Occupation numbers, nonintegral

When the variational energy is a functional of the reference state well-defined for nonintegral occupation numbers. This implies two chain rules, for m 0 ... 

The spin-polarized approach also clears up most problems with complex atoms. Of the transition-metal atoms only atomic iron and cobalt have nonintegral numbers of d and s electrons using the Perdew-Zunger [13] local-density functional. From a practical point of view fractional-occupation-number solutions should be avoided at all costs. There appears to be no SCF procedure that will work efficiently for spin-polarized atomic cobalt or iron. Instead one must do several SCF... 

Recalling that the occupation numbers kp of an ON vector are zero or one, we conclude that the occupation numbers cop of an electronic state (1.7.13) are real numbers between zero and one - zero for spin orbitals that are unoccupied in all ON vectors, one for spin orbitals that are occupied in all ON vectors, and nonintegral for spin orbitals that are occupied in some but not all ON vectors ... 

The actual Hill plot for Hb is far from a linear line with constant slope. The actual curve has a varying slope between one to three. Thermodynamically, Eq. (6.8.2) implies that all n ligands bind simultaneously to Hb. There is no provision in this model for intermediary occupancy states. Therefore, this model is thermodynamically unacceptable. This is true a fortiori when n, obtained by fitting the experimental data, turns out to be a nonintegral number. 

Thus the density functional 2[p], Eq. (45), does not exist at T = 0, and one cannot construct a Kohn-Sham potential v. This can be understood in the following way. The physical system that the ensemble describes is a physical mixture of distinct, noninteracting N0-and (No + l)-electron systems. The Kohn-Sham equations, were they to exist, would collapse these into a single pseudosystem with a nonintegral number of electrons. The notion of fractional occupancy, Eqs. (38), within the Kohn-Sham equations at T = 0 is simply incorrect at T = 0 for a nondegenerate HOMO [42],... 

Instabilities of valence (viz. atoms flipping from one state of valence to another as a function of changes in the environment) and mixed valence (an atom exhibits simultaneously two valences, or two valence states coexist on the same site) are both related to intermediate valence (the atom in the condensed phase exhibits some mean, nonintegral valence). The effects are usually encountered when dealing with 4/ and 5/ electrons, and it is therefore very relevant to determine the / count, or effective number of / electrons on a given site. Various core-level spectroscopies have been used to probe / electron occupancies, and there is a vast literature on this field (see the review by Fuggle [615]). 

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