First oder term: static dipole-dipole interaction

The first order term in $V_{\rm eff}$ in our expansion Eq. (IX.2.20) is

$$V_{\rm eff}^{(1)} = \sum_i \sum_{k} p_k \langle k\vert v_i \vert ... ...'\vert v_i' \vert k'\rangle = \sum_i \langle v_i \rangle \langle v_i' \rangle .$$ (2.23)

This is just given by the expectation value of the terms that make up the interaction potential $= \sum_i v_i \otimes v_i'$, Eq. (IX.2.22). For the dipole-dipole interaction, this gives

$$V_{\rm eff}^{(1)}({\bf R}) = \frac{\langle {\bf d}\rangle \langle... ...ngle {\bf d}\rangle ) ({\bf R}\langle {\bf d}'\rangle ) }{\vert{\bf R}\vert^5}.$$ (2.24)

Since the first order is linear in the interaction potential operator $V$, the effective $V_{\rm eff}^{(1)}$ is essential just the $V$ with all operators replaced by their expectation values. This is the static dipole-dipole interaction between the molecules. The corresponding force between the two dipoles is

$${\bf F}_{\rm eff}^{(1)}({\bf R})= - {\bf\nabla} V_{\rm eff}^{(1)}({\bf R}).$$ (2.25)

Its form is just as in the classical dipole-dipole interaction. However, this interaction is zero if one of the expectation values of the dipole moment operators vanishes. Such molecules are said to have no static dipole moment.