Relaxation in isolated molecules

We are quite used to these observations in macroscopic phenomena. What may appear as a surprise is that such situations are also encountered in microscopic systems, including single molecules. For example, an optical transition of a large molecule into an excited electronic state is often followed by relaxation of the electronic energy due to coupling to nuclear (vibrational) levels associated with lower electronic states, in a way which appears to be radiationless (no photon emitted) and collisionless (take place on a timescale shorter than collision times at the 

Another relaxation process encountered in isolated molecules is the phenomenon of intramolecular vibrational relaxation. Following excitation of a high-lying vibrational level associated with a particular molecular mode, the excitation energy can rapidly spread to other nuclear modes. This is again a case of an initially prepared single state decaying into an effective continuum. 

In both cases, because of restrictions imposed on the excitation process (e.g. optical selection rules), the initially excited state is not an exact eigenstate of the molecular Hamiltonian (see below). At the same time, if the molecule is large enough, this initially prepared zero-order excited state is embedded in a bath of a very large number of other states. Interaction between these zero-order states results from residual molecular interactions such as corrections to the Bom Oppenheimer approximation in the first example and anharmonic corrections to nuclear potential surfaces in the second. These exist even in the absence of interactions with other molecules, giving rise to relaxation even in isolated (large) molecules. The quasi-continuous manifolds of states are sometimes referred to as molecular heat baths. The fact that these states are initially not populated implies that these baths are at zero temperature. 

Problem 9.1. In the analysis that led to the result (9.24) for the decay of the initially prepared state 11) we have used the representation defined by the eigenstates of Ho. In the alternative representation defined by the full set of eigenstates [/) of H the initial state is given by 

Show that in terms of the coefficients Cj the probability Pi (z) that the system remains in the initial state is given by 

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Although the radiationless processes appeared to be intramolecular in character, the presence of the solvent or rigid matrix presented an essential complication to a theoretical description of the radiationless phenomena. A complete theory of radiationless processes must begin with a description of electronic relaxation in isolated molecules. Once the isolated molecule case is properly understood, the effects of the external media can be considered and the bulk of the experimental data concerning radiationless transitions can be confronted. 

A previous review provides a description of the theory of electronic relaxation in polyatomic molecules with particular emphasis on the vibronic state dependence of radiationless transition rates. A sequal review considers the general question of collisional effects on electronic relaxation, while the present one covers only the special phenomenon of collision-induced intersystem crossing. It departs from the other collisional effects review in presenting only a qualitative description of the theory the full theoretical details can be obtained from the previous review and the original papers.As a review of the basic concepts of radiationless transitions theory is necessary as a prelude to a discussion of collision-induced intersystem crossing, considerable overlap exists between this section and Section II of the previous collision effects review. However, since many concepts from radiationless transition theory, such as the nature and criteria for irreversible decay, the role of the preparation of the initial state, the occurrence of intramolecular vibrational relaxation, etc. pervade the other papers on laser chemistry in these volumes, it is useful to recall the primary results of the theory of electronic relaxation in isolated molecules and its relevance to the material in the present volume as well as to this review. 

It thus appears that, in general, the rates of electronic relaxation processes in dense media are of the same order of magnitude as those found for isolated molecules in the gas phase, assuming that the molecule has a sufficiently large number of vibrational degrees of freedom. Therefore it may be concluded that the mechanism which operates in isolated molecules must be responsible for the gross features of the phenomena observed in dense media. 

The photoinduced creation of an ion or exciton in condensed molecular media causes greater relaxation than in isolated molecules because the induced charge redistribution generates electronic and atomic relaxation in the other constitutents of the medium as well as in the excited molecule itself. The associated intermolecular contributions to the relaxation energy are comparable in magnitude to the intramolecular ones from the molecule itself, i.e., E (inter)=l-2eV, E (inter)=0.leV (4, 6, 10). 

Estimates of IVR rates have also been inferred from spectral studies. The principal spectral approach, the measurement of the emission spectra of isolated molecules (for a review, see, for example, Ref. 3), relies on the fact that the spectral characteristics of emission from a molecule will depend intimately on the vibrational character of the excited molecular state. If one prepares a gaseous molecule in a well-defined vibrational state and subsequently observes emission bands that would not be expected to arise from this initially prepared state, then some IVR process can be inferred. Moreover, a rate can be calculated for the process by comparing the intensities of expected emission bands (vibrationally unredistributed emission) and unexpected emission bands (vibrationally redistributed emission). Redistributed and unredistributed emission often are loosely termed relaxed and unrelaxed. Since energy is conserved in isolated molecules, there is no real energy relaxation, as occurs in solution or in solids. 

As mentioned in the introduction, the above discussion of the small-, large-, and intermediate-molecule limits of electronic relaxation processes can also be utilized with very minor modifications to discuss the phenomena of intramolecular vibrational relaxation in isolated polyatomic mole-cules. ° Figure 4 is still applicable to this situation. The basis functions are now taken to be either pure harmonic vibrational states, some local-mode vibrational eigenfunctions, or some alternative nonlinear mode-type wave-functions. In the following the nomenclature of vibrational modes is utilized, but its interpretation as normal or local can be chosen to suit the circumstances at hand. 

State now involves a dissipative irreversible decay into the effective continuum of levels < >/. Questions concerning the precise nature of the prepared state, the contribution from the reversible dephasing processes at intermediate energies, and the role of intramolecular vibrational relaxation at high enough energies are central to the understanding of infrared multiphoton dissociation processes in isolated molecules. ... 

There exists, in the past few years an increasing interest in the influence of external (magnetic and electric) fields on the dynamics of excited molecular states. This interest is not surprising if we are reminded of the role played by this kind of studies in the development of the atomic physics. We will limit our discussion to the phenomena related to the collisional electronic relaxation application of magnetic fields in the studies of predissociation and of dephasing processes in isolated molecules will not be treated here. 

On the theoretical front, it is possible to make a few simple assertions. We have already seen that a collisional component to the randomisation process may become faster the more dense are the states of the molecule. It is also obvious that the first order component will become slower as the states become further apart, but the molecular level density where this begins is not known a cut-off at about 1000 states per wavenumber has been suggested [82.S2] for intramolecular vibrational relaxation of isolated molecules in one kind of experiment. It is also obvious that there must be propensity rules for the occurrence of randomising transitions within any grain [81.P2] for example, transitions between states of... 

Hence, internal conversion is typically expected to occur in molecules with 4 atoms, and intersystem crossing sets in when N k, 10. These are rough guidelines for the large-molecule regime in which nonradiative relaxation is prevalent in isolated molecules. It includes all aromatic molecules (the smallest common one of which is benzene) formaldehyde and larger molecules with the carbonyl chromophore and all laser dyes such as rhodamines, oxazines and commarins (Chapter 9). 

Here p is the radius of the effective cross-section, (v) is the average velocity of colliding particles, and p is their reduced mass. When rotational relaxation of heavy molecules in a solution of light particles is considered, the above criterion is well satisfied. In the opposite case the situation is quite different. Even if the relaxation is induced by collisions of similar particles (as in a one-component system), the fraction of molecules which remain adiabatically isolated from the heat reservoir is fairly large. For such molecules energy relaxation is much slower than that of angular momentum, i.e. xe/xj > 1. 

Fig. 9.1 The internuclear transfer of magnetization via NOE cross-relaxation in an isolated spin-pair. (A) Build-up curves for the cross-peak intensity in a 2D NOESY experiment for various internuclear distances r. The dashed line indicates a typical mixing time tm = 300rns used for drug-like molecules.

For gases, n = e 1 is an excellent approximation. The easiest approach to condensed phases maintains this approximation, where calculations of the molecular first-order and response properties are performed for the isolated molecule, while accounting for the effect of intermolecular interactions through the number density N = Aa/Vm, and therefore by taking appropriate values of Vm. This rough, often at best qualitative, approach is somewhat relaxed by employing expansions of the birefringence constant with the density, that is in inverse powers of Vm. This introduces the appropriate virial coefficients [15,16]... 

But what is perhaps more interesting is that, at the time of his most important activity, Barriol enumerated the techniques used in the theoretical laboratory of Nancy to solve the problem of three-dimensional structural studies on isolated molecules, static studies on associated solutions, and dynamic studies of associations. They were five techniques explicitly mentioned by Barriol, four experimental and one theoretical. Theoretical calculations are situated on the same level as the four experimental techniques, which included dielectric polarization, nuclear magnetic resonance, microwave spectrography, and dielectric relaxation. [54]... 

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