Ab Initio Auxiliary-Field Quantum Monte Carlo in the Thermodynamic Limit

Abstract

Ab initio auxiliary-field quantum Monte Carlo (AFQMC) is a systematically improvable many-body method, but its application to extended solids has been severely limited by unfavorable computational scaling and memory requirements that obstruct direct access to the thermodynamic and complete-basis-set limits. By combining tensor hypercontraction via interpolative separable density fitting with $\mathbf{k}$-point symmetry, we reduce the computational and memory scaling of ab initio AFQMC for solids to $O(N^3)$ and $O(N^2)$ with arbitrary basis, respectively, comparable to diffusion Monte Carlo. This enables direct and simultaneous thermodynamic-limit and complete-basis-set AFQMC calculations across insulating, metallic, and strongly correlated solids, without embedding, local approximations, empirical finite-size corrections, or composite schemes. Our results establish AFQMC as a general-purpose, systematically improvable alternative to diffusion Monte Carlo and coupled-cluster methods for predictive ab initio simulations of solids, enabling accurate energies and magnetic observables within a unified framework.

Figures (19)

• Figure 1: Error in the cohesive energy of diamond crystal predicted by different methods relative to experimental values brewerCOHESIVE1977Schimka2011JanruscicIntroduction2004. MP2, LNO-CCSD, and LNO-CCSD(T) values are taken from Ref. Ye2024Oct; The size-consistent Brillouin–Wigner approach (BWs2) value is taken from chenRegularized2025; PBE and HSE correspond to the Heyd-Scuseria-Ernzerhof and Perdew-Burke-Ernzerhof functionals, respectively, with values from Ref. Schimka2011Jan; DMC result is from Ref. benaliSystematic2020. The previous AFQMC result (denoted as "prev. AFQMC") is taken from Ref. Malone2020May, the AFQMC result is from this work, which is corrected for atomic phaseless, crystalline phaseless and pseudopotential errors as described in the main text. The uncorrected AFQMC value is denoted as uncorr. AFQMC in the plot and is also given in Appendix \ref{['cohdata']}.
• Figure 2: Errors in AFQMC from different sources contributing to the cohesive energy of diamond, including the atomic phaseless error, crystalline phaseless error, and pseudopotential error. The atomic phaseless error is corrected using semistochastic heat-bath configuration interaction (SHCI) for TZ and QZ basis sets and extrapolated to the complete basis set (CBS) limit. The crystalline phaseless error is corrected using CISD-AFQMC on a $2\times2\times2$ supercell in the DZ basis. The pseudopotential error was corrected using a $3\times3\times3$ supercell at the DZ level. For clarity, the error bars of the corrections are not shown since each is smaller than 0.01 eV, which would increase the final uncertainty by 0.01 eV.
• Figure 3: Error in the cohesive energy of silicon crystal predicted by different methods relative to experiment brewerCOHESIVE1977Schimka2011JanruscicIntroduction2004. The MP2 value and the CCSD value are from Ref. McClain2017Mar; PBE and HSE values are taken from Ref. Schimka2011Jan; VMC and DMC/LDA results are from Ref. leungCalculations1999, and the DMC/PBE0 result is from Ref. Annaberdiyev2021May; prev. AFQMC denotes the previous AFQMC result from Ref. morales2020accelerate; prev. pw AFQMC denotes the previous plane-wave AFQMC result from zhangQuantumMonteCarlo2003. Note that we apply the ZPE correction to the experimental value, so we removed the ZPE correction from the pw AFQMC result. AFQMC result is from this work, which is corrected for atomic phaseless, crystalline phaseless and pseudopotential errors as described in the main text. The uncorrected AFQMC value is denoted as uncorr. AFQMC in the plot and is also given in Appendix \ref{['cohdata']}.
• Figure 4: Errors in AFQMC from different sources, including atomic phaseless, crystalline phaseless and pseudopotential error in the cohesive energy of silicon. The atomic phaseless error was corrected using SHCI for TZ and QZ basis sets and extrapolated to the complete basis set limit. The crystalline phaseless error was corrected using CISD-AFQMC on a $2\times2\times2$ supercell in the DZ basis. The pseudopotential error was corrected using a $3\times3\times3$ supercell at the DZ level. The error bars of the corrections are not shown since each is smaller than 0.01 eV, which would increase the final uncertainty by less than 0.01 eV.
• Figure 5: Error in the cohesive energy of BCC lithium predicted by different methods relative to experiment chase1982janafSchimka2011Jan. The BWs2 value is from Ref. chenRegularized2025; The dRPA value is from Ref. Ye2024Oct; PBE and HSE values are taken from Ref. Schimka2011Jan; The CCSD value is from Ref. Neufeld2022Aug; The CCSDT and DCSDT values are from Ref. Neufeld2023Oct, and the LNO-CCSD data is from Ref. Ye2024Oct; VMC is from Ref. Yao1996Sep, and the DMC result is from Ref. Rasch2015Jul. AFQMC result is from this work, which is corrected for atomic phaseless, crystalline phaseless and pseudopotential errors as described in the main text. The uncorrected AFQMC value is denoted as uncorr. AFQMC in the plot and is also given in Appendix \ref{['cohdata']}.