Molecular orbitals Kohn-Sham

The method for spectrum decomposition proposed above is certainly not unique, and other criteria and threshold values could be invoked. Besides that, it depends on the approximate validity of a few hypotheses. First, we assume the adequacy of CIS wavefunctions to describe the electronic density. Second, we also assume that the usage of time-dependent DPT amplitudes together with Kohn-Sham orbitals results in an acceptable representation of the CIS wavefunction. Third, we assume that the density partition among the units is uniquely defined, even though the molecular orbitals (Kohn-Sham or Hartree-Fock) are not unique and the MuUiken partition employed is somewhat arbitrary. Due to all these factors, we should take the decomposition as a qualitative analysis of the several contributions to each band, rather than an exact numerical analysis. 

To solve the Kohn-Sham equations a number of different approaches and strategies have been proposed. One important way in which these can differ is in the choice of basis set for expanding the Kohn-Sham orbitals. In most (but not all) DPT programs for calculating the properties of molecular systems (rather than for solid-state materials) the Kohn-Sham orbitals are expressed as a linear combination of atomic-centred basis functions ... 

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. 

Gritsenko, O. V., Baerends, E. J., 1997, Electron Correlation Effects on the Shape of the Kohn-Sham Molecular Orbitals , Theor. Chem. Acc., 96, 44. 

Figure 2.11. Kohn-Sham molecular correlation diagram of principal spin orbitals for the 7] -/V 2CuN( ) 11M5 complex (BP/DNP). Bending of the adduct results in a splitting of both 2tt levels giving rise to (a) SOMO and (b) LUMO (after [32]).

In a molecular-orbital-type (Hartree-Fock or Kohn-Sham density-functional) treatment of a three-dimensional atomic system, the field-free eigenfunctions ir e can be rigorously separated into radial (r) and angular (9) components, governed by respective quantum numbers n and l. In accordance with Sturm-Liouville theory, each increase of n (for... 

The basic idea underlying AIMD is to compute the forces acting on the nuclei by use of quantum mechanical DFT-based calculations. In the Car-Parrinello method [10], the electronic degrees of freedom (as described by the Kohn-Sham orbitals y/i(r)) are treated as dynamic classical variables. In this way, electronic-structure calculations are performed on-the-fly as the molecular dynamics trajectory is generated. Car and Parrinello specified system dynamics by postulating a classical Lagrangian ... 

Fukui functions and other response properties can also be derived from the one-electron Kohn-Sham orbitals of the unperturbed system [14]. Following Equation 12.9, Fukui functions can be connected and estimated within the molecular orbital picture as well. Under frozen orbital approximation (FOA of Fukui) and neglecting the second-order variations in the electron density, the Fukui function can be approximated as follows [15] ... 

Based on the foregoing discussion, one might suppose that the Fukui function is nothing more than a DFT-inspired restatement of frontier molecular orbital (FMO) theory. This is not quite true. Because DFT is, in principle, exact, the Fukui function includes effects—notably electron correlation and orbital relaxation—that are a priori neglected in an FMO approach. This is most clear when the electron density is expressed in terms of the occupied Kohn-Sham spin-orbitals [16],... 

Few years later, Fuentealba and Cedillo [23] has shown that the variation of the Kohn-Sham FF with respect to the external perturbation depends on the knowledge on the highest occupied molecular orbital (HOMO) density and a mean energy difference of all of the occupied and unoccupied orbital. The quantity, mean energy difference, has been approximately interpreted as hardness. Under this approximation, it has been stated that the greater the hardness, the smaller the variation of the FF, under the external perturbation. This statement then signifies that the system will become less reactive as the hardness of the system increases due to the external perturbation. 

The basic concepts of the one-electron Kohn-Sham theory have been presented and the structure, properties and approximations of the Kohn-Sham exchange-correlation potential have been overviewed. The discussion has been focused on the most recent developments in the theory, such as the construction of from the correlated densities, the methods to obtain total energy and energy differences from the potential, and the orbital dependent approximations to v. The recent achievements in analysis of the atomic shell and molecular bond midpoint structure of have been... 

Now we know that the asymptotic decay of the density far from atom B (but not so far that ps still dominates the molecular density p ) is determined by the ionization energy Ib- The asymptotic decay of the density pa is however also determined by the highest occupied Kohn-Sham orbital Kohn-Sham potential cannot go to zero in the asymptotic region of atom B otherwise the density decay around atom B would be determined by 1a- As the Hartree and the external potential go to zero in the outer asymptotic region it follows from Eq. (145) that the only possibility is that the molecular exchange-correlation potential around atom B is given by... 

In spite of the absence of a typical chromophore, 1,2-dithiin is a bright reddish-orange color. Absorption maxima were found at 451 (2.75 eV), 279 (4.36 eV), and 248 nm (5.00 eV), and the colored band was assigned to a A excitation <1991JST(230)287>. The main reason for the colored absorption of 1,2-dithiin is the low HOMO-LUMO gap of the KS orbitals which amounts to only 3.6 eV (HOMO = highest occupied molecular orbital LUMO = lowest unoccupied molecular orbital KS = Kohn-Sham) <2000JMM177>. By comparison, saturated 1,2-dithiane is colorless (290 nm). 

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). 

Due to the spin polarization effect, the magnetic orbitals can be difficult to identify from a spin-unrestricted calculation. Since the total energy of a Kohn—Sham determinant is invariant under unitary transformations between the spin-up orbitals among each other and spin-down orbitals among each other, one can arrange each spin-up orbital to overlap at most with each spin-down orbital on the basis of the corresponding orbital transformation (COT) (88—90). Then, the molecular orbitals (MOs) are ordered into pairs of maximum similarity between spin-up and spin-down orbitals and can be separated into three groups (i) the MOs with spatial overlap close to one (doubly occupied MOs),... 

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