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Variational transition state theory: theoretical framework and recent developments Cite this: Chem. Soc. Rev., 2017, 46,7548 Junwei Lucas Bao * and Donald G. Truhlar *

This article reviews the fundamentals of variational transition state theory (VTST), its recent theoretical development, and some modern applications. The theoretical methods reviewed here include multidimensional quantum mechanical tunneling, multistructural VTST (MS-VTST), multi-path VTST (MP-VTST), both Received 14th August 2017 reaction-path VTST (RP-VTST) and variable reaction coordinate VTST (VRC-VTST), system-specific DOI: 10.1039/c7cs00602k quantum Rice–Ramsperger–Kassel theory (SS-QRRK) for predicting pressure-dependent rate constants, and VTST in the solid phase, liquid phase, and enzymes. We also provide some perspectives regarding rsc.li/chem-soc-rev the general applicability of VTST.

1. Introduction profound influence on modern chemical kinetics. In the 1920s, the concept of a transition state emerged, and in the 1930s it Chemical kinetics has a long history in chemical science. In became a central theoretical foundation for quantitative theory, 1864, Waage and Guldberg formulated the law of mass action,1 especially because of the work of Eyring, Wigner, Polanyi, and which can be considered as the first published work developing Evans.3 The direct observation of transition state structures by the theory of chemical kinetics. About 25 years later, Arrhenius various spectroscopic techniques is considered to be one of the proposed his well known empirical relation for the temperature major challenges in modern chemical science.4 dependence of reaction rates.2 Both of these studies still exert a For a long time, transition state theory was viewed some- what skeptically as far as its relevance to quantitative work. For

Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing example, a group of distinguished researchers wrote in 1983 Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, USA. that ‘‘The overall picture is that the validity of the transition

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. E-mail: [email protected], [email protected] state theory has not yet been really proved and its success

Junwei Lucas Bao joined the Don Truhlar is native of Chicago. Department of Chemistry at the He received a PhD from Caltech and University of Minnesota Twin has been on the Chemistry faculty at Cities in 2013, and he is currently the University of Minnesota since pursuing a PhD degree in 1969, now as Regents Professor. theoretical chemistry under the He was honored to receive the ACS supervision of Professor Donald G. Award for Computers in Chemical Truhlar. Currently, his main and Pharmaceutical Research, NAS research interests include: (1) Award for Scientific Reviewing, ACS multi-structural and multi-path Peter Debye Award for Physical variational transition state theory Chemistry, WATOC Schro¨dinger with multidimensional tunneling, Medal, Dudley Herschbach Award Junwei Lucas Bao theory of pressure-dependent rate Donald G. Truhlar for Collision Dynamics, RSC constants, torsional anharmonicity, Chemical Dynamics Award, and and their applications in combustion and atmospheric chemistry; (2) APS Earle Plyler Prize for Molecular Spectroscopy and Dynamics. His multi-configuration pair-density functional theory (MC-PDFT), current research interests center on dynamics and electronic structure, including its separated-pair (SP) version with the correlating- especially photochemistry, transition metal catalysis, and density participating-orbital (CPO) scheme, and their applications in functional theory. modeling transition metal compounds and spin-splittings of divalent radicals.

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seems to be mysterious.’’5 This began to change though when a briefer summary including a few new developments. We start accurate quantum mechanical calculations became available by using classical mechanics to derive variational transition for prototype reactions of small molecules, and when accurate state theory for gas-phase reactions (Section 2.1), and then we semiclassical tunneling methods became available.6 Now transition introduce quantum effects into VTST, in particular vibrational state theory is understood on a firm basis, especially since the mode quantization in the partition functions (Section 2.2) and observation of quantized transition state level structure in quantum mechanical tunneling in the treatment of reaction- converged quantum mechanical dynamics calculations7–9 has coordinate motion, including multidimensional coupling of provided a solid quantum mechanical foundation for transition reaction coordinate motion to other coordinates (Section 2.3). state theory in terms of short-lived resonance states.10–13 We emphasize the importance of utilizing curvilinear coordinate Transition state theory is basically a theory for electronically for carrying out the VTST calculations in Section 2.5. State-selected adiabatic reactions, i.e., reactions that occur on a single Born– extensions of VTST are presented in Section 2.6. Oppenheimer potential energy surface. There are some extensions to multiple-surface reactions, but they involve additional assump- 2.1 Classical variational transition state theory tions that are not traditionally considered to be part of transition Transition state theory is first justified in classical mechanics; state theory, and we will not consider them here. Furthermore, then we add quantization effects. In classical mechanics, we transition state theory in its basic form predicts reaction rates in work with trajectories. At any given time, a trajectory is a point equilibrated ensembles, in particular canonical reaction rates in in phase space, which is the space of all the coordinates and thermal ensembles and microcanonical reaction rates in ensembles their conjugate momenta. A point in phase space is called a with fixed total energy or fixed total energy and total angular phase point. momentum; thus it is not designed for state-specific reaction Transition state theory may be applied to both thermal rates or for predicting product internal-state distributions, ensembles and microcanonical ensembles. In either case the although sometimes it is applied to such properties by making first assumption is that the reactants are in local equilibrium. additional assumptions. For thermal ensembles, this means that the population of Transition state theory is most powerful in the form of reactant states is given by a Boltzmann distribution, even if variational transition state theory14–25 (VTST), especially when reactants are not in equilibrium with products. For a micro- combined with modern electronic structure theory, multi- canonical ensemble, this means for classical systems that all dimensional tunneling methods, and statistical mechanical cells of equal volume in reactant phase space are equally and quantum mechanical theory for anharmonicity, and it populated; for microcanonical quantum systems it means that has become a powerful tool in understanding and predicting all reactant states of a given energy are equally populated. In the chemical reaction mechanisms and reaction rates.26–36 rest of this subsection we assume a classical mechanical world. We have reviewed VTST elsewhere in a variety of The maintenance of reactant equilibrium means that the ways,6,20–25,30,33,35,37–41 but the current review has a special rate of thermalizing collisions for maintaining the equilibrium focus on several modern enhancements that were not available states of these species is at least as fast as the rate at which

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. when those earlier reviews were written. In Section 2, we cover these states are depopulated by the chemical reaction. Since the the fundamental principles of variational transition state theory phase points in the reactant region are assumed to be in in the context of gas-phase reactions; this section uses a reaction thermal equilibrium, Liouville’s theorem guarantees that as coordinate associated with a reaction path, and it includes the time evolves, the Boltzmann distribution is also satisfied in tunneling. The next four sections continue the treatment of the transition state region except that if products are absent gas-phase reactions, covering multistructural transition states then phase points corresponding to trajectories that originated (Section 3), variable reaction coordinates (Section 4), pressure in the product region are missing.43,44 This local-equilibrium effects (Section 5), and tests of the applicability of the theory assumption in the reactant region is assumed valid for most (Section 6). gas-phase bimolecular reactions and reactions in the liquid Sections 1–3 apply to tight transition states, while Section 4 phase because the inelastic collisions of the reactants are applies for loose transition states. Section 7 discusses extension of efficient enough to repopulate their rovibrational states and the treatment to condensed-phase reactions, and Section 8 discusses maintain local equilibrium in phase space, but it is not applicable selected applications. We conclude this review in Section 9. for unimolecular reactions (and some bimolecular reactions) in Most of the computational methods reviewed here are the gas phase at low pressure. When the equilibrium assumption available in the Polyrate computer code.42 breaks down, ‘‘thermal’’ rate constants become pressure dependent, and the thermal rate constants calculated by using the transition state theory are the high-pressure limit rate constants. In the 2. Fundamentals of gas-phase high-pressure limit, the word ‘‘thermal’’ refers to rate constants variational transition state theory averaged over Boltzmann distributions of both reactants and bath gases; but at lower pressures, for reactions whose rate A detailed pedagogical review of the fundamental theory under- constants are pressure-dependent, ‘‘thermal’’ refers to a Boltzmann lying gas-phase variational transition state theory, including distribution of bath gases, but there is a pressure- and temperature- derivations, is available elsewhere.41 In this section we present dependent non-Boltzmann energy distribution of reactants.

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Thus, in the fall-off region, rate constants are functions of both where R indicates that the volume integral is carried out over pressure and temperature. We will discuss pressure dependence the reactant region. Differentiating eqn (4) with respect to t and in Section 5. plugging into eqn (3) yields 22,45 ð ð It is convenient to introduce mass-scaled coordinates, R dN d N N which may also be called isoinertial coordinates. With such a ¼ rd6 t ¼ = rvd6 t (5) dt dt R R coordinate system, the motion of an N-atom system in three dimensions can be reduced to that of a single-particle with a Then we can use Gauss’s theorem to rewrite the volume integral reduced mass m moving in 3N-dimensional space. In other in eqn (5) as a surface integral, ð words the reduced mass becomes the same for all directions dNR of motion (and hence the word ‘‘isoinertial’’). The reduced ¼ rv ndS (6) dt S mass m is usually chosen to be 1 atomic mass unit, or it can be set to be the reduced mass of the relative translational motion where n is a unit vector normal to the surface S pointing out of for a bimolecular reaction (one can use any value as long as one the volume R, and S is a (6N 1)-dimensional hypersurface that is consistent). The value of m sets the scale for the reaction separates the reactant region from the product region; this coordinate s; however, the value of any physical observable is hypersurface is called the dividing surface. independent of the value of m. If one sets m = 1 amu, the The local one-way flux of systems going from reactants to numerical value of s in Å is the same as the numerical value of s products through this dividing surface is ð in amu1/2 if one used the familiar mass-weighted coordinates46 F þ ¼ rv ndS (7) of spectroscopists. The motions of all the particles are governed Sþ by a potential energy surface V({ri}) that depends on the set of N where we consider the portion, S+, of the dividing surface for three-dimensional positions ri of the particles, each with its own mass. Equivalently speaking, the motion may be described which the flux is positive. We define the coordinate q3N as the as that of a single particle with reduced mass m governed by coordinate normal to surface S, and we call this coordinate the V(q), where q is a 3N-dimensional generalized mass-scaled local reaction coordinate z. Note that the surface S is locally coordinate, and its conjugate momentum is p. normal to some global reaction path from reactants to products. Now we present classical transition state theory. Consider an The dividing surface is put at the location where the global ensemble of systems in phase space. Each phase point (q,p)in reaction coordinate is s*, and the local reaction coordinate is z*. The local one-way flux is evaluated as: the 6N-dimensional phase space represents an N-atom system, ð ð and the motion of each point is governed by the following set of þ pz F ¼ rz_dS ¼ r dq1 ...dq3N1dp1 ...dp3N1dpz Hamilton equations: Sþ Sþ m ð ð (8) þ1 dq @H dp @H 6N2 pz ¼ ; ¼ (1) ¼ d t r dpz dt @p dt @q z¼z 0 m Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. where the classical Hamiltonian is As we mentioned at the beginning of this subsection, TST assumes the reactants are in local equilibrium at temperature T 1 X3N and the density of classical phase points satisfies the Boltzmann Hðq; pÞ¼ p 2 þ VðqÞ (2) 2m i distribution i¼1

H/kBT r = r0e (9) Liouville’s theorem states that the phase space volume is conserved as time evolves; it can be written in the form of a where H is the Hamiltonian, kB is Boltzmann constant, T is continuity equation as follows: temperature, and r0 is a constant. Plugging eqn (9) into eqn (8) and (4) yields @r ð ð = rv 0 þ1p þ ¼ (3) þ 6N2 z H=kBT @t F ¼ r0 d t e dpz (10) z¼z 0 m where = is the 6N-dimensional divergence operator, v is a ð r R H=kBT 6N 6N-dimensional velocity of a phase point, and is the density N ¼ r0 e d t (11) of points in phase space. Integrating r over the whole phase R space yields the total number of classical phase points (i.e., the Now, we define a generalized transition state (GT) as a (6N 2)- total number of systems in the ensemble). dimensional fixed-z hyperplane with the following Hamiltonian HGT: The number of systems located in the reactant region is NR, GT and it can be evaluated by integrating the density of phase H ðÞq1; ...; q3N1; p1; ...; p3N1; z ¼ z p 2 points over the phase space of the reactant: ¼ HqðÞ; ...; q ; p ; ...; p z (12) 1 3N 1 3N 2m ð Y3N R 6N 6N GT N ¼ rd t; d t ¼ dpidqi (4) The generalized-transition-state Hamiltonian H is defined by R i¼1 removing the kinetic energy associated with the motion of the

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local reaction coordinate z from the total Hamiltonian, and For the generalized transition state, we freeze one degree of fixing the value of z at some specific value z*. Notice that z is a freedom (the reaction coordinate) and take the local zero of GT GT parameter rather than a variable in H , i.e., H depends energy as the classical potential energy VRP evaluated at z = z* parametrically on z. Substituting eqn (12) into eqn (10), we on the reaction path (RP). The generalized transition state obtain the one-way flux FGT(T,z*) through a generalized transition classical partition function per volume for the transition state state fixed at z=z*, as becomes ð Ð F GTðÞ¼T; z r k T eH=kBT d6N2t (13) d6N2teðÞHVRP =kBT 0 B FGT ¼ GT (17) GT Classical Vh3N1 where GT indicates that the integral is carried out over the generalized transition state. The total Hamiltonian can be written as the sum of the The local equilibrium one-way flux FGT(T,z*) gives an upper Hamiltonians of the reactants, HA and HB, and using

bound to the global reactive flux F(T) if we arbitrarily put our R X 3NX X NX = r0h VFClassical (18) dividing surface at the saddle point in this classical dynamics R R R approach. There is no guarantee that the trajectories in phase where NX is the number of atoms in X (NA + NB = N), and the A B space that are originating from the reactant region toward the constant r0 = r0r0, we obtain product region will cross the dividing surface once and only Ð X 6NX H =kBT once. In the conventional transition state theory, one simply X X d te FClassical ¼ (19) puts the dividing surface at the saddle point and makes a non- Vh3NX recrossing assumption that there is no recrossing on this surface. In order to have the best estimation of the classical rate in which X is A or B, X represents that the volume integral is X constant, the ideal location to put the dividing surface S would carried out over reactant region, and H is the Hamiltonian of be the location at which FGT(T,z*) = F(T), which means all the the reactant X. trajectories originating from the reactant region cross the surface We then use eqn (15)–(18) to re-write eqn (14) as

only once (no recrossing) and end up in the product region. If one is GT kBT F willing to make the transition state hypersurface arbitrarily convo- kGT ðTÞ¼ Classical eVRPðÞz¼z =kBT (20) Classical h A B luted in shape, or if one considers simple enough systems at low FClassicalFClassical energy,47 one can make classical transition state theory exact. For realpotentialenergysurfacesanddividing surfaces that are simple Follow a similar derivation, the classical generalized TST rate enough to allow for the calculation of the flux, completely eliminat- constant for unimolecular reactions is:

ing all the recrossing is almost impossible, so we would put the GT kBT Q dividing surface at the so-called dynamical bottleneck, which is the kGT ðTÞ¼ ClassicaleVRPðÞz¼z =kBT (21) Classical h QA position at which the recrossing is minimized. Classical transition Classical Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. state theory will overestimate the classical rate constant, but the variational procedure will minimize this overestimation. A convenient choice for the reaction path is the minimum 49 For a bimolecular reaction (A + B), the classical generalized energy path (MEP) in isoinertial coordinates. The global GT reaction coordinate s is then the signed distance along the transition state theory rate constant kClassical(T), which is the flux per volume divided by the product of the concentration of MEP. An MEP is the union of the paths of steepest descent in reactants, obtained by the above procedure is isoinertial coordinates from the saddle point towards reactants Ð and products. The physical meaning of the MEP is that it is the r k T d6N2teH=kBT GT 0 B GT path followed by a classical trajectory that starts at the saddle kClassicalðTÞ¼ R R (14) NA NB V point but is continuously damped so as to always have only an infinitesimal velocity. A positive value of s represents for the where V is the three-dimensional real-space volume of the direction toward the product; s = 0 is the saddle point; and a system, and GT indicates that the integral is carried out over negative value of s represents the direction toward the reactant the generalized transition state. We can use the following side. Starting from the saddle point with coordinates x‡ (which definition of the classical partition function for A ð may also be called x(s = 0)), the geometry of a non-stationary 1 point that is close to the saddle point on the reaction path is A 6N H=kBT QClassicalðTÞ¼ 3N d te (15) h A computed using:

‡ where h is Planck’s constant, with an analogous expression for x(s1 = ds)=x(s =0) dsL(x ) (22) B. Recall that Planck’s constant occurs in the classical partition where ds is the step size (typically chosen to be 0.001 to 0.003 Å) function because we require the classical partition function to along the MEP, and L(x‡) is the normal-mode eigenvector equal the quantum one in the classical limit.48 We also define a associated with the imaginary frequency (it is a column of classical partition function per volume as the orthogonal matrix that diagonalizes the Hessian matrix

FClassical = QClassical(T)/V (16) evaluated at the saddle point). If we start from any non-stationary

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point x(sj) on MEP, then the next point can be computed by using obtained by firstly integrating the E, J-resolved rate constant the normalized gradient as: kGT(E,J,s) over the angular momentum J distribution) is rV s NGTðE; sÞ MEPj kGT E; s x sjþ1 ¼ sj ds ¼ xsj ds (23) ð Þ¼ (31) rVMEP sj rRðEÞh where NGT(E,s) is the phase-space volume of the generalized At each value of s, we can define a new coordinate system by transition state (it is, quantum mechanically, the number of rotating and translating the original coordinates so that the states for the generalized transition state), and r (E) is the coordinate z that we mentioned in the generalized TST is R density of states for reactant. The density of states is the inverse tangent to MEP at s, and any surface that is orthogonal to Laplace transform of the partition function. In microcanonical MEP can be described by z* = 0. The dividing surface intersects variational transition state theory (mVT), the variational rate MEP at the reaction coordinate s*. Note that s* depends on constant is obtained by minimizing eqn (31) with respect to s: temperature. In canonical variational transition state theory (CVT), we min NGTðE; sÞ kmVTðEÞ¼ s (32) variationally optimize the position of the dividing surface to r ðEÞh GT R minimize the generalized transition state rate constant kClassical (T) since the local one-way flux provides an upper bound of the and therefore the location of microcanonical variational transi- global reactive flux. The CVT rate constant is computed by tion state is energy-dependent. To obtain canonical mVT rate constant, one needs to perform the following integration: CVT GT CVT GT kClassicalðTÞ¼kClassical T; s ¼ min kClassicalðT; sÞ (24) ð s þ1 mVT mVT k ðTÞ¼ k ðEÞPRðEÞdE (33) Therefore 0 where PR(E) is the normalized population distribution of the reactant: @ GT kClassicalðT; sÞ ¼ 0 (25) @s CVT E=kBT s¼s rRðEÞe PRðEÞ¼ (34) QRðTÞ which leads to the following CVT rate constant for bimolecular reactions where QR(T) is the canonical partition function of reactant. If the dividing surface is located at the saddle point of the potential energy GT CVT kBT FClassicalðÞs ¼ s VCVT ðÞs¼s k T k ðTÞ¼ e MEP = B (26) surface, then eqn (33) is equivalent to the rate constant expression Classical h A B FClassicalFClassical of the conventional transition state theory. It can be shown that

and for unimolecular reactions kCVT(T) Z kmVT(T) (35)

GT kBT Q ðÞs ¼ s CVT CVT Classical VMEPðÞs¼s =kBT Inserting eqn (34) into (33) shows that the canonical rate kClassicalðTÞ¼ e (27)

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. A h QClassical constant may be considered as a Laplace transform of the microcanonical one, or the microcanonical rate constant may Another way to understand the canonical variational approach be considered to be an inverse Laplace transform of the is to re-write the classical generalized TST rate constant in canonical one. This will be a useful perspective in Section 5. terms of the standard-state Gibbs free energy of activation GT,0 DGact (T,s). 2.2 Quantization: quasiclassical variational transition state theory

GT kBT 0 DGGT;0 T;s k T The classical canonical variational transition state theory cannot k ðTÞ¼ K e act ð Þ= B (28) Classical h be rigorously quantized because in order to define the generalized transition state appropriately, we need to know the local reaction where K0 is the reaction quotient (for the formation of TS) coordinate z and its conjugate momentum pz simultaneously, evaluated at the standard-state concentration of each species: which is a violation of Heisenberg’s uncertainty principle. ( 1 ðunimolecularÞ However, the quantum mechanical effects along the reaction K0 ¼ (29) path usually cannot be ignored, and therefore we put in the 1 ðÞc ðbimolecularÞ quantum effects approximately to obtain what may be called quantized variational transition state theory. In order to quantize where, for gas-phase bimolecular reaction with unit cm3 classical variational transition state theory,38,50–56,77 we need to: (1) molecule1 s1,(c1)1 is k T/p1, where p1 is 1 bar; for liquid- B quantize the translational, rotational, and vibrational degrees of phase reaction, the standard-state concentration is 1 mole L1. freedom (the electronic degrees of freedom have already been CVT is equivalent to maximizing the free energy of activation quantized by the Born–Oppenheimer approximation); (2) quantize CVT;0 GT;0 the corresponding partition functions; and (3) quantize the DGact ðTÞ¼max DGact ðT; sÞ (30) s motion along the reaction coordinate, which includes quantum In a microcanonical ensemble (NVE ensemble), the micro- mechanical tunneling (including nonclassical reflection). We canonical energy-resolved classical rate constant (which is will discuss quantum mechanical tunneling in Section 2.3.

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First, we quantize all the degrees of freedom except for the errors in the electronic structure level and for the vibrational vibrational mode associated with the reaction coordinate s at a anharmonicity. The scale factors58 are developed based on a given generalized transition state. This degree of quantization database that consists of 15 representative small molecules, was called hybrid in early papers on VTST; now we call it whose experimental harmonic vibrational frequencies, funda- quasiclassical. In many cases we ignore the coupling between mental frequencies and zero-point vibrational energies are some of the degrees of the freedom; that is, we assume available. The ZPE scale factor (lZPE) can be written as the independent normal modes of vibration, and we neglect vibration– product of the harmonic frequency scale factor (lHO, which rotation coupling. VTST theory is not restricted to neglecting these corrects the systematic error of a given electronic structure couplings, but computations of the partition functions are more method) with the vibrational anharmonicity scale factor (lanh, complicated if they are included. (We do include such coupling in which corrects the vibrational anharmonicity). some of the approximations considered later in this section and in Instead of using the generic scale factors, the vibrational Sections 3.1 and 4.1.) anharmonicity scale factor for a given structure can be computed by The quantized translational motion is described by a particle using hybrid,59 degeneracy-corrected,60 second-order61–63 vibrational in 3-dimensional box; since box length is macroscopic, this agrees perturbation theory (HDCVPT2); and it is equal to the ratio of exactly with the classical result. For overall rotation, usually, it is HDCVPT2 computed anharmonic ZPE to the harmonic-oscillator sufficient to use the classical rotor approximation, provided one also ZPE. (The use of the HDCVPT2 method, like the use of vibrational correctly includes the rotational symmetry number (which is a scale factors, includes the effects of mode–mode coupling.) quantum effect); nevertheless, at accessible temperatures including To properly take account of anharmonicity, we have usually 300 K and below, all of the diatomic hydrides, many important free assumed that it is sufficient to include anharmonicity in all

radicals (e.g.,OH,CH,andCH2), and a few polyatomics have large bends and stretches by a single scaling factor. Note that the rotational constants, for which the classical approximation is not scale factors designed to treat the zero-point energy are primarily

always sufficiently accurate. For H2, it is advisable to always use the dictated by the stretching frequencies since they are larger than the quantal rotational partition function. bending and torsion frequencies and hence their contributions For vibrational motion, the traditional procedure is to use dominate the zero-point energy. Since stretches usually have the harmonic oscillator approximation, which assumes a quadratic negative anharmonicity, frequency-scaling factors are usually potential. A quadratic potential has only one minimum-energy smaller than unity, which means that partition functions are structure, whereas real molecules often have multiple structures increased by using a frequency-scaling factor. However, anharmoni- (conformations) due to torsions and/or rings. In some cases, the city of a bend mode could either increase or decrease the partition single-structure approximation causes large errors, and eliminating function. To achieve the most accurate possible thermodynamic and these errors is the subject of Section 3. kinetic calculations requires treating anharmonicity in each high- If the goal is a quantitative calculation, one should explicitly frequency mode/non-torsional mode individually. One approxi- consider anharmonicity even for stretches and bends in a single mation for computing such vibrational partition function with structure (this vibrational anharmonicity needs to be corrected vibrational anharmonicity is called simple perturbation theory

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. because the harmonic-oscillator approximation truncates the (SPT);64 it only requires calculation of the fundamental frequencies, series expansion of the potential energy at the second-order which can be computed by the above-mentioned HDCVPT2 method. terms). Vibrational anharmonicity can be included by vibrational The quantized total partition function of species X can be self-consistent-filed theory (VSCF), vibrational perturbation theory then written as (VPT), vibrational configurational interaction (VCI) and other Q(X) = Q (X)Q (X)Q (X)Q (X) (36) VSCF-based methods,57 but this can get expensive. To avoid the trans rot vib elec cost of those higher-level vibrational calculations, it is convenient with the vibrational partition function in the QHO approxi- 58 useascalefactor (for a given level of electronic structure theory) mation given by to scale all the computed harmonic vibrational frequencies in Yi¼F hni=2kBT order to account for vibrational anharmonicity; such scale factors HO e Qvib ðXÞ¼ (37) 1 ehni=kBT can also partially correct for systematic errors in the chosen level of i¼1 electronic structure theory. Using the harmonic oscillator formulas with effective frequencies (such as those obtained by scaling) is where for the reactant and product, the vibrational degree of called the quasi-harmonic oscillator (QHO) approximation. (Some- freedom F =3N 5 for linear molecule, or 3N 6 for nonlinear times it is just called the harmonic oscillator approximation, but molecule, 3N 6 for linear transition states, and 3N 7 for

that can lead to confusion.) Note that the scaled frequencies can nonlinear transition states. And ni is the vibrational frequency (and usually do) account for vibrational mode coupling as well as for the normal mode i. intramode anharmonicity, so the QHO approximation is a way to Note that we have introduced the zero point energy, we have to include non-separability in a separable-mode formalism. specify what we take as the zero of energy for partition functions. For the purpose of thermochemical kinetics computations, Textbooks routinely take the zero point level as the zero of energy; the scale factor we often choose is the one that is developed for then partition functions tend to unity (or, more generally, to the reproducing an accurate zero point energy; this is denoted degeneracy of the ground state) when the temperature goes to as lZPE, and it includes the corrections both for systematic zero. In our work (and in this whole article) the zero of energy for

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the vibrational partition function is chosen to be VMEP(s). We also make. (If, instead of using bottom of the potential well, one define the zero of energy for the electronic partition function as uses ZPE(s) of reactant(s) as the zero of energy, then the

VMEP(s), and hence, without considering spin–orbit coupling or numerator of the vibrational partition function is 1, and the G low-lying electronic states, the electronic partition function is just barrier height is ZPE-inclusive, i.e., it is based on V a .) equal to the degeneracy of the ground electronic state, but the The quantized CVT rate constant (which we denote as vibrational partition function is not (and we do include low-lying CVT/T, where ‘‘/T’’ means tunneling) can be then written as electronic states in the electronic partition function when they Bimolecular reaction : kCVT=TðTÞ are present – see the next paragraph). The rotational partition GT (40) function (which is approximated by the classical rotor approxi- kBT Q ðÞs ¼ s VT CVT ¼ kT eV ðÞs¼s =kBT mation) is determined by the principal moments of inertial of the h FR species. For the generalized transition state, QGT is a function of reaction coordinate s. Unimolecular reaction : kCVT=TðTÞ In thermochemical kinetics we usually assume the molecules GT (41) kBT Q ðÞs ¼ s VT CVT are in their ground electronic states. The zero of energy is set T V ðÞs¼s =kBT ¼ k A e equal to be electronic energy of the ground state, the electronic h Q partition function is T where k is the (temperature-dependent) tunneling transmission VT CVT E1/kBT V s s Qelec = g0 + g1e + (38) coefficient, which will be discussed in Section 2.3. ¼ is the classical barrier height evaluated as the difference between where g and g are the degeneracy of ground state and first 0 1 the ground state energy of the variational transition state and the excited state, and E is the adiabatic excitation energy of the 1 reactant(s), and because the latter is the zero of energy, first excited state (i.e., the first excited state energy relative to VVT s ¼ sCVT is simply the potential energy of the variational the ground state). Usually the first excited state is much higher transition state; QGT and QA are the translational-motion- than the ground state, and thus only including the first term is excluded quantized partition functions; FR is the product of usually sufficient, i.e., the electronic partition function is equal the reactants’ partition functions per volume, which is equal to to the ground-state degeneracy. Notable exceptions are reactions 3=2 involving halogen atoms and the OH radical. R A A A B B B 2pmAmBkBT F ¼ QeleQrotQvibQeleQrotQvib 2 (42) Next we need to consider the vibrational zero-point energy (ZPE), mGTh which is of central importance in the interpretation of the kinetic isotope effects.65–74 The generalized normal mode analysis is carried in which mAmB/mGT is just the reduced mass of relative translational out along the MEP, and the total zero-point energy at each s is motion of A and B (because mGT = mA + mB). An alternative computed as the summation of the ZPEs of all the vibrational (equivalent) way for computing partition functions is that, in GT A modes that are orthogonal to the reaction coordinate. We define the eqn (40) and (41), the partition functions Q and Q are vibrationally adiabatic ground-state potential energy VG as follows: the total partition function (including translational partition a R

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. function), and F is the product of reactants’ total partition G G Va = VMEP(s)+e (s) (39) functions per volume. where eG(s) is the zero-point energy of the transverse vibrational The above eqn (40) and (41) can also be re-written as: modes. The generalized normal mode analysis along the reaction kCVT/T(T)=kTGCVTkTST (43) coordinate is carried out with curvilinear coordinates rather than TST CVT with rectilinear coordinates or Cartesian coordinates. In order to where k is the conventional TST rate constant, and G is the eliminate or minimize the appearance of unphysical imaginary canonical variational transition state theory recrossing transmission frequencies for modes transverse to the reaction coordinate, one coefficient. (A pioneering physical picture for the transmission 245 should use curvilinear coordinates, which are physically more coefficients was developed in Hirschfelder and Wigner’s work T meaningful.75,76 Note that only at the stationary points are the in the very early days.) If we ignore tunneling (i.e.,wesetk =1), harmonic vibrational frequencies independent of the choice of then the CVT variational transmission coefficient is coordinates. GCVT = kCVT/kTST (44) One important practical issue about the formulas for computing rate constants is the choice of the zero of energy for the calculation, The value of GCVT characterizes the importance of variational which must be done consistently. If one chooses the zero of energy effects (recrossing) in the studied system, and it is smaller than as the bottom of the potential energy well, i.e.,theequilibrium unity. Under the harmonic-oscillator approximation, CVT variational potential energy, for the lowest-energy structure of the reactant in a transmission coefficient is computed as unimolecular reaction, or as the summation of the equilibrium QVT-HO exp VVT k T GCVT B potential energies of the two (infinitely separated) reactants in a ¼ z-HO z (45) Q expðÞV =kBT bimolecular reaction, then the vibrational partition function must be computed accordingly, and this is why we include ehni/2kBT in the where QVT-HO is the vibrational partition function of variational numerator of eqn (37). In the rate constant expressions, the barrier transition state, Q‡-HO is the vibrational partition function of the height is then ZPE-exclusive. As stated above, this is the choice we saddle point; VVT is the potential energy of the variational transition

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‡ state, and V is the potential energy of the saddle point. The location because the point group of the TS is C3v (for which the order of of the GT is temperature-dependent, and so does the GCVT. rotational subgroup is 3), the forward reaction symmetry number is Similarly, for microcanonical variational transition state 4, which coincidently happens to be equal to the number of

theory (mVT), we have equivalent H atoms on CH4. The reaction symmetry number should also include the contributions from the non-superimposable mirror NzðEÞ kmVTðEÞ¼GmVTðEÞ (46) images.80 However, if the MS-T method137,138 (which will be r ðEÞh R discussed in later section) has been applied, then the reaction where GmVT(E) is the microcanonical variational transmission symmetry number should exclude the mirror images, because coefficient. Microcanonical variational transition state theory they have already been included in MS-T partition function and provides a variational extension of the Rice–Ramsperger–Kassel– we should not double count them. Marcus (RRKM) theory, which is based on conventional transition We should remind ourselves that after this non-rigorous state theory. If quantum mechanical tunneling through rovibrational quantization of the classical canonical variational transition excited states of the transition state is considered, the number state theory, the CVT rate constant no longer necessarily of states of the generalized transition state would be replaced by provides an upper bound to the exact quantum rate constant. the cumulative reaction probability,77,78 which thereby becomes Nevertheless, the variationally optimized rate constants provide an effective number of reactive states. much better predictions than the ones obtained by putting Notice that in the above equations, the overall rotational dividing surface at the saddle point as in the conventional TST. symmetry number of the molecule is included in the rotational partition function. Sometimes, the rotational symmetry numbers 2.3 Quantum mechanical tunneling for TS and for reactant(s) in their rotational partition functions 2.3.1 Importance of tunneling. Tunneling is the quantum are factored out to the reaction symmetry number79 mechanical process by which a particle has a finite probability 8 < sR sz unimolecular reaction to pass through a barrier, even though its energy is lower than the barrier. Nonclassical reflection refers to the quantum srxn ¼ : (47) sR1sR2 sz bimolecular reaction diffraction phenomenon that when the incident particle’s energy is higher than the barrier, there is a finite probability of the particle and the corresponding VTST rate constants are expressed as (in being reflected. Because the treatment presented so far treats the which the rotational partition function excludes the rotational reaction-coordinate motion as classical, it does not include either symmetry number) quantum mechanical tunneling through the reaction barrier or Bimolecular reaction : kCVT=TðTÞ nonclassical reflection from the barrier. Tunneling increases the rate constant, and nonclassical reflection decreases it. Since GT (48) k T Q ðÞs ¼ s VT CVT T B V ðÞs¼s =kBT tunneling occurs at lower energies where the Boltzmann factors ¼ srxnk e h FR are larger, tunneling is usually the more important one of the two phenomena for thermal rate constants, and the net effect of CVT=T Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. Unimolecular reaction : k ðTÞ including these two quantum effects on reaction-coordinate GT (49) motion are usually lumped together in a transmission coefficient kBT Q ðÞs ¼ s VT CVT ¼ s kT eV ðÞs¼s =kBT rxn h QA that is simply called the tunneling transmission coefficient. Quantum mechanical tunneling is almost always significant R ‡ where s and s are the rotational symmetry number for the for reactions involving transfer of light atoms (such as hydrogen reactant and transition state structure, which are equal to the order atoms, protons, or hydride ions) at low temperatures. It is often of the rotational subgroup for polyatomic molecule; rotational most easily noticed in large values of H/D kinetic isotope effects symmetry number is 1 for CNv point group (for instance, a hetero- (KIEs). Sometimes it is also significant for heavier-atom motion, nuclear diatomic molecule) and 2 for DNh point group (for instance, for example, carbon tunneling. a homonuclear diatomic molecule); for monoatomic species, which 2.3.2 Tunneling transmission coefficient. The tunneling R do not have rotational motion, s is taken to be 1. transmission coefficient k(T) for VTST is defined as the ratio A common misunderstanding about the reaction symmetry of the thermally averaged quantum tunneling probability to the number is that it represents the number of equivalent ways (or quasiclassical transmission probability. The quasiclassical chemically equivalent positions) for the reactants to react; this is transmission probability is the one that is assumed in quasiclassical simply not true, although in some cases they (the number of VTST, and it is a Heaviside step function that increases from equivalent ways to react and the reaction symmetry number) might 0 to 1 at the maximum of the effective potential. The tunneling coincidentally have the same value. For instance, in the case of transmission coefficient is: - CH4 +OH CH3 +H2O, the number of equivalent ways for OH to Ð þ1 PTðEÞeE=kBT dE abstract the H atom from CH4 is 4 (CH4 has four chemically Ð E0 kðTÞ¼ þ1 (50) equivalent H atoms), but the reaction symmetry number for forward PCðEÞeE=kBT dE E0 reaction is 12 (the order of rotational subgroup for Td point group

(CH4) is 12, and rotational symmetry number for OH is 1 and for TS where E0 is the higher of either the ground-vibrational-state

(Cs point group) is 1); but in the case of CH4 +O- CH3 +OH, energy of the reactants or the ground-vibrational-state energy of

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the products (this is the lowest energy at which reaction can Tunneling models can also be divided into two classes in occur), PT(E) is the quantum tunneling probability, and PC(E)is another way, namely models with analytic results (Wigner the classical transmission probability. We assume that the model, parabolic models, and Eckart model) and models where effective potential for tunneling is the vibrationally adiabatic the tunneling probability must be calculated by numerical G ground-state potential energy curve Va of eqn (39), whose integration. The latter models are usually based on the semi- maximum value is called VAG. Then, if the transition state were classical WKB approximation94 or – to be more precise – on a at the maximum of the vibrationally adiabatic ground-state uniformized WKB approximation. The primitive WKB approxi- potential energy curve, we would have mation suffers from infinities and has poor behavior for energies 8 near the barrier top; uniformization95,96 is the process by which one < 0 E VAG C include higher-order effects to obtain more uniformly accurate P ðEÞ¼: (51) 1 E 4 VAG results, and when one does this, the results for tunneling through a barrier are often in good agreement (B15% or better) with a where VAG is the maximum of the vibrationally adiabatic quantal treatment of the same barrier model. But converting the G ground-state potential energy Va along the reaction coordinate. multidimensional problem into a tractable problem with an effective But in general, the CVT transition state is not at the maximum barrier as a function of a tunneling coordinate (or a set of such of the vibrationally adiabatic ground-state potential energy; tractable problems) is harder. We will begin with analytic one- AG G CVT CVT dimensional models. thus we have to replace V by Va s ðTÞ where s ðTÞ is the location of the CVT transition state at temperature T. 2.3.3.1 One-dimensional tunneling models. A popular approxi- The above definition of the PC(E) is consistent with the mation is to assume that the shape of the potential energy along classical picture of a moving particle (i.e., no tunneling under the reaction coordinate s can be approximated by an Eckart the barrier and no reflection above the barrier). Carrying out potential:82 the integral in the denominator of eqn (50), we get ð yDV By þ1 1 AG VðsÞ¼ þ (54) kTðTÞ¼ PTðEÞeðÞEV =kBT dE (52) 1 þ y ð1 þ yÞ2 kBT E0 where As for the quantum tunneling probability PT(E), it satisfies: 8 y = exp[a(s s0)] (55) T AG > 0 P ðEÞ o 1=2 E o V <> 1/2 1/2 2 B =[Vmax +(Vmax DV) ] (56) PTðEÞ¼ 1=2 E ¼ VAG (53) > N :> in which DV is the value of the potential at s =+ relative to T AG 1=2 o P ðEÞ1 E 4 V s = N, Vmax is the value of the potential at its maximum

relative to s = N. And the constant s0 is defined to give the The region below barrier is the tunneling region, and the region maximum value of V(s), (i.e., Vmax)ats =0: Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. above barrier is non-classical reflection region. The remaining T 1 problem is approximating the quantum tunneling probability P (E). s0 =(2a) ln(1 DV/Vmax) (57) 2.3.3 Semiclassical tunneling approximations. In this section, The intrinsic barrier height Vb described by the potential is: we review various models for approximating the accurate quantum tunneling in reaction rate constant computations. Commonly used Vb = Vmax max[0,DV] (58) tunneling models can be divided to two classes: one-dimensional The parameters in Eckart model are usually chosen to fit the and multidimensional tunneling models. The multidimensional enthalpy of reaction at 0 K and the enthalpy of activation at 0 K. tunneling models include the variation of the vibrational The parameter a determines the range of the potential, and it is frequencies along the reaction path, as included by the final related to the magnitude of the second derivative of the term of eqn (39). The one-dimensional tunneling models do potential at its maximum by not. The one-dimensional models reviewed in this article sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 81,82 include the Eckart tunneling model, the Wigner tunneling BFs 83 a ¼ (59) model, the one-dimensional truncated parabolic tunneling 2VmaxðÞVmax DV approximation;84 and multidimensional tunneling models include 49,50 zero-curvature tunneling (ZCT), the parabolic model with an where the second derivative Fs,inpractice,canonlybeapproximately effective frequency,85 small-curvature tunneling (SCT),86 large- computed by: curvature tunneling (LCT),71,87–90 microcanonically optimized multi- F =4p2m|n‡|2 (60) dimensional tunneling (mOMT),71 and least-action tunneling s (LAT).91–93 All of the one-dimensional tunneling models and where |n‡| is the magnitude of the imaginary vibrational the ZCT tunneling model ignore the reaction-path curvature; frequency of the saddle point, and m is the reduced mass this is not a good assumption for many cases. The effects of the associated with the imaginary-frequency vibrational mode. reaction-path curvature often dramatically increase the amount The reduced mass for reaction coordinate tunneling is often of tunneling. arbitrarily chosen to be 1 amu (atomic mass unit), or 1.008 amu

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for hydrogen atom tunneling (if we assume that the mass of the Plug the above tunneling probability into eqn (52), we can get heavy atom is infinitely large). This way of approximating F is ! s hnzx artificial, and it simply cannot be done unambiguously. The ð exp h nz þ1 2pkBT only advantages of the Eckart tunneling model are that one kTðTÞ¼ hipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx (70) does not need to compute a reaction path, and the tunneling 2pkBT g 1 þ exp 2g 1 1 þ x=g probability PT(E) can be analytically evaluated, which provides computational simplicity, but it may yield quantitatively in which unsatisfactory results. The PT(E) given by Eckart potential is: x = g(E Vmax)/Vmax (71) coshða þ bÞcoshða bÞ PTðEÞ¼ (61) c coshða þ bÞþcoshðdÞ Because of our previous assumption (g 1), we carry out the integral in eqn (70) from N to +N, with a further assumption where that the temperature T is very high,22 i.e., a =2p(2mE)1/2/(ha) (62) hnz 1 (72) b =2p[2m(E DV)]1/2/(ha) (63) 2pkBT

2 2 1/2 and finally an analytical expression can be arrived: d =2p[2mB h a /4] /(ha) (64) z h n ðÞ2kBT The Eckart tunneling model can be viewed as a crude approximation kT T ð Þ¼ z (73) to the zero-curvature tunneling (ZCT) approximation, which sin½hjjn =ðÞ2kBT systematically underestimates the tunneling due to neglecting This expression could at most be valid when the temperature T reaction path curvature; however, the Eckart tunneling model is very high and the barrier is very high or broad, but actually often overestimates the ZCT tunneling transmission coefficient, this simple expression for computing the tunneling transmis- which in turn partially compensates its intrinsic error. Eckart sion coefficient is basically never valid primarily because it uses tunneling model uses a fit to VMEP, in the sense that one fits it to G a parabolic fit to VMEP – not to Va . enthalpy at 0 K for the energetics, but the frequency is still fit to Eqn (73) can be further simplified since t/sin(t) E 1+t2/6 for the one based on VMEP, which is probably why it often over- t { 1, we obtain the simple Wigner tunneling approximation:83 estimates the tunneling probability compared to ZCT. (The VMEP ! 2 curve tends to be thinner for the nearly symmetric barriers that 1 h nz kTðTÞ¼1 þ (74) have the greatest tunneling.) 24 kBT Although PT(E) for Eckart tunneling can be expressed analytically, the tunneling transmission coefficient kT(T) needs to be computed The Wigner tunneling transmission coefficient is very simple to by numerical integration. To further simplify the computation of compute; it only requires the imaginary frequency of the saddle kT(T), one could make additional assumptions (which may not be point, but it is not useful for quantitative results.

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. appropriate though) based on Eckart tunneling. If we consider a It can be shown that, eqn (73) can also be derived from an 97 symmetric Eckart potential function, that is, we set DV =0inthe infinite untruncated parabolic barrier. An untruncated para- asymmetric potential given by eqn (54) and we can get bola has E0 = N, and when T is lowered, the tunneling through an untruncated parabola become so unphysically large Vmax VðsÞ¼ rffiffiffiffiffiffiffiffiffiffiffiffi (65) that the Boltzmann average in eqn (52) fails to converge and the Fs cosh 2 s transmission coefficient blows up when 2Vmax h|n‡|/(2k T)=p (75) The corresponding tunneling probability, which can be derived B from eqn (61), is Skodje and Truhlar84 have eliminated this issue by using the uniformedized WKB semi-classical tunneling approximation, cosh 2ge1=2 1 PTðEÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (66) and they have further generalized the result to be applicable for 1=2 2 2 coshðÞþ 2ge cosh 4g p any unsymmetrically truncated parabolic-type barrier that is defined as: where 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > Vmax e = E/Vmax > 0 s o > 2 z 2 (67) > 2p mjjn > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pVmax < g ¼ (68) 2 Vmax Vmax DV z V 2p2m nz s2 s hjjn VðsÞ¼> max 2 2 > 2p2mjjnz 2p2mjjnz > If we assume the barrier is very high or very broad such that > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > c > V DV g 1, then we have > max : DVs4 2 1 2p2mjjnz PTðEÞ¼ (69) 1 þ exp½ 2gðÞ1 e1=2 (76)

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and the intrinsic barrier Vb is modes that are orthogonal to the reaction coordinate s,andthis zero-point energy depends on s. The contribution of vibrations Vb = Vmax max[0,DV] (77) G transverse to the reaction path to Va (s) is the reason why the The Skodje–Truhlar one-dimensional truncated parabolic SCT approximation is multidimensional. The ZCT tunneling tunneling transmission coefficient kT(T) is given by: probability is given by: 8  8 > 1 2p >0 E E0 > exp V > > z z b > > h n ðÞ2kBT kBT hjjn 2pkBT <> 1 G G > 41 ½1 þ expð2yÞ Va ðs ¼ 1ÞoE Va ðÞs > z z z ZCT >sin½hjjn =ðÞ2kBT 2pkBT=ðÞhjjn 1 hjjn P ðEÞ¼ > > ZCT G G G < >1 P 2Va ðÞs E Va ðÞs oE 2Va ðÞs E0 Vb 2pkBT > kT T ¼1 : ð Þ¼> z G >kBT hjjn 12Va ðÞs E0 oE > >  > 1 2p (79) >exp V 1 > z b > kBT hjjn 2pkBT G : o1 where E0 is the maximum between the Va of the reactants and 12pk T=ðÞhjjnz hjjnz B of the products; s* is the location of the variational transition G (78) state, where the Va reaches its maximum. The magnitude of the imaginary action integral is which is a singularity-free generalization of the result given ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s4 by eqn (73). The above results (eqn (76)–(78)) are for one- y ¼ 2m VGðsÞE ds (80) h a dimensional tunneling; later, Skodje et al. also showed that so

this parabolic model could also be generalized so that it G where s and s4 are the two turning points, at which Va = E. includes the reaction-path curvature with the small-curvature o The integrand is the imaginary part of the momentum of mass approximation,85 however, the correct effective mass for the m (that is the reduced mass of the isoinertial coordinate SCT tunneling approximation was only derived later;86,98 it is system). The ZCT approximation underestimates the tunneling not the one that is presented in this earlier paper85 (which is transmission coefficient because it neglects reaction-path curvature; valid only if the reaction-path curvature is confined to a single there are shorter tunneling paths than the MEP that have the same transverse mode, which basically occurs only for atom–diatom effective potential,119 which is given by eqn (39), and those paths reactions with collinear reaction paths). We will present the dominate the tunneling. SCT tunneling model in the next section. In the small-curvature tunneling approximation (SCT), an We do not recommend using any of the one-dimensional effective reduced mass meff is introduced in computing the models for quantitative studies; and they were reviewed in tunneling probability as expressed in eqn (79). The value of meff this sub-section for completeness. Although these models are is smaller than m, which (as can be seen from eqn (80)) is computationally inexpensive and certainly very easy to use, equivalent to shortening the tunneling path. In the current their accuracy is generally not good, and therefore for quantitative Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. version of SCT (which is also called the centrifugal-dominant study, we recommend more accurate multidimensional tunneling small curvature approximation), which is used in modern VTST models. calculations, meff is determined by reaction-path curvature as 86,98 2.3.3.2 Multidimensional tunneling (MT) models. In real follows: ( ÈÉ dynamics, the motion of the reaction coordinate is not decoupled exp 2aðsÞ½aðsÞ2 þ½dtðsÞ=ds2 from the other degrees of freedom, and the curvature of the MEP in meff ðsÞ¼m min (81) the mass-scaled coordinate system is one manifestation of the 1 41,99 coupling. A major consequence of reaction-path curvature is in which a¯(s)=|k(s)%t(s)|, and the reaction-path curvature |k(s)| that tunneling occurs in a corner cutting fashion.6,40,41,71,84–93,98–118 FP1 1=2 49 2 That is, the tunneling path does not need to follow the MEP; is ½BmF ðsÞ , where BmF(s) is the curvature component internal centrifugal effects cause the dynamical path to be displaced m¼1 from the MEP; specifically, the quantum centrifugal effect makes of vector BF(s), which represents the coupling between the the system move toward the concave side (that is the inside) of the reaction coordinate s and a mode m that is perpendicular to it: X3N MEP, not the convex side (that is the outside) as the classical dn ðsÞ B ðsÞ¼signðsÞ i LGTðsÞ (82) centrifugal effect does. This is called the corner cutting effect, mF ds i;m i¼1 and it is due to the negative kinetic energy (imaginary momentum)

in the classically forbidden region leading to a negative internal where ni(s) is the component i of the unit vector perpendicular centrifugal effect. to the generalized transition state dividing surface at s and GT First, we begin with the zero-curvature tunneling approxi- Li,m(s) is the i-th component of the eigenvector for vibrational mation (ZCT). In ZCT, the reaction path curvature is neglected, mode m perpendicular to n(s)ats. however, the contributions of other degrees of freedom are The effective harmonic potential V for the vibrational mode G included in the vibrationally adiabatic potential V a (s), which u1 (which is constructed by a local rotation of the vibrational

contains the vibrational ground-state energies of all the vibrational axes such that BF(s) lies along with it, and by construction, the

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curvature coupling in all other vibrational coordinates u2 to and these tunneling paths can connect vibrational-ground-state

uF1 are zero in this rotated coordinate system) is turning points on the reactant side (in the exoergic direction "# XF1 2 of reaction) with vibrationally excited turning points on the 1 B ðsÞ V ¼ V þ m m;F o ðsÞ u 2 (83) product side. In the LCT method, the tunneling paths can be in MEP 2 kðsÞ m 1 m¼1 a wider region than the nearly quadratic valley around the MEP, and the region that they pass through is called the reaction where o (s) is the vibrational frequency of mode m at reaction m swath. The LCT tunneling probability includes the contribution coordinate s. The associated turning point %t(s) for zero-point from all the tunneling paths originating from the reactant motion in this harmonic potential is then: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi valley and some of these occur earlier than when the system u u h gets all the way in to where it hits the barrier. That is, in LCT, tðsÞ¼u "#(84) u 1=2 FP1 2 one can have a significant probability of tunneling before t Bm;F ðsÞ m omðsÞ reaching the reaction-coordinate turning point. The classical m¼1 kðsÞ reaction-coordinate turning points are the points at which the This form of turning point is able to avoid unphysically large ground-state vibrationally adiabatic potential is equal to the tunneling probability that existed in the old version of the total energy E in either the reactant and product valley. Unlike theory,85 which is caused by the assumption that all modes are the treatment in ZCT and SCT, the LCT tunneling at energy extended to their turning points along the tunneling path; we E takes place all along the entrance channel up to the turning no longer use the effective mass formula from the old version of point, and it is initiated by the vibrational modes that are the theory. orthogonal to the reaction coordinate rather than the motion along the reaction coordinate. For large reaction-path curvature, The turning-point derivative terms dtm(s)/ds are evaluated at s by a two-point central-difference method using the turning- the LCT approximation gives larger tunneling probability than point results obtained at adjacent locations at which the SCT due to the very significant shortening of the tunneling normal modes are found. Because the gradient of the potential paths; however, in the limit of small reaction-path curvature, the LCT SCT energy at the saddle point is zero, the effective reduced mass tunneling probability P (E) could be smaller than P (E), meff(s = 0) is computed by a linear interpolation between the two because SCT is a more accurate treatment in this limit region. closest values of meff(s) as the following At intermediate reaction-path curvature, SCT and LCT give similar results. 1 meff ðs ¼ 0Þ¼ meff ðs ¼þdsÞþmeff ðs ¼dsÞ (85) In the microcanonically optimized multidimensional tunneling 2 model (mOMT), the tunneling probability at energy E is simply set where ds is the step size of the reaction coordinate. to be the maximum between PLCT(E)andPSCT(E). When one uses The corner-cutting model that led to the effective reduced mOMT, one often finds some contribution form large-curvature mass meff(s) does not apply to energies above the maximum of tunneling at low energy, even if the more important higher-energy G V a , it is therefore inappropriate to compute the transmission tunneling is dominated by small-curvature tunneling. In some

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. probability at the non-classical reflection region based on older work, we used a method now called the canonically PSCT(E). The transmission probability in the non-classical optimized multidimensional tunneling model (OMT), in which reflection region is computed by the thermally averaged tunneling transmission coefficient at temperature T is set to be the maximum between kLCT(T) and PSCT(E)=1 PSCT(2VG E), VG(s ) o E r 2V G(s ) E a a * a * 0 kSCT(T). (86) The most complete approximation for the quantum tunneling which means that the effective reduced mass is only used for is the least-action tunneling model (LAT).91–93 In this model, a computing the transmission probability at tunneling energies. sequence of paths that are parameterized by a parameter a is SCT is currently the most widely used tunneling approximation considered for each tunneling energy E and each final vibrational among the many multidimensional tunneling models; and it is state (not necessarily the vibrational ground state). The para- usually able to give a satisfactory tunneling transmission meter a is defined so that these LAT paths are all located at or coefficient with a balanced computational cost (when compared between the MEP (a = 0) and LCT paths (a = 1). At every tunneling to other more complex multidimensional models, which could energy E, within the chosen sequence of paths, the tunneling be expensive for medium to large sized molecules). path that has the largest tunneling probability is chosen to be Nevertheless, the tunneling path can be further optimized, and the LAT tunneling path; that is, the LAT tunneling path is in some cases the system can tunnel directly into vibrationally variationally optimized to minimize the action integral at E, excited states (i.e., it is not necessary for tunneling to be governed with respect to the parameter a. The LAT tunneling was originally by the ground-state vibrationally adiabatic potential energy curve). computationally very demanding, because one has to compute The large-curvature tunneling approximation71,87–90 (LCT) was the information of huge number of points on potential energy developed for more appropriately (as compared to SCT) describing surface (not just on MEP) including Hessians, but we have reactions with large curvature of the reaction path. One of the key derived efficient algorithms for it.93 However, it is seldom used features in the LCT approximation is that the tunneling paths are because the results are not sufficiently more accurate than those set of straight lines that connect reactant and product sides, obtained by the mOMT method.

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In the computations of the multi-dimensional tunneling top of the effective barrier for tunneling is below the zero point transmission coefficients, it is very important for the computed energy of the reactants, then there is no tunneling because all reaction path (i.e., the range of s being computed) to be long systems have enough energy to pass over the barrier without T enough so that the quantum tunneling is converged; otherwise a tunneling. Thus k1 is unity (or less than unity if we include non-converged quantum tunneling calculation can yield erroneous nonclassical reflection). But if the pressure is large enough to tunneling transmission coefficients. In practice, one can extend the stabilize states of the intermediate that have energies lower range of the reaction path that was computed in the previous than the reactant zero point energy and lower than the top of calculation and then redo the computation (with some possible the effective barrier for tunneling, those low energy states can T restart options to take advantage of the information produced in the tunnel and k2 can be greater than unity. This shows that the previous run), and the tunneling transmission coefficients should well-known dictum that one can ignore an intermediate that remain the same at all temperatures. Extrapolation techniques occurs before the rate-determining transition state is only are available to minimize the needed length of reaction path partially true, and it is dangerous to ignore the existence of a computation.120 van der Waals intermediate in a bimolecular reaction (or 2.3.4 Tunneling and the treatment of reaction intermediates. similarly, in a two-step unimolecular reaction). This is a very An important issue involving tunneling is the treatment of reaction general issue since most bimolecular reactions have barrier, intermediates. Consider a two-step reaction: and a ground-electronic-state barrier is always preceded by a

k van der Waals minimum. However, the issue is usually important 1 k2 A þ B ! I ƒ! P (87) only for reactions with low barriers. For a high barrier, since the k 1 tunneling usually has important contributions only from energies The first step is the formation of an intermediate (I), with a at most a few kcal mol1 below the barrier top, the tunneling from

forward rate constant of k1; the intermediate I further proceeds intermediate states whose energy is below the reactant zero point

to yield the product P, with a rate constant of k2. The overall energy is expected to be negligible. reaction rate for the formation of product is: There is another complication, namely the assumption that the intermediate is in equilibrium with A and B. If one is not in d½P ¼ k2½I (88) the high-pressure limit, there may be inadequate collisional dt stabilization of the complex. In the low-pressure limit, there are and if the intermediate I is in equilibrium with A and B, we no stabilizing collisions, and therefore there are no tunneling would have contributions at energies below the ground-state energy of the 121 d½P reactants (which would be the lowest possible energy that a ¼ k K½A½B (89) dt 2 complex could have). The low-pressure limit and the high-pressure limit are easily handled as limiting cases, but at intermediate where K is the equilibrium constant for the first step, and pressures one must use the theory of pressure-dependent reactions, K = k /k . And thus the overall bimolecular rate constant is 1 1 which is treated in a later section. Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. k = k K (90) overall 2 2.4 Recrossing transmission coefficient With (variational) transition state theory, the rate constant k2 is The basic assumption of quasiclassical transition state theory, GT VGTVI TkBT Q which may be used to derive the equations in Section 2.2, is k ¼ k e kBT (91) 2 2 h QI that the systems with energy higher than the vibrationally adiabatic potential barrier (which may be called the quasiclassical T where k2 is the tunneling transmission coefficient computed by threshold energy) react with unit probability and systems with using the intermediate I as the reactant. The equilibrium lower energy do not react, i.e., the reaction probability is a constant K of the first step is Heaviside step function as a function of energy in the reaction I VIVAVB Q coordinate. Because the vibrationally adiabatic barrier involves K ¼ e kBT (92) QAQB=V vibrational frequencies and rotational moments of inertia that depend on the reaction coordinate, VTST goes beyond the and thus, the overall bimolecular rate constant is equivalent to assumption of a separable reaction coordinate that is assumed GT VGTVAVB in purely classical conventional transition state theory. TkBT Q k ¼ k e kBT (93) overall 2 h QAQB=V In Section 2.3, we saw the use of a tunneling transmission coefficient to account the breakdown of the step function If one ignores the intermediate, that is if one computes the probability due motion along the reaction coordinate being rate constant directly for A + B - P, the rate constant expres- quantum mechanical so there is a small probability that T sion is almost the same as eqn (93); the difference is that k2 in systems with higher energy than the effective barrier diffract T eqn (93) is replaced by k1, which is the tunneling transmission off the barrier top and do not react and a small probability that coefficient computed by using A and B as the reactants. To systems with less energy that the barrier tunnel through it. The discuss this, we consider the case of an exothermic reaction at latter events happen at lower energy where the Boltzmann low temperature where the reactant is in its ground state. If the factor is higher, and so their effect usually dominates over

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the effect of nonclassical reflection, and the tunneling trans- transition state. This would mean that 20% of the flux through mission coefficient is usually great than unity. We take account the conventional transition state is nonreactive because it does of tunneling by doing a multidimensional semiclassical calculation not pass through the variational transition state, which has a that accounts for two aspects of reaction-coordinate non- higher free energy of activation. separability, namely (i) the already-mentioned reaction- One can also define an exact recrossing transmission coordinate dependence of the vibration frequencies and (ii) coefficient as the corner cutting effect by which the tunneling path involves Gexact or gexact = kexact/kTST (98) displacement of the transverse vibrations to the concave side of the curved minimum energy path during the tunneling event. This is the definition implicitly used in original literature,3 There is another reason why the step function probability where it is simply called the transmission coefficient. assumption can break down, namely that even for classical reaction coordinate motion, the non-separability of the reaction 2.5 Importance of using curvilinear coordinates in VTST/MT path can lead to less than unit probability of reaction for calculations for vibrations transverse to the reaction coordinate systems with energy greater than the quasiclassical threshold In the VTST/MT calculations, the computations of both the because systems cross the transition state but then recross it, variational transition states and the multi-dimensional quantum making the net reactive flux smaller than the one-way flux. tunneling transmission coefficients rely on the computed We can account for this by including another transmission vibrationally adiabatic ground-state energy along the reaction coefficient in the rate constant; we call this the recrossing path; the generalized normal mode analysis is performed at transmission coefficient. Note that the recrossing transmission evenly spaced points along the minimum energy path. For non- coefficient could also be called the quasiclassical transmission stationary points (notice that the reactants, products, and the coefficient because the effect is present even for classical conventional transition state structures are stationary points) reaction-coordinate motion. along the reaction path, the frequency calculations have to be In general, then there are two contributions to the transmission carried out based on projected Hessians (which project out the coefficient, and we write translational modes, overall rotational modes and the mode corresponding to the reaction-coordinate degree of freedom). g = kG (94) One should keep in mind the fact that the projected Hessians where g is the full transmission coefficient, k is the tunneling in this generalized normal mode analysis are coordinate- transmission coefficient, and G is the recrossing transmission system-dependent,75,76 and so are the bound-mode vibrational coefficient. We can apply this to the quasiclassical VTST rate frequencies of non-stationary points. constant calculated without tunneling or recrossing to get a The reason that the vibrational frequencies along the MEP better answer. When we do this, we usually set G = 1. For are coordinate-dependent can be explained as follows. The choice example, using canonical variational transition state theory, we between the rectilinear coordinates (which are linear combinations write the rate constant with tunneling as of Cartesian coordinates) and the curvilinear coordinates (internal Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. coordinates) is equivalent to the choice between two different kCVT/T = kkCVT (95) definitions, s and s0, of the reaction coordinate for points off the where kCVT is the quasiclassical VTST rate constant. However, MEP; they have the same values for stationary points on the MEP, sometimes we use a different notation, namely but not for non-stationary points. These two different choices of the reaction coordinate are related by76,122 kCVT/T = kGCVTkTST = gkTST (96) XF1 XF1 TST 1 where k is the quasiclassical conventional VTST rate constant, s0 ¼ s þ b ðsÞq q þ oq3 (99) 2 ij i j i and i¼1 j¼1

CVT CVT TST G = k /k (97) in which F is the number of internal coordinates, and F 1 These different conventions should be clear from the context. excludes the reaction coordinate s (which is also called qF). The The quantity GCVT is sometimes called the variational effect qi represent curvilinear (internal) coordinates, which measure because it accounts for the difference between variational distortions away from the MEP, and by definition they vanish on 0 transition state theory and conventional transition state theory the MEP. The bij involve second-order partial derivatives of s when both are applied without tunneling. Its deviation from with respect to qi, and they are in general nonzero. The double ... unity is a measure of the extent to which VTST accounts for the summation runs over a complete, non-redundant set q1, q2, , breakdown of the no-recrossing assumption of quasiclassical qF1 of such coordinates that are orthogonal to the MEP. The transition state theory at the conventional transition state that matrix elements of the Hessians expressed using these two different reaction coordinate definitions are related by:123 passes through the saddle point. For example, a recrossing "# transmission coefficient of 0.8 means that 20% of the quasi- @2V @2V @V classical flux through the conventional transition state does not ¼ bij @qi@qj s0 q0 0;...;0;s0 @qi@qj s @s q contribute to the net one-way flux because the true dynamical ¼ð Þ q¼ð0;...;0;sÞ bottleneck is the variational transition state, not the conventional (100)

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0 0 in which q {q1,q2,..., qF1,s} and q {q1,q2,..., qF1,s }. reaction path (i.e.,therangeofs being computed) to be long Clearly, at stationary points, where the first-order derivative of enough so that the quantum tunneling is converged; otherwise a V with respect to s is zero, the two Hessians are the same; non-converged quantum tunneling calculation can yield erroneously Cartesian coordinates and internal coordinates yield the same large tunneling transmission coefficients. In practice, one can extend frequencies for the stationary points. However, at non-stationary the range of the reaction path that was computed in the previous points, where the (qV/qs) term does not vanish, the frequencies calculation and then redo the computation (with some possible are coordinate-system-dependent. restart options to take advantage of the information produced in And thus, the important consequence is that, the results the previous run), and the tunneling transmission coefficients of the VTST/MT calculations depend on the choice of the should remain the same at all temperatures. coordinates75,76 that are used for the generalized mode analysis along MEP. It is highly possible for a set of Cartesian coordinates (a 2.6 State-selected rate constants linear combination of which yields a set of rectilinear coordinates) Although transition state theory is formulated for the calculation to unphysically yield one or more imaginary frequencies along the of the reaction rates of thermally equilibrated or microcanonically reaction path75,76 (along the reaction path, in the absence of a equilibrated reagents, extensions have been proposed to real bifurcation, there should be no imaginary frequency after approximate vibrational-state-selected rate constants.126,127 the projection); and the resulting VTST/MT rate constants can Liu recently reviewed the experimental progress made in studying be significantly affected, and thus they could be unreliable or the mode-selected reaction dynamics.128 In this section, we even wrong. Along the reaction path, the free energy cannot be briefly review using VTST to study state-selected reaction rate computed at the regions where one has imaginary frequencies, constants. and thus Polyrate program42 arbitrarily sets the vibrational There are two basic ways to extend VTST to reactions in partition function to very large number (which decreases the which the initial vibrational state is specified to have some Gibbs free energy by a large amount) in order to ensure that the specific excitation. One is to assume vibrational adiabaticity of computed canonical variational transition state will never occur the excited mode;127 then, in the sums over vibrational quantum in these unphysical regions which contains imaginary frequencies. numbers for the state-selected mode, only one term contributes. Using Cartesian coordinates for generalized normal mode analysis Another is based on non-adiabaticity. We first review the former along the reaction path could possibly cause: (1) unphysical approach and then the latter one. discontinuity of the vibrationally adiabatic ground-state energy Consider the vibrational partition function with F vibrational G (Va ) along the reaction path; (2) unphysical sudden jump(s) on degrees of freedom assumed separable (for stationary points, F the Gibbs free energy at a given temperature along the reaction includes all the vibrational degrees of freedom; and for non- path, because of the above-mentioned arbitrary setting that the stationary points, F includes all the bound-mode vibrational program has to make; (3) in the case of microcanonical VTST degrees of freedom that are orthogonal to MEP, which excludes calculations, using Cartesian coordinate can cause unphysical the reaction coordinate),

sudden jump(s) in the computed vibrational number of states max YF nmX¼nm Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. mVT vib N (E,s) and thus the energy-dependent microcanonical variational QvibðTÞ¼ eem ðÞnm =kBT (101) transition state would be located in these unphysical regions, which m¼1 nm¼0 leads to unphysical results for the mVT rate constants. In order to obtain reliable VTST/MT rate constants, one where m labels a vibrational mode, and nm is the number of max should first choose a set of physically reasonable curvilinear quanta in that mode, and nm is the number of quanta used to coordinates that are capable of describing the atomic motion converge the summation. If the vibrational energy levels are along the whole reaction path, and then use this set of coordinates computed by harmonic oscillator (HO) approximation to carry out the generalized normal mode analysis and to further 1 evibðÞ¼n n þ hv (102) compute the variational transition states and multi-dimensional m m m 2 m tunneling. In Polyrate, one could choose either a set of 3N 6 (where N is total number of atoms) non-redundant internal where vm is the vibrational frequency of mode m. This yields the coordinates75 (via the ‘‘CURV2’’ option), or redundant internal HO vibrational partition function of eqn (37), which is used in coordinates124 (via the ‘‘CURV3’’ option), as curvilinear coordinates computing thermal VTST rate constants in the harmonic for performing the generalized normal mode analysis. A set of approximation. In mode-selected VTST, one modifies eqn (101) to

physically meaningful and well-chosen internal coordinates is "#max r n 0 ¼n mY¼F YF mXm0 ÈÉ vib r vib usually able to give an imaginary-frequency-free reaction path in vib r em ðÞnm =kBT e 0 ðÞnm0 kBT Q nm ;T ¼ e e m physically important regions (i.e., the regions where the variational 0 r m¼1 m ¼F þ1 nm0 ¼0 transition states are located and where the quantum tunneling (103) take place). In some cases, it isusefultoalsore-orientthe generalized-transition-state dividing surface (via the ‘‘RODS’’ in which the modes from m =1tom = Fr are the selected modes option) to eliminate the imaginary frequencies along the path.125 (i.e., each of these modes is restricted to have a specific number r r As for the computations of the multi-dimensional tunneling of excitation quanta equal to nm); and the remaining F–F modes transmission coefficients, it is very important for the computed are treated as usual with the summation over all the vibrational

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0 quantum numbers nm0 of modes m going up to a high enough State-selected VTST calculations have been tested for various value to converge the partition function, i.e., up to a large number systems,130–136 and good agreement with experimental values or that converges the summation. This restricted vibrational partition accurate quantum dynamics are confirmed. The most sophisticated function is a function of the selected vibrational quantum states, treatment131 assumes that the reaction is vibrationally adiabatic up which are characterized by the set of the specified vibrational to the first occurrence in proceeding from reactants to products of r quantum numbers {nm}. In adiabatic mode-selected VTST, the an appreciable local maximum in the reaction-path curvature; at locations of the variational transition states are determined based that point the reaction is treated as if all flux is suddenly diverted to on the vibrational partition functions of the generalized transition the ground vibrational channel. state using eqn (103) with the bound-mode vibrational frequencies along the reaction path. This same restriction is also used to generate state-selected conventional transition state theory, in which 3. Multistructural and multipath the generalized transition state is chosen to be at the saddle point variational transition state theory rather than variationally optimized. The multidimensional quantum tunneling transmission 3.1 Multistructural anharmonicity coefficients in state-selected VTST is computed based on the The textbook way of computing molecular partition function following state-selected vibrationally adiabatic potential: is based on a single lowest-energy structure and on the

r (quasi-)harmonic oscillator approximation. A harmonic oscillator ÈÉ mX¼F mX0¼F g r vib r vib;g has only one minimum-energy structure, but real species often Va nm ;s ¼ VMEPðsÞþ em nm;s þ em0 ðÞs;nm0 ¼ 0 m¼1 m0¼Frþ1 have multiple structures, e.g., gauche and trans 1,2-dichloro- (104) ethane. This section discusses how to include those features in VTST. Some of the material in this section applies to any reactant The effective potential of eqn (104) is based on the adiabatic or transition state that has multiple structures, but most of the approximation, which is a good approximation when the dynamical discussion is written in the framework of torsional multistructural bottleneck occurs prior to the region of large reaction-path anharmonicity, which has two components: multiple-structure curvature; large reaction-path curvature causes strong vibrational anharmonicity and torsional potential anharmonicity. The 129 nonadiabaticity, and in such a case, one should use the second multiple-structure and torsional potential anharmonicity effects 126 way to treat state-selected reactions, namely the diabatic one can be very important in the computation of partition functions, with the vibrationally diabatic potential given by especially at high temperatures. ÈÉ mX¼Fr mX0¼F If a reagent or a transition state being studied has one or r vib;d r vib;d Vd nm ;s ¼ VMEPðsÞþ em nm;s þ em0 ðÞs;nm0 ¼ 0 more torsional degrees of freedom (internal rotations), the harmonic m¼1 m0¼Frþ1 oscillator approximation fails because(a)thepotentialfunctionis (105) not quadratic and (b) torsions usually generate multiple structures, which is similar in form to the expression for the vibrationally each of which contributes to the partition function. Issue (a) is called

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. 137 g torsional potential anharmonicity, and issue (b) is called multiple adiabatic potential Va, but instead of using adiabatic frequencies, it uses diabatic frequencies which change smoothly along the structure anharmonicity. Multistructural torsion anharmonicity can be treated with the recently developed multistructural torsion (MS-T) reaction coordinate. When the adiabatic frequencies vm(s)show method, which has two versions: MS-T with uncoupled torsional an avoided crossing in the frequencies vm vs. s diagram (this is called the vibrational-mode correlation diagram, in which the potential method (MS-T(U) method), and MS-T with coupled curves corresponding to modes of the same symmetry do not torsional potential method (MS-T(C) method). We recommend cross), it is often a better approximation physically to assume that the second version of this method, which is based on a coupled 138 the modes preserve their character in passing through the avoided torsional potential. Including the MS-T treatment of anharmonicity in VTST will be called multistructural VTST crossing region. For instance, in the OH + H2 - H2O+H reaction,126 there are five bound vibrational modes along the (MS-VTST) or multipath VTST (MP-VTST), depending on how reaction coordinate s, in which the three modes with the lowest completely it is included. adiabatic vibrational frequencies (of which the modes are denoted A methyl group can usually be safely treated as nearly separable as m = 1, 2, 3) change smoothly along s, and the two highest- from other torsions, but more generally the torsional degrees of vibrational-frequency modes (which are denoted as m =4,5) freedom in a molecule are coupled; they interact with each other, and sometimes this coupling is strong. If we rotate each torsional with the same symmetry have an avoided crossing at s0 (which d bond to n different angular positions, and if we have t torsions is about 0.16 Å). The diabatic frequencies vm(s) are then defined as: (temporarily excluding methyl groups because rotating methyl ( groups does not generate new distinguishable structures; however vmðsÞðm¼1;2;3Þ we do consider all the torsions including methyl groups in the vd ðsÞ¼ m final MS-T partition function), we will generate nt initial structures. yðÞs0 s vmðsÞþyðÞss0 dm;4v5ðsÞþdm;5v4ðsÞ ðm¼4;5Þ However, after optimizing all these initial structures by some (106) electronic structure method (this step is called the conformational where y is the step function, and d is Kronecker delta. search),thedistinguishableconformersonefindswillnotbeequal

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t rot to n . All of the distinguishable conformers that are generated by where Qj is the rotational partition function of structure j, HO torsions are included in the computation of the MS-T partition Qj is the harmonic-oscillator (or quasi-harmonic-oscillator)

function, including non-superimposable mirror images. The vibrational partition function of structure j, Uj is the relative multiple structure effect can be very large when one is treating Born–Oppenheimer equilibrium energy of structure j with a long chain molecule. respect to the Born–Oppenheimer equilibrium energy of the One may propose to treat all the local minima (or in the case lowest-energy conformer (which is denoted as the global minimum

of transition state, all the first order saddle points on reaction or GM), t is the total number of torsions, and fj,Z is the torsional paths from reactants to products) with the harmonic oscillator correction factor for the coupled torsion Z of structure j.Ifwesetall formulas; and this is called the multistructural local harmonic the torsional correction factors in eqn (109) to unity, then this oscillator approximation (MS-LH).139 However, treating low- equation reduces to the single-structure harmonic-oscillator frequency modes that are torsions or have large contributions from rovibrational partition function (SS-HO) of structure j. torsions by the harmonic oscillator approximation can be very The rotational partition function for structure j is computed inaccurate. The harmonic potential is a good approximation to the using the classical rotor approximation, which is sufficient for real potential only around the equilibrium structure. The harmonic most purposes, and which yields: vibrational partition function has the high-temperature limit: pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2k T 3=2 Qrot B I rotI rotI rot k T j ¼ 2 j;1 j;2 j;3 (110) Qvib ¼ QB (107) srot;j h hni i where srot,j is the overall rotational symmetry number for and this is the vibrational partition function of a classical harmonic structure j (each conformer j has its own symmetry number), rot rot rot oscillator. and Ij,1 , Ij,2 , and Ij,3 are the three principal moment of inertia associated with the overall rotation of the rigid structure j. The At very low temperatures where kBT { hn, an internal rotation can be reasonably well treated with the harmonic- rotational symmetry number of conformer j is equal to the oscillator approximation at each minimum of the torsional order of the rotational subgroup for a polyatomic molecule (conformer j); the rotational symmetry number is 1 for a potential; at very high temperature where kBT is much larger than the internal rotational barrier W, and the internal rotation heteronuclear diatomic molecule and 2 for a homonuclear becomes a free rotor. At intermediate temperature where diatomic molecule. The torsional correction factors of all the coupled torsions hn o kBT o W, torsional degrees of freedom should be treated as hindered rotors. are treated together by Here we review the MS-T method based on a coupled Yt Yt qRCðCÞ f j;Z torsional potential (i.e., MS-T(C) method). By ‘‘coupled’’, we j;Z ¼ CHOðCÞ (111) mean that we are using consistently coupled torsional vibrational Z¼1 Z¼1 qj;Z

frequencies, coupled effective torsional barriers, and coupled RC(C) where qj,Z is the reference classical partition function of

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. internal moments of inertia; the kinetic couplings between the coupled torsional motion Z for structure j using the coupled torsional motions and the overall rotation of the molecule are effective torsional barrier, and q CHO(C) is the classical harmonic also fully treated. j,Z oscillator partition function using consistently coupled frequencies. The MS-T rotational–vibrational (which can be abbreviated The reference classical partition function of structure j is an as rovibrational) partition functioniscomputedasthesummation approximation to the fully coupled classical partition function of the single-structure torsional rovibrational partition function QFCC (where all t torsions and the overall rotation are coupled): (SS-T) of the unique structure j. In the equations used for the MS-T ð ð 2 ðtþ3Þ=2 2p=s1 2p=st ffiffiffiffiffiffiffiffiffiffiffi Yt partition function, a unique structure refers to a quantum 8p kBT p QFCC ¼ eV=kBT det S df mechanically distinguishable (both structurally and energetically) s 2ph2 Z rot 0 0 Z¼1 conformational structure, which is generated by a change in (112) vibrational coordinates, usually by a torsion; the unique structures are torsional conformers – not chemical isomers. And a distinguish- where srot is the overall rotational symmetry number, sZ is the able non-superimposable mirror image that can be generated torsional symmetry number for torsion Z (the symmetry number for via torsion also counts as a unique structure. The MS-T partition a torsion can be determined by treating the least symmetric of the function is: two rotating fragments as a fixed frame and counting the number of X MS-T SS-T identical structures obtained when the more symmetrical top is Q ¼ Qj (108) j rotated from 0 to 360 degrees with respect to this torsional bond and relaxed), V is the potential energy as function of torsional angles

where the SS-T rovibrational partition function for distinguish- fZ,anddet(S) is the determinant of the Kilpatrick and Pitzer able structure j is defined as moment of inertia matrix for overall and internal rotation,140 which is uniquely defined by the torsional coordinates (dihedral angles). U Yt QSS-T ¼ QrotQHO exp j f (109) The fully coupled classical partition function cannot be analytically j j j k T j;Z B Z¼1 integrated, and the computation of the whole potential energy V

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as function of torsional angles is generally unaffordable for systems space to be tessellated here is described by the dihedral angles f f ... f with multiple torsions, and therefore several approximations are 1, 2, , tSC, and the points correspond to structures. Each cell made in order to evaluate QFCC: (a) we approximate the determinant corresponds to a specific structure and consists of all torsional

of S(f1,...,ft) matrix in the vicinity of each conformational structure configurations closer to this structure than to any other structure

j by its value at a local minimum or the saddle point; (b) we use a when only the tSC strongly coupled degrees of freedom are congruent transformation to block diagonalize the S matrix so that considered. In Voronoi tessellation, all the distinguishable and it can be written as the product of the three principal moment of indistinguishable structures that can be interconverted via inertia for the overall rotation of a rigid structure, with the torsional torsions are included. The volume of the SC torsional subspace moment of inertial matrix D (which fully accounts for kinetic is then given by coupling of torsions to each other and to overall rotation); ð2pÞtSC f ...f OSC;tot ¼ (115) (c) we approximate the potential V( 1, , t) with the uncoupled tQSC local reference potential in the vicinity of structure j as st hi t¼1 W V ¼ U þ j;t 1 cos M f f (113) j;t j 2 j;t j;t j;t;eq where tSC is the number of torsions in the strongly coupled space, and st is the torsional symmetry number for torsion t. where, in the coupled MS-T calculation, Wj,t is the coupled Whenever we use Voronoi tessellation, we assume that the (C) effective torsional barrier Wj,t for torsion t, which is computed torsional subspace is so strongly coupled that we cannot assign by diagonalizing the torsional Wilson–Decius–Cross GF submatrix,46 Mj,tSC by considering each torsion separately; then we replace all and Mj,t is the local periodicity of torsion t for conformer j,whichis Mj,tSC for strongly coupled torsions of a given conformer j by a SC related to the number of structures (including both distinguishable single Mj equal to and indistinguishable ones) that can be generated by torsion t. 2p MSC  The local periodicity of an uncoupled torsional degree of j ¼ 1=t (116) SC SC freedom is the same in all unique conformers, and it is an Oj integer equal to the number of distinguishable conformers (including mirror images) that can be generated by this torsion, As we can see in eqn (111) and (114), it is the product of the local multiplied by the torsional symmetry number of this torsion. periodicities rather than the individual terms that are needed in the computation of the torsional correction. In the practical For instance, for a nearly-separable uncoupled –CH3 group, 141 there is only one distinguishable conformer that can be generated Voronoi algorithm, eqn (116) is computed by: by rotating this torsional bond, and the torsional symmetry 1=tSC SC Ntot number of methyl group is 3; and thus it has M =3.Instrongly Mj ¼ (117) Nj coupled systems, Mj,t is generally a conformer-dependent non- 141–143 integer, and it is assigned by using Voronoi tessellation. In this sampling algorithm, a total number of Ntot random As a result of putting all these elements together, our reference points in the torsional space are generated, and each point is

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. classical partition function for structure j, in which all the assigned to the structure that is the closest to it (in other words,

coupled torsions are treated together in product form, is each point is assigned to a particular Voronoi tile), and Nj is the ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !number of points that are assigned to structure j. Yt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Yt p Yt ðCÞ ðCÞ 2pk T W W SC qRCðCÞ ¼ det D B exp j;Z I j;Z Here we give some numerical examples of the Mj values for j;Z j hM 2k T 0 2k T Z¼1 t¼1 j;t Z¼1 B B the strongly coupled (SC) torsional degrees of freedom in some (114) systems; all the methyl groups in the following examples are SC excluded in the given Mj values, because they are treated as

where I0 is modified Bessel function. Notice that, for coupled nearly separable with M = 3. For 1-pentyl radical (for which the torsions, it is the product of local periodicities that appears conformational search is carried out by M06-2X/MG3S), the SC here, rather than a separate dependence on the local periodicity Mj values for the SC torsions in its eight distinguishable of each coupled torsion. conformers (which excludes 7 distinguishable mirror images) When torsions are strongly coupled together, it is some- are, from lowest-energy conformer to highest-energy conformer,

times impossible to assign integer Mj,t values for each torsion. respectively 2.67, 2.63, 2.73, 3.84, 3.82, 3.82, 3.36, and 3.39. The We divide the t torsions into two types: nearly separable (NS, for transition state structure of the hydrogen abstraction reaction

instance CH3 groups) and strongly coupled (SC). We use the from a methyl group of the tert-butanol by HO2 radical (the notation NS:SC = tNS:tSC to denote that tNS torsions are treated products are (CH3)2C(OH)CH2 radical and H2O2) possesses five

as nearly separable and tSC torsions are treated as strongly SC torsions and has 46 distinguishable conformers (i.e., 23 pairs coupled. In general, the strongly coupled coordinates may be of non-superimposable mirror images, which are found by SC further partitioned into two or more subspaces, with each M08-HX/MG3S); the Mj values range from 2.00 to 2.45, with subspace involving only those coordinates that are strongly a mean value of 2.17 and a standard deviation of 0.13. For coupled to each other. Each of the strongly coupled subspaces (S)-sec-butanol (for which the conformational search is carried out SC is treated by Voronoi tessellation separately. Voronoi tessellation by M08-HX/MG3S), the Mj values for the SC torsions in its nine divides a space into cells around a discrete set of points, and the distinguishable conformers (from the lowest-energy conformer to

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the highest-energy one) are respectively 3.01, 2.97, 3.01, 3.00, 3.00, 3.2 Multistructural variational transition state theory 2.95, 3.02, 3.05, and 3.00. In multistructural variational transition state theory146,150 (MS-VTST), CHO(C) The classical harmonic oscillator partition function q j,Z with we replace the single structure harmonic oscillator partition coupled frequencies is: functions in the expression of the VTST rate constant, by MS-T QF partition functions. And therefore, the multistructural and Yt t oj;m torsional anharmonicity is incorporated in the VTST theory. 1 h m¼1 CHOðCÞ ¼ F t (118) We now derive the MS-VTST rate constant for unimolecular kBT Q Z¼1 qj;Z o~ j;m reactions, for bimolecular reactions one can follow the similar m¼1 approach. For unimolecular reactions, the single structural

where F is the number of vibrational degrees of freedom, oj,m is variational transition state theory rate constant with multi- the vibrational frequency of normal mode m, and o~ j,m are the dimensional tunneling can be written as non-torsional vibrational frequencies; these frequencies are GT GT-SSHO kBT Q Q ðÞs ¼ s computed by diagonalizing respectively the full Wilson– kSS-VTST=TðTÞ¼kT ele eVsðÞ¼s =kBT h QR QR-SSHO Decius–Cross GF matrix and its non-torsional submatrix with ele non-redundant internal coordinates. (120) The final result is where QGT-SSHO and QR-SSHO are respectively the single-structural qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi QF (lowest-energy structure) harmonic-oscillator rovibrational partition ! ! Yt t=2 det Dj oj;m Yt ðCÞ ðCÞ function for the generalized transition state and for the reactant. 2p m¼1 Wj;Z Wj;Z fj;Z ¼ exp I0 k T Qt FQt 2k T 2k T The location of the variational transition state is denoted as s*; Z¼1 B Z¼1 B B Mj;t o~ j;m V(s = s*) is the potential energy of the variational transition t¼1 m¼1 state. We can re-write this expression in terms of the recrossing (119) transmission coefficient and the conventional (single-structural) In general, for a system with multiple coupled torsional TST rate constant degrees of freedom, the accurate partition function is unaffordable z z-SSHO SS-VTST=T T VTSTkBT QeleQ Vz k T to compute. Nevertheless, for two-dimensional systems, i.e.,mole- k ðTÞ¼k G e B h QR QR-SSHO cules with two coupled torsional degrees of freedom, where the ele (121) highly accurate partition functions are available via the extended ¼ kTGVTSTkSS-TSTðTÞ two-dimensional torsion method144 (E2DT) which is a method that VTST is based on the solution of the torsional Schro¨dinger equation and where G is the VTST recrossing transmission coefficient that includes full coupling in both the kinetic and potential energy, (which could be computed from canonical, microcanonical, or the satisfactory accuracy of the MS-T method with coupled any other version of variational transition state theory) resulting torsional potential, i.e., MS-T(C) method, has been confirmed; from variational effects. In CVT, the recrossing transmission VTST Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. and it is concluded that ‘‘If the E2DT method is not affordable, coefficient is a special case of G and is computed as the MS-T(C) method is the best option to calculate rovibrational CVT QGT-SSHO s ¼ sCVT eVsðÞ¼s =kBT partition functions including torsional anharmonicity’’. GCVT ¼ (122) Qz-SSHO Vz k T To carry out MS-T calculations, one has to do an exhaustive e B conformational search at the first step, which might be unafford- where Q‡-SSHO is the single-structural harmonic-oscillator able for structures with a large number of torsional degrees of rovibrational partition function for the lowest-energy saddle freedom, even when using economical density functional methods. point. In order to take the MS-T effects into VTST calculation, Recently we have proposed a much less expensive method called we replace the single-structural transition state theory rate dual-level MS-T,145 which requires the user to carry out a con- constant with the multistructural TST (MS-TST) rate constant formational search with a chosen low-level theory (for instance, a in eqn (121). The multistructural TST rate constant is defined by semiempirical molecular orbital method) and then re-optimize replacing the single-structural partition functions with MS-T the 30% energetically lowest low-level structures at a higher level. partition functions Dual-level MS-T is able to combine the information from a low- z MS T level theory with that from a higher-level theory to obtain an k T Q Qz- - z kMS-TSTðTÞ¼ B ele eV kBT (123) approximate MS-T partition function that agrees with the full R R-MS-T h QeleQ high-level MS-T partition function within a factor 2 to 3; an even more encouraging feature is that we have shown that when one is which can be written in terms of the multistructural and MS-T using dual-level MS-T partition functions in computing rate con- torsional anharmonicity factor Ffwd of forward reaction: stants or thermodynamic functions (in which cases one usually MS-TST MS-T SS-TST k (T)=F fwd k (T) (124) needs the ratio of the partition functions rather than an individual partition function, and the errors of the dual-level method canceled F MS-TðTSÞ Qz-MS-T Qz-SSHO out to a large extent), very satisfactoryresults(ascomparedtousing F MS-T fwd ¼ MS-T ¼ R-MS-T R-SSHO (125) full high-level MS-T partition functions) can be obtained. F ðRÞ Q =Q

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The multistructural canonical variational transition state state conformer, MS-VTST does not require this one-to-one theory (MS-CVT) rate constant is defined as the following: mapping; and actually based on the local equilibrium assumption in transition state theory, one should not associate a given transition kMS-CVT/T(T)=kTGCVTkMS-TST(T)=FMS-TkSS-CVT/T(T) fwd state conformer with a unique path originating from a specific (126) reactant conformer. We view this advantage of MS-VTST as the where the single-structural CVT rate constant is correct formulation of transition state theory with multiple conformers. kSS-CVT/T(T)=kTGCVTkSS-TST(T) (127)

Notice that, in the MS-CVT theory presented here, we approximate 3.3 Multipath variational transition state theory the ratio of the MS-T partition function to the SS-HO partition MS-VTST only considers one single lowest-energy path of the function at the variational transition state as the ratio at the reaction. The multistructural effect of the saddle point (i.e., the conventional transition state; and the variational transmission multiple conformers of the saddle point) can lead to different coefficients are computed based on single-structural (quasi-) reaction paths for the same chemical reaction (with different harmonic-oscillator approximation. This is usually a very good barrier heights). A more complete approach would be to approximation because usually the variational transition states explicitly consider the contributions from all these different are very close to the saddle point (at generally accessible reaction paths (of the same reaction) to the total reaction rate temperatures, the CVT variational transition state is usually constant, and treat the variational effects and quantum tunneling not more than 0.1 to 0.2 Å away from the saddle point). Also, of each reaction path individually. Multi-path variational transition by adopting this approximation, we completely eliminate the state theory149,150 (MP-VTST) is a generalization based on need for computing the multiple-structural anharmonicity MS-VTST. (which requires optimizations and generalized normal mode We now derive the MP-VTST rate constant for a unimolecular analysis for all the possible generalized conformers of the non- reaction (for a bimolecular reaction the derivation follows a stationary points) along the reaction path, which is simply similar manner). For a unimolecular reaction, the MP-CVT rate unaffordable for large or even medium-sized transition state constant is the sum of the rate constants of all the reaction MS-T structures. For bimolecular reactions, the factor F fwd for forward paths, which includes all the contributions from different paths reaction is defined as: into the total reactive flux: z-MS-T z-SSHO MS-T Q Q z F ¼ (128) X Q Qz-SS-T z fwd R-MS-T R-SSHO R-MS-T R-SSHO MP-CVT=T kBT T CVT ele p V k T Q Q Q Q k k G e 1 B A A B B ¼ p p R R-MS-T (129) h p QeleQ MS-T For the reverse reaction, we can also define the Frev factor ‡ for the reverse reaction, in which one replaces the reactant where V1 is the classical barrier height of the lowest-energy ‡-SS-T partition function in the denominator with the product partition reaction path; Qp is the single-structural rovibrational torsional

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. function. We can also define the MS-T standard-state reaction partition function for the saddle point of path p,whichrepresents ,MS-T MS-T equilibrium constant K1 being the product of Frxn factor the contribution from the p-th saddle point structure to the total MS-T MS-T for the reaction (which is equal to Ffwd /Frev ), with the conformational rovibrational torsional partition function of the single-structural harmonic-oscillator standard-state reaction transition state: equilibrium constant K1,SS-HO (which can be derived from ;SS-HO Yt SS-HO standard Gibbs free energy of reaction DGrxn ); zSS-T z z-HO Qp ¼ Qrot;pQvib;p exp Up kBT fp;Z (130) and the MS-T standard Gibbs free energy of reaction Z¼1 ;MS-T ;MS-T DGrxn ¼RT ln K . The basic assumption of transition state theory is that the where Up is the relative electronic energy of the saddle point p conformational structures of the reactants in phase space are in with respect to the zero of energy set to be the electronic energy local equilibrium with each other and with those of the transition of the lowest-energy saddle point structure. Notice that the state that correspond to motion toward the product. MS-VTST is differences between the barrier heights of various paths applied under this assumption, and it assumes that the inter- have already been included in the Boltzmann factor of the conversion between the conformers of the reactant is much faster single-structural rovibrational torsional partition function of 121 than the chemical reaction itself. However, if the interconversion the saddle point of path p in eqn (130), and therefore in is comparable or even slower than the reaction being considered, as ‡ eqn (129) only the V1 of the lowest-energy path was explicitly possibly in the case where the torsional barriers are high as written out. compared to the reaction barrier, then these different conformers Eqn (129) can be further modified as follows should be considered as different reactants. It is also interesting to point out that, as compared to the generalized Winstein–Holness z Q z X MP-CVT=T kBT ele V kBT T CVT z-SS-T (GWH) equation,147,148 or other similar treatments (such as the k ¼ e 1 k G Q h QR QR-MS-T p p p so-called multi-conformer TST), which requires one to associate ele p a unique reactant conformer with each corresponding transition (131)

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and it is equal to The microcanonical variational transition state theory can also be generalized to include MS-T effects; this is multistructural P Qz-MS-T T CVT z-SS-T microcanonical VTST (MS-mVT). And this is done by kp Gp Qp SS HO zSSHO z z- - z MP-CVT=T p Q1 kBT QeleQ1 V k T k P e 1 B ¼ SS T R-MS-T R R SSHO Qz- - Q h Q Q - Nz;MS-T p ele kMS-mVTðEÞ¼GmVTðEÞ (136) p QR-SSHO hrR;MS-T (132) in which the number of states for the saddle point is computed ‡-SSHO where Q1 is the single-structural rovibrational partition from the integration of density of states, and the density function of the saddle point of the lowest-energy path. Notice of states is computed from the inverse Laplace transform of that if we included all the reaction paths, then the denominator the QMS-T. of the first term in eqn (132) is equal to the MS-T conforma- ‡ tional rovibrational partition function Q -MS-T. Finally, eqn (129) 3.4 Truncation of paths included in MP-VTST can be expressed in the following form Ideally, all the reaction paths generated by the multiple con- MP-CVT/T MS-T TST k (T)=hgiF fwd k1 (133) formational structures of the transition state structure should be included in the computation of MP-VTST. However, for even TST where k1 is the conventional (single-structural) TST rate constant a medium-sized chain molecule, the number of conformers of computed based on the lowest-energy path. The averaged the transition state structure is very large, and it is simply generalized transmission coefficient hgi is defined as unaffordable to perform VTST calculations on all of the paths. P T CVT z-SS-T In eqn (134), formally, the summation is from p = 1 to the total kp Gp Qp p g P number of distinguishable reaction paths P (including paths hi¼ z-SS-T (134) Qp that are mirror images), which is equal to the number N of p distinguishable structures of the transition state (TS). Practically, MS-T we have to limit the number of paths included in MP-VTST The multistructural and torsional anharmonicity factor Ffwd is . calculation, that is to say, in the practical MP-VTST calculation, z-MS-T z-SSHO Q Q1 P o N.IfP = N, then the denominator of eqn (134) is exactly F MS-T ¼ (135) MS-T fwd QR-MS-T=QR-SSHO equal to Q (TS); and if P o N, then we have introduced truncation errors in the computed MP-VTST rate constant. As we can see from eqn (133), if we only include the single A recent study152 compares the MP-VTST results including lowest-energy path, the MP-VTST rate constant reduces to the different numbers of paths to the MP-VTST with all the paths MS-VTST rate constant. included; the comparison is for the hydrogen abstraction In the above approach, the torsional anharmonicity is only reaction of tert-butanol. In total there are 46 reaction paths. It included for the saddle point and the reactant(s); the torsional was found that the variational transmission coefficient could be

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. anharmonicity at the variational transition states are approxi- very different from path to path; for the lowest energy path, the mated as its values at the saddle point. We can also treat the variational transmission coefficients at all temperatures are torsional anharmonicity as a function of reaction coordinate s close to unity, while for high energy paths, the variational along the MEP by replacing the single-structural HO rovibrational transmission coefficient decreases almost linearly as temperature partition functions QX-SSHO (X is saddle point, reactant, or variational increases. The tunneling transmission coefficients are also transition state) in the above equations with the single-structural path-dependent, and higher-energy paths usually have larger rovibrational partition function with torsional anharmonicity tunneling transmission coefficients. A simple hypothesis to QX-SS-T;151 the computation of variational transmission coefficient explain this trend is that taller barrier tend to be thinner is also based on QX-SS-T. For the generalized transition state, barriers; interestingly, this explanation does not support the QGT-SS-T(s) treats the torsional anharmonicity as a function of observations in this study, where we examines the widths of the G the reaction coordinate s. The location of the dividing surface Va potential curves for different paths in the energy range that (or equivalently speaking, the value of variational transmission has significant tunneling contributions, and we found that the CVT coefficient Gp ) will be different from the previous approach, barriers have very similar widths. Therefore we must attribute because the generalized-transition-state Gibbs free energy of the difference to something else, and we concluded that GT,0 activation DGact (T,s) is now evaluated with the torsional because the higher-energy paths can tunnel at energies farther anharmonicity; and hence the position of the generalized- below the barrier top, they are able to gain more from tunneling GT,0 transition-state dividing surface, where DGact (T,s) reaches contributions. As for the error introduced by truncating the the maximum, could be different, i.e., the locations of the number of paths, it was found that at lower temperatures (less variational transition states are possibly different from the ones than 300 K), one typically needs to include more paths (about that are determined based on QX-SSHO. As a further complete 30% of the total number of paths) in order to have an error treatment, in the full MP-VTST, we use QX-MS-T(s) to compute smaller than 20%, whereas at higher temperatures, including the variational transmission coefficient, which is computationally only a few paths in MP-VTST is good enough (as compared to prohibitive. MP-VTST with all paths being included).

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4. Loose transition states modes of reactants are assumed to be conserved in the association reaction. The transitional modes consist of all of the vibrational We can divide transition states into two classes. In one class, modes except for conserved modes and the overall rotational and the transition state is loose. A loose transition state is composed of translational modes. During the association, the transitional fragments that rotate with respect to one another freely or almost modes are converted from the free rotational modes and freely in more than one dimension (note that the existence of translational modes of the infinitely separated reactants to torsions, which are one-dimensional internal rotations, is not vibrational modes and overall rotation of the associated species. enough to make a transition state loose). Typical loose transition The high-pressure-limit rate constant given by VRC-VTST for states are found in bond fission reactions that do not have an association (R1 +R2 - P) is intrinsic barrier, i.e., where the potential energy is in mono- ðð 2 tonically uphill or monotonically downhill in the reverse geh s1s2 k T eE=kBT N E; J; s dEdJ ð Þ¼ 3=2 z ð Þ association. ð2pmkBTÞ s Q1Q2 E;J In the second class, the transition state is tight, which (137) means it is not loose as defined above. Reactions with barriers

(saddle points) usually have tight transition states. Furthermore, where ge is the ratio of the electronic partition function of if a saddle point is present but it is too low in energy (e.g., below transition state to the product of the electronic partition functions ‡ the reactants of the downhill direction), then the dynamical of reactants; the ratio of the rotational symmetry numbers s1s2/s bottleneck might be a tight transition state near the saddle is equal to 0.5 if the reactants are the same, and it is equal to 1

point, but it also might be a loose transition state leading into a if the reactants are different; Q1 and Q2 are the rotational well between the reactants and the saddle point. In fact, there partition functions excluding rotational symmetry number for are probably two dynamical bottlenecks. (One can also find the reactants; m is the reduced mass for the relative translational successive transition states, i.e., transition states in series, when motion of the associating reactants; N(E,J,s) is the number of there is no barrier between reactants and products.153)Ifoneof accessible states at energy E and total angular momentum J,at them has a much higher free energy of activation than the other, reaction coordinate s. then that one should be used for transition state theory. If the The meaning of the reaction coordinate s in VRC-VTST is two free energies of activation are comparable, one can use the different from the one in reaction-path VTST. In VRC-VTST, the unified statistical theory or the canonical unified statistical reaction coordinate is defined by pivot points. Pivot points are theory, as discussed in Section 4.2. used to orientate the two fragments in the VRC-VTST calcula- The implementation of VTST that we have reviewed so far is tion. The locations of the pivot points are critical to the final for tight transition states, and this version of VTST is called computed rate constant; they should be chosen to yield the reaction-path VTST (RP-VTST) because we search along a reaction variationally lowest rate constants at each of the temperatures path for variationally optimal dividing surfaces that are trans- that one considers, and this is generally done by trial and error. verse to a reaction path; the accessible coordinate space can be Fixing the location of pivot point at the center of mass of the

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. described in terms of nearly harmonic vibrations and torsions fragment or at the atomic nucleus of the dissociating/forming around the reaction path, and the reaction coordinate is usually bond, does not necessarily lead to variationally lowest rate the distance along a reaction path, typically the MEP. Reaction- constant. Optimal locations of pivot points can be temperature- path VTST is not usually valid for a loose transition state dependent. The reader is referred to the SI of our paper161 on " because the accessible coordinate has wide-amplitude motion 2CF2 C2F4 for details of the way we use pivot points to define a that cannot be well descried by torsions and nearly harmonic reaction coordinate. stretches and bends. It is still possible to use reaction-path VTST In the single faceted VRC-VTST, each reactant has one single

for barrierless reactions by computing a reaction path without pivot point, and we denote their coordinates as P1 and P2; the

starting from the saddle point; however, this strategy is expected reaction coordinate s is defined as s =|P1 P2|. Depends on the to give realistic results only (if at all) at fairly high temperatures location of the pivot points, s may or may not be (and generally where the transition state may tighten up. A more appropriate way it is not) the distance between the two centers of mass of the to treat loose transition states is with loose transition state theory,154 two reactants. In multi-faceted VRC-VTST, each reactant can which has now been well developed in a form called variable- have more than one pivot points. We denote the distance reaction-coordinate VTST155–159 (VRC-VTST), as discussed next. between one pivot point on reactant 1 and another pivot point

on reactant 2 as rij, where i is the index of pivot points on 4.1 Variable-reaction-coordinate variational transition state theory reactant 1, and j is the index of pivot points on reactant 2. The Next we review the VRC-VTST theory. The VRC-VTST methodology reaction coordinate s is defined as the minimal value of the is available in the Polyrate code42 and also in the earlier Variflex rij set: code.160 The codes were written completely independently. The s = min{rij} (138) present article discusses the way that the calculations are done in Polyrate. According to variational transition state theory, the pivot In VRC-VTST, the vibrational modes are classified as conserved points should be defined so that the integral in eqn (137) is modes and transitional modes. The frequencies of the vibrational minimized.

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The number of states N(E,J,s) is computed as 4.2 Canonical unified statistical theory and related extensions Here we consider the case of two or more transition states in N(E,J,s)=hNq(E,J,s)iO (139) series. This should not be confused with multiple transition where the bracket means the average over all possible orientations states in parallel, which are treated by multistructural variational of the two reactants and of the orientation of the vector s.We transition state theory, multipath variational transition state define the vector s as a vector connecting two pivot points, one in theory, or ensemble-averaged variational transition state theory. each fragment. The subscript q means a specific orientation of the We will limit our discussion here to two transition states in whole system. And this average is carried out by Monte Carlo series, but the generalization to three or more is straight- sampling, where the orientations of the system are randomly forward. When there are two transition state regions in series, sampled over the whole phase space corresponding to a given s. we may call them inner and outer transition state regions.162 The practical implementation of the algorithm is presented in The outer region is where the centrifugal barrier is located, and " 161 detail in the SI of our paper on 2CF2 C2F4. The number of in some cases it may be treated using only the long-range form states at a given orientation is computed as of the interaction potential, although in general – using the qffiffiffiffiffiffiffiffiffiffiffiffi Q2 Qvk language of association reactions – one must also consider the 2pI ðkÞ ka1 i transformation of long-range rotations to librations (which is a 2 2 8p k¼1 i¼1 NqðE; J; sÞ¼FJ ms name for loose vibrations) and eventually to tight vibrations as Gðv=2 1Þ Q3 pffiffiffiffiffiffiffiffiffiffiffi 2pIi;q the fragments approach. The inner region is where chemical (140) i¼1 interactions (valence forces) have overtaken the long-range *+ ! v=22 X3 2 forces. In some cases, the inner region provides the highest Ji max E Vq ; 0 free energy barrier at high temperatures, and the outer region 2I i¼1 i;q OJ does so at low temperatures. 22,163 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The canonical unified statistical theory (CUS) has been u u X2 Xvk . developed as an extension to canonical ensembles of Miller’s t ð12Þ ðkÞ ðkÞ ðkÞ 164,246 F ¼ 1 þ m n d ni Ii (141) unified statistical theory (US) for microcanonical ensembles, k¼1 i¼1 which is based on the branching analysis of Hirschfelder and Wigner.245 In this theory, local fluxes are calculated for dividing where the bracket denotes the averaging over all possible surfaces at both bottlenecks of the chemical process and also at a orientations of the vector of total angular momentum J while dividing surface between them. This provides the simplest way to its length is fixed. v = v1 + v2 + 3, where v1 and v2 are the number allow for a statistical branching probability for the reaction (k) of orientations sampled for reactant 1 and 2; Ii is the ith complex to re-dissociate to reactants. In the case of two transition moment of inertia of the kth (k = 1, 2) reactant, and Ii,q is the states in series, the CUS rate constant is: ith moment of inertia for the system as a whole at a given CUS CVT CUS orientation q. n(12) is the unit vector pointed from one pivot k (T)=k (T)G (T) (142)

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. point on reactant 2 to the one on reactant 1; d(k) is the vector where GCUS(T) is the CUS recrossing transmission coefficient, connecting the center of mass of the kth reactant to its pivot which plays the role of including additional recrossing corrections point; n(k) is the unit vector directed along the ith principal axis i on the CVT rate constants. The GCUS(T) factor is calculated as: of the kth reactant; and ka is number of monoatomic reactant, which is equal to 0 or 1. qmaxðTÞ GCUSðTÞ¼ As in reaction path VTST, the variational calculation can be qmax ðTÞþqCVT ðTÞqmaxðTÞqCVTðTÞ=qminðTÞ done in different ensembles. For VRC-VTST, the optimization (143) of the transition-state dividing surface can be done by canonical CVT variational transition state theory (CVT), microcanonical where q (T) is the rovibrational partition function for the variational transition state theory (mVT), or energy and total canonical variational transition state (the location of which is CVT angular momentum resolved microcanonical variational theory denoted as s , which corresponds to the highest maximum (E, J-mVT). In CVT-VRC-VTST, N(E,J,s) is calculated for all E and J point on the generalized-transition-state Gibbs free energy max at a given value of s, then the rate constant k(T,s) is minimized surface), q (T) is for the second highest maximum point on with respect to s at each T. In microcanonical mVT-VRC-VTST, the generalized-transition-state Gibbs free energy surface (the max min the integral over J in eqn (137) is computed first to obtain location of which is denoted as s ), and q is for the lowest CVT max N(E,s); and this is done by integrating eqn (140) over J and then local minimum point that is between s and s on the averaging Nq(E,s) via eqn (139). Then N(E,s) is minimized with generalized-transition-state Gibbs free energy surface. All these respect to s at each E, and this optimized N(E) is used for the three partition functions are computed based on the zero-of- integral over E to obtain the final rate constant. In E, J-mVT, energy being at the minimum of reactants’ potential well (i.e.,

N(E,J) is minimized with respect to s for each E and J, and the value of VMEP at s = N); notice that, there is a relation then the integral is carried out to obtain final rate constant; E, between the generalized-transition-state vibrational partition J-mVT yields the smallest rate constant as compared to the function QGT(T,s) that is based on the zero of energy being

former two. at the value of VMEP(s) of the potential on MEP, and the

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vibrational partition function qGT(T,s) that is based on the equilibrium distribution of states, and hence the rate constants former definition of zero of energy: for bimolecular reaction can be treated as pressure independent. However, in general we must consider pressure dependence qGT(T,s) = exp[V (s)/k T]QGT(T,s) (144) MEP B when either the reactant or the product (or both) is unimolecular, CVT max min i.e., association reactions, unimolecular dissociation, and Clearly, if s , s , and s are the same, as in the case which there is only one local maximum on the generalized- unimolecular isomerization. Variational transition state theory transition-state Gibbs free energy surface along the reaction provides the high-pressure-limit rate constants. However, in coordinate, the CUS recrossing transmission coefficient GCUS(T)=1. practical laboratory measurements, the pressure is seldom high The CUS theory has been tested for various systems against enough to measure the high-pressure limit, and measured rate accurate quantum dynamics rate constants or the best available constants can be significantly different from the ones computed experimentally measured rate constants.165–170,361 It does not for the high-pressure-limit. For thermal unimolecular reactions, necessarily get the recrossing right, but it does provide a smooth the rate falls off as pressure decreases, and this is called the connection between the regime where the inner bottleneck is falloff effect. It is a consequence of a non-equilibrium state dominant and the one where the outer bottleneck is dominant. distribution in the reactant. 34,178–182 The CUS approximation is valid at high enough pressure A master equation may be used to accurately treat where the statistical dissociation of the complex is promoted by non-equilibrium effects if the full matrix of state-selected reaction 34,181,182 collision-induced energy redistribution, and it is valid at low rates and state-to-state energy transfer rates is available. pressure if intramolecular vibrational relaxation is fast enough. This is essentially never the case, but a master question can still But when tunneling is important, it has the same issues with be used if one makes additional assumptions. For example, one total energy conservation as those discussed in relation to CVT may assume that the unimolecular state-specific rate constants in Section 2.3.4. do not depend on the full state specification but rather only on The CUS rate constant kCUS can also be expressed by using the total energy or on the total energy E and total angular the rate constant (or the reactive flux coefficient) of passing the momentum J. We will make the former assumption. Then the highest variational transition state kI, the second highest master equation becomes one-dimensional, and it may be 183,184 variational transition state kII, and the formation of the lowest written: ð local minimum complex kC, on the Gibbs free energy surface 1 d 0 0 0 along the MEP, as follows: piðE;tÞ¼oiðT;PÞ PiðE;E ;TÞpiðE ;tÞdE oiðT;PÞpiðE;tÞ dt 0 1 1 1 1 NXisom NXisom CUS ¼ I þ II C (145) k k k k þ kijðEÞpjðE;tÞ kjiðEÞpiðE;tÞ jai jai And when there exists multiple parallel branching reactions after the complex, then it is straightforward to generalize NXreact þ y ðtÞy ðtÞk ðEÞb ðE;tÞ eqn (145) to the following: nA nB in n n¼1 Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. 1 1 1 1 P NreactXþNprod CCUS ¼ I þ II C (146) k k ki k k E p E;t i nið Þ ið Þ n¼1 II where ki is the rate constant for the i-th parallel reaction after (147) the complex; here the transition state theory gives the overall reactive flux. And this is called the competitive canonical d d y ðtÞ¼ y ðtÞ unified statistical (CCUS) model.248 The CCUS model has been dt nA dt nB widely used in studying reaction process with competitive ð NXisom 1 reaction paths171–177,248 with the VTST theoretical framework, ¼ kniðEÞpiðE; tÞdE (148) that is with the statistical model. One especially interesting i¼1 0 application is the use of CCUS to explain the experimental KIEs ð NXisom 1 247 for the SN2 ion–molecule reactions, and for the competition ynAðtÞynBðtÞ kinðEÞbnðE; TÞdE 248 0 between E2 and SN2 reactions. A non-statistical extension of i¼1 CCUS model will be discussed in Section 6. where pi(E,t) is the population of isomer i with energy E at time 1 t, oi(T,P) is the collision frequency (which is in the unit of s ;it 5. Pressure-dependent reaction rate is the bimolecular collision rate constant times the concentration of constants bath gas M), P(E,E0,t) is the probability density of collisional energy 0 transfer from energy E to E, ynA(t)andynB(t) are concentrations of 5.1 Overview the nth arrangement-channel reactants with A and B distinguishing

For reactions with two reactants and more than one product, it between the two reactants, kij(E), kin(E), and kni(E) are the is usually assumed that collisional repopulation of states microcanonical rate constants for isomerization, association

depleted by reaction is fast enough to maintain the thermal and dissociation reactions, bn(E,T) is the Boltzmann distribution

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for bimolecular reactant channel n, and Nisom, Nreact, and Nprod However, in thermally activated unimolecular reactions, the are the numbers of isomers, bimolecular reactant channels, and low-pressure rate constants are smaller than the high-pressure bimolecular product channels, respectively. equilibrium rate constant because collisional energy transfer is Popular master equation solvers are available, such as too slow to repopulate the reacting states of the reactant (i.e.,to Mesmer,185 Multiwell,186 and RMG.187 The computations of form the energized unimolecular reactive states), and hence – microcanonical rate constants in these codes do not explicitly as already mentioned – the pressure effect for this kind of include variational effects, microcanonical multidimensional reaction is called ‘‘falloff’’. In both mechanisms, the micro- tunneling, or multistructural and coupled torsional potential canonical rate constants can be obtained via quantum RRK anharmonicity However, in some cases variational effects are (QRRK) theory189 or via the full microcanonical variational included effectively, for example, many Mesmer calculations transition state theory (sometimes called variational RRKM). Barker, have been performed for loose TSs with good success by first Westmoreland, Dean, and Bozzelli and their coworkers190–197 have computing the parameters for a generalized three-parameter carried out important pioneering works in the developments and Arrhenius expression by using Variflex, which includes variational applications of QRRK theory. In many approaches, QRRK theory has effects, and then using the inverse Laplace transform method for been replaced by RRKM theory (i.e., transition state theory) or full the generalized 3-parameter Arrhenius expression to compute variational transition state theory, which are more complete theories, k(E). The inverse Laplace transform method can be viewed as an but QRRK theory can significantly lower the computational effort as approximate way to include various dynamics effects if the compared to full microcanonical VTST with multidimensional computed canonical rate constants include those effects. tunneling and anharmonicity. The recently proposed system- Multiwell calculations have also been reported using variational specific quantum RRK theory78,198,199 (SS-QRRK) is able to effects; Multiwell includes a code (Ktools) for explicitly calculating incorporate variational effects, multidimensional tunneling, k(E,J) (the rate constants as a function of energy and total angular and various anharmonicity effects into the microcanonical rate momentum) for loose TSs (even those with more than one constants, without any additional cost except for the computations bottleneck); Ktools also computes canonical VTST rate constants of the high-pressure-limit rate constants. In conventional in two ways: based on the k(E,J)s and based on a canonical VTST RRKM theory for k(E), energy-resolved variational effects, approach (however, in Multiwell, the canonical variation is car- multi-dimensional tunneling and multistructural and coupled ried out based on computing a large number of trial TST rate torsional potential anharmonicity are not included; some of constants at a given temperature along the reaction coordinate, these dynamic effects can be directly incorporated into micro- and its generalized normal mode analysis is based on Cartesian canonical rate constants in RRKM theory only with considerable rather than curvilinear coordinates, which, as we discussed in effort. SS-QRRK can be viewed as an effective way of recovering Section 2.5, are not well suited for VTST and should be used with the above-mentioned effects in k(E) from the high-pressure-limit caution). Neither Mesmer nor Multiwell has the capability to canonical k(T) that do include those effects in the canonical automatically compute multistructural effects and the effects of ensemble; and we use it to compute the pressure-dependence so coupled internal rotors. that the computed k(T,p) can connect with the correct high-

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. However, even when adequate dynamical methods can be pressure-limit. employed, master equation computations add an extra layer of complexity, and it would require a significant effort to compute 5.2 Thermal activation multidimensional tunneling by summation over calculations The thermal activation mechanism for unimolecular reaction for a large number of vibrationally excited states for each local X - Pis conformational structure. In many cases we would like to use kactðE;TÞ approximate chemical models to predict the pressure dependence XðTÞþM ! XðEÞþM (149) without solving a master equation. kdeactðTÞ Pressure-dependent elementary reactions can be classified, kcon;XðEÞ based on their mechanism, as chemically activated and thermally XðEÞ ƒƒƒƒ! P (150) activated. In a chemical activation mechanism, a reactive adduct

is generated via a chemical reaction; and in a thermal activation where X is the reactant, M is bath gas (such as He, Ar, N2, etc.), P mechanism, the rovibrationally excited unimolecular states are is the product, and X* is the rovibrationally excited (energized) generated via thermal collisions between the reactant and a bath complex, which is generated by collisions with a thermal gas. In the chemical activation mechanism, because collisional ensemble of bath molecules M. (Of course M could be X, i.e., energy transfer to stabilize the energized adduct is slow at low reaction could be occurring in neat X, and the equations can pressure, the low-pressure rate constants of the formation of easily be modified to treat that special case, so we need not the stabilized adduct are lower than the high-pressure-limit, and discuss it explicitly.) The parenthetical arguments following the rate constants of the further isomerization/dissociation of the X and X* denote that X is thermalized at temperature T, and X* formed adduct are larger than the high-pressure equilibrium rate has internal energy E, which is greater than the threshold

constant. Experimentally, Rabinovitch and co-workers are among energy E0 for the reaction step. (M is also thermal at temperature the pioneers who carried out various measurements in investi- T, and the translational energy of X* is thermal at temperature T, gating the kinetics of chemically activated reaction systems.188 but these conditions are not explicit in the notation.) The rate

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constant for the thermal activation step is kact, and we will call the RRK theory, which is retained in QRRK and SS-QRRK, is that we

reverse step de-activation and label it kdeact,althoughinthisstep neglect rotational energy and assume the vibrational energy E is the decrease of energy in X* could still yield a molecule that is randomly distributed among all the modes (nevertheless, in activated enough to react, except in the limit of so called strong SS-QRRK the rotational contribution is implicitly included in collisions, in which each collision fully de-energizes X*. The rate k(E) because the canonical high-pressure-limit includes rotational K constant for the step in which reactant is converted to product is modes, and thus they are implicitly in A and E0, which are

called kcon,X (this could be either isomerization or dissociation). discussed below). For simplicity we treat X and X* as multi-mode The phenomenological unimolecular reaction rate constant is harmonic oscillators with all frequencies the same (i.e.,an defined as effective geometric mean frequency); this will not be as serious 1 d½X an approximation as it might seem because the resulting formula k ðTÞ¼ (151) uni ½X dt will be used only to make a kinetics-specific inverse Laplace transform of a thermal rate constants calculated without this In the classical Lindemann theory, all of the rate constants assumption. In calculations, we will use the geometric mean are considered to be energy-independent. If we use steady-state frequency of X, and we will call in n (in wavenumbers). With all approximation to treat X*, i.e., if we assume the internal energy distributed among harmonic oscillators of the same frequency, we may change the independent variable from dX½ ¼ kact½X½Mkdeact½½X Mkcon;X½¼X 0 (152) internal energy E to number of vibrational quanta n: dt % then we get n = E/hcv (157) k ½X½M where c is speed of light. Following a similar derivation to that ½X ¼ act (153) kdeact½Mþkcon;X in classical Lindemann theory, but employing n-dependent rate constants, the unimolecular rate constant is

If we assume ideal-gas behavior, [M] can be replaced by p/RT, Xþ1 ðp=RTÞkactðn; TÞkcon;XðnÞ where p is pressure. Then, since kuni[X] = kcon,X[X*], we have kuniðT; pÞ¼ (158) n¼m ðp=RTÞkdeactðTÞþkcon;Xðn; mÞ ðp=RTÞkactkcon;X kuniðT; pÞ¼ (154) ðp=RTÞkdeact þ kcon;X in which m is the number of quanta at the threshold energy E0, % In the high-pressure limit, the Lindemann unimolecular m = E0/hcv (159) rate constant becomes and the determination of threshold energy in SS-QRRK will be N discussed later. kuni = kactkcon,X/kdeact (155) The ratio of the thermal activation and de-activation rate and the unimolecular reaction is of pseudo-first order; whereas constants in eqn (149) is a mixed-ensemble equilibrium constant, 0 in the limit of very low pressure, the low-pressure-limit kuni is representing the thermal equilibrium of species X* in a micro- N Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. much smaller than kuni: canonical ensemble at energy E with species X in a thermal 0 ensemble with temperature T. In the QRRK model, this is the kuni =(p/RT)kact (156) fraction of molecules at temperature T that have n quanta of which depends linearly on the pressure, and the reaction is of excitation, which is given by200 second order. The falling off of the k rate constant predicted by uni k ðn; TÞ classical Lindemann theory starts too early, i.e.,thecomputed K ðn; TÞ¼ act act;X k ðTÞ pressure at which significant falloff effect begins is too high as deact  compared to experiment. nhcn hcn sðn þ s 1Þ! ¼ exp 1 exp For higher accuracy, the energy dependence of kcon,X should kBT kBT n!ðs 1Þ! be considered. One could use microcanonical VTST to calculate (160) the energy-dependent rate constants, just as RRKM theory uses microcanonical conventional TST energy-dependent rate constants. where s is the number of vibrational degrees of freedom of X* Alternatively one could calculate thermal rate constants and obtain (which is 3N 6 for a nonlinear molecule). We assume that kdeact the fixed-energy ones by an inverse Laplace transform as mentioned does not depend on E, i.e., does not depend on number of at the end of Section 2.1, but inverse Laplace transform is often vibrational quanta n, but does depend on T;thenwesolve unstable. What we will do instead is to use SS-QRRK as an effective eqn (160) for kact,whichyields  kinetics-specific inverse Laplace transform to convert canonical nhcn hcn sðn þ s 1Þ! kactðn; TÞ¼kdeactðTÞ exp 1 exp kcon,X rate constants to microcanonical ones; SS-QRRK is a physical kBT kBT n!ðs 1Þ! model for achieving this goal, but without explicitly carrying out the (161) inverse Laplace transform in its actual computation. In the thermal activation step, the thermally equilibrated In QRRK, a unimolecular reaction happens when a specific reactant X at temperature T collides with bath gas M to produce vibrational mode associated with the reaction coordinate possesses

X* with internal energy E, and a key assumption of classical threshold energy E0. Therefore the QRRK rate constant equals a

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frequency factor AK (the superscript stands for Kassel) times the With this energy of activation, we have a local Arrhenius fit fraction of molecules that have at least m quanta of excitation; for temperature T given by 181,201 this is given by N k (T)=A(T)exp[Ea(T)/RT] n!ðn m þ s 1Þ! k ðn; mÞ¼AK (162) con;X ðn mÞ!ðn þ s 1Þ! with A(T) obtained by N A(T)=k (T)exp[Ea(T)/RT] (167) Substituting eqn (159), (161) and (162) into (158) yields the K QRRK rate constant, but in order to finish the calculation we must Then Ea(T) and A(T) are used for E0 and A in the QRRK K specify A , E0,andkdeact.Thecomputationofkdeact will be discussed expression given by eqn (158), (159), (161) and (162). This yields K K in Section 5.4, and next we turn our attention to A and E0. SS-QRRK. Notice that the parameters E0 and A are constants Taking the high-pressure-limit of eqn (158), using eqn (159), in conventional QRRK, but they depend on temperature in and summing the series yields the following pure Arrhenius form: SS-QRRK, which is a major change. This procedure allows one

N K to incorporate the full apparatus of VTST (including variational kuni(T)=A exp(E0/kBT) (163) optimization of the transition state, tunneling, and anharmonicity) and this motivates the approach we proposed to obtain the in the high-pressure-limit rate constants and approximately trans- parameters needed in SS-QRRK computations. The first step is fer it to the microcanonical energy-dependent rate constants, to calculate the high-pressure-limit canonical-ensemble VTST without actually doing the microcanonical computations. We view rate constant (for example, an MP-CVT/SCT calculation that the subsequently obtained SS-QRRK microcanonical rate constants includes variational effects, multidimensional tunneling, var- as an effective approximation to the microcanonical variational ious anharmonicity). Next, we fit the canonical-ensemble high- transition state theory rate constants including variational pressure-limit rate constant by the four-parameter formula202 optimization, tunneling, and anharmonicity. 8 0  If one wants to do the microcanonical computations (which > n 0 > T E ðÞT þT0 >B exp endothermic reaction could be expensive in order to fully account for MS-T anharmonicity < 300 RT2 T 2 ðÞþ 0 and microcanonical multidimensional tunneling), the sum in k1ðTÞ¼ > 0  > T þT n E0ðÞT þT eqn (158) can be replaced by an integration, which yields :> 0 0 ð B exp 2 2 exothermic reaction 1 300 RTðÞþT0 ðp=RTÞKact;XðE; TÞkcon;XðEÞkdeactðTÞ kuniðT; pÞ¼ dE (164) E¼Ethr ðp=RTÞkdeactðTÞþkcon;XðEÞ

0 0 (168) where B, T0, E , and n are fitting constants. In combustion community, the widely used fitting formulas are either conventional in which Arrhenius formula or the three-parameter fitting formula in the n Ð rðEÞ expðE=RTÞ rðEÞ expðE=RTÞ form BT exp(E/RT), which assumes a linear dependence of Kact;XðE; TÞ¼ ¼ 1dE0rðE0Þ expðE0=RTÞ QXðTÞ activation energy on temperature; however, this linear dependence 0 Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. can be very inaccurate at low temperatures, where the tunneling (169) 150 non-linearly reduce the activation energy significantly, and also X where r(E) is MS-T rovibrational density of states of X, Q (T)is at high temperature. Our four-parameter formulas are more the MS-T rovibrational partition function of reactant X, Ethr is physically sound, and they permits a nonzero rate constant for the threshold energy. Notice that, in this rigorous microcanonical an exothermic reaction in the low-temperature limit; furthermore treatment where the microcanonical rate constants are computed they are more accurate in terms of the fitting error.202 by mVT theory (which can be viewed as a variational RRKM Based on our four-parameter fitting formula, the local Arrhenius theory), the threshold energy is the ZPE-included electronic activation energy (which may also be called the Tolman activation potential-energy barrier height; this is different from the way of energy)203 for temperatures near a given temperature T0 is obtained determining the threshold energy in SS-QRRK theory, which is a from the negative slope of the Arrhenius plot, temperature-dependent activation energy. Also, the rotational 1 0 dlnk contributions are explicitly treated in this formalism. The micro- EaðT Þ¼R (165) dð1=TÞ T¼T0 canonical rate constants in eqn (168) can be computed by MS-mVT with multidimensional tunneling. In the case of small-curvature which yields tunneling (SCT), the MS-mVT/SCT rate constant is E T 0 að Þ Nz;MS-T=SCT 8 kMS-mVT=SCT ðEÞ¼GmVTðEÞ (170) 0 04 03 2 02 hrX;MS-T > E T þ 2T0T T0 T > þ n0RT 0 endothermic reaction <> 02 2 2 ‡,MS-T/SCT ðÞT þ T0 where the cumulative reaction probability N is the ¼ SCT > 0 04 03 2 02 0 02 convolution of the SCT tunneling probability P (E) with the > E T þ 2T0T T0 T n RT > exothermic reaction : 2 þ 0 density of states of the saddle point, in which the summation T 02 T 2 T þ T0 ðÞþ 0 goes over all rovibrational states of all conformations of the (166) transition state, from vibrational ground-state and continues

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up in energy until it converges or until one reaches the dissociation falloff region, eqn (176) involves the additional assumption that energy of the transition state. This is computationally demanding energy transfer competes with energy relaxation in the same even for medium-sized systems. way for all product channels. But this is not necessarily the case Finally, let us consider use SS-QRRK for treating unimolecular in the fall-off region if high-energy states are depleted more dissociation with multiple parallel dissociation reactions. rapidly by reactions with lower-energy thresholds than they are

kact ðE;TÞ by reactions with high-energy thresholds. This phenomenon XðTÞþM ! XðEÞþM (171) has been discussed in conjunction with the treatment of k ðTÞ deact the unimolecular reactions of 2-methylhexyl radicals.205 8 1 Thus eqn (176) should be used with caution in the falloff region > kð Þ ðEÞ > con;X > ƒƒƒƒ! P if the parallel reactions have a large difference in threshold > 1 > energies. > > ð2Þ < kcon;XðEÞ ƒƒƒƒ! P2 5.3 Chemical activation X ðEÞ> (172) > > For bimolecular association reaction or a two-step reaction with > > a unimolecular intermediate, the chemical activation mechanisms > N > kð Þ ðEÞ > con;X are: : ƒƒƒƒ! P N (a) association reaction (Y + Z - YZ)

where the reactant X has multiple parallel dissociation reactions, kadd þM Y þ Z ! YZ ! YZ (177) k of which the products are respectively P1,P2,...,PN (notice krev deact that here the notation P for the i-th dissociation reaction i - - represents the product(s) of this dissociation reaction). The (b) two-step reaction (Y + Z YZ P)

overall phenomenological unimolecular reaction rate constant kadd þM Y þ Z ! YZ ! YZ (for the total dissociation rate constant of the reactant X) is krev kdeact (178) defined as: # kcon;YZ P 1 d½X koverallðTÞ¼ (173) uni ½X dt where YZ* is the activated adduct generated by reaction between Y and Z, and YZ is the stabilized product (it is the which, by using the condition [X] = [P ]+[P]+ +[P ], can be 1 2 N intermediate in the case of two-step reaction), P is the product re-written as the summation of the rate constants of all the of further reaction of the intermediate YZ*. In chemical parallel dissociation reactions activation, the rate constant kadd of the formation of YZ* is XN not pressure-dependent because it is formed via a bimolecular koverallðTÞ¼ kðiÞ ðTÞ (174) uni con;X chemical reaction, of which the value is just the high-pressure- i 1 ¼ N Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. limit temperature-dependent bimolecular reaction rate kadd(T). where The rate constant for the dissociation of YZ* back to the

1 dP½ reactants Y and Z is krev, which is treated as energy-dependent; kðiÞ ðTÞ¼ i (175) con;X ½X dt the rate constant for the formation of the product(s) from YZ* is kcon,YZ(E). And the SS-QRRK pressure-dependent rate constant for the Following a similar treatment as for thermally activated i-th dissociation reaction, which is a straightforward generalization unimolecular reactions, and defining kstab as the phenomeno- of eqn (158), is: logical bimolecular stabilization rate constant, Xþ1 ðiÞ ðiÞ ðp=RTÞkactðn; TÞkcon;XðnÞ 1 d½YZ k ðT; pÞ¼ (176) kstab ¼ (179) con;X PN ½Y½Z dt n¼mi ðiÞ ðp=RTÞkdeactðTÞþ kcon;XðÞn; mi i¼1 and krxn as the phenomenological bimolecular rate constant for

where mi is the threshold number of quanta for dissociation reaction (i) reaction i, which is determined by using eqn (159), and kcon,X(n) 1 d½P k ¼ (180) is determined by eqn (162). Eqn (176) is equivalent to eqn (A5) of rxn ½Y½Z dt Dollet et al.,204 who applied it to study fall-off in the cases such as

SiH4 - SiH3 +HorSiH2 +H2,althoughtheirsworkwasbased We applied steady-state approximation to the activated on conventional TST without including the dynamic effects we adduct YZ*. Then the pressure-dependent rate constants given consider in SS-QRRK. by SS-QRRK theory are: The addition of rate constants for parallel reactions, as in (a) for association reaction

eqn (174), is clearly valid for microcanonical rate constants and Xþ1 1 ðp=RTÞkdeact f ðEÞ for high-pressure-limit thermal rate constants for which kstab ¼ kaddðTÞ (181) ðp=RTÞkdeact þ krevðEÞ the energy distribution is thermal. But for application to the E0

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(b) for two-step reaction undergoing multiple further parallel dissociation reactions. The mechanism is as follows: Xþ1 1 ðp=RTÞkdeact f ðEÞ kstab ¼ kaddðTÞ (182) kadd þM ðp=RTÞkdeact þ krevðEÞþkcon;YZðEÞ ! E0 Y þ Z YZ ! YZ (189) krev kdeact and 8 1 > kð Þ ðEÞ > ƒƒƒƒcon;YZƒ Xþ1 > ! P1 1 kcon;YZðEÞ f ðEÞ > krxn ¼ kaddðTÞ (183) > ðp=RTÞkdeact þ krevðEÞþkcon;YZðEÞ > E0 > ð2Þ < kcon;YZðEÞ ƒƒƒƒƒ! P2 YZ ðEÞ> (190) where f (E) is the fraction of energized adduct (YZ*) at energy E, > > which is given by > > > N > kð Þ ðEÞ > con;YZ krevðEÞKact;YZðEÞ : ƒƒƒƒƒ f E ! PN ð Þ¼þ1P (184) krevðEÞKact;YZðEÞ E0 The phenomenological bimolecular rate constant for the formation of the product(s) P of parallel reaction i in eqn (190) and K is given by eqn (157) and (160), and E is the i act,YZ 0 is defined as: SS-QRRK threshold energy for YZ dissociating back to reactants 1 dP Y and Z, and krev and kcon,YZ are computed by eqn (162) with the ðiÞ ½i krxn ¼ (191) appropriate m (number of quanta at threshold energy) and AK ½Y½Z dt

values (for krev, these parameters are computed from the The SS-QRRK rate constant for the stabilization of the reaction of YZ dissociating to Y and Z; and for kcon,YZ, from intermediate YZ, which is an extension based on eqn (182), is: the reaction of YZ dissociation to P). Xþ1 Instead of using SS-QRRK to approximate microcanonical ðp=RTÞk f ðEÞ k ¼ k1 ðTÞ deact rate constants, the above equations can be converted to the stab add PN i E0 ð Þ form needed to use microcanonical VTST rate constants ðp=RTÞkdeact þ krevðEÞþ kcon;YZðEÞ i¼1 (including microcanonical quantum tunneling): (192) (a) association reaction

ð where E0 is the SS-QRRK threshold energy for YZ dissociating þ1 p=RT k g E dE 1 ð Þ deact ð Þ back to reactants Y and Z. As we have discussed at the end of kstab ¼ kaddðTÞ MS-mVT (185) Ethr ðp=RTÞkdeact þ krev ðEÞ Section 5.2, if the depletions of the high-energy states by the low-threshold and high-threshold reactions are comparable, (b) two-step reaction then the current treatment for parallel channels provides a Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. ð good approximation. And the pressure-dependent rate constant þ1 ðp=RTÞk gðEÞdE k k1 T deact (i) stab ¼ addð Þ MS-mVT MS-mVT krxn for the formation of the product(s) Pi is: Ethr ðp=RTÞkdeact þ krev ðEÞþkcon;YZ ðEÞ Xþ1 ðiÞ (186) i kcon;YZðEÞf ðEÞ kð Þ ¼ k1 ðTÞ rxn add PN i E0 ð Þ ðp=RTÞkdeact þ krevðEÞþ kcon;YZðEÞ ð MS-mVT i¼1 þ1 k ðEÞgðEÞdE k k1 T con;YZ (193) rxn ¼ addð Þ MS-mVT MS-mVT Ethr ðp=RTÞkdeact þ krev ðEÞþkcon;YZ ðEÞ (187) 5.4 Collisional deactivation rate constant where the energy distribution function In the strong collision approximation, the collision efficiency is unity, which means that, in the strong collision limit, each kMSmVTðEÞrMS-TðEÞ expðÞE=k T gðE޼Рrev YZ B (188) collision produces a Boltzmann distribution of the energized þ1 MSmVT MS-T krev ðEÞr ðEÞ expðÞE=kBT dE complex (i.e., of the activated adduct in a chemically activated Ethr YZ system, and the rovibrationally excited unimolecular states in

and in mVT, the threshold energy is denoted as Ethr, and it is a thermally activated system), which usually has a much ZPE-included electronic potential-energy barrier height, which lower average energy than the energy prior to collision (i.e., is different from the threshold energy used in SS-QRRK theory de-activation has occurred). Note that de-activation refers to (which is a temperature-dependent activation energy). reducing the total energy below the threshold for reaction. As for the thermal activation process of unimolecular However, in reality it may takes multiple energy transfer collisions dissociation with multiple parallel dissociation reactions, which to fully de-activate the energized complex. At lower pressure, where we discussed in Section 5.2, here we consider using SS-QRRK there is more time between collisions, the energy relaxation time for chemical activation processes with the intermediate YZ may be slower than the reaction time. Troe developed approximate

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way to treat single-well, single-reaction-channel systems by a trajectory calculations, as done in ref. 214–216 (although the way master-equation-based weak-collision model,206–209 which is to extract information from trajectory calculation is not completely derived from the full master equation with an assumption that unambiguous). Most applications use a value that has been used the collisional energy transfer probability has an exponential in the literature for similar systems; or this quantity is simply form; Troe’s collision model has been tested against various treated as a fitting parameter to reproduce the experimental results bath gases, and its performance is well validated.215 In Troe’s in a certain range of temperature and pressure, and then one

model, the bimolecular collisional rate constant kdeact (in units hopes that the so-obtained pressure-dependence model becomes of cm3 molecule1 s1) is computed as: predictive for other temperatures and pressures.

For the numerical value of FE , Troe studied a set of small k = b O *k (194) 0 deact c 2,2 HS molecules and concluded that their values are all close to 212 where kHS is the hard-sphere collisional rate constant, bc is the unity. Using eqn (197), we also confirmed these observations

collision efficiency (dimensionless), and O2,2* is the reduced for some cases, although, for larger sized molecules and at high

collisional integral, for which the following approximation is temperatures, FE0 can deviate from 1 or 1.15 by quite a large available:206,207 amount; in fact, for a large-sized molecule with lots of vibrational 8  degrees of freedom at high temperatures, F grows exponentially. 1 E0 > kBT kBT 161 > For instance, for C2F4 dissociating to two CF2, FE is computed <> 0:697 þ 0:5185 log10 2½3300 0 eAM eAM to be 1.01 at 300 K, 1.12 at 400 K, and 1.39 at 1000 K; for O 2;2 ¼ >  199 > 1 (SiH3)2SiHSiH isomerization to (SiH3)2SiSiH3 , it is 1.43 at :> kBT kBT 0:636 þ 0:567 log10 2½= 3300 300 K, 1.63 at 400 K, and 6.54 at 1000 K; and for H addition on eAM eAM toluene,78 it is 1.31 at 300 K, 1.47 at 400 K, 4.02 at 1000 K. (195) Therefore it is not justified to use a constant value close to unity, 189,190 where eA–M is the Lennard-Jones interaction parameter between as has been done in the literature. When the density of molecule A and M and is computed as the geometric average of states is obtained via our MS-T partition function (see Section 3.1)

eA–A and eM–M. The collision efficiency is computed as: and FE0 is computed by eqn (197), the multiple structural and torsional anharmonicity (and also, the vibrational anharmonicity b jhDEij c for high-frequency modes when a vibrational scale factor is 1=2 ¼ (196) 1 bc FE0 ðTÞkBT applied) is built into FE0, and therefore the previously proposed 212 where |hDEi| is the value of the average energy transferred empirical internal rotor correction and anharmonicity correction to F should not be used. during both energization and de-energization processes (this E0 The collision efficiency b is smaller than 1 and it is quantity is also denoted as |hDEiall|, in order to be distinguish- c able from the energy transferred only for de-energization significantly smaller than 1 at high temperatures for small 179,210,211 molecules; furthermore b decreases with increasing temperature. hDEidown, which is a positive number) and c Ð The computed value of bc clearly depends on the value of the þ1 E=kBT Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. E rðEÞe dE energy transfer parameter |hDEi |,andthisisoftenthelargest F T 0 all E0 ð Þ¼ E =k T (197) kBTrðÞE0 e 0 B source of uncertainty in falloff computations. Typically one finds that larger molecules have higher values of the energy transfer is the thermal population of internal energies of unimolecular parameter;217,218 and a larger value of the energy transfer states above the threshold energy of the reactant normalized by parameter leads to a larger value of collision efficiency (i.e.,a a density of states factor at the threshold energy.212 If only the stronger collision). As mentioned above, a common practice in hDEi parameter is available, then the collision efficiency down the literature is to use a previously used value of the energy could be computed as:179 transfer parameter for similar systems. In principle, |hDEi | all 2 should depend on energy or temperature,218–221 but it is often hDEidown bc ¼ (198) approximated as a constant; to introduce the temperature hDEidown þ FE0 ðTÞkBT dependence, an additional empirical parameter is then needed.

The FE0 factor can be computed directly from the density of Furthermore, even when an experimental value is available, the states (which is the inverse Laplace transform of the rovibrational experiment may correspond to a different energy or temperature partition function), and in this case, the multistructural torsional than the one that is needed for the kinetics calculation. Yet another

anharmonicity could be included in the factor FE0 by using complication that occurs in chemical activation mechanisms is that the MS-T partition function, which is reviewed in Section 3.1, in the available experiments do not usually identify the adduct. For the inverse Laplace transform; if the single structure quasi- example, in our calculations on addition of H atoms to toluene, we

harmonic oscillator model is a good approximation, FE0 can also used an experimental value for disappearance of toluene, whereas be computed by direct counting algorithms, or (with much what is really needed is an experimental value for producing the less computational cost) the empirical Whitten–Rabinovitch adduct of H with toluene. The reader is directed to the 213 222–228 approximation, which is based on the classical harmonic literature for examples of experimental values of |hDEiall|.

oscillator model. In principle, the value of the energy transfer A number of collision efficiencies bc for a variety of chemical

parameter |hDEiall| could be determined from theoretical systems that are obtained in the experimental work (in some

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cases, they are not obtained directly or solely based on experi- where a1, a2, a3, l1, l2, T1, and T2 are the fitting parameters, in

mental measurements; they are derived with some information which a1, a2, and a3 are unitless, and l1, l2, T1, and T2 are in K. from theoretical models) are collected in Tardy and Rabinovitch’s The final fitted pressure-dependent rate constant k(T,p)is 229 N review; for the collected collision efficiencies in their review, expressed in terms of the fitted k (T), k0(T), and p1/2(T) (and the authors emphasized that ‘‘much of the presentation is thus it does not require any additional fitting) by the following qualitative in nature’’ and they noted a ‘‘lack of quantification’’. interpolatory equation: Here we give some examples of numerical values in some of the p2 þ BðTÞdðTÞp recent SS-QRRK calculations for the collision efficiency b .ForH kðT; pÞ¼ k1ðTÞ (201) c p2 þ dðTÞ addition on toluene (Ar as bath gas, with the energy transfer 1 78 parameter |hDEiall| being 130 cm ), it is 0.2 at 300 K, 0.05 at where 800 K, and 0.003 at 1500 K. For the SO and OH association 2 k ðTÞ B 0 reaction in the atmosphere (assuming N2 as bath gas, with the ¼ 1 (202) 1 232 k ðTÞkBT energy transfer parameter |hDEiall|being74cm ), it is 0.21 at 250 K, 0.18 at 298 K, and 0.14 at 400 K. For C2F4 dissociating to and two CF2 (with Ar as bath gas, with the energy transfer parameter 2 1 p1=2ðTÞ |hDEiall| being 250 cm ),161 it is 0.4 at 300 K, 0.2 at 800 K, and d ¼ (203) 0.1 at 1500 K. 1 2Bp1=2ðTÞ

5.5 Fitting the pressure-dependent rate constant k(T,p) in which the Boltzmann constant kB in eqn (202) is 1.38 1022 bar cm3 molecule1 K1, p is pressure in bar, B is a Functional forms have previously been proposed for fitting the derived parameter in units of bar1, and d is in bar2. Eqn (201) pressure-dependent rate constant with respect to both the has the following three properties, which ensure the fitted rate 230,231 temperature and pressure, butrecentlywehavesuggested constants have the right high-pressure-limit, low-pressure- a simpler formula for this purpose, and it has been shown that this limit, and the transition pressures: fitting yields quite satisfactory results.232 The formula involves N making three one-dimensional fits as a function of temperature. k(T,p = N)=k (T) (204) Step 1: fit the computed high-pressure-limit rate constant k(T,p =0)=k0(T)[M] (205) kN(T) (for a bimolecular association reaction, the units are cm3 molecule1 s1; and for a unimolecular reaction, the units s1), and which has already been done by using eqn (164). N k[T,p = p1/2(T)] = k (T)/2 (206) Step 2: fit the low-pressure-limit rate constant k0(T), which is

defined as the value of k(T,p)/[M] in the limit of p going to zero. The pressure-dependent activation energy Ea(T,p) at a given

The low-pressure-limit rate constant is a pseudo-third-order pressure p = p0 can then be determined by numerical differ- 6 2 1 rate constant with unit being cm molecule s for a bimolecular entiation by employing the following formula: Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. association, and it is a pseudo-second-order rate constant with  @ ln kðT; pÞ unit being cm3 molecule1 s1 for a unimolecular reaction. The E ðÞ¼T; p RT 2 (207) a 0 @T fitting formula used in this step has the same functional form p¼p0 as the one used in step 1, i.e., In the high-pressure limit, the above equation would yield 8  > n results that are identical to the ones given by eqn (166) at T = T0. > T ETðÞþ T0 <> A exp 2 2 endothermicreaction 300 RTðÞþ T0 k0 ¼ >  > T þ T n ETðÞþ T 6. Applicability of transition state :> 0 0 A exp 2 2 exothermic reaction 300 RTðÞþ T0 theory (199) In this section, we discuss the applicability of transition state

where A, n, E and T0 are the four fitting parameters, T is theory. When we say transition state theory in such a context, temperature in K, and R is the ideal gas constant (in the unit we mean variational transition state theory. of 1.9872 103 kcal mol1 K1). Classical transition state theory without a transmission 19,41 Step 3: fit the transition pressures p1/2(T), which have the coefficient involves two main assumptions. The first assumption units of bar. The transition pressure is defined as the pressure of transition state theory is that the states (states in a quantal at which the rate constant is half of the high-pressure-limit. The treatment, phase points in a classical one) of the reactants are in

p1/2(T) could be fit with many possible formulas, and we give equilibrium (thermal equilibrium when we discuss temperature- one of the fitting formulas here: dependent rate constants as in a canonical ensemble; micro- canonical equilibrium when we discuss energy-dependent rate a 1 a log p bar a a 3 3 constants). The question arises of whether it is an additional 10 1=2 ¼ 1 þ 2 ðÞl T =T þ ðÞl T =T 1 þ e 1 1 1 þ e 2 2 assumption to treat the transition states by equilibrium statistical (200) mechanics. This question has been answered, and in classical

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mechanical language we have the theorem:43,44 ‘‘When products the case.7–10 The levels can be observed in the accurate quantum are absent the distribution in the transition region is identical to mechanical cumulative reaction probabilities; the levels have an equilibrium distribution except that states [phase points] lying widths related to the lifetimes of individual quantized transition on trajectories originating in the products are missing.’’ Thus it is states. The widths can also be related to tunneling, i.e., tunnel- not a separate assumption to assume that states of the transition ing through the barrier in a vibrationally adiabatic potential state have a Boltzmann distribution. energy curve corresponds to reacting at an energy lower than the The second assumption of transition state theory is that the center of the broadened vibrational level, and thus it accounts states crossing the transition state dividing surface in the for the broadening. It is beyond the scope of the present paper direction toward the products are on trajectories that originated to go into this in more detail, but reviews are available.8,9 from the reactant region and that they will proceed to products In addition to the three basic assumptions of quasiclassical without recrossing the transition state.20 transition state theory (quasi-equilibrium, no recrossing, and These two basic assumptions are called the local-equilibrium or the quasiclassical postulate), there are additional approximations quasi-equilibrium assumption and the no-recrossing assumption. in the treatment of the transmission coefficient unless one uses We have stated the assumptions in classical mechanical language, eqn (98), but eqn (98) is not generally practical. but they can be rephrased quantum mechanically by using Confirmation of the quantitative accuracy of transition state Heisenberg-transformed projection operators in the flux calculations theory including tunneling transmission coefficients is compli- instead of the classical concept of where trajectories initiated.19,233 cated by the uncertainties in most available potential energy However, one cannot fully translate the classical justification of surfaces. Thus there are errors in the potential energy surface transition state theory into quantum mechanics234 (for example, and errors in the dynamics for the given surface, and these two one no longer has a practical variational bound on the rate constant) kinds of error can cancel or reinforce. One of the few systems because the uncertainty relations prevent us from simultaneously for which the potential energy surface is well enough known so

fixing the reaction coordinate at a definite value and specifying the as not to complicate tests of the dynamics is the H + H2 reaction, sign of the momentum in that coordinate. Thus, in order to pass and transition state theory with tunneling transmission coefficients from classical transition state theory to quasiclassical transition state that include corner cutting agree well with experiment for that theory, we make another assumption, and the assumption we case.106,239 It is encouraging that as the potential energy surfaces make is that we can replace the classical partition functions by have improved in accuracy in recent years, there are many other quantum mechanical ones in calculating the flux. We may call cases where VTST agrees well with experiment. Nevertheless the best this the quasiclassical postulate since this is the step that tests of the dynamics are still the tests against accurate quantum converts classical transition state theory to what we have called dynamics when the same potential energy surface is used because quasiclassical transition state theory. (In early papers we called then one can compare approximate and accurate dynamics on the it hybrid transition state theory, but now we recommend the same potential energy surface, and the difference cannot be due to quasiclassical notation. Note also that this is called semi- inaccuracy of the surface but must instead reflect a test of the theory classical transition state theory in the kinetic-isotope-effects used to calculate the approximate rate constants. Transition state

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. literature,235–237 but we prefer to save the word semiclassical theory has fared very well in such tests. For example, in a review for WKB-like treatments of tunneling. Thus when we say ‘‘semi- of 311 such rate constant comparisons for collinear and three- classical,’’ we refer to including tunneling, but when they say dimensional atom–diatom reactions in the temperature interval ‘‘semiclassical’’ in the kinetic-isotope-effects literature, they 200–2400 K, CVT/mOMT rate constants were found to agree with mean without tunneling, which is our quasiclassical.) accurate quantum mechanical rate constants within 22% on Although the quasiclassical postulate is far from rigorous, it average.6 Accurate quantal rate constants are also available for

can be justified in various ways. Wigner was the first to do this; H+CH4 - H2 +CH3 in the temperature range 200–1000 K; in he showed that such a substitution is justified to the lowest this case the average deviation is only 14%.240 order in h.83,238 Transition state theory is a dynamical theory because it Modern quantum dynamics provides an even better justifi- calculates the flux through a dividing surface. It is also a statistical cation for the quasiclassical postulate. A key aspect of the theory in that it provides a statistically averaged high-pressure- quasiclassical postulate is that systems cannot pass through limitrateconstantofanensemble,wherethehigh-pressurelimit the transition state with less than zero point energy except by is invoked to main the local equilibrium of reactant states. Strictly tunneling. If we take the quasiclassical postulate literally, it speaking, non-equilibrium effects are not included, although implies that systems actually pass through the transition state sometimes one extends the theory by including a non- not just in the quantized ground state (i.e., with zero point equilibrium factor in the transmission coefficient and still calls energy) but in all of the quantized vibrational levels (excited the result transition theory, so one must be careful to under- levels as well as the zero point level). Because the transition stand what an author means. When we calculate pressure- state is metastable (i.e., unlike stationary states in quantum dependent reactions, we still make the equilibrium assumption mechanics, it has a finite lifetime), the quantized levels are for microcanonical ensembles, but we no longer assume that broadened, i.e., they are quantum mechanical resonance states these microcanonical ensembles are weighted the same as in a with finite widths related to their lifetimes. Accurate quantum canonical ensemble (as we do for thermal equilibrium); so the mechanical calculations have confirmed that this is actually theory is still applied to equilibrium ensembles but they are not

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the same ones as for thermal rate constants (i.e., as for canonical unified statistical theory245,246) has been applied to study simi- rate constants). RRKM theory is transition state theory applied lar problems,247,248 and the computed kinetic isotopic effects to microcanonical ensembles for unimolecular reactions, so (KIEs) agree reasonably well with experimental observations for

discussions about whether microcanonical ensembles are gas-phase SN2 reactions. equilibrated during reactions apply to both transition state theory Consider, as an example, association reactions involving and RRKM theory. small molecules, where the system crosses a loose bottleneck. The local-equilibrium assumption is usually expected to If there are not enough low-frequency degrees of freedom to hold well for bimolecular reactions because the translational redistribute energy, the system can hit the repulsive wall energy, rotational energy, and vibrational energy of reactants behind the well, and bounce right back, and thus the capture are assumed to equilibrate on a faster time scale than the time probability is small. This fact has been known for a long scale for reaction. This is usually true, but there may be deviations time.26,249 For many years, there have been studies of whether for very fast reactants. Thus bimolecular reactions without an energy is randomized in unimolecular decay, as assumed in intermediate are usually treated as pressure independent. The RRK, RRKM, and TST theories. This is actually the central participation of an intermediate in a low-temperature reaction issue about the validity of RRKM theory since RRKM theory is involving tunneling, as discussed in Section 2.3.4, may, however, TST with the RRK assumption that reactants are equilibrated, cause pressure effects, and pressure effects are prominent in the which is also a fundamental assumption of TST. In some cases discussed in Section 5, namely (i) chemical activation, which cases,250 TST describes small-molecule association reasonably is a two step mechanism with the first step bimolecular well. and second being unimolecular and (ii) thermally activated Transition state theory does not predict product energy unimolecular reactions. Section 5 discusses the reaction of distributions,251 and it predicts chemical product distributions chemically activated intermediates and thermally activated only when they each have their own variational transition state intermediates as a function of total energy E (one can similarly dividing surface. Therefore it does not predict relative reaction treat their rates as functions of E and total angular momentum J). rates that are determined by bifurcation of a reaction path after a The rate at which the distribution of E becomes thermalized is single dynamical bottleneck that applies to both reactions.252–256 not a question that transition state theory answers or makes Similarly, if there is a high-free-energy bottleneck followed by two assumptions about; but the transition state quasi-equilibrium parallel lower ones, and the later dynamical bottlenecks are not assumption enters when we assume that for those species with separated from the earlier one by a thermalized intermediate, a given E or E and J, phase space is statistically occupied subject one should not try to predict relative reaction rates from the two only to these constraints. This follows if we assume fast lower ones by transition state theory – not because transition intramolecular vibrational-energy relaxation (‘‘fast IVR’’), and state theory is in error but because it does not address this issue. fast IVR may be considered part of the transition state theory When there is a bifurcation after the overall bottleneck, TST only quasi-equilibrium assumption in this context. Thus transition gives the sum of reaction rates along the two paths. One can state theory, and hence also RRKM theory, is not applicable to however, try to model such problems with beyond-transition-

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. very short-lived complexes that do not have time for full state-theory assumptions. For example, we combined VTST intramolecular energy redistribution241 or during the induction with non-statistical assumptions to study bifurcation in the

period of a long-lived complex, i.e., during that period before a BH3 + propene system, and we found that our computed steady-state phenomenological rate constant is established. selectivity agrees well with experimental observations.257 This Next we turn to the no-recrossing assumption. method is called the canonical competitive non-statistical Hase and co-workers discussed thepossiblebreakdownofTST (CCNM) model. based on trajectory computations for ion–molecule reactions.242,243 The dynamical problem that is modeled by CCNM is that of As Hase points out,244 ‘‘the RRKM model for the complexes may a reaction with an overall dynamical bottleneck (the maximum be invalid’’ and ‘‘recrossing of the central barrier may also be of the generalized transition state free energy along the MEP) important, violating the fundamental assumption of transition that is followed by a branching into two possible products. In state theory.’’ One issue is that classical trajectory computations such models, we have two types of reaction mechanisms.257 The could overestimate the recrossing because classical trajectory indirect mechanism, which is computed by VTST/MT, applies computations do not enforce zero point vibrational energy during to some fraction of the molecules in the ensemble that become the trajectory. Although conservation of zero point energy is not equilibrated in an intermediate complex (which is denoted as required during dynamics, we know – as discussed above – that ‘‘C’’), and the reaction of the equilibrated intermediate is cumulative reaction probabilities (and hence threshold energies) controlled by the second set of the dynamical bottlenecks (that are consistent with its approximate conservation. Therefore, it is is the two transition states, which are denoted by 2A and 2B, not clear to what extent these recrossing effects observed in leading to the two possible products); and the direct mechanism, trajectory computations are due to the classical nature of the which is modeled by non-statistical phase space theory,258,259 calculations. It has also been pointed though that the quantitative applies to the remaining portion of the ensemble members, which validity of transition-state theory may be quite different in keep a significant amount of energy in the reaction coordinate classical and quantum mechanics.26 Interestingly, a canonical degree of freedom until after passing the second bottleneck region unified statistical theory163 (which is an extension of the earlier (sometimes this is called a ‘‘hot’’ molecule reaction). The total rate

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262 constant ktot, which is the sum of the direct (kdir)andindirect(kind) Glowacki and co-workers presented an interesting study

components, is computed as: for the same BH3 + propene system by an alternative method. 1 1 1 1 They use a master equation approach with the microcanonical ¼ þ (208) rate constants needed in the model being computed by RRKM ktot k1 kC k2 theory, and they also reproduce the experimental selectivity. where k*2 =min(kC,k*2A + k*2B), and ki is the reactive flux coefficient However, the master equation (which is reviewed in Section 5.1) for a dividing surface at location i (i = *1, C, 2A, *2B; ‘‘*1’’ is for the was originally developed and used for studying the pressure first highest local maximum, ‘‘C’’ is at the location of the complex, and temperature-dependence of gas-phase reactions, and when ‘‘*2A’’ and ‘‘*2B’’ are for the two branching reaction A and B). The one tries to apply it in the liquid phase, there are huge direct component (kdir)iscomputedas: uncertainties in what to use for the average energy transfer parameter and the collision frequency. Glowacki et al. varied kdir = k*1k*2/kC (209) these parameters to reproduce the experimental selectivity, and and the total reaction rate constants for the two branches A their adjusted collisional frequencies are around 3–10 ps1, 1 and B are: and the adjusted hDEidown are 900–1100 cm . We may compare this to the gas phase where the usually used value of hDEi for CCNM RA=B k2A down kA ¼ kdir þ ðÞktot kdir (210) 1 1 þ RA=B k2A þ k2B small to medium sized molecules is around 100–500 cm . These authors consider their adjusted energy transfer para-

CCNM 1 k2B meters ‘‘are broadly consistent with gas phase studies’’, kB ¼ kdir þ ðÞktot kdir (211) 1 þ RA=B k2A þ k2B and their adjusted collisional frequencies are consistent with ‘‘experimental studies of solution phase intermolecular where RA/B is the branching ratio of the direct reactive components, vibrational relaxation show fast intermolecular energy transfer which is approximated as the equilibrium constant KA/B given by to the solvent.’’ 260,261 statistical phase space theory with additional non-statistical Another very important question is the accuracy of pressure- corrections: dependent VTST calculations with SS-QRRK theory. Rigorously Q speaking, the pressure-dependence calculation is beyond transition A ðÞVAVB =kBT KA=B ¼ e (212) QB state theory; transition state theory just provides the key ingredients (high-pressure-limit rate constant, and the effectively transformed where the rovibrational partition function Q (X = A or B, which X energy-resolved microcanonical SS-QRRK rate constants) that denote the final product of reaction A or reaction B) is "#are needed in the computation of pressure-dependent rate X  constant. tn =trelax en =kBT QX ¼ QX;? 1 e F e F (213) One important assumption about the non-statistical effects nF in such calculations is the model we adopted for treating the

in which QX,> is the rovibrational partition function that includes energy transfer. The collision efficiency we used (i.e., eqn (196)

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. all the bound-mode vibrations that are orthogonal to the reaction in Section 5) is derived from the solution of master equation coordinate s, but excluding the reaction coordinate s itself, which with an exponential-down model for vibrational energy transfer; becomes a bound-mode vibration in the product (and it is the probability of an energy transfer collision reducing the energy denoted as mode F, and its corresponding vibrational quantum from E to E0 is assumed to have an exponential functional form,

number is nF and vibrational energy is enF). The parameter trelax is with the larger the energy being transferred, the smaller the the characteristic relaxation time for intramolecular vibrational transfer probability. This has often been used satisfactorily, relaxation, which cannot be obtained with a statistical theory, especially if one takes the uncertainty of the energy transfer and it is treated as an empirical parameter in the CCNM model parameter itself into consideration; a test for such a model on 198 (although, some representative values, which are about 100 fs, are the simple CHF3 dissociation reaction (the dissociating 258 available in the literature). The tnF characterizes the duration products are HF and CF2) and the C2F4 dissociation reaction

of the collisional thermalization for a given vibrational state, and (which is a uphill from reactant to two CF2 on a PES without it is estimated in the model as the ratio of a characteristic saddle point) at combustion temperatures and over a wide

distance (sproduct–sTS1) in coordinates scaled to a reduced mass pressure range was consistent with the expected accuracy. to a characteristic velocity in the reaction coordinate for reaction Another deficiency in the current SS-QRRK treatment is that 257 coordinate quantum number nF. we did not explicitly consider the J (angular momentum) Although, we do not expect this proposed CCNM model to dependence of the microcanonical rate constant, since the be able to quantitatively treat any bifurcation systems without QRRK model is based on vibrational energy randomization; any further modification, our point is that the VTST theory but the necessity of introducing the J-dependence in such a itself (when applied with a high-level potential energy surface SS-QRRK treatment needs further investigation since: (a) the and appropriate treatment of anharmonicity and tunneling) appropriately adjusted energy transfer parameter (which is can be accurate for the total rate constant but it should not be essentially an empirical parameter in most common practices) applied to predict what happens after the system passes the is able to compensate such dependence in the final falloff overall dynamical bottleneck. behavior; (b) SS-QRRK is an efficient model to approximately

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capture all the physical effects contained in high-pressure-limit where d is the Dirac delta function, and W(R) is the PMF. thermal rate constants and transfer them into microcanonical The reason that W(R) is called the potential of mean force rate constants, and the resulting microcanonical rate constants is that, when we take the derivative of W(R) with respect to R, therefore implicitly also contain these physical effects and we have information about J.  @WðRÞ @UðR; rÞ ¼ (217) @R @R 7. VTST in condensed phases 7.1 Kinetics in liquid solutions where U(R,r) is the total potential energy of the entire system, The central concept for reactions in liquid solutions is the free and the kinetic energy has been separated from the total energy surface (FES), which is another name for the potential of Hamiltonian and has been integrated out over momenta. This mean force (PMF).180,263,264 The degrees of freedom for species result indicates that qW(R)/qR is the mean force acting on that are of explicit interest (such as reactants) are called the coordinate R that is averaged over all other coordinates. primary degrees of freedom (primary system), and the remaining In the implicit solvation models, the PMF (the free energy degrees of freedom (such as solvents) are called secondary surface) W(R)is degrees of freedom (secondary subsystem or environment). The W(R)=V(R)+DGS*(R) (218) PMF is a free energy function for the secondary subsystem, but an appropriately ensemble-averaged potential energy surface for the where R denotes the 3N 6 internal solute coordinate, V(R)is primary system. the gas-phase Born–Oppenheimer electronic potential energy In practice, the solvent degrees of freedom can be treated surface, and DGS*(R) is the fixed-concentration Gibbs free using an explicit solvation model, an implicit solvation model, energy of solvation; it can be interpreted as the work of or an explicit–implicit hybrid treatment.265,266 In the case of coupling a solute that is fixed in position to the equilibrated explicit solvation, the solvent molecules are treated at the full solvent at fixed temperature and pressure in an infinitely dilute atomistic level; the configuration space has to be statistically solution.277 The superscript ‘‘*’’ denotes a Gibbs free energy of sampled (via Monte Carlo or molecular dynamics) to provide an solvation that corresponds to the same concentration (e.g., ensemble average. In an implicit solvation model, the solute 1 mol L1) of the molecule both in the gas phase and in the molecules are embedded in the reaction field, which is the liquid-phase solution; that is an ideal gas at a gas-phase electrostatic field produced by the solvent when it is polarized concentration of 1 mol L1, dissolving as an ideal solution at by the solute; there are many reviews that cover implicit a liquid phase concentration of 1 mol L1. In a practical solvation models.267–272 In the explicit–implicit hybrid treatment calculations that employ an implicit solvation model (for 278 or the so-called mixed discrete-continuum models, one explicitly instance SMD ), DGS* has already been included in the SCF adds a few solvent molecules (first solvation shell) around the single-point energy, i.e., what we have calculated directly from a solute molecule and then embeds them in a reaction field due to self-consistent reaction field (SCRF) calculation is the free Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. the remaining environmental molecules;273 it has been found energy surface. that,274–276 in the case of strong interaction between solvent and The molar Gibbs free energy of a species i in liquid-phase solute (such as hydrogen bond, and molecular anions), adding a solution is:277 few explicit solvent molecules is able to yield better results as G = G * TS (219) compared to using purely implicit solvation treatment. i i lib,i

Let R denote the 3N 6 primary system coordinates, and P where Gi* is the chemical potential of a species whose center denote their conjugate momenta; the remaining degrees of of mass is constrained to a fixed position in the solution freedom of the whole system are denoted by r for internal (it is called as the pseudo-chemical potential, and it is coordinates and p for conjugate momenta. The free energy G(S) associated with the internal free energy of the species in of species S is solution including its coupling to the solvent); one may approximate ð Gi*inas: eGðSÞ=kBT ¼ C eHðR;P;r;pÞ=kBT dRdPdrdp (214) S Gi*=U0 + Gint (220) where the volume integral is carried out over the range of R where Gint is the internal thermal Gibbs free energy summed corresponding to species S; and C is a geometry-independent over all the configurations; U0 is the zero-point inclusive free constant. We can carry out this integral in two steps: first ð energy surface, given in the harmonic approximation by: 0 eWðRÞ=kBT ¼ C0 dðR R0ÞeHðR ;P;r;pÞ=kBT dR0dPdrdp (215) h XFvib S U ¼ W þ o (221) 0 2 m and then m¼1 ð eGðSÞ=kBT ¼ C00 eWðRÞ=kBT dR (216) where Fvib is the number of vibrational degrees of freedom, and S om is the vibrational frequency of mode m. The entropy term

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Slib,i in eqn (219) is the entropy of liberation of species i, which solvent motion, but this is not possible when one uses the is given by: PMF. We should keep in mind that specifying the transition "# state dividing surface is the same as specifying the reaction h 3 Slib;i ¼R ln ciNA pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (222) coordinate since the reaction coordinate is the degree of free- 2pm k T i B dom normal to this surface (the degree of freedom missing in this surface). If the best dividing surface has a reaction coordinate where R is the gas constant, ci is concentration of i in moles per that includes a component of solvent motion, then a reaction unit volume, NA is Avogadro’s constant. The last term of coordinate without solvent motion must correspond to a dividing eqn (219) is the free energy of liberation, which is formally surface that leads to a larger transition state theory rate. Therefore identical to the translational molar free energy in an ideal gas. the effect of including solvent motion in the reaction coordinate In order to obtain the thermodynamic standard-state Gibbs can be mimicked by adding a friction term that slows down the free energy (which is directly related to partition functions and rate. Such friction terms are present in Kramers theory,284 standard equilibrium constants), the standard-state Gibbs free 26,285–287 Grote–Hynes theory, and other stochastic dynamics energy of species B in solution G ðBÞ (i.e., the chemical soln theories.288 potential of B) should be computed as: There are three approaches for computing the reaction ! 281,289 GsolnðBÞ¼GgasðBÞþDGS ðBÞþDGS ðBÞ (223) rate constant for a liquid-phase reaction: (1) separable

equilibrium solvation (SES); (2) equilibrium solvation path where GgasðBÞ is the standard-state gas-phase Gibbs free energy (ESP); (3) non-equilibrium solvation (NES). Both SES and ESP of species B, which is usually computed based on gas-phase are based on the equilibrium solvation assumption, which computed frequencies of the gas-phase optimized geometries; assumes that the solvation is in equilibrium with solvent at but one can also use solution-phase computed frequencies of all points along the reaction path. In NES, on the other hand, 279 solution-phase optimized geometries. The superscript ‘‘1’’ in the solvent coordinates are explicitly required for defining the eqn (223) denotes the usual thermodynamic standard state, reaction path, and this is called dynamic or nonadiabatic which is defined as follows: (a) gas-phase molecules: all solvation. the molecules are ideal gases at a pressure of 1 bar, which In SES, one first finds the reaction path in gas-phase, and 1 corresponds to 0.0404 mol L at 298 K; (b) solutes in liquid- then adds the free energy of solvation (at a fixed solute 1 phase solution: 1 mol L ; (c) solvents: calculate from the geometry) to the gas-phase free energy for each point on the 1 density, e.g., 55.5 mol L for water at 298 K. The quantity gas-phase reaction path, using the gas-phase optimized geometries ! DGS ðBÞ in eqn (223) is the amount that one must add to the and gas-phase MEP. In ESP, the reactionpathisdeterminedby fixed-concentration Gibbs free energy of solvation to change the using the potential of mean force, which has been described in standard state for the left hand side of the solvation process eqn (218); the position of the variational transition state in ESP 1 (the gas phase) to a 1 bar standard state from the 1 mol L not only moves along the gas-phase reaction path but also standard state that would correspond to fixed concentration, perpendicular to it in the remaining 3N 7 degrees of freedom;

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. while keeping the standard state for the right-hand side of the thus the ESP method is more flexible and more realistic than 1 solvation process (the liquid-phase solution) at 1 mol L ; the the SES method. Although ESP is a more complete treatment as ! 1 value of DGS ðBÞ is +1.9 kcal mol for solutes at 298 K, and it compared to SES, the computationally less expensive SES often is +2.4 kcal mol1 for water (as a pure solvent) at 298 K. reasonably agrees with ESP281 (within a factor of 2, when The inclusion of tunneling in condensed-phase calculations the same solvation model269,273 is employed and quantum with explicit solvent does not require additional dynamical tunneling is considered). In NES, the specification of the reaction approximations although sampling raises new issues, as dis- path explicitly involves one or more solvent coordinate as well as cussed in Section 7.3. With implicit solvent though, one only the solute coordinates. has the PMF, which is a statistical quantity, whereas in principle Non-equilibrium solvation can be considered via a linear- one should carry out tunneling on the true potential energy response theory for solvent friction effects, which can be function. In other words, one should calculate the tunneling incorporated with VTST and tunneling;290,291 the key concept using the potential energy surface and average the dynamics, in such linear-response treatment, is the division of the system but with the PMF one has averaged the potential, and it is an into an explicit subspace and an implicit bath, and a solvent additional approximation to carry out tunneling on the averaged coordinate (which is chosen based on physical basis and it is potential. This additional approximation is called the canonical often preferable to use one or more collective solvent coordinates, mean shape (CMS) approximation.280 With this approximation, that is a coordinate representing the solvent electric polarization VTST calculations are readily interfaced with various solvation field) is introduced in the Hamiltonian using the approximation model calculations.21,23,273,281,282 of a single harmonic oscillator linearly coupled to the reaction Further complications could arise due to strong interactions coordinate. between solute and solvent, and the dynamic solvent effect becomes non-negligible. The key issue here is participation 7.2 Solid–vapor interface kinetics of the solvent in the reaction coordinate.263,283 With explicit Classical transition state theory has been applied to solids; we solvent the reaction coordinate can have a component of will limit our discussion to computing the migration and

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diffusion of atomic species on crystalline surfaces. Voter and VTST can be used, instead of conventional TST, to compute 292 Doll compared classical trajectory simulations results with the hopping rate khop, and therefore it can give a better classical TST results for diffusion constants. In trajectory prediction for diffusion constants. The practical computations simulations, the diffusion constants D are obtained by: can be carried out with an embedded-cluster model.296–300 In such model systems, there is only one non-metal atom. Dr2ðtÞ D ¼ lim (224) For instance, in the treatment of H atom diffusion on finite t!1 2dt copper(100) planes,297,298 the PES is approximated as the sum where d is the dimensionality of the system; it is 2 for most of finite range H–Cu and Cu–Cu pair potentials. The H atom surfaces and 1 for channeled surfaces like the (211) surface of and the near six Cu atoms are allowed to move in the calculation, 2 bcc crystals, and hDr (t)i is the mean-squared displacement of and other Cu atoms are fixed at experimental bulk geometry. the migrating atom, which can be obtained by averaging over The areas of these lattice planes and their number are taken 2 trajectories. In practice, D is obtained by plotting hDr (t)i large enough so that any atoms within the finite ranges of the with respect to time t, and fitting a line to the limiting slope. interaction potentials of the seven non-fixed atoms (i.e., H and The temperature dependence of the diffusion constant is the six near Cu atoms) are included. The reactant is H located in approximately described by an Arrhenius-type equation: one site (hollow formed by four Cu atoms) and the product is H

Ea/kBT in the next-door site; and the transition state structure is a D = D0e (225) bridge site between these two fourfold sites. Normal mode If the motions of the migrating atoms are assumed to be analysis is carried out along the minimum energy path, independent random hops between adjacent binding sites, and canonical variational transition state is determined by the diffusion constants can be computed by transition state maximizing the Gibbs free energy of activation; multidimensional theory as quantum mechanical tunneling can also be included in the computation of the hopping rate to study the diffusion of l2 D ¼ k (226) hydrogen on metal surface.301–304 2d hop For hydrogen diffusion on the Ni(100) surface, it has been where l is the distance (hop length) between adjacent experimentally observed305–307 that there exists a transition minimum-energy binding sites, and khop is the hopping rate. temperature, and for temperatures that are lower than the One-dimensional harmonic conventional classical transition transition temperature, the diffusion coefficients were inter- state theory yields preted experimentally as nearly temperature-independent due ‡ to tunneling. The quantitative definition of the transition k = n v eV /kBT (227) hop p 0 temperature is the temperature at which the Arrhenius plot where V‡ is the energy difference between saddle point has maximum curvature. This zero-temperature-limit, where (between two binding sites) and local binding site minimum; the rate coefficient becomes a constant, has also been observed 308 v0 is the harmonic frequency of the migrating atom moving in organic reactions. However, the transition temperature

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. along a pseudo-one-dimensional potential, in which all the was misinterpreted in the physics literature. Reanalysis300 of

other atoms are adiabatically relaxed; and np is the number of the low-temperature situation shows that the experimental ‡ binding sites accessible by a single hop. If we equate V to Ea ‘‘transition temperature’’ actually signifies the transition from

and equate eqn (225) and (226), the parameter D0 in the ground-state-dominated tunneling to thermally activated tunnel- Arrhenius-type equation for the diffusion constant is: ing. Thermally activated tunneling continues to dominate the rate 2 constant at temperatures far above the transition temperature. npv0l D0 ¼ (228) In low-temperature-limit, only the vibrational ground state is 2d populated, and the rate constant is constant because it simply Transition state theory can also be used to compute the corresponds to the rate of reaction by tunneling out of a single classical TST rate constant for transition from state A to state B state. Under the harmonic-oscillator approximation, the low- without invoking the harmonic approximation: temperature-limit diffusion coefficients are: rffiffiffiffiffiffiffiffiffiffiffiffi 1 2k T l2 k ¼ B hid½f ðRÞjrf j (229) DTðÞ¼scuRPG ER (231) A!B 2 pm A low 2d 0 where f (R) = 0 defines the dividing surface, and the ensemble where l is the lateral distance between the two minimum-energy

average hd[f (R)]|rf|iA is taken over the configuration space sites, d is the dimensionality of the systems (which is usually 2 belonging to state A (TST).293,294 Voter and Doll computed for surface diffusion), s is a symmetry factor that accounts for the ensemble average by the Monte Carlo method using the number of equivalent paths from reactants to products for the Metropolis algorithm as295 reaction (which is 4 for the (100) surface), c is the speed of light, uR is the vibrational frequency in wavenumber for the lowest- hd[f (R)]|rf|i = f /o (230) A B frequency hydrogenic vibration at the reactant well (which

where o is the width of the simulation box, and fB is the corresponds to the reaction coordinate over most of the reaction G R fraction of the Monte Carlo steps that lie inside the box. path away from the reactant well), and P (E0) is the quantum

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transmission probability at the harmonic vibrational ground- quantity is the generalized transition state classical free energy of state energy of the reactant well. At higher, but still low activation given by temperatures, where other states start to contribute, one can DG (T,z = z*) = DW (T,z = z*) G (T,z = z ) approximate the tunneling as occurring at a sequence of discrete GT,C GT,C C,F R (233) energies in the reactant well, rather than out of a translational 296 continuum. where GC,F(T,z = zR) is the classical free energy contribution of

the reaction coordinate at its value at the reactant state zR;it 7.3 Enzyme kinetics can be computed by calculating the free energy difference with Unlike in the solution phase, where the solute and solvent can and without this coordinate by not projecting and projecting usually be clearly separated, in enzymes the partition of the the reaction coordinate from the Hessian matrix respectively. entire system into primary and secondary is somewhat (Step 3) The last part of this stage is to replace classical arbitrary. In practical computations of enzyme kinetics, one vibrational partition functions by quantum mechanical vibrational often uses a combination of molecular mechanics (MM) and partition function, in order to include the quantization effects in quantum mechanics (QM), i.e., the QM/MM method. Enzyme– vibrational free energies. In practice, this quantization is only solvent systems can have a huge number of possible con- done for an N1-atom subsystem; in particular, only 3N1 7 figurations, and these configurations make significant con- modes are quantized at the transition state, and 3N1 6 modes tributions to the free energy of activation; the configurations are quantized at the reactant. The Gibbs free energy of are usually sampled by MC or MD techniques. Ensemble- activation obtained this way is called the single-reaction- averaged VTST309 (EA-VTST) with multidimensional tunneling coordinate (SRC) quasiclassical (QC) generalized-transition- was developed for modeling kinetics in enzymes, and we will state (GT) free energy of activation, which is denoted as SRC briefly review EA-VTST here; the readers who are interested in the DGGT,QC. The VTST rate constant for stage 1, which is denoted (1) detailed theory are directed to other previous publications with a as k , for a unimolecular enzymatic reaction (EP - ES, or higher concentration of focus on this topic.30,33,36,118,310–314 EP - E + P, where E, S, and P denote, respectively, the enzyme, EA-VTST calculations can be divided into three stages, and substrate, and product, and ES is a Michaelis adduct) is thus: . each of the stages corresponds to a different level of complete- k T kð1Þ ¼ B exp DGSRC RT (234) ness of the dynamical treatment. h GT;QC Stage 1. (Step 1) In this step we perform classical molecular dynamics (via, for instance, umbrella sampling) along a pre- Stage 2. In this stage, we calculate ensemble-averaged defined reaction coordinate z (which can also be called a recrossing and tunneling transmission coefficients in order distinguished reaction coordinate, with a good example being to obtain the ensemble-averaged quasiclassical variational the difference between internuclear distances of the forming transition state theory rate constant. For EA-VTST with tunnel- and breaking bonds, and a more sophisticated example being a ing (abbreviated as ‘‘T’’), the rate constant is: collective coordinate that is defined by the energy gap between Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. the valence bond states corresponding to the reactant and kEA-VTST/T = gk(1) (235) product states) to produce the classical potential of mean where k(1) is the VTST rate constant obtained in stage 1, and g is force W (z) (free energy profile, or – more technically – the C ensemble-averaged transmission coefficient generalized free energy of activation profile). Once the classical potential of mean force is available, the generalized-transition- 1 XM g ¼ G kT (236) state (GT) classical potential of mean force activation energy M i i i¼1 DWGT,C(T,z = z*) at temperature T is computed by: where M is the total number of ensemble members included in DWGT;CðÞ¼T; z ¼ z max WCðT; zÞWCðÞT; z ¼ zR (232) z the average, Gi is the recrossing transmission coefficient of T member i, and ki is the tunneling transmission coefficient of which is the difference between the maximum of classical member i. In principle, M is large enough to converge the potential of mean force and the reactant classical potential of summation; in practice one can use M of about 20. During stage mean force. Here ‘‘C’’ means classical. 2, the system evolves in a fixed field of its surroundings (these

(Step 2) In the first step, DWGT,C(T,z = z*) has already surroundings are called the secondary zone, i.e., solvent and included classical free energy contributions associated with part of the substrate and coenzyme); this is called the static all degrees of freedom that are orthogonal to the reaction secondary-zone approximation. This stage involves semiclassical coordinate; at a generalized transition state, the reaction reaction-path calculations for a subsystem (which is called coordinate itself does not contribute to the free energy (while the primary zone) within the secondary-zone being in frozen all the vibrational modes that are orthogonal to reaction configurations that are taken from a quasiclassical transition coordinate do contribute), but at the reactant it does. Thus, state ensemble in stage 1. Notice that, in eqn (236), averaging in the 2nd step, we add the missing contribution for the the transmission coefficient over M ensemble members is vibrational reaction-coordinate-mode free energies of the reactant equivalent to letting the secondary zone participate in the reaction

to the computed classical DWGT,C(T,z = z*); and the so-obtained coordinate.

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Stage 3. Usually, the static secondary-zone (also called study the kinetics of isomerization reactions between organic ‘‘the frozen bath’’) assumption works very well, and the non- radicals146,371–377 that are abundant in combustion flames. equilibrium solvation effects are negligible; it is sufficient Guan and co-workers have also applied MS-VTST to study to stop at stage 2. As a further improvement, the entropic hydrogen transfer between dimethyl ether and the methoxy contributions from the secondary-zone can be included in the radical,378 and combustion modeling of dimethyl ether.379 transmission coefficient. Then in the equilibrium secondary- Kinetics of the unimolecular reactions of ethoxyethylperoxy zone approximation (which can be viewed as an analogous to a radicals, which are main intermediates in the oxidation of liquid-phase kinetics calculation with the equilibrium solvation diethyl ether under engine-relevant conditions, have also been model), the free energy (potential of mean force) is further studied using VTST.380 Klippenstein and co-workers have corrected by the bath free energy (which is obtained by equilibrating combined VTST with the master equation approach to study the secondary-zone along the reaction path of stage 2, and the some of the key decomposition and association reactions in bath free energy as a function of reaction coordinate is obtained combustion.381–384 via free energy perturbation simulations), and the recrossing transmission coefficient is computed based on this corrected 8.2 Atmospheric chemistry free energy surface. Studying atmospherically important reactions is critical to our understanding of climate change and air pollution. VTST together with pressure-dependence theory is indispensible in 8. Selected applications providing accurate rate constants for global climate modeling. We have studied the chemistry of Criegee intermediates,385 In applications, one requires a potential energy surface. There whose importance in atmospheric science is widely appreciated, are two options. One option is to employ an analytic function, and understanding their fate is a prerequisite to modeling which may be obtained by a simple theory, such as the London climate-controlling atmospheric aerosol formation. 315 316–324 equation, or by a fit to electronic structure calculations. We have modeled the dissociation of C F with VRC-VTST 325,326 2 4 The second option is direct dynamics. In direct dynamics and SS-QRRK theory;161 this represents a class of reactions and ‘‘all required energies and forces for each geometry that is serves as an example for studying hydrofluorocarbon or Freon important for evaluating dynamical properties are obtained decomposition. The decomposition of hydrofluorocarbons or 71 directly from electronic structure calculations.’’ In modern Freon is related to the destruction of the ozone layer. work, one almost always uses direct dynamics. We have also studied the reaction of SO2 with OH in the atmosphere;232 this reaction plays an essential role in acid rain 8.1 Combustion chemistry and it is still attractive to experimentalists up-to-date,386 and VTST is a widely used tool for predicting the rate constants of our temperature- and pressure-dependent rate constant elementary reactions that are important in fuel combustion, computations help to resolve the discrepancies between various which is invaluable for selecting potential biofuels and fuel experimental studies. Cartoni et al. have studied the formation

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. + additives, for optimizing the combustion process, and for of HSO2 ion in the gas phase, which is of special interest designing next-generation combustion engines. It has been because of an interesting reverse temperature-dependent demonstrated that anharmonicity150,327–330 plays an important kinetics.387 In the study of the important roles played by role in controlling the VTST rate constants of hydrogen abstraction water and ammonia in promoting atmospheric reaction in

reactions. One interesting study that uses MS and MP-VTST in atmosphere: the rate constant of water-catalyzed H2SO4 +OH analyzing the effect of hydrogen-bonded transition states on reaction is about 1000 times larger than that of the same rate constants (for gas-phase hydrogen abstraction reactions) reaction without water as reported by Long et al.;388 it has also shows that,150 the conventional thinking of H-bonded TSs been found that ammonia can accelerate this reaction by increasing rate constants, which is solely based on the energetic enhancing quantum tunneling.389 Using VTST with multi- effects of the hydrogen bond that excludes the entropic effects, dimensional tunneling, the unusual temperature dependence

is not reliable; a hydrogen bond can reduce the entropy of the atmospherically important reaction OH + H2S reaction and thereby increase the free energy of the transition state has been understood, and predictions have been made for this (this is enthalpy–entropy compensation), and hence, a strong reaction at higher temperatures.390 Loerting and co-workers hydrogen bond interaction may lead to a slower reaction rate, have studied the proton transfer in small cyclic water clusters, and reliable conclusions about rate constants must be based on which are important in understanding water-cluster involved free energies of activation, not barrier heights or enthalpies of atmospheric reactions.391 activation. VTST is a powerful tool for filling the gap between limited Among the many reactions that are important for ignition experimental measurements and the need for rate constants are the hydrogen abstraction reactions from the fuel molecules in climate modeling at different altitudes, and in understand- 78,88,331–333 334–337 by various small radicals, such as H, HO2, ing previously proposed important atmospheric reaction OH,338–355 and O radicals.356–362 VTST can also supply rate mechanisms,392–399 and to provide rate constants that cover constants for newly proposed reaction mechanisms for oxidation the tropospheric and stratospheric temperature and pressure chemistry of fuels.363–370 VTST/MT has also been used to range. As an example for using VTST/MT in modeling complex

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reaction mechanisms in atmospheric science and geoscience, error introduced in the solvation model partially cancels the Molina and co-workers400 have studied the heterogeneous error in the less-accurate density functional. reactions for chlorine nitrate hydrolysis on water-ice surface When using VTST to compute rate constants in the liquid in polar stratospheric cloud; their study incorporates proposed phase, the accuracy of the final computed rate constants reaction mechanisms with reliable kinetics calculations, which depends on many factors, and the accuracy of the solvation lead to a good agreement of the estimated reaction probability model is one of the most critical factors. with the experientially reported one for such a complex system. 8.5 Enzyme reactions 8.3 Plasma chemistry Ensemble-averaged VTST with tunneling has been applied to Silicon-based chemically active nanodusty plasmas are an important study a number of enzyme reactions, including reactions 416 417 research subject in plasma physics and engineering.401–406 catalyzed by yeast enolase, triosephosphate isomerase, 418,419 309 Engineers need thermochemical properties and rate constants methylamine dehydrogenase, alcohol dehydrogenase, 420 421 to understand experimentally observed particle distributions, thermophilic alcohol dehydrogenase, haloalkane dehalogenase, 422,423 particle growth rates, and various transport phenomena in dihydrofolate reductase from E. coli, hyperthermophilic DHFR 424 425,426 plasma. We have used density functional theory and MS-T to from Thermotoga maritima, xylose isomerase, short-chain 427 428 study the thermodynamic properties for various branched acyl-CoA dehydrogenase, catechol o-methyltransferase, 429 430,431 432 silicon clusters,407,408 and we found that the contribution from glyoxalase I, coenzyme B12, HIV-1 protease, 433 multiple conformational structures and torsional anharmoni- 4-oxalocrotonate tautomerase enzyme, hydride transfer between + 434 city is significant; the conventional group-additivity based Anabaena Tyr303Ser FNRrd/FNRox and NADP /H, hydride transfer 435 estimations of thermodynamic properties were found to be from NADH to FMN in morphinone reductase. Some of this 30,33,36,310,311,314,436 unreliable. We have studied detailed chemical mechanisms work is reviewed elsewhere. for the growth of silicon hydride clusters,199,409 and we have One of the most important quantities in enzyme kinetics computed the rate constants for all elementary steps by using is the kinetic isotope effect (KIE), which is often used to VTST and multidimensional tunneling. We have also carried determine mechanisms (primarily by helping to identify the out a systematic benchmark study for testing various density slow step) and which often provides evidence for quantum functionals for their performance on elementary reactions mechanical tunneling. The ability for VTST to accurately in the polymerization of neutral silane with the silylene or computing such KIEs is strong evidence for the soundness of 31,36,419,425,437–445 silyl anion;199 this could serve as a reference for future the theory. silicon-based nanodusty plasma chemistry study. Oueslati One interesting finding is that, it is possible for the isotope 446,447 et al. have compared the VTST/MT computed rate constants D to tunnel more than H, and this can only be explained 239,448 for hydrogen abstraction from tetramethylsilane,410 which by multidimensional tunneling (as opposed to one- can be viewed as a prototypical species in silane-based dimensional tunneling) because if both isotopes tunnel along plasmas, and good agreements with experimental values was the same path with the same effective potential, H will always Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. achieved. tunnel more. However, in multidimensional tunneling, both the tunneling paths and the effective potentials depend on all 8.4 Organic chemistry in solution the masses in the system. VTST with state-of-the-art solvation models can be used to calculate reaction rates and elucidate reaction mechanisms in 9. Summary liquid-phase solutions. As a recent example, we have used SMD implicit solvation model to study the E1cb mechanism for In this review, we have discussed the fundamentals of elimination reaction in liquid phase.264 The paper reporting variational transition state theory, including multidimensional these results contains an explanation of free energy surfaces, tunneling and methods for incorporating multiple-structure and using these we are able to distinguish between the concerted and torsional-potential anharmonicity in VTST. We reviewed second-order mechanism for b eliminations and non-concerted variable-reaction-coordinate VTST as well as reaction-path VTST. mechanisms with discrete carbanion intermediates; distinguishing We have discussed the recent extension of VTST for the these experimentally is very difficult. convenient calculation of pressure-dependent rate constants Liquid-phase VTST calculations have also been carried out and falloff effects. We also briefly reviewed the use of VTST to investigate hydrogen migration in carbenes,411 hydrogen in condensed phases, including solid–gas interfaces, liquid addition on heterocyclic ring,412 and hydrogen donation in solutions, and enzyme kinetics. A number of recent applications reductive decarboxylation reactions.413 In the study of the of VTST with its various recent developments are briefly kinetics of hydrogen transfer reaction for phenolic compounds reviewed as well. in water,414 we have found that the M05 exchange–correlation Here, we summarize the major steps in kinetics studies. functional415 yields more accurate rate constants than does Please notice that for the sake of brevity, the following guide M08-HX, which is a density functional that is generally more does not include reaction systems that may require special accurate for gas-phase reactions; this is possibly because the theoretical treatment.

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Kinetics in a few words Doctoral Dissertation Fellowship (DDF) provided by University 1 Benchmark electronic structure methods (usually various density of Minnesota. functionals) against reaction energies and barrier heights for your system. A good summary of the accuracy and the computational cost of hundreds of commonly used electronic structure methods for kinetics is given in ref. 415. References 2 For reactions with barriers, use reaction-path variational transition state theory (RP-VTST) with multidimensional tunneling (MT); for 1 P. Waage and C. M. Guldberg, Forhandlinger: Videnskabs- barrierless reactions, use variable reaction coordinate variational transition state theory (VRC-VTST). Carry out direct dynamics rate Selskabet i Christiania, 1864, vol. 35 (see translation in constant calculations using one of these methods. J. Chem. Educ., 1986, 63, 1044). 2 S. A. Arrhenius, Z. Phys. Chem., 1889, 4, 96. Gas-phase reactions Liquid-phase reactions 3 S. Glasstone, K. Laidler and H. Eyring, The Theory of Rate 3-1 For RP-VTST, if needed, include For reactions in the condensed Processes, McGraw-Hill, New York, 1941. the effects of multistructural phase, employ the separable torsional anharmonicity (MS-T) in equilibrium solvation (SES) 4 J. C. Polanyi and A. H. Zewail, Acc. Chem. Res., 1995, 28, 119. the partition functions; this may VTST, the equilibrium solvation 5 R. Daudel, G. Leroy, D. Peeters and M. Sana, Quantum involve extensive conformational path (ESP) VTST, or the Chemistry, John Wiley & Sons, Chichester, 1983. searches. Compute MS-VTST/MT ensemble-average (EA) VTST. rate constants. If affordable, For enzyme reactions, employ 6 T. C. Allison and D. G. Truhlar, Testing the Accuracy of compute MP-VTST/MT rate ensemble-averaged VTST. Practical Semiclassical Methods: Variational Transition constants. State Theory with Optimized Multidimensional Tunneling, 3-2Forreactionswhoserateconstants are pressure-dependent, do in Modern Methods for Multidimensional Dynamics Compu- pressure-dependence calculations tations in Chemistry, ed. D. L. Thompson, World Scientific, either by the SS-QRRK method Singapore, 1998, p. 618. based on canonical VTST rate constants or by using a master 7 D. C. Chatfield, R. S. Friedman, D. G. Truhlar, B. C. Garrett equation solver with micro- and D. W. Schwenke, J. Am. Chem. Soc., 1991, 113, 486. canonical VTST rate constants. 8 D. C. Chatfield, R. S. Friedman, D. W. Schwenke and 4 Compare the computed rate constants with experiments and explain any differences. D. G. Truhlar, J. Phys. Chem., 1992, 96, 2414. 5 For the purpose of studying a complex reaction network, add the 9 D. C. Chatfield, R. S. Friedman, S. L. Mielke, G. C. Lynch, computed elementary reaction rate constants to the reaction model T. C. Allison, D. G. Truhlar and D. W. Schwenke, Computa- or update the older reaction kinetics data in the model. tional Spectroscopy of the Transition State, in Dynamics of Molecules and Chemical Reactions, ed. R. E. Wyatt and Variational transition state theory, combined with the power J. Z. H. Zhang, Marcel Dekker, New York, 1996, p. 323. of modern electronic structure methods, has been applied 10 D. C. Chatfield, R. S. Friedman, D. G. Truhlar and successfully to study the kinetics of a variety of chemical D. W. Schwenke, Faraday Discuss. Chem. Soc., 1991, 91, 289. systems. We believe that the recent extensions of VTST can be 11 R. S. Friedman and D. G. Truhlar, Chem. Phys. Lett., 1991,

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. useful tools for future research in combustion, atmospheric 183, 539. chemistry, environmental science, and other fields in which 12 R. S. Friedman, V. D. Hullinger and D. G. Truhlar, J. Phys. accurate kinetics data (which could be experimentally very hard Chem., 1995, 99, 3184. to measure) play an indispensible role in understanding the 13 R. S. Friedman and D. G. Truhlar, Barrier Resonances and detailed mechanisms of the chemical processes. VTST has also Chemical Reactivity, in Multiparticle Quantum Scattering been applied successfully to enzyme kinetics and other with Applications to Nuclear, Atomic, and Molecular Physics, condensed-phase reactions. With the recent progress of the ed. D. G. Truhlar and B. Simon, Springer, New York, 1997, development of more accurate and affordable electronic structure p. 243. methods for treating inherently multiconfigurational systems, 14 E. Wigner, J. Chem. Phys., 1937, 5, 720. VTST is readily applicable to study the detailed kinetics in 15 J. Horiuti, Bull. Chem. Soc. Jpn., 1938, 13, 210. modern organometallic and inorganic catalysis. 16 J. C. Keck, J. Chem. Phys., 1960, 32, 1035. 17 J. C. Keck, Adv. Chem. Phys., 1967, 13, 85. 18 B. C. Garrett and D. G. Truhlar, J. Chem. Phys., 1979, 70, 1593. Conflicts of interest 19 B. C. Garrett and D. G. Truhlar, J. Phys. Chem., 1979, 83, 1052. 20 D. G. Truhlar and B. C. Garrett, Acc. Chem. Res., 1980, 13,440. There are no conflicts to declare. 21 D. G. Truhlar and B. C. Garrett, Annu. Rev. Phys. Chem., 1984, 35, 159. 22 D. G. Truhlar, A. D. Isaacson and B. C. Garrett, Generalized Acknowledgements Transition State Theory, in Theory of Chemical Reaction Dynamics, ed. M. Baer, CRC Press, Boca Raton, 1985, p. 65. This work was supported in part by the U.S. Department of 23 B. C. Garrett and D. G. Truhlar, Transition State Theory, in Energy, Office of Science, Office of Basic Energy Sciences under Encyclopedia of Computational Chemistry, ed. P. V. R. Schleyer, Award Number DE-SC0015997. J. L. Bao also acknowledges a N.L.Allinger,T.Clark,J.Gasteiger,P.A.Kollmanand

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H. F. Schaefer III, John Wiley & Sons, Chichester, UK, 1998, Chemical Reaction Dynamics, ed. D. C. Clary and D. Reidel, vol. 5, p. 3094. Dordrecht, Holland, 1986, p. 285. 24 B. C. Garrett and D. G. Truhlar, Variational Transition State 41 S. C. Tucker and D. G. Truhlar, Dynamical Formulation of Theory, in Theory and Applications of Computational Chemistry: Transition State Theory: Variational Transition States and The First Forty Years,ed.C.E.Dykstra,G.Frenking,K.S.Kim Semiclassical Tunneling, in New Theoretical Concepts for Under- and G. E. Scuseria, Elsevier, Amsterdam, 2005, p. 67. standing Organic Reactions, ed. J. Bertran and I. G. Csizmadia, 25 A. Fernandez-Ramos, B. A. Ellingson, B. C. Garrett and Kluwer, Dordrecht, Netherlands, 1989, p. 291. D. G. Truhlar, Rev. Comput. Chem., 2007, 23, 125. 42 J. Zheng, J. L. Bao, R. Meana-Pan˜eda, S. Zhang, B. J. Lynch, 26 D. G. Truhlar, W. L. Hase and J. T. Hynes, J. Phys. Chem., J. C. Corchado, Y.-Y. Chuang, P. L. Fast, W.-P. Hu, Y.-P. Liu, 1983, 87, 2664. G. C. Lynch, K. A. Nguyen, C. F. Jackels, A. F. Ramos, 27 D. G. Truhlar, D.-h. Lu, S. C. Tucker, X. G. Zhao, A. Gonza`lez- B. A. Ellingson, V. S. Melissas, J. Villa`, I. Rossi, E. L. Coitino, Lafont,T.N.Truong,D.Maurice,Y.-P.LiuandG.C.Lynch, J. Pu, T. V. Albu, R. Steckler, B. C. Garrett, A. D. Isaacson ACS Symp. Ser., 1992, 502, 16. and D. G. Truhlar, Polyrate 2017 revision B, University of 28 D. G. Truhlar, B. C. Garrett and S. J. Klippenstein, J. Phys. Minnesota, Minneapolis, 2017, https://comp.chem.umn. Chem., 1996, 100, 12771. edu/polyrate/. 29 M. B. Hall, P. Margl, G. Na´ray-Szabo´, V. L. Schramm, D. G. 43 J. B. Anderson, J. Chem. Phys., 1973, 58, 4684. Truhlar, R. A. van Santen, A. Warshel and J. L. Whitten, 44 J. B. Anderson, Adv. Chem. Phys., 1995, 91, 381. ACS Symp. Ser., 1992, 721,2. 45 A. D. Isaacson and D. G. Truhlar, J. Chem. Phys., 1982, 76, 1380. 30 J. Gao and D. G. Truhlar, Annu. Rev. Phys. Chem., 2002, 53,467. 46 E. B. Wilson, Jr., J. C. Decius and P. C. Cross, Molecular 31 D. G. Truhlar, J. Gao, M. Garcia-Viloca, C. Alhambra, Vibrations, McGraw-Hill, New York, 1955. J.Corchado,M.L.SanchezandT.D.Poulsen,Int. J. Quantum 47 P. Pechukas and E. Pollak, J. Chem. Phys., 1979, 71, 2062. Chem., 2004, 100, 1136. 48 D. A. McQuarrie, Statistical Mechanics, Harper & Row, 32 M. Garcia-Viloca, J. Gao, M. Karplus and D. G. Truhlar, New York, 1973. Science, 2004, 303, 186. 49 D. G. Truhlar and A. Kuppermann, J. Am. Chem. Soc., 1971, 33 D. G. Truhlar, Variational Transition State Theory and 93, 1840. Multidimensional Tunneling for Simple and Complex 50 B. C. Garrett, D. G. Truhlar, R. S. Grev and A. W. Magnuson, Reactions in the Gas Phase, Solids, Liquids, and Enzymes, J. Phys. Chem., 1980, 84, 1730; erratum: 1983, 87, 4554. in Isotope Effects in Chemistry and Biology, ed. A. Kohen and 51 D. C. Chatfield, R. S. Friedman, D. G. Truhlar, B. C. Garrett H.-H. Limbach, Marcel Dekker, Inc., New York, 2006, and D. W. Schwenke, J. Am. Chem. Soc., 1991, 113, 486. p. 579. 52 D. C. Chatfield, R. S. Friedman, D. G. Truhlar and 34 A. Ferna´ndez-Ramos, J. A. Miller, S. J. Klippenstein and D. W. Schwenke, Faraday Discuss. Chem. Soc., 1991, 91, 289. D. G. Truhlar, Chem. Rev., 2006, 106, 4518. 53 R. S. Friedman and D. G. Truhlar, Chem. Phys. Lett., 1991, 35 D. G. Truhlar and B. C. Garrett, Variational Transition State 183, 539.

Published on 22 November 2017. Downloaded 19/12/2017 23:44:35. Theory, in The Treatment of Hydrogen Transfer Reactions, in 54 D. C. Chatfield, R. S. Friedman, G. C. Lynch, D. G. Truhlar HydrogenTransfer Reactions, ed. J. T. Hynes, J. P. Klinman, and D. W. Schwenke, J. Chem. Phys., 1993, 98, 342. H. H. Limbach and R. L. Schowen, Wiley-VCH, Weinheim, 55 R. S. Friedman, V. D. Hullinger and D. G. Truhlar, J. Phys. Germany, 2007, vol. 2, p. 833. Chem., 1995, 99, 3184. 36A.Dybala-Defratyka,P.Paneth and D. G. Truhlar, Quantum 56 R. S. Friedman and D. G. Truhlar, Barrier Resonances and Catalysis in Enzymes, in Quantum Tunneling in Enzyme- Chemical Reactivity, in Multiparticle Quantum Scattering Catalyzed Reactions,ed.R.K.AllemannandN.S.Scrutton, with Applications to Nuclear, Atomic, and Molecular Physics, RSC Publishing, Cambridge, UK, 2009, p. 36. ed. D. G. Truhlar and B. Simon, Springer, New York, 1997, 37 B. C. Garrett, D. G. Truhlar and R. S. Grev, Determination p. 243. of the Bottleneck Regions of Potential Energy Surfaces for 57 For some review articles, see for instances: (a)J.M.Bowman, Atom Transfer Reactions by Variational Transition State Acc. Chem. Res., 1986, 19, 202; (b)J.M.Bowman,S.Carterand Theory, in Potential Energy Surfaces and Dynamics Calculations, H. Xinchuan, Int. Rev. Phys. Chem., 2003, 22, 533; ed. D. G. Truhlar, Plenum Press, New York, 1981, p. 587. (c) J. M. Bowman, T. Carrington and H.-D. Meyer, Mol. 38 D. G. Truhlar, A. D. Isaacson, R. T. Skodje and B. C. Garrett, Phys., 2008, 106, 2145. J. Phys. Chem., 1982, 86, 2252; erratum: 1983, 87, 4554. 58 I. M. Alecu, J. Zheng, Y. Zhao and D. G. Truhlar, J. Chem. 39 M. M. Kreevoy and D. G. Truhlar, Transition State Theory, Theory Comput., 2010, 6, 2872. in Investigation of Rates and Mechanisms of Reactions, ed. 59 J. Bloino, M. Biczysko and V. Barone, J. Chem. Theory C. F. Bernasconi, John Wiley and Sons, New York, 4th edn, Comput., 2012, 8, 1015. 1986, Part 1, p. 13. 60 K. M. Kuhler, D. G. Truhlar and A. D. Isaacson, J. Chem. 40 D. G. Truhlar, F. B. Brown, R. Steckler and A. D. Isaacson, Phys., 1996, 104, 4664. The Representation and Use of Potential Energy Surfaces 61 H. H. Nielsen, Encycl. Phys., 1959, 37, 173. in the Wide Vicinity of a Reaction Path for Dynamics 62 I. M. Mills, Vibration-Rotation Structure in Asymmetric- Calculations on Polyatomic Reactions, in The Theory of and symmetric-Top Molecules, in Molecular Spectroscopy:

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