Editing Kinetic isotope effect (section)

== Theory ==
Theoretical treatment of isotope effects relies heavily on [[transition state theory]], which assumes a single potential energy surface for the reaction, and a barrier between the reactants and the products on this surface, on top of which resides the transition state.<ref name=Saunders>{{cite book | vauthors = Melander L, Saunders WH | title = Reaction Rates of Isotopic Molecules | publisher = Wiley | location = New York | date = 1980 }}</ref><ref>{{cite journal | vauthors = Bigeleisen J, Wolfsberg M | title = Theoretical and experimental aspects of isotope effects in chemical kinetics. | journal = Advances in Chemical Physics | date = January 1957  | volume = 1 | pages = 15–76 }}</ref> The KIE arises largely from the changes to vibrational ground states produced by the isotopic perturbation along the minimum energy pathway of the potential energy surface, which may only be accounted for with quantum mechanical treatments of the system. Depending on the mass of the atom that moves along the reaction coordinate and nature (width and height) of the energy barrier, [[quantum tunneling]] may also make a large contribution to an observed KIE and may need to be separately considered, in addition to the "semi-classical" transition state theory model.<ref name="Saunders" />

The deuterium kinetic isotope effect ({{sup|2}}H KIE) is by far the most common, useful, and well-understood type of KIE. The accurate prediction of the numerical value of a {{sup|2}}H KIE using [[density functional theory]] calculations is now fairly routine. Moreover, several qualitative and semi-quantitative models allow rough estimates of deuterium isotope effects to be made without calculations, often providing enough information to rationalize experimental data or even support or refute different mechanistic possibilities. Starting materials containing {{sup|2}}H are often commercially available, making the synthesis of isotopically enriched starting materials relatively straightforward. Also, due to the large relative difference in the mass of {{sup|2}}H and {{sup|1}}H and the attendant differences in vibrational frequency, the isotope effect is larger than for any other pair of isotopes except {{sup|1}}H and {{sup|3}}H,<ref>If [[muonium]] (μ{{sup|+}}e{{sup|–}}) is treated as an isotope of hydrogen, then even larger KIEs are possible, in principle. However, studies involving muonium are limited by the short half-life of the muon (22 microseconds) (see {{cite journal | vauthors = Villà J, Corchado JC, González-Lafont A, Lluch JM, Truhlar DG | title = Explanation of deuterium and muonium kinetic isotope effects for hydrogen atom addition to an olefin. | journal = Journal of the American Chemical Society | date = November 1998 | volume = 120 | issue = 46 | pages = 12141–2 | doi = 10.1021/ja982616i | bibcode = 1998JAChS.12012141V }} for an example of a ''k''{{sub|Mu}}/''k''{{sub|H}} isotope effect.)</ref> allowing both primary and secondary isotope effects to be easily measured and interpreted. In contrast, secondary effects are generally very small for heavier elements and close in magnitude to the experimental uncertainty, which complicates their interpretation and limits their utility. In the context of isotope effects, ''hydrogen'' often means the light isotope, protium ({{sup|1}}H), specifically. In the rest of this article, reference to ''hydrogen'' and ''deuterium'' in parallel grammatical constructions or direct comparisons between them should be taken to mean {{sup|1}}H and {{sup|2}}H.{{efn|This convention exists for both convenience in nomenclature and as a reflection of how {{sup|2}}H KIEs are generally studied experimentally: Though deuterium ({{sup|2}}H) has the IUPAC-sanctioned symbol D, there is no common symbol that specifically refers to protium ({{sup|1}}H). Still, it has proven useful to have labels to refer to the respective rate constants of protium- or deuterium-containing isotopologues, so ''k''{{sub|H}} and ''k''{{sub|D}}, respectively, have typically been used. Moreover, the magnitude of a KIE can then be expressed as ''k''{{sub|H}}/''k''{{sub|D}}. This notation is consistent with the fact that, experimentally, {{sup|2}}H KIEs are measured by comparing the reaction rate of a {{sup|2}}H-enriched starting material to that of an unenriched starting material containing hydrogen at natural abundance. This is almost always valid, since protium accounts for ~99.985% of natural hydrogen, so there is usually no need to further deplete the deuterium in the starting material to obtain a "protium-enriched" sample. Combined, the notation and experimental setup led to the common conceptualization of deuterium as a "substituent" that takes the place of normal hydrogen in an isotope effect study.}}

The theory of KIEs was first formulated by [[Jacob Bigeleisen]] in 1949.<ref>{{cite journal|last=Bigeleisen|first=Jacob | name-list-style = vanc |title=The Relative Reaction Velocities of Isotopic Molecules|journal=Journal of Chemical Physics|date=August 1949|volume=17|issue=8|pages=675–678|doi=10.1063/1.1747368|bibcode=1949JChPh..17..675B}}</ref><ref name="Laidler_1987">{{cite book | vauthors = Laidler KJ | title = Chemical Kinetics | edition = 3rd | publisher = Harper & Row | date = 1987 | isbn = 978-0-06-043862-3 }}</ref>{{rp|427}} Bigeleisen's general formula for {{sup|2}}H KIEs (which is also applicable to heavier elements) is given below. It employs transition state theory and a statistical mechanical treatment of translational, rotational, and vibrational levels for the calculation of rate constants ''k''{{sub|H}} and ''k''{{sub|D}}.  However, this formula is "semi-classical" in that it neglects the contribution from quantum tunneling, which is often introduced as a separate correction factor. Bigeleisen's formula also does not deal with differences in non-bonded repulsive interactions caused by the slightly shorter C–{{sup|2}}H bond compared to a C–H bond. In the equation, subscript H or D refer to the species with {{sup|1}}H or {{sup|2}}H, respectively; quantities with or without the double-dagger, ‡, refer to transition state or reactant ground state, respectively.<ref name="Buncel5" /><ref name=":4">{{Cite book | title = Mechanism and theory in organic chemistry | first1 = Thomas H | last1 = Lowry | first2 = Kathleen Schueller | last2 = Richardson | name-list-style = vanc | date = 1987 | publisher = Harper & Row | isbn = 978-0-06-044084-8 | edition = 3rd | location = New York | pages = [https://archive.org/details/mechanismtheoryi000321/page/256 256] | oclc = 14214254 | url-access = registration | url = https://archive.org/details/mechanismtheoryi000321/page/256 }}</ref> (Strictly speaking, a <math>
\kappa_\mathrm{H}/\kappa_{\mathrm{D}}
</math> term resulting from an isotopic difference in transmission coefficients should also be included.<ref>{{Cite book |title=Determination of organic reaction mechanisms| first =  Barry Keith | last = Carpenter | name-list-style = vanc |date=1984|publisher=Wiley |isbn=978-0-471-89369-1 |location=New York |pages=86 |oclc=9894996 }}</ref>)

:<math chem="">
\frac{k_\ce{H}}{k_\ce{D}} = \left(\frac{\sigma_\ce{H} \sigma^\ddagger_\ce{D}}{\sigma_\ce{D} \sigma^\ddagger_\ce{H}} \right) \left(\frac{M^\ddagger_\ce{H} M_\ce{D}}{M^\ddagger_\ce{D} M_\ce{H}}\right)^{\frac 3 2}\left(\frac{I^\ddagger_{x\ce{H}}I^\ddagger_{y\ce{H}}I^\ddagger_{z\ce{H}}}{I^\ddagger_{x\ce{D}}I^\ddagger_{y\ce{D}}I^\ddagger_{z\ce{D}}}\frac{I_{x\ce{D}}I_{y\ce{D}}I_{z\ce{D}}}{I_{x\ce{H}}I_{y\ce{H}}I_{z\ce{H}}}\right)^{\frac 1 2}
 \left(\frac{\prod\limits_{i=1}^{3N^\ddagger -7}\frac{1-e^{-u^\ddagger_{i\ce{D}}}}{1-e^{-u^\ddagger_{i\ce{H}}}}}{\prod\limits_{i=1}^{3N -6}\frac{1-e^{-u_{i\ce{D}}}}{1-e^{-u_{i\ce{H}}}}} \right) e^{-\frac 1 2 \left[\sum\limits_{i=1}^{3N^\ddagger-7}(u^\ddagger_{i\ce{H}}-u^\ddagger_{i\ce{D}})-\sum\limits_{i=1}^{3N-6}(u_{i\ce{H}}-u_{i\ce{D}})\right]}
</math>,

where we define

:<math chem="">
u_i:= \frac{h\nu_i}{k_\mathrm{B}T} =
\frac{hcN_\mathrm{A}\tilde{\nu}_i}{RT}
</math> and <math chem="">
u_i^\ddagger:= \frac{h\nu_i^\ddagger}{k_\mathrm{B}T} =
\frac{hcN_\mathrm{A}\tilde{\nu}_i^\ddagger}{RT}
</math>.

Here, ''h'' = [[Planck constant]]; ''k''{{sub|B}} = [[Boltzmann constant]]; <math>\tilde{\nu}_i</math> = frequency of vibration, expressed in [[wavenumber]]; ''c'' = [[speed of light]]; ''N''{{sub|A}} = [[Avogadro constant]]; and ''R'' = [[universal gas constant]]. The σ{{sub|X}} (X = H or D) are the symmetry numbers for the reactants and transition states. The ''M''{{sub|X}} are the molecular masses of the corresponding species, and the ''I{{sub|q}}''{{sub|X}} (''q'' = ''x'', ''y'', or ''z'') terms are the moments of inertia about the three principal axes. The ''u{{sub|i}}''{{sub|X}} are directly proportional to the corresponding vibrational frequencies, ''ν{{sub|i}}'', and the vibrational [[zero-point energy]] (ZPE) (see below). The integers ''N'' and ''N''{{sup|‡}} are the number of atoms in the reactants and the transition states, respectively.<ref name="Buncel5" /> The complicated expression given above can be represented as the product of four separate factors:<ref name="Buncel5" />

:<math chem="">\frac{k_\ce{H}}{k_\ce{D}} = \mathbf{S} \times \mathbf{MMI} \times \mathbf{EXC} \times \mathbf{ZPE}</math>.

For the special case of {{sup|2}}H isotope effects, we will argue that the first three terms can be treated as equal to or well approximated by unity. The first factor '''S''' (containing σ{{sub|X}}) is the ratio of the symmetry numbers for the various species. This will be a rational number (a ratio of integers) that depends on the number of molecular and bond rotations leading to the permutation of identical atoms or groups in the reactants and the transition state.<ref name=":4" /> For systems of low symmetry, all σ{{sub|X}} (reactant and transition state) will be unity; thus '''S''' can often be neglected. The '''MMI''' factor (containing the ''M''{{sub|X}} and ''I{{sub|q}}''{{sub|X}}) refers to the ratio of the molecular masses and the moments of inertia. Since hydrogen and deuterium tend to be much lighter than most reactants and transition states, there is little difference in the molecular masses and moments of inertia between H and D containing molecules, so the '''MMI''' factor is usually also approximated as unity. The '''EXC''' factor (containing the product of vibrational [[partition function (statistical mechanics)|partition function]]s) corrects for the KIE caused by the reactions of vibrationally excited molecules. The fraction of molecules with enough energy to have excited state A–H/D bond vibrations is generally small for reactions at or near room temperature (bonds to hydrogen usually vibrate at 1000&nbsp;cm{{sup|−1}} or higher, so exp(-''u{{sub|i}}'') = exp(-''hν{{sub|i}}''/''k''{{sub|B}}''T'') < 0.01 at 298 K, resulting in negligible contributions from the 1–exp(-''u{{sub|i}}'') factors). Hence, for hydrogen/deuterium KIEs, the observed values are typically dominated by the last factor, '''ZPE''' (an exponential function of vibrational ZPE differences), consisting of contributions from the ZPE differences for each of the vibrational modes of the reactants and transition state, which can be represented as follows:<ref name="Buncel5" />

:<math chem="">\begin{align}
\frac{k_\ce{H}}{k_\ce{D}} &\cong \exp\left\{-\frac 1 2\left[\sum\limits_{i=1}^{3N^\ddagger-7}(u^\ddagger_{i\ce{H}}-u^\ddagger_{i\ce{D}})-\sum\limits_{i=1}^{3N-6}(u_{i\ce{H}}-u_{i\ce{D}})\right]\right\}
\\ &\cong \exp\left[\sum_{i}^{\mathrm{(react.)}}\frac{1}{2}\Delta u_i-\sum_i^{\mathrm{(TS)}}\frac{1}{2}\Delta u_i^\ddagger\right]
\end{align}</math>,

where we define

:<math>\Delta u_i := u_{i\mathrm{H}}-u_{i\mathrm{D}}</math> and <math>\Delta u_i^\ddagger := u_{i\mathrm{H}}^\ddagger-u_{i\mathrm{D}}^\ddagger</math>.

The sums in the exponent of the second expression can be interpreted as running over all vibrational modes of the reactant ground state and the transition state. Or, one may interpret them as running over those modes unique to the reactant or the transition state or whose vibrational frequencies change substantially upon advancing along the reaction coordinate. The remaining pairs of reactant and transition state vibrational modes have very similar <math>\Delta u_i</math> and <math>\Delta u_i^\ddagger</math>, and cancellations occur when the sums in the exponent are calculated. Thus, in practice, {{sup|2}}H KIEs are often largely dependent on a handful of key vibrational modes because of this cancellation, making qualitative analyses of ''k''{{sub|H}}/''k''{{sub|D}} possible.<ref name=":4" />

As mentioned, especially for {{sup|1}}H/{{sup|2}}H substitution, most KIEs arise from the difference in ZPE between the reactants and the transition state of the isotopologues; this difference can be understood qualitatively as follows: in the [[Born–Oppenheimer approximation]], the potential energy surface is the same for both isotopic species. However, a quantum treatment of the energy introduces discrete vibrational levels onto this curve, and the lowest possible energy state of a molecule corresponds to the lowest vibrational energy level, which is slightly higher in energy than the minimum of the [[Energy profile (chemistry)|potential energy curve]]. This difference, known as the ZPE, is a manifestation of the uncertainty principle that necessitates an uncertainty in the C-H or C-D bond length. Since the heavier (in this case the deuterated) species behaves more "classically", its vibrational energy levels are closer to the classical potential energy curve, and it has a lower ZPE. The ZPE differences between the two isotopic species, at least in most cases, diminish in the transition state, since the bond force constant decreases during bond breaking. Hence, the lower ZPE of the deuterated species translates into a larger activation energy for its reaction, as shown in the following figure, leading to a normal KIE.<ref name="pmid21124393">{{cite journal | vauthors = Carpenter BK | title = Kinetic isotope effects: unearthing the unconventional | journal = Nature Chemistry | volume = 2 | issue = 2 | pages = 80–2 | date = February 2010 | pmid = 21124393 | doi = 10.1038/nchem.531 | bibcode = 2010NatCh...2...80C }}</ref> This effect should, in principle, be taken into account all 3''N−''6 vibrational modes for the starting material and 3''N''{{sup|‡}}''−''7 vibrational modes at the transition state (one mode, the one corresponding to the reaction coordinate, is missing at the transition state, since a bond breaks and there is no restorative force against the motion).  The [[harmonic oscillator]] is a good approximation for a vibrating bond, at least for low-energy vibrational states. Quantum mechanics gives the vibrational ZPE as <math>
\epsilon_i^{(0)}=\frac{1}{2}h\nu_i
</math>. Thus, we can readily interpret the factor of {{sfrac|1|2}} and the sums of <math>
u_i = h\nu_i/k_\mathrm{B}T
</math> terms over ground state and transition state vibrational modes in the exponent of the simplified formula above. For a harmonic oscillator, vibrational frequency is inversely proportional to the square root of the reduced mass of the vibrating system:

:<math>
\nu_{\mathrm{X}}=\frac{1}{2\pi }\sqrt{\frac{k_\mathrm{f}}{\mu_\mathrm{X}}}\cong
\frac{1}{2\pi }\sqrt{\frac{k_\mathrm{f}}{m_\mathrm{X}}}
</math>,

where ''k''{{sub|f}} is the [[Hooke's law|force constant]]. Moreover, the reduced mass is approximated by the mass of the light atom of the system, X = H or D. Because ''m''{{sub|D}} ≈ 2''m''{{sub|H}},

:<math>
\Delta u_i \cong \left(1-\frac{1}{\sqrt 2}\right)\frac{h\nu_{i\mathrm{H}}}{k_\mathrm{B}T}
</math>.

In the case of homolytic C–H/D bond dissociation, the transition state term disappears; and neglecting other vibrational modes, ''k''{{sub|H}}/''k''{{sub|D}} = exp({{sfrac|1|2}}Δ''u{{sub|i}}''). Thus, a larger isotope effect is observed for a stiffer ("stronger") C–H/D bond. For most reactions of interest, a hydrogen atom is transferred between two atoms, with a transition-state [A···H···B]{{sup|‡}} and vibrational modes at the transition state need to be accounted for.  Nevertheless, it is still generally true that cleavage of a bond with a higher vibrational frequency will give a larger isotope effect. [[File:C-H and C-D bond breaking.png|thumb|center|500px|Differences in ZPE and corresponding differences in activation energy for the breaking of analogous C-H and C-D bonds. In this schematic, the curves actually represent (3''N-''6)- and (3''N''{{sup|‡}}-7)-dimensional hypersurfaces, and the vibrational mode whose ZPE is illustrated at the transition state is ''not'' the same one as the reaction coordinate. The reaction coordinate represents a vibration with a negative force constant (and imaginary vibrational frequency) for the transition state. The ZPE shown for the ground state ''may'' refer to the vibration corresponding to the reaction coordinate in the case of a primary KIE.]]

To calculate the maximum possible value for a non-tunneling {{sup|2}}H KIE, we consider the case where the ZPE difference between the stretching vibrations of a C-{{sup|1}}H bond (3000&nbsp;cm{{sup|−1}}) and C-{{sup|2}}H bond (2200&nbsp;cm{{sup|−1}}) disappears in the transition state (an energy difference of [3000 – 2200&nbsp;cm{{sup|−1}}]/2 = 400&nbsp;cm{{sup|−1}} ≈ 1.15 kcal/mol), without any compensation from a ZPE difference at the transition state (e.g., from the symmetric A···H···B stretch, which is unique to the transition state). The simplified formula above, predicts a maximum for ''k''{{sub|H}}/''k''{{sub|D}} as 6.9. If the complete disappearance of two bending vibrations is also included, ''k''{{sub|H}}/''k''{{sub|D}} values as large as 15-20 can be predicted. Bending frequencies are very unlikely to vanish in the transition state, however, and there are only a few cases in which ''k''{{sub|H}}/''k''{{sub|D}} values exceed 7-8 near room temperature. Furthermore, it is often found that tunneling is a major factor when they do exceed such values. A value of ''k''{{sub|H}}/''k''{{sub|D}} ~ 10 is thought to be maximal for a semi-classical PKIE (no tunneling) for reactions at ≈298 K. (The formula for ''k''{{sub|H}}/''k''{{sub|D}} has a temperature dependence, so larger isotope effects are possible at lower temperatures.)<ref>{{Cite book |title=Perspectives on structure and mechanism in organic chemistry | first = Felix A | last = Carroll | name-list-style = vanc |date=2010|publisher=John Wiley|isbn=978-0-470-27610-5 |edition=2nd|location=Hoboken, N.J.|oclc=286483846}}</ref> Depending on the nature of the transition state of H-transfer (symmetric vs. "early" or "late" and linear vs. bent); the extent to which a primary {{sup|2}}H isotope effect approaches this maximum, varies. A model developed by [[Frank Westheimer|Westheimer]] predicted that symmetrical (thermoneutral, by [[Hammond's postulate]]), linear transition states have the largest isotope effects, while transition states that are "early" or "late" (for exothermic or endothermic reactions, respectively), or nonlinear (e.g. cyclic) exhibit smaller effects. These predictions have since received extensive experimental support.<ref>{{Cite journal|last=Kwart|first=Harold | name-list-style = vanc |date=1982-12-01|title=Temperature dependence of the primary kinetic hydrogen isotope effect as a mechanistic criterion |journal=Accounts of Chemical Research|volume=15|issue=12|pages=401–408|doi=10.1021/ar00084a004|issn=0001-4842}}</ref>

For secondary {{sup|2}}H isotope effects, [[Andrew Streitwieser|Streitwieser]] proposed that weakening (or strengthening, in the case of an inverse isotope effect) of bending modes from the reactant ground state to the transition state are largely responsible for observed isotope effects. These changes are attributed to a change in steric environment when the carbon bound to the H/D undergoes rehybridization from sp{{sup|3}} to sp{{sup|2}} or vice versa (an α SKIE), or bond weakening due to hyperconjugation in cases where a carbocation is being generated one carbon atom away (a β SKIE). These isotope effects have a theoretical maximum of ''k''{{sub|H}}/''k''{{sub|D}} = 2{{sup|0.5}} ≈ 1.4. For a SKIE at the α position, rehybridization from sp{{sup|3}} to sp{{sup|2}} produces a normal isotope effect, while rehybridization from sp{{sup|2}} to sp{{sup|3}} results in an inverse isotope effect with a theoretical minimum of ''k''{{sub|H}}/''k''{{sub|D}} = 2{{sup|-0.5}} ≈ 0.7.  In practice, ''k''{{sub|H}}/''k''{{sub|D}} ~ 1.1-1.2 and ''k''{{sub|H}}''/k''{{sub|D}} ~ 0.8-0.9 are typical for α SKIEs, while ''k''{{sub|H}}/''k''{{sub|D}} ~ 1.15-1.3 are typical for β SKIE. For reactants containing several isotopically substituted β-hydrogens, the observed isotope effect is often the result of several H/D's at the β position acting in concert. In these cases, the effect of each isotopically labeled atom is multiplicative, and cases where ''k''{{sub|H}}/''k''{{sub|D}} > 2 are not uncommon.<ref>{{cite journal | vauthors = Streitwieser A, Jagow RH, Fahey RC, Suzuki S | title = Kinetic isotope effects in the acetolyses of deuterated cyclopentyl tosylates1, 2. | journal = Journal of the American Chemical Society | date = May 1958 | volume = 80 | issue = 9 | pages = 2326–32 | doi = 10.1021/ja01542a075 | bibcode = 1958JAChS..80.2326S }}</ref>

The following simple expressions relating {{sup|2}}H and {{sup|3}}H KIEs, which are also known as the [[Swain equation]] (or the Swain-Schaad-Stivers equations), can be derived from the general expression given above using some simplifications:<ref name="Saunders" /><ref>{{cite journal| vauthors = Swain CG, Stivers EC, Reuwer Jr JF, Schaad LJ |title=Use of Hydrogen Isotope Effects to Identify the Attacking Nucleophile in the Enolization of Ketones Catalyzed by Acetic Acid|journal=Journal of the American Chemical Society|date=1 November 1958 |volume=80 |issue=21 |pages=5885–5893 |doi=10.1021/ja01554a077 |bibcode=1958JAChS..80.5885S }}</ref>

:<math chem="">\left(\frac{\ln\left(\frac{k_\ce{H}}{k_\ce{T}}\right)}{\ln\left(\frac{k_\ce{H}}{k_\ce{D}}\right)}\right)_s\cong\frac{1-\sqrt{m_\ce{H}/m_\ce{T}}}{1-\sqrt{m_\ce{H}/m_\ce{D}}}=\frac{1-\sqrt{1/3}}{1-\sqrt{1/2}}\cong1.44</math>;

i.e.,

:<math chem>\left(\frac{k_\ce{H}}{k_\ce{T}}\right)_s=\left(\frac{k_\ce{H}}{k_\ce{D}}\right)_s^{1.44}</math>.

In deriving these expressions, the reasonable approximation that reduced mass roughly equals the mass of the {{sup|1}}H, {{sup|2}}H, or {{sup|3}}H, was used. Also, the vibrational motion was assumed to be approximated by a harmonic oscillator, so that <math>
u_{i\mathrm{X}}\propto \mu_{\mathrm{X}}^{-1/2}\cong m_{\mathrm{X}}^{-1/2}
</math>; X = {{sup|1,2,3}}H. The subscript "''s''" refers to these "semi-classical" KIEs, which disregard quantum tunneling. Tunneling contributions must be treated separately as a correction factor.

For isotope effects involving elements other than hydrogen, many of these simplifications are not valid, and the magnitude of the isotope effect may depend strongly on some or all of the neglected factors. Thus, KIEs for elements other than hydrogen are often much harder to rationalize or interpret. In many cases and especially for hydrogen-transfer reactions, contributions to KIEs from tunneling are significant (see below).

=== Tunneling ===
In some cases, a further rate enhancement is seen for the lighter isotope, possibly due to [[quantum tunneling]]. This is typically only observed for reactions involving bonds to hydrogen. Tunneling occurs when a molecule penetrates through a potential energy barrier rather than over it.<ref name="anslyn">{{cite book | vauthors = Anslyn EV, Dougherty DA
 |year=2006
 |title=Modern Physical Organic Chemistry
 | url = https://archive.org/details/modernphysicalor00ansl
 | url-access = limited
 |publisher= University Science Books
 |pages=[https://archive.org/details/modernphysicalor00ansl/page/n455 428]–437
 |isbn=978-1-891389-31-3
}}</ref><ref name="razauy">{{cite book
 | vauthors = Razauy M
 |year=2003
 |title=Quantum Theory of Tunneling
 | url = https://archive.org/details/quantumtheoryoft0000raza
 | url-access = registration
 |publisher=[[World Scientific]]
 |isbn=978-981-238-019-7
}}</ref> Though not allowed by [[classical mechanics]], particles can pass through classically forbidden regions of space in quantum mechanics based on [[wave–particle duality]].<ref name="silbey">{{cite book
 | vauthors = Silbey RJ, Alberty RA, Bawendi MG
 |year=2005
 |title=Physical Chemistry
 |pages=326–338
 |publisher=[[John Wiley & Sons]]
 |isbn=978-0-471-21504-2
}}</ref>

[[File:Energy Well Model.png|thumb|550px|right|The potential energy well of a tunneling reaction. The dash-red arrow shows the classical activated process, while the solid-red arrow shows the tunneling path.<ref name="anslyn" />]]

Tunneling can be analyzed using Bell's modification of the [[Arrhenius equation]], which includes the addition of a tunneling factor, Q:
:<math>k=QAe^{-E/RT}</math>
where A is the Arrhenius parameter, E is the barrier height and
:<math>Q = \frac{e^\alpha}{\beta-\alpha}(\beta e^{-\alpha}-\alpha e^{- \beta})</math>
where <math>\alpha=\frac{E}{RT}</math> and <math>\beta=\frac{2a \pi ^2(2mE)^{1/2}}{h}</math>

Examination of the ''β'' term shows exponential dependence on the particle's mass. As a result, tunneling is much more likely for a lighter particle such as hydrogen. Simply doubling the mass of a tunneling proton by replacing it with a deuteron drastically reduces the rate of such reactions. As a result, very large KIEs are observed that can not be accounted for by differences in ZPEs.
[[File:Donor Acceptor Model.png|thumb|250px|center|Donor-acceptor model of a proton transfer.<ref name=Borgis>{{cite journal
 | vauthors = Borgis D, Hynes JT
 |year=1993
 |title=Dynamical theory of proton tunneling transfer rates in solution: General formulation
 |journal=[[Chemical Physics]]
 |volume=170 |issue=3 |pages=315–346
 |bibcode=1993CP....170..315B
 |doi=10.1016/0301-0104(93)85117-Q
}}</ref>]]

Also, the ''β'' term depends linearly with barrier width, 2a. As with mass, tunneling is greatest for small barrier widths. Optimal tunneling distances of protons between donor and acceptor atom is 40&nbsp;pm.<ref name="Krishtalik_2000">{{cite journal | vauthors = Krishtalik LI | title = The mechanism of the proton transfer: an outline | journal = Biochimica et Biophysica Acta (BBA) - Bioenergetics | volume = 1458 | issue = 1 | pages = 6–27 | date = May 2000 | pmid = 10812022 | doi = 10.1016/S0005-2728(00)00057-8 | doi-access = free }}</ref>

{{hidden|toggle=left|1=Temperature dependence in tunneling|2=
[[Quantum tunneling|Tunneling]] is a quantum effect tied to the laws of wave mechanics, not [[kinetics (physics)|kinetics]]. Therefore, tunneling tends to become more important at low temperatures, where even the smallest kinetic energy barriers may not be overcome but can be tunneled through.<ref name="anslyn" />

Peter S. Zuev et al. reported rate constants for the ring expansion of 1-methylcyclobutylfluorocarbene to be 4.0 × 10{{sup|−6}}/s in nitrogen and 4.0 × 10{{sup|−5}}/s in argon at 8 kelvin. They calculated that at 8 kelvin, the reaction would proceed via a single quantum state of the reactant so that the reported rate constant is temperature independent and the tunneling contribution to the rate was 152 orders of magnitude greater than the contribution of passage over the transition state energy barrier.<ref name=Zuev>{{cite journal | vauthors = Zuev PS, Sheridan RS, Albu TV, Truhlar DG, Hrovat DA, Borden WT | title = Carbon tunneling from a single quantum state | journal = Science | volume = 299 | issue = 5608 | pages = 867–70 | date = February 2003 | pmid = 12574623 | doi = 10.1126/science.1079294 | bibcode = 2003Sci...299..867Z | s2cid = 20959068 }}</ref>

So even though conventional chemical reactions tend to slow down dramatically as the temperature is lowered, tunneling reactions rarely change at all. Particles that tunnel through an activation barrier are a direct result of the fact that the wave function of an intermediate species, reactant or product is not confined to the energy well of a particular trough along the energy surface of a reaction but can "leak out" into the next energy minimum. In light of this, tunneling ''should'' be temperature independent.<ref name="anslyn" /><ref name="Atkins">{{cite book
 | vauthors = Atkins P, de Paula J
 |year=2006
 |edition = 8th
 |title=Atkins' Physical Chemistry
 | url = https://archive.org/details/atkinsphysicalch00atki
 | url-access = limited
 |pages=[https://archive.org/details/atkinsphysicalch00atki/page/n318 286]–288, 816–818
 |publisher=[[Oxford University Press]]
 |isbn=978-0-19-870072-2
}}</ref>

For the hydrogen abstraction from gaseous n-alkanes and cycloalkanes by hydrogen atoms over the temperature range 363–463 K, the H/D KIE data were characterized by small [[preexponential factor]] ratios ''A''{{sub|H}}/''A''{{sub|D}} ranging from 0.43 to 0.54 and large activation energy differences from 9.0 to 9.7 kJ/mol. Basing their arguments on [[transition state theory]], the small ''A'' factor ratios associated with the large activation energy differences (usually about 4.5 kJ/mol for C–H(D) bonds) provided strong evidence for tunneling. For the purpose of this discussion, it is important is that the ''A'' factor ratio for the various paraffins they used was roughly constant throughout the temperature range.<ref name=Fujisaki>{{cite journal
 | vauthors = Fujisaki N, Ruf A, Gaeumann T
 |year=1987
 |title=Tunnel effects in hydrogen-atom-transfer reactions as studied by the temperature dependence of the hydrogen deuterium kinetic isotope effects
 |journal=[[Journal of Physical Chemistry]]
 |volume=91 |issue=6 |pages=1602–1606
 |doi=10.1021/j100290a062
}}</ref>

The observation that tunneling is not entirely temperature independent can be explained by the fact that not all molecules of a given species occupy their vibrational ground state at varying temperatures. Adding thermal energy to a potential energy well could cause higher vibrational levels than the ground state to become populated. For a conventional kinetically driven reaction, this excitation would only have a small influence on the rate. However, for a tunneling reaction, the difference between the ZPE and the first vibrational energy level could be huge. The tunneling correction term ''Q'' is linearly dependent on barrier width and this width is significantly diminished as the number [[vibrational modes]] on the [[Morse potential]] increase. The decrease of the barrier width can have such a huge impact on the tunneling rate that even a small population of excited vibrational states would dominate this process.<ref name="anslyn" /><ref name="Atkins" />
}}

{{hidden|toggle=left|1=Criteria for KIE tunneling|2=
To determine if tunneling is involved in KIE of a reaction with H or D, a few criteria are considered:
# Δ(''E{{sub|a}}''{{sup|H}}-''E{{sub|a}}''{{sup|D}}) > Δ(''ZPE''{{sup|H}}-''ZPE''{{sup|D}}) (''E{{sub|a}}''=activation energy; ZPE=zero point energy)
# Reaction still proceeds at lower temperatures.
# The [[Arrhenius equation|Arrhenius]] pre-exponential factors ''A''{{sub|D}}/''A''{{sub|H}} is not equal to 1.
# A large negative [[entropy]] of activation.
# The geometries of the reactants and products are usually very similar.<ref name="anslyn" />
Also for reactions where isotopes include H, D and T, a criterion of tunneling is the Swain-Schaad relations which compare the rate constants (''k'') of the reactions where H, D or T are exchanged:
:''k''{{sub|H}}/''k''{{sub|T}}=(''k''{{sub|D}}/''k''{{sub|T}})''{{sup|X}}'' and  ''k''{{sub|H}}/''k''{{sub|T}}=(''k''{{sub|H}}/''k''{{sub|D}})''{{sup|Y}}''
Experimental values of X exceeding 3.26 and Y lower than 1.44 are evidence of a certain amount of contribution from tunneling.<ref name="Krishtalik_2000" /><ref name="Laidler_1987" />{{rp|437–8}}
}}

{{hidden|toggle=left|1=Examples for tunneling in KIE|2=
In organic reactions, this proton tunneling effect has been observed in such reactions as the [[deprotonation]] and iodination of [[nitropropane]] with hindered [[pyridine]] base<ref>{{cite journal
 | vauthors = Lewis ES, Funderburk L
 |year=1967
 |title=Rates and isotope effects in the proton transfers from 2-nitropropane to pyridine bases
 |journal=[[Journal of the American Chemical Society]]
 |volume=89 |issue=10 |pages=2322–2327
 |doi=10.1021/ja00986a013
|bibcode=1967JAChS..89.2322L
 }}</ref> with a reported KIE of 25 at 25°C:

:[[File:KIE effect iodination.png|400px|KIE effect iodination]]

and in a [[Sigmatropic reaction|1,5-sigmatropic hydrogen shift]],<ref>{{cite journal
 | vauthors = Dewar MJ, Healy EF, Ruiz JM
 |year=1988
 |title=Mechanism of the 1,5-sigmatropic hydrogen shift in 1,3-pentadiene
 |journal=[[Journal of the American Chemical Society]]
 |volume=110 |issue=8 |pages=2666–2667
 |doi=10.1021/ja00216a060
|bibcode=1988JAChS.110.2666D
 }}</ref> though it is observed that it is hard to extrapolate experimental values obtained at high temperature to lower temperatures:<ref>{{cite journal | vauthors = von Doering W, Zhao X | title = Effect on kinetics by deuterium in the 1,5-hydrogen shift of a cisoid-locked 1,3(Z)-pentadiene, 2-methyl-10-methylenebicyclo[4.4.0]dec-1-ene: evidence for tunneling? | journal = Journal of the American Chemical Society | volume = 128 | issue = 28 | pages = 9080–5 | date = July 2006 | pmid = 16834382 | doi = 10.1021/ja057377v | bibcode = 2006JAChS.128.9080D }}</ref><ref>In this study the KIE is measured by sensitive [[proton NMR]]. The extrapolated KIE at 25°C is 16.6 but the margin of error is high</ref>

:[[File:KIE effect sigmatropicReaction 2006.png|400px|KIE effect sigmatropic reaction]]
It has long been speculated that high efficiency of enzyme catalysis in proton or hydride ion transfer reactions could be due partly to the quantum mechanical tunneling effect. Environment at the active site of an enzyme positions the donor and acceptor atom close to the optimal tunneling distance, where the amino acid side chains can "force" the donor and acceptor atom closer together by electrostatic and noncovalent interactions. It is also possible that the enzyme and its unusual hydrophobic environment inside a reaction site provides tunneling-promoting vibration.<ref name=Kohen>{{cite journal | vauthors = Kohen A, Klinman JP | title = Hydrogen tunneling in biology | journal = Chemistry & Biology | volume = 6 | issue = 7 | pages = R191-8 | date = July 1999 | pmid = 10381408 | doi = 10.1016/S1074-5521(99)80058-1 | doi-access = free }}</ref> Studies on ketosteroid isomerase have provided experimental evidence that the enzyme actually enhances the coupled motion/hydrogen tunneling by comparing primary and secondary KIEs of the reaction under enzyme-catalyzed and non-enzyme-catalyzed conditions.<ref name=pollack>{{cite journal | vauthors = Wilde TC, Blotny G, Pollack RM | title = Experimental evidence for enzyme-enhanced coupled motion/quantum mechanical hydrogen tunneling by ketosteroid isomerase | journal = Journal of the American Chemical Society | volume = 130 | issue = 20 | pages = 6577–85 | date = May 2008 | pmid = 18426205 | doi = 10.1021/ja0732330 | bibcode = 2008JAChS.130.6577W }}</ref>

Many examples exist for proton tunneling in enzyme-catalyzed reactions that were discovered by KIE. A well-studied example is methylamine dehydrogenase, where large primary KIEs of 5–55 have been observed for the proton transfer step.<ref name=Villa>{{cite journal
 | vauthors = Truhlar DG, Gao J, Alhambra C, Garcia-Viloca M, Corchado J, Sánchez M, Villà J
 |year=2002
 |title=The Incorporation of Quantum Effects in Enzyme Kinetics Modeling
 |journal=[[Accounts of Chemical Research]]
 |volume=35 |issue=6 |pages=341–349
 |doi=10.1021/ar0100226
|pmid=12069618
 }}</ref>

[[File:Methylamine dehydrogenase.gif|thumb|2000px|center|Mechanism of methylamine dehydrogenase, the [[quinoprotein]] converts primary amines to aldehyde and ammonia.]]

Another example of tunneling contribution to proton transfer in enzymatic reactions is the reaction carried out by [[alcohol dehydrogenase]]. Competitive KIEs for the hydrogen transfer step at 25°C resulted in 3.6 and 10.2 for primary and secondary KIEs, respectively.<ref name=KlinmanKohen>{{cite journal
 | last1= Kohen |first1=A|last2=Klinman|first2=J. P|authorlink2=Judith Klinman
 |year=1998
 |title=Enzyme Catalysis: Beyond Classical Paradigms
 |journal=[[Accounts of Chemical Research]]
 |volume=31 |issue=7 |pages=397–404
 |doi=10.1021/ar9701225
}}</ref>
[[File:Alcohol dehydrogenase.png|thumb|800px|center|Mechanism of alcohol dehydrogenase. The rate-limiting step is the proton transfer.]]
}}

=== Transient kinetic isotope effect ===

{{Main|Transient kinetic isotope fractionation}}

Isotopic effect expressed with the equations given above only refer to reactions that can be described with [[first-order kinetics]]. In all instances in which this is not possible, [[Transient kinetic isotope fractionation|transient KIEs]] should be taken into account using the GEBIK and GEBIF equations.<ref name="Maggi">{{cite journal
 | vauthors = Maggi F, Riley WJ
 |year=2010
 |title=Mathematical treatment of isotopologue and isotopomer speciation and fractionation in biochemical kinetics
 |journal=[[Geochimica et Cosmochimica Acta]]
 |volume=74 |issue=6 |page=1823
 |bibcode=2010GeCoA..74.1823M
 |doi=10.1016/j.gca.2009.12.021
|s2cid=55422661
 |url=http://www.escholarship.org/uc/item/0x8927jx
 }}</ref>