Linear 1,3-dipoles

We now turn to the gas-phase 1,3-dipolar cycloaddition of fulminic acid to ethyne. The concerted, almost synchronous nature of this reaction might create the impression that the electronic mechanism of this process should be very similar to that of the Diels-Alder reaction. Such an expectation is reinforced by frontier orbital theory, which treats both reactions in very much the same way (see Ref. 32). The only significant differences are related to the fact that the lowest unoccupied MO (LUMO) for a linear 1,3-dipole... 

Figure D.7 illustrates the application of the rules to delocalized systems A8a-1, A8b-1, A8c-1, A9a-1, A9b-1, and A9c-1. In particular, 1,3-dipoles are prominent members of the tricentric cases. Azomethine ylids, azomethine imines, nitrones, carbonyl ylids (e.g. A8c), carbonyl imines, carbonyl oxides fit prototype A8 bent nitrile ylids, may also be treated as A8. The linear 1,3-dipoles are treated as a triply-bonded systems with two mutually orthogonal sets of paired faces.

Cycloadditions The [2+1] cycloaddition of sulfur dioxide to ketenes affords an unstable [2+1] cycloadduct, which is in equilibrium with the linear 1,3-dipole. The intermediate can be trapped with C=N double bond-containing substrates, such as azomethines, ketenimines and isocyanates . 

In linear dipole-dipole clusters, each LiF unit retains its identity along the chain, but the two end units acquire conspicuous ionic character (cationic at one end, anionic at the other), whereas interior units are nearly neutral. For example, in (LiF)4 the successive Quf charges along the chain are... 

This difference is particularly strong for end units, but diminishes toward the interior of linear dipole chains. In linear (LiF)4, for example, AR is 0.12 and 0.10 A for the two... 

The rotational quantum numbers of the free linear dipole are j and m (with m S j), and the rotational constant is denoted by B. Analogous to the strong-field limit of the Stark effect, for linear dipoles at small r, one has... 

More complicated anisotropies of the potential are, for example, encountered in the association of two linear dipoles. Adiabatic channel potential curves for this case have been calculated and expressed analytically in Ref. 16. More systematic studies, also comparing SACM and trajectory results, were reported in Ref. [36], One may as well consider open-shell effects for example, the association of two open-shell HO radicals in their lowest rotational state was treated in Ref. 37. Figure 12 shows the lowest rovi-bronic adiabatic channel potential curves for this system. The ultimate goal... 

J. Troe My answer to Prof. Herman is that the high-Stark-field description of the close approach of a dipole to an ion can very well be represented in terms of the relevant quantum numbers. The linear dipole-free rotor quantum numbers j and m are converted to the oscillating dipole quantum number v with the identity v - 2j - m. ... 

If only the first derivatives in tie dipole-moment function and the second derivatives (k=j) in the potential function are retained, the strict harmonic-oscillator-linear-dipole-moment approximation, the selection rules are strict ... 

Consequently, by equation (71), a non-uniform electric field acting on a molecule induces in graeral not the linear dipole (70a) but rather ... 

Non-linear Dipole Polarizabilities. In weak electric fields, the linear dipolar polarizability a appearing in equations (67) and (68) is a quantity specific to the atom or molecule (the volume polarizability Og corresponds to the spatial dimensions of the electron shell and is of the order and dimoision of 10 m ) in fact, it describes the linear distortion of the shell under the influence of the field strength E (in the case of atoms, the electrons are, classically, performing harmonic oscillations). In the general case, the polarizability can depend on the field strength, if the latter is sufficiently large. One thus has to re-write equation (67) as follows ... 

Non-linear Multipolar Polarizabilities. Just as the polarizability tensor a in equation (68) defined a linear dipole moment, or moment of order 1 (as due to the first power of E), so the tensors p and y appearing in Bom s expansions (76) can be taken to define respectively a dipole moment of order 2 (induced by the square of E) ... 

In the general case, in addition to the contribution (149a) or (149e) to Rm resulting from linear dipole polarizability, other contributions due to quadrupole polarizability and non-linear dipolar polarizabilities have to be taken into account. Also, contributions to have been calcxilated which result if the part dependent on molecular orientation (multipole interactions 0 is included in the correlation function... 

R. W. Hellwarth. Theory of molecular light scattering spectra using the linear dipole approximation. J. Chem. Phys., 52 2128-2138 (1970). 

A.I.Maergoiz, E.E.Nikitin, and J.Troe, Adiabatic chaimel potential curves for two linear dipole rotors I. Classifieation of states and numerical calculations for identical rotors, J. Chem. Phys. 9, 5117 (1991)... 

Refractive Index Experimental Data for Gas and Liquid. From a measured refractive index it is always possible to extract formally an average linear dipole polarizability, a, from the Lorentz-Lorenz equation,... 

Fig. 7 a Temporal evolution of the PL spectrum of a single 3-phase chromophore recorded during the rotation of a polariser placed in the emission light path as sketched in b. c PL spectra of the 0-0 transition taken at the polarisation angle marked in a by the respective coloured lines. Note that the PL intensity of the ZPL centred at 2.806 eV does not show a full modulation, as expected for emission from a linear dipole... 

The situation is somewhat different for the convergence with the wavefunction model, i.e. the treatment of electron correlation. As an anisotropic and nonlinear property the first dipole hyperpolarizability is considerably more sensitive to the correlation treatment than linear dipole polarizabilities. Uncorrelated methods like HF-SCF or CCS yield for /3 results which are for small molecules at most qualitatively correct. Also CC2 is for higher-order properties not accurate enough to allow for detailed quantitative studies. Thus the CCSD model is the lowest level which provides a consistent and accurate treatment of dynamic electron correlation effects for frequency-dependent properties. With the CC3 model which also includes the effects of connected triples the electronic structure problem for j8 seems to be solved with an accuracy that surpasses that of the latest experiments (vide infra). 

The rather small activation barrier of concerted cycloadditions involving bent dipoles leads us to exclude the two-step mechanism for these reactions. On the other hand, the quite large activation barrier of the cycloadditions involving linear dipoles would be in favor of the conclusion of Hyberty et al., i.e., for those reactions quite probably both mechanisms compete. . . . ... 

The objective is to stop the motion of the particle by an external field E(t) which interacts via a linear (dipole) coupling as... 

We now employ the ideas described in Section III.A. For example, if one aims at a preferential dissociation into the products H + OD, a heating in the O-H bond should be triggered. Within LCT this can be achieved if one couples the field to the momentum of the H atom, so that (in analogy to the one-dimensional case, Eq. (20), and for a linear dipole moment) one has... 

This packet, which is localized around Ri = 15 A, has a full width at halfmaximum (FWHM) of 5 A. The mean momentum of the approaching particles is P, which corresponds to an impact energy of Ep = P2/2m. Figure 9 exhibits the potential energy curve for the HF molecule [166] (left panel). Here, we present results for a linear dipole moment, for a more realistic description, see Ref. 107. As in the case of photodissociation, we use the energy rate d(Ho)t/dt, which leads to a field of the form Eq. (20) but now with a negative value of the parameter X. This ensures that the internal energy is decreased at any instant of time. 

A.I.Maergorz, E.E.Nikitin, and J.Troe, Calculation of cross sections and rate constants for capture of two identical linear dipole molecules, Khim. Fiz. 12, 841... 

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