KS orbitals

An AO basis is chosen in tenns of which the KS orbitals are to be expanded. 

This is because no four-indexed two-electron integral like expressions enter into the integrals needed to compute the energy. All such integrals involve p(r) or the product p(/)p(r) because p is itself expanded in a basis (say of M functions), even the term p(r)p(r) scales no worse than tvF. The solution of the KS equations for the KS orbitals ([). involves solving a matrix eigenvalue problem this... 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

The KS orbitals can be determined by a numerical procedure, analogous to numerical HF methods. In practice such procedures are limited to small systems, and essentially all calculations employ an expansion of the KS orbitals in an atomic basis set. 

In practice a DFT calculation involves an effort similar to that required for an HF calculation. Furthermore, DFT methods are one-dimensional just as HF methods are increasing the size of the basis set allows a better and better description of the KS orbitals. Since the DFT energy depends directly on the electron density, it is expected that it has basis set requirements similar to those for HF methods, i.e. close to converged with a TZ(2df) type basis. 

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,... 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

In the present approach, the KS orbitals are expanded in a set of functions related to atomic orbitals (Linear Combination of Atomic Orbitals, LCAO). These functions usually are optimized in atomic calculations. In our implementation a basis set of contracted Gaussians VF/ is used. The basis set is in general a truncated (finite) basis set reasonably selected . 

On the other hand, the calculation of the Fukui function with the HF frontier orbitals is, in general, qualitatively very similar to the one obtained through KS orbitals. However, there may be cases where the absence of correlation effects in HF may lead to large differences with respect to the KS description. 

Hence, ( ks hI Ks) - Ks(n 1) Hl Ks(rt )> j - /> where KS orbital energy is an approximation to the ionization potential of the jth electron. When j is the homo, we know that the difference above gives exactly Sj, since j K j) = (j Vjf j) is an exact condition. [48] For other orbitals, this tends to offer a reasonable approximation to most ionization potentials. (In practise, we actually improve this exact condition by shifting all orbital energies to make it true. [50])... 

This also ascribes a meaning to the other KS orbital energies. Without a reasonable description of the other ionization potentials, many excited state descriptions will fail because most excited states for molecules are of Rydberg character, or require some mixture of Rydberg character in their accurate description. 

Its first term, Emp2, has exactly the same functional form as the conventional second order Moller-Plesset correlation contribution (MP2). However, in Eqs (4 7 4.8) the ( )9 represent KS orbitals, which experience the multiplicative KS potential (2 7), rather... 

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