Toward quantum Monte Carlo forces on heavier ions: Scaling properties

Abstract

Quantum Monte Carlo (QMC) forces have been studied extensively in recent decades because of their importance with spectroscopic observables and geometry optimization. Here, we benchmark the accuracy and computational cost of QMC forces. The zero-variance zero-bias (ZVZB) force estimator is used in standard variational and diffusion Monte Carlo simulations with mean-field based trial wavefunctions and atomic pseudopotentials. Statistical force uncertainties are obtained with a recently developed regression technique for heavy tailed QMC data [P. Lopez Rios and G. J. Conduit, Phys. Rev. E 99, 063312 (2019)]. By considering selected atoms and dimers with elements ranging from H to Zn (1 ≤ Zeff ≤ 20), we assess the accuracy and the computational cost of ZVZB forces as the effective pseudopotential valence charge, Zeff, increases. We find that the costs of QMC energies and forces approximately follow simple power laws in Zeff. The force uncertainty grows more rapidly, leading to a best case cost scaling relationship of approximately Zeff6.5(3) for diffusion Monte Carlo. We find that the accessible system size at fixed computational cost scales as Zeff−2, insensitive to model assumptions or the use of the “space warp” variance-reduction technique. Our results predict the practical cost of obtaining forces for a range of materials, such as transition metal oxides where QMC forces have yet to be applied, and underscore the importance of further developing force variance-reduction techniques, particularly for atoms with high Zeff.

Candidate structure for the H$_2$-PRE phase of solid hydrogen

“…However, the difference of dynamic enthalpy between P2 1 /c-8 and the second most stable structure is, at most, just 0.049(15) mHartree/atom. Therefore, C2/c-24 or Cmca-12 can be predicted comparably stable as P2 1 /c-8, by taking into account anharmonic contributions to ZPE [17,23,24] or nuclear quantum effects [51], or using FNDMC forces [52,53]. Nevertheless, P21/c-8 would remain among the most stable structures.…”

Stochastic effective core potentials, improving efficiency using a spin-dependent core definition

“…Eqn (8) performs an update of the determinant of a product of two matrices, using the determinant of a reduced matrix. The last expression of F depends on X 0 and we can write…”

“…7 Furthermore, the unfavourable scaling with the effective nuclear charge Z eff remains, Z 6.5 eff for the forces in diffusion Monte Carlo (DMC). 8 Developments addressing this challenge have been mostly focusing on improving the sampling and reducing the correlation factor. For instance in variational Monte Carlo (VMC) the dynamic can be carried out very efficiently in spherical coordinates.…”

“…7 Furthermore, the unfavourable scaling with the effective nuclear charge Z eff remains, Z 6.5 eff for the forces in diffusion Monte Carlo (DMC). 8…”

Machine Learning Diffusion Monte Carlo Forces

“…The computational overhead associated with calculating forces is especially problematic for many QMC techniques, such as the Diffusion Monte Carlo (DMC) method. 10 Although QMC techniques are increasingly being used to provide computational ground truths regarding energies, wave functions, and electron densities in chemistry and condensed matter physics because of their exceptional accuracy 8,11,12 and modest O(N 3 )−O(N 4 ) scaling, 3 it has long been a challenge to calculate forces within most real-space QMC methods. 13−15 This is because the direct application of the Hellman− Feynman theorem, 16 which is employed to compute forces in most theories, results in an infinite variance problem that stems from the sampling of electron positions that approach the nucleus.…”