KS eigenfunctions

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

It should be noted that in the case of the helium isoelectronic series (singlet GS of a two-electron system), because only one orbital is sufficient to construct the determinantal wave function and to obtain the density as n(r) — 2[t)r(r)], the minimization over n in Eq. (64) is equivalent to the minimization over ip in Eq. (28). Therefore the HF-KS and HF equations and their eigenfunctions are the same, ipir) = and consequently AT n =... 

The differential virial theorem (169) for noninteracting systems can alternatively be obtained [31], [32] by summing (with the weights fj ) similar relations obtained for separate eigenfunctions 4>ja(r) of the one-electron Schrodinger equation (40) [in particular the KS equation (50)]. Just in that way one can obtain, from the one-electron HF equations (33), the differential virial theorem for the HF (approximate) solution of the GS problem, as is shown in Appendix B, Eq. (302), in a form ... 

If we refrain from such a restriction and consider a spin-operator-dependent Hamiltonian, such as the 4-component KS Hamiltonian or the Dirac-Coulomb Hamiltonian, the Hamiltonian does not commute with the square of the spin operator. The square of the spin operator and the Hamiltonian then do not share the same set of eigenfunctions, and hence spin is no longer a good quantum number. In this noncollinear framework we must therefore find a different solution and may define a spin density equal to the magnetization vector (32). 

We determine the ground-state electronic structure of solids within Density Functional Theory (DFT) and the usual KS variational procedure, all implemented in the computational package gtoff [14]. The results of the all-electron, full-potential calculations are Bloch eigenfunctions p,k(r), expressed as linear combinations of Gaussian Type Orbitals (GTOs), and KS eigenvalues p k-... 

As in the UHF method, allowing differing KS orbitals for electrons with different spins can produce a wave function for the reference system s that is not an eigenfunction of 5, but this spin-contamination is less of a problem in KS DFT than in the UHF method. 

Since spin-orbit coupling is not present in the nonrelativistic limit and the spin-orbit-coupling-free Hamiltonian commutes with the spin operators, the N KS spinors can be constructed from spatial 2-spinors ff tensorially multiplied with spin eigenfunctions a such that four-component spin orbitals ipi,x = cp1 OL and = respectively, are obtained. Then, the spin-orbit-coupling-free (SOfree) z-component of the magnetization resembles the nonrelativistic spin density. 

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