Born-Oppenheimer Approximation

We recall the Born-Oppenheimer Approximation for the total wave function $\Psi(q,X)$ of a molecule, cf. Eq. (V.2.15),

$$\Psi(q,X)=\sum_\alpha \phi_\alpha (X) \psi_\alpha(q,X),$$ (1.9)

where $q$ stands for the electronic, $X$ for the nuclear coordinates, and the sum is over a complete set of adiabatic electronic eigenstates with electronic quantum numbers $\alpha$. This form leads to the Schrödinger equation in the adiabatic approximation Eq. (V.2.17),

$$\left[\langle \psi_\alpha\vert \mathcal{H}_{\rm n}\vert\psi_\alph... ...ha(X)\right] \vert\phi_\alpha\rangle_n = \mathcal{E} \vert\phi_\alpha\rangle_n.$$ (1.10)

Here, $E_\alpha(X)$ is the potential acting on the nuclei. We now specify the kinetic energy of the nuclear part for a diatomic molecule,

$$\mathcal{H}_{\rm n} = \frac{{\bf P}^2}{2M} + \frac{{\bf p}^2}{2\mu}.$$ (1.11)

Exercise Check that $\mathcal{H}_{\rm n}=\frac{{\bf p}_1^2}{2m_1} + \frac{{\bf p}_2^2}{2m_2}$.

The effective nuclear Hamiltonian corresponding to an electronic eigenstate $\alpha$ thus is

$$\mathcal{H}_{{\rm n},\alpha} = \langle \psi_\alpha\vert \frac{{\b... ...e \psi_\alpha\vert \frac{{\bf p}^2}{2\mu}\vert\psi_\alpha\rangle + E_\alpha(r),$$ (1.12)

which is a sum of a center-of-mass Hamiltonian and a Hamiltonian for the relative motion of the two nuclei. Only the latter is interesting because it contains the potential $E_\alpha(r)$. Note that both center-of-mass and relative Hamiltonian still contain the geometrical phase terms, cf. Eq. (E.5.2), which however we will neglect in the following.