Vibronic dephasing of anharmonic molecules. II. Impurity molecules isolated in low-temperature matrices

The quantum‐mechanical theory of vibronic dephasing presented in the first paper of this series is applied to the case of a diatomic impurity dissolved in a solid rare‐gas host. An explicit expression for the pure dephasing rate T_2′^(−1) is derived in terms of microscopic properties of the impurity and host, and the effects of variations in the parameters characterizing these properties are investigated. The expression for T_2′^(−1) is applied specifically to the system Cl_2/Ar in order to relate the results to those of previous classical‐trajectory calculations and of experimental measurements. The significance of anharmonicity in the intramolecular potential curve (of the impurity) is demonstrated

Vibrational relaxation of diatomic molecules in solids at low temperatures

The application of a hemiquantal method to the specific problem of the vibrational relaxation of a diatomic molecule embedded in a one dimensional lattice is presented. The vectorization of a CYBER 205 algorithm which integrates the 1,000 to 10,000 simultaneous hemiquantal differential equations is examined with comments on optimization. Results of the simulations are briefly discussed

Quantum Mechanics of One‐Dimensional Two‐Particle Models. Electrons Interacting in an Infinite Square Well

Solutions of Schrödinger's equation for the system of two particles bound in a one‐dimensional infinite square well and repelling each other with a Coulomb force are obtained by the method of finite differences. For the case of a 4.0‐a.u. well, the energy levels are shifted above those of the noninteracting‐particle model by as much as a factor of 4 although the excitation energies are only about 50% greater. The analytical form of the solutions is also obtained and it is shown that every eigenstate is doubly degenerate due to the "pathological'' nature of the one‐dimensional Coulomb potential. This degeneracy is verified numerically by the finite‐difference method. The properties of the model system are compared with those of the free‐electron and hard‐sphere models; perturbation and variational treatments are also carried out using the hard‐sphere Hamiltonian as a zeroth‐order approximation. The lowest several finite‐difference eigenvalues converge from below with decreasing mesh size to energies below those of the "best'' linear variational function consisting of hard‐sphere eigenfunctions. The finite‐difference solutions in general give expectation values and matrix elements more accurately than do the other approximations

Quantum-Mechanical Treatment of Inelastic Collisions. II. Exchange Reactions

The theory for the quantum-mechanical treatment of inelastic collisions developed in the first paper of this series is extended to treat collinear, electronically adiabatic exchange reactions. The formalism is applied to three model potential energy surfaces for the exchange of identical particles. The calculated reaction probabilities are reasonable and two independent checks indicate that they are reliable

Quantum-Mechanical Treatment of Inelastic Collisions. I. General Theory and Application to Nonreactive Collisions

A general method for the quantum-mechanical treatment of the inelastic collision of composite particles is presented. The method, which is applicable to both nonreactive and reactive collisions, consists of constructing the total stationary scattering wavefunction describing the collision as a linear combination of linearly independent functions which satisfy the Schödinger equation and also arbitrary boundary conditions specified in the asymptotic region. The formalism is developed for nonreactive collisions using a collinear model to simplify the mathematical treatment. In this paper, it is applied to two examples of impulsive collisions. In one case, for which a comparison is possible, calculated transition probabilities agree well with previously published values

On the Sensitivity of Granger Causality to Errors-In-Variables, Linear Transformations and Subsampling

This article studies the sensitivity of Granger causality to the addition of noise, the introduction of subsampling, and the application of causal invertible filters to weakly stationary processes. Using canonical spectral factors and Wold decompositions, we give general conditions under which additive noise or filtering distorts Granger‐causal properties by inducing (spurious) Granger causality, as well as conditions under which it does not. For the errors‐in‐variables case, we give a continuity result, which implies that: a ‘small’ noise‐to‐signal ratio entails ‘small’ distortions in Granger causality. On filtering, we give general necessary and sufficient conditions under which ‘spurious’ causal relations between (vector) time series are not induced by linear transformations of the variables involved. This also yields transformations (or filters) which can eliminate Granger causality from one vector to another one. In a number of cases, we clarify results in the existing literature, with a number of calculations streamlining some existing approaches

Entropy and Energy Profiles of Chemical Reactions

The description of chemical processes at the molecular level is often facilitated by use of reaction coordinates, or collective variables (CVs). The CV measures the progress of the reaction and allows the construction of profiles that track the evolution of a specific property as the reaction progresses. Whereas CVs are routinely used, especially alongside enhanced sampling techniques, links between profiles and thermodynamic state functions and reaction rate constants are not rigorously exploited. Here, we report a unified treatment of such reaction profiles. Tractable expressions are derived for the free-energy, internal-energy, and entropy profiles as functions of only the CV. We demonstrate the ability of this treatment to extract quantitative insight from the entropy and internal-energy profiles of various real-world physicochemical processes, including intramolecular organic reactions, ionic transport in superionic electrolytes, and molecular transport in nanoporous materials

Entropy and Energy Profiles of Chemical Reactions

The description of chemical processes at the molecular level is often facilitated by use of reaction coordinates, or collective variables (CVs). The CV measures the progress of the reaction and allows the construction of profiles that track the evolution of a specific property as the reaction progresses. Whereas CVs are routinely used, especially alongside enhanced sampling techniques, links between profiles and thermodynamic state functions and reaction rate constants are not rigorously exploited. Here, we report a unified treatment of such reaction profiles. Tractable expressions are derived for the free-energy, internal-energy, and entropy profiles as functions of only the CV.We demonstrate the ability of this treatment to extract quantitative insight from the entropy and internal-energy profiles of various real-world physicochemical processes, including intramolecular organic reactions, ionic transport in superionic electrolytes, and molecular transport in nanoporous materials.Comment: 24 pages, 5 figures, 3 table