Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal block preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods

ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers

Solving the electronic structure from a generalized or standard eigenproblem is often the bottleneck in large scale calculations based on Kohn-Sham density-functional theory. This problem must be addressed by essentially all current electronic structure codes, based on similar matrix expressions, and by high-performance computation. We here present a unified software interface, ELSI, to access different strategies that address the Kohn-Sham eigenvalue problem. Currently supported algorithms include the dense generalized eigensolver library ELPA, the orbital minimization method implemented in libOMM, and the pole expansion and selected inversion (PEXSI) approach with lower computational complexity for semilocal density functionals. The ELSI interface aims to simplify the implementation and optimal use of the different strategies, by offering (a) a unified software framework designed for the electronic structure solvers in Kohn-Sham density-functional theory; (b) reasonable default parameters for a chosen solver; (c) automatic conversion between input and internal working matrix formats, and in the future (d) recommendation of the optimal solver depending on the specific problem. Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800 basis functions) on distributed memory supercomputing architectures.Comment: 55 pages, 14 figures, 2 table

Numerical Methods for Electronic Structure Calculations of Materials

This is the published version. Copyright 2010 Society for Industrial and Applied MathematicsThe goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scientific computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful but approximate versions of this equation, which allow one to study nontrivial systems, took about five or six decades to develop. In particular, the last two decades saw a flurry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as density functional theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials. This article will emphasize pseudopotential-density functional theory, but other techniques will be discussed as well

Shift-invert diagonalization of large many-body localizing spin chains

We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a many-body localization (MBL) transition, using shift-invert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl/. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for mid-spectrum eigenstates of spin chains of sizes up to $L=26$. This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem

Exploring potential energy surfaces in ground- and excited states

Chemical reactivity of atoms, molecules and ions is governed by their underlying potential energy surface. Calculating the whole potential energy surface within reasonable bounds, is impossible for all but the smallest molecules. Usually, only parts of the full potential energy surface can be studied, namely stationary points and the minimum energy paths connecting them. By comparing energies of stationary points and their separating barriers, conclusions regarding possible reactions mechanism, or their infeasibility, can be drawn. Taking excited states into account leads to further complications, as now multiple potential energy surfaces have to be considered and root flips between different excited states may occur, requiring effective state-tracking. Part II of this thesis describes the required methods to locate stationary points and minimum energy paths on potential energy surfaces, by using surface-walking, chain-of-states optimization and intrinsic reaction coordinate integration. Several approaches to state-tracking are presented in chapter 4. Results of this thesis are presented in Part III, containing two contributions to the field of photochemistry: chapter 12 provides a possible excited-state reaction mechanism for a biaryl cross-coupling reaction and offers a plausible explanation for its high regioselectivity. The second contribution is the development pysisyphus (chapter 13), an external optimizer implemented in python, aware of excited states and thus the core of this thesis. By implementing the state-tracking algorithms outlined in chapter 4 it allows effective and efficient optimizations of stationary points in ground- and excited-states. The performance of pysisyphus is verified for several established benchmark sets. Results for several excited-state optimizations are presented in section 13.3, where pysisyphus shows good performance for the optimization of sizeable transition-metal complexes

Daubechies Wavelets for Linear Scaling Density Functional Theory

We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized contracted basis functions in which the Kohn-Sham orbitals can be represented with an arbitrarily high, controllable precision. Ground state energies and the forces acting on the ions can be calculated in this basis with the same accuracy as if they were calculated directly in a Daubechies wavelets basis, provided that the amplitude of these contracted basis functions is sufficiently small on the surface of the localization region, which is guaranteed by the optimization procedure described in this work. This approach reduces the computational costs of DFT calculations, and can be combined with sparse matrix algebra to obtain linear scaling with respect to the number of electrons in the system. Calculations on systems of 10,000 atoms or more thus become feasible in a systematic basis set with moderate computational resources. Further computational savings can be achieved by exploiting the similarity of the contracted basis functions for closely related environments, e.g. in geometry optimizations or combined calculations of neutral and charged systems

Density functional theory for large molecular systems

Nøyaktige simuleringer av kjemiske og biologiske prosesser på molekylært nivå har lenge vært uoppnålig for en rekke molekylære systemer, og har nå blitt mulig for mange av disse systemene gjennom ny metodeutvikling av Simen Reine, Trygve Helgaker og medarbeidere ved Universitetet i Oslo. Datasimuleringer er utbredt innen kjemi og relaterte felt som biologi, farmasi og medisin. Kvantekjemiske metoder er fundamentale for de mest nøyaktig simuleringsteknikkene, og er til stor hjelp ved bestemmelse og prediksjon av molekylære egenskaper, som for eksempel molekylers struktur, og gir i tillegg viktig og detaljert innsikt i kjemiske reaksjoner - både kvalitativt og kvantitativt. Anvendelsesområdet er nært knyttet til metodenes nøyaktighet, effektivitet og brukervennelighet. Utviklingen av nye og forbedrede metoder gjør oss i stand til å studere molekylære systemer som foreløpig har vært utenfor rekkevidde, og gir oss mer nøyaktig beskrivelse av de systemene vi allerede behandler idag. Som en konsekvens vil man kunne redusere bruken av kostbare og tidkrevende eksperimenter og samtidig hjelpe forskere verden over til bedre å forstå kjemiske mekanismer. De fleste kvantekjemiske beregninger som utføres idag benytter tetthetsfunksjonalteori (DFT), da denne metoden utgjør et godt kompromiss mellom nøyaktighet og beregningstid. Selv om DFT er meget nyttig, er dagens metoder begrenset til systemer bestående av noen få hundre atomer, og utelukker derfor en rekke systemer, for eksempel proteiner. I doktorgraden "Tetthetsfunksjonalteori for store molekylære systemer" har nye metoder innen DFT blitt utviklet med tanke på rutinemessige beregninger for store systemer. Beregninger for systemer med 1400 atomer er rapportert og metodene er i etterkant blitt benyttet for systemer med opp til 4000 atomer