Generalized Fock matrix

Using, e.g., the NAO Fock-matrix elements for Li2+ quoted in note 11 (together with the corresponding value for /ss/ = (lsA F lsB) = —0.0668), we obtain the estimate Fab — —0.0910. This somewhat underestimates the magnitude of the actual interaction element (—0.1191), but shows the general direction and strength of the hybridization effect. 

NBO analysis can be used to quantify this phenomenon. Since tire NBOs do not diagonalize the Fock operator (or tire Kohn-Sham operator, if the analysis is carried out for DFT instead of HF), when the Fock matrix is formed in the NBO basis, off-diagonal elements will in general be non-zero. Second-order perturbation tlieory indicates that these off-diagonal elements between filled and empty NBOs can be interpreted as the stabilization energies... 

The Fock matrix (4 42) is in general not Hermitian for a non-converged MCSCF wave function. With optimized orbitals the gradient is zero. The MCSCF Fock matrix is thus Hermitian at this point on the energy surface. This condition has been used as a basis for optimization schemes in earlier developments of the MCSCF methodology. Convergence of such first order optimization schemes is, however, often poor, and they are not very much used today. 

MNDO [37], a modified NDDO (Section 6.2.5) method, was reported in 1977 [38]. MNDO is conveniently explained by reference to CNDO (Section 6.2.3). MNDO is a general geometry method with a minimal valence basis set of Slater-type orbitals. The Fock matrix elements are calculated using Eq. 6.1=5.82. We discuss the core and two-electron integrals in the same order as for CNDO. 

The matrices F(n) and aFM are easily calculated in the AO basis from the Roothaan-Bagus integrals. Conversely, the construction of Q(n> (and thus the total Fock matrix) requires MO integrals with one general and three active indices. The Q(n> matrix can be calculated as a matrix product if all density and integral elements for a given distribution vx are held in core at the same time. 

In general, Hartree-Fock calculations on molecules give molecular orbitals (MOs) that extend (i.e., are delocalized) over the entire molecular framework. These so-called canonical MOs (CMOs) may be transformed into localized orbitals, such as NBOs. Technically, the Fock matrix in the basis of CMOs is con-... 

For k = 0, the Fock matrix and its derivatives with respect to the displacements of the nuclei are always block diagonal. Then one can directly apply the analytical derivative methods developed for finite systems to extended systems [69,86,87,88]. But when the displacements break the translational symmetry, the Fock matrix and its derivatives are no longer block diagonal. To solve the CPHF equations, one needs to use the symmetrized (normal mode) coordinates instead of the Cartesian coordinates of the nuclei. Efficient analytical methods have been developed to calculate the energy derivatives for k / 0 with both plane wave [89-90] and general basis functions [85]. The latter can be functions of nuclear coordinates and have linear dependence. These methods reduce the computational cost required to calculate the phonon spectrum with k 7 0 to the same as that needed for the spectrum at k = 0. 

The generalized Fock matrix F, defined above, is a convenient partial sum because it can also be used in the Hessian matrix element construction. This matrix is not symmetric before convergence is reached in the MCSCF iterative procedure, but becomes symmetric at convergence , resulting in vanishing orbital gradient vector elements. 

Since the Fock matrix is dependent on the orbital coefficients, the Roothaan equations have to be repeatedly solved in an iterative process, the self-consistent field (SCF) procedure. One important step in the SCF procedure is the conversion of the general eigenvalue equation (7) into an ordinary one by an orthogonalization transformation... 

A generalization of the Hiickel method to nonplanar systems comprised of carbon and heteroatoms is the Extended Hiickel Theory (EHT) [27 30]. It takes explicitly into account all valence electrons, i.e., Is for H and 2s,2p for C, N, O, and F. Similar to the HMO method, the Fock matrix in EHT FEHT does not contain two-electron integrals. The diagonal elements F T are obtained from experimental ionization potentials (IPs) where the Koopmans theorem [31] has been used. 

---