Kohn-Sham orbital eigenvalues

The Janak s theorem (eq.17) and the hardness tensor definition (eq. 20) allows the calculations of Tiij as the first derivative of the Kohn-Sham orbital eigenvalues with respect to the orbital occupation numbers [17] ... 

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. 

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. 

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,. 

Thus, the response kernel for the interacting system can be obtained from that of the noninteracting system if one has a suitable functional form for the XC energy density functional for TD systems. The standard form for the kernel yo(r, r" Kohn Sham orbitals (/ (r), their energy eigenvalues sk, and the occupation numbers nk, is given [17,19] by... 

In Kohn-Sham density functional theory, the ionization potential is the negative of the eigenvalue of the highest occupied Kohn-Sham orbital. 86-88 The IP = —sH0M0 relation holds, however, only for the exact exchange-correlation potential. Numerical confirmations for this relation exist for model systems such as the... 

The relation between other than eHOMO eigenvalues of the exact Kohn-Sham orbitals and higher ionization potentials is currently an object of studies by Baerends and collaborators.95,96... 

Note that the summation in Eq. (344) extends over all single-particle transitions q a between occupied and unoccupied Kohn-Sham orbitals, including the continuum states. Up to this point, no approximations have been made. In order to actually calculate 2(o ), the eigenvalue problem (344) has to be truncated in one way or another. One possibility is to expand all quantities in Eq. (344) about one particular KS-orbital energy difference co... 

We emphasize that the calculation of excitation energies from Eqs. (362) and (363) involves only known ground-state quantities, i.e., the ordinary static Kohn-Sham orbitals and the corresponding Kohn-Sham eigenvalues. Thus the scheme described here requires only one selfconsistent Kohn-Sham calculation, whereas the so-called Ajcf procedure involves linear combinations of two or more selfconsistent total energies [209]. So far, the best results are obtained with the optimized effective potential for in the KLI x-only approximation. Further improvement is expected from the inclusion of correlation terms [6,225] in the OPM. 

Unfortunately, this does not give the correct answer, giving instead the state where all the electrons are in the lowest energy Kohn-Sham orbital this violates the Pauli exclusion principle. Satisfying the Pauli exclusion principle requires that every state of the system be occupied by no fewer than zero and no more than two electrons (one with spin a and one with spin ft). This indicates that the eigenvalues of the first-order density matrix [it follows from the defining Eq. (64) that the eigenvectors of y(r,r ) are the Kohn-Sham orbitals]... 

The eigenvalues of the first-order density matrix are identified with the occupation numbers for their associated Kohn-Sham orbitals. 

It should be understood that the Kohn-Sham orbitals are strictly good only for generating the electron densities that enter Eqs. [5] or [8], and the eigenvalues for the one-electron orbitals are not necessarily the same values that would be obtained from an exact solution for the many-body wavefunction. Nonetheless, the orbital energies are often used to estimate reasonable values for quantities such as electron affinities and electronegativities. 

---