Kohn-Sham framework

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

Here, we should mention that there exists an extensive discussion in the literature on the capabilities of spin-DFT regarding, for instance, the question whether the Kohn-Sham spin density has to be equal to the spin density of the fully interacting system of electrons (and in the case of open-shell singlet broken-symmetry (BS) determinants (see below) for binuclear transition-metal clusters this is certainly not the case see Ref. (33) for a more detailed discussion). But the situation is much more subtle and one may basically set up the variational procedure in a Kohn-Sham framework such that the spin density of the Kohn-Sham system of noninteracting fermions represents the true spin density. However, the frame of this review is not sufficient to present all details on this matter (34,35). 

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). 

Current hybrid functionals do not improve this situation. Their non-local component (Hartree-Fock exchange) cannot give rise to any attraction. To describe quantitatively the long-range interactions, either a non-local approximation to Exc[p] must be applied within the Kohn-Sham framework or methods using other-than-Kohn-Sham formalism should be used. Some of such approaches will be discussed in the last section of this review. Here, we mention an especially promising combination of symmetry adapted perturbation theory with of the Kohn-Sham orbitals.125... 

It is worthwhile to underline the key difference between the Cortona and Kohn-Sham frameworks to obtain ground-state energy and density in practical calculations... 

There are basically two methods to find the ground-state density within the Kohn-Sham framework for a given external potential, which will be introduced successively. The first one focuses on the self-consistency equation (58) that must be fullfilled by the ground-state density, while the second one aims at directly minimizing the energy functional at fixed external potential. 

O. L. Malkina, V. G. Malkin. Relativistic four-component calculations of electronic g-tensors in the matrix Dirac-Kohn-Sham framework. Chem. Phys. Lett., 488 (2010) 94-97. 

Restricted magnetically balanced basis has been applied by Malkin and co-workers for relativistic calculations of scalar nuclear spin-spin coupling tensors in the matrix Dirac-Kohn-Sham framework. Benchmark relativistic calculations have been carried out for the H-X and H-H couplings in the XH4 series where X = C, Si, Ge, Sn and Pb. One-bond couplings, X, in the gas-phase have been determined by Antusek et for CH4, /hc= 125.3 Hz, SiH4, /hsi = (-) 201.0 Hz, GeH4, /HOe = (-)96.7 Hz, and calculated theoretically. The calculations have been also performed for whose experimental value in SnH4 has been reported by Laaksonen and Wasylishen. ... 

We recall that in the time-dependent Kohn-Sham framework, the density of the interacting system of electrons is obtained from a fictitious system of non-interacting electrons. Clearly, we can also calculate the linear change of density using the Kohn-Sham system... 

To obtain excitation energies and properties within the time-dependent Kohn-Sham framework, it is possible to propagate in time the time-dependent electron density, through the solution of Eq. (4.60), and then extract energies and oscillator strengths from a Fourier analysis of the results [98-102]. Alternatively, the excited-state properties can be determined through the linear response theory. This is an efficient approach which avoids the direct solution of the time-dependent Kohn-Sham equations and is often used in practical applications. 

Villaume S, Saue T, Norman P. Linear complex polarization propagator in a four-component Kohn-Sham framework. J Chem Phys. 2010 133 064105. 

All calculations were performed within the Kohn-Sham framework. Geometries of all radical species were optimized in solvent at the B3LYP/6-311-l-G(d,p) level of theory [29-31] using both the polarizable continuum model through the integral equation formalism (lEF-PCM), as implemented in Gaussian09 [32], and the conductor-like screening model (COSMO), as implemented in MOLPRO... 

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