Orbital Methods

The second broad approach to the description of molecular structure which is of importance in organic chemistry is molecular orbital theory. Molecular orbital (MO) theory discards the idea that bonding electron pairs are localized between specific atoms in a molecule and instead pictures electrons as being distributed among a set of molecular orbitals of discrete energies. In contrast to the orbitals described by valence bond theory, which are centered on two specific atoms, these orbitals can extend over the entire molecule. The theory is based on the Schrodinger equation, 

In order to make the mathematics tractable, a number of approximations are made. The choice of approximations has produced a variety of molecular orbital methods, the judicious application of which can provide a valuable insight into questions of bonding, structure, and dynamics. The discussion that follows will not be 

All but the crudest molecular orbital calculations are done on computers and there is a necessary trade-off between the accuracy of the calculations and the expense in terms of computer time. In general, the more severe the approximations, the more limited is the range of applicability of the particular calculation. The organic chemist who wishes to make use of the results of molecular orbital calculations must therefore make a judgement about the applicability of the various methods to the particular problem. In general, the programs which are used for the calculations are available and the complexity of use tends to increase with the sophistication of the calculation.  

Mathematically, the molecular orbitals are treated as linear combinations of atomic orbitals, so that the wave function, (A, is expressed as a sum of individual atomic orbitals multiplied by appropriate weighting factors (coefficients)  

The coefficients indicate the contribution of each atomic orbital to the molecular orbital. This method of representing the molecular orbital wave function in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals-molecular orbital (LCAO-MO) approximation. The combination of atomic orbitals chosen is called the basis set. A minimal basis set for molecules containing C, H, O, and N would consist of Is, 2s, 2px, 2py, and 2p orbitals for C, O, and N, and a l5 orbital for hydrogen. Inclusion of additional orbitals in the basis set leads to an extended basis set. The basis sets are mathematical expressions describing the properties of the atomic orbitals. 

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Molecules. The electronic configurations of molecules can be built up by direct addition of atomic orbitals (LCAO method) or by considering molecular orbitals which occupy all of the space around the atoms of the molecule (molecular orbital method). 

When most quantum ohemists were pursuing improvements in the moleoular orbital method, he returned to the valenoe bond theory and developed the so-oalled GVB methods that allow eleotron oorrelation to be inoluded within a valenoe bond framework... 

Flehre W J, Ditchfieid R and Popie J A 1972 Self-consistent molecular-orbital methods XII. Further extension of Gaussian-type basis sets for use in molecular orbital studies of organic molecules J. Chem. Phys. 56 2257-61 Flariharan P C and Popie J A 1973 The influence of polarization functions on molecular orbital hydrogenation energies Theoret. Chim. Acta. 28 213-22... 

Krishnan R, Binkley J S, Seeger R and Popie J A 1980 Self-consistent molecular orbital methods XX. A basis set for correlated wave functions J. Chem. Phys. 72 650-4... 

Stewart J J P 1991 Semiempirical molecular orbital methods Reviews in Computationai Chemistry vo 1, ed K B Lipkowitz and D B Boyd (New York VCH) pp 45-81... 

To become familar with the algorithm for charge calculation by partial equalization of orbital electronegativity (PEOE) and by a modified Hiickel Molecular Orbital method... 

Most simple empirical or semi-empirical molecular orbital methods. including all ofthose ii sed in IlyperCh em, neglect inner sh ell orbitals and electrons and use a minimal basis se.i r>f valence Slater orbitals. 

Stewart)) P1990. Semi-empirical Molecular Orbital Methods. In Lipkowitz K B and D B Boyd (Editors). 

The simplest molecular orbital method to use, and the one involving the most drastic approximations and assumptions, is the Huckel method. One str ength of the Huckel method is that it provides a semiquantitative theoretical treatment of ground-state energies, bond orders, electron densities, and free valences that appeals to the pictorial sense of molecular structure and reactive affinity that most chemists use in their everyday work. Although one rarely sees Huckel calculations in the resear ch literature anymore, they introduce the reader to many of the concepts and much of the nomenclature used in more rigorous molecular orbital calculations. 

The Bom-Oppenheimer approximation is not peculiar to the Huckel molecular orbital method. It is used in virtually all molecular orbital calculations and most atomic energy calculations. It is an excellent approximation in the sense that the approximated energies are very close to the energies we get in test cases on simple systems where the approximation is not made. 

In summary, we have made three assumptions 1) the Bom-Oppenheimer approximation, 2) the independent particle assumption governing molecular orbitals, and 3) the assumption of n-molecular orbital theory, but the third is unique to the Huckel molecular orbital method. 

The Jacobi method is probably the simplest diagonalization method that is well adapted to computers. It is limited to real symmetric matrices, but that is the only kind we will get by the formula for generating simple Huckel molecular orbital method (HMO) matrices just described. A rotation matrix is defined, for example. 

The cornerstone of semiempirical and ab initio molecular orbital methods is the Harhee equation and its extensions and variants, the Harhee-Fock and Roothaan-Hall equations. We have seen that the Hamiltonian for the hydrogen atom. 

Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian H whose energy levels are sought and one needs to construct a trial wavefunction in which some flexibility exists (e.g., as in the linear variational method where the Cj coefficients can be varied). In Section 6 this tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry. 

We will use the valence bond approach extensively m our discussion of organic molecules and expand on it shortly First though let s introduce the molecular orbital method to see how it uses the Is orbitals of two hydrogen atoms to generate the orbitals of an H2 molecule... 

The molecular orbital approach to chemical bonding rests on the notion that as elec trons m atoms occupy atomic orbitals electrons m molecules occupy molecular orbitals Just as our first task m writing the electron configuration of an atom is to identify the atomic orbitals that are available to it so too must we first describe the orbitals avail able to a molecule In the molecular orbital method this is done by representing molec ular orbitals as combinations of atomic orbitals the linear combination of atomic orbitals molecular orbital (LCAO MO) method... 

Frontier orbital analysis is a powerful theory that aids our understanding of a great number of organic reactions Its early development is attributed to Professor Kenichi Fukui of Kyoto University Japan The application of frontier orbital methods to Diels-Alder reactions represents one part of what organic chemists refer to as the Woodward-Hoffmann rules a beautifully simple analysis of organic reactions by Professor R B Woodward of Harvard University and Professor Roald Hoffmann of Cornell University Professors Fukui and Hoffmann were corecipients of the 1981 Nobel Prize m chemistry for their work... 

When a molecular orbital method was used to calculate the charge distribution in cyclohexadienyl cation it gave the results indicated How does the charge at each carbon compare with that deduced by examining the res onance structures for cyclohexadienyl cation ... 

Hehre, W.J. Stewart, R.F. Pople, J.A. Self-consistent molecular-orbital methods. I. Use of Gaussian expressions of Slater-Type Atomic Orbitals 7 Chem. 51 2657-2664, 1969. 

Pietro, W.I. Francl, M.M. Hehre, W.I. Defrees, D.I. Pople, I.A. Binkley, I.S. Self-consistent molecular orbital methods. 24. Supplemented small split-valence basis sets for second-row elements J. Am. Chem. Soc. 104 5039-5048, 1982. 

Electrophilic Aromatic Substitution. The Tt-excessive character of the pyrrole ring makes the indole ring susceptible to electrophilic attack. The reactivity is greater at the 3-position than at the 2-position. This reactivity pattern is suggested both by electron density distributions calculated by molecular orbital methods and by the relative energies of the intermediates for electrophilic substitution, as represented by the protonated stmctures (7a) and (7b). Stmcture (7b) is more favorable than (7a) because it retains the ben2enoid character of the carbocycHc ring (12). 

In subsequent studies attempting to find a correlation of physicochemical properties and antimicrobial activity, other parameters have been employed, such as Hammett O values, electronic distribution calculated by molecular orbital methods, spectral characteristics, and hydrophobicity constants. No new insight on the role of physiochemical properties of the sulfonamides has resulted. Acid dissociation appears to play a predominant role, since it affects aqueous solubiUty, partition coefficient and transport across membranes, protein binding, tubular secretion, and reabsorption in the kidneys. An exhaustive discussion of these studies has been provided (10). 

Oxirane (1) and methyloxirane (3) are miscible with water, ethyloxirane is very soluble in water, while compounds such as cyclopentene oxide and higher oxiranes are essentially insoluble (B-73MI50501) (for a discussion of the solubilities of heterocycles, see (63PMH(l)l77)). Other physical properties of heterocycles, such as dipole moments and electrochemical properties, are discussed in various chapters of pmh. The optical activity of chiral oxiranes has been investigated by ab initio molecular orbital methods (8UA1023). 

Free Electron Molecular Orbital method colour and constitution, 1, 342 Freelingyne occurrence, 4, 706 Free radical processes in photography, 1, 387-389 Friedlander synthesis quinolines, 2, 443 thioindigo dyes, 4, 910 Fries rearrangement chroman-4-one synthesis from, 3, 850 Fructose, 1-deoxy- C NMR, 4, 575 Frusemide as diuretic, 1, 174 metabolism, 1, 245 FS-32 — see 1/f-Indazole, l-[3-... 

Hiickel linear combination of atomic orbitals pyridines and benzo derivatives, 2, 102 Hiickel molecular orbital method colour and constitution, 1, 342 Hugerschoff bases synthesis, 6, 475-477, 493 Humulene... 

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