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Title The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics

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Author Stuck, David

Publication Date 2015

Peer reviewed|Thesis/dissertation

eScholarship.org Powered by the California Digital Library University of California The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics

David Edward St¨uck

A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy

Chemistry

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Martin Head-Gordon, Chair Professor William H. Miller Professor Alexis T. Bell

Summer 2015 The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics

Abstract

The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics by David Edward St¨uck Doctor of Philosophy in Chemistry University of California, Berkeley Professor Martin Head-Gordon, Chair

The work herein is concerned with developing computational models to understand molecules. The underlying theme of this research is the reassessment of zeroth-order approximations for higher-level methods. For second-order Møller-Plesset theory (MP2), qualitative failures of the Hartree-Fock orbitals in the form of spin contamination can lead to catastrophic errors in the second order energies. By working with orbitals optimized in the presence of correlations, orbital-optimized MP2 can ﬁx the spin contamination problem that plague radicals, aromatics, and transition metal complexes. In path integral Monte Carlo for vibrational energies, the zeroth-order propagator is typically chosen to be the most general possible, the free particle propagator; we chose to be informed by the molecular structure we have already attained and apply a propagator based on the harmonic modes of the molecule, improving sampling eﬃciency and our Trotter approximation. i

Contents

Contents i

List of Figures iii

List of Tables vi

1 Introduction 1 1.1 Background ...... 1 1.2 Electron Correlation ...... 5 1.3 Statistical Quantum Thermodynamics ...... 8 1.4 Outline ...... 9 1.5 Additional Work ...... 11

2 On the Nature of Electron Correlation in C60 13 2.1 Introduction ...... 13 2.2 Results and Discussion ...... 15 2.3 Conclusion ...... 21

3 Regularized Orbital-Optimized MP2 22 3.1 Introduction ...... 22 3.2 Theory ...... 26 3.3 Results and Discussion ...... 28 3.4 Conclusion ...... 34

4 Stability Analysis without Analytical Hessians 35 4.1 Abstract ...... 35 4.2 Introduction ...... 35 4.3 Method ...... 37 4.4 Results ...... 39 4.5 Conclusions ...... 44 4.6 Acknowledgements ...... 45 ii

5 Exponential Regularized OOMP2 for Dissociations 46 5.1 Introduction ...... 46 5.2 Theory ...... 47 5.3 Results ...... 48 5.4 Conclusion ...... 50 5.5 Acknowledgements ...... 51

6 Regularized CC2 53 6.1 Introduction ...... 53 6.2 Computational Methods ...... 55 6.3 Results and Discussion ...... 56 6.4 Conclusion ...... 58 6.5 Acknowledgements ...... 59

7 Path Integrals for Anharmonic Vibrational Energy 60 7.1 Introduction ...... 60 7.2 Theory ...... 61 7.3 Results and Discussion ...... 65 7.4 Conclusion ...... 70 7.5 Acknowledgements ...... 71

References 74 iii

List of Figures

2.1 Natural orbital occupation numbers of UHF spincontaminated singlets for C36 i i and C60. Orbitals are numbered as a fraction of the total π space (i.e. 36 or 60 th for the i π orbital of C36 or C60 respectively)...... 17 2.2 Unpaired electron density of singlet (top) and triplet (bottom) C60 (left) and C36 (right) plotted at isovalue 0.006 A˚−3, with shading determined by the sign of the spin density as described in the text...... 18

2.3 Natural orbital occupation numbers from O2 calculations on singlet C36 and C60. Orbitals are numbered as a fraction of the total π space...... 20

3.1 Li2 dissociation curve for MP2 using restricted and unrestricted orbitals and for OOMP2 with a cc-pVDZ basis. RMP2 dissociates incorrectly and UMP2 distorts the equilibrium description while OOMP2 gets the best of both worlds by continuously connecting the two regimes, albeit with a kink due to a slight discontinuous change to the orbitals upon unrestriction...... 24 3.2 Dependence of the OOMP2 energy (the standard RIMP2 energy without singles contribution) on the two occupied-virtual mixing angles for the hydrogen molecule in the STO-3G basis at 0.74 A.˚ The region around the RHF minimum at (0◦, 0◦) is well behaved...... 29 3.3 Dependence of the OOMP2 energy on the two occupied-virtual mixing angles for the hydrogen molecule in the STO-3G basis at 4.0 A.˚ Divergences appear for orbitals with unfavorable HF energies but very large negative MP2 energy due to HOMO-LUMO energy coalescence. There is a stable minimum near the UHF solution around (140◦, 40◦), but it is not the global minimum due to the divergences. 29 3.4 δ-OOMP2 orbital energy surface with level shifts, δ, of 100 mEh (left) and 400 mEh (right) for the hydrogen molecule in the STO-3G basis at 4.0 A.˚ The level shift of 400 mEh has restored the solution near the UHF orbitals to be the global minimum and has removed the divergences...... 30 3.5 RMS error on the G2 test set of atomization energies for δ-OOMP2, δ-RIMP2, and correlation scaled RIMP2 and OOMP2 as a function of the regularization (2) parameter δ (bottom) or scaling parameter, s, given by Es = E0 + sE (top). . 31 iv

3.6 RMS errors of δ-OOMP2 relative to standard RIMP2 on various test sets. With- out regularization OOMP2 performs worse than RIMP2 for the G2 and S22 test sets but a level shift of 400 mEh improves δ-OOMP2 over RIMP2 and unregularized OOMP2 for all test sets...... 33 3.7 (a) Bond length errors vs. CCSD(T) of OOMP2, δ-OOMP2, and MP2 for ﬁve small radicals. (b) Harmonic frequencies plotted against CCSD(T) for the same ﬁve radicals. R2 values for frequencies are 0.979, 0.998, and -0.003 for OOMP2, δ-OOMP2, and MP2 respectively. MP2 and reference CCSD(T) values taken from the work of Bozkaya[100]...... 33

4.1 Potential curves (green for unrestricted and red for restricted, where it differs from unrestricted) for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix (purple for internal stability of the unrestricted solution, blue for external stability of the restricted solution, where it differs from unrestricted) at the Hartree-Fock (HF) level. The lowest energy solution changes character from restricted to unrestricted when the former becomes unstable...... 40 4.2 Potential curves for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix using orbital-optimized MP2 (OOMP2) in the cc-pVDZ basis. The format follows Figure 6.1. OOMP2 behaves qualitatively differently from HF (see Figure 4.1). The restricted solution is stable (positive eigenvalue) to spin-polarization at all bond-lengths, and a distinct stable unrestricted solution appears at partially stretched bondlengths...... 41 4.3 The dependence of the OOMP2 energy of H2 in a minimal basis on the spin polarization angle (see text for definition) at a series of bond-lengths around the critical value at which the character of the lowest energy solution changes. There are two local minima, one restricted and one unrestricted, at these bond-lengths, and at the critical bond-length the nature of the lowest energy solution switches discontinuously...... 42 4.4 Potential curves for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix using regularized orbital optimized MP2 (δ-OOMP2) in the cc-pVDZ basis. The format follows Figure 6.1. δ-OOMP2 behaves qualitatively differently from OOMP2 (see Figure 4.2), but is similar to HF (see Figure 4.1). The restricted solution becomes unstable at a critical bond-length, beyond which the unrestricted solution is lowest in energy...... 43 4.5 The dependence of the δ-OOMP2 energy of H2 in a minimal basis on the spin polarization angle (see text for definition) at a series of bond-lengths around the critical value at which the character of the lowest energy solution changes. For any given bond-length there is only one local minimum, which changes character from restricted to unrestricted at the critical bond-length...... 44 v

5.1 Dissociation curve of ethane in an aug-cc-pVTZ basis. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals...... 48 5.2 Dissociation curve of ethene in an aug-cc-pVTZ basis. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals...... 49 5.3 Dissociation curve of ethane in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals...... 50 5.4 Dissociation curve of ethene in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals...... 51 5.5 Dissociation curve of ethyne in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals...... 52

6.1 δ-CC2 RMSE for various ground state test sets divided by RIMP2 RMSE on the same sets for various values of δ...... 56 6.2 Ozone symmetric dissociation curve at angle 142.76◦ for CC2 with regularization parameters 0, 100, 150, and 200 mEh and CCSD in an aug-cc-pVTZ basis. . . . 57

2 7.1 Plot of h∆V iλ as a function of lambda for sampling with P = 200. R values for the ﬁts are 0.996, 0.984, and 0.998 for the monomer, dimer and sulfate cluster respectively...... 66

7.2 Errors in anharmonicity on a Morse potential of H2 using the free particle and harmonic propagator full energy approaches as well as thermodynamic integration. The Morse potential is parameterized with De = 0.176 and a = 1.4886. . . 67 7.3 Errors in anharmonicity for H2O monomer using FFP, HOP, and TI. Reference values from direct grid ﬁtting of CCSD(T) calculation[192]...... 69

7.4 Errors in anharmonicity for H2O dimer using HOP and TI. The reference here is based on CCSD(T) electronics, but only VPT2 for nuclear energies[193]. . . . . 70

7.5 Anharmonic ZPE for Sulfate 3 H2O cluster using TI...... 71 7.6 Relative Energies of Sulfate 3 H2O clusters using TI...... 72 7.7 Relative Energies of Sulfate 4 H2O clusters using TI...... 72 7.8 Relative Energies of Sulfate 5 H2O clusters using TI...... 73 7.9 Relative Energies of Sulfate 6 H2O clusters using TI...... 73 vi

List of Tables

2.1 Quantiﬁcation of spin symmetry breaking in fullerene systems. ∆E = ERHF − EUHF and number of unpaired electrons as described in the text...... 16 2.2 Calculated Etriplet − Esinglet from restricted and unrestricted HF and MP2 compared to experimental literature values of C60 and C36...... 19 5.1 Root mean square error (RMSE) in kcal/mol for σ-OOMP2 with various values of σ and δ-OOMP2 with the recommended parameterization of 400 mEh. . . . . 49 6.1 Excited state errors on the Thiel test set for regularized δ-EOM-CC2 vs. CC3 values in TZVP and aug-cc-pVTZ basis for varying values of δ. All errors in eV. 58 6.2 Excited state errors on the Wiberg test set for regularized δ-EOM-CC2 values in a 6-311(3+,3+)G** basis for varying values of δ vs. accurate experimental values. All errors in eV...... 58

7.1 Table of the number of samples required to reduce sampling error to within 5% of calculated anharmonic ZPE...... 68

7.2 Anharmonic ZPE for the lowest energy Sulfate-3 H2O cluster with CC-VSCF on MP2/TZP[194], TOSH and VPT2 on B3LYP/6-31+G*[179], and PIMC TI on polarizable force ﬁeld...... 69 vii

Acknowledgments

I thank Ellie for supporting me in every way while working on my PhD. I thank Martin for being engaging when I was learning new things, open when I had ideas, and kind when I made mistakes. I thank the group for being good colleagues, classmates, and friends; I don’t know how I’d have made it 5 years with out you all, Paul, Sam, Eric, Yuezhi, Jules, Narbe, Nick, Jonathan, Rostam, Westin, Kristi, Evgeny, Tom, Daniel, Fran, Shaama. 1

Chapter 1

Introduction

1.1 Background

The First Question: Why? Why do we study chemistry? We do so because it is useful and it is hard. It is useful because chemical processes take place on the energy scale of life. Interesting molecules change from stable to reactive depending on their environment, allowing humans to manipulate them into different forms. It is hard because it takes place on a length scale that is small enough to have no intuitive understanding by way of human senses. We can see a ball rolling, a lion hunting prey, or even an enzyme unzipping a DNA helix and we don’t need equations to at least get a sense of what’s happening. But even seeing the density of a benzene molecule as in recent experiments gives us no intuition about its behavior and reactivity, which brings us to the point: we can only ever understand chemistry through models. For many years these models have been purely heuristic or quantitative but disconnected (we could measure acidity and heat of formation, but had no link between the two models). In the early 20th century, the development of quantum mechanics revolutionized our understanding of the behavior of systems on an atomic length scale. We finally had the tools to describe chemical processes starting from the atomic pieces of electrons and nuclei. This knowledge allowed us to develop assessment tools that rely on the electronic and vibrational structure of a molecule (spectroscopy, NMR, . . . ) rather than simple reactivity measures. While spectra are useful, we can still (by definition really) only ever measure projections of the molecular wave function and can only do so for chemical systems we can access, i.e. have produced and stabilized in a measurable form and amount. The theoretician, aided by quantum theory, can now calculate the properties of molecules directly and quantitatively but must make major approximations to do so in a tractable way. In this way, the tools of experimentalist and theoretician are complementary: experimentalists know that what they do is real but can’t be sure of what they do, while computational chemists can know exactly what they’re modeling but not be sure of whether it’s modeling reality. CHAPTER 1. INTRODUCTION 2

The Second Question: What? What is quantum mechanics as it relates to molecular processes? Quantum mechanics (QM) is fundamentally different from classical mechanics (CM) in that particles are treated as fields to be characterized by multi-dimensional wave functions in a Hilbert spacerather than the discrete points (xi, pi) of CM. On large size or energy scales these fields become so localized as to be effectively points obeying CM, which is the correspondence principle. Molecular quantum mechanics is primarily governed by the Hamiltonian, given in atomic units by, ˆ ˆ ˆ ˆ ˆ ˆ Hmol = Tn + Te + Vnn + Vne + Vee n e n n e e X 1 X 1 X ZiZj X X Zi X 1 (1.1) = ∇ˆ 2 + ∇ˆ 2 + − + 2M Ri 2 ri ˆ ˆ ˆ |rˆ − rˆ | i i i j>i |Ri − Rj| i j |Ri − rˆj| j>i i j

− it Hˆ The dynamics of the system are determined by the time propagator e ~ . Considering the density operator, e−βHˆ , shows that at low enough temperatures, systems will be domi- nated by the lowest eigenstate of Hˆ , the ground state, assuming excited states are higher in energy than thermal ﬂuctuations (which for the electronic part will almost certainly be true in standard conditions). It is also interesting to consider molecules interacting with photons of light, which is not ˆ included in Hmol, but as far as this work is concerned we will focus on characterizing the molecular structure itself as a required ﬁrst step to studying interactions of light and matter.

The Third Question: How? ˆ Given Hmol for a chemical system, how do we determine the molecular wavefunction and its properties? The ﬁrst approximation that we will assume for all of our modeling herein is that the electronic states are not coupled through the kinetic energy operator–the Born- Oppenheimer approximation. We deﬁne, ˆ ˆ ˆ Hnuc = Tn + Vnn (1.2) ˆ ˆ ˆ ˆ Hel = Te + Vne + Vee

ˆ If we solve for the eigenvectors of Hel(r, R) over the space of electrons for given R, call el them φi(r, R) with eigenvalues Ei , we can without assumption write the wavefunction as a linear combination of the eigenvectors with R-dependent coeﬃcients giving Ψ(r, R) = χi(R)φi(r, R). The approximation we make then is that, ˆ ˆ hφj(r, R)|Hnuc|χi(R)i|φi(r, R)i = Hnuc|χj(R)iδi,j (1.3) CHAPTER 1. INTRODUCTION 3

ˆ Thus our nuclear wavefunctions are eigenfunctions of Hnuc plus a term that represents the electronic potential energy surface. We will often neglect the quantum nature of the nuclei and simply add the nuclear repulsion to our electronic energies or account for quantum effects through the harmonic approximation to Eel(R) to account for the fact that even the lowest energy vibrational state has zero-point energy (ZPE). We will also discuss higher level approximations later in this work. Now that we’ve waved away the nuclear part of the equations for the moment, we can discuss how we go about solving for the eigenvalues of the Hamiltonian. A simple approach is to choose a form for our wave function and minimize the energy (expectation value of Hˆ ) with respect to the wavefunction parameters. This will give us an upper bound to the energy by the variational principle–since all wavefunctions can be written as linear combinations of orthogonal eigenvectors, the expectation value of an operator can be written as a weighted average of its eigenvalues which is always greater than or equal to the lowest such eigenvalue (assuming bounded from below). The simplest many electron wavefunction that satisfies Fermi statistics is an antisymmetrized product of one electron functions called a Slater determinant. The remaining question then is how will we construct these one electron functions? It turns out that the realization that we should use Gaussians because it simplifies the mathematics was a foundational one for computational chemistry [1–3]. Now we pick a one electron basis, the atomic orbitals or AO’s, which are combinations of radial Gaussians and spherical harmonics that are preoptimized for chemical systems. Each occupied orbital, φi, will now be expressed in the basis of AO’s, χµ, as,

|ii = Cµi|µi (1.5)

We require, without loss of generality, that the {φi} to be orthogonal and deﬁne a set of unoccupied orbitals, {φa}, to form a complete orthogonal basis over the AO space. These virtual orbitals are a formality for now but will become signiﬁcant in higher level theories. The full wave function can be constructed as ψ(ri) = Det[φ(r1) . . . φ(rn)] to antisymmetrize the wavefunction. We can now formulate the energy in matrix form by taking advantage of the Slater-Condon rules for inner products of determinantal forms with one and two electron operators. CHAPTER 1. INTRODUCTION 4

E = hψ|Hˆ |ψi 1 = hi|Tˆ + Vˆ |ii + hij||iji ne 2 (1.6) 1 1 = h + J − K ii 2 ii 2 ii Since this energy is invariant to orbital rotations within the occupied subspace, we can minimize the energy with respect to occupied virtual mixing only. Parameterizing all unitary orbital rotations by U = eΘ where Θ is an antisymmetric matrix and taking the gradient of E(Θ) gives:

∂E = hia + Jia − Kia ∂θia (1.7) = Fia where F is the famous Fock matrix. We have now derived the Hartree-Fock (HF) method using a variational approach, but should note that the traditional derivation makes use of a mean-field approximation where electrons are repelled by only the density of the electrons[4]. These approaches give identical results; assuming independent electrons in our form for the wavefunction (excepting correlation through the antisymmetrizer) is equivalent to using a mean-field coulomb potential. While the mean-field approach gives better physical intuition, the variational method dresses the mathematics in a way that shows it clearly as a nonlinear optimization problem and invites us to consider the toolset of nonlinear optimization. We can implement this algorithm in O (n3) time by taking advantage of the spatial sparsity of the atomic orbital overlaps. What we have left out of the previous discussion is the spin component of the electrons. 1 As Fermions with with a spin of 2 , electrons have a spin degree of freedom that can be ˆ described in the basis of eigenstates of Sz, |αi and |βi. The first approximation we make is that each orbital has been projected onto either |αi or |βi, and thus Slater determinants will ˆ be eigenstates of Sz. In this unrestricted HF (UHF) approximation, single determinants are not necessarily eigenstates of the full Sˆ2 operator–a condition that the exact wavefunction can meet due to Sˆ2 commuting with Hˆ . To satisfy this condition we can require that for every α electron, we have a corresponding β electron that has the same spacial orbital. This new approximation is referred to as restricted HF (RHF). By the variational principle, each added constraint reduces the space we minimize over and thus can only raise the energy, bringing it farther from the true ground state energy, but our base wavefunction presumably is closer to the true eigenfunction in some sense as they share the correct spin symmetry. This tension between better energetics versus spin symmetry, referred to as the symmetry dilemma, will be fundamental to much of Chapters 2-4 on symmetry breaking and orbital-optimized methods. CHAPTER 1. INTRODUCTION 5

HF has given us an approach to optimize our one electron basis of MOs and construct a many electron function by taking the ﬁrst n of them. What’s more is that we can form a basis for the n-particle Fock space by taking all possible combinations of n orbitals, or more constructively, view each of these determinants as an excitation from the ground state HF determinant, |0i. We classify these determinants by excitation level so a state with orbital ab i and j replaced with a and b will be denoted as the doubly excited determinant |ij i.

1.2 Electron Correlation

Conﬁguration Interaction With only averaged Coulomb forces between electrons in the HF approach, the next level to improve our calculations is the account for the correlations between electrons. Although incredibly limited in practice, we will begin our discussion of electron correlation with con- ﬁguration interaction (CI) as a simple starting point. Given our may electron basis we’ve constructed using HF orbitals, if we want the lowest eigenvalue of Hˆ , we can simply construct the H matrix in a basis of our determinants and use any linear algebra approach to get the lowest eigenvalue (think of this as diagonalization but in practice something like Davidson[5]). Since the determinants form a complete basis over the Fock space determined by the AO’s, it actually doesn’t matter which set of MO’s we use to construct H; full CI (FCI) will always gives the exact energy for a given AO basis. So why do we need anything N other than FCI? Considering the size of H, we realize there are n determinants when we have N AO’s and n electrons. This factorial growth is explosive and limits FCI to only the smallest of atoms and diatomics. If the full matrix is too large, why not just truncate it at some point? we can do this and truncate at a given excitation level (i.e. CIS for singles, CISD for singles and doubles. . . ) guided by the fact that we consider the ground state determinant to be a zeroth order approximation and thus mostly correct. While not unreasonable for small systems, the fundamental failing of truncated CI methods is their failure to be size consistent–the property that a method gives the same result for two noninteracting systems as it would fro calculating them separately. The basic idea of why CISD fails this test, is that while it can describe double excitations on both independent molecules, when taken together, these uncoupled double excitations are formally quadruple excitations and can’t be described in the CISD framework. Another way we can view CI that will more naturally extend to other correlated methods is to view it as variationally minimizing a linearly parameterized combination of excitations. CHAPTER 1. INTRODUCTION 6

The wavefunction can then be written, ˆ ˆ ˆ ΨCI = (1 + T1 + T2 + ··· + Tn)|0i ˆ a † T1 = ti aâaî (1.8) ˆ ab † † T2 = tij aˆbaâaˆjaî

ˆ th † Where Tx is the x order excitation operator,a î destroys an electron in orbital i anda â creates an electron in orbital a. Minimizing the expectation value of the Hamiltonian with respect to the t-amplitudes is equivalent to finding the lowest eigenvector as before. The first lesson we take from CI is that the more approximate the method, the more sensitive the results will be to zeroth order orbitals. The second is that the “no free lunch” idea can be seen in the a tradeoff of cost versus accuracy. A third lesson is that we want methods that are size consistent to allow us to describe retains where bonds are formed or broken and to be able to apply our methods to large systems without accuracy degrading.

Coupled Cluster Whereas the linear parameterization of the truncated CI wavefunction led to its failure to describe independent correlations on diﬀerent subsystems, coupled clusters (CC) utilizes an exponential parameterization to build the separability into the method. The CC wavefunction is constructed as,

(Tˆ1+Tˆ2+... ) ΨCC = e |0i (1.9)

Now if we truncate at double excitations (referred to as CCSD) we see that the wavefunction can contain higher than double excitations (by considering the Taylor expansion), but only as products of the singles and double excitation amplitudes. Unfortunately, attempting to solve for the t-amplitudes using a variational approach leads to equations that don’t truncate, so we solve using a projective approach . The CCSD amplitudes are solved using the following projected equations: ¯ h0|H|0i = ECC a ¯ hi |H|0i = 0 (1.10) ab ¯ hij |H|0i = 0

ˆ ˆ ˆ ˆ Where H¯ is the similarity transformed Hamiltonian e−(T1+T2)Heˆ T1+T2 . We can simplify the equations by rewriting H¯ using the Baker-Campbell-Hausdorﬀ expansion which truncates after a ﬁnite number of terms. These equations can then be self-consistently iterated to solve for the amplitudes. To really make sense of CC we recommend learning about diagrammatic representations[6, 7] of the equations but do no use them here. CHAPTER 1. INTRODUCTION 7

CCSD is a O (N 6) method and CCSDT scales as O (N 8), but intermediate approximations come in a variety of flavors. CCSD(T)[8] is a O (N 7) approximation to CCSDT that avoids iterating the T3 equations with a perturbative approximation for the triples amplitudes and is considered the gold standard of quantum chemistry for many systems. Another perturbative approximation is CC2[9] which approximates the T2 equations by including only ˆ ˆ ˆ terms first order in (H − F ) (the fluctuation potential) and T2 in the T2 equations. This O (N 5) method is most widely used, however, for calculating excited states. One can study CC excited states through a response[10, 11] or equation of motion (EOM) approach[12]. While differing for properties, the two approaches yield the same energy, so we will chose to consider EOM-CC. In this approach one calculates excited states by forming a CI like linear expansion from the CC wavefunction. For EOM-CCSD we have,

ˆ ˆ |Ψi = Reˆ T1+T2 |0i (1.11) ˆ a † ab † † R = r0 + ri aâaî + rij aˆbaâaˆjaî And solving for the right eigenvectors of H¯ in the basis of zero, single, and double excitations out of eTˆ|0i. For CC2, the excitations are out of the CC2 wavefunction and similar approximations are made to the R2 amplitudes as the T2 amplitude equations. These will be discussed in more detail in Chapter 6.

Perturbation Theory Møller Plesset theory is a perturbative approach (specifically Rayleigh-Schrödingerpertur- bation theory) that splits the Hamiltonian into zeroth and first order parts as well as the wavefunction and energy,

Hˆ = Fˆ + λVˆ ψ = ψ(0) + λψ(1) + λ2ψ(2) + ... (1.12) E = E(0) + λE(1) + λ2E(2) + ...

The eigenvalue equation can than be broken into orders of λ, to give the following equations:

Fˆ|ψ(0)i = E(0)|ψ(0)i Fˆ|ψ(1)i + Vˆ |ψ(0)i = E(0)|ψ(1)i + E(1)|ψ(0)i (1.13) Fˆ|ψ(2)i + Vˆ |ψ(1)i = E(0)|ψ(2)i + E(1)|ψ(1)i + E(2)|ψ(0)i . .

(i) (j) We use intermediate normalization, the requirement that hψ |ψ i = δij, to allow us to solve for the wavefunction and energy projectively. Projecting the nth equation with hψ(0) gives us an equation for the nth order energy in terms of the lower order wavefunctions. CHAPTER 1. INTRODUCTION 8

Projecting with excited determinants allows us to solve for the wavefunctions. To calculate the energy up to second order (MP2) we have,

E(0) + E(1) = h0|Fˆ + Vˆ |0i (1.14) = EHF

hab|Vˆ |0i |ψ(1)i = ij |abi E(0) − hab|Fˆ|abi ij ij ij (1.15) hij||abi ab = |ij i i + j − a − b

E(2) = h0|Vˆ |ψ(1)i |hij||abi|2 (1.16) = i + j − a − b

Where we have used that fact that the for HF orbitals Fia = 0 and we can diagonalize the 5 occupied and virtual blocks of F to give diagonal elements p. This give us a O (N ) method that performs well for closed-shell molecular energies, geometries, and frequencies. The problems of MP2 that can occur when orbital symmetry breaking takes place for open-shell systems or aromatic molecules will be further studied in Chapters 2-4.

1.3 Statistical Quantum Thermodynamics

Even if electronics are solved for exactly they are still only part of the story. As discussed previously, the Born-Oppenheimer approximation allows us to separate the electronic problem from the nuclear problem, but not neglect it entirely. While nuclear contributions tend to be smaller and thus merit weaker approximations, often the harmonic level is not enough. One option is to study vibrations using a path integral (PI) formulation that allows us to turn problems of quantum mechancis into coupled classical thermodynamic problems[13, 14]. We begin with a fundamental quantity of thermodynamics, the partition function Q. We can translate the classical sum over states to the quantum trace over density operator and CHAPTER 1. INTRODUCTION 9 see how expectation values work in the quantum context in the following equations, X Q = e−βEi i X −βHˆ Q = hψi|e |ψii i Q = Tr e−βHˆ (1.17) Tr Aeˆ −βHˆ hAˆi = Tr e−βHˆ d hHˆ i = − ln Q dβ Where β is the inverse temperature. We get the path integral formulation by representing the trace in a spatial coordinate basis and inserting P − 1 sets of the identity,

Q = Tr e−βHˆ Z −βHˆ Q = dx1 hx1|e |x1i (1.18) Z Z − β Hˆ − β Hˆ Q = ... dx1 . . . dxP hx1|e P |x2i ... hxP |e P |x1

β If P is small enough, we can use a high temperature approximation to these density matrix elements by treating them with some zeroth order Hamiltonian, 3 ! − β (Hˆ +∆Vˆ ) − β Hˆ − β ∆Vˆ β h i e P 0 = e P 0 e P + O Hˆ , ∆Vˆ P o

P Z Z β β Y − Hˆ0 − ∆Vˆ Q ≈ ... dx1 . . . dxP hxi|e P |xi+1ihxi|e P |xi+1i (1.19) i P Z Z β β Y − Hˆ0 − ∆V (xi) Q ≈ ... dx1 . . . dxP hxi|e P |xi+1ie P i ˆ ˆ If we select H0 = T , the free particle approximation, we get a partition function that corresponds to a classical polymer of P systems connected to their neighbors with harmonic P ˆ potentials simulated at a temperature β . In chapter 7 we will discuss how using a better H0 can improve our results for calculating vibrational anharmonicities.

1.4 Outline

This thesis is organized as follows. CHAPTER 1. INTRODUCTION 10

Chapter 2

The ground state restricted Hartree Fock (RHF) wave function of C60 is found to be unstable with respect to spin symmetry breaking, and further minimization leads to a signiﬁcantly spin contaminated unrestricted (UHF) solution (hS2i = 7.5, 9.6 for singlet and triplet respec-

tively). The nature of the symmetry breaking in C60 relative to the radicaloid fullerene, C36, is assessed by energy lowering of the UHF solution, hS2i, and the unpaired electron number.

We conclude that the high value of each of these measures in C60 is not attributable to strong correlation behavior as is the case for C36. Instead, their origin is from the collective eﬀect of relatively weak, global correlations present in the π space of both fullerenes. Second order perturbation (MP2) calculations of the singlet triplet gap are signiﬁcantly more accurate with RHF orbitals than UHF orbitals, while orbital optimized opposite spin second order correlation (O2) performs even better.

Chapter 3 Orbital optimized second order perturbation theory (OOMP2) optimizes the zeroth order wave function in the presence of correlations, removing the dependence of the method on Hartree–Fock orbitals. This is particularly important for systems where mean field orbitals spin contaminate to artificially lower the zeroth order energy such as open shell molecules, highly conjugated systems, and organometallic compounds. Unfortunately, the promise of OOMP2 is hampered by the possibility of solutions being drawn into divergences, which can occur during the optimization procedure if HOMO and LUMO energies approach degeneracy. In this work, we regularize these divergences through the simple addition of a level shift parameter to the denominator of the MP2 amplitudes. We find that a large level shift parameter of 400 mEh removes divergent behavior while also improving the overall accuracy of the method for atomization energies, barrier heights, intermolecular interactions, radical stabilization energies, and metal binding energies.

Chapter 4 Following the lowest eigenvalue of the orbital-optimized second order Møller-Plesset pertur-

bation theory (OOMP2) hessian during H2 dissociation reveals the surprising stability of the spin-restricted solution at all separations, with a second independent unrestricted solution. We show that a single stable solution can be recovered by using the regularized OOMP2 method (δ-OOMP2), which contains a level shift.

Chapter ?? Previous work[15] established the unexpected behavior of orbital-optimized second-order perturbation theory (OOMP2) for bond dissociations wherein orbitals could change discon- CHAPTER 1. INTRODUCTION 11 tinuously at the unrestriction point. Level-shift regularization (δ-OOMP2) was able to ﬁx the problem for H2 but we ﬁnd this solution does not generalize to even other single bond dissociations. We implement a new regularization approach (σ-OOMP2) based on Evangelista’s similarity renormalization group theory[16] that we show to be more robust for describing even triple bond dissociations.

Chapter 6 We extend the family of semi-empirically modiﬁed methods based on the introduction of a regularization parameter to ground and excited state CC2. It is found that a value of 150 mEh reduces errors in energies across a broad spectrum of ground state chemical test sets and corrects the reported failure of CC2 for ozone. Similarly, a value of 150 mEh balances systematic errors for valence and Rydberg excited states in small molecule test sets. Based on the apparent robustness of these results we suggest the consideration of δ-CC2 as a semi-empirical, trivially modiﬁed CC2-based method.

Chapter 7 Electronic structure theory results are often limited in accuracy by their description of vibrational errors, which are challenging to calculate beyond the harmonic approximation. To calculate anharmonic vibrational energy corrections for low temperature molecules and clusters with systematically reducible errors, we propose a novel combination of using thermodynamic integration and a static harmonic propagator for path integral Monte Carlo. The method requires only electronic single point calculations for sampling as opposed to gradients or Hessians (beyond an initial frequency calculation), and requires a much smaller number of beads and steps due to its use of a more appropriate zeroth order approximation to the propagator. The method is applied to toy systems as well as reassessing the global minimum energy structure of low temperature sulfate-water clusters.

1.5 Additional Work

In addition to the chapters listed above, some work has not made it into the thesis (mostly due to the papers being written in Word rather than LaTex. . . ). Most notably, my work in collaboration with the Long group, modeling molecular organic frameworks (MOF) and their application for hydrogen storage. We published a paper comparing our models to the cite speciﬁc binding energies they got from temperature dependent IR on H2 in a BTT MOF[17]. We benchmarked our binding enthalpies against experimental numbers showing good agreement, and looked at metal and ion substituted MOFs. Our prediction that Br would make a stronger binding compound was conﬁrmed after publication, but was not studied further do to dangerous steps in synthesis due to the Br. CHAPTER 1. INTRODUCTION 12

A second project with the Long group involved trying to understand the differential binding of H2 in MOFs that only differed by an isomerization in the organic linker[18]. Modeling MOF-74 is difficult due to metal chains down the framework, but we did our best to create a model that isolated the differences in the linker by capping metals with CO and freezing them into experimental geometries. Our work showed a combination of increased electrons on the metal and an extra interaction with the ring present in only one isomer to be creating higher binding in the meta variant. Last I just wanted to document some work that never went anywhere do to convergence issues and early discouraging results. I proposed using orthogonalized Hartree product orbitals[19, 20] as a way to avoid artificial spin symmetry breaking for radicals and aromatic compounds. The hypothesis was that since artificial spin symmetry breaking could be caused by HF using Fermi correlation (from the antisymmetrization of the wavefunction) to com- pensate for a complete lack of Coulomb correlation. Whereas OOMP2 tries to account for Coulomb correlation in the optimization process to achieve balance, another option could be to just get rid of Fermi correlation by using Hartree product orbitals. The first problem is that they are not invariant to occupied-occupied rotations which makes the non-linear optimization process a huge pain. The second problem is that it appears that spin contaminated solutions still appear and were probably the global minima (although not for certain due to difficulty converging solutions); this may still be happening due to localization since we’re not allowing for Fermi correlation. 13

Chapter 2

On the Nature of Electron Correlation in C60

2.1 Introduction

Quantum chemists strive to model realistic chemical systems with a high degree of accuracy. The problem is that accurate methods such as multireference configuration interaction (MRCI) and even single reference coupled cluster methods such as CCSD(T) become rapidly unfeasible as system size increases[8, 21]. For this reason, whenever possible we would prefer to take advantage of simpler, single determinant methods such as Hartree Fock (HF)[22] or density functional theory (DFT)[23, 24]. To that end, we would like to first approximate the system as well as possible within the space of HF methods and second to gain some insight into when our approximations will be valid. By allowing different orbitals for different spins, unrestricted Hartree Fock (UHF)[25] can improve upon the energy of spin restricted HF (RHF). UHF thus incorporates some correlation between electrons of opposite spin in a single determinant wave function[26, 27] as a result of permitting spin contamination[28]. This increase in the expectation value of the total spin squared operator is due to the breaking of spin symmetry and the presence of higher spin states in the solution[29]. Although spin contamination is an unphysical aspect of the UHF wavefunction, it may imply that there is static correlation present in the system. By static correlation, we mean that the lowest energy HF (i.e. UHF) wave function is not an appropriate zero order wave function, and thus cannot be satisfactorily corrected by e.g. low order perturbation theory.

A simple but illustrative case of spin symmetry breaking is the dissociation of H2 into two + – H atoms[4]. RHF is insuﬃcient, and dissociates H2 into a superposition of 2 H and H + H . Allowing for the unrestriction of the spin orbitals leads to the correct products; however, as the bond length increases beyond the equilibrium value, at a certain point, hS2i becomes increasingly contaminated up to a ﬁnal value of 1 due to the wavefunction becoming a linear CHAPTER 2. ON THE NATURE OF ELECTRON CORRELATION IN C60 14

combination of a singlet and triplet. The energy is exact at dissociation (“asymptotic regime”), indicating that spin contamination is not necessarily a serious problem. The more challenging region of the surface is the so-called “recoupling regime” at bond lengths where the RHF solution is no longer the global minimum, but before the asymptotic regime is reached. This is the region where, unlike the equilibrium geometry, static correlation is becoming important, but unlike the “asymptotic regime,” the correlation energy is nonzero. As opposed to RHF, in a broken spin symmetry UHF solution, the natural orbitals can have fractional occupation numbers leading to a picture of a molecule with polyradical behavior. Measures of the extent of spin symmetry breaking should thus be signals of polyradical nature in a system. For example hS2i has been correlated to polyradicalism in polyacenes[30]; as the length of the acene increases, the singlet-triplet gap decreases and the spin contamination increases. Another study on the acenes used density matrix renormalization group theory to correlate the π space and showed the polyradicalism of the larger acenes through unpaired electron number as well as various correlation functions [31]. A more complex example of a polyradical and likely strongly correlated system is the

ground state of the fullerene, C36. On the experimental side, it has been found from NMR that solid C36 is of D6h symmetry due to the single peak present in the C13 spectra[32]. On the computational side, however, there has been signiﬁcant discrepancy between diﬀerent electronic structure methods on the optimized geometry[33–37] and even multiplicity[37, 38] of the ground state of this small fullerene. One explanation for the failing of simple methods to give the fully symmetric D6h symmetry is given by Fowler et al [39]. Their semi-empirical CI-based estimations of correlation

energy in the isomers of C36 show that the D6h symmetry structure has a signiﬁcantly larger correlation energy than any other isomer. They argue that methods which neglect the correlation, such as restricted HF (and perhaps restricted DFT with inexact functionals) therefore falsely give lower energies for the geometries with less correlation and thus misrepresent them as being similar in energy to the D6h isomer. Another computational study on C36 has found that the RHF ground state is unstable with respect to UHF leading to a spin contaminated solution[37]. This spin contamination has been seen as an indicator of the presence of radicaloid character in the electronic structure of the ground state, which would in turn emphasize the importance of using a higher level of theory than basic RHF for its description.

Based on the interesting nature of the HF results for the radicaloid C36, herein we investigate the larger, and experimentally more stable fullerene, C60, for similarities and diﬀerences in the Hartree-Fock descriptions of their ground state, and therefore the comparative character of electron correlations in the two fullerenes. CHAPTER 2. ON THE NATURE OF ELECTRON CORRELATION IN C60 15

2.2 Results and Discussion

HF calculations were run using QCHEM 3.0[40] with a 6-31G* basis and all geometries were optimized at the HF level of theory used for energies; for the purposes of this communication RHF implies restricted closed shell HF for singlets and restricted open shell HF for triplet calculations.

The ﬁrst surprise for C60, considering its molecular stability, is the discovery that the RHF ground state, although a minimum in the restricted space, is unstable with respect to UHF orbital rotations. This ﬁnding brings up the question of the extent of electron

correlations in the relatively stable C60 as compared to the radicaloid C36. Several metrics are used to compare the nature of the spin symmetry breaking in the two systems. The first metric used is the energy lowering due to breaking of spin symmetry defined as ∆E = ERHF − EUHF. As can be seen from the variational principle, this difference must be a nonnegative value, but the extent can give us a way to quantify the energetic gain from spin symmetry breaking. The second is hS2i which shows the extent of contamination by higher spin states. Of course, bearing in mind the simple H2 example discussed in the introduction, neither of these measures by themselves can conclusively indicate whether the correlations are of the strong static type (“recoupling regime”) or not (e.g. “asymptotic regime”). The next metric considered is the number of unpaired electrons. In restricted frameworks, the number of unpaired electrons is constrained to be an integer; however, in UHF there is no single definition for the number of unpaired electrons in, say, a singlet polyradicaloid. For the purposes of this paper, we will choose the definition given by Head-Gordon[41]:

M X nU = min(ni, 2 − ni) i=1

th where ni is the i natural orbital occupation number (NOON) and M is the dimension of the one particle basis. This definition is chosen for its straightforward interpretation and correct bounds on the maximum number of unpaired electrons. The value for each metric in the case of the two fullerenes has been compiled in Table 2.1 and we now look at the first two rows to analyze the spin symmetry breaking in C60. Each metric gives results which are, at least on first inspection, quite dramatic. The energy is lowered by a value even larger than the RHF singlet-triplet gap of 55 kcal/mol! The singlet and triplet values of hS2i are both about 7.5 higher than they should be, and there are about 9 more unpaired electrons than RHF. Do these data, particularly the unpaired electron numbers, indicate that C60 may be a strongly correlated molecule? We next look to C36, a system that is more definitively considered to be strongly correlated[33, 34, 37, 38]. When comparing the two systems it is important to take into consideration their size, since C60 has almost twice as many π electrons as C36. In light of this fact, viewing Table 2.1 shows the two fullerenes have similar values of spin contamination by looking at the hS2i values, but in terms of unpaired electrons per C atom (or π electron), CHAPTER 2. ON THE NATURE OF ELECTRON CORRELATION IN C60 16

∆E (kcal/mol) hS2i Unpaired e−

C60 singlet 59.62 7.5 8.8 triplet 85.77 9.6 10.6

C36 singlet 180.88 7.7 9.9 triplet 124.59 8.7 10.2

Table 2.1: Quantiﬁcation of spin symmetry breaking in fullerene systems. ∆E = ERHF − EUHF and number of unpaired electrons as described in the text.

1 − 1 − C36 (0.28 e /C) shows nearly twice as many as C60 (0.15 e /C). In terms of energy lowering, the difference is even more dramatic: the RHF-UHF energy lowering is 5 kcal/mol/C 1 1 for C36, but only about 1 kcal/mol/C for C60. When analyzed more carefully, the hS2i and unpaired electron numbers for the two fullerenes show an important difference. While the hS2i values are very large in both cases, for C60, the difference between triplet and singlet is about two, which is the correct difference. In C36, however, the difference is only one. Likewise, for unpaired electrons the difference of about two for C60 is the correct one for a triplet versus a singlet state, but in C36 the number of unpaired electrons in singlet and triplet are nearly identical. We see then, that while in

C60 the behavior of the triplet relative to the singlet is preserved upon spin unrestriction, in C36 spin symmetry breaking gives us a picture of nearly equally occupied HOMO and LUMO, characteristic of strongly correlated systems. We can gain more insight into the nature of the unpaired electrons by looking directly at the NOON of the two systems shown in Figure 2.1. There are a couple of interesting things about this plot. First, we see that in both systems, the entire π space is at least partially spin polarized. Thus spin polarization is a collective phenomenon in both fullerenes. It is therefore nearly certain that spin polarization will occur in all larger fullerenes as well.

Second, there is a large diﬀerence in the extent of unpairing in C60 compared to C36. We know from the number of unpaired electrons that there should be nearly twice the number of unpaired electrons per carbon in C36, but this plot shows that C60 doesn’t have any natural orbitals that are actually half occupied, while C36 has the HOMO and LUMO both with occupation number of nearly one. Apart from this pair, the C36 and C60 NOON distributions look qualitatively similar. We thus expect to see at least two strongly correlated electrons in C36. Another way to look at this unpairing is to plot the unpaired electron density. We can extend the idea of unpaired electrons to an unpaired electron density by adding densities of natural orbitals weighted by unpaired electron number. Likewise, we can create a spin density that is the diﬀerence between α and β densities. Figure 2.2 shows plots of the unpaired electron density, colored by the value of the spin density (light gray indicating excess α and dark gray excess β) of singlet and triplet C36 and C60. These plots show global spin polarization of the entire π space. Rather than seeing 4 CHAPTER 2. ON THE NATURE OF ELECTRON CORRELATION IN C60 17

Figure 2.1: Natural orbital occupation numbers of UHF spincontaminated singlets for C36 i i th and C60. Orbitals are numbered as a fraction of the total π space (i.e. 36 or 60 for the i π orbital of C36 or C60 respectively). or 5 localized areas of spin polarization as might be suggested by the value of 8.8 unpaired electrons, the entire π space polarizes. Similar results have been found in studies on graphene fragments[42–44], but with the key distinction that spin polarization primarily takes place on the edge carbons, which are bonded to hydrogens. In fullerenes, there are no edge sites so the entire molecule spin polarizes with an antiferromagnetic pattern that is frustrated by the presence of pentagons.

To further confirm the idea that C36 is strongly correlated while C60 is not, we have run CASSCF calculations using GAMESS [45] on C36 and C60 with [6,6] and [10,8] active spaces respectively chosen based on orbital symmetries. For C60 only one configuration gave significant weight and the HOMO and LUMO occupation numbers are nearly 2 and 0. The results for C36 on the other hand show significant static correlation and occupation numbers of 1.21 and 0.79 for HOMO and LUMO. These results are interesting in two ways. First, they qualitatively support the greater unpairing seen in the UHF wave function for C36 versus C60. Second, they do not support the observation of other NOON values larger than 0.5 seen in the UHF calculations for both C36 and C60. We may therefore conclude that the relative behavior of the highly spin contaminated

UHF wave functions for C36 versus C60 should alert us to diﬀerences in correlations in the π space of C36 and C60. However, from the second observation above, we do not have a clear picture of whether the UHF wave functions are particularly “better” than the RHF CHAPTER 2. ON THE NATURE OF ELECTRON CORRELATION IN C60 18

Figure 2.2: Unpaired electron density of singlet (top) and triplet (bottom) C60 (left) and ˚−3 C36 (right) plotted at isovalue 0.006 A , with shading determined by the sign of the spin density as described in the text. ones. The UHF energies are certainly much lower than the RHF ones, but RHF is a proper eigenfunction of S2. This is simply the much discussed symmetry dilemma[28], and therefore further assessment is needed. To gain some insight into the comparative quality of the two HF wave functions for the two fullerenes, we will consider how they handle the singlet-triplet gap, an observable property that typically depends significantly on correlation. This assessment can be guided by good gas phase experimental values for C60 from phosphorescence in rare gas matrices[46] and approximate numbers for C36 from anion photoelectron spectroscopy[47]. From Table 2.2 we see that for C60 UHF gives a significantly better singlet triplet gap than RHF. While RHF substantially overestimates the gap by about 18 kcal/mol, UHF underestimates it by less than 8 kcal/mol. What is more surprising is that for C36, RHF actually predicts a triplet ground state, whereas UHF at least gives the sign of the singlet triplet gap correctly. However the magnitude of the singlet-triplet gap errors are nearly equal for both RHF and UHF for C36. One explanation of this improvement is that the spin polarization is not just a spurious by-product brought about by the mathematical formalism, but can give a better description of properties (within the HF regime) by capturing part of the true electron correlation effect. This correlation is Hollett and Gill’s “Type A” static correlation[27] and, as Fukutome claims, can contain information about the spin correlation[26]. To confirm that the presence of spin contamination is not significantly due to basis set incompleteness, we have also calculated the spin contaminated singlet state of C60 in the CHAPTER 2. ON THE NATURE OF ELECTRON CORRELATION IN C60 19

Singlet-Triplet Gap (kcal/mol)

C60 RHF 55.16 UHF 29.01 RMP2 43.28 UMP2 80.20 O2 34.6 Experimental[46] 36.95 ± 0.02

C36 RHF -21.29 UHF 35.00 RMP2 18.26 UMP2 26.33 O2 22.07 Experimental[47] ∼ 8

Table 2.2: Calculated Etriplet−Esinglet from restricted and unrestricted HF and MP2 compared to experimental literature values of C60 and C36.

6-311G(2df) basis. At twice the basis size, we computed hS2i = 7.4, a value similar enough to the small basis that we can be conﬁdent that the basis set is not of key importance in our result.

With the discovery of the spin contaminated UHF solution for C60 we are left with the diﬃcult question of how to move forward to post-HF methods on these large molecules. It is well known that spin contaminated UHF orbitals typically recover signiﬁcantly less correlation energy than RHF orbitals at the MP2 level[48], but at the same time, spin contamination is an indicator of the poor quality of RHF orbitals. Table 2.2 shows how the two versions of MP2 handle the singlet-triplet gap of C36 and C60. For C60, RMP2 reduces the error of RHF by a factor of three to 6 kcal/mol. By contrast, UMP2 for C60 using the spin contaminated orbitals performs characteristically poorly, actually increasing the error. One way forward is to use a method that does not rely on the quality of HF orbitals, but reoptimizes them in the presence of electron correlation. Brueckner coupled cluster methods[49–51] would be ideal, but are presently too expensive for routine application to problems of this size. A less computationally demanding alternative is orbital optimized scaled opposite spin MP2 (O2)[52]. The O2 method is found to give the same results for

C60 whether initialized with RHF or spin contaminated UHF orbitals – yielding closed shell orbitals for the singlet and nearly uncontaminated orbitals for the triplet. For C60, O2 gives a reasonably accurate singlet-triplet gap result (within 2.4 kcal/mol or 7% of experiment), improving on RMP2. The MP2 and O2 results also give us more insight into the nature of correlation in

C60. The success of second order perturbation theory on observable properties using either restricted or optimized orbitals shows that strong correlation, which require multiple deter- CHAPTER 2. ON THE NATURE OF ELECTRON CORRELATION IN C60 20

minants to describe, must not be present in C60. The signs of strong correlation—very high spin contamination and unpaired electron number—must in fact be attributable to relatively small (on a per atom scale) but global electron correlations in the π space.

For C36 on the other hand, none of the MP2 methods, including O2, give an accurate value for the singlet-triplet gap and one must go to multireference MP2 to get values that match experiment[37]. For a strongly correlated system such as this one, it is as expected that to properly describe the system, we would need to use a method with more than one Slater determinant as the reference, or relatively high order single reference coupled cluster theory. This distinction between the two fullerenes can be made more clearly by analyzing the NOONs resulting from the O2 calculations as shown in Figure 2.3. We can see that the unpaired nature of C60’s π space has been tamed by stabilizing the orbitals with respect to the MP2 correlation. C36 on the other hand still has its highest occupied orbital signiﬁcantly unoccupied. These data give further support to the CASSCF results showing major static correlation present only in C36.

Figure 2.3: Natural orbital occupation numbers from O2 calculations on singlet C36 and C60. Orbitals are numbered as a fraction of the total π space. CHAPTER 2. ON THE NATURE OF ELECTRON CORRELATION IN C60 21

2.3 Conclusion

Perhaps surprisingly, we have found the RHF ground state of the very stable C60 molecule to be unstable with respect to spin symmetry breaking, and have compared the nature of its spin polarization to that of the much less stable fullerene, C36. From the analysis of the number of unpaired electrons and hS2i we are left with the picture of a global π correlation present in both fullerenes but with the addition of at least two strongly correlated electrons in C36. The global correlation is responsible for the spin polarization of the whole π space seen in Figure 2.2, the large value added to hS2i, and the number of unpaired electrons.

This correlation serves as a background to the strong correlation in C36 which is seen in the additional spin polarization energy and bringing together of hS2i and unpaired electron number for the singlet and triplet. These results form an interesting case study on the issues associated with simple Hartree- Fock based calculations on molecules with extended π systems. Even stable molecule like

C60 can have unstable RHF solutions due to an aggregate of weaker correlations, leading to dramatically diﬀerent, lower energy, UHF solutions. However, it is clear that large hS2i values do not necessarily imply the presence of strong, static correlations, and performing MP2 from the RHF orbitals is clearly preferable to using UHF orbitals. It is still better to use correlation optimized orbitals, as in the O2 method[52]. 22

Chapter 3

Regularized Orbital-Optimized MP2

3.1 Introduction

The simplest wave function ansatz that satisfies Fermi statistics is an antisymmetrized product of single electron orbitals: the Slater determinant. The Hartree–Fock (HF) method— defined by variationally minimizing the expectation value of the Hamiltonian within the Slater determinant ansatz—gives a mean-field description that obtains approximately 99% of the total energy but ultimately fails at describing most chemical processes, such as reaction energies, with any reasonable accuracy[22]. Excepting cases of strong/static correlation, where multi-reference methods are required to properly describe the physics, a primary assumption in electronic structure theory is that the HF method is a good zeroth order approximation for the true wave function. Møller–Plesset perturbation theory (MP), configuration interaction, and coupled cluster (CC) theories typically use HF orbitals as the starting point for building up to a more accurate wave function. Unfortunately, the assumption that HF orbitals are a good zeroth order approximation does fail, particularly in the case of significant spin contamination. Restricted HF (RHF) follows our chemical intuition that electrons are paired by requiring alpha and beta electrons to have the same spatial orbitals. Removing this requirement leads to unrestricted HF (UHF)[25], which allows for extra variational degrees of freedom that potentially lower the energy but lead to other unintended consequences. The most clear implication of this unrestriction of the wavefunction is the introduction of spin contamination, indicating that the wavefunction is no longer an eigenfunction of the spin squared operator[28]. While one might not be too concerned about getting the total spin of the wavefunction correct since the energy has no direct dependence on spin degrees of freedom, such broken symmetry solutions typically lead to very poor zeroth order wavefunctions for MP2 and CC theories[48, 53–55]. The point is made quite clearly in the dissociation of the lithium dimer where MP2 using UHF orbitals dissociates correctly but gives a very poor description of the ground state potential, while RHF orbitals lead to the correct equilibrium behavior but dissociate CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 23 wildly incorrectly (Figure 3.1). Artificial spin symmetry breaking occurs on a larger scale in

C60, leading UMP2 to fail at describing properties such as the single-triplet gap[56]. Spin contamination also leads to total delocalization of solitons in neutral polyenyl chains that experimentally are known to be localized over about 18 carbon atoms[57]. These examples illustrate the need for post-HF methods that are not tied to HF orbitals. One such option is approximating the Brueckner orbitals—defined as the orbitals that give a ground state determinant with maximal overlap with the exact wave function or equivalently, the orbitals for which there are no single excitations in full configuration interaction (FCI) wavefunction[58]. Within the framework of FCI, one can calculate the Brueckner orbitals using either a projective or variational approach[59]. In the projective approach, one would calculate singles amplitudes as a function of orbital rotations and minimize them to zero. For the variational approach, we constrain the singles amplitudes to be zero and minimize the CI energy with respect to orbitals, which gives the Brueckener orbitals (since they satisfy the constraint by construction). While these two approaches are identical in the FCI limit, introducing truncations to the CI expansion will lead to two distinct methods. These Brueckner orbital methods can also be applied to the size-consistent exponential ansatz of coupled-cluster approximations, although they are not exact in the full limit[60]. The projective approach has been implemented for CCSD as Brueckener doubles (BD)[49, 50] with the variational approach implemented for CCSD as orbital-optimized doubles (OD)[61] and for Møller Plesset theory as orbital-optimized MP2 (OOMP2)[52] and OOMP3[62]. While minimizing a non-variational method may sound troubling, the value being minimized is actually a constrained energy where the single’s contribution is set to zero, penalizing orbital solutions which have large singles contributions to the full energy. Many of the failings of HF orbitals can be mitigated by the use of approximate Brueckner orbital methods. Spin contamination is generally removed or significantly reduced leading to spin eigenfunctions without resorting to restricted constraints[52, 56, 59, 63]. In addition to rectifying spin properties, the use of approximate Brueckner orbitals has been shown to improve the description of bond lengths, frequencies, and relative energies of open shell systems[51, 52, 59, 61, 63, 64]. The use of the variational approach garners additional benefits due to the fact that the energy is made stable to orbital rotations. This fact gives rise to a Hellmann-Feynmann condition, simplifying response properties of the wavefunction and removing first derivative discontinuities present in UMP2 at the unrestriction point[65]. MP2 is the one of the simplest computational methods to account for electron correlation and naturally includes long-range dispersion interactions[4]. It is ab initio, systematically im- provable, and an important alternative to the more commonly used density functional theory (DFT) in cases where DFT self-interaction error is present[66]. For these reasons, OOMP2 is an important method to accurately model large, open-shell systems such as radicals or organometallic compounds, striking a balance between the speed of HF and the accuracy of CCSD. For the case of Li2 dissociation, it connects the spin pure equilibrium description to the unrestricted asymptotic limit as seen in Figure 3.1. Recently, variants of OOMP2 have been proposed that use a Thouless expansion representation of OOMP2[67] along with CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 24 another approach to orbital-optimization based on a one particle operator approximation to the MP2 energy[68].

Figure 3.1: Li2 dissociation curve for MP2 using restricted and unrestricted orbitals and for OOMP2 with a cc-pVDZ basis. RMP2 dissociates incorrectly and UMP2 distorts the equilibrium description while OOMP2 gets the best of both worlds by continuously connecting the two regimes, albeit with a kink due to a slight discontinuous change to the orbitals upon unrestriction.

While enabling many improvements to traditional MP2, OOMP2 brings with it the loom- ing concern of energy divergence. Inherent in Rayleigh-Schrödingerperturbation theory is the divergence for zeroth order states with nearly degenerate energies, which translates in MP2 to divergence as the HOMO-LUMO gap goes to zero. While there are classes of post- Kohn Sham theories that can properly describe small band gap systems such as RPA[69, 70] and GW theory[71, 72], in small molecular systems such a degeneracy in the HF orbital energies is a key indicator of the presence of static correlation requiring multireference techniques rather than MP2. For OOMP2, on the other hand, these divergences do not need to be present in the orbitals used to calculate the final energy to cause problems; due to the non-variational nature of the constrained energy, the method must only come across one of these mathematical artifacts during the optimization procedure to keep from finding a truly stable set of orbitals. In this case, by removing divergences, one could properly converge to a set of orbitals that do not contain orbital degeneracies. Thus, unlike standard MP2, the divergences present in CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 25

OOMP2 can occur in many more situations since they may arise during the optimization process even if they would have not appeared in the final energy expression. There have been many approaches taken to regularize standard MP2 theory for nearly degenerate zeroth order energies. Some apply methods of pseudo-degenerate perturbation theory where zeroth order subspaces are defined for applying perturbation theory followed by diagonalization[73]. Another approach is to treat each second order excited state contribution as uncoupled from all others and diagonalize as in degeneracy-corrected perturbation theory (DCPT2)[74, 75] or a generalized iterative approach to diagonalize a dressed Hamiltonian[76]. There have been many methods based on the repartitioning of the diagonal portion of the zeroth and first order Hamiltonian through level shifts, which leaves the energy unchanged up to first order but modifies higher order terms. One partitioning is to shift the degeneracy into the imaginary plane through a complex level shift parameter to damp out divergences[77, 78]. Other repartitioning approaches have focused on the convergence of the MP series[79, 80] and making low levels of theory stable to difficult correlations in single reference[81, 82] and multireference perturbation theory[83, 84]. In the context of complete active space second-order perturbation theory (CASPT2), a method has been developed to add a state independent level shift and then add a correction to remove the effect of the shift on the energy[85]. These approaches all have established merits, but are more complicated than the very simplest possibility, which is the introduction of a static level shift, which perhaps surprisingly, has not been carefully explored hitherto, to our knowledge. The simple addition of a single, state independent level shift is well suited to our situation since we do not intend to properly describe these degenerate cases, but simply remove them as minima from our orbital optimization space. It is a key point that we are not trying to develop a pseudo-degenerate perturbation theory but simply modify our OOMP2 energy functional in a way that avoids artificial minima. Accordingly, the purpose of this paper is to explore a modified OOMP2 theory with a level-shift parameter to regularize divergences that can arise during orbital optimization. We regard this parameter as potentially serving two purposes. First, regularization itself, and, second, since stability improvements should be related to accuracy improvements, the level shift parameter is also a degree of freedom with which to remove some of the systematic error of OOMP2. After discussing the theory, we investigate the magnitude of the level shift needed for regularization, and then explore how compatible (or incompatible) it is with training the level shift parameter to remove systematic OOMP2 errors in calculated atomization energies. The transferability of the optimized parameter is then further investigated on a range of other relative energies, and also on optimized bond lengths and harmonic vibrational frequencies for molecules that are sensitive to symmetry breaking in the MP2 wavefunction. CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 26

3.2 Theory

Using spin-orbital notation, the resolution of the identity[86] (RI) MP2 energy is given by

occ virt 1 X X E = (ia||jb) T ab RIMP2 4 RI ij ij ab where

ab (ia||jb)RI Tij = i + j − a − b AUX X P Q P Q (ia||jb)RI = Cia(P |Q)Cjb − Cib (P |Q)Cja PQ AUX P X −1 Cia = (ia|R)(R|P ) R

This is the standard MP2 energy expression where p are given by diagonal elements of the pseudocanonical (block diagonalized) Fock matrix and the two electron integrals have been expanded using the resolution of the identity. For simplicity, we assume all occupied ab orbitals are correlated. As usual, Tij is the coeﬃcient of the double excitation i → a, j → b of the ﬁrst order wavefunction. A subtlety to this equation is that the singles energy, which for Hartree Fock orbitals is strictly zero, is neglected even for non-HF orbitals when using OOMP2 as in OD. As mentioned above, the purpose of neglecting the singles contribution and minimizing the energy is to reach approximate Brueckner orbitals. To minimize the energy we need the electronic gradient which is expressed as

∂E X X ∂Ubj = (2Fbj + 2Lbj) ∂θai ∂θai j b with,

X (2) X (2) Lai = Pjk Aaijk + Pbc Aaibc jk bc X X ab − Tjk (ij||bk)RI jk b X X bc + Tij (ab||jc)RI j bc X (2) X (2) + FajPji + Pab F(bi) j b CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 27 and,

(2) −1 X X P = T abT ab ij 2 ik jk k ab (2) 1 X X P = T acT bc ab 2 ij ij ij c

Apqrs = (pq||rs) + (pq||sr) = 2(pq|rs) − (pr|ps) − (ps|qr)

(2) Standard notation is used, where Fpq are Fock matrix elements, Ppq are elements of the correction to the two particle density matrix, and Apqrs is from the HF orbital Hessian. Note that the last two terms of the Lagrangian (Lai) come from oﬀ-diagonal Fock matrix elements and appear since we are not using HF orbitals. Our proposal is to tame the divergence of the OOMP2 energy by modifying the T amplitudes which contain the energy denominators. The simplest place to start is to add a level shift to the zeroth order energies which takes the form of a small constant factor to the denominator, thus setting a lower limit to the divergence. Our new amplitudes are thus expressed,

ab (ia||jb)RI Tij (δ) = i + j − a − b − δ This choice gives the added beneﬁt of leaving the gradient equations unchanged except for the replacement of T with T (δ). The level shift can be theoretically justiﬁed as a repartitioning of the zeroth order Hamil- tonian as,

δ H0 = H0 + δ · 1 Vδ = V − δ · 1

which leaves the first order energy unchanged, but modifies the first order amplitudes as ab Tij (δ). Another way to derive δ-OOMP2 is to start from the Hylleraas functional[87] and penalize large amplitudes by including a third term:

† † † JH (T) = 2T V − T (H0 − E0)T + δ · T T

From here we minimize JH by diﬀerentiating with respect to T and setting equal to zero:

∂J H = 2V + 2(E + δ − H )T = 0 ∂T 0 0 CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 28 and we get, V T = (E0 + δ − H0) Thus we arrive at at the same equations by viewing δ as a level shift from repartitioning the Hamiltonian, or as a quadratic penalty function applied to the T amplitudes. More complicated (i.e. more non-linear) penalty functions are also possible[88].

3.3 Results and Discussion

Divergence While the possibility of the OOMP2 energy diverging is clear, what is unclear is under what circumstances these divergences will interfere with the optimization procedure. We are limited in our understanding of the energy as a function of orbitals due to the high

dimensionality of the problem. One exception, however, is the case of H2 in a minimal basis (in an unrestricted framework), for which the only degrees of freedom to which the energy is not invariant are the 2 rotations between occupied and virtual alpha and beta orbitals. Thus, for a given bond length, we can plot the OOMP2 energy landscape in three dimensions. Figures 3.2 and 3.3 plot the energy surface at equilibrium and stretched geometries as a function of Given’s rotations between occupied and virtual orbitals in α and β subspaces. There are a few points of note to help orient oneself in these plots. First, the (0◦, 0◦) point corresponds to the orbitals obtained by diagonalizing the core Hamiltonian and, for minimal

basis H2, corresponds quite nearly to RHF orbitals. Second, since the two axes correspond to mixing α and β orbitals independently, all points that lie off the central diagonal (θα = θβ) will correspond to spin contaminated (unrestricted) orbitals. The final point to note is that rotation by 180◦ corresponds to multiplying the molecular orbital by -1 and leaves the energy unchanged. Thus the plotted region contains points corresponding to the same orbitals, but we leave this degeneracy in the plot to get a clearer visual representation of the surface. When we look at the energy surface at equilibrium, there appears to be a clear, single minimum corresponding to the RHF solution at the origin. Although some “dents” appear near the top of the curve, we can safely say that they will have no effect on any optimization procedure since they appear near the top of a nearly 1000 kcal/mol high maximum. We can feel relatively sure that optimization on this energy surface will yield the global minimum solution without much difficulty. Once we stretch the bond to 4.0 A,˚ we get a qualitatively different picture. Now, the new unrestricted solution shows up as a wide minimum around the point (140◦, 40◦). Unlike the equilibrium case, there are points on the orbital surface where the energy diverges due to the coalescence of the HOMO and LUMO energies. In fact, the restricted solution is surrounded by these divergences. CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 29

Figure 3.2: Dependence of the OOMP2 energy (the standard RIMP2 energy without singles contribution) on the two occupied-virtual mixing angles for the hydrogen molecule in the STO-3G basis at 0.74 A.˚ The region around the RHF minimum at (0◦, 0◦) is well behaved.

Figure 3.3: Dependence of the OOMP2 energy on the two occupied-virtual mixing angles for the hydrogen molecule in the STO-3G basis at 4.0 A.˚ Divergences appear for orbitals with unfavorable HF energies but very large negative MP2 energy due to HOMO-LUMO energy coalescence. There is a stable minimum near the UHF solution around (140◦, 40◦), but it is not the global minimum due to the divergences.

To get a sense of the size of the regularization parameter needed to be to remove these pits, Figure 3.4 plots the energy surface for δ-OOMP2 for δ values of 100 and 400 mEh. It shows that in our toy case, one must go to values over 10 eV to tame the divergences of OOMP2. A value this high will certainly have a significant effect on absolute energies, but potentially less so on relative energies as we will see. In this case, the unrestricted solution has been restored as the global minimum and the divergences have nearly been reduced to saddle points. In general, we expect that level shifts of this magnitude should remove divergences as absolute minima, since there will be a large penalty from the first order energy for bringing the HOMO and LUMO orbital energies to degeneracy. Unfortunately, there may still be artificial, shallow local minima, but they will be easily identifiable by a HOMO-LUMO gap of zero. The difficulty of looking at divergences that arise in OOMP2 is that it is a problem that CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 30

Figure 3.4: δ-OOMP2 orbital energy surface with level shifts, δ, of 100 mEh (left) and 400 mEh (right) for the hydrogen molecule in the STO-3G basis at 4.0 A.˚ The level shift of 400 mEh has restored the solution near the UHF orbitals to be the global minimum and has removed the divergences. depends on the specifics of the optimization algorithm and can be fixed by knowing the right answer ahead of time (since presumably the final set of orbitals should have a non-zero HOMO-LUMO gap). Rather than immediately fixing our parameter by how “regularized” it makes the optimization, we choose to look at the problem from a different perspective, by viewing δ as a semi-empirical parameter and testing how it can improve systematic errors in OOMP2. We can then assess how compatible (or incompatible) the two perspectives are.

Test Sets We proceed by investigating the effect of the δ parameter on errors in calculated atomization energies compared to QCISD(T) [89] for the 148 small molecules of the G2 test set [90, 91] in the cc-pVTZ basis. The G2 test set is chosen as a fair testing grounds since thermochemistry of closed shell systems is definitely not the target of OOMP2; in fact, for such systems, standard RIMP2 will likely be faster and more accurate. In this sense, we hope to parametrize δ-OOMP2 for general improvement, rather than fit it to a specific problem. The moderately sized cc-pVTZ basis is used in both reference and OOMP2 calculations with the matching auxiliary basis set for the resolution of the identity. In this way, we are not compensating for basis set incompleteness. Figure 3.5 shows the root mean square error (RMSE) for δ-OOMP2 as well as the corresponding δ-RIMP2 and an RIMP2 and OOMP2 variant with directly scaled correlation energy (as has been previously applied to MP2 [92, 93]). The other two methods are briefly considered here as a way to provide a fair comparison: inclusion of a semi-empirical parameter will necessarily improve the statistical errors and we want to make sure that the parameterization we are working with gives comparable improvements to other simple, singly- parameterized variants of MP2. These results show that a significantly large regularization CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 31

parameter of around 300 or 400 mEh optimally reduces the systematic errors of OOMP2 in atomization energies. By comparing to the other two methods it seems that the improvements are typical of parameterizations that reduce the correlation energyof MP2.

Figure 3.5: RMS error on the G2 test set of atomization energies for δ-OOMP2, δ-RIMP2, and correlation scaled RIMP2 and OOMP2 as a function of the regularization parameter δ (2) (bottom) or scaling parameter, s, given by Es = E0 + sE (top).

While the size of the optimal level shift seems surprisingly large, parameters on the same order of magnitude have been shown to reduce errors in CASPT2 [85] (although the study was more focused on removing the effects of the parameter rather than exploiting its reduction of errors). It is also important to compare to the previous scaled MP2 results for bond dissociations[92] and atomization energies[93] of very small molecules which actually suggest scaling the correlation by a value larger than one. These studies, however, are comparing to experimental values and not to a higher level theory in the same basis set and are thus accounting for basis set incompleteness of their double and triple zeta MP2 calculations which becomes the major factor in the results. Another interesting point to note is that δ-OOMP2 has a larger optimal value of δ compared to δ-RIMP2, which reinforces the idea that while MP2 typically over correlates, OOMP2 over correlates even more. In this context, the discussion of over-correlation applies specifically to relative energies of chemical significance. Thus while it is recognized that MP2 typically under estimates absolute correlation energies (except in some recently understood cases for heavy atoms[94–96]), it tends to over emphasize the effects of correlation for relative energies as seen in the G2 results for scaled-RIMP2. Nonetheless it is important to note that CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 32 this over-correlation is not a universal rule but is a significant trend seen quite clearly in the 148 molecules of the G2 test set. Finally it is also encouraging that the optimal value seems quite compatible with the values inferred as suitable for regularization in the previous section. Performance of δ-OOMP2 on test sets that typically give standard MP2 trouble are shown in Fig 3.6. In addition to the G2 atomization energies, we have considered the S22 (weak interactions), RSE43 (radical stabilization energies), and BH76 (barrier heights) test sets from Grimme’s GMTKN30 database[97] as well as a subset of the SRMBE12 that excludes second row transition metals. Calculations on the S22 test set are run in basis sets matching the original CCSD(T) values[98] without extrapolation while RSE43, BH76, and SRMB9 are run using a T-Q basis set extrapolation[99] to compare to reference values. Since errors in the various test sets can be orders of magnitude different, all RMS errors are plotted relative to that of RIMP2 using unrestricted orbitals that have been confirmed as local minima by running a stability analysis. In all cases the regularization parameter improves the performance with a degree of insensitivity to the parameter that is surprising but very promising. While the S22 test set is not one particularly suited to orbital optimization, it is a sensitive and important case with systematic errors that, while not improved directly with orbital optimization, are reduced by scaling back the correlation energy. These improvements are seen across all subsets of interactions—hydrogen bonding, dispersion, and mixed–but most prominently improve the dispersion interactions. Radical stabilization is where OOMP2 really shines and it is good to see that the level shift reduces error in these systems as well. It is important to recognize that the major failing of RIMP2 in these cases is due to the spin contamination in the reference, and RIMP2 can be improved using ROHF orbitals which are, however, not local minima in the full orbital space of spin polarized orbitals (and hence curves that smoothly separate bonds to correct fragments cannot be obtained). Barrier heights are another case that requires balancing the description of two different types of systems, in this case ground and transition states. Here too, the largest errors come from cases where spin symmetry breaking is not present equally on either side of the reaction leading to cases with large errors; however, these systems can not be simply rescued by a restricting the reference as reducing the degrees of freedom leads to even worse errors. The significant improvement seen in the description of single reference metal containing compounds is very encouraging. These systems are better described by the orbital-optimized reference, but also show great improvement with respect to the level shift.

Frequencies Recent results[100] have shown that for a collection of small radicals, while standard MP2 fails dramatically for vibrational frequencies that involve symmetry breaking, OOMP2 systematically overestimates bond lengths and under estimates frequencies. This overestimation CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 33

Figure 3.6: RMS errors of δ-OOMP2 relative to standard RIMP2 on various test sets. With- out regularization OOMP2 performs worse than RIMP2 for the G2 and S22 test sets but a level shift of 400 mEh improves δ-OOMP2 over RIMP2 and unregularized OOMP2 for all test sets.

(a) 0.05 (b) Bond Length Errors 2500 Frequency Errors 0.04 * *25006 cm -1 2000 0.03 OOMP2 δ-OOMP2 )

MP2 -1 0.02 1500 δ-OOMP2 (cm

0.01 " OOMP2 1000 MP2 0

Error vs. CCSD(T) (Å) 500 -0.01 500 1000 1500 2000 + HOOH+ HOOH+ HOOH+ HOOH+ LiO2 C3 NO 2 cis cis trans trans -1 "CCSD(T) (cm ) -0.02

Figure 3.7: (a) Bond length errors vs. CCSD(T) of OOMP2, δ-OOMP2, and MP2 for ﬁve small radicals. (b) Harmonic frequencies plotted against CCSD(T) for the same ﬁve radicals. R2 values for frequencies are 0.979, 0.998, and -0.003 for OOMP2, δ-OOMP2, and MP2 respectively. MP2 and reference CCSD(T) values taken from the work of Bozkaya[100].

of bond lengths ﬁts with our understanding that OOMP2 over correlates: as a single bond is pulled apart from equilibrium, electron correlations tend to grow stronger in the intermediate regime before they die oﬀ as the systems become separated. Thus the decrease in correlation energy from including the level shift parameter should decrease bond lengths, reducing the systematic errors. CHAPTER 3. REGULARIZED ORBITAL-OPTIMIZED MP2 34

Figure 3.7(a) shows the results with a 400 mEh level shift parameter confirming the improvement. The large scale failure of MP2 and systematic improvement of δ-OOMP2 over standard OOMP2 for frequencies is seen in figure 3.7(b). The improvement is particularly promising since it shows that the parameter independently chosen based on properties at the externally fixed geometries of the test sets is transferable to describing the correct equilibrium placement and local environment on the potential energy surface.

3.4 Conclusion

We have presented a simple proposal for regularizing orbital optimized MP2 for near orbital degeneracy. Comparisons to standard OOMP2 and MP2 on various test sets have shown that choosing a large nonzero value for δ not only helps the method avoid diverging to artificial minima but also improves the method’s accuracy. We have selected a roughly optimal value of 400 mEh to use as the recommended value for δ-OOMP2 based initially on thermochemistry, but which shows improvements for all of the test sets studied in this work. While the cost of δ-OOMP2 is the introduction of semi-empiricism, the benefits extend beyond improved statistical errors to include the ultimate goal of stabilizing the optimization to the presence of divergences. Since one of the greatest drawback of OOMP2 is the computational time required to iteratively calculate the MP2 energy, we plan to apply a level shift to the iterative O(N 4) orbital-optimized opposite-spin scaled second-order correlation (O2)[52] to make for a more tractable method. There are also other interesting possibilities for related future work. First, there is great interest in double hybrid density functionals (DHDFs) [101] at present (for instance [102–105]), including the recent possibility that orbital-optimized DHDFs [106] can offer significant advantages. Very likely the inclusion of a regularization parameter as a component of an OO-DHDF would be useful both for accuracy and stability of the resulting functional. Separately, electronic attenuation has been shown to substantially increase the accuracy of MP2 theory for non-covalent interactions in finite basis sets [107, 108]. It may be that combining regularization and attenuation will further broaden the applicability of these MP2-derived methods. 35

Chapter 4

Stability Analysis without Analytical Hessians

4.1 Abstract

Wavefunction stability analysis is commonly applied to converged self-consistent field (SCF) solutions to verify whether the electronic energy is a local minimum with respect to second order variations in the orbitals. By iterative diagonalization, the procedure calculates the lowest eigenvalue of the stability matrix or electronic hessian. However, analytical expressions for the electronic hessian are unavailable for most advanced post-Hartree Fock (HF) wave function methods and even some Kohn-Sham (KS) density functionals. To address such cases, we formulate the hessian-vector product within the iterative diagonalization procedure as a finite difference of the electronic gradient with respect to orbital perturbations in the direction of the vector. As a model application, following the lowest eigenvalue of the orbital-optimized second order Møller–Plesset perturbation theory (OOMP2) hessian during H2 dissociation reveals the surprising stability of the spin-restricted solution at all separations, with a second independent unrestricted solution. We show that a single stable solution can be recovered by using the regularized OOMP2 method (δ-OOMP2), which contains a level shift. Internal and external stability analyses are also performed for SCF solutions of a recently developed range-separated hybrid density functional, ωB97X-V, for which the analytical hessian is not yet available due to the complexity of its long-range non-local VV10 correlation functional.

4.2 Introduction

Self-consistent field (SCF) solutions to wavefunction theory and Kohn-Sham (KS)[109, 110] formalism of density functional theory (DFT) are typically determined by imposing constraints on the spin orbitals. These constraints not only lower SCF costs, but also allow the CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 36 approximate wavefunction to share some properties in common with the exact wavefunction such as spin or spatial symmetry. Variational minimization ensures that the energy is stationary with respect to first order changes in the spin orbitals. Therefore, second derivatives with respect to spin orbital coefficients must be positive for the energy to be a true local minimum, and the procedure to verify this condition is termed stability analysis.

Thouless originally derived the conditions for stability of HF wavefunctions from second quantization[111]. This was followed by a density matrix-based approach[112], and a refor- mulation of the Thouless conditions to treat both closed and open-shell systems[113, 114]. Seeger and Pople[115] devised a systematic approach to treat HF instability beginning with real spin-restricted HF orbitals, and progressively removing each of these constraints. For each case, they obtained the conditions for internal stability, where spin orbitals are varied within the space of deﬁned constraints, as well as external stability where one constraint is removed at a time. Stability analysis for HF involves the calculation of the lowest eigenvalue of a stability matrix (or electronic hessian). Since diagonalization of the large stability matrix (whose elements form a fourth rank tensor) may be prohibitive, stability analysis employs iterative diagonalization techniques such as the Davidson method[116]. Fortunately, the critical step in iterative diagonalization, which involves contraction of the stability matrix with a trial vector, can be performed in a manner very similar to forming a Fock matrix. Therefore the cost of SCF stability analysis is comparable to SCF costs.

The HF solution is typically used as a reference for advanced methods that incorpo- rate correlation such as second order Møller– Plesset perturbation theory (MP2) and coupled cluster (CC) theory, although HF orbitals quite commonly suffer from spatial or spin symmetry-breaking. To address these problems, orbital-optimized second-order perturbation theory (OOMP2)[117] distinguishes itself from standard MP2 by optimizing the zeroth order orbitals in the presence of correlation in an approach based on approximate Brueck- ner orbitals[118]. By optimizing the single reference, artificial spin contamination can be removed[117–120] and energies as well as properties of open shell molecules can be significantly improved[117–119, 121–123]. Because the energy is made stationary to changes in the orbitals, a Hellman-Feynman condition applies and all first order properties will be continuous as the orbitals change continuously[124]. Recently, δ-OOMP2 has been developed as a simple way to regularize the method against small HOMO-LUMO gaps as well as removing systematic errors in the method[125]. While approximate forms have been applied in previous studies[119], full analytical expressions for the electronic hessian are unavailable and finite difference electronic hessians are intractable. As a result, the stability of spin-restricted and unrestricted formalisms of OOMP2 has not been properly investigated. For the same reason, stability analysis is not available for size-consistent, Brueckner orbital-based coupled cluster techniques such as Brueckner theory doubles (BD)[126] and optimized-orbital coupled cluster doubles (OD)[118, 121, 127]. CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 37

The stability conditions for density functionals are essentially analogous to HF, and have been derived by Bauernschmitt and Ahlrichs for internal (singlet) and external (triplet) stability of restricted KS-DFT [128]. The formalism, however, requires calculation of second derivatives of the exchange-correlation energy. Analytical expressions for the second derivative of the exchange correlation term in KS-DFT are not available for all functionals. ωB97X-V, for instance, is a minimally parameterized range-separated hybrid functional that can accurately capture both non-covalent interactions as well as thermochemistry[129]. The functional includes non-local correlation described by VV10[130], for which an analytical form of the hessian has not yet been derived. In such cases, stability analysis can prove intractable since calculation and diagonalization of the full ﬁnite diﬀerence electronic hessian is not feasible.

Our aim is to establish a technique for stability analysis that is readily applicable to any post-HF or KS-DFT method, regardless of the availability of analytical second derivatives of electronic energy. We have previously reported a finite differences implementation of the Davidson method to calculate the lowest eigenvalue of a nuclear hessian, which can determine whether a stationary point calculated using geometry optimization is a minimum or saddle point. The same approach can be extended to wavefunction space, where the finite differences Davidson method is applied to perturbations in the molecular orbitals in order to calculate the lowest eigenvalue of the electronic hessian[131]. Potential curves for dissociation of H2 are calculated to analyze the stability of SCF solutions for OOMP2 and δ-OOMP2 theory, with some interesting and in some ways remarkable results. Additionally, finite-difference based stability analysis is applied to the ωB97X-V functional in order to demonstrate the utility of this technique when second derivatives are unavailable.

4.3 Method

The Davidson method is an iterative diagonalization procedure to determine a few extreme eigenvalues of large symmetric matrices when full diagonalization is prohibitive. The algorithm is described in detail elsewhere[116, 132]. Brieﬂy, the procedure employs a small orthonormal subspace of vectors, Bk = [bi] at each iteration k, consisting of dominant com- T ponents of the desired eigenvector of a matrix, A. A smaller interaction matrix, Bk ABk, is constructed and diagonalized to obtain the lowest/highest eigenpair, (λk, yk). The Ritz vector, xk = Bkyk, is then used to estimate the residual error between the exact and approximate eigenvector, rk = −(λkI − A)xk. The initial subspace is augmented with a new vector that contains this information, and the procedure is iterated until convergence.

The Davidson method was originally applied to large-scale configurational interaction (CI) treatment of wavefunctions[116, 133]. The finite difference implementation of the David- son method can be used when the matrix calculation itself is intractable. For instance, if CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 38 the matrix A corresponds to the hessian of the energy with respect to nuclear displacements, the exact matrix-vector product, Ab1, is replaced with a finite difference approximation in terms of the gradient of the energy (∇E)

(∇E(X + ξb ) − ∇E(X − ξb )) Ab ≈ 0 1 0 1 (4.1) 1 2ξ

where b1 is the subspace guess, X0 corresponds to nuclear coordinates of a system, and ξ is the finite difference step. This expression can be used to calculate a few key eigenvectors as inputs to mode-following methods for transition state searches on nuclear potential energy surfaces[134–136]. The same principle can also be applied to selective mode tracking in vibrational analysis[137, 138], and characterization of stationary points[131, 139], where the lowest one or two eigenvalues of the nuclear hessian are sufficient to verify whether a geometry corresponds to a minimum or transition state, respectively.

Wavefunction stability analysis also requires only the lowest eigenvalue of the electronic hessian. Therefore, the finite difference Davidson approach can be extended to stability analysis in cases where analytical hessians are either expensive or unavailable. Since rotations between occupied-occupied or virtual-virtual orbitals do not affect the total energy, stability analysis is carried out in the space of occupied-virtual rotations. The most obvious choice for the initial subspace guess, therefore, corresponds to a HOMO-LUMO rotation. To avoid possible orthogonality between the guess and the exact eigenvector, a small amount of randomness is added in to the subspace guess.

Orbital perturbation in the occupied-virtual space along the subspace guess closely follows the procedure outlined by Van Voorhis and Head-Gordon[140]. A skew-symmetric unitary transformation matrix,U1± , is determined by ﬁrst scaling the guess,

∆1± = ±ξb1 (4.2)

where b1 is the subspace guess corresponding to HOMO-LUMO rotation, ξ(= 0.01) is the ﬁnite diﬀerence step, and the number in the subscript corresponds to the iteration. The transformation matrix is then given by

∆1± U1± = e (4.3)

The oﬀ-diagonal elements of this matrix correspond to rotations in the occupied-virtual space. The rotated orbitals are given by a unitary transformation of the converged SCF σ orbital coeﬃcients, C0 , where corresponds to α- or β-spin.

Rotations of α-spin and β-spin orbital coeﬃcients are identical during internal stability analysis of restricted or unrestricted spin orbitals. In order to examine external stability of CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 39 restricted spin orbitals, on the other hand, spin symmetry needs to be broken. Therefore, α-spin and β-spin orbital coeﬃcients are rotated in opposite directions.

β α β α C1+ = −C1+ and C1− = −C1− (4.4) The hessian-vector product Davidson iterations is then calculated similar to equation 4.1 using finite differences of gradients with respect to the rotated coefficients

" #T (∇E(Cα ) − ∇E(Cα )) (∇E(Cβ ) − ∇E(Cβ )) Ab ≈ 1+ 1− , 1+ 1− (4.5) 1 2ξ 2ξ

where A corresponds to the electronic hessian. The Davidson algorithm proposed by Sleijpen and van der Vorst[141] is then employed to iteratively calculate the lowest eigenvalue.

Convergence can be accelerated using a good preconditioner for the residual. In the th original Davidson algorithm, the preconditioner at the k iteration, Ξk, is given by

−1 Ξk = (λkI − D) (4.6)

where D is a matrix consisting of the diagonal elements of A. A reasonable guess for the diagonal hessian is the difference between orbital eigenvalues, , in the occupied-virtual space[140], Dia,jb = (a − i)δijδab (4.7) where subscripts (i, j) correspond to occupied orbitals and (a, b) to virtual orbitals. In order to ensure the convergence of the method to the lowest eigenvalue, the preconditioner must be negative definite[132]. In cases where preconditioning exceeds a certain cutoff, the cutoff value replaces the difference between the eigenvalue and diagonal element. The chosen value, ∆E = −0.1Eh, is determined using simple benchmarking of the H2 molecule at equilibrium separation with B3LYP[142, 143], and correlation-consistent basis sets. The technique is implemented in a developmental version of Q-Chem 4.2[144], in order to examine internal stability of real restricted or unrestricted orbitals, as well as external stability of restricted orbitals for OOMP2 theory and any KS-DFT.

4.4 Results

HF vs. orbital-optimized MP2 for bond dissociation Bond dissociation problems are an important application of stability analysis. The reason is that many orbital optimization methods will not automatically change the character of the orbitals from restricted to unrestricted as the bond is stretched, and therefore stability analysis is needed to detect such a change. Figure 4.1 illustrates the standard result seen for CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 40

Figure 4.1: Potential curves (green for unrestricted and red for restricted, where it diﬀers from unrestricted) for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix (purple for internal stability of the unrestricted solution, blue for external stability of the restricted solution, where it diﬀers from unrestricted) at the Hartree-Fock (HF) level. The lowest energy solution changes character from restricted to unrestricted when the former becomes unstable.

Hartree-Fock theory for the toy problem of H2 dissociation. The RHF to UHF instability is detected by a sign change of the smallest eigenvalue, which occurs at a bond-length of about 1.2A˚. Beyond this distance, the UHF solution exhibits an increasing positive smallest eigenvalue and becomes a distinct, lower energy solution, whilst the smallest eigenvalue of the RHF solution becomes steadily more negative.

How does the inclusion of electron correlation in the OOMP2 method affect this picture? The results are shown in Figure 4.2, and at first glance the ROOMP2 and UOOMP2 energy curves look qualitatively similar to the RHF and UHF ones. However the ROOMP2 energy reaches a maximum value around 2.8A˚ and then begins to turn over, as a result of the HOMO-LUMO gap decreasing. The ROOMP2 and UOOMP2 curves actually cross again at still larger separations than are shown on the figure. What are the implications for orbital stability analysis? Using the finite difference stability analysis code yields very interesting results. The ROOMP2 and UOOMP2 solutions are in fact both stable when they are distinct CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 41

Figure 4.2: Potential curves for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix using orbital-optimized MP2 (OOMP2) in the cc-pVDZ basis. The format follows Figure 6.1. OOMP2 behaves qualitatively diﬀerently from HF (see Figure 4.1). The restricted solution is stable (positive eigenvalue) to spin-polarization at all bond-lengths, and a distinct stable unrestricted solution appears at partially stretched bondlengths.

solutions. They apparently do not coalesce upon going to shorter bond-lengths.

As a surprising consequence, despite the Hellman-Feynman condition for OOMP2, there are still ﬁrst derivative discontinuities in the dissociation curve for single bond dissociations such as H2. It is scarcely visible in Figure 4.2, but this is nonetheless a real eﬀect. As a result of the ROOMP2 solution always being a true minimum in orbital space, the UOOMP2 solution must cross it in the energy coordinate without crossing in orbital space.

To better understand the topography of the solutions we look at the UOOMP2 energy for H2 as a function of spin-polarization from the ROOMP2 solution in the minimal basis case where there is only a single orbital rotation angle (θα and θβ) in each of the α and β spaces. A spin polarization angle, φ, can therefore be deﬁned such that θα = φ and θβ = −φ. Figure 4.3 shows the OOMP2 energy as a function of φ for a number of bond lengths close to the crossing, from ROOMP2 being lowest energy to UOOMP2 being lowest. The key observation from Figure 6.3 is the appearance of a second minimum at non-zero φ as the CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 42

Figure 4.3: The dependence of the OOMP2 energy of H2 in a minimal basis on the spin polarization angle (see text for deﬁnition) at a series of bond-lengths around the critical value at which the character of the lowest energy solution changes. There are two local minima, one restricted and one unrestricted, at these bond-lengths, and at the critical bond- length the nature of the lowest energy solution switches discontinuously. bond is stretched, whilst the ﬁrst stationary point (φ = 0) remains a minimum. As the bond-length increases, the second solution eventually becomes the global minimum leading to the discontinuous change in orbitals as we follow the lowest energy orbitals.

While there is no reason to assume that the global minimum of a nonlinear problem will not jump between multiple minima as parameters change, it is still surprising to see it here due to our experience with HF (as exempliﬁed by Figure 4.1). HF is a diagonalization-based approach, and so two states with the same energy that can couple through the Hamiltonian should split in energy. OOMP2 on the other hand adds a perturbative correction, which in this case preferentially stabilizes the restricted solution and lowers its energy relative to the unrestricted orbitals bringing their energies to coalescence. Similar observations have been made in the context of orbital optimization in active space methods[145, 146]. In cases such as these, as a consequence of the discontinuous change in orbitals, the potential energy surface exhibits a ﬁrst derivative discontinuity at the point of the jump in orbital solutions (here, the ROOMP2 to UOOMP2 transition). CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 43

Figure 4.4: Potential curves for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix using regularized orbital optimized MP2 (δ-OOMP2) in the cc-pVDZ basis. The format follows Figure 6.1. δ-OOMP2 behaves qualitatively diﬀerently from OOMP2 (see Figure 4.2), but is similar to HF (see Figure 4.1). The restricted solution becomes unstable at a critical bond-length, beyond which the unrestricted solution is lowest in energy.

How might one overcome this unphysical behavior of OOMP2, and recover smoother potential energy surfaces? We cannot give a complete answer here, but we can apply stability analysis to a modified form of OOMP2 that includes a fixed level shift of 0.4 a.u., termed δ-OOMP2. δ-OOMP2 has been shown to yield systematic improvements relative to OOMP2 across a broad range of properties while being robust to divergences during orbital optimization[125]. The performance of δ-OOMP2 for the dissociation of H2 is shown in Figure 4.4, and presents a striking contrast with OOMP2 shown in Figure 4.2. δ-OOMP2 shows only one stable solution at any geometry, like HF, and unlike OOMP2. As a consequence, as shown in Figure 4.5 for minimal basis H2, the optimized orbitals for the global minimum do not change discontinuously as the bond is stretched, and thus the potential energy surface is continuous through first derivatives. Further calculations on a much larger range of molecules are required to test the generality of the present positive result, and the stability analysis method introduced here is a crucial tool for this purpose. CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 44

Figure 4.5: The dependence of the δ-OOMP2 energy of H2 in a minimal basis on the spin polarization angle (see text for deﬁnition) at a series of bond-lengths around the critical value at which the character of the lowest energy solution changes. For any given bond-length there is only one local minimum, which changes character from restricted to unrestricted at the critical bond-length.

4.5 Conclusions

Stability analysis has thus far been limited to formalisms for which analytical second derivatives are available since the cost of full finite difference hessian calculation is prohibitive. We describe a hessian-free approach in which the hessian-vector product required for iterative diagonalization within the Davidson method is approximated by finite differences of the gradients with respect to rotation of molecular orbital coefficients in the occupied-virtual space. The procedure is implemented for both orbital-optimized post-HF methods such as OOMP2 as well as DFT, and can successfully examine internal and external stability with respect to spin symmetry constraints. In future, the implementation will also include internal and external stability analysis for complex as well as general spin orbitals. The technique will also be made available for other orbital-optimized methods such as coupled cluster-based BD and OD, for which stability analysis has hitherto not been performed. CHAPTER 4. STABILITY ANALYSIS WITHOUT ANALYTICAL HESSIANS 45

4.6 Acknowledgements

The development of the stability approach is done by the first author of this paper from which this chapter is cut from, Shaama Mallikarjun Sharada. This research was supported by a grant from Chevron Energy Technology Co., by the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC05- 06OR23100, and by the Office of Science, Office of Basic Energy Sciences, of the (U.S.) Department of Energy under Contract No. DE-AC02-05CH11231. We are also grateful to Narbe Mardirossian for providing instructive examples for the stability analysis with the ωB97X-V functional, and Dr. Julien Panetier for electrocatalyst geometries. We acknowledge computational resources obtained under National Science Foundation (NSF) Award No. CHE-1048789 and NSF CHE-0840505. 46

Chapter 5

Exponential Regularized OOMP2 for Dissociations

5.1 Introduction

Dissociation processes are good test cases for highlighting the problems associated with the symmetry dilemma[28]. For example, for Li2 dissociation, restricted MP2 gives a good description of the equilibrium geometry but qualitatively fails in the dissociation limit, while unrestricted MP2 can handle the dissociation limit properly but fails in the equilibrium regime. In our previous work, we have proposed orbital-optimized second-order perturbation theory (OOMP2) as a black box approach that connects the closed shell equilibrium description to the unrestricted dissociated state[52, 147]. The OOMP2 method, which can be thought of as a approximation to Brueckner orbitals[59], optimizes orbitals in the presence of correlation thereby reducing errors from spin contamination and improving descriptions of bond lengths, frequencies, and relative energies[52, 56, 57, 63, 64]. Another benefit of the method is that due to a Hellmann-Feynmann condition, first derivatives of the energy will be continuous as the orbitals change continuously[65]. The possibility of degeneracies leading to failures during optimization encouraged the development of δ-OOMP2, where a 400 Eh level-shift is added to the T amplitudes to regularize the method[147]. It was shown that this parameterization improved properties of the method while also stabilizing the orbital potential energy surface for H2. Previously, the development of a gradient-based stability analysis algorithm allowed us to calculate the stability of the OOMP2 solutions across dissociation curves and found the surprising result that for the dissociation of H2, while the energy was changing continuously from the restricted to unrestricted solution, the wavefunction was not[15]! The restricted solution remained stable throughout the curve meaning that instead of the unrestricted solution breaking off at the unrestriction point, it formed at a higher energy and the two crossed in energy but not wavefunction space. It was shown that the δ-OOMP2 is capable CHAPTER 5. EXPONENTIAL REGULARIZED OOMP2 FOR DISSOCIATIONS 47

of fixing this problem by pushing the unrestriction point closer to equilibrium for H2. Unfortunately, further tests revealed that the current regularization scheme does not fix the problem of discontinuous orbitals in several other dissociations. While increasing the level shift parameter is an option, a significant increase would lead to a degradation of the methods accuracy. Another consideration is an entirely different approach to regularization entirely. Inspired by Evangelista’s work on similarity renormalization group methods[16] we apply an exponential regularization to OOMP2.

5.2 Theory

We regularize the energy expression as follows,

(2) ab E (σ) = Teij (σ)hij||abi

hij||abi 2 ab2 ab − σ ∆ij Teij (σ) = ab 1 − e ∆ij ab ∆ij = i + j − a − b Unfortunately, the inclusion of the orbital energies in our regularization factor leads to additional terms in the orbital gradient.

(2) ∂E X X ∂Ubj = (2Fbj + 2Lbj) ∂θai ∂θai j b

X (2) X (2) X X ab Lai = Pejk Aaijk + Pebc Aaibc − Tejk (ij||bk) jk bc jk b X X bc X (2) X (2) + Teij (ab||jc) + FajPeji + Peab F(bi) j bc j b

Apqrs = 2(pq|rs) − (pr|ps) − (ps|qr)

bc bc ! (2) −1 X X ∆jk ∆ 1 P = T abT ab − ik eij eik ejk 2 ab2 2 ab2 2 − σ ∆jk − σ ∆ik i − j k ab 1 − e 1 − e

4 2 − 2 ∆ab2 + δ hik||abi e σ ik ij σ bc ac ! (2) 1 X X ∆ij ∆ij 1 P = T acT bc − eab eij eij 2 bc2 2 ac2 2 − ∆ − σ ∆ij − ij c 1 − e σ ij 1 − e a b

4 2 − 2 ∆ac2 − δ hij||aci e σ ij ab σ CHAPTER 5. EXPONENTIAL REGULARIZED OOMP2 FOR DISSOCIATIONS 48

5.3 Results

We ﬁrst look at the breakdown of δ-OOMP2 for dissociating single and double bonds. Fig 5.1 and 5.2 show that for ethane and ethene, δ-OOMP2 jumps discontinuously between orbitals. This discontinuity in the wavefunction is seen in the jump in hS2i at the unrestriction point and more deﬁnitively through the stability of the restricted solution as seen in the plotted lowest eigenvalue of the orbital Hessian. For ethane, the jump is small; the restricted solution remains stable for less than 0.1 A˚ after the unrestricted solution becomes the global minima. For ethene, we can see that the restricted solution remains stable throughout the entire dissociation process.

Figure 5.1: Dissociation curve of ethane in an aug-cc-pVTZ basis. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals.

To get an idea of how the new exponential regularization eﬀects general performance, we look to the W4-11 small molecule test set of atomization energies, reaction energies, heavy atom transfers, and isomerizations[148]. We see in Table 5.1 that errors are minimized by a value of σ about 3.2. Looking at the multireference (MR) and nonMR subsets (based on a %TAEe[T4 + T5] diagnostic with a cutoﬀ at ≥ 0.5%) we see that the new parameterization out performs δ-OOMP2 on non-MR but is poorer for the MR subset. This is promising as the method is fundamentally single reference and should thus be focused on non-MR problems. On the other hand, we don’t want to disregard the MR subset as some portion of the %TAEe[T4 + T5] diagnostic may be indicative of the single reference, poor zeroth order CHAPTER 5. EXPONENTIAL REGULARIZED OOMP2 FOR DISSOCIATIONS 49

Figure 5.2: Dissociation curve of ethene in an aug-cc-pVTZ basis. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals.

systems OOMP2 is intended to handle. Thus a value of 3.2 appears to be a reasonable compromise between the error on the subsets and a preference for a higher regularization parameter.

reg nonMR MR Total σ-OOMP2 0 7.85 10.63 9.76 2.4 7.11 6.15 6.49 3.2 6.93 7.01 6.98 4.0 6.60 8.57 8.14 δ-OOMP2 7.84 5.44 6.37

Table 5.1: Root mean square error (RMSE) in kcal/mol for σ-OOMP2 with various values of σ and δ-OOMP2 with the recommended parameterization of 400 mEh.

We now consider our new σ-OOMP2 approach on the highly problematic dissociation of non-trivial bonds. Across the board, we see the σ regularization ﬁx the qualitative failing of untampered OOMP2. Figures 5.3 and 5.4 show that double and single bonds are dissociated from a restricted equilibrium to the correct asymptotic limit. The clearest way to see that the orbitals unrestricted continuously, is that the lowest eigenvalue of the restricted solution crosses zero at the same point that the hS2i value becomes non-zero. This regularization CHAPTER 5. EXPONENTIAL REGULARIZED OOMP2 FOR DISSOCIATIONS 50 parameter is even capable of dissociating triple bonds continuously as shown in Figure 5.5 for ethyne.

Figure 5.3: Dissociation curve of ethane in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals.

With the the orbitals changing continuously across the potential surface due to the new regularization approach, we have restored the ﬁrst derivative continuity that makes OOMP2 particularly appealing. Unlike MP2, response properties of σ-OOMP2 will be continuous due to the optimization with respect to orbitals and now the important condition that orbitals change continuously.

5.4 Conclusion

We have further investigated the unexpected behavior of OOMP2 having discontinuous orbital changes during bond dissociations. Based on a recommendation from the work of Evangelista, we have implemented a new regularization approach, σ-OOMP2 that is capable of improving thermochemistry as well as ﬁxing the qualitative failures seen during dissociations by selecting a regularization parameter σ = 3.2. CHAPTER 5. EXPONENTIAL REGULARIZED OOMP2 FOR DISSOCIATIONS 51

Figure 5.4: Dissociation curve of ethene in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals.

5.5 Acknowledgements

D. S. is supported in part by the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC05-06OR23100. This work was also supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under contract no. DE-AC02-05CH11231. We acknowledge computational resources obtained under NSF award CHE-1048789. CHAPTER 5. EXPONENTIAL REGULARIZED OOMP2 FOR DISSOCIATIONS 52

Figure 5.5: Dissociation curve of ethyne in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals. 53

Chapter 6

Regularized CC2

6.1 Introduction

While mean field Hartree Fock (HF) theory often provides a good foundation for calculating molecular energies, it has been well established that one needs to go further and account for dynamic correlation in order to get energies with acceptable accuracy. Coupled cluster formalism has proven to be a valuable approach to describing dynamic correlation by truncating the expansion by level of excitation or based on ideas of perturbation theory[6]. CC2[9] and Møller Plesset second-order perturbation theory (MP2) can both be viewed as perturbative approximations to full coupled cluster singles and doubles (CCSD). In perturbation approaches, the molecular Hamiltonian is broken into a mean-field part, Fˆ, that is solved exactly within a given basis and the remaining correlated part, ∆V . In CC2, the doubles amplitude equations are treated to first order in ∆V where singles amplitudes are considered zeroth order and doubles are treated as first order. In MP2, by contrast, the amplitudes themselves are expanded in orders of ∆V and the energy is truncated at second order[6]. As approximations to CCSD, MP2 and CC2 share several properties; both methods are intermediate in accuracy between HF and CCSD and scale as O(N 5). These methods both have efficient resolution of the identity implementations[86, 149] and can have their scaling reduced to O(N 4) using a scaled opposite spin approximation[150, 151]. While the scaling of the two methods is equivalent, the typical performance is not, with MP2 being faster in practice due to CC2 involving an iterative O(N 5) step and requiring storage of the doubles amplitudes. Despite their similarities, the two methods have found favor in differing applications–MP2 for ground state energies, and CC2 for excited states. Excited states from linear response theory can only reliably be obtained for CC2 as MP2 will have second order poles which are inconsistent with the exact result[9, 152]. Methods such as CIS(D)[153, 154] serve as excited state analogs of MP2. On the other hand, the iterative nature and observed systematic CHAPTER 6. REGULARIZED CC2 54 errors of CC2, as well an increased sensitivity to strong correlation have led to it being seen as non-competitive with MP2 for ground state calculations[151, 155]. It can also correct Hilbert space topology by removing the artificial separate minima that may occur for both restricted and unrestricted orbitals upon bond stretching[15]. One major weakness of MP2 theory is its strong dependence on good HF reference orbitals. For HF solutions, MP2 theory only includes doubles corrections which is to say that it cannot correct singles amplitudes (which are associated with orbital rotations) beyond first order, the same as HF. By classifying single excitations as a zeroth order effect, CC2 solves the singles amplitude equations iteratively and to full order using approximate doubles equations. This extra freedom allows the method to correct for deficiencies in the HF orbitals but as mentioned previously, can lead to problems when strong correlations are present[155]. In simple cases such as closed shell organic molecules, the HF reference is qualitatively correct enough that MP2 is able to account for the correlation energy on top of the HF reference. However, in the more difficult cases of radicals, inorganics, aromatics, and transition states, HF orbitals can be qualitatively incorrect, as often signaled by spin contamination[55, 56, 156]. In light of these problematic references, orbital-optimized MP2 (OOMP2)[52, 64] was proposed as a way to introduce correlated reference orbitals. This improvement comes at the cost of OOMP2 becoming an iterative fifth order method. While solving the problem of poor references, OOMP2 created new issues of divergences appearing during optimization. We suggested a simple way to correct these divergences by adding a level-shift to the standard T2 equation leading to δ-OOMP2[147]. This level-shift parameter was chosen to be rather large (400 mEh) so as to make optimization more robust but also, importantly, because it removed systematic errors in atomization energies, radical stabilizations and geometries, reaction energies, and barrier heights. In light of the observed improvement of OOMP2 upon inclusion of a regularization parameter, one might wonder if CC2 could benefit similarly. Although there are no orbital stability reasons for considering a level-shift for CC2 since we don’t need to avoid divergences, we consider it valuable in and of itself to assess the performance of the parameterization on various methods to improve systematic errors. For CC2, it is better to view the parameterization as coming from a penalty on the ˆ norm of T2 which can be introduced in the Lagrangian; this term will be additive with the orbital energies and end up shifting the denominator in the T2 equations. Thus we can view the affect of the parameterization as damping the norm of the T2 operator which may be systematically overestimated similarly to MP2. In mathematical terms, the penalty function serves to regularize otherwise ill-conditioned equations for the CC2 doubles amplitudes. In this study, we are interested in the extensibility of this simple parameterization to, not only, ground state CC2 calculations, but also more interestingly to excited states. Reg- ularization can be naturally extended to excited state calculations through linear response, or by damping the R2 operator in the equation of motion (EOM) formulation. If damping T2 amplitudes is a robust approach, then improvements in ground state energies should similarly improve the transformed hamiltonian of EOM-CC2. CHAPTER 6. REGULARIZED CC2 55

6.2 Computational Methods

For this work, standard CC2 was implemented in QChem[40] taking advantage of the libten- sor library[40]. We can write out the CC2 Lagrangian with the The modiﬁcations to the standard equations[9] is simply the addition of a constant, δ, to the zeroth order matrix elements in the doubles equations: ˆ ˆ hµ1|H + [H, T2]|0i = 0 ˆ hµ2|H + [F + δ, T2]|0i = 0

The T2 amplitudes can then be solved as a function of δ as:

ab −T1 T1 ab hij |e He |0i tij = i − a + j − b + δ The standard EOM-CC2 equations are similarly recast with a modiﬁed energy denominator for the doubles:

ab −T1 T1 c c ab hij |e He |kirk rij = i − a + j − b + δ + ω For an implementation-level description of these equations see the supplemental information of Hohenstein et. al.[157]. All calculations are run with the frozen core approximation. Ground state benchmarks are similar to those considered for assessing δ-OOMP2[147] and include the 148 atomization energies from the G2 test set[90, 91] compared to QCISD(T)[89] values in a cc-pVTZ basis. From Grimme’s GMTKN30 database[97] we have considered barrier heights and reaction energies from the BH76 set and radical stabilization energies of the RSE43 set in a cc-pVTZ basis. Lastly, we’ve looked at the W4-11 small molecule test set which covers a range of atomization energies, reaction energies, heavy atom transfers, and isomerizations using an aug-cc-pVTZ basis set[148]. This set is split into multi reference (MR) and non-MR subsets based on the %TAEe[T4 + T5] diagnostic from the benchmark data where molecules with ≥ 0.5% being classiﬁed as MR which corresponds to a natural break in the data and selects approximately 30% of the molecules. Excited state calculations were benchmarked using the singlet excitation test sets of valence states from Thiel et. al.[158, 159] and both valence and Rydberg states from Wiberg et. al. [160]. Due to the presence of Rydberg states which are ignored in the Thiel benchmarks, excitations were matched up with benchmark values by recalculating CC2 values and matching energies within symmetry representation rather than by the order listed by the term symbol. Energies were calculated using the TZVP and aug-cc-pVTZ basis sets for the Thiel set and 6-311G(3+,3+)**[161] for the Wiberg set to compare to the benchmark results. CHAPTER 6. REGULARIZED CC2 56

6.3 Results and Discussion

Figure 6.1 plots the root mean square error (RMSE) of δ-CC2 for various values of δ from 0 to 400 mEh relative to the RMSE of RIMP2 on the same test sets. By standardizing to RIMP2, these RMSEs allow for the errors to be considered on the same scale and allows for a quick comparison of CC2 and MP2 performance. Before considering the parameterization of CC2 it is interesting to note that the standard method performs similarly to RIMP2 for the more well behaved systems (e.g. the RMS error for RIMP2 and CC2 is 13.44 and 12.72 kcal/mol for the G2 test set) but mirrors OOMP2 in its improvement on radicals and more difficult multireference systems (e.g. the RMS errors for RIMP2, OOMP2, and CC2 are 4.25, 1.48, and 2.06 kcal/mol for the RSE43 test set). Mean field HF has a difficult time describing radicals and often qualitatively fails as signaled by a spin contaminated reference; CC2 allows for a modified reference through the iterative inclusion of T1 amplitudes which can be viewed as allowing for a correlated reference. While CC2 cannot be expected to properly describe MR systems, by allowing for a modified reference, it can at least improve over MP2 which is stuck in the world of completely uncorrelated orbitals. The one exception to expectations here is the lack of improvement for barrier heights (RMS error is 4.47 kcal/mol for CC2 and 4.44 kcal/mol for RIMP2) which tend to be less well behaved and thus more poorly described by the mean-field approximation.

Figure 6.1: δ-CC2 RMSE for various ground state test sets divided by RIMP2 RMSE on the same sets for various values of δ. CHAPTER 6. REGULARIZED CC2 57

When we look at the eﬀect of regularization through the varying of δ, we see improvements for values of δ somewhere between 100 and 200 mEh or simply very little change. The greatest improvement is seen for atomization energies with most other properties relatively insensitive to even large values of the δ parameter. Based on these results we select a regularization parameter of 150 Eh for ground state δ-CC2. CC2 has been shown to qualitatively fail for calculating the equilibrium geometry of ozone due to its biradical nature[162]. Unlike MP2 and higher level coupled cluster theory, restricted CC2 predicts a barrierless symmetric dissociation due to an unbalanced description of correlation in the biradicaloid species. Turning to δ-CC2 we can see from Figure 6.2 that a regularization parameter of 150 Eh leads to a properly bound state and increasing it further primarily leads to a constant shift in the surface. The regularization appears to make the method less sensitive to problematic correlations when electrons are well paired.

Figure 6.2: Ozone symmetric dissociation curve at angle 142.76◦ for CC2 with regularization parameters 0, 100, 150, and 200 mEh and CCSD in an aug-cc-pVTZ basis.

Excited states for molecules in the Thiel test set are calculated in the TZVP and aug- cc-pVTZ basis sets. Table 6.1 shows an analysis of the errors with respect to the subset of 22 excitations which were calculated with CC3 as the more approximate CCSDR(3)[163] results have nearly the same systematic error as CC2 with respect to the higher level CC3 (mean error 0.08 vs 0.11)[159]. Although the performance on these valence states is not an improvement, the accuracy is only slightly degraded with the addition of the regularization parameter. To more completely understand the eﬀects of regularization, we also consider the excited state benchmarks of CHAPTER 6. REGULARIZED CC2 58

TZVP aug-cc-pVTZ δ 0 100 200 0 100 150 200 ME 0.14 0.21 0.27 0.11 0.20 0.24 0.27 RMSE 0.25 0.31 0.37 0.17 0.25 0.29 0.33 MAX 0.86 1.00 1.12 0.43 0.56 0.64 0.72

Table 6.1: Excited state errors on the Thiel test set for regularized δ-EOM-CC2 vs. CC3 values in TZVP and aug-cc-pVTZ basis for varying values of δ. All errors in eV.

Wiberg, which contain both valence and excited states. Table 6.2 shows the performance of δ-EOM-CC2 on the total test set as well as the 39 Rydberg and 30 valence states separately. Here we see that again, the systematic error for valence states is slightly increased, but now this degradation is compensated by improved performance on Rydberg states. Previous studies have characterized EOM-CC2 as having diﬃculties with Rydberg states compared to other methods[164], so here we can view the regularization as creating a more balanced description of two subsets.

Total Rydberg Valence δ 0 100 150 200 0 100 150 200 0 100 150 200 ME -0.18 -0.4 0.01 0.06 -0.48 -0.34 -0.28 -0.22 0.20 0.35 0.39 0.44 RMSE 0.53 0.48 0.46 0.45 0.60 0.45 0.38 0.32 0.42 0.52 0.54 0.57 MAX 1.25 1.29 1.31 1.33 1.25 1.04 0.95 0.86 1.00 1.29 1.31 1.33

Table 6.2: Excited state errors on the Wiberg test set for regularized δ-EOM-CC2 values in a 6-311(3+,3+)G** basis for varying values of δ vs. accurate experimental values. All errors in eV.

6.4 Conclusion

The regularization approach created to avoid divergences in OOMP2 has been extended to CC2 for the dual purpose of creating a method with lower systematic errors at no computational cost and assessing the robustness of this particular semi-emperical parameterization. Benchmarks on ground state systems show improvements that are maximized nearly across the board at around δ = 150 mEh which we suggest for all future work. For excited states, the same regularization parameter is seen to create a more balanced description of Rydberg and valence states. The improved performance of δ-CC2 across a wide range of ground state ˆ test sets shows the eﬃcacy of the simplistic regularized T2 approach to parameterization, while the excited state results show that these same ground state improvements might not always transfer to all types of excited states but may still create a more robust, balanced CHAPTER 6. REGULARIZED CC2 59 method, as evidenced by lower overall RMS error in the test sets containing both valence states (degraded slightly) and Rydberg states (improved signiﬁcantly).

6.5 Acknowledgements

Many thanks to Evgeny Epifanovsky for his work on the object-oriented ccman2 module and his assistance in developing within it. D. S. is supported in part by the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC05-06OR23100. This work was also supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under contract no. DE-AC02-05CH11231. We acknowledge computational resources obtained under NSF award CHE-1048789. 60

Chapter 7

Path Integrals for Anharmonic Vibrational Energy

7.1 Introduction

An electronic structure theorist’s ﬁrst approximation is to assume the nuclear and electronic problems are separable and then focus on the electrons. When accuracy is limited by neglect of nuclear vibrations or if nonzero temperature estimates of thermodynamic quantities are required, the vibration problem comes back to the forefront. The obvious starting point when dealing with vibrations is the harmonic approach which simply requires a calculation of the matrix of second derivatives of the electronic energy with respect to nuclear displacements– the Hessian. Diagonalizing the Hessian gives us the frequencies, reduced masses, and normal modes from which we can calculate harmonic approximations to zero-point vibration energy (ZPE) as well as any other thermodynamic quantities we desire because we have a simple analytic form for the partition function for a harmonic oscillator,

− 1 β ω e 2 ~ QHO = 1 − e−β~ω While simple in form and straightforwardly calculated, the harmonic approximation is still fundamentally limited in accuracy. This limitation can be a foundational one when comparing to high accuracy experimental results or if dealing with cold, light atoms. To go beyond this level of electronic structure theorist can turn to several methods which will be very familiar from experience with electrons, namely VSCF, VPT, and VCI. Like their electronic counterparts, accuracy comes at a cost. For VCI, reasonable accuracy can be obtained by including up to quadruples but doing so scales O(N 8), but the presence of Fermi resonances can lead to catastrophically bad performance. Another issue that arises with the wavefunction, based approaches to the vibrational problem is that higher derivatives with respect to nuclear displacements must be calculated either by having specially implemented code or using cumbersome ﬁnite diﬀerence. CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 61

Another way to calculate the full vibrational energy is to use a real space path integral approach with Monte Carlo sampling (PIMC). Broadly, the advantages are that the method can be carried out with only single-point energy evaluations and that the sampling is not inherently superlinearly scaling (although this depends very much on the efficiency of sampling). The drawback is in the huge prefactor that comes from gathering a statistically significant number of samples, and the fact that comes from gathering a statistically significant number of samples, and the fact that each sample still costs as much as the single point evaluations from your model for the electrons. What we propose then is to take the dearth of information about the system that we’ve gained from the Hessian and develop a PIMC method that efficiently samples just the remaining anharmonic part of the vibrational energy. To do this we apply a static harmonic propagator (as opposed to the more standard free particle propagator) sampled with Levy flights for large scale, global moves and in conjunction with thermodynamic integration so that we can estimate anharmonicity directly. The harmonic oscillator propagator has been used in the past statically for toy systems[165] as well as dynamically in several cases[166, 167]. Various other high level propagators have also been proposed[168–170]. We are interested primarily in the static harmonic propagator as the various others require calculations of the gradient or hessian of V at each step as opposed to just a single point value, which would make the method too costly to consider applying with electronic structure theory. Thermodynamic integration[171] gives us a direct way to calculate free energy differences between two given Hamiltonians and has been used on a broad class of problems, from reaction profiles[172–176], to isotopic substitution energies[177], to most relatedly anharmonic corrections for high temperature classical vibrational energy[178]. We were initially interested in high accuracy vibrational energies after a study on sulfate- water clusters found that their assignment of low energy conformers was limited by the accuracy of their zero-point energy which had significant anharmonicity[179, 180]. There are many low energy conformations for 6 water clusters, but their relative energies are dramatically affected by vibrational energies due to differing number of suflate-water and water-water hydrogen bonds.

7.2 Theory

Model As a quantum thermodynamics approach, PIMC[13, 181] starts with the partition function, which is the trace of the Boltzmann operator, and then expanding in a position basis, inserts P resolutions of the identity.

R R − β Hˆ − β Hˆ Q = ... dx1 . . . dxP hx1|e P |x2i ... hxP |e P |x1i CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 62

The purpose of inserting the P states is that now we can use high temperature approximations for each of the P density matrix elements. The Trotter approximation allows us to split the density matrix elements into two–one whose distribution we know and one which β is diagonal in the position basis, both at an eﬀective temperature = . P

− β (Hˆ +∆Vˆ ) β ˆ β ˆ β 2 h i P 0 − P H0 − P ∆V ˆ ˆ e = e e + O P Ho, ∆V

Thus if we pick P large enough, we can make the error as small as we want. It is also important to note that the error in the Trotter approximation can be reduced by reducing [H0, ∆V ], most obviously by reducing the magnitude of ∆V . It’s worth noting that in the context of PIMC the Boltzmann operator is also referred to as the propagator. This interpretation is built on the idea that the Boltzmann operator iτ ˆ ˆ H can equivalently be viewed as an imaginary time propagator through e−βH = e ~ with τ = −iβ~. Thus PIMC can be interpreted as approximating a cyclic path of time iβ~ by P short time steps. ˆ ˆ The standard, most general approach to PIMC involves taking H0 to be T , the free particle Hamiltonian, which means ∆Vˆ becomes the entire Vˆ . Since molecules are bound states we must have a small step size (large P ) to properly approximate the full propagator. The beneﬁts of the free particle propagator (FFP) are simplicity, spatial invariance, and model invariance (we don’t need any information about V to from FP density matrix elements). Given that we care about low T vibrational energies for systems where we already have calculated Hessians, it makes sense to use the system speciﬁc information to improve our sampling approach. After transforming from standard cartesians to normal modes, we now split the Hamil- tonian using the harmonic reference as our short time propagator,

1 2 − 2 β 2 β mω γ 2 2 2 − Hˆ 2π~ β − (tanh( )(x +x )+csch(γ)(xi−xi+1) )+∆V (xi) hx |e P |x i = γcsch(γ)e P 2γ 2 i i+1 i i+1 mP

β ω Where we use the variable γ = ~ , which should be viewed as the ratio of quantum P vibrational spacing versus thermal ﬂuctuations at our artiﬁcially high temperature, . We can calculate a primitive energy estimator in this representation to get,

P 1 P 1 1 2 2 2 2 εE(x1, . . . , xP ) = P 2 ~ω coth(γ) + 2 ~ω sech (γ)xi − coth(γ)csch(γ)(xi − xi+2) i=1 +∆V (xi)

The important points to know about this estimator are that it provides an estimate of the total vibrational energy (we get the anharmonicity by subtracting oﬀ the exact harmonic CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 63 energy) and that, similar to its free particle counterpart, its variance increases with P [182]. This increased variance is particularly troublesome since increasing P already grows our system size and frustrates sampling and now it also increases the number of samples we need to reduce sampling error! This variance is also the variance in the estimation of the entire vibrational energy which is potentially an order of magnitude larger than the anharmonic energy, which is all we actually care about calculating. One way to avoid these problems is to directly sample the anharmonicity using thermodynamic integration (TI). TI is a way to calculate free energy diﬀerences between two states given a parametrized Hamiltonian, Hλ, that continuously connects them as λ varies from 0 to 1.

Hλ = T + (1 − λ)V0 + λV1 log Q F = − λ λ β dF F − F = R 1 dλ λ 1 0 0 dλ R −βHλ R 1 dqβ(V1 − V0)e = 0 dλ βQλ R 1 = 0 dλhV1 − V0iλ

Where h·iλ denotes the thermodynamic average sampled with Hλ. Its worth noting that to estimate the anharmonicity we cannot simply sample ∆V using the full H since,

HO Efull − EHO = hεestifull − hεest iHO

6= h∆V ifull

This path from H0 to H1 can be any number of options but ideally we will pick the smoothest, most linear path to expedite the numerical quadrature. We apply Gaussian quadrature to approximate the integration over λ, which we mention primarily because the inferior trapezoid approximation is still commonly used by some in the field[176]. For our purposes, H0 is the harmonic reference and H1 is the full Hamiltonian so we can propose Hλ = H0 + λ∆V . For anharmonicity in clusters we will find this path gives us nearly linear transitions. One major issue that must be addressed is that in invoking the local harmonic reference, we lose the spatial invariance of our sampling and our choice of coordinate system becomes very important. The important factor is to select coordinates in which the potential written in normal modes will remain harmonic for as long as possible. While this is a hard problem to solve in general, we have a very good heuristic solution for molecular vibrations in internal coordinates selected based on chemical bonding[183, 184]. With the goal of using a non-redundant, black-box approach to selecting internal coordinates, we apply the delocalized internals of Baker et. al.[185, 186]. We use an atom CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 64 specific distance criterion to determine bonds (which may need to be modified by the user for unusual systems) as well as all possible bonds and torisions between bonds; for each separated cluster, we add in local translations and rotations and given all these primitive internal coordinates we transform to delocalized, non-redundant internals by doing a singular value decomposition on the B matrices. The last issue to consider when putting together our model is what we will use for the calculation of electronic energy to give us V . While there is nothing formally stopping us from using electronic structure theory, the large number of expected samples discourages us from doing so. We propose the use of parameterized force fields to calculate anharmonic corrections to higher accuracy harmonic results. Although the absolute energies from forcefields will have large errors and even the errors in the harmonic energy tend to be too large, by looking at the anharmonic corrections, errors are inherently on a smaller order of magnitude.

Sampling To evaluate the multidimensional integrals of our path integral model, we turn to the Metropolis Monte Carlo algorithm, taking adavantage of our known harmonic information to improve sampling eﬃciency. For sampling a value A(x) using the Hamiltonian H we have,

Z Z P ! −∆V (x) Y X A(xi) Y e hA(x)i = ... dx ρ (x , x , ) H i 0 i i+1 P Q i i i H0 ρ0(xi, xi+1, ) = hxi|e |xi+1i

To do this efficiently we propose moving a subsection of the P beads at a time using the −∆V analytic distribution from ρ0 and accept or reject them based on weights e , an approach referred to as Levy flights. Using the known ρHO distribution allows us to dramatically improve our sampling by taking steps correctly based on the harmonic potential and then accepting or rejecting based only on ∆V . This is much more efficient compared to the free particle approximation which only considers mass and temperature in determining steps and mush accept/reject based on the larger full V . With Levy flights the only parameters left to adjust our sampling are the number of degrees of freedom we sample at a time and the fraction of beads that we move per step; we choose these values to give us acceptance rates of near 60-70%. While we will have to look at test cases to assess model error, sampling error can be CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 65 much more easily understood. We can estimate the variance in the mean by,

N σ2 2 X h(h∆V i − ∆V )2i = + (h∆V ∆V i − h∆V ih∆V i) N N 2 i j i j i

Where ∆V and σ2 are the mean and variance of the estimator over all the sampling points. By keeping track of some ﬁnite portion of the autocorrelation function, A(k), say 1000 points, we can construct the variance in the mean precisely accounting for correlations. The next assumption that the mean value is normally distributed over our sampling points is fairly reasonable given the Central Limit Theorem and our the large sample sizes. This allows us to calculate a range that we are 95% conﬁdent contains the correct mean (for a given model physics) as,

r2τ 95% CI = ±1.96σ N Our implementation estimates the number of samples required to get the conﬁdence inter- val to be within a 5% relative error and sets the run length after running for an initialization period.

7.3 Results and Discussion

Model Systems Before attempting to study water-sulfate clusters we will study some well characterized model systems: a Morse potential model for H2,H2O, and H2O dimer. The methods we consider are full vibrational energy calculations using free particle propagator (FPP) and HO propagator, as well as TI using the HOP. Additionally, before we begin looking at systems, we need to determine the temperature for our sampling. Often specific temperatures are stipulated by experimental conditions but if ZPE is sought, low finite temperatures must be used as an approximation. A temperature of 60 K (corresponding to about 0.12 kcal/mol or 34 cm−1 thermal fluctuations) is used which is effectively 0 K on the energy scales we’re considering. PIMC sampling begins to CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 66 become much more difficult at lower temperatures and begins to require larger values for P but it possible. While 60 K is not zero, higher vibrational states are effectively unpopulated and we will consider energies and free energies at this temperature as ZPE. The last remaining consideration before applying TI to test systems is that we haven’t described yet is how many points we’ve chosen for our numeric integration of h∆V iλ. Figure ?? shows a plot of h∆V iλ versus λ for several of our test systems using 11 grid points. As hinted at earlier, the plots for these small molecules are very linear, which mean we can accurately integrate them with a single quadrature point! It is important to recognize that all these systems are of clusters of small molecules, notably without bond torsions, so this fortuitous linearity must be reconsidered before applying the TI approach to more complicated molecules. From these test systems, we see that our TI approach allows us to describe low temperature vibrations with a dramatically reduced P and N value compared to FPP and HOP approaches. It also appears that we can achieve nearly zero vibrational error by choosing P = 200 for all theses systems and reduce all errors except electronic error due to our force field to small, controlled values within 5% of the anharmonic ZPE we’re calculating.

2 Figure 7.1: Plot of h∆V iλ as a function of lambda for sampling with P = 200. R values for the ﬁts are 0.996, 0.984, and 0.998 for the monomer, dimer and sulfate cluster respectively.

From the Morse oscillator we see that sampling using the HOP and TI yields improvements on two fronts: the higher level description of the physics leads to a reduction in the number of beads, P , required to get the same model error and improvements to sampling lead to a reduction in sampling number, N to get the same statistical errors. Figure 7.2 CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 67 shows the dramatic improvement from using HOP and TI with respect to P ; each method improves on the former by a factor of 2. From Table 7.1 we can see that the number of sampling points needed to get the same level of statistical errors increases dramatically with P for FPP, slightly less so for HOP, and much less so for TI. There are a couple important facts to consider when looking at this Table. First, since these errors are considered relative to the total anharmonic ZPE we need to compare the methods across P values with similarly reduced model error: for H2 we need to compare P values of 400 for FPP to 200 for HOP 100 for TI since these all give similar errors with respect to P . This comparison shows how each new approach reduced sampling errors by a full order of magnitude. Second the number of single point calculations required is really P times N so higher P values additionally increase the number of sample points (and explains why required N values can decrease with P since NP is still increasing).

Figure 7.2: Errors in anharmonicity on a Morse potential of H2 using the free particle and harmonic propagator full energy approaches as well as thermodynamic integration. The Morse potential is parameterized with De = 0.176 and a = 1.4886.

Figures 7.3 and 7.4 show results for water monomer and dimer using the standard polarizable AMOEBA[187–189] force ﬁeld for water[190, 191]. We see similar multiplicative improvement of TI over HOP and HOP over FPP in the monomer case. Unlike the Morse potential, for which we are comparing to an exact result using the same model potential as

our path integrals, for the H2O cases and beyond, we will be using forceﬁelds for our PIMC but have reference values from electronic structure. This elucidates the three types of error in our method: statistical (controlled by N), vibrational (controlled by P), and electronic CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 68

P H2 Water Monomer Water Dimer Sulfate 3H2O Sulfate 6H2O FPP 50 10580 8502 100 24660 81668 200 1519848 636924 300 11581045 1574675 400 26368856 3462925 HOP 50 271278 36574 140365 100 427778 124347 222831 200 2429968 463723 711451 300 5334265 856822 761297 TI 50 158440 17276 17276 31056 25657 100 408121 15360 15360 31362 15070 200 338591 17107 17107 18349 31702 300 275426 11551 11576 19033 31311

Table 7.1: Table of the number of samples required to reduce sampling error to within 5% of calculated anharmonic ZPE.

(inherent in the method used for single point calculations). These errors are also illustrated well in Figure 7.3; statistical error is in the error bars, vibrational disappears as you reach the asymptote, and electronic is the diﬀerence between the asymptote and zero. Assumed

in the previous sentence is that the reference is exact, which for the H2O reference of a full grid approach using CCSD(T) results, is probably close enough to being true[192]. Not so clear is the reference value used for the water dimer[193]. Here electronic error is quite low as the calculations are run with CCSD(T) again, but here the anharmonicity is only accounted for by VPT2. Thus, the diﬀerence between the TI P = 200 values and the reference then are primarily due to the vibrational errors of the reference added to the electronic error from the PIMC calculation (plus smaller contributions from statistical and vibrational).

Sulfate-Water Clusters Before we begin calculating enharmonic ZPE for various sulfate clusters we will study the

lowest energy 3 H2O cluster further. Figure 7.5 shows that we get similar behavior with respect to P as in the small test cases. This cluster has also been studied with CC-VSCF with MP2 electronics and VPT2 and TOSH with B3LYP electronic energies[179, 194] and we compare to these results in Table 7.2. While there are deviations between the fairly high level CC-VSCF values and our PIMC results, these diﬀerences are not entirely due to the electronic error in the TI approach. The CC-VSCF reported values neglect the lowest 14 out of 36 modes. While low modes account CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 69

Figure 7.3: Errors in anharmonicity for H2O monomer using FFP, HOP, and TI. Reference values from direct grid ﬁtting of CCSD(T) calculation[192]. for less of the total vibrational energy, they are dramatically more anharmonic; as seen by their results, if the cutoﬀ had included 3 fewer modes (raised by 50 cm−1) the anharmonic correction would change by 15%! The ability to handle low modes is a major strength of the PIMC approach to anharmonicity.

Anharmonic ZPE CC-VSCF -1.98 TOSH -1.39 VPT2 -1.25 TI-PIMC -1.77

Table 7.2: Anharmonic ZPE for the lowest energy Sulfate-3 H2O cluster with CC-VSCF on MP2/TZP[194], TOSH and VPT2 on B3LYP/6-31+G*[179], and PIMC TI on polarizable force ﬁeld.

On to our original inspiration, we calculate anharmonic corrections to the sulfate clusters of Lambrecht et. al.[179, 180] in a way that is not susceptible to the dramatic failure that can show up in VPT2 and TOSH. Figures 7.6-7.9 show the eﬀect of our additional anharmonic term on the relative energy ordering of sulfate clusters. Electronic energies are CCSD(T) that are db-pV(TQ)Z extrapolated and harmonic vibrational energies are calculated at B3LYP/6-311++G(3df,3pd)[180]. The anharmonic correction causes several clusters CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 70

Figure 7.4: Errors in anharmonicity for H2O dimer using HOP and TI. The reference here is based on CCSD(T) electronics, but only VPT2 for nuclear energies[193].

to change ordering, including the global minimum of the 5 H2O clusters. The presence of error bars shows shows the statistical error, which could be further reduced through extra sampling, but it is important to realize that there are additional, non-quantiﬁed errors due to the DFT harmonic calculation as well as the use of forceﬁelds that are likely to be on a similar scale of 0.1 kcal/mol.

7.4 Conclusion

We have developed a new approach to calculating anharmonic vibrational energy corrections by combining the idea of using a local harmonic approach with thermodynamic integration, which allows us to systematically reduce vibrational errors for a given electronic model. The method can sample difficult, low temperatures by taking advantage of an improved zeroth order propagator, but the cost comes in the form of only applying at temperatures low enough that the sampled region of phase space is well characterized by the minimum energy potential well. The cost of Monte Carlo sampling is that a large amount of single points must be calculated, but due to improvements in efficiency, we’ve calculated accurate values with on the order of 4 × 105 single point calculations for systems of 63 modes. Given fast enough algorithms, the TI approach can be applied to electronic structure single points just as easily as force field calculations to further reduce errors. CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 71

Figure 7.5: Anharmonic ZPE for Sulfate 3 H2O cluster using TI.

7.5 Acknowledgements

D. S. is supported in part by the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC05-06OR23100. This work was also supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under contract no. DE-AC02-05CH11231. We acknowledge computational resources obtained under NSF award CHE-1048789. CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 72

Figure 7.6: Relative Energies of Sulfate 3 H2O clusters using TI.

Figure 7.7: Relative Energies of Sulfate 4 H2O clusters using TI. CHAPTER 7. PATH INTEGRALS FOR ANHARMONIC VIBRATIONAL ENERGY 73

Figure 7.8: Relative Energies of Sulfate 5 H2O clusters using TI.

Figure 7.9: Relative Energies of Sulfate 6 H2O clusters using TI. 74

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