Semiempirical approaches

It is natural that people should want to calculate properties of systems that are too complex to be treated by the ab initio techniques of the day, and so approximation procedures have grown up alongside ab initio methodology. These approximations neglect certain terms and adopt simply evaluated formulas for others. To achieve useful accuracy, at least some of the terms retained must be parameterized, with the parameters determined all, or in part, from experimental data. This has given rise to a variety of so-called semiempirical methods. 

The earliest semiempirical methods were well in advance of ab initio methods partly because of the limits on computing power. The first of these is the ir-molecular orbital (MO) theory of conjugated and aromatic molecules proposed by Hiickel in 1931. Most succeeding semiempirical theories are direct descendants of Hiickel s original approach wherein one or more of the approximations has been improved. It is interesting to note that Hiickel theory in its original form still sees infrequent application to properties such as electric susceptibilities, soliton dynamics, and others. 

Parr ° and by Pople (PPP) to explicitly include self-consistent coulombic electron repulsion. A second development, crucial for magnetic properties, was the inclusion of a-valence elearons in the late 1950s and early 1960s. This period gave birth to the extended Hiickel theory (EHT), which has proved extremely valuable for the analysis of ground-state reaaivities, and also to CNDO/1, the first in a series of all valence electron NDO (neglect of differential overlap) methods. 

A few years later, reparameterization of CNDO/1 (to give CNDO/2 with better bond lengths and dipole moments), followed by incorporation of atomic elearon exchange, led to the I(ntermediate)NDO method. This is the lowest semiempirical level of theory that can properly account for elearon spin polarization and, hence, for magnetic properties that depend on the latter. 

All of the semiempirical treatments described thus far fall under the rubric of the molecular orbital (MO) approximation. To charaaerize optical speara, one must, in general, also include the interaaion between excited elearon configurations obtained by promotion of one or more elearons from occupied to virtual orbitals. Configuration interaaion (Cl) was first introduced into the PPP method and, then, generalized to all valence electrons in 1968 by Del Bene and Jaffe. The latter method, called CNDO/S, is more similar to PPP in the calculation of elearon repulsion integrals and more flexible than either PPP or CNDO/2 in the treatment of resonance terms. The extension to INDO/S has also been made, and all three Cl approaches have achieved some important successes. 

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In order to include other interactions such as dipolar or hydrogen bonding, many semiempirical approaches have been tried [196, 197, 200], including adding terms to Eq. X-45 [198, 201] or modifying the definition of [202, 199]. Perhaps the most well-known of these approaches comes from Fowkes [203, 204] suggestion that the interactions across a water-hydrocarbon interface are dominated by dispersion forces such that Eq. X-45 could be modified as... 

Before discussing the kinds of kinetic information provided by potential energy surfaces we will briefly consider methods for calculating these surfaces, without going into detail, for theoretical calculations are outside the scope of this treatment. Detailed procedures are given by Eyring et ah There are three approaches to the problem. The most basic one is purely theoretical, in the sense that it uses only fundamental physical quantities, such as electronic charge. The next level is the semiempirical approach, which introduces experimental data into the calculations in a limited way. The third approach, the empirical one, makes extensive use of experimental results. 

A useful, semiempirical approach to noncatalytic surface reactions is to postulate a rate equation of the form... 

Amokrane and Badiali proposed a semiempirical approach to the determination of the solvent contribution C, to the capacitance of the double layer in aqueous and nonaqueous " solutions. They used the relation C = Cf - C m, where Q is the experimentally determined capacity of the inner layer and Cm is the contribution of the metal. The plots ofC, vs. (Tm were presented for various solvents and correlated with their properties.However, the problem of the supporting electrolyte was entirely neglected in the quoted papers. It was shown recently that the height and position of the maximum on the C, vs. Gm plots depend on the type of the supporting electrolyte. Experimental differential capacity data obtained on the Hg electrode in methanol and ethanol containing various electrolytes with nonadsorbing anions (F , PFg, ClOi) indicate that the type as well as concentration of the electrolyte influences the position and the height of the maximum on the C, vs. plots (Fig. 13). 

We now turn to the problem of simplifying the recovery of the dynamic correlation energy. We consider the simplest situation, viz., where the zeroth-order wavefunction can be chosen as the SCF approximation. A challenging disparity exists between the energetic smallness of these refinements and the complexity and magnitude of the computational efforts required for their variational determination. In order to reduce this disproportion, various semiempirical approaches have been proposed (56-61), notably in particular the introduction of semiempirical elements into MP2 theory which has led to the successful Gn methods (62-64). 

It has been suggested that a symmetrical intermediate or transition state as represented in Scheme IV.54 could be involved, in which the new and the old nitrogen-oxygen bonds are formed and broken to the same extent. A semiempirical approach for calculating the free energy of activation for the degenerate rearrangement of the anion of 3-acetylamino-5-methyl-l,2,4-... 

The second semiempirical approach to the evaluation of the activity coefficient is the use of the Davies equation which modifies the Bronsted extension of the limiting Debye-Hilckel expression and is given by... 

Two semiempirical approaches which to date seem to account best for the effects observed at high ionic istrength, will be discussed here. The first, due to Lietzke et at. 197), views the solution as a mixture of two components one has the limiting Debye-HOckel character apd the other component exhibits the behavior of fused salts. The activity coefficient for fused salts is given by... 

Starting with the semiempirical approach of Kauzmann et al. (16), Ruch and Schonhofer developed a theory of chirality functions (17,18). These amount to polynomials over a set of variables that correspond to the identity of substituents at various substitution positions on a particular achiral parent molecule. The values of the variables can be adjusted so that the polynomial evaluates to a good fit to the experimentally measured molar rotations of a homologous series of compounds (2). Thus, properties 1 and 2 are satisfied, but the variables are qualitatively distinct for the same substituent at different positions or different substituents at the same positions, violating property 3. Furthermore, there is a different polynomial for each symmetry class of base molecule. Thus, chirality functions are not continuous functions of atom properties and conformation (property 4). 

A semiempirical approach has been developed by Solomon et al. (1992) and applied to the estimation of ODPs on both short and long time scales. The ODP of compound X is calculated from Eq. (A) (Solomon and Albritton, 1992) ... 

Equation (4.70) is a starting point in the determination of diffusivities in liquid metal alloys, but in most real systems, experimental values are difficult to obtain to confirm theoretical expressions, and pair potentials and molecular interactions that exist in liquid alloys are not sufficiently quantified. Even semiempirical approaches do not fare well when applied to liquid alloy systems. There have been some attempts to correlate diffusivities with thermodynamic quantities such as partial molar enthalpy and free energy of solution, but their application has been limited to only a few systems. 

The Halpin-Tsai equations represent a semiempirical approach to the problem of the significant separation between the upper and lower bounds of elastic properties observed when the fiber and matrix elastic constants differ significantly. The equations employ the rule-of-mixture approximations for axial elastic modulus and Poisson s ratio [Equations. (5.119) and (5.120), respectively]. The expressions for the transverse elastic modulus, Et, and the axial and transverse shear moduli, Ga and Gf, are assumed to be of the general form... 

Ab Initio and Semiempirical Approaches to Donor-Acceptor Interactions in r-Stacks... 

In general, the semiempirical approach has been widely adopted in inorganic systems, and the most common approximation is either to assume that Hu is directly proportional to the overlap integral Sij9 or related to it by a Wolfsberg-Helmholz type of formula (43) ... 

In addition to the VSEPR theory8 mentioned above, other theoretical or semiempirical approaches have addressed the problem of the positions occupied by various ligands as a function of their nature, for comparison with the numerous experimental results now available molecular orbital calculations, (four-electron, three-centre model with neglect of the P d orbitals)18,19 semiempirical calculations20,21 non-empirical calculations22,23 and hybrid orbitals24,25. 

In this connection, it is very interesting that the volume and intrachain changes obtained by various experimental methods 24,29,85) [Eq. (101)] agree well with Eq. (56) following from the Tobolsky-Shen semiempirical equation of state or the related phenomenological Eq. (76). The values of y determined from the data are rather small (0.1-0.3). As has been mentioned above, according to the semiempirical approach by Tobolsky and Shen one can formally suggest that the front-factor in Eq. (28) is pressure dependent. If it is really so, then the parameter y for rubbers can be considered as an experimental coefficient similar to the coefficient of thermal expansion and compressibility 29). 

A linear correlation between 13C chemical shifts and local n electron densities has been reported for monocyclic (4n + 2) n electron systems such as benzene and nonbenzenoid aromatic ions [76] (Section 3.1.3, Fig. 3.2). In contrast to theoretical predictions (86.7 ppm per n electron [75]), the experimental slope is 160 ppm per it electron (Fig. 3.2), so that additional parameters such as o electron density and bond order have to be taken into account [381]. Another semiempirical approach based on perturbational MO theory predicts alkyl-induced 13C chemical shifts in aromatic hydrocarbons by means of a two-parameter equation parameters are the atom-atom polarizability nijt obtained from HMO calculations, and an empirically determined substituent constant [382]. 

Since the Eq must describe states ranging from the ideal gas to the dense compressed state, the G factor must reduce to the virial expansion at low density and must approach the value determined by the repulsive potential at the high density limit. Dr. Jacobs took a semiempirical approach to this problem and used the results of Monte Carlo (MC) and Lennard-Jones Devonshire (LJD) calculations to determine unknown parameters in theoretical expressions for p0(v) and G(v,T). ... 

Laricchia, G. and Wilkin, C. (1997). Semiempirical approach to positron annihilation in molecules. Phys. Rev. Lett. 79 2241-2244. 

Cottet, H., and Gareil, P. (2000). From small charged molecules to oligomers A semiempirical approach to the modeling of actual mobility in free solution. Electrophoresis 21, 1493-1504. 

We will take a semiempirical approach using numerous molecules, models, assumptions, and estimates rather than doing new calorimetric experiments and/or quantum chemical calculations. Indeed, we will also test what is probably the simplest assumption - that (4n + 2) n electrons found within a conjugated ring species is expected to result in enhanced stability and that this compound is called aromatic. We will consider the dihydroindene (indane) skeleton composed of a benzene ring fused to a nonaromatic five-membered ring that lacks additional double bonds, and will use this carbocyclic hydrocarbon with X = Y = Z = CH2 as a paradigm for many heterocyclic derivatives for which the possible aromaticity is of relevance to the current chapter. Similarly we use indene with -X-Y- = -CH=CH-, Z = CH2 for a variety of unsaturated heterocycles of interest here. 

Density functional calculations are usually faster than ab initio, but slower than semiempirical. DFT is relatively new (serious DFT computational chemistry goes back to the 1980s, while computational chemistry with the ab initio and semiempirical approaches was being done in the 1960s). 

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