Kohn-Sham self-consistent-field methods

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. 

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. 

Harrison, et al. have reported an efficient, accurate multiresolution solver for the Kohn-Sham and Hartree-Fock self-consistent field methods for general polyatomic molecules. The Hartree-Fock exchange is a nonlocal operator, whose evaluation has been a computational bottleneck for electronic structure calculations, scaling as for small molecules... 

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... 

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 numerical integration also can be used to calculate the matrix elements of the exchange-correlation potential. For the numerical integration, the atomic partition method proposed by Savin [392] and Becke [393] has been adopted and combined with Gauss-Legendre (radial) and Lebedev (angular) quadratures [394]. The Kohn-Sham LCAO periodic method based on numerical integration at each cycle of the self-consistent-field process is computationally more expensive than the periodic LCAO Hartree-Fock method that is almost fully analytical. 

Each one of the HF or Kohn-Sham MOs is expressed as a LCAO approach or, more generally, basis functions. Since the self-consistent field procedure in the Kohn-Sham formulation of DFT is very similar to that of the HF method, the choice of an appropriate basis set is equally important in DFT. 

The Kohn-Sham equations are solved in a self-consistent field fashion. Initially a charge density is needed so that Exc can be computed. To obtain the charge density, an initial guess to the Kohn-Sham orbitals is needed. This initial guess can be obtained from a set of basis functions whereby the coefficients of expansion of the basis functions can be optimized just like in the HF method. From the function of Exc in terms of the density, the term xc is computed. The Kohn-Sham equations (Equation 9-50) are then solved to obtain an improved set of Kohn-Sham orbitals. The improved set of Kohn-Sham orbitals is then used to calculate a better density. This iterative process is repeated until the exchange-correlation energy and the density converge to within some tolerance. 

The prime in Eq. (3-62) indicates that the sum is restricted to sites that do not belong to the same molecule. Depending on the specific implementation the tensors T(1) are multiplied with appropriate /e factors for the associated atoms. The last term in Eq. (3-59), efacM, is the macroscopic electric field. This completes the most usual form of vpo1, i.e., the potential of the dipoles due to the total field at the polarizable sites is made a part of the effective Hamiltonian and Eq.(3-24) is solved self-consistently. Since the induced dipoles M in the solvent (MM) part are self-consistent for any field E, i.e., also for intermediate fields during the iterative process for solving Eq. (3-24), in this way we obtain an overall self-consistent solution, similar to, e.g., the HF or Kohn-Sham procedure. Extension to post-HF methods are straightforward because the reaction potential (RP) is formally a one-particle... 

The Kohn-Sham-Dirac equation (28) has to be solved self consistently, since the crystal potential and the XC-field depend via the (magnetization) density on its solutions. For a local orbital method it is advantageous to use a strictly local language for all relevant quantities, so that computationally expensive transformations between different numerical representations are avoided during the self consistency cycle. In the (R)FPLO method, the density n(r) and the magnetization density m(r) = m r)z are represented as lattice sums... 

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