Molecular orbital method symmetry-based

Julg et al.24 employed an approximation based on the self consistent field molecular orbital method to evaluate the average energy per atom for various structures. They calculate that whereas the normal b.c.c. structure is more stable for clusters containing more than 106 atoms, smaller clusters prefer to take up pentagonal symmetry. However, these authors make an important point, namely, that the calculated energies for different structures are very similar. Interconversion of different structures will be facile, and external factors such as the method of deposition, level of impurities, support effects, etc., may cause the less stable structure to grow. For example, impurities on... 

The perfectly octahedral species conform to the expectations based on the simple MO derivation given above. The nonoctahedral fluoride species do not, but this difficulty is a result of the oversimplifications in the method. There is no inherent necessity for delocalized MOs to be restricted to octahedral symmetry. Furthermore, it is possible to transform delocalized molecular orbitals into localized molecular orbitals. Although the VSEPR theory is often couched in valence bond terms, it depends basically on the repulsion of electrons of like spins, and if these are in localized orbitals the results should be comparable. 

The CASSCF method has been mentioned at several places in these lecture notes. Here we shall give a more detailed account of this method, which is probably the most widely used MCSCF method today. It is based on a partitioning of the occupied molecular orbitals into subsets, corresponding to how they are used to build the wave function. We define for each symmetry block of MO s the following subsets ... 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

Although Otto Diels and Kurt Alder won the 1950 Nobel Prize in Chemistry for the Diels-Alder reaction, almost 20 years later R. Hoffmann and R. B. Woodward gave the explanation of this reaction. They published a classical textbook, The Conservation of Orbital Symmetry. K. Fukui (the co-recipient with R. Hoffmann of the 1981 Nobel Prize in Chemistry) gave the Frontier molecular orbital (FMO) theory, which also explains pericyclic reactions. Both theories allow us to predict the conditions under which a pericyclic reaction will occur and what the stereochemical outcome will be. Between these two fundamental approaches to pericyclic reactions, the FMO approach is simpler because it is based on a pictorial approach. Another method similar to the FMO approach of analyzing pericyclic reactions is the transition state aromaticity approach. 

Several researchers have recently devoted considerable effort to the derivation and efficient implementation of techniques based on spin-restricted reference determinants that reduce the computational discrepancy between closed- and open-shell systems. " This emphasis on spin-restricted techniques has resulted in part from a bias toward reference wavefunctions that maintain the spin symmetry of the exact wavefunction (such as the ROHF determinant), but also because of the possible efficiency advantages of spin-restricted methods over unrestricted techniques. Thus, since the component molecular orbitals are constrained to have identical spatial parts for each spin function, it should be possible to construct the correlated wavefunction in a manner that takes advantage of this symmetry. 

It is apparent that the methods discussed here are particularly effective for molecules whose MO s are largely determined by their point group symmetry because, in these cases, central AO s are good MO templates. Clearly, there are other types of molecules in which central AO s are less suitable and other simple orbitals are more effective templates. For general pi-electron systems, for example, free-elec-tron MO s would be more useful GO s (14). In its broadest sense, then, the generator orbital approach is based on the principle that major characteristics of complicated molecular orbitals can be semi-quantitatively anticipated by conceptual analogy to simpler orbitals with which they share essential physical features. We feel that there is an elucidating quality in this reduction of the more complex to the less complex, which is of value to theorists and non-theorists alike. 

The two books with Harry Gray [19, 32] are good pedagogical introductions to molecular orbital theory with emphasis on orbital symmetry and computational procedures. They are based on the lectures of the authors and the development of the extended Wolfsberg-Helmholz method. 

Consequently, the symmetry of the wave function which is intermolecular in nature may be conveniently evaluated in the approximation of a linear combination of frontier molecular orbitals of fragments (LCFMOF). It should be noted that the method of perturbations of molecular orbitals developed by Dewar [17] is actually based on the LCFMOF approximation. 

At first we would like to recall a few important conclusions from earlier studies. The electronic structure of the reactant CpML has been discussed by Hofmann and Padmanabhan [22] for various ligands L and M = Co, Rh, and Ir with the extended Huckel method. As shown in Fig. 5, the valence molecular orbitals are m, ma", n -I- l)a, (w -I- l)a", and (n + 2)a orbitals under Cj symmetry. One sees that the m and rm" orbitals are mainly d x yy and orbitals stabilized by interaction with the ir orbitals on L. At somewhat higher energy is the occupied metal-based d orbital, (n + l)a , with a weak M-L antibonding a interaction. The metal-based (m + l)fl" (dj,j) and (n -H 2)a (d y) orbitals are highest in energy, destabilized by interaction with occupied tt orbitals on the Cp ring. In addition, the (n -I- 2)a orbital destabilized by interaction with the a orbital on L. Thus, it... 

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