Kohn-Sham orbitals requirements

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. 

N (Kohn-Sham) orbitals can be expanded into atomic orbitals according to (4). Furthermore, the expansion coefficients C, can be determined by requiring that they optimize the total (Kohn-Sham) energy. This results in the (Kohn-Sham) matrix equation similar to (5)... 

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 physical description of the functional derivative Vee (r) requires knowledge of the wavefunction 4 for the determination of the electron-interaction component W e(r) = Wnlr) -i- W (r), and knowledge of both the wavefunction P and the Kohn-Sham orbitals < i(x) for the correlation-kinetic-energy component W, (r). The corresponding Kohn-Sham wavefunction is then a single Slater determinant. It has, however, also been proposed [42,52,53] that the wavefunction V be determined by solution of the Sturm-Liouville equation... 

Note that the determination of this field thus requires knowledge of the Kohn-Sham orbitals. 

Whereas the Kohn-Sham orbital approach is in principle exact, it requires knowing the exchange-correlation energy and potential. A widely used approximation is called the local density approximation (LDA) in which the ex-change-correlation functional is given by... 

Most molecular quantum-mechanical methods, whether SCF, Cl, perturbation theory (Section 16.3), coupled cluster (Section 16.4), or density functional (Section 16.5), begin the calculation with the choice of a set of basis functions Xn which are used to express the MOs (pi as = IiiCriXr [Eq. (14.33)]. (Density-functional theory uses orbitals called Kohn-Sham orbitals P that are expressed as (pf = 1,iCriXn see Section 16.5.) The use of an adequate basis set is an essential requirement for success of the calculation. 

The non-linear coupling of the Kohn-Sham wave function xl>) with the effective one particle potential in the Kohn-Sham equation requires an iterative solution of the KS equations (SCF treatment similar to the traditional Hartree-Fock scheme). As in the solution of the HF equations for molecules mainly the linear combination of atomic orbitals is used (HF-LCAO) ... 

By symmetry, the Kohn-Sham potential r s(r) must be uniform or constant, and we take it to be zero. We impose boundary conditions within a cube of volume V oo, i.e., we require that the orbitals repeat from one face of the cube to its opposite face. (Presumably any choice of boundary conditions would give the same answer as V oo.) The Kohn-Sham orbitals are then plane waves exp(ik r)/ /V, with momenta or wavevectors k and energies k /2. The number of orbitals of both spins in a volume d k of wavevector space is 2[V/(27r) ]d A , by an elementary geometrical argument [53]. 

The GGA to be derived in Sect. 1.6.4 wiU preserve all the good or mixed features of LSD listed above, while eliminating bad features (1) and (2) but not (3) (5). Elimination of (3) (5) wiU probably require the construction of E c fis[,n from the Kohn-Sham orbitals (which are themselves highly-nonlocal functionals of the density). For example, the self-interaction correction [9,68] to LSD eliminates most of the bad features (3) and (4), but not in an entirely satisfactory way. 

All of these approximations are density functionals, because the Kohn-Sham orbitals are implicit functionals of the density. Finding the exchange-correlation potential for nmgs (3) (5) requires the construction of the optimized effective potential [119], which is now practical even for fully three-dimensional densities [120]. For many purposes a non-selfconsistent implementation of nmgs (3)-(5) using GGA orbitals will suffice. 

While plane waves are a good representation of delocalized Kohn-Sham orbitals in metals, a huge number of them would be required in the expansion (O Eq. 7.67) to obtain a good approximation of atomic orbitals, in particular near the nucleus where they oscillate rapidly. Therefore, in order to reduce the size of the basis set, only the valence electrons are treated explicitly, while the core electrons (i.e., the inner shells) are taken into account implicitly through pseudopotentials combining their effect on the valence electrons with the nuclear Coulomb potential. This frozen core approximation is justified as typically only the valence electrons participate in chemical interactions. To minimize the number of basis functions the pseudopotentials are constructed in such a way as to produce nodeless atomic valence wavefunctions. Beyond a specified cutoff distance from the nucleus, Ecut the nodeless pseudo-wavefunctions are required to be identical to the reference all-electron wavefunctions. 

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