The energies and wavefunctions of stationary states of a system are given by the
solutions of the Schrödinger Equation :
In this equation is the Hamiltonian operator which in this case gives the kinetic and potential energies of a system of atomic nuclei and electrons. As we shall see below it is analogous to the classical kinetic energy of the particles and the Coulomb electrostatic interactions between the nuclei and electrons. is a wavefunction, one of the solutions of the eigenvalue equation. This wavefunction depends on the coordinates of the electrons and the nuclei. The Hamiltonian is composed of three parts : the kinetic energy of the nuclei, the kinetic energy of the electrons, and the potential energy of nuclei and electrons.
Schrödinger equation :
Four approximations are commonly (but not necessarily) made :
The motivation behind this is that the electrons are so much lighter than the nuclei that their motion can easily follow the nuclear motion. In practice, this approximation is usually valid. From this point we will look at the electronic wavefunction which is obtained by solving the electronic Schrödinger equation :
This equation still contains the positions of the nuclei, however not as variables but as parameters.
The electronic Hamiltonian contains three terms : kinetic energy, electrostatic
interaction between electrons and nuclei, and electrostatic repulsion between
electrons. In order to simplify expressions and to make the theory independent
of the experimental values of physical constants, atomic units are introduced :
e = 1 charge of electron
m = 1 mass of the electron
= 1 Planck's constant divided by 2 pi
Derived atomic units of length and energy are :
1 bohr =
1 hartree = J = 627.51 kcal/mol
With these units the electronic Hamiltonian is :
The symbol is the Laplace operator (also called "del-squared"). The total energy in the Born-Oppenheimer model is obtained by adding the nuclear repulsion energy to the electronic energy :
The total energy defines a potential energy hypersurface E=f(Q) which can be used to subsequently solve a Schrödinger equation for the nuclear motion :
In the following section we shall deal with the important problem of solving the electronic Schrödinger equation.