SCF LCAO CO equations

Let us write down the SCF LCAO CO equations as if they corresponded to a large molecule (Bloch functions will be used instead of atomic orbitals). Then the n-th 

The symbol x q means the q-ih atomic orbital (from the set we prepared for the unit cell motif) located in the cell indicated by vector Rj (j-th cell). 

Ill the cApicSSiuU fOi ipni have taken inm aeeuuui thai thCie is ii j leasuil whatsoever that the coefficients c were Jfc-independent, since the expansion functions 4 depend on k. This situation does not differ from that, which we encountered in the Hartree-Fock-Roothaan method (cf. p. 365), with one technical exception instead of the atomic orbitals we have q mmetry orbitals, in our case Bloch functions. 

Therefore, despite the fact that the secular determinant is of rather low rank (w), the infinity of the crystal, forces us to solve this equation an infinite number of times. For the time being, though, do not worry too much. 

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The set of SCF LCAO CO equations will be very similar to the set tor the moleeular orbital method (SCF LCAO MO). In prineiple, the only difference will be that, in the erystal ease, we will eonsequently use symmetry orbitals (Bloeh funetions) instead of atomie orbitals. 

The above relativistic ab initio SCF LCAO CO method (further details can be found elsewhere< >) has the shortcoming that, in the final equations (1.127) and (1.128), the states denoted by h are not specified. To do this, as in the case of atoms and molecules, one must classify the states (by applying advanced group-theoretical methods) not only in the usual way but also according to j (the internal quantum number). This classification must be carried out before performing actual calculations... 

Here is the number of AOs per unit cell in chain P and n/ is the number of filled bands in the same chain. All the other quantities in equations (6.36)-(6.38) were defined earlier in Section 1.1. In this way the ab initio SCF LCAO CO method (see Section 1.1) can be reformulated to obtain MCF charge densities of the interacting chains. By taking the electrostatic interactions between these chains [which can be easily obtained by the appropriate generalization of equation (6.32)], one obtains automatically the polarization contribution as well. For further details see Section 6.3, in which the interaction of a polyglycine chain with different polynucleotide chains is discussed. 

The ab initio SCF LCAO crystal orbital (CO) method (which applies a non-local exchange and keeps all the occurring three-and four center integrals if the number of neighbours to be taken into account has been chosen) was developed about twelve years ago (1 ). In this theory one has to solve the generalized eigenvalue equation... 

Since the ab initio calculations yield molecular orbitals (MO s) in terms of atom centered basis functions, it is straightforward to cast these mo s in the "LCAO" form of equations 20 and 21 (and their "primed" counterparts). These SCF MO s have been found to yield X values quite similar to those obtained from rigorous application of the corresponding orbital method to the ab initio wavefunctions. For the case of Co(NIfe )e, in which M (eg symmetry) is empty, X has been defined by using the "primed" counterpart of eq 22 and the y value obtained for the occupied L orbital of eg S3rmmetry. 

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