BO approximation

Because nuclei are much heavier than electrons, the Schrodinger equation can be approximately separated into the nuclear and the electron parts. Thus the electronic Schr”odinger equation for a molecule can be solved separately at each fixed nuclear configuration. This is called the Born-Oppenheimer approximation.

In the following, we will consider the simplest molecule \(H_2^+\), which contains only one electron. This simple system will demonstrate the basic concepts in chemical bonding. The Schr”odinger equation for H\(_2^+\) is:

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

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:

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

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 - B distance.

Note that the Hamiltonian includes also the quantum mechanical kinetic energy for the protons. As such the wavefunction depends on \(\vec{r}_1\), \(\vec{R}_A\) and \(\vec{R}_B\). Because the nuclear mass \(M\) is much larger than the electron mass \(m_e\), the wavefunction can be separated (Born-Oppenheimer approximation):

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

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 be often be separated into vibrational, rotational and translational parts. The electronic Schr”odinger equation can now be written as:

\[{\hat{H}_e\psi_e = E_e\psi_e}\]

Note that Eq. (\ref{eq11.4}) depends parametrically on \(R\) (``one equation for each value of \(R\)’’). The \textit{electronic Hamiltonian} is:

\[{\hat{H}_e = -\frac{\hbar^2}{2m_e}\Delta_e + \frac{e^2}{4\pi\epsilon_0} \left(\frac{1}{R} - \frac{1}{|r_1 - R_A|} - \frac{1}{|r_1 - R_B|}\right)}\]

Because \(R\) is a parameter, both \(E_e\) and \(\psi_e\) are functions of \(R\).