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 multiSlater-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 wavefunctions 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.

💡 Analysis

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 multiSlater-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 wavefunctions 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.

📄 Content

Computing the energy of a water molecule using MultiDeterminants: A simple, efficient algorithm Bryan K. Clark∗ Princeton Center For Theoretical Science, Princeton University, Princeton, NJ 08544 Department of Physics, Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 Miguel A. Morales† Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94550, U.S.A. Jeremy McMinis‡ Department of Physics, University of Illinois at Urbana Champaign, Urbana, IL 61801 Jeongnim Kim§ National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801 Gustavo E. Scuseria¶ Department of Chemistry and Department of Physics & Astronomy, Rice University, Houston, TX 77005-1892, USA 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 so- phisticated wave-functions are critical to ascertaining new physics. One such wave function is the multiSlater-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 wavefunctions 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. I. INTRODUCTION Being able to accurately calculate the material properties of molecules and solids is important for a variety of fields. There exist a variety of different methodologies to accomplish this including density functional theory [1], quantum chemistry [2] and quantum Monte Carlo [3]. All these methods have different regimes of applicability and different tradeoffs between accuracy and speed. For larger and more complicated systems, quantum Monte Carlo methods are often a good choice due to their polynomial scaling with particle number and high levels of accuracy. For ground state calculations, the most commonly used QMC method is fixed node diffusion Monte Carlo (FNDMC). FNDMC takes as input a trial wave function ΨT and returns the ground state energy of the fixed node wave-function ΨF N, the wave-function with the lowest ground state energy with the same nodes as ΨT (i.e. ΨT (R) = 0 ⇐⇒ΨF N(R) = 0). The accuracy of FNDMC is dependent on the quality of the trial wave function. Although the most commonly used form of a trial-wavefunction is the Slater-Jastrow form, acheiving higher levels of accuracy require the use of more sophisticated ansatz. This is especially true in intrinsically multi-reference problems. Such ansatz are only useful if they can be used with limited computational complexity and memory. A natural extension of the Slater-Jastrow wave-function is the multiSlater-Jastrow form. One advantage of this wavefunction is that any state can be represented in the limit of a large enough number of determinants. Therefore, an answer ∗Electronic address: bclark@princeton.edu †Electronic address: moralessilva2@llnl.gov ‡Electronic address: jmcminis@illinois.edu §Electronic address: jnkim@illinois.edu ¶Electronic address: guscus@rice.edu arXiv:1106.2456v1 [cond-mat.mtrl-sci] 13 Jun 2011 2 can be systematically improved upon by including more determinants in the ansatz. In this work, we describe a new algorithm that allows FNDMC to use the multiSlater-Jastrow form in an efficient manner. To use a wave-function in FNDMC, there are two important and computationally demanding requirements: the ability to compute ratios of the wavefunction evaluated in two configurations that differ by the location of a single particle and the ability to evaluate gradients and laplacians of the wavefunction. Here we will describe algorithms to accomplish both these tasks. These algorithms will require a minimal amount of additional memory with respect to the single determinant case and will scale (for a single step) with a computational complexity that goes as O(n2 + nsn + ne) where n is the total number of particles, ns the total number of single excitations and ne is the total number of excitations. This scaling is the same with respect to particle number as the single determinant case and scales linearly with the total number of excitations. Other recent work has also proposed an algorithm for computing ratios of multi-determinants [4]. The method described here scales better by a factor of the particle number n in almost all regimes of interest. In addition, it is significantly simpler requiring less book-keeping and eliminating the need for recursive trees. This simple structure also makes it transparent that the computation involved c

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