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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... [Pg.209]

According to the methods of statistical mechanics, we define the projection operator Pq complementary projector Qo as... [Pg.150]

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-... [Pg.2474]

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... [Pg.117]


See other pages where Projector operator method is mentioned: [Pg.41]    [Pg.167]    [Pg.103]    [Pg.264]    [Pg.262]    [Pg.307]    [Pg.55]    [Pg.415]    [Pg.1016]    [Pg.83]    [Pg.97]    [Pg.310]    [Pg.24]    [Pg.292]    [Pg.111]    [Pg.71]    [Pg.55]    [Pg.80]    [Pg.103]    [Pg.460]    [Pg.51]    [Pg.317]    [Pg.31]    [Pg.3016]    [Pg.39]    [Pg.81]   
See also in sourсe #XX -- [ Pg.152 , Pg.153 , Pg.154 ]




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Operating Methods

Projector

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