Approximate molecular orbital methods

Pople, J. A., and D. L. Beveridge. 1970. Approximate Molecular Orbital Methods. McGraw-Hill, New York. 

A wide range of theoretical methods has been applied to the study of the structure of small metal clusters. The extremes are represented on the one hand by semi-empirical molecular orbital (Extended Huckel) (8 ) and valence bond methods (Diatomics-In-Molecules) ( ) and on the other hand by rigorous initio calculations with large basis sets and extensive configuration interaction (Cl) (10). A number of approaches lying between these two extremes have been employed Including the X-a method (11), approximate molecular orbital methods such as CNDO (12) and PRDDO (13) and Hartree-Fock initio molecular orbital theory with moderate Cl. 

M. C. Zerner,/. Chem. Phys., 62. 27S8 (1975). An Approximate Molecular Orbital Method. 

Many approximate molecular orbital theories have been devised. Most of these methods are not in widespread use today in their original form. Nevertheless, the more widely used methods of today are derived from earlier formalisms, which we will therefore consider where appropriate. We will concentrate on the semi-empirical methods developed in the research groups of Pople and Dewar. The former pioneered the CNDO, INDO and NDDO methods, which are now relatively little used in their original form but provided the basis for subsequent work by the Dewar group, whose research resulted in the popular MINDO/3, MNDO and AMI methods. Our aim will be to show how the theory can be applied in a practical way, not only to highlight their successes but also to show where problems were encountered and how these problems were overcome. We will also consider the Hiickel molecular orbital approach and the extended Hiickel method Our discussion of the underlying theoretical background of the approximate molecular orbital methods will be based on the Roothaan-Hall framework we have already developed. This will help us to establish the similarities and the differences with the ab initio approach. 

One-determinant approximation One-electron approximation One-particle approximation Molecular orbital method Independent-particle approximation Mean field approximation Hartree-Fock method... 

For p orbitals, U corresponds to a pure rotation matrix however, this is not generally true for d orbitals. In both cases, the U transformation is unique in an anisotropic environment and eliminates most of the need for drastic spherical averaging typical of approximate molecular orbital methods. 

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. 

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. 

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... 

HyperChem currently supports one first-principle method ab initio theory), one independent-electron method (extended Hiickel theory), and eight semi-empirical SCFmethods (CNDO, INDO, MINDO/3, MNDO, AMI, PM3, ZINDO/1, and ZINDO/S). This section gives sufficient details on each method to serve as an introduction to approximate molecular orbital calculations. For further details, the original papers on each method should be consulted, as well as other research literature. References appear in the following sections. 

Use of high-level ab initio molecular orbital methods or density functional Hamiltonians to simulate the QM region through approximate solutions of the electronic Schrodinger equation. 

As noted eailier, the dynamic sensitivity of VCD carries it beyond the Bom-Oppenheimer approximation (17, 18). Even though non-Bom-Oppenheimer calculations have recently been initiated (116-119), the use of molecular orbital methods to describe these vibrationally induced currents is lagging behind the measurement and interpretation of VCD based on such currents. Nevertheless, the empirical basis that is now emerging for understanding VCD intensities based on chirally oriented oscillators and induced currents makes possible the immediate application of this methodology to stereochemical problems of widespread interest. 

The 5-position of the nonprotonated 1,2,4-thiadiazole system was calculated to be the most reactive in nucleophilic substitution reactions using a simple molecular orbital method with LCAO approximation (84CHEC-I(6)463>. 

Unfortunately, the Schrodinger equation can be solved exactly only for one-electron systems such as the hydrogen atom. If it could be solved exactly for molecules containing two or more electrons,3 we would have a precise picture of the shape of the orbitals available to each electron (especially for the important ground state) and the energy for each orbital. Since exact solutions are not available, drastic approximations must be made. There are two chief general methods of approximation the molecular-orbital method and the valence-bond method. 

Qualitatively, the resonance picture is often used to describe the structure of molecules, but quantitative valence-bond calculations become much more difficult as the structures become more complicated (e.g., naphthalene, pyridine, etc.). Therefore the molecular-orbital method is used much more often for the solution of wave equations.5 If we look at benzene by this method (qualitatively), we see that each carbon atom, being connected to three other atoms, uses sp1 orbitals to form a bonds, so that all 12 atoms are in one plane. Each carbon has a p orbital (containing one electron) remaining and each of these can overlap equally with the two adjacent p orbitals. This overlap of six orbitals (see Figure 2.1) produces six new orbitals, three of which (shown) are bonding. These three (called it orbitals) all occupy approximately the same space.6 One of the three is of lower energy than... 

These integrals are difficult to evaluate exactly and the Hiickel molecular orbital method centres on approximations to them. 

---