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Born Oppenheimer Approximation

The Born-Oppenheimer Approximation (BOA) is a central approximation in molecular spectroscopy. It is an assumption that the electronic and nuclear motion can be separated i.e. the wavefunctions can be separated into non-interacting parts. There are no interactions between the discrete adiabatic states within the Born-Oppenheimer approximation. This breaks down at large internuclear separations, where the separation between electronic states is the same order of magnitude as the coupling between them.

The total Hamiltonian can be expressed as:


\begin{displaymath}
{\rm {\bf H}= {\bf H}_{el} + {\bf H}_{rel} + {\bf H}_{rad} + {\bf H}_{rot}}
\end{displaymath} (3.4)

where H$_{\rm el}$ is the non-relativistic electronic Hamiltonian describing the Coulombic interactions between nuclei and electrons. The eigenvalues are the Born-Oppenheimer potentials and the eigenfunctions are the adiabatic wavefunctions,:


\begin{displaymath}
{\rm {\bf H}_{el}= -\frac{\hbar^{2}}{2m}\sum_{n}\nabla_{n}^{2}+ V(\hat{r},R)}
\end{displaymath} (3.5)

where m is the mass of an electron (labelled n), V is the electrostatic potential as a function of the electronic co-ordinates $\hat{r}$ and the nuclear co-ordinate R. H$_{\rm rel}$ is the relativistic interaction which is approximated to the spin-orbit interaction here (neglecting spin-spin and hyperfine interactions):


\begin{displaymath}
{\rm {\bf H}_{rel}= {\bf H}_{so} = \sum_{n}A_{n}{\bf\underline {L}_{n}.\underline{S}_{n}}}
\end{displaymath} (3.6)

This term is responsible for mixing between the $^{1}\Pi $ and both the $^{3}\Sigma$ and $^{3}\Pi$ states in SiH+ and GeH+.

H$_{\rm rad}$ and H$_{\rm rot}$ are the radial and rotational components of the nuclear kinetic energy operator T$_{\rm n}$:


\begin{displaymath}
{\rm {\bf T}_n = {\bf H}_{\rm rot}+{\bf H}_{\rm rad}}
\end{displaymath} (3.7)


\begin{displaymath}
{\rm {\bf H}_{rad}= - \frac{\hbar}{2\mu R^{2}} . \frac{\partial}{\partial R}.\left(R^{2}.\frac{\partial}{\partial R}\right)}
\end{displaymath} (3.8)

and

\begin{displaymath}
{\rm {\bf H}_{rot}= - \frac{\hbar}{2\mu R^{2}}.\left[\underline{\bf J}-\underline{\bf L}-\underline{\bf S}\right]^{2}}
\end{displaymath} (3.9)

where $\mu $ is the reduced mass. H$_{\rm rot}$ furthermore can be separated into a diagonal part (corresponding to the rotational energy of the state):


\begin{displaymath}
{\rm E_{rot}(R)= \frac{\hbar^{2}}{2\mu R^{2}}\left[J(J+1)- \Omega^{2} \right]}
\end{displaymath} (3.10)

and an off-diagonal part, the second term of which is responsible for rotational electronic (Coriolis) coupling between states of different $\Lambda$ but identical $\Sigma$. This is the origin of the lambda doubling, which is strongly R dependent,:


\begin{displaymath}
{\rm {\bf H}'_{rot}(R)= \frac{\hbar^{2}}{2\mu R^{2}}\left[(L_+S_--L_-S_+)-(J_+L_-+J_-L_+)-(J_+S_-+J_-S_+)\right]}
\end{displaymath} (3.11)

where

$\displaystyle {\bf J}_{\pm}={\bf J}_{x}\pm i{\bf J}_{y}$     (3.12)
$\displaystyle {\bf L}_{\pm}={\bf L}_{x}\pm i{\bf L}_{y}$     (3.13)
$\displaystyle {\bf S}_{\pm}={\bf S}_{x}\pm i{\bf S}_{y}$     (3.14)

The H $^{\prime}_{\rm rot}$ term couples the angular momenta and causes the interaction between different electronic states. Rotational-electronic coupling between electronic states manifests itself as Feshbach resonances.


next up previous contents
Next: Predissociation by tunnelling Up: Predissociation Previous: Predissociation   Contents
Tim Gibbon
1999-09-06