Kohn-Sham density functional theory KS-DFT

Due to its excellent balance between accuracy and computational cost, Kohn-Sham density functional theory (KS-DFT) [13,14] is usually the method of choice to investigate electronic ground states and their properties in chemistry and solid-state physics [15,16]. Hartree-Fock (HF) wavefunctions, on the other hand, are the starting point for ab initio electron correlation methods [4,15] which are discussed in Section 4. 

For variational methods, such as Hartree-Fock (HF), multi-configurational self-consistent field (MCSCF), and Kohn-Sham density functional theory (KS-DFT), the initial values of the parameters are equal to zero and 0) thus corresponds to the reference state in the absence of the perturbation. The A operators are the non-redundant state-transfer or orbital-transfer operators, and carries no time-dependence (the sole time-dependence lies in the complex A parameters). Furthermore, the operator A (t)A is anti-Hermitian, and tlie exponential operator is thus explicitly unitary so that the norm of the reference state is preserved. Perturbation theory is invoked in order to solve for the time-dependence of the parameters, and we expand the parameters in orders of the perturbation... 

We shall in this section give a historic overview of how the electronic structure theory for transition metal complexes in their ground state has evolved from the 1950s to the present time. The account will include a discussion of wave function methods based on Hartree Fock and post-Hartree Fock approaches as well as Kohn-Sham density functional theory (KS-DFT). 

A feasible compromise of accurate forces and available system size might be Kohn-Sham density functional theory (KS-DFT). Several approaches, especially for general gradient approximation (GGA) functionals, are known to reduce the computational cost much lower than for conventional correlation methods. Unfortunately, KS-DFT aeeounts for electrostatic, exchange and induction forces very well, but fails for the description of dispersion forces. Del Popolo et al. have attributed... 

Because the Fock matrix depends on the one-particle density matrix P constructed conventionally using the MO coefficient matrix C as the solution of the pseudo-eigenvalue problem (Eq. [7]), the SCF equation needs to be solved iteratively. The same holds for Kohn-Sham density functional theory (KS-DFT) where the exchange part in the Fock matrix (Eq. [9]) is at least partly replaced by a so-called exchange-correlation functional term. For both HF and DFT, Eq. [7] needs to be solved self-consistently, and accordingly, these methods are denoted as SCE methods. 

Instead of supposing there to be a single Kohn-Sham potential, one can think of it as a vector in Fock space. For each sheet ft = N of the latter, there is a component vKS(r,N) and a corresponding set of Kohn-Sham equations. Density functional theory and Kohn-Sham theory hold separately on each sheet. Ensemble-average properties are then composed of weighted contributions from each sheet, computable sheet by sheet via the techniques of DFT and the KS equations. Nevertheless, though completely valid, this procedure would yield for the reactivity indices f(r), s(r), and S the results already obtained directly from Eqs. (28). We are left without proper definitions of chemical-reactivity indices for systems with discrete spectra at T = 0 [43]. 

A popular alternative is to employ density functional theory (DFT) methods. Kohn-Sham (KS) orbital energies in the ground state are a more reliable predictor of the ion state ordering than Koopmans theorem. There are good theoretical reasons for interpreting them as approximate vertical IE." For transition metal compounds AE methods, the DFT equivalent... 

As an example of a mature topic, consider Density Functional Theory (DFT). DFT is far from new and can be traced back to the work of John Slater and other solid state physicists in the 1950 s, but it was ignored by chemists despite the famous papers by Hohenberg/ Kohn (1964) and Kohn/ Sham (KS) (1965). The HF-LCAO model dominated molecular structure theory from the 1960 s until the early 1990s and I guess the turning point was the release of the rather primitive KS-LCAO version of GAUSSIAN. DFT never looked back after that point, and it quickly became the standard for molecular structure calculations. So this Volume of the SPR doesn t have a self contained Chapter on DFT because the field is mature. 

---