Matrix elements connecting different electronic configurations

2 Matrix elements connecting different electronic configurations 

Such general expressions for matrix elements of electrostatic interactions, covering the cases of three and four open shells, may be found in Chapter 25 of [14]. However, they are rather cumbersome and, therefore of little use for practical applications. Quite often sets of simpler formulas, adopted for particular cases of configurations, are employed. Below we shall present such expressions only for the simplest interconfigurational matrix elements occurring while improving the description of a shell of equivalent electrons (the appropriate formulas for the more complex cases may be found in [14])  

A two-electron matrix element of the operator of the electrostatic interaction energy is equal to 

Here the most general case of radial integral in such a matrix element looks like 

If a matrix element connects two pairs of equivalent electrons, then instead of (29.17) we have 

This term occurs for matrix elements differing only by a principal quantum number of one electron. It describes terms corresponding to kinetic energy and electrostatic interaction of an electron with a nuclear field (integral of the type (19.23)) as well as to the interaction with closed shells (summation over nok). 

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Matrix elements connecting electronic configurations 351 29.2 Matrix elements connecting different electronic configurations... 

The energies of the unperturbed (sub)states have been defined above. Ha is the configuration interaction Hamiltonian, which describes essentially the electron-electron interaction [128]. The off-diagonal terms are taken as the perturbation. The aim of this discussion is, at least for this simple model, to present the structure of the corresponding perturbational formulas. For example, it will be shown that different energy denominators occur, which are connected to the different states involved. In this model, we neglect any diagonal contributions to the model Hamilton operator and treat the matrix elements as real for simplicity. 

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