The Born-Oppenheimer Approximation - Quantum Mechanics for Chemistry

The simplest molecule¶

Coordinates of the hydrogen molecule ion — Fig. 1 Coordinates used to describe the $H_2^+$ molecule.

In the following, we will consider the simplest molecule $H_2^+$ , which contains only one electron. This simple system demonstrates the basic concepts of chemical bonding. The Schrodinger equation for $H_2^+$ is:

{H\psi(\vec{r}_1,\vec{R}_A,\vec{R}_B) = E\psi(\vec{r}_1,\vec{R}_A,\vec{R}_B)}

(1)

where $\vec{r}_1$ is the vector locating the (only) electron and $\vec{R}_A$ and $\vec{R}_B$ are the positions of the two protons. The Hamiltonian for $H_2^+$ is:

{\hat{H} = -\frac{\hbar^2}{2M}(\Delta_A + \Delta_B) - \frac{\hbar^2}{2m_e}\Delta_e+ \frac{e^2}{4\pi\epsilon_0}\left(\frac{1}{R} - \frac{1}{r_{1A}} - \frac{1}{r_{1B}}\right)}

(2)

where $M$ is the proton mass, $m_e$ is the electron mass, $r_{1A}$ is the distance between the electron and nucleus A, $r_{1B}$ is the distance between the electron and nucleus B, and $R$ is the A to B distance.

Note that the Hamiltonian also includes the quantum mechanical kinetic energy of the protons. As such, the wavefunction depends on $\vec{r}_1$ , $\vec{R}_A$ , and $\vec{R}_B$ .

The Born-Oppenheimer approximation¶

Because the nuclear mass $M$ is much larger than the electron mass $m_e$ , the wavefunction can be separated (the Born-Oppenheimer approximation):

{\psi(\vec{r}_1,\vec{R}_A,\vec{R}_B) = \psi_e(\vec{r}_1, R)\psi_n(\vec{R}_A, \vec{R}_B)}

(3)

where $\psi_e$ is the electronic wavefunction that depends on the distance $R$ between the nuclei and $\psi_n$ is the nuclear wavefunction depending on $\vec{R}_A$ and $\vec{R}_B$ . It can be shown that the nuclear part can often be further separated into vibrational, rotational, and translational parts. The electronic Schrodinger equation can now be written as:

Because $R$ is a parameter, both $E_e$ and $\psi_e$ are functions of $R$ . Solving the electronic problem at a sequence of fixed nuclear geometries traces out the potential energy surface on which the nuclei move, which is the central idea that makes molecular electronic structure tractable.