Quantum Monte Carlo for Atoms, Molecules and Solids
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| Quantum Monte Carlo for Atoms, Molecules and Solids
W. A. Lester, Jr. Department of Chemistry, University of California, Berkeley, California Lubos Mitas Department of Physics, North Carolina State University, Raleigh, North Carolina 27695 Brian Hammond Microsoft Corporation, 45 Liberty Boulevard, Great Valley Corp Ctr., Malvern, Pennsylvania 19355 Abstract The quantum Monte Carlo (QMC) method has become an increasingly important method for the solution of the stationary Schrödinger equation for atoms, molecules and solids. The method has been established as an approach of high accuracy that scales better with system size than other ab initio methods. Further, the method as typically implemented, takes full advantage of parallel computing systems. These aspects for electronic structure will be described as well as recent applications that demonstrate the breadth of application of the method. In addition, frontier areas for further theoretical development are indicated. IntroductionIn the past decade the field of computational science, the ability to simulate physical processes based on fundamental physical laws, has undergone rapid growth. This progress has been achieved not only due to the exponential increase in computing power now available but also because of new generation of simulation algorithms. However, each new generation of high-performance computing platforms presents a renewed challenge to the developers of computational methods to take full advantage of the underlying computer architecture. In many cases the original computational method has not been well suited to both reduce the time to solution for a given physical system, and, more importantly to increase the accuracy of the simulation. The ab initio simulation of the electronic structure of atoms, molecules and solids has greatly benefited from an enormous increase in computing power, but it has also remained bound for the most part by approximations originally created to make the computations possible. The most popular of these approximations are those based on basis set expansions that have their origin with either the Hartree-Fock or Kohn-Sham equations. These methods have presented a number of challenges to the use of massively parallel computers as well as well-known limitations in their accuracy. A method that shows great promise both in its inherent accuracy as well as its ability to make efficient use of modern computing power is quantum Monte Carlo (QMC). QMC is a broad term that can refer to a number of techniques that use stochastic methods to simulate quantum systems. For electronic structure theory we are referring to the simulation of the electronic Schrödinger equation by the use of random numbers to sample the electronic wave function and its properties. For reviews since 2001, see Aspuru-Guzik 1, Aspuru-Guzik2, Anderson 1, Towler1. QMC has consistently shown the ability to recover 90-95 % of the electron correlation energy (the difference between the Hartree-Fock and exact energies), even in cases that are poorly described by the Hartree-Fock method even for large systems with hundreds of electrons. In a few-electron systems, the method is capable to solve the stationary Schrodinger problem virtually exactly. Until recently the success of the QMC method was limited to simulation and prediction of properties of small molecules consisting of low atomic weight atoms, owing to the amount of computer time required to reduce statistical error to physically significant levels. Great strides in methods combined with QMC’s inherent parallelism have removed these limits and made possible the accurate simulation of much larger molecules as well as solids [ref]. Advances in the method have widened the application of QMC to many electronic properties of these systems besides the energy. One of the greatest strengths of the approach lies in the ability to numerically sample quantum operators and wave functions that cannot be analytically integrated or represented. This capability has enabled the QMC method to make significant contributions to fundamental understanding of the electronic wave function and its properties. The QMC method has grown in interest for the study of a wide range of physical systems including atomic, molecular, condensed-matter, and nuclear. In this paper, we shall not address the latter area [ref], which is mentioned only to indicate the broad range of phenomena accessible by the method. The general applicability of the method has led us to limit consideration in this article to the electronic structure of the aforementioned systems due to space limitations and author interest. Path integral MC (PIMC) will not be discussed nor shall approaches for the determination of the internal energy of molecular systems including clusters [ref.] Certain properties make QMC particularly attractive for the treatment of electronic structure. These include limited dependence on basis set which is a feature that dominates all other ab initio methods, high accuracy of energy calculations that are typically as accurate as other state-of-the-art ab initio methods [refs], and dependence of computation on system size N or scaling [refs], as well as the ease of adaptation of QMC computer programs to parallel computers [refs]. This paper is organized as follows. Section II will summarize the main variants of QMC applied to atoms, molecules, and solids. These methods are variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). Section III turns to other issues that play an important role in the development of QMC. Section IV briefly summarizes selected applications that provide an indication of the breadth and level of accuracy for systems studied to date. Finally, Section V points to various new methods that hold promise for advancing the state-of-the-art. QMC MethodsThe QMC methods to be discussed solve the non-relativistic, time-independent, fixed-nuclei, electronic Schrödinger equation,
where  is the molecular Hamiltonian. The Hamiltonian contains the kinetic and potential energy operators for the electrons,  where is the Lapacian operator for the kinetic energy,
and  is the potential energy operator, which for a system comprised of  electrons and  nuclei with nuclear charges  and no external fields is,
where r is the distance between the particles. Atomic units have been used, in which the mass and the charge of the electron are set to unity. To solve this time-independent equation using Monte Carlo, we start with the time-dependent Schrödinger equation written in imaginary time by substituting τ = it,
that has the formal solution
The quantity ET is an energy offset chosen to reduce the impact of the exponential factor. As τ → ∞ this equation (2.5) converges exponentially to the lowest electronic state with non-zero Ck , which is usually the ground state Φ0. Equation (2.4) above has been written with the Hamiltonian expanded to show that it has the form of a diffusion equation (the kinetic energy term) and a rate equation (with “rate constant” 𝒱 – ET). Both these processes can be simulated using Monte Carlo sampling methods to propagate the solution to large imaginary time. (Note: since τ is imaginary time, no description of real time dynamics is possible; see, however, ref.[Mitas-Grossman].) The simulation is performed by constructing by choosing an initial ensemble of pseudo particles, each of which represents the positions of all the electrons in the system – R={x1, x2, …, xn}. Each of these electronic configurations then undergoes a random walk in imaginary time until sufficient time has elapsed for the excited states to decay away. The resulting density distribution is proportional to Φ0. Such a simulation, however, poses two serious problems. First, to simulate a diffusion equation one requires the distribution to represent a positive density, whereas for Fermion systems Φ0 necessarily changes sign. Second, the rate equation depends on the potential energy which varies widely and diverges when two particles approach, which will cause the Monte Carlo simulation to be unstable. Both these issues can be dealt with by the introduction of importance sampling. Choose a trial function, ΨT, which is a good approximation to the desired state Φ0. Define a new function,  , and then rewrite the imaginary time dependant Schrödinger equation in terms of this new function:
where  , and  . The quantity  is a vector field pointing towards regions of large  , and  is the local energy of . The effect of importance sampling is twofold: first to introduce the term  which is a directed drift moving the system to areas of large , and second, the rate term now depends on the local energy which will be much smoother than the potential energy for good trial function choices. It can be seen from the form of the local energy, as  becomes a constant, namely  the ground state energy.
Once  has converged to  , expectation values of Φ0 may be measured by continuing the Monte Carlo simulation and sampling various operators, the most important of these being the Hamiltonian. The energy of  is obtained from the local energy as follows,
The last step is obtained from the Hermitian property of ℋ. One difficulty imposed by importance sampling is that all properties involving an operator  which does not commute with the Hamiltonian will result in a mixed estimate,  rather than the “pure” estimate  . There are several methods to extract the pure  distribution needed for operators that do not commute with the Hamiltonian from the mixed distribution that will be addressed below.
Simulating the importance sampled Schrödinger equation is formally the same as before, with the diffusion process now having a directed drift, and the new “rate constant” being  The sampling procedure can be performed in a number of ways. Below we present the two QMC approaches that dominate interest and have been used for most calculations: Variational Monte Carlo (VMC) and Diffusion Monte Carlo (DMC). VMC is a method to sample from the  distribution yielding the energy and other properties of the trial function. DMC is a projector method which samples from the mixed distribution and in principle yields exact energies and properties.
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