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.

Shape and energy consistent pseudopotentials for correlated electron systems.

A method is developed for generating pseudopotentials for use in correlated-electron calculations. The paradigms of shape and energy consistency are combined and defined in terms of correlated-electron wave-functions. The resulting energy consistent correlated electron pseudopotentials (eCEPPs) are constructed for H, Li-F, Sc-Fe, and Cu. Their accuracy is quantified by comparing the relaxed molecular geometries and dissociation energies which they provide with all electron results, with all quantities evaluated using coupled cluster singles, doubles, and triples calculations. Errors inherent in the pseudopotentials are also compared with those arising from a number of approximations commonly used with pseudopotentials. The eCEPPs provide a significant improvement in optimised geometries and dissociation energies for small molecules, with errors for the latter being an order-of-magnitude smaller than for Hartree-Fock-based pseudopotentials available in the literature. Gaussian basis sets are optimised for use with these pseudopotentials.

Correlated electron pseudopotentials for 3d-transition metals.

A recently published correlated electron pseudopotentials (CEPPs) method has been adapted for application to the 3d-transition metals, and to include relativistic effects. New CEPPs are reported for the atoms Sc - Fe, constructed from atomic quantum chemical calculations that include an accurate description of correlated electrons. Dissociation energies, molecular geometries, and zero-point vibrational energies of small molecules are compared with all electron results, with all quantities evaluated using coupled cluster singles doubles and triples calculations. The CEPPs give better results in the correlated-electron calculations than Hartree-Fock-based pseudopotentials available in the literature.

Comparison of Smooth Hartree-Fock Pseudopotentials.

The accuracy of two widely used scalar relativistic Hartree-Fock pseudopotentials, the Trail-Needs-Dirac-Fock (TNDF) and the Burkatzki-Filippi-Dolg (BFD) pseudopotentials, is assessed. The performance of the pseudopotentials is tested for a chemically representative set of 34 first-row molecules. All comparisons are made at the Hartree-Fock level of theory, and both sets of pseudopotentials give good results. The all-electron equilibrium geometries, molecular dissociation energies, and zero-point vibrational energies are reproduced a little more accurately by the TNDF pseudopotentials than the BFD ones.

Pseudopotentials for correlated electron systems.

A scheme is developed for creating pseudopotentials for use in correlated-electron calculations. Pseudopotentials for the light elements H, Li, Be, B, C, N, O, and F, are reported, based on data from high-level quantum chemical calculations. Results obtained with these correlated electron pseudopotentials (CEPPs) are compared with data for atomic energy levels and the dissociation energies, molecular geometries, and zero-point vibrational energies of small molecules obtained from coupled cluster single double triple calculations with large basis sets. The CEPPs give better results in correlated-electron calculations than Hartree-Fock-based pseudopotentials available in the literature.

Optimum and efficient sampling for variational quantum Monte Carlo.

Quantum mechanics for many-body systems may be reduced to the evaluation of integrals in 3N dimensions using Monte Carlo, providing the Quantum Monte Carlo ab initio methods. Here we limit ourselves to expectation values for trial wave functions, that is to variational quantum Monte Carlo. Almost all previous implementations employ samples distributed as the physical probability density of the trial wave function, and assume the central limit theorem to be valid. In this paper we provide an analysis of random error in estimation and optimization that leads naturally to new sampling strategies with improved computational and statistical properties. A rigorous lower limit to the random error is derived, and an efficient sampling strategy presented that significantly increases computational efficiency. In addition the infinite variance heavy tailed random errors of optimum parameters in conventional methods are replaced with a Normal random error, strengthening the theoretical basis of optimization. The method is applied to a number of first row systems and compared with previously published results.

Methods for calculating forces within quantum Monte Carlo simulations.

Atomic force calculations within the variational and diffusion quantum Monte Carlo methods are described. The advantages of calculating diffusion quantum Monte Carlo forces with the 'pure' rather than the 'mixed' probability distribution are discussed. An accurate and practical method for calculating forces using the pure distribution is presented and tested for the SiH molecule. The statistics of force estimators are explored and violations of the central limit theorem are found in some cases.

Energy derivatives in quantum Monte Carlo involving the zero-variance property.

We give an exact expression for the nth derivative of the expectation value of the energy that satisfies the zero-variance (ZV) principle when the wave function and its first n derivatives are exact. The ZV principle was previously applied to the first energy derivative ("force") within the variational Monte Carlo and mixed-estimator diffusion Monte Carlo methods. We present a new expression for the force in pure-estimator diffusion Monte Carlo that satisfies the ZV principle and can be evaluated much more efficiently than previous expressions while maintaining comparable accuracy. This expression is the sum of a pure expectation value and a variational expectation value, which separately satisfy the ZV principle. The bias in this force estimator is second order in the deviation of the trial wave function from the diffusion Monte Carlo wave function. Results for small molecules demonstrate the accuracy of the method and its statistical efficiency.

Spectroscopic data for the LiH molecule from pseudopotential quantum Monte Carlo calculations.

Quantum Monte Carlo and quantum chemistry techniques are used to investigate pseudopotential models of the lithium hydride (LiH) molecule. Interatomic potentials are calculated and tested by comparing with the experimental spectroscopic constants and well depth. Two recently developed pseudopotentials are tested, and the effects of introducing a Li core polarization potential are investigated. The calculations are sufficiently accurate to isolate the errors from the pseudopotentials and core polarization potential. Core-valence correlation and core relaxation are found to be important in determining the interatomic potential.

Alternative sampling for variational quantum Monte Carlo.

Expectation values of physical quantities may accurately be obtained by the evaluation of integrals within many-body quantum mechanics, and these multidimensional integrals may be estimated using Monte Carlo methods. In a previous publication it has been shown that for the simplest, most commonly applied strategy in continuum quantum Monte Carlo, the random error in the resulting estimates is not well controlled. At best the central limit theorem is valid in its weakest form, and at worst it is invalid and replaced by an alternative generalized central limit theorem and non-normal random error. In both cases the random error is not controlled. Here we consider a new "residual sampling strategy" that reintroduces the central limit theorem in its strongest form, and provides full control of the random error in estimates. Estimates of the total energy and the variance of the local energy within variational Monte Carlo are considered in detail, and the approach presented may be generalized to expectation values of other operators, and to other variants of the quantum Monte Carlo method.

Heavy-tailed random error in quantum Monte Carlo.

The combination of continuum many-body quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the central limit theorem to hold true. In what follows it is demonstrated that, for the most popular implementation of the variational Monte Carlo method, the central limit theorem has limited validity, or is invalid and must be replaced by a generalized central limit theorem. Estimates of the total energy and the variance of the local energy are examined in detail, and shown to exhibit uncontrolled statistical errors through an explicit derivation of the distribution of the random error. Several examples are given of estimated quantities for which the central limit theorem is not valid. The approach used is generally applicable to characterizing the random error of estimates, and to quantum Monte Carlo methods beyond variational Monte Carlo.

Energies of the first row atoms from quantum Monte Carlo.

All-electron variational and diffusion quantum Monte Carlo calculations of the ground state energies of the first row atoms (from Li to Ne) are reported. The authors use trial wave functions of four types: single-determinant Slater-Jastrow wave functions, multideterminant Slater-Jastrow wave functions, single-determinant Slater-Jastrow wave functions with backflow transformations, and multideterminant Slater-Jastrow wave functions with backflow transformations. At the diffusion quantum Monte Carlo level and using their multideterminant Slater-Jastrow wave functions with backflow transformations, they recover 99% or more of the correlation energies for Li, Be, B, C, N, and Ne, 97% for O, and 98% for F.

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.

Smooth relativistic Hartree-Fock pseudopotentials for H to Ba and Lu to Hg.

We report smooth relativistic Hartree-Fock pseudopotentials (also known as averaged relativistic effective potentials) and spin-orbit operators for the atoms H to Ba and Lu to Hg. We remove the unphysical extremely nonlocal behavior resulting from the exchange interaction in a controlled manner, and represent the resulting pseudopotentials in an analytic form suitable for use within standard quantum chemistry codes. These pseudopotentials are suitable for use within Hartree-Fock and correlated wave function methods, including diffusion quantum Monte Carlo calculations.

Norm-conserving Hartree-Fock pseudopotentials and their asymptotic behavior.

We investigate the properties of norm-conserving pseudopotentials (effective core potentials) generated by inversion of the Hartree-Fock equations. In particular, we investigate the asymptotic behavior as r-->infinity and find that such pseudopotentials are nonlocal over all space, apart from a few special cases such as H and He. Such extreme nonlocality leads to a lack of transferability and, within periodic boundary conditions, an undefined total energy. The extreme nonlocality must therefore be removed, and we argue that the best way to accomplish this is a minor relaxation of the norm-conservation condition. This is implemented, and pseudopotentials for the atoms H-Ar are constructed and tested.

Energy loss of atoms at metal surfaces due to electron-hole pair excitations: first-principles theory of "chemicurrents".

A method is presented for calculating electron-hole pair excitation due to an incident atom or molecule interacting with a metal surface. Energy loss is described using an ab initio approach that obtains a position-dependent friction coefficient for an adsorbate moving near a metal surface from a total energy pseudopotential calculation. A semiclassical forced oscillator model is constructed to describe excitation of the electron gas due to the incident molecule. This approach is applied to H and D atoms incident on a Cu(111) surface, and we obtain theoretical estimates of the "chemicurrents" measured by Nienhaus et al. [Phys. Rev. Lett. 82, 446 (1999)] for these atoms incident on the surface of a Schottky diode.