Dipole superposition

State I ) m the electronic ground state. In principle, other possibilities may also be conceived for the preparation step, as discussed in section A3.13.1, section A3.13.2 and section A3.13.3. In order to detemiine superposition coefficients within a realistic experimental set-up using irradiation, the following questions need to be answered (1) Wliat are the eigenstates (2) What are the electric dipole transition matrix elements (3) What is the orientation of the molecule with respect to the laboratory fixed (Imearly or circularly) polarized electric field vector of the radiation The first question requires knowledge of the potential energy surface, or... 

Figure 15-3 shows the optical absorption spectrum of a MEH-PPV/C60 film with different C, content compared to the optical absorption spectrum of the components alone. The peak at 2.5 eV is identified as the n-n absorption of MEH-PPV and is clearly observed along with the first dipole-allowed transition in C(l0 (at 3.75 eV). The spectrum is a simple superposition of the two components. Further-... 

The electron-phonon operator is a tensor product between the electronic dipole and the nuclear dipole operators. A mixing between the AA and BB via the singlet-spin diradical AB state is possible now. A linear superposition of identical vibration states in AA and BB is performed by the excited state diradical AB. If the system started at cis state, after coupling may decohere by emission of a vibration photon in the trans state furthermore, relaxation to the trans... 

Stockmayer potential is considered as a superposition of a Lennard-Jones (6-12) potential and the interaction of two point dipoles. Many of the properties of gases and liquids have been calculated in terms of these two potential functions. It should be borne in mind, however, that Lennard-Jones and Stockmayer potentials are idealizations of the true energy of interaction and that they are reasonably accurate for a number of simple molecules. The interaction of long molecules, molecules in excited states, free radicals, and ions cannot be described by these two potential functions (Ref 8a, pp 23 35)... 

Let us return now to the problem of a sphere in a uniform field. We note from (5.13) and (5.14) that the field outside the sphere is the superposition of the applied field and the field of an ideal dipole at the origin with dipole moment... 

In the preceding sections the applied field was taken to be parallel to the principal axes of the ellipsoid. When the applied field E0 is arbitrarily directed, the induced dipole moment follows readily from superposition ... 

Energy barriers for internal rotation have been derived, especially during the 1950s, by analyzing (68M12 68M13) microwave spectra of molecules. The method works with molecules with a permanent dipole moment and in the gas phase. Limitations are dictated by the molecular size. The barriers are obtained from rotational energy levels of the molecule as a whole, perturbed by the internal rotor. When different conformers are present in the sample and their interconversion is slower than microwave absorption (barriers smaller than 20 kJ mol can be measured), the spectrum is just a superposition of the lines of the separate species which can be qualitatively and quantitatively determined. 

Nitrogen-bearing cyclophanes like 351 [16] and 352 [17] bind larger organic anions in water due to superposition of the hydrophobic effect and electrostatic attraction. The phenanthridinium hosts like 351 have been found to form the most stable nucleotide complexes known so far. On the other hand, free tetrapyrrolic porphyrins do not bind anions since their cavity is too small to take advantage of the convergent N-H dipoles for the complex stabilization [18]. However, expanded diprotonated porphyrins like sapphyrin 353 were shown to form stable complexes with phosphate [19a] and halide [19b] anions. 

Fig. 2.20 (a) Schematic diagram showing the field distribution above a protruding surface atom. This distribution is the superposition of a uniform applied field and a dipole field. 

Note that the above equations for the radiation of a dipole refer to a single dipolar source of radiation. For our spectroscopic interests, we more commonly are dealing with an assembly of many identical such systems. Usually, the total radiation of such an assembly of sources may be assumed to be the incoherent superposition of the individual intensities (see, however, the discussions of the intercollisional effect below, Chapters 3 and 5). We may then simply multiply the contributions of one source by the number iV of sources in the (initial) state i. 

This breakdown of the linear relationship between the absorption coefficient a and the product of densities, Q1Q2, indicates that the observed absorption is not a binary process. Specifically, for the case at hand, one can no longer assume that the measured absorption consists of an incoherent superposition of the pair contributions. Rather, the correlations of the dipoles that are induced in subsequent binary collisions lead to a partially destructive interference, an absorption defect that occurs if the product of the time T12 between Ne-Xe collisions, and microwave frequency, /, approaches unity [404], We note that for the spectra shown above, Figs. 3.1 and 3.2, the product fx 2 is substantially greater than unity at all frequencies where experimental data are shown and, consequently, incoherent superpositions of the waves arising from different induced dipoles occur. The intercollisional absorption defect is limited to low frequencies (Lewis 1980). 

Figure 3.4 is a schematic illustration of the fact that dipoles induced in successive collisions tend to be more or less antiparallel. This anticorrelation of dipoles induced in subsequent collisions leads to the absorption defect and causes the breakdown of the pair behavior illustrated in Fig. 3.3, if the product fx 2 is of the order of unity or less. If, on the other hand, fxn 1, superposition occurs with widely varying, random phase differences which render an interference effect inefficient. 

If at least one of the interacting particles is a molecule, further induction mechanisms arise. Molecules are surrounded by an electric field which may be viewed as a superposition of multipole fields. A collisional partner will be polarized in the multipole field and thus give rise to induced dipole components. In the case of symmetric diatoms like H2 or N2, the lowest-order multipole is a quadrupole and asymptotically, for R - 00, the quadrupole-induced dipole may be written as [288, 289]... 

Recent work improved earlier results and considered the effects of electron correlation and vibrational averaging [278], Especially the effects of intra-atomic correlation, which were seen to be significant for rare-gas pairs, have been studied for H2-He pairs and compared with interatomic electron correlation the contributions due to intra- and interatomic correlation are of opposite sign. Localized SCF orbitals were used again to reduce the basis set superposition error. Special care was taken to assure that the supermolecular wavefunctions separate correctly for R —> oo into a product of correlated H2 wavefunctions, and a correlated as well as polarized He wavefunction. At the Cl level, all atomic and molecular properties (polarizability, quadrupole moment) were found to be in agreement with the accurate values to within 1%. Various extensions of the basis set have resulted in variations of the induced dipole moment of less than 1% [279], Table 4.5 shows the computed dipole components, px, pz, as functions of separation, R, orientation (0°, 90°, 45° relative to the internuclear axis), and three vibrational spacings r, in 10-6 a.u. of dipole strength [279]. 

H2-H2 dipole. Early attempts to calculate the induced dipole moments from first principles were described elsewhere [281]. Only in recent times could the substantial problems of such computations be controlled and precise data be generated by SCF and Cl calculations, so that that the basis set superposition errors were small enough and the Cl excitation level is adequate for the long-range effects. The details of the computations are given elsewhere [282, 281],... 

The three-body spectra and their associated correlation functions may be considered to be a superposition of three components of different nature. One part arises from two-body dynamics where the third atom acts strictly as a perturbing field. The second part represents the contributions of the irreducible three-body dynamics to the pairwise-additive induction. The third part is due to the three-body induction mechanism and contains the irreducible dipole. These agents vary differently with temperature and could in principle be separated on that basis. 

Next we follow the history of one atom of type A over a time T tc in which the atom makes N collisions at times t with i = 1, N, where t, is the time of closest approach of a collision. The dipole moment MO may be written as the superposition of the moments induced in subsequent binary collisions, namely... 

The spectral profile, J(term depends on the product of real and imaginary parts of A, 91 A and 3 X, Eq. 5.92, which will be positive at low frequencies if both, the dipole function and the interatomic potential are short range [236]. In other words, the intercollisional absorption profile is negative under such conditions an absorption dip is obtained, in agreement with the observations. 

Thus, if van der Waals forces are responsible for the stability of the caffeine-pyrogallol complex we would expect the molecules to be oriented as in V. Bearing in mind, however, the uncertainty as to the exact orientation of caffeine s dipole vector, the superposition of molecular planes might in the final analysis not be so different from I. In that case one would have to look to other evidence to establish whether or not charge transfer forces were involved. 

The generalization to the control of the dynamics of a molecule with n electronic states is straightforward. For the purpose of deducing the control conditions we will examine the extreme case in which every possible pair of these electronic states is connected via the radiation field and a nonzero transition dipole moment. If the molecule is coupled to a radiation field that is a superposition of individual fields, each of which is resonant with a dipole allowed transition between two surfaces, the density operator of the system can be represented in the form... 

Using this approach the +) and —) states are not coupled by the field of the ion, but are only split in energy. At high collision velocities the initial state 0) is simply projected onto the 0 + 1) state, a coherent superposition of +) and -) states, by the dipole matrix element. However, at lower velocities the change in energy of the +) and -) states during the collision allows the +) and -) states themselves to be populated rather than only a coherent superposition. The latter feature allows nondipole transitions at lower collision velocities, as observed experimentally. 

Turning to the barriers we note that they are defined as the difference in enthalpy between the transition state and the separated reactants. This is important because in gas-phase ion-molecule reactions the transition state is typically preceded by an ion-dipole complex131 133 formed between the reactants, and the term barrier is sometimes used for the enthalpy difference between the transition state and the ion-dipole complex. However, these ion-dipole complexes have little relevance to the main topic discussed in this chapter and hence the chosen definition of AH is more appropriate. For reasons explained elsewhere,118 the barriers reported in Table 16 have not been corrected for the basis set superposition error (BSSE),134 although such corrected values are available.118... 

In this chapter, dielectric response of only isotropic medium is considered. However, in a local-order scale, such a medium is actually anisotropic. The anisotropy is characterized by a local axially symmetric potential. Spatial motion of a dipole in such a potential can be represented as a superposition of oscillations (librations) in a symmetry-axis plane and of a dipole s precession about this axis. In our theory this anisotropy is revealed as follows. The spectral function presents a linear combination of the transverse (K ) and the longitudinal (K ) spectral functions, which are found, respectively, for the parallel and the transverse orientations of the potential symmetry axis with... 

The latter is determined by the oscillation frequency, decaying coefficient, and vibration lifetime. This nonrigid dipole moment stipulates a Lorentz-like addition to the correlation function. As a result, the form of the calculated R-band substantially changes, if to compare it with this band described in terms of the pure hat-curved model. Application to ordinary and heavy water of the so-corrected hat-curved model is shown to improve description (given in terms of a simple analytical theory) of the far-infra red spectrum comprising superposition of the R- and librational bands. 

A two-dimensional rotation of a dipole we may represent as a superposition of (1) a deflection relative the symmetry axis and (2) a precession about this axis. The square of the polar velocity i) presents in Eqs. (42a) and (42b) the axial component of a kinetic energy and (1/ sini )2—the precessional component. Below we consider two limiting cases. 

Water has a complex structure determined by hydrogen bonding of molecules. In view of the above discussion we assume now that a dipole is not rigid. Introducing some changes in our recent works [6, 8], we represent a total dipole moment ptot as a superposition of the constant part p and of a small decaying component p(f) due to fast vibration of the H-bonded molecules ... 

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