Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem

The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized Bethe-Salpeter equation in the context of computing exciton energies and states. A computational challenge is that at least half of the eigenvalues and the associated eigenvectors are desired in practice. We establish the equivalence between Bethe-Salpeter eigenvalue problems and real Hamiltonian eigenvalue problems. Based on theoretical analysis, structure preserving algorithms for a class of Bethe-Salpeter eigenvalue problems are proposed. We also show that for this class of problems all eigenvalues obtained from the Tamm-Dancoff approximation are overestimated. In order to solve large scale problems of practical interest, we discuss parallel implementations of our algorithms targeting distributed memory systems. Several numerical examples are presented to demonstrate the efficiency and accuracy of our algorithms

Efficient Algorithms for Solving Structured Eigenvalue Problems Arising in the Description of Electronic Excitations

Matrices arising in linear-response time-dependent density functional theory and many-body perturbation theory, in particular in the Bethe-Salpeter approach, show a 2 × 2 block structure. The motivation to devise new algorithms, instead of using general purpose eigenvalue solvers, comes from the need to solve large problems on high performance computers. This requires parallelizable and communication-avoiding algorithms and implementations. We point out various novel directions for diagonalizing structured matrices. These include the solution of skew-symmetric eigenvalue problems in ELPA, as well as structure preserving spectral divide-and-conquer schemes employing generalized polar decompostions

Towards the Description of Core-Excited States within the Framework of Many-Body Perturbation Theory

Gegenstand dieser Arbeit ist die Entwicklung effizienter und robuster Methoden zur Beschreibung von Ladungs- und ladungsneutralen Anregungen von kernnahen Zuständen in molekularen Systemen mittels Vielteilchen-Störungstheorie. Um auch die Anwendbarkeit auf Systeme, die schwere Elemente beinhalten, zu gewährleisten, wird der Formalismus in einem zwei-komponentigen Kramers-symmetrischen Rahmen präsentiert. Dies erlaubt neben skalarrelativistischen Effekten auch die explizite Beschreibung der Spin-Bahn-Wechselwirkung. Die erarbeiteten Ansätze bleiben jedoch auch im ein-komponentigen, nichtrelativistischen Grenzfall gültig. Es wird eine neue Methode zur Berechnung von Quasiteilchenenergien in der GW-Näherung vorgestellt und bewertet. Dabei wird die Beschreibung von Ladungsanregungen, den Ionisierungsenergien und Elektronenaffinitäten, gleichermaßen für Valenz- und Rumpfelektronen ermöglicht. Diese Technik verringert gerade für die letztere Anwendung den Aufwand drastisch im Vergleich zu etablierten Methoden und besitzt zudem auch Vorteile für Systeme mit einer hohen Zustandsdichte. Die so erhaltenen Quasiteilchenenergien werden im Weiteren als Ausgangspunkt zur Berechnung von ladungsneutralen Anregungen, wie sie zur Beschreibung von Experimenten aus der Röntgenabsorptionsspektroskopie benötigt werden, mittels der Bethe-Salpeter Gleichung verwendet. Es werden zwei Ansätze diskutiert, die bereits im Rahmen der Dichtefunktionaltheorie und post-Hartree–Fock Methoden bekannt sind. Erstere folgt der gedämpften linearen Antworttheorie und führt eine künstliche Lebenszeit für angeregte Zustände ein. Dies ermöglicht die Berechnung dynamischer Polarisierbarkeiten für beliebige Frequenzen. Die zweite Methode nutzt die schwache Kopplung zwischen den Anregungen rumpfnaher und Valenzelektronen in der elektronischen Hesse-Matrix. Das Problem wird in einem entsprechenden Unterraum der Einfachanregungen von Rumpfelektronen gelöst. Beide Methoden werden im Folgenden zusammen mit den entsprechenden Implementierungen vorgestellt und auf die Möglichkeit zur Beschreibung von Anregungen von kernnahen Elektronen bewertet. Die Ergebnisse werden mit weiteren Methoden, sowie experimentellen Daten verglichen, und auf reale Fragestellungen angewandt

TurboRVB: A many-body toolkit for ab initio electronic simulations by quantum Monte Carlo

TurboRVB is a computational package for ab initio Quantum Monte Carlo (QMC) simulations of both molecular and bulk electronic systems. The code implements two types of well established QMC algorithms: Variational Monte Carlo (VMC) and diffusion Monte Carlo in its robust and efficient lattice regularized variant. A key feature of the code is the possibility of using strongly correlated many-body wave functions (WFs), capable of describing several materials with very high accuracy, even when standard mean-field approaches [e.g., density functional theory (DFT)] fail. The electronic WF is obtained by applying a Jastrow factor, which takes into account dynamical correlations, to the most general mean-field ground state, written either as an antisymmetrized geminal power with spin-singlet pairing or as a Pfaffian, including both singlet and triplet correlations. This WF can be viewed as an efficient implementation of the so-called resonating valence bond (RVB) Ansatz, first proposed by Pauling and Anderson in quantum chemistry [L. Pauling, The Nature of the Chemical Bond (Cornell University Press, 1960)] and condensed matter physics [P.W. Anderson, Mat. Res. Bull 8, 153 (1973)], respectively. The RVB Ansatz implemented in TurboRVB has a large variational freedom, including the Jastrow correlated Slater determinant as its simplest, but nontrivial case. Moreover, it has the remarkable advantage of remaining with an affordable computational cost, proportional to the one spent for the evaluation of a single Slater determinant. Therefore, its application to large systems is computationally feasible. The WF is expanded in a localized basis set. Several basis set functions are implemented, such as Gaussian, Slater, and mixed types, with no restriction on the choice of their contraction. The code implements the adjoint algorithmic differentiation that enables a very efficient evaluation of energy derivatives, comprising the ionic forces. Thus, one can perform structural optimizations and molecular dynamics in the canonical NVT ensemble at the VMC level. For the electronic part, a full WF optimization (Jastrow and antisymmetric parts together) is made possible, thanks to state-of-the-art stochastic algorithms for energy minimization. In the optimization procedure, the first guess can be obtained at the mean-field level by a built-in DFT driver. The code has been efficiently parallelized by using a hybrid MPI-OpenMP protocol, which is also an ideal environment for exploiting the computational power of modern Graphics Processing Unit accelerators

Nonadiabatic Dynamics: A Semiclassical Approach

Nonadiabatic dynamics has been an essential part of quantum chemistry since the 1930’s. Nonadiabatic effects play a crucial role in photo-physical and photo-chemical reactions for both small and large molecules in both gas and condensed phases. Modeling dynamics of photoinduced reactions has been a new frontier of chemistry. Many dynamical phenomena, such as intersystem crossing, non-radiative relaxation, and charge energy transfer, require a nonadiabatic description which incorporates transitions between electronic states. In Chapter 2, the property of scattering region in the semiclassical limit is investigated. We suggest that a nuclear wavepacket close enough to the conical intersection will propagate ballistically in a straight line through the scattering region with distance λ+, the impact parameter, away from the conical intersection. Upon taking the semiclassical limit, we have proven that in a certain neighborhood of the conical intersection, the adiabatic propagation and ballistic propagation are both valid. The resulted complete propagator is governed by the semiclassical propagation along the reference path which connects the initial and final points, and an integration over the impact parameter, hence only depends on the initial and final classical states of the system. In Chapter 3, we identify the main differences between the effects of Kramers symmetry on the systems with even and odd number of electrons, the ways how the aforementioned symmetry affects the structure of the Conical Seams (CSs), and how it shows up in semiclassical propagation of nuclear wavepackets, crossing the CSs. We identify the topological invariants, associated with CSs, in three cases: even and odd number of electrons with time-reversal symmetry, as well as absence of the latter. We obtain asymptotically exact semiclassical analytical solutions for wavepackets scattered on a CS for all three cases, identify topological features in a non-trivial shape of the scattered wavepacket, and connect them to the topological invariants, associated with CSs. We argue that, due to robustness of topology, the non-trivial wavepacket structure is a topologically protected evidence of a wavepacket having passed through a CS, rather than a feature of a semiclassical approximation. In Chapter 4, we present, in detail, an algorithm based on Monte-Carlo sampling of the semiclassical time-dependent wavefunction, that involves running simple surface hopping dynamics, followed by a post-processing step which adds little cost. The method requires only a few quantities from quantum chemistry calculations, can systematically be improved, and provides excellent agreement with exact quantum mechanical results. Here we show excellent agreement with exact solutions for scattering results of standard test problems. Additionally, we find that convergence of the wavefunction is controlled by complex valued phase factors, the size of the nonadiabatic coupling region, and the choice of sampling function. These results help in determining the range of applicability of the method, and provide a starting point for further improvement

A Multireference Density Functional Approach to the Calculation of the Excited States of Uranium Ions

An accurate and efficient hybrid Density Functional Theory (DFT)/Multireference Configuration Interaction (MRCI) model for computing electronic excitation energies in heavy element atoms and molecules was developed. This model incorporated relativistic effects essential for accurate qualitative and quantitative spectroscopic predictions on heavy elements, while simultaneously removing spin-multiplicity limitations inherent in the original model on which it is based. This model was used to successfully compute ground and low-lying electronic states for atoms in the first two rows of the period table, which were used for calibration. Once calibrated, calculations on carbon monoxide, bromine fluoride, the bromine atom, uranium +4 and +5 ions and the uranyl (UO22+) ion showed the model achieved reductions in relative error with respect to Time Dependent Density Functional Theory (TDDFT) of 11-42%, with a corresponding reduction in computational effort in terms of MRCI expansion sizes of a factor of 25-64