Gradient 4-spinor

If p and q are both occupied spinor indices, the first-order density matrix element is a delta function, Dpq = Spq, and the gradient matrix element is zero. This is in accord... 

A similar expression arises if the first index is a negative-energy spinor index, while the element with two negative-energy indices obviously vanishes. The gradient is then simply related to the one-electron matrix, which on diagonalization will make the gradient zero. 

The Fock matrix is identical in form to the electron Fock matrix given above, and the gradient is the same in form as the spinor gradient involving one unoccupied electron spinor. Thus, for the purposes of self-consistent field methods, we may treat the negative-energy spinors as simply an extension of the set of unoccupied spinors. The Hessian may similarly be defined using the principles and formulas set out above. 

We would expect the open-open Fock matrix elements to correspond to redundant spinor rotation parameters because the energy should be invariant to rotations of equivalent spinors. However, if we construct the spinor rotation gradient, we see that there is a term that survives in the diagonal of the gradient ... 

The imaginary part of the integral is not necessarily zero, so the diagonal spinor rotation gradient is not zero. To explain this rather peculiar situation, we must turn to the application of symmetry. 

Starting from the Dirac-Coulomb approximation, a set of Dirac-Kohn-Sham equations may again be derived. In chapter 8, a spinor-rotation procedure was used to derive the relativistic Fock operator. A similar procedure applied to the present case shows that the gradient of the energy has elements the form... 

For the optimal set of Dirac-Kohn-Sham spinors the gradient must disappear, and this can be achieved by rotating to the spinor set that diagonalizes the Dirac-Kohn-Sham matrix with elements... 

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

If we expand ( )r in the complete orthonormal set of spinors, the contributions to frr that come from core spinors are made positive by the terms from Brr, and the remaining contributions sum to a value that is less negative than It must also be true that fan > Cy. The terms from the residual Fock operator are small, so the second derivative is positive and the minimum, with a small gradient, must be nearby. In this case also, the deviation of the valence pseudospinor from the valence spinor causes mixing of other spinors to compensate for the change. 

In the third and most general case, where the valence pseudospinor is not orthogonal to the core or to the trial function, the gradient is again given by (20.103). For a core spinor ft, we can use the expansion of the valence pseudospinor in terms of the complete set of spinors... 

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