Matrix Elements

We recall the form of the interaction operator,

$$V = E^{\rm d-d}({\bf R}) = \frac{{\bf d} {\bf D}}{\vert{\bf R}\vert^3}-3 \frac{({\bf R} {\bf d}) ({\bf R}{\bf D}) }{\vert{\bf R}\vert^5},$$ (2.28)

where we now write ${\bf d}$ for ${\bf d}_1$ and ${\bf D}$ for ${\bf d}_2$. Choosing

$${\bf R} = R{\bf e}_z$$ (2.29)

in $z$-direction, we can write the interaction operator as a quadratic form,

$$V = {\bf d}\underline{\underline{M}}{\bf D},\quad \underline{\underline{M}} = \frac{1}{R^3} (1,1,-2).$$ (2.30)

We abbreviate the matrix elements of the dipole moment components as

$$\langle k \vert{\bf d}_\alpha\vert n\rangle \equiv d_\alpha^{kn},... ...le k \vert{\bf D}_\alpha\vert n\rangle \equiv D_\alpha^{kn},\quad \alpha=x,y,z.$$ (2.31)

This allows us to write the square in the numerator of Eq. (IX.2.27) as

$${\langle kk'\vert V\vert nn'\rangle \langle nn' \vert V \vert kk'\rangle} = \langle kk'\vert {\bf d}\underline{\underline{M}}{\bf D}\vert nn'\rangle \langle nn'\vert {\bf d}\underline{\underline{M}}{\bf D}\vert kk'\rangle$$ (2.32)

$$= \sum_{\alpha\beta\gamma\delta} d_\alpha^{kn} M_{\alpha\beta} D_\beta^{k'n'} d_\gamma^{nk} M_{\gamma\delta} D_\delta^{n'k'}$$

$$= \sum_{\alpha\gamma} d_\alpha^{kn} d_\gamma^{nk} D_\alpha^{k'n'} D_\gamma^{n'k'}M_{\alpha\alpha} M_{\gamma\gamma}.$$

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:

$$d_\alpha^{kn} d_\beta^{nk}=\frac{1}{3}\delta_{\alpha\beta} {\bf d... ...ha^{kn} D_\beta^{nk}=\frac{1}{3}\delta_{\alpha\beta} {\bf D}^{kn} {\bf D}^{nk}.$$ (2.33)

Then,

$${\langle kk'\vert V\vert nn'\rangle \langle nn' \vert V \vert kk'\rangle} = \frac{1}{9}{\bf d}^{kn} {\bf d}^{nk} {\bf D}^{k'n'} {\bf D}^{n'k'} \sum_{\alpha\beta} M_{\alpha\beta} M_{\alpha\beta}$$

$$= \frac{1}{9}{\bf d}^{kn} {\bf d}^{nk} {\bf D}^{k'n'} {\bf D}^{n'k'} \sum_{\alpha} M_{\alpha\alpha} M_{\alpha\alpha}$$

$$= \frac{2}{3}\frac{1}{R^6} {\bf d}^{kn} {\bf d}^{nk}{\bf D}^{k'n'} {\bf D}^{n'k'}.$$ (2.34)

The effective interaction therefore is

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

If the two molecules are in their groundstates labeled as $k=0$ and $k'=0'$, this becomes

$$V_{\rm eff}^{(2), {\rm GS}}(R) \equiv \frac{2}{3}\frac{1}{R^6}\sum_{nn'} \frac{ {\bf d}^{kn} {\bf d}^{nk}{\bf D}^{k'n'} {\bf D}^{n'k'} }{E_{0}+E_{0'} -E_{n}-E_{n'}}.$$ (2.36)

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