Self-consistent field atomic orbitals

Calculations of vibrational frequencies in a three-center bond as a function of Si—Si separation were performed by Zacher et al. (1986), using linear-combination-of-atomic-orbital/self-consistent field calculations on defect molecules (H3Si—H—SiH3). The value of Van de Walle et al. for H+ at a bond center in crystalline Si agrees well with the value predicted by Zacher et al. for a Si—H distance of 1.59 A. 

Linear combination of atomic orbitals-molecular-orbital-self-consistent field. 

The aim of this chapter is to review the current status of the quantum-mechanical calculation of electric and magnetic properties of isolated atoms and molecules. In view of the rapid advances made during the past decade in the calculation of ab initio molecular wavefunctions, we will clearly concentrate for the most part on the calculation of such properties using standard ab initio methods such as gaussian orbital LCAO-MO-SCF (linear combination of atomic orbital-miolecular orbital-self-consistent field), configuration interaction (Cl), coupled Hartree-Fock, and the like, but will also review similar calculations at the semi-empirical and empirical level where appropriate. For readers unfamiliar with the theory of electric and magnetic properties, the books by Davies and by Atkins review the subject thoroughly, whilst the more technical details of quantum-mechanical calculations on atoms and molecules have been described in many other places. ... 

For a many-electron atom, the self-consistent-field (SCF) method is used to construct an approximate wave function as a Slater determinant of (one-electron) spin-orbitals. The one-electron spatial part of a spin-orbital is an atomic orbital (AO). We took each AO as a product of a spherical harmonic and a radial factor. As an initial approximation to the radial factors, we can use hydrogenlike radial functions with effective nuclear charges. 

It is well known that the PES cannot be given analytically in quantum chemistry, but can be calculated point by point through iterative solution of matrix eigenvalue problems arising from the application of LCAO-MO SCF Cl (Linear Combination of Atomic Orbitals - Molecular Orbital Self-Consistent-Field Configuration Interaction) methods. If... 

Many problems in force field investigations arise from the calculation of Coulomb interactions with fixed charges, thereby neglecting possible mutual polarization. With that obvious drawback in mind, Ulrich Sternberg developed the COSMOS (Computer Simulation of Molecular Structures) force field [30], which extends a classical molecular mechanics force field by serai-empirical charge calculation based on bond polarization theory [31, 32]. This approach has the advantage that the atomic charges depend on the three-dimensional structure of the molecule. Parts of the functional form of COSMOS were taken from the PIMM force field of Lindner et al., which combines self-consistent field theory for r-orbitals ( nr-SCF) with molecular mechanics [33, 34]. 

Application of the variational self-consistent field method to the Haitiee-Fock equations with a linear combination of atomic orbitals leads to the Roothaan-Hall equation set published contemporaneously and independently by Roothaan and Hall in 1951. For a minimal basis set, there are as many matr ix elements as there are atoms, but there may be many more elements if the basis set is not minimal. 

The magnitude and "shape" of sueh a mean-field potential is shown below for the Beryllium atom. In this figure, the nueleus is at the origin, and one eleetron is plaeed at a distanee from the nueleus equal to the maximum of the Is orbital s radial probability density (near 0.13 A). The radial eoordinate of the seeond is plotted as the abseissa this seeond eleetron is arbitrarily eonstrained to lie on the line eonneeting the nueleus and the first eleetron (along this direetion, the inter-eleetronie interaetions are largest). On the ordinate, there are two quantities plotted (i) the Self-Consistent Field (SCF) mean-field... 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

The self-consistent field function for atoms with 2 to 36 electrons are computed with a minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are optimized so as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1 s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the ground state configuration and the positive ions. 

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

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