Compact and Flexible Basis Functions for Quantum Monte Carlo Calculations

Molecular calculations in quantum Monte Carlo frequently employ a mixed basis consisting of contracted and primitive Gaussian functions. While standard basis sets of varying size and accuracy are available in the literature, we demonstrate that reoptimizing the primitive function exponents within quantum Monte Carlo yields more compact basis sets for a given accuracy. Particularly large gains are achieved for highly excited states. For calculations requiring non-diverging pseudopotentials, we introduce Gauss-Slater basis functions that behave as Gaussians at short distances and Slaters at long distances. These basis functions further improve the energy and fluctuations of the local energy for a given basis size. Gains achieved by exponent optimization and Gauss-Slater basis use are exemplified by calculations for the ground state of carbon, the lowest lying excited states of carbon with $^5S^o$, $^3P^o$, $^1D^o$, $^3F^o$ symmetries, carbon dimer, and naphthalene. Basis size reduction enables quantum Monte Carlo treatment of larger molecules at high accuracy.Comment: 8 Pages, 2 Figures, 9 Table

Approaching Chemical Accuracy with Quantum Monte Carlo

A quantum Monte Carlo study of the atomization energies for the G2 set of molecules is presented. Basis size dependence of diffusion Monte Carlo atomization energies is studied with a single determinant Slater-Jastrow trial wavefunction formed from Hartree-Fock orbitals. With the largest basis set, the mean absolute deviation from experimental atomization energies for the G2 set is 3.0 kcal/mol. Optimizing the orbitals within variational Monte Carlo improves the agreement between diffusion Monte Carlo and experiment, reducing the mean absolute deviation to 2.1 kcal/mol. Moving beyond a single determinant Slater-Jastrow trial wavefunction, diffusion Monte Carlo with a small complete active space Slater-Jastrow trial wavefunction results in near chemical accuracy. In this case, the mean absolute deviation from experimental atomization energies is 1.2 kcal/mol. It is shown from calculations on systems containing phosphorus that the accuracy can be further improved by employing a larger active space.Comment: 6 pages, 5 figure

Semistochastic Projector Monte Carlo Method

We introduce a semistochastic implementation of the power method to compute, for very large matrices, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix multiplication is partially implemented numerically exactly and partially with respect to expectation values only. Compared to a fully stochastic method, the semistochastic approach significantly reduces the computational time required to obtain the eigenvalue to a specified statistical uncertainty. This is demonstrated by the application of the semistochastic quantum Monte Carlo method to systems with a sign problem: the fermion Hubbard model and the carbon dimer.Comment: 5 pages, 5 figure

Introduction to the variational and diffusion Monte Carlo methods

We provide a pedagogical introduction to the two main variants of real-space quantum Monte Carlo methods for electronic-structure calculations: variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). Assuming no prior knowledge on the subject, we review in depth the Metropolis-Hastings algorithm used in VMC for sampling the square of an approximate wave function, discussing details important for applications to electronic systems. We also review in detail the more sophisticated DMC algorithm within the fixed-node approximation, introduced to avoid the infamous Fermionic sign problem, which allows one to sample a more accurate approximation to the ground-state wave function. Throughout this review, we discuss the statistical methods used for evaluating expectation values and statistical uncertainties. In particular, we show how to estimate nonlinear functions of expectation values and their statistical uncertainties.Comment: Advances in Quantum Chemistry, 2015, Electron Correlation in Molecules -- ab initio Beyond Gaussian Quantum Chemistry, pp.000

Compositional Heterogeneity In Biologically Relevant Membrane Models

Lateral organization of the cellular plasma membrane promotes biological function by permitting regulation of cellular processes. Evidence now supports the hypothesis that the coexistence of a variety of lipid molecules, with different melting temperatures and acyl chain lengths, contribute to this lateral organization by forming lipid rafts, nanoscale domains with distinct biophysical properties. Phase separation into immiscible micron-sized domains is readily observed in model membranes, chemically simplified lipid mixtures that have been studied under equilibrium conditions to understand how composition and temperature affect domain properties. As lipid domains have yet to be imaged directly in live resting cells, the relevance of model membranes is uncertain. We have focused on characterizing the biologically relevant outer leaflet membrane model brain sphingomyelin (bSM)/1-palmitoyl-2-oleoyl-sn-glycero-3phosphocholine (POPC)/cholesterol (Chol) in which nanoscale domains challenge conventional imaging techniques. We have determined the temperature-dependent ternary phase diagrams for bSM/POPC/Chol and bSM/1,2-dioleoyl-sn-glycero-3phosphocholine (DOPC)/Chol using Förster resonance energy transfer (FRET) and differential scanning calorimetry, and we have confirmed the biologically relevant liquid-disordered (Ld) and liquid-ordered (Lo) coexistence region using electron spin resonance spectroscopy. We have determined that the size of coexisting Ld+Lo domains in bSM/POPC/Chol is 2-6 nm radius using FRET and small-angle neutron scattering. Ultimately, this careful characterization of model membrane will serve as a starting point for investigating the influence of peptides on domain size and other biophysical properties. Rafts can be stabilized to form larger platforms through protein-protein and protein-lipid interactions. Understanding how these domains form, grow, and stabilize in model systems is a first step toward elucidating their roles in important membrane-mediated processes

Quantum Monte Carlo Developments For Discrete And Continuous Spaces

This thesis details four research projects related to zero temperature quantum Monte Carlo. Chapters 2-4 focus on continuum quantum Monte Carlo, and specifically its application to molecular systems; whereas Chapter 5 focuses on quantum Monte Carlo in a discrete space. Chapter 2 focuses on improving upon the single-particle basis functions employed in quantum Monte Carlo calculations for molecular systems. For calculations requiring non-diverging pseudopotentials, a class of functions is introduced that is capable of producing the short- and long-range asymptotic behavior of the exact wavefunction. It is demonstrated that this form of basis function produces superior accuracy and efficiency when compared to the basis sets typically employed in quantum Monte Carlo. Although the basis functions introduced in Chapter 2 are capable of producing superior results, it is necessary that the parameters of the functional form are near-optimal for the full potential of the functions to be realized. Chapter 3 introduces a simple yet general method for constructing basis sets of a desired functional form appropriate for molecular electronic structure calculations. A standard basis set is created for each of the elements from hydrogen to argon. Chapter 4 explores the effect of different aspects of the trial wavefunction on the accuracy of quantum Monte Carlo. By systematically testing the effect of the basis size, orbital quality, and determinant expansion quality, this work offers guidance to quantum Monte Carlo practitioners for achieving results to within chemical accuracy of experiment. In Chapter 5, semistochastic projection, a hybrid of deterministic and stochastic projection, is introduced for finding the dominant eigenvalue and eigenvector of a matrix. This method, like stochastic projection, is applicable to matrices well beyond the size that can be handled by deterministic methods. Semistochastic projection improves over stochastic projection by significantly reducing the computational time required to obtain the eigenvalue within a specified statistical uncertainty. After the semistochastic projection method is introduced, it is applied to determine the ground state energy of the Hamiltonian in a discrete basis. This special case of semistochastic projection, dubbed semistochastic quantum Monte Carlo, is shown to be orders of magnitude more efficient than stochastic quantum Monte Carlo

Using small-angle neutron scattering to detect nanoscopic lipid domains

The cell plasma membrane is a complex system, which is thought to be capable of exhibiting non-random lateral organization. Studies of live cells and model membranes have yielded mechanisms responsible for the formation, growth, and maintenance of nanoscopic heterogeneities, although the existence and mechanisms that give rise to these heterogeneities remain controversial. Small-angle neutron scattering (SANS) is a tool ideally suited to interrogate lateral heterogeneity in model membranes, primarily due to its unique spatial resolution (i.e., 3c5-100 nm) and its ability to resolve structure with minimal perturbation to the membrane. In this review we examine several methods used to analyze the SANS signal arising from freely suspended unilamellar vesicles containing lateral heterogeneity. Specifically, we discuss an analytical model for a single, round domain on a spherical vesicle. We then discuss a numerical method that uses Monte Carlo simulation to describe systems with multiple domains and/or more complicated morphologies. Also discussed are several model-independent approaches that are sensitive to membrane heterogeneity. The review concludes with several recent applications of SANS to the study of membrane raft mixtures. \ua9 2013 Elsevier Ireland Ltd. All rights reserved.Peer reviewed: YesNRC publication: Ye

Phase behavior and domain size in sphingomyelin-containing lipid bilayers

Membrane raft size measurements are crucial to understanding the stability and functionality of rafts in cells. The challenge of accurately measuring raft size is evidenced by the disparate reports of domain sizes, which range from nanometers to microns for the ternary model membrane system sphingomyelin (SM)/1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC)/cholesterol (Chol). Using F\uf6rster resonance energy transfer (FRET) and differential scanning calorimetry (DSC), we established phase diagrams for porcine brain SM (bSM)/dioleoyl-sn-glycero-3-phosphocholine (DOPC)/Chol and bSM/POPC/Chol at 15 and 25 C. By combining two techniques with different spatial sensitivities, namely FRET and small-angle neutron scattering (SANS), we have significantly narrowed the uncertainty in domain size estimates for bSM/POPC/Chol mixtures. Compositional trends in FRET data revealed coexisting domains at 15 and 25 C for both mixtures, while SANS measurements detected no domain formation for bSM/POPC/Chol. Together these results indicate that liquid domains in bSM/POPC/Chol are between 2 and 7 nm in radius at 25 C: that is, domains must be on the order of the 2-6 nm F\uf6rster distance of the FRET probes, but smaller than the ~ 7 nm minimum cluster size detectable with SANS. However, for palmitoyl SM (PSM)/POPC/Chol at a similar composition, SANS detected coexisting liquid domains. This increase in domain size upon replacing the natural SM component (which consists of a mixture of chain lengths) with synthetic PSM, suggests a role for SM chain length in modulating raft size in vivo. \ua9 2012 Elsevier B.V. All rights reserved.Peer reviewed: YesNRC publication: Ye