Efficient Heat-Bath Sampling in Fock Space

Abstract

We introduce an algorithm for sampling many-body quantum states in Fock space. The algorithm efficiently samples states with probability approximately proportional to an arbitrary function of the second-quantized Hamiltonian matrix element connecting the sampled state to the current state. We apply the new sampling algorithm to the recently developed semistochastic full configuration interaction quantum Monte Carlo (S-FCIQMC) method, a semistochastic implementation of the power method for projecting out the ground state energy in a basis of Slater determinants. Our new sampling method requires modest additional computational time and memory compared to uniform sampling but results in newly spawned weights that are approximately of the same magnitude, thereby greatly improving the efficiency of projection. A comparison in efficiency between our sampling algorithm and uniform sampling is performed on the all-electron nitrogen dimer at equilibrium in Dunning's cc-pVXZ basis sets with X ∈ {D, T, Q, 5}, demonstrating a large gain in efficiency that increases with basis set size. In addition, a comparison in efficiency is performed on three all-electron first-row dimers, B2, N2, and F2, in a cc-pVQZ basis, demonstrating that the gain in efficiency compared to uniform sampling also increases dramatically with the number of electrons.

Many-Body Expanded Full Configuration Interaction. II. Strongly Correlated Regime

“…For this specific case, we assume that the restricted HF (RHF) determinant is indeed comprising a fair approximation to the FCI wave function, i.e., it has the dominant weight. The MBE-FCI decomposition of the FCI correlation energy will involve a total of 6 CASCI (6,4) calculations at order 1 (the (6, 3) RHF reference space augmented by all possible single virtual MOs of the expansion space), 15 CASCI(6, 5) calculations at order 2 (augmentation by all unique pair combinations of virtual MOs), etc., and culminate in a single CASCI (6,9) calculation at order 6 (i.e., FCI for system (a)). (6,9) system using the MBE-FCI method.…”

“…As an example of this, case (c) in Figure 2 (6,4) reference space calculation then precedes the actual MBE-FCI expansion. This is otherwise initiated at order 1 with 5 CASCI(6, 5) calculations (each involving the reference space and a single virtual MO of the expansion space) and ends in a single CASCI (6,9) calculation at order 5 (i.e., FCI for system (c)). In analogy with case (c), one may similarly choose to expand the reference space even further.…”

Projector Quantum Monte Carlo Method for Nonlinear Wave Functions

“…Each iteration consists of 100,000-200,000 random samples of the wavefunction. For each, a small selection of configurations, {m}, are sampled from the set of non-zero connections via H mn in the manner of FCIQMC, and unbiasing for the probability with a computed normalized generation probability [28,29]. Furthermore, the derivatives…”

Beyond Walkers in Stochastic Quantum Chemistry: Reducing Error Using Fast Randomized Iteration

“…The alias method is a more efficient implementation of multinomial sampling than the one described above. 34,36 The systematic sampling scheme typically achieves reduced variance in the vector n. The m random numbers {U k } used in the selection of elements are generated from a single random number r chosen uniformly on the interval (0, 1), as follows:…”