Basic Principles and
Hartree-Fock Theory		 previous next

The Schrödinger Equation and Molecular Hamiltonian

As explained in the introduction, ab initio methods primarily revolve around the solution of the non-realtivistic, time independent, Schrödinger wave equation[1,2,3]

\begin{displaymath}\hat{H} \Psi = E \Psi\end{displaymath} (1)

for the complex systems of electrons found in atoms, electrostatic complexes, ions, and molecules. Whereas some methods in current use may seek to provide a relativistic description of electronic behavior, the methods discussed in this dissertation will be entirely non-relativistic in nature. It is important, however, to note that relativistic effects are largely negligible for atoms in the first row of the periodic table but grow more prominent with increasing nuclear mass and can become substantial for compounds including elements in the third and subsequent rows. For these heavier compounds, the methods discussed in this dissertation may be modified to include relativistic effects through the use of relativistic core potentials or the inclusion of spin-orbit coupling into the electronic Hamiltonian.

Equation (1.1) is an eigenvalue equation in which the Hamiltonian ($\hat{H}$), a hermitian operator, acts upon the wave function ($\Psi$) and returns the wave function multiplied by the total energy of the system (E). The fact that the Hamiltonian is a hermitian operator guarantees that the energy is a real valued quantity and also has the consequence that wave functions which give different energies must be orthogonal. The Schrödinger equation can be solved exactly for both E and $\Psi$ provided the exact form of the time independent Hamiltonian is known. The exact Hamiltonian can frequently be ``derived'' from classical mechanics once the operator form of a few basic quantities such as linear and angular momentum are known. For a molecular system, the exact Hamiltonian in atomic units is given by

\begin{displaymath}-\sum_{i=1}^{n} \frac{1}{2} {\nabla_i}^2-\sum_{A=1}^{M} \fr......i}_j}+\sum_{A=1}^{M} \sum_{B>A}^{M} \frac{Z_A Z_B}{{R_A}_B}.\end{displaymath} (2)

In equation (1.2), the capital letters correspond to the M nuclei, whereas, the lower case letters correspond to the N electrons. Note that the masses of the nuclei are in atomic units and are thus taken relative to the mass of an electron. The physical significance of each term in the molecular Hamiltonian is broken down in Table 1.1 for easy inspection.


Table 1: The terms of the molecular Hamiltonian
${\displaystyle -\sum_{i=1}^{n} \frac{1}{2} {\nabla_i}^2}$ Kinetic energy of the electrons
${\displaystyle -\sum_{A=1}^{M} \frac{1}{2M_A}{\nabla_A}^2}$ Kinetic energy of the nuclei
${\displaystyle -\sum_{i=1}^{n} \sum_{A=1}^{M} \frac{Z_A}{{r_i}_A}}$ Potential energy resulting from the coulombic attraction of ith electron to Ath nucleus
${\displaystyle+\sum_{i=1}^{M} \sum_{i=1}^{n} \sum_{j>i}^{n}\frac{1}{{r_i}_j}}$ Inter-electron repulsion energy
${\displaystyle +\sum_{A=1}^{M} \sum_{B>A}^{M} \frac{Z_A Z_B}{{R_A}_B}}$ Inter-nuclear repulsion energy
This page maintained by Brian C. Hoffman
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