Kohn-Sham self-consistent-field

KOHN-SHAM SELF-CONSISTENT FIELD METHODOLOGY 255... 

Kohn-Sham Self-consistent Field Methodology... 

Garza, J., Vargas, R. and Vela, A. 1988. Numerical self-consistent-field method to solve the Kohn-Sham equations in confined many-electron atoms. Phys. Rev. E. 58 3949-54. 

As already mentioned, the MOs in Eqs. (10a) and (10b) are assumed to be exact, i.e., represented at the complete basis set limit. In practice, the Kohn-Sham equations are converted into the respective self-consistent-field matrix equations in the basis, and the perturbation treatment is carried out from there. Any dependence of the basis set (GIAOs) on the perturbation (Bext in the case of shielding tensors) is in this way naturally covered. 

For variational methods, such as Hartree-Fock (HF), multi-configurational self-consistent field (MCSCF), and Kohn-Sham density functional theory (KS-DFT), the initial values of the parameters are equal to zero and 0) thus corresponds to the reference state in the absence of the perturbation. The A operators are the non-redundant state-transfer or orbital-transfer operators, and carries no time-dependence (the sole time-dependence lies in the complex A parameters). Furthermore, the operator A (t)A is anti-Hermitian, and tlie exponential operator is thus explicitly unitary so that the norm of the reference state is preserved. Perturbation theory is invoked in order to solve for the time-dependence of the parameters, and we expand the parameters in orders of the perturbation... 

Hyperfine couplings, in particular the isotropic part which measures the spin density at the nuclei, puts special demands on spin-restricted wave-functions. For example, complete active space (CAS) approaches are designed for a correlated treatment of the valence orbitals, while the core orbitals are doubly occupied. This leaves little flexibility in the wave function for calculating properties of this kind that depend on the spin polarization near the nucleus. This is equally true for self-consistent field methods, like restricted open-shell Hartree-Fock (ROHF) or Kohn-Sham (ROKS) methods. On the other hand, unrestricted methods introduce spin contamination in the reference (ground) state resulting in overestimation of the spin-polarization. 

The most uniformly successful family of methods begins with the simplest possible n-electron wavefunction satisfying the Pauli antisymmetry principle - a Slater determinant [2] of one-electron functions % r.to) called spinorbitals. Each spinorbital is a product of a molecular orbital xpt(r) and a spinfunction a(to) or P(co). The V /.(r) are found by the self-consistent-field (SCF) procedure introduced [3] into quantum chemistry by Hartree. The Hartree-Fock (HF) [4] and Kohn-Sham density functional (KS) [5,6] theories are both of this type, as are their many simplified variants [7-16],... 

The self-consistent field procedure in Kohn-Sham DFT is very similar to that of the conventional Hartree-Fock method [269]. The main difference is that the functional Exc[p] and matrix elements of Vxc(r) have to be evaluated in Kohn-Sham DFT numerically, whereas the Hartree-Fock method is entirely analytic. Efficient formulas for computing matrix elements of Vxc(r) in finite basis sets have been developed [270, 271], along with accurate numerical integration grids [272-277] and techniques for real-space grid integration [278,279]. 

The method is an approximate self-consistent-field (SCF) ab initio method, as it contains no empirical parameters. All of the SCF matrix elements depend entirely on the geometry and basis set, which must be orthonormal atomic orbitals. Originally, the impetus for its development was to mimic Hartree-Fock-Roothaan [5] (HFR) calculations especially for large transition metal complexes where full HFR calculations were still impossible (40 years ago). However, as we will show here, the method may be better described as an approximate Kohn-Sham (KS) density functional theory (DFT)... 

The wave function of fragment A, ll a, can either be a single determinant from HF theory or Kohn-Sham DFT, or a multiconfiguration wave function derived from complete active space self-consistent field (CASSCF) or valence bond (VB) calculations. 

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