Ground-state electronic properties of LiH calculated from the “Bounce” version of quantum Monte Carlo
Introduction
Quantum Monte Carlo (QMC) is a set of versatile algorithms used to solve a modified time-independent Schrödinger equation for a system with many particles, such as atoms with one or more electrons or various molecules [1], [2], [3], [4], [5]. Starting with an inputted wave-function Ψ, that is used for importance sampling by guiding electrons from unimportant regions of space to important regions of space, QMC can sample the desired, unknown, putatively exact ground state wave-function Φ0 that describes the distribution of electrons in the system. Depending on the type of the QMC algorithm used, the solution to the Schrödinger equation can yield either a mixed distribution of electrons, Φ0Ψ, or their pure distribution, , albeit with a minimal bias originating from the guiding function's inexact, fixed-exchange nodes when using algorithms scalable to a large number of electrons [6].
The type of the distribution is important, because it dictates what observables can be obtained from its sampling without bias being introduced by the guiding function, Ψ. The mixed estimate of a property that corresponds to an operator is given by . For cases where does not commute with the Hamiltonian, this estimate yields degraded results due to its inherently large guiding function bias [7]. The actual desired result is given by , obtained by pure-sampling methods; see recent review [8].
The so-called “Bounce” algorithm [9] is a new, straight-forward and computationally efficient QMC method known to generate accurate estimates of the energy. The energy is calculated by sampling from the mixed distribution, but the algorithm's effectiveness with sampling from the pure distribution has not been thoroughly tested.
In previous work with reptation quantum Monte Carlo (RQMC) [10], an algorithm that is conceptually similar to the Bounce, we studied the behaviour of the energy and electronic properties for LiH ground-state. The properties we obtained in that work were of high quality, albeit through expenditure of many CPU hours. In the present study we aim to determine whether or not the Bounce is competitive with RQMC in terms of its computational efficiency and the accuracy of the calculated properties. Our approach is to perform calculations for short, medium and long reptiles. The most efficient calculations are those taken with short reptiles; the least efficient ones are done with long reptiles, the conditions chosen in our previous work.
In this paper we exploit commercial software (ADF [11], [12], [13]) generated wave-functions without resorting to an optimized, explicitly correlated Jastrow function. This has been shown to yield results in good agreement with the accuracy of previous works, albeit with a corresponding loss of efficiency [10].
This paper is organized as follows: in the next section we describe the theoretical basis of the Bounce, followed by a section with formulas for estimating the properties of interest to us. Next we describe the program's structure and some of its technical details. This is followed by sections with the results, their discussion and conclusions.
Section snippets
Theory
The goal of the Bounce is to simulate the solution to the following modified, imaginary-time dependent Schrödinger equation:where f is the mixed distribution given byHere, F(x) is the drift termEL(x) is the local energyx is the set of 3n coordinates of the n electrons, and E0 is the exact (true) ground state energy of the system. The method does not rest on the choice of E0. Ψ is called a “guiding function”, used to move
Electric moments
Electric multipole moments (dipole, quadrupole, octupole, etc.) are fundamental quantities that arise from the charge distribution inside molecules and also influence their interaction with external electric fields. These moments are useful because they provide an insight into the electrical and structural properties of molecules, such as charge distribution and symmetry, respectively [14]. For example, the dipole, quadrupole and octupole moments of the hydrogen atom are all zero because its
Program structure
The computation of the wave-function and its derivatives, the local energy and the drift is done using the method outlined by Bueckert et al. [17]. The distinct advantage of this method is that the derivatives are calculated analytically rather than numerically. This allows for a much more efficient and precise calculation of the aforementioned quantities when working with a large number of basis set parameters (atomic orbital exponents and the molecular orbital coefficients). Furthermore, the
Truncations
To control the drift velocity, Fi = ∇ Ψ(xi)/Ψ(xi), that can push the electron too far from a region with reasonable probability, we truncate the velocity components as follows [19]:
The local energy has a singularity for when an electron is close to a nucleus, leading to large fluctuations in the simulated energy, and contributing to the time-step bias of the Monte Carlo estimates. To ameliorate this, we truncate the local energies as follows [19]:
Technical details and results
Recently we computed physical properties of LiH using RQMC, an algorithm similar to the Bounce, but one where in each iteration M > 1 scales are chopped from the reptile [10]. That study used L0 = 121, M0 = 20 and τ = 0.012 ⋯ 0.002(0.002). Here, L0 corresponds to the (initial) length of the reptile at the largest time-step, τ0. Similarily, M0 corresponds to the (initial) RQMC chop-size at the largest time-step.
The importance sampling wave-function for LiH was a single determinant HF-SCF wave-function
Conclusions
To our knowledge, we have reported the first use of the Bounce algorithm to calculate ground-state molecular properties represented by operators that do not commute with the Hamiltonian. We conclude (at least with respect to LiH) that the quality of the results is strongly dependent on the length of the reptile at which the calculations are performed. The accuracy and precision of the energy and other properties calculated using the Bounce are consistently inferior to that obtained from the
Acknowledgments
This work was supported, in part, by grants from the Natural Sciences and Engineering Research Council of Canada. Mr. Ospadov gratefully acknowledges financial support from Brock University. Our results were generated by using SHARCNET high-performance computing facilities. Access to their resources is gratefully acknowledged.
Wai Kong Yuen is an associate professor of Mathematics at Brock University. He received his BSc at the University of Hong Kong, and his MSc and PhD at the University of Toronto, under the direction of Prof. J. S. Rosenthal. Previously, he held a faculty position at York University. Yuen's main research interests are in studying the efficiencies of various Markov chain Monte Carlo (MCMC) and nonparametric estimation methods, as well as their applications in natural sciences. Yuen has published a
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Wai Kong Yuen is an associate professor of Mathematics at Brock University. He received his BSc at the University of Hong Kong, and his MSc and PhD at the University of Toronto, under the direction of Prof. J. S. Rosenthal. Previously, he held a faculty position at York University. Yuen's main research interests are in studying the efficiencies of various Markov chain Monte Carlo (MCMC) and nonparametric estimation methods, as well as their applications in natural sciences. Yuen has published a significant number of papers on a wide range of topics, including the theory of MCMC, quantum Monte Carlo methods, nonparametric estimation methods and optimal robust designs.
Egor Ospadov received his BSc in Physics from University of Waterloo. He was recently awarded a MSc in Theoretical Physics from Brock University. Mr. Ospadov is currently pursuing a PhD in Theoretical Physics at Brock University under the supervision of Dr. Rothstein. His current research is focused on quantum Monte Carlo methods in physics and chemistry. Previously he had researched topics involved theoretical biophysics. Ospadov is an active consultant in the area of high-performance scientific computing. His consultations include assembly, installation and maintenance of computer clusters, and optimization of various molecular modeling and quantum chemistry software.
Stuart Rothstein is a professor of Chemistry and Physics at Brock University, Director of its Institute for Scientific Computation, and a member of the Centre for Biotechnology. His undergraduate training was at the University of Illinois, followed by graduate work at the University of Michigan, where he earned his PhD under the direction of Prof. S.M. Blinder. Rothstein's post-doctoral training with Harris J. Silverstone was taken at Johns Hopkins University, followed by a period of teaching at Swarthmore College. Rothstein's research interests are in computational chemistry, physics and biology. He has co-organized conferences the area of computational science and has several publications in that field.