Types of Molecular Orbital Calculations

The first section summarizes simply the essential features of the different types of molecular orbital calculations currently used to solve theoretical problems in organometallic chemistry. A critical comparison of these calculations is also given. The second section discusses the bonding in organometallic complexes and draws on recent computational results and develops the chemical and structural implications of bonding models based on perturbation theory arguments. 

Application of LC AO-MO ideas to transition metal complexes faces no special restrictions (13, 17, 25, 37, 69, 236). Molecular orbitals are expressed as linear combinations of 1-electron atomic wave functions, 

Except for 1-electron atoms, atomic orbitals are not simple functions of the distance between the electron and the nucleus. Accurate wave functions are generally expressed as a linear combination of simple algebraic functions. The most convenient functions from which to build up accurate atomic orbitals are the Slater orbitals (28, 29, 206)  

An alternative set of functions used to build up atomic orbitals are Gaussian functions that have a radical dependence expf-ftr2). A linear combination of these functions gives a reasonable representation of an atomic orbital. The functions are used for computational expediency in ab initio calculations, because four-center Gaussian integrals can be reduced to two-center integrals, which are relatively easy to calculate on a computer (21). 

The Hamiltonian, Hea, which is called the Hartree-Fock-Roothan operator is a 1-electron operator whose application yields the energy of an electron moving in the average field of the other electrons and nuclei. In principle an SCF theory approach will lead to a well-defined expression for Hett for closed and open shell systems (188, 189), and with the aid of modern computers Hm integrals can be evaluated numerically even for transition metal complexes. This type of ab initio calculation has been reported for a reasonable number of organometallic complexes of first-row transition elements by Hillier, Veillard, and their co-workers (48, 49, 102, 103, 111-115 58, 68, 70, 187, 228, 229). 

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Several types of molecular orbital calculation have been carried out on the H- and the 2.ff-structures of w-triazole, and on simple alkyl derivatives from these, estimates of several physical properties, in-... 

Two groups have studied the bonding in pentadienyl-metal-tricar-bonyl complexes (119, 238) and are agreed that effective overlap between the pentadienyl nonbonding orbital and an orbital of suitable symmetry on the metal (Fig. 17) makes a major contribution to the stability of these complexes. However, the two types of molecular orbital calculation [one an extended Hiickel (119) and the other a parameter-free approximate Hartree-Fock calculation (255)] disagree about the precise ordering of energy levels in this type of complex. 

The Hartree-Fock method is a molecular orbital theory method that is central to chemistry (11). The detailed theory of the classical Hartree-Fock type of molecular orbital calculation may be found in reviews (12,13) and books (11,14,15). Most of the equations and theoretical background are taken and adapted from Szabo and Ostlund (11). 

The kind of qualitative considerations which have been used to construct the ethylene molecular orbitals do not give an indication of how much each atomic orbital contributes to the individual molecular orbitals. These coefficients are obtained only by solution of one of the types of molecular orbital calculations. Without these coefficients we cannot specify the exact shapes of the molecular orbitals. However, the qualitative ideas do permit conclusions about the symmetry of the orbitals. As will be seen in Chapter 10, just knowing the symmetry of the molecular orbitals provides very useful insight into many chemical reactions. 

The considerable number of molecular orbital calculations which have recently been made for sandwich compounds are however considered in some detail in Section 6. This has been done in order to make clear the relationship between the ligand field and molecular orbital approaches, and also to indicate the need for the use of a more sophisticated molecular orbital scheme than that adopted in this Introduction, i.e. one in which the a-framework of the rings is specifically included in the basis set as well as the rr-type orbitals. 

For alkynes bonded to higher nuclearity clusters no overall molecular orbital treatment encompassing all the variations in geometry has appeared yet. However, there are a small number of examples of specific alkyne-substituted clusters which have been analyzed by one type of molecular orbital treatment or another, and a number of these have been mentioned in Section III,G because photoelectron spectroscopy has been used as an aid to assignments. CNDO calculations (397) on Fe3(CO)9(EtCCEt) (390) and M3(CO)9(/i-H)(CCR) (M = Ru, Os) (391) and Fenske-Hall calculations (398) on Co4(CO)10(PhCCH) (389) indicate that there is net back donation into alkyne n orbitals, which increases as the number of metal atoms to which the ligand is bonded increases. The normally accepted view of considering the interaction... 

Molecular orbital calculations have shown that closo boranes have two types of molecular orbitals that are instrumental in holding the boron framework... 

In principle we can perform some sort of molecular orbital calculation on molecules of almost any sort of complexity. It is, however, often extremely profitable in terms of understanding the orbital structure to relate the level arrangement in a complex system to that of a simpler one. 3.1-3.3 show three examples of different types of relationships which we will frequently use. We will be interested in seeing how the levels of the species at the left-hand side of these figures are altered electronically during these processes by using the powerful techniques of perturbation theory. We shall not derive the elements of the theory itself but will make use of its mathematical results/" which will very quickly show a striking resemblance to the orbital interaction results of Chapter 2. 

These amino reductones described above are usually very unstable reaction intermediate compounds and, therefore, isolation and elucidation of their precise chemical structures by ordinary experimental techniques are rather difficult. However, owing to the recent remarkable progress in computational chemistry, various types of molecular orbital methods are now applicable to obtain needed information about their precise structures and chemical reactivities. For instance, the optimized structure of L-ascqrbic acid, an important acid-reductone in food and biological systems, was obtained by both semi-empirical and ab initio molecular orbital methods (Abe et aL, 1987, 1992). Semi-empirical molecular orbital calculations were also used to elucidate the autoxidation mechanism of L-ascorbic acid (Kurata et aL, 1996a,b). 

An accurate analysis on this point was made by Richardson for the Ng molecule and leads to very interesting conclusions. This author calculated the field gradient by using the Ransil s molecular orbitals in the approximation (i) (zeta constant), (iii) (zeta varied in the molecule), and a new type of molecular orbital [approximation (iv)] with a more extended basis, in which each atomic orbital 2s and 2p of the nitrogen atom is composed of two Slater-type functions with different zeta parameters (double zeta approximation). 

Abstract This work reports the formulation of Shannon entropy indices in terms of seniority numbers of the Slater determinants expanding an A-electron wave fimc-tion. Numerical determinations of those indices prove that they provide a suitable quantitative procedure to evaluate compactness of wave functions and to describe their configurational structures. An analysis of the results, calculated for full configuration interaction wave functions in selected atomic and molecular systems, allows one to compare and to discuss the behavior of several types of molecular orbital basis sets in order to achieve more compact wave... 

Molecular orbital calculations have been used to estimate equilibrium constants, although up to the present these attempts have not met with much success. Using calculations of this type, 2- and 4-hydroxypyridine 1-oxide were predicted to be more stable than 1-hydroxypyrid-2- and -4-one by ca. 20 kcal/mole, which corresponds to a ratio of ca. 10 between the forms. It was later shown experimentally that, at least in the series of 4-substituted compounds, there is very little energy difference between the forms and that the ratio between them is about unity. Molecular orbital calculations for... 

Tsuda and Oikawa (1989) investigated the photolysis of the 1,2-isomer of 10.89 (1,2-benzoquinone diazide) by means of MINDO/3 molecular orbital calculations with configurational interaction. These authors came to the conclusion that no ketocarbene of the type of 10.90 is formed, but that the rearrangement into the cyclopentadienyl ketene 10.94 is a concerted reaction in which the elimination of nitrogen and the rearrangement take place simultaneously. In the opinion of the present author the theoretical result for 1,2-quinone diazide is not necessarily in contradiction to the experimental investigations of Sander, Yankelevich et al., and Nakamura et al., as the reagents used were not exactly the same. 

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