Stuart M. Rothstein

Rare gas interactions using an improved statistical method

New density functionals are employed to represent the correlation and exchange energies (per electron) in the calculation of rare gas interactions from the electron gas model. The correlation energy density functional is a rational function involving two parameters which were optimized to reproduce the correlation energy of He atom. Our results indicate that these parameters are ’’universal’’, i.e., they are accurate for all rare gas atoms. The exchange energy density functional is that recommended by Handler. Rare gas systems X2 and XY are investigated, where X and Y are He, Ne, Ar, and Kr. The results are a significant improvement over those available from competing electron gas models.

Reliable diffusion quantum Monte Carlo

By introducing perturbations of O(τ) type (where τ is the time step used in the simulation) to our diffusion quantum Monte Carlo algorithm we obtain a simulated energy which is reasonably constant over a wide range of τ values. Reliable estimation of the fixed‐node energy (τ=0 intercept) results, as extrapolation becomes more robust and a radical change in small τ behavior becomes less likely. We apply our techniques to the problem of estimating the ground‐state energy of LiH and H2O.

Diffusion quantum Monte Carlo calculation of nondifferential properties for atomic ground states

An algorithm to sample the exact (within the nodal error) ground state distribution to find nondifferential properties of multielectron systems is developed and applied to first-row atoms. Calculated properties are the distribution moments and the electronic density at the nucleus (expected value of the δ operator). For this purpose compact trial functions are developed and optimized, and a new estimator for the δ is formulated. A comparison is made with results of highly accurate post-Hartree-Fock calculations, thereby illuminating the nodal error in our estimates. In general, we obtain more accurate estimates for the distribution moments than those obtained previously using Monte Carlo methods, despite the relative crudeness of our trial functions. We confirm the literature values for the electron density at the nucleus for the lighter atoms (Li-C), but disagree with previous (Monte Carlo) estimates for the heavier ones (N-Ne).

Infinitesimal differential diffusion quantum Monte Carlo: diatomic molecular properties

We show how to estimate, for a given molecule, the first and higher derivatives of the expected value of an operator with respect to one or more physical parameters. This is done with high accuracy achieved by sampling to within a certain approximation from the exact electron distribution, compatible with the Hellmann–Feynman theorem. Finite difference approximations are avoided. The required derivatives of the unknown exact wave function are determined by averaging expressions involving only the total serial correlation of known quantities. The operator is not restricted to the case of the molecular Hamiltonian. This allows for computation of virtually all ground‐state properties of a molecule by a single, relatively trivial computer program. Our formulas are presented and applied in the context of a diatomic molecule (LiH), but they can be readily extended to polyatomics.

Molecular dynamics simulation study on the structural stabilities of polyglutamine peptides

It is known that Huntingtons disease patients commonly have glutamine (Q) repeat sequences longer than a critical length in the coding area of Huntingtin protein in their genes. As the polyglutamine (polyQ) region becomes longer than the critical length, the disease occurs and Huntingtin protein aggregates, both in vitro and in vivo, as suggested by experimental and clinical data. The determination of polyglutamine structure is thus very important for elucidation of the aggregation and disease mechanisms. Here, we perform molecular dynamics calculations on the stability of the structure based on the beta-helix structure suggested by Perutz et al. (2002) [Perutz, M.F., Finch, J.T., Berriman, J., Lesk, A., 2002. Amyloid fibers are water-filled nanotubes. Proc. Natl. Acad. Sci. USA 99, 5591]. We ensure that perfect hydrogen bonds are present between main chains of the beta-helix based on the previous studies, and perform simulations of stretches with 20, 25, 30, 37 and 40 glutamine residues (20Q, 25Q, 30Q, 37Q and 40Q) for the Perutz models with 18.5 and 20 residues per turn (one coil). Our results indicate that the structure becomes more stable with the increase of repeated number of Q, and there is a critical Q number of around 30, above which the structure of the Perutz model is kept stable. In contrast to previous studies, we started molecular dynamics simulations from conformations in which the hydrogen bonds are firmly formed between stacked main chains. This has rendered the initial beta-helix structures of polyQ much more stable for longer time, as compared to those proposed previously. Model calculations for the initial structures of polyQ dimer and tetramer have also been carried out to study a possible mechanism for aggregation.

Sampling the exact electron distribution by diffusion quantum Monte Carlo

By the accummulation of branching factors in diffusion quantum Monte Carlo (DQMC) and their use as statistical weights, instead of the standard deletion and replication of configurations, we can estimate the averages of (nondifferential) operators taken over the exact electron distribution. This requires only a trivial modification of existing DQMC codes. We illustrate our algorithm by computing ground‐state properties for H2 and LiH which are related to the interelectron distance. We also estimate the dipole moment of LiH.

New formula to estimate the ground state energy, E0

The one‐electron reduced local energy function, EL(1), is introduced which has the properties 〈EL〉=〈H〉, and 〈E2L〉 ⩽〈H2〉. It is suggested that the accuracy of EL reflects the local accuracy of an approximate wavefunction. From the properties of EL a modified form of Weinstein’s formula, Ew′, is defined which is bounded as follows: Ew⩽Ew′⩽〈H〉, where Ew is Weinstein’s lower bound formula for the ground state energy, E0. The estimate to E0 provided by Ew′ is not guaranteed as an upper or lower bound, but it provides a useful adjunct to 〈H〉 which does not require 〈H2〉. Applications to a variety of approximate wavefunctions for X 1Σ+g H2 and 1 1S He are presented.

Are properties derived from variance-optimized wave functions generally more accurate? Monte Carlo study of non-energy-related properties of H2, He, and LiH

It is commonly believed that variance-optimized wave functions yield “satisfactory” if not, in principle, better estimates of non-energy-related physical properties than their energy-optimized counterparts. We test this notion by calculating a number of ground-state physical properties using a variety of variance- and energy-optimized wave functions for He, H2, and LiH. We gauge the quality of the properties using as a “metric” the sum of absolute relative errors. Our results suggest that the energy-optimized wave functions consistently provide better estimates of non-energy-related properties than variance-optimized ones. We present qualitative arguments supporting these findings.

Quadratic accuracy diffusion Monte Carlo

Abstract We derive a Monte Carlo Green function with a quadratic time-step bias, and point out the importance of properly simulating the discontinuities of the drift function at the nuclei. We suggest that for small atoms and molecules, where the nodes in the trial function are well separated, our algorithm enables one to use large time steps, thus gaining in precision of the ground-energy estimate by dramatically reducing the serial correlation of consecutive iterations.

Infinitesimal differential diffusion quantum Monte Carlo study of CuH spectroscopic constants

We show how to extend the formalism of infinitesimal differential diffusion quantum Monte Carlo to the case of higher derivatives of the ground‐state energy of a molecule with respect to the molecular geometry for a large‐Z molecule. We propose a new all‐electron approach based on the idea of using different simulation time‐scales for different shells of the constituent atoms, viz., a substantially smaller time‐step for innermost shells, and relatively large time‐steps for the valence electrons. We avoid the approximations inherent in pseudopotential methods while at the same time, when used in conjunction with variational Monte Carlo, the sampling is done on a firm theoretical basis. We use CuH as an example, obtaining properties with an accuracy on par with those from SDCI, despite our using a crude single‐determinant wave function with only two adjustable parameters.