# Born-Oppenheimer approximation

by schafspelz
Tags: approximation, bornoppenheimer
 P: 5 Thanks for your interest! Just to make sure that there aren't any misunderstandings, I will also repeat the main definitions. As usual, the total wavefunction $\Psi({\bf r},{\bf R})$ expanded as a series of electronic wavefunctions $\chi_k({\bf r};{\bf R})$ $\Psi({\bf r},{\bf R}) = \sum\limits_k \chi_k({\bf r};{\bf R}) \eta_k({\bf R})$. The electronic Hamiltonian ${\cal H}_\mathrm{e}$ is expressed as ${\cal H}_\mathrm{e}=T_\mathrm{e}+V_\mathrm{ee}+V_\mathrm{en}+V_\mathrm{nn}$, which satisfy the electronic Schrödinger equation ${\cal H}_\mathrm{e} \chi_k({\bf r};{\bf R}) = E_k({\bf R}) \chi_k({\bf r};{\bf R})$. The full Hamiltonian is defined as ${\cal H}=T_\mathrm{n}+E_k({\bf R})$, whose matrix elements should be calculated in the basis of $\eta_k({\bf R})$. Then the Hamiltonian reads ${\cal H} = \left(\begin{array}{ccc} T_\mathrm{n}+E_1({\bf R}) & 0 & 0 & \cdots\\ 0 & T_\mathrm{n}+E_2({\bf R}) & 0 & \cdots\\ 0 & 0 & T_\mathrm{n}+E_3({\bf R}) & \cdots\\ \vdots & \vdots & \vdots & \ddots \end{array}\right) + \underbrace{\left(\begin{array}{ccc} \tilde{H}_{11} & \tilde{H}_{12} & \tilde{H}_{13} & \cdots\\ \tilde{H}_{21} & \tilde{H}_{22} & \tilde{H}_{23} & \cdots\\ \tilde{H}_{31} & \tilde{H}_{32} & \tilde{H}_{33} \cdots\\ \vdots & \vdots & \vdots & \ddots \end{array}\right)}_\text{Non-adiabacity operator}$. The non-adiabacity operator contains the following several elements. $\tilde{H}_{ij} = -\frac{\hbar}{2M}\Big( \underbrace{2\left\langle\chi_i({\bf r};{\bf R})\left|\nabla_\mathrm{n}\right|\chi_j({\bf r};{\bf R})\right\rangle \nabla_\mathrm{n}}_\text{first order} + \underbrace{\left\langle\chi_i({\bf r};{\bf R})\left|\nabla_\mathrm{n}^2\right|\chi_j({\bf r};{\bf R})\right\rangle}_\text{second order} \Big)$ If $i\neq j$, the elements of $\tilde{H}_{ij}$ only appear as off-diagonal elements and are neglected in the Born-Oppenheimer approximation. So my question is whether the $E_k({\bf R})$ is already the adiabatic potential energy surface with an avoided crossing OR does the avoided crossing occur only when the off-diagonals are accounted for (which seems to be analogous to the concept of the adiabatic theorem)?