Computing the energy of a water molecule using multideterminants: A simple, efficient algorithm

Abstract

Quantum Monte Carlo (QMC) methods such as variational Monte Carlo and fixed node diffusion Monte Carlo depend heavily on the quality of the trial wave function. Although Slater-Jastrow wave functions are the most commonly used variational ansatz in electronic structure, more sophisticated wave functions are critical to ascertaining new physics. One such wave function is the multi-Slater-Jastrow wave function which consists of a Jastrow function multiplied by the sum of Slater determinants. In this paper we describe a method for working with these wave functions in QMC codes that is easy to implement, efficient both in computational speed as well as memory, and easily parallelized. The computational cost scales quadratically with particle number making this scaling no worse than the single determinant case and linear with the total number of excitations. Additionally, we implement this method and use it to compute the ground state energy of a water molecule.

Using CIPSI Nodes in Diffusion Monte Carlo

“…To enhance prefetching the algorithm should allow the predictability of the data arrival in the CPU (that is, avoid random access as much as possible). It is this important aspect that has motivated us to use Sherman-Morrison updates, despite the fact that a method like the Table method [81] has formally a better scaling. Indeed, massive calculations of scalar products at the heart of repeated uses of SM updates are so ideally adapted to present-day processors that very high performances can be obtained.…”

“…Indeed, computing at each of Monte Carlo step the first and second derivatives of the trial wavefunction (drift vector and local energy) is the hot spot of DMC. However, efficient algorithms have been proposed to perform such calculations [80][81][82]. Here, we have briefly summarized our recently introduced algorithm allowing to compute N -determinant expansions issued from selected CI calculations with a computational cost roughly proportional to √ N (with a small prefactor).…”

“…In a given one-particle basis set, the wavefunction is improved by increasing the The main price to pay for such advantages is of course the need of considering much larger multideterminant expansions (from tens of thousands up to a few millions) than in standard DMC implementations where compactness of the trial wavefunction is searched for. However, efficient algorithms have been proposed to perform such calculations [80][81][82].…”

“…• Optimized Sherman-Morrison updates. As proposed in a number of works, [80][81][82] we calculate the determinants and their derivatives using the Sherman-Morrison (SM) formula for updating the inverse Slater matrices. However, in contrast with other implementations, we have found more efficient not to compare the Slater matrix to a common reference (typically, the Hartree-Fock determinant) but instead to perform the Sherman-Morrison updates with respect to the previously computed determinant D σ i−1 .…”

Variational solution of the congruently transformed Hamiltonian for many-electron systems using a full-configuration-interaction calculation

“…[9][10][11][12][13] However, unlike the electronnuclear cusp, atom-centered basis functions are not ideal for accurate description of the many-electron wavefunction near the electron-electron cusp. [14][15][16] Indeed it * corresponding author: archakra@syr.edu has been shown that the slow convergence of a full configuration interaction (FCI) calculation with respect to the one-particle basis is related to the inadequate treatment of the electron-electron cusp. [15] The solution is to include explicit r 12 dependence in the form of the wavefunction, and there is a large assortment of quantum chemical methods that have incorporated this approach.…”