Modeling the Non-Equilibrium Behavior of Chemically Reactive Atomistic Level Systems Using Steepest-Entropy-Ascent Quantum Thermodynamics

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1 Modeling the Non-Equilibrium Behavior of Chemically Reactive Atomistic Level Systems Using Steepest-Entropy-Ascent Quantum Thermodynamics Omar Al-Abbasi Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy In Mechanical Engineering Michael von Spakovsky, Chair Gian Paolo Beretta Daniel Crawford Douglas Nelson Mark Paul Scott Huxtable October 8, 3 Blacksburg, Virginia Keywords: Steepest entropy ascent, chemical reaction, quantum thermodynamics. Copyright 3, Omar Al-Abbasi

2 Abstract Predicting the kinetics of a chemical reaction is a challenging task, particularly for systems in states far from equilibrium. This work discusses the use of a relatively new theory known as intrinsic quantum thermodynamics (IQT) and its mathematical framework steepest-entropy-ascent quantum thermodynamics (SEA-QT) to predict the reaction kinetics at atomistic levels of chemically reactive systems in the non-equilibrium realm. IQT has emerged over the last three decades as the theory that not only unifies two of the three theories of physical reality, namely, quantum mechanics (QM), and thermodynamics but as well provides a physical basis for both the entropy and entropy production. The SEA-QT framework is able to describe the evolution in state of a system undergoing a dissipative process based on the principle of steepest-entropy ascent or locally-maximal-entropy generation. The work presented in this dissertation demonstrates for the first time the use of the SEA-QT framework to model the evolution in state of a chemically reactive system as its state relaxes to stable equilibrium. This framework brings a number of benefits to the field of reaction kinetics. Among these is the ability to predict the unique non-equilibrium (kinetic) thermodynamic path which the state of the system follows in relaxing to stable equilibrium. As a consequence, the reaction rate kinetics at every instant of time is known as are the chemical affinities, the reaction coordinates, the direction of reaction, the activation energies, the entropy, the entropy production, etc. All is accomplished without any limiting assumption of stable or pseudo-stable equilibrium. The objective of this work is to implement the SEA-QT framework to describe the chemical reaction process as a dissipative one governed by the laws of quantum mechanics and thermodynamics and to extract thermodynamic properties for states that are far from equilibrium. The F+H HF+H and H+F HF+F reaction mechanisms are used as model problems to implement this framework. ii

3 Acknowledgements It was a great opportunity to work with Dr. von Spakovsky in the field of non-equilibrium thermodynamics where I have been able to see thermodynamics from a broader perspective. I would like to express my sincere gratitude to my advisor whose encouragement and kind help has made this work possible. I would like to express my gratitude to Dr. Beretta as well, who is the inventor and the originator of the equation of motion used in this work and the person who is pushing this theory to a new and higher level. In addition, I would like to extend my gratefulness to all the members of my committee for their time and helpful comments: Dr. Daniel Crawford, Dr. Mark Paul, Dr. Douglas Nelson and Dr. Scott Huxtable. I wish to express my gratitude to the University of Bahrain for its support and sponsorship during my graduate studies, which greatly contributed to the successful completion of the present research. I would like to thank my colleagues, who are working in this theory as well, Dr. Charles E. Smith, Aimen Younis, Sergio Cano, Alejandro Fuentes and a special thanks goes to my friend Guanchen Li for his insightful comments and ideas. Also, I would like to thank my friend Dr. Irfan Zardadkhan for his help in developing some of the algorithms used in this work. My thanks go to my family for their love and continuous support throughout my time in graduate school. iii

4 Table of Contents Abstract... ii Acknowledgements... iii Table of Contents... iv Table of Figures... vii List of Tables... xiv Chapter Introduction.... Introduction.... Reaction Path and Steepest-Entropy-Ascent Quantum Thermodynamics (SEA-QT)....3 Kinetics and Entropy of a Reactive System at the Microscopic and Mesoscopic Scales Research Objectives and Original Contributions... 6 Chapter - Literature Review Chemical Reaction Path The Potential Energy Surface The Minimum Energy Path Principle Chemical Reaction Rate Simple Collision Theory Trajectory Calculation Method Transition State Theory Cumulative Reaction Probability Method Quantum Scattering Theory Quantum Effects and Dissipation Quantum Statistical Mechanics (QSM) Nonlinear Schrödinger Equation of Motion... 4 iv

5 .3.3. Dissipative Quantum Dynamics (DQD) Intrinsic Quantum Thermodynamics... 6 Chapter 3 Model and Theory Energy and Particle Occupation Number Eigenvalue Problems One-particle Energy Eigenvalue Problems Compatible Compositions Occupation Coefficients Hamiltonian Operator Occupation Number Operators Reaction Coordinate Operators Equation of Motion of SEA-QT Expectation Values of the Energy, the Occupation Number and the Reaction Coordinate Chapter 4 Numerical Approach The One-Particle Energy Eigenvalue Problem Occupation Coefficients System Initial Conditions Solution Procedure and Convergence Criteria Code Verification Verifying the Density Operator Verifying the Hamiltonian Operator Verifying the conservation of Number of Particles Reaction Rate Constant Thermodynamic Properties for Non-Equilibrium States Chapter 5 Results and Discussions... 6 v

6 5. Hydrogen Molecule and Fluorine Atom Reaction Mechanism One Reaction Mechanism Effect of the Number of Levels Occupied Initially Effect of the Number of Translation Degrees of Freedom Effect of the Number of Internal Degrees of Freedom Fitting the relaxation time τ to experiments Two Reaction Mechanisms The Activation Energy Energy Transfer between Energy Modes Thermodynamic Properties for Non-Equilibrium States Relaxation Time Functional Fluorine Molecule and Hydrogen Atom Reaction Mechanism... Chapter 6 Conclusions and Recommendations... References... vi

7 Table of Figures Figure. Two-dimensional cut in the energy (E)-entropy (S) plane of the hypersurface of stable equilibrium states (of which the curve is a two-dimensional cut) for a thermodynamic system with translational degrees of freedom; is the set of all relevant parameters (e.g., volume, electrostatic field strength, gravity, etc.) that define the system and n vii the set of constituent amounts.... Figure. Role of the Gibbs free energy and the progress of a chemical reaction Figure.3 Depiction of the possible progress of a chemical reaction, which SEA-QT captures without an a priori assumption of forward-backward progress Figure. Depiction of a simple reaction mechanism and its possible configurations on the PES Figure. Depiction of a MEP from reactants to products on a PES... Figure.3 Depiction of different images for the Nudged Elastic Band (NEB) method.... Figure.4 Depiction of two colliding particles as hard-spheres... 3 Figure.5 Three particles colliding in a chemical reaction Figure.6 Depiction of initial conditions and orientations of the A+BC reaction Figure.7 Depiction of a PES with different choices of dividing surface S and S Figure.8 A state of non-zero entropy represented in QSM by an ensemble of identical systems in different pure states Figure.9 Two-dimensional cut in the energy (E)-entropy (S) plane of the hypersurface of stable equilibrium states (of which the curve is a two-dimensional cut) for a thermodynamic system with translational degrees of freedom; is the set of all relevant parameters (e.g., volume, electrostatic field strength, gravity, etc.) that define the system and n the set of constituent amounts Figure 3. Depiction for the SEA-QT principle with b the gradient vector of the entropy change, bl the projection of vector b onto the manifold L that encompasses the constraints g and g on the system (i.e., conservation of the number of atoms and energy), and b the projection of b perpendicular to the manifold Figure 3. General overview of the implementation of the SEA-QT framework for a chemically reactive system Figure 3.3 Depiction of a possible occupation of particles in an internal energy configuration. 33 Figure 4. The algorithm for the restricted integer partitioning

8 Figure 4. Depiction of the construction the unique permutations of rows of is Figure 4.3 Partially canonical states for the initial composition of system A (dashed blue curve), stable equilibrium states for the initial composition of A in surrogate system B (green solid curve) and stable equilibrium states for the final composition of A in surrogate system B (black solid curve) Figure 5. Overlapping of the energies of diatomic molecules H and HF Figure 5. Initial and final and states of the chemically reactive system at 3 K for the F+H reaction mechanism and the parameter values given in Tables Figure 5.3 The instantaneous entropy values for the F+H reaction process which is at an initial stable equilibrium of 3 K Figure 5.4 The instantaneous entropy generation values for the F+H reactive system which is at an initial stable equilibrium temperature of 3 K Figure 5.5 The expectation values of the particle number operator for each species of the F+H reaction corresponding to an initial stable equilibrium of temperature at 3 K Figure 5.6 The expectation energies for each species of the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5.7 The expectation values for the system entropy and the energy, and number of particles for each species of the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5.8 The instantaneous expectation values of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5.9 The instantaneous expectation values of the rate of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5. The forward and backward reaction rate constants as well as the equilibrium constant as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5. The forward, backward, and net reaction rates as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5. Conservation of the number of particles in the F+H reaction viii

9 Figure 5.3 Initial and final states of the F+H reactive system corresponding to an initial stable equilibrium temperature of 3 K and a different number of system eigenlevels initially occupied Figure 5.4 The instantaneous entropy values for different values of for the F+H reaction process corresponding to an initial stable equilibrium of Figure 5.5 The instantaneous entropy generation values for different values of for the F+H reaction process corresponding to an initial stable equilibrium of Figure 5.6 The expectation values for different values of of the particle number operator for each species of the F+H reaction corresponding to an initial stable equilibrium of temperature at Figure 5.7 The expectation values for different values of of the energy for each species of the F+H reaction corresponding to an initial stable equilibrium of temperature at Figure 5.8 The instantaneous expectation values for different values of of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5.9 The instantaneous expectation values for different values of of the rate of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5. Forward and backward reaction rate constants for different values of as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K Figure 5. The forward, backward, and net reaction rates as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K for different number of occupied system eigenlevels initially Figure 5. Initial and final states of the F+H reactive system corresponding to an initial stable equilibrium temperature of 3 K for samplings of and 5 translational quantum given in Table Figure 5.3 The instantaneous expectation values of the entropy for the F+H reaction corresponding to at an initial stable equilibrium of 3 K are plotted for different sampling number of the translational quantum numbers ix

10 Figure 5.4 The instantaneous expectation values of the entropy generation for the F+H reaction corresponding to at an initial stable equilibrium of 3 K are plotted for different sampling number of the translational quantum numbers Figure 5.5 The expectation values of the particle number operator for each species for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K plotted for different sampling numbers of translational quantum numbers Figure 5.6 The instantaneous expectation values of the reaction coordinate operator for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K plotted for different sampling numbers of the translational quantum numbers Figure 5.7 The expectation values of the rate of reaction coordinate operator corresponding to an initial reaction temperature at 3 K plotted for different sampling numbers of translational quantum numbers Figure 5.8 Forward and backward reaction rate constants as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K plotted for different sampling numbers of the translational quantum numbers Figure 5.9 The instantaneous expectation values of the entropy for the F+H reaction corresponding to an initial equilibrium of 3 K and for the varying internal energy eigenstructure given in Table Figure 5.3 The instantaneous expectation values of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K and for the internal eigenstructures given in Table Figure 5.3 The instantaneous expectation values of the rate of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K and for the internal eigenstructures given in Table Figure 5.3 The instantaneous expectation values of the entropy for the F+H reaction corresponding to an initial stable equilibrium of 3 K and for two values of τ Figure 5.33 The instantaneous expectation values of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium of 3 K and for two values of τ Figure 5.34 The forward, backward, and net reaction rates as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K and for different values of τ x

11 Figure 5.35 Fitted values for τ for the F+H reaction corresponding to different initial stable equilibrium temperatures and for the forward reaction rate constant of Heidner et al. [89] Figure 5.36 The limits on the compatible compositions for the F+H two-reaction-mechanisms system Figure 5.37 Initial and finial states of the chemically reactive system and its state evolution for the F+H two-reaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K and the parameter values given in Tables 5., 5. and Figure 5.38 Expectation values of the entropy as a function in time for the F+H two-reactionmechanisms system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.39 Expectation values of the entropy generation rate as a function in time for the F+H two-reaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.4 Expectation values of the particle number operator for each species of the F+H tworeaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.4 Expectation energies for each species and the system of the F+H two-reactionmechanisms system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.4 Expectation values for the system entropy and energy and the number of particles and energy for each species of the F+H two-reaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.43 Instantaneous expectation values of the reaction coordinate of the F+H two-reactionmechanisms system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.44 Instantaneous expectation values of the rate of the reaction coordinate for the F+H two-reaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.45 Forward and backward reaction rate constants as well as the equilibrium constant as a function of time for the F H HHF reaction mechanisms corresponding to an initial stable equilibrium temperature at 3 K Figure 5.46 Forward and backward reaction rate constants as well as the equilibrium constant as a function of time for the HHF HF H reaction mechanism corresponding to an initial stable equilibrium temperature at 3 K xi

12 Figure 5.47 Forward, backward, and net reaction rates as a function of time for the F H HHF reaction mechanism corresponding to an initial stable equilibrium temperature of 3 K Figure 5.48 Forward, backward, and net reaction rates as a function of time for the HHF HF H reaction mechanism corresponding to an initial stable equilibrium temperature of 3 K Figure 5.49 Activation energy evolution for different initial stable equilibrium temperatures Figure 5.5 Activation energy for different quantum structures of the activated complex HHF corresponding to an initial stable equilibirum temperature of 3 K.... Figure 5.5 Energy transfer between the different energy eigenmodes at 3 K for the F+H onereaction-mechanism system of Section 5.. and an initial stable equilibrium temperature of 3 K.... Figure 5.5 Energy transfer between the different energy eigenmodes for H molecule during the F+H one-reaction-mechanism system evolution of Section 5.. and an initial stable equilibrium temperature of 3 K.... Figure 5.53 Energy transfer between the different energy eigenmodes for HF molecule during the F+H one-reaction-mechanism system evolution of Section 5.. and an initial stable equilibrium temperature of 3 K Figure 5.54 Temperature profiles in time for three different initial stable equilibrium temperatures for the F+H one-reaction-mechanism system Figure 5.55 Pressure profiles in time for three different initial stable equilibrium temperatures and for the F+H one-reaction-mechanism system Figure 5.56 Pressure profiles in time for three different cases given in Table 5.5 and for the F+H one-reaction-mechanism system Figure 5.57 Chemical potential profiles in time for the different species for the F+H one-reactionmechanism system corresponding to an initial stable equilibrium temperature of 3K Figure 5.58 The projection of the gradient of the entropy and the relaxation time functional against the expectation value of the entropy at an initial stable equilibrium temperature of 3K for the F+H reaction mechanism Figure 5.59 Projection of the gradient of the entropy (D D) and the entropy generation rate as a function of the expectation values of the entropy for the F+H one-reaction-mechanism system corresponding to an initial stable equilibrium temperature of 3 K xii

13 Figure 5.6 Instantaneous expectation values of the entropy for the H+F one-reaction-mechanism system at an initial stable equilibrium temperature of 3 K Figure 5.6 Instantaneous expectation values of the entropy generation rate for the H+F onereaction-mechanism system corresponding to at an initial stable equilibrium temperature of 3 K Figure 5.6 Expectation values of the particle number operator for each species of the H+F onereaction-mechanism system corresponding to an initial equilibrium temperature of 3 K Figure 5.63 Expectation energies for each species of the H+F one-reaction-mechanism system corresponding to an initial equilibrium temperature of 3 K Figure 5.64 Expectation values for the system entropy and the energy as well as the energy and number of particles for each species of the H+F one-reaction-mechanism system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.65 Instantaneous expectation values of the reaction coordinate for the H+F one-reactionmechanism system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.66 Expectation values of the rate of the reaction coordinate of the H+F one-reactionmechanism system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.67 Forward and backward reaction rate constants as well as the equilibrium constant as a function of time for the H+F one-reaction-mechanism system corresponding to an initial stable equilibrium temperature of 3 K... 7 Figure 5.68 The forward, backward, and net reaction rates as a function of time for the H+F onereaction-mechanism system corresponding to an initial stable equilibrium temperature of 3 K Figure 5.69 Comparison of the τ for H+F reaction fitted using equation (5.6) with that of the F+H reaction fitted based on Heidner et al. [89]... 8 xiii

14 List of Tables Table 3. Limits on the reaction coordinates for the reaction mechanism in equation (3.6)... 3 Table 3. Solution for the number of compatible compositions and the relation between the eigenvalues of ns and those of εs... 3 Table 3.3 Dimensionality of the occupation coefficient subspaces Table 3.4 Occupation coefficient matrix Table 3.5 Occupation coefficient matrix for two interacting species with 3 and internal energy modes for each, respectively Table 4. Three species with their corresponding initial number of particles and internal energy levels Table 5. Parameter values needed to calculate the one-particle energy structure for the hydrogen molecule and fluorine atom reaction mechanism Table 5. Internal structure of the diatomic molecules in the F + H reaction mechanism Table 5.3 Translational quantum numbers considered for each species in the F + H reaction mechanism Table 5.4 Limits on the reaction coordinates for the F + H reaction mechanism Table 5.5 The number of compatible compositions for the F + H reaction mechanism Table 5.6 The different translational quantum number used and the forward reaction rate constants which results Table 5.7 Different internal energy eigenstructure for the diatomic molecules in the F + H reaction Table 5.8 The reaction rate constant for the different cases of the internal structures Table 5.9 The forward reaction rates constant by Heidner et al. [89] (HBMG) and the corresponding fitted τ at different stable equilibrium temperatures Table 5. Internal eigenstructure of the polyatomic molecules for the two F+H reaction mechanism system Table 5. Translational energy eigenstructure for each species for the F+H two-reactionmechanisms system Table 5. Limits on the reaction coordinates for the F + H two-reaction-mechanisms system xiv

15 Table 5.3 The number of compatible compositions for the F+H two-reaction-mechanisms system Table 5.4 The different internal eigenstructure of the activated complex HHF.... Table 5.5 Translational quantum numbers considered for each species for the three different cases Table 5.6 Comparison of values of the forward reaction rate constant kf found in the literature with those predicted using SEA-QT and equation (5.) for τ(ρ).... Table 5.7 Constants used to calculate the one-particle energy eigenstructure for the H+ F onereaction-mechanism system.... Table 5.8 Internal structure of the diatomic molecules in the H+F reaction mechanism.... Table 5.9 Translational quantum number range considered for each species in the H+F reaction mechanism.... Table 5. Limits on the reaction coordinates for the H+F reaction mechanism... Table 5. The solution to the number of compatible compositions for the H + F reaction mechanism.... Table 5. Comparison of values of the forward reaction rate constant kf found in the literature with those predicted using SEA-QT and equation (5.) for τ(ρ) for both the F + H and H + F reaction mechanisms xv

16 Chapter Introduction. Introduction Over the last three decades, Intrinsic Quantum Thermodynamics (IQT) [-5] has emerged as the theory that unifies two of the three theories of physical reality, namely, quantum mechanics (QM), and thermodynamics. This theory views the entropy as a fundamental property of matter regardless of its size or state. This point of view is quite different from that of either QM or quantum statistical mechanics (QSM) in which the entropy either plays no role or is merely a statistical illusion. In other words, since the entropy is necessarily a consequence of the st and nd laws of thermodynamics and QM excludes the nd law of thermodynamics, the entropy is excluded as well, while in QSM bases the entropy on a set of exogenous statistics which have no basis in physical reality. Thus, in order to describe the dissipative processes observed in nature ad hoc additions are made to QM and/or QSM. More on this topic is provided in the next chapter. In contrast, IQT provides a mathematical framework in which the evolution of state of a system undergoing a dissipative process can be predicted based on the principle of steepest entropy ascent (SEA) or maximum local entropy production (MLEP). To do so, the IQT equation of motion, first introduced by Beretta [5] and subsequently modified to deal with a larger class of processes by Smith [6] and Smith and von Spakovsky [7] is used. Both versions of this equation have been used to predict the experimental results of dissipative, one-particle systems as they relax towards a state of stable equilibrium [7]. The work to date in applying the framework of IQT called SEA-QT to predicting the evolution in state of nanoscale systems has been exclusively focused on non-reactive systems (e.g., [7, 8]). Thus, the question posed here is the following: can this framework be extended to describing the dynamics of reactive nanoscale systems as they evolve from a state of non-equilibrium to one of stable equilibrium? Such an evolution as well as that of the non-reactive systems found in [7,8] include a set of states not envisioned in QM or classical thermodynamics (CT). As seen in Figure., thermodynamics encompasses a much broader class of states than those that form the basis of CT (i.e., stable equilibrium states) and of classical and quantum mechanics (i.e., pure or zero-entropy states). It includes these as well as the set of not-stable equilibrium states, which exist in the region between the zero-entropy axis of pure states and the hyper-surface of stable

17 equilibrium states of which the curve in Figure. is a two-dimensional cut in the energy (E)- entropy (S) plane. SEA-QT is able to describe the kinematics of these other states as well as predict the evolution of these states in time as the state of a system relaxes to stable equilibrium or to mutual stable equilibrium with a heat or mass reservoir. Figure. Two-dimensional cut in the energy (E)-entropy (S) plane of the hypersurface of stable equilibrium states (of which the curve is a two-dimensional cut) for a thermodynamic system with translational degrees of freedom; is the set of all relevant parameters (e.g., volume, electrostatic field strength, gravity, etc.) that define the system and n the set of constituent amounts.. Reaction Path and Steepest-Entropy-Ascent Quantum Thermodynamics (SEA- QT) For a long time, thermodynamics as a science has been thought of as thermostatics where just the initial and finial states of the system are known but not the unique thermodynamic path that the system undergoes. Clearly, the reason behind this has been the absence of a thermodynamic equation of motion, which describes the unique path of a process. This, however, changed with the introduction of SEA-QT, which with its equation of motion is able to predict the unique path that a system undergoing a process follows, while obeying the laws of thermodynamics. In SEA- QT, the states of a system evolve in the direction of SEP or MLEP. This evolution in effect maximizes the entropy and predicts the unique thermodynamics path, which the system takes. In contrast, a constrained maximization problem, which likewise maximizes the entropy, is unable to predict such a path. Nor is the flip side to this maximization problem, i.e., the constrained minimization of energy, able to do so. It is the latter which is used in chemistry and although called the minimum energy path (MEP) that a reactive system takes in transforming reactants into products, the path predicted is that followed on a minimum energy surface for which the underlying

18 assumption is that of stable equilibrium. Thus, this path is not the unique, non-equilibrium, thermodynamic path, which the system follows in transforming its reactants into products. Therefore, extending SEA-QT to deal with chemical reactions could provide the means for plugging a gap, which exists in the current paradigm, namely, that of knowing the unique nonequilibrium path that the system follows in its evolution in state..3 Kinetics and Entropy of a Reactive System at the Microscopic and Mesoscopic Scales In the macroscopic realm, the laws of thermodynamics and, in particular, the second law play an important role in determining whether a chemical reaction will spontaneously occur or not. In addition, an important question that the second law or more specifically the entropy answers is when does a chemical reaction stop or, said differently, when does the net rate of change equal zero (see Figure.)? However, at the mesoscopic and microscopic scales, the second law of thermodynamics is absent in all conventional theoretical models of physics and thermodynamics that predict the kinetics of chemical reactions. Some of the most commonly used methods for predicting the kinetics of chemical reactions are discussed in Chapter. The reason for this absence is that these models are primarily based on Classical Mechanics (CM) or QM which envision the microscopic and mesoscopic world as undergoing reversible processes only. Even when viewed from the standpoint of QSM, processes only appear to be irreversible due to the loss of information in the modeling process. Thus, the entropy from this standpoint is not physical but simply a measure of uncertainty or as J. Willard Gibbs called it "mixedupness" [9]. Furthermore, at these levels of description, chemical reactions in these conventional theories are seen as resulting from inelastic collisions between the constituents of the reactant mixture and the constituents of the product mixture. However, if entropy is truly a fundamental property of matter as suggested by IQT, then the question arises as to whether or not chemical reactions at these scales can practically be viewed in light of the second law? In other words, how can the laws of thermodynamics and the second law in particular be inserted in the kinetics and dynamics of a chemical reaction at the microscopic and mesoscopic levels? A depiction as seen in Figure. is problematic from the standpoint that the only defined point on this curve is that at stable equilibrium since all other points represent non-equilibrium states. 3

19 Gibbs free energy G = H T S Pure reactants Stable Equilibrium Pure products Extent of the reaction Figure. Role of the Gibbs free energy and the progress of a chemical reaction. It is suggested in this doctoral work that this can be done at the microscopic level by the application of SEA-QT to reactive systems. This would entail using the equation of motion of SEA-QT to model the chemical kinetics of the system as it relaxes from a state of non-equilibrium to one of stable equilibrium. This equation predicts the irreversible evolution in time of a density or state operator ρ that contains all the information about the state of the constituents of the reactant and product mixtures of the reactive system at every instant of time. The basis for this state operator is constructed from the energy eigenvalue problem for the system expressed by H E (.) i i i where H is the system Hamiltonian, i are the system eigenvectors and required basis vectors and Ei is the i th system energy eigenvalue. The density operator is then written in terms of this basis such that t t y (.) y i i i i t (.3) i i i Where is the density matrix and y i the occupation probabilities for each system energy eigenlevel. Given an initial state operator, these occupation probabilities can be determined at every instant of time for, for example, an isolated system consisting of a single particle, an assembly of indistinguishable particles, or a field using the following SEA-QT equation of motion: 4

20 d t dt i H, t D (.4) where H and ρ are the system Hamiltonian and state operator, respectively, and τ is the relaxation time, which is a functional of ρ. D is the dissipative operator and it acts on the state operator in a way, which pulls it in the direction of the projection of the gradient of the entropy functional k Tr( ln ) onto the hyper-plane of constant system energy. Once the above initial value b problem is solved, all information about the state of a system is at least in theory available at every instant of time. A more detailed description and discussion about the SEA-QT framework outlined above is given in succeeding chapters. If the SEA-QT framework is developed and used as proposed here, a problem long considered as one of the most open in reaction kinetics [], namely, that of the quantum rate dynamics far from equilibrium, becomes solvable. SEA-QT does not require the thermal (stable) equilibrium assumption, which is made in the conventional theories such as, for example, transition state theory (TST). Furthermore, solving the kinetic chemical reaction problem in the SEA-QT framework provides information about the dynamical behavior of the system at each instant of time. For instance, the rate of a reaction is not a constant value but a variable, which changes in time as the reaction proceeds forward and/or backwards. Therefore, as in Figure.3, reactions which decay back and forth between reactants and products can easly be captured by SEA-QT, without any a priori assumption such as that used by the Arrhenius formulation of a transmission coefficient, which indicates the extent to which the reaction describes a forward-backwork path between reactants and products. Reactants Products Figure.3 Depiction of the possible progress of a chemical reaction, which SEA-QT captures without an a priori assumption of forward-backward progress. 5

21 .4 Research Objectives and Original Contributions In this doctoral research, two major goals are envisioned. The first is to apply the SEA-QT framework to an atomistic-level reactive system in the region far from equilibrium to demonstrate the effectiveness of this approach in being able to predict the reaction kinetics of these systems. The second is to extract dynamical properties at each non-equilibrium state through which the system passes as it evolves towards stable equilibrium. These include the concentration of each species, the instantaneous reaction rate, chemical affinity, reaction coordinate, entropy generation, etc. An additional goal closely related to the first two is to link the results obtained from SEA-QT to data available in the literature for a specific chemical reaction. This will be done to the extent possible considering that the approaches in the literature used to establish this data both experimentally and numerically are limited due to the equilibrium as well as other limiting assumptions of existing methods. In order to achieve these goals, the following tasks need to be accomplished: Understand the underlying physics, mathematical, and thermodynamic background of IQT Understand the SEA-QT framework for describing chemically reactive systems Study and understand the features and limitations of the principal conventional theories for describing the reaction kinetics at the atomistic level of chemically reactive systems Choose a benchmark reactive system extensively studied both experimentally and numerically in the literature Develop and implement the energy eigenstructure for the benchmark system consistent with the literature; this will establish the basis for the kinematics of the system and provide the framework with which the dynamics of the system will operate Develop and implement a procedure for constructing the initial states far from equilibrium, which will be used by the SEA-QT equation of motion Implement the SEA-QT equation of motion and apply it to the chosen atomistic-level reactive system Establish and implement a method for extracting thermodynamic property data for the benchmark system (e.g., chemical potentials, chemical affinities, etc.) for each of the nonequilibrium states through which the system passes Link the results obtained from the SEA-QT simulations with the experimental or theoretical results for a benchmark reaction 6

22 Make predictions of other well-studied reactions in the literature There are three primary original contributions to the literature in this doctoral work. The first is the implementation of a general SEA-QT framework for establishing the reaction kinetics of nanoscale chemically reactive systems even far form equilibrium and this without the limiting assumption of stable or pseudo-stable equilibrium found in all conventional theories. The second closely related to the first is the incorporation of both the st and nd laws of thermodynamics into determining the reaction kinetics. The third original contribution is the development and implementation of a procedure for extracting thermodynamic property data for states other than that of stable equilibrium. Finally, this doctoral work will transform the reaction rate constant of the conventional theories into a variable rate with instantaneous values for all the states through which the system passes in relaxing to stable equilibrium. 7

23 Chapter - Literature Review.. Chemical Reaction Path The chemical reaction path is an important component of theoretical chemistry since it is the first step toward determining the rate of a reaction. This section discusses two foundational concepts, which undergird the determination of this path, namely, the potential energy surface and the minimum energy path principle....the Potential Energy Surface In a general sense, the potential energy surface (PES) represents the total energy that results from all possible configurations of the atoms for a given chemical reaction. The first surface was created by Eyring and Polanyi in 93 for the H H H H reaction []. The PES is based on the Born Oppenheimer [] approximation in which it is assumed that since electrons are much lighter than the nuclei and, thus, move much faster, the nuclei can be assumed to be stationary. This permits a separation of the two motions, that of the electrons and that of the nuclei. To determine this surface (i.e., hypersurface) which represents the total energy of the atomic arrangements or configurations with atomic positions as variables, the stationary electronic Schrödinger equation or energy eigenvalue problem is solved. The total energy in this problem is given by the electronic Hamiltonian operator for fixed nuclei, i.e., H e Electrons e Nuclei Electrons Z Electrons Electrons i e i me i 4 i j R 4 i r j i ji ri r j e 4 Nuclei Nuclei R ZZ i ji i j i j R (.) where ħ is Planck's modified constant, Zi is the atomic number of nucleus i, me is the mass of the electron, is the vacuum permittivity, e is the electronic charge, and r and R are the electronic and nuclear vector coordinates, respectively. The electronic energy eigenvalue problem is written as 8

24 H ( r, R) E ( R) ( r, R) (.) e el where He is the electronic Hamiltonian, Eel the electronic energy eigenvalue, and the stationary wavefunction. Several ab initio methods are available for solving this eigenvalue problem such as the Hartree-Fock approximation [3], Moller-Plesset perturbation theory [4], the multiconfigurational self-consistent field method [5], the coupled cluster method [6], density functional theory [7], etc. These methods vary in terms of their accuracy and computational cost. A discussion about these methods and their limitation is given by Young [8]. After solving the electronic energy eigenvalue problem, the E el found in equation (.) represent the heights of the PES relative to the atomic coordinates R that describe the possible different atomic configurations of a given reaction mechanism. A schematic of a simple chemical reaction mechanism and its possible configurations on the associated PES is shown in Figure.. Potential energy surface Product C Reactant B Product D Possible trajectory of a chemical reaction Reactant A Figure. Depiction of a simple reaction mechanism and its possible configurations on the PES.... The Minimum Energy Path Principle The minimum energy path (MEP) plays a crucial role in understanding the nature of a chemical reaction in which it tells whether the reaction is endothermic or exothermic. Furthermore, the activation energy, which is the energy needed for a reaction to occur, is known by determining the MEP. In addition, some methods, which predict the reaction rate, require the MEP to be known. The MEP, which is also known as the reaction coordinate or intrinsic reaction path, is defined by Quapp and Heidrich [9] as "a continuous line connecting, in any configuration space, reagents and products through all intermediate stationary points (transition-state saddle points and reaction- 9

25 intermediate minima)". Figure. shows a depiction of a MEP. On this figure and according to the previous definition of the MEP, one sees that a continuous line connects both reactants and products and that both reactants and products are located at two minima on the PES. In addition, the continuous line passes through a saddle point or what is known as the transition state that represents a local maximum along the MEP. The saddle point or the transition state is a stationary point where the force (i.e., the negative of the gradient of the PES relative to atomic coordinates) equals zero. Finding these saddle points is important to determining the activation energy of a chemical reaction as well as finding the transition structure, which is central to predicting the reaction rate for some theoretical models (e.g., TST). One should distinguish between the MEP and the reaction trajectory appearing in Figure. in which the former is hypothetical and unique for each PES and the latter an infinite possible set of reaction trajectories, depending on the initial conditions and configurations of reactants. Reactants Saddle point / Transition state Minimum Energy Path Products Figure. Depiction of a MEP from reactants to products on a PES. There are several methods to predict the reaction path MEP on the PES. The commonly used methods are discussed here. The first is the Nudged Elastic Band (NEB) method [], which is efficient and widely used. In this method, which is depicted in Figure.3, in order to find the MEP, several images (i.e., atomic configurations) are created connecting the reactants to the products by springs representing a band. An optimization algorithm is then applied to reduce the forces in those springs. Once those forces are minimized, the band represents the MEP. The Climbing Imagenudged Elastic Band (CI-NEB) method [] is an enhanced version of the NEB method where the saddle point is more rigorously found without adding significant computational burden.

26 Product Spring MEP Different images Reactant Figure.3 Depiction of different images for the Nudged Elastic Band (NEB) method. In order to distinguish between minimum, maximum, and saddle points on the PES, first derivatives as well as second derivatives (i.e., Hessian matrix) of the PES with respect to the atomic coordinates must be examined. Finding the Hessian matrix is a computationally expensive process. A method which does not require finding these Hessian matrices is the Ridge method [] which makes it a more computationally efficient method. Another feature that this method possesses is that it does not need any assumption about the geometrical configuration of the transition state. Another method which does not require evaluating the Hessian matrix is the Dimer method [3]. Again, this is an effective method in finding the transition structure as well as the MEP. A review of these methods and some additional ones can be found in [4] and [5]. The above methods relate solely to the PES, which represents the configuration space only. This means that the kinetic energy of the nuclei is ignored in the process of determining the reaction path [6]. A class of methods that does take the kinetic energy of the nuclei into account in determining the reaction path is one which is based on phase space rather than the configuration space and is known as the Reaction Path Hamiltonian (RPH) method [7]. In the RPH method, the reaction is envisioned as the vibrational motion of a supermolecule (i.e., all reactant constituents) along the constrained path of the reaction coordinate. In doing so, the coordinate system x j (i.e., j=,, N where N is the number of particles) is transformed into the mass-weighted coordinates q j x j mj. The kinetic (T) and potential (V) energy terms of the classical Hamiltonian (H) are Phase space in mathematics and physics consists of all possible values of position and momentum for all particles.

27 then written as T dqi i dt (.3) V E S v S S X S S k k k,3 N7 (.4) where ES is the potential energy at location S, vs and along the direction of the reaction coordinate, respectively, and S are the eigenvector and eigenvalue k the eigenvalues along all other directions. The RPH is then constructed by representing the kinetic energy T in terms of the eigenmodes along the reaction coordinate S and all other directions X k. Once the RPH is constructed, the evolution of the different initial conditions can be tested as the process of transforming reactants into products proceeds... Chemical Reaction Rate Obtaining the theoretical rate of a chemical reaction has been a challenging problem for a long time. Several theories/methods (e.g., TST [8], the trajectory calculation method [9], quantum scattering theory [3], etc.) have been developed seeking answers to this problem. Although some of these methods have advanced and matured over time, they have been developed for equilibrium conditions and, thus, cannot describe systems in states away from stable equilibrium, especially far from stable equilibrium. Thus, what the actual reaction rate is is still an open question. For a thorough presentation of the history and development of reaction rate theory, one can refer to the work of Pollak and Talkner [] and Hänggi, Talkner, and Borkovec [3]. The following discussion focuses on some of the most popular methods for predicting the reaction rate, which is expressed in general terms for a bimolecular reaction of the form as follows: A B C D (.5) n n r k( T)[ C ] A[ C ] B (.6) A where r is the reaction rate, k(t) the reaction rate constant, T the stable equilibrium temperature, CA and CB are the concentrations of species A and B, respectively, and B n A and n B the order of reaction for A and B based on the initial concentrations and reaction rates. The orders of reaction

28 are determined experimentally, while the reaction rate constant results, for example, from experiment or the various theories/methods represented in the following sections.... Simple Collision Theory Collision theory was first proposed by Max Trautz in 96 and William Lewis in 98 [3]. This theory is based solely on classical mechanics. The two primary parameters, which control the rate of reaction in this theory, are the concentration of reactants and their temperature. A reactant's concentration will affect the frequency of collision and temperature is tied to the particle kinetic energy. Several models are available in collision theory of which the simplest assumes reactants to be modeled as hard-spheres and collisions proceed if the impact parameter b is less than or equal to r A r given as B as in Figure.4. The reaction rate constant, which the hard-sphere model predicts, is kt B k( T) N A ( ra rb ) (.7) 8 where NA is Avogadro's number, kb Boltzmann constant, r A and r B are the collisional radii for reactants A and B, respectively, and the reduced mass. Due to the simplified assumptions made in this model, the rate constant is over estimated by several orders of magnitude [33]. Two of the principal assumptions in this model are that the relative velocities between particles are constant and every collision between reactants results in products. A Initial velocity vector U b r A r B B Figure.4 Depiction of two colliding particles as hard-spheres. A modified version of collision theory assumes that only certain particles that are colliding force the reaction to proceed. In this version, the particles having energies larger than the threshold energy of reaction (i.e., the activation energy) are successful in getting the reaction to proceed. 3

29 This modified version is tied to the Boltzmann distribution for the particle energies and predicts the bimolecular reaction rate constant as 8kT B Ea k( T) N A ( ra rb ) exp kt B (.8) where Ea is the activation energy of reaction and T the temperature. Even after this modification, the theory commonly over estimates the reaction rate constant. The weakness of this theory is due to the assumption that molecules are hard-spheres with no intermolecular interactions and that the vibrational and rotational energies of the molecules play no role in the reaction [34].... Trajectory Calculation Method An alternate method to the previous one is called the trajectory calculation method (TCM), which predicts the chemical reaction rate by investigating how many reaction paths transform reactants into products. In order to find these paths or trajectories, the PES must be known ahead of time. Classical trajectory calculations [35], which are based on Classical Mechanics, i.e., Newton's nd law, do not take quantum effects into consideration, while quasiclassical trajectory methods [36] consider these effects. The following discussion is limited to the quasiclassical method. A first step is to write the Hamiltonian (kinetic and potential energies) for a system, which consists of N particles, namely, p p H T V V ( r,..., r ) N i i i mi N (.9) where T and V are the kinetic and potential energies and pi the momentum vector of the i th particle. A common practice in trajectory calculations is to transform the coordinate system from the Cartesian coordinate frame where every particle is traced individually in space to the center-ofmass and relative-mass coordinate frame. In the latter, the whole system plus the relative distances between particles are traced as the system moves in space. To determine the trajectory, the following Hamiltonian equations of motion are solved simultaneously, i.e., dpi dt dh dq i (.) 4

30 dqi dt dh dp i (.) for i =,, N and where qi is the generalized coordinate vector. Note that the kinetic energy term of the Hamiltonian in equation (.9) is due to the translational motion of the particles, while the potential energy term depends on the intermolecular forces between particles and, thus, on the relative distances between particles. The latter, of course, depend on the positions and orientations of each particle, which in turn are affected by the internal modes of vibrational and rotational energy storage of any group of particles sufficiently close to form bonds. As an illustration, consider the reactants A and BC shown in Figure.5 relative to a xyz-coordinate frame. The potential energy term of the Hamiltonian depends on the relative distances Figure.5) between the particles, which in turn depends on the vibrational and rotational energies of BC. To determine these, it is assumed that BC is restricted to the quantum mechanical vibrational-rotational eigenstates characterized by the rotational quantum number J and the vibrational quantum number ν such that the rotational the rotational-vibrational energy eigenvalue is given by [36] i, j i E J Gi Fij J J (.) i j This series is usually truncated at a finite number of terms while the coefficients Gi and Fij can be found in the literature for a given BC. The value of Eν,J for a given J and ν is constrained by the limits (minimum and maximum) of the internuclear distance J J BC BC r r BC r AB [36] given by, r AC, and r BC (see BC e, J (.3) D exp r r E where the second term to the left of the equals is a Morse function used to represent the vibrational potential energy and D, α, and re (the equilibrium distance) are appropriate constants particular to BC found in the literature. 5

31 A r AB B C r AC r BC Figure.5 Three particles colliding in a chemical reaction. Now to generate the tens of thousands of trajectories via solutions of the Hamiltonian equation of motion (equations (.) and (.)) needed to determine the reaction rate constant k(t), values are picked for the impact factor b (see Figure.6) and for ν and J. Values for then chosen randomly (e.g., via a Monte Carlo simulation) and the distances r BC, θ, ϕ and η are determined. Hundreds or thousands of trajectories are then generated for the given set of values for b, η, and J. These values are then changed and the process is repeated for a new set of random values for r BC been generated., θ, ϕ, and η. The procedure then continues until tens of thousands of trajectories have r AC and r AB z b A U B η ϕ θ C y x Figure.6 Depiction of initial conditions and orientations of the A+BC reaction. With these many trajectories on the PES established, the probability of reaction for a given set of values of η, J, u, and b is calculated as follows: P R NR, J, U, b, J, U, b lim (.4) N N, J, U, b Tot 6

32 Here NR is the number of reactive trajectories, i.e., those that result in products, and NTot the total number of trajectories generated. The cross section of the reaction is then calculated via R b,, max,,, J U P J U b bdb (.5) R This cross section is then weighted by the Maxwell-Boltzmann distribution, F(U), for the relative velocity to obtain the microscopic rate coefficient given by k J,, J, U U F( U) du (.6) R Finally, the overall reaction rate constant is obtained by summing over all vibrational and rotational energies with the corresponding weight of each quantum level, i.e., the rotationalvibrational distribution function F(ν, J), i.e.,,, k T k J F J overall, J 7 (.7) Greater detail of this quasiclassical procedure of collision dynamics of atom and diatomic molecule reaction A BC AB C is outlined by Karplus, Porter, Sharma [36]. A more recent quasiclassical trajectory calculation study for predicting the reaction cross section and reaction rate constant is given by Aoiz et al. [37]. As should be obvious from the presentation above, the quasiclassical or for that matter classical trajectory calculation approach is a very computationally expensive method for finding the reaction rate constant [8]. It requires investigating ~ O( 4 ) paths in order to obtain accurate results and that is why alternative approaches have been sought. One such approach is presented next...3. Transition State Theory This next approach for predicting the chemical reaction rate constant is called Transition State Theory (TST), which is based on classical mechanics and is primarily used for bimolecular reactions. TST is thought of as an approximate approach for predicting the reaction rate constant, because it discards the information from the state-to-state kinetics of a reaction. TST focuses on the activated complex, which is also known as the supermolecule that for a bimolecular reaction is represented by AB ( AB). The activated complex connects reactants to products of the PES. The main assumptions of TST are ( AB ) is found at the saddle point that

33 The transition state, i.e., the activated complex, is in pseudo-equilibrium with the reactants No recrossing is allowed on the dividing surface that separates reactants and products According to Laidler and King [8], TST was developed by Eyring, Evans, and Polanyi in 935. Eyring formulated the reaction rate constant for a bimolecular reaction as k T kt B h A Q Q Q B exp E RT where h is Planck's constant, κ is the transmission coefficient, E (.8) is the activation energy, which is the difference in energy between reactants and activated complex, and Q is the canonical partition function [33] for the reactants A and B and for the activation complex ( AB). Each partition function can be thought of as the total number of particles to the number of particles in the ground state [38]. Thus, one way of thinking about the ratios of the partition functions is in terms of how many eigenstates are accessible for the reaction to happen [39]. The reaction rate constant for a general reaction when the Born-Oppenheimer approximation holds is given by [4]: The Q and Q k T kt B Q exp h Q E RT (.9) include rotational, vibrational, and translational contributions for the reactants and activated complex, respectively. In TST, the location of the dividing surface is kept fixed on the hypersurface (see Figure.7). This assumption does not lead to an accurate prediction of the reaction rate constant since the dividing surface between reactants and products should be the one, which minimizes the flux. S Product side Reactant side S Figure.7 Depiction of a PES with different choices of dividing surface S and S. 8

34 This limitation has been relaxed in Variational Transition State Theory (VTST). The intrinsic difference between VTST and TST is that partition functions are not evaluated at the saddle point of the PES but at the location which gives the minimum ratios for the partition functions [4]. The partition function for the transition structure or activated complex in VTST is given by [4]:,,,, rot vib el Q T s Q T s Q T s Q T s (.) where s is the location along the reaction coordinate (i.e., MEP) and T the equilibrium temperature. The translational partition function is omitted here because it cancels with the one in the denominator. The rotational partition function for the nonlinear transition state (i.e., three particles or more) is expressed as where rot 3 kt B Qrot T, s Is Is Is rot is the rotational symmetry number and I,,3 () s (.) are the principle moments of inertia. The vibrational transition partition function using the harmonic vibrational description is given by 3N6 3N6 3N7 3N7 n m ms k,,, BT Qvib T s Qvib m T s e (.) m m n m where 3N-6 is for harmonic motion, 3N-7 for anharmonic motion, n is the energy eigenlevel, m is the mode, and () the angular frequency of the normal mode for the dividing surface on the m s PES. The frequencies are obtained by diagonalizing the Hessian matrix, which is the second derivative of the potential function with respect to the isoinertial Cartesian coordinates. Elements of this matrix are written as H i, j i V (.3) q q Once the Hessian matrix is diagonalized, the quantities appearing on the diagonal become the eigenvalues λ. The normal frequencies are then computed from where μ is the reduced mass. m s j m s (.4) Although TST was developed based on classical mechanics, it has been improved to accommodate quantum effects. The difficulty is that the Heisenberg position-momentum uncertainty principle forbids specifying the trajectory path and its conjugate momentum exactly 9

35 [43]. An important contribution by Garrett et al. [44] was to develop a method which takes into account the threshold contributions (i.e., energy corresponding to the ground eigenstate of conventional TST []) in the quantized partition function and tunneling correction factors. Another important contribution by Marcus and Coltrin [45] was to introduce a new tunneling path (i.e., reactive trajectory tunneling through the potential energy barrier) for a certain class of reactions, which acts as the steepest ascent path on the PES or the minimum energy path in the classical sense. For a discussion about more recent developments of TST one can refer to the work by Truhlar, Garrett, and Klippenstein [43]...4. Cumulative Reaction Probability Method The cumulative reaction probability (CRP) method pioneered by Miller [46-49] is a relatively new method for calculating the reaction rate constant. This method is defined by Aoiz et al. [37] as "the sum of reaction probabilities summed over all initial and final states". The CRP method's advantage over the TCM is in how long each trajectory is followed. The classical TCM requires that each trajectory is followed from the reactant side to the product side, where as the CRP method only follows a trajectory from some arbitrary location to see if it will turn into a reactive or nonreactive trajectory. Also, the CRP method is reported to have more accurate results compared to TST, because it relaxes the assumption of no recrossing that is required by TST. The reaction rate constant giving in terms of the classical (no quantum effects) cumulative reaction probability N(E) is given by [48] where Q r k ( ) BT k T e N EdE (.5) Q T r E is the partition function of the reactants and E is the total energy eigenvalue. The classical cumulative reaction probability is found from N E E H, F,, F d d p q p q p q q p (.6) Q T r where F is the number of degrees of freedom (i.e., 3n-5 or 3n-6 for a linear or nonlinear molecule; respectively, and n is the number of reactant particles); H pq, is the total Hamiltonian of the molecular system as in equation (.9); F pq, is the flux operator, which is a hypersurface defined in terms of some coordinate space f ( q) that separates reactants from products; and pq, is the dynamical factor.

36 The CRP method has been modified to incorporate quantum effects as well. The cumulative reaction probability N(E) is given in terms of a scattering matrix S [5] in which its elements represent the reaction probabilities, i.e., where n p and n r N E np nr np, nr S (.7) are the quantum numbers of the reactants and products, respectively. The reaction rate constant is then expressed as E B k T k T N E e de (.8) Qr For more details one can refer to the work by Miller [48]...5. Quantum Scattering Theory Classically chemists have viewed a chemical reaction as a scattering problem across the saddle point on the PES. Although the classical description of a scattering problem allows one to predict useful information such as the reaction rate constant about a chemical reaction, its predictions are not accurate in some situations (e.g., for light atoms) because it does not allow for the tunneling phenomenon. Therefore, scattering theory has been extended into the quantum realm. This quasiclassical approach is the most expensive one in terms of computation of all the other methods for calculating the reaction rate constant. It has been limited to 3 and 4 atom systems and beyond that it is reported to be impractical [5] since the problem grows exponentially with the number of degrees of freedom for the discretized configuration space. Both time-independent and timedependent quantum methods have been developed to solve the scattering problem for a chemically reactive system. Among the popular methods for solving the time-independent scattering problem are the algebraic and coupled-channel methods [5]. In the time-dependent scattering problem, an initial wave packet governed by the time-dependent Schrödinger equation is allowed to propagate from the reactant to the product side. Althorpe and Clary [3] indicate that the key advantage for using the time-dependent method is that it "yields a physical picture of the dynamics". If so, as shown in the present doctoral work, this picture at best is incomplete. Another advantage of using the time-dependent method is that in a single calculation the reactivity for a wide range of S-matrix is a unitary matrix connecting asymptotic particle states in Hilbert space, which maps the incoming colliding particles to the resulting outgoing ones. Also, it is known as the evolution operator in the time interval minus infinity to plus infinity.

37 translational energies is obtained, which contrasts with the time-independent method in which each calculation is associated with a single translational energy. Furthermore, a scattering problem using the time-dependent method scales as ~n, whereas for the time-independent problem scales as ~n 3, where n is the number of basis functions. On the other hand, the time-independent method is reported to give results that are more accurate. For the time-dependent method, the process starts by selecting an initial state vector t and using the solution to the time-dependent Schrödinger equation after which the state vector is propagated in time until it passes out of the reactant region. i H t t exp (.9) Several propagation schemes are used, among the well-known ones are the Chebyshev and split operator methods [5]. To recover the scattering, amplitudes are integrated over time and transformed by Fourier transforms to the energy representation. An outline of this process can be found in the work done by Neuhauser et al. [53]. The efficiency of the time-dependent methods are directly related to the propagation scheme used. Once the scattering matrix is computed, the reaction probability can be found which in turn leads to the reaction rate constant..3. Quantum Effects and Dissipation As suggested above, the lack of a dissipated model at an atomistic level where quantum effects are important is at least in part the, if not the major, cause of the inaccuracy of predicted results, which proceed from the time-dependent Schrödinger equation of motion. QM suggests that only pure states exist at an atomistic level; and yet if dissipative or irreversible processes are indeed present, then a theory is needed which is able to describe a set of states not envisioned by QM for which the entropy is greater than zero. In a similar vein, the linear dynamics of the Schrödinger equation must be augmented by a non-linear dynamics, which accounts for a dissipative or irreversible evolution in state that the equation of motion of QM is unable to do. Thus, in this section, attempts to address these other states of non-zero entropy and the presence of irreversibilities and irreversible evolutions in states at this level are discussed in terms of the concepts of entropy, entropy generation, and dissipative models of physical reality.

38 .3.. Quantum Statistical Mechanics (QSM) QM deals with and describes systems in pure states for which the entropy of thermodynamics is always zero. To describe states for which the entropy is greater than zero, QM is supplemented with a non-physical theory, statistical mechanics, the result being Quantum Statistical Mechanics (QSM). In QSM, such states are described as a weighted statistical average of a so-called heterogeneous ensemble of an infinite number of identical systems each in its own pure state. This is illustrated conceptually in Figure.8. The ensemble QSM i i i QSM, and each state is weighted by a different weight of blocks as indicated in this figure. is formed from different pure states w i represented by the number QSM w QSM n n n QSM w n w QSM w 4 QSM w 3 Figure.8 A state of non-zero entropy represented in QSM by an ensemble of identical systems in different pure states. QSM Mathematically the state of the system is expressed by the state operator such that for which in general QSM i i QSM i w (.3) QSM QSM QSM QSM 3 (.3) and the weights wi satisfy the condition that N w i i 3

39 This approach for describing states of non-zero entropy suffers from at least three drawbacks. For example, at stable equilibrium, the adiabatic availability [54] of the system must be zero to be consistent with the nd law of thermodynamics, but the states of the members of the ensemble, which are necessarily not in states of stable equilibrium, implies that individual members of the ensemble have values of the adiabatic availability greater than zero. This leads to a perpetual motion machine of the second kind (PMM), a clear violation of the nd law. A second drawback is that, as shown by Park [55] and Park and Simmons [56], the state operator 4 QSM describes the state of the ensemble but not that of the system. Thus, the notion of "the state of the system", a bedrock of physical thought, is lost. A last point is that the Schrödinger equation or more precisely its equivalent, the von Neumann equation, is unable to predict the time evolution of this state operator to states of greater entropy. Thus, as a way of describing non-zero entropy states and irreversible evolutions in state, QSM falls short..3.. Nonlinear Schrödinger Equation of Motion In the quantum world, the time-dependent Schrödinger equation is the complete equation of motion that describes the evolution of a zero-entropy state, i.e., a pure state (see Figure.9). However, as already mentioned above, it is unable to describe the evolution of non-zero-entropy states. Several attempts have been made to complete the description of the Schrödinger equation among which are nonlinear Schrödinger equations (NLSEs). NLSE has been developed to describe both dissipative and non-dissipative systems. The nondissipative NLSE has been applied to a wide variety of applications, e.g., heat pulses in anharmonic crystals [57], Bose-condensed trapped neutral atoms [58], and nonlinear optics [59]. Dissipative versions of the NLSE have been introduced by Kostin [6] and later by Yasue [6] where the source of the dissipation is phenomenological and not fundamental. The dynamics that emerges from the use of this model is a non-unitary one and is not a priori subject to the irreversibility paradox, since the dissipative NLSE does not use statistical mechanics to describe the system. Conversely, this method is reported to violate the superposition principle and to produce some doubtful results. Moreover, as pointed out by Çubukçu [6], density operators emerging from this model are not unique. This may result in having several von Neumann equations describing the same evolution of system state. In general, this approach has been less successful when compared to other methods (e.g., the open system model ) in handling dissipative phenomena.

40 Figure.9 Two-dimensional cut in the energy (E)-entropy (S) plane of the hypersurface of stable equilibrium states (of which the curve is a two-dimensional cut) for a thermodynamic system with translational degrees of freedom; is the set of all relevant parameters (e.g., volume, electrostatic field strength, gravity, etc.) that define the system and n the set of constituent amounts Dissipative Quantum Dynamics (DQD) In an attempt to model dissipative processes at an atomistic level, Dissipative Quantum Dynamics (DQD) more commonly now called Quantum Thermodynamics proposes an "open system model", i.e., a closed-system-plus-environment model. With this model, DQD suggests that the microscopic mechanism for dissipation is an on-going loss of quantum correlations [63] between the environment and closed system which are coupled via weak interactions, i.e., statistical perturbations. This loss revolves around the Born-Markov approximation, which assumes a periodic build-up and then decay of correlations in a timeframe very short when compared to the timeframe required for the state of the system to relax to stable equilibrium. This effectively, it is argued, maintains the system and environment decoupled at all times. It is furthermore assumed that the number of degrees of freedom of the environment is much larger than that of the system. Under such a scenario, the reduced state of the system alone is governed by a quantum master equation (QME), e.g., the KSGL equation [64-67] or its equivalents. Thus, while the overall linear dynamics of the closed-system-plus-environment composite is reversible and unitary and governed by the time-dependent Schrödinger equation, that of the system alone follows an irreversible and non-unitary linear dynamics captured by the QME [5]. Thus, irreversible relaxation to stable equilibrium is accomplished at the expense of introducing the socalled Loschmidt paradox, which arises when trying to force irreversible, non-unitary behavior 5

41 from dynamics that is intrinsically reversible and unitary. Furthermore, as pointed out by Nakatani and Ogawa [68], the Born-Markov approximation for obtaining QMEs cannot be used for composite systems in the strong coupling regime, no matter how short the environment correlation time is. In addition, this approach does not provide a physical meaning for the entropy generation, since it results from the erasure of correlations that build up and disappear between the system and the environment. In other words, the entropy generation is due to a loss of information. Therefore, this approach is not only not general but non-physical as well Intrinsic Quantum Thermodynamics To overcome all of the drawbacks to the various approaches presented in the previous sections, IQT can be used and is, in fact, the underlying physical theory upon which this doctoral research is based. IQT was first introduced in 976 in a series of four papers [-4] by Hatsopoulos and Gyftopoulos, which lay the foundations for IQT. Initially called the "Unified Quantum Theory of Mechanics and Thermodynamics", IQT unifies the two physical theories of QM and thermodynamics into a single theory with a single kinematics and dynamics. Among their several contributions are rigorous definitions for thermodynamic states, system, properties and process that avoid the use of circular arguments and eliminate any ambiguities. In addition, these authors propose a density or state operator based on a homogeneous ensemble of identical systems identically prepared. This density operator is the ontological entity that completely describes any state of zero or non-zero entropy and avoids the PMM problem inherent to the density operator of QSM. Moreover, Hatsopoulos and Gyftopoulos formulate the nd law of thermodynamics at a fundamental level of description on par with the postulates of QM so that this law applies to systems of any size, even ones consisting of a single particle. In 98, Beretta [5] in a major contribution to IQT proposed a complete, non-linear equation of motion for a closed quantum system, which contains the von Neumann equation of motion or its equivalent, the Schrödinger equation, as a special case. Thus, not only is it able to describe the linear unitary evolutions in state envisioned by QM but as well the irreversible nonlinear, nonunitary evolutions envisioned by thermodynamics. Furthermore, this equation and the postulates of IQT resolve the "irreversibility paradox", providing a physical basis for the entropy and the entropy generation. Dissipation is, therefore, not viewed as a loss of information but rather as the process of redistributing the system energy among the degrees of freedom available to the system, i.e., among the modes of energy storage intrinsic to the system. The equation of motion itself is 6

42 formulated on the basis of the principle of steepest entropy ascent (SEP) or equivalently locallymaximal entropy production (LMEP). A brief description of this equation for a single constituent system consisting of a single particle, an assembly of indistinguishable particles, or a field is given in Chapter. A more detailed description of the IQT equation of motion is provided in Chapter 3. Finally, in 985, Beretta proposed an extension of this equation of motion to a general quantum system consisting of two or more distinguishable particles [69]. In 9, the IQT equation of motion was extended by Beretta to account for non-work interactions [7]. This was further developed by Smith [6] and Smith and von Spakovsky [7] to allow the system to come to mutual equilibrium with a heat or mass reservoir via a heat or mass interaction. It is worth mentioning that Beretta's original equation of motion has been independently derived via variational methods by Gheorghiu-Svirschevski [7]. Further details about the development of IQT can be found in [6]. In the next chapter, the mathematical framework of IQT called SEA-QT is given and discussed along with the dynamical features of its equation. In addition, the SEA-QT framework, describing the kinematics of an atomistic-level, chemically reactive system in states near or far from stable equilibrium, is presented and discussed in detail. It is the combination of these dynamics and kinematics, which provide the basis for the chemical kinetic predictions made in this doctoral research. 7

43 3 Chapter 3 Model and Theory This chapter describes the SEA-QT mathematical model and framework for the evolution dynamics for chemically reactive systems far from equilibrium. This framework is the one proposed and developed by Beretta and von Spakovsky [7]. In this chapter, a detailed description of the mathematical framework to model a chemically reactive system consisting of r species Ai contained in an isolated and fixed tank and evolving from a non-equilibrium state is provided. As mentioned earlier, this general framework is based on the steepest entropy ascent principle that governs the evolution of state of the system, which is depicted in Figure 3.. In this figure, the vector b represent the gradient of the entropy in state space. This vector can be decomposed into two perpendicular vectors, one bl which exists on the manifold L and the other b which is perpendicular to it. The manifold is spanned by the constraints of the problem, which are g and g (i.e., conservation of the number of atoms and of energy, respectively). The equation of motion of the SEA-QT framework determines the unique thermodynamic path or trajectory in state space that the dissipative system undergoes, which is always tangent to the b vector and perpendicular to the set of constraints g and g. Trajectory in state space b g b L g b b g g b L L g, g Figure 3. Depiction for the SEA-QT principle with b the gradient vector of the entropy change, b L the projection of vector b onto the manifold L that encompasses the constraints g and g on the system (i.e., conservation of the number of atoms and energy), and b the projection of b perpendicular to the manifold. 8

44 Briefly, the steps required to implement the current model is to construct the system-level Hamiltonian from the one-particle energy eigenvalue problem. Similarly, the occupation number operator is constructed from the information embedded in the occupation coefficients. Finally, the initial condition representing the state of the reactants in a non-equilibrium state is created and the state is evolved using the SEA-QT equation of motion. Figure 3. shows the steps required to implement this framework. A thorough and detailed discussion about this framework is found in [7]. Solve Proportionality relation Calculate Occupation Coefficients Solve One-Particle Energy Eigenvalue Problem Construct System-Level Eigenvectors Internal Energy Eigenvalues Translational Energy Eigenvalues Construct System- Level Occupation Number Operators Construct System- Level Hamiltonian Operator Implement SEA- QT Equation of Motion Calculate Expectation Value of: Energy Number Operator Reaction Coordinate Figure 3. General overview of the implementation of the SEA-QT framework for a chemically reactive system. 3. Energy and Particle Occupation Number Eigenvalue Problems The system-level energy eigenvalue problem is defined as follows: H E s,..., C q,..., L (3.) sqs sqs sqs s s where H is the Hamiltonian operator and Esq s and sq s the system-level eigenvalues and eigenvectors, respectively. C is the number of subspaces of the compatible compositions and Ls the dimension of each subspace s. The dimension of the overall Hilbert space H. of the system is L C s L s. Similarly, the particle occupation number eigenvalue problems are given by 9

45 where Aj i i sq Aii j sqs iji sqs s s s N s,..., C q,..., L i,..., r j,..., M N is the Ai-particles in the ji th sq internal level occupation number operator, s the eigenvalue for the qs th combination in the s th compatible composition, r the number of species, and Mi the number of eigenmodes for the one-ai-particle internal Hamiltonian operator associated with the internal modes (i.e., vibrational, rotational, etc. energy levels). In order to construct these system-level operators, information has to be built up from the single particle for each species Ai. To do so, the number of compatible compositions must be determined using the proportionality relations. This information is then used with the Ai-particle internal mode eigenvalues to construct the compositions is given in Section 3... sqs ij i i i (3.). Further clarification about the compatible iji 3.. One-particle Energy Eigenvalue Problems The first step in implementing the SEA-QT framework to describe the evolution of a chemically reactive system is to solve the one-particle eigenvalue problems. This step is separated further into two classes of problems, one that is associated with the internal energy modes (i.e., vibration, rotation, electronic, etc.) the other problem with the translation energy modes. The former is given by where Ai e j i and A i H int Ai Ai Ai Ai H e i,..., r j,..., M (3.3) int ji ji ji i i is the Hamiltonian operator associated with the internal energy modes of species Ai and Ai ji are the eigenvalues and eigenvectors for species Ai associated with the internal modes, respectively. The eigenvalue problem for translation is expressed by ˆ i rˆ i i i i ˆ ˆ ˆ ˆ Hˆ ˆ ˆ r V r ˆ r e ˆ r (3.4a) mˆ Hˆ ˆ e ˆ (3.4b) i i i where Ĥ is the Hamiltonian operator for the translation energy modes, ˆm the mass of the particle, the modified Planck constant, ˆr the vector coordinate, ei the eigenvalues, and ˆ i and ˆi the eigenfunctions and eigenvectors, respectively, expressed in the position and Hamiltonian basis, respectively. ˆ is the Laplacian operator and V the potential energy operator. rˆ 3

46 In order to construct the system-level eigenvector sqs in equation (3.), the one-ai-particle eigenvectors in equations (3.3) and (3.4) need to be expressed in the Hamiltonian basis. For example, if species Ai has four internal energy modes, then, Mi in equation (3.3) equals 4 and ji =,, 3, 4. Similarly, the eigenvectors in equation (3.3) expressed in the Hamiltonian basis are given by A A A A,, 3, 4 A j i 3.. Compatible Compositions A compatible composition can be thought of as the eigenvalues of the particle number operator in the proportionality relations. In order to determine the number of compatible compositions, one needs to know the initial amount nia of the reactant species Ai. The proportionality relations are written as n n ( ) n is i s ia il ls l n ia i s (3.5) Where the ni are the eigenvalues of the Ai-particle number operator, τ is the number of reaction mechanisms, ls the eigenvalue of the reaction coordinate operator for reaction "l" corresponding to the s th compatible composition, and "l". A compact way of expressing il ls and the stoichiometric coefficient of species Ai for reaction il is by using s and i to represent a set of values for both the reaction coordinate and the stoichiometric coefficient, respectively. The implementation of the compatible compositions is discussed in the following example. Example 3.: This example illustrates the concept of compatible composition. Consider the following chemical reaction: A B C D (3.6) Since there is one reaction mechanism, Equation (3.5) simplifies to n n since τ =. is ia i s The initial amount of reacting species A and B is then fixed and in this example, the number of 3

47 reacting particles is assumed to be naa 3 and nba. Next, limits on the reaction coordinate eigenvalue are found using Equation (3.5). The results are given in Table 3.. Table 3. Limits on the reaction coordinates for the reaction mechanism in equation (3.6) i Ai i n ia A - 3 B - 3 C 4 D n is 3s s s s n is 3 s s s s The solution of the set of inequalities in the last column of Table 3. determines the number of compatible composition. Table 3. shows the number of compatible composition as well as the relation between the eigenvalues of the number operator (nis) are those of the reaction coordinate operator (εs). Table 3. Solution for the number of compatible compositions and the relation between the eigenvalues of n s and those of ε s s s Ai n is n i n i ns 3 ns 3 n3s 4 n4s This table shows that εs takes the values {, } and, thus, the number of compatible compositions in this case equals, 3,,,,,,, s n n n, and the number of Hilbert subspaces s is. Table 3. also shows that the number of particles is conserved for each compatible r i n is composition s, i.e., the number of particles in each subspace s is given by 4. 3

48 3..3 Occupation Coefficients The next task is to determine the occupation coefficients sqs ij i, which describe the relation between the particles and their energy eigenlevel, i.e., how a particle occupies or is distributed among the various internal (vibrational, rotational, electronic, etc.) energy eigenlevels. In order Species Energy level A B C D Figure 3.3 Depiction of a possible occupation of particles in an internal energy configuration. to find the occupation coefficients, the number of internal degrees of freedom (vibration, rotation, electronic, etc.) needs to be determined for each species of the reactant mixture. Again, for illustrational purposes, each species is assumed to have two internal degrees of freedom. The dimensionality of each subspace is determined by where Mi L n is L n M Mi is i n is nis i!! M! (3.7) B r int Mi s Ln is i (3.8) int L L s s (3.9) L is the number of possible ways that nis indistinguishable Ai particles can be distributed on the Mi internal energy eigenlevels, L B s int is the dimension of each subspaces s associated with the internal degrees of freedom, and L is the total dimension for the internal degrees of freedom. For example 3., the dimensionality of each internal subspace is as shown in Table 3.3. int sq iji Now the occupation coefficients s for the reaction mechanism of equation (3.6) is a matrix of size 8 and contains two smaller matrices each of size 88for H and 8 for B int B int H, where H and H 3 B int occupation coefficients are given as in Table 3.4. B int are the Hilbert spaces of each internal subspace s. The 33

49 Table 3.3 Dimensionality of the occupation coefficient subspaces. i Mi 3 4 L s Mi nis M L n s M L n s M3 n3s L Mi ni L Mi ni 4 3 L M4 L n 4s L B s int 8 L= Table 3.4 Occupation coefficient matrix. s = B int L 8 s = B int L ji M= M= M3= M4= k qs

50 3. Hamiltonian Operator As mentioned earlier in this chapter, the system-level Hamiltonian is constructed from the one-ai-particle Hamiltonians; and following what is described in Section 3.., the Hilbert space A i H associated with each species Ai is factored into translational and internal energy modes. Now the Hilbert spaces Ai Ai Ai Ai Ai Ai A i tr rot vib el tr int H H H H H H H (3.) H s associated with each compatible composition s can be written as i i i r is r is r A n A n A nis tr int s tr int s s i i i H H H H H H (3.) and similarly the Hamiltonian H s associated with each compatible composition s is given by int I s and tr I s tr int tr int tr int s s s s s s H H I I H V (3.) are the internal and translational identity operators, respectively, the interaction between the translational and internal energy modes. tr int V s represents The first step in constructing the system-level eigenvector is to build the system-level eigenvectors associated with the internal modes of the one-particle eigenvectors given in equation (3.3). This set of eigenvectors are related to a fixed compatible composition s and are associated with the occupation coefficients by the following expression: sq int r Mi s A int i ij i int (3.3) sq j s i i ji The corresponding eigenvalues are given by Example 3.: E int int sqs r Mi i ji Ai ji int sqs iji e (3.4) To illustrate the use of equations (3.3) and (3.4), it is assumed that particle of species A interacts with particles of species B (i.e., r = ). In addition, species A has 3 internal energy modes and species B has internal energy modes. The occupation coefficients for this example are given in Table 3.5. The eigenvectors and eigenvalues are as follow: and A A A,, 3 35 B B and,

51 A ji B,, and e b, b e a a a 3 Table 3.5 Occupation coefficient matrix for two interacting species with 3 and internal energy modes for each, respectively. ji s = B int 3 L ji M=3 M= qs 3 3 Since this compatible composition space s has eigenvectors for the internal energy modes. Thus, int B int 3 L, there must be 3 system-level int int 3 and the corresponding eigenvalues, int () () () 3 () () E a a a b b a b int () () () 3 () () E a a a b b a b int 3 () () () 3 () () E a a a b b a b b Due to the intermolecular forces, the translational Hamiltonian tr Hs is generally non-separable. Nevertheless, using a unitary transformation T separating the center of mass coordinate from the relative coordinates, allows one to account for the overall translational Hamiltonian as a sum of several Hamiltonians corresponding to a group of Hamiltonian in center-of-mass coordinates is given as n s particles. Therefore, the translational 36

52 ns ˆ tr tr ˆ s ˆ s s j s js js The corresponding translational Hilbert subspace H TH T H I (3.5) n js tr H s is factored as follows s tr H H ˆ (3.6) s The translational Hamiltonian is now decoupled and equation (3.4b) can be written as follows, js where K sjs ˆ s sj ˆ s sjs sj ˆ s H e s,..., C j,..., n k,..., K (3.7) js ksj k s sj k s sj s s sj s s sjs is the number of translation eigenvalues. Furthermore, Ksj s s sjs sjs sjs j s k sj k sj k sj s s k s s s sj s Hˆ e ˆ ˆ s,..., C j,..., n (3.8) Using equation (3.8) and the unitary transformation T, the translational Hamiltonian in the original coordinate frame is found to be tr s ˆ tr s and the corresponding energy eigenvalues for this Hamiltonian are tr tr sqs H T H T (3.9) s sjs sns k ss ksj k s sns E e e e (3.) All the elements required to construct the system-level eigenvectors are now in place. The system-level eigenvectors corresponding to the Hamiltonian composition s are given as tr int sqs tr sqs int sqs and the corresponding system-level eigenvalues are The Hilbert subspace H s for a fixed compatible (3.) tr int sqs tr sqs int sqs E E E (3.) H s associated with the Hamiltonian operator ofh. This allows writing the Hilbert subspace H s in the following form: sq s s Ls qs sqs H s is spanned by the set H H (3.3) A crucial element in the development of this framework is the D projector P H sq s. The set of these projectors is used in construct all the operators defined in subspaceh. These D projectors are written as s 37

53 so that projectors P H s for subspaces PH (3.4) sq sq s s sqs H s is given by s Ls PH P (3.5) H qs where Ls is the dimension of each subspace s. The overall system Hilbert space H can now be expressed by where the identity operator on H is C Ls sqs sq s H H (3.6) C C Ls sqs I PH P (3.7) H s s s qs sq s Finally, the system-level Hamiltonian is 3.3 Occupation Number Operators C C Ls H H E P (3.8) H s s s qs Information about the number of particles is embedded in the occupation coefficients sqs sq s sqs ij i However, since the occupation coefficients relate the number particles to the internal energy eigenmodes only, the occupation number operators are first constructed for the internal energy modes in subspace s using the occupation coefficients s, int Aj i i N sqs ij i int Ls int s, int sqs Ai j i ij P int i sq q s int s as follows: H (3.9) N represents the occupation number operator of species Ai at the ji internal energy level for a fixed subspace s. Now, to construct the occupation number operator that accounts for both internal and translational energy modes on a specific subspace s, equation (3.9) is multiplied by identity operator for translational for subspace s, i.e., or the occupation coefficients sqs iji I tr s tr L s P H (3.3) tr qs sq tr s can directly be multiplied by projectors P H as follow: sq s. 38

54 tr int Ls Ls int s s, int tr sqs Ai j i Ai j i s ij tr int i H sq int H sq tr q s s s qs N N I P P Ls qs sqs iji P H sqs 39 (3.3) The occupation number operator for species Ai at the ji energy level over all the subspaces s is then given by, N Ai ji C s s Ai ji N (3.3) Finally, the occupation number operator for species Ai over all subspaces s and energy levels ji is given by, 3.4 Reaction Coordinate Operators Mi C Mi Ls sqs A i Ai j i iji sq j s i s ji qs N N P (3.33) H The reaction coordinate is another quantity that can be measured in the SEA-QT framework. Since the aim of the reaction coordinate is quantify the extent of a given chemical reaction, the reaction coordinate operator represents a different way of constructing the occupation number operator given in equation (3.33). The reaction coordinate operator for a given reaction l and the set of reaction coordinate operators for all τ reactions are given by C E P l,..., (3.34a) l ls s H s C E P s s H s and the occupation number operator in terms of the reaction coordinate operator is, C Ai is s ia i s (3.34b) N n PH n I E (3.35) To demonstrate the use of equation (3.35), example 3. of Section 3.., which describe the chemical reaction mechanism A B C D, is used again. The compatible compositions are E, as in Table 3.. Substituting E () P () P P H H H D are as follows: NA 3I E 3I P H NB I E I P H E into equation (3.34b) yields,. Therefore, the occupation number operators for species A, B, C and

55 NC E P H ND E P H 3.5 Equation of Motion of SEA-QT The next element in the SEA-QT framework for describing chemically reactive systems is the SEA-QT equation of motion, which for a single particle, an assembly of indistinguishable particles, or a field is expressed as d i H, M, (3.36) dt k B where ρ is the density or state operator that describes the state of the system at every instant of time, the modified Planck constant, τ the internal-relaxation time for the dissipation. Note that τ can be a constant or a functional of ρ, The operator defined for M, as,, Here M is the variance, i.e., of the non-equilibrium Massieu operator defined as in equation (3.36) represent the anti-commutator M M M (3.37) M M M (3.38) M S H H (3.39) where S and H are the entropy and Hamiltonian operators, respectively, and θh is the so-called non-equilibrium temperature given in terms of the variance of the entropy and Hamiltonian operators by where H HH (3.4) S H HH Tr H Tr H Tr H (3.4) SH Tr SH Tr S H Tr S Tr H (3.4) H and S are the variances of the S and H operators expressed as H H I H (3.43) 4

56 S S I S (3.44) The entropy operator S given in equations (3.39), (3.4) and (3.44) is given by S k Bln k ln I B (3.45) B B where kb is Boltzmann s constant and B is an idempotent operator which is the projection operator onto the range of ρ. Now returning to the SEA-QT equation of motion, equation (3.4), the first term on the right governs the linear Hamiltonian dynamics of the state evolution, and the second, the so-called dissipation term, governs the nonlinear non-hamiltonian steepest-entropy-ascent dynamics. The non-hamiltonian dissipative term acts on the state operator ρ in a way which pulls it in the direction of the projection of the gradient of the entropy functional onto the hyper-plane of constant system energy and conserved ρ (i.e., ). All of the operators introduced in Sections 3., 3.3, and 3.4, are constructed on the basis of each subspace s and its corresponding dimensionality (i.e., qs). Equation (3.36) is, thus, reformulated in terms of the system-level occupation probabilities dimensionalities. In order to do so, ρ, H, S, occupation probabilities y sqs as follows: S sqs C sqs C Ls H H, S H Ls sqs sq s y sqs, eigenvalues, and subspace, and θh are rewritten using the y PH (3.46) H E PH (3.47) k sqs B Tr P sqs H sq s y sq s ln sqs (3.48) C Ls C Ls HH ysqs Esqs ysq E (3.49) s sqs s qs s qs C Ls C Ls C Ls SH ysq k ln ln s B ysq E s sq s kb ysq y s sqs ysq E s sqs (3.5) s qs s qs s qs H C Ls C Ls ysq E s sq s ysq E s sqs s qs s qs C Ls C Ls C Ls ysq E ln ln s sq k s B ysq s ysq E s sq k s B ysq y s sqs s qs s qs s qs (3.5) 4

57 Upon substitution of equations (3.46) to (3.5) into equation (3.36), the SEA-QT equation of motion is cast in terms of the occupation probabilities and is expressed as. C Ls C Ls kb ysq y ln ln s sq s kb ysq k s B ysq y s sq s Esq y s sq E s sqs (3.5) s qs H s qs Note that equation (3.5) represents the case where ρ and H commutes. Once this equation is solved, information about all the states through which the system passes is known and all the dynamics about the process captured. 3.6 Expectation Values of the Energy, the Occupation Number and the Reaction Coordinate As mentioned in Chapter, the merit of the SEA-QT framework is that the full picture of the dynamical evolution of the system is captured including all the states which the system exhibits from the initial state (i.e., regardless of how far from equilibrium the system is) until the final state of stable equilibrium. To the properties of each system state throughout the evolution process, the expectation value of the operators constructed in the previous sections need to be determined. Thus, the expectation value of the system energy is given by C Ls H Tr H y E (3.53) sqs while that of the occupation number operator is obtained by, 4 sqs sqs N n Tr P (3.54) H Ai C is s C s s N n Tr P (3.55) H The former results can be expressed in terms of the occupation probabilities coefficients sqs ij i as follows: s Mi C Ls C Ls Mi sqs A i Ai j i sqs is sqs iji ji s qs s qs ji s ysq s and the occupation N N y n y (3.56) The expectation value of the reaction coordinates is expressed as C s C E Tr P (3.57) ls s ls H s E Tr P (3.58) Again, these expectation values can be given in terms of the occupation probabilities follows: s H s ysq s as

58 C Ls E y (3.59) s s qs The occupation number and the reaction coordinate operators expressed in equations (3.54) to (3.59) can be reformulated in a more compact way by defining the reaction coordinate occupation probabilities ws, which represent the probability of the system being in a particular s compatible composition subspace s. Thus, and sqs Ls Tr ws Tr PH PH y (3.6) Ls s sq sqs q s s qs C w Then, rewriting equations (3.56) and (3.59) using equation (3.6) which yields, s s (3.6) C N n w (3.6) s C s s s s s E w (3.63) Example 3. in Section 3.. is now used to illustrate the use of the reaction coordinate occupation probabilities ws. For the chemical reaction A B C D and for the solution of the compatible compositions given in Table 3., the expectation values of the occupation number operators are as follows: N 3w w A NB NC ND w w w Finally, the expectation value rates of the occupation number and reaction coordinate operators can now be expressed as C Ls N n y (3.64) s s qs C Ls y s s k sqs E (3.65) sqs This last expression is important for determining the rate of the chemical reaction as well as the reaction rate constant k which will be discussed further in the next chapters. 43

59 4 Chapter 4 Numerical Approach Several different topics are discussed in this chapter. These topics can be classified into two primary categories. The first is to solve and verify the developed numerical code and the second is to determine some reaction kinetics and thermodynamic properties. The discussion starts by describing the models for the one-particle energy eigenvalue problems. Some of the important aspects about the schemes needed in the development of the numerical code are then presented. After this, a description of the method used to create the initial conditions is given. The numerical solver as well as the convergence criteria are then described. The last item in the first category is the tests performed to verify the results obtained by the code developed. For the second category, the procedures to calculate the reaction rate constant as well as the rate of the chemical reaction are described. Finally, the method for predicting thermodynamic properties for non-equilibrium states is provided. 4. The One-Particle Energy Eigenvalue Problem The reacting mixture considered in this work is assumed to behave as a Gibbs-Dalton mixture of ideal gases. For that reason, the energy eigenvalues for translation, vibration, and rotation for the species involved are given by a set of closed-form relations. The translational energy eigenvalue expression is given by e tr h nx ny nz k 8m L x L y Lz tr where k is the one-particle translational energy eigenvalue; h is Planck's constant; m is the mass of the particle; k=,, is the principal quantum number; nx, ny and nz are the quantum numbers in the x, y and z directions, respectively; and Lx, Ly, and Lz are the dimensions for the system volume in the x, y and z directions, respectively. The expression for the vibrational energy eigenvalues is vib or e hc K v vib v (4.) e h v (4.) v (4.3) 44

60 where ν is the vibrational quantum number which takes values of ν =,,, ; ω is the vibrational frequency, c the speed of light and K the wavenumber. Finally, the rotational energy eigenvalue expression for a polyatomic molecule is given by the following equation: l j j e rot j (4.4) I A IC I A where j and l are the rotational quantum numbers with j taking values of,,, and l values of j to j. In equation (4.4), ħ the Planck s modified constant, IA and IC are moments of inertia of a symmetric top polyatomic molecule. This equation does not hold for the case of non-symmetric top polyatomic molecules where IA IB IC. For the case of a diatomic molecule such as the ones considered in this work, equation (4.4) reduces to the following form: j j e rot j (4.5) r where µ is the reduced mass and r the distance between two atoms. 4. Occupation Coefficients As discussed earlier in Section 3..3, the occupation coefficients are an essential element in the development of the current SEA-QT framework since they encompass the information of the distribution of particles among the available internal energy eigenlevels. In order to construct these coefficients, the following steps need to be carried out:. Determine how many ways a particle or group of particles of a certain species Ai can be distributed or partitioned among the internal energy levels ji for that species.. Determine the unique set of permutations for step and for each species. 3. Construct the unique permutations for all the species combined in order to arrive at the occupation coefficients. To complete the first step, the integer n that represents the number of particles of species Ai must be partitioned into k parts representing the available internal energy eigenlevels of species Ai. This process is known as integer partitioning or more precisely as restricted integer partitioning. A comprehensive overview about the partitioning theory can be found in [73]. Different algorithms are available to carry out the process, and the one adopted in this work is a recursive one that is described by the algorithm called Partition_nk, given in Figure

61 if n is scalar if n = return γ ={empty set} if k = return γ = n if n = return γ= {n,,,, } else a = {n-, } both of length k Initialize γ {n,,,, } and,,., temp l = length(a) temp from to l a.append( temp ) if a() > a() if l < k n { a(), a(:end), } Partition_nk( n, k) if a() > a() and l > or a(end-) a(end) > n { a(), a(:end), a(end) } Partition_nk( n, k) Figure 4. The algorithm for the restricted integer partitioning. The algorithm in Figure 4. is applied to all the species Ai in all the compatible subspaces s as follows: for s to C for i to r int Partition_nk( n, M ) is Ai Ai Here C is the number of subspaces, r the number of species in the chemical reaction, of particles of species Ai, and int M A i n Ai the number the number of internal energy eigenlevels associated with species Ai. For each set of s and i, there exists a solution block from the partitioning process. The following example illustrates the implementation of partitioning algorithm: Example 4.: Table 4. describes the number of particles for species A, B and C with their corresponding available number of internal energy eigenlevels. The outcomes of the 46

62 Table 4. Three species with their corresponding initial number of particles and internal energy levels Ai n Ai int M Ai A 5 3 B 4 C Partition_nk algorithm with inputs from Table 4. are as follows As Bs 4 3 Cs Now, step of the process for determining the occupation coefficients is to find all the permutations of the solution given by the partitioning algorithm. In doing so, the results of this step will to be consistent with the dimensions given by equation (3.7) in Chapter 3. However, the results given by the partitioning algorithm does not represent all the possible ways of distributing the particles among the available internal energy eigenlevels. To do so, the following steps are taken: Thus, continuing with example 4., the complete set of permutations for each row of and Cs for j to number of rows of is = all unique permutations of row j that are compatible with equation (3.7) is as follows: As, Bs As B s Cs 47

63 Where for As, Bs respectively 3 5 3! L5 5! 3! L 4 L 4! 5 4!!!!!, and Cs, Finally, in step 3 for each compatible subspace s, each row of the block the other rows of the other blocks is the depiction in Figure 4. is written as follows: is is combined with all. This process is illustrated in Figure 4.. In algorithmic form, Figure 4. Depiction of the construction the unique permutations of rows of. is 48

64 for i to number of rows of for j to number of rows of s s... for k to number of rows of i, j,, k is = concatenate rows s s rs Although the above algorithm is simple and easy to implement, it is not especially efficient if the βis's have large number of rows. A better scheme, which is implemented in this work, is to generate the row permutations based on the fact that each block L is B int s Mi Ln is r Mi Ln is i Mi Ln is The occupation coefficient for each subspace s is be given by, is is repeated is times, i.e., (4.6) where is and ks j (4.7) s is js ks i k j are vectors for all ones expressed as r is, ks Mi M Dim i n L n Dim L is ks (4.8) By implementing equation (4.7) the loops needed in the former scheme are eliminated making it a fast method for building these occupation coefficients. Moreover, equation (4.6) is used only as a check, if each block of is is repeated is times or not by equation (4.7). Finally, note that, given in equation (4.7), is equivalent to the dimensionality of for each subspace s, the number of rows of s equals number of columns equals M. s r i i sqs iji int L B s introduced in Section Where, (given by equation (3.8)) and the 49

65 4.3 System Initial Conditions It is important to emphasize that the operators constructed in this work have a very sparse structure, since they are constructed using a Hamiltonian basis set as is explained in Section 3. of Chapter 3. Moreover, due to the assumption made earlier in this chapter that the reactant mixture behaves as a Gibbs-Dalton mixture of ideal gases, the operators are only diagonally occupied as illustrated below for the occupation number operator constructed using equation (3.33). where x, x, and x i N A i x x are the number of particles of species Ai in the system-levels,, and i, respectively. In order to implement the SEA-QT equation of motion, the density operator for an initial non-equilibrium state needs to be created. The steps needed to initialize the density operator are as follow: Step Find which system-levels where reactant species occupies. To illustrate this, suppose for species A, the occupation number operator is given as N A x x Then it is said that particles of species A occupy system eigenlevels and. The occupied system eigenlevels for species A is expressed as, A x i. This process is done for all reactant species as follows: R A A An (4.9) 5

66 where n is the number of reactant species in the chemical reaction and R is all the system eigenlevels occupied by reactant species. The same procedure is repeated to find the occupied system eigenlevels by product species, P B B Bm (4.) where m is the number of product species in the chemical reaction, m + n = r is the total number of species in the chemical reaction and species. P are all the system eigenlevels occupied by product Next, is to find to intersection of the two sets found in equations (4.9) and (4.), i.e., R P (4.) This step is essential since some species of reactants and products may share the same system level entries as illustrated below, N Reactant x x x 3 y 3 x, N 4 Product y 4 y i In the example above, the third entry of the occupation number operator for both, reactant and product is occupied. Thus, it is necessary to exclude this level in the initialization of the nonequilibrium density operator to insure that only reactant species exists in the initial state. Step The set given by equation (4.), determines the levels of the partially canonical density pe operator or matrix that can be occupied, this density operator need to be constructed and then perturbed. In order to do so, the following set of equations for the occupation probabilities pe y j must be solved: y pe j pe E j j e (4.) Ek e k k 5

67 subject to the constraints The E j pe j pe j Tr H y E H (4.3) pe pe A j j A,..., i j i j Tr N y n N i r (4.4) in these equations are the system-level energy eigenvalues and the i n j i the system-level particle number eigenvalues for the i th species. is either or values and only the j entries corresponding to operator or matrix, the of equation (4.) can be set to. To find the initial non-equilibrium density pe y j are perturbed as follow: f j pe y j (4.5) se y y j i se j j se fi yi j f y (4.6) where λ is an arbitrary perturbation parameter constrained by < λ <, and the occupation probabilities for the stable equilibrium density matrix given by y se j e k E j kb T e Ek kb T Here T is the stable equilibrium temperature. Once the occupation probabilities (4.6) are determined, the initial state operator y y init init is written as, 4.4 Solution Procedure and Convergence Criteria y i y j pe y j are the (4.7) in equation (4.8) In this section, the required inputs to fully describe the problem using the SEA-QT framework are first discussed, and then the scheme implemented to evolve the state of the system is described In order to run the equation of motion of SEA-QT to the chemically reactive system, the following inputs are needed in order to fully describe the problem: 5

68 Mass of each species Quantum numbers associated with each mode of energy storage for each species Distance between atoms for each molecule Vibrational frequencies of the interacting molecules Dissociation energy associated with each molecule Number of particles interacting initially Relaxation time τ of the equation of motion In order to evolve the initial state of the system using the SEA-QT equation of motion, equation (3.5), the initial density operator init given by equation (4.8), is used as the starting point for the SEA-QT equation of motion, which is solved for the evolution of the state of the system. This equation is solved using a forth order Runge Kutta explicit scheme. The problem is solved and coded in Matlab environment using the "ode45" command and the relative error tolerance set to -7. Since the entropy of the stable equilibrium state can be calculated directly by se se se b S k Tr ln (4.9) the solution of the SEA-QT equation of motion is said to reach stable equilibrium when the relative difference of the entropy between the current state and the stable equilibrium state is less than -7 as follows: where S t S se S se 7 (4.) S is the entropy calculated at every time step. t An important point to emphasise is that the time required to solve the SEA-QT model on an Intel Duo Core CPU with.3 GHz workstation is on the order of seconds to minutes for the chemical reactions considered in Chapter 5. This is a huge advantage compared to the computational cost required by, for example, the scattering calculations [5]. Furthermore, since the solution is obtained via solving a set of first order ordinary differential equations, the memory requirement for solving the system models formulated using this framework is minimal, which contrasts with methods that depend on 3D grids of the configurational space where the dimensionality grows exponentially []. 53

69 4.5 Code Verification In order to verify that the code developed is producing results that are physically correct, the following series of test are conducted Verifying the Density Operator As described earlier, the density operator is the operator that holds all the information about the state of the system in every time step of the evolution. In order for the generated by the SEA- QT equation of motion to be that of a well-behaved solution, the following items are checked in every time step, namely, that [74]: Also, t t is Hermitian is non-negative t Tr t i is not idempotent property so i i i must be satisfied Verifying the Hamiltonian Operator Since the system under consideration is a closed one, the energy of the system is conserved if it remains constant throughout the chemical reaction process. Thus, the expectation value of the system's Hamiltonian must satisfy H Tr H constant t t Although the above constraint is necessary it is not sufficient. Thus, in addition to the above constraint, the summation of the energies of all the species in the reaction must be equal to the same system energy. This statement is expressed mathematically by r Ai H H Tr H i t constant In order to construct the system Hamiltonian t t Ai t H that only corresponds to the energy contributions of species Ai, equation (3.4) for the system-level internal degree energy eigenvalues is truncated to yield E M int i int i s int e A sq j sqs i iji ji (4.) 54

70 In a similar fashion, equation (3.) is truncated to determine the system-level translational energy eigenvalues for each system level sq s (3.8) are used to construct the operator for species Ai. Then, equations (3.), (3.), (3.4) and A i H t Verifying the conservation of Number of Particles The expectation value N Ai. of the occupation number operator is not a conserved quantity. However, the number of atoms involved in the chemical reaction is conserved. For example, for the following reaction: the number of atoms for species A at every time step must satisfy A B AB B (4.) Similarly, for species B, N A N constant (4.3) AB N N N constant (4.4) B AB B 4.6 Reaction Rate Constant An important quantity in the field of reaction kinetics is the reaction rate constant k. This quantity is necessary for determining the rate of a chemical reaction. Although the classical and quasi-classical methods discussed in Chapter can predict a value at a given temperature for k, the SEA-QT framework has the advantage of being able to predict the k for the full kinetics of the chemical reaction rather than only for the initial kinetic state. It does so without any classical assumptions eschewing Newton s nd Law as the driving mechanism for the chemical reactions and replacing it with the nd Law of thermodynamics and its principle of SEA. To illustrate the use of the reaction rate constant, consider the following reaction AB C. Conventionally for this reaction, the rate equation is given by na 55 nb r k T A B (4.5) where r is the net rate of the chemical reaction and has units of mol/(l sec) or in atomistic level units, it can be expressed, for example as particles/(cm 3 sec). In equation (4.5), [A] and [B] are the concentrations for species A and B, respectively; and n A and n B are the reaction orders for species A and B, respectively. Note that this equation assumes that the backward reaction rate is negligible which, in fact, is a good assumption only at the beginning of the reaction.

71 As is shown in equation (4.5), the role of the reaction rate constant k is to quantify the kinetics of the chemical reaction. This kinetic information is directly available in the SEA-QT framework through the use of the rate of the reaction coordinate rate of the chemical reaction in the SEA-QT framework is expressed as where r N V E n I E Ai ia i V given in equation (3.65). Indeed, the net (4.6) N A i is the time rate of change for expectation operator for species Ai, nia the initial number of particles for species Ai, V the volume and i the set of stoichiometric coefficients for species Ai. In addition, the concentrations for species [A] and [B] are given using equations (3.56) or (3.6) as N A and N B, respectively. Note that unlike equation (4.5), equation (4.6) applies not just at the beginning of the reaction but throughout the entire kinetic evolution of state. The SEA-QT model provides much more information about the dynamics of the reaction process. In fact, k is not a constant but instead a function of time t, and k and the net rate must be split into two parts, one corresponding to the forward rate and where k f and r f k b and the other to the backward rate r b such that r t r t r t (4.7) f,, b na nb nc b r t k f t T A t B t k t T C t (4.8) are the forward and backward reaction rate constants, respectively, which are, in fact, no longer constants but instead functions of t. In order to calculate both and k b, equation (4.8) is coupled with the zero rate condition at stable equilibrium and the assumption that the detailed balance condition also holds for the time-dependent rate constants such that the ratio of the reaction rate constants is given by k f t, T k f tse, T A B se se (4.9) k t, T k t, T C b b se se where t se and se denotes the time and concentration at the stable equilibrium state, respectively. Therefore, k f and k b are found by solving equations (4.8) and (4.9) simultaneously. Note that the temperature T in the functional expression for k f and k b at times t is the stable equilibrium 56

72 temperature at time t se and not the non-equilibrium temperature H (equation (3.4)) which characterize the states at times t. 4.7 Thermodynamic Properties for Non-Equilibrium States The intensive thermodynamics properties of temperature T, pressure P and chemical potential i are well-defined properties for stable equilibrium states. These properties are given by U T S U P V U i N V, N S, N i V, S, Nj ( ji) (4.3) (4.3) (4.3) where U is the energy, V the volume, N the composition. In order to predict those properties and others for states, which are not in stable equilibrium (i.e., partially canonical as well as nonequilibrium states), a surrogate system as described in [75] is used. Figure 4.3 illustrates the use of the surrogate system. In this figure, the surrogate system is denoted as system B and is comprised Partially canonical states for the initial composition of system A Stable equilibrium states of the initial composition (Surrogate system B) E Stable equilibrium states for the final composition (Surrogate system B) E Plane of constant energy E S, n T final, n f fixed E S, n S T initial, n i fixed S Figure 4.3 Partially canonical states for the initial composition of system A (dashed blue curve), stable equilibrium states for the initial composition of A in surrogate system B (green solid curve) and stable equilibrium states for the final composition of A in surrogate system B (black solid curve). 57

73 of the same species as system A. System B is always in some stable equilibrium state, while system A starts and ends in a partially canonical state, passing through successive non-equilibrium states as a result of the chemical kinetic behavior of the system. Thus, surrogate system B has no active chemical reaction mechanisms while system A does. Although the two systems are different systems, they have the same values U, V and N. Since, each state of the surrogate system is a stable equilibrium state, the following expression derived from SEA-QT [76, 77] using the Maximum Entropy Principle is used to calculate the temperature T for the surrogate system given where q si U and N at a given instant of time: r r 3 ln q s U i Ni ui Ni kb T Ni kbt (4.33) i i ln T is the one-particle partition function for the internal degrees of freedom for species i and to simplify the notation the subscript i is dropped. This function given by and qs v one, and where q q q q q (4.34) s sv sr sd se is the one-particle vibrational partition function, v q se qs r the rotational one, qs D the dissociation the electronic one. The first of these is expressed as follows: qs v (4.35) exp / T is the vibrational characteristic temperature, i.e., v where is the vibrational frequency. That for rotation is written as h (4.36) k B where r qs g exp r l l l (4.37) l T g l l (4.37) l r (4.37) Ik B Here, r is the rotational characteristic temperature, while I is the moment of inertia. As for dissociation, 58

74 D qs exp D (4.38) kb T where D is equal to the first vibrational quantum level, e j qs g exp e e j (4.39) j kb T The ej in this last equation are the electronic energy levels. Once T has been found the entropy is determined from where q c r E S n s kb ln qc (4.4) i T is the total partition function given as, where and q t q e i q q exp (4.4) i kbt c t s V 3 qt mk 3 bt (4.4) h is the translational partition function, m the particle mass, h the Planck's constant. Now, using the definition given in equation (4.3), the pressure is expressed as follows, ln qc P kb T V (4.43) ei e i P exp q c i k B T V (4.44) The only degrees of freedom, i.e., energy eigenvalues, dependent on the volume are the ones associated with the translational degrees of freedom. Therefore, substituting equation (4.) into the above equation results in e i kb T n h nx y nz P exp (4.45) q c V i 8 m L x L y L z Now substituting back the expression for the translational energy eigenvalue, equation (4.), into the last equation yields ei exp ei i kb T P V q c tr (4.46) 59

75 Finally, the chemical potential at stable equilibrium can be calculated from the following characteristic function for an ideal gas [76, 77]: 5 kb 3 kb T ln P kb T ln T kb T ln qs kb T ln mk 3 B (4.47) h 6

76 5 Chapter 5 Results and Discussions In this chapter, the extensively studied reaction mechanism of the fluorine atom reacting with the hydrogen molecule is modeled and studied using SEA-QT. Several parameters affecting the evolution of the chemical reaction are studied such as the effect of the number and type of translational and internal energy eigenlevels present. The relaxation time of the equation of motion is studied and fitted to experimental data available in the literature. The fluorine atom and hydrogen molecule reaction mechanism is then split into two reaction mechanisms. One that describes the reactants and the activated complex, and the second describing the activated complex and the products. The interesting phenomenon of transferring energy between the available energy modes is presented next. In addition, the non-equilibrium thermodynamic properties derived in Chapter 4 are predicted for the one-reaction-mechanism of fluorine and hydrogen. The relaxation time is then studied further in an attempt to search for a general functional which can be used to predict this parameter. Finally, another reaction mechanism, namely, the hydrogen atom reacting with the fluorine molecule, is modeled and studied using the developed time functional at different temperatures, and the results are matched to data in the literature. 5. Hydrogen Molecule and Fluorine Atom Reaction Mechanism The chemical reaction mechanism of a hydrogen molecule with a fluorine atom, i.e., F H HF H (5.) has been considered as the benchmark exothermic reaction and for decades has been under intensive investigation both theoretically and experimentally. The characteristic kinetics of this reaction has been studied theoretically using different approaches. Wilkins [35], Muckerman [78], and Hutchinson and Wyatt [79] have investigated the reaction rate constant classically using the trajectory calculation approach. Several quasi-classical studies has been conducted for this reaction as well among which are the ones done by Feng, Grant, and Root [8] and Aoiz et al. [8, 8]. This chemical reaction is considered as one of the few reactions that can be studied using extensive quantum models due to the simple geometrical configurations of the constituents in this reaction. The literature is rich with studies that investigate several aspects of the F+H reaction 6

77 Energy from the quantum mechanical point of view. In particular, Wu, Johnson, and Levine [83] study the effects of the location of the energy barrier on the characteristics of the reaction dynamics. Redmon and Wyatt [84] and Rosenman et al. [85] predict the reaction probabilities and the cross sections for the three dimensional quantum model at a low region of electronic energies. Castillo et al., [86] investigate the effect of collision angle on the cross section of the reaction. A more recent study is that Moix and Huarte-Larrañaga [87], who investigate the rate constant based on flux correlation functions. This reaction has also been under intensive experimental investigation. Among the many experimental studies reported in the literature to investigate the reaction kinetics is the one conducted by Wurzberg and Houston [88] who study the reaction rate constant in the temperature range from 9 K to 373 K. Heidner et al. [89] investigate the reaction rate constant over a wider temperature range (i.e., from 95 K to 765 K), while Chapman et at. [9] look at the state-to-state reactive scattering process at collision energies of.4 kcal/mole. Other studies are devoted to estimating the activation energy (i.e., the height of the reaction barrier) of the chemical reaction [9-94]. For a review about this chemical reaction, one can refer to the work by Persky and Kornweitz [95]. 5.. One Reaction Mechanism In this section, the reaction mechanism given in equation (5.) is modeled using the SEA-QT framework outlined in Chapter 3. The choice for the translational energy eigenvalues is such that energies eigenstructure of reactant and product are overlapping as depicted in Figure 5.. H HF Overlap region Intermolecular distance Figure 5. Overlapping of the energies of diatomic molecules H and HF. 6

78 In order to construct the energy structure associated with each one-particle species and given by equations (4.) to (4.4), the parameter values given in Table 5. are used. For the first set of results presented here, the initial amount of each reacting species is fixed at and n a H na F particle particle. The initial temperature of the reaction is chosen to be 3 K. The relaxation time τ appearing in equation (3.36) is set at 5. - sec, this choice is made to get kinetic results that are comparable to that available in the literature, such that the reaction rate constant predicted by the SEA-QT model is of the same order of magnitude to that in the literature. The quantum numbers of the internal degrees for the freedom for the diatomic constituents are those shown in Table 5.. Again this choice is made so that the internal eigenlevels associated with each diatomic species is comparable in terms of the number of vibrational and rotational eigenlevels used to define the energy eigenstructure as in [96-98]. Table 5. Parameter values needed to calculate the one-particle energy structure for the hydrogen molecule and fluorine atom reaction mechanism. F H H HF Mass (kg) Bond length ( Å ) Dissociation energy (ev) [99] Wavenumber (cm - ) [] Table 5. Internal structure of the diatomic molecules in the F + H reaction mechanism. Vibrational quantum # Rotational quantum # H HF,,, 3,,, 3,,, Table 5.3 Translational quantum numbers considered for each species in the F + H reaction mechanism. Ai F H HF H Translational quantum # ( evenly spaced samples) k=,...,55 k=,..., k=,...,5 k=,..., 63

79 As discussed earlier, the quantum translational degrees are chosen such that the range of energies on the reactant side overlap those on the product side. Moreover, the translational quantum numbers are only sampled evenly, which in future work will be done in a more robust way based on density of state calculations. For the results shown in this section, the choice of translational quantum numbers is as shown in Table 5.3. The same analysis done in Section 3.. of Chapter 3 to determine the number of compatible subspaces as well as the limits of the reaction coordinates is done next. The results for the latter for the given amounts of n a F and n a H and the F + H reaction mechanism are shown in Table 5.4. Table 5.4 Limits on the reaction coordinates for the F + H reaction mechanism. Ai i n ia F - n is s n is H - s s HF H s s s s The solution of the set of inequalities given by in Table 5.4 as well as the relation between the eigenvalues of the number operator ( n is s ) to the eigenvalues of the reaction coordinates ( shown in Table 5.5. The number of compatible compositions is again. Table 5.5 The number of compatible compositions for the F + H reaction mechanism. s ) are Ai F H HF H s s n is n i n i n s n s n 3s n 4s Equation (3.34b) is now used to construct the reaction coordinate operator as follows: 64 E () P () P P (5.) H H H

80 The expectation value of the reaction coordinate is next calculated using equation (3.63) and is given in terms of the reaction coordinate occupation probabilities ws such that E Tr P w (5.3) H The occupation number operators are then found using equation (3.35) with the result that yielding NF I E I P H N I E I P H H NHF NH E E P H P H The expectation values of these number operators are determined with equation (3.6) NF N NHF NH w w H w w Thus, a way of verifying the results generated by the SEA-QT model and code is to check that the expectation values of the occupation number operators for F and H as well as those for H and HF are always equal. (5.4) (5.5) E vs S "partially stable" E vs S "stable" E Energy (ev) Entropy (ev/k) x -4 Figure 5. Initial and final and states of the chemically reactive system at 3 K for the F+H reaction mechanism and the parameter values given in Tables Figure 5. shows the initially perturbed starting state far from equilibrium for the F+H reactive system. The initial unperturbed state of these reactions is one of stable equilibrium at 3 65

81 K. Also seen in this figure is the state evolution path (i.e., blue curve) as well as the final stable chemical equilibrium state. Since the driving principle for this model is that of SEA the prediction of the entropy and entropy generation rate at every instant of time is a direct outcome of the SEA- QT framework. Figure 5.3 and Figure 5.4 show the instantaneous values for both entropy and entropy generation. The former indicates that the entropy increases rapidly at the beginning of the reaction and then slows as the reaction process approaches stable equilibrium. Moreover, Figure 5.4 shows that the bulk of the entropy generated in the process occurs during the first third of the process. 6 x -4 5 Entropy (ev/k) Time (sec) x - Figure 5.3 The instantaneous entropy values for the F+H reaction process which is at an initial stable equilibrium of 3 K. x 7 Entropy Generation Rate (ev/k sec) Time (sec) x - Figure 5.4 The instantaneous entropy generation values for the F+H reactive system which is at an initial stable equilibrium temperature of 3 K. 66

82 One of the distinct features that the current SEA-QT framework provides is the a detailed dynamics of the reaction process. Thus, information such as the expectation values of the occupation number operator or that of the number of particles/moles of each species are known at every instant of time. For example, the expectation value of the particle number operator is shown for each species in Figure 5.5. The results are consistent with equations (5.4) and (5.5) which predict that at every instant of time species F and H have the same composition as do species H and HF. Thus, mass is conserved at ever instant of time. Number of Particles F H H HF.5.5 Time (sec) x - Figure 5.5 The expectation values of the particle number operator for each species of the F+H reaction corresponding to an initial stable equilibrium of temperature at 3 K. Figure 5.6 shows the expectation energy of each species throughout the reaction process as well as that of the system. Clearly, energy conservation is satisfied at ever instant of time..5 F Energy (ev).5 H H HF System.5.5 Time (sec) x - Figure 5.6 The expectation energies for each species of the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K. 67

83 The expectation values for the entropy, energy, and number of particles plotted above can be combined in a 3D plot to show the behavior of the system for each species along the reaction path. This behavior is shown in Figure 5.7. The reactant species (F and H) decrease in terms of both their energy and the number of particles from right to left, while the products (H and HF) increase from left to right in terms of their energy and the number of particles. The entropy plotted is not that of the individual species but instead that for the reacting system as a whole, i.e., the entropy is a surrogate for time in this figure. Figure 5.7 The expectation values for the system entropy and the energy, and number of particles for each species of the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K. Reaction Coordinate Time (sec) x - Figure 5.8 The instantaneous expectation values of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K. 68

84 The instantaneous expectation values of the reaction coordinate and the rate of reaction coordinate are shown in Figure 5.8 and Figure 5.9, respectively. The trend of the reaction coordinate and the rate of reaction coordinate closely follows the trend given by the entropy and entropy generation in Figure 5.3 and Figure 5.4. This behavior is expected since the entire reaction process is directly tied to the evolution of entropy. 3.5 x 3 Rate of Reaction Coordinate Time (sec) x - Figure 5.9 The instantaneous expectation values of the rate of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K. 5 k f Reaction Rate Constants -5 - k b K Time (sec) x - Figure 5. The forward and backward reaction rate constants as well as the equilibrium constant as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K. 69

85 The forward and backward reaction rate constants kf and kb as well as the equilibrium constant K as a ratio of kf and kb are plotted in Figure 5.. The figure shows that the forward reaction rate constant is significantly higher than the backward one throughout the evolution of state. Furthermore, the initial values for each are considerable different that later values which are greater but plateau out and even decrease somewhat as the reaction proceeds. x 3 r f Reaction Rates.5.5 r b r.5.5 Time (sec) x - Figure 5. The forward, backward, and net reaction rates as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K. Figure 5. shows the forward, backward, and net reaction rates for the F+H the reaction. Based on the detailed balance assumption [98], the forward and the backward reaction rates are equivalent at stable equilibrium as indicated in the figure. As can be seen the backward reaction as expected plays no role initially but then gradually increases its contribution as that of the forward reaction peaks and then decreases. At about a third of the way into the reaction, these rates have already approached each other very closely, a trend which continues until they are identical at stable equilibrium. Note that though these rates differ by as much as an order of magnitude early on, their associated reaction rate constants kf and kb consistently to 3 orders of magnitude different with the latter always the smallest. As part of the code verification outlined in Section 4.5 conservation of both the density operator and the number of particles is verified here. The density operator is checked to be Hermitian, non-negative with its trace equal to at every instant of time. Similarly, the number of particles is always conserved as shown in Figure 5. for the atomic constituents of the F+H reaction, namely, F and H. As can be seen, these particles are conserved at every instant of time. 7

86 Number of particles Hydrogen Fluorine Time (sec) x - Figure 5. Conservation of the number of particles in the F+H reaction Effect of the Number of Levels Occupied Initially The results reported in the previous sections are for the case when the number of system eigenlevels occupied at the beginning of the chemical reaction is =. In this section, the number of initially occupied eigenlevels ; is altered to assess the effects on the kinetics results generated earlier. A comparison between three different is made, i.e., for =,, E vs S "partially stable" E vs S "stable" E Energy (ev) 5 = = = Entropy (ev/k) x -4 Figure 5.3 Initial and final states of the F+H reactive system corresponding to an initial stable equilibrium temperature of 3 K and a different number of system eigenlevels initially occupied. Figure 5.3 shows the three different initial states considered (i.e., the green circles on the left) the state evolution path, and the final stable chemical equilibrium state. The results presented 7

87 in this figure shows that the more system eigenlevels initially occupied, the closer the state is to the stable equilibrium curve of the final state (i.e., the curve where all the energy levels are occupied). Equivalently, the more system energy eigenlevels occupied initially, the higher the expectation value of entropy of the system is. Figure 5.4 and Figure 5.5 show the expectation values of the entropy and entropy generation rate in time. As can be seen in Figure 5.4, the more system eigenlevels are initially occupied, the slower the evolution of system state is as indicated by the decrease in slope of the entropy versus time curve as increases. Furthermore, as expected the total entropy generated as presented by the area under each curve in Figure 5.5 decreases as increases since the reaction starts at states successively closer to the final state of stable equilibrium. In addition, one can see that the number of initially occupied eigenlevels does not affect the final stable equilibrium state, since the latter only depends on the expectation energy and the energy structure of the system which is the same for each values of used here. In other words, the location of the initial state only affects the dynamics of the reaction. 6 x -4 Entropy (ev/k) = = =3.5.5 Time (sec) x - Figure 5.4 The instantaneous entropy values for different values of for the F+H reaction process corresponding to an initial stable equilibrium of 3. 7

88 Entropy Generation Rate (ev/k sec) x = = =3.5.5 Time (sec) x - Figure 5.5 The instantaneous entropy generation values for different values of for the F+H reaction process corresponding to an initial stable equilibrium of 3. The expectation values of the particle number operator for each species as a function in time for different are presented in Figure 5.6. The only effect of the number of initially occupied energy eigenlevels is the speed at which the stable equilibrium composition is approached as represented by the decreased slopes of the particle number curves as increases. In a similar fashion, the speed of evolution of the energy expectation value associated with each species shown in Figure 5.7 decreases as increase..8 F Number of Particles.6.4. increase in this direction H H HF.5.5 Time (sec) x - Figure 5.6 The expectation values for different values of of the particle number operator for each species of the F+H reaction corresponding to an initial stable equilibrium of temperature at 3. 73

89 .5 F Energy (ev).5 H H HF System.5.5 Time (sec) x - Figure 5.7 The expectation values for different values of of the energy for each species of the F+H reaction corresponding to an initial stable equilibrium of temperature at 3. The expectation values of the reaction coordinate and the rate of the reaction coordinate are plotted in Figure 5.7 and Figure 5.8 for different values of. These figures again show how the dynamics of the problem are changed by altering the number of initially occupied system eigenlevels. As seen in Figure 5.7, the reaction coordinate value increases much more rapidly at lower values of than at higher ones. Furthermore, the rate of the reaction coordinate seen infigure 5.8 is notably higher at small than at larger ones, showing how the speed of the reaction noticeably decreases with increases in. Reaction Coordinate =. = =3.5 Time (sec).5 x - Figure 5.8 The instantaneous expectation values for different values of of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K. 74

90 Rate of Reaction Coordinate 3 x = = =3.5.5 Time (sec) x - Figure 5.9 The instantaneous expectation values for different values of of the rate of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K. The forward and backward reaction rate constants for the different are plotted in Figure 5.. Although the number of system eigenlevels initially occupied changes the dynamics of the chemical reaction as confirmed previously and in this figure as well, the effect of, though noticeable for the first third of the reaction, is not significant. In fact as the reaction progresses towards stable equilibrium, the effect of becomes insignificant since the speed of the evolution in that region is already slow. Thus, the difference in reaction rate constants almost vanishes. Reaction Rate Constants Reaction Rate Constants increases in this direction increases in this direction.5.5 Time (sec) x - Figure 5. Forward and backward reaction rate constants for different values of as a function of time - for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K k f k b K.5.5 Time (sec) x -

91 x 3 r f Reaction Rates.5.5 decreases in this direction r b r Figure 5. The forward, backward, and net reaction rates as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K for different number of occupied system eigenlevels initially. Figure 5. shows that the reaction rate noticeably slows by starting closer to the final stable equilibrium state. The rate increases by around 3% for the case of = compared to that for =3, yet the reaction rates at the initial state are very close for the 3 different values of. This figure furthermore shows that the number of initially occupied system eigenlevels, mainly affects the rate of the chemical reaction in the first third of the reaction process after which the rate converges to a common value..5.5 Time (sec) x Effect of the Number of Translation Degrees of Freedom For the results presented in the previous sections, the sampling number for the translational energy eigenvalues associated with each species is set to Mtr =. The samples are equally spaced. In this section, the effect of changing the sampling number as well as the spectrum of the translational quantum numbers is studied. With respect to the former, a system with a sampling of translational degrees of freedom per species is compared to one with 5. In each case, the sampling is evenly spaced with the same quantum number range as indicated in Table 5.3. Figure 5. shows the initially perturbed starting state far from equilibrium as well as the state evolution path (i.e., the blue curve). Moreover, two different final stable chemical equilibrium states are indicated, one corresponding to a thermodynamics system where for every species Mtr = (i.e., the green circle on the solid red curve) and a different system where for each species Mtr =5 (i.e., the green circle on the solid 76

92 green curve). The first noticeable thing is that increasing sampling number of the translational quantum numbers does not affect the shape of the curve of stable equilibrium states. However, it does push the whole curve to the right. This means that the stable equilibrium state of the reaction process happens at a higher expectation value of the entropy when the sampling is larger. Energy (ev) E vs S partially stable E vs S stable M tr = E vs S stable M tr = 5 E Entropy (ev/k) x -3 Figure 5. Initial and final states of the F+H reactive system corresponding to an initial stable equilibrium temperature of 3 K for samplings of and 5 translational quantum given in Table 5.3. x -3.8 Entropy (ev/k).6.4. M tr = M tr =5.5.5 Time (sec) x - Figure 5.3 The instantaneous expectation values of the entropy for the F+H reaction corresponding to at an initial stable equilibrium of 3 K are plotted for different sampling number of the translational quantum numbers. Figure 5.3 shows the expectation values of the entropy for different sampling number of the translational quantum numbers. This figure confirms the statement made earlier that the final stable equilibrium state is attained at higher value of the entropy for the case of Mtr =5. By increasing 77

93 the sampling of translatin quantum numbers associated with each species from to 5, the entropy at the final stable equilibrium state increases by around 5% when compared to that for Mtr =. Therefore, the sampling number of translation quantum numbers associated with each species plays an important role in determining the final stable equilibrium state of the chemical reaction. As noted earlier, a method to deal with this deficiency of the framework used here, i.e., the susceptibility of the results to the number and spacing of the samples used, has already been developed and will be reported in future work. That method, which is based on the density of states can be applied not just to the translational degrees of freedom but to internal degrees of freedom (e.g., rotation) as needed. Entropy Generation Rate (ev/k sec) 4 x 7 3 M tr = M tr =5.5.5 Time (sec) x - Figure 5.4 The instantaneous expectation values of the entropy generation for the F+H reaction corresponding to at an initial stable equilibrium of 3 K are plotted for different sampling number of the translational quantum numbers. The expectation values of the entropy generation plotted in Figure 5.4 have higher values for the case where Mtr =5. Again, a 5% increase is seen for the entropy generation with an increase in the sampling number of the translation quantum numbers from to 5 for each species. Although, the initial values for the two cases are close to each other, the entropy generation rate is significantly larger at - sec for the case of Mtr =5. Furthermore, the entropy generation rate exhibits a steeper drop for this case once it reaches its peak. Thus, this should be an indicator of having higher reaction rates associated with the case for Mtr =5. 78

94 .8 F Number of Particles.6.4. M tr decreases in this direction H H HF Figure 5.5 The expectation values of the particle number operator for each species for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K plotted for different sampling numbers of translational quantum numbers. Results for the expectation values of the number operator corresponding to each species in the reaction are plotted in Figure 5.5. The direct effect of increasing the sampling number is the speed of the chemical reaction. The more translational energy eigenvalues considered, the faster the reaction is. Also noted in this figure is the fact that although the final state occurs at a larger value of the entropy, the final composition barely changes, suggesting that the translation eigenlevels have little effect on final compositions..5.5 Time (sec) x - Reaction Coordinate M tr = M tr =5.5.5 Time (sec) x - Figure 5.6 The instantaneous expectation values of the reaction coordinate operator for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K plotted for different sampling numbers of the translational quantum numbers. 79

95 The expectation values of the reaction coordinate corresponding to the two cases of Mtr are plotted in Figure 5.6. Again, one can see a faster dynamics for the case of more samples (i.e., Mtr=5). Note that although the reaction coordinate curve follows a similar curvature as that for the entropy curve, altering Mtr does not change the final value of the reaction coordinate, which is consistent with what was seen earlier with the expectation values of the particle number operators. In a similar fashion, expectation values of the rate of the reaction coordinate operator presented in Figure 5.7 show as expected a faster dynamics when more translational degrees of freedom are considered in the eigenstructure of the system. Rate of Reaction Coordinate 5 x 4 3 M tr = M tr =5.5.5 Time (sec) x - Figure 5.7 The expectation values of the rate of reaction coordinate operator corresponding to an initial reaction temperature at 3 K plotted for different sampling numbers of translational quantum numbers. The forward and backward reaction rate constants for the two cases of Mtr are plotted in Figure 5.8. The rate constants again are higher for the case the higher sampling of the translational quantum numbers. The value for the forward reaction rates at the initial state for the case of Mtr = is 3. - cm 3 molecule - sec - and for Mtr = 5 is cm 3 molecule - sec -. Although it is higher for the second case it is still on the same order of magnitude. The difference in both the forward and backward rate constants relative to the value of Mtr is highest about quarter of the way into the reaction, decreasing significantly afterwards and following the trends of the expectation values of the entropy generation and rate of reaction coordinate. 8

96 Figure 5.8 Forward and backward reaction rate constants as a function of time for the F+H reaction - corresponding to an initial stable equilibrium temperature of 3 K plotted for different sampling numbers of the translational quantum numbers. The effect of the spectrum or range.5 of translational quantum.5 numbers is examined next. For this study, the sampling number of translational quantum numbers is kept constant at samples, while the range of quantum numbers for each species is varied. Table 5.6 lists the different translational quantum number ranges used and the forward reaction rate constant which results. Four cases are considered. The reaction rate constant is that found at the start of the reaction. Table 5.6 The different translational quantum number used and the forward reaction rate constants which results. Case F H HF H kf (cm 3 / molecule sec) k=,..., k=,..., k=,...,5 k=,..., k=,...,55 k=,...,5 k=,...,5 k=,..., k=,...,55 k=,..., k=,..., k=,...,.96-4 k=,...,55 k=,..., k=,...,5 k=,..., The first thing to notice is that when a larger range of translational quantum numbers is available in the energy eigenstructure of the reactant side, the reaction tends to gain some speed. Similarly higher translation energies on the product side slow the reaction rate. The second thing to notice is that though translation energies are affecting the rate of the reaction, their contribution is not significant. For example, when the quantum number range increases by a factor of 5 to, the energy eigenstructure for translation is significantly but the reaction kinetics is only somewhat affected. Reaction Rate Constants Reaction Rate Constants - - M tr decreases in this direction M tr decreases in this direction Time (sec) x - Time (sec) 8 k f k b K x -

97 5...3 Effect of the Number of Internal Degrees of Freedom In this section, the effect of the internal degrees of freedom (i.e., vibrational and rotational energy eigenvalues) for the diatomic molecules is studied. Again all other parameter values are set to the same ones used to generate results given in Section 5... Three cases are carried out here, comparing the effect of changes in the vibrational and rotational eigenstructure on both the reaction dynamics and the extent (i.e., reaction coordinate) of the chemical reaction. The vibrational and rotational quantum numbers diatomic reactant molecules (i.e., H) are not changed, following the eigenstructure given in the literature [96, 97], where the ones associated with product molecules (i.e., HF) are changed as presented in Table 5.7. Table 5.7 Different internal energy eigenstructure for the diatomic molecules in the F + H reaction. Case Vibrational quantum # H Rotational quantum # Vibrational quantum # HF Rotational quantum #,,,,,, 3, 3,,,, 4 6 x -4 5 Entropy (ev/k) 4 3 v HF = j HF =, v HF =..4 j HF =, v HF = j HF = Time (sec).5 3 x - Figure 5.9 The instantaneous expectation values of the entropy for the F+H reaction corresponding to an initial equilibrium of 3 K and for the varying internal energy eigenstructure given in Table 5.7. The expectation value of the entropy as a function in time for the three different cases outlined above is plotted in Figure 5.9. The first thing to notice is that increasing the internal energy eigenstructure of the one-particle species pushes the stable equilibrium state to one of higher 8

98 entropy and that is because the more available internal energy eigenvalues there are, the farther the curve of stable equilibrium states moves to the right as seen earlier in Figure 5.. The second thing closely related to the first is that the available vibrational quantum numbers exert a greater influence on the final stable equilibrium state than that of the rotational ones. This is due to the fact that the vibrational energy eigenvalues are higher than that of their rotational counterparts. Table 5.8 The reaction rate constant for the different cases of the internal structures. Reaction Rate Constant Case kf (cm 3 / molecule sec) Reaction Coordinate v HF = j HF =, v HF =..4 j HF =, v HF = j HF = Time (sec) x - Figure 5.3 The instantaneous expectation values of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K and for the internal eigenstructures given in Table 5.7. The quite interesting result of this study is that the addition of only few rotational quantum states has a greater effect on the dynamics of the reaction than increasing the number of vibrational levels. This is reflected in the higher expectation values for the entropy seen in Figure 5.9 for case 3 and is born out with the values of the forward reaction rate constants shown in Table 5.8. Although the changes do not affect the order of the magnitude of these values, the difference between the three cases are nonetheless significant with again case 3 showing the grease of change. 83

99 This latter results is undoubtedly due to the fact that the higher vibrational energy eigenlevels are too high to play an important role in the dynamics of the reaction. Rate of Reaction Coordinate.5 x.5.5 v HF = j HF =, v HF =..4 j HF =, v HF = j HF = Time (sec) x - Figure 5.3 The instantaneous expectation values of the rate of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K and for the internal eigenstructures given in Table 5.7. The expectation values of the reaction coordinate and the rate of the reaction coordinate for the three different internal eigenstructure are plotted in Figure 5.3 and Figure 5.3, respectively. Clearly, the chemical reaction is closer to being complete at stable equilibrium (see Figure 5.3) when more internal quantum numbers are available. As seen in Figure 5.3, the expectation values for the rate of the reaction for the case with more rotational quantum numbers is slightly higher than that with more vibrational quantum numbers. Moreover, when compared to the addition of more translational levels, the effects of increasing the rotational and vibrational levels plays a much greater role in the kinetics of the reaction process Fitting the relaxation time τ to experiments It is clear from the results presented so far how the energy eigenstructure as well as the choice of initial state for the reaction influence the kinetics of the chemical reaction. Another important parameter which affects these kinetics is the choice of relaxation time τ. To show this effect the F+H reactive system of Section 5.. which was modeled with τ = 5. - sec, is now modeled with τ one third the previous value, i.e., τ =.7 - sec, and the results compared with the earlier 84

100 results. The choice for the latter τ was to see how reducing this quantity by one third is going to affect the kinetics of the reaction process. Figure 5.3 shows the effect of the relaxation time τ on the expectation value of the entropy. As mentioned earlier, τ affects the speed at which the final state is reached, but it does not change the characteristics of the final stable equilibrium state of the system. The slope of the entropy curve 6 x -4 5 Entropy (ev/k) 4 3 = 5. - = Time (sec) x - Figure 5.3 The instantaneous expectation values of the entropy for the F+H reaction corresponding to an initial stable equilibrium of 3 K and for two values of τ. Reaction Coordinate = 5. - = Time (sec) x - Figure 5.33 The instantaneous expectation values of the reaction coordinate for the F+H reaction corresponding to an initial stable equilibrium of 3 K and for two values of τ. 85

101 in this figure is considerably steeper for the case of τ =.7 - even though the final value of the entropy is the same for both cases. The implication of course, is that as the relaxation time decreases the rate of the kinetics of the reaction increases. x 3 5 =.7 - r f r b Reaction rates 4 3 = 5. - r r f r b r Figure 5.34 The forward, backward, and net reaction rates as a function of time for the F+H reaction corresponding to an initial stable equilibrium temperature of 3 K and for different values of τ. The expectation values for the reaction coordinate for the different τ are plotted in Figure The reaction coordinate closely follows the shape of the entropy curve with the reaction coordinate for the smaller τ having a steeper slope than that for the larger τ. Again, it is obvious that τ does not affect the extent of the chemical reaction but simply the speed of the kinetics of the chemical reaction. The forward, backward, and net reaction rates for the two values of τ are plotted in Figure Confirming the results of the previous two figures the reaction rates shown in Figure 5.34 increase by a factor of about 3 which is directly proportional to the 3-fold decrease in the values of τ. One can also observed that the bell shape of the curve for the faster reaction narrows when compared to that for the slower reaction Time (sec) x - Now, in order to have an idea of what the value of τ should be, it is fit to reproduce experimentally derived data available in the literature at different initial stable equilibrium temperatures. The numerical fit is done with the experimental data for kf of Heidner et al. [89]. To fit this data, the number of vibrational and rotational energy eigenlevels, which make up the energy eigenstructure of the system, must be set to those considered by the experiment. In Section 5..., these were arbitrary sets. Six different initial states, each at a different stable equilibrium 86

102 temperature corresponding to those reported in Heidner et al. [89], are considered, and a different value of τ found for each based on the kinetics of each initial perturbed state. The results for each temperature are given in Table 5.9 with the middle column containing the experimental values of Heidner et al. [89] and the column on the right the fitted values for τ. Table 5.9 The forward reaction rates constant by Heidner et al. [89] (HBMG) and the corresponding fitted τ at different stable equilibrium temperatures. T (K) kf(t) / - (cm 3 /molecule-sec) τ / - (sec) Results for the fitted τ are plotted against their corresponding temperatures in Figure Notably the behavior of τ is not linear, especially at low temperatures where the curve for τ is steeper and more curved, becoming less so as the temperature increases. This behavior is consistent with what is obtained for the forward reaction rate constant using quasi-classical methods for which the treatment of the vibrational and rotational degrees of freedom is quantum mechanical, i.e., for these methods, the behavior is also not linear especially at low temperatures. x (sec) Temperature (K) Figure 5.35 Fitted values for τ for the F+H reaction corresponding to different initial stable equilibrium temperatures and for the forward reaction rate constant of Heidner et al. [89]. 87

103 5.. Two Reaction Mechanisms In this section, the reaction mechanism of equation (5.) is separated into two-reaction- mechanisms. The first one is from the reactant side to the activated complex, while the second is from the activated complex to the product side as follows: Again, the initial composition of the reacting mixture contains F H HHF (5.6) HHF HF H (5.7) na F particle and particle. The initial stable equilibrium temperature at which the reaction takes place is 3 K. The relaxation time τ of the equation of motion (i.e., equation (3.36)) is kept constant and equal to 5. - sec. In addition to the values used for the one reaction mechanism presented in Table 5., the wavenumbers of the activated complex HHF used in the following calculations are 397 cm -, 39 cm -, 47 cm - []. The quantum numbers for the internal degrees of freedom of the polyatomic constituents those given in Table 5. and following those of energy eigenstructure giving in the [96]. The translational quantum number for each of the constituents of the F+H two- reaction-mechanism system are described in Table 5.. Again, the choice for such translational energy eigenvalues is that the energy eigenstructure for the three parts of this reaction, namely, reactants, activated complex, and products, are overlapping as depicted in Figure 5.. Table 5. Internal eigenstructure of the polyatomic molecules for the two F+H reaction mechanism system. n a H Vibrational quantum # Rotational quantum # H HF HHF,,, 3,,,,, 3,,,, The same procedure carried out in Section 5.. is repeated here to determine the number of compatible compositions, i.e., subspaces, as well as the limits on the reaction coordinates for the initial amounts of n a F and n a H. The solution set of inequalities of the last column of Table 5.4 is shown graphically in Figure Table 5.3 shows the compatible compositions as well as the relationship between the eigenvalues ( n is ) of the particle number operator and the eigenvalues ( s ) of the reaction coordinates. 88

104 Table 5. Translational energy eigenstructure for each species for the F+H two-reaction-mechanisms system. Ai F H HHF HF H Translational quantum # ( evenly spaced samples) k=,...,55 k=,..., k=,...,65 k=,...,5 k=,..., Table 5. Limits on the reaction coordinates for the F + H two-reaction-mechanisms system. Ai i i n ia n is n is F - s s s H - s HHF - s s s s HF H s s s s s.5.5 s Figure 5.36 The limits on the compatible compositions for the F+H two-reaction-mechanisms system. 89

105 Table 5.3 The number of compatible compositions for the F+H two-reaction-mechanisms system. s 3 s 3 s s Ai n is n i n i n i3 F H HHF HF H n s n s n 3s n 4s n 5s Again, equation (3.34b) is used to construct the reaction coordinate operator as follows: 9 E () PH () P () H P P P H 3 H H 3 E (5.8) E () P () P () P P H H H 3 H 3 The expectation values of each reaction coordinate is calculated using equation (3.63) and is given in terms of the reaction coordinate occupation probabilities ws by H 3 3 E Tr PH P w H 3 w3 E (5.9) E Tr P w The occupation number operators are expressed as a function of the reaction coordinate operator using equation (3.35) such that N I E I P P F H H3 N I E I P P H H H3 E E P H E P H 3 E P H 3 NHHF NHF NH (5.) The expectation value of each number operator can then be determined with equation (3.6) yielding

106 NF N w w H NHHF NHF NH w w w 3 3 (5.) E vs S "partially stable" E vs S "stable" E Energy (ev) Entropy (ev/k) x -4 Figure 5.37 Initial and finial states of the chemically reactive system and its state evolution for the F+H two-reaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K and the parameter values given in Tables 5., 5. and 5.. Based on these last expressions, the expectation values for the number operators for of species F and H are identical as are those for species HF and H. The results for the F+H two-reactionmechanisms system presented on the following pages correspond to an initial stable equilibrium temperature of 3 K and an initially perturbed state in which only two energy eigenlevels are occupied, both on the reactant side (i.e., = ). This initially perturbed starting state far from equilibrium is shown in Figure This figure is similar to that for the single reaction mechanism results, i.e., Figure 5., with the stable equilibrium curve no further to the right. The reason is that even though the two-reaction-mechanisms system in comparison to the one-reaction-mechanism system has more degrees of freedom (i.e., energy eigenlevels), the effect is one of simply increasing the number of different ways the system energy can be stored or distributed among the available energy eigenlevels. However, the final value of the system entropy is not changed. This must be true since the two-reaction-mechanisms system reflects the same reactive system as that 9

107 of the one-reaction-mechanism system. Thus, the total entropy generated in both is the same, leading to the same final entropy. Figure 5.38 and Figure 5.39 show the expectation values of the entropy and entropy generation rate for the F+H two-reaction-mechanisms system, respectively. In comparison with the-one reaction-mechanism system, the final stable equilibrium state is attained as already mentioned at the same value of the entropy. The entropy generation rate curve is the same as well since it is the only source of entropy increase. Therefore, adding the activation complex to the reaction 6 x -4 5 Entropy (ev/k) Time (sec) x - Figure 5.38 Expectation values of the entropy as a function in time for the F+H two-reaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K. x 7 Entropy Generation Rate (ev/k sec) Time (sec) x - Figure 5.39 Expectation values of the entropy generation rate as a function in time for the F+H tworeaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K. 9

108 mechanism picture does not change the kinetics of the process but simply makes it possible to extract more information as will be discussed later. The curve for the entropy generation peaks at the beginning of the reaction process and then decreases as the state gets closer to that of stable equilibrium. The expectation values of the particle number operators are shown in Figure 5.4. Again, this figure looks very similar to that for the case of one-reaction-mechanism system given in Figure 5.5. The amounts of the final products seem to be the same or almost so to those for the one-reactionmechanism. However, the amounts of the reactants has decreased slightly, compensating for the creation of the activated complex HHF. The activated complex HHF is always present in some present in some small throughout the reaction process. The results here are again consistent with equations (5.) and (5.), which predict that at every instant of time, species F and H have the same composition as do species H and HF and, therefore, mass is conserved at ever instant of time. Number of Particles F H H HF HHF.5.5 Time (sec) x - Figure 5.4 Expectation values of the particle number operator for each species of the F+H two-reactionmechanisms system corresponding to an initial stable equilibrium temperature of 3 K. The expectation values of the energy of the system well as those of each species throughout the reaction process are plotted in Figure 5.4. Again, the expectation energy of the system is constant throughout the reaction process, while the expectation energies of the product species are the same or almost the same as those for the one-reaction-mechanism system. The reactant species energies in contrast are somewhat smaller, compensating for the energy possessed by the activated complex. Energy is conserved at every instant of time. 93

109 Energy (ev).5.5 F H H FH HHF System.5.5 Time (sec) x - Figure 5.4 Expectation energies for each species and the system of the F+H two-reaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K. Figure 5.4 shows a 3D plot for the expectation values for the system s entropy and energy and the energy and number of particles associated with each species. As shown in this figure, the reactant species (F and H) decrease in terms of both their energy and the number of particles from right to left. At the same time, the products (H and HF) increase from left to right in terms of their Figure 5.4 Expectation values for the system entropy and energy and the number of particles and energy for each species of the F+H two-reaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K. energy and the number of particles, while only a small amount of the activated complex is accumulated in the system as the reaction moves towards stable equilibrium. Again, the entropy 94

110 axis can be thought of as a time axis where the higher the value of the entropy is, the more time has elapsed in the reaction. Reaction Coordinate h" i h" i.5.5 Time (sec) x - Figure 5.43 Instantaneous expectation values of the reaction coordinate of the F+H two-reactionmechanisms system corresponding to an initial stable equilibrium temperature of 3 K. Rate of Reaction Coordinate 3 x h "_ i h "_ i.5.5 Time (sec) x - Figure 5.44 Instantaneous expectation values of the rate of the reaction coordinate for the F+H tworeaction-mechanisms system corresponding to an initial stable equilibrium temperature of 3 K. The expectation values of the reaction coordinate and the rate of the reaction coordinate are plotted in Figure 5.43 and Figure 5.44, respectively. The value of the reaction coordinate of the first reaction goes further towards completion than for the second reaction mechanism. This result is anticipated since a smaller amount for each reactant of the first reaction mechanism exists at 95

111 stable equilibrium as seen in Figure 5.4. The values of the rate of the reaction coordinates exhibit higher peaks at beginning of the reaction than what is seen for the one-reaction-mechanism system. It is only at the first third of the reaction process that a difference in rates between the two mechanisms appears, with the first being somewhat faster. The time to reach equilibrium is about the same for the two-reaction-mechanism systems as it is for the one-reaction-mechanism system. The forward and backward reaction rate constants as well as the equilibrium constant for the first reaction mechanism (equation (5.6)) are plotted in Figure Here, the forward reaction rate constant is comparable to that obtained for the one-reaction-mechanism system plotted in Figure 5.. In contrast, the backward reaction rate constant is orders of magnitude higher than that for the forward reaction, the exact opposite of what is seen in Figure 5.. The reason is that as in equation (4.8), where for the case here r k FH k HHF, the concentration of the activated complex HHF is much smaller compared to the concentration of reactants. Therefore, in order to satisfy the rate, the backward rate constant kb must maintain a very high value. k f f b Reaction Rate Constants - k b K Time (sec) x - Figure 5.45 Forward and backward reaction rate constants as well as the equilibrium constant as a function of time for the F H HHF reaction mechanisms corresponding to an initial stable equilibrium temperature at 3 K. The forward and backward reaction rate constants as well as to the equilibrium constant of the second reaction mechanism (i.e., equation (5.7)) are shown in Figure This time the forward reaction rate constant has higher values for the same reason discussed above with regards to kb in the first reaction mechanism, i.e., kf is now associated with the concentration of the activated complex. Note that the values of kf and kb for the second reaction exhibit smaller variation 96

112 compared to the first reaction, which is explained below in light of the reaction rates plotted in Figure k f Reaction Rate Constants k b K Time (sec) x - Figure 5.46 Forward and backward reaction rate constants as well as the equilibrium constant as a function of time for the HHF HF H reaction mechanism corresponding to an initial stable equilibrium temperature at 3 K. The forward, backward, and net reaction rates for the first reaction mechanism (equation (5.6)) are presented in Figure The values and trends are comparable to the results found for the one-reaction-mechanism system in Figure 5.. Initially the forward reaction rate dominates and peaks at the same time that the entropy generation rate and the rate of the reaction coordinate do. In contrast, the backward reaction rate only becomes noticeable once the forward reaction rate has already peaked. x 3 r f Reaction Rate.5.5 r b r.5.5 Time (sec) x - Figure 5.47 Forward, backward, and net reaction rates as a function of time for the F H HHF reaction mechanism corresponding to an initial stable equilibrium temperature of 3 K. 97

113 5 x 3 Reaction Rate 4 3 r f r b r Figure 5.48 Forward, backward, and net reaction rates as a function of time for the HHF HF H reaction mechanism corresponding to an initial stable equilibrium temperature of 3 K. Figure 5.48 represents the results for the forward, backward, and net reaction rates of the second reaction mechanism (equation (5.7)), shows a markedly different behavior. Notably, both the forward and backward reaction rates are relatively close in values even early in the reaction and continue to be throughout the reaction. This explains why the reaction rate constants in Figure 5.46 are more flattened when compared to their values in the first reaction mechanism The Activation Energy The idea behind separating the one reaction mechanism of fluorine and hydrogen molecule into the two-reaction-mechanisms given in equations (5.6) and (5.7) is that it allows one to calculate an activation energy of the reaction E A such that A HHF F H E H H H (5.) Thus, in the following, the difference between the energies possessed by the activated complex and the reactants (i.e., F and H) are reported at different system expectation energies (i.e., initial stable equilibrium temperatures) as well as for different energy eigenstructure for the activated complex. Again the parameters used for the analysis in this section matches that described in Section Time (sec) x - The results plotted in Figure 5.49 show the value of the activation energy given by equation (5.) for the initial reaction temperatures of 3 K, 5 K, and 7 K. The internal and 98

114 translational energy structures of the species in the reaction mechanisms those given in Table 5. and Table 5., respectively. Activation Energy (ev) T = 3 K T = 5 K T = 7 K Time (sec) Time (sec) x - - Figure 5.49 Activation energy evolution for different initial stable equilibrium temperatures. The first thing to notice here is that the activation energy in this analysis is a dynamic property as opposed to the static one calculated from information of the potential energy surface alone. Second, the height of the barrier in this analysis is proportional to the initial stable equilibrium temperature of the reaction or said differently to the expectation energy of the system. Thus, the activation energies at each final stable equilibrium state (i.e., at. - sec) are 9.9 mev for an initial stable equilibrium temperature of 3 K, 3.3 mev for an initial stable equilibrium temperature at 5 K, and 3.8 mev for an initial stable equilibrium temperature of 7 K. Clearly, as the expectation energy of the system increases, the activation energy barrier of the final stable equilibrium state increases as well. Note that the variation in the values seen for 99 E A in Figure 5.49 are dependent on the amount of HHF present at any given instant of time. Clearly, when initially there is little HHF present, the value of Activation Energy (ev).35.5 EA simple reflects the negative of the sum of the reactant (F+H) energies which, of course, places the curve for 3 K above that for 5 K and in turn that curve above that for 7 K. Between. - and.4 - sec this picture begins to change as the three curves cross over each other due to the appearance of a sufficient amount of the activated complex HHF, which for the higher temperatures is more significant. The amount of HHF for the three cases continues to increase driving the activation energy towards positive values which culminate in the values listed above for the final stable equilibrium states..3

115 Next, the effect of the energy eigenstructure of the activated complex HHF on the height of the energy barrier is investigated. The results presented are calculated at an initial stable equilibrium temperature of 3 K and for the four cases of internal quantum numbers given in to Table 5.4. In this table, HHF and jhhf are the vibration and rotation quantum numbers of the HHF. Both the internal and translational quantum numbers for the other species of the reaction are equivalent to the one given in Table 5. and Table 5., respectively. Table 5.4 The different internal eigenstructure of the activated complex HHF.. HHF j HHF Case,,, 3,,, 4,,, 3, Activation Energy (ev) HHF = -. HHF =, HHF =,, HHF =,,, 3 Time (sec) x Time (sec) x - Activation Energy (ev) Figure 5.5 Activation energy for different quantum structures of the activated complex HHF corresponding to an initial stable equilibirum temperature of 3 K. To determine the vibrational eigenenergies of a molecule, which for the case of an ideal gas is given by equations (4.) or (4.3), the vibrational frequency must be identified. Moreover, the dissociation energy of that particular molecule needs to be known in order to reference the vibrational eigenenergies to their datum. The assumption that is made here is that the dissociation energy of HHF is equivalent to that of H. This assumption is used since a dissociation energy of HHF was not found in the literature and it is uncertain whether or not this quantity is even defined for activated complex structures. With this assumption and the frequency indicate, results are

116 plotted in Figure 5.5 for the four cases of Table 5.4 to see the effect of changing the internal energy eigenstructure of the activated complex HHF on the height of the activation. The first thing to notice in Figure 5.5 is that the profiles closely follow those for the expectation values of the entropy of the system. Furthermore, this figure shows that the more internal modes available for the activated complex, the higher the activation energy is. For the case in which HHF has 4 vibration energy eigenlevels the activation energy at the final stable equilibrium state is around 5 mev. Note that unlike in Figure 5.49, there is no cross over of the curves at any point during the dynamic evolution in state of the system. 5. Energy Transfer between Energy Modes One of the great advantages of using the equation of motion of the SEA-QT framework is the picture that it provides of how energy is transferred or swapped between the available energy modes of the system. In order to demonstrate this, three different system Hamiltonians are constructed from the original Hamiltonian for the F+H one-reaction-mechanism system of Section 5.., i.e., one with only vibrational modes, another with only rotational, and a third with only translational. The evolutions of the energies of the different system Hamiltonians using the density operator () t determined in Section 5.., i.e., tr tr H Tr( H ) (5.3) vib vib H Tr( H ) (5.4) rot rot H Tr( H ) (5.5) As can be seen in Figure 5.5, the energy available in the vibrational mode reduces during the reaction or said differently is transferred to both rotational and translational modes as the reaction proceeds. The expectation value of the rotational system energy does not increase significantly, since the energy eigenlevels available in this mode are already small compared to the other modes. As this figure suggest, the translational eigenenergies play an important role in the evolution towards the stable equilibrium state.

117 Energy (ev) translational energy vibrational energy rotational energy.5.5 Time (sec) x - Figure 5.5 Energy transfer between the different energy eigenmodes at 3 K for the F+H one-reactionmechanism system of Section 5.. and an initial stable equilibrium temperature of 3 K. A similar procedure is followed to construct three different Hamiltonians for which only the translational, vibrational, and rotational energy eigenmodes of H are included for in each Hamiltonian. The density operator evolution utilized for Figure 5.5 is used to show how the transfer of energy between the available energy eigenmodes of the H molecule is affected. Energy (ev) H translational energy H vibrational energy H rotational energy.5.5 Time (sec) x - Figure 5.5 Energy transfer between the different energy eigenmodes for H molecule during the F+H one-reaction-mechanism system evolution of Section 5.. and an initial stable equilibrium temperature of 3 K. The results are plotted in Figure 5.5. Initially, at the beginning of the reaction process, the dominant energy mode for the H molecule is the vibrational mode. As the reaction proceeds toward the final stable equilibrium state, the vibrational energy decreases in time since H is a

118 reactant that depletes in time. One can see that both the translational and rotational energies are not that significant for the H molecule when compared to that of the vibration..5 Energy (ev).4.3. HF translational energy HF vibrational vnergy HF rotational energy..5.5 Time (sec) x - Figure 5.53 Energy transfer between the different energy eigenmodes for HF molecule during the F+H one-reaction-mechanism system evolution of Section 5.. and an initial stable equilibrium temperature of 3 K. In Figure 5.53 the energy transfer between the different energy eigenmodes for the HF molecule is presented. Initially, the energies for all modes are zero since this species does not initially exist the state evolves towards that of stable equilibrium, the translational and vibrational eigenmodes begin to dominate the energy possessed by the HF molecule. Again in this figure the dynamics of the energy swap is connected to the evolution of the entropy, that is to say, when the curve for the expectation of the entropy flattens then similar behavior happens to the mechanism of the swap in the energy. 5.3 Thermodynamic Properties for Non-Equilibrium States As described earlier in Section 4.7 of Chapter 4, the thermodynamic properties (i.e., temperature, pressure, chemical potential, etc.) can be predicted along the thermodynamic path, which the system takes, by using the surrogate system approach outlined in [54, 75]. The thermodynamic properties of interest in this work are the temperature, pressure and chemical potential. In this section, the thermodynamic property evolutions presented are those which correspond to the F+H one-reaction-mechanism system of Section

119 The temperature is found by finding the root that satisfies the non-linear equation (4.33). The expectation value of the system energy as well as the constituent mixture predicted at each instant is found using the equation of motion is used in equation (4.33) to predict the temperature at each instant of time. The results are plotted in Figure 5.54 for three cases (3 K, 5 K, and 7K) and show the temperature profile in time for the three different initial stable equilibrium temperatures. The high temperatures predicted at the final stable equilibrium state for each case are consistent with what is expected from the MEP model of Section 4.7 and the given compositions and energies. As can be seen the temperatures begin to increase steeply early on in the reaction but then begin to plateau out about halfway through the reaction. The highest temperatures are reached by the system initially at the highest temperature as would be expected. Note that the profiles follow that of the evolution of the system s entropy. 5 4 Temperature (K) 3 T o = 3K T o = 5K T o = 7K.5 Time (sec).5 x - Figure 5.54 Temperature profiles in time for three different initial stable equilibrium temperatures for the F+H one-reaction-mechanism system. In a similar fashion, the pressure is predicted in the non-equilibrium region using equation (4.46). Figure 5.55 shows the pressure profile for the same three different cases plotted in Figure Note that when the initial stable equilibrium temperature of the reaction increases, the ratio between the pressure at the final stable equilibrium state and that of the initial one decreases. For example, the pressure ratio for the 3 K case is about fifteen whereas that for the 7 K case is about five. The pressure profiles as expected have the same trends as those of the temperature profiles with the highest pressures both initially and finally for the 7 K case. 4

120 5 Pressure (MPa) 5 5 T o = 3K T o = 5K T o = 7K.5.5 Time (sec) x - Figure 5.55 Pressure profiles in time for three different initial stable equilibrium temperatures and for the F+H one-reaction-mechanism system. As can be seen in equation (4.46), the pressure is sensitive to the translational eigenenergies available in defining the eigenstructure of the thermodynamic system. In order to investigate the relationship between the pressure and the translational available eigenenergies, the F+H onereaction-mechanism system of Section 5.. is compared to two other reactive systems, which have the same eigenstructure characteristics except for those of translation. Three cases are considered in which the eigenstructure is varied relative to the translational quantum number ranges described in Table 5.5. Table 5.5 Translational quantum numbers considered for each species for the three different cases. Case F H HF H k=,...,55 k=,..., k=,...,5 k=,..., k=,...,5 k=,...,5 k=,...,5 k=,...,5 3 k=,...,3 k=,...,3 k=,...,3 k=,...,3 The pressure profiles along the reaction path are plotted in Figure Note that the three different profiles correspond to an initial stable equilibrium temperature of 3 K. Clearly, the pressure is sensitive to the available translational eigenenergies for the system, and as these energy eigenlevels increase so does the pressure. Again, the point is made here that the choice of translational eigenenergy ranges as well as sampling is somewhat arbitrary in this work and that 5

121 future work will entail implementing a more robust way of selecting these ranges and samplings based on the density of states. Pressure (MPa) 5 5 Case Case Case 3.5 Time (sec).5 x - Figure 5.56 Pressure profiles in time for three different cases given in Table 5.5 and for the F+H onereaction-mechanism system. The third thermodynamic property of interest is the chemical potential, which is given by equation (4.47). The chemical potential in that equation is not referenced to any datum state. Thus, for the following results, μ is referenced to the initial stable equilibrium state of the chemical reaction according to the following expression: where c ' c c c k ref b To T T T T T To P o 5 T o q s ln ln ln Po To qs o T 3 To kb ln m k 3 b h is the chemical potential for a given state, unperturbed state of the chemical reaction, q s and c ref (5.6) the chemical potential at the initial q s o are the partition functions associated with the internal energy eigenmodes at the given state and the initial stable equilibrium state, respectively, Po and To are the initial pressure and temperature of the system, kb is the Boltzmann constant, h the Planck s constant, and m is the mass of the species. Figure 5.57 shows the evolution of the chemical potential for the different species of the F+H one-reaction-mechanism system. Not surprisingly, the chemical potential for all the species 6

122 increases in time since c is a function of both T and P and both of these properties increase in time. Also, note that the c for H has the highest values, followed by F, H and HF. This in part is Chemical Potential (ev) 5 F 5 H H HF.5 Time (sec).5 x - Figure 5.57 Chemical potential profiles in time for the different species for the F+H one-reactionmechanism system corresponding to an initial stable equilibrium temperature of 3K. due to their masses simple as the mass decreases, the contribution of the last term in equation (5.6) increases. As with the profiles for T and P, that for the chemical potential closely follows that for the entropy. 5.4 Relaxation Time Functional Up until now, the results presented in the previous sections have been based on a relaxation time τ the value for which has either been arbitrarily chosen or has been fitted to reproduce results found in the literature. In this section, an effort is made to understand this critical parameter. As described earlier, τ can be a either a constant or a functional of the density operator ρ. Although the exact form of this functional is not yet known, a lower bound based on the time-energy uncertainty principle is suggested by Beretta [7, ] and is expressed as follows: Here ( ) 4 D D HH * * X Y Tr X Y X Y (5.7) (5.8) 7

123 and * X D D is the conjugate transpose of X and is the projection of the gradient of the entropy functional onto the hyperplane of the generators of the motion, which are Tr( ) and Tr( H) where D is defined as D ln ln... H H ln H H HH (D D) x -5 (5.9) (D D) x Entropy (ev/k) -4 6 x -4 Figure 5.58 The projection of the gradient of the entropy and the relaxation time functional against the expectation value of the entropy at an initial stable equilibrium temperature of 3K for the F+H reaction mechanism. In order to have a visual picture of the behavior of τ and D, the inner product ( D D ) and equation (5.7) are plotted against the entropy in Figure 5.58 for the F+H one-reaction-mechanism system with an initial stable equilibrium temperature of 3 K. In this figure, the term D D and τ have similar trends at the beginning of the evolution and at the end, but the two quantities do not peak at the same time. Moreover, if the functional for τ suggested by the lower limit of equation (5.7) is used in the equation of motion, the numerical solution becomes unstable near stable equilibrium because τ becomes too small. Away around this is to calculate the value of τ at the initial state using the lower limit of equation (5.7) and to then fix τ at this value in the equation of motion. This is a reasonable assumption when comparing the SEA-QT predictions for the forward rate constant to those given in the literature since the latter are precisely those predicted for the initial state of the reaction. 8

124 D D Now, Figure 5.58 indicates that which the entropy generation rate peaks. By plotting the two results (i.e., peaks at around.4-4 ev/k, which is the value at D D and the entropy generation rate) versus the expectation value of the entropy as in Figure 5.59, one can see that the entropy generation rate follows the very same trend as that of the gradient of the projection of the entropy functional. Here the final stable equilibrium state occurs at about ev/k at which point both quantities are effectively zero. Both curves exhibit a smooth almost symmetric shape x 5 (D D) Entropy Generation (D D) 5 Entropy Generation (ev/k sec) Figure 5.59 Projection of the gradient of the entropy (D D) and the entropy generation rate as a function of the expectation values of the entropy for the F+H one-reaction-mechanism system corresponding to an initial stable equilibrium temperature of 3 K Entropy (ev/k) x -4 withd D having the steeper slope. Clearly, a functional for 9 ( ) based on D D would exhibit the appropriate trends, increasing as the intensity of the reaction peaks and then decreasing as the reaction descends from its peak and approaches stable equilibrium. However, the problem with using the lower limit of equation (5.7) is two-fold: the magnitude for τ is several orders of magnitude smaller and as stable equilibrium is approached the value becomes so small that it causes numerical instabilities in the solution of the SEA-QT equation of motion. To address these two deficiencies and provide a reasonable estimation of τ so that SEA-QT predictions for values of kf are on the order of those found in the literature for a given reaction mechanism, the equality in equation (5.7) is modified to the following form: D D ( ) c 4 HH (5.)

125 where c is a dimensionless constant whose magnitude is that of the Boltzmann constant. The introduction of c, which is based on an a priori knowledge of the order of magnitude needed, addresses both deficiencies outlined in the previous paragraph. It is, thus, this last expression which is explored below. The forward reaction rate constant is calculated for different initial stable equilibrium temperatures using equation (5.). Table 5.6 shows the results for kf using the suggested functional, and clearly the comparison with the literature is relatively close especially at the higher temperatures. However, as is seen at the end of the following section, equation (5.) is not sufficiently general to address the value of τ needed for other reaction mechanisms. Table 5.6 Comparison of values of the forward reaction rate constant k f found in the literature with those predicted using SEA-QT and equation (5.) for τ(ρ). T kf(t) / - kf * (T) / * Forward reaction rate constant from Heidner et al. [89] 5.5 Fluorine Molecule and Hydrogen Atom Reaction Mechanism In this section, the reaction mechanism of the fluorine molecule and the hydrogen atom, i.e., H F H HF (5.) using the SEA-QT framework is modeled. The same procedure as outlined in Section 5.. is utilized with the values of the constants related to the energy eigenstructure of each particle presented in Table 5.7; Table 5.7 Constants used to calculate the one-particle energy eigenstructure for the H+ F one-reactionmechanism system. H F F HF Mass (kg) Bond length ( Å ) Dissociation energy (ev) [99] Wavenumber (cm - )

126 na H The reaction is modeled with the following initial compositions for hydrogen and fluorine, particle and n a F particle. Note that the literature is not as rich in terms of studies for this reaction as for the F + H reaction mechanism. Therefore, the choice of quantum number range of the internal degrees of freedom for the diatomic species is somewhat problematic and is, thus, made to match that of the F + H reaction mechanism presented earlier. These quantum number ranges are described in Table 5.8. Table 5.8 Internal structure of the diatomic molecules in the H+F reaction mechanism. F HF Vibrational quantum #,,, 3 Rotational quantum #,,, 3,,, As before, the translational quantum numbers are sampled evenly across the quantum number ranges given in Table 5.9. Table 5.9 Translational quantum number range considered for each species in the H+F reaction mechanism. Ai H F F HF Translational quantum # ( evenly spaced samples) k=,..., k=,..., k=,...,3 k=,...,4 Now, the number of compatible subspaces as well as the limits on the reaction coordinates are determined with the procedure of Section 5... The results for these limits appear in Table 5.. Furthermore, the solution of the set of inequalities given in the last column of this table as well as Table 5. Limits on the reaction coordinates for the H+F reaction mechanism. Ai i n ia n n H - s s F - s s F s s HF s s the relation between the eigenvalues ( n is ) of the particle number operator to the eigenvalues ( s ) of the reaction coordinate are shown in the Table 5.. is is

127 Table 5. The solution to the number of compatible compositions for the H + F reaction mechanism. s s Ai n is n i n i3 H F H HF n s n s n 3s n 4s The reaction coordinate operator is now given as follows, E () PH () PH PH (5.) while the expectation of the reaction coordinate is expressed in terms of the reaction coordinate occupation probability w such that E Tr P w (5.3) H Next, the occupation number operators are written as a function of the reaction coordinate operator with the result that N I I P H N I I P F NF NHF E E E H E H P H P H (5.4) The expectation values of the number operators are then given in terms of the reaction coordinate occupation probabilities ws as follows: NH N NF w w F NHF w w (5.5) As for the case of the F+H reaction, it is assumed here that all the energy eigenlevels associated with the reactants are occupied initially. In addition, the relaxation time is calculated

128 such that the initial forward reaction rate constant predicted by SEA-QT is equivalent to found from [3] kt ( ) 4.8 T exp T The uncertainty in this equation as suggested in [3] varies from temperature range of 5-5 K increasing to ±.4 at K. 7 x -4 log( k) (5.6) +.3 to -. in the Entropy (ev/k) Time (sec) x -9 Figure 5.6 Instantaneous expectation values of the entropy for the H+F one-reaction-mechanism system at an initial stable equilibrium temperature of 3 K. 4 x 5 Entropy Generation Rate (ev/k sec) Time (sec) x -9 Figure 5.6 Instantaneous expectation values of the entropy generation rate for the H+F one-reactionmechanism system corresponding to at an initial stable equilibrium temperature of 3 K. The expectation values the entropy for the H+F one-reaction-mechanism system corresponding to an initial stable equilibrium temperature of 3 K are plotted in Figure 5.6. The 3

129 final value for this reaction is more than 5% higher than that found for the H+F reaction. Also, note that this reaction starts at a value of the entropy significantly higher (4-4 ev/k versus. -4 ev/k) than for the F+H reaction. The expectation values of the entropy generation rate of the process are shown in Figure 5.6. As for the previous reaction the entropy generation rate peaks at the beginning of the reaction then decreases as the system approaches stable equilibrium. It is furthermore interesting to note that as a consequence of this reaction being an order of magnitude slower than the F+H reaction, i.e., it is less facile, the amount of entropy generated is quite a bit less. Number of Particles H F H HF Time (sec) x -9 Figure 5.6 Expectation values of the particle number operator for each species of the H+F one-reactionmechanism system corresponding to an initial equilibrium temperature of 3 K. Energy (ev) H F F HF System Time (sec) x -9 Figure 5.63 Expectation energies for each species of the H+F one-reaction-mechanism system corresponding to an initial equilibrium temperature of 3 K. 4

130 The expectation values of the particle number operators are plotted in Figure 5.6. The profile in this figure is close to that for the F+H reaction mechanism. The steepest slope on these curves is reached when the entropy generation rate peaks. After that point, the profile eventually becomes less steep as the state evolves towards stable equilibrium. The results shown are consistent with equations (5.4) and (5.5). Figure 5.63 shows the expectation values for the energies associated with each species as well as that for the reacting system. This figure confirms that the energy of the system is conserved throughout the reaction process. The expectation values for the system entropy and energy as well as energies and numbers of particles for all the species are plotted in 3D in Figure This figure shows the behavior of these quantities as the system evolves from a state far from equilibrium to that of stable equilibrium. Reactants (H and F) decrease in terms of both their energies and numbers of particles from right to left, while the products (F and HF) increase these quantities from left to right. The arrow in this figure shows the direction of the evolution of system state for which the total entropy of the system is increasing. Figure 5.64 Expectation values for the system entropy and the energy as well as the energy and number of particles for each species of the H+F one-reaction-mechanism system corresponding to an initial stable equilibrium temperature of 3 K. 5