Matrix Representation of the Propagator

In the last section we have derived an expression for the linear response function or polarization propagator in the frequency domain, Eq. (3.110). However, application of this expression requires that one knows all unperturbed excited states of the system and their energies En or the excitation energies En — od corresponding 

We start by taking the time derivative of the time-dependent linear response function, Eq. (3.107), 

we use the fact that the time derivative of an operator in the interaction picture is the commutator of this operator with the Hamiltonian [see Exercise 3. 10], i.e. 

The Dirac 6 function is sjonmetric and the second term on the right-hand side is again a linear response function, so that we can write 

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