258 research outputs found
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Implementation and benchmark of a long-range corrected functional in the density functional based tight-binding method
Bridging the gap between first principles methods and empirical schemes, the
density functional based tight-binding method (DFTB) has become a versatile
tool in predictive atomistic simulations over the past years. One of the major
restrictions of this method is the limitation to local or gradient corrected
exchange-correlation functionals. This excludes the important class of hybrid
or long-range corrected functionals, which are advantageous in thermochemistry,
as well as in the computation of vibrational, photoelectron and optical
spectra. The present work provides a detailed account of the implementation of
DFTB for a long-range corrected functional in generalized Kohn-Sham theory. We
apply the method to a set of organic molecules and compare ionization
potentials and electron affinities with the original DFTB method and higher
level theory. The new scheme cures the significant overpolarization in electric
fields found for local DFTB, which parallels the functional dependence in first
principles density functional theory (DFT). At the same time the computational
savings with respect to full DFT calculations are not compromised as evidenced
by numerical benchmark data
Fitting Effective Diffusion Models to Data Associated with a "Glassy Potential": Estimation, Classical Inference Procedures and Some Heuristics
A variety of researchers have successfully obtained the parameters of low
dimensional diffusion models using the data that comes out of atomistic
simulations. This naturally raises a variety of questions about efficient
estimation, goodness-of-fit tests, and confidence interval estimation. The
first part of this article uses maximum likelihood estimation to obtain the
parameters of a diffusion model from a scalar time series. I address numerical
issues associated with attempting to realize asymptotic statistics results with
moderate sample sizes in the presence of exact and approximated transition
densities. Approximate transition densities are used because the analytic
solution of a transition density associated with a parametric diffusion model
is often unknown.I am primarily interested in how well the deterministic
transition density expansions of Ait-Sahalia capture the curvature of the
transition density in (idealized) situations that occur when one carries out
simulations in the presence of a "glassy" interaction potential. Accurate
approximation of the curvature of the transition density is desirable because
it can be used to quantify the goodness-of-fit of the model and to calculate
asymptotic confidence intervals of the estimated parameters. The second part of
this paper contributes a heuristic estimation technique for approximating a
nonlinear diffusion model. A "global" nonlinear model is obtained by taking a
batch of time series and applying simple local models to portions of the data.
I demonstrate the technique on a diffusion model with a known transition
density and on data generated by the Stochastic Simulation Algorithm.Comment: 30 pages 10 figures Submitted to SIAM MMS (typos removed and slightly
shortened
All-Electron Ground-State and Time-Dependent Density Functional Theory: Fast Algorithms and Better Approximations
Density functional theory (DFT), in its ground-state as well as time-dependent variant, have enjoyed incredible success in predicting a range of physical, chemical and materials properties. Although a formally exact theory, in practice DFT entails two key approximations---(a) the pseudopotential approximation, and (b) the exchange-correlation approximation. The pseudopotential approximation models the effect of sharply varying core-electrons along with the singular nuclear potential into a smooth effective potential called the pseudopotential, thereby mitigating the need for a highly refined spatial discretization. The exchange-correlation approximation, on the other hand, models the quantum many-electron interactions into an effective mean-field of the electron density (), and, remains an unavoidable approximation in DFT.
The overarching goal of this dissertation work is ---(a) to develop efficient numerical methods for all-electron DFT and TDDFT calculations which can dispense with the pseudopotentials without incurring huge computational cost, and (b) to provide key insights into the nature of the exchange-correlation potential that can later constitute a route to systematic improvement of the exchange-correlation approximation through machine learning algorithms (i.e., which can learn these functionals using training data from wavefunction-based methods). This, in turn, involves---(a) obtaining training data mapping to , and (b) using machine learning on the training data ( maps) to obtain the functional form of , with conformity to the known exact conditions.
The research efforts, in this thesis, constitute significant steps towards both the aforementioned goals. To begin with, we have developed a computationally efficient approach to perform large-scale all-electron DFT calculations by augmenting the classical finite element basis with compactly supported atom-centered numerical basis functions. We term the resultant basis as enriched finite element basis. Our numerical investigations show an extraordinary -fold and -fold speedup afforded by the enriched finite element basis over classical finite element and Gaussian basis, respectively. In the case of TDDFT, we have developed an efficient emph{a priori} spatio-temporal discretization scheme guided by rigorous error estimates based on the time-dependent Kohn-Sham equations. Our numerical studies show a staggering -fold speedup afforded by higher-order finite elements over linear finite elements. Furthermore, for pseudopotential calculations, our approach achieve a -fold speedup over finite difference based approaches. The aforementioned emph{a priori} spatio-temporal discretization strategy forms an important foundation for extending the key ideas of the enriched finite element basis to TDDFT. Lastly, as a first step towards the goal of machine-learned exchange-correlation functionals, we have addressed the challenge of obtaining the training data mapping to . This constitute generating accurate ground-state density, , from wavefunction-based calculations, and then inverting the Kohn-Sham eigenvalue problem to obtain the that yields the same . This is otherwise known as the emph{inverse} DFT problem. Heretofore, this remained an open challenge owing lack of accurate and systematically convergent numerical techniques. To this end, we have provided a robust and systematically convergent scheme to solve the inverse DFT problem, employing finite element basis. We obtained the exact corresponding to ground-state densities obtained from configuration interaction calculations, to unprecedented accuracy, for both weak and strongly correlated polyatomic systems ranging up to 40 electrons. This ability to evaluate exact 's from ground-state densities provides a powerful tool in the future testing and development of approximate exchange-correlation functionals.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/153371/1/bikash_1.pd
Development of efficient electronic-structure methods based on the adiabatic-connection fluctuation-dissipation theorem and Møller–Plesset perturbation theory
One of the major goals of quantum chemistry is to develop electronic-structure methods, which are not only highly accurate in the evaluation of electronic ground-state properties, but also computationally tractable and versatile in their application. A theory with great potential in this respect, however, without being free from shortcomings is the random phase approximation (RPA).
In this work, developments are presented, which address the most important of these shortcomings subject to the constraint to obtain low- and linear-scaling electronic-structure methods. A scheme combining an elegant way to introduce local orbitals and multi-node parallelism is put forward, which not only allows to evaluate the RPA correlation energy in a fraction of the time of former theories, but also enables a scalable decrease of the high memory requirements. Furthermore, a quadratic-scaling self-consistent minimization of the total RPA energy with respect to the one-particle density matrix in the atomic-orbital space is introduced, making the RPA energy variationally stable and independent of the quality of the reference calculation. To address the slow convergence with respect to the size of the basis set and the self-correlation inherent in the RPA functional, range-separation of the electron-electron interaction is exploited for atomic-orbital RPA, yielding a linear-scaling range-separated RPA method with consistent performance over a broad range of chemical problems. As a natural extension, the concepts including local orbitals, self-consistency, and range-separation are further combined in a RPA-based generalized Kohn–Sham method, which not only shows a balanced performance in general main group thermochemistry, kinetics, and noncovalent interactions, but also yields accurate ionization potentials and fundamental gaps.
The origin of the self-correlation error within RPA lies in the neglect of exchange-effects in the calculation of the interacting density-density response functions. While range-separation is a reasonable approach to counteract this shortcoming — since self-correlation is pronounced at short interelectronic distances — a more rigorous but computationally sophisticated approach is to introduce the missing exchange-effects, at least to some extent. To make RPA with exchange methods applicable to systems containing hundreds of atoms and hence a suitable choice for practical applications, a framework is developed, which allows to devise highly efficient low- and linear-scaling RPA with exchange methods.
The developments presented in this work, however, are not only limited to RPA and beyond-RPA methods. The connection between RPA and many-body perturbation theory is further used to present a second-order Møller–Plesset perturbation theory method, which combines the tools to obtain low- and linear-scaling RPA and beyond-RPA methods with efficient linear-algebra routines, making it highly efficient and applicable to large molecular systems comprising several thousand of basis functions
Parameter Estimation for Stable Distributions with Application to Commodity Futures Log-Returns
This paper explores the theory behind the rich and robust family of α-stable distributions to estimate parameters from financial asset log-returns data. We discuss four-parameter estimation methods including the quantiles, logarithmic moments method, maximum likelihood (ML), and the empirical characteristics function (ECF) method. The contribution of the paper is two-fold: first, we discuss the above parametric approaches and investigate their performance through error analysis. Moreover, we argue that the ECF performs better than the ML over a wide range of shape parameter values, α including values closest to 0 and 2 and that the ECF has a better convergence rate than the ML. Secondly, we compare the t location-scale distribution to the general stable distribution and show that the former fails to capture skewness which might exist in the data. This is observed through applying the ECF to commodity futures log-returns data to obtain the skewness parameter
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