Fock operator spin contamination

Because the convenience of the one-electron formalism is retained, DFT methods can easily take into account the scalar relativistic effects and spin-orbit effects, via either perturbation or variational methods. The retention of the one-electron picture provides a convenient means of analyzing the effects of relativity on specific orbitals of a molecule. Spin-unrestricted Hartree-Fock (UHF) calculations usually suffer from spin contamination, particularly in systems that have low-lying excited states (such as metal-containing systems). By contrast, in spin-unrestricted Kohn-Sham (UKS) DFT calculations the spin-contamination problem is generally less significant for many open-shell systems (39). For example, for transition metal methyl complexes, the deviation of the calculated UKS expectation values S (S = spin angular momentum operator) from the contamination-free theoretical values are all less than 5% (32). 

Typical structures are specified in Table 1 which uses the labelling of carbon atoms in Cjo defined in Fig. 1. The restricted open-shell Hartree-Fock (ROHF) method was used in all geometry optimizations using a minimal basis set of orbitals (STO-3G) [13]. These calculations are therefore exploratory in nature. Here we have chosen to use the standard ab initio ROHF method since it is well-known that the UHF method (as used in the PRDDO approximation [9]) does not give wave functions which are eigenstates of the total spin operator S. The effect of spin contamination on molecular properties is uncertain, particularly if the contamination is high (the... 

Abstract An expression for the square of the spin operator expectation valne, S, is obtained for a general complex Hartree-Fock wave fnnction and decomposed into four contributions the main one whose expression is formally identical to the restricted (open-shell) Hartree-Fock expression. A spin contamination one formally analogous to that found for spin nnrestricted Hartree-Fock wave functions. A noncollinearity contribntion related to the fact that the wave fnnction is not an eigenfunction of the spin- S operator. A perpendicularity contribution related to the fact that the spin density is not constrained to be zero in the xy-plane. All these contributions are evaluated and compared for the H2O+ system. The optimization of the collinearity axis is also considered. 

Equation (17) means that the excited Slater determinant must be an eigenvector of the operator. As shown by Fock [39], the condition (17) is fulfilled if the set of orbitals associated with the spin functions lies completely within the space defined by the set associated with the a spin functions. This condition eliminates spin contamination and can be written as the orthogonality constraint [40] ... 

A component, Ms, of the angular spin moment S along an arbitrary axis (usually chosen as the z-ditection), Ms can take values between —S and S —S, —S -(-1,..., S — 1, S. The term is also used to denote an operation (spin symmetry projection) to eliminate spin contamination (wavefunctions are not eigenfunctions of the operators of and 5 associated mostly with the case of triplet instability. See Instability of the Hartree-Fock Solution. 

The downside to the (spin)-unrestricted Hartree-Fock (UHF) method is that the unrestricted wavefunction usually will not be an eigenfunction of the operator. Since the Hamiltonian and operators commnte, the true wavefunction must be an eigenfunction of both of these operators. The UHF wavefunction is typically contaminated with higher spin states for singlet states, the most important contaminant is the triplet state. A procedure called spin projection can be used to remove much of this contamination. However, geometry optimization is difficult to perform with spin projection. Therefore, great care is needed when an unrestricted wavefunction is utilized, as it must be when the molecule of interest is inherently open shell, like in radicals. 

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. 

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