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Lambda doubling

For all states with $\Lambda > 0$, there is a degeneracy associated with the two possible values $\pm\vert\Lambda\vert$. This arises due to the interaction between the rotational and electronic motion and is particularly significant for $\Pi$  electronic states.

Consider the case of a $^{1}\Pi $ state. The $^{1}\Pi^{+}$ state can interact with a nearby $^{1}\Sigma ^{+}$ state, leading to a shift in the energies for both the $^{1}\Sigma ^{+}$ and $^{1}\Pi^{+}$ electronic states. The degeneracy of the $^{1}\Pi^{+}$ and $^{1}\Pi^{-}$ levels is then removed. It can be considered that this splitting arises due to the inability of the electrons to follow the nuclear motion as the rotation increases, leading to a mixture between the $^{1}\Pi $ state with electronic states of different $\Lambda$. The two electronic states, $^{1}\Pi^{+}$ and $^{1}\Pi^{-}$ have different symmetries and are labelled accordingly [72] :


$\displaystyle +(-1)^{J} ~[{\rm ~e~ levels}]$      
$\displaystyle -(-1)^{J} ~[{\rm ~f~ levels}]$      

For example, consider the origin of the lambda doubling in GeH+ pictorially. Figure 3.4 shows interactions between the possible orientations of the p orbital of the germanium ion and the hydrogen s orbital [48]. The germanium and hydrogen nuclei lie along the z axis. If the electrons were able to follow the motion of the nuclei exactly with no `slip', the two $\pi$ orbitals would be degenerate. However, with increasing molecular rotation, the electrons tend to `lag' behind the nuclei. The py orbital (e-component) takes on a partly $\sigma $, whereas the pz orbital has $\pi$ character. The distinction between these characteristics becomes impossible with increasing rotation, thus the degeneracy of the $^{1}\Pi $ state is lifted. The px orbital (f-component) remains unaffected with increased molecular rotation as it lies out of the plane of rotation.

The selection rules for electric dipole transitions obey the selection rule ${\rm\Delta J=0, \pm 1}$ which in terms of e and f levels correlates to:


$\displaystyle \Delta {\rm J}=0, {\rm ~e\leftrightarrow f}$     (3.1)
$\displaystyle \Delta {\rm J}=\pm1,~ {\rm e\leftrightarrow e ~and~ f\leftrightarrow f}$     (3.2)

For GeH+ the e component of the $^{1}\Pi $ state is shifted (by rotational-electronic coupling to the $^{1}\Sigma ^{+}$ state) with respect to the f component and can be expressed as:


\begin{displaymath}
{\rm E_{v}(e)= E_{v}(f)+q_{v}(J(J+1))-q_{d}[J(J+1)]^{2}+\dots}
\end{displaymath} (3.3)

Figure 3.4: Origin of the $^{1}\Pi -^{1}\Sigma $ lambda doubling interaction in GeH+. Note that the $\Pi ^{f}$ orbital lies out of the plane of rotation (yz).
\resizebox{5in}{!}{
\includegraphics{figures/lamborigin.eps}}


next up previous contents
Next: Predissociation Up: Spectroscopic theory Previous: Coupling of Angular Momentum   Contents
Tim Gibbon
1999-09-06