Semi-empirical MO theory

The theoretical methods used commonly can be divided into three main categories, semi-empirical MO theory, DFT and ab-initio MO theory. Although it is no longer applied often, Hiickel molecular orbital (HMO) theory will be employed to introduce some of the principles used by the more modem techniques. 

Population anaiysis methods of assigning charges rely on the LCAO approximation and express the numbers of electrons assigned to an atom as the sum of the populations of the AOs centered at its nucleus. The simplest of these methods is the Coulson analysis usually used in semi-empirical MO theory. This analysis assumes that the orbitals are orthogonal, which leads to the very simple expression for the electronic population of atom i that is given by Eq. (53), where Natomic orbitals centered... 

Extended Hiickel Theory (EHT) uses the highest degree of approximation of any of the approaches we have already considered. The Hamiltonian operator is the least complex and the basis set of orbitals includes only pure outer atomic orbitals for each atom in the molecule. Many of the interactions that would be considered in semi-empirical MO theory are ignored in EHT. EHT calculations are the least computationally expensive of all, which means that the method is often used as a quick and dirty means of obtaining electronic information about a molecule. EHT is suitable for all elements in the periodic table, so it may be applied to organometallic chemistry. Although molecular orbital energy values and thermodynamic information about a molecule are not accessible from EHT calculations, the method does provide useful information about the shape and contour of molecular orbitals. 

It is noteworthy that the LUMO of a Fischer type carbene resembles the LUMO (determined using semi-empirical MO theory) of a typical ester such as methyl acetate (Figure 10-9a) or a ketone such as acetone (Figure 10-9b). In fact, the analogy between Fischer carbenes and carbonyl compounds should be useful to readers already familiar with organic chemistry. Indeed, the chemistry of these two types of compounds, seemingly so different in structure, is similar in several ways. 

Whereas it is generally sufficient (at least for the pubhshed methods) to specify the semi-empirical MO technique used in order to define the exact method used for the calculations, ab-initio theory offers far more variations, so that the exact level of the calculation must be specified. The starting point of most ab-initio jobs is an SCF calculation analogous to those discussed above for semi-empirical MO calculations. In ab-initio theory, however, all necessary integrals are calculated correctly, so that the calculations are very much (by a factor of about 1000) more time-consuming than their semi-empirical counterparts. 

UPS and MO theory are inextricably linked. It is impossible to interpret UPS measurements without at least a qualitative MO treatment, and it is now commonplace to perform quantitative calculations for comparison with the UPS data. The development of UPS techniques since about 1960 helped to popularise MO theory, and semi-empirical MO methods are often calibrated by appeal to UPS data. Thus an approximation which greatly simplifies an MO calculation is held to be justifiable if, over a fair range of molecules, the calculated orbital energies are in good agreement with UPS binding energies. 

Despite the VB-theoretic successes of recent decades there still seems a domination by MO theory. Ab-initio VB theory seems weak in comparison to ab-initio MO-based theory in consideration of general availability of canned programs. Semi-empirical VB theory (the simplest versions of which are the focus of the present chapter) seems weak in comparison to semi-empirical MO-based theory when consideration is made of a comprehensive semiempirical model to make quantitative estimates for heats of formation, geometries, or excitation spectra, all for a diversity of chemical species. 

Professor Houk and I are coevals and we embarked on our research careers at about the same time. In the beginning of the 1970s, both he and I were independently working on mechanistic aspects of pericyclic reactions, using a combination of experiment, simple perturbational MO theory and semi-empirical MO calculations. My published work in this area was of variable quality whereas Ken s was uniformly... 

Semi-empirical MO methods address electron correlation implicitly they simply adjust parameters until the calculations give the correct answer compared with experiment. EHT does not address electron correlation at all, so quantitative results from such calculations are almost always wrong unless fortuitous. There are, however, several approaches to explicitly account for electron correlation. One approach is to perform post-ab initio (post-H-F) calculations that in effect mix different electronic configurations involving the ground state and several excited states of the molecule. Such calculations are quite computationally intensive and can be performed only on relatively small molecules. Two commonly-seen acronyms associated with the post H-F approach to electron correlation are MP2 and Cl, which stand for Mpller-Plesset theory at the level of second-order and configuration interaction, respectively. 

Semi-empirical MO Theory3 Now good for most organometallic molecules Same as ab initio MO theory AfH° values Good most transition metals are now parameterized Medium Upto thousands of atoms... 

Schulman and Venanzi (20) have evaluated a=s (0) n(0) and b=(r 3)c(r 3)N. For other than triple bonds they find that a is 13 79 a.u. and b is l -7 a.u., while for triple bonds a is 13 10 a.u. and b is 20 85 a.u. In the opinion of the reviewer the large difference calculated for b in these two cases is not reasonable. Furthermore, if one assumes a value of 2 5 a.u. or 2 88 a.u. for (r 3)c, as obtained earlier by Schulman and Newton (42) and by Blizzard and Santry (16, 17) from their studies of C-C coupling constants, then one obtains (r 3)N <0 71 a.u. when the N atom is singly bonded to carbon. The conclusion that (r 3)N is less than one-half (r 3)c also seems unreasonable (see Table I). One must ask how much agreement should be expected between observed /(C-X) values and those obtained from semi-empirical MO calculations at the INDO level of approximation, and to what extent the disagreement between theory and experiment is due to inadequacy of the wave functions. 

The so-called heavy-atom chemical shift of light nuclei in nuclear magnetic resonance (NMR) had been identified as a spin-orbit effect early on by Nomura etal. (1969). The theory had been formulated by Pyykktt (1983) and Pyper (1983), and was previously treated in the framework of semi-empirical MO studies (PyykktJ et al. 1987). The basis for the interpretation of these spin-orbit effects in analogy to the Fermi contact mechanism of spin-spin coupling has been discussed by Kaupp et al. (1998b). 

As the model of solvent effect on carbon-13 chemical shifts, we used one developed by Ando et al.[6] based on a Klopman s "solvaton" theory[11], been successfully applied to interpret the dielectric solvent effect on carbon-13 chemical shifts of many organic compounds. According to this model, the interaction of solute with solvent molecules is incorporated into semi-empirical MO calculations by an assumption of virtual particle called a solvaton. The details of theory and calculation have been reported[7]. Carbon-13 NMR spectra were recorded at 30 C on a JEOL PS-100 spectrometer operated at 25 MHz. 

It is relevant that the microwave spectrum of o-fluorotoluene leads to a value of 2.72 kJ mol" for the 3-fold barrier. Another recent determination is that by n.m.r. spectroscopy of a barrier of 5 kJ mol" for methyl rotation in u-chlorotoluene dissolved in the nematic phase of 4-amino-4-methoxybenzylidene-a-methylcinnamic acid n-propyl ester. The position of minimal energy was inferred to be that in which the chlorine is staggered to the methyl group. This conclusion has also been reached from semi-empirical MO calculations by the extended Hiickel theory, which give a barrier height of 7.41 kJ mol. ... 

FE profiles can be expressed as functions of certain reaction coordinates, which are defined by Rc-c -Rn-c nd / n h Menshutkin and ammonia ionization reaction, respectively. We used the QM/MM-MD method to describe both solution systems for the purpose to include the solvent molecular strucmres explicitly. In the original smdies [7-9], the QM parts of the whole solution systems were treated by semi-empirical MO methods, i.e., AMI and PM3 level of theory. 

Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Divide and Conquer for Semi-empirical MO Methods Localized MO SCF Methods. 

Murrell, J. N. Harget, A. J. Semi-empirical Self-consistent-field Molecular Orbital Theory of Mo/ecw/e Wiley Interscience, New York, 1971. 

More sophisticated procedures involve taking the start MO coefficients from a semi-empirical calculation, such as Extended HUckel Theory (EHT) or Intermediate Neglect of Differential Overlap (INDO) (Sections 3.12 and 3.9). The EHT method has the advantage that it is readily parameterized for all elements, and it can provide start orbitals for systems involving elements from essentially the whole periodic table. An INDO calculation normally provides better start orbitals, but at a price. The INDO... 

The fact that features in the total electron density are closely related to the shapes of the HOMO and LUMO provides a much better rationale of why FMO theory works as well as it does, than does the perturbation derivation. It should be noted, however, that improvements in the wave function do not necessarily lead to a better performance of the FMO method. Indeed the use of MOs from semi-empirical methods usually works better than data from ab initio wave functions. Furthermore it should be kept in mind that only the HOMO orbital converges to a specific shape and energy as the basis set is... 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

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