By Per-Olov Lowdin
ADVANCES IN QUANTUM CHEMISTRY VOL 1.
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This potential, we note, has the desired property that it is a function only of position, not of k. A simpler potential which has the same property is the other one suggested by the present writer (Slater 1951a), formed from the complete Coulomb potential of all electrons and nuclei, corrected by an exchange correction which is the average of all Hartree-Fock potentials at the given point of space, averaged over the occupied orbitals, or, even more crudely, which is proportional to the 1/3 power of the electron density.
Results of this calculation are shown in Fig. 6. This case is considerably 05 04 03 - 02 ; L3 > 01 ‘L $ W z 0 w-01 -02 -03 -04 r X L Fig. 6. Energy bands of gallium, from Wood (1963). more complicated than the others that have been discussed, for two reasons. In the first place, there are four atoms per unit cell, whereas in the other elements so far considered we do not have more than two. Second, there is much more empty space in the crystal between the spheres, so that a much larger part of the wave function must be expanded in plane waves.
HO - E $ = $0 E = Eo + (66) E$l The following equations have to be satisfied separately (Hylleraas, 1930d): (HO- EJ$O = 0, (HO- EOMl (H'- El)$, = 0, (HO- EOW2 ( H ' - E1)$1 - E2$0 = 0, + ( H o - E0)$, + (H' - + - E2$n-2 - ... - En$o= 0. , up to E2, and E2n+1- Applying the variational principle we have Hylleraas 30 separately for any power of E. , normalized $o, and collecting terms of E ~ " ' , one finds E2n+t - ElNnn - E2(Nn(n-1) + N
Advances in Quantum Chemistry by Per-Olov Lowdin