Polarization propagator spectral representation

Excited triplet states n) with energy E are included in the sum for the FC and SD terms, while excited singlet states contribute to the OP term. Recalling the spectral representation of the polarization propagator for zero frequency w [60]... 

Polarization propagators or linear response functions are normally not calculated from their spectral representation but from an alternative matrix representation [25,46-48] which avoids the explicit calculation of the excited states. 

In addition to the linear response function r r we have also introduced the quadratic r r r , and the cubic r r r r , j, 3 response functions. The relation (9) only ensures that the real part of the second term is correct. In fact it turns out (Zubarev, 1974, Chap. 15 Oddershede et al., 1984, Chaps 2.1 and 2.2) that r r is the spectral representation of the retarded polarization propagator... 

However, Eq. (3.108) is only the definition of the Fourier transform, which then has to be applied to the expression for the time-dependent polarization propagator in Eq. (3.107) with t —t replaced by t. This leads us [see Exercise 3.8] to the spectral representation of the polarization propagator or hnear response function... 

From the spectral representation, Eq. (3.110), we can easily verify the general symmetry property of the polarization propagator... 

In the discussion of the spectral representation of the polarization propagator in Section 3.11 we have seen that the electronic vertical excitation energies of the system En —Eq ) are the poles of the polarization propagator. In the matrix representation Eq. (3.159) a polarization propagator has a pole, if the principal propagator matrix E — hujS) becomes singular. This leads to the homogeneous linear equations... 

In Section 9.2 it was mentioned that the simplest approximation for an excited state 4 ° ) is to represent it by one singly excited determinant Approximating at the same time the groimd-state wavefunction with the Hartree-Fock determinant 0 and the Hamiltonian by the Hartree-Fock Hamiltonian F, Eq. (9.15), the excitation energies En — E become equal to orbital energy differences ta — and the transition moments (4 q° O 4 ° ) become simple matrix elements of the corresponding one-electron operator d in the molecular orbital basis ( d ) [see Exercise 10.ll. The spectral representation of the polarization propagator, Eq. (3.110), thus becomes approximated as... 

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