Quantum Monte Carlo Calculations for Minimum Energy Structures

Abstract

We present an efficient method to find minimum energy structures using energy estimates from accurate quantum Monte Carlo calculations. This method involves a stochastic process formed from the stochastic energy estimates from Monte Carlo that can be averaged to find precise structural minima while using inexpensive calculations with moderate statistical uncertainty. We demonstrate the applicability of the algorithm by minimizing the energy of the H2O-OH − complex and showing that the structural minima from quantum Monte Carlo calculations affect the qualitative behavior of the potential energy surface substantially.First principles electronic structure methods have been used to describe and explain a wide range of properties for different condensed matter systems. A critical step is the accurate determination of the ground state atomic structure, since many important properties of a material can change dramatically depending on the structure. Because of the balance between accuracy and computational cost, density functional theory (DFT) has become a commonly used method to find equilibrium geometries of both molecules and extended systems. The primary reason for this is the availablility of forces with little extra computational cost over the energy calculation. Using a typical quasi-Newton minimization algorithm, the local minimum of a potential energy surface can be found in O(N DOF ), where N DOF is the number of degrees of freedom to be optimized. This favorable scaling has made it possible to find minima for many systems of interest. However, in many situations, including transition metals, excited states, and weak binding, current density functional theories may not be accurate enough even for structures, and more accurate post-Hartree-Fock methods that scale as O(N 5−7 e ), where N e is the number of electrons, can often be too computationally expensive.Quantum Monte Carlo (QMC), a stochastic approach to solving the many-body Schrödinger equation, offers favorable scaling to obtain the total energy, O(N 2−3 e ), and has been shown to provide near chemical accuracy in many different systems [1,2]. However, there are two major challenges in using QMC methods to obtain high precision minimum energy structures. The first is that the techniques so far proposed to calculate forces in diffusion Monte Carlo all have large variances and error terms that depend on the quality of the trial wave function, which is often poor in systems where DFT fails and one would like to apply QMC methods. In fact, despite much work in recent years [3][4][5][6][7][8], QMC forces using the highly accurate diffusion Monte Carlo method have not been applied to more than a few degrees of freedom, although the simpler variational Monte Carlo technique has been applied to more [9]. The second challenge is the stochastic nature of Monte Carlo algorithms, which provides uncertainty in any estimator that only decreases as the square root of the computer time. Reducing the uncertainty enough to resolve the minimum structure accurately using...

Simulation and understanding of atomic and molecular quantum crystals

“…In spite of this, thanks to the escalating increase in computing efficiency and recent algorithmic advances, the use of eQMC methods is transitioning from that of benchmark calculations in few-atoms systems to that of production runs in hundreds-of-atoms systems Esler et al, 2012;Wagner, 2014). Actually, efficient QMCbased methods have been already developed that allow to simulate both the electrons and nuclei in crystals quantum mechanically (Grossman and Mitas, 2005;Wagner and Grossman, 2010). Among those, we highlight the coupled electron-ion Monte Carlo method due to Pierleoni, Ceperley, and collaborators (Pierleoni, Ceperley, and Holzmann, 2004;Pierleoni and Ceperley, 2005;Pierleoni and Ceperley, 2006), for its special relevance to the field of quantum solids (see, for instance, Sec.…”

Optimized Structure and Vibrational Properties by Error Affected Potential Energy Surfaces

“…Due to this major drawback, structural optimizations of molecules using QMC methods have been efficiently implemented only in very recent years. 42,[53][54][55][56] To tackle this problem, in a recent work Sorella and co-workers 56 proposed a combination of the reweighting method, 57 the correlated sampling technique, 58 and the space warp coordinate transformation. [59][60][61] With the help of the algorithmic adjoint differentiation, all the components of the ionic VMC forces can be calculated with and without pseudopotentials in a computational time that is only about 4 times that of an ordinary energy calculation.…”

The role of density functional theory methods in the prediction of nanostructured gas-adsorbent materials

“…It is worth noticing, however, that QMC methods are neither exempt of some important technical problems. These are essentially related to the tediousness found in the computation of atomic forces [291,292] and the correction of numerical bias introduced by the nodal surface approximation [58].…”