LCAO-MO-SCF equation

Semirigorous LCAO-MO-SCF methods start with the complete many-electron Hamiltonian and make certain approximations for the integrals and for the form of the matrices to be solved. Several years ago, such a method was derived starting with the correct many electron Hamiltonian (in which interelectronic interactions are included explicitly) and the LCAO-MO-SCF equations of Roothaan and then making a consistent series of systematic... 

In general, the orbitals in this method are expanded as linear combinations of a basis set of Slater functions in the same spirit as in the LCAO-MO-SCF method. However, in the present case the orbitals are essentially localized, and a description such as equation (67) is clearly equivalent to a linear combination of a great many VB structures, both covalent and ionic. Thus, in the case of methane this should provide a very good description of the ground state, particularly of the potential surfaces for such processes as... 

Secondly, one can obtain the value of K approximately by use of Hiickel or SCF LCAO MO coefficients (Equation 11)65). We note that... 

A LCAO-MO-SCF calculation of equation (45) suggests that the reaction is best described as olefin insertion and not hydride migration. Good TT-donor ligands are predicted to promote the reaction. 

The SCF-Xa Scattered-wave Method.—An alternative to the LCAO-MO method for molecules is the Xa scattered wave or multiple-scattering Xa method, which was suggested less than 10 years ago114 and now has an extensive literative. The basic theory has been refined in several directions115 and there are several excellent reviews.116-180 The method is based on the scattered-wave method in solids181 and aims to solve the Fock equation,... 

A solution is ffie use of the LCAO MO method (alge-braization of the Fock equations). It leads to simplffieation of the computational scheme of ffie Hartree-Fock meffiod. In ffie SCF LCAO MO meffiod, ffie Fock equations (eompli-eated differential-integral equations) are solved in a very simple way. From Eqs. (8.53) and (8.32), we have... 

The set of SCF LCAO CO equations will be very similar to the set tor the moleeular orbital method (SCF LCAO MO). In prineiple, the only difference will be that, in the erystal ease, we will eonsequently use symmetry orbitals (Bloeh funetions) instead of atomie orbitals. 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

Formally, these equations are similar to the SCF LCAO MO equations. Thus, a good definition of //effective may yield satisfactory solutions, without the tedious iteration procedure of the SCF method. In practice, two calculation procedures have evolved from these equations, the Hiickel approximation and the Wheland-Mulliken approximation. Neither of them specifies the analytical form of H, but instead treats some of the matrix elements H,. as adjustable parameters. Their main difference resides in the neglect (Hiickel) or non-neglect (Wheland-Mulliken) of overlap. Their common feature is the tight-binding approximation, namely the neglect of all matrix elements which involve non-bonded atoms. 

In the SCF LCAO MO method, the Fock equations (complicated differential-integral equations) are solved in a very simple way. From (8.49) and (8.30) we have... 

An ab initio SCF-LCAO-MO all electron calculation has been performed for the Ziegler-Natta catalyst related reaction, equation (9). During the reaction... 

Since the ab initio calculations yield molecular orbitals (MO s) in terms of atom centered basis functions, it is straightforward to cast these mo s in the "LCAO" form of equations 20 and 21 (and their "primed" counterparts). These SCF MO s have been found to yield X values quite similar to those obtained from rigorous application of the corresponding orbital method to the ab initio wavefunctions. For the case of Co(NIfe )e, in which M (eg symmetry) is empty, X has been defined by using the "primed" counterpart of eq 22 and the y value obtained for the occupied L orbital of eg S3rmmetry. 

Pople-Pariser-Parr PPP (CNDO) method. MO s from LCAO SCF, i.e. electron correlation taken into account. Equations to derive the LCAO coefficients by the variational procedure were proposed by Roothaan... 

The numerical atomic orbital generated by solving the Schrodinger equation for "atom in molecule" is employed as a basis function of LCAO, which is renewed for each SCF iteration taking a modification of charge density into account. The atomic orbitals up to nd, (n+l)s and n+l)p are utilized for nd transition elements, where n is 3,4 and 5. The MO energy and the wave function are obtained by solving the secular equation. 

The IGLO method starts with the MO-LCAO-SCF calculation which produces the conventional (canonical) molecular orbitals [fa). These satisfy Hartree-Fock equations written in the form... 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

---