Orbital functionals external potential

In Ecjuation (3.47) we have written the external potential in the form appropriate to the interaction with M nuclei. , are the orbital energies and Vxc is known as the exchange-correlation functional, related to the exchange-correlation energy by ... 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

This says that/(r) is the functional derivative (Section 1.23.2, The KohnSham equations) of the chemical potential with respect to the external potential (i.e. the potential caused by the nuclear framework), at constant electron number and that it is also the derivative of the electron density with respect to electron number at constant external potential. The second equality shows fir) to be the sensitivity of p(r) to a change in N, at constant geometry. A change in electron density should be primarily electron withdrawal from or addition to the HOMO or LUMO, the frontier orbitals of Fukui [154] (hence the name bestowed on the function by Parr and Yang). Since p(r) varies from point to point in a molecule, so does the Fukui... 

A very important aspect of DFPT is the extension to perturbations of the variational principle. In a given external potential v r) the charge density and Kohn-Sham orbitals are obtained by minimizing the functional (see Sect. 4)... 

Given the V-electron Hamiltonian H = T + U + V, and postulating some rule 1fr— that determines a reference state, this defines an orbital functional E=T+U+V + EC, where T + U + V = (P H 4>) is explicit, as defined previously, and Ec — /iexac( — (

orbital functional defined for a particular approximate model. It is assumed that E is minimized in the ground state of the model. V = Y,i nfilvli) if v is a general nonlocal external potential. 

Symmetrised density-functionals, which have been proposed recently [88] as the correct solution of the symmetry dilemma in Kohn-Sham theory, also naturally lead to fractional occupations. The symmetry dilemma occurs because the density or spin-density of KS theory may exhibit lower symmetry them the external potential due to the nuclear conformation. This in turn leads to a KS Hamiltonian with broken symmetry, leading to electronic orbitals that cannot be assigned to an irreducible representation... 

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