Spin Projection Methods

To reduce or eliminate spin contamination problems in unrestricted wave functions, spin projection methods have been developed diat annihilate the contributions of certain spin states higher than the desired one. As derived in Appendix C, the PUHF energy for a wave function that has had contamination from the next higher spin state annihilated is computed as... 

The BS plus spin projection method discussed here is closely connected to the simple open-shell singlet method for optical excitations based on the Slater sum rule and ASCF (self-consistent-field total energy difference method). The mixed spin excited state is like the BS state, also of mixed spin. The Slater sum rule method" " is also quite effective for multiplet problems for excited states of transition metal complexes as shown in the work of Dahl and Baerends. ... 

The matrix elements for 5 needed for the spin projection methods discussed below involve multiple products of the a- overlap matrix and can be computed readily for UHF wavefunctions. Since the Hartree-Fock energy is invariant to unitary transformations among occupied orbitals of the same spin, the a and P orbitals can be rotated so that the a-fi overlap matrix is diagonal. These are termed corresponding orbitals, and their use greatly simplifies expressions for matrix elements ofS. ... 

The spin projection methods discussed so far apply the projection after the wavefunction has been determined. A second approach is to construct the wavefunction so that there is no spin contamination from the beginning. This leads to methods such as ROHF and restricted open-shell M0ller-Plesset perturbation theory (ROMPn)," and to valence bond, MCSCF, and Cl and MRDCI methods that use spin-adapted configurations. 

To overcome some of the problems inlierent in the UFIF method, it is possible to derive SCF equations based on minimizing the energy of a wavefiinction fomied by spin projecting a single Slater detemiinant starting... 

If spin contamination is small, continue to use unrestricted methods, preferably with spin-annihilated wave functions and spin projected energies. Do not use spin projection with DFT methods. When the amount of spin contamination is more significant, use restricted open-shell methods. If all else fails, use highly correlated methods. 

Analogously to MP methods, coupled cluster theory may also be based on a UFIF reference wave function. The resulting UCC methods again suffer from spin contamination of the underlying UHF, but the infinite nature of coupled cluster methods is substantially better at reducing spin contamination relative to UMP. Projection methods analogous to those of the PUMP case have been considered but are not commonly used. ROHF based coupled cluster methods have also been proposed, but appear to give results very similar to UCC, especially at the CCSD(T) level. 

The corresponding curves based on a UHF reference are shown in Figure 11.13. It is immediately clear that DFT methods do not have the spin contamination problem in the intermediate region, indeed spin contamination is not well defined in DFT." Removing the spin contamination by projection methods gives rise to discontinuous derivatives and artificial minima, analogously to the PUFIF case in Figures 11.3 and... 

Once a description of the electronic structure has been obtained in these terms, it is possible to proceed with the evaluation of spectroscopic properties. Specifically, the hyperfine coupling constants for oligonuclear systems can be calculated through spin projection of site-specific expectation values. A full derivation of the method has been reported recently (105) and a general outline will only be presented here. For the calculation of the hyperfine coupling constants, the total system of IV transition metal centers is viewed as composed of IV subsystems, each of which is assumed to have definite properties. Here the isotropic hyperfine is considered, but similar considerations apply for the anisotropic hyperfine coupling constants. For the nucleus in subsystem A, it can be... 

The method described above is of general validity and can be applied to transition metal clusters of arbitrary shape, size, and nucleanty. It should be noted that in the specific case of a system comprising only two interacting exchanged coupled centers, our general treatment yields the same result as that of Bencini and Gatteschi (121), which was specifically formulated for dimers. In this case, the relation between the spin-projection coefficient and the on-site spin expectation value is simply given by... 

The largest drawback of the spin annihilation procedure is similar to that of the sum method. That is, while the spin-annihilated wave function which results from the application of Aj+i to the 50 50 antecedent is in principle spin pure, it is expressed in the MOs that were optimized for the 50 50 case. These MOs minimize the energy of the contaminated state, but not that of the spin pure state, and errors can be significant. Nevertheless, the speed of the sum and projection methods, and their utihty in many if not all instances, makes them useful for rough applications prior to resort to more expensive and sophisticated models. 

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. 

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