THEORETICAL METHODS CALCULATIONS

For unsubstituted aromatic hydrocarbons all the carbon atoms are assigned the same Coulomb integral (a°) and all C—C bonds are assigned the same resonance integral 0°). In heteroaromatic molecules the approximate Coulomb integral for heteroatom ax is expressed as in Eq. (1) in terms of a0 and (3° and the electronegativity parameter h. 

Resonance integrals of bonds between atoms X and Y, XY, are expressed as defined in Eq. (2), where kXY depends on the bond length. There has been considerable variation in the values taken for the Coulomb and resonance integrals for heterocyclic molecules. One of the best available set of parameters is still that originally suggested by A. Streitwieser (Molecular orbital theory. J. Wiley Sons, Inc., N.Y.-L., 1961)  

In this notation, heteroatoms which contribute one and two ir-electrons to the aromatic system are designed accordingly. 

A semi-empirical ir-electron theory that takes electron repulsion into account is the Pariser-Parr-Pople (PPP) method (R.G. Parr. Quantum Theory of Molecular Electronic Structure, Benjamin, N.Y., 1963). 

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Initially, most theoretical methods calculated the properties of molecules in the gas phase as isolated species, but chemical reactions are most often carried out in solution. Biochemical reactions normally take place in water. Consequently, there is increasing interest in methods for including solvents in the calculations. In the simplest approach, solvents are treated as a continuum, whose average properties are included in the calculation. Explicit inclusion of solvent molecules in the calculation greatly expands the size of the problem, but newer approaches do this for at least those solvent molecules next to the dissolved species of interest. The detailed structures and properties of these solvent molecules affect their direct interaction with the dissolved species. Reactions at catalytic surfaces present an additional challenge, as the theoretical techniques must be able to handle the reactants and the atoms in the surface, as well as possible solvent species. The first concrete examples of computationally based rational catalyst design have begun to appear in publications and to have impact in industry. 

The data collected are subjected to Fourier transformation yielding a peak at the frequency of each sine wave component in the EXAFS. The sine wave frequencies are proportional to the absorber-scatterer (a-s) distance /7IS. Each peak in the display represents a particular shell of atoms. To answer the question of how many of what kind of atom, one must do curve fitting. This requires a reliance on chemical intuition, experience, and adherence to reasonable chemical bond distances expected for the molecule under study. In practice, two methods are used to determine what the back-scattered EXAFS data for a given system should look like. The first, an empirical method, compares the unknown system to known models the second, a theoretical method, calculates the expected behavior of the a-s pair. The empirical method depends on having information on a suitable model, whereas the theoretical method is dependent on having good wave function descriptions of both absorber and scatterer. 

Although the recently observed discrepancy between the transition state structure predicted by theory and by interpreting the experimental KIEs using the traditional methods appears to favor the transition state structure predicted by theory, it is worth noting that none of the 39 theoretical methods calculated all six KIEs that were measured for the ethyl chloride-cyanide ion reaction within the experimental error. Obviously there is a need for further investigations of the relationship between observed KIEs and transition state structure. 

Figure Al.3.16. Reflectivity of silicon. The theoretical curve is from an empirical pseudopotential method calculation [25], The experimental curve is from [31],...

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

Let us illustrate this with the example of the bromination of monosubstituted benzene derivatives. Observations on the product distributions and relative reaction rates compared with unsubstituted benzene led chemists to conceive the notion of inductive and resonance effects that made it possible to explain" the experimental observations. On an even more quantitative basis, linear free energy relationships of the form of the Hammett equation allowed the estimation of relative rates. It has to be emphasized that inductive and resonance effects were conceived, not from theoretical calculations, but as constructs to order observations. The explanation" is built on analogy, not on any theoretical method. 

Computer graphics has had a dramatic impact upon molecular modelling. It should always be remembered, however, that there is much more to molecular modelling than computer graphics. It is the interaction between molecular graphics and the imderlying theoretical methods that has enhanced the accessibility of molecular modelling methods and assisted the analysis and interpretation of such calculations. 

Theoretical methods ranging from the now obsolete HMO studies to ab initio calculations have been used extensively on pyrazoles. Although not emphasized in earlier reviews (66AHC(6)347,67HC(22)l), the most recent publications (B-76MI40402,79RCR289) contain several references to theoretical studies. Some publications related to structural studies are to be found in the following sections, especially in connection with NMR spectroscopy (Section 4.04.1.3.4), UV spectroscopy (Section 4.04.1.3.6), PE spectroscopy (Section 4.04.1.3.9) and tautomerism (Section 4.04.1.5). 

On the other hand, theoretical methods allow an insight into the structure of non-existent molecules like 2//-indazole (37) or the anion of indazole (38). INDO calculations have been performed by Palmer et al. on the anion of indazole (38) (75JCS(P1)1695). The optimized geometry obtained by them is shown in Figure 7. The N—N bond distance is longer in the... 

In this chapter, we will consider the other half of a model chemistry definition the theoretical method used to model the molecular system. This chapter will serve as an introductory survey of the major classes of electronic structure calculations. The examples and exercises will compare the strengths and weaknesses of various specific methods in more detail. The final section of the chapter considers the CPU, memory and disk resource requirements of the various methods. 

The table on the next page indicates the relationship between problem size and resource requirements for various theoretical methods. Problem size is measured primarily as the total number of basis functions (N) involved in a calculation, which itself depends on both the system size and the basis set chosen some items depend also on the number of occupied and virtual (unoccupied) orbitals (O and V respectively), which again depend on both the molecular system and the basis set. The table lists both the formal, algorithmic dependence and the actual dependence as implemented in Gaussian (as of this writing), which may be somewhat better due to various computational techniques... 

Optimize the structure of acetyl radical using the 6-31G(d) basis set at the HF, MP2, B3LYP and QCISD levels of theory. We chose to perform an Opt Freq calculation at the Flartree-Fock level in order to produce initial force constants for the later optimizations (retrieved from the checkpoint file via OptsReadFC). Compare the predicted spin polarizations (listed as part of the population analysis output) for the carbon and oxygen atoms for the various methods to one another and to the experimental values of 0.7 for the C2 carbon atom and 0.2 for the oxygen atom. Note that for the MP2 and QCISD calculations you will need to include the keyword Density=Current in the job s route section, which specifies that the population analysis be performed using the electron density computed by the current theoretical method (the default is to use the Hartree-Fock density). 

Calculation of thertnochetnical quantities like those we have just considered are a widely-used method for evaluating the accuracy of theoretical methods and models. In this section, we will look at the Gaussian-2 molecule set and then consider how well a variety of model chemistries perform on it. Note that our consideration of the G2 method itself will come later in this chapter. 

Chapter 1, Computational Models and Model Chemistries, provides an overview of the computational chemistry field and where electronic structure theory fits within it. It also discusses the general theoretical methods and procedures employed in electronic structure calculations (a more detailed treatment of the underlying quantum mechanical theory is given in Appendix A). 

The performance is (as expected) very good. MMX provides relative (and absolute) stabilities with a MAD of only 1.2 kcal/mol, which is better than the estimates from the combined theoretical methods in Table 11.31. Considering that force field calculations require a factor of 10 less computer time for these systems than the ab initio methods combined in Table 11.31, this clearly shows that knowledge of the strengths and weakness of different theoretical tools is important in selecting a proper model for answering a given question. 

Tliis interpretation is based only upon the structural and electronic properties of the pyridinium cations. Tire calculation of relative activation Ijarri-ers for the competing substitution reactions will give more reliable results —especially if solvent effects are included in the calculations. In order to assess the reliability of actual theoretical methods as applied to model sys-... 

The first systematic theoretical study on dihydro-1,2,4-triazines was recently carried out (98JOC5824) the stabilities of all the possible unsubstituted dihydro-1,2,4-triazines were calculated using various theoretical methods, all reliable calculation methods consistently show that the 2,5-dihydro isomer 98 is the most stable. This is in perfect agreement with the experimental observations all the synthetic methods used for the preparation of dihydro-1,2,4-triazines result in 2,5-dihydro isomer 98, provided the structures of the reactants and the reaction mechanism allow its formation. Thus, although Metze and Scherowsky (59CB2481) claimed the formation of 1,2-dihydro-1,2,4-triazine 92 (R = = Ph) in the reduction... 

This chapter will try to cover some developments in the theoretical understanding of metal-catalyzed cycloaddition reactions. The reactions to be discussed below are related to the other chapters in this book in an attempt to obtain a coherent picture of the metal-catalyzed reactions discussed. The intention with this chapter is not to go into details of the theoretical methods used for the calculations - the reader must go to the original literature to obtain this information. The examples chosen are related to the different chapters, i.e. this chapter will cover carbo-Diels-Alder, hetero-Diels-Alder and 1,3-dipolar cycloaddition reactions. Each section will start with a description of the reactions considered, based on the frontier molecular orbital approach, in an attempt for the reader to understand the basis molecular orbital concepts for the reaction. 

We also showed that the photoemission intensities of such a complex system as CusPts on a platinum substrate can be calculated in good agreement with experiments and this renders confidence into the power of the theoretical methods and the underlying principles. 

Pd4oCu4oP2o, Pd5oCu3oP2o, and Pd6oCu2oP20 alloys were measured by resonant ultrasound spectroscopy (RUS). In this technique, the spectrum of mechanical resonances for a parallelepiped sample is measured and compared with a theoretical spectrum calculated for a given set of elastic constants. The true set of elastic constants is calculated by a recursive regression method that matches the two spectra [28,29]. 

During the last decade, the progress in theoretical methods and the access to quantum-chemical calculations has become more available. Nowadays, the use of quantum-chemical calculations in the interpretation of experimental UPS valence band spectra is a common approach [26-29]. 

Only the structures of di- and trisulfane have been determined experimentally. For a number of other sulfanes structural information is available from theoretical calculations using either density functional theory or ab initio molecular orbital theory. In all cases the unbranched chain has been confirmed as the most stable structure but these chains can exist as different ro-tamers and, in some cases, as enantiomers. However, by theoretical methods information about the structures and stabilities of additional isomeric sul-fane molecules with branched sulfur chains and cluster-like structures was obtained which were identified as local minima on the potential energy hypersurface (see later). 

By ab initio MO and density functional theoretical (DPT) calculations it has been shown that the branched isomers of the sulfanes are local minima on the particular potential energy hypersurface. In the case of disulfane the thiosulfoxide isomer H2S=S of Cg symmetry is by 138 kj mol less stable than the chain-like molecule of C2 symmetry at the QCISD(T)/6-31+G // MP2/6-31G level of theory at 0 K [49]. At the MP2/6-311G //MP2/6-3110 level the energy difference is 143 kJ mol" and the activation energy for the isomerization is 210 kJ mol at 0 K [50]. Somewhat smaller values (117/195 kJ mor ) have been calculated with the more elaborate CCSD(T)/ ANO-L method [50]. The high barrier of ca. 80 kJ mol" for the isomerization of the pyramidal H2S=S back to the screw-like disulfane structure means that the thiosulfoxide, once it has been formed, will not decompose in an unimolecular reaction at low temperature, e.g., in a matrix-isolation experiment. The transition state structure is characterized by a hydrogen atom bridging the two sulfur atoms. 

Recent theoretical studies have demonstrated that it is possible to calculate accurately adsorbate stmcture and energy levels, to explain trends with variations in metal composition, and to interpret and predict the influence of promoters and poisons on the adsorption of reactants. Additional efforts along these lines will contribute greatly to understanding how catalyst stmcture and composition influence catalyst-adsorbate interactions and the reactions of adsorbed species on a catalyst surface. With sufficient development of theoretical methods, it should be possible to predict the desired catalyst composition and stmcture to catalyze specific reactions prior to formulation and testing of new catalysts. 

Table 1. The 72-atom model examined by different theoretical methods. The energy differences (AE in kcal/mol) are calculated with respect to the lowest SCF energy. q(Fe) stands for Mulliken population charges on the Fe atoms q(S) and SS(b.i.) are the Mulliken population charges and the bond index for the bridging S atoms, respectively AEq is the calculated Mossbauer quadrupole splitting constant [mm/sec]. The PUHF spin states are those projected from the UHF wavefunction with 5 = 5,. 

Theoretical Methods and Results for Electronic Structure Calculations on Very Large Systems... 

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