Kohn-Sham DFT

The remaining contribution to the kinetic energy is due to electron correlation 

The Kohn-Sham approximation to the kinetic energy rjp] is extremely accurate. The correlation kinetic energy r .[p] is usually smaller than the magnitude of the correlation energy. 

In Kohn-Sham DFT (KS-DFT), the only portion of the total energy 

Substituting Equation 1.27 into Equation 1.32 and invoking the variational principle give rise to the Kohn-Sham equation 

The potential that enters into this equation is called the Kohn-Sham potential 

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In actual practice, self-consistent Kohn-Sham DFT calculations are performed in an iterative manner that is analogous to an SCF computation. This simiBarity to the methodology of Hartree-Fock theory was pointed out by Kohn and Sham. 

Over the past decade, Kohn-Sham density functional theory (DFT) has evolved into what is now one of the major approaches in quantum chemistry.1-20 It is routinely applied to various problems concerning, among other matters, chemical structure and reactivity in such diverse fields as organic, organometallic, and inorganic chemistry, covering the gas and condensed phases as well as the solid state. What is it that makes Kohn-Sham DFT so attractive Certainly, an important reason is that it represents a first-principles... 

This qualitative model, based on semiempirical MO theory, focuses entirely on the so-called electronic effects, as the A—H bonding orbital interactions are often called. However, steric repulsion (i.e., the destabilizing orbital interactions) between the hydrogen substituents in AH3 is just as important in the interplay of mechanisms that determine whether the molecule adopts a planar or a pyramidal shape. In fact, as will become clear from the following discussion, which is based on a Kohn-Sham DFT study at the BP86/TZ2P level,107 108 steric repulsion turns out to be the decisive factor in determining the pucker of our example.133... 

The Exchange-Correlation Energy Functional Various Levels of Kohn-Sham DFT... 

The only cases for which one might anticipate differences between DFT and wavefunction theory as regards visualization (Sections 5.5.6 and 6.3.6) are those involving orbitals as explained in Section 7.2.3.2, The Kohn-Sham equations, the orbitals of currently popular DFT methods were introduced to make the calculation of the electron density tractable, but in pure DFT theory orbitals would not exist. Thus electron density, spin density, and electrostatic potential can be visualized in Kohn-Sham DFT calculations just as in ab initio or semiempirical work. However, visualization of orbitals, so important in wavefunction work (especially the HOMO and FUMO, which in frontier orbital theory [154] strongly influence reactivity) does not seem possible in a pure DFT approach, one in which wavefunctions are not invoked. In currently popular DFT calculations one can visualize the Kohn-Sham orbitals, which are qualitatively much like wavefunction orbitals [130] (Section 7.3.5, Ionization energies and electron affinities). 

In recent years, density functional theory (DFT) has become the most widely used electronic structure method for large molecular systems. The Kohn-Sham DFT method accounts for exchange and correlation effects via a particular exchange correlation functional. In its present form, Kohn-Sham DFT is not, strictly speaking, an ab initio method, since the functionals contain empirical parameters. 

In Kohn-Sham DFT based approaches, expressions that are of similar structure as Eqs. (9a) and (9b) are obtained, but in the form of contributions from all occupied Kohn-Sham MOs The excited-state wavefunctions are at the same time formally replaced by the unoccupied MOs, and the many-electron perturbation operators /T(M41, etc. by their one-electron counterparts //(M-41, etc. Orbital energies e and ea formally substitute the total energies of the states (see later). Thus, similar interpretations of NMR parameters can be worked out in which the highest occupied MO-lowest unoccupied MO gap (HLG) plays a highly important role. It must be emphasized, though, that there is no one-to-one correspondence between the excited states of the SOS equations and the unoccupied orbitals which enter the DFT expressions, nor between excitation energies and orbital energy differences, i.e., there are no one-determinantal wavefunctions in Kohn-Sham DFT perturbation theory which approximate the reference and excited states. 

If the Kohn-Sham orbitals [52] of density functional theory (DFT) [53] are used instead of Hartree-Fock orbitals in the reference state [54], the RI can become essential for the realization of electron propagator calculations. Modern implementations of Kohn-Sham DFT [55] use the variational approximation of the Coulomb potential [45,46] (which is mathematically equivalent to the RI as presented above), and four-index integrals are not used at all. A very interesting example of this combination is the use of the GW approximation [56] for molecular systems [54],... 

Since DFT calculations are in principle only applicable for the electronic ground state, they cannot be used in order to describe electronic excitations. Still it is possible to treat electronic exciations from first principles by either using quantum chemistry methods [114] or time-dependent density-functional theory (TDDFT) [115,116], First attempts have been done in order to calculate the chemicurrent created by an atom incident on a metal surface based on time-dependent density functional theory [117, 118]. In this approach, three independent steps are preformed. First, a conventional Kohn-Sham DFT calculation is performed in order to evaluate the ground state potential energy surface. Then, the resulting Kohn-Sham states are used in the framework of time-dependent DFT in order to obtain a position dependent friction coefficient. Finally, this friction coefficient is used in a forced oscillator model in which the probability density of electron-hole pair excitations caused by the classical motion of the incident atom is estimated. 

They took the standard energy functional in DFT theory but, rather than simply solving this, they sought a way of achieving the solution of the Kohn-Sham DFT equations, geometry relaxation and volume and strain relaxation simultaneously. 

Kohn-Sham DFT with Real-Space Grids... 

Though a number of approaches have been made for incorporating dispersion into DFT, the most widely utilized method is to append a semiclassical correction. This approach, called DFT-D, was developed by Grimme. A dispersion term is added to the Kohn-Sham DFT energy (Eq. (1.47)) of the form... 

For our purposes here, it is only important to know that Kohn-Sham DFT is algorithmically very similar to Hartree-Fock theory. Consequently, it is computa-... 

The main complication associated with extending a Hartree-Fock quadratic response code to Kohn-Sham DFT is the evaluation of the exchange-correlation contribution to We refer to [41] for detailed expressions and Section 3 for a discussion... 

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