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Physical Review B

Density matrix-based variational quantum Monte Carlo providing an asymptotically linear scaling behavior for the local energy

Jörg Kußmann, Hella Riede, Christian Ochsenfeld

Published Apr 11, 2007• 14 citations
Publisher

Abstract

We present a reformulation of the variational quantum Monte Carlo (VQMC) method using the $N$-particle density matrix ($N$-PDM VQMC), which allows us to reduce the conventional cubic scaling behavior to linear for the calculation of the local energy with the number of electrons for systems with a nonvanishing highest occupied molecular orbital--lowest unoccupied molecular orbital gap. The ``perturbation'' of the electron distribution, caused by the truncation of the self-consistent-field (SCF) one-electron density matrix, is compensated by the use of the McWeeny purification transformation in the presented scheme. The performance and accuracy of the presented method are illustrated through exemplary calculations.

DOI
10.1103/physrevb.75.165107

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We present a reformulation of the variational quantum Monte Carlo (VQMC) method using the NNN-particle density matrix (NNN-PDM VQMC), which allows us to reduce the conventional cubic scaling behavior to linear for the calculation of the local energy with the number of electrons for systems with a nonvanishing highest occupied molecular orbital--lowest unoccupied molecular orbital gap. The ``perturbation'' of the electron distribution, caused by the truncation of the self-consistent-field (SCF) one-electron density matrix, is compensated by the use of the McWeeny purification transformation in the presented scheme. The performance and accuracy of the presented method are illustrated through exemplary calculations.
A reformulation of the fixed-node diffusion quantum Monte Carlo method (FN-DQMC) in terms of the N-particle density matrix is presented, which allows us to reduce the computational effort to linear for the evaluation of the local energy. The reformulation is based on our recently introduced density matrix-based approach for a linear-scaling variational QMC method [J. Kussmann et al., Phys. Rev. B. 75, 165107 (2007)]. However, within the latter approach of using the positive semi-definite N-particle trial density (rhoN T(R)=mid R:Psi(T)(R)mid R:(2)), the nodal information of the trial function is lost. Therefore, a straightforward application to the FN-DQMC method is not possible, in which the sign of the trial function is usually traced in order to confine the random walkers to their nodal pockets. As a solution, we reformulate the FN-DQMC approach in terms of off-diagonal elements of the N-particle density matrix rhoN T(R;R'), so that the nodal information of the trial density matrix is obtained. Besides all-electron moves, a scheme to perform single-electron moves within N-PDM QMC is described in detail. The efficiency of our method is illustrated for exemplary calculations.
A scheme for imposing the electron-nuclear cusp condition in molecular quantum Monte Carlo (QMC) calculations using Gaussian basis functions is presented. While the constraints proposed by Ma et al. [J. Chem. Phys. 122, 224322 (2005)] have been used, in our scheme the basis functions are modified instead of the molecular orbitals. This method can also be directly applied to density matrix-based QMC calculations whose effort scales linearly for calculating the local energy [J. Kussmann et al., Phys. Rev. B 75, 165107 (2007)]. Furthermore, virtually no additional computational effort in the evaluation of the local energy arises within QMC calculations.
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.
For atomic and molecular quantum Monte Carlo calculations, most of the computational effort is spent in the evaluation of the local energy. We describe a scheme for reducing the computational cost of the evaluation of the Slater determinants and correlation function for the correlated molecular orbital (CMO) ansatz. A sparse representation of the Slater determinants makes possible efficient evaluation of molecular orbitals. A modification to the scaled distance function facilitates a linear scaling implementation of the Schmidt- Moskowitz-Boys-Handy (SMBH) correlation function that preserves the efficient matrix multiplication structure of the SMBH function. For the evaluation of the local energy, these two methods lead to asymptotic linear scaling with respect to the molecule size.
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.
For atomic and molecular quantum Monte Carlo calculations, most of the computational effort is spent in the evaluation of the local energy. We describe a scheme for reducing the computational cost of the evaluation of the Slater determinants and correlation function for the correlated molecular orbital (CMO) ansatz. A sparse representation of the Slater determinants makes possible efficient evaluation of molecular orbitals. A modification to the scaled distance function facilitates a linear scaling implementation of the Schmidt-Moskowitz-Boys-Handy (SMBH) correlation function that preserves the efficient matrix multiplication structure of the SMBH function. For the evaluation of the local energy, these two methods lead to asymptotic linear scaling with respect to the molecule size.