Unperturbed molecule

If the perturbations thus caused are relatively slight, the accepted perturbation theory can be used to interpret observed spectral changes (3,10,39). The spectral effect is calculated as the difference of the long-wavelength band positions for the perturbed and the initial dyes. In a general form, the band maximum shift, AX, can be derived from equation 4 analogous to the weU-known Hammett equation. Here p is a characteristic of an unperturbed molecule, eg, the electron density or bond order change on excitation or the difference between the frontier level and the level of the substitution. The other parameter. O, is an estimate of the perturbation. 

One aspect of the mathematical treatment of the quantum mechanical theory is of particular interest. The wavefunction of the perturbed molecule (i.e. the molecule after the radiation is switched on ) involves a summation over all the stationary states of the unperturbed molecule (i.e. the molecule before the radiation is switched on ). The expression for intensity of the line arising from the transition k —> n involves a product of transition moments, MkrMrn, where r is any one of the stationary states and is often referred to as the third common level in the scattering act. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

Proceeding in a manner paralleling the derivation of the excluded volume for a pair of molecules, we consider the polymer molecule to consist of a swarm of segments distributed on the average about the molecular center of gravity in accordance with the Gaussian formula (see Eq. XII-51). This spatial distribution in the unperturbed molecule, as it would exist on the average in the total absence of inter-... 

In network BI in which chain expansion was the greatest, the measured results show more chain swelling than a network with f=3 but less than a network with f=4. Chain swelling was less than that of the phantom network model for the other two networks, and in one case, the chains coiled to a size slightly less than that of the unperturbed molecule. 

A technique which is not a laser method but which is most useful when combined with laser spectroscopy (LA/LIF) is that of supersonic molecular beams (27). If a molecule can be coaxed into the gas phase, it can be expanded through a supersonic nozzle at fairly high flux into a supersonic beam. The apparatus for this is fairly simple, in molecular beam terms. The result of the supersonic expansion is to cool the molecules rotationally to a few degrees Kelvin and vibrationally to a few tens of degrees, eliminating almost all thermal population of vibrational and rotational states and enormously simplifying the LA/LIF spectra that are observed. It is then possible, even for complex molecules, to make reliable vibronic assignments and infer structural parameters of the unperturbed molecule therefrom. Molecules as complex as metal phthalocyanines have been examined by this technique. 

Mononuclear ER4 and simple four-coordinate compounds of E(IV) states are the baseline for viewing the other coordination numbers, the effect of bulky ligands, bonds to other E or metals, E(II) compounds, multiple bonds and other phenomena discussed in later sections. Basic parameters for some simple compounds are presented in Table 1, taken from the gas-phase data summarized by Molloy and Zuckerman5 and Haaland6. These data show the unperturbed molecules in the gas phase and provide the base for... 

A satisfactory feature of (116) is that the first-order term, which describes the interaction between the external charge and the unperturbed molecule, continues to disappear if the electron distribution is highly uniform as in an alternant hydrocarbon in such cases the propensities for reaction are still dominated by the polarization term, but this now has a less simple form, depending upon all atom-atom polarizabilities and on the position of the attacking ion with respect to all conjugated centres. 

According to frontier molecular orbital theory, the strongest interactions are between those orbitals that have coefficients with similar magnitudes relative to the unperturbed molecules, t.e. the interaction is between the small coefficient on the dienophile and the small coefficient on the diene [16], [17]. 

For this case, the primary change that is observable in the IR spectrum is due to changes in the vibrahonal frequencies of the probe molecule due to modificahons in bond energies. This can lead to changes in bond force constants and the normal mode frequencies of the probe molecule. In some cases, where the symmetry of the molecule is perturbed, un-allowed vibrational modes in the unperturbed molecule can be come allowed and therefore observed. A good example of this effect is with the adsorption of homonuclear diatomic molecules, such as N2 and H2 (see Section 4.5.6.8). 

Phosphorescence is defined as the emission of light due to a transition between states of different spin multiplicities. Because this is theoretically not an allowed transition for an ideal unperturbed molecule, phosphorescence lifetimes tend to be relatively long, typically 10 3-10-2 s. 

So far, only transitions involving unperturbed molecules have been considered. If collisions between molecules are allowed, then electric dipole transitions of otherwise forbidden systems become more probable. From the measurements of the absorption in oxygen, Badger et al.13 predict that the effective first-order transition probability, A, for the O-st Aj, -> 3S5") system, will be given by... 

The strength of the interaction in the course of a collision leading to population of vibrational states of a certain electronic state can be judged from the widths of the individual vibrational lines in a measured spectrum. If the lines are narrow and well separated, the average interaction is weak, and we can expect that the approximations leading to (III.4) are valid. In all these cases, however, it is also expected that the Z>,(t>) deviate very little from the Franck-Condon factors for the same ionizing transition in the unperturbed molecule caused by photoionization. 

The ESIPT fluorescence of 3-hydroxyflavone (3HF) (12) and 2,5-bis-(2-benzoxzoyl)-4-methoxyphenol (BBMP) (52) were studied in polyethylene films [146], It was found that perturbations due to H-bonded impurities are largely suppressed in the polymer matrix as compared with other media. Thus, the fluo-rophores behaved almost as unperturbed molecules, with spectral quality similar to those obtained from extensively purified solvents or embedded in rare matrices, even if they are introduced into the film from unpurified and undried chloroform. 

In this section we outline the coupled cluster-molecular mechanics response method, the CC/MM response method. Ref. [51] considers CC response functions for molecular systems in vacuum and for further details we refer to this article. The identification of response functions is closely connected to time-dependent perturbation theory [51,65,66,67,68,69,70], Our starting point is the quasienergy and we identify the response functions as simple derivatives of the quasienergy. For a molecular system in vacuum where Hqm is the vacuum Hamiltonian for the unperturbed molecule and V" is a time-dependent perturbation we have the following time-dependent Hamiltonian, H,... 

Over the past few years, sophisticated techniques have been developed to investigate reactive, unstable. species even in the gas phase, e.g., molecular beam experiments, monitored by laser spectroscopy. As a specific example, the vibrational frequency of the high-temperature Na Cl molecule was measured at 364.6985(25) cm (Horiai et al., 1988). Although this value is not very different from the frequency of 335.9 cm found for the argon matrix-isolated molecule (Ismail et al, 1975), the higher accuracy and knowledge of the exact absolute wave number of the unperturbed molecule are sometimes essential in order to elucidate its molecular physics. On the other hand, more chemically related problems (e.g., reactions of NaCl in solid noble gases) can be solved in a much simpler and more economical way by the matrix technique. This is demonstrated in thi.s chapter. 

The interpretation of Eq. (6.1-1) is that Raman scattering is a two-photon process involving virtual absorption from the initial state / to the entire manifold of eigenstates of the unperturbed molecule, followed by virtual emission to the final state /. 

In the adiabatic Bom-Oppenheimer approximation, the eigenstates of the unperturbed molecule, i. e. i >, / >, and r >, may be expressed as products of electronic, vibrational, and rotational states. In the following we restrict our discussion to the vibrational resonance Raman effect and hence, assuming that the molecule is initially in the vibrational state u, > and that the Raman transition starts and terminates in the ground electronic state g >, we may write... 

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