Partitioning and Inner Projections

The size of the f a) subspace is quite large for reasonable basis sets. For N electrons and a spin orbital basis of rank K, there are K — N) K — 2)N/2 elements in the orthogonal complement space. The solution of the Dyson-like equation would require the inversion of a matrix of this dimension for every value of the energy parameter. This approximation of the self-energy first introduced by J. Linderberg and Y. Ohrn,, has been discussed by G. D. Purvis III, and Y. Ohrn, and J. Schirmer and L. S. Cederbaum . A simplified version of this approach is the so-called Diagonal 2p-h TDA , which becomes 

Starting with the matrix electron propagator in Eq. (9.12), an alternative formulation of approximation can be introduced. One employs an inner projection 

It is convenient to use an orthonormal set of inner projection basis elements so that (hj hj) = 1 and (hj hj) = 0 for i j. A first step in seeking adequate approximation schemes for the electron propagator is a partitioning of the inner projection manifold. When the aim is to obtain a theoretical photoelectron spectrum, it is convenient to choose the partitioning 

The Rayleigh-Schrodinger perturbation expansion for the reference state, 

The concept of order in the perturbation expansion of the electron propagator ultimately means order in terms of the electron-electron interaction, or equivalently, two-electron integrals. The inclusion of electron correlation through first order in the reference state is achieved with the double excitation terms K2, whereas the Ki terms are also needed for second-order corrections. 

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E. Brandas, R.J. Bartlett, Reduced Partitioning Technique for Configuration Interaction Calculations Using Pade Approximants and Inner Projections, Chem. Phys. Lett. 8 (1971) 153. 

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