Unrestricted Kohn-Sham approach

The exact energy functional (and the exchange correlation functional) are indeed functionals of the total density, even for open-shell systems [47]. However, for the construction of approximate functionals of closed as well as open-shell systems, it has been advantageous to consider functionals with more flexibility, where the a- and j8-densities can be varied separately, i.e. E[p, p ]. The variational search for a minimum of tire E[p, p ] functional can be carried out by unrestricted and spin-restricted approaches. The two methods differ only by the conditions of constraint imposed in minimization and lead to different sets of Kohn-Sham equations for the spin orbitals. The unrestricted Kohn-Sham approach is the one most commonly used and is implemented in various standard quantum chemistry software packages. However, this method has a major disadvantage, namely a spin contamination problem, and in recent years the alternative spin-restricted Kohn-Sham approach has become a popular contester [48-50]. 

All the quantum-mechanical computations have been based on the unrestricted Kohn-Sham (UKS) approach to the DFT [13], using the B3LYP hybrid functional both for the geometry optimizations and for the evaluation of magnetic properties. This functional combines Hartree-Fock and Becke exchange terms with the Lee-Young-Parr correlation functional, [26] in the same ratios as in the Becke three parameters model and, when applied to open shell systems, it has provided... 

Hyperfine couplings, in particular the isotropic part which measures the spin density at the nuclei, puts special demands on spin-restricted wave-functions. For example, complete active space (CAS) approaches are designed for a correlated treatment of the valence orbitals, while the core orbitals are doubly occupied. This leaves little flexibility in the wave function for calculating properties of this kind that depend on the spin polarization near the nucleus. This is equally true for self-consistent field methods, like restricted open-shell Hartree-Fock (ROHF) or Kohn-Sham (ROKS) methods. On the other hand, unrestricted methods introduce spin contamination in the reference (ground) state resulting in overestimation of the spin-polarization. 

Hanson and Martin applied the same approach to investigate the rupture of isoprene and butadiene oligomers in order to investigate covalent bond rupture in rubber.Using density functional theory, they identified the point of rupture where the unrestricted solution to the Kohn-Sham electronic wave functions falls below the restricted solution. This implies that the rupture process should be heterolytic and so this concept can only be applied for instances in which radical species are formed in the initial rupture event. 

The ESR hyperfine coupling is determined by triplet perturbations. Thus, in principle one should use an unrestricted wave function to describe the reference state. However, it is also possible to use a spin-restricted wave function (Fernandez et al. 1992) and take into account the triplet nature of the perturbation in the definition of the response. Within such a (e.g., SCF or MCSCF) restricted-unrestricted approach, first-order properties are given as the sum of the usual expectation value term and a response correction that takes into account the change of the wave function induced by the perturbation (of the type (0 H° 0)). This restricted-unrestricted approach has also been extended to restricted Kohn-Sham density functional theory (Rinkevicius et al. 2004). 

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