Variational Monte Carlo studies of atoms

We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps $τ\sim O(1)$ and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps $τ\sim O(1)$ and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.

arXiv:1810.00864Oct 1, 20182 citations

Monte Carlo Simulations in Quantum Mechanics

Andreas Wipf

Lecture notes in physicsJan 1, 20210 citations

Variance minimization variational Monte Carlo method

Imran Khan, Bo Gao

We present a variational Monte Carlo (VMC) method that works equally well for the ground and the excited states of a quantum system. The method is based on the minimization of the variance of energy, as opposed to the energy itself in standard methods. As a test, it is applied to the investigation of the universal spectrum at the van der Waals length scale for two identical Bose atoms in a symmetric harmonic trap, with results compared to the basically exact results obtained from a multiscale quantum-defect theory.We present a variational Monte Carlo (VMC) method that works equally well for the ground and the excited states of a quantum system. The method is based on the minimization of the variance of energy, as opposed to the energy itself in standard methods. As a test, it is applied to the investigation of the universal spectrum at the van der Waals length scale for two identical Bose atoms in a symmetric harmonic trap, with results compared to the basically exact results obtained from a multiscale quantum-defect theory.

arXiv:physics/0701223APSJan 19, 20070 citations

INVESTIGATIONS IN QUANTUM MONTE CARLO

Matthew Clifford Wilson

Feb 2, 20070 citations

Electronic Parameter Updates and Atomic Forces in Quantum Monte Carlo

Myung Won Lee, Massimo Mella, Andrew M. Rappe

An improved method of moving the parameters in variational quantum Monte Carlo (VMC) calculations is described, and then applied to the calculation of atomic forces for first-row hydrides. We propose use of Newton's method with singular-value decomposition (SVD) to update the variational parameters, combined with steepest descent (SD) updates along directions rejected by the SVD. Using this approach to minimize the energy expectation value, we compute atomic forces [J. Chem. Phys. 118, 7193 (2003)] for first-row monohydrides in VMC and fixed-node diffusion Monte Carlo (DMC). The forces agree quantitatively with potential-energy surfaces, demonstrating the accuracy of this force procedure. Properties obtained from these calculations, such as harmonic vibrational frequencies and anharmonicity constants, agree well with experimental values.An improved method of moving the parameters in variational quantum Monte Carlo (VMC) calculations is described, and then applied to the calculation of atomic forces for first-row hydrides. We propose use of Newton's method with singular-value decomposition (SVD) to update the variational parameters, combined with steepest descent (SD) updates along directions rejected by the SVD. Using this approach to minimize the energy expectation value, we compute atomic forces [J. Chem. Phys. 118, 7193 (2003)] for first-row monohydrides in VMC and fixed-node diffusion Monte Carlo (DMC). The forces agree quantitatively with potential-energy surfaces, demonstrating the accuracy of this force procedure. Properties obtained from these calculations, such as harmonic vibrational frequencies and anharmonicity constants, agree well with experimental values.

arXiv:physics/0411209Nov 23, 20040 citations

τ\sim O(1)

and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.