Quantum Monte Carlo calculations of energy gaps from first principles

Abstract

We review the use of continuum quantum Monte Carlo (QMC) methods for the calculation of energy gaps from first principles, and present a broad set of excited-state calculations carried out with the variational and fixed-node diffusion QMC methods on atoms, molecules, and solids. We propose a finite-size-error correction scheme for bulk energy gaps calculated in finite cells subject to periodic boundary conditions. We show that finite-size effects are qualitatively different in twodimensional materials, demonstrating the effect in a QMC calculation of the band gap and exciton binding energy of monolayer phosphorene. We investigate the fixed-node errors in diffusion Monte Carlo gaps evaluated with Slater-Jastrow trial wave functions by examining the effects of backflow transformations, and also by considering the formation of restricted multideterminant expansions for excited-state wave functions. For several molecules, we examine the importance of structural relaxation in the excited state in determining excited-state energies. We study the feasibility of using variational Monte Carlo with backflow correlations to obtain accurate excited-state energies at reduced computational cost, finding that this approach can be valid. We find that diffusion Monte Carlo gap calculations can be performed with much larger time steps than are typically required to converge the total energy, at significantly diminished computational expense, but that in order to alleviate fixed-node errors in calculations on solids the inclusion of backflow correlations is sometimes necessary.PACS numbers: 31.15. A-, 31.15.vj, 31.50.Df, 71.15.Qe, 71.35.-y arXiv:1806.04750v4 [cond-mat.mtrl-sci]

Variational and diffusion quantum Monte Carlo calculations with the CASINO code

“…Time-step bias vanishes in the limit of zero time step and is linear in the time step for sufficiently small τ . Figure 5 demonstrates that the time-step bias per particle is relatively small in this linear regime, does not get more severe in larger systems, is greatly reduced if the trial wave function is good, and largely cancels out of energy differences 69,73 if trial wave functions of similar quality are used. In order to obtain accurate total energies one must either (i) use a sufficiently small time step that the bias is negligible or (ii) perform simulations at different time steps and extrapolate to zero time step.…”

“…So the leading-order finitesize correction to the quasiparticle gap ∆ qp = E N +1 + E N −1 − 2E N in a finite cell is that the supercell Madelung constant v M must be subtracted from the gap. 73 Note that the supercell Madelung constant must be evaluated using the screened Coulomb interaction. In the case of a crystal of cubic symmetry the screened Madelung constant is sim-…”

“…Hence a suggested procedure for calculating QMC quasiparticle gaps is to perform calculations at a range of system sizes, correct the leading-order errors by subtracting the screened Madelung constant, then extrapolate the resulting gaps to infinite system size assuming a 1/N error; this simultaneously removes residual systematic effects and averages out quasirandom finitesize errors. 73 To calculate an ab initio exciton binding energy, the supercell must be significantly larger than the exciton Bohr radius. In smaller cells, the quasielectron and quasihole are effectively unbound, and hence the excitonic gap is similar to the quasiparticle gap.…”

“…The conclusion is modified in 2D or layered systems, where the interaction is of Rytova-Keldysh form, leading to O(N −2/3 ) scaling of the finite-size error that eventually crosses over to O(N −1 ) behavior on near-macroscopic length scales. 73…”

Excited States with Selected Configuration Interaction-Quantum Monte Carlo: Chemically Accurate Excitation Energies and Geometries

“…We note that the "truncation" scheme should be applied to a relatively large expansion since, during the CIPSI iterations, the largest coefficients vary much at the beginning but tend to stabilize as the size of the expansion grows. If the size of the initial expansion is such that the determinants kept after truncation have converged coefficients relative to 3 FIG. 2.…”