Attila Szabo

First passage time approach to diffusion controlled reactions

Association reactions involving diffusion in one, two, and three‐dimensional finite domains governed by Smoluchowski‐type equations (e.g., interchain reaction of macromolecules, ligand binding to receptors, repressor–operator association of DNA strand) are shown to be often well described by first‐order kinetics and characterized by an average reaction (passage) time τ. An inhomogeneous differential equation is derived which, for problems with high symmetry, yields τ by simple quadrature without taking recourse to detailed cumbersome time‐dependent solutions of the original Smoluchowski equation. The cases of diffusion and nondiffusion controlled processes are included in the treatment. For reaction processes involving free diffusion and intramolecular chain motion, the validity of the passage time approximation is analyzed.

Theory of NMR relaxation in macromolecules: Restricted diffusion and jump models for multiple internal rotations in amino acid side chains

The theory required to extract detailed motional information from NMR relaxation times of nuclei in an amino acid side chain containing multiple internal rotation axes attached to a large macromolecule of at least cylindrical symmetry is developed. Emphasis is placed on the analysis of 13C‐NMR of protonated carbons where dipolar relaxation is predominant. Extension to other relaxation processes is straightforward. The spectral density from which the relaxation times can be calculated is obtained for various models for the motion of the side chain. The existing theory which assumes that internal rotations are both independent and free is generalized to incorporate excluded volume effects in a heuristic way by restricting the amplitude of the internal rotations. It is found that small amplitude motions are ineffective in causing relaxation. Thus jump models involving a relatively small number of configurations are appropriate to describe the motion of the side chain. The advantage of jump models is twofold:...

Dynamics of reactions involving diffusive barrier crossing

We develop a first passage time description for the kinetics of reactions involving diffusive barrier crossing in a bistable (and also in a more general) potential, a situation realized, for example, in some photoisomerization processes. In case the reactant is in thermal equilibrium, the first passage times account well for the reaction dynamics as shown by comparison with exact numerical calculations. A simple integral expression for the rate constants is presented. For a case involving a reactant initially far off equilibrium, a two relaxation time description for the particle number N(t) is derived and compared with the results of an ’’exact’’ calculation. This description results from a knowledge of N(t = 0), Ṅ(t = 0), F∞0dt N(t), i.e., the first passage time, and F∞0dt t N(t).

Theory of polarized fluorescent emission in uniaxial liquid crystals

The theory for the fluorescence emission anisotropy [r(t)] of a cylindrical probe in a macroscopically aligned uniaxial liquid crystal (e.g., an oriented membrane) is developed for the situation where the equilibrium orientational distribution of the probes is not random. Expressions are derived for r(0) and r(∞) involving equilibrium averages for the general case where the emission and absorption dipoles and the major axis of the probe are not parallel. The rotational motion of the probe is described as diffusion in a potential which is consistent with the equilibrium distribution (i.e., using the Smoluchowski equation). Both the short time behavior and the time integral of r(t) are explicitly expressed in terms of equilibrium averages. It is shown how these limits can be used to construct useful approximations for the time dependence of r(t), and how the diffusion constant can be obtained for the short time behavior of r(t).

The correlation energy in the random phase approximation: Intermolecular forces between closed‐shell systems

A new expression for the correlation energy within the random phase approximation (RPA) is presented. It has the following properties: it is (1) size consistent, (2) invariant to unitary transformations of degenerate orbitals, (3) correct to second order in perturbation theory, and (4) when applied to a supermolecule comprised of two interacting closed‐shells, it describes the dispersive part of the interaction at the coupled Hartree–Fock (HF) level, i.e., the van der Waals’ coefficient extracted from its long‐range behavior is identical to that obtained from the Casimir–Polder expression using the dynamic coupled Hartree–Fock polarizabilities of the isolated systems. This expression, which requires only particle–hole two‐electron integrals for its evaluation, is expected to yield considerably more accurate potential energy curves between closed‐shell systems than second‐order Moller–Plesset perturbation theory which, as is shown, describes dispersion forces at the less accurate uncoupled HF level. In add...

Localized partial traps in diffusion processes and random walks

Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Greens function can be analytically expressed in terms of the Greens function for diffusion in the absence of reaction. Moreover, a simple relation between the Greens functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.

On the second-moment condition of Stillinger and Lovett

The Stillinger-Lovett second-moment condition of electrolyte solutions is derived rigorously and simply from only some reasonable (but apparently never proven rigorously) assumptions concerning the asymptotic form of the direct correlation function and the Ornstein-Zernike equation. The derivation suggests that this condition is not the first member of a hierarchy of moment conditions and that there exists no simple result for a fourth-moment condition.

Length dependence of excitation energies in linear polyenes: Localized and delocalized descriptions

Abstract The length dependence of the lowest allowed transition energy of linear polyenes is studied using delocalized SCF and localized excitonic approaches. Within the PPP SCF approximations the calculated transition energies converge to a finite values as N −1 as the number of double bonds ( N ) becomes large, when the excited state contains all singly excited configurations. On the other hand, the fully localized excitonic method at the level of single excitations, although it predicts a gap in the excitation spectrum of an infinite polyene, gives results which converge to this value as N −2 . The inclusion of double and triple excitations into the excitonic method by means of perturbation theory does not appear to change this behavior. The reasons for the discrepancy between the two approaches is analyzed. Experimentally, the transition energies in solution converge to a finite value as N −1 to a good degree of approximation. If it is assumed that the solvent shift is constant for long polyenes, the available experimental results favour the delocalized approach as the starting point in describing the length dependence of the excitation energy of long polyenes.

Analytic Coulomb approximations for dynamic multipole polarizabilities and dispersion forces

This paper presents a comprehensive and unified treatment of atomic multipole oscillator strengths, dynamic multipole polarizabilities, and dispersion force constants in a variety of Coulomb‐like approximations. A theoretically and computationally superior modification of the original Bates–Damgaard (BD) procedure, referred to here simply as the Coulomb approximation (CA), is introduced. An analytic expression for the dynamic multipole polarizability is found which contains as special cases this quantity within the CA, the extended Coulomb approximation (ECA) of Adelman and Szabo, and the quantum defect orbital (QDO) method of Simons. This expression contains model‐dependent parameters determined from ground and excited state ionization potentials and is derived using a powerful approach based on the sturmian representation of a generalized Coulomb Green’s function. In addition, this result is obtained within the ECA and QDO models through an extension of the novel algebraic procedure previously used in o...

Reactions governed by a binomial redistribution process—The ehrenfest urn problem☆

A distributive process of the binomial type in a one-dimensional discrete space with an absorbing barrier is studied. A simple expression for the particle number Σ(t) is derived. The analysis is based on recursion relationships and sum rules for the underlying eigenvectors, the Krawtchouk polynomials. The first passage time is determined, and the validity of the passage time approximation to Σ(t) tested. The continuous limit, corresponding to the diffusion and reaction of a harmonically bound particle, is briefly described.