Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo

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Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo Edgar JosuéLandinez Borda,1 John Gomez,2 and Miguel A. Morales1, a) 1)Lawrence Livermore National Laboratory, Livermore California 94550, United States 2)Applied Physics Program and Department of Chemistry, Rice University, Houston Texas 77005-1892, United States (Dated: July 23, 2018) The Auxiliary-Field Quantum Monte Carlo (AFQMC) algorithm is a powerful quantum many-body method that can be used successfully as an alternative to standard quantum chemistry approaches to compute the ground state of many body systems, such as molecules and solids, with high accuracy. In this article we use AFQMC with trial wave-functions built from non-orthogonal multi Slater determinant expansions to study the energetics of molecular systems, including the 55 molecules of the G1 test set and the isomerization 2+ path of the [Cu2O2] molecule. The main goal of this study is to show the ability of non-orthogonal multi Slater determinant expansions to produce high-quality, compact trial wave-functions for quantum Monte Carlo methods. We obtain systematically improvable results as the number of determinants is increased, with high accuracy typically obtained with tens of determinants. Great reduction in the average error and traditional statistical indicators are observed in the total and absorption energies of the molecules in the G1 test set with as few as 10-20 determinants. In the case of the relative energies along the isomerization path of 2+ the [Cu2O2] , our results compare favorably with other advanced quantum many-body methods, including DMRG and complete-renormalized CCSD(T). Discrepancies in previous studies for this molecular problem are identified and attributed to the differences in the number of electrons and active spaces considered in such calculations. Keywords: AFQMC, Atomization energies, Non Orthogonal Determinants

I. INTRODUCTION with applicability even to strongly correlated materials. Traditional quantum chemistry methods, like Many- Advances in the comprehension and predictive capabil- Body Perturbation Theory (MBPT), Coupled Cluster ities of electronic properties of matter, from single atoms (CC) and Configuration Interaction (CI), can offer ac- to condensed matter systems, are a major quest that per- curate solutions to the many-electron problem but their meates many scientific and technological fields. Due to computational cost typically scales unfavorably with sys- the complexity of the fundamental equations of matter tem size (N6−7 for CC methods). While their ex- at the atomic scale, over the last several decades, com- tension to systems with periodic boundary conditions putational methods have become a valuable tool in the has been slow, implementations in standard computa- discovery, characterization and optimization of new ma- tional packages are more common3–6 and applications to terials. First-principles computational methods, those solids are appearing more frequently in the literature, that do not rely on empirical or experimental param- including calculations based on second-order Moller- eters and attempt a direct solution to the fundamen- Plesser perturbation theory (MP2)7–9, Random Phase tal equations, have been mostly based on density func- Approximation10,11, Coupled Cluster Singles-Doubles tional theory (DFT)1, due to its good predictive capabil- (CCSD)12, among others. ity and modest computational cost. Unfortunately, DFT Quantum Monte Carlo (QMC) methods13 offer an im- is based on approximations to electronic exchange and portant alternative to traditional quantum chemistry ap- correlation which are known to be unreliable in many proaches for the study of many-electron problems, with materials where these effects dominate or are difficult both finite and periodic boundary conditions. They offer to approximate2, so called strongly correlated materials. a favorable scaling with system size, typically between As computer power increases and numerical algorithms N3-N4, offer excellent parallel efficiency14,15, and are ca-

arXiv:1801.10307v2 [physics.chem-ph] 19 Jul 2018 improve, we are quickly approaching a point where the pable of treating correlated electron systems with few use of accurate quantum many-body approaches for the approximations. Most QMC methods used in the study study of material properties is becoming feasible. Quan- of realistic materials rely on a trial wave-function to con- tum many-body methods are typically orders of magni- trol the notorious sign problem that plagues all fermionic tude more computationally expensive than DFT, which Monte Carlo methods16. The trial wave-function not has prevented their widespread application to bulk ma- only controls the magnitude of the resulting approxi- terials in the past, but could offer an accurate alternative mation, but also the sampling efficiency and the magnitude of statistical uncertainties. Hence, accurate and efficient wave-function ansatz are important to the success of QMC methods in their application to realistic a)Electronic mail: [email protected] problems in physics, chemistry and material science. Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 2

In this article we examine non-orthogonal multi-Slater from selected CI calculations37, leading to expansions in determinant (NOMSD) expansions as an accurate and orthogonal determinants connected by particle-hole ex- efficient trial wave-function ansatz for QMC simula- citations. While these expansions lead to systematically tions. We test the efficiency and accuracy of these improvable results with reasonable stability and robust- wave-functions in combination with the Auxiliary-Field ness, the expansions are typically very large requiring quantum Monte Carlo (AFQMC) method, as imple- thousands of terms in order to reach high accuracy28. mented in the QMCPACK14,15,17,18 simulation pack- In this article, we propose the use of non-orthogonal age. We show how these wave-functions have the ca- Slater determinant expansions, where no orthogonal- pacity to systematically reduce errors associated with ity constraint is imposed between determinants, hence 19 the phaseless approximation in AFQMC, employed hϕi|ϕji 6= 0. In fact, each Slater determinant is rep- to control the sign problem, with highly compact ex- resented as an orbital rotation from a given reference, P i † Z c cq pansions. These wave-functions have been used in the |ϕii = e pq p |ϕref i, where Z is a Unitary matrix. past to study strongly correlated lattice Hamiltonians 20–24 The trial wave-function is generated using a version of with great success . They have also been recently the projected Hartree-Fock (PHF) algorithm developed popularized in connection with symmetry projection in 25,38,39 25,26 in the Scuseria group at Rice University . Trial Generalized Hartree-Fock theories . While this ar- wave-functions are obtained by a direct minimization of ticle focuses on calculations of small molecular sys- the energy, E = hΦ|Hˆ |Φi/hΦ|Φi, using a BFGS-like al- tems, as a first application of the wave-function ansatz gorithm and analytical energy gradients, see Jimenez- in AFQMC calculations of realistic Hamiltonians, sim- Hoyos, C., et al.,39 for the relevant equations. We ilar improvements are expected when NOMSD wave- use 2 different approaches, the few-determinant (FED) functions are employed in other situations, including algorithm25,26 and the resonating Hartree-Fock (ResHF) calculations of solids/extended systems, strongly corre- approach20,21,40. In the FED algorithm, the Slater de- lated problems and other QMC approaches like Diffu- terminant expansion is generated iteratively, adding and sion Monte Carlo. For example, we have successfully optimizing one determinant in each iteration to an al- employed these wave-functions in studies of strongly cor- ready existing expansion. During each iteration, deter- related, periodic lattice Hamiltonians27. The NOMSD minants |ϕii (i = 1, 2, . . . , nd − 1) obtained from previ- wave-function ansatz now extends the arsenal of trial ous iterations are kept fixed25,26 and the energy is mini- wave-functions employed in QMC calculations, includ- 28,29 mized with respect to the orbital rotation matrix of the ing truncated CI expansions , Antisymmetric Gemi- new determinant and all linear coefficients. This pro- nal Powers (AGP)30–32, Pfaffians33,34, Bardeen-Cooper- 35,36 cess continues until a given number of determinants is Schrieffer (BCS) , among many others. generated. At this point, the linear coefficients are re- The structure of this paper is as follows: in the section optimized by solving the associated eigenvalue problem. II we briefly describe the wave function ansatz and its op- In the FED theory, symmetry projectors can be incorpo- timization method. Section III shows the application to rated straightforwardly. The resulting single- or multi- the approach to the calculation of total and atomization reference symmetry-projected FED wave functions have energies for a subset of molecules of the G1 set. In section been shown to be quite accurate. However, we will not IV we describe the study of the isomerization path of the focus on symmetry restoration in this work. [Cu O ]2+ molecule, a scenario where different contribu- 2 2 In the ResHF approach, the energy is minimized with tions to the correlation energy are significant along the respect to all variational parameters in the trial wave- path making the calculation quite challenging even for function, including the rotation matrices of all determi- traditional methods. Finally, we discuss clear limitations nants and all linear coefficients. In this work, we use with this approach and make concluding remarks. a combination of both approaches to generate our trial wave-functions. We first generate a NOMSD expansion of a given length (e.g. 20 determinants) using the FED II. NON-ORTHOGONAL MULTI-SLATER DETERMINANT TRIAL WAVE FUNCTIONS approach. The resulting expansion is then optimized using the ResHF approach, but without new determinants. In other words, we use the FED approach to generate an The trial wave-functions used in this work have the initial set of parameters for the ResHF approach. We find typical form: this to be successful in most situations studied so far. In

nd some situations, particularly when generating long ex- X pansions, we insert ResHF steps in between blocks of |Φi = ci |ϕii, (1) i=1 FED steps to improve the stability and efficiency of the optimization at later stages of the iterative cycle. We where |ϕii are Slater determinants and ci are linear vari- must mention that we almost never converge the ResHF ational parameters. In traditional quantum Monte Carlo calculations to high precision. This is typically expen- calculations, multi-determinants trial wave-functions of sive to do, but also not necessary in AFQMC. The trial this form have been produced from truncated Configu- wave-function in AFQMC does not need to satisfy any ration Interaction (CI) calculations28, or more recently stationary properties, so there is no need for high con- Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 3 vergence. mization (for example, low rank update schemes29 can The mean-field orbitals used in the FED and ResHF be used to speed up the evaluation of orthogonal ex- theories could be obtained from restricted-HF (RHF), pansions), we have decided to present such comparisons unrestricted-HF (UHF), or even generalized-HF (GHF) and details of the implementation in a future publica- wave functions. The three cases represent different levels tion focused on those aspects of the calculation. From of symmetry-breaking of the Hartree-Fock determinant. the results presented above, it is clear that the NOMSD In this work, we use orbitals from RHF wave-functions in approach leads to a compact and systematically improv- closed-shell systems and from UHF in open-shell cases. able wave-function which can succesfully compete with While the use of GHF wave-functions can lead to sig- state-of-the-art alternatives in the field. nificant improvements in our calculations in combination with NOMSD, we have not explored this in this work and will be the focus of a future publication. III. SIMULATION DETAILS

SCI The MOLPRO43 software package was used to per- NOMSD form all Hartree-Fock and CCSD(T) calculations. In ad- )

a dition, it was also used to generate all the integral ﬁles H

( 3 10 (matrix elements of the Hamiltonian in the HF basis) E

necessary for both AFQMC calculations and for the iterative Hartree-Fock method used to generate the NOMSD wave-functions. The PHF code25,38,39 was used to generate NOMSD wave-functions. CCSDTQ calculations were performed with the AQUARIUS package44. All -621.589 AFQMC calculations were performed with the QMC- PACK software package18. QMCPACK offers a gen- -621.591 eral implementation of the AFQMC algorithm, including standard improvements to the method like mean-field -621.593 45 45 46 Energy (Ha) subtraction , force bias , hybrid propagation , single and multi-determinant wave-functions with both orthog- -621.595 onal and non-orthogonal multi-Slater determinants, pop- 100 101 102 103 104 N Determinants ulation control, large scale parallelization, multi-core implementations, among many others. The code can be Figure 1. Performance comparison of multi-determinant trial used to study both finite as well as extended systems. In- wave functions in AFQMC calculations for the NaCl molecule terfaces to the code exists for Molpro, GAMESS, PySCF at the equilibrium geometry with a cc-pVDZ basis set. Re- and VASP, the latter 2 codes able to generate input for sults for a NOMSD expansion are shown in blue (dashed bars calculations with periodic boundary conditions. Both on top and triangles in bottom figures) and for a orthogonal free projection47 and propagation with the phase-less expansion obtained from a selected Configuration Interaction constraint48 are implemented. For additional implemen- calculation shown red (clear bars on top and circles in bottom tation details we refer the reader to the QMCPACK refer- figures). The top figure shows the standard deviation of the ence article18 and to the code’s manual17. For additional energy estimator, while the bottom figure shows the total en- details about the AFQMC method, we refer the reader ergy, both as a function of the number of determinants in the 45–49 expansion.The horizontal line in the bottom figure indicates to the many excellent articles in the literature . the CCSDTQ result. Orthogonal expansions require expan- All calculations presented in this article employed the sions with approximately 2 orders of magnitude more terms phase-less approximation48. Single determinant trial to reach similar values of the energy and standard deviation. wave-functions were generated with Restricted Hartree- Error bars are smaller than the symbol size. Fock for closed-shell systems and with Restricted Open- Shell Harthree-Fock for open shell cases. Unless other- Figure 1 shows a comparison of the performance of wise stated, we employed a time-step of 0.005 ha−1 and AFQMC calculations for the NaCl molecule using a 576 walkers, both choices leading to systematic errors NOMSD wave-function and an orthogonal multi-Slater below ' 0.2 mHa. Due to the complexity of some of determinant expansion obtained from selected Configu- the figures, we typically avoid plotting error bars to im- ration Interaction (selCI), for which we used the DICE prove the clarity of the data presented. Unless otherwise code41,42. The NOMSD wave-function requires approxi- stated, all error bars have been converged to less than mately 2 orders of magnitude less determinants in the ex- 0.4 mHa in single determinant calculations (average er- pansion to reach a similar reduction in phase-less bias and ror of ' 0.2 mHa) and to less than ' 0.2 mHa in NOMSD energy fluctuations compared to the selCI wave-function. calculations (average error below ' 0.1 mHa). All calcu- We have found this to be the case in all situations we lations employ the frozen-core approximation. We chose have studied so far. Since actual execution times will be the standard cores used by MOLPRO, namely He cores very sensitive to implementation details and code opti- are frozen in 1st row atoms while Ne cores are frozen in Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 4 the case of 2nd row atoms. For a detailed analysis of the CCSDTQ), obtained by AFQMC, as a function of the influence of frozen-core approximations on the total and number of determinants in the NOMSD expansion, for 50 atomization energies, see Ref. . Correlation-consistent the molecules H2O, O2, and NaCl. This figure shows atomic basis sets were used and geometries were taken several features of the use of NOMSD wave-functions in from Ref.51 for the molecules in the G1 test set and from AFQMC. First notice that, since all 3 molecules possess Ref.52 for the study of the isomerization path of the cop- weakly correlated electronic structures, single determi- per oxide molecule. nant calculations already produce reasonable results with errors around ∼1%-4%. In all cases the amount of correlation energy approaches systematically 100% as the IV. G1 TEST SET determinant expansion in increased, even though the convergence can be from above since AFQMC is not a vari- In this section we study the energies of 55 molecules ational method. In the case of H2O and O2, less than 10 of the G1 test set53 to analyze the accuracy and rate determinants are needed to obtain 99.5% of the corre- of convergence of the AFQMC method when NOMSD lation energy, while at least 50 determinants are needed trial wave-functions are used. This set has a long history to obtain a comparable accuracy in NaCl. As will be in both quantum chemistry and quantum Monte Carlo seen below, molecules with a strong ionic character tend communities, often to have a slower convergence rate with respect to the used as benchmark to test new developments and number of determinants. Nonetheless, even though the capabilities28,51,54. Accurate reference data exists for all specific rate of convergence clearly depends on the spe- molecules in the set, including atomization energies50.A cific molecule, rapid convergence is observed in all three large literature exists with careful analysis of the influ- cases. ence and magnitude of the various approximations employed in theoretical calculations50,55–57, including effects associated with the choice of basis set and approximations to electronic correlation.

A. Total Energies

As an illustrative example, Figure 2 shows the percent- age of the correlation energy (measured with respect to

Table I. Raw performance statistics of the difference of the total energies between AFQMC with the NOMSD expansion and CCSDTQ. All calculations were performed with Dunning’s cc-pVDZ basis set and employed the frozen-core approximation. Column definitions -Max. neg.: maximum negative difference; Max. pos: Maximum positive difference; ME: mean error; MAE: mean absolute error; STDEV: standard deviation. All quantities are in mHa.

AFQMC Ndet Max. neg. Max. pos. ME MAE STDEV 1 det −8.00 6.27 1.01 2.27 2.72 5 det −1.8 3.42 0.11 0.65 0.95 20 det −1.07 2.00 −0.03 0.31 0.54

In order to have a more quantitative characterization the mean absolute error decreases from 2.27 mHa to 0.31 of the accuracy of the AFQMC-NOMSD approach, we mHa by going from single determinant to a 5 determi- performed CCSDTQ calculations for a subset of the 55 nant expansion, with a further reduction to 0.34 mHa molecules considered in this work using Dunning’s58 cc- when the expansion is increased to 20 determinants. In pVDZ basis set. We were forced to limit our analysis to general, all statistical measures of the energy difference a subset of the 42 molecules in the test set due to dif- show a significant and systematic reduction between sin- ficulties with the convergence of CCSDTQ for some of gle determinant calculations and NOMSD with increas- the molecules. We expect CCSDTQ to provide reliable ingly larger expansion sizes. Notice that the choice of results for these molecules, with errors below 1 mHa50. a 20 determinant expansion is somewhat arbitrary, with Figure 3 shows the error in the total energy calculated by the intention of showing a very significant reduction on AFQMC-NOMSD when measured with respect to CCS- the errors with just a handful of determinants. While the DTQ. We present results using 1, 5, and 20 determinants magnitude of the errors is dependent on the molecule, the in the NOMSD expansions to show the convergence of capacity for significant improvement with a very compact the correlation energy as the expansion is increased. Ta- expansion is universally observed. ble IV A shows the associated performance statistics for the energy differences. For the selected set of molecules, Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 5

B. Atomization Energies

Figure 4 shows the error in the atomization energy, calculated with AFQMC-NOMSD, for the 55 molecules considered in this work. Table II shows the associated performance statistics. Reference values have been obtained from Feller et al.50,55 by subtracting the Zero Point En- ergy (ZPE), Spin Orbit (SO), Core Valence (CV), Scalar Relativistic (SR) and the Diagonal Born-Oppenheinmer Correction (DBOC) contributions from the experimental values. Atomization energies have been extrapolated to the Complete Basis Set (CBS) limit using results with Dunning’s cc-pV(X)Z basis sets, with X = {D,T,Q}. We used the mixed Gaussian/Exponential −(n−1) −(n−1)2 formula, E(n) = ECBS + Ae + Be , which Figure 2. Convergence of AFQMC correlation energy with has been shown to perform well for CCSD(T)56,57. Simi- the number of determinants in a non-orthogonal expansion. lar to the previous section, we show results for expansions Equilibrium geometries and a cc-pVDZ basis set has been of size 1, 5, and 20, with the goal of showing the capacity used in all cases. The inset shows the convergence of the of this approach for error reduction with short, compact correlation energy of the trial wave function. expansions.

Table II. Raw performance statistics of the AFQMC atomization energies error with the NOMSD expansion respect to the reference values. All quantities are in mHa.

AFQMC Ndet Max. neg. Max. pos. ME MAE STDEV 1 det −23.78 2.66 −6.32 6.68 5.78 5 det −17.15 3.04 −2.93 3.38 3.59 20 det −7.81 2.42 −0.52 1.17 1.78

As expected from the discussion above, the NOMSD V. COPPER OXIDE ISOMERIZATION PATH expansion is able to reduce signiﬁcantly the errors of

AFQMC throughout the test set. Overall, the mean ab- 2+ solute error is reduced from 6.68 mHa to 3.38 mHa when The study of the isomerization path of the [Cu2O2] going from a single determinant to a 5 determinant ex- molecule has become a common practice in the last pansion, and to 1.17 mHa when 20 determinants are used. decade to test the capacity of ab initio and quantum Several observations can be made upon careful inspection chemistry methods to accurately describe the energetics of correlated molecular problems. As pointed out by K. of the various molecules in the set. First, we notice that 52 for many molecules a 5 determinant expansion is enough Samanta et al., , the interest comes from the dissimilar to drastically reduce the error in the atomization ener- nature of the correlation energy in the molecule through gies, often with error reductions of over 90%. We can also the path. A successful description of the relative energy notice a set of molecules for which convergence is slow, of the molecule along the path requires methods that including for example LiF and NaCl. These molecules offer a balanced treatment of both static and dynamic have a strong ionic character, which seems to lead to slow components of the correlation energy, since both terms convergence with respect to expansion length. We must have dominant contributions at different points along the point out that the errors in the atomization energies dis- path. cussed above contain, in addition to any bias associated Figure 5 shows a schematic representation of the stable with the AFQMC method, a contribution from basis set isomers at the end points of the isomerization path. Fol- extrapolation since the largest basis set we considered lowing previous studies, we define a linearized isomeriza- was cc-pVQZ. Since the focus of the article is the char- tion path between isomers A and B through the quantity acterization of the NOMSD ansatz, we decided to avoid f f defined by the equation qi(f) = qi(f = 0) + 100 (qi(f = a careful analysis of basis set extrapolation errors in the 100) − qi(f = 0)); where qi(f) are the coordinates of results presented below. For a detail analysis of basis set the atoms in the molecule along the isomerization path, effects in atomization energies of the G1 test set, we refer qi(f = 0) are the atomic coordinates for isomer B and 56 the reader to Ref. . qi(f = 100) are the atomic coordinates for isomer A. We use the same geometry, pseudopotential and basis set employed in several previous studies of this molecule in order to be able to draw a direct comparison with Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 6

Figure 3. Comparison of AFQMC total energies diﬀerences, employing NOMSD trial wave functions with 1,5 and 20 determinants, against Coupled-Cluster calculations. The Coupled-Cluster calculations include single, double, triple and quadruple excitations (CCSDTQ). A cc-pVDZ basis set has been used in all cases. The red arrows indicate that the values of that diﬀerence are greater than 5 mHa and point to its explicit value

Figure 4. Atomization energies for the set of molecules, comparison of the extrapolated data of AFQMC for the cc-pVXZ, X = {D,T,Q}, displaying the mean absolute error in comparison to the reference values other correlated electronic structure approaches. For a calculations correlated explicitly the 2s and 2p orbitals of detailed description and a summary from previous theo- oxygen in addition to the 4s and 3d orbitals of Cu lead- retical calculations, we refer the reader to Refs.59,52 and ing to 32 correlated electrons, while PHF calculations in references therein. Ref.52 correlated all 52 electrons in the calculation. As Previous studies of this molecule using advanced will be shown, the resulting differences in the number electronic structure methods including Density Matrix of core electrons in each of the calculations leads to the Renormalization Group (DMRG)60, Complete Renor- observed discrepancies in previous results, rather than malized Coupled Cluster (CR-CC)61, and Projected strong inaccuracies in the predicted relative energies. Hartree-Fock (PHF)52, appear to lead to discrepancies Figure 6 shows the dependence of the AFQMC- on the relative energy differences across the isomerization NOMSD total energy with the number of terms in the path. In particular, PHF calculations reported in Ref.52 expansion for the 2 endpoints of the isomerization path 2+ seem to lead to larger energy differences when compared of the [Cu2O2] molecule. to DMRG and CR-CC. As shown below, these discrepan- In this case 32 electrons (CR-CC) are directly cor- cies seem to be related to the different choices of core and related in the calculation. As we can see, single de- active spaces selected in each of the calculations. While terminant trial wave-functions lead to poor results for all these calculations appear to use the same atomic basis this molecule with errors in the relative energy on the set and the same molecular geometries, DMRG calcula- order of 35 (Kcal/mol). Convergence in the relative tions reported in Ref.60 used an active space of (28e,32o) energy between the endpoints of the path are reached leading a calculation with 28 correlated electrons, CR-CC with NOMSD expansions with several hundred deter- Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 7

Figure 5. Schematic representation of the geometry of the 2+ [Cu2O2] molecule at the endpoints the isomerization path. Configuration A (f = 100) is characterized by atomic dis- Figure 7. Relative energy of the isomerization path comparing tances ROO ' 1.4A˚ and RCuCu ' 3.6A˚, while configuration NOMSD-AFQMC with previous results. B (f = 0) is characterized by distances ROO ' 2.3A˚ and RCuCu ' 2.8A˚. Similar to previous studies, a linear interpo- lation between the two configurations defines the path. the path, relative to configuration B. We show results for AFQMC-NOMSD calculations with 28, 32 and 52 correlated electrons in order to mimic the choices made by previous calculations, which we also show. AFQMC calculations with 28 active electrons used 100 determinants, while calculations with 32 and 52 active electrons used 250 determinants. Larger active spaces required larger expansions in order to reached the similar levels of accuracy. Excellent agreement is observed between AFQMC, CR-CC and DMRG, when the appropriate number of active electrons is considered. This is a very strong in- dication that the discrepancies between these methods is mostly attributable to the differences in the number of correlated electrons.

VI. DISCUSSION

Several observations about the NOMSD approach are Figure 6. Convergence of the AFQMC-NOMSD total energy appropriate at this point. First we must point out that, 2+ for the [Cu2O2] molecule as a function of the number of similar to truncated CI expansions, the NOMSD ex- determinants in the expansion for the endpoints of the iso- pansion is not size extensive nor size consistent. This merization path. means that the number of terms in the expansion necessary for an equivalent description of the system will ultimately scale exponentially as a function of the sys- minants. This is to be expected since the electronic tem size. While this creates serious problems for direct structure of configuration B has significant static corre- use of truncated CI (or non-orthogonal) expansions in lation, requiring expansions with over 200 determinants quantum chemistry methods for periodic systems, the for reasonable convergence. The total energy of config- relevant question in our case is not whether the wave- uration A converges quickly, requiring only 75 determi- function is size-extensive but whether the reduction of nants for an accuracy of 1 kcal/mol, since in this case dy- the phase-less error is. While a study of the scaling of namic correlation is dominant and is efficiently captured the phase-less error with expansion length is outside the by the AFQMC method. Even in this complicated sce- scope of the work presented in this article, it is unlikely nario, NOMSD offers a systematically convergent, com- that in the thermodynamic limit non-exponential behav- pact choice for trial wave-functions. ior is obtained. Nonetheless, for systems of typical inter- Figure 7 shows the total energy of the molecule along est (up to 100 atoms), this approach leads to a simple Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 8 and well-defined path for error reduction. For larger cal- associated with basis set extrapolation. In addition, we culations, it is possible to employ the NOMSD approach obtain an mean absolute error of 0.31 mHa in the total in an active space approach, where only a small subset of energies when comparing against CCSDTQ, for a subset the orbitals are included in the construction of the corre- of the 55 molecules in the cc-pVDZ basis. These errors lated trial wave-function. This will lead to better scaling can easily be reduced further with longer expansions if and higher efficiency without sacrificing too much accu- desired. 2+ racy. A future publication will discuss modifications and For the correlated molecular system, [Cu2O2] , we improvements of the method needed in its application to showed the vast improvement obtained in the relative correlated materials in periodic boundary conditions. energies along the isomerization path from the use of Another important consideration is the evaluation ef- NOMSD expansions in combination with AFQMC. In ficiency of the NOMSD wave-function compared to tra- this case, AFQMC calculations with single determinant ditional expansions based on truncated or selected CI. trial wave-functions lead to large errors in the relative en- Since different configurations in orthogonal expansions ergies along the path, with errors as large as 30 kcal/mol differ by a finite number of orbitals, fast low-rank up- when compared to advanced electronic structure meth- date schemes can be used29 which leads to highly effi- ods like DMRG and CR-CC. In this case, NOMSD ex- cient evaluation routines. In the case of NOMSD expan- pansions of several hundred determinants were necessary sions, no such optimizations have been devised leading to in the AFQMC calculations to obtain well converged re- much higher evaluation cost as a function of expansion sults. We also showed that discrepancies in previous cal- length. A proper comparison of the two methods requires culations can be well explained by the differences in the a careful analysis of implementation details, which is out- number of core electrons used, obtaining excellent agree- side the scope of this work. Nonetheless, we believe that ment with CR-CC and DMRG when consistent choices the current approach offers sufficiently compact expan- are made. We believe that the work presented in this sions to compensate for the lower evaluation efficiency. article will lead to a further examination and optimiza- A detailed analysis of the computational aspects of the tion of the use of non-orthogonal compact expansions in method will be presented in a separate publication. quantum Monte Carlo and will further contribute to the quest for accurate and efficient wave-function ansatz for correlated quantum many-body systems. VII. CONCLUSIONS

The AFQMC method offers a powerful alternative for VIII. ACKNOWLEDGEMENTS the study of quantum many-body problems in physics, chemistry and material science. Its ability to handle both We would like to thank Chia-Chen Chang, Matthew weakly and strongly correlated electronic structure prob- Otten and Gustavo E. Scuseria for useful discussions. We lems with favorable scaling offers an excellent alterna- would also like to thank Carlos Jimenez-Hoyos for provid- tive to traditional methods in quantum chemistry and ing us access to the PHF code used to generate NOMSD ab initio electronic structure. When combined with flex- expansions. The PHF code used in this work was de- ible and efficient trial wave-functions, the method has the veloped in the Scuseria group at Rice University25,38,39. potential to become the method of choice for the study The contribution of John Gomez to this work was car- of realistic quantum systems. In this article, we present ried out during his summer internship at LLNL in 2016. non-orthogonal Multi-Slater determinant expansions as This work was performed under the auspices of the U.S. an accurate and compact choice for trial wave-functions Department of Energy by Lawrence Livermore National in AFQMC. They offer the flexibility, simplicity and com- Laboratory and supported by LLNL-LDRD project No. pact representation needed for realistic calculations of 15-ERD-013. Livermore National Laboratory is operated materials and complicated molecular systems. NOMSD by Lawrence Livermore National Security, LLC, for the expansions have the capability to recover significant frac- U.S. Department of Energy, National Nuclear Security tions of the correlation energy with a modest number Administration under Contract DE-AC52-07NA27344. of terms, compared to traditional orthogonal expansions J. A. G. acknowledges the support of the National Sci- based on configuration interaction which typically have ence Foundation Graduate Research Fellowship Program slowly decaying tails which require orders of magnitude (DGE-1450681). more terms for similar accuracy. For weakly correlated systems, including the molecules in the G1 test set, we showed the capacity of the NOMSD approach to systematically reduce the errors in total and REFERENCES atomization energies with short expansions, even as small 1 as 5 determinants in many cases. We obtain a mean Richard M. Martin. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, 2004. doi: absolute error of 1.17 mHa in the atomization energies 10.1017/CBO9780511805769. of the 55 molecules considered in this work with a 20 2Aron J. Cohen, Paula Mori-Sanchez, and Weitao Yang. Chal- determinant expansion, which includes potential errors lenges for density functional theory. Chemical Reviews, 112(1): Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 9

289–320, 2012. doi:10.1021/cr200107z. URL http://dx.doi. 15Kenneth P. Esler, Jeongnim Kim, David M. Ceperley, and Luke org/10.1021/cr200107z. PMID: 22191548. Shulenburger. Accelerating quantum monte carlo simulations of 3Cesare Pisani, Martin Schutz, Silvia Casassa, Denis Usvyat, real materials on gpu clusters. Computing in Science & Engi- Lorenzo Maschio, Marco Lorenz, and Alessandro Erba. Cryscor: neering, 14(1):40–51, 2012. doi:10.1109/MCSE.2010.122. URL a program for the post-hartree-fock treatment of periodic http://aip.scitation.org/doi/abs/10.1109/MCSE.2010.122. systems. Phys. Chem. Chem. Phys., 14:7615–7628, 2012. 16Matthias Troyer and Uwe-Jens Wiese. Computational complex- doi:10.1039/C2CP23927B. URL http://dx.doi.org/10.1039/ ity and fundamental limitations to fermionic quantum monte C2CP23927B. carlo simulations. Phys. Rev. Lett., 94:170201, May 2005. doi: 4JürgHutter, Marcella Iannuzzi, Florian Schiffmann, and Joost 10.1103/PhysRevLett.94.170201. URL https://link.aps.org/ VandeVondele. cp2k: atomistic simulations of condensed mat- doi/10.1103/PhysRevLett.94.170201. ter systems. Wiley Interdisciplinary Reviews: Computational 17Qmcpack. URL http://qmcpack.org/. Molecular Science, 4(1):15–25, 2014. ISSN 1759-0884. doi: 18Jeongnim Kim, Andrew D Baczewski, Todd D Beaudet, Anouar 10.1002/wcms.1159. URL http://dx.doi.org/10.1002/wcms. Benali, M Chandler Bennett, Mark A Berrill, Nick S Blunt, 1159. Edgar JosuéLandinez Borda, Michele Casula, David M Ceper- 5G. Kresse and J. Furthmuller. Efficient iterative schemes ley, Simone Chiesa, Bryan K Clark, Raymond C Clay III, Kris T for ab initio total-energy calculations using a plane-wave ba- Delaney, Mark Dewing, Kenneth P Esler, Hongxia Hao, Olle sis set. Phys. Rev. B, 54:11169–11186, Oct 1996. doi: Heinonen, Paul R C Kent, Jaron T Krogel, Ilkka Kylänpää, 10.1103/PhysRevB.54.11169. URL https://link.aps.org/doi/ Ying Wai Li, M Graham Lopez, Ye Luo, Fionn D Malone, 10.1103/PhysRevB.54.11169. Richard M Martin, Amrita Mathuriya, Jeremy McMinis, Cody A 6Qiming Sun, Timothy C. Berkelbach, Nick S. Blunt, George H. Melton, Lubos Mitas, Miguel A Morales, Eric Neuscamman, Booth, Sheng Guo, Zhendong Li, Junzi Liu, James D. McClain, William D Parker, Sergio D Pineda Flores, Nichols A Romero, Elvira R. Sayfutyarova, Sandeep Sharma, Sebastian Wouters, Brenda M Rubenstein, Jacqueline A R Shea, Hyeondeok Shin, and Garnet Kin-Lic Chan. Pyscf: the python-based simula- Luke Shulenburger, Andreas F Tillack, Joshua P Townsend, tions of chemistry framework. Wiley Interdisciplinary Reviews: Norm M Tubman, Brett Van Der Goetz, Jordan E Vincent, Computational Molecular Science, 8(1):e1340–n/a, 2018. ISSN D ChangMo Yang, Yubo Yang, Shuai Zhang, and Luning Zhao. 1759-0884. doi:10.1002/wcms.1340. URL http://dx.doi.org/ Qmcpack : an open source ab initio quantum monte carlo package 10.1002/wcms.1340. e1340. for the electronic structure of atoms, molecules and solids. Jour- 7M. Marsman, A. Grüneis,J. Paier, and G. Kresse. Second- nal of Physics: Condensed Matter, 30(19):195901, 2018. URL order møller–plesset perturbation theory applied to extended sys- http://stacks.iop.org/0953-8984/30/i=19/a=195901. tems. i. within the projector-augmented-wave formalism using a 19Malliga Suewattana, Wirawan Purwanto, Shiwei Zhang, Henry plane wave basis set. The Journal of Chemical Physics, 130 Krakauer, and Eric J. Walter. Phaseless auxiliary-field quan- (18):184103, 2009. doi:10.1063/1.3126249. URL http://aip. tum monte carlo calculations with plane waves and pseudopo- scitation.org/doi/abs/10.1063/1.3126249. tentials: Applications to atoms and molecules. Phys. Rev. B, 8Andreas Grüneis,Martijn Marsman, and Georg Kresse. Second- 75:245123, Jun 2007. doi:10.1103/PhysRevB.75.245123. URL order møller–plesset perturbation theory applied to extended sys- https://link.aps.org/doi/10.1103/PhysRevB.75.245123. tems. ii. structural and energetic properties. The Journal of 20Atsushi Ikawa, Shoji Yamamoto, and Hideo Fukutome. Orbital Chemical Physics, 133(7):074107, 2010. doi:10.1063/1.3466765. optimization in the resonating hartree-fock approximation and URL https://doi.org/10.1063/1.3466765. its application to the one dimensional hubbard model. Journal 9Mauro Del Ben, Mandes Schönherr, JürgHutter, and Joost of the Physical Society of Japan, 62(5):1653–1668, 1993. doi: VandeVondele. Bulk liquid water at ambient temperature and 10.1143/JPSJ.62.1653. URL https://doi.org/10.1143/JPSJ. pressure from mp2 theory. The Journal of Physical Chemistry 62.1653. Letters, 4(21):3753–3759, 2013. doi:10.1021/jz401931f. URL 21Hideo Fukutome. Theory of resonating quantum fluctua- http://dx.doi.org/10.1021/jz401931f. tions in a fermion systemresonating hartree-fock approxima- 10Joachim Paier, Xinguo Ren, Patrick Rinke, Gustavo E Scuse- tion. Progress of Theoretical Physics, 80(3):417–432, 1988. doi: ria, Andreas Grüneis,Georg Kresse, and Matthias Scheffler. As- 10.1143/PTP.80.417. URL +http://dx.doi.org/10.1143/PTP. sessment of correlation energies based on the random-phase ap- 80.417. proximation. New Journal of Physics, 14(4):043002, 2012. URL 22Norikazu Tomita, Seiichiro Ten-no, and Yoshitaka Tanimura. Ab http://stacks.iop.org/1367-2630/14/i=4/a=043002. initio molecular orbital calculations by the resonating hartree- 11Mauro Del Ben, JürgHutter, and Joost VandeVondele. Electron fock approach: superposition of non-orthogonal slater determi- correlation in the condensed phase from a resolution of iden- nants. Chemical Physics Letters, 263(5):687 – 690, 1996. ISSN tity approach based on the gaussian and plane waves scheme. 0009-2614. doi:https://doi.org/10.1016/S0009-2614(96)01266- Journal of Chemical Theory and Computation, 9(6):2654–2671, 3. URL http://www.sciencedirect.com/science/article/pii/ 2013. doi:10.1021/ct4002202. URL http://dx.doi.org/10. S0009261496012663. 1021/ct4002202. PMID: 26583860. 23Masatoshi Imada and Tsuyoshi Kashima. Path-integral renor- 12James McClain, Qiming Sun, Garnet Kin-Lic Chan, and Tim- malization group method for numerical study of strongly cor- othy C. Berkelbach. Gaussian-based coupled-cluster theory for related electron systems. Journal of the Physical Society of the ground-state and band structure of solids. Journal of Chem- Japan, 69(9):2723–2726, 2000. doi:10.1143/JPSJ.69.2723. URL ical Theory and Computation, 13(3):1209–1218, 2017. doi: https://doi.org/10.1143/JPSJ.69.2723. 10.1021/acs.jctc.7b00049. URL http://dx.doi.org/10.1021/ 24Yoshiki Imai, Yuichi Otsuka, and Masatoshi Imada. Applica- acs.jctc.7b00049. PMID: 28218843. tions of path-integral renormalization group method combined 13W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal. with density functional theory. Journal of Physics: Condensed Quantum monte carlo simulations of solids. Rev. Mod. Phys., 73: Matter, 19(36):365230, 2007. URL http://stacks.iop.org/ 33–83, Jan 2001. doi:10.1103/RevModPhys.73.33. URL https: 0953-8984/19/i=36/a=365230. //link.aps.org/doi/10.1103/RevModPhys.73.33. 25Carlos A. Jiménez-Hoyos, R. Rodr´ıguez-Guzmán, and Gus- 14Jeongnim Kim, Kenneth P Esler, Jeremy McMinis, Miguel A tavo E. Scuseria. Multi-component symmetry-projected ap- Morales, Bryan K Clark, Luke Shulenburger, and David M proach for molecular ground state correlations. The Journal of Ceperley. Hybrid algorithms in quantum monte carlo. Jour- Chemical Physics, 139(20):204102, 2013. doi:10.1063/1.4832476. nal of Physics: Conference Series, 402(1):012008, 2012. URL URL https://doi.org/10.1063/1.4832476. http://stacks.iop.org/1742-6596/402/i=1/a=012008. 26Hao Shi, Carlos A. Jiménez-Hoyos, R. Rodr´ıguez-Guzmán,Gus- tavo E. Scuseria, and Shiwei Zhang. Symmetry-projected wave Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 10

functions in quantum monte carlo calculations. Phys. Rev. B, 40Norikazu Tomita. Many-body wave functions approximated by 89:125129, Mar 2014. doi:10.1103/PhysRevB.89.125129. URL the superposition of spin-projected nonorthogonal slater deter- https://link.aps.org/doi/10.1103/PhysRevB.89.125129. minants in the resonating hartree-fock method. Phys. Rev. B, 27Chia-Chen Chang, Brenda M. Rubenstein, and Miguel A. 69:045110, Jan 2004. doi:10.1103/PhysRevB.69.045110. URL Morales. Auxiliary-field-based trial wave functions in quantum https://link.aps.org/doi/10.1103/PhysRevB.69.045110. monte carlo calculations. Phys. Rev. B, 94:235144, Dec 2016. 41Adam A. Holmes, Norm M. Tubman, and C. J. Umrigar. Heat- doi:10.1103/PhysRevB.94.235144. URL https://link.aps.org/ bath configuration interaction: An efficient selected configuration doi/10.1103/PhysRevB.94.235144. interaction algorithm inspired by heat-bath sampling. Journal of 28Miguel A. Morales, Jeremy McMinis, Bryan K. Clark, Jeongnim Chemical Theory and Computation, 12(8):3674–3680, 2016. doi: Kim, and Gustavo E. Scuseria. Multideterminant wave functions 10.1021/acs.jctc.6b00407. URL https://doi.org/10.1021/acs. in quantum monte carlo. Journal of Chemical Theory and Com- jctc.6b00407. PMID: 27428771. putation, 8(7):2181–2188, 2012. doi:10.1021/ct3003404. URL 42Sandeep Sharma, Adam A. Holmes, Guillaume Jeanmairet, Ali http://dx.doi.org/10.1021/ct3003404. PMID: 26588949. Alavi, and C. J. Umrigar. Semistochastic heat-bath config- 29Bryan K. Clark, Miguel A. Morales, Jeremy McMinis, Jeongnim uration interaction method: Selected configuration interaction Kim, and Gustavo E. Scuseria. Computing the energy of a water with semistochastic perturbation theory. Journal of Chem- molecule using multideterminants: A simple, efficient algorithm. ical Theory and Computation, 13(4):1595–1604, 2017. doi: The Journal of Chemical Physics, 135(24):244105, 2011. doi: 10.1021/acs.jctc.6b01028. URL https://doi.org/10.1021/acs. 10.1063/1.3665391. URL https://doi.org/10.1063/1.3665391. jctc.6b01028. PMID: 28263594. 30Michele Casula, Claudio Attaccalite, and Sandro Sorella. Corre- 43Hans-Joachim Werner, Peter J. Knowles, Gerald Knizia, Fred- lated geminal wave function for molecules: An efficient resonat- erick R. Manby, and Martin Schütz.Molpro: a general-purpose ing valence bond approach. The Journal of Chemical Physics, quantum chemistry program package. Wiley Interdisciplinary 121(15):7110–7126, 2004. doi:10.1063/1.1794632. URL https: Reviews: Computational Molecular Science, 2(2):242–253, 2012. //doi.org/10.1063/1.1794632. ISSN 1759-0884. doi:10.1002/wcms.82. URL http://dx.doi. 31Eric Neuscamman. Size consistency error in the antisym- org/10.1002/wcms.82. metric geminal power wave function can be completely re- 44Edgar Solomonik, Devin Matthews, Jeff R. Hammond, John F. moved. Phys. Rev. Lett., 109:203001, Nov 2012. doi: Stanton, and James Demmel. A massively parallel tensor con- 10.1103/PhysRevLett.109.203001. URL https://link.aps.org/ traction framework for coupled-cluster computations. Journal of doi/10.1103/PhysRevLett.109.203001. Parallel and Distributed Computing, 74(12):3176 – 3190, 2014. 32Eric Neuscamman. The jastrow antisymmetric geminal power ISSN 0743-7315. doi:https://doi.org/10.1016/j.jpdc.2014.06.002. in hilbert space: Theory, benchmarking, and application to a URL http://www.sciencedirect.com/science/article/pii/ novel transition state. The Journal of Chemical Physics, 139 S074373151400104X. Domain-Specific Languages and High-Level (19):194105, 2013. doi:10.1063/1.4829835. URL https://doi. Frameworks for High-Performance Computing. org/10.1063/1.4829835. 45Wirawan Purwanto and Shiwei Zhang. Quantum monte carlo 33M. Bajdich, L. Mitas, G. Drobný,L. K. Wagner, and K. E. method for the ground state of many-boson systems. Phys. Rev. Schmidt. Pfaffian pairing wave functions in electronic-structure E, 70:056702, Nov 2004. doi:10.1103/PhysRevE.70.056702. URL quantum monte carlo simulations. Phys. Rev. Lett., 96:130201, https://link.aps.org/doi/10.1103/PhysRevE.70.056702. Apr 2006. doi:10.1103/PhysRevLett.96.130201. URL https: 46Wirawan Purwanto, Henry Krakauer, and Shiwei Zhang. //link.aps.org/doi/10.1103/PhysRevLett.96.130201. Pressure-induced diamond to β-tin transition in bulk silicon: A 34M. Bajdich, L. K. Mitas, L. andgubernatis2 Wagner, and K. E. quantum monte carlo study. Phys. Rev. B, 80:214116, Dec 2009. Schmidt. Pfaffian pairing and backflow wavefunctions for elec- doi:10.1103/PhysRevB.80.214116. URL https://link.aps.org/ tronic structure quantum monte carlo methods. Phys. Rev. B, doi/10.1103/PhysRevB.80.214116. 77:115112, Mar 2008. doi:10.1103/PhysRevB.77.115112. URL 47W. A. Al-Saidi, Shiwei Zhang, and Henry Krakauer. Auxiliary- https://link.aps.org/doi/10.1103/PhysRevB.77.115112. field quantum monte carlo calculations of molecular systems with 35M. Guerrero, G. Ortiz, and J.E. Gubernatis. Search for a gaussian basis. The Journal of Chemical Physics, 124(22): superconductivity in the two-dimensional hubbard model us- 224101, 2006. doi:10.1063/1.2200885. URL https://doi.org/ ing the constrained path monte carlo. Computer Physics 10.1063/1.2200885. Communications, 127(1):143 – 149, 2000. ISSN 0010- 48Shiwei Zhang and Henry Krakauer. Quantum monte carlo 4655. doi:https://doi.org/10.1016/S0010-4655(00)00024-2. method using phase-free random walks with slater deter- URL http://www.sciencedirect.com/science/article/pii/ minants. Phys. Rev. Lett., 90:136401, Apr 2003. doi: S0010465500000242. 10.1103/PhysRevLett.90.136401. URL https://link.aps.org/ 36J. Carlson, Stefano Gandolfi, Kevin E. Schmidt, and Shi- doi/10.1103/PhysRevLett.90.136401. wei Zhang. Auxiliary-field quantum monte carlo method for 49W. A. Al-Saidi, Shiwei Zhang, and Henry Krakauer. Auxiliary- strongly paired fermions. Phys. Rev. A, 84:061602, Dec 2011. field quantum monte carlo calculations of molecular systems with doi:10.1103/PhysRevA.84.061602. URL https://link.aps.org/ a gaussian basis. The Journal of Chemical Physics, 124(22): doi/10.1103/PhysRevA.84.061602. 224101, 2006. doi:10.1063/1.2200885. URL https://doi.org/ 37Michel Caffarel, Thomas Applencourt, Emmanuel Giner, and 10.1063/1.2200885. Anthony Scemama. Using CIPSI Nodes in Diffusion Monte 50David Feller and Kirk A. Peterson. Re-examination of at- Carlo, chapter 2, pages 15–46. American Chemical Society, omization energies for the gaussian-2 set of molecules. The 2016. doi:10.1021/bk-2016-1234.ch002. URL http://pubs.acs. Journal of Chemical Physics, 110(17):8384–8396, 1999. doi: org/doi/abs/10.1021/bk-2016-1234.ch002. 10.1063/1.478747. URL https://doi.org/10.1063/1.478747. 38Carlos A. Jiménez-Hoyos, Thomas M. Henderson, Takashi 51Jeffrey C. Grossman. Benchmark quantum monte carlo calcu- Tsuchimochi, and Gustavo E. Scuseria. Projected hartree–fock lations. The Journal of Chemical Physics, 117(4):1434–1440, theory. The Journal of Chemical Physics, 136(16):164109, 2002. doi:10.1063/1.1487829. URL https://doi.org/10.1063/ 2012. doi:10.1063/1.4705280. URL https://doi.org/10.1063/ 1.1487829. 1.4705280. 52Kousik Samanta, Carlos A. Jiménez-Hoyos, and Gustavo E. 39Roman Schutski, Carlos A. Jiménez-Hoyos, and Gustavo E. Scuseria. Exploring copper oxide cores using the projected Scuseria. Analytic energy gradient for the projected hartree– hartree–fock method. Journal of Chemical Theory and Com- fock method. The Journal of Chemical Physics, 140(20):204101, putation, 8(12):4944–4949, 2012. doi:10.1021/ct300689e. URL 2014. doi:10.1063/1.4876490. URL https://doi.org/10.1063/ http://dx.doi.org/10.1021/ct300689e. PMID: 26593188. 1.4876490. Non-Orthogonal Multi-Slater Determinant Expansions in Auxiliary Field Quantum Monte Carlo 11

53Larry A. Curtiss, Christopher Jones, Gary W. Trucks, Krishnan tion effects: The second row atoms al–ar, and the first row Raghavachari, and John A. Pople. Gaussian-1 theory of molecu- atoms b–ne revisited. The Journal of Chemical Physics, 117 lar energies for second-row compounds. The Journal of Chemi- (23):10548–10560, 2002. doi:10.1063/1.1520138. URL https: cal Physics, 93(4):2537–2545, 1990. doi:10.1063/1.458892. URL //doi.org/10.1063/1.1520138. https://doi.org/10.1063/1.458892. 59Benjamin F. Gherman and Christopher J. Cramer. Quantum 54Norbert Nemec, Michael D. Towler, and R. J. Needs. Benchmark chemical studies of molecules incorporating a cu2o22+ core. all-electron ab initio quantum monte carlo calculations for small Coordination Chemistry Reviews, 253(5):723 – 753, 2009. molecules. The Journal of Chemical Physics, 132(3):034111, ISSN 0010-8545. doi:https://doi.org/10.1016/j.ccr.2007.11.018. 2010. doi:10.1063/1.3288054. URL https://doi.org/10.1063/ URL http://www.sciencedirect.com/science/article/pii/ 1.3288054. S0010854507002731. Theory and Computing in Contemporary 55David Feller, Kirk A. Peterson, and David A. Dixon. A sur- Coordination Chemistry. vey of factors contributing to accurate theoretical predictions of 60Takeshi Yanai, Yuki Kurashige, Eric Neuscamman, and Garnet atomization energies and molecular structures. The Journal of Kin-Lic Chan. Multireference quantum chemistry through a joint Chemical Physics, 129(20):204105, 2008. doi:10.1063/1.3008061. density matrix renormalization group and canonical transforma- URL https://doi.org/10.1063/1.3008061. tion theory. The Journal of Chemical Physics, 132(2):024105, 56David Feller, Kirk A. Peterson, and J. Grant Hill. On the effec- 2010. doi:10.1063/1.3275806. URL https://doi.org/10.1063/ tiveness of ccsd(t) complete basis set extrapolations for atomiza- 1.3275806. tion energies. The Journal of Chemical Physics, 135(4):044102, 61Christopher J. Cramer, Marta W loch, Piotr Piecuch, Cristina 2011. doi:10.1063/1.3613639. URL https://doi.org/10.1063/ Puzzarini, and Laura Gagliardi. Theoretical models on the 1.3613639. cu2o2 torture track: Mechanistic implications for oxytyrosinase 57David Feller. Benchmarks of improved complete basis set extrap- and small-molecule analogues. The Journal of Physical Chem- olation schemes designed for standard ccsd(t) atomization ener- istry A, 110(5):1991–2004, 2006. doi:10.1021/jp056791e. URL gies. The Journal of Chemical Physics, 138(7):074103, 2013. doi: http://dx.doi.org/10.1021/jp056791e. PMID: 16451035. 10.1063/1.4791560. URL https://doi.org/10.1063/1.4791560. 58Kirk A. Peterson and Thom H. Dunning Jr. Accurate correlation consistent basis sets for molecular core–valence correla-