Spin projection operator

A further alternative is to remove the higher-spin states from the unrestricted wavefunction by means of a spin-projection operator. Spin-projected energies are designated by a P prefix (e.g. PHF, PMP). 

UHF Methods. A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this leads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,40 where two sets of singly occupied orbitals are constructed instead of the doubly occupied set. The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minimization,41 arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. 

The determinantal functions must be linearly independent and eigenfunctions of the spin operators S2 and Sz, and preferably they belong to a specified row of a specified irreducible representation of the symmetry group of the molecule [10, 11]. Definite spin states can be obtained by applying a spin projection operator to the spin-orbital product defining a configuration [12]. Suppose d>0 to be the solution of the Hartree-Fock equation. From functions of the same symmetry as d>0 one can build a wave function d>,... 

A is the antisymmetrizer, ensuring that the wavefunction changes sign on interchange of two electrons (and thus the wavefunction obeys the Pauli exclusion principle), and 0(S) is a spin projection operator " that ensures that the wavefunction remains an eigenfunction of the spin-squared operator,... 

The UHF method many-electron wavefunction is the eigenfunction of the spin projection operator Sz with zero eigenvalue, the ROHF method many-electron function... 

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). 

Different schemes have been devised to remove contaminants from higher spin states from UHF wavefunctions by means of the spin projection operators, which were introduced originally by Lbwdin. Removing high-energy spin states does lower the energies of UHF wavefunctions. However, these procedures lead to projected UHF (PUHF) wavefunctions that consist of several Slater determinants whose coefficients (cf. Eq, [4]) in been optimized variationally. 

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... 

A is the antisymmetrizing operator, C)g is the spin-projection operator for spin quantum number S, and 0] is a product of a and 3 one-electron spin functions of magnetic quantum number Mg. The spin projection yields two separate functions for spatial orbital products not restricted to double occupancy ... 

One important observation that should be made about the spin tensor operators is that any singlet operator commutes with both the shift curators S and the spin-projection operator 5 see (2.3.1) and (2.3.2). It therefcM e follows that singlet opo tors also commute with 5 since this operator may be expressed in terms of the shift operators and the spin-piojection operator (2.2.38) ... 

We should therefore be able to establish the general spin properties of determinants by examining the commutators between the spin operators and the spin-orbital ON operators. We note that the spin-orbital ON operators commute with the spin-projection operator ... 

This relation follows from the observation that the spin-projection operator (2.2.30) is a linear combination of spin-orbital ON operators. Thus, since the ON operators commute among themselves (1.3.4), they must also commute with the spin-projection operator. From the commutation relations (2.4.5) and from the observation that there are no degeneracies among the spin-orbital occupation numbers (2.4.4), we conclude that the Slater determinants are eigenfunctions of the projected spin ... 

The effect of the spin-projection operator in (2.4.26) is easily obtained from (2.4.24) ... 

The expectation values of the remaining two terms in (2.7.21) vanish from spin-symmetry considerations since the reference wave function is an eigenfunction of the spin-projection operator (2.7.4) ... 

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