K theory

With these motivational remarks, we now recover standard KS theory (in the particular instance just defined) from Eq. (3.16) for this specific case but with an important new constraint. Let Sjvi/p(p) the set of all Independent Particle FORDO s corresponding to p with precisely N non-zero terms ... 

Whereas the classic Kohn-Sham (KS) formulation of DFT is restricted to the time-independent case, the formalism of TD-DFT generalizes KS theory to include the case of a time-dependent, local external potential w(t) [27]. 

The complete treatment of solvation effects, including the solute selfpolarization contribution was developed in the frame of the DFT-KS formalism. Within this self consistent field like formulation, the fundamental expressions (96) and (97) provide an appropriate scheme for the variational treatment of solvent effects in the context of the KS theory. The effective KS potential naturally appears as a sum of three contributions the effective KS potential of the isolated solute, the electrostatic correction which is identified with the RF potential and an exchange-correlation correction. Simple formulae for these quantities have been presented within the LDA approximation. There is however, another alternative to express the solva-... 

A complete treatment, including the solute self-polarization contribution, may be developped in the context of the KS theory. It was shown that within the LDA approximation, simple expressions for the effective KS potential may be obtained. 

Pitzer KS. Theory ion interaction approach. In Pytkowicz RM, ed. Activity Coefficients in Electrolyte Solutions. Vol. 1. Boca Raton, FL CRC Press, 1979 157-208. 

The KS theory exchange-correlation energy functional Exc [p] and its derivative (potential) vxc (r) can both be expressed as a sum of two terms, one representative of the quantum-mechanical Pauli and Coulomb correlations and the other of the correlation-kinetic effects. Thus, we may write... 

With the KS theory Fermi and Coulomb holes defined by Eq. (27), the derivative vxc(r) can also be expressed in terms of its separate Pauli, Coulomb and correlation-kinetic contributions as... 

Time-dependent density functional theory (TDDFT) as a complete formalism [7] is a more recent development, although the historical roots date back to the time-dependent Thomas-Fermi model proposed by Bloch [8] as early as 1933. The first and rather successful steps towards a time-dependent Kohn-Sham (TDKS) scheme were taken by Peuckert [9] and by Zangwill and Soven [10]. These authors treated the linear density response of rare-gas atoms to a time-dependent external potential as the response of non-interacting electrons to an effective time-dependent potential. In analogy to stationary KS theory, this effective potential was assumed to contain an exchange-correlation (xc) part, r,c(r, t), in addition to the time-dependent external and Hartree terms ... 

It is universally agreed in KS theory and practice that the electric field due to... 

In KS theory [4], the occupied orbital functions of the model state are determined by the KS equations... 

Density functional theory also offers an attractive computational scheme, the Kohn-Sham (KS) theory [2], similar to the Hartree-Fock (HF) approach, which in principle takes into account both the electron exchange and correlation effects. The canonical KS orbitals thus offer certain interpretative advantages over the widely used HF orbitals, especially for describing the bond dissociation and the open system characteristics, when the electrons are added or removed from the system [3,82,126-130]. For this reason, a determined effort has been made to calculate the reactivity indices from the KS DFT calculations [3,82,83,112,118,119,121, 131-136]. 

The transformation (42) is a mapping from R4 to R3 it therefore leaves one degree of freedom in the parametric space undetermined. In KS theory (Kustaanheimo and Stiefel 1965, Stiefel and Scheifele 1971), this freedom is taken advantage of by trying to inherit as much as... 

It is certainly interesting that the working equations of KS theory are so similar to those of HF theory, but the extremely important differences between the two theories are even more intriguing. The most profound of these is, of course, that KS theory is in principle capable of yielding the exact Schrodinger energy, which is certainly not the case with HF theory. That such a simple approach has such an enormous potential immediately makes the prospect of its application to chemical systems attractive, especially when contrasted with the traditional ab initio approaches. 

As we noted in the introductory section, the working equations in KS theory are of the same form as those in HF theory, merely with the HF exchange potential redefined. In deriving an expression for the DFT XC potential corresponding to Eq. (15), the calculus of variations may be used to obtain... 

From the preceding two examples, it is evident that KS DFT has serious problems with radical abstraction reactions. It is also clear that, as before, gradient corrections to the exchange functional play an important role, though this is by no means sufficient for good performance. These results do not bode well for the application of KS theory with current functionals to other, more complicated radical abstraction reactions. 

Symmetrised density-functionals, which have been proposed recently [88] as the correct solution of the symmetry dilemma in Kohn-Sham theory, also naturally lead to fractional occupations. The symmetry dilemma occurs because the density or spin-density of KS theory may exhibit lower symmetry them the external potential due to the nuclear conformation. This in turn leads to a KS Hamiltonian with broken symmetry, leading to electronic orbitals that cannot be assigned to an irreducible representation... 

The key-problem in KS theory is to determine an analytical form for the term. The knowledge of the exact expression of the term leads, through eqn (5), to the exact total energy. Unfortunately, this expression is unknown, and approximations must be used. The most common approach to the problem of the representation of E c is to separate the correlation contribution from the exchange counterpart. This distinction is somewhat artificial in the context of DF theory, but the separation between these two terms considerably simplifies the discussion. Let us start with the simplest DF approach to the problem of exchange functional, i.e. the local spin density approximation (LSD), in which the functional for the uniform electron gas of density p is integrated over the whole space ... 

---