Timescales

The various physical techniques that we might use to study molecular species depend on a variety of proeesses. The conclusions we could draw about structures are related to the timescales associated with these proeesses, and it is important for us to understand these if we are to avoid making erroneous deductions. In relation to any one type of experiment, there are in fact four different times for us to consider the time during which a quantum of radiation or a particle can interact with a molecule the lifetime of any excited state of the molecule the minimum lifetime that the species being studied must have to allow it to be seen as a distinct species and the total duration of an experiment in which the species is observed, which may be as much as several hours or as little as 10 s. Before we consider these further, we must look at the timescales of typical molecular processes so that we can relate them to timescales associated with structural techniques. Typical vibrational frequencies are of the order of lO to 10 Hz, while rotational frequencies are around 10 ° to 10 Hz. The inversion of ammonia has a rate of about 10 Hz at room temperature, while the corresponding rate for phosphine is 10 Hz. The inversion rate for methane is 10 Hz, so any one molecule inverts, on average, once every 100 million years But remember that there are 6 x 10 molecules in a mole of gas, so in fact the inversion is by no means a rare occurrence. Pseudorotation in PF5, which switches axial and equatorial fluorine atoms, has a rate of about 10 Hz at room temperature, while the rate for PCI5 is 10 Hz. 

The time during which radiation can interact with a molecule is essentially the time taken for photons to pass by the molecule or relevant part of it. X-rays travel at the speed of light electrons are a little slower and neutrons considerably slower. This gives us maximum interaction times in diffraction experiments of around 10 to 10 s. This is very much less than the time taken for molecular vibrations, rotations or rearrangements, and so each particle or photon sees a molecule with an instantaneous structure, and in a fixed electronic, vibrational and rotational state. These extremely short timescales have been exploited in studies of the movements of atoms during dissociation reactions (Section 2.8.1). 

Technique Energy of excited state/Hz Typical relaxation time/s Typical line widtb/Hz 

When relaxation times are short, the Uncertainty Principle becomes important, because the lifetime t of an excited state and the uncertainty in its energy, A , are related by rA (where S = A/2 n). Spectra consist of lines representing transitions, and if the uncertainty in the upper state becomes large, these hnes could be broadened, so that resolution is lost and, in extreme cases, the whole spectrum might become just a single, extremely broad hump. The constant Ti is very small, about 10 J s, but for electronic spectra of transition-metal complexes in solution, relaxation times are typically around 10 s, so that A is of the order of 10 J per molecule, or 60kJ mol. This is comparable with the transition energies involved. Most such electronic spectra therefore consist of a few broad hnes, and much potentially useful information is lost. 

It is important to realize that the relaxation times might depend on some factors that are properties of the atom or molecule itself and on others that are related to its environment. Thus rotational spectra of gases have linewidths (related to the rotational relaxation times) that depend on the mean times between coUisions for the molecules, which in turn depend on the gas pressure. In liquids, the collision lifetimes are much shorter, and so rotational energy is effectively non-quantized. On the other hand, if the probability of collisions is reduced, as in a molecular beam, we can increase the relaxation time, reduce linewidths, and so improve resolution. Of course, the relaxation time only defines a minimum width of spectral lines, which may be broadened by other experimental factors. 

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Structural investigations of metal-ion hydration have been carried out by spectroscopic, scadering and diffraction teclmiques, but these teclmiques do not always give identical results since they measure in different timescales. There are tliree distinct types of measurement ... 

The hydration of more inert ions has been studied by O labelling mass spectrometry. 0-emiched water is used, and an equilibrium between the solvent and the hydration around the central ion is first attained, after which the cation is extracted rapidly and analysed. The method essentially reveals the number of oxygen atoms that exchange slowly on the timescale of the extraction, and has been used to establish the existence of the stable [1 10304] cluster in aqueous solution. 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

Berne B J 1985 Molecular dynamics and Monte Carlo simulations of rare events Multiple Timescales ed J V Brackbill and B I Cohen (New York Academic Press)... 

The first two of these we can readily approach with the knowledge gained from the studies of trappmg and sticking of rare-gas atoms, but the long timescales involved in the third process may perhaps more usefiilly be addressed by kinetics and transition state theory [35]. 

The thennalization stage of this dissociation reaction is not amenable to modelling at the molecular dynamics level becanse of the long timescales required. For some systems, snch as O2 /Pt(l 11), a kinetic treatment is very snccessfiil [77]. However, in others, thennalization is not complete, and the internal energy of the molecnle can still enliance reaction, as observed for N2 /Fe(l 11) [78, 79] and in tlie dissociation of some small hydrocarbons on metal snrfaces [M]- A detailed explanation of these systems is presently not available. 

In the statistical description of ununolecular kinetics, known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory [4,7,8], it is assumed that complete IVR occurs on a timescale much shorter than that for the unimolecular reaction [9]. Furdiemiore, to identify states of the system as those for the reactant, a dividing surface [10], called a transition state, is placed at the potential energy barrier region of the potential energy surface. The assumption implicit m RRKM theory is described in the next section. 

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

In the above discussion it was assumed that the barriers are low for transitions between the different confonnations of the fluxional molecule, as depicted in figure A3.12.5 and therefore the transitions occur on a timescale much shorter than the RRKM lifetime. This is the rapid IVR assumption of RRKM theory discussed in section A3.12.2. Accordingly, an initial microcanonical ensemble over all the confonnations decays exponentially. However, for some fluxional molecules, transitions between the different confonnations may be slower than the RRKM rate, giving rise to bottlenecks in the unimolecular dissociation [4, ]. The ensuing lifetime distribution, equation (A3.12.7), will be non-exponential, as is the case for intrinsic non-RRKM dynamics, for an mitial microcanonical ensemble of molecular states. 

Furthemiore, IVR is not rapid between the C2H4 intramolecular modes and different excitation patterns of these modes result in different dissociation rates. As a result of these different timescales for dissociation, the relative populations of the vibrational modes of the C2H4 dimer change with time. 

To be effective, it is necessary for the ions traversing the instniment to experience several RF cycles. Thus, unlike magnetic sector instmments, the ions fonned in the ion source of a quadnipole mass filter apparatus are accelerated to only a few eV kinetic energy (typically 5-10 eV). The timescale of the experiment is therefore... 

The principal dilTerence from liquid-state NMR is that the interactions which are averaged by molecular motion on the NMR timescale in liquids lead, because of their anisotropic nature, to much wider lines in solids. Extra infonnation is, in principle, available but is often masked by the lower resolution. Thus, many of the teclmiques developed for liquid-state NMR are not currently feasible in the solid state. Furthemiore, the increased linewidth and the methods used to achieve high resolution put more demands on the spectrometer. Nevertheless, the field of solid-state NMR is advancing rapidly, with a steady stream of new experiments forthcoming. 

STM has not as yet proved to be easily applicable to the area of ultrafast surface phenomena. Nevertheless, some success has been achieved in the direct observation of dynamic processes with a larger timescale. Kitamura et al [23], using a high-temperature STM to scan single lines repeatedly and to display the results as a time-ver.sn.s-position pseudoimage, were able to follow the difflision of atomic-scale vacancies on a heated Si(OOl) surface in real time. They were able to show that vacancy diffusion proceeds exclusively in one dimension, along the dimer row. 

The timescale is just one sub-classification of chemical exchange. It can be further divided into coupled versus uncoupled systems, mutual or non-mutual exchange, inter- or intra-molecular processes and solids versus liquids. However, all of these can be treated in a consistent and clear fashion. 

These experiments yield T2 which, in the case of fast exchange, gives the ratio (Aoi) /k. However, since the experiments themselves have an implicit timescale, absolute rates can be obtained in favourable circumstances. For the CPMG experiment, the timescale is the repetition time of the refocusing pulse for the Tjp experiment, it is the rate of precession around the effective RF field. If this timescale is fast witli respect to the exchange rate, then the experiment effectively measures T2 in the absence of exchange. If the timescale is slow, the apparent T2 contains the effects of exchange. Therefore, the apparent T2 shows a dispersion as the... 

The approach is ideally suited to the study of IVR on fast timescales, which is the most important primary process in imimolecular reactions. The application of high-resolution rovibrational overtone spectroscopy to this problem has been extensively demonstrated. Effective Hamiltonian analyses alone are insufficient, as has been demonstrated by explicit quantum dynamical models based on ab initio theory [95]. The fast IVR characteristic of the CH cliromophore in various molecular environments is probably the most comprehensively studied example of the kind [96] (see chapter A3.13). The importance of this question to chemical kinetics can perhaps best be illustrated with the following examples. The atom recombination reaction... 

Both MD and MC teclmiques evolve a finite-sized molecular configuration forward in time, in a step-by-step fashion. (In this context, MC simulation time has to be interpreted liberally, but there is a broad coimection between real time and simulation time (see [1, chapter 2]).) Connnon features of MD and MC simulation teclmiques are that there are limits on the typical timescales and length scales that can be investigated. The consequences of finite size must be considered both in specifying the molecular mteractions, and in analysing the results. 

The vibrationally excited states of H2-OH have enough energy to decay either to H2 and OH or to cross the barrier to reaction. Time-dependent experiments have been carried out to monitor the non-reactive decay (to H2 + OH), which occurs on a timescale of microseconds for H2-OH but nanoseconds for D2-OH [52, 58]. Analogous experiments have also been carried out for complexes in which the H2 vibration is excited [59]. The reactive decay products have not yet been detected, but it is probably only a matter of time. Even if it proves impossible for H2-OH, there are plenty of other pre-reactive complexes that can be produced. There is little doubt that the spectroscopy of such species will be a rich source of infonnation on reactive potential energy surfaces in the fairly near future. 

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