Predicting thermal rate constants of chemical reactions is a major goal in computational chemistry. Such information is required for simulations of complex reacting systems in many areas of science and technologies.
The calculation of rate constants requires a delicate balance between the accuracy of the dynamical theory and the efficiency in obtaining accurate potential energy information. In the extreme of rigorous dynamical treatment, accurate quantum dynamics calculations yield detailed state-to-state reactive cross sections or rate constants with full consideration of quantum effects. However, such calculations are currently limited to systems having very small number of atoms with the use of global analytical potential energy functions. At the other extreme, transition state theory (TST) has been practical for a wide range of chemical processes due to its simplicity. The basic model only requires potential energy information at the reactant(s) and transition state, since it treats many dynamical effects only approximately. Thus, such information can be obtained from accurate electronic structure calculations. There are, however, many reactions in which variational effects and/or tunneling are important and for these systems more accurate dynamical treatments are desirable. Variational transition state theory (CVT) provides a well-established methodology to bridge this gap. In this case, additional information on the potential energy surface along the Minimum Energy Path (MEP) connecting the reactants and products is also needed and obtained from electronic structure calculations.
2. Potential Energy Surface
For TST calculations, geometries, energies, and vibrational frequencies at the reactants and transition state are needed. For CVT, similar information along the MEP is also required.
As illustrated in the figure above, the reactants or products are local minima on the potential energy surface. Normal mode analyses at these points should have all real frequencies. Transition states on the other hand are local maxima, which have one or more imaginary frequencies. PES is a complicated 3N-6 dimention surface for a polyatomic N-atom system and thus characterizing these stationary points are important prior to any rate calculation. In particular, it is crucial to perform normal mode analysis of the transition state to confirm that it has only one imaginary frequency whose eigenvector shows the motion that connects the corresponding reactants and products. This can be done by animating the vibrational mode of the imaginary frequency. For some complex reactions, such animation sometime is not clear. In such cases, calculations for MEP (or IRC) as shown by the red curves in the above figure are recommended.
For the reaction from the reactant to Product A, both TST and CVT methods are applicable for all temperature range. For the reaction from the reactant to Product B, TST is also applicable for all temperature range whereas CVT is only applicable in the temperature range where the temperature-dependent variational transition states, s* as discussed below, are within the valley-ridge inflection points from the transition state.
3. Transition State Theory
Transition state theory or activated complex theory provides a simple formalism for obtaining thermal rate constant by mixing the important features of the potential energy surface with a statistical representation of the dynamics. In addition to the Born- Oppenheimer approximation, TST is based on three assumptions:
Classically there exists a surface in phase space that divides it into a reactant region and a product region. It is assumed that this dividing surface is located at the transition state, which is defined as the maximum value on the minimumenergy path (MEP) of the potential energy surface that connects the reactant(s) and product(s). Any trajectory passing through the dividing surface (or bottleneck) from the reactant side is assumed to eventually form products. This is often referred to as the nonrecrossing rule.
The reactant equilibrium is assumed to maintain a Boltzmann energy distribution.
Activated complexes are assumed to have Boltzmann energy distributions corresponding to the temperature of the reacting system. These activated complexes are defined as super-molecules having configurations located in the vicinity of the transition state.
Let us consider the reaction .
Applying the previous assumptions, we can express the quasi-equilibrium constant between the reactants and the activated complexes as
Here is where is Boltzmann's constant and T is the temperature. is the classical barrier height, i.e. the potential energy difference between the reactants and transition state. and represent the concentrations of their respective species while is the concentration of the activated complexes. is Avogadro number. V is the volume. The terms denote their respective total partition functions.
Due to the non-recrossing assumption, every activated complex that originates on the reactant side of the MEP eventually forms products; therefore the rate of product formation may be expressed as
where is the average velocity of such a complex moving towards products along the reaction coordinate through the region of width around the transition state. The term is given by
Here m is the reduced mass of the activated complex. Note that from the non-recrossing assumption, only forward velocities in this region are considered.
In the activated complex region, the force acting on the system is nearly zero. Thus, we may approximate the motion of the reaction coordinate in this region as a particle moving in a 1-dimensional box of length . The energy levels of a 1-dimensional box are given by
A factor of 1/2 was added to exclude the reverse motion since, we only consider activated complexes that originate from the reactant side.
Since the motion along the reaction coordinate has been treated separately, the partition function of the activated complex has one less degree of vibrational freedom and will be denoted as .
Putting Eqs. (1) - (5), and (7) together with yields
Usually rotational partition functions have an associated symmetry number to account for indistinguishable orientations. These symmetry numbers are combined into a single term called the reaction symmetry number and denoted as which represents the number of indistinguishable ways the reactants may approach the activated complex region. We will discuss forward and reverse reaction symmetry numbers in more detail in a later subsection. The final transition state rate equation is then given by
where the translational partition functions are expressed in per unit volume.
4. Variational Transition State Theory
Canonical variational transition state theory (CVT) is an extension of transition state theory. It minimizes the recrossing effects and provides a framework for a more accurate description of quantum tunneling effects to be considered. CVT minimizes the recrossing effects by effectively moving the dividing surface along the MEP between the reactants and products so as to minimize the rate. The reaction coordinate s is defined as the distance along the MEP with the origin located at the saddle point and is positive on the product side and negative on the reactant side. For a canonical ensemble at a given temperature T, the canonical variational theory (CVT) rate constant for a bimolecular reaction is given by
In these equations, is the generalized transition state theory rate constant where the dividing surface is located at s on the MEP. The term is the partition function of the generalized transition state at s with the motion along the reaction coordinate removed. is the classical potential energy (the Born-Oppenheimer potential energy) along the MEP with the zero of energy at the reactants. Note that if the generalized transition state is located at the saddle point (s=0), Eq. (11) reduces to that of conventional TST. CVT yields hybrid rate constants which treat motion along the reaction coordinate classically. Quantum effects in this degree of freedom are included by multiplying the CVT rate constant by a transmission coefficient . Thus, the final CVT rate constant is given by
For CVT rate calculations the partition functionsmust be calculated at the stationary points and at points along the MEP. We assume the partition functions are separable products of the translational, rotational, vibrational, and electronic partition functions.
For bimolecular reactions, the ratio of the translational partition functions may be simplified to yield the relative translational partition in per unit volume as
where is the reduced mass given by . For unimolecular reactions, the ratio of translational partitions function is unity, .
5.2 Rotational Partition Function
Since rotational energy levels are closely spaced, the rotational partition functions may be approximated by the classical form. The rotational partition function for a polyatomic molecule is then given by
where , , and are the moments of inertia for the molecular system. Rotational symmetry numbers are included in the reaction symmetry number discussed in a later section.
5.3 Vibrational Partition Function
In the present version of TheRate in CSE-Online, the vibrational partition functions are calculated quantum mechanically within the framework of the harmonic approximation. The harmonic oscillator partition function is given by:
where is the vibrational frequency in for mode i. The product is over all vibrational modes.
5.4 Electronic Partition Function
For the electronic partition function, an adiabatic potential energy surface is assumed. The electronic degeneracies along the MEP are assumed to be the same as at the transition state. The formula employed is
where is the energy of the electronic level relative to the ground state and is the corresponding degeneracy.
5.5 Forward and Reverse Reaction Symmetries
The reaction symmetry number which appeared in the rate expression is the ratio of the rotational symmetry numbers and is also known as the statistical factor. It describes the number of symmetry equivalent reaction paths. For bimolecular reactions, the forward reaction symmetry number is computed using the following formula,
where is the symmetry number associated with the point group to which the considered molecular configuration belongs. Similarly, the reverse symmetry number is given by
For optically active molecules there is an extra correction since each optically active isomer represents distinct but energetically equivalent states.Each optically active configuration should be given an extra factor of 1/2.
Table: Symmetry numbers (NSYM) for various point groups.
Point Group Symmetry
NSYM
Point Group Symmetry
NSYM
, ,
1
Atom
1
, ,
2
, ,
4
, ,
3
, ,
6
, ,
4
, ,
8
, ,
5
, ,
10
, ,
6
, ,
12
7
8
16
2
3
4
1
2
, ,
12
,
24
60
Example:
We use the reaction as an illustrative example. From Table above we find that the complex belongs to the point group with . The reactant belongs to the point group with . The reactant belongs to the point group with . The product atom belongs to the spherical orthogonal point group with 1. The product belongs to the point group with . The forward reaction symmetry is then
For some reactions the transition state belongs to a higher symmetry point group than other points along the MEP. Consider for example the reaction, the transition state has symmetry while points along the MEP only have symmetry. In this case, the lower symmetry point group should be used, even in TST calculations, since the activated complexes in general have lower symmetry with the transition state representing a special case.
6. Tunneling Methods
The current TheRate version includes four different semi-classical methods for calculating the transmission coefficients : namely the one-dimensional Wigner and Eckart methods and the multi-dimensional zero-curvature and centrifugal-dominant small-curvature methods. For convenience, we label them as W, ECK, ZCT and SCT, respectively. For TST rates, the Wigner and Eckart transmission probabilities are used. For CVT rates the ZCT and SCT transmission probabilities are utilized. The Eckart method is a special case of the ZCT calculation where the potential energy for tunneling is fitted by an Eckart function. Also, the ZCT method is a limiting case of the SCT method. These methods are described in detail below.
6.1 Wigner Tunneling Correction
The Wigner correction for tunneling assumes a parabolic potential for the nuclear motion near the transition state
This methodology requires no ab initio calculations at points other than reactants, products and saddle point. It uses the zero-curvature methodology developed in the previous section. The term is evaluated with an approximated adiabatic ground-state potential energy curve based on an Eckart function.
To do this, first we approximate the classical potential energy by an Eckart function whose parameters are calculated from the classical potential energies at the reactants , saddle point , and products and from the imaginary frequency as follows:
is the classical endoergicity, is the classical barrier height, is the imaginary frequency, and is the reduced mass. By convention, is set equal to zero at the reactants.
This fit yields the range parameter
is then approximated by another Eckart function which is assumed to have the same range parameter and location of the maximum as the classical approximation. This Eckart function goes through the zero-point corrected energies at the reactants, saddle point, and products. This yields
Here is the zero-point energy corrected barrier at the saddle point, relative to reactants, and is the sum of the zero-point energies of the two reactants.
For most reactions, the zero-point energy correction often lowers the barrier. In this case, the range parameter for the curve should be larger; that is, the width of the curve is wider than that of the curve. Thus, the approximation of using the same for and curves often overestimates the tunneling probability. We have found that, in many cases, this error is approximately compensated for by the ``corner cutting'' effects (see SCT methodology) that are not included in this methodology.
6.3 Small-Curvature Tunneling
First, we approximate the effective potential for tunneling to be
where is the vibrational state of mode orthogonal to the MEP. If all vibrational modes are in their ground state, is equivalent to the vibrationally adiabatic ground state potential curve , which is used in calculations of tunneling contributions to thermal rate constants. The transmission coefficient, , is then approximated as the ratio of the thermally averaged multidimensional semi-classical transmission probability, , to the thermally averaged classical transmission probability for scattering by the effective potential . If we denote the CVT transition state at the temperature T as , then the quasi-classical threshold energy can be denoted as . The equation for then becomes
Notice that the integral in the numerator of Eq. (38) involves E above (T) as well as tunneling energies below this. Thus, the semi-classical transmission probability P(E) accounts for both non-classical reflection at energies above the quasi-classical threshold and non-classical transmission, i.e. tunneling, at energies below that threshold. Because of the Boltzmann factor in Eq. (38), tunneling is by far the more important of these two quantum effects. The centrifugal-dominant small-curvature semi-classical approximation (SCT) is a generalization of the Marcus–Coltrin approximation in which the tunneling path is distorted from the MEP out to a concave-side vibrational turning point in the direction of the internal centrifugal force. The figure below illustrates the MEP in the red curve and SCT tunneling paths in the green and light blue curves.
Instead of explicitly defining this tunneling path, the centrifugal effect is included by replacing the reduced mass by an effective reduced mass, , which is used to evaluate the imaginary action integral and thereby tunneling probabilities. Note that in the mass-weighted Cartesian coordinate system, the reduced mass is set equal to 1 amu. The transmission probability at energy E is
Note that the ZCT results can be obtained by setting equal to in Eq. (40). The effect of the reaction-path curvature included in the effective reduced mass is explained below.
The small-curvature tunneling amplitude corresponds approximately to an implicit tunneling path that follows the line of concave-side vibrational turning point at a distance from the MEP in the direction of the reaction-path curvature vector. Let the distance along the small-curvature tunneling path be and the curvature at s be ; then it can be shown by analytical geometry that
However, to make the method generally applicable even when is greater than or equal to the radius of curvature of the reaction path, we include only the leading terms of Eq. (45) but not singularities by the approximated form below
where is the transpose of the generalized normal mode eigenvector of mode i, is the force constant matrix (Hessian matrix), and is the gradient. Finally, within the harmonic approximation, is given by
Main Web
As illustrated in the figure above, the reactants or products are local minima on the potential energy surface. Normal mode analyses at these points should have all real frequencies. Transition states on the other hand are local maxima, which have one or more imaginary frequencies. PES is a complicated 3N-6 dimention surface for a polyatomic N-atom system and thus characterizing these stationary points are important prior to any rate calculation. In particular, it is crucial to perform normal mode analysis of the transition state to confirm that it has only one imaginary frequency whose eigenvector shows the motion that connects the corresponding reactants and products. This can be done by animating the vibrational mode of the imaginary frequency. For some complex reactions, such animation sometime is not clear. In such cases, calculations for MEP (or IRC) as shown by the red curves in the above figure are recommended.
For the reaction from the reactant to Product A, both TST and CVT methods are applicable for all temperature range. For the reaction from the reactant to Product B, TST is also applicable for all temperature range whereas CVT is only applicable in the temperature range where the temperature-dependent variational transition states, s* as discussed below, are within the valley-ridge inflection points from the transition state.
![$A + B \rightarrow [C^{\ddagger}] \rightarrow {\it products}$](/twiki/pub/Main/KineticTheory/latexa109344863a84523bffeae7a6faec8a2.png)
.
Applying the previous assumptions, we can express the quasi-equilibrium constant between the reactants and the activated complexes as
![\begin{equation*} \[{[N_{C}^{\ddagger}] \over [A] [B]} = {{q_{\ddagger} \over N_{a} V} \over {q_{A} \over N_{a} V} {q_{B} \over N_{a} V}} \exp (-\beta \Delta V^{\ddagger})\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex846f70bab7a3fec3f6dcf28a35d42887.png)
is 
where 
is Boltzmann's constant and T is the temperature. 
is the classical barrier height, i.e. the potential energy difference between the reactants and transition state. ![$[A]$](/twiki/pub/Main/KineticTheory/latexb173b52774806d75a71c06ca8281b7bb.png)
and ![$[B]$](/twiki/pub/Main/KineticTheory/latex66184fb63a81a55ef471eaa5f7b1700b.png)
represent the concentrations of their respective species while ![$[N^{\ddagger}_{C}]$](/twiki/pub/Main/KineticTheory/latex5692cc0bc8f995f4543f3e7ed027b9be.png)
is the concentration of the activated complexes. 
is Avogadro number. V is the volume. The terms 
denote their respective total partition functions.
Due to the non-recrossing assumption, every activated complex that originates on the reactant side of the MEP eventually forms products; therefore the rate of product formation may be expressed as
![\begin{equation*} \[{r= {[N_{C}^{\ddagger}] \over \tau}\] \] \end{equation*}](/twiki/pub/Main/KineticTheory/latex4d75acb0f00f47f7d599b33632688571.png)
is the average lifetime of an activated complex originating from the reactant side. The term ![\begin{equation*} \[\tau = {\delta \over <\nu> } \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex5713e8c1df6fa6a7f78a3596c7ce9f19.png)
is the average velocity of such a complex moving towards products along the reaction coordinate through the region of width 
around the transition state. The term ![\begin{equation*} \[<\nu> = \int_{0}^{\infty} \nu g(\nu)d\nu = \sqrt{2 k_{b}T \over \pi m} \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex326740a67da01aff737900d2d842f3b5.png)
is the 1-way Maxwell-Boltzmann speed distribution
![\begin{equation*} \[g(\nu) = 2 \sqrt{m \over 2 \pi k_{b} T} \exp \left ({-m {\nu}^{2} \over 2 k_{b} T} \right )\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex9ffa3b2936df0b13598ead6faaefefe6.png)
![\begin{equation*} \[\varepsilon_{n} = {h^{2} n^{2} \over 8 m {\delta}^{2}}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latexb902b1c9e363c95c850474be7e86a0ac.png)
![\begin{equation*} \[q_{rc} = {1 \over 2} \sum_{n=1}^{\infty} \exp (-\beta \varepsilon_{n}) = {\left ( 2 \pi m k_{b} T \right )^{1/2} \over 2h} \delta \label{eq:1d_part_fct} \]\end{equation*}](/twiki/pub/Main/KineticTheory/latexf0f27fce907077a301e509fd6591e857.png)
.
Putting Eqs. (1) - (5), and (7) together with ![$r = k[A][B]$](/twiki/pub/Main/KineticTheory/latex944dc710a0e3fe045ccfa124fb9650ae.png)
yields
![\begin{eqnarray*} \[k & = & {[N_{C}^{\ddagger}] \over \tau [A] [B]} \\ & = & {<\nu> \over \delta} { \left ( q_{rc}^{f} q_{\ddagger}^{'} / N_{a} V \right ) \over \left (q_{A} / N_{a} V \right ) \left (q_{B} / N_{a} V \right )} \exp (-\beta \Delta V^{\ddagger}) \\ & = & {\sqrt{2 \pi m k_{b} T} \over N_{a} V} {\delta q_{\ddagger}^{'} \over h} {N_{a} V \over q_{A}} {N_{a} V \over q_{B}} {\sqrt{k_{b}T \over 2 \pi m} \over \delta} \exp (-\beta \Delta V^{\ddagger}) \]\end{eqnarray*}](/twiki/pub/Main/KineticTheory/latex33cda44489aadb5667e2bd5c66bdaa9e.png)
which represents the number of indistinguishable ways the reactants may approach the activated complex region. We will discuss forward and reverse reaction symmetry numbers in more detail in a later subsection. The final transition state rate equation is then given by
![\begin{equation*}\[ k = \sigma {k T \over h} {q_{\ddagger}^{'} \over N_{a} q_{A} q_{B}} \exp (-\beta \Delta V^{\ddagger}) \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex8e8095541c41baa6efa5af9faa97439e.png)
![\begin{equation*} \[k^{GT}(T,s)= \sigma{k_{b}T \over h}{q^{GT}_{\ddagger}(T,s) \over N_{a}q_{A} q_{B}} \exp({-\beta V_{MEP}(s)}).\]\end{equation*}](/twiki/pub/Main/KineticTheory/latex959118213e4417e724e1605c8cac1bde.png)
is the generalized transition state theory rate constant where the dividing surface is located at s on the MEP. The term 
is the partition function of the generalized transition state at s with the motion along the reaction coordinate removed. 
is the classical potential energy (the Born-Oppenheimer potential energy) along the MEP with the zero of energy at the reactants. Note that if the generalized transition state is located at the saddle point (s=0), Eq. (11) reduces to that of conventional TST. CVT yields hybrid rate constants which treat motion along the reaction coordinate classically. Quantum effects in this degree of freedom are included by multiplying the CVT rate constant by a transmission coefficient 
. Thus, the final CVT rate constant is given by
![\begin{equation*} \[k(T) = \kappa(T) k^{CVT}(T). \]\end{equation*}](/twiki/pub/Main/KineticTheory/latexc60ee4ea99b9e8a863b854c7adbff155.png)
![\begin{equation*}\[ q = q_{trans} q_{vib} q_{rot} q_{elect} \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex58af259c77285334256aff85b47f77fa.png)
![\begin{equation*} \[{q_{\ddagger,trans} \over q_{A,trans} q_{B,trans}} = q_{trans} = {2 \pi \mu k_{b} T \over h^{2}}^{3/2} \]\end{equation*}](/twiki/pub/Main/KineticTheory/latexb2c05d947577441814ebed6352e3b52b.png)
is the reduced mass given by 
. For unimolecular reactions, the ratio of translational partitions function is unity, 
.
![\begin{equation*} \[q_{rot} = \left ( {2 k_{b} T \over \hbar^{2}} \right )^{3/2} \sqrt {\pi I_{A} I_{B} I_{C}}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex336eed492d1b2b8bd817b4e2e6629573.png)
, 
, and 
are the moments of inertia for the molecular system. Rotational symmetry numbers are included in the reaction symmetry number discussed in a later section.
![\begin{equation*} \[q_{vib} = \prod{1 \over 1 - e^{- \beta hc \tilde \nu_{i}}}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex9fc55336f97de983b225c7572a3c2895.png)
is the vibrational frequency in 
for mode i. The product is over all vibrational modes.
![\begin{equation*} \[q_{elect} = \omega _{e1} + \omega _{e2}\exp (-\beta \Delta \varepsilon _{12}) + \ldots \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex8be1beef16f164eedacc9c3f9b2dcfcb.png)
is the energy of the 
electronic level relative to the ground state and 
is the corresponding degeneracy.
![\begin{equation*} \[\sigma_{f} = {\text{NSYM}(Reactant1) * NYSM(Reactant2) \over \text{NSYM}(GTS)}.\] \end{equation*}}](/twiki/pub/Main/KineticTheory/latex226e204bd81e4aec3299789105babb5d.png)
is the symmetry number associated with the point group to which the considered molecular configuration belongs. Similarly, the reverse symmetry number is given by
![\begin{equation*} \[\sigma_{r} = {\text{NSYM}(Product1) * \text{NSYM}(Product2) \over \text{NSYM}(GTS)}.\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex623f5a6feced45cb2bc2d214f16e3c78.png)
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
reaction as an illustrative example. From Table above we find that the 
complex belongs to the 
. The 
reactant belongs to the 
. The 
reactant belongs to the 
. The 
product atom belongs to the spherical orthogonal point group with 
product belongs to the 
point group with 
. The forward reaction symmetry is then
![\begin{equation*} \[\sigma_{f} = {6 * 1 \over 3} = 2 \]\end{equation*}](/twiki/pub/Main/KineticTheory/latexba958f80aeb33998546f025ba6018122.png)
![\begin{equation*} \[\sigma_{r} = {12 * 1 \over 3} = 4. \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex00249de460060f78b044424e91409adf.png)
reaction, the transition state has ![\begin{equation*} \[V(x) = V_{\circ}-{1 \over 2} m \omega_{\ddagger} x^{2} \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex0bd4a2522baae0898e56b26e2c25b836.png)
is the energy at the top of the barrier, and 
is the imaginary frequency of the transition state. The Wigner correction is then given by
![\begin{equation*} \[{\kappa}(T) = 1 + {1 \over 24}[\hbar {\omega}^{\ddagger}\beta]^{2} \]\end{equation*}](/twiki/pub/Main/KineticTheory/latexe05849d51cdfa9950b944f2a6f10015b.png)
is evaluated with an approximated adiabatic ground-state potential energy curve 
based on an Eckart function.
To do this, first we approximate the classical potential energy 
, saddle point 
, and products 
and from the imaginary frequency as follows:
![\begin{equation*} \[V_{MEP}(s) = {AY \over 1 + Y} + {BY \over (1 + Y)^2} \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex9ca97b6f34c5c8c5851d228dd1ae384e.png)
![\begin{equation*} \[Y = e^{\alpha(s - s_{\circ})}, \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex77061f96c700da149e31c20766a7521d.png)
![\begin{equation*} \[A = \Delta E_{C} = V_{MEP}(s = + \infty), \]\end{equation*}](/twiki/pub/Main/KineticTheory/latexfcdfb6de49c047263132de151e1a516f.png)
![\begin{equation*} \[B = (2V^{\ddagger} - A) + 2\sqrt{V^{\ddagger}(V^{\ddagger} - A)}, \]\end{equation*}](/twiki/pub/Main/KineticTheory/latex0dc84a841c16df181a914eb854e3a14c.png)
![\begin{equation*} \[s_{\circ} = - {1 \over \alpha} \ln \left ({A + B \over B - Z} \right ) ,\] \end{equation*}](/twiki/pub/Main/KineticTheory/latexfb3de18d4240a26e855d7121d49c81b6.png)
![\begin{equation*}\[ \alpha^{2} = -{\mu (\omega^{\ddagger})^{2}B \over 2 V^{\ddagger} (V^{\ddagger} - A)}.\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex648e75049dc450bdbe5e52cf69847ee6.png)
is the classical endoergicity, 
is the classical barrier height, 
is the imaginary frequency, and 
is set equal to zero at the reactants.
This fit yields the range parameter ![\begin{equation*} \[\alpha^{2} = -{\mu (\omega^{\ddagger})^{2}B \over 2 V^{\ddagger} (V^{\ddagger} - A)}.\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex963558da535ff6707627da0efdf425b5.png)
is then approximated by another Eckart function which is assumed to have the same range parameter 
and location of the maximum 
as the classical ![\begin{equation*} \[V^{G}_{a}(s) = {ay \over 1 + y} + {by \over (1 + y)^{2}} + c\] \end{equation*}](/twiki/pub/Main/KineticTheory/latexb7cd62c24815b77b4c2c171adde0c977.png)
![\begin{equation*} \[y = e^{\alpha(s - s_{\circ})},\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex5760105fc32dcad510bd3fa9f2c92fde.png)
![\begin{equation*} \[a = V^{G}_{a}(s = + \infty) - V^{G}_{a}(s = -\infty), \]\end{equation*}](/twiki/pub/Main/KineticTheory/latexa56111689f80b5c7bd991cc2267eec15.png)
![\begin{equation*} \[b = (2\Delta V^{\ddagger G}_{a} - a) + 2 \sqrt{\Delta V^{\ddagger G}_{a}(\Delta V^{\ddagger G}_{a} - a)},\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex068b3187c7a911a27eabd6c4bfee0d11.png)
![\begin{equation*} \[c = \varepsilon^{G}_{int}(s = -\infty),\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex4fe115a87030bac0c05f212f8373e869.png)
![\begin{equation*} \[s_{\circ} = - {1 \over \alpha} \ln \left ({a + b \over b - a} \right).\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex8c0656c0de7a1c8efd00bc00b53aee52.png)
is the zero-point energy corrected barrier at the saddle point, relative to reactants, and 
is the sum of the zero-point energies of the two reactants.
For most reactions, the zero-point energy correction often lowers the barrier. In this case, the range parameter 
for the ![\begin{equation*} \[V_{d}(s) = V_{MEP}(s) + \sum _{i=1}^{3N-7} \left ({1 \over 2} + m_{i} \right ) \hbar \omega _{i}(s), \label {eq:Vd}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex3642163b117f24e7a73e98b6fc714853.png)
is the vibrational state of mode 
orthogonal to the MEP. If all vibrational modes are in their ground state, 
is equivalent to the vibrationally adiabatic ground state potential curve 
, which is used in calculations of tunneling contributions to thermal rate constants. The transmission coefficient, 
, to the thermally averaged classical transmission probability for scattering by the effective potential 
, then the quasi-classical threshold energy 
can be denoted as 
. The equation for 
then becomes
![\begin{equation*} \[\kappa(T) = {\int ^{\infty} _{0} P(E)e^{-E/k_{b}T}\,dE \over \int ^{\infty} _{E_{*}(T)} e ^{-E/k_{b}T} \, dE} \label{eq:kappa}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex876df0906892a6a617455b7810b0d79d.png)
(T) as well as tunneling energies below this. Thus, the semi-classical transmission probability P(E) accounts for both non-classical reflection at energies above the quasi-classical threshold and non-classical transmission, i.e. tunneling, at energies below that threshold. Because of the Boltzmann factor in Eq. (38), tunneling is by far the more important of these two quantum effects. The centrifugal-dominant small-curvature semi-classical approximation (SCT) is a generalization of the Marcus–Coltrin approximation in which the tunneling path is distorted from the MEP out to a concave-side vibrational turning point in the direction of the internal centrifugal force. The figure below illustrates the MEP in the red curve and SCT tunneling paths in the green and light blue curves.
Instead of explicitly defining this tunneling path, the centrifugal effect is included by replacing the reduced mass by an effective reduced mass, 
, which is used to evaluate the imaginary action integral and thereby tunneling probabilities. Note that in the mass-weighted Cartesian coordinate system, the reduced mass ![\begin{equation*}\[ P(E) = {1 \over \lbrace 1 + e^{2\theta (E)} \rbrace }\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex052bd3ac4266cbf902796bb4765259c3.png)
is the imaginary action integral evaluated along the tunneling path,
![\begin{equation*} \[\theta (E) = {2 \pi \over h} \int ^{s_{r}}_{s_{l}} {\sqrt{2 \mu _{eff}(s)|E-V_{d}(s)|}} \, ds. \label{eq:theta}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex063ebf062aa6c2d513ad9b721a5af989.png)
and 
, are the reaction coordinate classical turning points defined by
![\begin{equation*} \[V_{d} \lbrack s_{l}(E) \rbrack = V_{d} \lbrack s_{r} (E) \rbrack = E.\] \end{equation*}](/twiki/pub/Main/KineticTheory/latexe24ccc85baca174ffca971a710af155b.png)
equal to 
from the MEP in the direction of the reaction-path curvature vector. Let the distance along the small-curvature tunneling path be 
and the curvature at s be 
; then it can be shown by analytical geometry that
![\begin{equation*} \[d\xi = \sqrt{\lbrace \lbrack 1 - \bar{a}(s) \rbrack ^{2} + \left \lbrack {d \bar{t} (s) \over ds} \right \rbrack ^{2} \rbrace } \, ds,\] \end{equation*}](/twiki/pub/Main/KineticTheory/latexc673b4da0fb4464218ace31140071086.png)
![\begin{equation*} \[\bar {a}(s) = |\kappa (s) \bar{t} (s)|. \label {eq:abar(s)}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex39a683996e3c045e0c3ec1752eabf57e.png)
![\begin{equation*}\[ \theta (E) = {2 \pi \over h} \int {\sqrt{2 \mu |E-V_{d}\lbrack s(\xi)\rbrack |} \, d\xi}. \label{eq:theta1}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latexa5e8cad44df8676191c43c94b2a26f4c.png)
![\begin{equation*} \[\mu_{eff}(s) = \mu \left \lbrace \left \lbrack 1 - \bar{a}(s) \right \rbrack^{2}+ \left \lbrack {d\bar{t}(s) \over ds} \right \rbrack ^{2} \right \rbrace , \label {eq:mueff}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex4db2be6cccf8b6f006ab4216c77012b5.png)
is greater than or equal to the radius of curvature of the reaction path, we include only the leading terms of Eq. (45) but not singularities by the approximated form below
![\begin{equation*} \[\mu _{eff}(s) = \mu \times min \left \lbrace \begin{array}{c} exp \lbrace - 2 \bar{a}(s) - \lbrack \bar{a}(s) \rbrack ^{2} + (d\bar{t}/ds)^{2} \rbrace \\ 1 \end{array} \right . \label{eq:mu_eff}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex5710afa5a53921f590c07e5a51fc1192.png)
![\begin{equation*}\[ \kappa (s) = \left \lbrace \sum _{i=1}^{F-1} \lbrack \kappa_{i} (s) \rbrack ^{2} \right \rbrace ^{1/2},\] \end{equation*}](/twiki/pub/Main/KineticTheory/latexd033443932cf9e2220bbd3ce86946ff2.png)
, and 
is the reaction-path curvature component of mode i given by
![\begin{equation*}\[ \kappa_{i}(s) = - \mathbf{L}_{i}^{T} \mathbf{F} {\nabla \mathbf{V} \over |\nabla \mathbf{V}|^{2}}, \label{eq:bmf}\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex07ea70939833ece9aaea746c219bec30.png)
is the transpose of the generalized normal mode eigenvector of mode i, 
is the force constant matrix (Hessian matrix), and 
is the gradient. Finally, within the harmonic approximation, ![\begin{equation*} \[\bar{t}(s)= \left ( {\kappa \hbar \over \mu} \right )^{1/2} \left \lbrace {\sum {i}^{F-1} \lbrack \kappa _{i}(s) \rbrack ^{2} \omega _{i}^{2}(s) \over (1+2m_{i})} \right \rbrace ^{-1/4},\] \end{equation*}](/twiki/pub/Main/KineticTheory/latex6600301d84b1617dcd6f1aa271e04c13.png)
is the generalized vibrational frequency of mode i, in state 
gif 
JPG 
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