In the previous post I was interested in how the force constants change with an applied field. In order to deal with that question, the dipole moment and polarizability were defined as electric field derivatives.
In this post I expand on where these definitions come from, and how such polarizabilities may be calculated – at least, within HartreeFock theory.
HelmannFeynman theorem
The dipole moment and polarizability definitions ultimately arise from the HelmannFeynman theorem.
The HelmannFeynman theorem states that if \( \lambda \) is a parameter on which the Hamiltonian depends then
The parameter \( \lambda \) may be

The coordinate of a particular nucleus (in which case the lefthandside represents the force on that nucleus along that coordinate), or

It may represent a component of an applied electric field (in which case the lefthand side represents a component of the dipole moment).
Proof
The proof is trivial:
The last term is zero because of the normalization condition \( \braket{\Psi}{\Psi} = 1 \).
Dipole and polarizabilities as energy derivatives
In order to understand how the dipole moment arises from the HelmannFeynman theorem we need to know what the Hamiltonian for a system looks like in the presence of an electric field.