Orbitals Kohn-Sham

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 ... 

Equation (3.74) is the exact exchange energy (obtained from the Slater determinant the Kohn-Sham orbitals), is the exchange energy under the local spin densit) ... 

Properties can be computed by finding the expectation value of the property operator with the natural orbitals weighted by the occupation number of each orbital. This is a much faster way to compute properties than trying to use the expectation value of a multiple-determinant wave function. Natural orbitals are not equivalent to HF or Kohn-Sham orbitals, although the same symmetry properties are present. 

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. 

Kohn-Sham orbitals functions for describing the electron density in density functional theory calculations... 

The Slater—Condon integrals Ft(ff), Ft(fd), and Gj-(fd), which represent the static electron correlation within the 4f" and 4f 15d1 configurations. They are obtained from the radial wave functions R, of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23,31... 

The Wyboume crystal field parameters B (f, f), B (d, d), and Bjj(f, d), which describe the interaction due to the presence of the ligands onto the electrons of the lanthanide center. They are deduced from the ligand field energies and wave functions obtained from Kohn—Sham orbitals of restricted DFT calculations within the average of configuration (AOC) reference by placing evenly n — 1 electrons in the 4f orbitals and one electron in the 5d.33... 

Figure 2. Representation of the energies of the 4f Kohn— Sham orbitals of Eu2+ in CsMgBr3 obtained from the output of AOC-type calculation. The two diagrams represent the cases of geometries (Table 2) at ground and excited configurations (GC versus EC).

In order to distinguish these orbitals from their Hartree-Fock counterparts, they are usually termed Kohn-Sham orbitals, or briefly KS orbitals. The connection of this artificial system to the one we are really interested in is now established by choosing the effective potential Vs such that the density resulting from the summation of the moduli of the squared orbitals tpj exactly equals the ground state density of our real target system of interacting electrons,... 

Note again the formal simplicity of equation (7-17) as compared to equation (7-18) in spite of the fact that the former is exact provided the correct Vxc is inserted, while the latter is inherently an approximation. The calculation of the formally L2/2 one-electron integrals contained in hllv, equation (7-13) is a fairly simple task compared to the determination of the classical Coulomb and the exchange-correlation contributions. However, before we turn to the question, how to deal with the Coulomb and Vxc integrals, we want to discuss what kind of basis functions are nowadays used in equation (7-4) to express the Kohn-Sham orbitals. 

Pople, J. A., Gill, P. M. W., Handy, N. C., 1995, Spin-Unrestricted Character of Kohn-Sham Orbitals for Open-Shell Systems , Int. J. Quant. Chem., 56, 303. 

Tozer, D. J., Handy, N. C., 1998, Improving Virtual Kohn-Sham Orbitals and Eigenvalues Apphcation to Excitation Energies and Static Polarizabilities , 7. Chem. Phys., 109, 10180. 

The Kohn-Sham determinant is the single determinant which reproduces the electron density and minimises the kinetic energy [1,9].) They observed that for the Be atom, the Kohn-Sham orbitals were nearly indistinguishable from the HF orbitals, and on this evidence they claim that the problem of finding a physically meaningful wave function from an electron density is solved . Here, we merely note that there are a number of desirable features for our model ... 

Provided the potential t) is local in r, in the limit that X - oo we will have p - p, independent of the choice of t). In this limit then, Equation (5) gives the Kohn-Sham orbitals and eigenvalues. The determinant formed from these orbitals is a wave function obtained from the density p,. 

However, one feature of the HF potential is that it is not a local potential. In the case of perfect data (i.e. zero experimental error), the fitted orbitals obtained are no longer Kohn-Sham orbitals, as they would have been if a local potential (for example, the local exchange approximation [27]) had been used. Since the fitted orbitals can be described as orbitals which minimise the HF energy and are constrained produce the real density , they are obviously quite closely related to the Kohn-Sham orbitals, which are orbitals which minimise the kinetic energy and produce the real density . In fact, Levy [16] has already considered these kind of orbitals within the context of hybrid density functional theories. 

A new and accurate quantum mechanical model for charge densities obtained from X-ray experiments has been proposed. This model yields an approximate experimental single determinant wave function. The orbitals for this wave function are best described as HF orbitals constrained to give the experimental density to a prescribed accuracy, and they are closely related to the Kohn-Sham orbitals of density functional theory. The model has been demonstrated with calculations on the beryllium crystal. 

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