The electrical dipole moment and polarizability
As an example of a molecular property, when we apply a uniform electric field \(\require{physics} \vb{F}\), the energy of the system becomes \[\begin{equation} E(\vb{F}) % = E^{(0)} % - \sum_m \mu_m F_m % - \frac{1}{2} \sum_{mn} \alpha_{mn} F_m F_n % + \order{\qty(\vb{F}^{c})^3} % \thinspace , \end{equation}\] in which \(\boldsymbol{\mu}\) is the (permanent) dipole moment \[\begin{equation} \mu_m = % - % \eval{ % \dv{ % E(\vb{F}) % }{F_m} % }_{\vb{F}_0} \end{equation}\] and \(\boldsymbol{\alpha}\) is the dipole-polarizability tensor(T. Helgaker 1998) \[\begin{equation} \alpha_{mn} = % - % \eval{ % \frac{ % \dd{^2}{E(\vb{F})} % }{\dd{F_m} \dd{F_n}} % }_{\vb{F}_0} % \thinspace , % \label{eq:electric_polarizability} \end{equation}\] both calculated at zero field \(\vb{F} = \vb{F}_0 = \vb{0}\).
The electric field \(\vb{F}\) does not cause a change of AO metric, so we can immediately write for the second-quantized Hamiltonian: \[\begin{equation} \hat{\mathcal{H}} % = % \sum_{pq}^K % h_{pq}(\vb{F}) % \hat{E}_{pq} % + \frac{1}{2} \sum_{pqrs}^K % g_{pqrs} % \hat{e}_{pqrs} % + h_{\text{nuc}}(\vb{F}) \thinspace . \end{equation}\]
Now, how do the one-electron integrals and the scalar term look like, in the presence of that uniform electric field? The external potential at a point \(\vb{r}\) can be written as \[\begin{equation} \phi_{\text{ext}}(\vb{r}) = - \vb{F} \vdot \vb{r} \thinspace , \end{equation}\] such that the one-electron integrals are \[\begin{equation} h_{pq}(\vb{F}) % = \int \dd{\vb{r}} % \phi_p^*(\vb{r}) % \qty( % - \frac{1}{2} \laplacian % - \sum_K \frac{Z_K}{|\vb{r} - \vb{R}_K|} % + \vb{F} \vdot \vb{r} % ) % \phi_q(\vb{r}) \end{equation}\] and the scalar term becomes \[\begin{equation} h_{\text{nuc}}(\vb{F}) % = % \frac{1}{2} \sum_{K \neq L} % \frac{ Z_K Z_L }{ \norm{\vb{R}_K - \vb{R}_L} } % - \sum_K Z_K \thinspace \vb{F} \vdot \vb{R}_K % \thinspace . \end{equation}\] Therefore, we can write the perturbed Hamiltonian in a uniform electric field \(\vb{F}\) as \[\begin{equation} \hat{\mathcal{H}}(\vb{F}) % = % \hat{\mathcal{H}}(\vb{F}_0) % - \vb{F} \vdot \qty( % \hat{\boldsymbol{\mu}} + \sum_K Z_K \vb{R}_K % ) % \thinspace , \end{equation}\] in which we have defined the electronic dipole operator \[\begin{equation} \hat{\mu}_m % = \sum_{pq}^K \mu_{m, pq} \hat{E}_{pq} % \thinspace , \end{equation}\] with \(\boldsymbol{\mu}_{pq}\) the electronic dipole matrix elements: \[\begin{align} \mu_{m, pq} % & = % - \pdv{ % h_{pq}(\vb{F}) }{ F_m } \\ & = % - \int \dd{\vb{r}} % \phi_p^*(\vb{r}) \thinspace % r_m \thinspace % \phi_q(\vb{r}) % \thinspace . \end{align}\]
The first-order partial perturbation derivative of the Hamiltonian then becomes particularly simple: \[\begin{equation} \eval{ % \pdv{ % \hat{\mathcal{H}}(\vb{F}) % }{F_m} % }_{\vb{F}_0} % = - \hat{\mu}_m - \sum_K Z_K R_{K, m} \end{equation}\] and the second-order partial perturbation derivative even vanishes: \[\begin{equation} \eval{ % \pdv{ % \hat{\mathcal{H}}(\vb{F}) % }{F_m}{F_n} % }_{\vb{F}_0} % = 0 % \thinspace . \end{equation}\]
For wave function models using the Rayleigh-Ritz quotient, the molecular dipole moment then becomes the (negative of the) expectation value over the first-order perturbation partial derivative, which becomes \[\begin{align} \mu_m % = \sum_K Z_K \thinspace R_{K, m} % + \sum_{pq}^K \mu_{m, pq} D_{pq} % \thinspace . % \label{eq:total_dipole_moment} \end{align}\] These expressions are also found in (Cramer 2004, Levine2014).
Only the first non-zero electric multipole moment is independent of origin (Levine 2014), so we will adopt the best practice to always specify the origin of our molecular calculations. If we change the origin of our molecular coordinate system, i.e. \[\begin{equation} \vb{r}' = \vb{r} - \vb{o} \thinspace , \end{equation}\] the dipole moment becomes \[\begin{align} \mu'_m % &= % \sum_K Z_K (R_{K, m} - o_m) % - \sum_{pq}^K (r_m - o_m)_{pq} D_{pq} \\ &= % \sum_K Z_K R_{K, m} % - \sum_{pq}^K r_{m, pq} D_{pq} % - o_m \qty( \sum_K Z_K - N) % \thinspace , \end{align}\] where we have used the trace property of the 1-DM. This means that only for neutral molecules, the dipole moment is independent of the choice of origin.