Introduction to Thermodynamics of Irreversible Processes.

Metastable electronic states decaying via autoionization or autodetachment are common gateway states for chemical processes initiated by electron-molecule interactions or photo-excitation and are ubiquitous in highly energetic environments. We present a robust theoretical approach for calculating positions and widths of electronic resonances. The method is based on the extended multiconfigurational quasidegenerate perturbation theory combined with complex absorbing potential technique (CAP-XMCQDPT2). The theory is capable of describing the resonance position and width for shape and Feshbach resonances with high accuracy and low computational cost. Importantly, the resonance parameters are extracted at a cost of a single electronic structure calculation. Resonances positions and widths computed for shape and Feshbach molecular resonances are in a good agreement with the experimental data and with the previous theoretical estimates.

CAP-XMCQDPT2 method for molecular electronic resonances

We consider a quantum-mechanical analysis of spontaneous emission in terms of an effective two-level system with a vacuum decay rate ${\ensuremath{\Gamma}}_{0}$ and transition angular frequency ${\ensuremath{\omega}}_{A}$. Our analysis is in principle exact, even though presented as a numerical solution of the time evolution including memory effects. The results so obtained are confronted with previous discussions in the literature. In terms of the dimensionless lifetime $\ensuremath{\tau}=t{\ensuremath{\Gamma}}_{0}$ of spontaneous emission, we obtain deviations from exponential decay of the form $O(1/\ensuremath{\tau})$ for the decay amplitude as well as the previously obtained asymptotic behaviors of the form $O(1/{\ensuremath{\tau}}^{2})$ or $O(1/\ensuremath{\tau}{\mathrm{ln}}^{2}\ensuremath{\tau})$ for $\ensuremath{\tau}\ensuremath{\gg}1$. The actual asymptotic behavior depends on the adopted regularization procedure as well as on the physical parameters at hand. We show that for any reasonable range of $\ensuremath{\tau}$ and for a sufficiently large value of the required angular frequency cutoff ${\ensuremath{\omega}}_{c}$ of the electromagnetic fluctuations, i.e., ${\ensuremath{\omega}}_{c}\ensuremath{\gg}{\ensuremath{\omega}}_{A}$, one obtains either a $O(1/\ensuremath{\tau})$ or a $O(1/{\ensuremath{\tau}}^{2})$ dependence. In the presence of physical boundaries, which can change the decay rate with many orders of magnitude, the conclusions remains the same after a suitable rescaling of parameters.

Memory effects in spontaneous emission processes

Influenced by the ideas of Jaynes and Prigogine from the mid-1950s, we present a unified formulation of dynamics and thermodynamics of irreversible processes. Our approach originates in the quantum theory of resonances described by effective Hamiltonians. The concept of effective Hamiltonian is extended to the concept of effective Liouvillian that deals with macroscopic observables and brings insight into the dissipative nonequilibrium thermodynamics. The time-energy/frequency Fourier-Laplace transformation and the use of projectors focus on the variables of interest. The line profiles and dynamics in quantum mechanics are treated on the same footing. The long macroscopic times in statistical physics are derived from short microscopic times by means of hierarchies of effective Liouvillians and perturbation theory in the complex plane. The theory is illustrated on solvable models of quasi-continua and continua related to fluctuations and dissipation and on a model of kinetics of a chemical reaction implying a short-lived (resonance) transition state.

Unstable States: From Quantum Mechanics to Statistical Physics

The dynamics of quantum resonances, their energies and their widths, are investigated by means of time-dependent unitary and non-unitary transformations of the Hamiltonian. The aim of the change of representation of the Schrodinger equation is to shape the wave function of the resonances to transform non-square integrable wave functions into localized square-integrable functions. The approach has two advantages: first, it closely relates the energy of the resonances to their dynamics, second, the computational scheme should be compatible with the standard codes of quantum chemistry. The real energies are directly converted into the complex energies of the resonances by analytical continuation. The approach is applied to the discrete Fano model, to a simple model of shape resonance and to the molecular anion H.

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Extracting complex resonance energies from real calculations

A simple model of dissipative processes originally developed for magnetic materials is extended to ferroelastic materials, in particular to shape memory alloys that display a pseudoelastic behaviour. The model based on non-equilibrium dissipative dynamics leads to kinetic equations. The characteristic feature of the model is a three-level response to the acting force field describing the transition between two opposite states of saturation through the intermediate parent structural state.

A Simple Model of Dissipative Processes in Ferroelastics

The mean values of the macroscopic observables determined in the first paper of the series are used to estimate the intensive parameters of extended Gibbs probability densities. Assuming that the standard deviations of the observables are constant leads to simple exact formulae for the entropy production, thus, extending far from equilibrium, the range of validity of the classical Onsager–Prigogine theory. The regression of a fluctuation and a model of chemical reaction allow to quantify the concept of time-dependent accompanying entropy. The analysis is especially relevant for oscillating regimes. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Towards a unified formulation of dynamics and thermodynamics II. Dissipative processes and entropy production

We propose a contribution to the theory of quantum resonances that combines complex absorbing potentials (CAP) with standard perturbation theory. We start from resolvents that depend on two variables, the complex energy z and a perturbation parameter $$\lambda$$
 . The wave functions and the energies of the resonances are expanded in powers of $$\lambda$$
 . It is shown that the zero-order terms correspond to the standard CAP method and that higher-order corrections are significant. The introduction of a convergence operator allows to control the convergence of the perturbation series. Due to the discretization of the continuum, the series are generally asymptotic. Finally, we relate the perturbation series to numerically convenient Taylor series. The theory is illustrated on two model examples.

Perturbation theory of quantum resonances

We go a step further on the path opened in the mid-1950s by Prigogine and his collaborators towards a unified description of dynamics and thermodynamics of irreversible processes. Here, the theory is developed from concepts and methods originating in the quantum theory of resonances. The Liouville-von Neumann equation is solved by means of effective Liouvillians, which are similar to the effective Hamiltonians used in quantum mechanics. Hierarchies of effective Liouvillians allow us to determine the long macroscopic time scales from short microscopic characteristic times. For that purpose, standard perturbation theory is used in the complex plane. Damped and oscillating irreversible decays of a fluctuation are described by means of a two-dimensional matrix representation of the Liouvillian. Finally, we derive a kinetic equation for a simple model of a chemical reaction proceeding towards equilibrium. The model implies a transition state assimilated to a short-lived resonance

Towards a unified formulation of dynamics and thermodynamics. I. From microscopic to macroscopic time scales

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