Spin-unrestricted Kohn—Sham equations

Similar to the spin-compensated case, the solution of the unrestricted Kohn-Sham equations starts with the external potential and the number of spin a and spin ji electrons in the state of interest (denoted Na and Np, respectively) then Eqs. (48) and (49) are solved until consistency is achieved. Using the Kohn-Sham orbitals and orbital energies, one then computes the total energy of the system using the spin-dependent generalization of Eq. (47),... 

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]. 

The one-electron Kohn-Sham equations were solved using the Vosko-Wilk-Nusair (VWN) functional [27] to obtain the local potential. Gradient correlations for the exchange (Becke fimctional) [28] and correlation (Perdew functional) [29] energy terms were included self-consistently. ADF represents molecular orbitals as linear combinations of Slater-type atomic orbitals. The double- basis set was employed and all calculations were spin unrestricted. Integration accuracies of 10 -10 and 10 were used during the single-point and vibrational frequency calculations, respectively. The cluster size chosen for Ag or any bimetallic was... 

The rectangular B matrix is used in the SCF procedure instead of the usual Ldwdin matrix. Hence this method can be employed either for Hartree-Fock or Kohn-Sham equations resolution in the Roothaan formalism. The extension to the unrestricted spin case is trivial. 

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