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Physical Review B

Optimized nonorthogonal localized orbitals for linear scaling quantum Monte Carlo calculations

Fernando Agustín Reboredo, Andrew Williamson

Published Mar 23, 2005• 39 citations
Publisher

Abstract

We derive an automatic procedure for generating a set of highly localized, nonorthogonal orbitals for linear scaling quantum Monte Carlo (QMC) calculations. We demonstrate the advantage of these orbitals for calculating the total energy of both semiconducting and metallic systems by studying bulk silicon and the homogeneous electron gas. For silicon, the improved localization of these orbitals reduces the computational time by a factor of 5 and the memory by a factor of 6 compared to localized, orthogonal orbitals. For jellium at typical metallic densities, we demonstrate that the total energy is converged to 3 meV per electron for orbitals truncated within spheres with radii $7{r}_{s}$, opening the possibility of linear scaling QMC calculations for realistic metallic systems.

DOI
10.1103/physrevb.71.121105

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We derive an automatic procedure for generating a set of highly localized, nonorthogonal orbitals for linear scaling quantum Monte Carlo (QMC) calculations. We demonstrate the advantage of these orbitals for calculating the total energy of both semiconducting and metallic systems by studying bulk silicon and the homogeneous electron gas. For silicon, the improved localization of these orbitals reduces the computational time by a factor of 5 and the memory by a factor of 6 compared to localized, orthogonal orbitals. For jellium at typical metallic densities, we demonstrate that the total energy is converged to 3 meV per electron for orbitals truncated within spheres with radii 7rs7{r}_{s}7rs​, opening the possibility of linear scaling QMC calculations for realistic metallic systems.
We derive an automatic procedure for generating a set of highly localized, non-orthogonal orbitals for linear scaling quantum Monte Carlo calculations. We demonstrate the advantage of these orbitals in calculations of the total energy of both semiconducting and metallic systems by studying bulk silicon and the homogeneous electron gas. For silicon, the improved localization of these orbitals reduces the computational time by a factor five and the memory by a factor of six compared to localized, orthogonal orbitals. For jellium, we demonstrate that the total energy is converged for orbitals truncated within spheres with radii 7-8 rsr_srs​, opening the possibility of linear scaling QMC calculations for realistic metallic systems.
A method is presented for using truncated, maximally localized Wannier functions to introduce sparsity into the Slater determinant part of the trial wave function in quantum Monte Carlo calculations. When combined with an efficient numerical evaluation of these localized orbitals, the dominant cost in the calculation, namely, the evaluation of the Slater determinant, scales linearly with system size. This technique is applied to accurate total energy calculation of hydrogenated silicon clusters and carbon fullerenes containing 20-1000 valence electrons.
We develop a local correlation variant of auxiliary-field quantum Monte Carlo (AFQMC) based on local natural orbitals (LNO-AFQMC). In LNO-AFQMC, independent AFQMC calculations are performed for each localized occupied orbital using a truncated set of tailored orbitals. Because the size of this space does not grow with the system size for a target accuracy, the method has linear scaling. Applying LNO-AFQMC to molecular problems containing a few hundred to a thousand orbitals, we demonstrate convergence of total energies with significantly reduced costs. The savings are more significant for larger systems and larger basis sets. However, even for our smallest system studied, we find that LNO-AFQMC is cheaper than canonical AFQMC, in contrast with many other reduced-scaling methods. Perhaps most significantly, we show that energy differences converge much more quickly than total energies, making the method ideal for applications in chemistry and material science. Our work paves the way for linear scaling AFQMC calculations of strongly correlated systems, which would have a transformative effect on ab initio quantum chemistry.
Abstract This paper describes the use of localized orbitals and shortrange correlation functions to achieve near linear scaling in fixednode diffusion QMC calculations. The success of the method is demonstrated with a series of all-electron calculations for linear hydrocarbons in the range of C2H6 to C2sHs2 with 18 to 202 electrons. The procedure used was not very different from that of earlier all-electron calculations for systems of these sizes. Some changes were needed for evaluation of the trial wave functions and their derivatives. The molecular orbitals for the determinantal wavefunction were composed of contracted Gaussian functions decaying rapidly with distance but limited by a cut-off radius. The resulting determinants were sparse and could be evaluated by a standard matrix LU decomposition package. The standard Schmidt Moskowitz correlation function was modified with use of a short-range scaled distance to allow short cut-off radii.
Quantum Monte Carlo (QMC) techniques are used to calculate the one-body density matrix and excitation energies for the valence electrons of bulk silicon. The one-body density matrix and energies are obtained from a Slater-Jastrow wave function with a determinant of local-density approximation (LDA) orbitals. The QMC density matrix evaluated in a basis of LDA orbitals is strongly diagonally dominant. The natural orbitals obtained by diagonalizing the QMC density matrix resemble the LDA orbitals very closely. Replacing the determinant of LDA orbitals in the wave function by a determinant of natural orbitals makes no significant difference to the quality of the wave function's nodal surface, leaving the diffusion Monte Carlo energy unchanged. The extended Koopmans's theorem for correlated wave functions is used to calculate excitation energies for silicon, which are in reasonable agreement with the available experimental data. A diagonal approximation to the theorem, evaluated in the basis of LDA orbitals, works quite well for both the quasihole and quasielectron states. We have found that this approximation has an advantageous scaling with system size, allowing more efficient studies of larger systems.
We have reformulated the quantum Monte Carlo (QMC) technique so that a large part of the calculation scales linearly with the number of atoms. The reformulation is related to a recent alternative proposal for achieving linear-scaling QMC, based on maximally localized Wannier orbitals (MLWO), but has the advantage of greater simplicity. The technique we propose draws on methods recently developed for linear-scaling density functional theory. We report tests of the new technique on the insulator MgO, and show that its linear-scaling performance is somewhat better than that achieved by the MLWO approach. Implications for the application of QMC to large complex systems are pointed out.