Theoretical methods

Early theoretical models were based on fractional energy loss 2m/M per elastic collision (for details, see LaVeme and Mozumder, 1984, Sect. 3, and references therein). Thus, frequently, the energy loss rate was written as —d (E)/dt = (2m/M)((E)-3feBT/2)vc, where vc is the collision frequency and (E) is the mean electron energy over an unspecified distribution. The heuristic inclusion of the term 3feBT/2 allowed the mean energy to attain the asymptotic thermal 

Two other attempts, without the use of a distribution function, are worth mentioning, as these are operationally related to experiments and serve to give a rough estimate of the thermalization time. Christophorou et al. (1975) note that in the presence of a relatively weak external field E, the rate of energy input to an electron by that field is (0 = eEvd, where vd is the drift velocity in the stationary state. Under equilibrium, it must be equal to the difference between the energy loss and gain rates by an electron s interaction with the medium. The mean electron energy is now approximated as (E) = (3eD )/(2p), where fl = vd /E is the drift mobility and D is the perpendicular diffusion coefficient (this approximation is actually valid for a Maxwellian distribution). Thus, from measurements of fl and D the thermalization time is estimated to be 

The thusly-obtained thermalization time depends weakly on the initial energy, for which a value 1 eV has been used in the irradiation case. Taking n = 1 gives T(h = 3.0, 1.5, and 0.5 ns respectively for LXe, LKr, and LAr and the values 10.0, 0.9, and 0.6 ps respectively for methane, neopentane, and tetram-ethylsilane, all liquids at their triple points. In these estimates, Schmidt s (1977) data were used for ng and E10. However, taking n = 1 can be very crude, as certain theories and experiments give n = -0.5. On the other hand, the use of 10% nonlinearity of mobility may seem arbitrary, but it has partial compensation in the definition of E10. 

FIGURE 8.1 Evolution of effective electron temperature (T) in helium at 290 K vs. density-normalized time. Reproduced from LaVerne and Mozumder (1984), with the permission of Elsevier . 

At long times the excess temperature, (T) - T, decays exponentially, as can be shown from the preceding equation. The relaxation rate has independent, additive contributions from momentum transfer collisions (as in the case of rare gases) and from each pair of states connected by inelastic collision. Thus the net relaxation rate is given by 

Theoretical methods ranging from the simple Hiickel (HMO) method to ab initio calculations have been used extensively to study the title compounds 84CHEC-i(4)i037, 84CHEC-I(6)1027 . The subject of these studies was mainly the molecules (36)—(39), which cannot be represented by classical Kekule structures. These heterocyclic systems can be represented by dipolar structures, or by 

Gimarc (83JA1979,86JA4303) proposed a rule of topological charge stabilization which states that heteroatoms prefer to be located at sites that conform to the pattern of relative electron densities determined by connectivity or topology in an isoelectronic, isostructural, homoatomic system that is called the uniform reference frame 83JA1979 . For the series of thienothiophene positional isomers, the pentalene dianion (5) serves as the uniform reference frame ((55) represents pentalene). 

The method of the local approach in combination with a PPP Hamiltonian has been used to investigate the many-particle character of 7c-electron bonding in all the four isomeric diaz-apentalenes, i.e., pyrrolo[3,2-Z ]pyrrole (56), pyrrolo[3,4-/ ]pyrrole (57), pyrrolo[2,3-Z ]pyrrole (58) and pyrrolo[3,4-c]pyrrole (44) 93JPC11427 . 

Ab initio and/or semiempirical (PM3) quantum-chemical calculation on the parent thieno[2,3-cjfuran (15) and furo[3,4-6]furan (20) systems in the ground and excited states have been performed 91CB2481 . 

Theoretical methods used for the calculation of BDE and RSE values in recent studies can be divided into two larger groups. The first corresponds to the class of density functional theory (DFT) methods. The most commonly used functional is the B3LYP 

A comparison of the stabihty values for aUyl radical (10) and benzyl radical (11) will be used here to illustrate the performance of these methods (Table 5.2). Earlier compilations of C-H BDE data put the aUyhc C-H bond in propene (369.0 8.8 kJ/ mol) and the benzylic C-H bond in toluene (370.3 6.3 kJ/mol) at almost identical values. Together with the C-H BDE in methane of438.9 0.4 kJ/mol this equates to RSE(3) values of -1-68.6 6.7 and 69.9 9.2 kJ/mol for the allyl and benzyl radical, respectively. New measurements of C-H bond strengths in the gas phase and in solution led to a shghtly larger difference with a C-H BDE in propene of 371.5 1.7 kJ/molandaC-HBDEin toluene of 375.7 2.5 kJ/mol.  

TABLE 5.2 RSEs at 298.15K Calculated According to Equation 5.3 of Allyl Radical (10) and Benzyl Radical (11) (in kj/mol) at Various Levels of Theory 

In combination with the unchanged C-H BDE in CH4 this equates to RSE(3) values of + 67.4 2.1 and +63.2 2.9 kJ/mol for the allyl and benzyl radicals, respectively. This implies a stability difference of these two systems of just over 4 kJ/mol. 

we review briefly the methods used most commonly for calculating molecules of heavy group 14 elements. More details about the methods can be found in Reference 25, which give an overall view of the field, and in the more specific references given in the discussion below. 

The use of computational methods in structural chemistry has changed focus in recent years, with less emphasis on the determination of absolute structure and greater emphasis on the application of these methods to explain topics such as conformation, tautomerism, and the source of any preferences for given forms. This change dominates the section below though the computations of some simple structures have been carried out. 

Density functional theory (DFT) calculations have also been used to calculate the position of tautomeric equilibria of heteroatom-substituted pyridines in both the gas-phase and under solvation conditions. For the former, in the case of the oxygen and sulfur substituents, the energy difference between the tautomers is reasonably small (in the order 

2 -bipyridine when compared to the trans-form this effect was attributed to the chelating effect of the two nitrogen atoms in that conformation 1998JST(427)87 . 

The half-way house between the accuracy of quantum mechanical methods and the speed of classical simulations belongs to semi-empirical calculations. These are more correctly classified as approximate quantum mechanical methods, meaning that while we still seek solutions to Schrodinger s equation, a wide series of approximations and parameterization schemes (often derived from experimental data) 

Structural Methods in Molecular Inorganic Chemistry, First Edition. David W. H. Rankin, Norbert W. Mitzel and Carole A. Morrison. 2013 John Wiley Sons, Ltd. Published 2013 by John Wiley Sons, Ltd. 

From this point, we restrict our discussion to quantum mechanical calculations. Quantum mechanics gives us electronic structure, and electronic strucmre, in effect, gives us chemistry. This approach therefore allows us to follow chemical reaction profiles that involve bond-making/breaking processes, and to calculate thermodynamic properties, along with the properties of molecular orbitals, electron densities, and just about any type of spectroscopic property you can think of. It is no small wonder, then, that computational chemistry is now an essential technique to aid and guide the interpretation of experimental results. 

Our discussions focus on the concepts behind quanmm mechanical modeling of gas- and solid-phase systems in the context of structural chemistry. Fuller accounts, including complete mathematical derivations, are available in [3-5]. 

In CHEC-I 84CHEC-l(3B)l , theoretical methods, particularly the calculation of bond lengths, bond angles, and electron densities were discussed alongside consideration of structure, and physical and spectroscopic properties. Theoretical aspects since 1984 are summarized below. 

Principal component analysis has been used to investigate aromaticity as a quantitative concept and revealed a dichotomy between classical and magnetic aromaticity 90JPR853, 90JPR870 . Principal component analysis has also been used to demonstrate a correlation between aromaticity and IR parameters obtained by ab initio 4-31G MO calculations 93JST(10l)97 . 

Ab initio calculations have been performed on the 1,4-dihydropyridazine nucleus which indicate considerable enamine character for this system 83JHC855 , and MO calculations have been carried out for different geometries of 1,2-dihydropyridazine at the 6-3IG level using structures optimised at the 4-31G level 89JST(60)17 . 

Whereas oxaziridine and diaziridine were partial subjects of comprehensive theoretical studies on cyclic compounds (73MI50800), diazirine and some of its simple derivatives were the special target of quantum chemical investigations. Since diazirine, the lowest molecular weight heterocycle, has only five atoms and is of high symmetry, there was a chance for ab initio calculations, which followed some semiempirical studies. 

In an early investigation (66T539) the two highest occupied and the two lowest unoccupied orbitals were calculated on the basis of an extended Hiickel theory to determine the electron transition responsible for the long wavelength UV absorption. An Ai- Bi, r— transition was discussed. 

Later there was an attempt by ab initio calculation to fit the electron structure of diazirine into the Walsh model of cyclopropane (69MI50800). According to these SCF-LCAO-MO calculations three MOs add to the description of the lone electron pairs, all of which also contribute to some extent to ring bonding. As to strain, 7r-character and conjugative effect, the term pseudo-rr-character was used. 

There was less agreement between calculated and experimental energy values. The use of 6-3IG, the best procedure in energy calculations of three-membered rings, yielded a value too low by more than 40 kJ moF in the case of diazirine bond separation energy was calculated as -45 kJ moF the experimental value is +0.4 kJ moF . Vibrational correction and extrapolation to 0 K would reduce this difference by several kJ moF . 

The heat of formation (332 kJ moF ) was calculated to be 376 kJ moF Since a correction of about 35 kJ moF is not unusual in calculations of three-membered rings, a corrected value of 340 kJ moF appears reasonable. 

In Sect. 2 we hence give only a rather selective overview of theoretical methods and results - in such a rapidly developing field with still many controversial aspects the selection clearly is biased by the knowledge and interests of the author, but it is hoped that this survey nevertheless is a useful introduction to the field, and should stimulate further developments. The selection of experimental results that will be discussed in Sect. 3 again cannot aim at an authorative review of all work that has been done in the field, but should be rather viewed as an introduction again, based on some key examples. In Sect. 4, a brief summary will be given. 

However, before any simulations can be undertaken one needs to describe the interactions between, for example, the Ce and O atoms, coined the forcefield or potential model , and also a way of moving the atoms into a low-energy configuration (theoretical method). 

There is a wealth of functional forms to describe the interactions between ions, which generally vary according to the class of material metal, semiconductor, insulator, ionic or covalent. Ceria is an ionic material and is well suited to being described using formal charges, which are balanced by repulsive interactions, the latter associated with neighbouring electron clouds. A popular functional form used to describe the interaction between ions i and j, is the Born model of an ionic solid, which can be written  

Theoretical methods may take one of two forms. The first, such as molecular dynamics (MD), allows atoms to move in time into different (possibly low-energy) positions. The other type is stochastic in that mathematical algorithms are used to determine low-energy ( stable ) configurations for the atoms. 

In molecular dynamics the forces acting upon the atoms are calculated and used to determine the acceleration, velocity and positions of all the atoms (according to Newtonian mechanics). Accordingly, one is able to simulate the system as it evolves as a function of time.  

Once we have chosen a model potential to describe our system of real particles, e.g., Lennard-Jones particles, the functional equation is viewed as an equation involving g R) as the unknown function. This equation can be solved by either analytical or numerical methods. One version of the PY equation is 

Equation (5.5.9) is an integral equation for y, provided we have chosen a pair potential U. Once we have solved (5.5.9) for y, we can compute g through (5.5.10). 

As an illustration of the content of the integral equation, let us substitute on the rhs of (5.5.9) y which is the solution for y at extremely low densities, p- 0. (See also the discussion in section 5.3.) This is also a common first step in an iterational procedure for solving Eq. (5.5.9)  

A detailed treatment of the theoretical approach used in treating LSV and CV boundary value problems can be found in the monograph by MacDonald [23], More specific information on the numerical solution of integral equations common to electrochemical methods is available in the chapter by Nicholson [30]. The most commonly used method for the calculation of the theoretical electrochemical response, at the present time, is digital simulation which has been well reviewed by Feldberg [31, 32], Prater [33], Maloy [34], and Britz [35]. 

CHEC-I covered the literature to 1982. The second edition covers the literature from 1982 to early 1994. This chapter relates to Chapter 4.11 in CHEC-1. Section 4.01.3 (Structural Methods) in CHEC-1 now becomes two different Sections (4.01.1.4 Theoretical Methods and 4.01.2 Experimental Structural Methods). Section 4.01.3 (Thermodynamic Aspects) is new. Sections 4.01.5-4.01.8 relate to 4.01.4 in CHEC-1. Sections 4.01.9 and 4.01.9 correspond to 4.01.4 in CHEC-T. Section 4.01.10 is new and 4.01.11 relates to 4.01.6 in CHEC-T. 

The following are the most important reviews on 1,2,3-triazoles and benzotriazoles since 1982  

This chapter covers literature published since 1982 early materials are included here only where needed as a basis for describing further developments or where not previously mentioned in CHEC-1 84CHEC-l(5)669 . The structural types surveyed here include 1,2,3-triazoles, benzotriazoles, their dihydro derivatives and carbocyclic fused compounds. Compounds with heterocyclic fused rings are not included. The nomenclature system was discussed in CHEC-T 84CHEC-I(5)670 . 

Current computing resources often limit the range of properties and reactions that can be modeled on semiconducting mineral surfaces. Most molecular modeling applications are restricted to address properties and processes at a thin near-surface region of the material. Here, some flexibility in the number of atoms to be considered is available based on the particular semiconductor and the property being addressed. Although the focus of this review is on ab initio methods, it should be mentioned that molecular mechanics calculations (interactions based on parameterized potentials) have the ability to treat certain problems at semiconductor surfaces (see Rustad, this volume). For this application, the accuracy can be expected to depend on whether or not the 

In this chapter, the most important quantum-mechanical methods that can be applied to geological materials are described briefly. The approach used follows that of modern quantum-chemistry textbooks rather than being a historical account of the development of quantum theory and the derivation of the Schrodinger equation from the classical wave equation. The latter approach may serve as a better introduction to the field for those readers with a more limited theoretical background and has recently been well presented in a chapter by McMillan and Hess (1988), which such readers are advised to study initially. Computational aspects of quantum chemistry are also well treated by Hinchliffe (1988). 

In the section that follows this introduction, the fundamentals of the quantum mechanics of molecules are presented first that is, the localized side of Fig. 1.1 is examined, basing the discussion on that of Levine (1983), a standard quantum-chemistry text. Details of the calculation of molecular wave functions using the standard Hartree-Fock methods are then discussed, drawing upon Schaefer (1972), Szabo and Ostlund (1989), and Hehre et al. (1986), particularly in the discussion of the agreement between calculated versus experimental properties as a function of the size of the expansion basis set. Improvements on the Hartree-Fock wave function using configuration-interaction (Cl) or many-body perturbation theory (MBPT), evaluation of properties from Hartree-Fock wave functions, and approximate Hartree-Fock methods are then discussed. 

The focus then shifts to the delocalized side of Fig. 1.1, first discussing Hartree-Fock band-structure studies, that is, calculations in which the full translational symmetry of a solid is exploited rather than the point-group symmetry of a molecule. A good general reference for such studies is Ashcroft and Mermin (1976). Density-functional theory is then discussed, based on a review by von Barth (1986), and including both the multiple-scattering self-consistent-field method (MS-SCF-ATa) and more accurate basis-function-density-functional approaches. We then describe the success of these methods in calculations on molecules and molecular clusters. Advances in density-functional band theory are then considered, with a presentation based on Srivastava and Weaire (1987). A discussion of the purely theoretical modified electron-gas ionic models is 

Qualitative molecular-orbital theory approaches (and related qualitative treatments) are discussed throughout the text (particularly in Chapters 4 and 6), and a more detailed discussion of the contributions of such approaches presented in Chapter 8. As with the experimental methods discussed in Chapter 2, the topics presented in the present chapter are associated with numerous abbreviations and acronyms (and alternative titles). Both to serve as a key to these abbreviations, and as a source of reference to the numerous theoretical approaches now available, they are listed along with brief descriptions and references to further information in Appendix C. 

Theoretical and structural studies have been briefly reviewed as late as 1979 (79AHC(25)147 (discussed were the aromaticity, basicity, thermodynamic properties, molecular dimensions and tautomeric properties ) and also in the early 1960s 63AHC(2)365, 62HC(17)1, p. 117). Significant new data have not been added but refinements in the data have been recorded. Tables on electron density, density, refractive indexes, molar refractivity, surface data and dissociation constants of isoxazole and its derivatives have been compiled (62HC(17)l,p. 177). Short reviews on all aspects of the physical properties as applied to isoxazoles have appeared in the series Physical Methods in Heterocyclic Chemistry (1963-1976, vols. 1-6). 

Structure of Five-membered Rings with One Heteroatom 

The much less sophisticated PPP approximation has been shown to well reproduce the electronic spectral features not only of the monocyclic furan, pyrrole, thiophene, selenophene and tellurophene but also many of the benzo fused derivatives as well (79MI30101, 68JPC3975, 68MI30100). 

All indications are that computation facilities (better programs, faster computers) will continue to progress at a steady pace and render such a book easier to write and more comprehensive. However, there are two neg- 

The second aspect is more fundamental. It is related to the very nature of chemistry (quantum chemistry is physics). Chemistry deals with fuzzy objects, like solvent or substituent effects, that are of paramount importance in tautomerism. These effects can be modeled using LFER (Linear Free Energy Relationships), like the famous Hammett and Taft equations, with considerable success. Quantum calculations apply to individual molecules and perturbations remain relatively difficult to consider (an exception is general solvation using an Onsager-type approach). However, preliminary attempts have been made to treat families of compounds in a variational way [81AQ(C)105]. 

Since the domain explored will always be a very small part of the possible cases of tautomerism, it is essential to have general rules for families of compounds, substituents, and solvents. This chemical approach is maintained in this chapter, although the importance of the calculations is recognized. The following discussion begins with calculation of tautomeric equilibrium constants, followed by the combined use of theoretical calculations and experimental results (an increasingly expanding field) and ends with the calculations of the mechanisms of proton transfer between tautomers. 

Bicyclic 5-5 Systems with One Bridgehead (Ring Junction) Nitrogen Atom No Extra Heteroatom 

Schmitz et al. 2001JP090 developed two group increment schemes for converting HF/6-31G(d) and B3LYP/6-31G(d) calculated energies of aliphatic amines to estimations of heats of formation. Application to the pyrrolizidine yielded calculated values of —1.09 (HF) and —1.02 (B3LYP) instead of — 0.93 kcal mob1 (experimental). 

Next we may consider the various quantum mechanical procedures that have been used to calculate potential surfaces for organic reactions. 

If it could be shown that ab initio SCF calculations were effective in at least certain connections, they would of course present obvious advantages in that they are based on a rigorous solution of a specific mathematical problem and so involve no parameters. Consequently they can be applied equally well to systems of all kinds, containing any elements. Semiempirical treatments are limited to systems for which parameters have been determined. Even if computation time presents an inseparable barrier to ab initio treatments of systems large enough to be of chemical interest, such calculations for simpler systems might prove useful as an aid in developing semiempirical treatments. 

At the opposite extreme from the ab initio SCF methods is the Wolfberg-Helmholtz approximation which Hoffmann 6 has applied extensively to organic problems under the term extended Hiickel method . While this has the advantage of requiring very little computation time, the results are so unreliable that the method is essentially useless for the calculation of potential surfaces. Not only are the errors in heats of atomization comparable with those given by ab initio SCF but they are not even the same for isomers. A good example is provided by cyclopropanone (1) which is predicted 7 to be less stable than the isomeric zwitterion 2, a result at variance with the available evidence ) concerning the 

This semiempirical treatment (CNDO = Complete Neglect of Differential Overlap), introduced by Pople et al. 9 is derived from the full Roothaan 3 LCAO SCF MO treatment by making the following approximations )  

Since this treatment is parametrized to mimic the results of ab initio calculations, it is not surprising to find that it gives equally inaccurate estimates of heats of atomization. The errors in bond lengths and bond angles are also greater, while force constants are in error by a factor of two or three. While the computation time required is much less than for the ab initio methods, 

Molecular mechanics calculations on two 6-arylpyrrolo[2,l-d][l,5]benzothia-zepines (1995JMC4730) confirmed that binding to a mitochondrial benzodiazepine receptor depends on conformational strain as well as on specific repulsive interactions involving their side-chains. 

Semiempirical molecular orbital calculations performed for a set of pyrrolo-benzodiazepines using MNDO and AMI were used for the interpretation of their mass-spectroscopic data (1996MI653). 

The TT-electron excess of the five-membered rings is accompanied by a high rr-donor character. The best measure of rr-donation is the value of first ionization potential, IP, which for all aromatic heterocycles with one heteroatom of pyrrole type reflects the energy of highest occupied rr-orbital. IP, values decrease in the sequence pyrrole indole carbazole furan benzo[/ ]furan dibenzofuran thiophene benzo[/ ]thiophene (Section 2.3.3.9, Tables 21 and 23). Thus, the more extensive the rr-system, the stronger is its electron donor ability. Furan and thiophene possess almost equal rr-donation, which is considerably lower than that of pyrrole. 

The relative importance of through-bond (hyperconjugative) and through-space (homoconjugative) interactions between the heteroatom and the double bond in 2,5-dihydro-furan, -thiophene and -pyrrole has been the subject of a CNDO/S study (76ZN(A)215). This analysis concluded that the proportion of through-space interaction increased from 19% in the dihydrofuran and 20% for dihydrothiophene to 83% for the dihydropyrrole (cf. Section 2.3.3.9). 

The ab initio STO-3G basis set has been used (after MNDO) to calculate electron densities, Mulliken Tr-overlap populations, and ionization potentials for pyrrole (85JCS(P2)97). As expected, the nitrogen atom has the highest electron density at 7.31, with the C-3 at 6.10 only slightly higher than the C-2 at 5.98. 

AMI studies for pyrrole, indole, and carbazole, and their A-deprotonated analogues have been reported (90JCS(P2)65, 90JCS(P2)i88l). Of note are the geometrical changes brought about by anion 

The strain in azetidines influences the tendency for ring formation enormously 1996CHEC-II(1B)507 . Within the homologous series of azaheterocycles, the tendency for cyclization is smallest for the nitrogen-containing four-membered ring (5 3 6 7 4). For some other studies on gas phase proton affinity and ab initio calculations on the azetidin-yl radical, CHEC-II(1996) 1996CHEC-II(1B)507 should be consulted. 

Ground-state conformations for cycloadditions involving 5 and 6 were analyzed by energy minimization of reactant conformations using MM2 force field to explain the effect on unsaturation in the stereoselectivity of cycloaddition 1998T7045 . 

4 Reaction of 2H-Azirines with Electrophiles and Metal-Induced Reactions 

A number of the theoretical issues dealing with monocyclic aziridines were discussed in CHEC-II(1996) and CHEC(1984) 1996CHEC-II(1A)1, 1984CHEC(7)47 . 

This effect is known as surface enhanced Raman scattering (SERS) [10-12]. The SERS effect has been widely investigated for various molecules adsorbed on rough metallic surfaces or on metallic clusters in colloids. Reviews on this topic can be found in Refs. [12-15]. The enhancement of normally Raman-active modes is a consequence of the enhancement of the electric field of the incoming and scattered radiation in the vicinity of a rough metal film upon coupling with the dipolar plasmon resonances in the metal clusters. This enhancement affects molecules located up to even 10 nm away from the metal surface [11, 12]. The enhancement factors are essentially determined by the electronic properties of the metal and by the morphology of the metal film. 

This is in contradiction to previous findings of Hirose et al. [18] and Kera et al. [19] who proposed the formation of a In4PTCDA complex. 

In order to assess the effect of a metal-PTCDA complex formation on the vibrational frequencies, theoretical calculations were preformed with the Gaus-sian 98 package on the I114PTCDA complex using the B3LYP functional and 

The most dramatic effect of the complex formation resides in the two-fold splitting of the breathing mode at 233 cm (see the elongation patterns in Fig- 

Frequeney ealculations performed with the same basis set and density functional methods in Gaussian 98 as in Ref. [4] but for a modified PTCDA mole- 

According to HMO as well as SCF calculations, electrophilic attack should occur at the 3-position of the pyrrolizine anion, a conclusion that fits experimental findings (see Section 8.01.4.2). 

A calculation using Hess and Schaad parameters on pyrrolizinones, yields resonance energies REPE = 0.0155/ for (2) and 0.0110 for (3) thus explaining the remarkable stability of these compounds 84ACH(37)l). 

We use a periodic supercell model based on the large unit cell (LUC) method [22] which is free from the limitations of different cluster models applicable mainly to ionic solids, e.g., alkali halides. The main computational equations for calculating the total energy of the crystal within the framework of the LUC have been given in Refs. [22-24]. Here, we shall outline some key elements of the method. The basic idea of the LUC is in computing the electronic structure of the unit cell extended in a special manner at k = 0 in the reduced Brillouin zone (BZ), which is equivalent to a band structure calculation at those BZ k points, which transform to the reduced BZ center on extending the unit cell [22]. The total energy of the crystal is 

The total energy equation (2) is obtained after the introduction of the cut-off function . 

The theoretical method described above has been used successfully in investigations of various perfect and defective oxide crystals (see Refs. [24, 26-29] and references therein) as well as a number of ionic crystals and different semiconductors. 

A study comparing the bond orders and 7t-localization of pyrrole, indole, and isoindole has been performed 84KGS1497 . The calculated bond orders are summarized in Tables 2, 3 (SCF method), and 4 (CNDO/2 method), and the calculated atomic charges are listed in Tables 5 and 6. 

C-l(C-3) and C-4(C-7). It has been suggested that viewing the n system of indole as three weakly interacting systems (benzene, N lone pair, and 2—3 double bond) while taking isoindole as a true 10 n system explains the differences in stability and reactivity between indole and isoindole 84KGS1497 . 

A scaled AM 1 field has been used successfully to help identify the fundamental vibrational bands of indole 88JCP(88)7295 . Pyrrole and benzene were used to generate a set of scale factors which were then used to simulate a complete vibrational spectrum (included in Table 40, see Section 2.01.3.5) which in turn was used to identify the fundamental vibrations of indole, and to calculate the derived force constants. 

Although as we shall see there is much experimental work that bears on the particle size dependence of catalytic behaviour, its interpretation is speculative, and there is little that informs directly on how surface atoms differ from 

The molecular orbital approach to describing free valencies at metal surfaces (Section 1.23) has been extended to treat small particles.  

The purpose of the last sections was to create an overview of the changes which occur in the physical properties of metal particles as their sizes alter, and the ways in which these may be explained, in order to provide a basis for understanding particle size effects in chemisorption and catalysis. The complications that may arise from the having a binodal distribution of particles size have however always to be kept in mind. The main conclusions have been summarised by Burch in the following way. 

To this catalogue may be added the worry (7) that the size distribution may be binodal. 

It is usual but not necessarily sensible to regard geometric and electronic structures as quite separate things, whereas in reality they are closely connected. The form adopted by a metallic or bimetallic particle will depend on the bonds formed between the component atoms, that is to say, on its electronic structure and, because of the mobility of surface atoms, the properties of very small particles (dispersion 50%) will be dominated by electronic factors. 

In principle, these calculations could be performed by solving the Hartree-Fock equations with the modified Fock operator. 

In the case of long (infinite) chains, the difficulty arises that the component along the long (say z) axis of 

10 Magnetic, Electrical, and Mechanical Properties of Polymers 

The first calculations of the r-electron polarizabilities of long-chain molecules employed the free-electron model and simple Huckel theoryIn both cases the computed polarizability tensor elements were proportional to the cube of the length of the chain, a / P. In these calculations (and in the calculation of p by Zyss, who showed also a definite dependence of the elements of P on chain length), no electron-electron interaction was taken into account. In order to take Coulomb interaction at least partially into account one must perform Hartree-Fock calculations on the ab initio level. Such calculations would be expected to affect (most probably decrease somewhat) the dependence on /. 

Unfortunately, the elements of the polarizability and hyperpolarizability tensors are very sensitive to the choice of basis set and therefore high-quality basis sets are needed to obtain reliable results. It has been shown, however, on a series of large hydrocarbons, that if a minimal basis set is used one can obtain rather good agreement with experiment (in cases for which experimental data are available) if one scales the minimal basis results with respect to results obtained with better (double-C) basis sets computed for the first members of the hydrocarbon series. This is true, however, only for the physically interesting longitudinal component of a, namely a , and not for the other elements, owing to the lack of flexibility of the basis sets in the other two directions. 

A new and powerful approach to the problem of hydrogen bonding is also provided by the technique of proton resonance. 

Let us now consider the statistical problem involved in the treatment of solutions containing such molecules. 

There exists at present no satisfactory theory of strong orientational effects from which one may deduce the thermodynamical properties and especially the excess functions from intermolecular forces and properties of the pure components. The main difficulty is that in this case the rotational partition function is no longer independent of the translational partition function. 

The existing approximate methods fall into two classes  

Sample airangements of molecules on a plane square lattice if the energy of unlike pairs is (a) unfavourable, (b) favourable 

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The main theoretical methods have in connnon the detemiination of the stability of steady-state or other... 

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. 

We now discuss the most important theoretical methods developed thus far the augmented plane wave (APW) and the Korringa-Kolm-Rostoker (KKR) methods, as well as the linear methods (linear APW (LAPW), the linear miiflfm-tin orbital [LMTO] and the projector-augmented wave [PAW]) methods. 

Wang Y A and Carter E A 2000 Orbital-free kinetic-energy density functional theory Theoretical Methods in Condensed Phase Chemistry (Progress in Theoretical Chemistry and Physics Series) ed S D Schwartz (Boston Kluwer) pp 117-84... 

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography. 

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

PDB files were designed for storage of crystal structures and related experimental information on biological macromolecules, primarily proteins, nucleic acids, and their complexes. Over the years the PDB file format was extended to handle results from other experimental (NM.R, cryoelectron microscopy) and theoretical methods... 

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. 

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. 

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. 

The input to a minimisation program consists of a set of initial coordinates for the system. The initial coordinates may come from a variety of sources. They may be obtained from an experimental technique, such as X-ray crystallography or NMR. In other cases a theoretical method is employed, such as a conformational search algorithm. A combination of experimenfal and theoretical approaches may also be used. For example, to study the behaviour of a protein in water one may take an X-ray structure of the protein and immerse it in a solvent bath, where the coordinates of the solvent molecules have been obtained from a Monte Carlo or molecular dynamics simulation. 

Each of these tools has advantages and limitations. Ab initio methods involve intensive computation and therefore tend to be limited, for practical reasons of computer time, to smaller atoms, molecules, radicals, and ions. Their CPU time needs usually vary with basis set size (M) as at least M correlated methods require time proportional to at least M because they involve transformation of the atomic-orbital-based two-electron integrals to the molecular orbital basis. As computers continue to advance in power and memory size, and as theoretical methods and algorithms continue to improve, ab initio techniques will be applied to larger and more complex species. When dealing with systems in which qualitatively new electronic environments and/or new bonding types arise, or excited electronic states that are unusual, ab initio methods are essential. Semi-empirical or empirical methods would be of little use on systems whose electronic properties have not been included in the data base used to construct the parameters of such models. 

These last three theoretical methods have no real kinship to any of the major categories or precursors. The first is the one that has caused Strike more despair than any other. It was once a Top Ten recipe in the first edition. But it was a reckless gamble on Strike s part and Strike paid for it. Here is what Strike wrote last year... 

In the first chapter, devoted to thiazole itself, specific emphasis has been given to the structure and mechanistic aspects of the reactivity of the molecule most of the theoretical methods and physical techniques available to date have been applied in the study of thiazole and its derivatives, and the results are discussed in detail The chapter devoted to methods of synthesis is especially detailed and traces the way for the preparation of any monocyclic thiazole derivative. Three chapters concern the non-tautomeric functional derivatives, and two are devoted to amino-, hydroxy- and mercaptothiazoles these chapters constitute the core of the book. All discussion of chemical properties is complemented by tables in which all the known derivatives are inventoried and characterized by their usual physical properties. This information should be of particular value to organic chemists in identifying natural or Synthetic thiazoles. Two brief chapters concern mesoionic thiazoles and selenazoles. Finally, an important chapter is devoted to cyanine dyes derived from thiazolium salts, completing some classical reviews on the subject and discussing recent developments in the studies of the reaction mechanisms involved in their synthesis. 

TABLE I-1. SYNOPSIS OF VARIOUS THEORETICAL METHODS THI AZOLE... 

Although the techniques described have resulted in the determination of many protein stmctures, the number is only a small fraction of the available protein sequences. Theoretical methods aimed at predicting the 3-D stmcture of a protein from its sequence therefore form a very active area of research. This is important both to understanding proteins and to the practical appHcations in biotechnology and the pharmaceutical industries. 

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... 

THEORETICAL METHODS 4.17.2.1 Charge Densities and Bond Orders... 

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