Auxiliary field quantum Monte Carlo at the basis set limit: application to lattice constants

TL;DR

The paper develops a cubic-scaling, plane-wave AFQMC implementation within the PAW framework in VASP, enabling calculations at the basis-set limit and avoiding basis-set extrapolation errors. By hierarchically embedding AFQMC within MP2 and RPA workflows, it shows that AFQMC corrections recover missing short- and long-range correlation effects, yielding lattice constants with a mean absolute error of $0.14\%$ relative to zero-point corrected experiments for C, BN, BP, and Si. The results highlight RPA as the most efficient reference for solids due to faster finite-size convergence, and demonstrate robust performance against time-step and trial-wavefunction choices. This work provides a scalable, benchmark-quality method for structural properties in condensed matter and clarifies the role of core correlations and PAW treatment in high-accuracy QMC calculations.

Abstract

We present a plane-wave (PW) implementation of the auxiliary-field quantum Monte Carlo (AFQMC) method within the projector augmented-wave (PAW) formalism in the Vienna ab initio Simulation Package (VASP). By employing an exact inversion of the PAW overlap operator, our approach maintains cubic scaling while naturally operating at the complete basis set limit defined by the PW cutoff. We benchmark this framework by calculating the equilibrium lattice constants and bulk moduli of C, BN, BP, and Si. Our analysis demonstrates that AFQMC systematically corrects the lack of long-range screening in MP2 and the missing higher-order exchange in RPA. We identify RPA as the optimal reference method due to the rapid convergence of the remaining short-range correlations with respect to supercell size. The resulting lattice constants exhibit a mean absolute relative error of 0.14 % relative to experiment, establishing the method as a rigorous benchmark tool for structural properties in condensed matter systems.

Figures (4)

• Figure 1: Equation of state of diamond calculated with AFQMC for three different imaginary time steps $\tau$. Total energies were obtained by combining twist-averaged AFQMC correlation energies (computed in an 8-atom cell and normalized to the primitive cell) with k-point converged HF energies from the primitive cell. The vertical lines connecting to the fitted curves mark the respective equilibrium lattice constants. The inset shows the total-energy differences between the largest and smallest time steps.
• Figure 2: Energy difference per atom between AFQMC and MP2 total energies for (a) C and (b) Si, using trial wavefunctions from different exchange-correlation functionals. Calculations were performed in an 8-atom supercell using twist averaging.
• Figure 3: Convergence of the MP2+AFQMC and RPA+AFQMC lattice constants (in ) with respect to the number of atoms in the supercell for (a) C, (b) Si, (c) BN, and (d) BP. Each point corresponds to a twist-averaged AFQMC calculation embedded in MP2 or RPA calculations obtained in the primitive cell. The yellow dashed lines and the green dashed-dotted lines denote the MP2 and RPA reference lattice constants, respectively.
• Figure 4: Relative percentage error in the MP2, RPA, AFQMC equilibrium lattice constants with respect to the ZPV-corrected experimental lattice constants at $T=$ 0K. MP2 and RPA results were calculated in the primitive cell utilizing k-point sampling. The AFQMC lattice constants were obtained in 32-atom cells, along with twist averaging, and using an RPA reference.