The Adiabatic Connection Formula

Similarly, taking as an approximate wave function for H yields 

Addition of these two inequalities gives Eq + Eo Eq + Eo, showing that the assumption was wrong. In other words, for the ground state there is a one-to-one correspondence between the electron density and the nuclear potential, and thereby also with the Hamilton operator and tlie energy. In the language of Density Functional Theory, the energy is a unique functional of the electron density, [p]. 

Litegrating over A between tlie limits 0 and 1 corresponds to smoothly transforming the non-interacting reference to the real system. This integration is done under the assumption that the density remains constant. 

Note that it is the same density that appears in the two integrals over Vext(O) and Vext(l), 

Using that Vext(l) = Vne, separating out the Coulomb part of Vge and using the definition of Ei (eq. (6.7)), gives the adiabatic connection formula (eq. (6.32)). 

Theory, the energy is a unique functional of the electron density, E[p]. 

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Although the adiabatic connection formula of Eq. (8) justifies a certain amount of Hartree-Fock mixing, there are situations in which a should vanish. In a spin-restricted description of the molecule Hj at infinite bond length (Sect. 4) the Hartree-Fock or A = 0 hole is equally distributed over both atoms, and is independent of the electron s position. But the hole for any finite A, however small, is entirely localized on the electron s atom, so no amount of Hartree-Fock mixing is acceptable in this case. 

The starting point for the proposed new approach is an exact formula [238], [239], based on the adiabatic connection formula and the zero-temperature fluctuation-dissipation theorem, relating the groundstate xc energy to the inter-... 

Substitution of these expressions into the adiabatic connection formula of Eq. (29)... 

The idea of mixing density functional approximations with exact (Hartree-Fock-Uke) exchange rests on theoretical considerations involving the adiabatic connection formula... 

The adiabatic connection formula is at the root of the ACMs. This formula is usually expressed in the form ... 

Several gradient corrected exchange [28-30] and correlation [31-34] functionals have been proposed. In our study we use different combinations of those of Becke [28] and Perdew and Wang [30] for the exchange and the functionals proposed by Perdew [31] and Proynov [34] for the correlation. Recently, Becke [29] has introduced the so-called hybrid functional that is based on the "adiabatic connection" formula [35], Because of its reliability, largely validated in the literature [36-38], we employ in our work also the B3LYP functional which uses the Becke gradient... 

The Variational Principle The Hohenberg-Kohn Theorems The Adiabatic Connection Formula Reference... 

The philosophy behind the hybrid functionals is simple and rooted in the adiabatic connection formula, which is a rigorous ab initio formula for the exchange-... 

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