Kohn-Sham representation

Further pragmatic moves are described in details in numerous books and reviews of which we cite the most concise and recent Ref. [82], Two further hypotheses are an important complement to the above cited theorems. One is the locality hypothesis, another is the Kohn-Sham representation of the single determinant reference state in terms of orbitals. The locality has been seriously questioned by Nesbet in recent papers [83,84], however, it remains the only practically implemented solution for the DFT. The single determinant form of the reference state in its turn guarantees that all the averages of the electron-electron interaction appearing in this context are in fact calculated with the two-electron density given by the determinant term in Eq. (5) with no cumulant. 

Two further hypotheses are an important complement to the theorems cited above. One is the locality hypothesis, and the other is the Kohn-Sham representation of the... 

Figure 2. Representation of the energies of the 4f Kohn— Sham orbitals of Eu2+ in CsMgBr3 obtained from the output of AOC-type calculation. The two diagrams represent the cases of geometries (Table 2) at ground and excited configurations (GC versus EC).

Are there any 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 many 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-... 

This step is similar to what we have done in equation (7-7) where we obtained the matrix representation of the Kohn-Sham operator. If we insert expression (7-14) for the charge density in terms of the LCAO functions and make use of the density matrix P defined in equation (7-15), we arrive at... 

The difficulty of this problem can be appreciated by noticing that in order to solve the Kohn-Sham equations exactly, one must have the exact exchange-correlation potential which, moreover, must be obtained from the exact exchange-correlation functional c[p( )] given by Eq. (160). As this functional is not known, the attempts to obtain a direct solution to the Kohn-Sham equations have had to rely on the use of approximate exchange-correlation functionals. This approximate direct method, however, does not satisfy the requirement of functional iV-representability,... 

The variational procedure in Eq. (100) is in the spirit of the Kohn-Sham ansatz. Since satisfies the (g, K) conditions, it is A-representable. In general, Pij ig corresponds to many different A-electron ensembles and one of them,, corresponds to the ground state of interest. However, for computational expediency in computing the energy, a Slater determinantal density matrix,... 

The preceding approach can be viewed as an orbital representation analogue for a recently proposed Kohn-Sham-based pair-density functional theory [17],... 

The second approach to this problem is to derive orbital-based reformulations of existing algorithms based on the spatial representation of the g-density. The resulting formulations are in the spirit of the orbital-resolved Kohn-Sham approach to density functional theory. 

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

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