Mean-Field Formalism in Second Quantization

In deriving a set of relativistic one-particle functions, that is, spinors, from a mean-field approach, we typically start by making a simple guess at these functions (or the electron density), and then try to refine them iteratively. The refinement can be done by diagonalizing a suitable Hamiltonian (or Fock) matrix, which defines a rotation of the spinors in the entire function space available. Normally, this iterative process reaches a stage where further rotations do not change the spinors, that is, they are self-consistent. Provided we have chosen our sequence of rotations carefully, this should correspond to the optimal set of spinors from the mean field. For the present chapter our main concern is the rotation of the set of one-particle functions, and how this can be cast in a consistent theoretical framework that also accounts for the positron contributions. 

The second-quantized Hamiltonian that defines our many-electron system is 

Most of the theory can be developed using only the one-electron part of this operator, with rather straightforward extensions to include the two-electron part. To effect the rotations in the function space we employ the exponential rotation opaator U = exp(iX), introduced in (5.35), but parametrized in terms of the operator k = iX. We want the rotations to preserve orthonormality in the set of one-particle functions, and therefore require that U he a unitary operator, that is 

From this it is easy to see that the unitarity condition is fiilfilled if 

The first step is to choose a reference state, or initial guess, which we will denote 0). To this reference state we apply the general rotation to obtain a new reference state. 

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