Karl F. Freed

Multiphonon Processes in the Nonradiative Decay of Large Molecules

Radiationless transition rates for polyatomic molecules are investigated in the simplest case of the statistical limit, i.e., large energy gaps between the two electronic states. By analogy with the theory of optical line shapes in solids, the “golden rule” rate expression for the nonradiative decay, which is usually written as a double sum over initial and final vibronic states, is equivalently represented in closed form as a single Fourier integral. For the case in which the vibrations are assumed to be harmonic, but may have different frequencies and equilibrium positions in the two electronic states, general closed‐form analytic expressions, which include all of the vibrational modes, are obtained for the transition rates. The energy gap law is again obtained in the weak coupling statistical limit along with a proper description of the propensity rules for the promoting mode. For the case of the aromatic hydrobarbons, the effective energy gap is found to be in agreement with Siebrands analysis, and e...

Theory of the Hyperfine Structure of Molecules: Application to 3Π States of Diatomic Molecules Intermediate between Hund's Cases (a) and (b)

In order to fully utilize the experimental accuracy of the high resolution afforded by radio‐frequency spectroscopy, a fourth‐order treatment of the energy is required. Especially, this fourth‐order treatment is expected to be necessary in order to separate the quadrupolar interactions of light atoms in molecules from pseudoquadrupolar contributions to the energy due to magnetic hyperfine interactions which are off diagonal in the rotational state. The example of 3Π states intermediate between Hunds Cases (a) and (b), where J is still a good quantum number, is chosen to demonstrate the general techniques of such a fourth‐order calculation. (Particular reference is made to the a 3Π state of isotopically substituted CO.) Van Vlecks method of reversed angular momentum is generalized to spherical tensor form in anticipation of future needs in the calculation of hyperfine structures of more complex systems. The spectroscopic parameters are explicitly written as matrix elements of the nonrelativistic wavefunc...

Intramolecular vibrational energy redistribution and the time evolution of molecular fluorescence

We note the presence of contradictory estimates of intramolecular vibrational relaxation rates in the literature where large molecules in high energy states, corresponding to huge densities of vibrational levels, have been ascribed relaxation rates orders of magnitude smaller than those assigned to smaller molecules with much smaller densities of vibrational levels. This unphysical disparity is explained as arising from vague (or undefined) definitions of intramolecular vibrational relaxation and/or from a consideration of quantities which are not directly measured or measurable. A resolution of a portion of the problem is already well known for electronic relaxation, but the application of those results to a description of the time evolution of the molecular fluorescence, produced during the intramolecular vibrational relaxation of the electronically excited molecules, requires a generalization of the electronic relaxation theory to separate and describe the ’’unrelaxed’’ and ’’relaxed’’ emission spectra...

Intramolecular perturbations and the quenching of luminescence in small molecules

Abstract We consider the problem of interpreting quenching kinetics in small molecules [e.g., CN, CH 2 , NO 2 and SO 2 ] which show significant intrinsic interactions, and argue the importance of analyzing these data in terms of collisional transitions between perturbed (mixed electronic parentage) as opposed to pure (non-spectroscopic) vibronic states. Time resolved experiments are discussed which could determine more unambiguous deactivation cross sections and spectral assignments.

Polymer viscosity in concentrated solutions

Our previous theory of polymer viscosity (which completely incorporates hydrodynamic interactions) is generalized to the case of polymer solutions at finite concentration. The specific viscosity of the polymer solution is expressed in a multiple scattering type cluster expansion which is formally in powers of the polymer concentration. We explicitly treat the case of concentrated solutions or the case of undiluted polymers in the absence of entanglements. It is shown that the presence of the other polymers leads to a screening of the intrachain hydrodynamics interactions when the natural size of a free polymer chain becomes on the order of the average volume available per chain in solution. We show how the screening is responsible for the transition from a polymer intrinsic viscosity which varies as M1/2 (M is the polymer molecular weight) in the infinitely dilute limit to a Rouse‐like specific viscosity which varies as M in the concentrated limit. It should be noted that the screening cannot be described...

Internal Rotation and the Breakdown of the Adiabatic Approximation: Many‐Phonon Radiationless Transitions

In this paper we consider the role of hindered internal rotation in the breakdown of the Born–Oppenheimer approximation. The intramolecular coupling matrix elements are derived in a form which allows the theory of many‐phonon radiationless transitions to be extended to include torsional motion. It is shown that the strongly coupled degree of freedom (that mode whose equilibrium positions differ greatly in the initial and final electronic states) can be separated out from the remaining vibrations in the general rate expressions. We treat, in particular, a model of photoisomerism in which the adiabatic potential curves describing the zero‐order states have certain simplifying features. We are able to derive in this case a simple rule governing the distribution of electronic energy amongst the internal modes in the radiationless deactivation of the lowest (twisted) triplet. In marked contrast to the usual situation encountered in the literature, this intersystem mixing with the ground state is predicted, und...

Systematic corrections to Flory–Huggins theory: Polymer–solvent–void systems and binary blend–void systems

A field theory, presented earlier by us, which is formally an exact mathematical solution of the Flory–Huggins lattice model, is used to evaluate corrections to Flory–Huggins mean field theory in a systematic series expansion in the inverse of the lattice coordination number and in the nearest‐neighbor interaction energies. We explicitly determine the first few corrections to the free energy of mixing for polymer–solvent–void systems and for systems containing two kinds of polymers and voids (binary blend–void systems). Applications of the theory to the calculation of equations of state and effective Flory χ parameters are discussed. We compare the result of our theory with the lattice Monte Carlo data of Dickman and Hall for the chain insertion probability and for the pressure in a system of athermal chains and voids. Good quantitative agreement is found. We discuss shortcomings of the lattice model in representing real polymers as well as possible extensions of the model to remedy these deficiencies.

Dissociation processes of polyatomic molecules

A general quantum mechanical theory is considered for the diverse dissociation processes of polyatomic molecules. The theory emphasizes the use of the proper normal vibrational coordinates for both the initial state of the polyatomic molecule and the final state of the dissociation fragments. The present theory provides a significant advance over all previous ones that invoke the quasidiatomic hypothesis, wherein the reaction coordinate is taken to be an initial normal mode, the remaining fragment normal modes are assumed to be identical to initial state normal modes, and the dissociation into vibrationally excited fragments is assumed to arise from the final state interactions occurring during the recoil of the fragments. Our more general theory of the vibrational and translational energy distributions in the fragments leads to a generalized Franck–Condon description of dissociation processes in terms of multidimensional bound‐continuum matrix elements which can be reduced to one‐dimensional ones that ar...

Theory of diatomic molecule photodissociation: Electronic angular momentum influence on fragment and fluorescence cross sections

We present a general theory of the photodissociation of diatomic molecules in the presence of nonadiabatic interactions between dissociative electronic states. Nonadiabatic couplings exert a profound influence on the photodissociation process when the molecule dissociates to atoms with nonvanishing electronic angular momentum, even if there are no ordinary curve crossings. Nonadiabatic interactions in this latter situation couple adiabatic molecular states which would otherwise be degenerate at infinite internuclear separation. Methods are developed for properly including nonadiabatic interactions in an exact or approximate calculation of photodissociation transition amplitudes for the production of the atomic fragments in particular fine structure levels. We derive differential cross sections for photofragment production and general expressions describing the detection of fragments by any secondary process, such as spontaneous emission, occurring during or after the breakup of the diatom. These expressio...

Investigations into Sequence and Conformational Dependence of Backbone Entropy, Inter-basin Dynamics and the Flory Isolated-pair Hypothesis for Peptides

The populations and transitions between Ramachandran basins are studied for combinations of the standard 20 amino acids in monomers, dimers and trimers using an implicit solvent Langevin dynamics algorithm and employing seven commonly used force-fields. Both the basin populations and inter-conversion rates are influenced by the nearest neighbors conformation and identity, contrary to the Flory isolated-pair hypothesis. This conclusion is robust to the choice of force-field, even though the use of different force-fields produces large variations in the populations and inter-conversion rates between the dominant helical, extended beta, and polyproline II basins. The computed variation of conformational and dynamical properties with different force-fields exceeds the difference between explicit and implicit solvent calculations using the same force-field. For all force-fields, the inter-basin transitions exhibit a directional dependence, with most transitions going through extended beta conformation, even when it is the least populated basin. The implications of these results are discussed in the context of estimates for the backbone entropy of single residues, and for the ability of all-atom simulations to reproduce experimental protein folding data.