September 26 - 27, 2008
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 Schrödinger 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 Schrödinger 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).
In my first lecture (Mathematical modelling of electronic structures), I will present the mathematical properties of the time-independent electronic Schrödinger equation, and show how to construct variational approximations of this equation, in the framework of wavefunction methods. I will mainly deal with the Hartree-Fock approximation; more advanced wavefunction methods will then be presented in the lectures by L. Slipchenko and A. Krylov.
In my second lecture (Mathematical aspects of density functional theory), I will examine the mathematical foundations of DFT. I will compare the constrained-search approach proposed by Levy and involving pure states, with the one proposed by Lieb and involving mixed states. These two approaches lead to Kohn-Sham and extended Kohn-Sham models respectively. I will then review the mathematical properties of the Kohn-Sham LDA and GGA models (corresponding to the first two rungs of the ladder of approximations previously presented by J. Perdew). Lastly, I will introduce the concept of bulk (or thermodynamic) limit, which allows one to rigorously derive DFT models for the condensed phase from molecular DFT models by letting the number of nuclei and electrons go to infinity in an appropriate way.
Coupled-cluster (CC) and equation-of-motion coupled-cluster (EOM-CC)
methods are the most reliable and versatile tools of electronic structure
theory. The exponential CC ansatz ensures size-extensivity. By increasing
the excitation level, systematic approximations approaching the exact
many-body solution are possible. EOM extends the CC methodology
(applicable to the wave functions dominated by a single Slater
determinant) to the open-shell and electronically excited species
with multi-configurational wave-functions. The lecture will present an
overview of CC and EOM-CC methods and highlight their important formal
properties.
Suggested reading:
1. T. Helgaker, P. Jorgensen, and J. Olsen, Molecular electronic structure
theory; Wiley & Sons, 2000.
2. A. I. Krylov, Equation-of-motion coupled-cluster methods for open-shell
and electronically excited species: The hitchhiker's guide to Fock space
Ann. Rev. Phys. Chem. v. 59, 433 (2008).
3. D. Mukherjee and S. Pal, Use of cluster expansion methods in the
open-shell correlation problem, Adv. Quantum Chem. v. 20, 291 (1989).
4. R.J. Bartlett and J.F. Stanton, Applications of post-Hartree-Fock
methods: A tutorial, Rev. Comp. Chem. v. 5, 65 (1994).
Density Functional Theory is one of the most successful approaches for computing the electronic structure of materials and is currently used to study thousand-atom systems today. The goals of this tutorial are two-fold. First, I will present the basic equations and ideas behind the solution of the many-body electronic Schrodinger equation through the Density Functional approach that leads to the Kohn-Sham equations. I will then discuss the most commonly used approximations and how they translate into the two main types of numerical algorithms used to solve the Kohn-Sham equations. In the second part of this talk, I will outline the major computational components of plane wave DFT codes, which represent the dominant approach for the simulation of materials science problems. Finally, I will discuss some of the computational challenges, including parallelization, of the resulting large-scale simulations.
Combined with 9/26 abstract.
My task is to discuss the basic principles of Quantum Mechanics
which are crucial for the electronic structure theory. The following
topics will be covered: the correspondence principle which connects
Classical Mechanics and Quantum Mechanics; the uncertainty principle and
related questions; the superposition principle. We shall discuss the
Hilbert space of wavefunctions, and the operators associated with the
observables. We shall illustrate the theory by considering the properties
of angular momentum (orbital and spin). The theory of hydrogen atom will
constitute the important part of the lecture.
Electronic structure theory predicts what atoms, molecules and solids can exist, and with what properties. The density functional theory (DFT) of Hohenberg, Kohn and Sham 1964-65 is now the most widely used method of electronic structure calculation in both condensed matter physics and quantum chemistry. Walter Kohn and John Pople shared the 1998 Nobel Prize in Chemistry for this theory and for its computational implementation. This tutorial will begin by explaining why the computational demands are far smaller in DFT than in many-electron wavefunction theory, especially for large molecules and for solids. Two fundamental theorems of DFT will be presented and proven by the transparent constrained-search approach of Levy: (1) The ground-state electron density n of a system of N electrons in the presence of an external scalar multiplicative potential v determines v and hence all properties of the system. (2)
There exists a universal density functional Q[n] such that minimization of Q[n] +
for given N and v yields the ground-state density n and energy E. For accurate
approximation of Q[n], one needs to introduce single-electron Kohn-Sham orbitals that yield the non-interacting kinetic energy part of Q[n] exactly, reducing the ground-state
problem to a problem of noninteracting electrons moving in a selfconsistent density-dependent effective potential. Then the only part of Q[n] that must be approximated is the many-body exchange-correlation energy E_xc[n], which will be defined precisely. From the definition, many exact properties of E_xc[n] can be derived and employed as constraints to construct nonempirical or empirical approximations to E_xc[n]. Some important constraints will be reviewed. A ladder of approximations, from the simplest
local density approximation to the most elaborate nonlocal approximations, will be reviewed, the surprising success of even the simplest one will be explained, and error estimates for each level will be presented. It will become apparent that local or semilocal approximations can suffice for most properties of most systems at or near the equilibrium nuclear positions, but that full nonlocality is needed to describe situations in which electrons are shared between separated subsystems, with noninteger average electron number in each subsystem. The exact density functional theory of such systems, as presented by Perdew, Parr, Levy and Balduz 1982, will be presented. The time-dependent density functional theory, which can describe time-dependent and excited
states, will be briefly reviewed. Some outstanding physical and mathematical problems of DFT will be summarized.
After separating the electronic end nuclear coordinates through the Born-Oppenheimer approximation, one may attempt to solve the electronic Schrodinger equation by a hierarchy of wave function techniques. The lowest level in this hierarchy and the core method of the wave function quantum chemistry is the Hartree-Fock (HF) model, in which each electron moves in a mean field created by all other electrons. In order to get a chemical accuracy, one needs to correlate the motion of the electrons. This may be achieved by either the perturbation theory or by the configuration interaction procedure, employed on the top of the HF wave function. Basic ideas and approximations used in these wave function methods, as well as numerical approaches, challenges, and limitations will be discussed.