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Matrix Elements

We recall the form of the interaction operator,
$\displaystyle V = E^{\rm d-d}({\bf R})$ $\displaystyle =$ $\displaystyle \frac{{\bf d} {\bf D}}{\vert{\bf R}\vert^3}-3
\frac{({\bf R} {\bf d}) ({\bf R}{\bf D}) }{\vert{\bf R}\vert^5},$ (2.28)

where we now write \bgroup\color{col1}$ {\bf d}$\egroup for \bgroup\color{col1}$ {\bf d}_1$\egroup and \bgroup\color{col1}$ {\bf D}$\egroup for \bgroup\color{col1}$ {\bf d}_2$\egroup. Choosing
$\displaystyle {\bf R} = R{\bf e}_z$     (2.29)

in \bgroup\color{col1}$ z$\egroup-direction, we can write the interaction operator as a quadratic form,
$\displaystyle V = {\bf d}\underline{\underline{M}}{\bf D},\quad \underline{\underline{M}}
= \frac{1}{R^3}$diag$\displaystyle (1,1,-2).$     (2.30)

We abbreviate the matrix elements of the dipole moment components as
$\displaystyle \langle k \vert{\bf d}_\alpha\vert n\rangle \equiv d_\alpha^{kn},...
...le k \vert{\bf D}_\alpha\vert n\rangle \equiv D_\alpha^{kn},\quad \alpha=x,y,z.$     (2.31)

This allows us to write the square in the numerator of Eq. (IX.2.27) as
$\displaystyle {\langle kk'\vert V\vert nn'\rangle \langle nn' \vert V \vert kk'\rangle}$ $\displaystyle =$ $\displaystyle \langle kk'\vert {\bf d}\underline{\underline{M}}{\bf D}\vert nn'\rangle
\langle nn'\vert {\bf d}\underline{\underline{M}}{\bf D}\vert kk'\rangle$ (2.32)
  $\displaystyle =$ $\displaystyle \sum_{\alpha\beta\gamma\delta} d_\alpha^{kn} M_{\alpha\beta} D_\beta^{k'n'}
d_\gamma^{nk} M_{\gamma\delta} D_\delta^{n'k'}$  
  $\displaystyle =$ $\displaystyle \sum_{\alpha\gamma} d_\alpha^{kn} d_\gamma^{nk} D_\alpha^{k'n'}
D_\gamma^{n'k'}M_{\alpha\alpha} M_{\gamma\gamma}.$  

For simplicity, we now assume spherical symmetry for both molecules (which is OK if they are 1-atom molecules, i.e. atoms, but not very realistic otherwise although the following calculations can be generalised to that case as well.) The following property of products of dipole moment operators then holds:
$\displaystyle d_\alpha^{kn} d_\beta^{nk}=\frac{1}{3}\delta_{\alpha\beta} {\bf d...
...ha^{kn} D_\beta^{nk}=\frac{1}{3}\delta_{\alpha\beta} {\bf D}^{kn} {\bf D}^{nk}.$     (2.33)

Then,
$\displaystyle {\langle kk'\vert V\vert nn'\rangle \langle nn' \vert V \vert kk'\rangle}$ $\displaystyle =$ $\displaystyle \frac{1}{9}{\bf d}^{kn} {\bf d}^{nk}
{\bf D}^{k'n'} {\bf D}^{n'k'} \sum_{\alpha\beta} M_{\alpha\beta} M_{\alpha\beta}$  
  $\displaystyle =$ $\displaystyle \frac{1}{9}{\bf d}^{kn} {\bf d}^{nk}
{\bf D}^{k'n'} {\bf D}^{n'k'} \sum_{\alpha} M_{\alpha\alpha} M_{\alpha\alpha}$  
  $\displaystyle =$ $\displaystyle \frac{2}{3}\frac{1}{R^6} {\bf d}^{kn} {\bf d}^{nk}{\bf D}^{k'n'} {\bf D}^{n'k'}.$ (2.34)

The effective interaction therefore is
$\displaystyle V_{\rm eff}^{(2)}(R)$ $\displaystyle \equiv$ $\displaystyle \frac{2}{3}\frac{1}{R^6}\sum_{kk'nn'} p_{k}p_{k'}
\frac{ {\bf d}^{kn} {\bf d}^{nk}{\bf D}^{k'n'} {\bf D}^{n'k'} }{E_{k}+E_{k'} -E_{n}-E_{n'}}.$ (2.35)

If the two molecules are in their groundstates labeled as \bgroup\color{col1}$ k=0$\egroup and \bgroup\color{col1}$ k'=0'$\egroup, this becomes
$\displaystyle V_{\rm eff}^{(2), {\rm GS}}(R)$ $\displaystyle \equiv$ $\displaystyle \frac{2}{3}\frac{1}{R^6}\sum_{nn'}
\frac{ {\bf d}^{kn} {\bf d}^{nk}{\bf D}^{k'n'} {\bf D}^{n'k'} }{E_{0}+E_{0'} -E_{n}-E_{n'}}.$ (2.36)

The interaction potential therefore is negative, corresponding to an attractive interaction, and falls of as \bgroup\color{col1}$ R^{-6}$\egroup.


next up previous contents index
Next: Examples Up: Second oder term: (London) Previous: Derivation from Second Order   Contents   Index
Tobias Brandes 2005-04-26