Energetics of Model Potentials

Using the formalism developed in section 20.4, we can analyze the behavior of the approximations to the valence spinors in the AIMP method. The purpose of this exercise is to determine whether the approximations to the one-electron Hamiltonian and the pseudospinors are likely to cause any undesirable behavior, such as collapse when a core spinor is mixed in. As before, the model we use is the simplest an atom with two valence electrons. We have three separate cases to consider. 

In the first case, no approximation is made to the valence spinor. Then there is no approximation to the Fock matrix apart from the addition of the projector term. The gradient for mixing an arbitrary function into the valence spinor is deduced from (20.64) 

The matrix elements of B are zero, and / r = Scy, so the gradient is zero and X = 0 is a stationary point. The terms that are of zeroth order in S in the second derivative expression, equation (20.67), can be written 

In the second case, the core orthogonality is kept but approximated. The valence pseudospinor is no longer an eigenfunction of the Fock operator or the model potential. We can write the gradient as 

The correction is essentially the difference in Coulomb potential between the valence spinor and the valence pseudospinor. Now the gradient can be written 

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The behavior of the SCF solutions is not the only issue of interest in the energetics of model potentials. Because Bk is finite, the core spinors are not moved an infinite distance from the valence space, but a finite distance. Any representation of the core spinors remains in the molecular basis set, and can contaminate a correlated calculation, if they are not removed (Klobukowski 1990). The result can be an overestimation of correlation effects. Diagonalizing B in the virtual spinor space should provide a means of recognizing and removing core-like spinors. 

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