Density matrices for atoms and solids. I. Effective potential matrix and the Bloch equation

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Abstract
By recognizing that the canonical density matrix C can be expressed in terms of an effective potential matrix, U , intimately related to the potential V in which the particles move, a powerful approximate analytic solution of the Bloch equation has been obtained. This solution for U reduces ( a ) to first-order perturbation theory on C when V is weak and ( b ) to the correct Thomas–Fermi result when V is almost constant in space. For an attractive defect centre in a metal, represented by a screened Coulomb potential which is not strong enough to bind an electron, it is shown that this approximate analytic solution may be used successfully in a numerical iterative solution of the Bloch equation and final numerical results are presented. For more strongly attractive centres, however, where bound states appear, the same numerical iterative scheme proves inadequate. A method is developed which orthogonalizes the approximate analytical density matrix to the wave function product for the lowest bound state. The new density matrix thereby formed is tested, and found to work successfully for an unscreened Coulomb field. This approach is then worked out for a screened potential created by a charge Z = 4 in a Fermi gas of density equal to that in Cu metal. Such a charged centre brings down a bound state from the conduction band and it is shown that the method employed successfully for the bare Coulomb field also leads to an accurate solution of the Bloch equation in this case. It is concluded that we have here a sufficiently powerful iterative scheme to carry out Hartree self-consistent calculations based on the Dirac density matrix, both for atoms, where the Fermi level lies in the bound state region, and for defects in metals, where the Fermi energy falls in the continuum. Such calculations are now in progress.

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