An Auxiliary-Field Quantum Monte Carlo Perspective on the Ground State of the Dense Uniform Electron Gas: An Investigation with Hartree-Fock Trial Wavefunctions
AAn Auxiliary-Field Quantum Monte Carlo Perspective on the Ground State of theDense Uniform Electron Gas: An Investigation with Hartree-Fock Trial Wavefunctions
Joonho Lee ∗ College of Chemistry, University of California, Berkeley, California 94720, USA.
Fionn D. Malone † and Miguel A. Morales ‡ Quantum Simulations Group, Lawrence Livermore National Laboratory,7000 East Avenue, Livermore, CA, 94551 USA.
We assess the utility of Hartree-Fock (HF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. The combina-tion of ph-AFQMC with spin-restricted HF (RHF+ph-AFQMC), was found to be highly accurateand efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for r s ≤ . r s ≤ . r s ≤ . r s ≤ . r s = 5 . r s = 5 .
0. We emphasize the need for a better trial wavefunction for ph-AFQMC in simulat-ing strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress thepotential utility of RHF+ph-AFQMC for simulating dense solids.
I. INTRODUCTION
Describing electron correlation in a scalable waythat can handle hundreds of electrons is a grandchallenge in quantum chemistry and condensed mat-ter physics. State-of-the-art methods include coupled-cluster (CC) methods , density matrix renormalizationgroup (DMRG) methods , and quantum Monte Carlo(QMC) approaches . Each method exhibits differentweaknesses and strengths and therefore they have beenapplied to solve a different class of problems in chemistryand condensed matter physics.In this work, we will focus on a projector QMCmethod, namely, the auxiliary-field QMC (AFQMC)approach . Projector Monte Carlo methods, whileformally exact, typically impose a constraint in the imag-inary time propagation in order to overcome the fermionsign problem and achieve a polynomial scaling algorithm.Both, diffusion Monte Carlo (DMC) and AFQMC en-force this constraint using a trial wavefunction, whichcan in principle be systematically improved towards theexact result. These constraints lead to the phaseless-AFQMC (ph-AFQMC) and fixed-node (FN-DMC) algo-rithms both of which scale like O ( N ) − O ( N ) with thenumber of electrons N .Although the formalism of DMC and AFQMC are verysimilar, there are some key differences between the two.First, AFQMC works in the second-quantized frameworkcommon to most quantum chemical methods, and intro-duces a finite basis set. Therefore, AFQMC energies needto be extrapolated to the complete basis set (CBS) limit in order to compare directly with experiments. This isin contrast with DMC which works in real space and di-rectly in the CBS limit. Second, incorporating widelyused Jastrow factors (JFs) into AFQMC is quite chal-lenging. JFs are economical ways to incorporate residualelectron correlation by enforcing cusp conditions eitherbetween electrons and nuclei or among electrons. Lastly,unlike FN-DMC, ph-AFQMC is not variational .Despite these issues, AFQMC offers a number ofpromising advantages precisely because it works di-rectly in an orbital-based basis. In particular, all-electron, frozen core and non-local pseudopotential cal-culations can be performed with no additional approx-imations. Furthermore, as most quantum chemistrymethods are performed with a finite basis set, manytricks used in quantum chemistry can be used to im-prove AFQMC as well. For instance, tensor hyper con-traction approaches have recently been employed toreduce the memory requirement of AFQMC . Em-ploying explicitly correlated basis functions (similar inspirit to JFs for DMC) should also be possible to re-duce the basis set incompleteness error of AFQMC . Inaddition to this, computing properties other than the to-tal energy, which has historically been a challenge forprojector QMC methods, can be more straightforwardlyachieved in AFQMC . Recent examples include one-and two-particle reduced density matrices , imaginarytime correlation functions as well as forces .AFQMC has been successfully applied in recent yearsto a number of challenging problems in both quantumchemistry and solid state physics . However, thebroad applicability of the method is not as well under- a r X i v : . [ phy s i c s . c h e m - ph ] J u l stood as more traditional quantum chemistry approacheswhich have seen decades worth of sustained develop-ment and benchmarking. The primary limiting factorof AFQMC is the choice of trial wavefunction. Single-determinant trial wavefunctions from Hartree–Fock (HF)or density functional theory calculations have shown re-markable accuracy for a broad range of applications in-cluding the two-dimensional Hubbard model , dipole-bound anions and solid state applications , withtotal energies often approaching the accuracy of cou-pled cluster singles and doubles with perturbative triplesCCSD(T) . However, for more strongly correlated sys-tems such as transition metal containing complexes singledeterminant trial wavefunctions are not sufficiently accu-rate and multi determinant trial wavefunctions becomenecessary .Often short determinantal expansions from com-plete active space self-consistent field (CASSCF)wavefunctions or non-orthogonal multi-Slater de-teraminant trial wavefunction can help to restorethe accuracy of the method. However, since multi-determinant wavefunctions scale exponentially with sys-tem size, this approach to improving the trial wavefunc-tion is ultimately limited, particularly for large scale ap-plications. The search for more economical and accuratetrial wavefunctions for AFQMC (and also FN-DMC) isan active area of research and no single approach canachieve both polynomial scaling and broadly consistentaccuracy at the same time.Our goal is to assess the quality of HF trial wave-functions for the Uniform Electron Gas (UEG) model .HF wavefunctions are the simplest possible referencestates to perform subsequent correlation calculations inquantum chemistry. Often, artificial symmetry break-ing caused by HF wavefunctions causes confusion in un-derstanding electron correlation. It is artificial be-cause the symmetry breaking occurs due to the lack oftreatment for weak correlation, not strong correlation.Artificial symmetry breaking affects the performance ofsubsequent correlation calculations greatly (especially forweakly correlated systems) . In such cases, it is pre-ferred to use spin-restricted HF (RHF) orbitals as op-posed to variationally preferred broken symmetry HF or-bitals. Therefore, one has to be cautious when choosing aproper HF state for correlation calculations. Nonetheless,HF is not only the simplest but also scalable to hundredsof electrons. With HF trials (which we name as HF+ph-AFQMC), AFQMC scales strictly as O ( N ) − O ( N ) fora fixed statistical accuracy. Therefore, it is crucial toassess the accuracy of HF+ph-AFQMC and understandthe scope of it in simulating large-scale chemical andsolid-state systems. After all, the gold-standard quan-tum chemistry method, CCSD(T) is performed on topof HF states.In the context of this paper, the UEG model providesus with a simplified version of the full ab-initio Hamil-tonian for a solid, essentially omitting the electron-ioninteraction term and all of the material complications it entails. One can tune the magnitude of dynamic andstatic correlation at the Hamiltonian level using a singleparameter (for a fixed number of electrons) through thedimensionless Wigner-Seitz radius r s . For low r s values,the electron density is high, the distance between elec-trons is short, and electrons are all paired. In this regime,diagrammatic resummation techniques such as randomphase approximation (RPA) and low-order perturbationtheory based on RHF references are highly accurate .We call this regime “weakly correlated”. On the otherhand, for high r s values, electrons are unpaired and spa-tially well-separated and closed-shell RHF references pro-vide qualitatively wrong picture (i.e., it does not describethe open-shell nature of the system). In this case, RPA orother low-order perturbation theory based on RHF refer-ences fails. Furthermore, the use of broken symmetry HFreferences does not provide accurate description eitherdue to spin contamination. We call this regime “stronglycorrelated”. This tunability allows for a unambiguouscomparison between the strengths and weaknesses of var-ious methods. Moreover, there exist a number of bench-mark results both for intermediate system sizes withinthe reach of traditional quantum chemistry approaches,as well as results for much larger system sizes, and alsoresults extrapolated to the thermodynamic limit.Recently, the formally exact full configuration interac-tion quantum Monte Carlo method (FCIQMC) pro-vided benchmark results for a range of densities for a14- and 54-electron system . These FCIQMC studiesalso motivated recent coupled-cluster Monte Carlo studies on the 14-electron UEG model by Neufeld andThom where they provided CCSD, CCSD and triples(CCSDT), CCSDT and quadruples (CCSDTQ) withRHF references results for a wide range of r s and ba-sis sets . As the scope of truncated spin-restricted CC(RCC) approaches is relatively well understood, compar-ing HF+ph-AFQMC against these results will lead us toa better understanding of the scope of HF+ph-AFQMC.Although the ph-AFQMC has been applied to 3D UEGbefore to construct the KZK functional and to small2D UEG models , we believe this is the first publishedextensive benchmarking study of the 3D UEG using ph-AFQMC.This paper is organized as follows: (1) we briefly reviewthe formalism of ph-AFQMC and the UEG model, (2)we analyze the basis set convergence of AFQMC in the14-electron UEG model and compare its result againstFCIQMC and CCMC , and (3) we study larger systems(54-electron and 114-electron) and discuss the AFQMCperspectives on simulating the ground state of solids. II. METHODS
In this section we briefly summarize the basics of theAFQMC method and the phaseless approximation whichleads to the phaseless AFQMC algorithm (ph-AFQMC).We use n occ to denote the number of occupied molecularorbitals (MOs) and n vir to denote the number of unoc-cupied MOs. A. AFQMC
1. Free-Projection AFQMC
The zero-temperature AFQMC algorithm is a stochas-tic realization of power methods that target the lowestroot of the Hamiltonian ˆ H . The algorithm is based onthe following identity: | Ψ (cid:105) ∝ lim τ →∞ exp (cid:16) − τ ˆ H (cid:17) | Φ (cid:105) = lim τ →∞ | Ψ( τ ) (cid:105) , (1)where | Ψ (cid:105) is the exact ground state and | Φ (cid:105) is an initialstarting wavefunction satisfying (cid:104) Φ | Ψ (cid:105) (cid:54) = 0. Althoughthe initial wavefunction | Ψ (cid:105) can differ from the trialwavefunction | Ψ T (cid:105) , for the purpose of this work we willassume | Ψ (cid:105) = | Φ T (cid:105) unless mentioned otherwise. Eq. (1)is implemented stochastically by repeatedly applying apropagator, exp( − ∆ τ ˆ H ), to a set of random walkers un-til the ground state is reached. Each walker is comprisedof a Slater determinant, | ψ n ( τ ) (cid:105) , and a weight w n ( τ ) suchthat the statistical representation of the wavefunction isgiven by | Ψ( τ ) (cid:105) = (cid:80) n w n ( τ ) | ψ n ( τ ) (cid:105) .In order to practically realize the projection, we firstsplit the Hamiltonian into one-body and two-body oper-ators (i.e., ˆ H = ˆ H + ˆ H ). For the two-body terms, wewrite them in the sum of squared operators,ˆ H = − N α (cid:88) α ˆ v α . (2)Then, we apply the Hubbard-Stratonovich transforma-tion to rewrite the imaginary-time propagator in termsof only one-body operators. With the symmetric Trotterdecomposition, the propagator readsexp( − ∆ τ ˆ H ) = (cid:90) d x p ( x ) ˆ B (∆ τ, x ) , (3)where p ( x ) is the standard normal distribution, x is avector of N α auxiliary fields and ˆ B is defined asˆ B (∆ τ, x ) = e − ∆ τ ˆ H e −√ ∆ τ x · ˆ v e − ∆ τ ˆ H . (4)At each time step, each walker draws Gaussian randomnumbers to sample one instance of x and provides a sam-ple to the HS transformation in Eq. (3). The applicationof a one-body operator such as Eq. (4) to a Slater deter-minant yields yet another single Slater determinant .For a generic ab-initio Hamiltonian the propagatorsappearing in Eq. (4) will in general be complex and theweights of the walkers will acquire a phase that will bedistributed uniformly in the complex plane in the longimaginary time limit . This ‘phase problem’ is analo-gous to the notorious fermion sign problem encountered in DMC and has no known solution in general. The phaseproblem can be somewhat mitigated through mean-fieldsubtraction (i.e., redefining ˆ v (cid:48) α = ˆ v α − (cid:104) ˆ v α (cid:105) ) in Eq.(4), but the statistics will be eventually swamped by thephase problem. Note that mean field subtraction is es-sentially identical to normal-ordering ˆ v α which ensuresˆ v (cid:48) α | Φ (cid:105) = 0 for all α .
2. Phaseless AFQMC
It is possible to eliminate this phase problem entirelyat the sake of introducing biases into the results using theso-called phaseless approximation . This is achieved byfirst performing an importance sampling transformationto the propagator such that walkers now undergo themodified propagation: w n ( τ + ∆ τ ) | ψ n ( τ + ∆ τ ) (cid:105) = (cid:104) I ( x n , ¯x n , τ, ∆ τ ) ˆ B (∆ τ, x n − ¯x n ) (cid:105) w n ( τ ) | ψ n ( τ ) (cid:105) , (5)where the importance function (in hybrid form) is definedas I ( x n , ¯x n , τ, ∆ τ ) = S n ( τ, ∆ τ ) e x n · ¯x n − ¯x n · ¯x n / , (6) S n is the overlap ratio of the n -th walker S n ( τ, ∆ τ ) = (cid:104) Ψ T | ˆ B (∆ τ, x n − ¯x n ) | ψ n ( τ ) (cid:105)(cid:104) Ψ T | ψ n ( τ ) (cid:105) , (7)and ¯x n is an “optimal” force bias which is a shift to theGaussian distribution, given as ¯x n (∆ τ, τ ) = −√ ∆ τ (cid:104) Ψ T | ˆ v (cid:48) | ψ n ( τ ) (cid:105)(cid:104) Ψ T | ψ n ( τ ) (cid:105) . (8)The phaseless approximation (ph) is then defined as amodification to this importance function I ph ( x n , ¯x n , τ, ∆ τ ) = | I ( x n , ¯x n , τ, ∆ τ ) |× max(0 , cos( θ n ( τ )))(9)where the phase θ n ( τ ) is given by θ n ( τ ) = arg ( S n ( τ, ∆ τ )) . (10)The walker weights and Slater determinants are then up-dated as w n ( τ + ∆ τ ) = I ph ( x n , ¯x n , τ, ∆ τ ) × w n ( τ ) (11) | ψ n ( τ + ∆ τ ) (cid:105) = ˆ B (∆ τ, x n − ¯x n ) | ψ n ( τ ) (cid:105) . (12)Evidently, the phaseless approximation ensures that thewalker weights remain real and non-negative throughoutthe simulation and therefore removes the phase problemcompletely.The mixed estimate for the local energy estimator canbe computed with the generalized Green’s function (orone-particle reduced density matrix) P , P pq = (cid:104) Ψ T | ˆ a † p ˆ a q | ψ n ( τ ) (cid:105)(cid:104) Ψ T | ψ n ( τ ) (cid:105) = (cid:18) C ψ n (cid:16) C † Ψ T C ψ n (cid:17) − C † Ψ T (cid:19) qp (13)where C ψ n is the occupied MO coefficient of | ψ n ( τ ) (cid:105) and C Ψ T the occupied MO coefficient of | Ψ T (cid:105) . Once the sim-ulation has equilibrated, we will have a statistical repre-sentation of the ground state wavefunction given by | Ψ( τ ) (cid:105) = (cid:88) n w n ( τ ) | ψ n ( τ ) (cid:105)(cid:104) Ψ T | ψ n ( τ ) (cid:105) , (14)from which we can compute the mixed estimator for theenergy as E ( τ ) = (cid:104) Ψ T | ˆ H | Ψ( τ ) (cid:105)(cid:104) Ψ T | Ψ( τ ) (cid:105) = (cid:80) n w n ( τ ) (cid:15) n ( τ ) (cid:80) n w n ( τ ) , (15)where (cid:15) n ( τ ) is the local energy of a walker, (cid:15) n ( τ ) = (cid:104) Ψ T | ˆ H | ψ n ( τ ) (cid:105)(cid:104) Ψ T | ψ n ( τ ) (cid:105) . (16)We will see how the local energy evaluation is donespecifically for the UEG model later.
3. Size-consistency of ph-AFQMC
Size-consistency is a property of a wavefunction for iso-lated systems A and B that asserts the product separabil-ity of a supersystem wavefunction ( | Ψ AB (cid:105) = | Ψ A (cid:105)| Ψ B (cid:105) )and also the additive separability of energy ( E AB = E A + E B ). Configuration interaction (CI) based quantumchemistry methods are in general not size-consistent. In particular, the only size-consistent CI methods are CIwith singles (CIS) and FCI. On the other hand, single-reference CC methods are size-consistent as long as theform of wavefunction is parametrized by an exponentialof the cluster operator. To reliably obtain the thermo-dynamic limit of large systems, size-consistency is nec-essary. At the thermodynamic limit, size-inconsistentmethods approach just mean-field total energy and es-timate no correlation energy. Therefore, CC methods,due to size-consistency, have stood out as a unique toolfor simulating bulk systems .We will show that ph-AFQMC is also size-consistentas long as the trial wavefunction is product separable.For isolated systems A and B , the supersystem Hamil-tonian separates into ˆ H A and ˆ H B . Furthermore, thesetwo operators commute since these systems are isolated.Therefore, the propagator is also product separable,exp( − ∆ τ ˆ H AB ) = exp( − ∆ τ ˆ H A ) exp( − ∆ τ ˆ H B ) (17)The HS transformation can be performed onexp( − ∆ τ ˆ H A ) and exp( − ∆ τ ˆ H B ) separately so that we have ˆ B AB = ˆ B A ˆ B B . This proves the size-consistencyof free-projection AFQMC.It can be also shown that the phaseless constraint isproduct separable. The overlap function in Eq. (7) canbe written as S ABn = (cid:104) Ψ AT | ˆ B A | ψ An (cid:105)(cid:104) Ψ AT | ψ An (cid:105) (cid:104) Ψ BT | ˆ B B | ψ Bn (cid:105)(cid:104) Ψ BT | ψ Bn (cid:105) = S An S Bn (18)where the only assumptions we are making are (1) theproduct separability of the trial wavefunction: | Ψ ABT (cid:105) = | Ψ AT (cid:105)| Ψ BT (cid:105) and (2) the product separability of the slaterdeterminant of n -th walker: | ψ ABn (cid:105) = | ψ An (cid:105)| ψ Bn (cid:105) . Theassumption (2) can be satisfied as long as we start froma product separable wavefunction since the propagator isproduct separable. With this overlap function, one canshow that the importance function also obeys the productseparability and therefore we conclude that ph-AFQMCis size-consistent . B. Uniform Electron Gas
The Hamiltonian for the uniform electron gas (UEG)is given simply as the sum of the kinetic energy andelectron-electron interaction operator (up to a constant):ˆ H = ˆ T + ˆ V ee + E M . (19)We will work with a basis of planewave spin orbitals (cid:104) r σ | G i σ i (cid:105) = L / e i G i · r δ σ,σ i , where L is the length of thesimulation cell, G i = πL n i for n i a vector of integers and σ i is a spin index (either α or β ). We impose a kineticenergy cutoff E cut and work with a finite basis of 2 M spin orbitals. In this basis the kinetic energy is writtenas ˆ T = (cid:88) G | G | a † G a G , (20)and the electron-electron interaction operator is given byˆ V ee = 12Ω (cid:88) Q (cid:54) = , G , G π | Q | a † G + Q a † G − Q a G a G , (21)where Ω = L is the simulation cell volume, Q is amomentum transfer vector that lives in an enlarged ba-sis of size 4 E cut and we have dropped the subscript in-dex on G for simplicity. Lastly, the Madelung energy E M is included to account for the self-interaction of theEwald sum under periodic boundary conditions . Forsimplicity we use the formula proposed by Schoof andco-workers E M ≈ − . × (cid:18) π (cid:19) / N / r − s , (22)where N is the number of electrons in the simulation cellcell and r s = (cid:16) L πN (cid:17) / is the dimensionless Wigner-Seitzradius.The local energy (cid:15) n ( τ ) for the UEG then reads (cid:15) n ( τ ) = E M + (cid:88) G | G | P GG + 12Ω (cid:88) Q (cid:54) = π | Q | (cid:0) Γ Q − Λ Q (cid:1) , (23)where the Coulomb two-body density matrix Γ Q isΓ Q = (cid:88) G P G + Q , G (cid:88) G P G − Q , G (24)and the exchange two-body density matrix Λ Q isΛ Q = (cid:88) G G P G + Q , G P G − Q , G (25)The formation of Γ Q costs O ( M ) whereas the forma-tion of Λ Q takes O ( M ) amount of work. Therefore, theevaluation of the exchange contribution is the bottleneckin the local energy evaluation. As noted in Ref. 39 theevaluation of the energy (and propagation) can be accel-erated using fast Fourier transforms, however we did notuse this optimization here.The two-body Hamiltonian ˆ V ee needs to be rewrittenas a sum of squares to employ the AFQMC algorithm. Itwas shown in Ref. 39 thatˆ V ee = 14 (cid:88) Q (cid:54) = (cid:104) ˆ A ( Q ) + ˆ B ( Q ) (cid:105) , (26)where ˆ A ( Q ) = (cid:115) π Ω | Q | (cid:16) ˆ ρ ( Q ) + ˆ ρ † ( Q ) (cid:17) , (27)and ˆ B ( Q ) = i (cid:115) π Ω | Q | , (cid:16) ˆ ρ ( Q ) − ˆ ρ † ( Q ) (cid:17) , (28)with the momentum transfer operator ˆ ρ defined asˆ ρ ( Q ) = (cid:88) G a † G + Q a G Θ (cid:32) E cut − | G + Q | (cid:33) , (29)where Θ is the Heaviside step function. The Hubbard-Stratonovich operators ˆ v are now ˆ A ( Q ) and ˆ B ( Q ), andthe rest of the AFQMC algorithm follows straightfor-wardly. C. Hartree-Fock Trial Wavefunctions
In ph-AFQMC, the main source of error is the bias in-troduced by the phaseless constraint. The magnitude ofthis bias is heavily dependent on the quality of trial wave-functions. Although there are advanced options avail-able for these such as multideterminantal trials and self-consistently determined single-determinantal trials , wewill employ a simple single determinant RHF trial wave-function in most cases. In the UEG model, this is an M × N matrix (where N is the number of electrons and M is the number of planewaves) with 1’s on the diagonalentries.Typically, for strongly correlated systems it is useful toexploit essential symmetry breaking with HF wavefunc-tions. It is essential (as opposed to artificial) in the sensethat the property of a single determinant wavefunction isqualitatively wrong without it. An attempt to exploit es-sential symmetry breaking typically leads to either spin-unrestricted HF (UHF) or spin-generalized HF (GHF)trial wavefunctions which have a lower energy than RHF.Indeed, such essential symmetry breaking was shown tobe powerful when applying ph-AFQMC to bond dissoci-ation of F .An example of artificial symmetry breaking is buck-minsterfullerene (C ), a stable, electron paramagneticresonance silent (EPR-silent) molecule . There is acomplex, GHF (cGHF) solution for C which wascharacterized to be an artifact due to the lack of treat-ment for weak correlation at the HF level . In otherwords, orbital optimization in the presence of weak cor-relation such as the second-order Møller-Plesset theorywould restore artificial symmetry breaking. Since bothexperiments and computations suggest that C isa stable closed-shell molecule, the RHF state is morequalitatively correct than other broken-symmetry HFstates. A detailed study of artificial versus essential sym-metry breaking in ph-AFQMC will be published in ourforthcoming paper.The instability of RHF solutions is expected at all r s values of the UEG model at the thermodynamic limitas proven by Overhauser. As mentioned in Ref. 85,however, the R to U spin-symmetry breaking may notoccur in the UEG model with a finite number of elec-trons. Instead, there is a critical Wigner-Seitz radius ( r cs )below which no UHF solution exists. This is not surpris-ing in the context of quantum chemistry since this is thesame concept as “Coulson-Fischer points” in molecules.Namely, when dissociating molecules there exists a criti-cal bond length where the R to U instability occurs. Atbond distances closer than this, there is no genuine UHFsolution.In order to perform HF calculations, one must com-pute the effective one-body operator called the “Fock”operator defined as F = ∂E∂ D (30)where D = C occ C † occ with C occ being the occupied MOcoefficient matrix. After some straightforward algebra,we find F = T + J − K (31)where the kinetic energy matrix, T , reads T G , G (cid:48) = 12 | G | δ G , G (cid:48) , (32)the Coulomb matrix, J , is J G , G (cid:48) = 12Ω (cid:88) Q (cid:54) =0 π | Q | ∂ Γ Q ∂D G , G (cid:48) (33)with ∂ Γ Q ∂D G , G (cid:48) = (cid:88) G δ G , G + Q δ G (cid:48) , G (cid:88) G D G − Q , G + (cid:88) G D G + Q , G (cid:88) G δ G , G − Q δ G (cid:48) G (34)and the exchange matrix is given by K G , G (cid:48) = 12Ω (cid:88) Q (cid:54) =0 π | Q | ∂ Λ Q ∂D G , G (cid:48) (35)with ∂ Λ Q ∂D G , G (cid:48) = (cid:88) G G (cid:18) δ G , G + Q δ G (cid:48) , G D G − Q , G + D G + Q , G δ G , G − Q δ G (cid:48) , G (cid:19) (36)A similar derivation can be found in Ref. 85.With the above Fock matrix, one can perform an HFcalculation by optimizing the HF energy expression withrespect to the orbital rotation parameter Θ vo (a matrix of n vir -by- n occ ) which relates two different MO coefficientsvia a unitary transformation, C (cid:48) = C exp ( ∆ ) (37)where the antihermition matrix ∆, which is parametrizedby Θ vo , ∆ = (cid:20) oo − Θ † vo Θ vo vv (cid:21) (38)The subscript of each matrix block denotes the dimensionof the corresponding block, o = n nocc and v = n nvir . AnHF solution is defined as a stationary point that satisfies ∂E HF ∂ Θ vo = 0 (39) where E HF = (cid:104) Φ HF | ˆ T + ˆ V ee | Φ HF (cid:105) . (40)The local stability of a given stationary point can thenbe tested by diagonalizing the orbital Hessian, M , M ai,bj = ∂ E∂ Θ ai ∂ Θ bj (41)We will see whether there is essential symmetry break-ing in the low r s regime in the UEG model and try toutilize this essential symmetry breaking when appropri-ate. III. COMPUTATIONAL DETAILS
Unless otherwise noted, the AFQMC calculations inthis work were performed by a development version of
QMCPACK . PAUXY was also used in the initial testingstages. Unless noted otherwise, AFQMC results beloware obtained using QMCPACK . HANDE was used to cross-check our numbers for small systems that are not pre-sented in this work. We used 0.005 a.u. for the time step∆ τ throughout the paper. This was found to be enoughfor systems we considered here. A total of 2880 walkerswere used and the population bias from this was foundto be negligible in the results reported here. The comb and pair branching population control algorithms areused in PAUXY and
QMCPACK respectively. The demonstra-tion of the convergence of these parameters is availablein the Supplementary Materials.All broken symmetry HF calculations were performedwith a development version of Q-Chem and the detailsfor the implementation can be found in refs. 49,94,95.The optimizer used for those HF calculations is geometrydirect minimization (GDM) developed by Van Voorhisand Head-Gordon . The internal stability analysis wasperformed for all HF solutions to ensure the local sta-bility of each solution where we used a finite-differenceorbital Hessian using analytic orbital gradient . All cal-culations were performed with periodic boundary condi-tions; no twist averaging was performed. IV. RESULTS
The UEG model has been explored by multiple meth-ods at T = 0 and an extensive amount of benchmarkdata are already available. We compare our ph-AFQMCresults against other methods and discuss whether theuse of RHF trial wavefunction is reliable for r s ≤ . r s increases (approaching the atomic limit)since electrons tend to localize. r s = 5 . A. Broken-Symmetry HF States
We summarize some interesting aspects in the HF solu-tions of the UEG model due to its simple form of Hamil-tonian:1. The MO coefficient matrix of an RHF solution isan identity matrix. This makes obtaining an UHFsolution from solving an eigenvalue equation for F difficult in the following sense. C from diagonaliz-ing F is always unitary and the subsequent densitymatrix D is therefore identity. Since D is identi-cal to the density matrix of RHF, one obtains anRHF solution immediately after one single diago-nalization of F . A direct energy minimization orthe use of the HF projector is necessary to obtainbroken-symmetry HF states.2. The RHF energy does not depend on the basis setsize. However, broken-symmetry solutions dependon the basis set size. It is important to convergetheir energies with respect to M when discussingtheir existence.3. Both Coulomb and kinetic energies are minimized in an RHF solution. In particular, the Coulombenergy is always zero in RHF.4. The R to U symmetry breaking is driven by thelowering of exchange energy which compensates theincrease in the Coulomb and kinetic energies.We are interested in the paramagnetic phase of the UEG.As r s increases, the ferromagnetic (i.e., spin-polarized)phase becomes the ground state . The GHF solutioncan appear in the transition between these two phasesat quite high r s values. Other than this transition, agenuine GHF solution does not appear and therefore westudy only the UHF solutions for the purpose of thisstudy.We discuss the UHF solutions in the 14-electron UEGmodel. In Fig. 1 (a), we show the basis set convergencebehavior of UHF. Unlike RHF, the UHF energy doesdepend on the basis set size. For the 14-electron UEGmodel, it is sufficient to converge the UHF energies over r s ≤ . M = 925. Based on Fig. 1 (a), we see thatthe energy lowering from RHF to UHF starts to appearfor r s > .
5. The critical Wigner-Seitz radius for the14-electron model is r cs ∈ (3 . , . (cid:104) ˆ S (cid:105) value of the UHF solutions for M = 943 as a function of r s as shown in Fig. 1 (b). Non-zero (cid:104) ˆ S (cid:105) indicates the appearance of a UHF solution. Itis clear that the RHF solution becomes unstable above N Range14 r cs ∈ (3 . , . r cs ∈ (4 . , . r cs ∈ (2 . , . r cs ,for the 14-, 54-, and 114-electron UEG models. Above r cs , theR to U instability occurs and thus UHF solutions appear. r s = 3 .
5. The emergent strong correlation as increasing r s is most obvious from looking at the momentum distri-bution (MD) (Fig. 1 (c)) and natural orbital occupationnumbers (NOONs) (Fig. 1 (d)). The MD is defined asthe diagonal elements of a one-particle reduced densitymatrix, n k = (cid:104) a † k α a k α + a † k β a k β (cid:105) (42)The MD for the UEG model was throughly studied inRef. 100. NOONs are the eigenvalues of a one-particledensity matrix which is closely related to the MD. BothMD and NOONs show the increasing number of open-shell electrons as increasing the r s values. The open-shell electrons appear for r s > . r s ≤ .
5, the54-electron model at r s ≤ .
5, and the 114-electron modelat r s ≤ .
5. We will study the 14-electron model at r s =0 . , . , . , . , . r s = 5 . r s = 0 . , . , . r s = 5 .
0, there is a UHF solution with (cid:104) ˆ S (cid:105) = 0 . B. The 14-Electron UEG Model
We begin by studying the 14-electron UEG which wasstudied in detail by Shepherd and co-workers in Ref. 62.This small benchmark system is helpful as it is accessible r s − − − − − − E UH F − E R H F ( E h ) ( a ) M = 57 M = 93 M = 179 M = 389 M = 925 r s h ˆ S i ( b )0 . . . . . . . | k | / k F . . . . . . . . . n k ( c ) r s = 1.0 r s = 2.0 r s = 3.0 r s = 4.0 r s = 5.0 r s = 6.0 r s = 7.0 r s = 8.0 r s = 9.0 r s = 10.0 . . . . . . . . . N OO N s ( d ) r s = 1.0 r s = 2.0 r s = 3.0 r s = 4.0 r s = 5.0 r s = 6.0 r s = 7.0 r s = 8.0 r s = 9.0 r s = 10.0 FIG. 1. Results of UHF solutions found in the 14-electron UEG model: (a) the basis set convergence behavior of the energylowering from RHF to UHF with M = 57 , , , ,
925 over r s ∈ [0 . , . (cid:104) ˆ S (cid:105) as a function of r s with M = 925, (c)the momentum distribution n k for various r s with M = 925, and (d) the natural orbital occupation numbers (NOONs) forvarious r s with M = 925. In (a), M = 389 and M = 925 are more or less on top of each other and visually indistinguishable.In (c) and (d), curves for r s ≤ . to most quantum chemistry methods, whilst still exhibit-ing the typical challenges one faces when simulating realsolids, namely basis set incompleteness error, and strongcorrelation (when r s is large). In addition, it has of lateemerged as a standard benchmark system for the UEG .
1. Basis Set Convergence
The basis set convergence of wavefunction based quan-tum chemistry methods for the UEG has been explored anumber of times by various methods and we willonly briefly comment on it here. For our purposes, it issufficient to note that the convergence to the CBS limit isslowest at high densities (low r s ) and thus it is sufficientto converge the basis set error here. This slower con-vergence can be understood simply because the electron-electron cusp is more pronounced at high densities (theelectron are more likely to coalesce). This is seen in Fig. 2and Table II where on the order of 2000 PWs are nec-essary to converge the total energy to within 1 mHa inabsolute energy (not per electron). Similar to previous studies we observe a more or less linear relationship be-tween E c and 1 /M for M greater than 925 . Thelinear extrapolation to the CBS limit when using PWswas thoroughly studied and understood by Shepherd andco-workers in Ref. 101. We point out that the use oftranscorrelated approaches in AFQMC could greatly ac-celerate the convergence to the CBS limit .FCIQMC with the initiator approximation ( i -FCIQMC) is a formally exact approach as long as thereis no initiator bias. Therefore, comparing ph-AFQMCwith i -FCIQMC is a good way to assess the accuracy ofph-AFQMC. We see that ph-AFQMC agrees well with i -FCIQMC within the error bar up to M = 179 and itstarts to deviate from i -FCIQMC beyond that. However, i -FCIQMC numbers for the larger M values should betaken cautiously as it has been noted elsewhere that thebias from the initiator approximation was not completelyremoved and that the i -FCIQMC results for r s = 0 . E h .Comparing ph-AFQMC and CC methods is perhapsmore relevant for the purpose of this paper. The UEGmodel r s = 0 . .
000 0 .
005 0 .
010 0 .
015 0 . /M − . − . − . − . − . E c ( E h ) . . − . − . − . E c , of the 14-electron UEG modelat r s = 0 .
5. The basis sets considered are M =57 , , , , , , , , M ≥ /M dependence occurs at least past M = 925. M ph-AFQMC i -FCIQMC RCCSD RCCSDT57 -0.5173(1) -0.5169(1) N/A N/A93 -0.5592(2) -0.5589(1) N/A N/A179 -0.5794(2) -0.5797(3) -0.57365(1) -0.57971(3)389 -0.5881(2) -0.5893(3) N/A N/A925 -0.5920(8) -0.5936(3) -0.58626(4) -0.5923(1)1189 -0.5921(2) -0.5939(4) N/A N/A1213 -0.5926(8) N/A N/A N/A1419 -0.5925(4) N/A -0.5872(1) -0.5930(2)2109 -0.5931(6) N/A -0.5875(1) -0.5938(1)TABLE II. The correlation energy comparison between ph-AFQMC, i -FCIQMC, RCCSD, and RCCSDT for the 14-electron UEG model at r s = 0 .
5. The i -FCIQMC numberswere taken from Ref. 61 and CC numbers were taken fromRef. 66. N/A means that the data is not available. Thesecalculations were performed using the PAUXY package. Errorbars were estimated using reblocking as implemented inthe pyblock package . CC methods on top of an RHF reference work very well.Neufeld and Thom found that for r s = 0 .
5, CCSDT isenough to converge the correlation energy with respect tothe excitation levels . Therefore, the CCSDT numbersin Table II should be considered to be exact for a givenbasis set. As it is clear from Table II, the CCSD cor-relation energies are all above those of ph-AFQMC andph-AFQMC agrees with CCSDT up to sub millihartree.These results are particularly encouraging for the fol-lowing reasons. This dense UEG model may be analogousto a weakly correlated molecular system, in the sensethat for a finite number of electrons it is relatively welldescribed by HF theory. In such a system, CCSDT (orCCSD(T)) should be more or less exact. Even if their ab-solute energies were not exact, the relative energies suchas barrier heights and interaction energies should be close M ph-AFQMC i -FCIQMC RCCSD RCCSDT RCCSDTQ r s =1.0179 -0.5187(6) N/A -0.50250(7) -0.51819(3) -0.51856(7)1189 -0.5302(3) -0.5305(5) N/A N/A N/A2109 -0.5298(8) N/A -0.5133(3) -0.5290(4) N/A r s =2.081 -0.4181(2) N/A -0.40181(4) -0.41339(3) -0.41579(2)925 -0.4438(6) -0.4431(5) -0.4077(1) -0.4388(1) N/A2109 -0.4420(9) N/A -0.4089(2) N/A N/A r s =3.081 -0.3590(2) N/A -0.32208(3) -0.35671(3) -0.36141(5)179 -0.3723(4) N/A -0.3347(1) -0.37246(5) N/A925 -0.3725(5) N/A -0.3389(2) N/A N/A r s =5.0179 -0.2701(1) -0.3017(7) -0.2510(1) -0.2925(1) N/ATABLE III. The correlation energy comparison between ph-AFQMC, i -FCIQMC, RCCSD, and RCCSDT for the 14-electron UEG model at r s = 1 . , . , . .
0. The i -FCIQMC numbers were taken from Ref. 62 and CC numberswere taken from Ref. 66. N/A means that the data is notavailable. to exact. The results here suggest that ph-AFQMC is apotentially powerful tool to handle such weakly corre-lated systems. For the rest of this section, we will assessthe accuracy of ph-AFQMC for higher r s where therecan be a good mixture of weak and strong correlation.Furthermore, the quality of the RHF trial wavefunctionwill start to degrade so we will show how this affectsph-AFQMC.
2. Assessment for lower densities
In Table III, we present the comparison of ph-AFQMC, i -FCIQMC, RCCSD, RCCSDT, and RCCSDTQ for se-lected basis sets. All of our ph-AFQMC is available inthe Supplementary Materials. At r s = 1 .
0, ph-AFQMCagrees with i -FCIQMC within the error bar of each resultwhen M = 1189. Small basis set ( M = 179) results sug-gest that from RCCSDT to RCCSDTQ only small corre-lation energy is gained. Therefore, we consider RCCSDTto be near-exact for larger basis sets. Near the CBS limit( M = 2109), we found that the ph-AFQMC energy is 15-16 m E h lower than RCCSD and is within the error barof RCCSDT.At r s = 2 . i -FCIQMC withineach error bar. However, RCCSDT struggles to obtainquantitatively accurate results for M = 925. RCCSDis about 36 m E h above and RCCSDT is about 5 m E h above the i -FCIQMC (and ph-AFQMC) correlation ener-gies. Like usual strongly correlated systems, the effect ofquadruples is not negligible here and it accounts for about2 m E h correlation energy in a small basis ( M = 81). Asshown in Table III, ph-AFQMC provides a lower corre-lation energy than even RCCSDTQ in the M = 81 basisset. Since neither ph-AFQMC nor RCCSDTQ is vari-0ational, such correlation energy comparisons should betaken with caution. Nevertheless, since ph-AFQMC and i -FCIQMC agree for M = 925 we expect ph-AFQMC tobe accurate (i.e., near-exact) for M = 81. This resulthighlights the utility of RHF+ph-AFQMC. Namely, itcan provide quantitatively accurate results when the roleof quadruples is not negligible and yet still small enoughfor the RHF trial wavefunction to behave well.Although at r s = 0 . , . , . E h at r s = 3 .
0, but the stability of thesimulations suffers noticeably. Nevertheless, at r s = 3 . r s = 5 . (cid:104) Ψ T | φ (cid:105) approaches zero. Although these rareevents can be effectively controlled by the use of boundson the local (and/or hybrid energy) they neverthelesssignify a worsening in the quality of trial wavefunctionfor a fixed system size. To demonstrate this, in Fig. 3 weplot the convergence of the ph-AFQMC energy with pro-jection time for a range of densities with M = 93 as wellas an estimate for the overlap (cid:80) n w n |(cid:104) Ψ T | φ n (cid:105)| / (cid:80) n w n .We see from Fig. 3 (a) that as r s increases the projectiontime necessary to converge to the ground state increasesas well as the frequency of rare events. This is correlatedwith a decrease in the magnitude in the overlap as is seenfrom Fig. 3 (b).Indeed, at r s = 5 . M = 389. Rather, the ph-AFQMC correla-tion energy decreases in magnitude with increasing basisset size. This signals a complete breakdown of the phase-less constraint with this trial wavefunction. We note thata similar effect can be observed in i -FCIQMC when theinitiator error is not fully converged for increased basisset sizes, where one finds that the correlation energy be-gins to plateau as a function of basis set. This suggeststhat an improved trial wavefunction is necessary to attainsensible results for this system.It is noteworthy to point out that this unusual behav-ior of ph-AFQMC energy with respect to the basis setsize could indicate the “non-variational” failure of ph-AFQMC. ph-AFQMC is formally non-variational in thesense that a variational energy estimator of a given ph-AFQMC wavefunction can be above the mixed energyestimator in Eq. (15). Similarly, CC methods are alsoformally non-variational due to their projective nature.With an RHF reference, it has shown catastrophic non-variationality for strongly correlated systems.
Itis possible that RCCSD (and even RCCSDT) is also ex- 246 ( E − ¯ E ) r s + r s ( E h ) (a)0 10 20 30 40 50 τ (a.u.)10 − − − | O v e r l a p | (b) r s =0.5 r s =1.0 r s =2.0 r s =3.0 r s =4.0 r s =5.0 FIG. 3. Panel (a) shows the convergence of the ph-AFQMCtotal energy to its equilibrated mean value ( ¯ E ) as a functionof projection time for a variety of values of r s . Note thatthe data have been shifted and scaled by r s for clarity. Notethe occurrence of spikes in the local energy increases with r s .Here we bounded the local energy during propagation but notwhen printing the estimator to reveal the degradation in theresults. Panel (b) plots the reduction in the magnitude inthe overlap between the walkers and the trial wavefunction(see main text for definition) with decreasing r s . The slowerequilibration of the overlap compared to the local energy hasbeen noted previously in Ref. 31. hibiting non-variationality for this r s value. This can beconfirmed with more sophisticated CC methods. The investigation of the non-variationality of ph-AFQMCand CC methods in the context of strong correlation willbe an interesting subject for future study.As discussed in Section IV A, above r s = 3 .
5, UHFsolutions appear and they can be often powerful trialwavefunctions for ph-AFQMC . As in Ref. 78, we em-ploy the spin-projection technique to remove the spin-contamination completely. That is, we use the RHFwavefunction as the initial wavefunction while using theUHF wavefunction for the constraints. As shown inSection II A 3, because both RHF and UHF are size-consistent the resulting UHF+ph-AFQMC performedwith an RHF initial wavefunction must be also size-consistent. In Table IV, we present the correlation ener-gies from UHF+ph-AFQMC. Given its substantial sym-metry breaking at r s = 5 . i -FCIQMC1 M RHF UHF NOMSD i -FCIQMC57 -0.2422(8) -0.24371(9) -0.2511(3) -0.2645(3)93 -0.2677(5) -0.26837(8) -0.2786(2) -0.2928(4)389 -0.2674(6) -0.2654(2) -0.2794(4) -0.304(1)TABLE IV. Comparison between ph-AFQMC correlation en-ergies using a RHF, a UHF and ten determinant NOMSD trialwavefunction at r s = 5 .
0. Correlation energies are measuredrelative to the RHF total energy. i -FCIQMC energies weretaken from Ref. 62. ph-AFQMC calculations were performedusing the development version of QMCPACK. Note that the M = 389 RHF+ph-AFQMC energy is above the M = 179RHF+ph-AFQMC energy in Table III and this is an artifactof the breakdown of the RHF trial wavefunction. benchmark energies.To investigate the ph-AFQMC results at r s = 5 . . Interested readers are referred to Ref.32 for further details. In Fig. 4, We find an initial rapiddecrease in the error of the ph-AFQMC correlation en-ergy and correspondingly a reduction in the ph-AFQMCstatistical variance in the local energy estimator. Thestatistical variance in the estimator should not be con-fused with the wavefunction variance in variational MC.Therefore, it is possible to observe some improvement inthe statistical error bar while the energy estimator doesnot improve noticeably as observed before . This longtail in the convergence of the ph-AFQMC energy is in-dicative that the system is strongly correlated. We notethat the FCI space for M = 57 contains on the order of10 determinants and therefore the improvement in thetrial wavefunction via determinantal expansions is even-tually limited by the exponential wall. We find similarbehavior for larger basis sets.Table IV summarizes our ph-AFQMC results using a10 determinant expansion. Note that the M = 93 and M = 389 values are roughly within error bars of eachother, despite the i -FCIQMC correlation energy decreas-ing by approximately 10 m E h . We found that for evenlarger basis sets the RHF+ph-AFQMC correlation en-ergies lay above those at M = 93. Nevertheless we seethat by improving the trial wavefunction the ph-AFQMCcorrelation energies begin to slowly approach those of i -FCIQMC values. The slow convergence is evidence of thelimitation of using multi Slater determinant trial wave-functions in strongly correlated systems. C. Larger Supercells
As explained in Section II A 3, ph-AFQMC is size-consistent and thus can reliably reach the thermody-namic limit using larger super cells along combined withfinite size corrections and twist averaging . As 0 . . . . . R e l a t i v e E rr o r ( % ) M=57M=93 N D . . . V a r i a n ce R e du c t i o n FIG. 4. Panel (a) shows the behavior of the relative errorin the ph-AFQMC correlation energy as a function of thenumber of determinants in the trial wavefunction expansion, N D . The relative error is measured with respect to the i -FCIQMC value. Panel (b) shows the corresponding reductionin statistical variance defined as Var( N D = 1) / Var( N D ). the total energy of the UEG model in the thermodynamiclimit is already well understood for solid state densities here we instead just study finite-sized UEG models andcompare with other available methods when applicable.Following the sequence of “magic numbers” in theUEG model, we study larger supercells (54 electrons and114 electrons) with ph-AFQMC for r s = 0 . , . , .
0. Inthe 14-electron UEG model, we obtained energies witherror bars of the order of 1 m E h . This is important formolecular applications where we aim for energy differ-ences between two finite systems. On the other hand,the cost of achieving the same statistical error for largersystems adds an extra O ( N ) to the computational cost ofph-AFQMC. This extra cost for sampling may be avoidedby the correlated sampling technique , but here weinstead compare the total energy per electron . This met-ric is well-suited for ab-initio solids (or extended systems)in general.High-quality DMC numbers are available for the54-electron UEG model and we compare ph-AFQMCagainst this. For the 114-electron UEG model, there areonly variational MC (VMC) results available so we willcompare against these.2 r s ph-AFQMC i -FCIQMC FN-DMC-SJ FN-DMC-BF0.5 3.22087(2) 3.22086(2) 3.22245(9) 3.22112(4)1 0.52967(2) 0.53073(4) 0.53089(9) 0.52989(4)2 -0.01429(3) N/A -0.01311(2) -0.013966(9)5 -0.07589(5) N/A -0.078649(7) -0.079036(3)TABLE V. The total energy per electron ( E h /e) comparisonbetween ph-AFQMC, i -FCIQMC, FN-DMC with a Slater-Jastrow (SJ) trial wavefunction, and DMC with a backflow(BF) trial wavefunction for the 54-electron UEG model at r s = 0 . , . , . , .
0. Both ph-AFQMC and i -FCIQMC num-bers are obtained from M = 1419. The i -FCIQMC numberswere taken from Ref. 61 and DMC numbers were taken fromRef. 121. N/A means that the data is not available.
1. The 54-Electron UEG Model
We found that from M = 1419 to M = 2109 thechange in E tot /N at r s = 0 . E h . We therefore do not perform the CBS extrapola-tion for the comparison between ph-AFQMC and DMC.The reported ph-AFQMC numbers are obtained from M = 1419 which enables a direct comparison betweenph-AFQMC and i -FCIQMC at r s = 0 . r s = 1 . M = 2109 at r s = 0 . , . , . i -FCIQMC, and DMC for the 54-electronUEG model. At r s = 0 .
5, ph-AFQMC and i -FCIQMCagree with each other within the error bar. Shepherd andco-workers found that the DMC-BF energy is somewhathigher than i -FCIQMC and suggested that the fixed-nodeerror with the backflow (BF) trial wavefunction may notbe small. Indeed, we reach the same conclusion withph-AFQMC. As explained in Section I, the differencebetween DMC and AFQMC is mainly the discretiza-tion (or basis set) we work with. It is interesting thatthe fixed-node error in FN-DMC can be non-negligibleeven with more sophisticated trial wavefunctions such asSlater-Jastrow (SJ) and BF. It is encouraging that we canachieve near-exact accuracy with ph-AFQMC at r s = 0 . r s = 1 .
0, ph-AFQMC is in a better agreement (thedifference is about 0.2 m E h /e) with FN-DMC-BF than i -FCIQMC is. In this case, i -FCIQMC suffers from theinitiator bias and results into about 1 m E h /e above theFN-DMC-BF energy. In fact, the ph-AFQMC energyis lower than that of FN-DMC-BF, which may indicatenon-negligible fixed-node errors even in FN-DMC-BF.Since ph-AFQMC is not variational while FN-DMC-BFis, more careful calibration is highly desirable to see ifthere are indeed fixed-node errors in FN-DMC-BF. ph-AFQMC agrees better with FN-DMC-BF than does FN-DMC-SJ by 1 m E h /e similarly to the r s = 0 . i -FCIQMC results at r s = 2 . E h /e below the FN-DMC-SJ energy and 0.3 m E h /e r s ph-AFQMC0.5 3.48453(8)1.0 0.59877(6)2.0 0.00487(6)TABLE VI. The total energy per electron ( E h /e) of ph-AFQMC for the 114-electron UEG model at r s = 0 . , . , . M = 2109. The VMC (Slater-Jastrow) energy at r s = 1 . E h /e. below the FN-DMC-BF energy. With a larger basis set M = 2109, the ph-AFQMC energy lies 0.4 m E h /e be-low the FN-DMC-BF energy as shown in the Supplemen-tary Materials. Further increasing r s to 5.0, we observethat ph-AFQMC is no longer comparable to FN-DMC-SJ and FN-DMC-BF as expected. As the use of UHFand NOMSD trials was found to be ineffective in the14-electron model at r s = 5 .
0, we did not perform suchcalculations here.In summary, for the 54-electron UEG model at r s =0 . , . , .
0, we observe that ph-AFQMC can obtainnearly exact E tot /N . In particular, its accuracy iscomparable to other state-of-the-art methods such as i -FCIQMC and FN-DMC-BF. The general conclusions aresimilar to the 14-electron UEG model: ph-AFQMC isparticularly well-suited for r s values smaller than 5 wherethere exists moderate strong correlation. It is encourag-ing that ph-AFQMC achieved these highly accurate re-sults using the simplest trial wavefunction, RHF.
2. The 114-Electron UEG Model
Encouraged by the near-exact accuracy of ph-AFQMCfor low r s values in the 14- and 54-electron UEG models,we used ph-AFQMC to provide benchmark numbers forthe 114-electron UEG model for future method develop-ment. The 114-electron UEG model has been relativelyless explored. For determinant-based algorithms like i -FCIQMC the sign problem is likely to preclude its appli-cation except for very high densities. On the other hand,for ph-AFQMC this does not pose a significant challengeespecially when considering r s ≤ . r s = 0 .
5, the total energy per electron changesby 0.5 m E h /e when increasing M from 1419 to 2109.For higher r s values, we expect this energy change tobe smaller. We will present the ph-AFQMC energies at r s = 0 . , . , . M = 2109. We expectthat our ph-AFQMC energies reported here have the ba-sis set incompleteness error of the order of 0.5 m E h /e perelectron. Therefore, the numbers reported here may beconsidered as an upper bound to the ph-AFQMC ener-gies at the CBS limit.The ph-AFQMC results are presented in Table VI. Theonly data available in literature is r s = 1 . E h /e which is at least 5 m E h /ehigher than our ph-AFQMC energies. Comparing ph-3AFQMC energies in Table V and Table VI, we note thatthe finite-size effect is still very large. Namely, the energyper electron is far from the convergence with respect tothe system size. It will be interesting to investigate finite-size effects with ph-AFQMC in more realistic systems inthe future. Although further comparisons are not possi-ble due to the lack of benchmark data, we believe that theph-AFQMC numbers in Table VI are close to the exactenergies and the correlation energy error is smaller than1 m E h per electron given the results for the 54-electronmodel. V. CONCLUSIONS
In this paper, we examined the performance of phase-less auxiliary-field quantum Monte Carlo (ph-AFQMC)with the spin-restricted Hartree-Fock (RHF) trial wave-function (i.e., RHF+ph-AFQMC) on the uniform elec-tron gas (UEG) problem. We considered the 14-electron,54-electron, and 114-electron UEG model. Throughthese studies, we found the following conclusions:1. In the 14-electron case, we compared RHF+ph-AFQMC with spin-restricted coupled-cluster(RCC) methods. Compared to RCC with singlesand doubles (RCCSD) and CC with singles, dou-bles, and triples (RCCSDT), RHF+ph-AFQMCperforms better than RCCSDT and similarly to orslightly worse than RCCSDTQ for r s ≤ . r s = 5 . i -FCIQMC) and fixed-node diffusionMC (FN-DMC) suggested that RHF+ph-AFQMCis a promising tool for simulating dense solids.Such connections between dense solids and theUEG model were previously made in Ref. 123–125.RHF+ph-AFQMC confirmed that the fixed-nodeerror in FN-DMC for r s = 0 . i -FCIQMC study. Moreover, RHF+ph-AFQMCrevealed that the initiator bias in i -FCIQMC for r s = 1 . E h per electron) andthe fixed-node error in FN-DMC with a back flowtrial wavefunction (FN-DMC-BF) may not be neg-ligible (0.3 m E h per electron). A smilar trend wasobserved in the case of r s = 2 .
0. Lastly, r s = 5 . r s up to 2.0.4. We produced RHF+ph-AFQMC energies of the114-electron problem for r s ≤ . E h per electron.It is the central message of this paper that evenwith the simplest trial wavefunction (RHF) ph-AFQMCis a powerful tool for simulating molecules and solidswhere there is no noticeable strong correlation betweenelectrons. In particular, its scope lies between CCSDand CCSDT. Given its low scaling ( O ( N ) − O ( N )),RHF+ph-AFQMC remains a promising tool.The future study should include a more extensivebenchmark of RHF+ph-AFQMC on more chemically rel-evant systems such as the W4-11 set as well as de-signing better and yet compact trial wavefunctions forAFQMC. Using dynamically correlated orbitals such asthose from orbital-optimized Møller-Plesset perturbationtheory can be an economical way to go beyond HF trialwavefunctions. Some essential symmetry break-ing in the HF trial wavefunction can potentially im-prove the performance of ph-AFQMC greatly such asusing complex, restricted HF orbitals.
Lastly, thefinite-temperature extension of ph-AFQMC has been wellestablished . The assessment of ph-AFQMC for thewarm dense UEG model , which has been the subjectof intense research of late is currently work inprogress.
VI. ACKNOWLEDGEMENT
The work of J.L. was partly supported by the CCMSsummer internship in 2018 at the Lawrence LivermoreNational Lab. J.L. thanks Martin Head-Gordon andSoojin Lee for consistent encouragement. We would liketo thank Carlos Jimenez-Hoyos for providing us accessto the PHF code used to generate NOMSD expansions.The PHF code used in this work was developed in theScuseria group at Rice University . This work wasperformed under the auspices of the U.S. Departmentof Energy (DOE) by LLNL under Contract No. DE-AC52-07NA27344. Funding support was from the U.S.DOE, Office of Science, Basic Energy Sciences, MaterialsSciences and Engineering Division, as part of the Com-putational Materials Sciences Program and Center forPredictive Simulation of Functional Materials (CPSFM).Computer time was provided by the Livermore Comput-ing Facilities.4 ∗ [email protected] † [email protected] ‡ [email protected] L. M. Huntington and M. Nooijen, J. Chem. Phys. ,184109 (2010). R. J. Bartlett and M. Musia(cid:32)l, Rev. Mod. Phys. , 291(2007). D. W. Small and M. Head-Gordon, J. Chem. Phys. ,114103 (2012). D. Kats and F. R. Manby, J. Chem. Phys. , 021102(2013). D. W. Small, K. V. Lawler, and M. Head-Gordon, J.Chem. Theory Comput. , 2027 (2014). J. Lee, D. W. Small, E. Epifanovsky, and M. Head-Gordon, J. Chem. Theory Comput. , 602 (2017). J. Lee, D. W. Small, and M. Head-Gordon, J. Chem.Phys. , 244121 (2018). S. R. White and R. L. Martin, J. Chem. Phys. , 4127(1999). G. K.-L. Chan and S. Sharma, Ann. Rev. Phys. Chem. , 465 (2011). W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Ra-jagopal, Rev. Mod. Phys. , 33 (2001). S. Zhang and H. Krakauer, Phys. Rev. Lett. , 136401(2003). G. H. Booth, A. J. W. Thom, and A. Alavi, J. Chem.Phys. , 054106 (2009). S. Zhang, J. Carlson, and J. E. Gubernatis, Phys. Rev.B , 7464 (1997). J. Carlson, J. E. Gubernatis, G. Ortiz, and S. Zhang,Phys. Rev. B , 12788 (1999). E. G. Hohenstein, R. M. Parrish, and T. J. Mart´ınez, J.Chem. Phys. , 044103 (2012). R. M. Parrish, E. G. Hohenstein, T. J. Mart´ınez, andC. D. Sherrill, J. Chem. Phys. , 224106 (2012). E. G. Hohenstein, R. M. Parrish, C. D. Sherrill, and T. J.Mart´ınez, J. Chem. Phys. , 221101 (2012). J. Lu and L. Ying, J. Comput. Phys. , 329 (2015). W. Hu, L. Lin, and C. Yang, J. Chem. Theory Comput. , 5420 (2017). K. Dong, W. Hu, and L. Lin, J. Chem. Theory Comput. , 1311 (2018). F. D. Malone, S. Zhang, and M. A. Morales, J. Chem.Theory. Comput. , 256 (2019). M. Motta, J. Shee, S. Zhang, and G. K. Chan, arXivpreprint arXiv:1810.01549 (2018). C. H¨attig, W. Klopper, A. K¨ohn, and D. P. Tew, Chem.Rev. , 4 (2012). M. Motta and S. Zhang, J. Chem. Theory Comput. ,5367 (2017). M. Motta, D. E. Galli, S. Moroni, and E. Vitali, J. Chem.Phys. , 024107 (2014). M. Motta, D. E. Galli, S. Moroni, and E. Vitali, J. Chem.Phys. , 164108 (2015). E. Vitali, H. Shi, M. Qin, and S. Zhang, Phys. Rev. B , 085140 (2016). M. Motta and S. Zhang, J. Chem. Phys. , 181101(2018). M. Motta and S. Zhang, WIREs Comput. Mol. Sci. ,e1364 (2018). S. Zhang, Emergent Phenomena in Correlated Matter:Autumn School Organized by the ForschungszentrumJ¨ulich and the German Research School for SimulationSciences at Forschungszentrum J¨ulich 23-27 September2013; Lecture Notes of the Autumn School CorrelatedElectrons 2013 (2013). W. Purwanto, S. Zhang, and H. Krakauer, J. Chem. Phys. , 094107 (2009). E. J. L. Borda, J. A. Gomez, and M. A. Morales, J. Chem.Phys. , 074105 (2019). J. Shee, B. Rudshteyn, E. J. Arthur, S. Zhang, D. R.Reichman, and R. A. Friesner, J. Chem. Theory Comput. , 2346 (2019). W. Purwanto, H. Krakauer, Y. Virgus, and S. Zhang, J.Chem. Phys. , 164105 (2011). F. Ma, S. Zhang, and H. Krakauer, New J. Phys. ,093017 (2013). S. Zhang, F. D. Malone, and M. A. Morales, J. Chem.Phys. , 164102 (2018). J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik,G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M.Henderson, C. A. Jim´enez-Hoyos, E. Kozik, X.-W. Liu,A. J. Millis, N. V. Prokof’ev, M. Qin, G. E. Scuseria,H. Shi, B. V. Svistunov, L. F. Tocchio, I. S. Tupitsyn,S. R. White, S. Zhang, B.-X. Zheng, Z. Zhu, and E. Gull(Simons Collaboration on the Many-Electron Problem),Phys. Rev. X , 041041 (2015). H. Hao, J. Shee, S. Upadhyay, C. Ataca, K. D. Jordan,and B. M. Rubenstein, J. Phys. Chem. Lett. , 6185(2018). M. Suewattana, W. Purwanto, S. Zhang, H. Krakauer,and E. J. Walter, Phys. Rev. B , 245123 (2007). F. Ma, S. Zhang, and H. Krakauer, Phys. Rev. B ,165103 (2017). F. Ma, W. Purwanto, S. Zhang, and H. Krakauer, Phys.Rev. Lett. , 226401 (2015). W. Purwanto, S. Zhang, and H. Krakauer, J. Chem. The-ory. Comput. , 4825 (2013). W. Purwanto, S. Zhang, and H. Krakauer, J. Chem. Phys. , 064302 (2015). H. Shi and S. Zhang, Phys. Rev. B , 125132 (2013). H. Shi, C. A. Jim´enez-Hoyos, R. Rodr´ıguez-Guzm´an,G. E. Scuseria, and S. Zhang, Phys. Rev. B , 125129(2014). C.-C. Chang and M. A. Morales, arXiv preprintarXiv:1711.02154 (2017). G. Giuliani and G. Vignale,
Quantum theory of the elec-tron liquid (Cambridge university press, 2005). C. A. Jim´enez-Hoyos, R. Rodr´ıguez-Guzm´an, and G. E.Scuseria, J. Phys. Chem. A , 9925 (2014). J. Lee and M. Head-Gordon, Phys. Chem. Chem. Phys. , 4763 (2019). C. F. Jackels and E. R. Davidson, J. Chem. Phys. ,2908 (1976). E. R. Davidson and W. T. Borden, J. Phys. Chem. ,4783 (1983). J. S. Andrews, D. Jayatilaka, R. G. Bone, N. C. Handy,and R. D. Amos, Chem. Phys. Lett. , 423 (1991). P. Y. Ayala and H. B. Schlegel, J. Chem. Phys. , 7560(1998). A. D. McLean, B. H. Lengsfield, J. Pacansky, andY. Ellinger, J. Chem. Phys. , 3567 (1985). C. Sherrill, M. S. Lee, and M. Head-Gordon, Chem. Phys.Lett. , 425 (1999). T. D. Crawford and J. F. Stanton, J. Chem. Phys. ,7873 (2000). J. Paldus and G. Thiamov´a, J. Math. Chem. , 88(2007). K. Raghavachari, G. W. Trucks, J. A. Pople, andM. Head-Gordon, Chem. Phys. Lett. , 479 (1989). S.-K. Ma and K. A. Brueckner, Phys. Rev. , 18 (1968). D. Cleland, G. H. Booth, and A. Alavi, J. Chem. Phys. , 041103 (2010). J. J. Shepherd, G. Booth, A. Gr¨uneis, and A. Alavi, Phys.Rev. B , 081103 (2012). J. J. Shepherd, G. H. Booth, and A. Alavi, J. Chem.Phys. , 244101 (2012). A. J. W. Thom, Phys. Rev. Lett. , 263004 (2010). J. S. Spencer and A. J. W. Thom, J. Chem. Phys. ,084108 (2016). R. S. T. Franklin, J. S. Spencer, A. Zoccante, andA. J. W. Thom, J. Chem. Phys. , 044111 (2016). V. A. Neufeld and A. J. W. Thom, J. Chem. Phys. ,194105 (2017). H. Kwee, S. Zhang, and H. Krakauer, Phys. Rev. Lett. , 126404 (2008). F. Ma, S. Zhang, and H. Krakauer, Phys. Rev. B ,155130 (2011). J. Hubbard, Phys. Rev. Lett. , 77 (1959). D. J. Thouless, Nuc. Phys. , 225 (1960). D. J. Thouless, Nuc. Phys. , 78 (1961). W. A. Al-Saidi, S. Zhang, and H. Krakauer, J. Chem.Phys. , 224101 (2006). R. J. Bartlett and I. Shavitt, Int. J. Quantum Chem. ,165 (1977). T. Gruber, K. Liao, T. Tsatsoulis, F. Hummel, andA. Gr¨uneis, Phys. Rev. X , 021043 (2018). L. M. Fraser, W. M. C. Foulkes, G. Rajagopal, R. J.Needs, S. D. Kenny, and A. J. Williamson, Phys. Rev. B , 1814 (1996). T. Schoof, S. Groth, J. Vorberger, and M. Bonitz, Phys.Rev. Lett. , 130402 (2015). M. Qin, H. Shi, and S. Zhang, Phys. Rev. B , 235119(2016). W. Purwanto, W. Al-Saidi, H. Krakauer, and S. Zhang,J. Chem. Phys. , 114309 (2008). P. Paul, K.-C. Kim, D. Sun, P. D. W. Boyd, and C. A.Reed, J. Am. Chem. Soc. , 4394 (2002). H.-D. Beckhaus, C. R¨uchardt, M. Kao, F. Diederich, andC. S. Foote, Angew. Chem. Int. Edit. , 63 (1992). S. Tomita, J. Andersen, K. Hansen, and P. Hvelplund,Chem. Phys. Lett. , 120 (2003). A. Overhauser, Phys. Rev. Lett. , 462 (1960). A. Overhauser, Phys. Rev. , 1437 (1962). A. Overhauser, Phys. Rev. , 691 (1968). S. Zhang and D. M. Ceperley, Phys. Rev. Lett. ,236404 (2008). C. Coulson and I. Fischer, Philos. Mag. , 386 (1949). J. Kim, A. T. Baczewski, T. D. Beaudet, A. Benali, M. C.Bennett, M. A. Berrill, N. S. Blunt, E. J. L. Borda, M. Ca-sula, D. M. Ceperley, S. Chiesa, B. K. Clark, R. C. C.III, K. T. Delaney, M. Dewing, K. P. Esler, H. Hao,O. Heinonen, P. R. C. Kent, J. T. Krogel, I. Kyl¨anp¨a¨a,Y. W. Li, M. G. Lopez, Y. Luo, F. D. Malone, R. M. Mar-tin, A. Mathuriya, J. McMinis, C. A. Melton, L. Mitas,M. A. Morales, E. Neuscamman, W. D. Parker, S. D. P.Flores, N. A. Romero, B. M. Rubenstein, J. A. R. Shea,H. Shin, L. Shulenburger, A. F. Tillack, J. P. Townsend,N. M. Tubman, B. V. D. Goetz, J. E. Vincent, D. C. Yang,Y. Yang, S. Zhang, and L. Zhao, J. Phys.: Cond. Mat. , 195901 (2018). See https://github.com/fdmalone/pauxy for details onhow to obtain the source code. J. S. Spencer, N. S. Blunt, W. A. Vigor, F. D. Malone,W. M. C. Foulkes, J. J. Shepherd, and A. J. W. Thom,J. Open Res. Softw. , 1 (2015). J. S. Spencer, N. S. Blunt, S. Choi, J. Etrych, M.-A. Filip,W. M. C. Foulkes, R. S. T. Franklin, W. J. Handley, F. D.Malone, V. A. Neufeld, R. Di Remigio, T. W. Rogers,C. J. C. Scott, J. J. Shepherd, W. A. Vigor, J. Weston,R. Xu, and A. J. W. Thom, J. Chem. Theory. Comput. , 1728 (2019). T. E. Booth and J. E. Gubernatis, Phys. Rev. E ,046704 (2009). L. K. Wagner, M. Bajdich, and L. Mitas, J. Comput.Phys. , 3390 (2009). Y. Shao, Z. Gan, E. Epifanovsky, A. T. Gilbert, M. Wor-mit, J. Kussmann, A. W. Lange, A. Behn, J. Deng,X. Feng, D. Ghosh, M. Goldey, P. R. Horn, L. D. Ja-cobson, I. Kaliman, R. Z. Khaliullin, T. Ku´s, A. Landau,J. Liu, E. I. Proynov, Y. M. Rhee, R. M. Richard, M. A.Rohrdanz, R. P. Steele, E. J. Sundstrom, H. L. Wood-cock, P. M. Zimmerman, D. Zuev, B. Albrecht, E. Al-guire, B. Austin, G. J. Beran, Y. A. Bernard, E. Berquist,K. Brandhorst, K. B. Bravaya, S. T. Brown, D. Casanova,C. M. Chang, Y. Chen, S. H. Chien, K. D. Closser,D. L. Crittenden, M. Diedenhofen, R. A. Distasio, H. Do,A. D. Dutoi, R. G. Edgar, S. Fatehi, L. Fusti-Molnar,A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M. W.Hanson-Heine, P. H. Harbach, A. W. Hauser, E. G. Ho-henstein, Z. C. Holden, T. C. Jagau, H. Ji, B. Kaduk,K. Khistyaev, J. Kim, J. Kim, R. A. King, P. Klun-zinger, D. Kosenkov, T. Kowalczyk, C. M. Krauter, K. U.Lao, A. D. Laurent, K. V. Lawler, S. V. Levchenko,C. Y. Lin, F. Liu, E. Livshits, R. C. Lochan, A. Luenser,P. Manohar, S. F. Manzer, S. P. Mao, N. Mardirossian,A. V. Marenich, S. A. Maurer, N. J. Mayhall, E. Neuscam-man, C. M. Oana, R. Olivares-Amaya, D. P. Oneill, J. A.Parkhill, T. M. Perrine, R. Peverati, A. Prociuk, D. R.Rehn, E. Rosta, N. J. Russ, S. M. Sharada, S. Sharma,D. W. Small, A. Sodt, T. Stein, D. St¨uck, Y. C. Su, A. J.Thom, T. Tsuchimochi, V. Vanovschi, L. Vogt, O. Vy-drov, T. Wang, M. A. Watson, J. Wenzel, A. White, C. F.Williams, J. Yang, S. Yeganeh, S. R. Yost, Z. Q. You, I. Y.Zhang, X. Zhang, Y. Zhao, B. R. Brooks, G. K. Chan,D. M. Chipman, C. J. Cramer, W. A. Goddard, M. S.Gordon, W. J. Hehre, A. Klamt, H. F. Schaefer, M. W.Schmidt, C. D. Sherrill, D. G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A. T. Bell, N. A. Besley, J. D.Chai, A. Dreuw, B. D. Dunietz, T. R. Furlani, S. R.Gwaltney, C. P. Hsu, Y. Jung, J. Kong, D. S. Lam-brecht, W. Liang, C. Ochsenfeld, V. A. Rassolov, L. V.Slipchenko, J. E. Subotnik, T. Van Voorhis, J. M. Her-bert, A. I. Krylov, P. M. Gill, and M. Head-Gordon, Mol.Phys. , 184 (2015). J. Lee and M. Head-Gordon, J. Chem. Theory Comput. , 5203 (2018). J. Lee and M. Head-Gordon, The Journal of ChemicalPhysics , 244106 (2019). T. Van Voorhis and M. Head-Gordon, Mol. Phys. ,1713 (2002). R. Seeger and J. A. Pople, J. Chem. Phys. , 3045 (1977),https://doi.org/10.1063/1.434318. S. M. Sharada, D. St¨uck, E. J. Sundstrom, A. T. Bell,and M. Head-Gordon, Mol. Phys. , 1802 (2015). F. H. Zong, C. Lin, and D. M. Ceperley, Phys. Rev. E , 036703 (2002). M. Holzmann, B. Bernu, C. Pierleoni, J. McMinis, D. M.Ceperley, V. Olevano, and L. Delle Site, Phys. Rev. Lett. , 110402 (2011).
J. J. Shepherd, A. Gr¨uneis, G. H. Booth, G. Kresse, andA. Alavi, Phys. Rev. B , 035111 (2012). J. J. Shepherd and A. Gr¨uneis, Phys. Rev. Lett. ,226401 (2013).
H. Luo and A. Alavi, J. Chem. Theory. Comput. , 1403(2018). H. Flyvbjerg and H. G. Petersen, J. Chem. Phys. , 461(1989). See https://github.com/jsspencer/pyblock for details onhow to obtain the source code.
W. Purwanto, H. Krakauer, and S. Zhang, Phys. Rev. B , 214116 (2009). T. Van Voorhis and M. Head-Gordon, J. Chem. Phys. , 8873 (2000).
B. Cooper and P. J. Knowles, J. Chem. Phys. , 234102(2010).
T. Van Voorhis and M. Head-Gordon, Chem. Phys. Lett. , 585 (2000).
J. B. Robinson and P. J. Knowles, J. Chem. Phys. ,044113 (2011).
C. A. Jim´enez-Hoyos, T. M. Henderson, T. Tsuchimochi,and G. E. Scuseria, J. Chem. Phys. , 164109 (2012).
C. A. Jim´enez-Hoyos, R. Rodr´ıguez-Guzm´an, and G. E.Scuseria, J. Chem. Phys. , 204102 (2013).
R. Schutski, C. A. Jim´enez-Hoyos, and G. E. Scuseria, J.Chem. Phys. , 204101 (2014).
S. Chiesa, D. M. Ceperley, R. M. Martin, and M. Holz-mann, Phys. Rev. Lett. , 076404 (2006). N. D. Drummond, R. J. Needs, A. Sorouri, and W. M. C.Foulkes, Phys. Rev. B , 125106 (2008). M. Holzmann, R. C. Clay, M. A. Morales, N. M. Tub-man, D. M. Ceperley, and C. Pierleoni, Phys. Rev. B ,035126 (2016). C. Lin, F. H. Zong, and D. M. Ceperley, Phys. Rev. E , 016702 (2001). D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. , 566(1980). J. Shee, S. Zhang, D. R. Reichman, and R. A. Friesner,J. Chem. Theory Comput. , 2667 (2017). J. Shee, E. J. Arthur, S. Zhang, D. R. Reichman, andR. A. Friesner, J. Chem. Theory Comput. , 4109 (2018). P. L. R´ıos, A. Ma, N. Drummond, M. Towler, andR. Needs, Phys. Rev. E , 066701 (2006). Y. Kwon, D. Ceperley, and R. M. Martin, Phys. Rev. B , 6800 (1998). E. E. Salpeter, Astrophys. J. , 669 (1961).
M. Baus and J.-P. Hansen, Phys. Rep. , 1 (1980). S. Huotari, J. A. Soininen, T. Pylkk¨anen, K. H¨am¨al¨ainen,A. Issolah, A. Titov, J. McMinis, J. Kim, K. Esler, D. M.Ceperley, et al. , Phys. Rev. Lett. , 086403 (2010).
A. Karton, S. Daon, and J. M. Martin, Chem. Phys. Lett. , 165 (2011).
D. W. Small, E. J. Sundstrom, and M. Head-Gordon, J.Chem. Phys. , 024104 (2015).
J. Lee, L. W. Bertels, and M. Head-Gordon, “Kohn-shamdensity functional theory with complex, spin-restrictedorbitals: Accessing a new class of densities without thesymmetry dilemma,” (2019), arXiv:1904.08093.
S. Zhang, Phys. Rev. Lett. , 2777 (1999). B. M. Rubenstein, S. Zhang, and D. R. Reichman, Phys.Rev. A , 053606 (2012). Y. Liu, M. Cho, and B. Rubenstein, J. Chem. Theory.Comput. , 4722 (2018). T. Dornheim, S. Groth, and M. Bonitz, Physics Reports , 1 (2018).
E. W. Brown, B. K. Clark, J. L. DuBois, and D. M.Ceperley, Phys. Rev. Lett. , 146405 (2013).
F. D. Malone, N. Blunt, E. W. Brown, D. Lee, J. Spencer,W. Foulkes, and J. J. Shepherd, Phys. Rev. Lett. ,115701 (2016).
T. Dornheim, S. Groth, F. D. Malone, T. Schoof,T. Sjostrom, W. M. C. Foulkes, and M. Bonitz, Phys.Plasmas , 056303 (2017). T. Dornheim, S. Groth, J. Vorberger, and M. Bonitz,Phys. Rev. Lett.121