Molecular orbital theory approximate methods

Presell is the basic theory of tjuaiiHim mechanics, particularly, semi-empirical molecular orbital theory. The authors detail and justify the approximations inherent in the semi-empirical Ham illoTi ian s. Includes useful discussion s of th e appiicaliori s of these methods to specific research problems. 

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. 

Because of its severe approximations, in using the Huckel method (1932) one ignores most of the real problems of molecular orbital theory. This is not because Huckel, a first-rate mathematician, did not see them clearly they were simply beyond the power of primitive mechanical calculators of his day. Huckel theory provided the foundation and stimulus for a generation s research, most notably in organic chemistry. Then, about 1960, digital computers became widely available to the scientific community. 

As presented, semi-empirieal methods are based on a single-eonfiguration pieture of eleetronie strueture. Extensions of sueh approaehes to permit eonsideration of more than a single important eonfiguration have been made (for exeellent overviews, see Approximate Molecular Orbital Theory by J. A. Pople and D. E. Beveridge, McGraw-Hill, New York... 

Energy, geometry, dipole moment, and the electrostatic potential all have a clear relation to experimental values. Calculated atomic charges are a different matter. There are various ways to define atomic charges. HyperChem uses Mulliken atomic charges, which are commonly used in Molecular Orbital theory. These quantities have only an approximate relation to experiment their values are sensitive to the basis set and to the method of calculation. 

The simplest approximation to the Schrodinger equation is an independent-electron approximation, such as the Hiickel method for Jt-electron systems, developed by E. Hiickel. Later, others, principally Roald Hoffmann of Cornell University, extended the Hiickel approximations to arbitrary systems having both n and a electrons—the Extended Hiickel Theory (EHT) approximation. This chapter describes some of the basics of molecular orbital theory with a view to later explaining the specifics of HyperChem EHT calculations. 

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods. The final section provides a similar overview of the theory underlying Density Functional Theory methods. 

Craig, D. P., Proc. Roy. Soc. London) A200, 474, Configurational interaction in molecular orbital theory. A higher approximation in the non-empirical method." (i)... 

Parallel to this use of relatively simple approximations of the molecular orbital theory to the study of complex molecules Berthier has investigated the possible utilization of more refined molecular orbital procedures in the study of necessarily smaller molecules. We owe him the first application of the SCF method to the study of fulvene and azulene and also a pioneering extension, presented in 1953, of the SCF method to the study of molecules with incomplete electronic shells. 

The variation method is usually employed to determine an approximate value of the lowest eneigy state (the ground state) of a given atomic or molecular system. It can, furthermore, be extended to the calculation of energy levels of excited stales. It forms the basis of molecular orbital theory and that which is often referred to (incorrectly) as theoretical chemistry". 

When the Hartree-Fock method is applied to molecules, molecular orbitals are used instead of atomic orbitals. To construct the molecular orbitals, one widely used approximation is LCAO (linear combinations of atomic orbitals). According to molecular orbital theory, the total wave function of the system is written as a combination of molecular orbitals, spin functions describing electrons in terms of spin j(a) or — j p). 

This theory proves to be remarkably useful in rationalizing the whole set of general rules and mechanistic aspects described in the previous section as characteristic features of the Diels-Alder reaction. The application of perturbation molecular orbital theory as an approximate quantum mechanical method forms the theoretical basis of Fukui s FMO theory. Perturbation theory predicts a net stabilization for the intermolecular interaction between a diene and a dienophile as a consequence of the interaction of an occupied molecular orbital of one reaction partner with an unoccupied molecular orbital of the other reaction partner. 

The simple, or Hiickel based, molecular orbital theory (HMO and PPP methods) frequently provides useful qualitative insights but cannot be used reliably in a quantitative manner. For this purpose it is necessary to use a method which takes account of all the electrons as well as their mutual repulsions. A major bottleneck in such calculations is in the computation and storage of the enormous number of electron-repulsion integrals involved. Early efforts to reduce this problem led Hoffmann to the EH approximation (I.N. Levine, Quantum Chemistry, 4-th ed., 1991, Prentice-Hall, Inc., Ch. 16, 17), and Pople and co-workers to the CNDO, INDO and NDDO-approximations (B-70MI40100). 

Ligand field theory may be taken to be the subject which attempts to rationalize and account for the physical properties of transition metal complexes in fairly simple-minded ways. It ranges from the simplest approach, crystal field theory, where ligands are represented by point charges, through to elementary forms of molecular orbital theory, where at least some attempt at a quantum mechanical treatment is involved. The aims of ligand field theory can be treated as essentially empirical in nature ab initio and even approximate proper quantum mechanical treatments are not considered to be part of the subject, although the simpler empirical methods may be. 

Exact solutions such as those given above have not yet been obtained for the usual many-electron molecules encountered by chemists. The approximate method which retains tile idea of orbitals for individual electrons is called molecular-orbital theory (M. O. theory). Its approach to the problem is similar to that used to describe atomic orbitals in the many-electron atom. Electrons are assumed to occupy the lowest energy orbitals with a maximum population of two electrons per orbital (to satisfy the Pauli exclusion principle). Furthermore, just as in the case of atoms, electron-electron repulsion is considered to cause degenerate (of equal energy) orbitals to be singly occupied before pairing occurs. 

It is apparent that the molecular orbital theory is a very useful method of classifying the ground and excited states of small molecules. The transition metal complexes occupy a special place here, and the last chapter is devoted entirely to this subject. We believe that modem inorganic chemists should be acquainted with the methods of the theory, and that they will find approximate one-electron calculations as helpful as the organic chemists have found simple Hiickel calculations. For this reason, we have included a calculation of the permanganate ion in Chapter 8. On the other hand, we have not considered conjugated pi systems because they are excellently discussed in a number of books. 

The AMI (8) approximation to molecular orbital theory has been used for these studies. This method overcomes the problems that previous semiempirical methods (notably, MNDO) (9) have in describing hydrogen-bonds. It has been used with success in several hydrogen-bonding studies. (10-12) Ab initio studies of H-bonding systems are very sensitive to basis set and correction for electron-... 

The aim of molecular orbital theory is to provide a complete description of the energies of electrons and nuclei in molecules. The principles of the method are simple a partial differential equation is set up, the solutions to which are the allowed energy levels of the system. However, the practice is rather different, and, just as it is impossible (at present) to obtain exact solutions to the wave equations for polyelectronic atoms, so it is not possible to obtain exact solutions for molecular species. Accordingly, the application of molecular orbital theory to molecules is in a regime of successive approximations. Numerous rigorous mathematical methods have been utilised in the effort to obtain ever more accurate solutions to the wave equations. This book is not concerned with the details of the methods which have been used, but only with their results. 

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