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Derivator

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could someone explain the following quote to me, please? It explains, why the Kohn-Sham-Scheme of DFT fails to describe transition states.

Failure to describe transition states:

The total energies predicted for transition states are very unreliable. Both in this

case and in the case of strongly correlated systems the true wave-function has not

only one dominating determinant but several important determinants. This multi-

configuration character of the wave-function becomes visible in the occupation

numbers. Since the occupation number of a certain natural orbital is equal to the

sum of the squared coefficients of all the determinants that contain this natural or-

bital, there will be occupation numbers that are neither close to one or zero. If

there is only one dominating determinant, the occupation number of the occupied

natural orbitals contained in this determinant will be close to one whereas the occu-

pation numbers of all the other virtual orbitals are very small. That the Kohn-Sham

methods fail in this case is not so surprising.

My questions:

- If [itex]N_i[/itex] ist the occupation number and the wavefunction may be given by [itex]\Psi = \sum_I{c_I \Phi_I}[/itex] where [itex]\Phi_I[/itex] are Slater-Determinants, why does from [itex]N_i = \sum{C_I^2}[/itex] follow that if many [itex]C_I \neq 0[/itex] there will be occupation numbers [itex]N_i[/itex] that are not close to 1 and not close to 0? (this question relates to the red colored text)
- If there is only one [itex]C_I[/itex] that is not vanishing and all other coefficients are small, why will be the occupation number of the natural orbital that is contained in the Slater-Determinant belonging to the [itex]C_I[/itex] that is not vanishing close to 1? Why are all occupation numbers of the virtual orbitals very small? (this question relates to the blue colored text)