Electron propagator spectral representation

After Fourier transformation, the time domain of the Green s function is translated to frequency dependency. We start with the resulting spectral representation of the one-electron propagator... 

Ionization energy and electron affinity are critical discriminants of systemic reactivity and the one electron propagator theory /1,2/ has provided an effective route to their accurate calculation. The spectral representation of the matrix electron propagator... 

Finally, in terms of the eigenfunctions and eigenvalues of L, the spectral representation of the dilated electron propagator is... 

The spectral representation of the dilated electron propagator in strict analogy to eqn. (1) is given by /20-23/ ... 

The form of the matrix elements in Eq. (4.93) permits partial summations in the spectral representation of the electron propagator. Thus, one obtains explicitly that... 

EVom the definition of the spectral representation, it follows that the elements of the electron propagator matrix G( ) become infinite when E equals an electron binding energy. Then, the inverse G E) has a zero eigenvalue at such an energy. This result can be used to devise iterative methods to find the electron propagator poles and residues at a given level of perturbation expansion. 

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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