Variational and diffusion quantum Monte Carlo calculations with the CASINO code.

We present an overview of the variational and diffusion quantum Monte Carlo methods as implemented in the casino program. We particularly focus on developments made in the last decade, describing state-of-the-art quantum Monte Carlo algorithms and software and discussing their strengths and weaknesses. We review a range of recent applications of casino.

Assessing the accuracy of quantum Monte Carlo and density functional theory for energetics of small water clusters.

We present a detailed study of the energetics of water clusters (H(2)O)(n) with n ≤ 6, comparing diffusion Monte Carlo (DMC) and approximate density functional theory (DFT) with well converged coupled-cluster benchmarks. We use the many-body decomposition of the total energy to classify the errors of DMC and DFT into 1-body, 2-body and beyond-2-body components. Using both equilibrium cluster configurations and thermal ensembles of configurations, we find DMC to be uniformly much more accurate than DFT, partly because some of the approximate functionals give poor 1-body distortion energies. Even when these are corrected, DFT remains considerably less accurate than DMC. When both 1- and 2-body errors of DFT are corrected, some functionals compete in accuracy with DMC; however, other functionals remain worse, showing that they suffer from significant beyond-2-body errors. Combining the evidence presented here with the recently demonstrated high accuracy of DMC for ice structures, we suggest how DMC can now be used to provide benchmarks for larger clusters and for bulk liquid water.

Fundamental high-pressure calibration from all-electron quantum Monte Carlo calculations.

We develop an all-electron quantum Monte Carlo (QMC) method for solids that does not rely on pseudopotentials, and use it to construct a primary ultra-high-pressure calibration based on the equation of state of cubic boron nitride. We compute the static contribution to the free energy with the QMC method and obtain the phonon contribution from density functional theory, yielding a high-accuracy calibration up to 900 GPa usable directly in experiment. We compute the anharmonic Raman frequency shift with QMC simulations as a function of pressure and temperature, allowing optical pressure calibration. In contrast to present experimental approaches, small systematic errors in the theoretical EOS do not increase with pressure, and no extrapolation is needed. This all-electron method is applicable to first-row solids, providing a new reference for ab initio calculations of solids and benchmarks for pseudopotential accuracy.

Quantum Monte Carlo computations of phase stability, equations of state, and elasticity of high-pressure silica.

Silica (SiO(2)) is an abundant component of the Earth whose crystalline polymorphs play key roles in its structure and dynamics. First principle density functional theory (DFT) methods have often been used to accurately predict properties of silicates, but fundamental failures occur. Such failures occur even in silica, the simplest silicate, and understanding pure silica is a prerequisite to understanding the rocky part of the Earth. Here, we study silica with quantum Monte Carlo (QMC), which until now was not computationally possible for such complex materials, and find that QMC overcomes the failures of DFT. QMC is a benchmark method that does not rely on density functionals but rather explicitly treats the electrons and their interactions via a stochastic solution of Schrödinger's equation. Using ground-state QMC plus phonons within the quasiharmonic approximation of density functional perturbation theory, we obtain the thermal pressure and equations of state of silica phases up to Earth's core-mantle boundary. Our results provide the best constrained equations of state and phase boundaries available for silica. QMC indicates a transition to the dense alpha-PbO(2) structure above the core-insulating D" layer, but the absence of a seismic signature suggests the transition does not contribute significantly to global seismic discontinuities in the lower mantle. However, the transition could still provide seismic signals from deeply subducted oceanic crust. We also find an accurate shear elastic constant for stishovite and its geophysically important softening with pressure.

Continuum variational and diffusion quantum Monte Carlo calculations.

This topical review describes the methodology of continuum variational and diffusion quantum Monte Carlo calculations. These stochastic methods are based on many-body wavefunctions and are capable of achieving very high accuracy. The algorithms are intrinsically parallel and well suited to implementation on petascale computers, and the computational cost scales as a polynomial in the number of particles. A guide to the systems and topics which have been investigated using these methods is given. The bulk of the article is devoted to an overview of the basic quantum Monte Carlo methods, the forms and optimization of wavefunctions, performing calculations under periodic boundary conditions, using pseudopotentials, excited-state calculations, sources of calculational inaccuracy, and calculating energy differences and forces.

Quantum Monte Carlo study of porphyrin transition metal complexes.

Diffusion quantum Monte Carlo (DMC) calculations for transition metal (M) porphyrin complexes (MPo, M=Ni,Cu,Zn) are reported. We calculate the binding energies of the transition metal atoms to the porphin molecule. Our DMC results are in reasonable agreement with those obtained from density functional theory calculations using the B3LYP hybrid exchange-correlation functional. Our study shows that such calculations are feasible with the DMC method.

Equation of state and Raman frequency of diamond from quantum Monte Carlo simulations.

We report variational and diffusion quantum Monte Carlo (VMC and DMC) calculations of the equation of state and Raman frequency of diamond. The pressure derivative of the bulk modulus, which has not been accurately determined experimentally, is calculated to be 3.8 (VMC) and 3.7 (DMC). The values of the Raman frequency calculated at the experimental volume are 1373 cm-1(VMC) and 1359 cm-1 (DMC), in good agreement with the experimental value of 1333 cm-1.

Quantum Monte Carlo calculations of the dissociation energy of the water dimer.

We report diffusion quantum Monte Carlo (DMC) calculations of the equilibrium dissociation energy D(e) of the water dimer. The dissociation energy measured experimentally, D(0), can be estimated from D(e) by adding a correction for vibrational effects. Using the measured dissociation energy and the modern value of the vibrational energy Mas et al., [J. Chem. Phys. 113, 6687 (2000)] leads to D(e)=5.00+/-0.7 kcal mol(-1), although the result Curtiss et al., [J. Chem. Phys. 71, 2703 (1979)] D(e)=5.44+/-0.7 kcal mol(-1), which uses an earlier estimate of the vibrational energy, has been widely quoted. High-level coupled cluster calculations Klopper et al., [Phys. Chem. Chem. Phys. 2, 2227 (2000)] have yielded D(e)=5.02+/-0.05 kcal mol(-1). In an attempt to shed new light on this old problem, we have performed all-electron DMC calculations on the water monomer and dimer using Slater-Jastrow wave functions with both Hartree-Fock approximation (HF) and B3LYP density functional theory single-particle orbitals. We obtain equilibrium dissociation energies for the dimer of 5.02+/-0.18 kcal mol(-1) (HF orbitals) and 5.21+/-0.18 kcal mol(-1) (B3LYP orbitals), in good agreement with the coupled cluster results.

Quantum Monte Carlo study of the Ne atom and the Ne+ ion.

We report all-electron and pseudopotential calculations of the ground-state energies of the neutral Ne atom and the Ne(+) ion using the variational and diffusion quantum Monte Carlo (DMC) methods. We investigate different levels of Slater-Jastrow trial wave function: (i) using Hartree-Fock orbitals, (ii) using orbitals optimized within a Monte Carlo procedure in the presence of a Jastrow factor, and (iii) including backflow correlations in the wave function. Small reductions in the total energy are obtained by optimizing the orbitals, while more significant reductions are obtained by incorporating backflow correlations. We study the finite-time-step and fixed-node biases in the DMC energy and show that there is a strong tendency for these errors to cancel when the first ionization potential (IP) is calculated. DMC gives highly accurate values for the IP of Ne at all the levels of trial wave function that we have considered.

Quantum Monte Carlo calculations of the dissociation energies of three-electron hemibonded radical cationic dimers.

We report variational and diffusion quantum Monte Carlo (VMC and DMC) calculations of the dissociation energies of the three-electron hemibonded radical cationic dimers of He, NH3, H2O, HF, and Ne. These systems are particularly difficult for standard density-functional methods such as the local-density approximation and the generalized gradient approximation. We have performed both all-electron (AE) and pseudopotential (PP) calculations using Slater-Jastrow wave functions with Hartree-Fock single-particle orbitals. Our results are in good agreement with coupled-cluster CCSD(T) calculations. We have also studied the relative stability of the hemibonded and hydrogen-bonded water radical dimer isomers. Our calculations indicate that the latter isomer is more stable, in agreement with post-Hartree-Fock methods. The excellent agreement between our AE and PP results demonstrates the high quality of the PPs used within our VMC and DMC calculations.

Inhomogeneous backflow transformations in quantum Monte Carlo calculations.

An inhomogeneous backflow transformation for many-particle wave functions is presented and applied to electrons in atoms, molecules, and solids. We report variational and diffusion quantum Monte Carlo (VMC and DMC) energies for various systems and study the computational cost of using backflow wave functions. We find that inhomogeneous backflow transformations can provide a substantial increase in the amount of correlation energy retrieved within VMC and DMC calculations. The backflow transformations significantly improve the wave functions and their nodal surfaces.

All-electron quantum Monte Carlo calculations for the noble gas atoms He to Xe.

We report all-electron variational and diffusion quantum Monte Carlo (VMC and DMC) calculations for the noble gas atoms He, Ne, Ar, Kr, and Xe. The calculations were performed using Slater-Jastrow wave functions with Hartree-Fock single-particle orbitals. The quality of both the optimized Jastrow factors and the nodal surfaces of the wave functions declines with increasing atomic number Z, but the DMC calculations are tractable and well behaved in all cases. We discuss the scaling of the computational cost of DMC calculations with Z.

Scheme for adding electron-nucleus cusps to Gaussian orbitals.

A simple scheme is described for introducing the correct cusps at nuclei into orbitals obtained from Gaussian basis set electronic structure calculations. The scheme is tested with all-electron variational quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) methods for the Ne atom, the H2 molecule, and 55 molecules from a standard benchmark set. It greatly reduces the variance of the local energy in all cases and slightly improves the variational energy. This scheme yields a general improvement in the efficiency of all-electron VMC and DMC calculations using Gaussian basis sets.

Interpretation of Hund's multiplicity rule for the carbon atom.

Hund's multiplicity rule is investigated for the carbon atom using quantum Monte Carlo methods. Our calculations give an accurate account of electronic correlation and obey the virial theorem to high accuracy. This allows us to obtain accurate values for each of the energy terms and therefore to give a convincing explanation of the mechanism by which Hund's rule operates in carbon. We find that the energy gain in the triplet with respect to the singlet state is due to the greater electron-nucleus attraction in the higher spin state, in accordance with Hartree-Fock calculations and studies including correlation. The method used here can easily be extended to heavier atoms.