Variational methods projection operator

The existing SCF procedures are of two types in restricted methods, the MO s, except for the hipest (singly) occupied MO, are filled by two electrons with antiparallel spin, while in unrestricted methods, the variation procedure is performed with individual spin orbitals. In the latter, a total wave function is not an eigenvalue of the spin operator S, which is disadvantageous in many applications because of a necessary annihilation of higher multiplets by the projection operator. Since in practical applications the unrestricted methods have not proved to be remarkably superior, we shall call our attention in this review mainly to the restricted methods. 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

The UHF ansatz is necessary because in case of neutral solitons one has to deal with a doublet state. Thus a DODS (different orbitals for different spins) ansatz, as the UHF one, is necessary to describe the system. However, in the UHF method described so far, one Slater determinant with different spatial orbitals for electrons of different spins is applied, which is not an eigenfunction of S2, i.e. S(S+l)h2. The best way to overcome this difficulty would be to use the PHF (Projected Hartree Fock) method, also called EHF method (Extended Hartree Fock) where before the variation the correct spin eigenfunction is projected out of the DODS ansatz Slater determinant [66,67a]. Unfortunately numerical solution of the rather complicated EHF equations in each time step seems to be too tedious at present. Moreover for large systems the EHF wavefunction approaches the UHF one [68], however, this might be due to the approximations used in [67a]. Another possibility is to apply the projection after the variation using again Lowdin s projection operator [66]. Projection and annihilation techniques were... 

To achieve symmetrization, a direct product of the space operator and the spin operator was constructed as a new operator, under which the variational expression of Loucks method is kept invariant (Yamagami and Hasegawa 1990). By the projection operator technique, it is straightforward to derive a symmetrized form of the relativistic APW method, which covers both symmorphic and non-symmorphic space group. 

This difficulty is overcome with the aid of a projection operator by projecting out from the Slater determinant the component with the desired multiplicity 25+1, annihilating all other contaminating components. This can be done either after an already performed calculation (spin projection after variation, UHF with annihilation), or, as Lowdin has pointed out, one would expect a more negative total energy if the variation is performed with an already spin-projected Slater determinant [spin projection before variation, spin-projected extended Hartree-Fock (EHF) method]. The reason is that a spin-projected Slater determinant is a given linear combination of different Slater determinants. The variation in the expectation value of the Hamiltonian formed with a spin-projected Sater determinant thus provides equations (EHF equations), whose solutions represent the solution of this particular multiconfigura-tional SCF problem. 

Dirac equation. This method of eliminating the small component is not a procedure that leads to a simplification. It does, however, have some motivation, both physical and practical. First, it projects out the negative-energy states, and leaves a Hamiltonian that may have a variational lower bound, avoiding the potential problem of variational collapse. Second, it removes from explicit consideration the small component, and with the use of the Dirac relation (4.14) it yields a one-component operator that can be used in nonrelativistic computer programs. 

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