Calculations theoretical

Several calculations on the electronic structure of pyridazine are avail-able. ° ° Calculations of the polarizability and rt-bond orders were 

600 j Tsuchiya, H. Arai, M. Hasebe, and H. Igeta, Chem. Pharm. Bull. 22, 2301 (1974). 

Ohsawa, K. Saiki, H. Igeta, A. Tsuji and T. Akimoto, Chem. Commun. 856 

Theoretical calculations are at present not used extensively in the study of isotope effects on CS. However, as this is a field that moves very quickly, new possibilities appear as shown at the end of this paragraph. 

Calculations in the nonadiabatic approximation are demonstrated by Vener [1, 59]. The fit is not as good as to be used in reality. 

Density functional theory (DFT) calculations can very easily be used to calculate nuclear shieldings. This is very useful for obtaining chemical shift information for tautomeric species that cannot be isolated. 

Recently, it has also been shown that deuterium isotope effects on chemical shifts can be correlated with OH stretching frequencies calculated in the B3YP/6-31G(d) approximation [60]. 

Very recently, Singleton has suggested a new way of calculating isotope effects illustrated by the bromonium ion [61]. This method could possibly also be used for tautomeric systems. Another new approach is the multicomponent molecular orbital method for direct treatment of nuclear quantum effects [62]. The basic idea is to incorporate the nuclear wave function and in particular the proton wave function directly into the electronic structure calculation. This approach has great potential but has so far been tested only for secondary isotope effects on chemical shifts [63]. The geometric isotope effect has also been looked into based on Pauling valence-bond orders [20]. 

Significant progress has been achieved in the theoretical calculation of these interactions. The most advanced theoretical approach to the problem relies upon the use of the Poisson-Debye equation for polarizable solutes of known structure embedded in a dielectric medium (Klapper et al., 1986 Sharp and Honig, 1990 Bashford and Karplus, 1990 Bajorath et al., 1991, Aqvist et al., 1991 Tidor and Karplus,1991 Sharp et al., 1992 Gilson, 1993 Loewenthal et al., 1993 Yang et al., 1993 Scott et al., 1994 Anni et al., 1994, Hechtetal., 1995 Honig and Nicholls, 1995  

For the classical treatment of electrostatic interaction in solution the Poison-Boltzman equation (PBS) is commonly used 

The physical basis of the second type of approach rests upon the effect of the local electrostatic potential upon dynamic interactions at encounters with charged quenching molecules resulting in fluorescence (phosphorescence) (Vogel et al., 1986 Anni et al., 1994) or between a stable radical, e.g. nitroxide, and another charged paramagnetic species (Likhtenshtein et al., 1972 Likhtenshtein, 1976, 1988,1993). In such cases, the relaxation parameters, i.e. the life-time of the fluorescence (phosphorescence) chromophore or spin-spin and spin-lattice relaxation rates of paramagnetic species are dependent upon the frequency of encounters, and, therefore, on local electrostatic fields 

Calculations of local charge Zx in the vicinity of a paramagnetic particle (such as the active site of metalloprotein or a spin label) colliding with a nitroxide or metallocomplex with known charge Zp can be carried out with the use of the Debye equation  

The equation 4.2 allows one to calculate the ZpZ product using the appropriate experimentally-measured log (k+/k°) or log (k7k°) value and an r value estimated for the encounter complex. From the value of ZxZp one can ascertain the Zx charge in the vicinity of the region of the paramagnetic complex encountered by the charged nitroxide spin-probe since the Zp charge of the nitroxide spin-probe is known. 

c-llfl,c-llfe diastereomer (21) [92JST(274)259], as is the O-outside cis conformation of the heterocycles for the r-4a,t-lla,t-lld isomer (24), which is consistent with qualitative experimental estimates (70T1217). It was calculated that for 21 the trans conformation is ca. 3.7 kcal mol more stable than the O-outside cis conformation. For the r-4a,c-lla,t-llh isomer (22) and the r-4a,t-lla,c-llh isomer (23), it was predicted that the O-inside ds conformations are ca. 1.7 kcal moF and 1.4 kcal mol , respectively, more stable than the respective trans conformations, which is the opposite of the experimentally estimated orders (70T1217). 

The lack of correlation between cyclization stereoselectivity and the steric energy of the products (calculated by the MMX force field) 

The G2 and G2(MP2) computational methods used here provide molecular energies with deviations from the experimental data similar, in many cases, to the 

5-bis(methylthio)thiophen, and all of their possible C-protonated forms, in 

CaMnOji calculated, orthorhombic a = 0.53298nm, (7 = 0.74837mn, c = 0.52828mn experimental, orthorhombic fl = 0.5281mn, = 0.74542mn, c = 0.52758mn 

The calculated stability order for CaMnOj is orthorhombic GeFeOj, 4H-, cubic, 2H, in agreement with the experimental findings. 

The calculated stability order is 4H, 2H, cubic so that the 4H structure is most stable, as found. 

The calculated stability order is 2H, 4H, cubic in agreement with experimental data. (Data from Spndena et al. (2007).) 

Many other examples of density functional calculations mirror this example. These give good descriptions of the electronic properties of ground-state solids but still remain less successful at accurately predicting excited state properties. 

The simplest model of the charge distribution that can be used for calculating the EFG tensor is the point charge model (PCM). In the PCM, the atoms in a solid or a molecule are treated as ions with fixed charge. This oversimplified approach often fails to produce the accurate EFGs because it does not consider the details of the charge distribution near the nucleus of interest [107]. 

The Zn EFG tensors of zinc-containing sohds have been calculated on isolated molecules and model clusters using Hartree—Fock and hybrid DFT methods [16,30,42—44,46,49-54,115-118] and directly on the periodic systems [21,22,31,43,55,119] with WIEN2k [21,22,119], CASTEP [31,43] and other codes [55] using periodic boundary conditions. The calculations were utilized to compare the predicted Zn EFGs with experimental data, assist in interpretation of the observed Zn spectra, confirm the reported crystal structures, refine the structural model proposed firom the powder XRD data and gain information on host—guest interactions. 

We have surveyed the apphcations of Zn SSNMR spectroscopy in the literature. The results are presented later and the selected examples are discussed. The data are also summarized in Table 1.1. 

Panels (e-h) showing the calculated excitation patterns. (Reproduced from Chizhik et al. [56]. Copyright (2011), with permission of Royal Society of Chemistry.) 

As it turns out, the T-2H form is dominant but the dipole moment is very small 0.218 D compared to the T-IH form 4.38D. It turned out to be necessary to synthesize both the tri labeled compound as well as the ND species. A combined microwave and electron diffraction study revealed that the ratio between the T-IH and T-2H species in the gas phase at room temperature is 1 1000. For a full account of all previous studies see Ref [57]. 

Density Functional calculations have now become very common and a number of entities such as structures, NMR chemical shifts and coupling constants, isotope effects on nuclear shielding (chemical shifts), IR frequencies, dipole moments, and energies can be calculated quite easily. The ones mentioned are useful in establishing tautomeric equilibria. Throughout the text some of these calculations have already been alluded to. 

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From downhole pressure drawdown and build-up surveys the reservoir permeability, the well productivity index and completion skin can be measured. Any deviation from previous measurements or from the theoretically calculated values should be investigated to determine whether the cause should be treated. 

Eddy currents and the magnetic flux that is associated to them are proportional to the radial distance of the coil center. The magnetic flux is proportional to the probe induction and consequently to the passing current. The theoretic calculation of this induction is given by the following equation ... 

Fig. VI-6. The force between two crossed cylinders coated with mica and carrying adsorbed bilayers of phosphatidylcholine lipids at 22°C. The solid symbols are for 1.2 mM salt while the open circles are for 10.9 roM salt. The solid curves are the DLVO theoretical calculations. The inset shows the effect of the van der Waals force at small separations the Hamaker constant is estimated from this to be 7 1 x 10 erg. In the absence of salt there is no double-layer force and the adhesive force is -1.0 mN/m. (From Ref. 66.)...

The reports were that water condensed from the vapor phase into 10-100-/im quartz or pyrex capillaries had physical properties distinctly different from those of bulk liquid water. Confirmations came from a variety of laboratories around the world (see the August 1971 issue of Journal of Colloid Interface Science), and it was proposed that a new phase of water had been found many called this water polywater rather than the original Deijaguin term, anomalous water. There were confirming theoretical calculations (see Refs. 121, 122) Eventually, however, it was determined that the micro-amoimts of water that could be isolated from small capillaries was always contaminated by salts and other impurities leached from the walls. The nonexistence of anomalous or poly water as a new, pure phase of water was acknowledged in 1974 by Deijaguin and co-workers [123]. There is a mass of fascinating anecdotal history omitted here for lack of space but told very well by Frank [124]. 

Small metal clusters are also of interest because of their importance in catalysis. Despite the fact that small clusters should consist of mostly surface atoms, measurement of the photon ionization threshold for Hg clusters suggest that a transition from van der Waals to metallic properties occurs in the range of 20-70 atoms per cluster [88] and near-bulk magnetic properties are expected for Ni, Pd, and Pt clusters of only 13 atoms [89] Theoretical calculations on Sin and other semiconductors predict that the stmcture reflects the bulk lattice for 1000 atoms but the bulk electronic wave functions are not obtained [90]. Bartell and co-workers [91] study beams of molecular clusters with electron dirfraction and molecular dynamics simulations and find new phases not observed in the bulk. Bulk models appear to be valid for their clusters of several thousand atoms (see Section IX-3). 

The numerical illustration given above is so highly idealized that any agreement with experiment could hardly be more than coincidental. A number of caveats have been mentioned, including the inadequacies of theoretical calculations, the various surface distortions and reconstructions expected, surface... 

On the other hand, in the theoretical calculations of statistical mechanics, it is frequently more convenient to use volume as an independent variable, so it is important to preserve the general importance of the chemical potential as something more than a quantity GTwhose usefulness is restricted to conditions of constant temperature and pressure. 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

There is no reason to doubt that the inequalities of section A2.5.4.5(e) are other than equalities. The equalities are assumed in most of the theoretical calculations of exponents, but they are confmned (within experimental error) by the experiments. 

Several instniments have been developed for measuring kinetics at temperatures below that of liquid nitrogen [81]. Liquid helium cooled drift tubes and ion traps have been employed, but this apparatus is of limited use since most gases freeze at temperatures below about 80 K. Molecules can be maintained in the gas phase at low temperatures in a free jet expansion. The CRESU apparatus (acronym for the French translation of reaction kinetics at supersonic conditions) uses a Laval nozzle expansion to obtain temperatures of 8-160 K. The merged ion beam and molecular beam apparatus are described above. These teclmiques have provided important infonnation on reactions pertinent to interstellar-cloud chemistry as well as the temperature dependence of reactions in a regime not otherwise accessible. In particular, infonnation on ion-molecule collision rates as a ftmction of temperature has proven valuable m refining theoretical calculations. 

Equation (B1.1.10) and equation (B1.1.11) are the critical ones for comparing observed intensities of electronic transitions with theoretical calculations using the electronic wavefiinctions. The transition moment integral... 

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

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. 

Fig. 11.6 The SC24/halide system. (Figure adapted from Lybrand T P, ] A McCammon and G Wipff 1986. Theoretical Calculation of Relative Binding Affinity in Host-Guest Systems. Proceedings of the National Al adeniy of Sciences USA 83 833-835.)...

As mentioned, we also carried out IR studies (a fast vibrational spectroscopy) early in our work on carbocations. In our studies of the norbornyl cation we obtained Raman spectra as well, although at the time it was not possible to theoretically calculate the spectra. Comparison with model compounds (the 2-norbornyl system and nortri-cyclane, respectively) indicated the symmetrical, bridged nature of the ion. In recent years, Sunko and Schleyer were able, using the since-developed Fourier transform-infrared (FT-IR) method, to obtain the spectrum of the norbornyl cation and to compare it with the theoretically calculated one. Again, it was rewarding that their data were in excellent accord with our earlier work. 

The discovery of a significant number of hypercoordinate carboca-tions ( nonclassical ions), initially based on solvolytic studies and subsequently as observable, stable ions in superacidic media as well as on theoretical calculations, showed that carbon hypercoordination is a general phenomenon in electron-deficient hydrocarbon systems. Some characteristic nonclassical carbocations are the following. 

According to early theoretical calculations Kloptnan and I carried out in 1971, the parent molecular ions of alkanes, such as CH4, observed in mass spectrometry, also prefer a planar hypercarbon structure. 

Acetic acid and other carboxylic acids are protonated in superacids to form stable carboxonium ions at low temperatures. Cleavage to related acyl cations is observed (by NMR) upon raising the temperature of the solutions. In excess superacids a diprotonation equilibrium, indicated by theoretical calculations, can play a role in the ionization process. 

Whereas the proton (H ) can be considered the ultimate Bronsted acid (having no electron), the helium dication (He ) is an even stronger, doubly electron-deficient eleetron aceeptor. In a theoretical, calculational study we found that the helionitronium trication (NOaHe" ) has a minimum structure isoelectronic and isostructural... 

Most of the envisioned practical applications for nonlinear optical materials would require solid materials. Unfortunately, only gas-phase calculations have been developed to a reliable level. Most often, the relationship between gas-phase and condensed-phase behavior for a particular class of compounds is determined experimentally. Theoretical calculations for the gas phase are then scaled accordingly. 

Numerous m.o.-theoretical calculations have been made on quinoline and quinolinium. Comparisons of the experimental results with the theoretical predictions reveals that, as expected (see 7.2), localisation energies give the best correlation. jr-Electron densities are a poor criterion of reactivity in electrophilic substitution the most reactive sites for both the quinolinium ion and the neutral molecule are predicted to be the 3-, 6- and 8-positions. ... 

X-ray analysis of 2-phenylmethylaraino-5-phenyl-A-2-thiazoline-4-one (176), which exists in the keto form in the solid state (420), and of 4,4 -diacetoxy-5,5 -dimethy]-2,2 bithiazolyl (177) (419) are available as model compounds for theoretical calculations (Scheme 92). 

The first empirical and qualitative approach to the electronic structure of thiazole appeared in 1931 in a paper entitled Aspects of the chemistry of the thiazole group (115). In this historical review. Hunter showed the technical importance of the group, especially of the benzothiazole derivatives, and correlated the observed reactivity with the mobility of the electronic system. In 1943, Jensen et al. (116) explained the low value observed for the dipole moment of thiazole (1.64D in benzene) by the small contribution of the polar-limiting structures and thus by an essentially dienic character of the v system of thiazole. The first theoretical calculation of the electronic structure of thiazole. benzothiazole, and their methyl derivatives was performed by Pullman and Metzger using the Huckel method (5, 6, 8). 

Table I-l lists the various theoretical treatments published on the thiazole molecule for each the type of approximation, the mode of parametrization. and, eventually, the geometry employed are given net charges and bond orders for various theoretical calculations are listed in Tables 1-2 and 1-3.

D. Comparison of Free-Radical Reactivity with Theoretical Calculations... 

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