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Transition state theory and its extension to include quantum mechanical tunneling Introduction - Review rate theory, i.e. transition state theory, full (TST) and a harmonic approximation (HTST) - How to extend to quantum mechanical systems? Methodology a. b. c. d. e. Simple extension of HTST to quantum mechanical systems Quantum statistical mechanics via Feynman path integrals Extension of transition state theory to quantum systems Harmonic, quantum TST (instantons, ImF) Open questions and challenges Applications H2 desorption / H2 dissociative sticking at surfaces Motivation The tunneling mechanism will dominate the rate of atomic rearrangements (chemical reactions, diffusion) at low enough temperature Typical temperature dependence of a rate constant T k high T, classical low T, tunneling 1/T 1/Tc cross over temp. How can the cross over temperature, Tc, be estimated and the contribution of tunneling to the transition rate evaluated? Want to find the rate constant for a given temperature Time dependent (wave packet) or time independent Schrödinger equation can, in principle, be used to calculate the reaction probability (transmission coefficient) as a function of energy, P(E), and rate constant then found by taking a Boltzmann average k(T) = But, this is difficult to carry out. - hard to solve the Shcrödinger equation for many degrees of freedom (max. around 6, typically). Select? - the range in energy that is relevant at around room temperature is low, which makes wave packet calculations even harder (more on that later) and extrapolations are needed. Instead, apply a statistical approach to find k(T) directly Even for a classical system (where Newton is OK), a direct dynamical calculation of a rate constant is hard Time scale problem: • Most interesting transitions are rare events (ie, much slower than vibrations). Typically there is clear separation of time scales! • A transition with an energy barrier of 0.5 eV and a typical prefactor occurs 1000 times per second at room temperature. A direct classical dynamics simulation would require 1012 force evaluations and thousands of years of CPU time to cover the average time period between such events. 1000/s 0.5 eV Cannot simply heat the system, the mechanism can change! Again, a statistical approach is needed Transition State Theory (Wigner, Eyring 1930s) Identify a 3N-1 dimensional dividing surface, a Transition State (TS), that represents a bottle neck for going from an initial to a final state: Final state Initial state 3N-1 dimensional dividing surface, TS The bottle neck can be due to an energy barrier and/or entropy barrier kTST = (probability of being in TS) • (flux out of TS) WARNING! Beware of P-Chem textbooks Bruce Mahan, J. Chem. Education 51, 709 (1974): “In view of [its] success, it is unfortunate that the theory [TST] does not enjoy a better understanding and confidence among non-specialists. Some of this difficulty can be traced to the rather unconvincing derivations of the [TST] expression for the rate constant which are found in many physical chemistry texts and monographs on chemical kinetics.” Not much seems to have changed since then. Textbooks on Physical Chemistry that are commonly used today, illustrate the assumptions of TST by showing a dip in potential energy corresponding to ‘activated complex’ - Wrong message AC R P Transition State Theory 1. 2. 3. 4. Born-Oppenheimer Classical dynamics of nuclei (need to extend to quantum systems…) Boltzmann distribution in R (OK if slow enough, kBT < Ea/5 ) No recrossings of TS, (often most serious, but can be fixed from trajectories) Note: - TST gives the lifetime, τ=1/k, of a given initial state - no knowledge of final state(s) Can run short time scale dynamics starting from TS to find the final state(s) Such trajectories can be used to take recrossings into account dynamical corrections kexact = κ kTST Need to create a TS dividing surface that encloses the initial state, R The probability of being in some subspace, S, of R is: e"V (x ) / kB T dx PS = ! #S Z $ S "V (x ) / k T e dx Z R #R configuration integrals ‡ B A hyperplane ax+b=0 is a particularly simple (but not necessarily good) choice for the dividing surface P v" R thickness !σ kTST = (probability of being in TS) • (flux out of TS) " = v" = % ! So, k ! e#V (x) / kB T dx ' v% ‡ e#V (x ) / kB T dx " $ $R 1 2 $ #&' 2 µ i v i / kB T ' v" e i dv " 0 #$ 1 2 1 2 $ # µ " v " / kB T $ #&' 2 µ i v i / kB T ' e 2 dv" e i dv #$ #$ $ 1 # µ " v "2 / kB T e 2 dv % TST = % % k BT Z‡ 2 "µ# Z R $ = % 0 v" $ %#$ 1 # µ " v "2 / kB T e 2 dv 1 # µ " v "2 / kB T e 2 dv " 3N-1 dimensional! 1 dimensional! " = k BT 2'µ" Harmonic approximation to TST Typically a good approximation for solids (low enough temperature, but not too low). When energy of second order saddle points is much higher than kBT over the energy of first order saddle points and when the potential is smooth enough that a second order Taylor approximation to the PES is good enough in the region with large statistical weight. Analogous to the standard approximation for diatomic molecules: 1 2 V (r) " V (r0 ) + k ( r # r0 ) 2 2nd order SP > kBT 1st order SP R V(r) r Derivation of HTST: Taylor expand PES around minimum, find normal modes 3N VR (q) " Vmin + # i=1 1 2 k R, i qR, i 2 R Similarly, Taylor expand PES around SP: 3N #1 $ V‡ (q) " VSP + ! i=1 1 k ‡ , i q‡2, i 2 Then, the TST rate constant becomes: k HTST = ! = k BT 2"µ# k BT Z ‡ 2 "µ# Z R , 3N $1 1 & ) $(VSP + % k‡, i q‡2, i + / kB T ' * i =1 2 , 3N 1 & ) $(Vmin + % k R, i q 2R, i + / kB T ' * i =1 2 -$, e -$, e Define: " = # / 2$ = ! which finally gives: ! 1 2$ k µ ‡ ' dq‡ dq R 3N " R, i # $(V HTST i=1 k = e 3N $1 #i=1"‡ , i SP $Vmin ) / kB T Agrees with the empirical Arrhenius law • Need to find all relevant saddle points on the potential energy rim surrounding the energy basin corresponding to the initial state. • The transition state is approximated as a set of hyperplanes going through the saddle points with the unstable mode normal to the hyperplane, and a second order Taylor approximation to the PES at minimum and saddle points is used. • For each saddle point region: TS × Saddle point D k ! HTST " R, i # $(V i=1 = e D$1 #i=1"‡ , i SP $Vmin ) / kB T Temperature and entropy are taken into account within the normal mode approximation HTST is more than 103 faster than full TST! How can HTST be extended to include quantum mechanical effects? Replace νi with qcl h/kBT to introduce the classical vibrational partition function, qcl, and then replace it with the quantum mechanical vibrational partition function for each normal D mode, to get " R, i # $(V $V ) / k T i=1 k HTST = e D$1 " ,i # i=1not tunneling. This will This includes zero-point energy effects, but SP min B ‡ be referred to as ʻquasi-quantumʼ approximation. Note: A simple zero point energy ʻcorrectionʼ, i.e. simply replacing ΔE = VSP - Vmin ! by ΔEZPC in the classical HTST expression is not appropriate at high T and can be a worse approximation than no quantum correction at all! Example: CH4 dissociative adsorption on Ni, activation energy (eV) Classical cl. with ZPC 0.82 0.66 quasi-quantum (at 500 K) 0.72 Extend TST to quantum mechanical systems using Feynman Path Integrals Assume thermalization and decoherence after each transition * Statistical mechanics of a quantum particle is mathematically equivalent (isomorphic) to classical statistical mechanics of a distribution of images of the particle connected with springs with stiffness proportional to T2 ksp = mP (2πkBT/h)2 * Instead of the potential surface, work with an effective potential V(r) Derivation of FPI representation of quantum partition function The thermally averaged expectation value for an observable is Where Q is the partition function is Use Trotter formula for operators and insert the identity operator to rewrite the partition function as With the solutions for a free particle going from xi to xi+1, it becomes Veff Equivalent to a classical configurational partition function for a ring polymer with P particles connected by springs. Isosurfaces of the effective potential, Veff: Example 1D Eckart potential Consider Feynman paths of the form , , only the first two Fourier components included q(!) R P ! At high T, T>Tc , quantum delocalization (non-zero x1) increases the effective potential The contour plot shows the value of the effective potential as a function of (x0,x1) Isosurfaces of the effective potential, Veff Consider Feynman paths of the form q(τ) = x0 + x1sin(2π τ/Τ) T=βh/2π, only first two Fourier components q(τ) x1 R P x0 At high T, T>Tc , quantum delocalization (non-zero x1) increases the effective potential τ Contour plot shows the value of the effective potential as a function of (x0,x1) At lower temperature new features appear - tunneling Summarize the temperature dependence of the effective QM potential surface, Veff for an Eckart potential barrier Example: Eckart potential High T, classical overthe-barrier transitions Below cross over T, new saddle points form instantons instanton RAW-QTST: an extension of TST for quantum mechanical systems, includes tunneling mechanism Quantum transition state theory based on Feynman path integrals Estimate the rate at which closed Feynman paths move from the reactant region, R, to the product region, P. The definition of the quantum transition state needs to involve both position and shape of the closed Feynman paths in order to confine the system to the bottle neck region around the instanton. RAW-QTST: a full free energy calculation of such a transition state. Harmonic QTST: make a second order expansion of the effective potential around the instanton Final expression for the rate constant (next slide) is equivalent to Im F (Langer, 1969) and instanton theory (Miller, 1975, Callan and Coleman, 1977) Analogy between HTST and HQTST HTST (N dimensions) Vsp HQTST (NP dimensions) Vins Spreading of replicas lowers effective activation energy - Accounts for tunneling Calculated sticking coefficient for H2 on Cu(110) Use EAM potential function with little input on the H-Cu interaction, dynamical surface atoms min. energy path NEB method 100 K instanton path While wave packet calculations for energy that is large compared to T=500 K show clear quantum effects in state-to-state cross sections, the thermal average that determines S is classical at T>300K Mills et al. Surf. Sci. 324, 305 (1995) Compare with the Harmonic approximation to QTST The full RAW-QTST calculation requires a large number of force evaluations. So, it can currently only be applied to systems described by analytical potential functions. Implemented using minimum mode following method (Henkelman & J, ‘99). Requires 500-1000 force evaluations per system replica and can be used with ab initio calculations of the atomic forces. A very brief intro to H2-Cu studies The sticking of H2 on Cu surfaces has become a benchmark problem in surface science due to extensive experimental and theoretical studies. Molecular beam studies: Balooch, Cardillo, Miller, Stickney (Surf. Sci. 1974) ….. Auerbach, Rettner, Michelsen (Surf. Sci. 1993) 375 citations Sitz and coworkers Wave packet calculations: Jackson and Metiu (Surf. Sci. 1974) Holloway and coworkers ….. Kroes, Olsen and coworkers (JCP 2005) 6D calculations with PES fitted to DFT calculations, frozen surface Compare wave packet and HQTST calculations H2 dissociative sticking on Cu(100) QD: 6-d wave packet calculations by Roar Olsen and Geert-Jan Kroes quasi-quantum classical Extrapolation to the relevant energy range QD: 6-d wave packet calculations by Roar Olsen and Geert-Jan Kroes. They had to work hard and use much computer time to push the calculations to this low energy. different extrapolations Some open questions A more general prefactor needs to be derived for RAW-QTST, so far only simple transitions with a simple energy barrier can be treated. How can dynamical corrections be evaluated given a RAW quantum mechanical transition state? Is there a variational principle for quantum transition state theory? Is there perhaps an even better way to approach the problem of estimating tunneling rates in complex systems, does a transition state theory approach optimal? Perhaps go back to ? k(T) = Summary RAW-QTST is an extension of classical transition state theory which includes quantum mechanical effects, such as tunneling by using closed Feynman paths. It can be applied to large systems with hundreds of degrees of freedom and there is no need to choose beforehand special treatment for some but not others. The definition of the transition state needs to be generalized, no transition state defined in the space of 3N coordinates will give a good rate estimate, a quantum mechanical transition state needs to be defined in the 3NP coordinate space of closed Feynman paths. A harmonic approximation to RAW-QTST can be used when the transition involves a significant energy barrier. A quadratic approximation around the first order saddle point, the instanton, is then used for the energy surface of the Feynman paths. Applications to H2/Cu dissociative adsorption and associative desorption show that quantum mechanical transition state theory can be used to estimate rates, even sticking probability of molecules at surface (good agreement with wave packet propagation results). The harmonic approximation even gives good results in these test problems. Several open questions remain in the formulation of quantum transition state theory and several improvements in the methodology could be, and should be, pursued. Acknowledgements Andri Arnaldsson and Greg Mills, former Ph.D. students Contributions were also made by Thomas Bligaard and Stefan Andersson (postdocs) Some of the calculations are in collaboration with Greg Schenter (PNNL), and with Roar Olsen and Geert-Jan Kroes (University of Leiden) Funded by European network on ʻPredicting catalysisʼ and European integrated project ʻNESSHyʼ on hydrogen storage.