Hartree-Fock wave functions multiple electronic states

The minimutn in the Ai singlet Hessian eigenvalue at 3.9ao arises because of an avoided crossing with a second Hartree-Fock state as indicated by the dotted line terminated by an arrow in Figure 10.3. This second RHF solution represents an excited (albeit unphysical) state of /4 symmetry. (For distances shorter than 4.0oo, the calculation of the excited RHF state becomes difficult and no plot has been attempted in this region.) Multiple RHF solutions close in energy are often found in regions where the Hartree-Fock wave function provides an inadequate description of the electronic system and where it is necessary to go beyond the Hartree-Fock model for a proper description of the electronic system. 

The few attempts at describing excited states in transition metal complexes within the Restricted Hartree Fock (RHF) formalism were rapidly abandoned due to the computational difficulties (convergence of the low-lying states in the open-shell formalism) and theoretical deficiencies (inherent lack of electronic correlation, inconsistent treatment of states of different multiplicities and d shell occupations). The simplest and most straightforward method to deal with correlation energy errors is the Configuration Interaction (Cl) approach where the single determinant HF wave function is extended to a wave function composed of a linear combination of many de-... 

The wave function obtained corresponds to the Unrestricted Hartree-Fock scheme and becomes equivalent to the RHF case if the orbitals < )a and < )p are the same. In this UHF form, the UHF wave function obeys the Pauli principle but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-electron case, an alternative form of the wave function which has the same total energy, which is a pure singlet state, but which is no longer antisymmetric as required by thePauli principle, is ... 

The spin dependency of the UHF Fockian and the corresponding eigenvectors has the unfortunate consequence that the resulting many-electron wave function is usually not a pure spin state, rather a mixture of states of different spin multiplicities. The state of a definite multiplicity can be selected by the appropriate spin-projection operator. The thorough investigation of this problem results in the spin-projected extended Hartree-Fock equations (Mayer 1980). 

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