Kohn-Sham procedure

Relation with the Conventional Kohn-Sham Procedure. 

At least for the case of a non-degenerate ground state of a closed shell system, it is possible to delineate the standard Kohn-Sham procedure quite sharply. (The caveat is directed toward issues of degeneracy at the Fermi level, fractional occupation, continuous non-integer electron number, and the like. In many but of course not all works, these aspects of the theory seem to be... 

The ensemble search in Eq. (82) is the Kohn-Sham procedure, generalized to allow fractional orbital occupation numbers [55, 57-59]. Equation (82) can... 

The prime in Eq. (3-62) indicates that the sum is restricted to sites that do not belong to the same molecule. Depending on the specific implementation the tensors T(1) are multiplied with appropriate /e factors for the associated atoms. The last term in Eq. (3-59), efacM, is the macroscopic electric field. This completes the most usual form of vpo1, i.e., the potential of the dipoles due to the total field at the polarizable sites is made a part of the effective Hamiltonian and Eq.(3-24) is solved self-consistently. Since the induced dipoles M in the solvent (MM) part are self-consistent for any field E, i.e., also for intermediate fields during the iterative process for solving Eq. (3-24), in this way we obtain an overall self-consistent solution, similar to, e.g., the HF or Kohn-Sham procedure. Extension to post-HF methods are straightforward because the reaction potential (RP) is formally a one-particle... 

The Kohn-Sham procedure is to solve for the orbitals that minimize the energy, which reduces to the set of pseudoeigenvalue equations... 

In practice, it has been found that HF and typical Kohn-Sham procedures (e.g. BP86, B3PW91, etc.) produce valence orbital energies having magnitudes that tend to be larger and smaller, respectively, than the experimental ionization energies of the electrons [4,8,10,14,16] ... 

We have now come to the discussion of the central equations which form the basis of almost any practical application of density functional theory the Kohn-Sham equations. Kohn and Sham [8] introduced an auxiliary noninteracting system of particles with the property that it yields the same ground state density as the real interacting system. In order to put the Kohn-Sham procedure on a rigorous basis we introduce the functional... 

The Hohenberg-Kohn Theorems and the Kohn-Sham procedure... 

Another interesting point is the nature of Ts. The use of Ts in the Kohn-Sham procedure comes from the availability of solutions for the noninteractive case. This quantity is the kinetic energy of the imaginary system of noninteracting electrons. Ts is an unknown functional of the density, but it can be obtained exactly using the wavefunction of the noninteracting system. Therefore, it is not required to have a functional, either in the interactive or noninteractive systems, for its exact calculation. 

The electronic case, which has been treated in all previous chapters of this book is essentially based on the Hohenberg-Kohn theorems [1] and the Kohn-Sham procedure [2]. This is the most modern use of DFT, to solve the Schrodinger equation of an electronic system, and it is the most precise first principles method for the calculation of energies of chemical interest. Although there have been, for more than 100 years, other series of developments that also use the electronic density, we will avoid using DFT to... 

Density functional techniques are available for the calculation of the molecular and electronic structures of ground state systems. Techniques of this kind are also applied extensively in the study of classical liquids, where applications cover a broad spectrum ranging from fluids at interfaces to theories of freezing and nucleation. The area of nuclear physics is still in a very early stage of development in the use of DFT, mainly because there is not yet a complete theory, as there is in the molecular and atomic cases. This chapter has focused on the theoretical aspects of other applications not related to the use of the Kohn-Sham procedure for electronic systems, with the hope that a better understanding of the problems and successes in the respective areas would help in the development of improved functionals. 

Without going into much detail, it is noted that the so-called Kohn-Sham procedure allows one to solve a set of pseudo-single-particle equations that would provide the exact ground state energy if the exact would be known. This... 

A particularly popular Kohn-Sham procedure is the local spin density approximation, or LSDA, which is obtained by adding the jellium correlation functional to HFS theory to yield... 

Since the 1980s, a wide variety of Kohn-Sham procedures with even better predictive power have been proposed. One of the most popular of these, which uses the B88 and LYP functionals, is the BLYP method... 

Becke and others have also introduced hybrid or adiabatic connection methods, in which a part of F30 is added to a Kohn-Sham procedure. One of these is the popular B3LYP method... 

Excitation energies can also be calculated from excited-state densities. A mapping between excited-state densities and the external potential, and a Kohn-Sham-like scheme for calculating the density and energy of a specific excited state of interest have been constructed, but the definition of a proper Kohn-Sham procedure is not trivial. ... 

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