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The following theorem (Ref.) implies that the functional Ev [γ] is stationary, i.e., δ Ev =0 with respect to all variations in the VR space. Theorem-A necessary condition for the differentiable functional J [y] to have an extremum (minimum) for y= y0 is that its first variation vanish for y= y0, i.e., that δJ [h] =0 for y= y0 and all admissible variations h. Thus, the functional Ev [γ] must be stationary because it is differentiable in the VR space and the Rayleigh-Ritz variational principle implies that it is minimum at the ground state one-matrix γ= γgs. The differentiability of Ev [γ] follows from the differentiability of the mapping γ→ Ψ0 in the VR space, if the ground state Ψ0 is nondegenerate.
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The following theorem (Ref.) implies that the functional Ev [γ] is stationary, i.e., δ Ev =0 with respect to all variations in the VR space. Theorem-A necessary condition for the differentiable functional J [y] to have an extremum (minimum) for y= y0 is that its first variation vanish for y= y0, i.e., that δJ [h] =0 for y= y0 and all admissible variations h. Thus, the functional Ev [γ] must be stationary because it is differentiable in the VR space and the Rayleigh-Ritz variational principle implies that it is minimum at the ground state one-matrix γ= γgs. The differentiability of Ev [γ] follows from the differentiability of the mapping γ→ Ψ0 in the VR space, if the ground state Ψ0 is nondegenerate.
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If the DFT-KS system is degenerate, the occupation numbers of the degenerate KS orbitals are not determined by the Aufbau principle. In this case, the KS system adopts an ensemble state, and the occupation numbers of the degenerate orbitals are chosen such that the KS system is self-consistent and reproduces the density of the interacting system (Refs.).
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If the DFT-KS system is degenerate, the occupation numbers of the degenerate KS orbitals are not determined by the Aufbau principle. In this case, the KS system adopts an ensemble state, and the occupation numbers of the degenerate orbitals are chosen such that the KS system is self-consistent and reproduces the density of the interacting system (Refs.).
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We do not demonstrate this statement for χ as an operator on an infinite-dimensional space. However, it is valid in a finite-dimensional basis (see Ref. for a proof in DFT).
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We do not demonstrate this statement for χ as an operator on an infinite-dimensional space. However, it is valid in a finite-dimensional basis (see Ref. for a proof in DFT).
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For convenience we assume here that the system has no pinned states.
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Generally, the degenerate 1MFT-KS eigenvalues can give only the chemical potential.
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Generally, the degenerate 1MFT-KS eigenvalues can give only the chemical potential.
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