Alternate derivation of linear response function

I’ve given a derivation before of the density-density response function, but today I want to give an alternate derivation of the more general linear-response function, which will prove to be useful in the derivation of the time-dependent Hartree-Fock equations (TDHF), also known in the nuclear physics community as the Random Phase Approximation (RPA). This derivation is largely taken from McWeeny (1989).

In order to derive the response function — also called frequency-dependent polarizability — we must first partition the Hamiltonian into two parts: the time-independent Hamiltonian, and the time dependent response:

Furthermore, the time-dependent Schrodinger equation is given as:

In the interaction picture, . Thus we can partition the time-dependent wavefunction (expanding over basis of complete eigenstates) as

Where

are real, positive, exact excitation frequencies of unperturbed system. Note that . We are assuming that is turned on slowly at time . Substitute 1 and 3 into 2, separate the orders, and impose the boundary conditions that and at . This gives

If we let

Where a ‘fixed’ Hermitian operator determines the `shape’ of the perturbation, while time dependence is confined to the (real) ‘strength’ factor . For a perturbation beginning at time up to time ,

Which, to first order, determines the perturbed wavefunction. Now we are interested not in the perturbed wavefunction per se, but rather in the response of an observable to the perturbation.

where

This is a time correlation function, relating fluctuation of at time t to the strength of the perturbation at some earlier time . is defined only for , in accordance with the principle of causality. Thus, it is a function only of the difference . Recalling the definitions of the Fourier transform :

Then instead of 8, we have:

Requiring to be Hermitian,

Now,

Which, upon combining the expressions for so as to `Hermitize’ the expression:

Thus

with real. Instead of working in the time domain, we may also consider the response in terms of a single oscillatory perturbation. This means that

To ensure builds smoothly from zero at , we can introduce a convergence factor with the initial condition and , which gives:

Then, collecting terms of :

Finally:

Which is the response function, or frequency-dependent polarizability.