Preconditioning and Perturbative Estimators in Full Configuration Interaction Quantum Monte Carlo

Abstract

We propose the use of preconditioning in FCIQMC which, in combination with perturbative estimators, greatly increases the efficiency of the algorithm. The use of preconditioning allows a time step close to unity to be used (without time-step errors), provided that multiple spawning attempts are made per walker. We show that this approach substantially reduces statistical noise on perturbative corrections to initiator error, which improve the accuracy of FCIQMC but which can suffer from significant noise in the original scheme. Therefore, the use of preconditioning and perturbatively-corrected estimators in combination leads to a significantly more efficient algorithm. In addition, a simpler approach to sampling variational and perturbative estimators in FCIQMC is presented, which also allows the variance of the energy to be calculated. These developments are investigated and applied to benzene (30e, 108o), an example where accurate treatment is not possible with the original method.

A Guided Tour of ab initio Nuclear Many-Body Theory

“…While QMC applications in ab initio nuclear structure have been focused on coordinate space, there are a wide variety of approaches that merge QMC techniques with the configuration space approaches discussed in previous sections. Examples include sampling the intermediate-state summations in MBPT [164], diagrammatic expansions [165][166][167], or the coefficients of correlated CC [168] or (No-Core) CI wave functions [137,138,[169][170][171].…”

A hybrid approach to extending selected configuration interaction and full configuration interaction quantum Monte Carlo

“…This approach speeds up the calculation of the PT2 correction by a factor of ∼ N spawn , as argued in Ref. (40), so that this not essential but certainly helpful. Timings are challenging to compare because they are highly implementation-dependent, and also highly-dependent on the simulation parameters used, including the amount of memory assigned for this task.…”

“…We will now describe how to calculate a second-order perturbative (PT2) correction from the rejected spawnings in the above approach. This is similar to the PT2 correction in SCI+PT2, but will sample a much larger space for a given V -up to the second-order interacting space beyond V. It is also the same estimator that has been applied recently to the previous i-FCIQMC method 40,43 , but will now sample a larger space for a given walker population.…”

“…Instead, we show how to calculate E var without the 2-RDM, as first suggested in Ref. (40). We do not calculate RDMs for any results in this article, although it is straightforward if desired.…”

“…If the PT2 correction is included then this error is reduced to less than 0.2 mHa, with a statistical error of 0.17 mHa. i-FCIQMC(SCI) simulations used preconditioning 40 and a larger spawned vector list in order to reduce statistical errors on the PT2 correction. They also used a larger deterministic space for the semi-stochastic adaptation, requiring extra memory for the deterministic Hamiltonian.…”