An analysis of the rate of isomerization in molecules with an asymmetric double well potential

The classical theory of the rate of unimolecular isomerization developed by Gray andRice as extended by Zhao and Rice is applied to the calculation of the rate of isomerization in modelsystems which have linear asymmetric double well potentials. We are interested in this system fortwo reasons. First, we are interested in the detailed dynamical processes for the mentioned systembecause it is widely related to practical chemical reactions. Second, the present model systemhas an asymmetric double well potential, which provides a different test of the accuracy of theapproximations used in the Gray-Zhao-Rice theory than posed by previous applications. We havecalculated relaxation rates and relaxation times for the model systems on different time scales.We find that for the systems under studies the Gray-Zhao-Rice version of the classical theory ofisomerization rate yields values in good agreement with those generated from trajectory calculationsand from the Reactive Island theory of De Leon et al.     

Relaxation in a double potential well

Kinetic equations are deduced for the density matrix describing the relaxation in a two-level system interacting with a heat reservoir. It is assumed that the frequency of transition between the levels is small relative to the characteristic frequency of fluctuation in the reservoir; the interaction may be of any strength. The equations are used to discuss the relaxation in such a system.     

Exact solutions for interacting finite potential wells

Exact wave functions and eigenenergy relations are obtained for a biwell potential as a function of interwell displacement and single well parameters. The proliferation of energies is shown explicitly as the wells are brought into close proximity. The model is compared with the opaque division-wall model. These results find important application in laser and nuclear physics.     

Exact solutions of a new Coulomb ring-shaped potential

A new Coulomb ring-shaped potential is proposed, which results from adding a potential proportional to(cos θ/r ² sin² θ)to a Coulomb potential. The Schrödinger equation with this new model potential is separated into angular and radial components. The exactly wavefunctions and the spectrum equation for bound state are presented by the standard approach.     

Exact solutions and coherent states for a generalized potential

Exact solutions of the vibrational Schrödinger equation for a generalized potential energy function \(\hbox {V(R)}=\hbox {C}_{0}(\mathrm{{R}-\mathrm {R}}_{\mathrm{e}})^{2}/[\hbox {aR}\,+\,(\mathrm{{b}-\mathrm {a}})\hbox {R}_{\mathrm{e}}]^{2}\) are obtained. It includes those of Dunham, Ogilvie and Simons–Parr–Finlan potentials as special cases corresponding to b \(=\) 1, a \(=\) 0, 1/2, 1, respectively. The analytical wave functions derived are useful to test the quality of numerical methods or to perform perturbative or variational calculations for the problems that cannot be solved exactly. Coherent states for generalized potential, which minimize the position–momentum uncertainty relation are also constructed.     

Resonance enhanced tunneling in an asymmetric double well potential: The 2B1Σ+ state of HCl

We investigate the exact quantum tunneling dynamics using a density matrix approach in the 2B¹Σ⁺ state of HCl which has recently been calculated to be an asymmetric double well potential. With the exact dynamics and a simple model of the collisional interaction process the transfer rate shows strong resonance enhancement for particular rotational states. The transmission coefficient approach, commonly used in tunneling calculations, shows no such features. This system suggests the possibility of the experimental observation of such rotationally induced tunneling resonances (perhaps in HCl in the gas phase). Although not strongly coupled to the environment itself the large resonance effects, caused by near degeneracies, in this system suggest that in treating systems which are strongly coupled to their environment (such as in the solid state) the exact quantum dynamics should be used rather than the transmission coefficient model.     

Exact analytical expressions for the potential of electrical double layer interactions for a sphere-plate system

Exact, closed-form analytical expressions are presented for evaluating the potential energy of electrical double layer (EDL) interactions between a sphere and an infinite flat plate for three different types of interactions: constant potential, constant charge, and an intermediate case as given by the linear superposition approximation (LSA). By taking advantage of the simpler sphere-plate geometry, simplifying assumptions used in the original Derjaguin approximation (DA) for sphere-sphere interaction are avoided, yielding expressions that are more accurate and applicable over the full range of κa. These analytical expressions are significant improvements over the existing equations in the literature that are valid only for large κa because the new equations facilitate the modeling of EDL interactions between nanoscale particles and surfaces over a wide range of ionic strength.     

Slightly asymmetric double minimum potentials

Although the introduction of a small amount of asymmetry into a symmetric double minimum potential soon switches the infrared selection rule from even ↔ odd to well ↔ well, it is shown that the sum of the intensities of the transitions starting from the pair of lowest levels is insensitive to the anharmonicity. Hence if these transitions are not separately resolved, the vibrational spectrum will evolve smoothly from the symmetric to the asymmetric case. The data relating to a series of hydrogen bonded cations (B¹HB²)⁺ are used as an example.     

Exact solutions of lattice polymer models

We consider directed path models of a selection of polymer and vesicle problems. Each model is used to illustrate an important method of solving lattice path enumeration problems. In particular, the Temperley method is used for the polymer collapse problem. The ZL method is used to solve the semi-continuous vesicle model. The Constant Term method is used to solve a set of partial difference equations for the polymer adsorption problem. The Kernel method is used to solve the functional equation that arises in the polymer force problem. Finally, the Transfer Matrix method is used to solve a problem in colloid dispersions. All these methods are combinatorially similar as they all construct equations by considering the action of adding an additional column to the set of objects.     

A simple and effective technique to locate quasi-degeneracy in a symmetric double well potential

Perturbation theory based model can be used to locate the quasi-degeneracy in an arbitrary double well potential. This method, extensively explain the effect of the coupling term on pair of states called quasi-degenerate. This model helps us to calculate the energy of the pair of quasi-degenerate states using appreciably small basis. Dispersion equation corresponding to the split energy levels are presented in a very explicit form. Numerical calculation shows that the proposed method can give extremely accurate results for symmetric double-well potentials.     

Extended diffusion in a double well potential: Transition from classical to quantum regime

The transition between the classical and quantum regimes in the diffusion of a particle in a 2-4 double-well potential is treated via the strong collision model in the high-temperature limit. Both the classical and semiclassical position correlation functions, their spectra, and correlation times are evaluated using the memory function formalism. It is shown that even in the high temperature limit, marked classical-quantum transition effects appear in the observables when collisions are rare.     

Theory of transport in nanofluidic channels with moderately thin electrical double layers: effect of the wall potential modulation on solutions of symmetric and asymmetric electrolytes

Electrokinetic phenomena play an important role for the transport in submicrometer-size channels since the electric double layers formed at the walls can occupy a substantial part of the channel volume. This presents a theoretical difficulty and specific problems are usually treated numerically or not comprehensively. In our work we present a theoretical model that allows one to obtain analytical expressions for the transport of fluid (electro-osmotic flow), ions (electric current), and dissolved charged molecules (analytes). The model is based on the weak double layer approximation and has a wide range of validity. An important feature of this theoretical approach is that it is applicable not only to symmetric but also to asymmetric 2:1 and 1:2 electrolytes which exhibit very interesting properties in nanoscale channels. The possibility of affecting the wall electrokinetic zeta potential by applying a transverse voltage bias is analyzed. This transverse bias is used in an attempt to control the transport in the channel and such devices are often called "fluidic field-effect transistors." Our model quantifies the effect of the voltage bias on the zeta potential of the channel wall and therefore can be used for prediction of transport and optimization of separations in such fluidic devices.     

Sorting of brownian rods by the use of an asymmetric potential

We present here a method for sorting nanometer scale brownian rods by using a switching asymmetric periodic potential. A two stage sorting process is used to isolate particles with specific dimensions, with acceptable sorting times as well as realizable potential barrier lengths. The method was tested using computer simulations. The ability to sort the nanometer scale anisotropic particles, such as gold nanorods, portends important applications in large scale data recording, photothermal surgery, and bioimaging.     

Kinetic analysis of the thermal isomerisation pathways in an asymmetric double azobenzene switch

Here we report a photochemical and kinetic study of the thermal relaxation reaction of a double azobenzene system, in which two azobenzene photochromic units are connected via a phenyl ring. Upon UV irradiation, three thermally unstable isomers are formed. Kinetic studies using arrayed (1)H-NMR spectroscopy revealed four distinct barriers for the thermal reversion to the stable isomer. The double isomerised Z,Z-2 can revert thermally to the E,E-2 isomer via either of two isomerisation pathways. The thermal Z to E isomerisations are not significantly affected by the state of the neighbouring azo-switching unit in the meta position. These findings are supported by quantum chemical calculations on the thermal Z to E isomerisation.     

Relativistic and nonrelativistic solutions for diatomic molecules in the presence of double ring-shaped Kratzer potential

The authors investigate solutions of the three dimensional Klein-Gordon and Schrodinger equations in the presence of a new exactly solvable potential of V(r,theta)=-2De(re/r-(1/2)(re2/r2))+b/r2 sin2 theta+a/r2 cos2 theta type, the so-called double ring-shaped Kratzer potential. For a diatomic molecule system in double ring-shaped Kratzer potential, the exact bound state energy eigenvalues and corresponding wave functions have been determined within the framework of the asymptotic iteration method. Bound state eigenfunction solutions used in applications related to molecular spectroscopy are obtained in terms of confluent hypergeometric function and Jacobi polynomial. This new formulation is tested by calculating the energies of rovibrational states of a number of diatomic molecules. Also, the author-prove that in the nonrelativistic limit c-->infinity, where c is the speed of light, solutions of the Klein-Gordon system converge to those of the Schrodinger system.     

Exact solutions for kinetic models of macromolecular dynamics

Dynamic biological processes such as enzyme catalysis, molecular motor translocation, and protein and nucleic acid conformational dynamics are inherently stochastic processes. However, when such processes are studied on a nonsynchronized ensemble, the inherent fluctuations are lost, and only the average rate of the process can be measured. With the recent development of methods of single-molecule manipulation and detection, it is now possible to follow the progress of an individual molecule, measuring not just the average rate but the fluctuations in this rate as well. These fluctuations can provide a great deal of detail about the underlying kinetic cycle that governs the dynamical behavior of the system. However, extracting this information from experiments requires the ability to calculate the general properties of arbitrarily complex theoretical kinetic schemes. We present here a general technique that determines the exact analytical solution for the mean velocity and for measures of the fluctuations. We adopt a formalism based on the master equation and show how the probability density for the position of a molecular motor at a given time can be solved exactly in Fourier-Laplace space. With this analytic solution, we can then calculate the mean velocity and fluctuation-related parameters, such as the randomness parameter (a dimensionless ratio of the diffusion constant and the velocity) and the dwell time distributions, which fully characterize the fluctuations of the system, both commonly used kinetic parameters in single-molecule measurements. Furthermore, we show that this formalism allows calculation of these parameters for a much wider class of general kinetic models than demonstrated with previous methods.     

Eigenstates of double wells at varying well depths

A recipe for the accurate calculation of eigenstates of quartic double-well oscillators is presented. The behavior of eigenenergies as a function of the coupling strength is also studied. The quality of the semiclassical (WKB ) expression for eigenvalues is critically assessed in this context. The analysis also includes a discussion on the possibility of degeneracy in such one-dimensional systems. © 1995 John Wiley & Sons, Inc.     

Exact on-event expressions for discrete potential systems

The properties of systems composed of atoms interacting though discrete potentials are dictated by a series of events which occur between pairs of atoms. There are only four basic event types for pairwise discrete potentials and the square-well/shoulder systems studied here exhibit them all. Closed analytical expressions are derived for the on-event kinetic energy distribution functions for an atom, which are distinct from the Maxwell-Boltzmann distribution function. Exact expressions are derived that directly relate the pressure and temperature of equilibrium discrete potential systems to the rates of each type of event. The pressure can be determined from knowledge of only the rate of core and bounce events. The temperature is given by the ratio of the number of bounce events to the number of disassociation/association events. All these expressions are validated with event-driven molecular dynamics simulations and agree with the data within the statistical precision of the simulations.     

Exact solutions to magnetogasdynamic equations in Lagrangian coordinates

The Lie group of point transformations, which leave the equations for a simplified model of one dimensional ideal gas in magnetogasdynamics invariant, are used to obtain some exact solutions for the governing system of hyperbolic partial differential equations (PDEs). Similarity variables which reduces the governing system of PDEs into system of ordinary differential equations (ODEs) are determined through the transformations. The resulting ODEs are solved analytically to obtain some exact solutions that exhibits space-time dependence. Further, we study the propagation of weak discontinuity through a state characterized by one of the solutions.