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 Josu´eLandinez 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 mag- nitude of statistical uncertainties. Hence, accurate and efficient wave-function ansatz are important to the suc- cess 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 h'ij'ji 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 y Z c cq pansions. These wave-functions have been used in the j'ii = e pq p j'ref 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 = hΦjH^ jΦi=hΦjΦi, 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 j'ii (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