Polarization propagator matrix representation

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. 

The follow ing step consists in a second change of matrix representation, by introducing a more convenient basis whose eigenvectors are linearly polarized waves propagating within the medium in the forward and backward directions. 

Fig. 2.14 Schematic diagram of a rotating analyzCT ellipsometer (RAE) and the Jones matrix representation for each comptment. The blue lines repiestait the trajectory of the electric field vector of the propagating light. The difference in the representations of the analyzer and polarizer is due to the angles a and ra being measured in opposite directions...

Before we continue in the derivation of a matrix representation of the polarization propagator, we want to mention that by taking the zero-frequency limit of the equation of motion in the frequency domain, we obtain the following relation between a polarization propagator and a ground-state expectation value... 

Using this relation in Eq. (3.152) leads us to an exact matrix representation of the polarization propagator in the superoperator formalism... 

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 10.1 we will illustrate this for ground-state expectation values such as Eq. (4.25) and many others and in Section 10.2 for sum-over-states expressions such as Eq. (4.74) and many others. In the rest of the chapter we will discuss methods in which approximations are made to the exact matrix representation of the linear response function or polarization propagator given in Eq. (3.159). This equation is exact as long as a complete set of excitation and de-excitation operators hn is used and the reference state is an eigenfunction of the imperturbed Hamiltonian. 

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... 

Exercise 10.3 Derive the partitioned form of the matrix representation of the polarization propagator, Eq. (10.15), using the relation for the inverse of a blocked matrix, Eq. (10.14). 

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