Basis Quantum Monte Carlo 法を用いた調和振動子内フェルミ粒子の計算

YAGI Toru

八木 徹

Fionn D. Malone, Anouar Benali, Miguel A. Morales +3 more

Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths, and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multideterminant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis-set full configuration interaction energies estimated with CIPSI are used as a reference in this comparative study between DMC and AFQMC. By focusing on a set of canonical finite--size solid-state systems, we show that both QMC methods can be made to systematically converge towards the same energy once basis-set effects and systematic biases have been removed. AFQMC shows a much smaller dependence on the trial wave function than DMC while simultaneously exhibiting a much larger basis-set dependence. We outline some of the remaining challenges and opportunities for improving these approaches.

F. R. Petruzielo

This thesis details four research projects related to zero temperature quantum Monte Carlo. Chapters 2-4 focus on continuum quantum Monte Carlo, and specifically its application to molecular systems; whereas Chapter 5 focuses on quantum Monte Carlo in a discrete space. Chapter 2 focuses on improving upon the single-particle basis functions employed in quantum Monte Carlo calculations for molecular systems. For calculations requiring non-diverging pseudopotentials, a class of functions is introduced that is capable of producing the short- and long-range asymptotic behavior of the exact wavefunction. It is demonstrated that this form of basis function produces superior accuracy and efficiency when compared to the basis sets typically employed in quantum Monte Carlo. Although the basis functions introduced in Chapter 2 are capable of producing superior results, it is necessary that the parameters of the functional form are near-optimal for the full potential of the functions to be realized. Chapter 3 introduces a simple yet general method for constructing basis sets of a desired functional form appropriate for molecular electronic structure calculations. A standard basis set is created for each of the elements from hydrogen to argon. Chapter 4 explores the effect of different aspects of the trial wavefunction on the accuracy of quantum Monte Carlo. By systematically testing the effect of the basis size, orbital quality, and determinant expansion quality, this work offers guidance to quantum Monte Carlo practitioners for achieving results to within chemical accuracy of experiment. In Chapter 5, semistochastic projection, a hybrid of deterministic and stochastic projection, is introduced for finding the dominant eigenvalue and eigenvector of a matrix. This method, like stochastic projection, is applicable to matrices well beyond the size that can be handled by deterministic methods. Semistochastic projection improves over stochastic projection by significantly reducing the computational time required to obtain the eigenvalue within a specified statistical uncertainty. After the semistochastic projection method is introduced, it is applied to determine the ground state energy of the Hamiltonian in a discrete basis. This special case of semistochastic projection, dubbed semistochastic quantum Monte Carlo, is shown to be orders of magnitude more efficient than stochastic quantum Monte Carlo.

徹 八木

Jörg Kußmann, Christian Ochsenfeld

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.