Efficient Two-Component Relativistic Method to Obtain Electronic

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Eﬃcient Two-Component Relativistic Method to Obtain Electronic and Molecular Structures of Heavy-Element Systems

ͷిࢠঢ়ଶ͓Αͼ෼ࢠߏ଄ͷͨΊͷܥॏݩૉ ޮ཰తͳ 2 ੒෼૬ର࿦๏

February 2017

Waseda University Graduate School of Advanced Science and Engineering Department of Chemistry and Biochemistry, Research on Electronic State Theory

Yuya NAKAJIMA தౢɹ༟໵

Contents

Chapter 1 General introduction 1

References ...... 6

Chapter 2 Theoretical background 7

2.1 Dirac Hamiltonian ...... 7

2.2 The IODKH method ...... 9

2.3 The LUT scheme ...... 12

2.4 Spin-free and spin-dependent formalisms ...... 14

2.5 The FCP method ...... 16

2.6 The GHF method ...... 19

2.7 The Analytical energy derivative for the GHF method . . . 21

References ...... 24

Chapter 3 Analytical energy gradient for spin-free inﬁnite-order

Douglas–Kroll–Hess method with local unitary transfor-

mation 25

3.1 Introduction ...... 25

3.2 Theory and implementation ...... 27

3.2.1 Energy gradient for IODKH ...... 27

3.2.2 Analytical derivative for the space transformation

matrices ...... 30

3.2.3 Energy gradient for LUT-IODKH ...... 32

3.2.4 Implementation ...... 33

i 3.3 Numerical assessments ...... 34

3.3.1 Computational details ...... 34

3.3.2 Numerical gradient values ...... 37

3.3.3 Accuracies of IODKH/C and LUT-IODKH/C methods 38

3.3.4 Computational cost of the LUT scheme ...... 45

3.3.5 Metal complexes ...... 47

3.3.6 Heavier analogues of ethylene ...... 48

3.3.7 Harmonic frequencies of diatomic molecules . . . . 49

3.4 Conclusion ...... 50

References ...... 52

Chapter 4 Implementation of spin-dependent relativistic analytical

energy gradient 57

4.1 Introduction ...... 57

4.2 Implementation ...... 60

4.3 Numerical assessments ...... 61

4.3.1 Computational details ...... 61

4.3.2 Accuracy of SD-IODKH and LUT-SD-IODKH . . . . 63

4.3.3 Computational cost of (LUT-)SD-IODKH method . . 67 − 4.3.4 Application in fac Ir(ppy3)...... 70 4.4 Conclusion ...... 72

References ...... 74

Chapter 5 Implementation of LUT-IODKH in GAMESS program 79

5.1 Introduction ...... 79

5.2 Implementation ...... 79

5.2.1 Relativistic correction in GAMESS program . . . . . 79

5.2.2 Threshold setting in the IOTC method ...... 80

ii 5.2.3 Integral evaluation at quadruple precision ...... 81

5.2.4 Combination with DC ...... 82

5.2.5 Input options for LUT-IOTC ...... 83

5.2.6 Major capabilities of LUT-IOTC ...... 83

5.3 Numerical assessment ...... 84

5.3.1 Computational details ...... 84

5.3.2 Total energies of heavy atoms and molecules . . . . 85

5.3.3 Threshold dependence of IOTC and LUT-IOTC . . . 86

5.3.4 Computational cost of LUT-IOTC ...... 87

5.4 Conclusion ...... 89

References ...... 91

Chapter 6 Relaxation of core orbitals in the frozen core potential treat-

ment 93

6.1 Introduction ...... 93

6.2 Theory and implementation ...... 94

6.2.1 FCP with relaxation of core electrons ...... 94

6.2.2 Implementation ...... 96

6.3 Numerical assessments ...... 98

6.3.1 Computational details ...... 98

6.3.2 Computational cost of FCP-CR ...... 99

6.3.3 Accuracy of FCP-CR ...... 103

6.3.4 Core ionization energy and core level shift ...... 104

6.3.5 Accuracy of an iterative procedure between valence

and core calculations ...... 107

6.4 Conclusion ...... 108

Appendix 6.A Dependence of core ionization potential energies 111

iii References ...... 113

Chapter 7 Relativistic eﬀect on enthalpy of formation for transition

metal complexes 117

7.1 Introduction ...... 117

7.2 Computational details ...... 119

7.3 Results and discussion ...... 121

7.3.1 Accuracy of WFT ...... 121

7.3.2 Functional dependence ...... 123

7.3.3 Geometry diﬀerence between PP and AE methods . 124

7.3.4 Eﬀect of the levels of relativistic Hamiltonians . . . . 124

7.3.5 Contribution of frozen core orbitals ...... 126

7.4 Conclusion ...... 128

References ...... 130

Chapter 8 General Conclusion 133

Acknowledgments 137

List of Achievements 139

iv List of abbreviations

2c two-component

4c four-component

AE all-electron

AO atomic orbital

BSS Barysz-Sadlej-Snijders

CA composite approach

CCSD coupled cluster singles and doubles

CCSD(T) coupled cluster singles, doubles, and perturbative triples

CIE core ionization energy

CLS core level shift

CPU central processing unit

DC divide-and-conquer

DFT density functional theory

DLU local approximation to the unitary decoupling transformation dmpe 1,2-bis(dimethylphosphino)ethane dOEI derivative of one-electron integral dppe 1,2-bis((pentaﬂuorophenyl)phosphino)ethane dTEI derivative of two-electron integral

ECP eﬀective core potential

ESC elimination of small components

F-dppe 1,2-bis-((pentaﬂuorophenyl)phosphino)ethane

FCP frozen core potential

v FCP-CR frozen-core potential with relaxation of core electrons

FW Fouldy-Wouthuysen

GAMESS General Atomic and Molecular Electronic Structure System

GHF general Hartree–Fock

HF Hartree–Fock

HFR Hartree–Fock–Roothaan

IODK inﬁnite-order Douglas–Kroll–Hess

IOTC inﬁnite-order two-component

LUT local unitary tranformation

MAE mean absolute error

MaxE maximum error

MCP model core potential

MP model potential

MP2 second-order Møller–Plesset perturbation

NESC normalized elimination of the small component

NMR nuclear magnetic resonance

NR non-relativistic

OEI one-electron integral

PP pseudo-potential ppy 2–phenylpyridine

RA regular approximation

RECP relativistic eﬀective core potential

RI resolution of identity

SAC-CI symmetry-adapted cluster conﬁguration interaction

SCF self-consistent ﬁeld

SD spin-dependent

SF spin-free

vi TEI two-electron integral

UT unitary transformation

WFT wave function theory

X2C exact two-component

vii

Chapter 1 General introduction

Quantum chemistry plays an essential role in the qualitative and/or quantitative prediction and analysis of energetics and molecular properties such as geometries, spectra, and reactivities. To date, the basic, commonly used equation of quantum chemistry has been the NR Schrodinger¨ equation. In significant cases involving relativistic effects, such as orbital contractions and splittings in heavy-element systems, intersystem crossing, and core-electron related properties, relativistic effects are mostly accounted for through corrections to the NR treatment. On the other hand, using the Dirac equation, as a basic equation that satisfies the Lorentz invariance for electron motion, can account for relativistic effects. In 2002, Barysz et al. proposed a rigorous 2c method, referred to as the IODKH method1, that treats only electronic states. Seino and Nakai proposed the LUT technique that is able to perform efficient calculations without the loss of accuracy, using the original IODKH method2–6. The construction and extension of a theory based on both IODKH and LUT methods would enable the efficient treatment of relativistic effects using basic equations.

Another eﬃcient method for the treatment of heavy elements is the ECP method, which reduces the number of electrons that are treated explicitly7. The

ECP method replaces the eﬀect of core orbitals on the valence electrons with a potential. Constructing this potential to include relativistic eﬀects provides a relativistic treatment in a convenient manner. In 2014, Seino et al. proposed the FCP

1 method8, which describes core potentials by utilizing explicit core-orbital information obtained through atomic calculations. However, the effects of the chemical environment are not taken into account in the FCP method for molecules. This thesis extends the LUT-IODKH and FCP methods from a theoretical perspective, and applies the extended method to the determination of relativistic effects in significant systems. In order to perform geometry optimizations and frequency calculations, the analytical energy gradient for LUT-IODKH was developed and extended to the spin-dependent method9,10. This gradient method and the LUT scheme were then implemented in the GAMESS quantum chemical package11.

Furthermore, FCP-CR, which relaxes the core orbitals that were treated as frozen orbitals by FCP, was also developed12. These methods were utilized for enthalpy of formation calculations, in the gas phase, of transition metal complexes.

This thesis includes seven chapters, in addition this general introduction chapter (Chapter 1).

Chapter 2 summarizes the theoretical background of the 2c relativistic method, as well as the IODKH, LUT, and FCP methods. Moreover, the energy and gradient expressions for the GHF method are provided.

Chapter 3 extends the IODKH method to include analytical energy gradients, and gradients combined with the LUT scheme. The energy gradient of the

IODKH Hamiltonian, with respect to nuclear coordinates, is analytically derived.

Numerical assessments of equilibrium bond lengths in diatomic molecules containing heavy elements indicate that the accuracy of this method is close to that of 4c. When compared to electron-correlation methods such as MP2 and DFT, good agreements with equilibrium bond lengths are observed. Especially for heavy-element systems, the simultaneous consideration of both electron correlation and relativistic eﬀects was found to be required for accurate calculations.

For harmonic frequency and force constant calculations, relativistic eﬀects were

2 found to contribute to 10% of the obtained values, even for ﬁrst-row transition metals. In order to treat larger-sized molecules, the LUT technique is applied to the gradient calculation. The resulting CPU time required for relativistic unitary transformations in 1d- to 3d-silver clusters was found to be similar to those of NR methods, and scaled quasi-linearly with respect to the number of atoms.

Following on from the previous chapter, Chapter 4 extends and implements the SD energy gradient. The NR and SF relativistic methods are only described within real number space because no spin operator, represented by Pauli spin matrices, are included. To treat the spin operator explicitly, the GHF method, and its energy gradient with respect to nuclear coordinates, is derived from the viewpoint of its implementation within complex number space. In addition, the IODKH and LUT-IODKH methods are extended to the SD treatment. These enable eﬃcient, self-consistent, relativistic geometry optimizations that include

SD relativistic effects. The atomic forces obtained by the analytical SD-IODKH gradient method are enough close to the corresponding numerical gradients in atomic units. Equilibrium bond lengths of diatomic molecules containing fifth- and sixth-row elements are close to those calculated using 4c methods. A comparison of the SF and SD relativistic effects reveals that the contributions of these effects depend on the type of orbital partitioning within the bond. In addition, the CPU time for the relativistic transformation scales quasi-linearly with respect to the number of atoms.

Chapter 5 implements the LUT-IODKH method into the GAMESS quantum chemical package and assesses its performance numerically. GAMESS includes relativistic correction functionalities based on 2c methods such as NESC, RESC,

ﬁrst- to third-order DKH, and IOTC. One of them, IOTC, is equivalent to IODKH from a theoretical point of view. Based on the original code for IOTC, the LUT technique, and its gradient extension, has been implemented in order to perform

3 LUT-IOTC calculations, such as excited-state calculations, in combination with

GAMESS functionalities. In order to verify the integrity of this implementation, the total energies of heavy atoms and molecules are calculated. Several of the

IOTC program-default settings were altered to obtain numerically reliable results.

Furthermore, when combined with the DC method, which has been implemented since 2009 in GAMESS, overall computational linear-scaling is achieved, from relativistic transformations to electron correlation method such as CCSD(T).

Chapter 6 describes studies into theoretically improving FCP for describing the core orbitals of molecules. Conventional FCP only describes valence orbitals of molecules accurately. In this chapter, FCP is extended by relaxing the core orbitals that have been treated as frozen orbitals, by means of valence-orbital potentials, for the accurate descriptions of molecular core orbitals. This enables accurate AE calculations using core potentials. Numerical assessments of gold clusters, composed of up to 86 atoms, indicate that FCP-CR is faster than AE. The orbital energies and total energies of coinage metal dimers are as accurate as those obtained by AE. Furthermore, because FCP-CR possesses core-orbital wavefunctions, it can provide core-electron related properties. Using these wavefunctions the CIEs of the 1s orbitals of C in vinyl acetate, and of the 4f orbitals of W in tungsten complexes, are qualitatively reproduced.

Chapter 7 discusses the evaluation of the gas-phase enthalpies of formation of transition metal complexes using the methods developed in the previous chapters.

These complexes include early and late transition metals, light and heavy metals, low- and high-valency complexes, soft and hard ligands, and organometallic complexes. Contributions of relativistic eﬀects to the enthalpies of formation, calculated using NR, ECP and FCP methods, are assessed and compared. In addition, the correlation between experimental and DFT-calculated data is examined.

Finally, future prospects are discussed after the results of the individual

4 chapters are brieﬂy summarized.

5 References

(1) Barysz, M.; Sadlej, A. J. J. Chem. Phys. 2002, 116, 2696.

(2) Seino, J.; Nakai, H. J. Chem. Phys. 2012, 136, 244102.

(3) Seino, J.; Nakai, H. J. Chem. Phys. 2012, 137, 144101.

(4) Seino, J.; Nakai, H. J. Chem. Phys. 2013, 139, 034109.

(5) Seino, J.; Nakai, H. J. Comput. Chem. Jpn. 2014, 13,1.

(6) Seino, J.; Nakai, H. Int. J. Quantum Chem. 2015, 115, 253.

(7) Schwerdtfeger, P. Chem. Phys. Chem 2011, 12, 3143.

(8) Seino, J.; Tarumi, M.; Nakai, H. Chem. Phys. Lett. 2014, 592, 341.

(9) Nakajima, Y.; Seino, J.; Nakai, H. J. Chem. Phys. 2013, 139, 244107.

(10) Nakajima, Y.; Seino, J.; Nakai, H. J. Chem. Theory Comput. 2016, 15, 68.

(11) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.;

Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.;

Dupuis, M.; Montgomery Jr., J. A. J. Comput. Chem. 1993, 14, 1347.

(12) Nakajima, Y.; Seino, J.; Nakai, H. Chem. Phys. Lett. 2016, 663, 97.

6 Chapter 2 Theoretical background

Relativistic quantum chemistry is based on the Dirac Hamiltonian, which contains electronic and positronic states. In the 2c relativistic treatment, the electronic state (or large component thereof), which is essential for chemistry, is only treated explicitly after several Hamiltonian manipulations, so as to ESCs and apply UTs that are directly related to the accuracy and eﬃciency of the approximations.

Moreover, the relativistic 2c Hamiltonian includes spin operators, or Pauli spin matrices, which require a generalized treatment of SCF-calculated wavefunctions in order to achieve highly accurate results. In this chapter, brief reviews of theoretical backgrounds are provided and include the IODKH Hamiltonian1, for accurate relativistic transformations, and its derivatives; the LUT2 and FCP3 schemes for eﬃcient relativistic transformations without the loss of accuracy; and the GHF method, and its derivatives, for explicit use with spin operators. In addition, methods for splitting the SF and SD terms are also presented.

2.1 Dirac Hamiltonian

Relativistic quantum chemistry is based on the Dirac equation in order to include relativistic eﬀects. The Dirac Hamiltonian is a one-particle Hamiltonian, which is

7 expressed as follows: ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ψ ⎟ ⎜ψ ⎟ ⎜ 1⎟ ⎜ 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ψ ⎟ ⎜ψ ⎟ ⎜ 2⎟ = ⎜ 2⎟ h4 ⎜ ⎟ E ⎜ ⎟ (2.1) ⎜ ⎟ ⎜ ⎟ ⎜ψ ⎟ ⎜ψ ⎟ ⎜ 3⎟ ⎜ 3⎟ ⎜ ⎟ ⎜ ⎟ ⎝ψ ⎠ ⎝ψ ⎠ 4 4 with ⎛ ⎞ ⎜ ⎟ ⎜ − ⎟ ⎜ V 0 cpz c px ipy ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 Vc(p + ip ) −cp ⎟ = ⎜ x y z ⎟ , h4 ⎜ ⎟ (2.2) ⎜ ⎟ ⎜ cp c p − ip V − 2c2 0 ⎟ ⎜ z x y ⎟ ⎜ ⎟ ⎝ + − − 2 ⎠ c(px ipy) cpz 0 V 2c where V is a scalar potential, px, py, and pz are momentum operators along with x-, y-, and z-axis, respectively, and c is the speed of light. Note that atomic units ψ are used hereafter. The upper two wavefunction components in Eq. (2.1), 1 and ψ ψ 2, are referred to as the large components, while the lower two components, 3 ψ and 4, are referred to as the small components. These two types of wavefunction are interpreted to be the electronic and positronic wavefunctions, respectively.

Individual wavefunction components correspond to α- and β-spin, respectively.

Furthermore, using Pauli spin matrices, the Dirac Hamiltonian can be rewritten as follows: ⎛ ⎞ ⎜ σ · p ⎟ ⎜ V12 c ⎟ = ⎜ ⎟ h4 ⎜ ⎟ (2.3) ⎝ σ · p − 2 ⎠ c V12 2c 12 with σ = σ ,σ ,σ , x y z (2.4) ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜01⎟ σ = ⎜ ⎟ , (2.5) x ⎝⎜ ⎠⎟ 10 ⎛ ⎞ ⎜ ⎟ ⎜ − ⎟ ⎜0 i⎟ σ = ⎜ ⎟ , (2.6) y ⎝⎜ ⎠⎟ i 0

8 and ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜10⎟ σ = ⎜ ⎟ , (2.7) z ⎝⎜ ⎠⎟ 0 −1

× where 1n is the n n identity matrix.

2.2 The IODKH method

As mentioned above, the Dirac equation includes both electronic and positronic information. In general chemistry however, more attention is given to the electronic information, which, conceptually, leads to a decoupling of the electronic and positronic states. The Dirac Hamiltonian is block-diagonalized using a unitary transformation U, which is expressed as: ⎛ ⎞ ⎜ + ⎟ ⎜h 0 ⎟ = † = ⎜ 2 ⎟ , h2 U h4U ⎜ ⎟ (2.8) ⎝⎜ −⎠⎟ 0h2

+ − where h2 and h2 are the electronic and positronic Hamiltonians, respectively. Practical calculations use only the electronic Hamiltonian. In this thesis, the

IODKH method, which is one of the most accurate relativistic transformation schemes, is used. Many electronic, two-component, relativistic Hamiltonians are generally expressed by:

= + + ++ , , H2 h2 (i) g2 (i j) (2.9) i i>j

+ ++ where h2 and g2 are one- and two-particle Hamiltonians, respectively. For the + one-particle Hamiltonian, h2 , the IODKH Hamiltonian was adopted, and is given by: + = Ω† Ω, h2 G (2.10) where the second-step transformation, Ω, is deﬁned by

Ω = + † −1/2, (12 Y Y) (2.11)

9 and the Hamiltonian transformed in Eq. (2.10), G, is expressed as

p2 = 2 + † + † + Π† σ · p σ · p Π, G p b12 Y − 12 Y M VM ( V ) (2.12) 1 ep with = + , M K(12 bpY) (2.13)

Π = α − −1 , K( b12 p Y) (2.14)

+ 1/2 ep 1 K = , (2.15) 2ep

= 1 , b + (2.16) ep 1 1/2 = + α2 2 , ep 1 p (2.17) and

α = 1/c. (2.18)

Here, p is the momentum operator and V is the sum of the nucleus-electron

= attraction operators with respect to an atom A, i.e., V A VA. The operator Y is obtained by the decoupling conditions: + = α3 − −1 σ · p σ · p epY Yep pKbVK p K V bK − − + α2 p 1Kσ · pVσ · pp 1KY − YKVK (2.19) + α4 pKbVbKpY − YKbσ · pVσ · pbK − + α3Y Kbσ · pVσ · pKp 1 − KVKbp Y.

The present thesis focuses on the relativistic eﬀect on the one-particle interac-

++ tion. Hence, the two-particle Hamiltonian g2 is replaced as the NR Coulomb interaction: 1 g(i, j) = . (2.20) rij

10 This treatment is denoted as IODKH/C. It should be noted that relativistic two- particle interactions can be included using the Breit-Pauli approximation,4 as well as the FW5 and IODKH transformations.6

In the pragmatic evaluation of Eq. (2.10),the matrix transformation method and the RI technique proposed by Hess are adopted.7 The ﬁrst step in the technique of Hess is the determination of the eigenvectors, {k},ofp2, which are expanded by the linear combination of the primitive Gaussian functions {φ},

| = |φ , k r Crk (2.21) r ﬃ where the coe cient Crk is determined by diagonalizing a matrix whose element φ 2 φ { } is r p s . The resulting (pseudo) eigenvectors, k , satisfy: 2 = ω δ , k p k k kk (2.22)

ω { } where k is a (pseudo) eigenvalue. The explicit matrix expressions, in k -space, for Eqs. (2.10)–(2.19) can be written as: + † = Ω Ω , h Gk k k k (2.23) 2 kk kk k,k − / † 1 2 Ω = + , kk k 12 Y Y k (2.24) 2 2 † p = + Gkk k p b12 k Ykk k − 12 k Yk k 1 ep k ,k (2.25) † † + | | +Π | σ · p σ · p | Π , Mkk k V k Mk k kk k V k k k k,k with = + , Mkk Kk 1 bkpkYkk (2.26) − Π = α − 1 , kk Kk bk pk Ykk (2.27)

(2.28)

Here, the approximate RI relationship is inserted between the individual operators, which is expressed as

|kk|≈1. (2.29) k The matrix elements, which are the functions of p2, f p2 , such as the operators

K, b, and p operators in Eqs. (2.15)–(2.17) become: 2 = ω δ . k f (p ) k f ( k) kk (2.30)

2.3 The LUT scheme

The IODKH method achieves high accuracy compared to the 4c method although the computational cost associated with the relativistic transformation is considerably large. On the basis of the nature of unitary transformations and relativistic eﬀects, approximated unitary transformations can be constructed in order to im- prove the eﬃciency. In this section, the concept of the LUT scheme and the

LUT-IODKH Hamiltonian are provided.

The LUT scheme arises from the localization of the relativistic eﬀect. The

12 entire one-particle unitary transformation U can be approximated in a block- diagonal matrix form consisting of the subsystem contributions:

UTotal ≈ UA ⊕ UB ⊕ UC ··· , (2.31) where {A, B, C, ···}denotes the subsystems. Here, atomic partitioning is adopted.

The relativistic eﬀect on the kinetic energy is dominant at each atom while the eﬀect on the nuclear attraction is dominant not only at each atom, but also on its interactions within a small distances τ. The matrix representation of the one-particle Hamiltonian with basis function {χ} is expressed as χA + χB ≈ χA LUT χB μ h2 ν μ h2 ν ⎧ ⎪ ⎪ A + + NR B ⎪ χμ T + V + V χν (A = B) ⎪ A C ⎪ CA ⎪ ⎨⎪ (2.32) χA + + + + NR + NR χB , ≤ τ = ⎪ μ V V T V ν (A B RAB ) ⎪ A B C ⎪ CA,B ⎪ ⎪ ⎪ A NR NR B ⎪ χμ T + V χν (A B, R >τ) ⎩ C AB C

NR NR where T and VC denote the standard NR kinetic and nuclear-attraction operators: 1 TNR = p21 , (2.33) 2 2 and NR = . VC VC (2.34)

+ + T and VC are the relativistic operators given by: p2 + = Ω† 2 + † Ω, T p b12 Y − 12 Y (2.35) 1 ep and + = Ω† † + Π† σ · p σ · p Π Ω. VC M VCM ( VC ) (2.36)

13 Note that the operators Y, Ω, M, and Π are determined within individual subsystems because U contains only single-subsystem information. Hence, the computational cost of the relativistic transformation using the LUT scheme scales linearly with respect to the system size.

2.4 Spin-free and spin-dependent formalisms

SF and SD terms are considered separately in standard relativistic quantum chemistry. One of the reasons for this is that the SF formalism is a straightforward extension to NR theory. This subsection explains how the SF and SD formalisms are obtained.

In order to obtain the SF formalism, the spin-related terms containing Pauli spin matrices are separated by the Dirac relationship8,

σ · pVσ · p = pV · p + iσ · (pV × p) (2.37) where i denotes the imaginary unit. Since the dot products and cross products can be evaluated in {χ}-space, the explicit expression containing these terms is given by: χ | p · p | χ = χ · + · + · χ V pxV px pyV py pzV pz (2.38) and ⎛ ⎞ ⎜ ⎟ ⎜ 0 χ (pV × p) χ ⎟ χ | σ · p × p | χ = ⎜ y ⎟ i ( V ) ⎜ ⎟ ⎝ χ p × p χ ⎠ ( V )y 0 ⎛ ⎞ (2.39) ⎜ ⎟ ⎜ χ (pV × p) χ χ (pV × p) χ ⎟ + ⎜ z x ⎟ i ⎜ ⎟ ⎝ χ p × p χ − χ p × p χ ⎠ ( V )x ( V )z with χ p × p χ = χ · − · χ , ( V )x pyV pz pzV py (2.40)

χ p × p χ = χ · − · χ , ( V )y pxV pz pzV px (2.41)

14 and χ p × p χ = χ · − · χ . ( V )z pxV py pyV px (2.42)

The momentum operator p is performed on the basis set {χ}. Partial integrations of Eqs. (2.40)–(2.42) leads to: ∂χ ∂χ ∂χ ∂χ χ (pV × p) χ = V − V , (2.43) x ∂y ∂z ∂z ∂y ∂χ ∂χ ∂χ ∂χ χ (pV × p) χ = V − V , (2.44) y ∂x ∂z ∂z ∂x and ∂χ ∂χ ∂χ ∂χ χ (pV × p) χ = V − V . (2.45) z ∂x ∂y ∂y ∂x

In these equations, x, y, and z refer to electronic coordinates of electrons. In the SF formalism, the IODKH and LUT-IODKH Hamiltonians are expressed as follows: + † = Ω Ω , h Gk k k k (2.46) 2 kk kk k,k − / † 1 2 Ω = + , kk k 12 Y Y k (2.47) 2 2 † p = + Gkk k p b12 k Ykk k − 12 k Yk k 1 ep k ,k (2.48) † † + | | +Π | p · p | Π , Mkk k V k Mk k kk k V k k k k,k with = + , Mkk Kk 1 bkpkYkk (2.49) − Π = α − 1 , kk Kk bk pk Ykk (2.50)

15 and ⎡ ⎢ 1 ⎢ − = ⎢α3 | | − 1 | p · p | Ykk ⎢ pkKkbk k V k Kk p Kk k V k bk Kk + ⎣ k ek ek − − + α2 1 | p · p | 1 − | | pk Kk k V k pk Kk Yk k Ykk Kk k V k Kk k + α4 | | pkKkbk k V k bk Kk pk Yk k k (2.51) − | p · p | Ykk Kk bk k V k bk Kk ) − + α3 | p · p | 1 Ykk Kk bk k V k Kk pk k,k − | | , Kk k V k Kk bk pk Yk k and ⎧ ⎪ ⎪ A + + NR B ⎪ χμ T + V + V χν (A = B) ⎪ A C ⎪ CA ⎪ ⎨⎪ A LUT B χA + + + + NR + NR χB , ≤ τ χμ h χν = ⎪ μ V V T V ν (A B RAB ) 2 ⎪ A B C ⎪ CA,B ⎪ ⎪ ⎪ A NR NR B ⎪ χμ T + V χν (A B, R >τ) ⎩ C AB C (2.52) with + = Ω† † + Π† p · p Π Ω. VC M VCM ( VC ) (2.53)

2.5 The FCP method

The IODKH method, when combined with the LUT scheme, extends the applicability of relativistic quantum chemistry to the calculation of relatively large molecules without loss of accuracy. However, all-electron calculations still have problems that include large basis set requirements for heavy elements, prob- lematic convergence during SCF calculations, and the demand for high memory and disk capacities. For the further extension of relativistic quantum chemistry, decreasing the number of electrons treated explicitly oﬀers a solution to these

16 problems. In this subsection, the FCP method, that provides a solution to the problems mentioned above, is explained.

All-electron calculations are based on the HFR equation:

F(D)C = SCE (2.54) with

Fμν(D) = Hμν + Gμν(D), (2.55) 1 Gμν(D) = Dλρ χμχρ|g|χνχλ− χμχρ|g|χλχν , (2.56) λρ 2 and occ = , Dμν 2 CμiCνi (2.57) i where F is the Fock matrix; D is the density matrix, C is the orbital coeﬃcient matrix, S is the overlap matrix, E is the orbital energy matrix, H is the one-electron integral, G is the two-electron integral, g is the two-electron Hamiltonian, χ are basis functions, while “occ” denotes occupied orbitals, and C is the complex conjugate of C.

The density matrix is divided into core and valence contributions as follows:

D = DC + DV (2.58) with core C = Dμν 2 CμcCνc (2.59) c and valence V = . Dμν 2 CμvCνv (2.60) v In the case of basis sets that segmented and contracted, the contributions from core and valence AOs to the core-density matrix, are large (μ and ν in core AOs) and small (μ or ν in valence AOs), respectively. The core AOs contribute moderately

17 to the valence-density matrix. In the case of generally contracted basis sets, the core and valence density matrices are negligibly aﬀected by valence and core

AOs, respectively, since the individual core and valence regions are suﬃciently described by core and valence AOs. When generally contracted basis sets are adopted, the entire density matrix can be approximately represented in a block- diagonalized form, as follows:

D ≈ dC ⊕ dV, (2.61) where dC and dV are the core- and valence-contributing density matrices, respectively, which include only core and valence AO indices. Since the individual atomic core-density matrices contribute mainly to the molecular core-density matrix, the introduction of the direct sum approximation to the core-density matrix suﬃciently reproduces the original molecular core-density matrix:

C ≈ ˜ C = ˜ C ⊕ ˜ C ⊕··· D D DA DB (2.62) and C ≈ ˜ C = ˜ C ⊕ ˜ C ⊕···, d d dA dB (2.63)

C where {A, B, ···}are individual atoms. The atomic core-density matrices, D˜ and C d˜ , correspond to the core-density matrices obtained from the individual atomic calculations, which are frozen for molecular calculations.

These two approximations lead to the Huzinaga-Cantu equation9, which is expressed as:

fV(dV)cV = sVcVεV (2.64) with V V V V C C = μν + − + − + μν, , fμν(d ) H 2Jμν Kμν 2Jμν,A Kμν,A P A (2.65) A A

18 valence V V V V V V V 1 V V V V 2Jμν − Kμν = dλρ χμ χρ g χν χλ − χμ χρ g χλ χν , (2.66) λ,ρ 2 core C C C V C V C 1 V C C V 2Jμν, − Kμν, = d˜λρ χμ χρ g χν χλ − χμ χρ g χλ χν , (2.67) A A A 2 λ,ρ∈A and

core = − ε χV ϕ ϕ χV . Pμν,A 2 c,A μ c,A c,A ν (2.68) c∈A ϕ ε C C Here, c,A is the core orbital, c,A is the core orbital energy, J and K are Coulomb and exchange terms for the core region obtained by atomic all-electron calculations, JV and KV are Coulomb and exchange terms for the valence region, and P represents projection/shift terms. Finally, the total, core, and valence energies are expressed as follows:

ETotal = EC + EV (2.69) with C − C + C 1 nA ZB nB nB ZA EC = tr DC H + F DC + (2.70) 2 RAB A

C where ZA is the nuclear charge, nA is the number of core electrons in atom A, and

RAB is the distance between atoms A and B.

2.6 The GHF method

The explicit consideration of the eﬀect of SD in quantum chemical calculation requires a description of spin mixing. The standard HF method only describes the spin parallel to the z-axis. In order to describe an arbitrary direction of

19 electronic spin, the GHF method should be adopted. Hence, in this section, the energy expression using the GHF method is presented.

In the GHF method, the wave function is expressed using a linear- combination of α- and β-spin orbitals:

α β ω ψ = φμα + φμβ = φμω, i Cμi Cμi Cμi (2.73) μ μ ω=α,β μ where C is the coeﬃcient matrix, ω is the spin index, and φ is the spacial orbital.

The HF energy is generally expressed as: 1 EHF = [i | h | i] + ij ij − ij ji + V . (2.74) 2 nuc i i,j

Here, EHF denotes the HF electronic energy, i and j are the electronic indices, [i | h | i] is a one-electron Hamiltonian matrix, and ij ij and ij ji are the NR two- electron interactions, i.e., Coulomb and exchange terms, respectively, while Vnuc is the nuclear repulsion potential. Note that the one-electron Hamiltonian contains the spin operator, or Pauli spin matrices in the case of the relativistic Hamiltonian.

Using Eq. (2.73), the energy expression for the one-electron Hamiltonian matrix can be rewritten as follows: ωω [i | h | i] = Dμν φμω h φνω (2.75) i ω=α,β ω=α,β μ,ν with

ωω = ω ω , Dμν CμiCνi (2.76) i where D is the density matrix. Likewise, the Coulomb and exchange terms can be obtained: 1 1 ωω ττ ij ij = Dμν Dλρ φμωφνω φλτφρτ (2.77) 2 2 i,j ω,ω τ,τ μ,ν,λ,ρ and 1 1 ωτ τω ij ji = DμλDνρ φμωφντ φλτφρω , (2.78) 2 2 i,j ω,ω τ,τ μ,ν,λ,ρ

20 where τ and τ are spin indices. By the integration of Eqs. (2.77) and (2.78) over spin functions, the entire two-electron integrals become: 1 ωω ττ ωτ τω Dμν Dλρ δωω δττ − DμλDνρ δωτ δτω φμφν φλφρ , (2.79) 2 ω,ω τ,τ μ,ν,λ,ρ where δ is the Kronecker delta and φμφν φλφρ is the two-electron integral in a given basis set. Finally the GHF energy can be provided: HF ωω E = Dμν φμω h φνω ω,ω μ,ν 1 ωω ττ ωτ τω + Dμν Dλρ δωω δττ − DμλDνρ δωτ δτω φμφν φλφρ (2.80) 2 ω,ω τ,τ μ,ν,λ,ρ

+ . Vnuc

2.7 The Analytical energy derivative for the GHF method

In the previous section, the energy expression for the GHF method, which contains a summation over electronic spins, is provided. To perform the geometry optimization based on the GHF method requires a diﬀerentiated GHF expression.

This section presents the analytical energy gradient for the GHF method.

The total energy from the GHF method is expressed by Eq. (2.80). Diﬀer- entiating the total energy with respect to a nuclear coordinate RA provides the

21 following expression: ⎛ ⎞ ⎛ ⎞ ∂ ωω ∂ ωω ∂ HF ⎜ hμν ⎟ ⎜ Dμν ⎟ E = ωω ⎜ ⎟ + ⎜ ⎟ ωω ∂ Dμν ⎝ ∂ ⎠ ⎝ ∂ ⎠ hμν RA ω,ω μ,ν RA ω,ω μ,ν RA 1 ωω ττ ωτ τω ∂ + δ δ − δ δ φ φ φ φ Dμν Dλρ ωω ττ DμλDνρ ωτ τω ∂ μ ν λ ρ 2 ω,ω τ,τ μ,ν,λ,ρ RA ⎛ ⎞ ⎜∂ ωω ⎟ 1 ⎜ Dμν ττ ωτ τω ⎟ + ⎜ δ δ − δ δ ⎟ φ φ φ φ ⎝ ∂ Dλρ ωω ττ DμλDνρ ωτ τω ⎠ μ ν λ ρ 2 ω,ω τ,τ μ,ν,λ,ρ RA ⎛ ττ ⎞ ⎜ ∂D ⎟ 1 ⎜ ωω λρ ωτ τω ⎟ + ⎜ δωω δττ − δωτ δτω ⎟ φμφν φλφρ ⎝Dμν ∂ DμλDνρ ⎠ 2 ω,ω τ,τ μ,ν,λ,ρ RA ⎛ ωτ ⎞ ⎜ ∂D ⎟ 1 ⎜ ωω ττ μλ τω ⎟ + ⎜ δ δ − δ δ ⎟ φ φ φ φ ⎝Dμν Dλρ ωω ττ ∂ Dνρ ωτ τω ⎠ μ ν λ ρ 2 ω,ω τ,τ μ,ν,λ,ρ RA ⎛ ⎞ ⎜ ∂ τω ⎟ 1 ⎜ ωω ττ ωτ Dνρ ⎟ + ⎜ δ δ − δ δ ⎟ φ φ φ φ ⎝Dμν Dλρ ωω ττ Dμλ ∂ ωτ τω ⎠ μ ν λ ρ 2 ω,ω τ,τ μ,ν,λ,ρ RA ∂ + Vnuc . ∂ RA (2.81)

The GHF-SCF Fock matrix, and its derivative for molecular orbitals at i = j under orthonormal conditions are as follows: ⎛ ⎞ ⎜ ⎟ ωω ωω ⎜ ττ τ ω ⎟ = + ⎜ δωω δττ − δωτ δτω ⎟ φμφν φλφρ Fμν hμν ⎝⎜Dλρ CλiCjρ ⎠⎟ (2.82) τ,τ λ,ρ i,j and ⎛ ⎞ ⎛ ⎞ ⎜∂ ω ⎟ ∂ ωω ⎜ C μ ⎟ ⎜ Sμν ⎟ ⎜ i ⎟ ωω ω + + ω ⎜ ⎟ ω = , ⎝⎜ ∂ ⎠⎟ Sμν Ciν c.c. Ciμ ⎝ ∂ ⎠ Ciν 0 (2.83) μ,ν RA μ,ν RA where S is the overlap matrix and c.c. is the complex conjugate. Eq. (2.81) can be

22 rewritten as: ⎛ ⎞ ∂ ωω ∂ HF ⎜ hμν ⎟ E = ωω ⎜ ⎟ ∂ Dμν ⎝ ∂ ⎠ RA ω,ω RA 1 ωω ττ ωτ τω ∂ + δ δ − δ δ φ φ φ φ Dμν Dλρ ωω ττ DμλDνρ ωτ τω ∂ μ ν λ ρ 2 ω,ω τ,τ μ,ν,λ,ρ RA ⎛ ⎞ ωω ⎜∂ μν ⎟ ∂ ωω ⎜ S ⎟ Vnuc + ⎜ ⎟ δ + , Wμν ⎝ ∂ ⎠ ωω ∂ ω,ω RA RA (2.84)

ε where W is the energy-weighted density matrix with the orbital energy i,

ωω = ε ω ω . Wμν iCμiCiν (2.85) i

23 References

(1) Barysz, M.; Sadlej, A. J. J. Chem. Phys. 2002, 116, 2696.

(2) Seino, J.; Nakai, H. J. Chem. Phys. 2012, 136, 244102.

(3) Seino, J.; Tarumi, M.; Nakai, H. Chem. Phys. Lett. 2014, 592, 341.

(4) Bethe, H. A.; Salpeter, E. E., Quantum Mechanics of One- and Two-electron

Atoms; Springer, Berlin: 1957.

(5) Foldy, L. L.; Wouthuysen, S. A. Phys. Rev. 1950, 78, 29.

(6) Seino, J.; Hada, M. Chem. Phys. Lett. 2008, 461, 327.

(7) Hess, B. A. Phys. Rev. A 1985, 32, 756.

(8) Reiher, M.; Wolf, A., Relativistic Quantum Chemistry; Wiley-VCH, Germany:

2009.

(9) Bonifacic, V.; Huzinaga, S. J. Chem. Phys. 1974, 60, 2779.

24 Chapter 3 Analytical energy gradient for spin-free inﬁnite-order Douglas–Kroll–Hess method with local unitary transformation

3.1 Introduction

Reliable predictions of geometries and properties for molecules including heavy elements are some of the most important tasks in quantum chemical calculations. Notably, in case of calculations for heavy (and super heavy) elements and their compounds, the inclusion of relativistic eﬀect is essential because electrons, especially core electrons, have considerably high speeds. For example, bond contractions/elongations due to relativistic shrinking/expansion of molecular or- bitals1–4 were observed in several diatomic molecules such as halogen dimers and

PtAu.5,6 The evaluations of ﬁrst- and higher-order derivatives of the molecular energy enables computations of optimized geometries, various molecular properties such as molecular vibrations observed in infrared and Raman spectra and chemical shifts observed in NMR spectra. Accordingly, the development of analytical energy derivatives with the relativistic Hamiltonian is desired in order to extend the applicability of the quantum chemical method.

A highly accurate treatment in relativistic quantum chemistry is based on the 4c relativistic theory, which adopts a one-particle Dirac operator that satisﬁes

25 Lorentz invariance only in terms of the motion of each electron. Numerical applications of the analytical energy gradient method with the 4c treatments conﬁrmed the high accuracy in determining molecular geometries,1 and electric and magnetic properties.7,8 Such applications, however, were suitable only in the case small- or medium-sized molecules because of the practical problems involving computational costs and/or variational conditions9 due to the existence of small-component Dirac spinors or positronic-state spinors. There have been several attempts to resolve these problems.10–12

An alternative approach is based on the 2c relativistic theory, which treats only electronic-state (or large-component) information. Recent developments in the 2c treatment of many-electron molecular systems have exhibited a higher de- gree of accuracy, in comparison with 4c treatment.13–18 The energy gradient methods for accurate 2c relativistic schemes are under development. The 2c relativistic theories are categorized into two types. The ﬁrst approach uses an equation for

ESC, such as RA,19–21 RESC,22 and NESC.23 The analytical energy gradient schemes have been made available for the zeroth-order RA and its scaling schemes,24–26

RESC,27 and NESC.28,29 The second approach uses the decoupling matrix transformation of 4c Dirac Hamiltonian and includes approaches such as the exact quasi-relativistic method.30 In this type of method we separate the electronic and positronic Hamiltonian from 4c Dirac Hamiltonian or Fock matrix using unitary transformation. The widely used schemes are the FW transformation31 and DKH transformation.32 These were extended to higher-order DKH33–36 and IODKH37 methods. The molecular property calculation for this type of scheme is based on the second-order38–40 and extended35,41,42 DKH Hamiltonian.

Nakai and co-workers have worked on the development of an accurate and eﬃcient 2c relativistic scheme, termed LUT, at the one- and two-particle SF-

IODKH levels.43,44 The LUT method utilizes the locality of the relativistic eﬀect.

26 Accordingly, the computational costs become drastically lower than those of conventional methods and scale linearly with respect to the system size with a small prefactor. In addition, the LUT scheme clariﬁes the primary relativistic eﬀect in molecular systems, because the deviations in total energy are less than milli- hartrees. Recently, the LUT scheme has been combined with the linear-scaling

DC based on HF45–47 and electron correlation48–53 methods, such as MP2 and

CCSD.54 The combination of LUT and DC techniques leads to the ﬁrst approach that achieves overall (quasi-)linear scaling with a small prefactor for relativistic electron correlation calculations.

In this chapter, the author derives the analytical energy gradient of the one- particle IODKH Hamiltonian for the accurate calculations. Furthermore, the

LUT scheme is introduced for eﬃcient calculations for large-sized molecules.

Theoretical aspects are explained in Section 3.2. In Section 3.3, the present scheme is numerically assessed from the viewpoints in accuracy and eﬃciency. The concluding remarks are ﬁnally provided in Section 3.4.

3.2 Theory and implementation 3.2.1 Energy gradient for IODKH

In the HF method, the ﬁrst derivative of the electronic energy with respect to a coordinate RA of a nucleus A is written as ⎛ ⎞ ⎜∂ + ⎟ ∂ ⎜ h μν ⎟ ∂ μλ||νρ ∂ ∂ μν E ⎜ 2 ⎟ 1 ( ) Vnuc S = μν ⎜ ⎟+ μν λρ + − ε μν , ∂ D ⎜ ∂ ⎟ D D ∂ ∂ iD ∂ RA ⎝ RA ⎠ 2 RA RA RA μ,ν μ,ν,λ,ρ i μ,ν (3.1) where μ, ν, λ, and ρ are the indices of {χ};(μλ||νρ), the antisymmetric electron {χ} repulsion integral in ; D, the density matrix; Vnuc, the nuclear repulsion potential; ε, the orbital energy; and S, the overlap matrix. In the expression of Eq. (3.1), ∂ μλ||νρ /∂ ∂ /∂ ∂ /∂ ( ) RA, Vnuc RA, and Sμν RA are identical to those in the NR theory.

27 The IODKH Hamiltonian in {χ} and that in {k} have the following relation

+ {χ} = † † + { } , h2 ( ) d X h2 ( k ) Xd (3.2) with

X = ΓΛ, (3.3)

ﬃ + {χ} + { } where d denotes the contraction coe cients. h2 ( ) and h2 ( k ) are the IODKH Hamiltonians in {χ} and {k}, respectively. Γ is the transformation matrix from space {χ} to the orthonormalized space, while Λ is the transformation matrix { } + {χ} from the orthonormalized space to k . The analytical derivative for h2 ( ) is expressed as ∂ + {χ} ∂ † ∂ + { } ∂ h2 ( ) = † X + + † h2 ( k ) + † + X , ∂ d ∂ h2 X X ∂ X X h2 ∂ d (3.4) RA RA RA RA with ∂ + { } ∂Ω† ∂ ∂Ω h2 ( k ) = Ω + Ω† G Ω + Ω† . ∂ ∂ G ∂ G∂ (3.5) RA RA RA RA

Here, we can obtain the derivatives of the matrix elements of Ω and G by diﬀer- entiating individual matrix elements with respect to {k} in Eqs. (2.24)–(2.28), − / † ∂Ω 1 † 3 2 ∂Y † ∂Y kk = − + + , ∂ k 12 Y Y ∂ Y Y ∂ k (3.6) RA 2 RA RA and ∂ ∂ ∂ 2 Gkk 2 † p = + ∂ ∂ k p b12 k ∂ Ykk k − 12 k Yk k RA RA RA 1 ep k ,k ∂ † + | | ∂ Mkk k V k Mk k (3.7) RA k ,k ∂ † + Π | σ · p σ · p | Π , ∂ kk k V k k k RA k ,k with

∂ ∂ ∂ Mkk Kk = + + , ∂ ∂ 1 bkpkYkk Kk ∂ bkpkYkk (3.8) RA RA RA

Note that vectors such as K, b, and p in Eqs. (3.6)–(3.10) have the information about molecular structure through the function {k}. Thus, the elements should be diﬀerentiated as ⎛ ⎞ ∂ ⎜ ∂p2 ⎟ Kk = −1α2 −3 −1 ⎜ k ⎟ , ∂ ek Kk ⎝∂ ⎠ (3.11) RA 8 RA ⎛ ⎞ ∂ ⎜ ∂p2 ⎟ bk = −1α2 −1 2 ⎜ k ⎟ , ∂ ek bk ⎝∂ ⎠ (3.12) RA 2 RA ⎛ ⎞ ∂ ⎜ ∂p2 ⎟ ek = 1α2 −1 ⎜ k ⎟ , ∂ ek ⎝∂ ⎠ (3.13) RA 2 RA and ⎛ ⎞ − ∂p 1 ⎜ ∂p2 ⎟ k = −1 −3 ⎜ k ⎟ . ∂ pk ⎝∂ ⎠ (3.14) RA 2 RA

29 3.2.2 Analytical derivative for the space transformation matrices

In order to obtain explicit derivatives of the space transformation matrix X and

∂ 2/∂ pk RA, the same procedure as that in Ref. [38] is adopted. First, the procedure for the derivative of Γ in Eq. (3.3) is explained. Γ is determined by the condition

ΓTSΓ = 1, (3.15) with the overlap matrix S. In more detail, S is ﬁrst diagonalized as

WTSW = s. (3.16)

Γ is obtained using the unitary transformation W and the eigenvalues of s

− / Γ = Ws 1 2. (3.17)

The explicit expression of the derivative of Γ is as follows:

∂Γ ∂ ∂ = W −1/2 − 1 −3/2 s . ∂ ∂ s Ws ∂ (3.18) RA RA 2 RA

The derivative of s is given by

∂ ∂ T ∂ ∂ s = W + T S + T W ∂ ∂ SW W ∂ W W S ∂ (3.19) RA R A RA RA ∂ = T S − , , W ∂ W [PW s]− (3.20) RA with ∂ = T W . PW W ∂ (3.21) RA

Here, square brackets represent the anti-commuter/commuter of matrices as

[A, B]± = AB ± BA. (3.22)

ﬀ ∂ /∂ Note that the o diagonal elements of s RA and the diagonal elements of PW are zero because s is a diagonal matrix and PW is an antisymmetric matrix. From Eq.

30 (3.20) and these conditions, the matrix expression of PW is obtained as T ∂ /∂ W ( S RA)W kk = − δ . (PW) (1 kk ) (3.23) kk − sk sk ∂ /∂ ∂ /∂ Thus, s RA is obtained by substituting Eq. (3.23) to Eq. (3.20). Since W RA is also given by ∂ W = , ∂ WPW (3.24) RA ∂Γ/∂ ∂ /∂ ∂ /∂ RA is determined from Eq. (3.18) using s RA and W RA.

The procedure for the derivative of Λ in Eq is explained. (3.3). The ﬁrst step is a transformation of TNR from {φ} to the orthonormalized space as

† T({φ}) = Γ TNRΓ. (3.25)

By diagonalizing T({φ}), Λ is obtained:

† T({k}) = Λ T({φ})Λ. (3.26)

Here, the derivative of T({k}) is expressed as

∂ { } ∂ {φ} T( k ) = Λ† T( ) Λ − , { } , ∂ ∂ [PV T( k )]− (3.27) RA RA with ∂Λ = Λ† . PV ∂ (3.28) RA ∂ { } /∂ ∂ /∂ Note that T( k ) RA has a relationship with p RA:

∂ ∂ T = p . ∂ 2∂ (3.29) RA RA ∂ {φ} /∂ ﬀ To obtain T( ) RA, Eq. (3.25) is di erentiated with respect to RA : ∂T({φ}) ∂ ∂ = −1/2 TW −1/2 − 1 −1 s , {χ} , ∂ s ∂ s s ∂ T( ) (3.30) RA RA 2 RA + where = T NR . TW W T W (3.31)

31 Eq. (3.30) is rewritten using the derivative of TW:

∂ ∂ NR TW = T T − , . ∂ W ∂ W [PW TW]− (3.32) RA RA

For the properties of Eq. (3.27), oﬀdiagonal elements on the left-hand side in { } Eq. (3.27) are zero because T( k ) is diagonal. The diagonal elements of PV are zero due to the antisymmetry of PV. These properties provide Λ† ∂ {φ} /∂ Λ T( ) RA kk = − δ . (PV) (1 kk ) (3.33) kk { } − { } T( k )k T( k )k

Finally, the derivative of Λ becomes

∂Λ = Λ . ∂ PV (3.34) RA

Using the derivatives of Γ and Λ, the derivative of the space transformation matrix X can be obtained:

∂ ∂Γ ∂Λ X = Λ + Γ . ∂ ∂ ∂ (3.35) RA RA RA

3.2.3 Energy gradient for LUT-IODKH

This subsection explains the analytical derivative of the LUT-SF-IODKH Hamil- tonian for eﬃcient gradient evaluations. The derivative expression of the LUT-

SF-IODKH matrix of Eq. (2.32) is given by ⎧ ⎪ ⎪ ∂ ⎪ χA NR χB = ⎪∂ μ VC ν (A B) ⎪ RA ⎪ CA ⎪ ⎪ ∂ ⎪ χA + + + χB ⎪ μ V V ν ∂ ⎨⎪∂R A B χA LUT χB = ⎪ A ∂ μ h2 ν ⎪ ∂ RA ⎪ A NR NR B ⎪+ χμ T + V χν (A B, R ≤ τ) ⎪ ∂ C AB ⎪ RA , ⎪ C A B ⎪ ⎪ ∂ ⎪ χA NR + NR χB , >τ, ⎩∂ μ T VC ν (A B RAB ) RA C (3.36)

32 where + ∂V ∂ ∂ C = Ω† † VC + Π† p · p Π Ω. ∂ M ∂ M ∂ VC (3.37) RA RA RA

Note that the IODKH transformation is only applied to the nucleus-electron interaction for diﬀerent atoms with a small distance τ.IfA = B, the derivative of + + the IODKH transformed matrices such as T and VC become zero because the transformations of the LUT-IODKH method have closure forms within individ- ≤ τ + ual atoms. If A B with RAB , the nuclear attraction elements VC for the ﬀ LUT-IODKH transformation are obtained by di erentiation with respect to RA as well as the NR kinetic terms TNR and the NR nuclear attraction ones VNR.

3.2.4 Implementation

This subsection presents the scheme for evaluating the energy gradient with the

(LUT-) IODKH Hamiltonian. Table 3.1 summarizes the procedures for (a) NR, (b)

IODKH, and (c) LUT-IODKH. For the NR/C case, the density matrix D and orbital energy ε obtained from energy calculation are preserved in the energy gradient calculation. Then the derivatives of NR one-electron integrals TNR and VNR are calculated in contracted formalism.

For the conventional IODKH case, integrals and momentum factors in uncon-

NR NR Ω Π −1 tracted formalism such as S, T , V , , Y, G, h, M, , K, b, ep, and p as well as the density matrix D and orbital energy ε obtained from energy calculation are preserved in energy gradient calculation. Furthermore, the derivatives of NR and relativistic integral, S, TNR, VNR, and σ · pVσ · p, are calculated in uncontracted formalism. Using S and TNR, the derivatives of transformation matrices Γ and

Λ are calculated using Eqs. (3.15)-(3.34). At the same time, the derivatives of kinematic factors of Eqs. (3.11)-(3.14) can be calculated. Using kinematic factors and their derivatives, the derivatives of Ω, G, and Y in {k} are calculated using

Eqs. (3.6), (3.7), and (3.10), respectively. Finally the derivative of relativistic one-

33 particle Hamiltonian in Eq. (3.5) is obtained by back transformation into {χ} using the derivatives of Γ and Λ.

For the LUT-IODKH case, the procedure becomes simpler than the procedure Ω Π of the conventional IODKH. First, the transformation factors , Y, M, , K, b, ep, − and p 1 in each subsystem as well as D and ε are preserved in energy calculation.

Then, the NR integrals for TNR and VNR are calculated in contracted formalism.

In order to obtain the transformed integrals for interatomic interactions with ≤ τ NR σ · p σ · p RAB in Eq. (3.36), the derivatives of V and V are calculated in uncontracted formalism. After that, the transformation is performed using Eq.

(3.37). Thus, the derivative of LUT-IODKH Hamiltonian matrix can be obtained.

Note that this complicated procedure involving Eqs. (3.15)-(3.34) is unnecessary in the case of the LUT-IODKH scheme. In addition, the range of transformation is extremely small. Consequently, the LUT-IODKH scheme is suitable for the eﬃcient calculation for analytical derivative of one-electron integral.

3.3 Numerical assessments 3.3.1 Computational details

This section presents the assessment of the performance of the analytical energy gradient for the IODKH and LUT-IODKH schemes. We adopted the spin-free formalism of IODKH/C: the one-electron Dirac Hamiltonian and two-electron non- relativistic Coulomb interaction. For LUT, the threshold for cutoﬀ of relativistic interaction, τ, was set to 3.5 Å. For comparison, the spin-free 4c Dirac-Coulomb and non-relativistic (NR/C) methods were also calculated. The calculations of parts of small-component integrals, i.e., (SS|SS)-type ones, were skipped in the 4c method.55

Numerical tests were performed for various closed-shell diatomic molecules: = hydrogen halides HX, halogen dimers X2 (X F, Cl, Br, I, and At), coinage metal

34 Table 3.1 Procedures for computing the analytical energy gradient of (a) NR, (b) IODKH, and (c) LUT-IODKH. (a) NR (b) IODKH (c) LUT-IODKH ε NR NR Ω ε D, , S, T , V , , Y, G, h, ε Ω Π −1 (I) From energy calculation D, −1 D, , , Y, M, , K, b, ep, p M, Π, K, b, ep, p (for whole subsystem) (for each system)

∂TNR ∂VNR ∂TNR ∂VNR (II) Derivatives of NR one-electrton integrals ∂ , ∂ – ∂ , ∂ RA RA RA RA (except for interatomic interac- ≤ τ tion with RAB ) Derivatives of NR one-electron integrals ∂S ∂TNR ∂VNR ∂(pV·p) ∂VNR ∂(pV·p) (III) for realtivistic transformation (uncon- – ∂ , ∂ , ∂ , ∂ ∂ , ∂ RA RA RA RA RA RA 35 tracted form) (for interatomic interaction ≤ τ with RAB ) ∂Γ ∂Λ (IV) Derivatives of transformation matrices – ∂ , ∂ [from Eqs. (3.18) and (3.34)] – RA RA ∂ ∂ −1 + ∂K ∂b ep p ∂V (V) Derivatives of relativistic one-electron integrals – ∂ , ∂ , ∂ , ∂ [from Eqs. (3.11)-(3.14)] ∂ [from Eq. (3.37)] RA RA RA RA RA ∂Y (for interatomic interaction ∂ [from Eq. (3.10)] ≤ τ RA with RAB ) ∂M ∂Π ∂ , ∂ [from Eqs. (3.8) and (3.9)] RA RA ∂Ω ∂G ∂ , ∂ [from Eqs. (3.6) and (3.7)] RA RA ∂(μλ||νρ) (VI) Derivatives of NR two-electron integral ∂ RA (VII) Energy gradient calculation from Eq. (3.1). = hydrides MH, and coinage metal dimers M2 (M Cu, Ag, and Au). In addition, one-, two-, and three-dimentional silver clusters Agn were calculated to investi- ﬃ gate the e ciency of the LUT scheme. Figure 3.1 illustrates the structures of Ann = , , ··· , clusters. For one-dimensional silver clusters with Agn (n 1 2 10), the Ag-

Ag bond distance used here was 2.706 Å. For two- and three-dimensional silver clusters, a face-centered cubic structure with the experimental lattice constant,

4.086 Å56, was adopted. The unit cells for two- and three-dimensional clusters contain ﬁve and 14 atoms, respectively. We adopted several sizes of Ann clusters; the total numbers of atoms in the two-dimensional clusters were 5, 13, 25, 41, 61,

85, 113, 145, and 181, which correspond to 1 × 1, 2 × 2, ···, and 9 × 9 unit cells, respectively. The total numbers of atoms in the three-dimensional clusters were

14, 23, 32, 38, 53, 63, 74, 88, 123, and 172, which correspond to 1 × 1 × 1, 2 × 1 × 1,

3×1×1, 2×2×1, 3×2×1, 2×2×2, 3×3×1, 3×2×2, 3×3×2, and 3×3×3. For more practical systems, 20 metal complexes including the fourth–sixth row-transition metals were examined. Heavier analogues of ethylene were employed to investigate the relativistic eﬀect in geometry. For harmonic frequency calculation, MH and M2 were utilized. The calculations were performed at the HF, MP2, and density functional theory (DFT) with B3LYP functional levels.57 The basis sets for all-electron calculation in diatomic molecules and 20 metal complexes are summarized in Table 3.2. All basis sets were used in uncontracted form. For the 4c method, the kinetically bal- anced small-component basis sets were adopted. MCP/NOSeC-V-TZP58–68 and

Stuttgart/Dresden (SDD69–72) type eﬀective core potential are used in RECP. In

SDD calculation, the ﬁrst–third row atoms were described with cc-pVTZ.73,74 The

IODKH energy gradient method with/without the LUT scheme was implemented in the GAMESS program.75 The 4c calculations were performed in the DIRAC12 program.76 The Gaussian09 program was used in SDD calculations.77

36 Figure 3.1 Geometry of (a) one-dimensional silver cluster Ag10, (b) two-dimensional silver × × × cluster Ag13 (2 2), and (c) three-dimensional silver cluster Ag63 (2 2 2).

Table 3.2 Basis sets for diatomic molecules and 20 metal complexes used at the levels of HF, MP2, and B3LYP in NR, IODKH, and large component of 4c. Period Element Description Basis sets 1H(6s3p2d) Ref. [78] 2 C,N,O,F (12s8p3d2 f ) Refs. [59, 79] 3 P, Cl (16s11p4d2 f ) Refs. [60, 80] 4 Ti, V (19s13p7d3 f ) Refs. [62, 80] Cr (20s13p8d3 f ) Refs. [62, 80] Co, Ni (20s12p9d3 f ) Refs. [62, 80] Cu (20s12p8d3 f ) Refs. [62, 80] Zn (20s13p9d3 f ) Refs. [62, 80] Br (20s15p10d2 f ) Refs. [63, 79] 5 Zr (22s17p13d3 f ) Refs. [64, 79] Nb (22s17p12d3 f ) Refs. [64, 79] Mo (22s16p11d3 f ) Refs. [64, 79] Ru (21s16p12d3 f ) Refs. [64, 79] Ag (22s16p12d3 f ) Refs. [64, 79] Cd (22s16p13d3 f ) Refs. [64, 79] I (22s18p13d2 f ) Refs. [63, 79] 6 Ta, W, Re (26s22p16d12 f ) Refs. [66, 79] Os, Pt, Au, Hg (26s21p16d12 f ) Refs. [66, 79] At (26s23p17d13 f ) Refs. [65, 79]

3.3.2 Numerical gradient values

This subsection investigates the accuracy of the analytical gradient values for

IODKH/C and LUT-IODKH/C Hamiltonians by comparing with the numerical

37 ones in HAt, At2, AuH, and Au2 molecules. Table 3.3 compares between the analytical and numerical energy gradient values calculated by the NR/C, IODKH/C,

LUT-IODKH/C, and 4c at the HF level. We examined the gradient values in 1.0,

1.5, 2.0, and 3.0 times of the experimental bond length, re, which were 1.742, 3.046,

81 5 82 83 1.5324, and 2.4719 Å for HAt ,At2 , AuH , and Au2 , respectively. The numerical gradient calculations were performed using a central diﬀerence method with

0.01 Å stepsize. The diﬀerences between the analytical gana and numerical gnum energy gradient values, that is, Δgana, are shown in parentheses, which represent the correctness of the analytical scheme. The deviations from 4c, Δg4c, are given in square brackets, which represent the two-electron relativistic eﬀect of contribution of (SS|SS)-type integrals.

In IODKH/C, the absolute deviations from the numerical results, i.e., Δgana, are less than 0.0004 Hartree/Bohr. Such small deviations conﬁrm the correctness of the present implementation for the analytical energy gradient of IODKH/C.

| Δ 4c | The deviations from 4c, i.e., g , are comparatively large in At2 and ﬀ | Au2. This is due to the lack of two-electron relativistic e ect in 2c or (SS SS)-type integral in 4c. In LUT-IODKH/C, | Δgana | are also small. The largest deviation is 0.0001 Hartree/Bohr. In addition, the absolute values of diﬀerences between

IODKH/C and LUT-IODKH/C, which are given in | ΔgLUT |, are less than 0.0003

Hartree/Bohr. This means that the LUT treatment evaluates the relativistic energy gradient without loss of accuracy.

3.3.3 Accuracies of IODKH/C and LUT-IODKH/C methods

This subsection examines the accuracy of the analytical energy gradient scheme for IODKH/C Hamiltonian. Table 3.4 shows the bond lengths (Å) of diatomic molecules calculated by the NR/C, IODKH/C, and 4c methods at the HF level.

The diﬀerence Δrrel of IODKH/C and 4c methods from NR/C are given in square

38 / / / / / Table 3.3 Analytical and or numerical gradient values (Hartree Bohr) of HAt, At2, AuH, and Au2 calculated by NR C, IODKH C, LUT-IODKH C, and 4c at HF level. NR/C IODKH/C LUT-IODKH/C4c Mole. lengtha) gana gana gnum Δganab) Δg4cc) gana gnum Δganab) ΔgLUTd) gana

HAt 1.0re 0.0113 0.0192 0.0195 (-0.0003) [-0.0002] 0.0194 0.0193 ( 0.0001) [ 0.0002] 0.0194 1.5re 0.0675 0.0657 0.0654 ( 0.0003) [ 0.0001] 0.0655 0.0655 ( 0.0000) [-0.0002] 0.0656 2.0re 0.0379 0.0366 0.0365 ( 0.0001) [-0.0001] 0.0365 0.0366 (-0.0001) [-0.0001] 0.0367 e) 3.0re 0.0098 0.0096 0.0097 (-0.0001) – 0.0096 0.0096 ( 0.0000) [ 0.0000] N. C. At2 1.0re 0.0256 0.0304 0.0305 (-0.0001) [-0.0110] 0.0304 0.0305 (-0.0001) [ 0.0000] 0.0414 1.5re 0.0273 0.0259 0.0255 ( 0.0004) [-0.0047] 0.0256 0.0256 ( 0.0000) [-0.0003] 0.0306 2.0re 0.0082 0.0079 0.0078 ( 0.0001) [-0.0027] 0.0078 0.0078 ( 0.0000) [-0.0001] 0.0106 39 3.0re 0.0019 0.0019 0.0015 ( 0.0004) [-0.0012] 0.0019 0.0020 (-0.0001) [ 0.0000] 0.0031 AuH 1.0re 0.0823 0.0135 0.0135 ( 0.0000) [ 0.0011] 0.0135 0.0135 ( 0.0000) [ 0.0000] 0.0124 1.5re 0.0336 0.0540 0.0540 ( 0.0000) [ 0.0000] 0.0540 0.0541 (-0.0001) [ 0.0000] 0.0540 2.0re 0.0313 0.0365 0.0365 ( 0.0000) [-0.0001] 0.0365 0.0365 ( 0.0000) [ 0.0000] 0.0367 3.0re 0.0149 0.0138 0.0138 ( 0.0000) [-0.0002] 0.0138 0.0138 ( 0.0000) [ 0.0000] 0.0140 Au2 1.0re 0.0798 0.0305 0.0304 ( 0.0001) [ 0.0117] 0.0305 0.0305 ( 0.0000) [ 0.0000] 0.0188 1.5re 0.0170 0.0270 0.0270 ( 0.0000) [-0.0053] 0.0268 0.0269 (-0.0001) [-0.0002] 0.0323 2.0re 0.0127 0.0134 0.0133 ( 0.0001) [-0.0031] 0.0133 0.0133 ( 0.0000) [-0.0001] 0.0165 3.0re 0.0045 0.0036 0.0037 (-0.0001) [-0.0013] 0.0036 0.0035 ( 0.0001) [ 0.0000] 0.0048 a) re denotes the experimental bond length taken from [5, 81–83]. b) Difference in analytical gradient value from the numerical one is given in parentheses. c) Difference in analytical gradient value from 4c is given in square brackets. d) Difference between IODKH and LUT-IODKH in analytical gradient value is given in square brackets. e) SCF calculations were not converged. brackets, representing the relativistic effect accounted by IODKH/C and 4c. The deviations Δr4c of IODKH/C from 4c are shown in parentheses, indicating the accuracy of the IODKH/C treatment.

ﬀ Δ rel The relativistic e ect on bond lengths, r , is negative except for F2 and Cl2.

The absolute values increase as heavier elements are added to the systems. For example, the values in halogen dimers are 0.000, 0.001, -0.002, -0.012, and -0.036

Å for F2,Cl2,Br2,I2, and At2, respectively. This might be the common tendency in the relativistic eﬀect. Furthermore, the deviations in halogen molecules are smaller than those in coinage metal molecules. This is because the coinage metal molecules have σ-bonding formed by each s valence orbital, which is largely shrunken by the relativistic eﬀect in comparison with σ-bonding formed by p valence orbitals seen in halogen molecules.

The deviations of IODKH/C from 4c, Δr4c, are comparatively small. The largest deviation is 0.010 Å for At2. This indicates that the spin-free two-electron relativistic corrections reasonably demonstrate the bond lengths of elements, at least, up to the sixth row of the periodic table.

Table 3.5 shows the bond lengths (Å) of diatomic molecules calculated by the NR/C and IODKH/C at the HF, MP2, and B3LYP levels. The diﬀerences Δrcorr of NR/C and IODKH/C methods with respect to the corresponding HF results are given in square brackets, representing the correlation eﬀect accounted by

NR/C and IODKH/C. The deviations Δrexp of NR/C and IODKH/C methods from reference are shown in parentheses, indicating the accuracy of the NR/C and

IODKH/C treatment. In additions, the mean absolute error (MAE) and the mean error (ME) are given for Δrexp.

In the MP2 results, the correlation eﬀect on bond lengths, Δrcorr, of IODKH/C shows the same tendency as NR/C except for HBr and HAt. Hydrogen halides except for HAt and halogen dimers have positive values, whereas coinage metal

40 = = / / Table 3.4 Bond lengths (Å) of HX, X2 (X F, Cl, Br, I, and At), MH, and M2 (M Cu, Ag, and Au) molecules obtained by NR C, IODKH C and 4c methods at HF level. NR IODKH/C4c Mole. rNR rIODKH Δrrela) Δr4cb) r4c Δrrela) HF 0.897 0.896 [-0.001] (-0.001) 0.897 [ 0.000] HCl 1.264 1.263 [-0.001] (-0.001) 1.264 [ 0.000] HBr 1.470 1.404 [-0.066] (-0.001) 1.405 [-0.065] HI 1.608 1.600 [-0.008] ( 0.000) 1.600 [-0.008] HAt 1.711 1.684 [-0.027] ( 0.001) 1.683 [-0.028] F 1.328 1.328 [ 0.000] ( 0.000) 1.328 [ 0.000]

41 2 Cl2 1.975 1.976 [ 0.001] ( 0.000) 1.976 [ 0.001] Br2 2.274 2.272 [-0.002] (-0.002) 2.274 [ 0.000] I2 2.675 2.663 [-0.012] (-0.002) 2.665 [-0.010] At2 2.890 2.854 [-0.036] ( 0.010) 2.844 [-0.046] CuH 1.568 1.540 [-0.028] ( 0.000) 1.540 [-0.028] AgH 1.778 1.700 [-0.078] ( 0.001) 1.699 [-0.079] AuH 1.828 1.568 [-0.260] (-0.001) 1.569 [-0.259] Cu2 2.440 2.401 [-0.039] (-0.001) 2.402 [-0.038] Ag2 2.818 2.706 [-0.112] ( 0.000) 2.706 [-0.112] Au2 2.924 2.604 [-0.320] ( 0.002) 2.602 [-0.322] a) Relativistic correction is in square brackets. b) Diﬀerence in bond length from that obtained using 4c is in parentheses. halides and coinage metal dimers have negative values. The deviations of NR/C from reference, Δrexp, increase as the heavier elements are present in the molecules.

On the other hand, the deviations of IODKH/C from reference remain approximately constant. For example, the values in coinage metal halides are -0.042,

-0.034, and -0.037 Å for CuH, AgH, and AuH, respectively. Furthermore, MAE and ME of IODKH/C are relatively smaller than those of NR/C. This indicates that both correlation and relativistic eﬀects are essential for the molecules containing heavier elements.

In the B3LYP results, the correlation eﬀects on bond lengths, Δrcorr,of

IODKH/C and NR/C show similar trends except for HBr. The deviations of

/ Δ exp NR C from reference, r , are positive except for F2. The values increase as the elements in the system are heavier. For example, the values in coinage dimers are 0.062, 0.151, and 0.332 Å for Cu2,Ag2, and Au2, respectively. The deviations / of IODKH C are positive except for F2 and CuH and slightly increases in the case of heavier elements. MAE and ME of IODKH/C are smaller than those of NR/C because both correlation and relativistic eﬀects are considered.

Table 3.6 shows the bond length (Å) of diatomic molecules calculated by the

IODKH/C and LUT-IODKH/C at the HF, MP2, and B3LYP levels. The deviation

ΔrLUT of the LUT-IODKH/C method from IODKH/C is shown in parentheses, indicating the accuracy of the LUT-IODKH/C treatment.

The deviations are both positive and negative. The largest absolute deviations in HF, MP2, and B3LYP are 0.014 Å for At2, 0.013 Å for At2, and 0.009 Å for

HAt, respectively. Since MAE and ME are reasonably small, the LUT-IODKH/C treatment can suﬃciently describe the relativistic eﬀect for estimating the bond lengths.

42 = = / Table 3.5 Bond lengths (Å) of HX, X2 (X F, Cl, Br, I, and At), MH, and M2 (M Cu, Ag, and Au) molecules calculated by the NR C and IODKH/C at HF, MP2 and B3LYP level. HF MP2 B3LYP Exptl. Ref. NR IODKH NR IODKH NR IODKH Mole. rNR rIODKH rNR Δrcorra) Δrexpb) rIODKH Δrcorra) Δrexpb) rNR Δrcorra) Δrexpb) rIODKH Δrcorra) Δrexpb) HF 0.897 0.896 0.917 [ 0.020] (-0.001) 0.917 [ 0.021] (-0.001) 0.922 [ 0.025] ( 0.004) 0.922 [ 0.026] ( 0.004) 0.9176 [83] HCl 1.264 1.263 1.270 [ 0.006] (-0.005) 1.270 [ 0.007] (-0.005) 1.281 [ 0.017] ( 0.006) 1.281 [ 0.018] ( 0.006) 1.2749 [83] HBr 1.470 1.404 1.412 [-0.058] (-0.003) 1.410 [ 0.006] (-0.005) 1.425 [-0.045] ( 0.010) 1.423 [ 0.019] ( 0.008) 1.4146 [83] HI 1.608 1.600 1.611 [ 0.003] ( 0.001) 1.603 [ 0.003] (-0.007) 1.624 [ 0.016] ( 0.014) 1.618 [ 0.018] ( 0.008) 1.6099 [83] HAt 1.711 1.684 1.714 [ 0.003] — 1.682 [-0.002] — 1.725 [ 0.014] — 1.696 [ 0.012] — — —

43 F2 1.328 1.328 1.400 [ 0.072] (-0.012) 1.400 [ 0.072] (-0.012) 1.398 [ 0.070] (-0.014) 1.398 [ 0.070] (-0.014) 1.4119 [83] Cl2 1.975 1.976 1.985 [ 0.010] (-0.003) 1.984 [ 0.008] (-0.004) 2.012 [ 0.037] ( 0.024) 2.012 [ 0.036] ( 0.024) 1.9879 [83] Br2 2.274 2.272 2.284 [ 0.010] ( 0.003) 2.281 [ 0.009] (-0.000) 2.316 [ 0.042] ( 0.035) 2.313 [ 0.041] ( 0.032) 2.2811 [83] I2 2.675 2.663 2.677 [ 0.002] ( 0.011) 2.669 [ 0.006] ( 0.003) 2.708 [ 0.033] ( 0.042) 2.702 [ 0.039] ( 0.036) 2.6663 [83] At2 2.890 2.854 2.892 [ 0.002] — 2.860 [ 0.006] — 2.919 [ 0.029] — 2.883 [ 0.029] — — — CuH 1.568 1.540 1.451 [-0.117] (-0.015) 1.424 [-0.116] (-0.042) 1.482 [-0.086] ( 0.016) 1.459 [-0.081] (-0.007) 1.4658 [82] AgH 1.778 1.700 1.662 [-0.116] ( 0.044) 1.584 [-0.116] (-0.034) 1.697 [-0.081] ( 0.079) 1.625 [-0.075] ( 0.007) 1.6179 [82] AuH 1.828 1.568 1.709 [-0.119] ( 0.177) 1.495 [-0.073] (-0.037) 1.746 [-0.082] ( 0.214) 1.538 [-0.030] ( 0.006) 1.5324 [82] Cu2 2.440 2.401 2.240 [-0.200] ( 0.021) 2.203 [-0.198] (-0.016) 2.281 [-0.159] ( 0.062) 2.248 [-0.153] ( 0.029) 2.2192 [84] Ag2 2.818 2.706 2.591 [-0.227] ( 0.061) 2.505 [-0.201] (-0.025) 2.681 [-0.137] ( 0.151) 2.595 [-0.111] ( 0.065) 2.5303 [85] Au2 2.924 2.604 2.710 [-0.214] ( 0.238) 2.468 [-0.136] (-0.004) 2.804 [-0.120] ( 0.332) 2.547 [-0.057] ( 0.075) 2.4719 [83] MAE 0.042 0.014 0.072 0.023 ME 0.037 -0.013 0.070 0.020 a) Diﬀerence in bond length from the corresponding HF result is given in squre brakets. b) Diﬀerence in bond length from the experimental result is given in parentheses. = = Table 3.6 Bond length (Å) of HX, X2 (X F, Cl, Br, I, and At), MH, and M2 (M Cu, Ag, and Au) molecules calculated by the IODKH and LUT-IODKH at HF, MP2 and B3LYP level. HF MP2 B3LYP Molecule rIODKH rLUT ΔrLUTa) rIODKH rLUT ΔrLUTa) rIODKH rLUT ΔrLUTa) HF 0.896 0.896 ( 0.000) 0.917 0.917 ( 0.000) 0.922 0.922 ( 0.000) HCl 1.263 1.264 ( 0.001) 1.270 1.270 ( 0.000) 1.281 1.281 ( 0.000) HBr 1.404 1.404 ( 0.000) 1.410 1.410 ( 0.000) 1.423 1.423 ( 0.000) HI 1.600 1.601 ( 0.001) 1.603 1.604 ( 0.001) 1.618 1.618 ( 0.000) HAt 1.684 1.682 (-0.002) 1.682 1.689 ( 0.007) 1.696 1.705 ( 0.009)

F2 1.328 1.328 ( 0.000) 1.400 1.400 ( 0.000) 1.398 1.398 ( 0.000) 44 Cl2 1.976 1.976 ( 0.000) 1.984 1.985 ( 0.001) 2.012 2.013 ( 0.001) Br 2.272 2.273 ( 0.001) 2.281 2.281 ( 0.000) 2.313 2.313 ( 0.000)

I2 2.663 2.664 ( 0.001) 2.669 2.667 (-0.002) 2.702 2.700 (-0.002) At2 2.854 2.840 (-0.014) 2.860 2.847 (-0.013) 2.883 2.879 (-0.004) CuH 1.540 1.541 ( 0.001) 1.424 1.425 ( 0.001) 1.459 1.460 ( 0.001) AgH 1.700 1.700 ( 0.000) 1.584 1.584 ( 0.000) 1.625 1.626 ( 0.001) AuH 1.568 1.570 ( 0.002) 1.495 1.491 (-0.004) 1.538 1.536 (-0.002)

Cu2 2.401 2.401 ( 0.000) 2.203 2.203 ( 0.000) 2.248 2.247 (-0.001) Ag2 2.706 2.705 (-0.001) 2.505 2.506 ( 0.001) 2.595 2.595 ( 0.000) Au2 2.604 2.600 (-0.004) 2.468 2.462 (-0.006) 2.547 2.540 (-0.007) MAE 0.002 0.003 0.002 ME -0.001 -0.001 0.000 a) Diﬀerence in bond length from that obtained using the IODKH method. 3.3.4 Computational cost of the LUT scheme

This subsection investigates the eﬃciency of the analytical energy gradient in the

LUT scheme using multi-dimensional silver clusters Agn. Figure 3.2 shows the system-size dependence of the CPU time for the one-electron analytical energy gradient calculation in the one-electron IODKH and LUT-IODKH transforma- = tions of one-dimensional silver clusters Agn (n 1, 2, ..., 10). The vertical axis stands for the CPU time measured with the Xeon X5470/3.33 GHz (Quad core) on a single core in seconds while the horizontal axis shows the system size n.

The computational cost of the LUT-IODKH transformation is small with a small prefactor compared with the conventional IODKH transformation. The scalings

. . are O(n1 31) and O(n2 52), respectively, which are in reasonable agreement with theoretical ones, namely, O(n1) and O(n3). In addition, Figure 3.3 shows the system size dependence of the CPU time in the LUT-IODKH transformations of two- and three- dimensional silver clusters. In multi-dimensional systems, the LUT

. treatment scales quasi-linear: O(n1 21) for both two- and three-dimensional cases.

Thus, the LUT scheme for the analytical energy gradient method can eﬃciently evaluate the gradient for any chemical compound in the relativistic realm.

Table 3.7 shows the CPU time for the seven steps in the analytical energy gradient calculations at the HF level in an Ag10 cluster: OEI, UT of OEI, an initial guess obtained with the Huckel¨ calculation, TEI, SCF, dOEI, and dTEI. The NR,

IODKH, and LUT-IODKH Hamiltonian were used. The percentage of the CPU time for each step is given in parentheses. In energy calculations from OEI to

SCF, the time for the unitary transformation is considerably reduced in LUT-

IODKH compared with IODKH. The conventional IODKH requires 53.31% for the unitary transformation in the analytical energy gradient calculation. The LUT-

IODKH method, however, can reduce CPU time. Note that the total CPU time

45 60000

50000 IODKH LUT-IODKH 40000

2.52 30000 O(n ) CPU time [s] 20000

10000 O(n1.31) 0 0246810 n Figure 3.2 System-size dependence of CPU time (s) of the one-electron analytical energy = , , ··· , gradient in Agn (n 1 2 10) obtained with conventional IODKH and LUT-IODKH methods. A Xeon X5470/3.33GHz (Quad Core) processor was used on a single core.

10.0

9.0 two-dimension 8.0 three-dimension 7.0 6.0 O(n1.21) 5.0 4.0 O(n1.21) CPU time [s] 3.0 2.0 1.0 0 0 50 100 150 200 n Figure 3.3 System-size dependence of CPU time (s) of the one-electron analytical energy gradient in multi-dimensional silver clusters Agn obtained with LUT-IODKH methods. A Xeon X5470/3.33GHz (Quad Core) processor was used on a single core.

46 Table 3.7 CPU time (s) for seven steps in Hartree-Fock calculation obtained with NR,

IODKH, and LUT-IODKH in Ag10 linear cluster: OEI, UT of OEI, initial guess (Guess) calculated with Huckel¨ scheme, TEI, SCF, dOEI, and dTEI. A Quad core Xeon/3.3 GHz processor was used on a single core. NR IODKH LUT-IODKH Time % Time % Time % OEI 0.22 ( 0.01) 0.22 ( 0.00) 0.23 ( 0.01) UT of OEI 0.00 ( 0.00) 41902.32 ( 41.56) 12.54 ( 0.31) Guess 254.07 ( 6.52) 262.55 ( 0.26) 267.26 ( 6.50) TEI 886.03 ( 22.74) 928.95 ( 0.92) 886.19 ( 21.57) SCF 1242.38 ( 31.89) 2446.54 ( 2.43) 1328.72 ( 32.34) dOEI 2.31 ( 0.06) 53755.54 ( 53.31) 15.91 ( 0.39) dTEI 1510.97 ( 38.78) 1536.95 ( 1.52) 1598.19 ( 38.89) Total 3895.98 (100.00) 100833.07 (100.00) 4109.04 (100.00) for the LUT-IODKH method is close to those of the NR method, indicating that the LUT-IODKH scheme achieve both high accuracy close to the conventional

IODKH method as well as the 4c one and low computational cost comparable with NR method.

3.3.5 Metal complexes

This subsection provides the geometry parameters for 20 transition-metal complexes. Table 3.8 shows the complexes calculated by the NR, RECP, MCP, and

LUT-IODKH methods at the B3LYP level. A-B indicates the bond length between

A and B. The differences Δrrel of the RECP, MCP, and LUT-SF-IODKH methods from the NR method are given in brackets, representing the relativistic effect accounted by the RECP, MCP, and LUT-IODKH methods. Figure 3.4 shows the difference in the bond length from that obtained using the NR method. The bond lengths in the sixth-row metal complexes calculated by relativistic methods are shorter than that of the NR method. Especially in the twelfth group, Zn, Cd, and

Hg, the bond lengths are shorter than those in the other groups.

47 Table 3.8 Bond lengths (Å) for the 20 transition-metal complexes calculated by NR, RECP, MCP, and LUT-IODKH at the B3LYP level. Entry Bond NR RECP MCP LUT-IODKH

TiOCl2 Ti-O 1.598 1.593 1.601 1.595 Ti-Cl 2.239 2.232 2.237 2.236

TiF4 Ti-F 1.754 1.749 1.755 1.752 VO(OEt)3 V-O 1.577 1.576 1.587 1.576 V-OEt 1.768 1.763 1.768 1.764

Cr(CO)6 Cr-C 1.926 1.920 1.931 1.920 Co(CO)4H Co-H 1.472 1.475 1.478 1.468 Co-C(ax.) 1.822 1.802 1.804 1.811 Co-C(eq.) 1.809 1.816 1.824 1.798

Ni(PF3)4 Ni-P 2.131 2.128 2.119 2.115 Ni(CO)4 Ni-C 1.847 1.832 1.820 1.830 ZnMe2 Zn-C 1.958 1.945 1.933 1.942 ZrF4 Zr-F 1.906 1.906 1.907 1.905 ZrBr4 Zr-Br 2.495 2.495 2.494 2.488 NbCl5 Nb-Cl(ax.) 2.342 2.339 2.342 2.336 Nb-Cl(eq.) 2.234 2.289 2.298 2.288

MoBr4 Mo-Br 2.400 2.406 2.402 2.390 RuO4 Ru-O 1.693 1.682 1.687 1.681 CdEt2 Cd-C 2.201 2.153 2.171 2.158 TaCl5 Ta-Cl(ax.) 2.361 2.337 2.304 2.306 Ta-Cl(eq.) 2.312 2.288 2.264 2.260

WOCl4 W-O 1.721 1.696 1.689 1.700 W-Cl(ax.) 2.431 2.404 2.365 2.405 W-Cl(eq.) 2.299 2.266 2.241 2.264

Re(CO)5Br Re-C(ax.) 1.994 1.944 1.931 1.944 Re-C(eq.) 2.077 2.023 2.006 2.022 Re-Br 2.703 2.656 2.662 2.652

OsO4 Os-O 1.737 1.697 1.684 1.700 cis-Platin Pt-N 2.195 2.106 2.057 2.086 Pt-Cl 2.370 2.307 2.294 2.303

HgMe2 Hg-C 2.247 2.123 2.111 2.118

3.3.6 Heavier analogues of ethylene

This subsection provides the geometry parameters for heavier analogues of ethy- = lene H2MMH2 (M C, Si, Ge, Sn, and Pb). Figure 3.5 shows M-M bond length and H-M-H angle calculated by the NR and LUT-SF-IODKH methods at the HF level. M-M bond length is longer in both methods as the heavier atom is included. In the case of Pb, the diﬀerences in the bond length and angle between

48 0.10 RECP 0.05 MCP LUT-IODKH 0.00 [Å] rel r ¨ -0.05

-0.10 Forth period Fifth period Sixth period -0.15 Ti-F V-O Zr-F Ni-P Pt-N Ti-O W-O Ni-C Cr-C Zn-C Pt-Cl Ti-Cl Cd-C Os-O Ru-O Co-H Hg-C Zr-Br Re-Br V-OEt Mo-Br Re-C (ap.) Re-C (eq.) Co-C (ap.) Co-C (eq.) W-Cl (ap.) W-Cl (eq.) Ta-Cl (ap.) Ta-Cl (eq.) Nb-Cl (ap.) Nb-Cl (eq.) Bond Figure 3.4 Diﬀerence in bond length calculated by the RECP, MCP, and LUT-SF-IODKH from that obtained using the NR method.

the LUT-SF-IODKH method and the NR one are largest in the other cases since the hybridization of s and p orbitals in valence ones is difficult due to the relativistic effect, which increases their orbital energy difference. In addition, the optimized geometry obtained by the LUT-IODKH method is close to PbH2 dimer rather than H2PbPbH2 molecule. Molecular orbitals for individual PbH2 dimer obtained by the LUT-IODKH method are not hybridized but the same as that of PbH2 monomer, while molecular orbitals obtained by the NR method are hybridized and contributed to M-M bonding.

3.3.7 Harmonic frequencies of diatomic molecules

This subsection provides the harmonic frequencies for heavy diatomic molecules = MH and M2 (M Cu, Ag, and Au). Table 3.9 shows the harmonic frequencies for heavy diatomic molecules by the NR and LUT-IODKH methods at the DFT with PBE0 functional. The difference between NR and LUT-IODKH, Δrel, is given in brackets indicates the relativistic effect, while the difference Δ of LUT-IODKH from experimental value is given in square brackets. The Hessian matrices were

49 3.5 120 (a) (b) 3.0 115 2.5 110 ]

Å 2.0 105 1.5 M-M [ H-M-H [°] 100 1.0 NR NR 0.5 95 LUT-IODKH LUT-IODKH 0.0 90 CSiGeSn Pb CSiGe Sn Pb Figure 3.5 Geometry parameters of (a) bond length (M-M) and (b) bond angle (H-M-H) for = heavier analogues of ethylene H2MMH2 (M C, Si, Ge, Sn, and Pb).

= Table 3.9 Harmonic frequencies of MH and M2 (M Cu, Ag, and Au) obtained by the NR and LUT-IODKH method at the DFT with PBE0 functional. Molecule NR Δrel LUT-IODKH Δ Exptl. CuH 1886.92 (−81.31) 1968.23 [26.97] 1941.26 AgH 1600.98 (−165.97) 1766.95 [7.05] 1759.90 AuH 1610.00 (−645.62) 2255.62 [−49.39] 2305.01 . − . . − . . Cu2 242 85 ( 20 12) 262 97 [ 1 58] 264 55 . − . . − . . Ag2 158 94 ( 21 32) 180 26 [ 12 14] 192 40 . − . . − . . Au2 118 02 ( 61 27) 179 29 [ 11 61] 190 90 obtained by numerical differentiations of analytical gradient values. The relativistic effects on harmonic frequencies are more than 8% even at lighter diatomic molecule of Cu2;AtAu2, the contribution reaches to 38%. Resultant deviation in LUT-IODKH may be originated mainly from the limitation of electron correlation in DFT and partly from the SD relativistic and two-electron relativistic effects.

3.4 Conclusion

In this Chapter, the analytical energy gradient for the LUT-IODKH method is provided. Since the size of the relativistic transformation matrix is smaller than that of the conventional IODKH method and the deﬁnition is closure within individual atoms, the derivative of the LUT-IODKH Hamiltonian can be obtained

50 eﬃciently. Numerical assessments were performed using several heavy-atom diatomic molecules for accuracy, of which bond lengths were in good agreement with both the 4c method at the HF level and the experimental values at the MP2 and DFT with B3LYP levels. Eﬃciency for the derivative calculation of the LUT-

IODKH Hamiltonian is high compared to that of the conventional IODKH method using multi-dimensional silver clusters. This scheme achieved quasi-linear scaling in one-, two-, and three-dimensional clusters with small prefactors, which enables the efficient calculation in as much CPU time a geometry optimization as that of the NR method. In addition, 20 metal complexes and heavier analogues of ethylene were optimized as more practical system, which indicated that the relativistic effect plays an important role in bonding. As well as those bonding, in terms of the harmonic frequency, the relativistic effect is non-negligibly contributed even at the forth row element.

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56 Chapter 4 Implementation of spin-dependent relativistic analytical energy gradient

4.1 Introduction

A practical computational method that includes high-level relativistic eﬀects has become one of the attracting goals for theoretical chemists. A fundamental treatment is based on a 4c Dirac theory, which adopts a one-particle Dirac operator.

For many electron systems, the 4c theory normally adopts the NR Coulomb in- teraction1 or a lower order of quantum electrodynamic effects such as the Breit2 or Gaunt3 interaction. Although the 4c relativistic calculations succeeded in accurately predicting molecular geometries4 and electronic/magnetic properties5,6 for small molecules, including at most 50 atoms, their application to more real- istic systems is difficult. Variational collapse arising from the existence of small- component Dirac spinors and/or positronic-state spinors has been a computational problem in the 4c calculations. The kinetic balance condition, which is a technique to resolve the variational problem, requires a large number of basis functions for the small-component spinors in comparison to those for the large-component spinors. To address the expensive computational cost in the 4c calculations, the basis spinor scheme has been proposed by Nakajima et al.7, which uses common expansion coefficients in each spinor. Recently, Kelley and Shiozaki have improved the algorithm using density fitting technique combined with a highly

57 parallelized implementation for a large-scale calculation and have extended the algorithm to an analytical nuclear gradients.8,9

An alternative approach is a 2c relativistic theory, which treats only electronic- state (or large-component) information. Recent development in 2c treatment for many-electron systems has exhibited high accuracy, even when compared to the

4c theory. In particular, the Dirac-exact approaches, which give equivalent results to the 4c treatments, have been proposed for the one-electron Dirac Hamiltonian such as NESC10, X2C11, and IODKH12 methods, and the many-electron Dirac-

Coulomb(-Gaunt) ones13–18. Expensive computations for the decoupling between electronic and positronic states were also accelerated by introducing the approximate local transformations19–26. Among the approximations, LUT by Seino and co-workers and DLU by Reiher and co-workers can accurately and eﬃciently reproduce the results of the original 2c treatments since they include not only the diagonal relativistic components but also oﬀ-diagonal ones between adjacent atoms for the one-electron Dirac Hamiltonian. The LUT scheme, furthermore, was successfully extended to the many-electron Dirac-Coulomb Hamiltonian22,23.

Analytical energy gradient schemes for relativistic theories are essential for the geometry optimization of heavy-element systems. The relativistic effects are roughly categorized into SF and SD effects. The SF effect directly contributes to bond contractions/elongations due to the relativistic shrinking/expansion of molecular orbitals4,27–29. The SD effect, which indirectly contributes to molecular structures due to energy-level splittings of degenerate orbitals, is non-negligible in heavy-element chemistry. For example, the bond lengths are shortened/elongated

30 31 by approximately 0.1 Å in At2 and 0.2 Å in both (113)H and (115)H by including the SD relativistic eﬀect. A contribution of a two-electron spin-orbit coupling is signiﬁcant to describe molecular properties such as magnetic shielding con- stants32,33, excitation energies34–36, and spectroscopic constants37. The analytical

58 energy gradient schemes for the SF relativistic eﬀect have been made available in

Dirac-exact 2c approaches such as NESC38–40, X2C41,42, and IODKH43, as well as in the approximate 2c approaches such as RA schemes44–46, RESC method47, and the

finite-order DKH methods48–54. Note that these three Dirac-exact 2c approaches give identical results although they need different procedures to obtain energies and properties. Recently, the analytical energy gradient for the 2c NESC method with a screened nucleus potential using effective nuclear charges has been proposed by Zou et al.55 The scheme has shown that a contribution of a spin-orbit coupling to bond lengths is small in general, but relatively large for van der Waals complexes. The progress of the analytical energy derivative in the relativistic quantum chemistry has been well described in recent review papers40,56,57.

From the computational point of view, the SF effect is more easily taken into account compared to the SD effect because modifications from the NR quantum chemical program are limited. In fact, the SF relativistic theory for the one-electron

Dirac Hamiltonian requires only the implementation related to the one-electron integrals. On the contrary, the implementation of the SD relativistic theory needs a considerable change of the NR program. For example, many variables such as orbital coeﬃcients, one- and two-electron integrals for the SD operators, and Fock matrices, should be treated as complex numbers. In consequence, the quantum chemical programs that can deal with the analytical energy gradient of the SD relativistic theories are very limited.

In this Chapter, the author implemented a computational program for the analytical energy gradient, as well as the energy at the SD relativistic level. The previous Chapter derived the formulation of the analytical energy gradient for the IODKH method with/without the LUT treatment, but the author implemented only the SF computation. The local approximation schemes are eﬀective to describe chemical bonds in both light- and heavy-element systems, especially for

59 comparatively strong bonds such as covalent bond, metallic bond, and ionic bond because the schemes reproduce the total energy obtained by corresponding conventional methods will less than 1 millihartree deviations. Note that an extension of the approximate local transformations to other Dirac-exact methods are possi- ble, whose derivation and implementation might be easy. The IODKH method denoted here is equivalent to IOTC12 method or BSS58,59 method. The present chapter is organized as follows. Implementation is explained in Section 4.2. Nu- merical assessments in terms of both accuracy and eﬃciency are presented in

Section 4.3. Finally, concluding remarks are given in Section 4.4.

4.2 Implementation

The theoretical aspects of (LUT-)IODKH is the same as Section 3.2. This section presents the implementation of evaluating the analytical energy gradient of the two-component (LUT-)IODKH method. To clarify the diﬀerence in implementation among NR, SF, and SD relativistic methods, we show the evaluation processes for all the methods in detail. Table 4.1 summarizes the procedures of energy and energy gradient calculations and data types. In the energy calculations, the procedures are divided into three parts: one-electron integral (including TNR, VNR,

σ · pVσ · p, matrix transformation, and relativistic transformation), two-electron integral, and SCF (including construction and diagonalization of Fock matrix). In the gradient calculations, the procedures are also divided into three part: one- electron integral, two-electron integral, and the energy derivative deﬁned in Eq.

(2.84).

In detail for the NR and SF calculations, all integrals, their derivatives without relativistic integrals, and transformations are evaluated in a real form. For the SF-

IODKH calculation, steps for the spin-free parts of the σ ·pVσ ·p integrals, matrix transformation, relativistic transformation, and these derivatives are added to

60 the procedure in NR. On the other hand, the LUT-IODKH method simpliﬁes the calculation using a relativistic transformation determined in each atom in the energy calculation.

For the SD calculations, the spin-dependent parts of the σ · pVσ · p integrals in a complex form are added to the SF procedure. Furthermore, the atomic orbital size increases to twice its original size in order to describe the α- and β-spin mixing.

Therefore, the relativistic transformation, SCF procedure, and these derivatives require complex calculations with twice the number of basis sets.

4.3 Numerical assessments 4.3.1 Computational details

This section examines the performance of the analytical energy gradients for the

2c IODKH and LUT-IODKH methods in terms of accuracy and eﬃciency. As the

2c relativistic Hamiltonian, the one-electron SD-IODKH Hamiltonian and two- electron Coulomb interaction, denoted by SD-IODKH, were adopted. In the LUT scheme, the threshold for cutoﬀ of relativistic interaction, τ, was set to 3.5 Å. For comparison, the 4c Dirac-Coulomb, (LUT-) SF-IODKH, and NR methods were also employed.

Twelve closed-shell diatomic molecules containing ﬁfth- and sixth-period atoms were calculated as numerical tests: InH, SnO, Sb2, HI, I2, AuH, Au2, TlH,

PbO, Bi2, HAt, and At2. Five van der Waals complexes, Ne2,Ar2,Kr2,Xe2, and ff Hg2, were used in order to test cuto radius. Furthermore, multi-dimensional silver clusters were employed to investigate the efficiency in LUT-SD-IODKH. = , ··· , The Ag-Ag bond length in one-dimensional (1D) Agn clusters (n 2 10) / was fixed at 2.706 Å. In two- and three-dimensional (2D 3D) Agn clusters, an experiment lattice constant of 4.806 Å with a face-centered cubic structure was

60 adopted . The total numbers of atoms in the 2D Agn clusters were 5, 8, 11, 13,

61 Table 4.1 Procedures of energy and energy gradient calculations and data types in NR, (LUT-)SF-IODKH, and (LUT-)SD-IODKH. SF SD NR IODKH LUT-IODKH IODKH LUT-IODKH Energy One-electron integral TNR real real real real real VNR real real real real real σ · pVσ · p – real real complex complex Matrix transformation X – real real (atom) real real (atom) Relativistic transformation – real real complex complex 62 / Two-electron integral 1 rij real real real real real SCF Fock real real real complex complex Diagonalization real real real complex complex Energy gradient Derivative of one-electron integral TNR real real real real real VNR real real real real real σ · pVσ · p – real real complex complex Matrix transformation X – real – real – Relativistic transformation – real real complex complex / Derivative of two-electron integral 1 rij real real real real real Summation for total gradient real real real complex complex 14, 17, 18, 23, 25, 28, 32, 39, 41, and 50, which correspond to 1 × 1, 1 × 2, 1 × 3,

2 × 2, 1 × 4, 1 × 5, 2 × 3, 2 × 4, 3 × 3, 2 × 5, 3 × 4, 3 × 5, 4 × 4, and 4 × 5 unit cells, respectively. The total numbers in the 3D Agn clusters were 14, 23, 32, 41, and 53, which correspond to 1×1×1, 1×1×2, 1×3×1, 1×1×4, and 1×2×3, respectively. − For a practical calculation, fac Ir(ppy)3 was also examined. All the NR, SF, and SD calculations were performed at the RHF, RHF, and GHF levels, respectively. For the diatomic molecules and Agn clusters, the Sapporo- DZP-201261 and DKH3-Gen-TK/NOSeC-V-TZP62–64 basis sets with uncontracted forms were employed in the H atom and other atoms, respectively. The basis sets used for van der Waals complexes were DKH3-Gen-TK/NOSeC-V-TZP in an uncontracted form for noble gas atoms (Ne2,Ar2,Kr2, and Xe2). For Hg2, two basis sets were adopted; DKH3-Gen-TK/NOSeC-V-TZP in an uncontracted form with a large exponent (> 1.2 × 109) and SARC-DKH in an uncontracted form with

< . × 6 − a moderate exponent ( 1 8 10 ). In fac Ir(ppy)3, the contracted DKH3-Gen- TK/NOSeC-V-TZP for Ir and cc-pVDZ65 for H, C, and N atoms were adopted.

− 66 Note that in the NR calculation for fac Ir(ppy)3, the basis set for Ir, TZP , was adopted. The analytical energy gradient for the (LUT-)SD-IODKH method was implemented in the GAMESS program67. The DIRAC12 program was utilized for the 4c calculations68.

4.3.2 Accuracy of SD-IODKH and LUT-SD-IODKH

This section examines the accuracy of the analytical energy gradient method for the 2c IODKH method with/without the LUT scheme. Table 4.2 shows the analytical and numerical gradient values (in hartree/bohr) at equilibrium lengths of re obtained by NR, SD-IODKH, LUT-SD-IODKH, and 4c at the HF level. Here,

= 31 the experimental bond lengths were used for re, namely, re 1.742 Å for HAt ,

69 70 71 3.046 Å for At2 , 1.5324 Å for AuH , and 2.4719 Å for Au2 . Furthermore, 1.5re

63 and 2.0re for AuH and HAt were also calculated. The numerical gradient values were obtained using a ﬁve point numerical diﬀerentiation method at the 0.001

Å step size. The inherent accuracy of this numerical diﬀerentiation is a fourth-

− order accuracy. The convergence threshold of the SCF calculation is 5.0 × 10 6 for the density changes. A quadruple precision routines for evaluating one-electron integrals including overlap and kinetic energy, and for diagonalizing the integral matrices were used to keep accuracy in the Hess’ numerical technique72. The deviation of the analytical gradient with respect to the numerical gradient, Δgana, shows the reliability of the present implementation.

The maximum absolute deviations of NR, SD-IODKH, LUT-SD-IODKH, and

4c are 0.000002, 0.000013, 0.000008, and 0.000002, respectively. The deviations of

SD-IODKH and LUT-SD-IODKH are slightly larger than that of NR. This is due to the resolution-of-identity (RI) approximation, which is essential for Hess’ numerical calculation72 in the IODKH method. Considering these theoretical demands, the results indicate the correct implementation of the present analytical energy gradients in SD-IODKH and LUT-SD-IODKH within the Hess approximation.

Table 4.3 presents the bond lengths (Å) of diatomic molecules containing

ﬁfth- and sixth-period elements calculated by the NR, SF-IODK/C, SD-IODKH,

LUT-SD-IODKH, and 4c methods. The deviations of the NR, SF-IODKH, and

SD-IODKH methods from 4c, Δr4c, indicates the fully relativistic effect, the SD relativistic effect in one-electron terms, and the fully relativistic effect in two- electron terms, respectively. The deviation of SD-IODKH from SF-IODKH, ΔrSD, is shown in square brackets, representing the SD relativistic effect in one-electron terms. The deviation of LUT-SD-IODKH from SD-IODKH, ΔrLUT, indicates the accuracy of the LUT scheme.

The fully relativistic eﬀect on bond lengths, Δr4c of NR, is mostly positive whereas I2,Bi2, and At2 are opposite. The absolute values are large in AuH and

64 / / Table 4.2 Analytical and or numerical gradient values (hartree bohr) of HAt, At2, AuH, and Au2 calculated by NR, SD-IODKH, LUT-SD-IODKH, and 4c at the HF level. NR SD-IODKH LUT-SD-IODKH 4c Molecules Length g ΔggΔggΔggΔg

AuH 1.0re 0.082251 0.000001 0.012531 0.000013 0.012518 0.000007 0.011711 0.000000 65 1.5re 0.033559 0.000000 0.054765 0.000007 0.054768 0.000001 0.054581 0.000000 2.0re 0.031298 0.000000 0.036769 0.000000 0.036770 0.000002 0.036850 0.000000 HAt 1.0re 0.011267 0.000002 0.008891 0.000009 0.008908 0.000001 0.009392 0.000000 − 1.5re 0.067515 0.000001 0.054690 0.000005 0.054686 0.000000 0.055145 0.000001 2.0re 0.037875 0.000000 0.000126 0.000000 0.000126 0.000008 0.031849 0.000000 Au2 1.0re 0.079816 0.000002 0.028908 0.000006 0.028880 0.000004 0.029315 0.000000 − − At2 1.0re 0.025557 -0.000001 0.007775 0.000002 0.007796 0.000000 0.008351 0.000002 Au2: 0.263 Å for AuH and 0.330 Å for Au2. The large deviations are reduced by the SF relativistic eﬀect. Actually, Δr4c of SF-IODKH are 0.007 Å for AuH and

0.008 Å for Au2. This indicates that AuH and Au2 have an s-bonding nature formed by each s valence orbital, which is largely contracted by the relativistic effect. However, the absolute values |Δr4c| of SF-IODKH exhibit large differences, − − especially in Bi2 and At2: 0.095 Å for Bi2 and 0.129 Å for At2. On the other hand, |Δr4c| of SD-IODKH are small in all molecules. This indicates that the SD relativistic effect in heavy-element systems is important when the bond has non-zero angular momentum such as p, d, and f orbitals. The contributions of two-electron relativistic effects can be small in bond lengths of the systems used here, although the contributions sometimes can be important.55 Fur- ff thermore, TlH has a negative value by the SD relativistic e ect, even though Bi2, ff ff HAt, and At2 have positive values. The SD relativistic e ect gives two di erent contributions to bond length: a splitting of degenerate orbitals and an overlap of orbitals by a second-order spin-orbit interaction. The former is when p, d, { } { } { } and f orbitals are split into p1/2,p3/2 , d3/2,d5/2 , and f5/2,f7/2 , respectively.

The orbitals with less (more) total angular momentum are stabilized and contracted (destabilized and expanded). The latter contributes to the stabilization of a molecule by an overlap of orbitals with the same total angular momentum σ π π σ and spatial symmetry, i.e., 1/2g and 1/2g or 1/2u and 1/2u, which are split by a spin-orbit interaction. In the case of TlH, the former contribution is dominant σ because two electrons occupy the 1/2g orbital, which is stabilized by the SD rela- ff tivistic e ect. In the case of Bi2, HAt, and At2, the latter contribution is dominant because electrons occupy bonding and/or anti-bonding orbitals. In this situation, the SD relativistic effect stabilizes a molecule largely when the bond length is long, although the opposite is true for the SF relativistic effect. As a result, the bonds are elongated by the SD relativistic effect.

66 In all cases, ΔrLUT is considerably small. This represents that the LUT treatment can suﬃciently describe both SF and SD relativistic eﬀects for estimating the bond lengths.

Note that in systems whose bond lengths are longer such as van der Waals complexes, the cutoﬀ radius including nearest neighbor atoms is set carefully in order to keep accuracy. Table 4.4 shows the bond lengths (Å) of van der Waals complexes obtained by the LUT-IODKH and IODKH methods at the MP2 level.

Two cutoﬀ radii set to 3.5 and 5.0 Å. In noble gas dimers, the deviations of LUT-

IODKH from IODKH with 5.0 Å of cutoﬀ radius were within 0.001 Å. On the / other hand, in Hg2, the deviations were 0.013 for DKH3-Gen-TK NOSeC-V-TZP and 0.006 for SARC-DKH. The large deviations are related with the large exponent values because the deviations increase when the exponent value increase. One reason of the deviations can be due to the dropping terms in the LUT scheme.

Another reason is due to the numerical error by large values in the integrals when we adopted a basis sets with an extremely large exponent value. These results indicate that van der Waals radius should be included at least in cutoﬀ radius for keeping accuracy when we use the LUT scheme in a geometry optimization.

4.3.3 Computational cost of (LUT-)SD-IODKH method

This subsection examines the computational cost of the analytical energy gradient for the SD-IODKH method with/without the LUT scheme on 1D, 2D, and 3D silver clusters. Figure 4.1 shows the system-size dependence of the central processing unit (CPU) time for the analytical gradient of the one-electron integrals in 1D, 2D, and 3D silver clusters obtained by the SD-IODKH and LUT-SD-IODKH methods.

The horizontal axis indicates the total numbers of the atoms n. The vertical axis indicates the CPU time in seconds. A Xeon E5-2690/2.90 GHz (Octa core) processor on a single core was utilized to measure the CPU time. In 1D cluster, the CPU time

67 Table 4.3 Bond lengths (Å) of diatomic molecules containing ﬁfth- and sixth-period elements calculated by the NR, SF-IODKH, SD-IODKH, LUT-SD-IODKH, and 4c methods at the HF level. IODKH NR 4c Molecule SF SD LUT-SD rNR Δr4c rSF Δr4c rSD Δr4c ΔrSD rLUT ΔrLUT r4c InH 1.856 0.012 1.845 0.001 1.843 −0.001 −0.002 1.843 0.000 1.844 SnO 1.798 0.008 1.789 −0.001 1.790 0.000 0.001 1.790 0.000 1.790 68 − Sb2 2.460 0.018 2.436 0.006 2.443 0.001 0.007 2.443 0.000 2.442 HI 1.608 0.005 1.600 −0.003 1.603 0.000 0.003 1.603 0.000 1.603 − − I2 2.675 0.003 2.664 0.014 2.679 0.001 0.015 2.679 0.000 2.678 AuH 1.828 0.263 1.572 0.007 1.569 0.004 −0.003 1.568 −0.001 1.565 − − Au2 2.924 0.330 2.602 0.008 2.593 0.001 0.009 2.593 0.000 2.594 TlH 1.932 0.069 1.890 0.027 1.859 −0.004 −0.031 1.859 0.000 1.863 PbO 1.890 0.019 1.864 −0.007 1.873 0.002 0.009 1.873 0.000 1.871 − − Bi2 2.647 0.015 2.567 0.095 2.667 0.005 0.100 2.667 0.000 2.662 HAt 1.711 0.001 1.683 −0.027 1.712 0.002 0.029 1.712 0.000 1.710 − − At2 2.890 0.081 2.842 0.129 2.976 0.005 0.134 2.978 0.002 2.971 Table 4.4 Bond lengths (Å) of van der Waals complexes obtained by the LUT-IODKH and IODKH methods at the MP2 level. LUT-IODKH Moleucle τ = 3.5 τ = 5.0 IODKH Δ

Ne2 3.278 3.278 3.278 0.000 Ar2 3.913 3.913 3.913 0.000 Kr2 4.170 4.087 4.087 0.000 Xe2 4.613 4.559 4.558 0.001 Hg2 3.495 3.495 3.501 0.006 Hg2 3.509 3.551 3.538 0.013 a) Results of the DKH3-Gen-TK/NOSeC-V-TZP basis set in an uncontracted form. b) Results of the SARC-DKH basis set in an uncontracted form.

. in SD-IODKH is long; the scaling is O(n2 98), which corresponds to an order in a matrix multiplication. On the other hand, the LUT scheme can drastically reduce

. the CPU time; the scaling is quasi-linear, O(n1 33). The results also represent that the LUT-SD-IODKH method achieves quasi-linear scaling for 2D and 3D clusters:

. . O(n1 35) for 2D and O(n1 36) for 3D.

Table 4.5 shows the CPU time for an analytical energy gradient calculation of Ag10. The NR, SF-IODKH, LUT-SF-IODKH, SD-IODKH, and LUT-SD-IODKH methods were adopted. The calculation is divided into six steps: namely, one- electron integrals including the 2c transformation (OEI), an initial guess (Guess) by Huckel¨ scheme, TEI, the SCF procedure, derivatives of OEI including the 2c transformation (dOEI), and dTEI. The percentage of CPU time in each step is given in parentheses.

In single-point energy calculations including OEI, Guess, TEI, and SCF, SF- and SD-IODKH for OEI require long CPU times in comparison with NR: 0.12 s for NR, 75.72 s for SF-IODKH, and 1487.83 s for SD-IODKH. The long CPU times for OEI are reduced by the LUT schemes: 4.47 s for LUT-SF-IODKH and 24.69 s for LUT-SD-IODKH. In the SCF step, the CPU time at the SD level is ≥10 times as much as that at the SF level. This is because the GHF method for the SD eﬀect

69 7000 70 (a) (b) 6000 60 SD-IODKH 2d 5000 LUT-SD-IODKH 50 3d 4000 40 O(n1.36) 3000 O(n2.98) 30 CPU time [s] CPU time [s] 2000 20 O(n1.35) 1000 10 O(n1.33) 0 0 020246810 0 40 60 n n Figure 4.1 System-size dependence of CPU time (s) of the one-electron analytical energy gradient obtained with the SD-IODKH and LUT-SD-IODKH methods: (a) 1D Agn cluster / and (b) 2D and 3D Agn cluster. A Xeon E5-2690 2.90 GHz (Octa Core) processor was used on a single core.

requires a calculation with a complex form and twice as large basis sets as the

RHF one.

In gradient calculations, including dOEI and dTEI, SF- and SD-IODKH for dOEI demand large CPU time in comparison to NR: 1.11 s for NR, 2652.28 s for SF-

IODKH, and 61247.55 s for SD-IODKH. The LUT scheme reduces the CPU times:

129.36 s for LUT-SF-IODKH and 263.35 s for LUT-SD-IODKH. Furthermore, in dTEI, the CPU times at the SD level are longer than those at the SF one. This is because additional CPU times are required to multiply the density matrix with the

GHF formalism to the derivative of TEI, although the integral calculations are the same. These results indicate that the LUT scheme resolves one of the bottlenecks, i.e., long CPU times in the OEI and dOEI steps.

− 4.3.4 Application in fac Ir(ppy3)

This subsection discusses the performance of the LUT-SF-IODKH and LUT-SD-

IODKH methods in fac-Ir(ppy)3. It should be noted that this molecule cannot be treated without the LUT technique for the relativistic calculation. Table 4.6 shows

70 Table 4.5 CPU time for six steps in the restricted or general Hartree-Fock calculation obtained with NR, SF-IODKH, LUT-SF-IODKH, SD-

IODKH, and LUT-SD-IODKH in Ag10 linear cluster: OEI, initial guess (Guess) calculated with the Huckel¨ scheme, TEI, SCF, dOEI, and dTEI. A Xeon E5-2690/2.90 GHz processor (Octa core) was used on a single core. NR SF-IODKH LUT-SF-IODKH SD-IODKH LUT-SD-IODKH Time (s) % Time (s) % Time (s) % Time (s) % Time (s) % 71 OEI 0.12 (0.00) 75.72 (1.35) 4.47 (0.15) 1487.83 (1.95) 24.69 (0.18) Guess 20.55 (0.75) 20.63 (0.37) 19.52 (0.67) 20.45 (0.03) 20.39 (0.15) TEI 675.33 (24.66) 725.13 (12.91) 687.98 (23.65) 690.26 (0.91) 693.53 (5.04) SCF 817.20 (29.84) 904.83 (16.11) 814.73 (28.01) 10507.45 (13.79) 10687.16 (77.74) dOEI 1.11 (0.04) 2652.28 (47.21) 129.36 (4.45) 61427.55 (80.64) 263.35 (1.92) dTEI 1224.57 (44.71) 1238.95 (22.06) 1253.02 (43.07) 2046.20 (2.69) 2058.81 (14.98) Total 2738.88 (100.00) 5617.54 (100.00) 2909.08 (100.00) 76179.74 (100.00) 13747.93 (100.00) Table 4.6 Geometric parameters of Ir(ppy)3 obtained by NR, LUT-SF-IODKH and LUT-SD- IODKH at the Hartree-Fock level. Parameters NR SF SD ΔSF ΔSD ΔRel Ir-N 2.225 2.198 2.149 −0.027 −0.049 −0.076 Ir-C 2.031 2.051 1.964 0.020 −0.087 −0.067 C-Ir-N 172.2 175.4 173.23.2 −2.21.0 C-Ir-C 95.396.992.51.6 −4.4 −2.8 N-Ir-N 97.796.596.5 −1.20.0 −1.2 the geometric parameters calculated at the Hartree-Fock level of theory. A-B and

A-B-C denote bond lengths between atoms A and B and angles among atoms A, B, and C, respectively. ΔSF is the diﬀerence of the geometrical parameters between

NR and LUT-SF-IODKH, indicating the SF relativistic eﬀect on the geometry.

ΔSD is the diﬀerence of the geometrical parameters between LUT-SF-IODKH and

LUT-SD-IODKH, indicating the SD relativistic eﬀect. ΔRel is the summation of

ΔSF and ΔSD, indicating that total relativistic effect. Note that we used the nearest neighboring atom from the central metal Ir to estimate the bond lengths and angles. ff In the bond lengths of fac-Ir(ppy)3, the SF relativistic e ect is smaller than the SD relativistic effect. For Ir-N and Ir-C, {ΔSF, ΔSD} are {−0.027, −0.049} and {0.020,

−0.087} in Å, respectively. On the other hand, the three bond angles, C-Ir-N,

C-Ir-C, and N-Ir-N, are slightly aﬀected by both SF and SD eﬀects. For C-Ir-N,

C-Ir-C, and N-Ir-N, {ΔSF, ΔSD} are {3.2, −2.2}, {1.6, −4.4}, and {−1.2, 0.0} in degrees, respectively.

4.4 Conclusion

The present study develops the analytical energy gradient for the GHF method based on the 2c IODKH method with/without the eﬃcient LUT scheme in a 2c relativistic transformation. The accuracy of the present method was assessed using several diatomic molecules containing ﬁfth- and sixth-period elements.

72 The optimized bond lengths are in good agreement with those of the 4c method.

Furthermore, the deviations of the LUT scheme are less than 0.003 Å in bond length. The eﬃciency of the LUT scheme was also examined using 1D, 2D, and

3D silver clusters. The CPU time for a one-electron relativistic transformation scales quasi-linear in these clusters. A practical calculation using a fac-Ir(ppy)3 molecule was performed to estimate the accuracy and efficiency of the proposed method. The results show that the SD relativistic effect shortens the bond length by approximately 0.1 Å. In addition, although the SD relativistic program is much more complicated than that of NR, the LUT-SD-IODKH scheme demands only about twice as much CPU time as NR. The present analytical energy gradient method for LUT-IODKH with the SD relativistic effect was found to be applicable to practical systems.

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