Spin Unrestricted Methods

It has been argued [12] that DFT calculations on open-shell systems should always be performed with spin-unrestricted methods. However, it is still of practical interest to compare the performance of procedures such as UB3LYP and RB3LYP in thermochemical predictions [13]. 

Technically, it is possible to define methods in which other restrictions than those on spin are lifted, but the most common unrestricted method is the spin-unrestricted method, and we follow the common usage of UHF for the spin-unrestricted Hartree-Fock method. 

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. 

UFF (universal force field) a molecular mechanics force field unrestricted (spin unrestricted) calculation in which particles of different spins are described by different spatial functions VTST (variational transition state theory) method for predicting rate constants... 

The existing SCF procedures are of two types in restricted methods, the MO s, except for the hipest (singly) occupied MO, are filled by two electrons with antiparallel spin, while in unrestricted methods, the variation procedure is performed with individual spin orbitals. In the latter, a total wave function is not an eigenvalue of the spin operator S, which is disadvantageous in many applications because of a necessary annihilation of higher multiplets by the projection operator. Since in practical applications the unrestricted methods have not proved to be remarkably superior, we shall call our attention in this review mainly to the restricted methods. 

Because of cancellation of errors in reactions such as (4.5), reasonable results are often obtained, even at quite simple levels of theory. However, it has been found [21, 34] that larger errors may occur with unrestricted methods if there is a significant difference between the degrees of spin contamination for the two radicals in the reaction. 

Several of the methods referred to in this chapter use the URCCSD(T) procedure in which a spin-unrestricted CCSD(T) calculation is performed on a high-spin RHF reference wavefunction, as implemented in the MOLPRO program. H. J. Werner, P. J. Knowles, R. D. Amos, A. Bemhardsson, A. Beming, P. Celani, D. 

The challenge with unrestricted methods is tire simultaneous minimization of spin contamination and accurate prediction of spin polarization . The projected UHF (PUHF, see Appendix C) spin density matrix catr be employed in Eq. (9.36), usually with somewhat improved results. 

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... 

Up to this point, we restricted ourselves to the simple case of determinants involving no more than two orbitals. However, the MO-VB correspondence is general, and in fact, any MO or MO—Cl wave function can be exactly transformed into a VB wave function, provided it is a spin-eigenfunction (i.e., not a spin-unrestricted wave function). While this is a trivial matter for small determinants, larger ones require a bit of algebra and a systematic method is discussed in Chapter 4 for the interested or advanced reader. 

When the second of the equivalence restrictions is removed, a single determinant wavefunction of lower energy is usually obtained. In fact, it is possible for a wave-function obtained in this way, a so-called unrestricted Hartree-Fock (UHF) wavefunction191 (perhaps more properly called a spin-unrestricted Hartree-Fock wavefunction) to go beyond the Hartree-Fock approximation and thus include some of the correlation energy. Lowdin192 describes this as a method for introducing a Coulomb hole to supplement the Fermi hole already accounted for in the RHF wavefunction. 

Often, TDDFT studies are targeted at closed-shell systems. However, many metal complexes are paramagnetic, and the open d- or /-shells may pose additional challenges for TDDFT spectral computations. Fan et al. [338] have studied the CD of high-spin trigonal dihedral chromium complexes, and developed a spin-unrestricted TDDFT method for the computations of the CD spectra. When possible, an analogous closed-shell cobalt(III) complex was calculated as well for comparison. 

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. 

There are a number of slightly more approximate methods for determining the electron affinity (EA) based on the restricted Hartree-Fock (RHF) and spin-unrestricted Hartree-Fock (UHF) methods. For closed shell anions, molecules which dissociate to... 

Fig. 4.27. Molecular-orbital energy-level diagram for the (FeO,) " cluster calculated using the MS-SCF-Jfa method. The highest-energy occupied orbital is the 2 2g i containing one electron. Also shown are the energies of Fe and O atomic orbitals. Spin-up (f ) and spin-down ( i ) molecular orbitals are shown in this spin-unrestricted calculation on a regular octahedral (O ) cluster at an iron-oxygen distance of 2.17 A (after Tossell et al., 1974 reproduced with the publisher s permission).

In this method the ionization energy is calculated from the weighted mean of the one-electron energies of the ground state and of a transition state in which 2/3 of an electron is removed from the spin orbital considered. The transition state calculations should be carried out through spin unrestricted calculations. 

Several researchers have recently devoted considerable effort to the derivation and efficient implementation of techniques based on spin-restricted reference determinants that reduce the computational discrepancy between closed- and open-shell systems. " This emphasis on spin-restricted techniques has resulted in part from a bias toward reference wavefunctions that maintain the spin symmetry of the exact wavefunction (such as the ROHF determinant), but also because of the possible efficiency advantages of spin-restricted methods over unrestricted techniques. Thus, since the component molecular orbitals are constrained to have identical spatial parts for each spin function, it should be possible to construct the correlated wavefunction in a manner that takes advantage of this symmetry. 

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