Katharine L. C. Hunt

Long-range dipoles, quadrupoles, and hyperpolarizabilities of interacting inert-gas atoms

Abstract A model relating the long-range dipoles of inert-gas heterodiatoms to the distortion of the electronic structure associated with dispersion forces is presented. Expressions are given for the contributions to the dipole varying as the inverse seventh power of the atomic separation R . The model is also used to determine the long-range quadrupoles of homonuclear atomic pairs. Pair hyperpolarizabilities are obtained in the model by adding contributions of classical multipolar interactions and contributions associated with the distortion of the electronic structure. An independent test shows the consistency of the model and provides values for the effective mean-square dipoles and quadrupoles of the inert-gas atoms.

Path integral solutions of stochastic equations for nonlinear irreversible processes: The uniqueness of the thermodynamic Lagrangian

The conditional propagator associated with a generalized Langevin equation or Fokker–Planck equation is represented as a functional integral over paths, each exponentially weighted according to the time integral of the thermodynamic Lagrangian along the path. A principal result of this paper is the proof that the thermodynamic Lagrangian is unique for single‐variable processes or for multivariable processes in flat spaces, contrary to earlier statements that the Lagrangian depends on an arbitrary parameter, often denoted α. Fixing the parameter α is equivalent to choosing a stochastic calculus (α = 0 corresponds to the Ito calculus and α = 1/2 to the Stratonovich calculus), but consistent operations with any stochastic calculus selected lead to the same Lagrangian. The Euler–Lagrange equations derived by a variational minimization of the thermodynamic action (the time integral of the Lagrangian) are also unique; these equations give the most heavily weighted path between any two fixed points in the space ...

Nonlocal polarizability densities and van der Waals interactions

The van der Waals interaction energy of a molecular pair is derived within a simple physical model that represents the distribution of polarizable matter in each molecule by the use of nonlocal polarizability densities. In the model the field due to the instantaneous polarization of one molecule polarizes the second molecule nonlocally, thereby producing a reaction field that acts on the first molecule. The resulting energy change depends upon the averaged product of the fluctuating polarization at different points in the first molecule; by the fluctuation–dissipation theorem, this product is related to the imaginary part of the nonlocal polarizability density. The model gives the van der Waals energy as an integral of a contracted tensor product of two dipole propagators and the imaginary‐frequency nonlocal polarizabilities of the interacting molecules. The validity of the model is established by comparison with results from second‐order perturbation theory.

Thermodynamics far from equilibrium: Reactions with multiple stationary states

We present a thermodynamic analysis of global validity for effectively one‐variable, irreversible chemical systems with multiple steady states. A hypothetical reaction chamber is held at constant temperature and volume and is connected by selectively permeable membranes to reservoirs of reactant(s) and product(s), each at constant selected pressures. An appropriate free energy function, which yields criteria of evolution to equilibrium for the composite system of reaction chamber and reservoirs, is a hybrid of Gibbs and Helmholtz free energies. The one variable in the reaction chamber is the pressure of a chemical intermediate which varies in time according to a given reaction mechanism. With the hybrid free energy, the kinetics for a given mechanism, and a concept of instantaneous indistinguishability of systems with different mechanisms, we establish a thermodynamic driving force, or species‐specific affinity, for the intermediate. The species‐specific affinity vanishes at steady states, and upon its di...

Thermodynamic and stochastic theory for nonequilibrium systems with multiple reactive intermediates: The concept and role of excess work

We continue our development of a global thermodynamic and stochastic theory of open chemical systems far from equilibrium with an analysis of a broad class of isothermal, multicomponent reaction mechanisms with multiple steady states, studied under the assumption of local equilibrium. We generalize species‐specific affinities of reaction intermediates, obtained in prior work for nonautocatalytic reaction mechanisms, to autocatalytic kinetics and define with these affinities an excess free energy differential Fφ. The quantity Fφ is the difference between the work required to reverse a spontaneous concentration change and the work available when the same concentration change is imposed on a system in a reference steady state. The integral of Fφ is in general not a state function; in contrast, the function φdet obtained by integrating Fφ along deterministic kinetic trajectories is a state function, as well as an identifiable term in the time‐integrated dissipation. Unlike the total integrated dissipation, φd...

Long‐range, collision‐induced hyperpolarizabilities of atoms or centrosymmetric linear molecules: Theory and numerical results for pairs containing H or He

For atoms or molecules of D∞h or higher symmetry, this work gives equations for the long‐range, collision‐induced changes in the first (Δβ) and second (Δγ) hyperpolarizabilities, complete to order R−7 in the intermolecular separation R for Δβ, and order R−6 for Δγ. The results include nonlinear dipole‐induced‐dipole (DID) interactions, higher multipole induction, induction due to the nonuniformity of the local fields, back induction, and dispersion. For pairs containing H or He, we have used ab initio values of the static (hyper)polarizabilities to obtain numerical results for the induction terms in Δβ and Δγ. For dispersion effects, we have derived analytic results in the form of integrals of the dynamic (hyper)polarizabilities over imaginary frequencies, and we have evaluated these numerically for the pairs H...H, H...He, and He...He using the values of the fourth dipole hyperpolarizability e(−iω; iω, 0, 0, 0, 0) obtained in this work, along with other hyperpolarizabilities calculated previously by Bish...

Thermodynamic and stochastic theory for nonequilibrium systems with more than one reactive intermediate: Nonautocatalytic or equilibrating systems

We consider chemical reactions occurring in a compartment separated by semipermeable membranes from reservoirs of reactant and product, both held at constant pressure. In earlier work, we have developed a nonequilibrium thermodynamic theory applicable to systems with a single reactive intermediate, and we have established a link between the thermodynamic and stochastic analyses of such systems. Here we show that our results generalize directly to cases with two or more reactive intermediates, if the reaction mechanism is nonautocatalytic, or if the system is evolving toward an equilibrium steady state in the reaction compartment without first exhausting the reactant or product reservoir. Starting with nonautocatalytic mechanisms, we identify effective driving forces for each intermediate; we then obtain the driving force for an arbitrary system by mapping to an instantaneously equivalent nonautocatalytic system. The driving force can be cast thermodynamically in terms of the difference between the actual ...

Dissipation in steady states of chemical systems and deviations from minimum entropy production

According to the linear thermodynamic theory of irreversible processes, entropy production is minimized at the steady states of nonequilibrium systems. The principle of minimum entropy production is obtained if the dissipation is expanded in the deviation δ from equilibrium, truncated to lowest order (δ2), and then differentiated. At this level of apprximation, the derivative of the dissipation is linear in δ and vanishes at the steady state. For a simple chemical reaction mechanism in a well-defined model system, we have derived the corrections to this approximation, by first evaluating the dissipation and its derivative exactly, and then expanding as a series in δ. To leading order in δ, the ratio of the derivative to the dissipation at the steady state is 12 (in dimensionless units), and this ratio does not decrease as equilibrium is approached. The steady state coincides with the state of minimum entropy production only at equilibrium. Given sufficient accuracy in measurements of species concentrations, the breakdown of the principle of minimum entropy production can be detected experimentally when the relative standard deviation in measurements of the dissipation is less than δ. Our example shows that the dissipation in a reaching chemical system may increase in time, in the later stages of relaxation toward a near-equilibrium steady state.

Stationary solutions of the master equation for single and multi-intermediate autocatalytic chemical systems

In this work, we test a hypothesized form for the stationary solution Ps(X,Y) of the stochastic master equation for a reacting chemical system with two reactive intermediates X and Y, and multiple steady states. Thermodynamic analyses and the exact results for nonautocatalytic or equilibrating systems suggest an approximation of the form Pas(X,Y)=N exp(−φ/kT), where the function φ is a line integral of a differential ‘‘excess’’ work Fφ, which depends on species‐specific affinities. The differential Fφ is inexact. In a preceding paper, we have given an analytic argument for the use of the deterministic kinetic trajectory, connecting (X,Y) to the steady state (Xs,Ys) as the path of integration for Fφ. Here, we show that use of the deterministic trajectories leads to a potential φdet which is continuous across the separatrix between the domains of attraction of the two stable steady states in the model studied.We compare the approximate form of Ps(X,Y) thus generated with numerical solutions of the time‐depe...

Infrared atmospheric emission and absorption by simple molecular complexes, from first principles

Quantum chemical methods are used to obtain the interaction-induced dipole surfaces (IDS) of complexes of two interacting (i.e. colliding) molecules, for example H2–H2, H2–He, etc., collisional complexes, along with their potential energy surfaces (PES). Eight H2 bond distances, from 0.942 to 2.801 bohr, are chosen for each H2 molecule to account for rotovibrational excitations. Rotovibrational matrix elements of these ID and PE surfaces are computed as necessary for the study of supermolecular (‘collision-induced’) absorption spectra of dense hydrogen gas, and of gaseous mixtures of hydrogen and helium, at temperatures up to several thousand kelvin and for frequencies from 0 to those of several H2 overtone bands. Rotovibrational state to state scattering calculations couple the collisional complex perturbatively to single photons. The absorption process causes rotovibrational transitions in one molecule, or simultaneous transitions in both molecules (when H2–H2 collisional complexes are considered). The spectral profiles of tens of thousands of such transitions are computed from first principles. Individual ‘lines’ are very broad so that they overlap substantially, forming a supermolecular quasi-continuum. The comparison of the computed collision-induced absorption (CIA) spectra with existing laboratory measurements at low temperatures (≤ 300 K) shows close agreement so that our results for higher temperatures, where laboratory experiments do not exist, may be used with confidence. Similar calculations of CIA spectra at high temperatures and frequencies are underway for other collisional systems (e.g. H2–H) of interest in astrophysical applications (e.g. ‘cool’ stellar atmospheres). Collision-induced Raman spectra (CIRS) have been similarly obtained; computed Raman spectra also compare favourably with existing laboratory measurements.