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 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. 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

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

Calculation of transition probabilities for collinear atom–diatom and diatom–diatom collisions with Lennard-Jones interaction

Numerical integration of the close coupled scattering equations is performed to obtain vibrational transition probabilities for three models of the electronically adiabatic H2-H2 collision. All three models use a Lennard-Jones interaction potential between the nearest atoms in the collision partners. The results are analyzed for some insight into the vibrational excitation process, including the effects of anharmonicities in the molecular vibration and of the internal structure (or lack of it) in one of the molecules. Conclusions are drawn on the value of similar model calculations. Among them is the conclusion that the replacement of earlier and simpler models of the interaction potential by the Lennard-Jones potential adds very little realism for all the complication it introduces

From free-energy profiles to activation free energies

Given a chemical reaction going from reactant (R) to the product (P) on a potential energy surface (PES) and a collective variable (CV) discriminating between R and P, we define the free-energy profile (FEP) as the logarithm of the marginal Boltzmann distribution of the CV. This FEP is not a true free energy. Nevertheless, it is common to treat the FEP as the “free-energy” analog of the minimum potential energy path and to take the activation free energy, ΔF‡ RP, as the difference between the maximum at the transition state and the minimum at R. We show that this approximation can result in large errors. The FEP depends on the CV and is, therefore, not unique. For the same reaction, different discriminating CVs can yield different ΔF‡ RP. We derive an exact expression for the activation free energy that avoids this ambiguity. We find ΔF‡ RP to be a combination of the probability of the system being in the reactant state, the probability density on the dividing surface, and the thermal de Broglie wavelength associated with the transition. We apply our formalism to simple analytic models and realistic chemical systems and show that the FEP-based approximation applies only at low temperatures for CVs with a small effective mass. Most chemical reactions occur on complex, high-dimensional PES that cannot be treated analytically and pose the added challenge of choosing a good CV. We study the influence of that choice and find that, while the reaction free energy is largely unaffected, ΔF‡ RP is quite sensitive

Quantum theory of concerted electronic and nuclear fluxes associated with adiabatic intramolecular processes

An elementary molecular process can be characterized by the flow of particles (i.e., electrons and nuclei) that compose the system. The flow, in turn, is quantitatively described by the flux (i.e., the time-sequence of maps of the rate of flow of particles though specified surfaces of observation) or, in more detail, by the flux density. The quantum theory of concerted electronic and nuclear fluxes (CENFs) associated with electronically adiabatic intramolecular processes is presented. In particular, it is emphasized how the electronic continuity equation can be employed to circumvent the failure of the Born–Oppenheimer approximation, which always predicts a vanishing electronic flux density (EFD). It is also shown that all CENFs accompanying coherent tunnelling between equivalent “reactant” and “product” configurations of isolated molecules are synchronous. The theory is applied to three systems of increasing complexity. The first application is to vibrating, aligned H2+(2Σg+), or vibrating and dissociating H2+(2Σg+, J = 0, M = 0). The EFD maps manifest a rich and surprising structure in this simplest of systems; for example, they show that the EFD is not necessarily synchronous with the nuclear flux density and can alternate in direction several times over the length of the molecule. The second application is to coherent tunnelling isomerization in the model inorganic system B4, in which all CENFs are synchronous. The contributions of core and valence electrons to the EFD are separately computed and it is found that core electrons flow with the nuclei, whereas the valence electrons flow obliquely to the core electrons in distinctive patterns. The third application is to the Cope rearrangement of semibullvalene, which also involves coherent tunnelling. An especially interesting discovery is that the so-called “pericyclic” electrons do not behave in the manner typically portrayed by the traditional Lewis structures with appended arrows. Indeed, it is found that only about 3 pericyclic electrons flow, in contrast to the 6 predicted by the Lewis picture. It is remarkable that the time scales of these three processes vary by 18 orders of magnitude: femtoseconds (H2+(2Σg+)); picoseconds (B4); kilosceconds (semibullvalene). It is emphasized that results presented herein are appearing in the literature for the first time