Basic Principles and
Hartree-Fock Theory		 previous next

The Born-Oppenheimer Approximation

Knowing the exact form of the Hamiltonian for a molecular system and the methods for solving an eigenvalue equation, it is theoretically possible to exactly determine the total energy and wave function for that system. However, as pointed out by Dirac, the equations resulting from the interactions of more than two bodies (a single electron and a nucleus) are much too complicated to be solved with current mathematical methods. Theorists are, therefore, forced to resort to a number of approximations that simplify the equations while retaining as much of the physical picture as possible. One of the most commonly used approximations in theoretical chemistry is the Born-Oppenheimer (BO) approximation. [4] 

In a molecular system, the electrons are orbiting very rapidly around the much slower and more massive nuclei and, thus, are able to adjust almost instantaneously to the motions of the nuclei. Therefore, when considering an electronic system, it is a reasonable first approximation to treat the nuclei as fixed in comparison to the motions of the electrons. The effect of fixing the nuclear positions is to eliminate the sum of the nuclear kinetic energies, which are now all zero, from the Hamiltonian and to fix the energy contribution from the repulsions of the nuclei, Enuc(RA). Because Enuc(RA) is an additive constant, it can be pulled outside the molecular Hamiltonian and added to the electronic energy, Ee(RA), resulting from the solution of the Schrödinger wave equation with the remaining Hamiltonian, $\hat{H_e}$, known as the electronic Hamiltonian. Written in atomic units, the electronic Hamiltonian is 

\begin{displaymath}\hat{H_e} = -\sum_{i=1}^{n} \frac{1}{2} {\nabla_i}^2 -\sum_{i......}{{r_i}_A} + \sum_{i=1}^{n} \sum_{j>i}^{n}\frac{1}{{r_i}_j}.\end{displaymath} (3)

Notice that both $\hat{H_e}$ and Ee(RA) depend parametrically on the positions of the nuclei; therefore, the effect of this approximation, the BO approximation, is to factor the full molecular Hamiltonian into both electronic and nuclear parts. For the remainder of this dissertation, we will be concerned with solutions of the wave equation with the electronic Hamiltonian and, as is customary, the subscript e will be dropped in subsequent sections. Within the BO approximation the total electronic energy of the molecular system Etot(RA) is the sum of both Enuc(RA) and Ee(RA). 

The rationale behind the BO approximation can also be exploited to construct a Hamiltonian describing the motion of the nuclei. In a molecule the electrons move much faster than the nuclei; therefore, from the viewpoint of the nuclei it is a reasonable approximation to replace the electronic coordinates by their averaged values, averaged over the electronic wave function. Within this approximation, the molecular Hamiltonian becomes

$\displaystyle \hat{H}$ = $\displaystyle -\sum_{A=1}^{M} \frac{1}{2M_A} {\nabla_A}^2 + \left<-\sum_{i=1}^......{1}{2} {\nabla_i}^2 - \sum_{i=1}^{n}\sum_{A=1}^{M} \frac{Z_A}{{r_i}_A} \right.$ (4)
    $\displaystyle \left. +\sum_{i=1}^{M} \sum_{i=1}^{n} \sum_{j>i}^{n}\frac{1}{{r_i}_j} \right> + \sum_{A=1}^{M} \sum_{B>A}^{M}\frac{Z_A Z_B}{{R_A}_B}$  
  = $\displaystyle -\sum_{A=1}^{M} \frac{1}{2M_A}{\nabla_A}^2 + E_e(r_a, R_A)+\sum_{A=1}^{M} \sum_{B>A}^{M} \frac{Z_A Z_B}{{R_A}_B}$  
  = $\displaystyle -\sum_{A=1}^{M} \frac{1}{2M_A}{\nabla_A}^2 + E_{tot}(r_a, R_A)$  

and depends only on the nuclear positions. This nuclear Hamiltonian depends on the total electronic energy at each choice of nuclear configurations. In the context of the BO approximation, the nuclei move on a potential energy surface obtained by solving the electronic wave equation. 

This page maintained by Brian C. Hoffman
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