Method, Hartree-Fock-Roothaan

Theoretical quantum-chemical study of pyridine adsorption at Hg electrode (including its charged surface) has been described by Man ko et al. [137,138]. An ab initio Hartree-Fock-Roothaan method has been employed. The electrode was modeled as a planar seven-atomic Hg-7 cluster. The deepest minimum of the total energy of the adsorption system was found for positive charge density and Py interacting with the metal through the lone electron... 

Lee, C.M. (1974). Spectroscopy and collision theory. III. Atomic eigenchannel calculation by a Hartree-Fock-Roothaan method, Phys. Rev. A 10, 584-600. 

The idea to employ a finite basis set of AOs to represent the MOs as linear combinations of the former apparently belongs to Lennard-Jones [68] and had been employed by Hiickel [37] and had been systematically explored by Roothaan [38]. That is why the combination of the Hartree-Fock approximation with the LCAO representation of MOs is called the Hartree-Fock-Roothaan method. 

This is closely analogous to the Hartree equations (Eq. (1.7)). The Kohn-Sham orbitals are separable by definition (the electrons they describe are noninteracting) analogous to the HF MOs. Eq. (1.50) can, therefore, be solved using a similar set of steps as was done in the Hartree-Fock-Roothaan method. 

Table 3.5. Comparison of metal-water bond lengths and binding energies calculated using (SCF) Hartree-Fock-Roothaan methods with...

Hartree-Fock-Roothaan methods have often been quite successful in the calculation of properties despite the fact that the variational principle upon which they are based ensures only the best total energy. In particular, other energetic properties such as force constants and charge-distribution properties such as electron-density distributions and electric-field gradients are well reproduced. 

Table 4.1. Comparison of orbital ionization potentials (in eV) for SiO observed experimentally from uv photoelectron spectroscopy with those calculated by the MS-SCF-Ya method, ab initio Hartree-Fock-Roothaan method, incorporating the ASCF approach (HFR ASCF), and with perturbation theory (HFR + PT)...

Fig. 4.8. Total deformation electron densities (Ap) in 4-membered silicate rings (a) experimental Ap in 0(3)-0(4) ring of coesite (b) theoretical Ap for HjSi O, calculated using Hartree-Fock-Roothaan method with a 6-31G basis set and (c) experimental Ap in 0(3)-0(5) ring of coesite (after Geisinger et al., 1987 reproduced with the publisher s permission). 

Table 5.18. Calculated and experimental geometries in borate polyhedra. Bond lengths and bond angles for the borate clusters shown calculated using ab initio SCF Hartree-Fock-Roothaan methods with various basis sets, and the modified electron-gas and modified neglect of differential overlap method, and compared with experimental values (see text for data sources)...

The quantum-mechanical description of minerals containing transition metals is at a less advanced stage. The accuracy of simple Hartree-Fock-Roothaan methods has not been fully determined for such systems. Local-density-functional methods have been successful for calculating the structural properties of high-symmetry materials, but excitation energies are still poorly reproduced. Local-density-functional cluster calculations have so far been restricted mostly to model potentials (e.g., muffin-tin potentials) so that their full power has not been utilized. We need to determine the accuracy and efficiency of Hartree-Fock-Roothaan (or Har-... 

In full analogy with molecules, we can formulate the SCF LCAO CO Hartree-Fock-Roothaan method (a CO instead of an MO). Each CO is characterized by a vector k e FEZ and is a linear combination of the Bloch functions with the same k. 

Hartree-Fock-Roothaan method, 548, 552 Heisenberg uncertainty... 

LCAO (p. 360) atomic basis set (p. 363) Hartree-Fock-Roothaan method (p. 364) bonding orbital (p. 371) antibonding orbital (p. 371) instabihty (p. 372)... 

Ill the cApicSSiuU fOi ipni have taken inm aeeuuui thai thCie is ii j leasuil whatsoever that the coefficients c were Jfc-independent, since the expansion functions 4> depend on k. This situation does not differ from that, which we encountered in the Hartree-Fock-Roothaan method (cf. p. 365), with one technical exception instead of the atomic orbitals we have q mmetry orbitals, in our case Bloch functions. 

M. N. Adamov, YuB Malykhanov, V.V. Meshkov, R.M. Chadin, Calculation of optical characteristics of atoms with a closed shell by the Hartree-Fock-Roothaan method. Opt Spectrosc. 96(2), 192-194 (2004)... 

Only basis orbitals of the same symmetry can be included in any one LCAO molecular orbital if it is to be an eigenfunction of the symmetry operators. The a basis function can combine with the Is, 2s, and 2p functions on the oxygen. The 2 basis function can combine with the 2py function on the oxygen, and the,2px function on the oxygen cannot combine with any of the other basis functions. Table 21.2 contains the values of the coefficients determined by the Hartree-Fock-Roothaan method for the seven canonical molecular orbitals, using Slater-type orbitals (STOs) as basis functions, with... 

Most of the methods we have mentioned rely on the Hartree-Fock-Roothaan method. The principal shortcoming of this method is that the only way to eliminate the correlation error is by using configuration interaction, which is rather inefficient. We now mention some computational methods that have been devised to provide a more efficient way to eliminate the correlation error. 

This method was mentioned in Chapter 19, and has become a common method in quantum chemical research. Detailed discussion of it is beyond the scope of this book. It has been found that the approximation schemes that have been developed work at least as well as Hartree-Fock-Roothaan methods with configuration interaction for most molecular properties such as bond lengths and energies of molecular ground states. 

As a result of spin orthogonality, only (1 /2) of the exchange terms are nonzero but they are there This was first pointed out by Fock [6] and was added as a correction to the method then developed by Hartree [7]. The combined method is now called the Hartree-Fock method if tabulated numerical orbitals are used, but the Hartree-Fock-Roothaan method in an LCAO basis. Today this method is... 

---