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Über dieses Buch

On the occasion of the fourth International Conference on Industrial and Applied Mathematics!, we decided to organize a sequence of 4 minisymposia devoted to the mathematical aspects and the numerical aspects of Quantum Chemistry. Our goal was to bring together scientists from different communities, namely mathematicians, experts at numerical analysis and computer science, chemists, just to see whether this heterogeneous set of lecturers can produce a rather homogeneous presentation of the domain to an uninitiated audience. To the best of our knowledgde, nothing of this kind had never been tempted so far. It seemed to us that it was the good time for doing it, both . because the interest of applied mathematicians into the world of computational chemistry has exponentially increased in the past few years, and because the community of chemists feels more and more concerned with the numerical issues. Indeed, in the early years of Quantum Chemistry, the pioneers (Coulson, Mac Weeny, just to quote two of them) used to solve fundamental equations modelling toy systems which could be simply numerically handled in view of their very limited size. The true difficulty arose with the need to model larger systems while possibly taking into account their interaction with their environment. Hand calculations were no longer possible, and computing science came into the picture.



General topics


Chapter 1. Is a molecule in chemistry explicable as a broken symmetry in quantum mechanics?

It is argued that incorporating the molecular geometry into the transformation specifications for deriving the standard (Eckart) Hamiltonian used to describe molecular spectra, cannot generally be accomplished without breaking the permutational symmetry requirements on identical nuclei.
Brian Sutcliffe

Chapter 2. SCF algorithms for HF electronic calculations

This paper presents some mathematical results on SCF algorithms for solving the Hartree-Fock problem. In the first part of the article the focus is on two classical SCF procedures, namely the Roothaan algorithm and the level-shifting algorithm. It is demonstrated that the Roothaan algorithm either converges towards a solution to the Hartree-Fock equations or oscillates between two states which are not solution to the Hartree-Fock equations, any other behavior (oscillations between more than two states, “chaotic” behavior, ... ) being excluded. The level-shifting algorithm is then proved to converge for large enough shift parameter, whatever the initial guess. The second part of the article details the convergence properties of a new algorithm recently introduced by Le Bris and the author, the so-called Optimal Damping Algorithm (ODA). Basic numerical simulations pointing out the principal features of the various algorithms under study are also provided.
Eric Cancès

Chapter 3. A pedagogical introduction to Quantum Monte-Carlo

Quantum Monte Carlo (QMC) methods are powerful stochastic approaches to calculate ground-state properties of quantum systems. They have been applied with success to a great variety of problems described by a Schrodinger-like Hamiltonian (quantum liquids and solids, spin systems, nuclear matter, ab initio quantum chemistry, etc ... ). In this paper we give a pedagogical presentation of the main ideas of QMC. We develop and exemplify the various concepts on the simplest system treatable by QMC, namely a 2 x 2 matrix. First, we discuss the Pure Diffusion Monte Carlo (PDMC) method which we consider to be the most natural implementation of QMC concepts. Then, we discuss the Diffusion Monte Carlo (DMC) algorithms based on a branching (birth-death) process. Next, we present the Stochastic Reconfiguration Monte Carlo (SRMC) method which combines the advantages of both PDMC and DMC approaches. A very recently introduced optimal version of SRMC is also discussed. Finally, two methods for accelerating QMC calculations are sketched: (a) the use of integrated Poisson processes to speed up the dynamics (b) the introduction of “renormalized” or “improved” estimators to decrease the statistical fluctuations.
Michel Caffarel, Roland Assaraf

Chapter 4. On the controllability of bilinear quantum systems

We present in this paper controllability results for quantum systems interacting with lasers. A negative result for infinite dimensional spaces serves as a starting point for a finite dimensional analysis. We show that under physically reasonable hypothesis in such systems we can control the population of the eigenstates. Applications are given for a five-level system.
Gabriel Turinici

Condensed phases


Chapter 5. Recent mathematical results on the quantum modeling of crystals

We describe in this paper a strategy, the so-called thermodynamic limit process, to build in a rigorous mathematical manner the quantum-mechanical models for the ground-state energy of solid crystals. These models are the analogues for the solid state of well-known models issued from Quantum Chemistry, namely Thomas-Fermi, Hartree and Hartree-Fock type models. We shall present a broad overview on recent mathematical studies on this topic.
I. Catto, P.-L. Lions, C. Le Bris

Chapter 6. Local density approximations for the energy of a periodic Coulomb model

We deal with local density approximations for the kinetic and exchange energy terms of a periodic Coulomb model. For the kinetic energy, we give a rigorous derivation of the usual combination of the von-Weizsäcker term and the Thomas-Fermi term in the “high density” limit. Furthermore, we justify the inclusion of the Dirac term for the exchange energy and the Slater term for the local exchange potential. Our method is based on deformations (local scaling transformat ions) of plane waves in a periodic box.
O. Bokanowski, B. Grébert, N. J. Mauser

Chapter 7. A mathematical insight into ab initio simulations of the solid phase

We give here an introduction to the language of ab initio solid-state theory. Starting from the intrinseque symmetries of a perfect crystal, we show how it is related to band structure theory and give a brief overview of the techniques in use to simulate such systems.
X. Blanc

Chapter 8. Examples of hidden numerical tricks in a solid state determination of electronic structure.

The paper focuses on the numerical aspects of the atomistic modelling of materials in the Density Functional formalism. As usual in such a modelling, calculations can be run at various levels of theory and using different numerical options. After a review of the approximations underlying the Local Density Approximation (LDA), the Local Spin Density Approximation (LSDA), etc. the numerical options regarding the size of the basis functions, the use of pseudo-potentials or the selection of k-points are examined in details as well as the subsequent numerical tricks occuring in the calculations, e.g. in geometry optimizations. Finally, results regarding an apatitic mineral are reported and orders of magnitudE' of the expected accuracy are provided.
Mireille Defranceschi, Vanina Louis-Achille

Chapter 9. Quantum mechanical models for systems in solution

An overview of modern theories for the modelling of solvent effects on the state and the properties of quantum mechanical molecular systems is presented. The emphasis here is on the models that exploit a continuum description of the solvent and introduce effective Hamiltonians to represent intermolecular interactions. The main theoretical and numerical aspects of these methods, in which mutual interactions between solute and solvent are included, are presented and discussed. As more specialistic feature, we also analyze their extension to the derivative theory, presenting some selected applications such as the search for the best geometry and the evaluation of molecular response properties in solution. In this context some comments on the eventual inclusion of dynamical aspects are also reported.
Benedetta Mennucci

Relativistic models


Chapter 10. Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues

The main goal of this paper is to describe some new variational methods for the characterization and computation of the eigenvalues and the eigenstates of Dirac operators. Our methods are all based on exact variational principles, both of min-max and of minimization types. The minimization procedure that we introduce is done in a particular set offunctions satisfying a nonlinear constraint. Finally, we present several numerical methods that we have implemented in particular cases, in order to construct approximate solutions of that minimization problem.
Jean Dolbeault, Maria J. Esteban, Eric Séré

Chapter 11. Quaternion symmetry of the Dirac equation

Following van der Waerden, the Dirac equation is derived from linearization of the Klein-Gordon equation using the algebraic properties of the Pauli spin matrices. As the algebra of these matrices is identical to that of quaternions, the Dirac equation can be reformulated in terms of quaternion algebra and therefore without reference to a specific spin quantization axis. In this paper we consider the symmetry content of the Dirac equation. It is found that the basic binary symmetry operations in spin space map onto the unit vectors of complex quaternions. We argue that a consistent choice of the inversion operator in spin space is of order four. We furthermore show that quaternion algebra is the natural language for time reversal symmetry. These considerations lead to the formulation of a symmetry scheme that automatically provides maximum point group and time reversal symmetry reduction in the solution of the Dirac equation in the finite basis approximation.
T. Saue, H. J. Aa. Jensen


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