Definition of the response kernel

Because G itself is defined by an orbital functional derivative, the increment AGs is proportional to a functional second derivative. It is convenient to define a response kernel f such that A O) induces Av = I] /l - nb)[(j f b)cbj(t) + 

The response kernel fc is a linear operator such that fc pj = j v. Variations of unoccupied orbitals 8 pa (V a) in the functional Ec are induced fly variations of occupied orbitals 8 / (i V) through unitarity. The combined total response kernel is the linear operator fh + fx + fc = fhxc = u + fc. 

In the exchange-only limit, these are the TDHF or RPA equations (Thouless [407], p. 89). If Aw = 0, excitation energies are values of tico for which the determinant of the residual homogeneous equations vanishes. 

The action integral defined in standard V-electron theory is 

While F[ t ] = [ ], the mean values of the time derivative are not equal. Even if H is independent of time, these values are constants J2, n and E, respectively. These constants can be reconciled by inserting a gauge potential y = E — JA nif-i into A, determined by equating the time-dependent phase factors. This also preserves the normalization (T T) = (4 4 ) = 1 for the time-dependent wave functions, so that the definition of correlation energy = ( t H T — t ) remains un- 

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