The Kohn-Sham Molecular Orbital Model

The MO concept is directly related to an approximate wavefunction consisting of a Slater determinant of occupied one-particle wavefunctions, or molecular orbitals. The Hartree-Fock orbitals are by definition the ones that minimize the expectation value of the Hamiltonian for this Slater determinant. They are usually considered to be the best orbitals, although it should not be forgotten that they are only optimal in the sense of energy minimization. 

The QMO approach has been mostly applied in the context of semiem-pirical calculations, although as a matter of fact many ab initio calculations, even those that use sophisticated techniques for the inclusion of correlation, very often cast their explanations somewhat illogically in simple MO language. Nevertheless, although MO theory has become a workhorse for everyday explanatory activity of chemists, it still suffers from the double odium of inaccuracy and frequent semiempiricism. 

When forming the one-determinantal wavefunction with Kohn-Sham orbitals instead of Hartree-Fock orbitals, and taking the expectation value of the Hamiltonian, one obtains 

HF orbitals. Of course, EKS is not the exact energy. It does not play a significant role in DFT calculations, where one focuses on approximations of the exact energy E, which are obtained by approximations to the exchange-correlation functional Exc[p], 

The differences between the KS and HF models can further be highlighted with the help of the correlation corrections to the various energy components defined above. The correlation correction is always defined as the difference between the exact quantity and either the KS or the HF one. In Eq. [8] below, the exact quantities are unsubscripted, the correlation correction is denoted by a subscript c if the difference is with respect to the KS quantity, and a subscript c plus superscript HF if the difference is with respect to Hartree-Fock (note that TKS aTJ  

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It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

An important aspect of the MO model is the choice of the molecular orbitals that are used for discussing the properties of molecules. Early work considered either qualitative sketches of MOs or molecular orbitals, which came from EHT calculations. Later work used Hartree-Fock (HF) orbitals for the MO models. The latter have the disadvantage that the correlation energy is not considered although correlation effects can be very important for the stmcture and reactivity of molecules. The advent of DFT calculations introduced also the use of Kohn-Sham (KS) orbitals for MO models. The advantage of KS orbitals is that correlation effects are... 

Localized orbitals have also been used as a tool to extract the infrared spectrum of a solute in solution [194,195,202] or to decompose the IR spetrum in intramolecular and intermolecular contributions [202]. Model electrostatics of solute molecules was also based on localized orbitals [242, 243], not only at the dipolar level [244]. As an extension we also defined molecular states from localized orbitals to study the electronic states of liquid water [245], or of solvated ions [47]. It is also possible to perform CP-MD propagating the Wannier orbitals, by constraining the Kohn-Sham orbitals to stay in a Wannier gauge [246]. 

The basis of DFT is that the ground-state energy of a molecular system is a function of the electron density [47]. The Kohn-Sham equations provide a rigorous theoretical model for the all-electron correlation effects within a one-electron, orbital-based scheme [48]. Therefore, DFT is similar to the one-electron HF approach, but the exchange-correlation term. Vex, is different in DFT it is created by the functional Exc(C) and in real applications we need approximations for this functional. The quality of DFT calculations depends heavily on the functional. The simplest approximate... 

Abstract. The paper by Kohn and Sham (KS) is important for at least two reasons. First, it is the basis for practical methods for density functional calculations. Second, it has endowed chemistry and physics with an independent particle model with very appealing features. As expressed in the title of the KS paper, correlation effects are included at the level of one-electron equations, the practical advantages of which have often been stressed. An implication that has been less widely recognized is that the KS molecular orbital model is physically well-founded and has certain advantages over the Hartree-Fock model. It provides an excellent basis for molecular orbital theoretical interpretation and prediction in chemistry. 

Since orbitals are model dependent, different models will have different orbitals. The basic distinction between DFT fi -orbitals and LFT fi -orbitals arises from their respective treatments of interelectron repulsions. In LFT, d-d repulsion is treated within a spherical approximation. For d and d configurations, there is a single free-ion term and hence no need to consider d-d interelectron repulsion at all. In contrast, the Kohn-Sham orbitals in DFT are computed relative to the total molecular potential. For a tetragonal d copper(II) complex, dx -y is singly occupied while the remaining -functions are doubly occupied. Hence, to a first approximation, the hole in the equatorial plane results in less d-d repulsion in the plane than perpendicular to the plane with the result that the in-plane dxy orbital falls relative to the out-of-plane dxzjdyz pair. 

Modern DFT no longer needs to approximate the molecular potential and calculations on [PdCU] " place d y below d z and dy. Note that both LFT and DFT predict the same ground state configuration, viz, d -y d dx dx dyf, that both models place the d -y orbital highest of the d set, and that the Kohn-Sham orbitals look as expected. However, the ground state DFT sequence is clearly not the same as that obtained from LFT with regard to the order of d y and dxzjdyz-... 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

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