Decaying states, theory

Decaying state theory and complex poles of the resolvent... 

The problem of resonance sfafes, or of decaying sfafes, normally involves only the continuous spectrum, in which case the energy distribution is, according to the above use of fhe Hermitian I, a real function, given by < To E>I dE = fl(E) dE. The explicit form of fhis disfribution in terms of computable matrix elements is given by Fano s theory or by decaying-state theory, as... 

Miller W H, Hernandez R, Moore C B and Polik W F A 1990 Transition state theory-based statistical distribution of unimolecular decay rates with application to unimolecular decomposition of formaldehyde J. Chem. Phys. 93 5657-66... 

Both unimolecular and bimolecular reactions are common throughout chemistry and biochemistry. Binding of a hormone to a reactor is a bimolecular process as is a substrate binding to an enzyme. Radioactive decay is often used as an example of a unimolecular reaction. However, this is a nuclear reaction rather than a chemical reaction. Examples of chemical unimolecular reactions would include isomerizations, decompositions, and dis-associations. See also Chemical Kinetics Elementary Reaction Unimolecular Bimolecular Transition-State Theory Elementary Reaction... 

Fig. I. Illustration of the relationship between reactants (designated as 2-A-B), products (A-A and B-B), and the activated complex. According to transition state theory, reaction kinetics is limited by the irreversible decay of the activated complex minus the rate at which the activated complex reversibly breaks down to reactants.

A theory of photochemical processes which relates macroscopic observables to molecular properties should have great appeal to the physical chemist, since in these cases, unlike the case of thermal reactions, the nature of the metastable decaying states can be unambiguously defined. Furthermore, since these states are routinely prepared in photochemical experimentation, they are worthy of extensive study. Perhaps a more complete understanding of the properties of isolated molecule metastable states can play the role of precursor to the understanding of the states involved in thermal reactions. 

RRKM theory, an approach to the calculation of the rate constant of indirect reactions that, essentially, is equivalent to transition-state theory. The reaction coordinate is identified as being the coordinate associated with the decay of an activated complex. It is a statistical theory based on the assumption that every state, within a narrow energy range of the activated complex, is populated with the same probability prior to the unimolecular reaction. The microcanonical rate constant k(E) is given by an expression that contains the ratio of the sum of states for the activated complex (with the reaction coordinate omitted) and the total density of states of the reactant. The canonical k(T) unimolecular rate constant is given by an expression that is similar to the transition-state theory expression of bimolecular reactions. 

Transition State theory (TST) is an extremely successful theory for chemical physics. It is simple to understand in its elementary versions and is appealing for its intuitiveness. It has been developed over the years into a whole series of theories or branches of theories, in order to make it more apt to calculate and predict kinetic factors of various chemical reactions or half reactions. Very many references appeared, dealing with transition state theories, in its many variants, and some of the articles in this series of reviews contain many of those. In particular, unimolecular decays are treated here in great detail by Rice [1]. A general review, exhaustive at its time of appearance, is Ref. 2. Some particularly stimulating references may be found in the work by Gaspard [3]. 

If colliding reactant molecules are to form products, they must first reach the top of the potential-energy barrier illustrated in Figure B.5. Transition-state theory assumes that the reacting system at the top of the barrier is a molecule (to which thermodynamics may be applied) and that this molecule, which is called the activated complex, is in chemical equilibrium with the reactants. The rate at which the activated complex decays to products then equals the reaction rate. 

From the preceding paragraph, the reader will note that many assumptions are involved in transition-state theory. Alternative derivations exhibit differing hypotheses. In a quicker but perhaps less intuitive derivation, translation in the reaction coordinate is treated formally as the low-frequency limit of a vibrational mode. Expansion of the vibrational partition function given in Section A.2.3 then yields Q = Q (k T/hv), which is substituted into equation (A-24), to be used directly in equation (66), thereby producing equation (69) when v = 1/t. The decay time thus is identified as the reciprocal of the small frequency of vibration in the direction of the reaction coordinate. 

Values for the transmission coefficient (k) at 275 K were 1.00, 1.02, 1.00 and 0.58 for FDS, D12KF1, R6D6KF1 and R12KF1, respectively. Transition-state theory assumes unity for K. Deviations of K from unity indicated poor approximation of the various transition-state thermodynamic parameters. Thus all complex decays were adequately described by transition-state theory, except for the R12KF1 peptide. 

This transition-state theory has been applied to exchange reactions of Li atoms with alkali halides (Kwei et al, 1971 Lees and Kwei, 1973) where examples of short-lived, or osculating, complexes may be found, and to beam studies of unimolecular decay (Lee et al, 1972), in particular on the extent of internal equilibration. 

In order to elucidate a mechanism, one must first consider the nature of the states initially formed by photoexcitation as well as the natures of other expected states eventually populated by internal conversion/intersystem crossing. Although it is by no means universally true, many transition metal complexes, when excited, undergo efficient relaxation to a bound, lowest energy excited state (LEES) or an ensemble of thermally equilibrated LEESs from which the various chemical processes lead to photoproducts. In such systems, the simplest model of which is illustrated by Figure 9, one can comfortably apply transition state theory to the rates and consider pressure effects in terms of the mechanisms of the individual decay LEES processes. In this case, the quantum yield of product formation would be defined by the ratio of rate constants by which the various chemical and photophysical paths for ES decay are partitioned. For Figure 9, in the absence of a bimolecular quencher Q, this would be... 

In simplistic terms, we can separate the photosubstitution chemistry according to the nature of the excited states from which this reaction occurs. First, one must consider where the ES responsible for the photochemistry is a bound, thermally relaxed state, that is, one for which a transition state theory treatment is relevant. In that case, one could, in principle, elucidate the individual rate constants kt for ES decay and determine AV values for these. This state need not be the LEES, although where this treatment proves valid, it is usually the case. 

Note that 4T/h has units of s and that the exponential is dimensionless. Thus, the expression in (3.1.17) is dimensionally correct for a first-order rate constant. For a second-order reaction, the equilibrium corresponding to (3.1.11) would have the concentrations of two reactants in the denominator on the left side and the activity coefficient for each of those species divided by the standard-state concentration, C, in the numerator on the right. Thus, C no longer divides out altogether and is carried to the first power into the denominator of the final expression. Since it normally has a unit value (usually 1 M ), its presence has no effect numerically, but it does dimensionally. The overall result is to create a prefactor having a numeric value equal to 4T/h but having units of M s as required. This point is often omitted in applications of transition state theory to processes more complicated than unimolecular decay. See Section 2.1.5 and reference 5. 

Transition state theory (also known as activated-complex theory) assumes that the transition state is much more likely to decay back to the original reactants than proceed to the stable products if this is the case, then first two reactions can be assumed to be in equilibrium. The reactive process can then be represented as... 

Multiphoton processes taking place in atoms in strong laser fields can be investigated by the non-Hermitian Floquet formalism (69-71,12). This time-independent theory is based on the equivalence of the time-dependent Schrodin-ger description to a time-independent field-dressed-atom picture, under assumption of monochromaticity, periodicity and adiabaticity (69,72). Implementation of complex coordinates within the Floquet formalism allows direct determination of the complex energy associated with the decaying state. The... 

The luminescence decay rate of ruthenium complexes doped in sol-gels does not follow a simple exponential decay. In theory, the interpretation of the excited-state dynamics of [Ru(bpy)3] has the potential to offer a wealth of information about the distribution of environments within sol-gels. Ironically, the same nonexponential behavior in decay rate constants that offers the potential for this information makes attaining detailed information all the more difficult. In general, decay rate constants are fit assuming a bimodal distribution where there is a long component that represents the bulk of the distribution and a short component that represents a different and smaller... 

In the context of the theory of decaying states for time-independenf Hamiltonians [6,37,89], the time-dependent wavefunction is in harmony with the form (1),... 

The continuous spectrum is also present, both in physical processes and in the quantum mechanical formalism, when an atomic (molecular) state is made to interact with an external electromagnetic field of appropriate frequency and strength. In conjunction with energy shifts, the normal processes involve ionization, or electron detachment, or molecular dissociation by absorption of one or more photons, or electron tunneling. Treated as stationary systems with time-independent atom - - field Hamiltonians, these problems are equivalent to the CESE scheme of a decaying state with a complex eigenvalue. For the treatment of the related MEPs, the implementation of the CESE approach has led to the state-specific, nonperturbative many-electron, many-photon (MEMP) theory [179-190] which was presented in Section 11. Its various applications include the ab initio calculation of properties from the interaction with electric and magnetic fields, of multiphoton above threshold ionization and detachment, of analysis of path interference in the ionization by di- and tri-chromatic ac-fields, of cross-sections for double electron photoionization and photodetachment, etc. 

Aspects of analysis of decaying states based on Eq. (6c) can be found in Refs. [3,10,11]. The basic formal theory concerning the connection between the time-evolution operator and the resolvent of the Hamiltonian goes back to the 1951 publication of Schbnberg [2,12]. 

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