Hartree-Fock method spin projected extended

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... 

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. 

In addition, there are two methods that involve spin projection operators. The spin extended Hartree-Fock (EHF) method starts with a single determinant of unrestricted spin orbitals and applies a spin projection operator before the molecular orbital coefficients are calculated in an SCF procedure. The half projected Hartree-Fock (HPHF) method uses a two determi-nantal wavefunction composed of spin-unrestricted orbitals to represent singlet and = 0 triplet states for systems with an equal number of a and electrons. The second determinant i.s... 

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