Quantum Monte Carlo calculations on Be and LiH

Quantum Monte Carlo (QMC) methods, as recently developed for molecular systems, are applied to Be and LiH. The importance of the trial wavefunction, written as a product of a correlation factor and an orbital part, is emphasised. It is shown that significant improvements in the accuracy of the approach are achieved if multi-configuration wavefunctions are used in preference to self-consistent field wavefunctions. Various forms of the correlation factor are investigated.     

Energy-consistent pseudopotentials for quantum Monte Carlo calculations

The authors present scalar-relativistic energy-consistent Hartree-Fock pseudopotentials for the main-group elements. The pseudopotentials do not exhibit a singularity at the nucleus and are therefore suitable for quantum Monte Carlo (QMC) calculations. They demonstrate their transferability through extensive benchmark calculations of atomic excitation spectra as well as molecular properties. In particular, they compute the vibrational frequencies and binding energies of 26 first- and second-row diatomic molecules using post-Hartree-Fock methods, finding excellent agreement with the corresponding all-electron values. They also show their pseudopotentials give superior accuracy than other existing pseudopotentials constructed specifically for QMC. Finally, valence basis sets of different sizes (VnZ with n=D,T,Q,5 for first and second rows, and n=D,T for third to fifth rows) optimized for our pseudopotentials are also presented.     

Ground-state properties of LiH by reptation quantum Monte Carlo methods

We apply reptation quantum Monte Carlo to calculate one- and two-electron properties for ground-state LiH, including all tensor components for static polarizabilities and hyperpolarizabilities to fourth-order in the field. The importance sampling is performed with a large (QZ4P) STO basis set single determinant, directly obtained from commercial software, without incurring the overhead of optimizing many-parameter Jastrow-type functions of the inter-electronic and internuclear distances. We present formulas for the electrical response properties free from the finite-field approximation, which can be problematic for the purposes of stochastic estimation. The α, γ, A and C polarizability values are reasonably consistent with recent determinations reported in the literature, where they exist. A sum rule is obeyed for components of the B tensor, but B(zz,zz) as well as β(zzz) differ from what was reported in the literature.     

Dissociation energy of the water dimer from quantum Monte Carlo calculations

We report a study of the electronic dissociation energy of the water dimer using quantum Monte Carlo techniques. We have performed variational quantum Monte Carlo and diffusion quantum Monte Carlo (DMC) calculations of the electronic ground state of the water monomer and dimer using all-electron and pseudopotential approaches. We have used Slater-Jastrow trial wave functions with B3LYP type single-particle orbitals, into which we have incorporated backflow correlations. When backflow correlations are introduced, the total energy of the water monomer decreases by about 4-5 mhartree, yielding a DMC energy of -76.428 30(5) hartree, which is only 10 mhartree above the experimental value. In our pseudopotential DMC calculations, we have compared the total energies of the water monomer and dimer obtained using the locality approximation with those from the variational scheme recently proposed by Casula [Phys. Rev. B 74, 161102(R) (2006)]. The time step errors in the Casula scheme are larger, and the extrapolation of the energy to zero time step always lies above the result obtained with the locality approximation. However, the errors cancel when energy differences are taken, yielding electronic dissociation energies within error bars of each other. The dissociation energies obtained in our various all-electron and pseudopotential calculations range between 5.03(7) and 5.47(9) kcalmol and are in good agreement with experiment. Our calculations give monomer dipole moments which range between 1.897(2) and 1.909(4) D and dimer dipole moments which range between 2.628(6) and 2.672(5) D.     

Jastrow correlated and quantum Monte Carlo calculations for the low-lying states of the carbon atom

Different computational methods are employed to calculate excitation energies of the carbon atom. Explicitly correlated wave functions have been obtained in a Variational Monte Carlo calculation. Fixed node Diffusion Monte Carlo calculations for the lowest energy excited states of a given symmetry are reported. A systematic and quantitative analysis of the performance of the different schemes in the calculation of the excitation energy of up to 27 excited states of the carbon atom is carried out. The quality of the different methods have been studied in terms of the deviation with respect to the experimental excitation energies. A good agreement with the experimental values has been reached.     

Maximum probability domains from Quantum Monte Carlo calculations

Although it would be tempting to associate the Lewis structures to the maxima of the squared wave function |Psi|2, we prefer in this paper the use of domains of the three-dimensional space, which maximize the probability of containing opposite-spin electron pairs. We find for simple systems (CH4, H2O, Ne, N2, C2H2) domains comparable to those obtained with the electron localization function (ELF) or by localizing molecular orbitals. The different domains we define can overlap, and this gives an interesting physical picture of the floppiness of CH5+ and of the symmetric hydrogen bond in FHF-. The presence of multiple solutions has an analogy with resonant structures, as shown in the trans-bent structure of Si2H2. Correlated wave functions were used (MCSCF or Slater-Jastrow) in the Variational Quantum Monte Carlo framework.     

Prospects for release-node quantum Monte Carlo

We perform release-node quantum Monte Carlo simulations on the first row diatomic molecules in order to assess how accurately their ground-state energies can be obtained. An analysis of the fermion-boson energy difference is shown to be strongly dependent on the nuclear charge, Z, which in turn determines the growth of variance of the release-node energy. It is possible to use maximum entropy analysis to extrapolate to ground-state energies only for the low Z elements. For the higher Z dimers beyond boron, the error growth is too large to allow accurate data for long enough imaginary times. Within the limit of our statistics we were able to estimate, in atomic units, the ground-state energy of Li(2) (-14.9947(1)), Be(2) (-29.3367(7)), and B(2)(-49.410(2)).     

Binding of hydrogen on benzene, coronene, and graphene from quantum Monte Carlo calculations

Quantum Monte Carlo calculations with the diffusion Monte Carlo (DMC) method have been used to compute the binding energy curves of hydrogen on benzene, coronene, and graphene. The DMC results on benzene agree with both M?ller-Plessett second order perturbation theory (MP2) and coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] calculations, giving an adsorption energy of ～25 meV. For coronene, DMC agrees well with MP2, giving an adsorption energy of ～40 meV. For physisorbed hydrogen on graphene, DMC predicts a very small adsorption energy of only 5 ± 5 meV. Density functional theory (DFT) calculations with various exchange-correlation functionals, including van der Waals corrected functionals, predict a wide range of binding energies on all three systems. The present DMC results are a step toward filling the gap in accurate benchmark data on weakly bound systems. These results can help us to understand the performance of current DFT based methods, and may aid in the development of improved approaches.     

Interaction between LiH molecule and Li atom from state-of-the-art electronic structure calculations

State-of-the-art ab initio techniques have been applied to compute the potential energy surface for the lithium atom interacting with the lithium hydride molecule in the Born-Oppenheimer approximation. The interaction potential was obtained using a combination of the explicitly correlated unrestricted coupled-cluster method with single, double, and noniterative triple excitations [UCCSD(T)-F12] for the core-core and core-valence correlation and full configuration interaction for the valence-valence correlation. The potential energy surface has a global minimum 8743 cm(-1) deep if the Li-H bond length is held fixed at the monomer equilibrium distance or 8825 cm(-1) deep if it is allowed to vary. In order to evaluate the performance of the conventional CCSD(T) approach, calculations were carried out using correlation-consistent polarized valence X-tuple-zeta basis sets, with X ranging from 2 to 5, and a very large set of bond functions. Using simple two-point extrapolations based on the single-power laws X(-2) and X(-3) for the orbital basis sets, we were able to reproduce the CCSD(T)-F12 results for the characteristic points of the potential with an error of 0.49% at worst. The contribution beyond the CCSD(T)-F12 model, obtained from full configuration interaction calculations for the valence-valence correlation, was shown to be very small, and the error bars on the potential were estimated. At linear LiH-Li geometries, the ground-state potential shows an avoided crossing with an ion-pair potential. The energy difference between the ground-state and excited-state potentials at the avoided crossing is only 94 cm(-1). Using both adiabatic and diabatic pictures, we analyze the interaction between the two potential energy surfaces and its possible impact on the collisional dynamics. When the Li-H bond is allowed to vary, a seam of conical intersections appears at C(2v) geometries. At the linear LiH-Li geometry, the conical intersection is at a Li-H distance which is only slightly larger than the monomer equilibrium distance, but for nonlinear geometries it quickly shifts to Li-H distances that are well outside the classical turning points of the ground-state potential of LiH. This suggests that the conical intersection will have little impact on the dynamics of Li-LiH collisions at ultralow temperatures. Finally, the reaction channels for the exchange and insertion reactions are also analyzed and found to be unimportant for the dynamics.     

An optimized initialization algorithm to ensure accuracy in quantum Monte Carlo calculations

Quantum Monte Carlo (QMC) calculations require the generation of random electronic configurations with respect to a desired probability density, usually the square of the magnitude of the wavefunction. In most cases, the Metropolis algorithm is used to generate a sequence of configurations in a Markov chain. This method has an inherent equilibration phase, during which the configurations are not representative of the desired density and must be discarded. If statistics are gathered before the walkers have equilibrated, contamination by nonequilibrated configurations can greatly reduce the accuracy of the results. Because separate Markov chains must be equilibrated for the walkers on each processor, the use of a long equilibration phase has a profoundly detrimental effect on the efficiency of large parallel calculations. The stratified atomic walker initialization (STRAW) shortens the equilibration phase of QMC calculations by generating statistically independent electronic configurations in regions of high probability density. This ensures the accuracy of calculations by avoiding contamination by nonequilibrated configurations. Shortening the length of the equilibration phase also results in significant improvements in the efficiency of parallel calculations, which reduces the total computational run time. For example, using STRAW rather than a standard initialization method in 512 processor calculations reduces the amount of time needed to calculate the energy expectation value of a trial function for a molecule of the energetic material RDX to within 0.01 au by 33%.     

Auxiliary-field quantum Monte Carlo calculations of molecular systems with a Gaussian basis

We extend the recently introduced phaseless auxiliary-field quantum Monte Carlo (QMC) approach to any single-particle basis and apply it to molecular systems with Gaussian basis sets. QMC methods in general scale favorably with the system size as a low power. A QMC approach with auxiliary fields, in principle, allows an exact solution of the Schrodinger equation in the chosen basis. However, the well-known sign/phase problem causes the statistical noise to increase exponentially. The phaseless method controls this problem by constraining the paths in the auxiliary-field path integrals with an approximate phase condition that depends on a trial wave function. In the present calculations, the trial wave function is a single Slater determinant from a Hartree-Fock calculation. The calculated all-electron total energies show typical systematic errors of no more than a few millihartrees compared to exact results. At equilibrium geometries in the molecules we studied, this accuracy is roughly comparable to that of coupled cluster with single and double excitations and with noniterative triples [CCSD(T)]. For stretched bonds in H(2)O, our method exhibits a better overall accuracy and a more uniform behavior than CCSD(T).     

Reliable radical stabilization energies from diffusion Monte Carlo calculations

We assess the performance of variational (VMC) and diffusion (DMC) quantum Monte Carlo methods for calculating the radical stabilization energies of a set of 43 carbon-centered radical species. Even using simple single-determinant trial wavefunctions, both methods perform exceptionally well, with mean absolute deviations from reference values well under the chemical accuracy standard of 1 kcal/mol. In addition, the use of DMC results in a highly concentrated spread of errors, with all 43 results within chemical accuracy at the 95% confidence level. These results indicate that DMC is an extremely reliable method for calculating radical stabilization energies and could be used as a benchmark method for larger systems in future.     

Efficient protocol for quantum Monte Carlo calculations of hydrogen abstraction barriers: Application to methanol

Accurate calculation of hydrogen abstraction reaction barriers is a challenging problem, often requiring high level quantum chemistry methods that scale poorly with system size. Quantum Monte Carlo (QMC) methods provide an alternative approach that exhibit much better scaling, but these methods are still computationally expensive. We describe approaches that can significantly reduce the cost of QMC calculations of barrier heights, using the hydrogen abstraction of methanol by a hydrogen atom as an illustrative example. By analysing the combined influence of trial wavefunctions and pseudopotential quadrature settings on the barrier heights, variance, and time‐step errors, we devise a simple protocol that minimizes the cost of the QMC calculations while retaining accuracy comparable to large‐basis coupled cluster theory. We demonstrate that this protocol is transferable to other hydrogen abstraction reactions.     

Electronic quantum Monte Carlo calculations of atomic forces, vibrations, and anharmonicities

Atomic forces are calculated for first-row monohydrides and carbon monoxide within electronic quantum Monte Carlo (QMC). Accurate and efficient forces are achieved by using an improved method for moving variational parameters in variational QMC. Newton's method with singular value decomposition (SVD) is combined with steepest-descent (SD) updates along directions rejected by the SVD, after initial SD steps. Dissociation energies in variational and diffusion QMC agree well with the experiment. The atomic forces agree quantitatively with potential-energy surfaces, demonstrating the accuracy of this force procedure. The harmonic vibrational frequencies and anharmonicity constants, derived from the QMC energies and atomic forces, also agree well with the experimental values.     

Monte Carlo methods for small molecule high-throughput experimentation

By analogy with Monte Carlo algorithms, we propose new strategies for design and redesign of small molecule libraries in high-throughput experimentation, or combinatorial chemistry. Several Monte Carlo methods are examined, including Metropolis, three types of biased schemes, and composite moves that include swapping or parallel tempering. Among them, the biased Monte Carlo schemes exhibit particularly high efficiency in locating optimal compounds. The Monte Carlo strategies are compared to a genetic algorithm approach. Although the best compounds identified by the genetic algorithm are comparable to those from the better Monte Carlo schemes, the diversity of favorable compounds identified is reduced by roughly 60%.     

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.     

Split orbitals for the LiH molecule

The four-electron problem in LiH has been treated by use of Slater type orbitals for the 1s electrons on the Li atom and split molecular orbitals for the two valence electrons. Some properties of the two dimensional spin space present in the case of four different space-functions are discussed. Total electronic energies and electric dipole moments have been calculated.

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Zusammenfassung LiH wird mit Slaterfunktionen für die 1s-Elektronen von Li und SMO's für die beiden Valenzelektronen behandelt. Einige Eigenschaften des zweidimensionalen Spinraums werden diskutiert. Elektronenenergien und Dipolmomente werden berechnet.

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Résumé Nous avons traité le problème à quatre électrons de LiH en utilisant des orbitales de Slater pour les électrons 1s de Li et des orbitales moléculaires distinctes pour les deux électrons de valence. Certaines propriétés de l'espace de spin à 2 dimensions existant dans le cas de quatre fonctions d'espace différentes sont discutées. On a calculé les énergies électroniques totales et les moments dipolaires.

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The work reported in this paper has been sponsored in part by the King Gustaf VI Adolf's 70-Years Fund for Swedish Culture, Knut and Alice Wallenberg's Foundation, and in part by the Aerospace Research Laboratories, OAR, through the European Office of Aerospace Research (OAR), United States Air Force.     

Quantum Monte Carlo calculations for ground and excited states

A brief overview of the diffusion quantum Monte Carlo method is given. We illustrate the application to ground‐state calculations by a study of the relative stability of carbon clusters near the crossover to fullerene stability, thereby determining the smallest stable fullerene. The application to excited states is illustrated via a study of excitonic states in small hydrogenated silicon clusters. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001