Spin-restricted open-shell Kohn-Sham

Filatov, M., Shaik, S., 1999, Application of Spin-Restricted Open-Shell Kohn-Sham Method to Atomic and Molecular Multiplet States , J. Chem. Phys., 110, 116. 

Kohn-Sham formalism, which suffers from the spin contamination problem, and on the sum-over-states or coupled perturbed Kohn-Sham approaches. In recent articles devoted to computations of elechonic g-tensors we advocated the use of an alternative approach, namely linear response theory based on the spin-restricted open-shell Kohn-Sham formalism, which is free from spin contamination problem (see Theory section). In the following we briefly review the applicability of this approach for some paramagnetic compounds. 

The analysis starts with a restricted open-shell Kohn-Sham (ROKS) calculation on the HS state. If necessary, the magnetic orbitals are transformed to the representation with local orthogonal orbitals a and b, as shown in the first column of Fig. 5.11. Staying within the spin-restricted formalism makes that for each a orbital a orbital can be found which has the same spatial part. In the first step, the direct exchange is estimated from the energy difference of the HS(ROKS) and a BS determinant in which only the spin of one of the unpaired electrons is inverted, but neither the core nor the magnetic orbitals are optimized. 

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]. 

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. 

For closed-shell and open-shell molecules, spin-restricted Kohn-Sham (RKS) and spin-unrestricted Kohn-Sham (UKS) density functional calculations were employed, respectively. Except for the calculations of excited states and the cases where pure states are sought, we have employed an approximation in which electron density is smeared among the closely spaced orbitals near the Fermi levels. In this procedure, fractional occupations are allowed for those frontier orbitals with energy difference within 0.01 hartree to avoid the violation of the Aufbau principle (46). 

The additional information of the spin density p (r) can then be directly exploited in the exchange-correlation functionals. For open-shell systems, two different restrictions are possible when introducing the noninteracting reference system [109, 111]. We can require (i) that only the total electron density of the fully interacting and of the reference system agree or (ii) that, in addition, the spin densities of the two systems are exactly the same. The first condition leads to a spin-restricted Kohn-Sham DFT formulation, while for the latter a spin-unrestricted Kohn-Sham DFT framework is required [109]. 

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