Hartree-Fock method real operator

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... 

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. 

The scheme would be to construct each term in the effect of the Hartree-Fock operator on a real-space representation of an MO, using either that spatial representation or the linear expansion representation, whichever is appropriate, easier or more tractable, and transform the expansion-method terms to the spatial representation by means of or its inverse and solve the resulting equation on a chosen grid of points. 

Alternative approaches to the many-electron problem, working in real space rather than in Hilbert space and with the electron density playing the major role, are provided by Bader s atoms in molecule [11, 12], which partitions the molecular space into basins associated with each atom and density-functional methods [3,13]. These latter are based on a modified Kohn-Sham form of the one-electron effective Hamiltonian, differing from the Hartree-Fock operator for the inclusion of a correlation potential. In these methods, it is possible to mimic correlated natural orbitals, as eigenvectors of the first-order reduced density operator, directly... 

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