# Introduction to Quantum Chemistry and ab initio Calculations of Interatomic Potentials<br/><br/>

Monday, July 30, 2007 - 9:30am - 10:45am

EE/CS 3-180

Eric Cances (CERMICS)

Quantum Chemistry aims at understanding the properties of matter through the modelling of its behaviour at a subatomic scale, where matter is described as an assembly of nuclei and electrons.

At this scale, the equation that rules the interactions between these constitutive elements is the Schrdinger equation. It can be considered (except in few special cases notably those involving relativistic phenomena or nuclear reactions) as a universal model for at least three reasons. First it contains all the physical information of the system under consideration so that any of the properties of this system can be deduced in theory from the Schrdinger equation associated to it. Second, the Schrdinger equation does not involve any empirical parameter, except some fundamental constants of Physics (the Planck constant, the mass and charge of the electron, ...); it can thus be written for any kind of molecular system provided its chemical composition, in terms of natures of nuclei and number of electrons, is known. Third, this model enjoys remarkable predictive capabilities, as confirmed by comparisons with a large amount of experimental data of various types.

Unfortunately, the Schrdinger equation cannot be directly simulated, except for very small chemical systems. It indeed reads as a time-dependent 3(M+N)-dimensional partial differential equation, where M is the number of nuclei and N the number of the electrons in the system under consideration. On the basis of asymptotic and semiclassical limit arguments, it is however often possible to approximate the Schrdinger dynamics by the so-called Born-Oppenheimer dynamics, in which nuclei behave as classical point-like particles. The internuclei (or interatomic) potential can be computed ab initio, by solving the time-independent electronic Schrdinger equation.

The latter equation is a 3N-dimensional partial differential equation (it is in fact a spectral problem), for which several approximation methods are available. The main of them are the wavefunction methods and the Density Functional Theory (DFT). They lead in particular to the Hartree-Fock model and to the Kohn-Sham model, respectively.

In this introductory lecture, I will first present the electronic Schrdinger equation, and show how to derive from this equation the Hartree-Fock and Kohn-Sham models. Although obtained from totally different approaches, these models have similar mathematical structures. They read as constrained optimization problems, whose Euler-Lagrange equations are nonlinear eigenvalue problems. In the second part of the lecture, I will introduce the Linear Combination of Atomic Orbitals (LCAO) discretization method, and briefly discuss two important numerical issues: ensuring self-consistent field (SCF) convergence, and reducing computational scaling.

At this scale, the equation that rules the interactions between these constitutive elements is the Schrdinger equation. It can be considered (except in few special cases notably those involving relativistic phenomena or nuclear reactions) as a universal model for at least three reasons. First it contains all the physical information of the system under consideration so that any of the properties of this system can be deduced in theory from the Schrdinger equation associated to it. Second, the Schrdinger equation does not involve any empirical parameter, except some fundamental constants of Physics (the Planck constant, the mass and charge of the electron, ...); it can thus be written for any kind of molecular system provided its chemical composition, in terms of natures of nuclei and number of electrons, is known. Third, this model enjoys remarkable predictive capabilities, as confirmed by comparisons with a large amount of experimental data of various types.

Unfortunately, the Schrdinger equation cannot be directly simulated, except for very small chemical systems. It indeed reads as a time-dependent 3(M+N)-dimensional partial differential equation, where M is the number of nuclei and N the number of the electrons in the system under consideration. On the basis of asymptotic and semiclassical limit arguments, it is however often possible to approximate the Schrdinger dynamics by the so-called Born-Oppenheimer dynamics, in which nuclei behave as classical point-like particles. The internuclei (or interatomic) potential can be computed ab initio, by solving the time-independent electronic Schrdinger equation.

The latter equation is a 3N-dimensional partial differential equation (it is in fact a spectral problem), for which several approximation methods are available. The main of them are the wavefunction methods and the Density Functional Theory (DFT). They lead in particular to the Hartree-Fock model and to the Kohn-Sham model, respectively.

In this introductory lecture, I will first present the electronic Schrdinger equation, and show how to derive from this equation the Hartree-Fock and Kohn-Sham models. Although obtained from totally different approaches, these models have similar mathematical structures. They read as constrained optimization problems, whose Euler-Lagrange equations are nonlinear eigenvalue problems. In the second part of the lecture, I will introduce the Linear Combination of Atomic Orbitals (LCAO) discretization method, and briefly discuss two important numerical issues: ensuring self-consistent field (SCF) convergence, and reducing computational scaling.