuclear dimer.

IV. Conclusion In this study, the coupled cluster method and the new developed DKH3 correlation consistent basis sets were used to study the highest spin diatomic molecule and its isoelectronic molecules. Different size of basis sets were used to eliminate the basis set

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incompleteness error by basis set extrapolation. The relatively small T1 diagnostics shows the reliability of the coupled cluster method and justify the chosen states are well separated from the excited states. The outer core electron correlation was added at a second step using the

FPD methodology to achieve better accuracy. The study provides the most accurate results up to date.

From this study, the Gd2 isoelectronic molecules were studied for the first time. The series of molecules form a progressive pattern in term of the spin multiplicity. The unexpected but not exceptional electronic structure is responsible for this unique pattern.

With the different occupancy of f orbitals, the molecules can be divided into 3 subgroups and the molecular properties, i.e. bond length, frequency, electronic state, etc. support such a categorization. The unique pattern of molecules is expected to help design and tune the properties of Ln dimer complex in medicine chemistry.

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Table. 1 The T1 diagnostics and electronic energies of valence only and core-valence calculations. The atoms in red indicate they have either 0, 7 or 14 f electrons. Due to large T1 diagnostic in its WCVQZ calculation, the PmHo energy is fitted using a 6-point grid, leaving out the furthest point from equilibrium geometry.

UCCSD(T) LaLu CeYb PrTm NdEr

T1 diag. 0.02~0.03 0.03 0.04~0.05 0.02~0.03

VDZ -23051.773610 -22914.823339 -22704.389389

VTZ -23052.182767 -22915.253902 -22799.420982 -22704.778435

VQZ -23052.316341 22915.389155 -22799.550478 -22704.900971

CBS -23052.392801 -22915.466465 -22799.624261 -22704.970829

CBS+CV(DZ) -23053.609073 -22916.670856 -22706.170151

CBS+CV(TZ) -23054.193378 -22917.291856 -22801.481730 -22706.840048

CBS+CV(QZ) -23054.434191 -22917.539210 -22801.736913 -22707.101230

CBS+CV(CBS) -23054.573179 -22917.681974 -22801.884195 -22707.251974

UCCSD(T) PmHo SmDy EuTb GdGd

T1 diag. 0.02~0.06 0.02~0.04 0.03~0.04 0.02~0.04

VDZ -22630.874300 -22578.382323 -22547.144136 -22536.800874

VTZ -22631.251088 -22578.741270 -22547.489770 -22537.096908

VQZ -22631.367819 -22578.852762 -22547.593110 -22537.220774

CBS -22631.434377 -22578.916252 -22547.652009 -22537.291431

CBS+CV(DZ) -22632.642012 -22580.121197 -22548.865458 -22538.496928

CBS+CV(TZ) -22633.319718 -22580.806202 -22549.543584 -22539.165489

CBS+CV(QZ) -22633.586646 -22581.077992 -22549.815341 -22539.444005

CBS+CV(CBS) -22633.740715 -22581.234859 -22549.972190 -22539.604756

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Table 2. The thermodynamic and spectroscopic data for Gd2 isoelectronic molelcules at UCCSD(T)/Val(CBS)+CV(CBS) level of theory. The atoms in red indicate they have either 0, 7 or 14 f electrons.

LaLu CeYb PrTm NdEr

Electronic State 5Σ 7Δ 9Δ 11Δ

Configuration 4f04f14ζ2ζ1ζ1π2 4f14f14ζ1ζ1ζ1π2 4f3413ζ1ζ1π2 4f4412ζ1ζ1π2

Re (ang.) 3.0127 3.1524 3.3431 3.3221

Binding energy 41.8 57.8 53.3 44.3 (kcal/mol)

ω (cm-1) 132.23 100.18 83.61 86.01

PmHo SmDy EuTb GdGd

Electronic State 13Δ 15Δ 17Δ 19Σ

5 11 1 1 2 6 10 1 1 2 7 8 2 1 1 1 7 7 2 1 1 2 Configuration 4f 4 ζ ζ π 4f 4 ζ ζ π 4f 4 ζ ζ ζ π 4f 4f ζ gζ g ζ u π u

Re (ang.) 3.3231 3.2199 3.3459 2.8980

Binding energy 58.2 23.3 40.1 38.3 (kcal/mol)

ω (cm-1) 83.83 89.67 95 143.96

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Table 3. The natural population analysis of isoelectronic molecules.

Natural electron configuration Ground atomic configuration

GdGd 4f7.005d1.396s1.48 4f75d16s2

Eu 4f 7.005d0.376s1.14 4f76s2 EuTb Tb 4f8.005d1.376s1.81 4f96s2

Sm 4f6.005d0.836s0.96 4f66s2 SmDy Dy 4f10.005d0.646s0.98 4f106s2

Pm 4f 5.005d0.896s0.97 4f56s2 PmHo Ho 4f11.005d0.546s0.98 4f116s2

Nd 4f4.005d0.906s0.94 4f46s2 NdEr Er 4f12.005d0.516s0.98 4f126s2

Pr 4f2.995d1.006s0.93 4f36s2 PrTm Tm 4f13.005d0.486s0.98 4f136s2

Ce 4f1.015d2.156s0.94 4f15d16s2 CeYb Yb 4f14.005d0.406s0.99 4f146s2

La 4f0.085d2.026s1.00 4f05d16s2 LaLu Lu 5d0.656p0.136s1.96 4f145d16s2

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Figure.1 Potential energy scan of EuTb molecule at UCCSD(T)/vdz level of theory. The top black curve is from 4f74f9ζ2ζ2 configuration. The bottom red curve is from 4f748ζ2ζ1ζ1π1 configuration.

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Figure 2. The bond length convergence trends of the isoelectronic molecules with respect to the valence basis sets. The DZ data for PrTm was leaving blank due to large T1 diagnostics.

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Figure 3. The binding energy convergence trends of the isoelectronic molecules with respect to the valence basis sets. The DZ data for PrTm, was leaving blank due to large T1 diagnostics.

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Figure 4. The frequency convergence trends of the isoelectronic molecules with respect to the valence basis sets. The DZ data for PrTm was leaving blank due to large T1 diagnostics.

154

b1u(1σ*) n.b. (La-6s)

ag(2σ) e(1π) a1(1σ)

eu(1π) n.b. ag(1σ) (Lu-6s)

Gd2 EuTb

Figure 5. Molecular orbital diagrams of valence shell for Gd2 and LaLu molecules.

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a1(1σ*) a1(1σ*) e(1π) e(1π) n.b. ) n.b.

(Ce-5d) (Tb-5d) )

a1(1σ)

) CeYb EuTb

) Figure 6. Molecular orbital diagrams of valence shell for CeYb and EuTb molecules.

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a1(1σ*) e(1π)

)

a1(1σ) PrTm NdEr PmHo SmDy

Figure 7. Molecular) orbital diagrams of valence shell for PrTm, NdEr, PmHo and SmDy molecules. Due to similarity, only one set of MO was drawn.

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Chapter 7 A coupled cluster study of coinage metal nitrosyls, M-NO0/+ (M = Cu, Ag, Au)

Abstract The ab initio potential energy surfaces and dipole moment surfaces for the bent CuNO,

AgNO, AuNO molecules and their cations have been calculated under the

Feller-Peterson-Dixon (FPD) framework at the coupled cluster with singles, doubles and perturbative triples, CCSD(T), level of theory including complete basis set extrapolation, outer-core correlation, scalar relativistic effect, spin-orbit coupling and higher level electron correlation. The Brueckner coupled cluster doubles with perturbative triples method,

BCCD(T), is used for the closed-shell molecules to obtain the contribution from orbital relaxation. Upon comparison to the available experimental values, the resulting spectroscopic constants showed significant improvements over those obtained with standard CCSD(T) method.

In addition to the spectroscopic studies, the thermodynamics of the molecules, including metal-NO bond strength and vertical transition energies are also calculated under the FPD framework.

In this work, it shows that the BCC method provides more accurate results for closed shell molecules than the conventional CCSD(T) method. And it provides the first wave-function based study for the AgNO molecule. The scalar relativistic contribution to the harmonic frequencies of the AuNO molecule is re-examined. The ground state of CuNO is determined as 1A’. The thermodynamic and spectroscopic data are predicted for the cations.

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I. Introduction The study of Cu and NO interaction has found its application in many different areas: such as NO decomposition1, corrosion process1, and biochemical catalysis2. Since 1991, various experimental techniques have been applied to characterize the CuNO molecule. The gas-phase existence of the molecule was first identified in mass spectrum by Schwarz and co-workers3. Other experimental studies include matrix FTIR4-8, electron spin resonance spectroscopy9,10, and UV-Vis spectroscopy4.

However, the determination of the ab initio ground state for the bent CuNO molecule is not unambiguous. In Schwarz’s pioneer study of the CuNO molecule3, the 3A” state was found to be lower in energy than the 1A’ state with the Hartree-Fock (HF), coupled cluster singles and doubles (CCSD) and configuration interaction (CI) methods. CCSD with perturbative triples, CCSD(T), however, gave the opposite conclusion that the singlet should be the ground state. Eighteen years later, Uzunova12 used complete active space 2nd order perturbative theory (CASPT2) with unrestricted natural orbitals (UNO) and found that the triplet state was the ground state, but in the same study CCSD(T) calculations again favored a singlet ground state. In 2012, Marquardt and co-workers13,14 used multi-reference configuration interaction (MRCI) method, with several different sizes of the reference spaces, as well as CCSD(T) to study this molecule. Unlike the CASPT2 study, the MRCI and

CCSD(T) calculations both supported a singlet ground state. Meanwhile, several DFT methods12,15 were also applied to study the CuNO molecule, but failed to achieve consensus.

The pure functionals gave a singlet ground state, while the hybrid functionals preferred a triplet ground state.

The difficulty of assigning the ground state might result from the near-degeneracy of the two states (ca. 3 kcal.mol)16, where accurate treatment of correlation energy or even the spin-orbit coupling is necessary. Different wave function theories (WFT) or DFT methods

161

have been applied but only accumulated the discrepancy as briefed above. From the view of

WFT, the determination of the ground state has discussed a lot on the methods11-13, but little discussion was given to the basis sets. In fact, in the early CCSD(T) study11, the basis sets were self-defined and the relativistic effective core potential (ECP) for Cu in the CuNO calculation included 18 electrons. For the CASPT2 study12, the 6-311G and 6-31G basis sets were used for Cu and non metal elements, respectively, with no diffuse functions included.

On the other hand, for the latest MRCI calculation13, correlation consistent basis sets were used. These basis sets systematically recover the correlation energy and give a well spanning of basis sets in the Hilbert space. With considering of their effort of MRCI and CCSD(T) calculations, it should be more plausible that 1A’ is the ground state.

So far the confirmation of the ground state assignment from previous theoretical studies was solely based on the comparison between calculated IR frequencies and the experimental ones. However, among the three vibration modes, the Cu-N stretching frequencies are poorly reproduced, deviating about 50 cm-1 for certain cases. As an alternative for the determination, there is also an electronic spectrum for the molecule4, which gives a recognizable peak at

5444.8 cm-1 (0.675 eV) with an absorbance about 0.34. This absorbance indicates a spin-allowed transition since the spin-orbit coupling is calculated to be small by Uzunova and this work. Therefore, a vertical transition calculation may help to determine the ground state by comparing the excitation energies of singlet and triplet states to the experimental value.

An extension of the CuNO molecule along the periodic table includes AgNO17,18 and

AuNO19-24 and their corresponding cations11,17,18,20,24-30 since they are proposed to have similar catalytic applications with CuNO. However, limited studies were performed on these molecules and conclusions are not without flaws. For example, the AgNO molecule was solely studied by DFT methods. No wave function based methods have been applied so far.

For the AuNO molecule, a comparison23 between all-electron scalar relativistic calculation,

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ECP calculation and all-electron non-relativistic calculation was performed with the PW91 functional. The results show that the inclusion of scalar relativistic effect could change the harmonic frequency nearly 200 cm-1, which is quite extraordinary.

Therefore, a systematic study of the coinage metal nitroxyls (MNO) and their cations is desired to fully understand their electronic structure as well as other molecular properties. As discussed by Marquardt13, the multi-reference method provides highly consistent results with the CCSD(T) method. This indicates that a single-reference method is well capable of describing the CuNO molecule near equilibrium. But the initial attempts with the conventional CCSD(T) method in this study shows that there is weak multi-reference

31 character for the neutral MNO molecules through the T1 diagnostic . And this weak multi-reference character generates some noise to the harmonic frequencies comparing to the experimental values. Therefore, a modified coupled cluster method, the Brueckner coupled cluster (BCC) method is used in this work to calculate the molecules.

The BCC theory with double and perturbative triple excitations (BCCD(T)) has been commonly considered as a better method than the conventional CCSD(T) only when dealing with the symmetry-breaking problems32,33. Coming to the non symmetry-breaking molecules, the BCCD(T) calculations give little difference from the CCSD(T) calculations32,33 and it is widely believed that the BCCD(T) method makes no improvement over the conventional

CCSD(T) method, or vice versa. In this work, however, it shows that for systems such as coinage metal nitroxyl molecules, the BCCD(T) method does provide a more accurate result than the conventional CCSD(T).

In this study, the coinage metal nitroxyls (CuNO, AgNO, AuNO) and their cations

(CuNO+, AgNO+, AuNO+) are studied systematically. The examination of contributions from different type of correlation energies and relativistic effects was performed within the

Feller-Peterson-Dixon (FPD)34 framework. The BCCD(T) and CCSD(T) methods are used to

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calculate the neutral closed-shell molecules and open-shell cations, respectively. The equation of motion (EOM) CCSD is used to help determine the ground state of the CuNO molecule.

The coupled cluster dipole moment surfaces are constructed to obtain the IR intensities. With this work, (1) the MNO molecules are studied by the BCC method for the first time and it shows an example which BCCD(T) method provides more accurate results than the conventional CCSD(T) method; (2) the EOM-CCSD calculation helps to clarify the controversy of the ground state determination of the CuNO molecule; (3) the FPD framework provides the most accurate spectroscopic and thermodynamic results for coinage metal nitroxyl molecules and the properties of cations are predicted.

II. Computational details The systematic study of the coinage metal nitroxyl molecules was performed under the

Feller-Peterson-Dixon (FPD) framework34.

The spectroscopic information was derived from the near equilibrium potential energy surface (PES) of each molecule. The near-equilibrium PES was constructed by fitting a grid of 84 points using the least square method in displacement coordinates. The grid points were sampled through a systematic deviation of the equilibrium geometry by a unit displacement along each coordinate. The geometrical parameters covered the range from -0.3 r0 to 0.5 r0 for bond lengths and 90o to 150o for bond angle in Cartesian coordinates. The setup was done by the Surgen code36.

The energy calculation for each grid point adopted the FPD strategy, where the composite energy was expressed as:

E(r1,r2,θ)=CBS+ΔCV+ΔDKH+ΔSO(+ΔT) (1)

In equation (1), the first term on the right-hand side represents the valence electronic energies extrapolated to the complete basis set limit (CBS). The energy calculation with

164

different size of basis sets used BCCD(T), or CCSD(T) method37 for closed-shell molecules or open-shell molecules, respectively. The correlation-consistent basis sets, aug-cc-pVnZ

(n=D,T,Q,5)38 for non-metal elements and aug-cc-pVnZ-PP (n=D,T,Q,5) with small core relativistic pseudo-potential for coinage metal elements were used39. For convenience they are abbreviated as aVnZ and aVnZ-PP basis sets. The Hartree-Fock energies were extrapolated by Karton and Martin’s formula40:

-6.57 n En=EHF-Limit+A(n+1)e (2) where En are the HF energies calculated by aVQZ and aV5Z sets and n is the cardinal number in the basis sets notation. The correlation energies were extrapolated by a two-parameter formula, Eq. (3), using aVQZ and aV5Z data41:

1 4 En E CBS  A/() n  (3) 2 where En are the energies calculated with corresponding basis sets and Ecbs is the extrapolated energy at the CBS limit.

The second term in Eq. 1 recovers the correlation energy from outer core electrons. For

N and O, the core electrons include 1s electrons. For Cu, Ag, and Au atom, the outer core electrons outside ECP are 3s3p, 4s4p, and 5s5p, respectively. The energies were calculated with the CCSD(T) method and aug-cc-pwCVQZ(-PP) basis sets39,42. The correlation energies out of outer core electrons were added to the CBS results to improve the theoretical accuracy.

The third term provides the scalar relativistic contribution from light atoms and a correction for the use of the ECP. The all-electron calculation with second-order

Douglas-Kroll-Hess (DKH) Hamiltonian43 was carried out with aVTZ-DK basis sets for metal and light atoms42,44,45. The CCSD(T) method was used to calculate the energy. The energy difference between calculations using aVTZ-DK sets and aVTZ(-PP) sets was taken as the contribution and correction for the scalar relativistic effects.

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The fourth term is the spin-orbit correction. For the neutral molecules, the 2-component

DFT method using B3LYP functional was used, while for the cations, the 2-component

CCSD(T) calculation was carried out. In both cases, the 2-component ECP Hamiltonian46 was applied. The uncontracted aVDZ and aVDZ-PP basis sets were used for non-metal and metal elements, respectively. The cutoff of the virtual orbital energies was set as 20 Hartrees, and the frozen core approximation was adopted same as the non-relativistic calculations.

The last term is the high-order correction, which takes the difference between a full

CCSDT calculation and the CCSD(T) calculation at the DZ quality of basis set. Since the

BCCD(T) method was not available for the open-shell molecules, this term was calculated for cations to provide corrections to the perturbative triples calculations used in the conventional

CCSD(T) method. The CCSDT method would be the highest level of theory used to study the

MNO cations.

After calculating the composite FPD energies for each grid point, the PES was fitted by a

5th order polynomial as in Eq. (4):

i j k V  cijk R1 R 2 R 3 (4) i, j , k  1 where R1, R2 and R3 corresponds to the internal displacement coordinates in term of rM-N, rN-O and θ, respectively. Additional 6th-order diagonal terms were included during the fitting process.

For the neutral molecules, the angle displacement was further expanded into a cubic polynomial as Eq. (5) to minimize the root-mean-square (RMS) error.

23 RAAA3 0   1    2   (5)

47 The value of A1 and A2 was determined by the boundary conditions , while A0 was roughly optimized in the fitting process.

The PES fitting as well as calculation of spectroscopic constants via 2nd order

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perturbation theory48 were carried out by the Surfit36 program. In all PES fitting, the RMS value was targeted to be less than 3 cm-1, typically less than 1 cm-1.

The dipole moment surfaces are calculated using the BCCD(T) and CCSD(T) method for the neutral molecules and cations, respectively with the aVTZ-PP basis sets. The dipole moments are calculated numerically with static electric field set as ±0.002 a.u. The obtained dipole moment surfaces in Cartesian coordinates are then rotated into the Eckart frame49 and fitted to the 4th-order polynomials. The new derivatives of dipole moments corresponding to each normal mode are obtained by the Surfit program and are used to calculate the intensities for each mode by the formula50:

('')dd 2 I( cm21 atm at 300 K) 65.785   i,, x i y (6) ii2 where ωi is the harmonic frequency for each mode. The d’i,x and d’i,x are the first derivatives of dipole moments along the x and y principal axes, respectively for each normal mode.

To estimate the vertical transition energies of CuNO at its composite geometry, the

EOM-CCSD51 approach was used for the molecules with 6 states evenly spanned in the A’ and A” symmetry. The frozen core approximation and the aVTZ-PP basis sets were used.

To calculate the dissociation energies of the title molecules, the FPD scheme similar to the spectroscopic calculations is adopted as well. But instead of using BCCD(T) method to calculate the neutral molecules, the conventional CCSD(T) method is used in order to compare the molecules and atoms at the same level of theory. For the sake of the same reason, the SO calculations were all carried out at the average-of-configuration HF level of theory.

For all the other selections such as basis sets, etc used in the thermochemistry calculations, the same choice was made as the spectroscopy calculations. The molecular energies were calculated at the VTZ-fitted geometries.

For the spin-orbit calculations, the DIRAC52 program was used. The high order calculation of CCSDT was performed by the MRCC54 code interfaced to MOLPRO. All the

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other calculations were carried out using the MOLPRO55 package.

III. Results and Discussion A. CuNO

The fitting results of CuNO 1A’ state are summarized in Table 1 at the BCCD(T) level of theory. Following the FPD framework, the CBS limit results were extrapolated first using the

QZ and 5Z data. It can be seen that as the basis sets become more complete, the calculated bond lengths, bond angle, and fundamental frequencies, are approaching to the CBS limit and the spacing between the neighboring basis set results is getting smaller and smaller. This complies with the general trend of correlation consistent basis sets. At the second stage, the outer-core electron correlation (ΔCV) was calculated. The calculation shows that the ΔCV correction has a larger contribution to the N-O stretching mode (ν1) than the M-N stretching mode (ν2) and bending mode (ν3). The ΔCV correction is also the major correction beyond the CBS limit comparing to the ΔDK correction and ΔSO correction, which are expected to be small, since the pseudo potential is well defined for the Cu atom and the relativistic effects are less significant for light atoms. It should be noted that for the neutral molecules, the

B3LYP functional was used to calculate the spin-orbit coupling contribution instead of the coupled cluster method. This choice was made since the CCSD(T) method suffers a large

RMS error as well as large deviations of harmonic frequencies. This situation magnifies when the spin-orbit coupling operator is turned on, thus the B3LYP method as a compromise has to be used for the spin-orbit calculations.

The composite results are compared to the experimental values at the end of the table. It can be seen that the agreement between this work and experiments is very good. For all of the

-1 three normal modes, the difference is about 30 cm at most. Comparing the same experimental technique using different matrix gases, it can be found that the matrix effect can

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cause a peak shift about 20 cm-1. Especially, the frequencies measured in the Ne matrix are closer than those measured in the Ar matrix with respect to the calculated results. While in term of the isotope effect, it is showed experimentally4 that it is negligible. Hence it can be concluded that the matrix effect should be the main source for the discrepancy between the experimental values and the theoretical values.

In term of the thermal chemistry, the Cu-NO homo cleavage bond dissociation energies are listed with respect to different sizes of basis sets and FPD terms. In order to compare the

MNO, metal and NO molecules at the same level of theory for the SO term, the average-of configuration HF calculations are carried out to obtain the SO contribution. It can be seen that at the DZ quality of basis set, the basis set incompleteness error (BSIE) from the CBS limit has fallen within the chemical accuracy of 1kcal/mol. The better agreement of the TZ value than the QZ and 5Z values, comparing to the CBS value, should merely come from fortunate error cancellation. Moreover, it is worthwhile to notice that the BSIE, CV contribution and the SO contribution are on the same order of magnitude, indicating the necessity to include these FPD terms to obtain accurate dissociation energy. The DK contribution, on the other hand, is much less important, as expected so.

B. AgNO

Table 2 shows the composite results for the AgNO molecule. Slightly different from the

CuNO molecule, the ν1 mode frequency calculated with the avdz basis set doesn’t follow the general convergence trend, which the frequencies converge gradually towards to the CBS limit. This is not uncommon among the calculations using correlation consistent basis sets since the DZ basis set has the largest basis set incompleteness error. Other than this, it can be seen that other properties follow the same converging pattern with the CuNO molecule. In term of the bond lengths, the AgNO molecule has a longer M-N bond length and shorter N-O

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bond length by ~0.4 angstrom and 0.01 angstrom, respectively than the CuNO molecule. In term of the frequencies, the ν1 frequency, which corresponds to the N-O stretching mode, is correspondingly larger than that of CuNO by ~100 cm-1. While for the other two modes, which involve the Ag atom, the frequencies get a blue shift by ~100 cm-1, too. These indicate that the Ag atom and the NO fragment have a weaker bond comparing to the CuNO molecule.

It should be also noted that for the AgNO molecule, the CV correction has a larger contribution to the ν2 and ν3 modes, in contrast to the CuNO molecule. For the CuNO

-1 molecule, the CV contribution for ν1 , ν2 and ν3 modes are 10.0, 5.4 and 2.8 cm , respectively. While for the AgNO molecule, they are -0.4, 16.3 and 15.2 cm-1, respectively.

The similar observation can also be found for the bond lengths, which the CV correction has a lager influence to the Ag-N bond length than the Cu-N bond length. For the relativistic effects (DK and SO), the contributions are both negligible same as the CuNO molecule.

Comparing to the experimental values, it can be found that the matrix effect plays a non-negligible role, too, in the frequency measurement, which the ν1 peak shifts about 30 cm-1 in different inert gas environments. Assuming the matrix effect dominantly consists of the experimental errors, the calculated ν1 frequency well falls into the range of the measured value ± matrix effect. Thus it is confident to conclude that the ν2 and ν3 frequencies are accurate and credible, which are not obtained experimentally.

In term of the thermal chemistry, the dissociation energy is well converged to the CBS limit with respect to different sizes of basis sets. Similar conclusions to the CuNO thermal chemistry can be drawn to the AgNO molecule, where at the DZ basis set, the dissociation energy has already reached to the chemical accuracy in regards to the CBS limit. The CV correction as well as the SO correction are one order of magnitude larger than the DK correction. It is worthy to note that the dissociation energy for the AgNO molecule is much less than that of the CuNO molecule by about 10 kcal/mol. This is in consistence with the

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calculated results of longer Ag-N bond lengths and lower Ag-N stretching frequencies. Thus it can be concluded that the AgNO molecule has a weaker metal-N bond than the CuNO molecule.

C. AuNO

In Table 3 there shows the composite results for the AuNO molecule. The convergence trend of basis sets is similar to the CuNO molecule. Comparing the bond length to the CuNO and AgNO molecules, the Au-N bond length is shorter than Ag-N but longer than the Cu-N bond length. This shortening of the Au-N bond length than the Ag-N bond length is mainly resulting from the relativistic effects of the Au atom. In term of the M-N-O bond angle, all the metal nitroxyl molecules have a bond angle close to 120 degree. This indicates that the

2 center atom N adopts an sp configuration, in the language of classic valence shell electron pair repulsion theory.

With respect to the fundamental frequencies, the AuNO ν1 N-O stretching mode has the highest frequency among the three coinage metal nitroxyl molecules, corresponding to its shortest N-O bond length. For the ν3 Au-N stretching mode, its frequency is close to the

Cu-N stretching mode, while the bending mode has the largest frequency among the three molecules.

The CV contribution for the AuNO molecule is also the most important term beyond the

CBS limit as the CuNO and AgNO molecules, and its importance to the three vibration modes is similar to the AgNO molecule, where the CV correction is more important for the

ν2 and ν3 modes than the ν1.mode. For the relativistic effects, the DK and SO corrections become more significant, one order of magnitude larger than those for the CuNO and AgNO molecules. Although saying so, yet the DK correction contributes no larger than 2 cm-1, in sharp contrast to the previous study which claims that the scalar relativistic effect could

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change the harmonic frequency about 200 cm-1. Due to the lack of experimental studies, it is not able to make a direct comparison. But based on the well agreement of experimental and theoretical studies for the CuNO molecule and the AgNO molecule, and their scalar relativistic effect is negligible, it is confident to conclude that the DK correction shouldn’t shift the peaks as much as 200 cm-1.

For the dissociation energy of the AuNO molecules, the BSIE is relatively more significant. The DZ value deviates from the CBS limit by about 2.5 kcal/mol while the TZ basis set is good enough to obtain the accurate value. Beyond the CBS limit, the CV correction is more important than the DK and SO correction by one order of magnitude. The

DK correction for the AuNO molecule, on the other hand, is one order of magnitude than that of the CuNO and AgNO molecules. It may also be interesting to note that the relativistic effects have a positive contribution to the dissociation energy for the AuNO molecule while a negative contribution to the CuNO and AgNO molecules. In the end, the AuNO molecule has strongest metal-N bond. The Au-N bond is about 10 kcal/mol stronger than the Cu-N bond, which is about 10 kcal/mol stronger than the Ag-N bond.

D. CuNO+

In Table 4 there shows the composite results for the CuNO cation. Different from the neutral species, the cations cannot perform the BCCD(T) calculation due to the technical limitation that the current version of Molpro doesn’t support BCCD(T) calculations for the open shell systems. Therefore, the conventional CCSD(T) method is used instead. Yet on the other hand, for the conventional coupled cluster method, the full triple excitation correlation is available. Thus an additional correction, the high-order correction ΔT is included into the

FPD equation as the difference between full CCSDT calculations and perturbative CCSD(T) calculations. For the cations, the spin-orbit coupling correction is calculated by the

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2-component CCSD(T) method as there is no nearby state disturbing the single reference character.

Comparing the CuNO+ cation and the CuNO molecule, the first difference may be the larger bond angle for the cation by about 10 degree. Concertedly, the cation has a shorter N-O bond length and longer Cu-N bond length than its neutral counterpart, both by ~0.04 angstrom. In term of the fundamental frequencies, the N-O stretching mode is about 300 cm-1 higher while the bending and Cu-N stretching modes about nearly 100 cm-1 lower than the corresponding modes for CuNO molecule.

Beyond the CBS limit, it can be found that the CV correction is generally more important than the DK and SO correction. And similar to the CuNO neutral molecule, the CV correction has a lager influence to the ν1 mode than the other two modes. While for the DK correction, the cation happens to be more influenced by the scalar relativistic effect than the neutral molecule. In term of the SO correction, it is totally negligible as the CuNO molecule. It is worthwhile to note that the ΔT correction is as important as the CV correction, indicating the necessity to include the higher-order correction in the FPD equation.

While looking at the dissociation energy of the CuNO+ cation, which is dissociated into

+ + the Cu and NO fragments, it can be seen that the CuNO dissociation energy is approximately 10 kcal/mol larger than the neutral CuNO molecule. Recalling that the dissociation energy of AuNO is ~10 kcal/mol larger than the CuNO molecule, which is also

~10 kcal/mol larger than the AgNO molecule, it seems that the 10 kcal/mol is the magic number for the dissociation energy among the coinage metal nitroxyl molecules and their cations. And this can be verified by comparing the AgNO+ and AuNO+ cations in the next sections. It should be noted, however, that although the SO correction is totally negligible for spectroscopy, it does not remain silence for thermal chemistry. In fact, it is as important as the

CV correction. The ΔT correction is left out in order to compare the dissociation energy with

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the neutral molecules as the same level of theory.

E. AgNO+

In Table 5 there shows the composite results for the AgNO+ cation. The bond angle for the cation is close to that of the CuNO+ cation and is ~10 degrees larger than AgNO. The N-O bond length is shorter and the Ag-N bond length is longer both by about 0.02 angstrom. This may be an interesting coincidence with the facts that between the CuNO and CuNO+ pair, the decrease and increase of bond length is both about 0.04 angstrom and as will be discussed in the next section, the bond length change between the AuNO and AuNO+ pair is both about

0.03 angstrom.

+ The fundamental frequencies of AgNO cation has a larger ν1 frequency and smaller ν2 and ν3 frequencies than AgNO. The same observation is found between the CuNO and

CuNO+ pair. Comparing to the experimental values, the difference is only about 10 cm-1,

-1 comparable to the matrix effect, which can shift the ν1 frequency by about 10 cm , too. For the CV correction, it has similar contribution to all three vibration modes and the contribution is at the same order of magnitude with the CuNO+ cation. The higher-order correction, on the other hand, is not as important as in the CuNO+ case. The relativistic effects (DK and SO) are still small to negligible.

In term of the thermal chemistry, as discussed before, the dissociation energy is about 10 kcal/mol larger than that of the AgNO molecule, as strong as the Cu-NO bond. The CV correction again dominates the contributions beyond the CBS limit.

F. AuNO+

Table 6 shows the composite results of the AuNO+ cation. Comparing to the neutral

AuNO molecule, the cation has a larger bond angle, shorter N-O bond length and longer

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Au-N bond length. The change of the bond lengths is both about 0.03 angstrom, and it happens to be in between the change of CuNO/CuNO+ pair and AgNO/AguNO+. The change of the fundamental frequencies is similar to the CuNO+ and AgNO+ cations, which the

ν1 mode has a red shift while the ν2 and ν3 modes experience a blue shift comparing to the neutral molecule.

Among the FPD terms beyond the CBS limit, the CV correction and the ΔT correction are the most important correction terms. For the relativistic effects, the DK and SO correction is comparable to the corresponding corrections for the AuNO molecules, and are larger than those of the CuNO+ and AgNO+ cations. Especially, it should be noted that the DK correction shifts the peaks no larger than 2 cm-1, well supporting the remarks in the AuNO section which conclude that the scalar relativistic effect shouldn’t cause a peak shift about 200 cm-1.

The dissociation energy of the AuNO+ cation is calculated to be about 35 kcal/mol, about

10 kcal/mol larger than the neutral molecule. In the end, the following order can be established among the studied molecules in term of the dissociation energy, from the largest to smallest:

AuNO+ (35) > AuNO (25)~CuNO+ (26) > CuNO (17)~AgNO+ (18) > AgNO (7) where the numbers in the parentheses are the dissociation energy rounded up to the closest integer in the unit of kcal/mol.

G. Vertical transition energies (VTE) and determination of ground state

As discussed in the introduction, the ground state determination of the CuNO molecule is not unambiguous, namely it is not clear whether the singlet or the triplet is the ground state.

Summarizing the experimental facts of the electronic spectrum, two low-lying electronic states (A and B) have been detected with the same conversion rate constant. Two sets of corresponding vibration frequencies were also measured in Ar matrix. Especially, the state B

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is only observed in the Ar matrix. A third state C can be found with excitation energy nearly

0.68 eV. The transition C←A was observed while the transition C←B was not. This implies A and C have the same spin multiplicity. But whether the transition is between singlets or between triplets is not known from that experiment.

Regarding the previous theoretical studies, a singly excited configuration interaction with 6-311G* basis set calculation56 was performed and predicted the first excitation energy as 0.31 eV from 1A’ to 1A”. An MRCI study with the correlation consistent basis sets, however, also failed to reproduce the vertical transition energy.

Based on these results, an EOM-CCSD calculation was carried out in this work on the 1A’ state for the CuNO molecule at its composite equilibrium geometry. The excitation energies of the same spin multiplicity were calculated. Using the aVTZ(-PP) basis sets, the first VTE was calculated as 0.76 eV, comparing to the experimental value of 0.68 eV. The good agreement indicates that the 1A’ state should be the ground state for the CuNO molecule. To be as rigorous as possible, it would be desirable to calculate the VTE for the triplet state using the same method and basis sets. Yet this is not approachable due to the technical limitations.

But considering the good agreement of other observables, it can be confidently concluded that the 1A’ state is the ground state for the CuNO molecule.

G. BCCD(T) vs. CCSD(T)

As discussed in previous sections, the BCCD(T) method is capable of providing accurate results comparing to the experimental values. Take the CuNO and AgNO molecules as example where experimental data is available. It can be seen from Table 7 that the conventional CCSD(T) results can be ~50 cm-1 off the experimental values including all the

FPD terms, whereas the BCCD(T) results match the experiment very well at the same composite level of theory. This is surprising since it has been long considered that the two

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methods should give very similar results except for symmetry-breaking systems.

But for the system of coinage metal nitroxyl molecules, which has weak multi-reference character, the BCCD(T) method does show a better performance. To explore a reasonable explanation, it may be helpful to review the algorithm of the BCCD(T) method. For the

Brueckner reference determinant, YB , it is established by rotating HF orbitals to vanish the T1 amplitudes in the conventional CCSD(T) method. As a result, the orbitals are relaxed and the

Brueckner reference determinant has the maximum overlap with the exact wave function57:

=max (8) B exact or in the language of functional analysis, the function distance between and in Yexact the Hilbert space is minimized:

||Y -Y ||=min exact B (9)

In the coupled cluster ansatz, the single amplitudes, T1, result in the orbital relaxation.

Thus the T1 diagnostic can be viewed as a measurement of how close the HF determinant is to the Brueckner determinant58. If the energy minimized determinant (HF) is very different from the distance minimized determinant (Brueckner), one may expect the multi-reference character for the studied molecules. For example, the HF energy for the H2O molecule only differs 0.7 mHartree from the Brueckner reference energy. While for the CuNO molecule, the energy difference is about 40 mHartree.

Therefore, the conventional CCSD method fails relative to the BCCD method because it doesn’t have a proper reference function. The singly excited determinants do not have direct contribution to the energy calculations and thus the frequencies are calculated with large errors. While for the BCCD method, it benefits from the proper reference determinant, and thus it provides more accurate results than the conventional CCSD method.

G. Dipole moment surfaces and normal mode intensities

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Molecular properties such as dipole moments may be expressed in different coordinate systems. In this study, the calculated dipole moment surface using the internal coordinates is transformed into Eckart frame coordinates as it allows the maximum separation of the vibration motion from the translation and rotation motions.

The calculated dipole moments for the CuNO, AgNO and AuNO molecules are 0.19,

0.01, 0.28 a.u. (1 a.u. = 2.5417 Debye), respectively. And for the CuNO+, AgNO+ and AuNO+ cations, the dipole moments are 2.86, 3.45, 2.66 a.u., respectively.

The intensities of each vibrational normal mode for each molecule are listed in Table 8.

For the neutral molecules, the N-O stretching modes have the largest intensity, following by the bending modes and then the M-N stretching modes. The major reason for small intensities along M-N modes is that the component dipole moment derivatives with respect to the normal mode the along x and y principal axis have close values but with different signs. As a result, when applying Eq. 5 to calculate the intensities, the dipole moment derivatives along x and y axes tend to cancel each other, leading to a small intensity for the M-N mode. This is in consistence with the experiments that the N-O stretching mode has the strongest intensity.

The similar observation can be made for the N-O mode and the bending mode intensities of the cations, which are much smaller than the corresponding modes in neutral molecules. A distinction between the neutral molecules and cations is that for the neutral molecules, it is the M-N mode that has the cancellation of dipole moment derivatives of x and y principal axes, while for the cations, it is the other two modes that have the cancellation of dipole moment derivatives of x and y principal axis.

IV. Conclusions In this study, the coinage metal nitroxyls and their cations are studied. The singlet state is determined to be the ground state for the neutral molecules based on the good agreement of

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thermal chemistry as well as spectroscopy. The AgNO molecule is studied by a wave function theory method for the first time and the relativistic effects of the AuNO molecule are discussed.

The FPD methodology is adopted to systematically study the molecules and provide accurate theoretical results. The dipole moments as well as intensities of vibration modes are calculated and the difference between calculated frequencies matches experimental values well. The main discrepancy should result from the matrix effect.

The BCCD(T) method was used and it is found to be more accurate than CCSD(T) for the coinage metal nitroxyl molecules. It is interpreted that BCCD(T) method has a better reference determinant and the HF determinant. As the multi-reference character is negligible, the Brueckner determinant is very close to the HF determinant so that the two correlation methods are nearly equivalent. When the system has a weak multi-reference character, the two methods become non-equivalent and the analysis in this study indicates that the BCCD(T) method should be universally better than the CCSD(T) method for weak multi-reference systems, as supported by examples of coinage metal nitroxyl molecules. While for the heavy multi-reference systems, both methods should fail.

The dissociation energies of the studied molecules are calculated. It is found that the energy difference between each neutral/cation pair is all about 10 kcal/mol, with the cations having larger dissociation energies.

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Table 1. The convergence trend of BCCD(T) for CuNO. r(N-O) r(Cu-N) 훉 훎 훎 훎 ퟏ ퟐ ퟑ De avdz 1.1875 1.9124 119.1 1590.2 429.0 262.7 16.37 avtz 1.1774 1.8887 119.3 1589.8 451.5 275.9 17.03 avqz 1.1740 1.8824 119.3 1603.2 457.4 279.1 17.21 av5z 1.1732 1.8825 119.2 1604.5 457.3 279.1 17.15 CBS 1.1727 1.8827 119.1 1605.5 457.1 278.9 17.08 횫퐂퐕 -0.0020 -0.0021 0.0 10.0 5.4 2.8 -0.23 횫퐃퐊 -0.0003 0.0031 -0.1 0.1 1.1 -0.2 -0.06 횫퐒퐎 0.0000 -0.0001 0.0 0.0 0.0 0.0 -0.19 Composite 1.1704 1.8836 119.0 1615.6 463.6 281.5 16.60 Exp4. (Ar) 1587.4 452.6 278.2 Exp7. (Ne) 1602.2 Exp5. (Ar) 1610.5 The bond lengths are in angstrom, frequencies are in wave numbers and the dissociation energies are in kcal/mol. Through the study, ν1 stands for the N-O stretching mode. ν2 stands for the M-N stretching mode. ν3 stands for the bending mode. The SO correction for the dissociation energy is calculated at the average of configuration HF level of theory, while for all the other observables the SO correction is calculated at the B3LYP level of theory.

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Table 2. The convergence trend of BCCD(T) for AgNO.

r(N-O) r(Ag-N) 훉 훎ퟏ 훎ퟐ 훎ퟑ De avdz 1.1746 2.3828 117.4 1707.5 325.7 134.0 6.13 avtz 1.1651 2.3041 117.8 1686.3 348.4 154.8 6.28 avqz 1.1617 2.2899 117.7 1697.6 356.0 161.0 6.37 av5z 1.1608 2.2889 117.6 1698.8 355.9 160.3 6.35 CBS 1.1603 2.2885 117.5 1700.2 356.6 160.4 6.34 횫퐂퐕 -0.0013 -0.0450 0.2 -0.4 16.3 15.1 0.50 횫퐃퐊 -0.0002 0.0076 -0.1 1.1 -0.2 -0.7 -0.04 횫퐒퐎 0.0001 0.0009 0.0 0.2 -0.4 -0.2 -0.19 Composite 1.1589 2.2520 117.6 1701.1 372.3 174.6 6.61 Exp18. (Ar) 1680.3 Exp18. (Ne) 1711.8/1707.3 The bond lengths are in angstrom, frequencies are in wave numbers and the dissociation energies are

in kcal/mol. The SO correction for the dissociation energy is calculated at the average of configuration

HF level of theory, while for all the other observables the SO correction is calculated at the B3LYP

level of theory.

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Table 3. The convergence trend of BCCD(T) for AuNO.

r(N-O) r(Au-N) 훉 훎ퟏ 훎ퟐ 훎ퟑ De avdz 1.1706 2.0705 117.7 1708.0 491.5 256.2 18.84 avtz 1.1602 2.0369 117.9 1707.3 517.0 273.6 20.77 avqz 1.1565 2.0319 117.9 1722.9 520.8 275.8 21.16 av5z 1.1558 2.0297 117.8 1724.5 521.9 276.5 21.25 CBS 1.1552 2.0283 117.8 1725.9 522.6 277.1 21.31 횫퐂퐕 -0.0017 -0.0171 0.1 7.6 15.4 11.8 2.19 횫퐃퐊 0.0000 -0.0031 0.0 -1.8 1.3 1.1 0.44 횫퐒퐎 0.0006 -0.0052 0.1 -3.2 -1.1 2.7 0.85 Composite 1.1541 2.0030 118.1 1728.5 538.2 292.7 24.91 The bond lengths are in angstrom, frequencies are in wave numbers and the dissociation energies are in kcal/mol. The SO correction for the dissociation energy is calculated at the average of configuration HF level of theory, while for all the other observables the SO correction is calculated at the B3LYP level of theory.

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Table 4. The convergence trend of CCSD(T) for CuNO+.

r(N-O) r(Cu-N) 훉 훎ퟏ 훎ퟐ 훎ퟑ De avdz 1.1556 1.9655 133.5 1896.2 303.2 189.2 23.91 avtz 1.1441 1.9319 135.1 1906.4 323.6 193.2 25.33 avqz 1.1406 1.9240 134.9 1922.4 331.3 195.7 25.85 av5z 1.1397 1.9244 134.6 1924.8 330.3 195.9 25.82 CBS 1.1391 1.9245 134.4 1926.7 329.4 196.0 25.79 횫퐂퐕 -0.0020 0.0000 0.1 7.7 1.9 0.6 0.16 횫퐃퐊 0.0000 0.0036 -0.4 -2.2 2.1 0.5 0.06 횫퐒퐎 0.0000 -0.0002 0.0 -0.1 0.1 0.0 -0.19 횫퐓 -0.0009 0.0000 -0.4 5.5 4.2 3.3 Composite 1.1362 1.9280 133.6 1937.6 337.7 200.4 25.82 The bond lengths are in angstrom, frequencies are in wave numbers and the dissociation energies are in kcal/mol. The SO correction for the dissociation energy is calculated at the average of configuration HF level of theory, while for all the other observables the SO correction is calculated at the CCSD(T) level of theory.

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Table 5. The convergence trend of CCSD(T) for AgNO+.

r(N-O) r(Ag-N) 훉 훎ퟏ 훎ퟐ 훎ퟑ De avdz 1.1561 2.3494 128.8 1884.8 235.6 138.2 14.80 avtz 1.1443 2.3004 130.3 1896.3 243.7 145.5 15.82 avqz 1.1407 2.2899 130.1 1912.6 249.3 147.3 16.12 av5z 1.1398 2.2906 129.8 1916.0 248.9 147.4 16.10 cbs 1.1392 2.2909 129.6 1918.6 248.3 147.4 16.07 횫퐂퐕 -0.0023 -0.0264 0.1 7.7 12.1 6.1 0.90 횫퐃퐊 0.0000 0.0048 -0.3 -1.9 -1.4 -0.9 -0.05 횫퐒퐎 0.0000 0.0009 0.0 0.2 -0.7 -0.3 -0.19 횫퐓 -0.0005 0.0024 -0.1 2.0 1.1 -0.1 Composite 1.1365 2.2725 129.3 1926.6 259.4 152.2 16.73 Exp18. (Ne) 1912.4/1910.9 Exp18. (Ar) 1904.3/1900.2 The bond lengths are in angstrom, frequencies are in wave numbers and the dissociation energies are in kcal/mol.The SO correction for the dissociation energy is calculated at the average of configuration HF level of theory, while for all the other observables the SO correction is calculated at the CCSD(T) level of theory.

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Table 6. The Convergence trend of CCSD(T) for AuNO+.

r(N-O) r(Au-N) 훉 훎ퟏ 훎ퟐ 훎ퟑ De avdz 1.1455 2.1219 124.3 1864.8 348.3 210.4 29.22 avtz 1.1345 2.0784 125.2 1874.7 370.9 223.6 31.20 avqz 1.1309 2.0721 125.1 1891.1 375.4 225.8 31.78 av5z 1.1301 2.0698 125.0 1893.8 376.3 226.4 31.83 CBS 1.1296 2.0682 124.9 1895.9 377.1 226.8 31.86 횫퐂퐕 -0.0022 -0.0256 0.5 6. 10.4 10.7 2.11 횫퐃퐊 -0.0001 -0.0032 0.1 -1.7 1.7 1.2 0.47 횫퐒퐎 0.0002 -0.0103 0.4 -1.0 2.2 4.0 0.49 횫퐓 -0.0014 0.0041 -0.2 12.1 -0.2 1.0 Composite 1.1261 2.0332 125.7 1911.9 391.2 243.7 34.93 The bond lengths are in angstrom, frequencies are in wave numbers and the dissociation energies are in kcal/mol.The SO correction for the dissociation energy is calculated at the average of configuration HF level of theory, while for all the other observables the SO correction is calculated at the CCSD(T) level of theory.

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Table 7. The composite BCCD(T) and CCSD(T) fundamental frequencies for CuNO and AgNO molecules comparing to the experimental values.

CuNO 훎ퟏ 훎ퟐ 훎ퟑ CCSD(T) 1666.3 435.3 266.4 BCCD(T) 1615.6 463.6 281.5 Exp.4 1587.4 452.6 278.2 Exp.7 1602.2 Exp.5 1610.5

AgNO CCSD(T) 1744.4 324.5 172.2 BCCD(T) 1701.1 372.3 174.6 Exp.18 1680.3 Exp.18 1711.8/1707.3

AuNO

CCSD(T) 1728.0 528.5 287.9

BCCD(T) 1728.5 538.2 292.7

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Table 8. The calculated intensities of each vibrational normal mode for the MNO and MNO+ molecules.

훎ퟏ 훎ퟐ 훎ퟑ CuNO 5474.9 113.5 2.32 AgNO 4923.8 109.3 5.15 AuNO 3304.1 124.9 6.10

CuNO+ 1.9 0.0 206.2 AgNO+ 9.0 2.0 174.1 AuNO+ 472.1 48.3 94.2 The unit is in cm-2*atm-1.

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Chapter 8 Accurate ab initio vibronic spectroscopy of the CCP and CCAs radicals Abstract

Explicitly correlated MRCI-F12 calculations have been carried out for the CCP and

CCAs radicals with systematic sequences of basis sets to determine their accurate near-equilibrium potential energy surfaces. The Feller-Peterson-Dixon (FPD) composite methodology was applied to rule out significant possible errors and provide the most accurate theoretical results. The complete basis set (CBS) limit is approached with the explicitly correlated correlation consistent basis set to eliminate the basis set incompleteness error. The outer-core electrons are correlated at the second stage to improve the valence results at the

CBS limit. The scalar relativistic effects are accounted by using the second-order

Douglas-Kroll-Hess Hamiltonian and the spin-orbit coupling contribution is calculated by the

Breit-Pauli Hamiltonian.

The title radicals are calculated to be linear in degenerate Π ground states. The

Renner-Teller (RT) effect is observed when the molecular symmetry is broken. Harmonic frequencies are calculated by the second-order vibrational perturbation theory and the RT parameter is calculated. The fundamental frequencies and anharmonic constants are calculated for the first time, which makes it able to compare with the experimental values directly.

I. Introduction It has been a long time for human kinds wondering the origin of universe and the alien life in the outer space. Since 18th centuries, people has observed and identified many particles in the interstellar medium. The observation of particles and radiation in universe is critical,

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since the confirmation of presence of certain particles helps people to infer the physical and chemical environments of the remote celestial bodies. One example of this kind of applications lies in the prediction and then the confirmation of the existence of the CCP radical.

The existence of the CCP radical in the interstellar medium was proposed more than 20 years ago, along with the search of other phosphorus-containing molecules1. It was assumed that the radical might exist in the molecular hot core of star-forming regions provided the absence of oxygen atoms2. In 2008, the radical was first detected in the circumstellar gas of

IRC +10216, as the fifth phosphorus-containing molecule found in interstellar space3. In the academic field, the CCP radical is of interest in the sense that the radical exhibits the

Renner-Teller (RT) effect. Telling from its name, the RT effect is similar to the Jahn-Teller effect, but only for the linear molecules. From previous studies, the CCP radical is found to be linear and in the ground 2Π state.

However, not much experimental work has been published to characterize the molecule.

The most important 2 experimental papers come from the molecule’s observer Clouthier in

20084 and 20093. By using the laser-induced fluorescence (LIF) spectrum and Fourier transform microwave (FTMW) technique at microwave, millimeter or submillimeter wavelengths, the C-C bond length was fitted as 1.29 angstrom and the C-P bond length was fitted as 1.62 angstrom. In term of the harmonic frequencies, the C-C stretching mode ω1,

-1 bending mode ω2 and C-P stretching mode ω3 were fitted to 1646.36, 211.34 and 837.75 cm , respectively. Other molecular constants derived from these techniques include spin-orbit constant, anharmonic constant and centrifugal distortion constant, etc. With respect to the thermochemistry, the atomization energy was determined as 268.7±3.5 kcal/mol by the

Knudsen cell mass spectrometry5.

For the theoretical studies, on the other hand, the calculation with high level theory and

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basis sets is still not available. This limits the comparison with the experimental studies.

Currently, the density functional theory (DFT) studies with the B3LYP functional1,4,6 give the

C-C stretching frequency as 1727 cm-1. The MP27 and QCISD6 calculations give the frequency as 2056 and 1694 cm-1, respectively. It can be seen that there is a large discrepancy between these theoretical work. As a consequence, it is hard to conclude which theoretical work is accurate and it is hard to estimate whether the fitting method for harmonic frequencies used in experimental studies is proper or not. In addition, these theoretical studies were carried out under the harmonic oscillator approximation, thus the fundamental frequencies including anharmonic corrections have never been calculated. A direct comparison to the observed frequencies hence has never been made. What is more, it was experimentally found that the spin-orbit coupling (SOC) constant is large for the CCP radical4.

But how the SOC effect interacting with the RT effect through shifting the frequencies is not documented. Therefore it would be interesting to examine the SOC effect explicitly and conduct a calculation with high level theory and basis sets to help analyze the complex spectra.

A natural extension to the CCP radical is its isovalent molecule of CCAs , which is of potential interest in the formation of durable films8. Similar to the CCP radical, the CCAs radical may also exhibit the RT effect and it has been experimentally characterized by LIF8,9 and FTMW10. By LIF, the fundamental C-C stretching frequency was determined as 1692 cm-1. In comparison, previous theoretical studies11-14 shows the C-C stretching frequency as

1749 cm-1 by B3LYP8, and the wave function based methods, QCISD(T)11 and MRCI12 give the frequency as 1712 and 1685 cm-1, respectively. Same with the CCP studies, no anharmonic correction has been calculated yet, and no SOC calculation has been carried out for the frequency calculations. In term of the thermochemistry, such as atomization energy, no studies have been carried out at all.

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Therefore, in this study, the CCP and CCAs molecules are studied theoretically with the aim to reach the possible highest theoretical accuracy. The explicitly correlated coupled cluster method has been applied to an analog molecule, the CCN15 radical. It has shown that the explicit correlation technique dramatically improves the basis sets quality comparing to the conventional basis sets with the same size. It also shows that the multi-reference character may not be ignored for the CCN molecule. Therefore, the combination of the explicit correlation method and the multi-reference configuration interaction (MRCI) method,

MRCI-F12, should be the most suitable method to recover the correlation energies for the title molecules.

Thus, the ground 2Π state near-equilibrium potential energy surfaces of the CCP and

CCAs radicals are calculated with the MRCI-F12 method. The two degenerate components of the ground state are calculated separately to investigate the RT effect, especially on the bending mode. The Feller-Peterson-Dixon (FPD) methodology16 is used to achieve the best accuracy and rule out significant possible errors. Under the FPD framework, the basis set incompleteness error, outer-core electron correlation, scalar relativistic effects as well as the spin-orbit effects are taken into consideration. The results are expected to provide the most accurate theoretical spectroscopic and thermodynamic data for CCP and CCAs as well as exposing the complex nature of the radicals.

II. Computational details

The spectroscopic properties of the CCP and CCAs radicals were obtained from the near-equilibrium potential energy surface (PES). The near-equilibrium PES was constructed by fitting a grid of 50 points using the least squares method in displacement coordinates. The

o geometrical parameters covered the range from r0-0.3 to r0+0.5 for bond lengths and 180 to

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210o for the bond angle.

For each grid point, the Feller-Peterson-Dixon (FPD)16 framework was carried out to obtain the composite energy, where the total energies of each grid point are expressed as a sum of component terms:

E(r1,r2,θ)=EVQZ+ΔCBS+ΔCV+ΔDKH+ΔSO (1) where the first term on the right-hand side represents the valence electronic energies extrapolated to the complete basis set (CBS) limit. The state-averaged (2A’+2A”) full complete active space self-consistent field (CASSCF) calculation (13e, 12o) was applied to serve as the references for the explicitly correlated multi-reference configuration interaction

(MRCI-F12) method17-19. To account for the size-consistence problem, the Davidson correction (+Q)20 was employed. The orbital basis sets adopted explicit correlated correlation consistent basis sets, cc-pVnZ-F12 (n=D,T,Q), for C and P21, and cc-pVnZ-PP-F12 for As22 using pseudo potential (PP). For convenience, they are abbreviated as VnZ-F12(-PP) basis sets. For the auxiliary basis sets (ABS) required in the F12 calculations, the cc-pVnZ/JKFIT23, aug-cc-pVnZ/MP2FIT24 and cc-pVnZ-F12/OPTRI25 sets were used for evaluation of Fock and exchange matrices (JKFIT), density fitting of remaining two-electron integrals (MP2FIT) and the resolution of identity (OPTRI), respectively, where n is the same cardinal number as used in the orbital basis sets. An exception should be noted that for the VDZ-F12 orbital basis sets, the cc-pVTZ/JKFIT and aug-cc-pVTZ/MP2FIT ABS were used instead of the DZ quality ABS. The germinal Slater exponent γ was chosen to be 0.9 for VDZ-F12 and 1.0 for all other valence orbital basis sets.

The Hartree-Fock energies were extrapolated by Karton-Martin’s formula26:

6.57 n En E HF Limit  A( n  1) e (2) where

En is the VTZ-F12 and VQZ-F12 HF energies and n is the cardinal number in the basis sets.

And the correlation energies were extrapolated by a two-parameter formula using VTZ-F12

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and VQZ-F12 energies:

1 E E  A/() n  4 (3) n CBS 2 where En are the energies calculated with corresponding basis sets and Ecbs is the extrapolated correlation energy at the CBS limit.

The second term in Eq. 1 recovers the correlation energy from outer core electrons. For C,

P and As, the core electrons include 1s, 2s2p and 3s3p3d electrons, respectively. The energies were calculated with the explicitly correlated coupled cluster method, CCSD(T)-F12b27,28, instead of the MRCI-F12 method due to the sharp scaling of the computational cost of the outer-core electron correlation. The corresponding weighted core-valence correlation consistent F12 basis sets, cc-pCVTZ-F1221,29, were used as the orbital basis sets and the

VTZ/JKFIT, aug-cc-pwCVTZ/MP2FIT and cc-pVTZ-F12/OPTRI basis sets were used as the auxiliary basis sets. The reference wave function was taken from the state-averaged full

CASSCF wave function. The Slater exponent γ was chosen to be 1.4.

The third term accounts for the scalar relativistic contributions from light atoms as well as a correction for the PP approximation in the case of CCAs. It is calculated as the difference between a calculation with the second-order Douglas-Kroll-Hess (DKH) Hamiltonian30 and a calculation using the non-relativistic Hamiltonian. The C atom used VTZ-DK and VTZ sets for the relativistic and non-relativistic calculation, respectively. The P atom used V(T+d)Z and V(T+d)Z-DK sets, respectively. And the As atom used VTZ-DK and VTZ-PP set, respectively. The reference function was taken from the state-averaged full CASSCF wave function and the conventional coupled cluster method with singles doubles and perturbative triples excitation, CCSD(T), was used to recover the correlation energy since the F12

Hamiltonian does not commute with the DKH Hamiltonian.

The last term is the spin-orbit contribution. The SOC correction energy was obtained by

CASSCF calculations with the Breit-Pauli Hamiltonian31. The VTZ(-PP)32,33 basis sets were

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used but for the C atom an additional d function with an exponent of 14.839 was added to its

VTZ set, following the study of the CCN radical15.

The spectroscopic information was derived from the fitting of the near equilibrium PES of each radical. The PES was fitted to a 5th order polynomial as in Eq. (4).

i j k V  cijk Q1 Q 2 Q 3 (4) i, j , k  1 where Q1, Q2 and Q3 corresponds to the internal displacement coordinates rcc-rcc,e, rcp-rcp,e and

th θccp-θccp,e, respectively. Additional 6 order diagonal terms were included during the fitting process.

The PES fitting as well as calculations of spectroscopic constants via 2nd order perturbation theory were carried out by the Surfit program34. In all PES fittings, the root mean square (RMS) value was less than 1 cm-1.

For the thermodynamic calculations, the energies of the radicals were obtained by fitting the PES at different level of theories under the FPD methodology. The energies of the atoms were calculated using the same methods and basis sets as the radicals at each level of theories.

III. Discussion

A. CCP

Table 1 shows the equilibrium bond lengths, harmonic frequencies and atomization energies derived from the CCP averaged PES, (2A’+2A”)/2, with respect to different size of basis sets and different correction terms in Equation 1. It can be seen that the calculated properties has a fast convergence with respect to the basis sets. For example, at the DZ quality basis sets, the bond lengths are within in 0.002 angstrom of the CBS limit. While for

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the harmonic frequencies, the difference between DZ results and CBS results are within 5 cm-1. These indicate that the bond lengths and frequencies systematically converged to the

CBS limit, following the general trend of the correlation consistent basis set, and the DZ basis sets are good enough to calculate the bond length and frequencies. In term of the atomization energies, however, it can be seen that the DZ value is about 4 kcal/mol from the

CBS value. Using the TZ quality, the atomization energy is barely falling in the range of chemical accuracy. Thus although the DZ sets have a good performance to calculate the bond lengths and frequencies with the F12 method, the larger basis sets are still in necessary for atomization energy to achieve chemical accuracy. Lastly, when compare the QZ results and the CBS results, it can be seen that the differences are negligible in term of all observables.

Therefore it can be concluded that the basis set incompleteness error is eliminated in this work.

To include the contribution out of outer-core electrons, the outer-core (CV) correlation calculations are carried out. It can be seen that in term of the bond lengths and harmonic frequencies, the CV contribution is very small or negligible. The bond lengths change at most

0.0052 angstrom while the frequencies change at most 9.7 cm-1. It should be noted, however, the CV correction is still important for the atomization energy that it contributes near 2 kcal/mol.

Going beyond the CV correction, the scalar relativistic effects (DK) are included. It can be seen that the DK correction is negligible in term of all calculated properties. This is as expected since neither C nor P atoms are very heavy, thus the small DK correction is not surprising.

In the end, the composite FPD results are listed to compare with the experimental values in Table 1 and Table 3. It should first be noted that the experimental bond lengths are obtained by fitting the FTMW data, where three different fitting methods were used and three

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sets of bond lengths were provided. To make the tables compact, these sets of bond lengths are rounded up to the second decimal and achieving one same value. Thus it can be seen that the composite bond lengths are in a good agreement with experiments, by differing 0.02 and

0.01 angstrom for the C-C bond length and C-P bond length, respectively.

In term of the frequencies, it may be more helpful to look at Table 3 first and compare the fundamental frequencies. It can be seen that the agreement between the experimental frequencies and the calculated frequencies derived from the averaged PES is fairly good. For the ν3 mode the values are almost identical while for the ν1 mode the difference is only about

27 cm-1. When compare the isotope effect, it can be found that the isotope effect shifts the

-1 observed peaks by about -50 and -20 cm for the ν1 and ν3 modes, respectively. This is in high consistence with the calculated results, which demonstrate that the isotope effect shift

-1 -49.6 and -19.6 cm to the ν1 and ν3 modes, respectively. In term of the harmonic frequencies, it should be aware that the experimental values were obtained by fitting other directly measured observables. When compare the experimental and theoretical values, the ω2 and ω3

12 13 modes still have a good agreement for CCP and CCP. But for the ω1, mode, the theoretical-experimental gap is relatively large. This is believed resulting from the fitting

12 13 errors of the experimental method. Comparing the ν1-ω1 gap between CCP and CCP firstly, it can be found that the differences are 29 and 14 cm-1 respectively by calculation while they

-1 13 are 2 and 15 cm respectively by experiment. It can be seen that the ν1-ω1 gap for CCP is fairly consistent between calculation and experiment but not for 12CCP. Secondly comparing

12 the theoretical-experimental gap, the CCP ν1 and ω1 theoretical-experimental gaps are 27 and 54 cm-1 respectively while for 13CCP they are 27 and 26 cm-1, respectively. It can be seen

12 that the theoretical-experimental gap is quite consistent only except for CCP ω1. Lastly, the anharmonic constant x11 is calculated to be relatively large. The large x11 value indicates a large shift between ν1 and ω1, matching the theoretical results. Although the experimental x11

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12 13 for CCP is not available, yet if one can approximate this value by using x11 of CCP, as

12 13 supported by the calculation that the theoretical x11 is close between CCP and CCP, then it

-1 would expect a large shift between ν1 and ω1, rather than a 2 cm difference. Thus it is

12 concluded that the experimental ω1 for CCP has some fitting errors and the theoretical value should be more reliable. In term of the rotational constant (Be) as well as centrifugal distortion constant (D), it can be seen from Table 3 that they also have a very good agreement with the experiment.

Finally, it can be seen that the calculated atomization energy (De) matches with the

Knudsen cell mass spectrometry (MS) very well. This agreement on one hand confirms the conclusion of the MS study that their proposed structure for the C2P molecule is correct and it is indeed linear CCP molecule. On the other hand, this agreement together with the agreement of bond lengths and spectroscopic data indicate the accuracy of the methods applied to describe the CCP radical.

B. CCAs

Table 2 shows the bond lengths, harmonic frequencies and the predicted atomization energies derived from the averaged PES, (2A’+2A”)/2 for the CCAs radical. Similar to the

CCP case, the basis set convergence is quite fast with the explicit correlation method and basis sets. The bond lengths differ at most 0.0003 angstrom between QZ results and the CBS limit. The corresponding difference of frequencies differs no more than 3 cm-1. For the atomization energy, the difference is also negligible. Therefore it is similar to the CCP case to conclude that the basis set incompleteness error is eliminated.

Beyond the CBS limit, the CV correction is included and it has much more significant influence to the C-As bong length. This should mainly result from the inclusion of 3d electrons of As into the correlation space. In term of the frequencies, the CV contribution is

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comparable to the CCP radical for each vibration mode. And for the atomization energy, the

CV correction is also similar to that of the CCP molecule by nearly 2 kcal/mol.

For the scalar relativistic correction (DK), it is as small as that of the CCP radical. This is not surprising because the pseudo-potential used in the As basis sets recovers most of the scalar relativistic effects while the C atoms are not heavy enough to exhibit a significant contribution.

In Table 3, the full FPD composite calculation determines the C-C bond length, C-As bond length as 1.3005 and 1.7329 angstrom, respectively. Comparing to the experimental values from the FTMW technique, the agreement between the theoretical and experimental methods is fairly good. It should be mentioned that for the sake of conciseness of the table, like the CCP case, the experimental values are also rounded up to the second decimal. Yet slightly different from CCP, the round-up of the C-As bond length didn’t become one same value. Thus the C-As bond length is denoted as a range between 1.73 to 1.75 angstrom.

In term of the frequencies, it can be seen that CCAs radical has a slightly larger isotope

-1 -1 effect than CCP, about 65 cm for ν1 and 20 cm for ν3. This is in high consistence with the calculations in this work, where different isotopes of C lead to a difference about 64 and 20

-1 cm , respectively for ν1 and ν3 modes. The comparison between experimental and theoretical work also achieves a very good agreement for both harmonic frequencies and fundamental

13 frequencies only except for the CCAs ω1 mode, which is quite close to its ν1 value. Similar

12 to the analysis of CCP, it can be found that the experimental x11 value for CCAs is -14.7

-1 13 -1 cm , almost identical to theoretical value. Giving the theoretical x11 for CCAs as -13.7 cm , it is reasonable to expect a similar experimental value. Thus similar to the conclusion in CCP, the fitted harmonic ω1 frequency is less reliable than the theoretical work. In term of the other two modes, the isotope effect is not as significant as ν1, as in the CCP case. Regarding to the rational constant and centrifugal distortion constant, the calculated values match with

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experimental values very well.

For the atomization energy, it is predicted to be 251.2 kcal/mol for CCAs, nearly 20 kcal/mol less than CCP. Considering the good agreement between this theoretical work and the experiment, it is confident to remark that the prediction of the CCAs atomization energy is accurate.

C. Renner Teller Effect

Figure 1 shows potential energy curves of the CCP and CCAs radicals along the bending mode. The two components of A’ and A” PES are plotted. It can be seen that without SOC, the two component curve A’ and A” are exactly degenerate at the 180 degree. As the radical becomes more and more bent, the A’ and A” curves begin to split and the splitting becomes larger and larger. This shows the Renner-Teller effect, as the electronic motion couples with the nuclear motion.

In terms of the harmonic frequencies, it can be seen from Table 3 that the C-C stretching mode (ω1) as well as the C-P stretching mode (ω3) are identical between the two states, since these two vibration motions do not involve any bending motion. For the ω2 mode there is a gap between A’ and A” about 126 and 144 cm-1, respectively for CCP and CCAs. For the fundamental frequencies, on the other hand, since the anharmonic constants x23 and x12 are involved, the ν1 and ν3 mode become slightly nondegenerate.

To quantify the RT effect, the RT parameter is calculated by:

22 ()2,AA ' 2, "   22 (5) ()2,AA ' 2, " which involves the differences in the bending harmonic frequencies on the A’ and A” PES.

Putting the composite bending harmonic frequencies into Eq. 5, the RT parameters for

CCP and CCAs are calculated as 0.57 and 0.78, respectively. The positive signs of the RT parameters indicate that the A’ state lies higher in energy than the A” curve upon bending.

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This is in consistence with the potential energy curve along the bending coordinate as shown in Fig. 1. Comparing to the CCN radical15, it can be found that its RT parameter is calculated as 0.43. Thus it may conclude that as the terminal pnictogen becomes heavier and heavier, the

RT parameter becomes larger and larger.

D. Spin-Orbit Coupling Effect

The spin-orbit coupling effect upon the CCP and CCAs radicals are summarized in Table

4 and the potential energy curves with SOC along the bending mode are plotted in Fig. 2. It can be seen from Table 4 that when the SOC is included, the harmonic frequencies of ω1 and

ω3 mode between the A’ state and A” state are not equal anymore. This is because the A’ and

A” states become nondegenerate after turning on the SO operator. In term of the bending mode, the frequency difference between A’ and A” states becomes smaller. In other word, the

RT parameter ϵ is approaching 0. These changes indicate that the SOC partly quenches the RT effect.

Tracing the root of the RT effect, it results from the coupling of the orbital angular momentum and the vibrational angular momentum. For the SO effect, on the other hand, it results from the coupling of the orbital angular momentum with the spin angular momentum. When the SO operator is turned on, the orbital angular momentum is no longer a good quantum number and thus no orbital angular momentum is available to couple with the vibrational angular momentum. Thus the competition of the orbital angular momentum between two effects makes them mutually exclusive. And how much the SOC quenches the RT effect will depend on how large the SOC is.

To approximately quantify the magnitude of the SOC, the SOC constant is calculated by approximating the SO operator as:

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(6) where A is the SO constant and and are spin and orbital angular momentum operator, respectively. The Eq. 6 is of the same functional form with the Breit-Pauli (BP) operator used in this work. Since the product can be written in term of J2, L2 and S2, the SO energy can thus be expressed in the atomic unit as:

1 EAJJLLSS [ (  1)  (  1)  (  1)] (7) SO 2

3 1 where J, L and S are , 1 and , respectively for the 2Π state of the CCP or CCAs radicals. 2 2

4 The ESO was calculated as 3.242 10 a.u. for the CCP radical using the BP operator in this work. Therefore, the SO constant is calculated to be 142.3 cm-1 by Eq. 7, in a good agreement with the experimental value4 of 140.5 cm-1. For the CCAs radical, the corresponding SO constant is calculated as 770.8 cm-1, comparing to the experimental value10 of 875.4 cm-1. The relatively large discrepancy in the CCAs case indicates that the perturbation approximation used in the BP operator for CCAs is not as good as in the CCP case, suggesting a variational method for the SO calculation may be needed for better accuracy for CCAs.

Comparing Fig. 1 and Fig. 2, there show the bending potential energy curves with and without the SOC for the CCP and CCAs radicals. For both radicals, the A’-A” gap gets narrowed when the SO operator is turned on. This follows the previous analysis that the SOC can quench the RT effect. In addition, the CCAs A’-A” gap gets narrowed at a larger extent than CCP. The RT parameters become 0.045 and -0.040 respectively for CCP and CCAs after turning on the SO operator, in comparison of 0.57 and 0.78 for CCP and CCAs before turning on the SO operator. It should be noted lastly that the SO RT parameter of CCAs is -0.040, indicating the A’ curve should lie below the A” curve, but this is against the plot in Fig.2.

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Considering calculated results are in good agreement with experiments, this negative RT parameter might suggest a new formula is needed to calculate the RT parameter.

In the end, several vibronic energy levels along the bending modes are listed in Table 5.

The vibronic levels are calculated using Peric and Peyerimhoff’s formula35:

KK( 1)22 E( v  1)   A  ) (8) 8( A

1 (v K 1)( v K 1) E( v  1)   A  22  [ 1,2 8 4( A ) (9) (v K )2  1 ( v  1) 2  K 2  ] 4( AA ) where v is the bending vibrational quantum , K is the non-negative vibronic angular momentum quantum number (K=v+1, v+1-2,…,1 or 0), Σ is the spin angular momentum along the linear axes of the molecule, ϵ is the SO-RT parameter, A is the SO constant, and ω is the bending harmonic frequency for the (A’+A”) averaged PES at the CBS+CV+DK level of theory, as shown in Table 3, using the 12C isotope. When the case K= v+1 is met, the Eq. 7 is adopted to calculate the vibronic energy levels. Otherwise when K< v+1, the Eq. 8 is used.

It can be seen that for the fundamental bending transitions (v=1←0),the vibronic splitting of the CCP and CCAs radicals are 212.74 and 167.86 cm-1, respectively, in a good agreement with harmonic frequencies in Table 3 as 212.9 and 167.9 cm-1, also noting that the anharmonicity is very small for both molecules.

IV. Conclusion To conclude, in this study the FPD methodology is applied on the CCP and CCAs radicals to achieve high-accuracy spectroscopic and thermodynamic data. This is the first time for the molecules to be studied with the high-level theory of MRCI-F12 or

CCSD(T)-F12 and high-quality basis sets of explicitly correlated correlation consistent basis

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sets. The basis set incompleteness error is eliminated by basis set extrapolation. The outer-core correlation and scalar relativistic effects are calculated and found to be small. The

RT effect as well as the SOC are discussed and it demonstrated that the two effects are in competition of orbital angular momentum and thus the SOC can partly quench the RT effect.

The composite FPD results have used the most sophiscated methods and basis sets and have included significant possible contributions. Thus it is believed to provide the most accurate theoretical results up to date.

Comparing to the experimental values, the calculations have a very good agreement with respect to both spectroscopic and thermodynamic properties. It is the first time to calculate the anharmonic constants as well as spin-orbit coupling effect for the vibration frequencies for both molecules. The atomization energy for the CCAs radical is predicted to be 251.2 kcal/mol.

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Table 1 The molecular constants of the CCP radical

MRCI(Q) r(C-C) r(C-P) ퟏ ퟐ ퟑ De VDZ-F12 1.3110 1.6157 1687.3 208.1 823.0 263.5 VTZ-F12 1.3120 1.6148 1691.2 203.7 824.5 266.2 VQZ-F12 1.3119 1.6143 1691.7 207.5 825.2 267.0 횫CBS 0.0000 -0.0002 0.5 3.1 0.3 0.4 횫퐂퐕 -0.0037 -0.0052 9.7 2.7 6.1 1.9 횫퐃퐊 -0.0002 -0.0002 -0.8 -0.4 -0.1 -0.3 Total-12CCPa 1.3080 1.6087 1700.6 212.9 830.6 269.0 LIF-12CCP4 1646.4 211.3 837.8 FTMW3 1.29 1.62 MS5 268.7±3.5 Total-13CCPb 1.3080 1.6087 1621.9 205.2 813.0 LIF-13CCP4 1609.6 204.1 819.0 a) composite results 12C isotope; b) composite results with 13C isotope; The bond lengths, frequencies and atomization energies are in the unit of angstrom, cm-1 and kcal/mol, respectively. The ω1, ω2 and ω3 modes stand for C-C stretching, bending and C-P stretching, respectively.

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Table 2 The molecular constants for the CCAs radical

MRCI(Q) r(C-C) r(C-As) ퟏ ퟐ ퟑ De VDZ-F12 1.3053 1.7454 1707.5 160.3 644.1 246.0 VTZ-F12 1.3044 1.7437 1711.6 158.3 646.8 248.8 VQZ-F12 1.3043 1.7429 1712.0 161.6 648.2 249.5 횫CBS 0.0000 0.0003 0.3 2.9 0.4 0.4 횫퐂퐕 -0.0036 -0.0106 7.2 4.1 4.95 1.7 횫퐃퐊 -0.0002 0.0009 -0.2 -0.7 -1.0 -0.4 Total-12CCAsa 1.3005 1.7329 1719.3 167.9 652.5 251.2 LIF-12CCAs8 1704.8 161.6 663.6 FTMW10 1.29 1.73~1.75 Total-13CCAsb 1652.2 161.5 633.1 LIF-13CCAs8 1627.2 157.9 642.1 a) composite results 12C isotope; b) composite results with 13C isotope; The bond lengths, frequencies and atomization energies are in the unit of angstrom, cm-1 and kcal/mol, respectively. The ω 1, ω 2 and ω 3 modes stand for C-C stretching, bending and C-As stretching, respectively.

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Table 3 The molecular constants obtained from A’ PES, A” PES and the averaged PES at the CBS+CV+DK level of accuracy

12CCP 12CCAs constants 2A’ 2A” Avg. Exp. 2A’ 2A” Avg. Exp.

re(CC) (Å) 1.3080 1.3080 1.3080 1.29 1.3005 1.3005 1.3005 1.29 re(CX) (Å) 1.6087 1.6087 1.6087 1.62 1.7329 1.7329 1.7329 1.73~1.75 Be(MHz) 6356.0 6356.0 6356.0 6392.4 4457.4 4457.4 4457.4 4474.6 D(KHz) 1.63 1.63 1.63 2.260 0.91 0.91 0.91 1.121 -1 ω1(cm ) 1700.6 1700.6 1700.6 (1635.5) 1646.4 (1609.6) 1719.3 1719.3 1719.3 (1652.2) 1704.8 (1627.2) -1 ω2(cm ) 266.4 140.3 212.9 (205.0) 211.3 (204.1) 223.8 79.6 167.9 (161.5) 161.6 (157.9) -1 ω3(cm ) 830.6 830.6 830.6 (811.6) 837.8 (819.0) 652.5 652.5 652.5 (633.1) 663.6 (642.1) -1 21 x11(cm ) -14.7 -14.7 -14.7 (-13.4) (-15.7) -14.9 -14.9 -14.9 (-13.7) -14.7

-1 x22(cm ) -2.0 -1.2 -1.6 (-1.4) -2.4 1.2 -1.6 (-1.5) -1 x33(cm ) -5.8 -5.8 -5.8 (-12.2) (-1.2) -2.5 -2.5 -2.5 (-2.3) -3.2 (-2.5) -1 x12(cm ) -3.2 -4.8 -3.6 (-3.4) (-2.24) -2.4 -6.7 -3.2 (-3.0) -1 x13(cm ) 7.7 7.7 7.7 (34.2) -29.2 (-21.8) -1.3 -1.3 -1.3 (-1.2) (-2.3) -1 x23(cm ) 8.5 14.0 9.7 (8.7) 6.44 (6.04) 10.3 21.2 11.2 (10.2) 7.5 (5.0) -1 ν1(cm ) 1671.9 1670.4 1671.5 (1621.9) 1644 (1594) 1686.60 1682.20 1685.7 (1621.2) 1692 (1627) -1 ν2(cm ) 265.4 143.4 213.2 (205.2) 223.20 91.50 169.3 (162.7) -1 ν3(cm ) 831.4 836.9 832.6 (813.0) 833 (813) 657.3 668.1 658.2 (638.1) 661 (641)

The numbers in parenthesis are fundamental frequencies using 13C isotope.

Table 4 The molecular constants obtained from A’ PES and A” PES and the averaged PES at the CBS+CV+DK+SO level of accuracy

+SOC CCP CCAs constants 2A’ 2A” 2A’ 2A”

re(CC) (Å) 1.3082 1.3078 1.3012 1.2998 re(CX) (Å) 1.6086 1.6088 1.7329 1.7330 -1 ω1(cm ) 1699.17 1702.02 1712.51 1726.08

-1 ω3(cm ) 830.59 830.66 651.60 653.48 -1 x11(cm ) -14.66 -14.66 -15.05 -14.71 -1 2 x (cm ) 1 22 19.65 -24.95 9.45 -11.93

-1 x33(cm ) -5.92 -5.66 -2.50 -2.40 -1 x12(cm ) -3.00 -4.20 -3.23 -3.11 -1 x13(cm ) 8.24 7.25 -1.11 -1.44 -1 x23(cm ) 9.00 10.36 10.67 11.68 -1 ν1(cm ) 1671.00 1672.10 1678.60 1692.80 -1 ν2(cm ) 274.60 146.20 195.10 145.50 -1 ν3(cm ) 831.90 833.3 656.7 659.6

Table 5. Selected bending mode vibronic energy levels (in cm-1) for CCP and CCAs and their correlation to the harmonic vibrational level. State CCP CCAs State

2 2 v=2 Φ7/2 567.21 1106.93 Φ7/2 v=2

2 2 κ Π3/2 566.97 1106.44 κ Π1/2 v=2

2 2 κ Π1/2 566.65 1106.43 κ Π3/2 v=2

2 2 μ Π1/2 427.09 938.88 Δ5/2 v=1

2 2 μ Π3/2 426.69 938.64 κ Σ1/2 v=1

2 2 Φ5/2 425.40 770.87 Π3/2 v=0

2 2 v=1 Δ5/2 354.80 336.09 μ Π3/2 v=2

2 2 κ Σ1/2 354.56 336.01 μ Π1/2 v=2

2 2 μ Σ1/2 213.46 335.70 Φ5/2 v=2

2 2 Δ3/2 212.74 168.03 μ Σ1/2 v=1

2 2 v=0 Π3/2 142.22 167.86 Δ3/2 v=1

2 2 Π1/2 0.00 0.00 Π1/2 v=0

The labels κ and μ indicate the upper and lower energy levels in the vibronic level pairs when the non-negative vibronic angular momentum quantum number K is greater than 0.

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Figure 1 Potential energy curves along the bending mode of the CCP and CCAs radicals without SOC. The energies at linear geometry are scaled to 0.

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Figure 2 Potential energy curves along the bending mode of the CCP and CCAs radical with SOC. The energies at linear geometry are scaled to 0.

214

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Chapter 9 Conclusions In this study, new correlation consistent basis sets were developed in conjunction with the DKH3 scalar relativistic Hamiltonian for the lanthanide elements La through Lu. The basis sets range from double- to quadruple-zeta, encompassing those for valence correlation

(4f5s5p5d6s, denoted cc-pVnZ-DK3) and outer-core correlation (valence + 4s4p4d, denoted cc-pwCVnZ-DK3). Both series of basis sets systematically converge the HF and correlation energies towards their respective CBS limits, and both atomic and molecular benchmark calculations demonstrate the robustness and reliability of the new sets.

The LuF molecule was studied as the first application of the new basis sets. The calculated spectroscopic constants have an excellent agreement with experiments. The discrepancy of the dissociation energy between previous calculations and experiments is examinated and found that the experimental value has never been measured directly.

Therefore, the calculated dissociation energy is believed to be the correct value and it is close to most previous theoretical studies.

The second application of the new basis sets involved the study of lanthanide trihalide molecules (LnX3, Ln=La, Nd, Gd, Dy, Lu; X=F, Cl, Br). The harmonic frequencies are calculated with the CCSD(T) method for the first time and thus provide the most accurate theoretical values. Among the selected molecules, only the LaF3, NdF3 and GdF3 molecules have a C3v structure while others are in the D3h point group. The bond length between Ln and

X decrease linearly as the Ln atom becomes heavier, indicating the lanthanide contraction.

This good linear-fitting function shows that previous study claiming a quadratic function for lanthanide contraction has an overfitting problem.

The third application included the study of the highest spin diatomic molecule Gd2 and the prediction of its isoelectronic analogs. The calculated spectroscopic constants of Gd2 have

217

a good agreement with experiments and the dissociation energy is calculated at the lower bound of the experimental value plus the error range. The isoelectronic analogs of Gd2 molecules are predicted. The most probable ground states for the bounded molecules are determined and their spectroscopic constants, spin multiplicities as well as thermodynamic properties are calculated.

Besides calculations involving lanthanide elements, the potential energy surfaces involving transition metals and main group elements were also constructed.

For the coinage metal nitroxyl molecules (MNO, M=Cu, Ag, Au), the BCCD(T) methods were used for the closed-shell molecules. The results show that the BCCD(T) method has a better performance than the conventional CCSD(T) method by improving the orbital quality.

The calculated spectroscopic and thermodynamic constants have an overall good agreement with experiment for those that are available. The relativistic effects of the AuNO molecule were examined and found to be small, contrary to previous studies.

For the CCP and CCAs radicals, the RT effect, as well as its interaction with spin-orbit coupling is studied through the construction of potential energy surfaces. It quantitatively shows that spin-orbit coupling can quench the RT effect. The calculated results agree with experimental values well and provide the most accurate theoretical values.

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Appendix A

Supplemental information for A coupled cluster study of coinage metal nitrosyls, M-NO0/+ (M = Cu, Ag, Au)

Qing Lu and Kirk A. Peterson Department of Chemistry, Washington State University, Pullman, Washington 99164-4630 USA

Contents:

Table S1: Molecular constants obtained at the final composite PES for the CuNO, AgNO and AuNO molecules

Table S2: Molecular constants obtained at the final composite PES for the CuNO+, AgNO+ and AuNO+ molecules.

Table S3: The expansion coefficients (in a.u.) of the final composite surface for the CuNO, AgNO and AuNO molecules.

Table S4: The expansion coefficients (in a.u.) of the final composite surface for the CuNO+, AgNO+ and AuNO+ molecules.

Table S5: The expansion coefficients (in a.u.) of the electric dipole moment surface calculated at the BCCD(T)/avtz-pp level of theory for the CuNO, AgNO and AuNO molecules in the Eckart-frame.

Table S6: The expansion coefficients (in a.u.) of the electric dipole moment surface calculated at the UCCSD(T)/avtz-pp level of theory for the CuNO+, AgNO+ and AuNO+ molecules in the Eckart-frame.

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Table S1: Molecular constants obtained at the final composite PES for the CuNO, AgNO and

AuNO molecules. The molecule is on the XZ plane. Ae Be Ce are rotational constants at Re;

αi βi γi are vibrational dependence of rotational constants; Dj is the first order centrifugal distortion constant. Vibration modes: 1=NO stretch; 2=M-N stretch; 3=bend

Constant CuNO AgNO AuNO

Re(M-N, Å) 1.8836 2.2520 2.0030 Re(N-O, Å) 1.1704 1.1589 1.1541 훉(∠MNO) 119.0 117.6 118.1

Ae-Z axis (MHz) 97025.5 87692.2 93250.3 Be-X axis (MHz) 4671.8 3115.1 3390.4 Ce-Y axis (MHz) 4457.2 3008.2 3271.5 훂ퟏ(MHz) 1949.1 1258.2 1226.8 훂ퟐ(MHz) -2085.9 -2159.7 -1633.3 훂ퟑ(MHz) -619.4 -231.1 -514.3 훃ퟏ(MHz) -9.6 -28.7 -3.9 훃ퟐ(MHz) 39.8 18.6 20.3 훃ퟑ(MHz) 10.2 34.2 12.9 후ퟏ(MHz) -4.9 -25.3 -2.3 후ퟐ(MHz) 40.1 20.8 21.6 후ퟑ(MHz) 14.2 33.0 13.3 Dj(kHz) 4.4 3.9 1.7 -1 ퟏ (cm ) 1645.0 1735.8 1757.8 -1 ퟐ (cm ) 472.3 377.2 548.7 -1 ퟑ (cm ) 286.6 177.3 296.5 -1 Χ11 (cm ) -16.0 -19.5 -15.5 -1 Χ22 (cm ) -4.2 -3.3 -4.1 -1 Χ33 (cm ) -0.7 -3.3 -0.1 -1 Χ12 (cm ) 6.1 2.2 2.9 -1 Χ13 (cm ) -0.7 6.3 0.5 -1 Χ23 (cm ) -6.7 1.4 -7.7 -1 훎ퟏ (cm ) 1615.0 1701.1 1728.5 -1 훎ퟐ (cm ) 463.6 372.3 538.2 -1 훎ퟑ (cm ) 281.5 174.6 292.7

220

Table S2: Molecular constants obtained at the final composite PES for the CuNO+, AgNO+ + and AuNO molecules. The molecule is on the XZ plane. Ae Be Ce are rotational constants at

Re; αi βi γi are vibrational dependence of rotational constants; Dj is the first order centrifugal distortion constant. Vibration modes: 1=NO stretch; 2=M-N stretch; 3=bend

Constant CuNO+ AgNO+ AuNO+

Re(M-N, Å) 1.9280 2.2725 2.0332 Re(N-O, Å) 1.1362 1.1365 1.1261 훉(∠MNO) 133.6 129.3 125.7

Ae-Z axis (MHz) 162067.2 127520.0 119711.6 Be-X axis (MHz) 4134.6 2869.3 3172.3 Ce-Y axis (MHz) 4031.3 2806.2 3090.4 훂ퟏ(MHz) 4853.9 3159.2 2009.5 훂ퟐ(MHz) -6780.4 -8051.9 -4023.7 훂ퟑ(MHz) -14767.9 -6474.2 -1890.7 훃ퟏ(MHz) 0.6 2.2 3.2 훃ퟐ(MHz) 54.0 34.3 25.1 훃ퟑ(MHz) -2.8 8.5 6.6 후ퟏ(MHz) 3.6 3.6 4.4 후ퟐ(MHz) 51.7 33.9 25.7 후ퟑ(MHz) 2.0 10.0 8.2 Dj(kHz) 4.5 3.4 1.9 -1 ퟏ (cm ) 1966.8 1954.4 1941.4 -1 ퟐ (cm ) 349.0 269.7 401.6 -1 ퟑ (cm ) 206.4 155.6 248.3 -1 Χ11 (cm ) -14.5 -13.8 -14.0 -1 Χ22 (cm ) -4.7 -3.9 -4.3 -1 Χ33 (cm ) -1.4 -0.6 -0.5 -1 Χ12 (cm ) 1.0 -0.3 0.4 -1 Χ13 (cm ) -1.5 0.0 -3.1 -1 Χ23 (cm ) -4.9 -4.4 -3.9 -1 훎ퟏ (cm ) 1937.6 1926.6 1911.9 -1 훎ퟐ (cm ) 337.7 259.4 391.2 -1 훎ퟑ (cm ) 200.4 152.2 243.7

221

Table S3: The expansion coefficients (in a.u.) of the final composite surface for the CuNO, AgNO and AuNO molecules.

i j k CuNO AgNO AuNO 0 0 0 -1784.13450755 -5442.755694 -19145.41943315 2 0 0 0.39502949 0.43475415 0.45018288 1 1 0 0.04274651 0.03102025 0.05573858 0 2 0 0.04291348 0.01514771 0.05528502 1 0 1 0.05037237 0.01875999 0.02696660 0 1 1 0.01910106 0.00419684 0.01327970 0 0 2 0.07487268 0.02202415 0.03816074 3 0 0 -0.56205280 -0.62076986 -0.61489065 2 1 0 -0.01772969 0.02038701 -0.03420001 1 2 0 -0.00469087 -0.01050267 -0.03164370 0 3 0 -0.04932487 -0.01382058 -0.05942143 2 0 1 0.00682292 0.00756297 0.00268914 1 1 1 -0.03372051 -0.01264457 -0.02982651 0 2 1 -0.02005367 -0.00734874 -0.01469260 1 0 2 -0.04653727 -0.00570003 -0.01267661 0 1 2 -0.01662162 -0.00924427 -0.01750552 0 0 3 -0.02222519 0.00290876 0.00419522 4 0 0 0.51871152 0.52575072 0.51037951 3 1 0 -0.01192185 -0.02619455 0.02209821 2 2 0 0.03590010 0.01203533 0.02541836 1 3 0 -0.00022676 -0.00915742 -0.00059235 0 4 0 0.03418020 0.00673922 0.03715687 3 0 1 -0.03376864 -0.01517346 -0.01539515 2 1 1 0.02089163 0.01010851 0.02014262 1 2 1 0.01274420 -0.00543240 0.00910689 0 3 1 0.00515218 0.00660784 0.00716270 2 0 2 -0.02236193 -0.00304916 -0.00617410 1 1 2 0.04183212 0.00777922 0.00766277 0 2 2 -0.00246736 -0.00493593 0.00037320 1 0 3 0.02758606 0.00079263 -0.00170371 0 1 3 0.00212774 -0.00127305 0.00144465 0 0 4 -0.01068905 0.00153061 0.00481399 5 0 0 -0.30426661 -0.24272214 -0.32586425 4 1 0 -0.01543704 -0.05471685 -0.04953120 3 2 0 -0.04819820 0.02314668 0.02084722 2 3 0 -0.00895037 -0.00469822 -0.00007878 1 4 0 -0.01911507 0.00749242 0.00509872 0 5 0 -0.01599753 -0.00219517 -0.01180347 4 0 1 -0.02314119 0.00513053 0.00525173 3 1 1 -0.00358710 0.02064103 0.00362475 2 2 1 -0.04498926 -0.01703703 -0.01805873 1 3 1 -0.03006374 0.00679435 0.00424300 0 4 1 0.00320685 -0.00299073 -0.00271593 3 0 2 0.01616825 -0.02600204 0.00611054 2 1 2 -0.07348768 0.02932759 0.01569557 1 2 2 -0.02023552 -0.00200437 0.00070683 0 3 2 -0.02306883 0.00472633 0.00103689 2 0 3 0.01543789 0.00565449 0.00652002 1 1 3 -0.02778832 -0.00092768 -0.00078146 0 2 3 -0.00255549 -0.00144019 -0.00555406 1 0 4 -0.02043344 0.00162846 -0.00048686

222

0 1 4 -0.00758048 -0.00137098 -0.00332155 0 0 5 0.00735043 0.00190062 0.00495921 6 0 0 0.07352974 0.05035666 0.15719136 0 6 0 0.00243785 0.00050050 -0.00077240 0 0 6 0.00109941 0.00131461 0.00335975

223

Table S4: The expansion coefficients (in a.u.) of the final composite surface for the CuNO+, AgNO+ and AuNO+ molecules.

i j k CuNO+ AgNO+ AuNO+ 0 0 0 -1783.86498242 -5442.49594654 -19145.09502872 2 0 0 0.54737675 0.53941932 0.54011378 1 1 0 0.01293390 0.00382533 0.03383972 0 2 0 0.03447361 0.01771023 0.04187337 1 0 1 0.01559738 0.00934103 0.02337782 0 1 1 0.01331523 0.00494232 0.02171632 0 0 2 0.02406604 0.01969058 0.05209531 3 0 0 -0.71523265 -0.70333360 -0.68739413 2 1 0 -0.01106845 -0.01272938 -0.05449158 1 2 0 -0.01620839 -0.00740455 -0.01333176 0 3 0 -0.03777765 -0.01658876 -0.04730281 2 0 1 0.00854430 0.00902764 0.00293326 1 1 1 -0.02895147 -0.01484127 -0.05361207 0 2 1 -0.01274866 -0.00653541 -0.02331110 1 0 2 -0.04069490 -0.03877222 -0.06491968 0 1 2 -0.01131194 -0.01048830 -0.02582896 0 0 3 -0.01623358 -0.01310164 -0.02420255 4 0 0 0.54695562 0.54509471 0.49373627 3 1 0 0.03269299 0.04161529 0.09451380 2 2 0 -0.01022150 0.01828829 -0.02555868 1 3 0 0.01488900 0.00152934 0.00193094 0 4 0 0.02661765 0.01223666 0.03211348 3 0 1 -0.02627873 -0.03829718 0.01089146 2 1 1 0.02423253 0.06497274 0.02876930 1 2 1 0.00333258 0.01011208 0.04599243 0 3 1 0.00475898 0.00745692 0.02114708 2 0 2 0.03535870 0.05581203 -0.00671808 1 1 2 0.01014565 -0.00647994 0.01889262 0 2 2 0.00556465 0.00806923 0.01063272 1 0 3 0.01573156 0.01696602 0.01747695 0 1 3 0.00058692 0.01019999 0.00648314 0 0 4 -0.00722781 -0.00504564 0.00127175 5 0 0 -0.28271139 -0.15363725 -0.44356279 4 1 0 -0.09330276 -0.16900513 -0.03612515 3 2 0 0.03032118 0.02223712 0.02766939 2 3 0 -0.00916515 -0.01848926 -0.00702405 1 4 0 -0.01765873 0.00109391 0.00074341 0 5 0 -0.01091225 -0.00801179 -0.01044909 4 0 1 0.01513866 0.04716120 -0.05152583 3 1 1 -0.02558195 -0.03834770 -0.00395463 2 2 1 -0.02005726 -0.10006184 0.00389639 1 3 1 0.00209377 -0.01144477 -0.04290052 0 4 1 -0.00174462 -0.00545134 -0.02021862 3 0 2 -0.02285082 -0.04942521 0.06527454 2 1 2 0.00914461 -0.01055567 -0.01904661 1 2 2 -0.02023817 0.02451766 -0.07033613 0 3 2 -0.00810470 -0.00563409 -0.00542298 2 0 3 -0.00496723 -0.03362308 0.09207716 1 1 3 0.01177263 0.00302749 -0.01858364 0 2 3 0.00359587 -0.00745367 0.00099041 1 0 4 0.01134539 0.01126784 -0.03648798

224

0 1 4 -0.00376686 -0.00132621 -0.00674607 0 0 5 0.00364291 0.00383598 -0.00075554 6 0 0 0.12076257 -0.06910557 0.48610132 0 6 0 -0.00087302 0.00270748 0.00020845 0 0 6 -0.00073404 -0.00230371 -0.00977343

225

Table S5: The expansion coefficients (in a.u.) of the electric dipole moment surface calculated at the BCCD(T)/avtz-pp level of theory for the CuNO, AgNO and AuNO molecules in the Eckart-frame.

CuNO AgNO AuNO i j k x y x y x y 0 0 0 -0.18717 -0.01611 0.00653 0.00312 0.28291 -0.03301 1 0 0 -3.58812 0.34105 -3.16168 0.11184 -2.50823 0.06886 0 1 0 0.15266 0.04943 0.26101 -0.00656 0.39804 0.03560 0 0 1 0.03993 -0.71244 0.07822 -0.63928 0.00935 -0.37443 2 0 0 0.02492 0.16510 -1.32978 0.27963 0.56624 0.03613 1 1 0 0.12859 -0.20661 1.65104 -0.38884 -0.06047 -0.12857 0 2 0 0.32525 -0.09921 -0.10216 0.03020 -0.09138 -0.03847 1 0 1 -0.17647 -0.38812 0.10297 -0.26025 -0.11191 -0.12821 0 1 1 0.24508 0.21054 0.28656 0.23942 0.16175 0.14829 0 0 2 0.60015 0.08341 0.81094 0.07098 0.83487 0.02942 3 0 0 3.14934 -0.47951 0.34210 -0.03238 -0.08970 0.08354 2 1 0 -0.94693 0.11238 1.55119 -0.25279 0.27786 -0.05356 1 2 0 0.94532 -0.23152 0.35031 0.18064 0.27873 -0.04085 0 3 0 -0.13134 0.06548 0.04218 -0.04562 -0.14116 0.03555 2 0 1 -0.93889 0.70946 -0.68711 0.28639 0.11738 0.02681 1 1 1 0.54549 0.07060 0.29815 0.29845 0.39773 0.09089 0 2 1 -0.05258 -0.03448 -0.60476 0.09473 -0.23849 -0.08434 1 0 2 -1.52748 0.11556 -0.11259 0.28023 -0.21001 0.15017 0 1 2 0.62102 -0.18808 0.44577 -0.16332 -0.48073 -0.07583 0 0 3 0.40937 -0.28604 0.08576 0.03896 -0.20513 -0.03687 4 0 0 -0.81691 0.03022 4.61091 -0.41526 1.07105 -0.18652 3 1 0 -0.77354 0.22677 -5.15243 0.38908 -2.22787 0.27514 2 2 0 -5.48961 0.64492 -1.31484 0.31539 1.14299 -0.05424 1 3 0 0.34033 0.20794 -0.37320 -0.07788 0.01741 -0.00473 0 4 0 -0.18892 -0.00246 -0.04569 0.02498 0.02100 -0.01036 3 0 1 -0.74066 0.63428 -0.69596 0.70384 -0.58884 0.06414 2 1 1 4.66116 -0.80847 2.90963 -1.42140 0.45017 -0.26588 1 2 1 0.90441 -0.11485 -0.29030 -0.10675 -0.22455 0.13972 0 3 1 0.00505 0.07751 0.39355 -0.06464 0.13241 0.03535 2 0 2 -2.95762 -0.45146 -5.24386 0.16469 -0.80275 -0.25096 1 1 2 -0.04127 0.80405 3.41897 -0.50608 1.25617 0.04700 0 2 2 0.22424 0.07583 -0.55456 0.10733 -0.13547 0.06009 1 0 3 1.43602 -0.69389 0.40204 -0.49309 0.76351 -0.39059 0 1 3 -0.17902 0.40553 -0.22464 0.37627 0.21463 0.14923 0 0 4 -0.41614 0.35296 0.16270 0.11464 -0.03318 0.15708

226

CuNO+ AgNO+ AuNO+ i j k x y x y x y 0 0 0 -2.73676 0.82673 -3.32609 0.89062 -2.50320 0.88117 1 0 0 -0.53753 0.24486 -0.75312 0.21704 -1.57373 0.38049 0 1 0 -0.66053 0.05077 -0.92847 0.05583 -0.41298 0.03352 0 0 1 -0.17145 -1.05425 -0.14580 -0.99222 -0.14728 -0.97124 2 0 0 1.13979 -0.18389 1.13285 -0.09768 2.87834 -0.26891 1 1 0 0.01363 -0.05822 0.34065 -0.11706 -1.20707 0.03351 0 2 0 -0.31354 0.00711 -0.37459 0.03538 -0.11040 -0.01918 1 0 1 -0.03318 -0.18000 -0.17024 -0.12596 -0.18618 -0.18384 0 1 1 -0.05543 0.10187 -0.02488 0.04716 -0.11064 0.10642 0 0 2 -0.10393 -0.07535 -0.30716 -0.13602 -0.31735 -0.15702 3 0 0 -2.68365 0.51147 -4.88112 0.66528 -2.41527 0.34839 2 1 0 1.56370 -0.16313 6.18136 -0.75202 1.55989 -0.03381 1 2 0 -0.99822 0.12771 -1.80046 0.26757 0.13109 -0.08260 0 3 0 0.15025 0.00331 0.53116 -0.07716 -0.21610 0.05276 2 0 1 -0.71433 -0.29340 -2.41783 -0.16778 1.58351 -0.22892 1 1 1 0.34230 -0.00799 0.89284 0.08246 -0.41268 0.07424 0 2 1 -0.24850 -0.05132 -0.36476 0.06928 0.07624 -0.06976 1 0 2 0.47037 -0.40277 0.52933 -0.37351 0.53071 -0.28853 0 1 2 -0.21683 -0.00791 -0.15999 -0.03543 -0.76582 0.06314 0 0 3 0.07278 0.16076 0.08275 0.18822 -0.20919 0.09986 4 0 0 2.54344 -0.71845 7.01704 -1.09321 0.54268 -0.40100 3 1 0 -2.21823 0.07358 -6.22811 0.50527 -3.16417 0.21841 2 2 0 -1.74974 0.19024 -6.35970 0.97433 7.28946 -1.10604 1 3 0 0.25136 -0.07005 0.83444 -0.14240 -1.61958 0.14969 0 4 0 -0.13034 0.00579 -0.33107 0.05311 0.24849 -0.04565 3 0 1 0.56600 0.27327 1.96333 0.45631 -4.41055 -0.34542 2 1 1 2.08664 -0.41644 4.58449 -0.86561 -4.23337 1.04266 1 2 1 -0.05349 -0.18664 -0.14007 -0.29695 -0.44654 0.25512 0 3 1 0.33319 0.01057 0.39949 -0.08630 -0.31412 0.08695 2 0 2 -1.94296 0.51610 -4.53239 0.73638 5.37881 -0.96997 1 1 2 0.51365 -0.05285 1.14600 -0.18361 -1.56081 0.23159 0 2 2 -0.23867 0.10467 0.04107 0.12104 0.07978 0.03772 1 0 3 0.08084 -0.00814 -0.22543 0.00402 1.56583 0.19687 0 1 3 0.15600 -0.08029 0.31869 -0.12075 0.20286 -0.00066 0 0 4 0.02860 -0.03554 0.10292 -0.05650 -0.13339 0.10243

227

Appendix B

Supplemental information for Accurate ab initio vibronic spectroscopy of the CCP and CCAs radicals

Qing Lu and Kirk A. Peterson Department of Chemistry, Washington State University, Pullman, Washington 99164, USA

Contents:

Table S1: The expansion coefficients (in a.u.) for the CCP radical of the A’, A” and (A’+A”)/2 PES computed at the CBS+CV+DK level of theory.

Table S2: The expansion coefficients (in a.u.) for the CCP radical of the A’, A” and (A’+A”)/2 PES computed at the CBS+CV+DK+SO level of theory.

Table S3: The expansion coefficients (in a.u.) for the CCAs radical of the A’, A” and (A’+A”)/2 PES computed at the CBS+CV+DK level of theory.

Table S4: The expansion coefficients (in a.u.) for the CCAs radical of the A’, A” and (A’+A”)/2 PES computed at the CBS+CV+DK+SO level of theory.

228

Table S1: The expansion coefficients (in a.u.) for the CCP radical of the A’, A” and (A’+A”)/2 PES computed at the CBS+CV+DK level of theory.

i j k A’ A” (A’+A”)/2 0 0 0 -418.12449 -418.12449 -418.12449 2 0 0 0.28106 0.28107 0.28106 0 0 2 0.02166 0.00602 0.01384 1 2 0 -0.03993 -0.03995 -0.03994 0 1 2 -0.03127 -0.01477 -0.02302 2 2 0 -0.00352 -0.00356 -0.00355 2 0 2 0.00150 -0.00507 -0.00179 0 0 4 0.00051 0.00066 0.00059 3 2 0 0.04011 0.04010 0.04011 0 5 0 -0.05635 -0.05632 -0.05633 1 2 2 -0.00228 0.03061 0.01418 0 1 4 -0.00325 0.00107 -0.00110 0 0 6 -0.00041 0.00082 0.00020 1 0 0 0.00000 0.00000 0.00000 1 1 0 0.03662 0.03662 0.03662 3 0 0 -0.33372 -0.33364 -0.33368 0 3 0 -0.20033 -0.20035 -0.20034 4 0 0 0.21448 0.21422 0.21435 1 3 0 0.00737 0.00734 0.00736 1 1 2 -0.01673 -0.01091 -0.01382 5 0 0 -0.10679 -0.10746 -0.10711 2 3 0 -0.01641 -0.01635 -0.01636 3 0 2 -0.00946 0.00209 -0.00366 0 3 2 -0.00034 -0.00384 -0.00209 6 0 0 0.04939 0.05107 0.05022 0 1 0 0.00000 0.00000 0.00000 0 2 0 0.20744 0.20745 0.20744 2 1 0 0.02898 0.02896 0.02897 1 0 2 0.00006 0.00673 0.00339 3 1 0 -0.00475 -0.00475 -0.00475 0 4 0 0.11625 0.11615 0.11621 0 2 2 0.01830 0.00561 0.01196 4 1 0 -0.02485 -0.02471 -0.02479 1 4 0 0.01090 0.01121 0.01103 2 1 2 0.02956 -0.00593 0.01178 1 0 4 0.00387 -0.00688 -0.00150 0 6 0 0.02033 0.02063 0.02045

229

Table S2: The expansion coefficients (in a.u.) for the CCP radical of the A’, A” and (A’+A”)/2 PES computed at the CBS+CV+DK+SO level of theory.

i j k A’ A” (A’+A”)/2 0 0 0 -418.12417 -418.12482 -418.12449 2 0 0 0.28067 0.28145 0.28106 0 0 2 0.01446 0.01322 0.01384 1 2 0 -0.03994 -0.03993 -0.03994 0 1 2 -0.02810 -0.01794 -0.02302 2 2 0 -0.00296 -0.00413 -0.00355 2 0 2 0.00053 -0.00412 -0.00179 0 0 4 0.05580 -0.05463 0.00059 3 2 0 0.03974 0.04044 0.04011 0 5 0 -0.05684 -0.05584 -0.05633 1 2 2 0.00929 0.01898 0.01418 0 1 4 -0.02844 0.02623 -0.00110 0 0 6 -0.12082 0.12123 0.00020 1 0 0 0.00000 0.00000 0.00000 1 1 0 0.03687 0.03637 0.03662 3 0 0 -0.33337 -0.33398 -0.33368 0 3 0 -0.20040 -0.20028 -0.20034 4 0 0 0.21408 0.21461 0.21435 1 3 0 0.00705 0.00766 0.00736 1 1 2 -0.01600 -0.01163 -0.01382 5 0 0 -0.10785 -0.10641 -0.10711 2 3 0 -0.01733 -0.01544 -0.01636 3 0 2 -0.00858 0.00120 -0.00366 0 3 2 -0.00519 0.00099 -0.00209 6 0 0 0.05200 0.04849 0.05022 0 1 0 0.00000 0.00000 0.00000 0 2 0 0.20739 0.20750 0.20744 2 1 0 0.02904 0.02890 0.02897 1 0 2 0.00372 0.00307 0.00339 3 1 0 -0.00515 -0.00435 -0.00475 0 4 0 0.11618 0.11621 0.11621 0 2 2 0.01833 0.00558 0.01196 4 1 0 -0.02456 -0.02501 -0.02479 1 4 0 0.01143 0.01063 0.01103 2 1 2 0.02228 0.00128 0.01178 1 0 4 -0.01679 0.01377 -0.00150 0 6 0 0.02141 0.01958 0.02045

230

Table S3: The expansion coefficients (in a.u.) for the CCAs radical of the A’, A” and (A’+A”)/2 PES computed at the CBS+CV+DK level of theory.

i j k A’ A” (A’+A”)/2 0 0 0 -2336.05479 -2336.05655 -2336.05479 2 0 0 0.29922 0.29920 0.29921 0 0 2 0.01653 0.00209 0.00931 1 2 0 -0.03291 -0.03286 -0.03289 0 1 2 -0.02523 -0.01099 -0.01811 2 2 0 -0.01138 -0.01146 -0.01142 2 0 2 0.00374 -0.00587 -0.00106 0 0 4 0.00033 0.00133 0.00083 3 2 0 0.01634 0.01540 0.01589 0 5 0 -0.03764 -0.03763 -0.03763 1 2 2 0.01050 0.03870 0.02465 0 1 4 0.00237 0.00438 0.00336 0 0 6 -0.00097 0.00004 -0.00047 1 0 0 0.00000 0.00000 0.00000 1 1 0 0.02158 0.02158 0.02158 3 0 0 -0.34519 -0.34518 -0.34518 0 3 0 -0.15505 -0.15506 -0.15506 4 0 0 0.21803 0.21825 0.21814 1 3 0 0.01136 0.01146 0.01141 1 1 2 -0.01522 -0.00182 -0.00852 5 0 0 -0.11540 -0.11543 -0.11543 2 3 0 -0.00671 -0.00491 -0.00581 3 0 2 -0.00416 0.00479 0.00029 0 3 2 -0.00057 -0.00580 -0.00322 6 0 0 0.05287 0.05220 0.05256 0 1 0 0.00000 0.00000 0.00000 0 2 0 0.16463 0.16464 0.16464 2 1 0 0.02836 0.02823 0.02829 1 0 2 -0.00019 0.00626 0.00303 3 1 0 0.00621 0.00614 0.00617 0 4 0 0.08507 0.08487 0.08498 0 2 2 0.01544 0.00195 0.00870 4 1 0 -0.01252 -0.01112 -0.01181 1 4 0 0.00617 0.00555 0.00590 2 1 2 0.00813 -0.01710 -0.00452 1 0 4 0.00120 -0.01017 -0.00449 0 6 0 0.01242 0.01301 0.01269

231

Table S4: The expansion coefficients (in a.u.) for the CCAs radical of the A’, A” and (A’+A”)/2 PES computed at the CBS+CV+DK+SO level of theory. i j k A’ A” (A’+A”)/2 0 0 0 -2336.05304 -2336.05655 -2336.05479 2 0 0 0.29747 0.30094 0.29921 0 0 2 0.00894 0.00968 0.00931 1 2 0 -0.03343 -0.03234 -0.03289 0 1 2 -0.02100 -0.01521 -0.01811 2 2 0 -0.00989 -0.01294 -0.01142 2 0 2 -0.00518 0.00307 -0.00106 0 0 4 0.02007 -0.01846 0.00083 3 2 0 0.01648 0.01526 0.01589 0 5 0 -0.04015 -0.03513 -0.03763 1 2 2 0.04051 0.00868 0.02465 0 1 4 -0.01086 0.01762 0.00336 0 0 6 -0.02376 0.02282 -0.00047 1 0 0 0.00000 0.00000 0.00000 1 1 0 0.02279 0.02038 0.02158 3 0 0 -0.34408 -0.34630 -0.34518 0 3 0 -0.15500 -0.15512 -0.15506 4 0 0 0.21622 0.22001 0.21814 1 3 0 0.01022 0.01257 0.01141 1 1 2 -0.00306 -0.01407 -0.00852 5 0 0 -0.11869 -0.11199 -0.11543 2 3 0 -0.01109 -0.00054 -0.00581 3 0 2 0.00301 -0.00237 0.00029 0 3 2 -0.00720 0.00083 -0.00322 6 0 0 0.06184 0.04323 0.05256 0 1 0 0.00000 0.00000 0.00000 0 2 0 0.16387 0.16541 0.16464 2 1 0 0.02928 0.02730 0.02829 1 0 2 0.00638 -0.00034 0.00303 3 1 0 0.00540 0.00695 0.00617 0 4 0 0.08482 0.08514 0.08498 0 2 2 0.00862 0.00882 0.00870 4 1 0 -0.01199 -0.01165 -0.01181 1 4 0 0.00909 0.00263 0.00590 2 1 2 -0.01957 0.01060 -0.00452 1 0 4 -0.02238 0.01341 -0.00449 0 6 0 0.01701 0.00842 0.01269

232