Theorem Kohn-Sham

The first Kohn-Sham theorem tells us that it is worth looking for a way to calculate molecular properties from the electron density. The second theorem suggests that a... 

Using Kohn-Sham theorem one can solve equation (4) by obtaining the self-consistent solution of the following Hamiltonian ... 

The first Kohn-Sham theorem tells us that it is worth looking for a way to calculate molecular properties from the electron density. The second theorem suggests that a variational approach might yield a way to calculate the energy and electron density (the electron density, in turn, could be used to calculate other properties). Recall that in wavefunction theory, the Hartree-Fock variational approach (section 5.2.3.4) led to the HF equations, which are used to calculate the energy and the wavefunction. An analogous variational approach led (1965) to the KS equations [26], the basis of current... 

The Hohenberg-Kohn and Kohn-Sham theorems simply state that the total energy can be obtained by applying the variation principle to the total energy density functional and suggest that the one-electron equations obtained in this way also account for electronic correlation. HKS derive another theorem that the total energy is uniquely determined by the density. 

The Kohn-Sham theorem then allows to obtain a more tractable single-particle Schrodinger equation. Many-particle effects are included by an effective exchange and correlation potential, which is derived variation-ally. 

Theory (DFT). The basic ideas of Density Functional Theory are contained in the two original papers of Hohenberg, Kohn and Sham, [22, 23] and are referred to as the Hohenberg-Kohn-Sham theorem. This theory has had a tremendous impact on realistic calculations of the properties of molecules and solids, and its applications to different problems continue to expand. A measure of its importance and success is that its main developer, W. Kohn (a theoretical physicist) shared the 1998 Nobel prize for Chemistry with J.A. Pople (a computational chemist). We will review here the essential ideas behind Density Functional Theory. 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

Our aim here is to apply the differential virial theorem to get an expression for the Kohn-Sham XC potential. To this end, we assume that a noninteracting system giving the same density as that of the interacting system exists. This system satisfies Equation 7.4, i.e., the Kohn-Sham equation. Since the total potential term of Kohn-Sham equation is the external potential for the noninteracting system, application of the differential virial relationship of Equation 7.41 to this system gives... 

Now we discuss the differential virial theorem for HF theory and the corresponding Kohn-Sham system. The Kohn-Sham system in this case is constructed [41] to... 

In summary, the original Thomas-Fermi-Dirac DFT was unable to give binding in molecules. This was corrected by Kohn-Sham, [11] who chose to use an orbital rather than density evaluation of the kinetic energy. By the virial theorem, = —E, so this was a necessity to obtain realistic results for energies. Next, it was shown that the exact exchange requires an orbital-dependent form, too. [47,48] The future seems to demand an orbital-dependent form for the correlation. 

We have applied a slight variation of this general idea to the exchange-correlation potential of the He atom [18], The virial theorem applied to the Kohn-Sham system yields [39] ... 

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