Inverting Fock equations

Workers in this field use several methods to derive the optimized parameters for the pseudopotentials and the pseudo-orbitals. Generally, the parameters can be obtained by a fit procedure taking the shape or the norm of the orbitals as a reference function. The pseudo-orbitals are derived by fitting them to numerical valence orbitals of all-electron calculations. Some methods generate the potentials on a numerical grid by inverting Fock equations for pseudo-orbitals derived from numerical atomic wavefunctions. The numerically tabulated potentials are then least-squares fit with analytical Gaussian... 

Once a nodeless orbital has been generated the one-electron atomic Fock equation is easily inverted to produce a (radially) local operator, the EP, which represents the core-valence interactions (22,23). 

L, F. Pacios and P. A. Christiansen, /. Chem. Phys., 82, 2664 (1985). Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. I. Li Through Ar. See also, P. A. Christiansen, Y, S. Lee, and K. S. Pitzer, /. Chem. Phys. 71, 4445 (1979). Improved Ab Initio Effective Core Potentials for Molecular Calculations. In this latter paper are found equations pertinent to inverting the Fock equations, i.e., solving for the effective potentials from the Fock equations of the atoms. 

In iteration n the A matrix has dimension (n -1- 1) x (n -t 1), where n usually is less than 20. The coefficients c can be obtained by directly inverting the A matrix and multiplying it onto the b vector, i.e. in the subspace of the iterations the linear equations are solved by direct inversion , thus the name DIIS. Having obtained the coefficients that minimize the error function at iteration n, the same set of coefficients is used for generating an extrapolated Fock matrix (F ) at iteration n, which is used in place of F for generating the new density matrix. 

An alternative to the operator approach is to start from the matrix equations (Filatov 2002). Then the elimination the small-component, the construction of the transformation and the transformed Fock matrix are all straightforward. There is no difficulty with interpretation because the inverse of a matrix is well defined. The matrix to be inverted is positive definite so it presents no numerical problems. The drawback of a matrix method is that the basis set for the small component must be used, at least to construct the potentials that appear in the inverse. In that case, the same number of integrals is required as in the full Dirac-Hartree-Fock method, and there is no reduction in the integral work or the construction of the Fock matrix. 

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