Kohn-Sham approximation

An alternative approach to conventional methods is the density functional theory (DFT). This theory is based on the fact that the ground state energy of a system can be expressed as a functional of the electron density of that system. This theory can be applied to chemical systems through the Kohn-Sham approximation, which is based, as the Hartree-Fock approximation, on an independent electron model. However, the electron correlation is included as a functional of the density. The exact form of this functional is not known, so that several functionals have been developed. 

The method of electronegativity equalization is used to fix the charges, with Mulliken definition of electronegativity. This is related by Parr [175] to the chemical potential of an electron gas, using Kohn-Sham approximation in the framework of density functional theory. Hence, electronegativity equalization corresponds to equating chemical potentials. 

In the Haitree-Fock or Kohn-Sham approximation (as discussed in Chapter 11, p. 667, we assume the Wf-tuple occupation of the molecular orbital (pAj, ni = 0,1,2),... 

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. 

J. Henriksson, T. Saue, P. Norman. Quadratic response functions in the relativistic four-component Kohn-Sham approximation. /. Chem. Phys., 128 (2008) 024105. 

Dash-dotted line Kohn-Sham approximation. 

K one would like to introduce an effective potential to describe deviations of the electron gas from uniformity, one would intuitively expect that Ffc , rather than the Slater potential, gives the most accurate description, since density fluctuations in the first place are expected at the Fermi sphere. This is confirmed in the Kohn-Sham approximation. Indeed, the total exchange energy for the homogeneous electron gas is given by ... 

Quadratic response function in the adiabatic four-component Kohn-Sham approximation has been formulated and implemented by Henriksson et al Applications to dihalosubstituted benzenes illustrate the significance of this work to describe heavy atom effects on the polarizability and first hyperpolarizability. In particular, using the CAM-B3LYP XC functional, relativity reduces the EFISHG nP response by 62% and 75% for meta- and urt/ju-dibromobenzene, and enhances the same response by 17% and 21% for meta- and or/ o-diiodobenzene, respectively. Moreover, these results have further evidenced that correlation and relativistic effects are not additive. 

Unlike the true propagator, the UCHF approximation is given by a simple closed formula and reqnires only minimum computational effort to evalnate on the fly if the orbitals are available. The nnconpled Hartree-Fock/Kohn-Sham approximation has almost completely vanished from the chemistry literature about 40 years ago when modem derivative techniques became available because of the poor results it produced for second-order properties. Some systematic expositions of analytical derivative methods still use it as a starting point, but it is in our opinion pedagogi-cally inappropriate, as it requires considerable effort to recover the coupled-perturbed Hartree-Fock results which can be derived in a simpler way. UCHF/UCKS is still used in some approximate theories, but we suspect that its only merit is easy computability. According to Geerlings et al. [29], the polarizabilities derived from the uncoupled density response function correlate well with accurate results but can be off by up to a factor of 2, and thus they are only qualitatively useful. Our results in Table 1 confirm this. 

Here p r) is the electron density, T[p] is the kinetic energy, and Vee[p] collects aU electron-electron interaction energies. A major technical problem is the accurate description of the kinetic energy functional T[p], In the Kohn-Sham approximation this problem is avoided by introducing orbitals of a non-interacting reference system. Levy and Perdew (Levy 1979 Levy and Perdew 1985) have shown that these orbitals are delivered by the following minimization procedure ... 

In the case of Hartree-Fock and formally also Kohn-Sham approximation and integer orbital occupations the above formula reduces to the following equation, which also shows the close link of the hole analysis to the 2-center Wiberg-Mayer bond indices ... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

In the preceding paragraph we have given a detailed survey of the Kohn-Sham approach to density functional theory. Now, we need to discuss some of the relevant properties pertaining to this scheme and how we have to interpret the various quantities it produces. We also will mention some areas connected to Kohn-Sham density functional theory which are still problematic. Before we enter this discussion the reader should be reminded to differentiate carefully between results that apply to the hypothetical situation in which the exact functional ExC and the corresponding potential Vxc are known and the real world in which we have to use approximations to these quantities. 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

Are there any remedies in sight within approximate Kohn-Sham density functional theory to get correct energies connected with physically reasonable densities, i. e., without having to use wrong, that is symmetry broken, densities In many cases the answer is indeed yes. But before we consider the answer further, we should point out that the question only needs to be asked in the context of the approximate functionals for degenerate states and related problems outlined above, an exact density functional in principle also exists. The real-life solution is to employ the non-interacting ensemble-Vs representable densities p intro-... 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

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