Projector operator method

Special Case when T = T = T. Let us now consider the special case when a complex symmetric operator is real, so that T = T. In this case, the operator T is also self-adjoint, T = T, and one can use the results of the conventional Hartree-Fock method 7. The eigenvalues are real, X = X. and - if an eigenvalue X is non-degenerate, the associated eigenfunction C is necessarily real or a real function multiplied by a constant phase factor exp(i a). In both cases, one has D = C 1 = C. In the conventional Hartree-Fock theory, the one-particle projector p takes the form... 

According to the methods of statistical mechanics, we define the projection operator Pq complementary projector Qo as... 

The Pauli approximation may be used in conjunction with this method by neglecting the small component spinors Q) of the Dirac equation, leading to RECPs expressed in terms of two-component spinors. The use of a nonrelativistic kinetic energy operator for the valence region, and two-component spinors leads to Hartree-Fock-like expressions for the pseudoorbitals. Note that the V s (effective potentials) in this expression are not the same for pseudo-orbitals of different symmetry. Thus the RECPs are expressed as products of angular projectors and radial functions. In the Dirac-Fock approximation, the orbitals with different total j quantum numbers, but which have the same / values are not degenerate, and thus the potentials derived from the Dirac-Fock calculations would be y-dependent. Consequently, the RECPs can be expressed in terms of the /y-dependent radial potentials by equa-... 

In spite of its computational intractability the full configuration interaction method forms the basis of a many-body theory. As stated above, the implementation of the algebraic approximation restricts the domain of the Hamiltonian operator, Jf, to a finite dimensional subspace 8 of the Hilbert space 1). If tP denotes the projector onto the subspace 8, then the Hamiltonian operator, Jf, is replaced by... 

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