dftatom: A robust and general Schr\"odinger and Dirac solver for atomic structure calculations

A robust and general solver for the radial Schr\"odinger, Dirac, and Kohn--Sham equations is presented. The formulation admits general potentials and meshes: uniform, exponential, or other defined by nodal distribution and derivative functions. For a given mesh type, convergence can be controlled systematically by increasing the number of grid points. Radial integrations are carried out using a combination of asymptotic forms, Runge-Kutta, and implicit Adams methods. Eigenfunctions are determined by a combination of bisection and perturbation methods for robustness and speed. An outward Poisson integration is employed to increase accuracy in the core region, allowing absolute accuracies of $10^{-8}$ Hartree to be attained for total energies of heavy atoms such as uranium. Detailed convergence studies are presented and computational parameters are provided to achieve accuracies commonly required in practice. Comparisons to analytic and current-benchmark density-functional results for atomic number $Z$ = 1--92 are presented, verifying and providing a refinement to current benchmarks. An efficient, modular Fortran 95 implementation, \ttt{dftatom}, is provided as open source, including examples, tests, and wrappers for interface to other languages; wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines.Comment: Submitted to Computer Physics Communication on August 27, 2012, revised February 1, 201

Atomistic Models of Materials: Mathematical Challenges

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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 ($rho(mathbf{r})$), 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 $rho(mathbf{r})$ to $v_text{xc}(mathbf{r})$, and (b) using machine learning on the training data ($rho(mathbf{r}) Leftrightarrow v_text{xc}(mathbf{r})$ maps) to obtain the functional form of $v_text{xc}[rho(mathbf{r})]$, 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 $50-300$-fold and $5-8$-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 $100$-fold speedup afforded by higher-order finite elements over linear finite elements. Furthermore, for pseudopotential calculations, our approach achieve a $3-60$-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 $rho(mathbf{r})$ to $v_text{xc}(mathbf{r})$. This constitute generating accurate ground-state density, $rho(mathbf{r})$, from wavefunction-based calculations, and then inverting the Kohn-Sham eigenvalue problem to obtain the $v_text{xc}(mathbf{r})$ that yields the same $rho(mathbf{r})$. 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 $v_text{xc}$ 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 $v_text{xc}$'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

Mathematical Methods in Quantum Chemistry

The field of quantum chemistry is concerned with the analysis and simulation of chemical phenomena on the basis of the fundamental equations of quantum mechanics. Since the ‘exact’ electronic Schrödinger equation for a molecule with $N$ electrons is a partial differential equation in 3$N$ dimension, direct discretization of each coordinate direction into $K$ gridpoints yields $K^{3N}$ gridpoints. Thus a single Carbon atom ($N = 6$) on a coarse ten point grid in each direction ($K = 10$) already has a prohibitive $10^{18}$ degrees of freedom. Hence quantum chemical simulations require highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together quantum chemists and the emerging and fast growing community of mathematicians working in the area, to assess recent advances and discuss long term prospects regarding the overarching challenges of (1) developing accurate reduced models at moderate computational cost, (2) developing more systematic ways to understand and exploit the multiscale nature of quantum chemistry problems. Topics of the workshop included: • wave function based electronic structure methods, • density functional theory, and • quantum molecular dynamics. Within these central and well established areas of quantum chemistry, the workshop focused on recent conceptual ideas and (where available) emerging mathematical results

Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.Comment: 63 pages, 13 figure

Mathematical and Numerical Aspects of Quantum Chemistry Problems

This workshop was aimed at strengthtening the interactions between well established experts in quantum chemistry, mathematical analysis, numerical analysis and computational metodology. Most of the mathematicians present in the worskhop have already contributed to the theoretical and numerical study of models in quantum physics and chemistry. Some others, familiar with contiguous fiels, were new to chemistry. Several distinguished researchers in theoretical chemistry participated in the workshop, and presented the mathematical and computational challenges of the field