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tags: Theory
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# Born–Oppenheimer approximation

$$\hat H \Psi(\vec{R})= E\Psi(\vec{R})$$

For multi-electrons and nucleus

$$\hat H = \hat{V}_{ee} +\hat{V}_{eN} +\hat{V}_{NN} +\hat{T}_{e} +\hat{T}_{N}$$

$V$ for potential energy

electron-electron interaction
$$\hat V_{ee}(\vec{r}) = \sum_{i>j}\frac{1}{r_{ij}}$$
electron-nucleus interaction
$$\hat V_{eN}(\vec{r},\vec{R}) = \sum_{i,A}\frac{1}{r_{iA}}$$
nucleus-nucleus interaction
$$\hat V_{NN}(\vec{R}) = \sum_{A>B}\frac{1}{r_{AB}}$$

$T$ for kinetic energy

electrons kinetic energy
$$\hat T_{e}(\vec{R}) = \sum_{i}\nabla_{r_i}^2$$
nucleus kinetic energy
$$\hat T_{N}(\vec{R}) = \sum_{A}\nabla_{R_A}^2$$
$\vec{r}=(\vec{r}_1,\vec{r}_2,\cdots)$ for electrons position set
$\vec{R}=(\vec{R}_1,\vec{R}_2,\cdots)$ for nucleus position set

Separate Hamiltonian operator into two parts

$$\hat H = \hat H_e(\vec{r}) + \hat H_N(\vec{R})$$

$$\hat{H}_e(\vec{r};\vec{R})=\hat{H}_e(\vec{r}) = \hat{V}_{ee}(\vec{r}) + \hat{V}_{eN}(\vec{r};\vec{R})+\hat{T}_{e}(\vec{r})$$

$$\hat{H}_N(\vec{R})=\hat{T}_{N}(\vec{R})+\hat{V}_{NN}(\vec{R})$$

Assume nucleus don't move when we calculat the energy of electrons
$$\Psi(\vec{r},\vec{R})=\Phi_e(\vec{r};\vec{R})\Phi_N(\vec{R})$$

Get the potential energy of the nucleus in the positions

$$\hat H \Psi(\vec{r},\vec{R})= \hat H_e(\vec{r})\Phi_e(\vec{r};\vec{R})\Phi_N(\vec{R}) + \hat H_N(\vec{R})\Phi_e(\vec{r};\vec{R})\Phi_N(\vec{R})$$

$$E\Phi_N(\vec{R})=(V
_e(\vec{R}) +\hat H_N(\vec{R})) \Phi_N(\vec{R})$$

$$E\Phi_N(\vec{R})=(V
(\vec{R}) +\hat T_N(\vec{R})) \Phi_N(\vec{R})$$
Wiki Link : https://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation