Orbital-Density Formalism

Adopting the constrained search procedure, a time-dependent formulation of KS theory is also possible. 5 Here we outline the hydrodynamic formulation of time-dependent KS theory. - For an N-electron system with a periodic time-dependent potential, we define 

The minimization of Eq. [76] is equivalent to performing the constrained search minimization, 

The set of time-dependent KS amplitudes ) (r,t) and phases S (r,t) are real single-particle functions of time and space. They are the solutions of the coupled nonlinear di rential equations 

The KS Lagrange multipliers = -aS (r,f)/dt are functions of both time and spatial coordinates and not constants as in time-independent KS theory. These Euler-Lagrange equations are obtained from the stationary principle 

Actually, the time-independent KS equations were derived earlier in Eqs. [34] and [35]. The above equations, which must be solved self-consistently, are reminiscent Hartree and Hartree—Fock theory, but they are fundamentally different because they yield (and are defined in terms ff) the exact ground state electron density. Payne has used the constrained search procedure to develop a DFT for the Hartree-Fbck model. 

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Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

Similar diagrams can be obtained for MLCT-states. In the 2t + 9t. case, the formal orbital density difference does corres-... 

At this point we want to emphasize that, by virtue of the Hohenberg-Kohn theorem applied to non-interacting systems, all single-particle orbitals are formally functionals of the densities, i.e. 

For covalently bonded atoms the overlap density is effectively projected into the terms of the one-center expansion. Any attempt to refine on an overlap population leads to large correlations between p>arameters, except when the overlap population is related to the one-center terms through an LCAO expansion as discussed in the last section of this article. When the overlap population is very small, the atomic multipole description reduces to the d-orbital product formalism. The relation becomes evident when the products of the spherical harmonic d-orbital functions are written as linear combinations of spherical harmonics ( ). 

Calculating xw within the framework of plain spin density functional theory (SDFT), there is no modification of the electronic potential due to the induced orbital magnetization. Working instead within the more appropriate current density functional theory, however, there would be a correction to the exchange correlation potential just as in the case of the spin susceptibility giving rise to a Stoner-like enhancement. Alternatively, this effect can be accounted for by adopting Brooks s orbital polarization formalism (Brooks 1985). 

Although the effect of the spin-orbit interaction was included for the band peak, it was not for the supercell peak since the effect was thought to be too small to be worth the effort (the two f-electrons in the core used to construct the potential were treated as j = 5/2 states, though). Multiplet dfects were also not taken into account. In principle, this can be done in the local-density formalism (von Barth 1979), but has not been applied to solids to the authors knowledge. If multiplet effects turn out to be crucial for interpreting the BIS spectra of cerium pnictides, then an effort could be made to incorporate those effects into the calculation. 

Band structure calculations of fee crystalline Cgo using a pseudopotential local density formalism predict C(io to be a semiconductor with a band gap in the range 0.9-1.5 eV, and widths of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) derived bands in the range 0.4-... 

It is clear that the density matrix formalism renders a considerable simplification of the basis for the quantum theory of many-particle systems. It emphasizes points of essential physical and chemical interests, and it avoids more artificial or conventional ideas, as for instance different types of basic orbitals. The question is, however, whether this formalism can be separated from the wave function idea itself as a fundament. Research on this point is in progress, and one can expect some interesting results within the next few years. 

We encounter a different type of bond in a nitrogen molecule, N2. There is a single electron in each of the three 2p-orbitals on each atom (33). However, when we try to pair them and form three bonds, only one of the three orbitals on each atom can overlap end to end to form a (T-bond (Fig. 3.10). Two of the 2/7-orbitals on each atom (2px and 2py) are perpendicular to the internuclear axis, and each one contains an unpaired electron (Fig. 3.11, top). When the electrons in one of these p-orbitals on each N atom pair, the orbitals can overlap only in a side-by-side arrangement. This overlap results in a TT-bond, a bond in which the two electrons lie in two lobes, one on each side of the internuclear axis (Fig. 3.11, bottom). More formally, a 7T-bond has a single nodal plane containing the internuclear axis. Although a TT-bond has electron density on each side of the internuclear axis, it is only one bond, with the electron cloud in the form of two lobes, just as a p-orbital is one orbital with two lobes. In a molecule with two Tr-bonds, such as N2, the... 

The F matrix elements in eqs. (15) and (16) are formally the same as for closed-shell systems, the only difference being the definition of the density matrix in eq. (17), where the singly occupied orbital (m) has also to be taken into account. The total electronic energy (not including core-core repulsions) is given by... 

The valence DOS has been computed for Ni and Ag clusters within the CNDO formalism. Blyholder [54] examined the Nis and M13 clusters. In both cases of s- and p-orbitals are occupied and lie well below the d-orbitals. Most of the intensity is near the middle of the d-orbitals with a fall-off in intensity as the HOMO is approached. Density of states for Agv, Agio, Agi3, and Agig clusters shows a strong d-component cc. 3.5 eV wide. The... 

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