An Analysis for some Methods and Algorithms of Quantum Chemistry vorgelegt von Dipl. Math. Thorsten Rohwedder aus Preetz, Holstein Von der Fakult¨atII - Mathematik und Naturwissenschaften - der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. rer. nat. Martin Skutella (TU Berlin) Berichter/Gutachter: Prof. Dr. rer. nat. Christian Lubich (Univ. T¨ubingen) Prof. Dr. rer. nat. Reinhold Schneider (TU Berlin) Prof. Dr. rer. nat. Harry Yserentant (TU Berlin) Tag der wissenschaftlichen Aussprache: 15.11. 2010 Berlin 2010 D 83 Dedicated to the people and things without whom this work would not have been possible: To my parents, without whom I would not be where I am now, to the people who have guided my way through science, in particular to Reinhold Schneider, Alexander Auer and Etienne Emmrich, to the Universities of Kiel and Berlin for providing the necessary financial support, to all the friends who have accompanied me on the way, and last, but not least, to Rock'n'Roll. Preface and overview More than 80 years after Paul Dirac stated that \the underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are [...] completely known"[62], the development of \approximate practical methods of applying quantum mechanics", demanded by Dirac in the same breath, is still a highly active field of research at the crossroads of physics, chemistry, applied mathematics and computer science. This circumstance is mainly owed to the interplay of two facts: On the one hand, the development of modern day computers has seen a phase of almost exponential growth at the end of the last century, so that calculations of theoretical chemistry and molecular physics have become competitive with practical experiments or at least often allow use- ful predictions of empirical parameters that can assist practical investigations. On the other hand, the solution of equations formulated in quantum mechanics is an exceedingly high-dimensional and thus computationally demanding problem, while at the same time, an extremely high accuracy is needed in order to obtain results utilizable in practice. Even nowadays, small to medium-sized quantum chemical problems push the limits of commonly available computational resources. To efficiently treat the variety of practical problems covered by the formalism of quantum mechanics, it is therefore indispensable to design highly problem-adapted methods and algorithms that balance the available com- putational resources against the respective required accuracy. These prerequisites have lead to a \zoo" of extremely sophisticated and well-developed methods and algorithms commonly used in quantum chemistry. Partly, the respective approaches are ab initio, i.e. the working equations are derived directly from the Schr¨odingerequation, as is for instance the case for the various variants of the Hartree-Fock method over perturbational methods, the Configuration Interaction (CI) and Coupled Cluster (CC) method and the recently revived CEPA method to reduced density matrix methods, to mention but the probably most important ones; to another part, they also integrate empirical parameters, as for instance in the successful Kohn-Sham model of density functional theory and the stochastic methods of Quantum Monte Carlo techniques do.1 Although the development of formal quantum mechanics and that of functional analysis are deeply interwoven, and although the theoretical properties of the Schr¨odingerequation and the Hamiltonian are quite well understood from a mathematical point of view (see Section 1), most of the practically relevant computational schemes mentioned above were introduced by physicists or chemists, and the actual algorithmic treatment of the elec- tronic Schr¨odingerequation does only recently seem to have aroused the broader attention of the mathematical community. Therefore, although there have been various efforts in 1For an introduction and references to the respective methods, see e.g. [103, 201] for Hartree-Fock, [201, 142] for perturbational approaches, Section 2.1 of this work for references for the CI method and den- sity functional theory, Section 3 for the Coupled Cluster method, [133, 208] for the CEPA method, [148] for the reduced density matrix methods and [78, 144] for a review of Quantum Monte Carlo techniques. understanding the methods of quantum chemistry from a mathematical point of view2 and to approach general problems in the numerical treatment of the electronic Schr¨odinger equation by means of concepts from mathematics,3 the stock of available mathematically rigorous analysis of the present practically relevant methods of quantum mechanics and of the convergence behaviour of the algorithms used for their treatment is on the whole still relatively scarce. It will be subject of the present work to approach this shortcoming, that is, to provide a numerical analysis for certain aspects of some well-known methods of quantum chemistry. The work is organized in four parts. The first part (Section 1) is an attempt to connect the world of mathematical physics to that of computational chemistry: Starting from the necessities imposed by the postulates of quantum mechanics, we introduce the operators and spaces needed to embed the main task of electronic structure calculation, i.e. the calculation of electronic states and energies, into a sound mathematical background; we review known theoretical results, prove some results needed later and derive the (Galerkin) framework that is in a wider sense the basis to all methods used in practical calculations in quantum chemistry. From Section 2 onwards, we turn towards the actual algorithmic treatment of the equa- tions derived in Section 1: Section 2, parts of which have already been published in [191], first provides a short introduction to the methods of Hartree-Fock, Kohn-Sham and CI; we then give a convergence analysis for a preconditioned steepest descent algorithm under orthogonality constraints, taylor-made for and commonly used in the context of Hartree- Fock and density functional theory calculations, but also providing a sensible algorithm for implementation of the CI method. Section 3, featuring some of the main achievements of this work, is dedicated to lifting the Coupled Cluster method, usually formulated in a finite dimensional, discretised subspace of a suitable Sobolev space H1, to the continuous space H1, resulting in what we will call the continuous Coupled Cluster method. To de- fine the continuous method, some formal problems have to be overcome; afterwards, the results for the continuous methods will be used to derive existence and (local) uniqueness statements for discretisations and to establish goal-oriented a-posteriori error estimators for the Coupled Cluster method. The last part of this work (Section 4) features an anal- ysis for the acceleration technique DIIS that is commonly used in quantum chemistry codes. To derive some (positive as well as negative) convergence results for DIIS, we establish connections to the well-known GMRES solver for linear systems as well as to quasi-Newton methods. At the beginning of each of the sections, a more thorough introduction to their respective subject is given. Also, in Sections 2 { 4, the main results of the respective section will explicitly be referenced there. Mathematical objects and notions used in this work are either introduced explicitly, or the reader is referred to according literature. Please also note that the most frequently used notations are compiled in the list of symbols at the end of this work. 2Cf. e.g. [7, 42, 43, 186, 190] for recent works and also [15, 139, 140, 143, 153] for the properties of the Hartree-Fock method, already analysed to some extent in the 1970-80's. 3See e.g. [45, 79, 92, 214]. 5 Preface and overview Contents 1 A mathematical framework for electronic structure calculation 1 1.1 General setting . .3 1.2 The Pauli principle and invariant subspaces of the Hamiltonian . .9 1.3 Strong and weak form of the electronic Schr¨odingerequation . 13 1.4 Orbitals, bases for tensor spaces and the Slater basis . 18 1.5 The electronic Schr¨odingerequation in Second Quantization . 22 1.6 Ellipticity results for the Hamiltonian and for Hamiltonian-like operators . 27 1.7 Conclusions - Towards discretisation . 32 2 Analysis of a \direct minimization" algorithm used in Hartree-Fock, DFT and CI calculations 35 2.1 Overview: The Hartree-Fock/Kohn-Sham models and the CI method . 36 2.2 Minimization problems on Grassmann manifolds . 41 2.3 Convergence analysis for a \Direct Minimization" algorithm . 48 2.4 Concluding remarks . 57 6 3 The continuous Coupled Cluster method 59 3.1 Notations, basic assumptions and definitions . 60 3.2 Continuity properties of cluster operators; the Coupled Cluster equations . 64 3.3 Analytical properties of the Coupled Cluster function . 75 3.4 Existence and uniqueness statements and error estimates . 79 3.5 Simplification and evaluation of Coupled Cluster function . 85 3.6 Concluding remarks . 91 4 The DIIS acceleration method 93 4.1 Notations and basic facts about DIIS . 95 4.2 Equivalence of DIIS to a projected Broyden's method . 96 4.3 DIIS applied to linear problems . 104 4.4 Convergence analysis for DIIS . 110 4.5 Concluding remarks . 125 Conclusion and outlook 127 Notation i References iv 1 A mathematical framework for electronic structure calculation Since the hour of