State density projected

Often one is not interested in the dispersion of the energy bands but merely in their state density (1.13) or in their Ji-projected state density (per spin) ... 

Fig.2.13. Illustration of how a projected state density N, which includes hybridisation, may be scaled into a hybridised canonical fc-state density No by the potential function P (E). The relevant equations are (2.12), which relates energy and structure constants, and (2.40), which relates Jl-projected energy and canonical state densities...

In the approximation that hybridisation may be neglected, the -projected state density may, of course, be obtained from the state density... 

One may then prove [2.15] that the -projected state density (2.37) for a given potential, as specified by a path P(E) may be obtained from the canonical number-of-states function through... 

Let us assume that we have calculated the energy bands and corresponding eigenvectors, and hence obtained the -projected state densities N°(E) for fee Ni. Now we want the -projected state densities for a slightly different atomic-sphere potential, say for the next iteration towards a self-consistent Ni potential, or for Ni at a different lattice constant, or for fee Pd or Rh. If the new potential v(r) is so similar to the original potential v°(r) that... 

With the properties of the character in mind one may obtain the -projected state density from the analogue of (2.37), i.e. 

Consequently, we need not truncate (6.41) after the first two terms, even though the projected state densities have been obtained by means of the linear method. Since (6.41), on account of the , term, is the more accurate expression, we prefer to use it instead of (6.39) to construct charge densities. It follows from the above that only the first term in (6.41) contributes net charge to the sphere, and hence the second and third terms represent the radial displacement of charge due to the broadening of the band around E fl. 

The change in the sum of the one-electron energies (7.33) has now been written in terms of the projected state density, the partial wave evaluated at the sphere, and the change in boundary condition of the solutions in the sphere. This boundary condition was imposed by the KKR-ASA equations, Sect. 108... 

The DDNS programme is designed to evaluate til-projected state densities and corresponding number of state functions by means of the tetrahedron technique [9.4,5]. The basic input is the eigenvalues and til characters generated by LMTO. The calculated functions are stored on disk, and may be retrieved later for use in the ASA self-consistency procedure, or simply for plotting purposes.1 The programme is based on the paper by Jepsen and Andersen [9.4], the unpublished thesis by Jepsen [9.6] and some private notes by 0. K. Andersen. 

The execution of the projected state density programme DDNS requires a data set with energy bands and til characters in the format described in Sect.9.4.5. The additional input shown in Table 9.6 consists of a few control parameters. 

The basic input includes the projected state densities and number-of-states functions generated by DDNS, and the atomic charge densities calculated by RHFS. The main output is the self-consistent potential parameters and the electronic pressure. 

The fourth step is to construct a new trial charge density. To this end the programme evaluates the moments of the projected state densities and inserts these together with the partial waves and their energy derivatives into (6.41, 8.30). The moment calculation is based on a set of projected state densities obtained in previous runs of LMTO and DDNS where the potential parameters used in LMTO could be those given in step three, i.e. constructed from a renormalised atomic potential, in a previous separate run of SCFC. The moments are evaluated by... 

The execution of SCFC requires a data set with total and projected state densities and number-of-states functions as generated by DDNS and stored in DOS/YY/B. An example of the additional input, which consists of control variables and the atomic charge densities, is shown in Table 9.7. 

N (E) o,e", (6.38) Jl-projected state-density function Terms of order higher than en... 

The immediate effect of separate j-projected state density is that this leads to distinct occupation numbers for the two j-bands with a given /. In the limit of vanishing spin-orbit splitting, A, , of the two j-bands, the ratio of their populations becomes equal to the ratio of their degeneracies. But in the other limit, when is much... 

---