Hartree-Fock method complex symmetric

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... 

It is interesting to observe that, if one limits oneself to study the original real and self-adjoint Hamiltonian with H+= H = H and applies the complex symmetric Hartree-Fock method to this particular case without any transformations whatsoever, one... 

The puzzle depended on the simple fact that most physicists using the method of complex scaling had not realized that the associated operator u - the so-called dilatation operator - was an unbounded operator, and that the change of spectra -e.g. the occurrence of complex eigenvalues - was due to a change of the boundary conditions. Some of these features have been clarified in reference A, and in this paper we will discuss how these properties will influence the Hartree-Fock scheme. The existence of the numerical examples finally convinced us that the Hartree-Fock scheme in the complex symmetric case would not automatically reduce to the ordinary Hartree-Fock scheme in the case when the many-electron Hamiltonian became real and self-adjoint. Some aspects of this problem have been briefly discussed at the 1987 Sanibel Symposium, and a preliminary report has been given in a paper4 which will be referred to as reference D. 

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