Density Kohn-Sham formulation

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

Nonetheless, Eq. (95) is perhaps the most natural generalization of the Kohn-Sham formulation to g-density functional theory. Indeed, Ziesche s first papers on 2-density functional theory feature an algorithm based on Eq. (95), although he did not write his equations in the potential functional formulation [1, 4]. The early work of Gonis and co-workers [68, 69] is also of this form. 

In the Kohn-Sham formulation of DFT the problem of finding the ground state energy of this system is exactly mapped onto one of finding the electron density which minimizes the total energy functional... 

In practical calculations, it is never possible to obtain the exact solution to eqn (1). Instead, approximate methods are used that are based on either the Hartree-Fock approximation or the density-functional formalism in the Kohn-Sham formulation. In the case of the Hartree-Fock approximation one may add correlation elfects, which, however, is beyond the scope of the present discussion (for details, see, e.g., ref. 1). Then, in both cases the problem of calculating the best approximation to the ground-state electronic energy is transformed into that of solving a set of singleparticle equations,... 

A simple estimate of the computational difficulties involved with the customary quantum mechanical approach to the many-electron problem illustrates vividly the point [255]. Consider a real-space representation of ( ii 2, , at) on a mesh in which each coordinate is discretized by using 20 mesh points (which is not very much). For N electrons, becomes a variable of 3N coordinates (ignoring spin), and 20 values are required to describe on the mesh. The density n(r) is a function of three coordinates and requires only 20 values on the same mesh. Cl and the Kohn-Sham formulation of DFT (see below) additionally employ sets of single-particle orbitals. N such orbitals, used to build the density, require 20 values on the same mesh. (A Cl calculation employs in addition unoccupied orbitals and requires more values.) For = 10 electrons, the many-body wave function thus requires 20 °/20 10 times more storage space than the density and sets of single-particle orbitals 20 °/10x 20 10 times more. Clever use of symmetries can reduce these ratios, but the full many-body wave function remains inaccessible for real systems with more than a few electrons. 

An approach for tracking electronic degrees of freedom in parallel with a numerical integration of the classical equations of motion for the nuclei, and therefore determining V(r ) on the fly, has been devised by Car and Parrinello [27]. This extended ensemble molecular dynamics method, termed ab initio molecular dynamics, solves the electronic problem approximately using the Kohn Sham formulation of Density Functional Theory. This approach proved useful for covalent systems it still has to be applied to the systems where the properties of interest are defined by Lennard-Jones interactions. 

Density functional theory in its Kohn-Sham formulation (KS) has recently been successfully applied for deriving the hyperfine interaction tensor for radical systems (for review, see Compared to the conventional post-SCF... 

Basic density-functional theory in the Hohenberg-Kohn and Kohn-Sham formulations " " is a time-independent, i.e. a static, approach. To remind the reader, the two Hohenberg-Kohn theorems state and prove (i) that there is a one-to-one mapping between the real system of interest and the artificial system of non-interacting particles that is described, and (ii) that the variational principle holds for this system. These two theorems and the Kohn-Sham equations that are used to perform the actual calculation need to be derived for time-dependent processes as well. 

In practice, the search for the optimum density Pa, defined by Eq. 13.7, is performed by exploiting the Kohn-Sham formulation [67] of DPT [68] to solve Eq. 13.4, in which is the environment-free Hamiltonian of the isolated system A and = X iVgmfc(r() is the potential energy operator describing the effect of environment B on system A, where has the form of a local, orbital-... 

In practice, the Kohn-Sham formulation [5] of DFT is almost always used. This overcomes the major difficulty with finding a density-functional for the kinetic energy by introducing a set of orthonormal auxiliary functions (i.e. the Kohn-Sham orbitals), with occupation numbers / , which sum to the ground state density,... 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

The second approach to this problem is to derive orbital-based reformulations of existing algorithms based on the spatial representation of the g-density. The resulting formulations are in the spirit of the orbital-resolved Kohn-Sham approach to density functional theory. 

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