Ground-state energy functional

In the next step one decomposes the ground state energy functional (3.3) in the standard fashion,... 

In Kohn-Sham (KS) density functional theory (DFT), the occupied orbital functions of a model state are derived by minimizing the ground-state energy functionals of Hohenberg and Kohn. It has been assumed for some time that effective potentials in the orbital KS equations are always equivalent to local potential functions. When tested by accurate model calculations, this locality assumption is found to fail for more than two electrons. Here this failure is explored in detail. The sources of the locality hypothesis in current DFT thinking are examined, and it is shown how the theory can be extended to an orbital functional theory (OFT) that removes the inconsistencies and paradoxes. 

Such an ensemble generalized ground-state energy functional, E = E[N, v] = E[p[N. i l- represents the thermodynamic potential of the N, v -representation, with the corresponding generalized Hellmann-Feynman expression for its differential (see equations (17), (22) and (27)) ... 

With [n,mj) one can then define the ground state energy functional As Etot[n,ni depends on Bg t only via (< >[ ,w] / the limit... 

The basic idea underlying Kohn-Sham theory [4] is the construction of a mode system of noninteracting quasi-particles for which the density is the same as that of the interacting system. As such the ground-state energy functional E[p] is partitioned as... 

The application of the variational principle to the ground-state energy functional of Eq. (27) for arbitrary norm conserving variations of the density leads to the Kohn-Sham equation... 

According to the Kohn-Sham approach the ground-state energy functional Eks is rewritten as... 

It is also possible to calculate excitation energies from the ground-state energy functional. In fact, it was proved by Perdew and Levy [30] that every extremum density rii r) of the ground-state energy functional Ey[n] yields the energy Ei of a stationary state of the system. The problem is that not every excited-state density, rii r), corresponds to an extremum of Ev n], which implies that not all excitation energies can be obtained from this procedure. 

To construct the ground-state energy functional we use the long known adiabatic connection fluctuation-dissipation (ACFD) formula ... 

The kinetic-energy discontinuity is thus simply the Kohn-Sham single-particle gap Aks, or HOMO-LUMO gap, whereas the exchange-correlation discontinuity Axe is a many-body elfect. The true fundamental gap A = (AT- -1)- - (7V—1) — 2 (N) is the discontinuity of the total ground-state energy functional, ... 

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is... 

There are two functions, so we shall obtain two eigenvalues. The ground-state energy will be the lower of the two. The full secular matrix is... 

Computations done in imaginary time can yield an excited-state energy by a transformation of the energy decay curve. If an accurate description of the ground state is already available, an excited-state description can be obtained by forcing the wave function to be orthogonal to the ground-state wave function. 

Such methods owe their modern origins to the Fiohenberg-Kohn theorem, published in 1964, which demonstrated the existence of a unique functional which determines the ground state energy and density exactly. The theorem does not provide the form of this functional, however. 

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

This proof shows that any approximate wave function will have an energy above or equal to the exact ground-state energy. There is a related theorem, known as MacDonald s Theorem, which states that the nth root of a set of secular equations (e.g. a Cl matrix) is an upper limit to the n — l)th excited exact state, within the given symmetry subclass. In other words, the lowest root obtained by diagonalizing a Cl matrix is an upper limit to the lowest exact wave functions, the 2nd root is an upper limit to the exact energy of the first excited state, the 3rd root is an upper limit to the exact second excited state and so on. 

Within the Bom-Oppenheimer approximation, the last term is a constant. It is seen that the Hamilton operator is uniquely determined by the number of electrons and the potential created by the nuclei, V e, i.e. the nuclear charges and positions. This means that the ground-state wave function (and thereby the electron density) and ground state energy are also given uniquely by these quantities. 

Lead, excess entropy of solution of noble metals in, 133 Lead-thalium, solid solution, 126 Lead-tin, system, energy of solution, 143 solution, enthalpy of formation, 143 Lead-zinc, alloy (Pb8Zn2), calculation of thermodynamic quantities, 136 Legendre expansion in total ground state wave function of helium, 294 Lennard-Jones 6-12 potential, in analy-... 

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