On Degeneracy, Ensembles and other Oddities

Another typical class of examples is given by the dissociation of diatomic molecules as already alluded to above in the case of the H2 molecule where the correct dissociation behavior was only achieved by allowing for symmetry broken spin densities. This problem 

Are there ary remedies in sight within approximate Kohn-Sham density functional theory to get correct energies connected with physically reasonable densities, i. e., without having to use wrong, that is symmetry broken, densities In maty cases the answer is indeed yes. But before we consider the answer further, we should point out that the question only needs to be asked in the context of the approximate functionals for degenerate states and related problems outlined above, an exact density functional in principle also exists. The real-life solution is to employ the non-interacting ensemble-Vs representable densities p intro- 

The densities p are obtained from a set of degenerate KS wave functions and the w are the corresponding weights. Without going into details we note that regular density functional theory can be extended to such ensembles. For our problems at hand, we can write down the energy expression as 

are the equivalent densities obtained from symmetry breaking. Let us clarify this concept by using the examples given above. In the B2 case, the two equivalent symmetry broken Kohn-Sham Slater determinants are 

These two determinants produce equivalent, but asymmetric densities. In addition, the energies obtained from these densities are the same, i. e. E[pj] = E[p2], If we now insert these two densities in equation (5-24) it is clear that the energy will be invariant to the choice of wt and w2. If we choose Wj = w2 = 1/2 we will also arrive at the physically correct, i. e. symmetric density. A very similar reasoning can be used for the H2 dissociation. We again have two equivalent Kohn-Sham spin densities corresponding to 

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