D-slates

A con figuration interaction calctilaliori is available on ly for single poiri ts wli en th e refereri ce groiiri d slate is obtained from an KHF calcti lation, ... 

In Section 19-E we found contributions to the effective charge 7 of the oxygen due to the matrix clement between an sp hybrid and a d slate and due to the matrix element For an oxygen displacement perpendicular to the Ti—O axis, the predicted transverse charge is simply this Z. However, for displacements along the Ti—O axis there are contributions from the transfer of charge between ions just as there were in the alkali halides, for which we obtained the transverse charge in Section 14-C. 

Exactly how this is done is st en in Fig. 20-11, where the model density of states from Fig. 20-8,b is redrawn (rotated 90") with the free-clectron and r/-like densities shown. separately. For a metal in a particular Dn column of the Solid Stale Table (3 < J7 < II), t l should be picked. so that the integral over the occupied portions of the two densities of states shown—and given in Eqs. (20-10) and (20-11)—comes to n electrons. This is carried out in Problem 20-3 to obtain the number Z, of free electrons per atom. The results arc shown in Table 20-4. We shall. sec in Section 20-F that the magnetic properties suggest a value for the 3d series of Z, = 0.6. The discrepancy is not large in view of the assumption of a uniform density of d slates. 

Andersen s assumption of a muffin-tin potential equal to the energy of the d slate, which was used in the preceding section to obtain muffin-tin orbitals, is used here in two ways. (This analysis has only very recently been made see Harrison and Froyen, 1979.) First, it is used to obtain the matrix elements and second, it is used in writing F, - = in Eq. (20-35). We begin with... 

In order to write the matrix element, we select a coordinate system with z-axis along the vector from the (/ > atom to the d> atom, and write the d slate as d> = 12, w> = E 2(r)Y"2(()i, plane wave, normalized in the volume Q of the system, can also be written in terms of spherical harmonics centered on the same atom (Schiff, 1968, p. 119) thus... 

Atoms in 5. P. D, F,. . . states have the same orbital angular momentum as a hydrogen atom with its single electron in an, v. p, d.f,. . . orbital. Thus we can define a quantum number L. which has the same relationship to the atomic state as / has to an atomic orbital (e.g.. L = 2 describes a D slate). L is given by ... 

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