Configuration interaction and stationary wavefunctions

1 Configuration interaction and stationary wavefunctions In order to simplify the treatment, a correlated wavefunction P(r)corr built by Cl between two uncorrelated wavefunctions will be considered. (To shorten the notations, the tilde describing the antisymmetric character of the wavefunctions and the spin of the electron have been omitted, and the spatial vectors of all electrons are indicated by the symbol r only.) One has 

The basis functions (r) and vP2(r) are solutions within the independent-particle model (operator H°, equ. (1.3))  

Because the (Oj values are different for different basis functions j, the time-dependent correlated wavefunction built from such basis functions cannot be a stationary wavefunction of the full Hamiltonian. Instead, the stationary 

Multiplication of the differential equation (7.77) by T r, t) or f(r, t), respectively, and integration over the coordinates r then leads to the two equations 

This shows that for the selected two-state system exactly two stationary states exist for the correlated wavefunction, and these are given by 

Hjk = (T/r) H % r) = 5 jkWj + H (for simplicity these matrix elements are assumed to be real). The general solution for the coefficients eft), and therefore also for the function F(r, t)corr, then becomes (see [FMa52, FLS65]) 

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