INDEX Canonical form

Let us now consider the special case when = T = T, i.e. when the original operator T is self-adjoint and real. In the conventional Hartree-Fock scheme - in the following indicated by an index c (=conventional) - which is based on relation (3.48), one has pc = pc and Teff c = Teff c, and it is obvious that the classical canonical form tc must be diagonal with only real eigenvalues. 

The index i is a superindex that refers to a particular realization of both E and N. If we now follow the same procedure used earlier to derive the canonical distribution by maximizing the entropy, it is found that the resulting probabilities are of the form... 

We first form the canonical index ((a6)(cd)) from the atomic centre labels. Then apply the permutations induced by the symmetry operations (summarised in the above table) and form the canonical index ((a 6 )(c d )) for each permutation there are three possibilities ... 

We now form the canonical index of an integral involving a basis function on each of the centres let ((ij)(k()) be such an index, it is only necessary that one of 4>i, 4>j, k and (j>i be on each of the centres a,b,c and d, not that they be on the respective centres. Now apply the transformations contained in the coded linear transformation array to this set of basis-function labels there are, this time, four possibilities ... 

We assume that the canonical orbitals 95m (k,r) = (pbasis vectors of irrep 7 with wavevector k, n discriminates between the independent bases of equivalent irreps the index a replaces 7, m,ij, and s replaces a, k) of an energy band under consideration form a basis in the space < (q,/3) of some indrep of the group G > of a crystal. The localized functions are defined by a unitary transformation in the space of indrep ... 

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