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We contrast Dirac's theory of transition probabilities and the theory of nonadiabatic transition probabilities, applied to a perturbed system that is coupled to a bath. In Dirac's analysis, the presence of an excited state |k0⟩ in the time-dependent wave function constitutes a transition. In the nonadiabatic theory, a transition occurs when the wave function develops a term that is not adiabatically connected to the initial state. Landau and Lifshitz separated Dirac's excited-state coefficients into a term that follows the adiabatic theorem of Born and Fock and a nonadiabatic term that represents excitation across an energy gap. If the system remains coherent, the two approaches are equivalent. However, differences between the two approaches arise when coupling to a bath causes dephasing, a situation that was not treated by Dirac. For two-level model systems in static electric fields, we add relaxation terms to the Liouville equation for the time derivative of the density matrix. We contrast the results obtained from the two theories. In the analysis based on Dirac's transition probabilities, the steady state of the system is not an equilibrium state; also, the steady-state population ρkk,s increases with increasing strength of the perturbation and its value depends on the dephasing time T2. In the nonadiabatic theory, the system evolves to the thermal equilibrium with the bath. The difference is not simply due to the choice of basis because the difference remains when the results are transformed to a common basis. [ABSTRACT FROM AUTHOR]

The probability of transition to an excited state of a quantum system in a time-dependent electromagnetic field determines the energy uptake from the field. The standard expression for the transition probability has been given by Dirac. Landau and Lifshitz suggested, instead, that the adiabatic effects of a perturbation should be excluded from the transition probability, leaving an expression in terms of the nonadiabatic response. In our previous work, we have found that these two approaches yield different results while a perturbing field is acting on the system. Here, we prove, for the first time, that differences between the two approaches may persist after the perturbing fields have been completely turned off. We have designed a pair of overlapping pulses in order to establish the possibility of lasting differences, in a case with dephasing. Our work goes beyond the analysis presented by Landau and Lifshitz, since they considered only linear response and required that a constant perturbation must remain as t → ∞. First, a "plateau" pulse populates an excited rotational state and produces coherences between the ground and excited states. Then, an infrared pulse acts while the electric field of the first pulse is constant, but after dephasing has occurred. The nonadiabatic perturbation theory permits dephasing, but dephasing of the perturbed part of the wave function cannot occur within Dirac's method. When the frequencies in both pulses are on resonance, the lasting differences in the calculated transition probabilities may exceed 35%. The predicted differences are larger for off-resonant perturbations. [ABSTRACT FROM AUTHOR]

For a quantum system in a time-dependent perturbation, we prove that the variance in the energy depends entirely on the nonadiabatic transition probability amplitudes bk(t). Landau and Lifshitz introduced the nonadiabatic coefficients for the excited states of a perturbed quantum system by integrating by parts in Dirac's expressions for the coefficients ck(1)(t) of the excited states to first order in the perturbation. This separates ck(1)(t) for each state into an adiabatic term ak(1)(t) and a nonadiabatic term bk(1)(t). The adiabatic term follows the adiabatic theorem of Born and Fock; it reflects the adjustment of the initial state to the perturbation without transitions. If the response to a time-dependent perturbation is entirely adiabatic, the variance in the energy is zero. The nonadiabatic term bk(1)(t) represents actual excitations away from the initial state. As a key result of the current work, we derive the variance in the energy of the quantum system and all of the higher moments of the energy distribution using the values of |bk(t)|2 for each of the excited states along with the energy differences between the excited states and the ground state. We prove that the same variance (through second order) is obtained in terms of Dirac's excited-state coefficients ck(t). We show that the results from a standard statistical analysis of the variance are consistent with the quantum results if the probability of excitation Pk is set equal to |bk(t)|2, but not if the probability of excitation is set equal to |ck(t)|2. We illustrate the differences between the variances calculated with the two different forms of Pk for vibration–rotation transitions of HCl in the gas phase. [ABSTRACT FROM AUTHOR]

We present numerical results for the dipole induced by interactions between a hydrogen molecule and a hydrogen atom, obtained from finite-field calculations in an aug-cc-pV5Z basis at the unrestricted coupled-cluster level including all single and double excitations in the exponential operator applied to a restricted Hartree–Fock reference state, with the triple excitations treated perturbatively, i.e., UCCSD(T) level. The Cartesian components of the dipole have been computed for nine different bond lengths r of H2 ranging from 0.942 a.u. to 2.801 a.u., for 16 different separations R between the centers of mass of H2 and H between 3.0 a.u. and 10.0 a.u., and for 19 angles θ between the H2 bond vector r and the vector R from the H2 center of mass to the nucleus of the H atom, ranging from 0° to 90° in intervals of 5°. We have expanded the interaction-induced dipole as a series in the spherical harmonics of the orientation angles of the H2 bond axis and of the intermolecular vector, with coefficients DλL(r, R). For the geometrical configurations that we have studied in this work, the most important coefficients DλL(r, R) in the series expansion are D01(r, R), D21(r, R), D23(r, R), D43(r, R), and D45(r, R). We show that the ab initio results for D23(r, R) and D45(r, R) converge to the classical induction forms at large R. The convergence of D45(r, R) to the hexadecapolar induction form is demonstrated for the first time. Close agreement between the long-range ab initio values of D01(r0 = 1.449 a.u., R) and the known analytical values due to van der Waals dispersion and back induction is also demonstrated for the first time. At shorter range, D01(r, R) characterizes isotropic overlap and exchange effects, as well as dispersion. The coefficients D21(r, R) and D43(r, R) represent anisotropic overlap effects. Our results for the DλL(r, R) coefficients are useful for calculations of the line shapes for collision-induced absorption and collision-induced emission in the infrared and far-infrared by gas mixtures containing both H2 molecules and H atoms. [ABSTRACT FROM AUTHOR]

In this work, we provide values for the quadrupole moment Θ, the hexadecapole moment Φ, the dipole polarizability α, the quadrupole polarizability C , the dipole-octopole polarizability E , the second dipole hyperpolarizability γ, and the dipole-dipole-quadrupole hyperpolarizability B for the hydrogen molecule in the ground singlet state, evaluated by finite-field configuration interaction singles and doubles (CISD) and coupled-cluster singles and doubles (CCSD) methods for 26 different H–H separations r, ranging from 0.567 a.u. to 10.0 a.u. Results obtained with various large correlation-consistent basis sets are compared at the vibrationally averaged bond length r0 in the ground state. Results over the full range of r values are presented at the CISD/d-aug-cc-pV6Z level for all of the independent components of the property tensors. In general, our values agree well with previous ab initio results of high accuracy for the ranges of H–H distances that have been treated in common. To our knowledge, for H2 in the ground state, our results are the first to be reported in the literature for Φ for r > 7.0 a.u., γ and B for r > 6.0 a.u., and C and E for any H–H separation outside a narrow range around the potential minimum. Quantum Monte Carlo values of Θ have been given previously for H–H distances out to 10.0 a.u., but the statistical error is relatively large for r > 7.0 a.u. At the larger r values in this work, αxx and αzz show the expected functional forms, to leading order in r−1. As r increases further, Θ and Φ vanish, while α, γ, and the components of B converge to twice the isolated-atom values. Components of C and E diverge as r increases. Vibrationally averaged values of the properties are reported for all of the bound states (vibrational quantum numbers υ = 0–14) with rotational quantum numbers J = 0–3. [ABSTRACT FROM AUTHOR]

For a quantum system subject to a time-dependent perturbing field, Dirac's analysis gives the probability of transition to an excited state |k⟩ in terms of the norm square of the entire excited-state coefficient ck(t) in the wave function. By integrating by parts in Dirac's equation for ck(t) at first order, Landau and Lifshitz separated ck(1)(t) into an adiabatic term ak(1)(t) that characterizes the gradual adjustment of the ground state to the perturbation without transitions and a nonadiabatic term bk(1)(t) that depends explicitly on the time derivative of the perturbation at times t′ ≤ t. Landau and Lifshitz stated that the probability of transition in a pulsed perturbation is given by |bk(t)|2, rather than by |ck(t)|2. We use the term "transition probability" to refer to the probability that a true excited-state component is present in the time-evolved wave function, as opposed to a smooth modification of the initial state. In recent work, we have examined the differences between |bk(t)|2 and |ck(t)|2 when a system is perturbed by a harmonic wave in a Gaussian envelope. We showed that significant differences exist when the frequency of the harmonic wave is off-resonance with the transition frequency. In this paper, we consider Gaussian perturbations and pulses that rise via a half Gaussian shoulder to a level plateau and later return to zero via a down-going half Gaussian. While the perturbation is constant, the transition probability |bk(t)|2 does not change. By contrast, |ck(t)|2 continues to oscillate while the perturbation is constant, and its time averaged value differs from |bk(t)|2. We suggest a general type of experiment to prove that the transition probability is given by |bk(t)|2, not |ck(t)|2. We propose a ratio test that does not require accurate knowledge of transition matrix elements or absolute field intensities. [ABSTRACT FROM AUTHOR]

For a perturbed quantum system initially in the ground state, the coefficient ck(t) of excited state k in the time-dependent wave function separates into adiabatic and nonadiabatic terms. The adiabatic term ak(t) accounts for the adjustment of the original ground state to form the new ground state of the instantaneous Hamiltonian H(t), by incorporating excited states of the unperturbed Hamiltonian H0 without transitions; ak(t) follows the adiabatic theorem of Born and Fock. The nonadiabatic term bk(t) describes excitation into another quantum state k; bk(t) is obtained as an integral containing the time derivative of the perturbation. The true transition probability is given by bk(t) 2, as first stated by Landau and Lifshitz. In this work, we contrast bk(t) 2 and ck(t) 2. The latter is the norm-square of the entire excited-state coefficient which is used for the transition probability within Fermi’s golden rule. Calculations are performed for a perturbing pulse consisting of a cosine or sine wave in a Gaussian envelope. When the transition frequency ωk0 is on resonance with the frequency ω of the cosine wave, bk(t) 2 and ck(t) 2 rise almost monotonically to the same final value; the two are intertwined, but they are out of phase with each other. Off resonance (when ωk0 ≠ ω), bk(t) 2 and ck(t) 2 differ significantly during the pulse. They oscillate out of phase and reach different maxima but then fall off to equal final values after the pulse has ended, when ak(t) ≡ 0. If ωk0 < ω, bk(t) 2 generally exceeds ck(t) 2, while the opposite is true when ωk0 > ω. While the transition probability is rising, the midpoints between successive maxima and minima fit Gaussian functions of the form a exp[−b(t − d)2]. To our knowledge, this is the first analysis of nonadiabatic transition probabilities during a perturbing pulse. [ABSTRACT FROM AUTHOR]

In this paper, we show that the full Hamiltonian for a molecule in an electromagnetic field can be separated into a molecular Hamiltonian and a field Hamiltonian, both with gauge-invariant expectation values. The expectation value of the molecular Hamiltonian gives physically meaningful results for the energy of a molecule in a time-dependent applied field. In contrast, the usual partitioning of the full Hamiltonian into molecular and field terms introduces an arbitrary gauge-dependent potential into the molecular Hamiltonian and leaves a gauge-dependent form of the Hamiltonian for the field. With the usual partitioning of the Hamiltonian, this same problem of gauge dependence arises even in the absence of an applied field, as we show explicitly by considering a gauge transformation from zero applied field and zero external potentials to zero applied field, but non-zero external vector and scalar potentials. We resolve this problem and also remove the gauge dependence from the Hamiltonian for a molecule in a non-zero applied field and from the field Hamiltonian, by repartitioning the full Hamiltonian. It is possible to remove the gauge dependence because the interaction of the molecular charges with the gauge potential cancels identically with a gauge-dependent term in the usual form of the field Hamiltonian. We treat the electromagnetic field classically and treat the molecule quantum mechanically, but nonrelativistically. Our derivation starts from the Lagrangian for a set of charged particles and an electromagnetic field, with the particle coordinates, the vector potential, the scalar potential, and their time derivatives treated as the variables in the Lagrangian. We construct the full Hamiltonian using a Lagrange multiplier method originally suggested by Dirac, partition this Hamiltonian into a molecular term Hm and a field term Hf, and show that both Hm and Hf have gauge-independent expectation values. Any gauge may be chosen for the calculations; but following our partitioning, the expectation values of the molecular Hamiltonian are identical to those obtained directly in the Coulomb gauge. As a corollary of this result, the power absorbed by a molecule from a time-dependent, applied electromagnetic field is equal to the time derivative of the non-adiabatic term in the molecular energy, in any gauge. [ABSTRACT FROM AUTHOR]

The energy of a molecule subject to a time-dependent perturbation separates completely into adiabatic and non-adiabatic terms, where the adiabatic term reflects the adjustment of the ground state to the perturbation, while the non-adiabatic term accounts for the transition energy [A. Mandal and K. L. C. Hunt, J. Chem. Phys. 137, 164109 (2012)]. For a molecule perturbed by a time-dependent electromagnetic field, in this work, we show that the expectation value of the power absorbed by the molecule is equal to the time rate of change of the non-adiabatic term in the energy. The non-adiabatic term is given by the transition probability to an excited state k, multiplied by the transition energy from the ground state to k, and then summed over the excited states. The expectation value of the power absorbed by the molecule is derived from the integral over space of the scalar product of the applied electric field and the non-adiabatic current density induced in the molecule by the field. No net power is absorbed due to the action of the applied electric field on the adiabatic current density. The work done on the molecule by the applied field is the time integral of the power absorbed. The result established here shows that work done on the molecule by the applied field changes the populations of the molecular states. [ABSTRACT FROM AUTHOR]

We present quantum mechanical calculations of the collision-induced absorption spectra of nitrogen molecules, using ab initio dipole moment and potential energy surfaces. Collision-induced spectra are first calculated using the isotropic interaction approximation. Then, we improve upon these results by considering the full anisotropic interaction potential. We also develop the computationally less expensive coupled-states approximation for calculating collision-induced spectra and validate this approximation by comparing the results to numerically exact close-coupling calculations for low energies. Angular localization of the scattering wave functions due to anisotropic interactions affects the line strength at low energies by two orders of magnitude. The effect of anisotropy decreases at higher energy, which validates the isotropic interaction approximation as a high-temperature approximation for calculating collision-induced spectra. Agreement with experimental data is reasonable in the isotropic interaction approximation, and improves when the full anisotropic potential is considered. Calculated absorption coefficients are tabulated for application in atmospheric modeling. [ABSTRACT FROM AUTHOR]

Quantum chemical methods have been used elsewhere to obtain the potential energy surface (PES) and the induced dipole surface (IDS) of H2-He collisional complexes at eight different H-H bond distances, fifteen atom-molecule separations, and 19 angular orientations each [X. Li, A. Mandal, E. Miliordos, and K. L. C. Hunt, J. Chem. Phys. 136, 044320 (2012)]. An atom-molecule state-to-state scattering formalism is employed, which couples the collisional molecular complex to the electromagnetic radiation field. In this way, we obtain theoretical collision-induced absorption (CIA) spectra of H2-He complexes for frequencies from 0 to 20 000 cm-1 and temperatures up to 9000 K. The work is based on the fundamental theory and is motivated by current research of certain astronomical objects, such as cool white dwarf stars, cool main sequence stars, M dwarfs, exoplanets, so-called 'first' stars. We compare our theoretical results to existing laboratory measurements of CIA spectra; very close agreement of theory and measurement is observed. We also discuss similar previous theoretical efforts. [ABSTRACT FROM AUTHOR]

We report new ab initio results for the interaction-induced dipole moments Δμ of hydrogen molecules colliding with helium atoms. These results are needed in order to calculate collision-induced absorption spectra at high temperatures; applications include modeling the radiative profiles of very cool white dwarf stars, with temperatures from 3500 K to 9000 K. We have evaluated the dipoles based on finite-field calculations, with coupled cluster methods in MOLPRO 2006 and aug-cc-pV5Z (spdfg) basis sets for both the H and He centers. We have obtained values of Δμ for eight H2 bond lengths ranging from 0.942 a.u. to 2.801 a.u., for 15 intermolecular separations R ranging from 2.0 a.u. to 10.0 a.u., and for 19 different relative orientations. In general, our values agree well with earlier ab initio results, for the geometrical configurations that are treated in common, but we have determined more points on the collision-induced dipole surface by an order of magnitude. These results make it possible to calculate transition probabilities for molecules in excited vibrational states, overtones, and rotational transitions with ΔJ > 4. We have cast our results in the symmetry-adapted form needed for absorption line shape calculations, by expressing Δμ as a series in the spherical harmonics of the orientation angles of the intermolecular vector and of a unit vector along the H2 bond axis. The expansion coefficients depend on the H2 bond length and the intermolecular distance R. For large separations R, we show that the ab initio values of the leading coefficients converge to the predictions from perturbation theory, including both classical multipole polarization and dispersion effects. [ABSTRACT FROM AUTHOR]

In this article, we relate derivatives of the polarization propagator used in many-body theory to the nonlinear (quadratic) polarization propagator, and we relate derivatives of the quadratic polarization propagator to the nonlinear propagator of the next higher order, the cubic polarization propagator. We restrict the analysis to differentiation with respect to parameters η for which the derivative of the Hamiltonian can be written as a sum of one-electron operators. Geometrical derivatives are obtained by specializing to the parameter η to the α coordinate of nucleus I. We treat orbital relaxation explicitly by allowing for the η dependence of creation and annihilation operators in the propagators. This treatment entails an extension of the geometrical derivative relations among response functions proven by Olsen and Jo\rgensen [J. Chem. Phys. 82, 3235 (1985)], because the propagator derivatives may involve changes in the one-electron orbitals that do not appear in the susceptibility derivatives. These results underlie the relations between Raman intensities and electric-field shielding tensors, which have been explained in terms of nonlocal polarizability and hyperpolarizability densities. The results suggest an alternative computational route to geometrical or other derivatives of both linear- and nonlinear-response functions: these derivatives can be evaluated without numerical differentiation, directly from the propagator of the next higher order. [ABSTRACT FROM AUTHOR]

In this work, we prove that an intramolecular dielectric model yields accurate results for the forces between nonoverlapping molecules, at first order in the intermolecular interaction. The analysis is valid within the Born–Oppenheimer approximation. Within any perturbed molecule, a nonlocal dielectric function [variant_greek_epsilon][sub v][sup -1](r,r[sup ′]) describes the screening of external potentials due to the induced redistribution of electronic charge, i.e., this function acts as the integral kernel that determines the effective potential at point r (within linear response), when an external potential φ[sub ex](r[sup ′]) acts on the molecule, at other points r[sup ′]. The dielectric function [variant_greek_epsilon][sub v][sup -1](r,r[sup ′]) depends on the nonlocal charge-density susceptibility, which can be calculated ab initio or by density functional techniques. From quantum mechanical perturbation theory, at first order the interaction energy of two molecules is determined by the unscreened Coulomb interaction energy of the unperturbed molecular charge distributions. Yet the first-order forces on the nuclei include dielectric screening effects, due to the redistribution of the electronic charge density of each molecule in the presence of the other. This counterintuitive result follows from a relation between the charge-density susceptibility and the derivatives of the electronic charge density with respect to nuclear coordinates. The derivation provides a quantum mechanical validation for dielectric screening models on the nanoscale, when the dielectric function for electronic response is nonlocal. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Simple, new expressions relate the electronic potential energy 〈V〉 and the total electronic energy E of a molecule to its averaged electron density 〈ρ[sub e](r)〉, the nonlocal charge-density susceptibility χ[sub e](r, r′;iω), the nuclear positions {R[sub N]}, and the nuclear charges {Z[sub N]}. The expressions derived in this work are exact nonrelativistically, within the Born-Oppenheimer approximation. The results give a nonlocal form for the electronic energy in density functional theory. The virial theorem for a system with Coulomb forces is used to derive the expectation value of the kinetic energy in terms of the expectation values of the potential energy and the derivatives of the potential energy operator with respect to nuclear coordinates; gradient expansions of the kinetic energy functional are not needed. Exchange and correlation effects on 〈V〉 and E are determined by an integral of the charge-density susceptibility χ[sub e](r, r′;iω), over imaginary frequencies. The results for 〈V〈 and E are first derived by use of the fluctuation-dissipation theorem and the symmetry properties of the charge-density susceptibility with respect to a change in the sign of ω. Identical results are derived by integration of χ[sub e](r, r′, iω) over imaginary frequencies and use of the closure relation. [ABSTRACT FROM AUTHOR]

For atoms or molecules of D[SUB∞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[SUP-7] in the intermolecular separation R for Δβ, and order R[SUP-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 ∊(-Iω; Iω, 0, 0, 0, 0) obtained in this work, along with other hyperpolarizabilities calculated previously by Bishop and Pipin. For later numerical applications to molecular pairs, we have developed constant ratio approximations (CRA1 and CRA2) to estimate the dispersion effects in terms of static (hyper)polarizabilities and van der Waals energy or polarizability coefficients. Tests of the approximations against accurate results for the pairs H…H, H…He, and He…He show that the root mean square (rms) error in CRA1 is ∼20%-25% for Δβ and Δγ ; for CRA2 the error in Δβ is similar, but the rms error in Δγ is less than 4%. At separations ∼1.0 a.u. outside the van der Waals minima of the pair potentials for H…H, H…He, and He…He, the nonlinear DID interactions make the dominant contributions to Δγ[SUBzzzz] where z is the interatomic axis) and to &Delta... [ABSTRACT FROM AUTHOR]

For a nonequilibrium system described at the mesoscopic level by the master equation, we prove that the probability of fluctuations about a steady state is governed by a thermodynamic function, the ‘‘excess work.’’ The theory applies to systems with one or more nonequilibrium steady states, for reactions in a compartment that contains intermediates Xj of variable concentration, along with a reactant A and product B whose concentrations are held constant by connection of the reaction chamber to external reservoirs. We use a known relation between the stationary solution Ps(X) of the master equation and an underlying stochastic Hamiltonian H: to logarithmic accuracy, the potential that gives Ps(X) is the stochastic action S evaluated along fluctuational trajectories, obtained by solving Hamilton’s equations of motion starting at a steady state. We prove that the differential action dS equals a differential excess work d[lowercase_phi_synonym]0, and show that d[lowercase_phi_synonym]0 can be measured experimentally in terms of total free energy changes for the reaction compartment and the reservoirs. Thus we connect the probability of concentration fluctuations in an open reaction compartment to thermodynamic functions for the entire closed system containing the compartment. The excess work d[lowercase_phi_synonym]0 is the difference between the total free energy change for a specified change in the quantities of A, X, Y, and B in the state of interest, and the free energy change for the same changes in species numbers, imposed on the same system in a reference state (A,X0,Y0,B).The reference-state concentration for species Xj is derived from the momentum pj canonically conjugate to Xj along the fluctuational trajectory. For systems with linear rate laws, the reference state (A,X0,Y0,B) is the steady state, and [lowercase_phi_synonym]0 is equivalent to the deterministic excess work [lowercase&#95... [ABSTRACT FROM AUTHOR]

Coupling between continuous-flow, stirred tank reactors (CSTR’s), each having multiple steady states, can produce new steady states with different concentrations of the chemical species in each of the coupled tanks. In this work, we identify a kinetic potential ψ that governs the deterministic time evolution of coupled tank reactors, when the reaction mechanism permits a single-variable description of the states of the individual tanks; examples include the iodate-arsenous acid reaction, a cubic model suggested by Noyes, and two quintic models. Stable steady states correspond to minima of ψ, and unstable steady states to maxima or saddle points; marginally stable states typically correspond to saddle-node points. We illustrate the variation in ψ due to changes in the rate constant for external material intake (k0) and for exchange between tanks (kx). For fixed k0 values, we analyze the changes in numbers and types of steady states as kx increases from zero. We show that steady states disappear by pairwise coalescence; we also show that new steady states may appear with increasing kx, when the reaction mechanism is sufficiently complex. For fixed initial conditions, the steady state ultimately reached in a mixing experiment may depend on the exchange rate constant as a function of time, kx(t) : Adiabatic mixing is obtained in the limit of slow changes in kx(t) and instantaneous mixing in the limit as kx(t)→∞ while t remains small. Analyses based on the potential ψ predict the outcome of mixing experiments for arbitrary kx(t). We show by explicit counterexamples that a prior theory developed by Noyes does not correctly predict the instability points or the transitions between steady states of coupled tanks, to be expected in mixing experiments. We further show that the outcome of such experiments is not connected to the relative stability of steady states in individual tank reactors. We find that coupling may effectively... [ABSTRACT FROM AUTHOR]

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 F[lowercase_phi_synonym]. The quantity F[lowercase_phi_synonym] 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 F[lowercase_phi_synonym] is in general not a state function; in contrast, the function [lowercase_phi_synonym]det obtained by integrating F[lowercase_phi_synonym] along deterministic kinetic trajectories is a state function, as well as an identifiable term in the time-integrated dissipation. Unlike the total integrated dissipation, [lowercase_phi_synonym]det remains finite during the infinite duration of the system’s relaxation to a steady state and hence [lowercase_phi_synonym]det can be used to characterize that process. The variational relation δ[lowercase_phi_synonym]≥0 is shown to be a necessary and sufficient thermodynamic criterion for a stable steady state in terms of the excess work of displacement of the intermediates and [lowercase_phi_synonym]det is a Liapunov function in the domain of attraction of such steady states. Based on these results and earlier work with nonautocatalytic and equilibrating systems, we hypothesize that the stationary distribution of the master equation may be obtained in the form Ps=N exp(-[lowercase_phi_synonym]det/kT) and provide an analytical... [ABSTRACT FROM AUTHOR]

Motion of nuclei within a molecule induces a magnetic moment me in the electronic charge distribution, giving a nonzero electronic contribution to the magnetic transition dipole that produces vibrational circular dichroism. In this paper, we develop a new susceptibility density theory for the induced magnetic moment. The theory is based on the response of the electrons to changes in the nuclear Coulomb field, due to shifts in nuclear positions. The electronic response to these changes depends on the same susceptibility densities that determine response to external fields. Our analysis suggests a new physical picture of vibrational circular dichroism. It yields an equation for the density of the induced electronic magnetic moment within a molecule; it also yields a new relation connecting the electric-field shielding at nucleus I of a molecule in an applied magnetic field of frequency ω to the derivative of me with respect to the velocity of nucleus I, regarded as a parameter in the electronic wave function. Within our theory, the derivative of me with respect to nuclear velocity separates into quantum-mechanical and classical components in close analogy with the Hellmann–Feynman theorem for forces on nuclei. In matrix-element form, results from our theory are identical to those obtained with nonadiabatic perturbation theory, to leading order. In general, the leading nonadiabatic corrections to electronic properties are determined directly by the electrons’ response to the changes in the nuclear Coulomb field, when the nuclei move. [ABSTRACT FROM AUTHOR]

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 chemical potential of the intermediate and its chemical potential at a reference state (the steady state of the instantaneously equivalent nonautocatalytic system); it can also be cast kinetically in terms of reactive fluxes in the instantaneously equivalent system. Taking the product of the driving force and the net flux of each intermediate and then summing over the species gives a term in the dissipation that is specific to the intermediates. This term is minimized at nonequilibrium steady states, unlike the total dissipation (or entropy production). For the nonautocatalytic or equilibrating systems, an integral of the driving forces yields a Liapunov function for the evolution of the reaction chamber toward the steady state. The same integral also determines the stationary solution of the birth–death master equation for the species numbers of intermediates in the reaction compartment; this generalizes the Einstein relation for the probability of... [ABSTRACT FROM AUTHOR]

We prove that Liapunov functions for a single reactive intermediate evolving toward a nonequilibrium steady state can be obtained from a global potential [lowercase_phi_synonym]. We consider reactions occurring in a chamber containing a reactant, the intermediate, and a product. Reservoirs connected to the chamber serve to hold the reactant and product concentrations constant, in nonequilibrium proportions. The Liapunov property of [lowercase_phi_synonym] is significant because of the role it plays in the thermodynamic and stochastic analysis of nonequilibrium systems: [lowercase_phi_synonym] is defined in terms of the reactive flux to produce the intermediate and the flux to remove the intermediate. The derivative of [lowercase_phi_synonym] with respect to the concentration of the intermediate yields an effective chemical driving force that is specific to the intermediate, and its time derivative yields a species-specific component of the dissipation that is minimized at steady states. These results hold both near to equilibrium and far from equilibrium for systems with one intermediate, independent of the number of steady states. Local Liapunov functions are also provided by the ‘‘excess dissipation,’’ the second variation in the entropy or in the Helmholtz free energy for the reaction chamber, and quadratic functions introduced in Keizer’s fluctuation–dissipation theory. Linearization of the force and flux expansions for nonequilibrium systems yields an idealized model in which the dissipation decreases monotonically in time and thus provides a Liapunov function for evolution to steady states. This result does not hold for a chemical system approaching a steady state with an arbitrarily small, but macroscopic displacement from equilibrium, even though the series expansions of the reactive fluxes and conjugate thermodynamic forces are closely approximated by truncation at the linear... [ABSTRACT FROM AUTHOR]

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 differentiation we obtain necessary and sufficient conditions for the stability of steady states and for critical points. The integral of the species-specific affinity globally provides valid Liapunov functions for the evolution of the intermediate. These results are independent of the number of steady states of the system, and they hold both near to and far from equilibrium. For a large class of mechanisms with a single intermediate, the integral of the species-specific affinity appears in the irreversible partof the time-dependent transition probability of the single-variable Master equation and in its stationary solution. Hence for these mechanisms we obtain a direct interpretation of the stochastic results in terms of thermodynamic quantities. The time rate of change in the pressure of the intermediate multiplied by its species-specific affinity yields a species-specific term in the dissipation. The total system dissipation (or entropy production) is not in... [ABSTRACT FROM AUTHOR]

The dynamics of fluctuations in systems approaching a nonequilibrium steady state, with or without detailed balance, are investigated by means of a Lagrangian function, which is derived from the generator of time displacement (Hamiltonian) of the mesoscopic evolution equation. In the thermodynamic limit, the stationary probability distribution for the fluctuating variables is expressed in terms of the action of this stochastic Lagrangian along the fluctuational trajectory, the most probable path of infinite duration for the generation of a particular fluctuation away from the steady state. The fluctuational trajectory is related by a gaugelike transformation to the deterministic trajectory, which is the most probable path for the relaxation of the macroscopic system to the steady state. This framework is applied to the analysis of one-variable chemical reactions modeled by a constant step master equation, and to two-variable systems in the linearized region around the steady state, where the fluctuations are described by a linear Fokker–Planck equation. In these examples, the thermodynamic significance of the action along the fluctuational trajectory is established by relating the irreversible (odd under time inversion) part of the Lagrangian and the time derivative of a deterministic excess work. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

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(-[lowercase_phi_synonym]/kT), where the function [lowercase_phi_synonym] is a line integral of a differential ‘‘excess’’ work F[lowercase_phi_synonym], which depends on species-specific affinities. The differential F[lowercase_phi_synonym] 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 F[lowercase_phi_synonym]. Here, we show that use of the deterministic trajectories leads to a potential [lowercase_phi_synonym]det 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-dependent master equation in the limit of attainment of a stationary distribution. Because the time required for convergence to the stationary distribution scales as eN with the particle number N in cases with two stable steady states, the numerical work is limited to systems with O(10–100) X and Y particles. System size affects the accuracy of the approximation. To isolate system-size effects, we compare numerical solutions and the corresponding approximations to Ps(X) for two single-intermediate master equations, since the approximation becomes exact in the limit of large particle number for such equations. Based on these comparisons, for the systems with two intermediates, the agreement between the approximation and the numerical solutions is... [ABSTRACT FROM AUTHOR]