Quantum resonances theory

Likewise, quantum mechanical calculation succeeds in giving a theoretical explanation of some facts that the resonance theory could not explain, for example, why bis(pyridine-2)monomethine cyanine and bis(pyridine-4)monomethine cyanine possess the same lowest energy transition contrary to the 2,2 - and 2,4 -quinoline monomethine dyes, together with a molecular coefficient extinction lower than that of the 4,4 -quinoline dye (11). Calculation shows also that there is no theoretical reason for observing a relationship between and pK in a large series of dyes with different nuclei as it has been postulated, even if limited observations and calculations in short homogeneous series could lead to this conclusion (105). 

The radical is much more stable if both stmctures exist. Quantum mechanical theory implies that the radical exists in both states separated by a small potential. Moreover, both molecular orbital theory and resonance theory show that the allyl carbocation is relatively stable. 

The error in Hiickel s treatment lies not in the quantum mechanical calculations themselves, which are correct as far as they go, but in the oversimplification of the problem and in the incorrect interpretation of the results. Consequently it has seemed desirable to us to make the necessary extensions and corrections in order to see if the theory can lead to a consistent picture. In the following discussion we have found it necessary to consider all of the different factors mentioned heretofore the resonance effect, the inductive effect, and the effect of polarization by the attacking group. The inclusion of these several effects in the theory has led to the introduction of a number of more or less arbitrary parameters, and has thus tended to remove significance from the agreement with experiment which is achieved. We feel, however, that the effects included are all justified empirically and must be considered in any satisfactory theory, and that the values used for the arbitrary parameters are reasonable. The results communicated in this paper show that the quantum mechanical theory of the structure of aromatic molecules can account for the phenomenon of directed substitution in a reasonable way. 

Fig. 3. The normalized excitation functions in A2 versus collision energy for the two isotopic channels for the F+HD reaction. The solid line is the result of quantum scattering theory using the SW-PES. The QCT simulations from Ref. 71 are plotted for comparison. The experiment, shown with points, is normalized to theory by a single scaling factor for both channels. Also shown in (a) is the theoretical decomposition of the excitation function into direct and resonant contributions using the J-shifting procedure.

Resonance such as (5.28a)-(5.28c) is inherently a quantal phenomenon, with no classical counterpart. In NBO language, each of the resonance interactions (5.28a)-(5.28c) corresponds to a donor-acceptor interaction between a nominally filled (donor Lewis-type) and unfilled (acceptor non-Lewis-type) orbital, the orbital counterpart of G. N. Lewis s general acid-base concept. As mentioned above, Lewis and Werner (among others) had well recognized the presence of such valence-like forces in the dative or coordinative binding of free molecular species. Thus, the advent of quantum mechanics and Pauling s resonance theory served to secure and justify chemical concepts that had previously been established on the basis of compelling chemical evidence. 

See Lead zirconate titanate ceramics Quality number 219 Quantum transmission 59 Reciprocal space 123, 353 Reciprocity principle 88 Reconstruction 14, 327 Au(lll) 327 DAS model 16 Si(lll)-2X1 14 Recursion relations 352 Repulsive atomic force 185, 192 Resonance frequency 234, 241 piezoelectric scanners 234 vibration isolation system 241 Resonance interactions 171, 177 and tunneling 177 Resonance theory of the chemical bond 172... 

More recently, Mies and Kraus have presented a quantum mechanical theory of the unimolecular decay of activated molecules.13 Because of the similarity between this process and autoionization they used the Fano theory of resonant scattering.2 Their theory provides a detailed description of the relationships between level widths, matrix elements coupling discrete levels to the translational continuum, and the rate of fragmentation of the molecule. 

We have used the concepts of the resonance methods many times in previous chapters to explain the chemical behavior of compounds and to describe the structures of compounds that cannot be represented satisfactorily by a single valence-bond structure (e.g., benzene, Section 6-5). We shall assume, therefore, that you are familiar with the qualitative ideas of resonance theory, and that you are aware that the so-called resonance and valence-bond methods are in fact synonymous. The further treatment given here emphasizes more directly the quantum-mechanical nature of valence-bond theory. The basis of molecular-orbital theory also is described and compared with valence-bond theory. First, however, we shall discuss general characteristics of simple covalent bonds that we would expect either theory to explain. 

Fig. 6.20 Ionization width vs electric field for the Na (20,19,0,0) level near its crossing with the (21,17,3,0) level from experiment (data points) and from WKB-quantum defect theory (solid line). The levels are specified as (n./q.ni.M) Because the lineshapes are quite asymmetric (except for very narrow lines), the width in this figure is taken to be the FWHM of the dominant feature corresponding to the (20,19,0,0) level in the photoionization cross section. For the narrowest line, experimental widths are limited by the 0.7 GHz laser linewidth. Error limits are asymmetric because of the peculiar fine shapes and because of uncertainties due to the overlapping m = 1 resonance (from ref. 37).

The selection rules for fi hyper-Raman scattering were derived by Cyvin, Rauch, and Decius, 02) and those for y hyper-Raman scattering, which has not yet been detected experimentally, by Christie and Lockwood 103>. From their tables one can see that silent modes become 3-active for such important point groups as C6, D6, C3v, C6v, C. D, 0 and Oh. Examples of additional 7 activity can be found in the point groups C4v, C. D. D, and Oh. Long and Stanton 104) have derived a quantum-mechanical theory of the hyper-Raman effect which indicates several possibilities for resonance enhancement of hyper-Raman intensities. Iha and Woo 105) extended the theory of nonlinear... 

Quantum-chemical calculations for pyrylium including one, two, or three water molecules using DFT and 6-31 + G(d,p) basis set revealed that the aromaticity (estimated by harmonic oscillator stabilization energy, HOSE natural resonance theory, NRT harmonic oscillator model of aromaticity, HOMA and nucleus-independent chemical shifts, NICS) is not influenced by water molecules [82],... 

I think that the theory of resonance is independent of the valence-bond method of approximate solution of the Schrodinger wave equation for molecules. I think that it was an accident in the development of the sciences of physics and chemistry that resonance theory was not completely formulated before quantum mechanics. It was, of course, partially formulated before quantum mechanics was discovered and the aspects of resonance theory that were introduced after quantum mechanics, and as a result of quantum mechanical argument, might well have been induced from chemical facts a number of years earlier. [25]... 

The Sn2 reactions are good examples to highlight the different merits of VB and MO theories in quantum chemistry [65]. In the VB method, the atomic features are preserved and the focus is the two-electron-two-center bonds, and each molecule is formed with bonds (plus the lone and core pairs). Whereas one resonance structure is not enough to describe a molecule, multi-resonance structures are adopted. In fact, the resonance theory can also be applied to illustrate the reactions in an intuitive way. For example, for the chloride-exchange reaction... 

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