research article calculation of the quantum-mechanical...

Research ArticleCalculation of the Quantum-MechanicalTunneling in Bound Potentials

Sophya Garashchuk Bing Gu and James Mazzuca

Department of Chemistry and Biochemistry University of South Carolina Columbia SC 29208 USA

Correspondence should be addressed to Sophya Garashchuk sgarashcmailchemscedu

Received 28 January 2014 Accepted 1 April 2014 Published 24 April 2014

Academic Editor Anton Kokalj

Copyright copy 2014 Sophya Garashchuk et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The quantum-mechanical tunneling is often important in low-energy reactions which involve motion of light nuclei occurringin condensed phase The potential energy profile for such processes is typically represented as a double-well potential along thereaction coordinate In a potential of this type defining reaction probabilities rigorously formulated only for unbound potentials interms of the scattering states with incomingoutgoing scattering boundary conditions becomes ambiguous Based on the analysisof a rectangular double-well potential a modified expression for the reaction probabilities and rate constants suitable for arbitrarydouble- (or multiple-) well potentials is developed with the goal of quantifying tunneling The proposed definition involves energyeigenstates of the bound potential and exact quantum-mechanical transmission probability through the barrier region of thecorresponding scattering potential Applications are given for several model systems including proton transfer in a HOndashHndashCH

3

model and the differences between the quantum-mechanical and quasiclassical tunneling probabilities are examined

1 Introduction

There is great interest in understanding the role of quantum-mechanical (QM) effects on reactivity at low temperature incomplex systems such as liquids [1] proteins [2] at surfaces[3] and so forth One example which motivated many theo-retical investigations [4] is the unusually high experimentalkinetic isotope effect (KIE = 81) of the transferring hydro-gendeuterium substitution on the reaction rate constant insoybean lipoxygenase-1 [5] attributed to QM tunneling asthe dominant reaction mechanism A few other exampleswhere tunneling defines the reaction rate are isomerization ofmethylhydroxycarbene [6] and reaction of hydroxyl radicalwith methanol [7] To understand the role of tunneling for aspecific system a typical approach is to construct an electronicpotential energy profile of a double-well character along one-dimensional reaction coordinate and to compute tunnel-ing probabilities using the quasiclassical Wentzel-Kramers-Brillouin (WKB) approximationThis was done for examplein recent studies of the barrier shape effect on the classicaland QM contributions to reactivity in enzyme catalysis [8ndash10] and in a study of the tunneling control of reactivity in

carbenes [6] Our goal here is to define and analyze QMreaction probabilities and thermal reaction rate constants inthe bound potentials in a more rigorous way than the WKBtheory There are two QM effects that should be considered(i) decay of the wavefunction inside a barrier for a range ofenergies treated approximately in the WKB approach and(ii) quantization of the energy levels due to the bound char-acter of a potential (The latter is not included in the WKBexcept via the harmonic zero-point energy of the reactantwell)

The ultimate information about a reactive process in gasphase is contained in the scatteringmatrix defined for a givencollision energy 119864 Its elements 119878

119903119901(119864) subscripts 119903 and 119901

label specific internal states of reactants and products aredefined in terms of the energy eigenstates with incom-ingoutgoing boundary conditions in the reaction coordinatefor a potential which vanishes at large reactant-product sep-aration [11] From the scattering matrix the cumulative reac-tion probability119873(119864) may be computed as

119873(119864) = sum

119903119901

10038161003816100381610038161003816

119878119903119901(119864)

10038161003816100381610038161003816

2

(1)

Hindawi Publishing CorporationJournal of eoretical ChemistryVolume 2014 Article ID 240491 11 pageshttpdxdoiorg1011552014240491

2 Journal of Theoretical Chemistry

Furthermore 119873(119864) determines the thermal reaction rateconstant as the Boltzmann average over the collision energy[12]

119896 (120573) =

1

2120587

int

infin

minusinfin119873(119864) 119890

minus120573119864119889119864

int

infin

minusinfin119890

minus120573119864119889119864

(2)

where 120573 is the inverse temperature 120573 = (119896119861119879)

minus1 (The atomicunits ℎ = 1 are used throughout) The cumulative reactionprobability can also be equivalently determined from expres-sions involving correlation functions of the flux operatoreigenstates evolved in time [12]These flux-based approachesnaturally connect to quantum and classical transition statetheories (beyond the scope of this paper)

Reactions in condensed phase are typically described bythe potential energy profile along the reaction coordinate of adouble-well character Therefore definitions of probabilitiesor rate constants related to scattering eigenstates are inappli-cable The flux-based approaches involving time-correlationfunctions are not formally guaranteed to reach a plateau (anddo not reach it in one-dimensional models) at long timeswithout inclusion of other degrees of freedom In the full-dimensional treatment the other ldquobathrdquo degrees of freedomare responsible for the wavefunction decoherence and forthe energy dissipation from the reactive degree of freedomThe plateau behavior of correlation functions is required fora rigorous definition of thermal reaction rate constants Ithas been argued though not formally proven that alreadyin two degrees of freedom oscillatory behavior of correlationfunctions is sufficiently damped with time to make the flux-based approaches work [13] Of course high-dimensionalmodels give more realistic representation of reactions incondensed phase but are often impractical If the couplingof the reaction coordinate to the bath degrees of freedom isweak or the bath degrees of freedom are not involved in 119905rearrangement of atoms then a reliable estimate of quantumtunneling probability as a characteristic of the barrier and ofthe potential energy profile along the reaction coordinate isimportant in itself Given that significance of QM tunnelingis the highest in the regime of just a few reactantproducteigenstates contributing to reactivity a simple expression forthe QM tunneling probabilities which can be implementedby solving the time-independent Schrodinger equation isuseful In the time domain treating a chemical reaction asan event proceeding from reactants to products rather thanthe transition state flux can be useful as well especially if thepotential depends on time for example mimicking the effectof substrate motion in reactions on surfaces [14]

For bound potentials the QM reaction probabilities andrate constants are usually estimated using the transition statetheory [15] or using the quasiclassicalWKB expression for thetunneling probability evaluated at the zero-point energy ofthe reactant well 119864 = 119864

0[16]

119879

QC(119864) = exp(minus2int

1199092

1199091

radic2119898 (119881 (119909) minus 119864)119889119909)

119881 (1199091) = 119881 (119909

2) = 119864

(3)

The transition state estimate is accurate for a single barrier atenergies comparable to its height while the WKB expressionis accurate at low energies when transmission from productsto reactants is justifiably neglected However the product re-gion and a possibility of a reverse reaction should be incor-porated for a more rigorous treatment the energy level split-tings measured spectroscopically are due to tunneling anddepend on all regions of the potential

Even though the approximate treatments of tunneling canbe surprisingly accurate the search for more conceptuallyrigorous expressions led to the development of less traditionalapproaches such as the stationary state decomposition forquadratic potentials [17] via the quantum Hamilton-Jacobiequation adapted for energy eigenstates [18 19] the exte-rior complex scaling method [20] and more recently thequantum instanton theory [21] ring polymer dynamics [22]and perturbative ldquobeyond instantonsrdquo correction to theWKBapproximation [23] The quantum tunneling is known to bevery sensitive to the barrier shape and energy as shownfor example in the context of enzyme catalysis by Hayand coworkers [9] Thus to assess the importance of QMtunneling we propose an adaptation of definitions of the QMtunneling probability and reaction rate constant based onexact energy levels of the bound system and exact QM tun-neling through the barrier of the corresponding scatteringsystem The expressions are obtained from the analysis ofa rectangular double-well potential (Section 2) and its unam-biguous mapping onto a scattering problem (For generalpotentials there is some ambiguity associated with the def-inition of the reactant and product regions) After modelapplications to rectangular and piecewise quadratic potentialwe use the approach to compute KIE on the reaction rateconstant of protondeuteron transfer in a model HOndashHndashCH3along a linear reaction path with realistic potential

(Section 3) Section 4 concludes

2 Formalism

The modified expression for the reaction rate constants isbased on the common functional form of the energy eigen-states for the open scattering system and the double-wellsystem (or any sequence of wells and barriers) consistingof rectangular potential steps Let us consider a one-dimen-sional open scattering system with a potential being constantin the asymptotic regions as 119909 rarr plusmninfin and arbitrary in theinteraction region referred to as the ldquobarrierrdquo region

119881 (119909) =

0 119909 lt 1199091reactant region

119881119887 (119909) 119909

1le 119909 le 119909

2barrier region

1198810

119909 gt 1199092product region

(4)

Describing scattering in the energy domain the energy eigen-states with appropriate boundary conditions such as a wave-function incoming from the left in the direction from

Journal of Theoretical Chemistry 3

reactants to products have the following asymptotic form for119864 gt 119881

0(119898 is the particle mass)

120595119864(119909)

=

exp (120580119896119903119909)

+119903 (119864) exp (minus120580119896119903119909) 119909 lt 1199091 119896119903=radic2119898119864

119905 (119864) exp (120580119896119901119909) 119909 gt 119909

2 119896119901= radic2119898 (119864 minus 119881

0)

(5)

The wavefunction in the barrier region

120595119864(119909) = 119891

119864(119909) 119909

1le 119909 le 119909

2 (6)

is an unspecified solution to the one-dimensional Schro-dinger equation which smoothly connects solutions in thetwo asymptotic regions and does not explicitly enter the ex-pression for the transmission probability 119879(119864)

119879 (119864) equiv 119873 (119864) = |119905 (119864)|

2119896119901

119896119903

(7)

For the double-well potential the energy eigenstateswith scat-tering boundary conditions [11] are undefined making (5)and (7) inapplicable Weiner [17 24] extended the conceptof scattering states to a piecewise quadratic double-wellpotential defining the incoming and outgoing boundary con-ditions in the parabolic wells These asymptotically divergingsolutions were determined from the quantum Hamilton-Jac-obi equation [25] resulting in impractical implementationbesides being limited by the functional form of the potential

Dynamics the probability of finding a particle initiallylocalized in the reactant well on the product side oscillatesin time without converging to a well-defined value unlessenergy dissipation and interaction of the reactive systemswith the environment are fully included

We will consider the rectangular double-well potentialwhich is easy to analyze in terms of the plane-wave solutionsFor such a potential 119881 shown in Figure 1

119881 (119909) =

infin 119909 le 1199090 119909 ge 119909

3walls

0 1199090lt 119909 lt 119909

1reactant region

119881119887 1199091le 119909 le 119909

2barrier region

1198810 1199092lt 119909 lt 119909

3product region

(8)

the energy eigenstates and the corresponding transmissionprobabilities for the open scattering system defined by (4)and for the full double-well potential of (8) are the sameat the energy levels 119864

119899of the full potential in the region

of finite 119881 Another feature of this bound potential is thatthe reactant and product regions are uniquely defined Thusthe expression for the thermal reaction rates of (2) can bemodified (i) using the transmission probability in the opensystem having the same barrier region as the double-wellpotential and (ii) using the energy eigenstates of the double-well potential

119896 (120573) =

1

2120587

sum119899120588119899119879 (119864119899) 119890

minus120573119864119899

sum119899120588119899119890

minus120573119864119899

(9)

minus4 minus2 0 2 4

Coordinate x [a0]

Ener

gyE

[har

tree]

4

2

0

x0 x1 x2 x3

n = 0dwn = 1dwn = 2dw

n = 0 rwn = 1 rw

Ψ1

Ψ0

Ψ2

Figure 1 The asymmetric rectangular double-well potential ofSection 31 Its three lowest eigenfunctions are shown as dashesThehorizontal lines mark the corresponding three energy levels (labeled119899 = 0 1 2 dw) and also two lowest energy levels of the reactant wellwith infinitely high walls (labeled 1198991015840 = 0 1 rw)

The summation goes over the energy eigenstates 120601119899of the

bound system

119867120601119899= 119864119899120601119899 (10)

and 120588119899is the probability of finding a particle on the reactant

side for each state 119899

120588119899= ⟨120601119899

10038161003816100381610038161003816

119875119903

10038161003816100381610038161003816

120601119899⟩ = int

119909119903

minusinfin

1003816100381610038161003816

120601119899(119909)

1003816100381610038161003816

2119889119909 (11)

Operator 119875119903projects a wavefunction onto the reactant region

defined as 119909 lt 119909119903

In case of the rectangular double well of (8) the projec-tions of the eigenstates on the reactant well are unambiguous(119909119903= 1199091)

120588119899= int

1199091

1199090

1003816100381610038161003816

120601119899 (119909)

1003816100381610038161003816

2119889119909 (12)

Generally a definition of 119875119903requires specification of the

reactant region Some possibilities are as follows (i) 119909119903is a

position of the barrier top for any 119899 and (ii) 119909119903is a classical

turning point that is 119909(119899)119903

is a root of 119881(119909(119899)119903) = 119864

119899in the

reactant wellThe first option is reasonable for simple barrierswith a singlemaximum while the second one is more generaland consistent with the WKB tunneling expression

To summarize the formalism for general potentials withreactant and product well regions the transmission probabil-ity is defined by the properties of the open scattering systemthat is by the barrier with the 119881(119909) set to constants beyondthe reactantproduct well minima 119909

0and 119909

3 respectively


minus2 minus1 0 1 2

0

05

x1 xL xR x2

Vb

V0

Pote

ntia

l [Eh

]

Coordinate [a0]

Figure 2 A double-well potential (solid line) and the correspondingto it scattering potential (dash) Coordinates 119909

1and 119909

2are the

positions of the minima in the reactant and product regionsrespectively and also define the barrier region 119909

119871and 119909

119877are

inflection points of the potential given by (20)

This is shown in Figure 2 for a piecewise quadratic continuouspotential The double-well potential simply specifies whichenergies out of the continuum of the scattering energy statesto include into the Boltzmann averaging in (9) For anasymmetric potential the reactant well may have eigenstatesbelow the asymptote119881

0of the product wellThose eigenstates

should be included into the Boltzmann averaging in (9)with the zero transmission probability The same argumentsand expressions are applicable to more complicated boundpotentials as long as the reactant and product regions arespecified

3 Examples and Discussion

In this section we examine reaction probabilities and rateconstants computed from (9) for several model systems withthe emphasis on comparison of the QM andWKB tunnelingprobabilities We also consider better-than-WKB approxi-mations relevant to each model a ldquominimalisticrdquo two-statedescription for a symmetric rectangular well (Section 31)a mixed QCQM calculation for an asymmetric well (Sec-tion 32) and a parabolic barrier approximation for a piece-wise quadratic potential (Section 33) Accuracy of transmis-sion probabilities and the energy level splittings followingfrom the quasiclassical WKB expression is examined Exam-ple of the KIE calculation for a model proton transfer systemis described in Section 34

31 Comparison of the QMWKB and Two-State Model Prob-abilities for a Symmetric Double Well Let us apply (9) toa symmetric rectangular double well and compare reactionprobabilities computed using the exact QM and approximateenergy levels and transmission probabilities For a general

rectangular barrier given by (8) without the infinitely highwalls the transmission probability is

119879 (119864)

=

4119896119903119896

2

119887119896119901

(119896119903+ 119896119901)

2

119896

2

119887+[119896

2

119903119896

2

119901+119896

2

119887(119896

2

119887minus 119896

2

119903minus 119896

2

119901)] sin2 (119896

119887119871)

for 119864 gt 119881119887

4119896119903119896

2

119887119896119901

(119896119903+ 119896119901)

2

119896

2

119887+[119896

2

119903119896

2

119901+119896

2

119887(119896

2

119887+ 119896

2

119903+ 119896

2

119901)] sinh2 (119896

119887119871)

for 1198810lt 119864 lt 119881

119887

(13)

In the expression above 119896119887= radic2119898|119881

119887minus 119864| and 119871 = 119909

2minus 1199091

is the barrier width 119896119903and 119896

119901are given in (5) Analytical

expression for QM transmission probability 119879(119864) allowshigh precision calculations performed with Maple [26] Theparticle mass is119898 = 1 a u

We are considering a symmetric barrier of fixed height119881119887

and variablewidthThewidth of thewells119889119889 = 1199091minus1199090= 1199093minus

1199092 is fixed at 119889 = 2119886

0 The barrier height 119881

119887is set to 4119864

ℎ The

barrier width 119871 is in the range of 119871 = [025 80]1198860 In the limit

of infinitely wide barrier the reactant and product wells havetwo bound states The transmission probabilities 119879(119864) andthe energy level splitting Δ associated with both asymptoticenergy levels are computed by considering two lowest pairsof energy eigenstates of the finite-width potentials ExactQM transmission probabilities are compared with thoseobtained using the quasiclassical WKB approach (labeledQC) and the two-state description (labeled 2119904) of this systemThe WKB tunneling probabilities are used in comparisoneven though the WKB method is inaccurate for potentialswith discontinuous derivatives because of its routine use intunneling calculations for chemical systems

In a symmetric potential the ground and first excitedstates have evenodd symmetry so the half-sumhalf-difference of the two states represents a particle located on thereactantproduct side Therefore for consistent comparisonof exact and approximate expressions the QM transmission119879

QM is defined as

119879

QM=

1205880119879 (1198640) + 1205881119879 (1198641)

1205880+ 1205881

(14)

to account for localization of the ldquoinitialrdquo wavefunction inthe reactant well The difference between (9) and (14) isthat the latter neglects the effect of temperature on theoccupation of 119899 = 0 and 119899 = 1 states This omission isaccurate in the limit of small energy level splittings Equation(14) is also consistent with the two-state representation of awavefunction in the basis formed by the ground eigenstatesof reactant and product wells 120601

119903and 120601

119901 respectively These

functions are the lowest energy eigenfunctions of the reactantand product wells in the limit of an infinitely wide barrier(The eigenfunctions of the isolated wells with infinitely highwalls cannot be used for a rectangular potential becausesuch eigenfunctions will have zero overlap) In the two-state


6 80

0

05

1

15

2

25

42Barrier width L [a0]

TQCTQM

T2sTQM

Figure 3 Tunneling in the symmetric rectangular double-wellpotential of Section 31 Ratios of approximate transmission prob-abilities that is the quasiclassical WKB (squaresdash) and the two-state representation (circlessolid lines) to the exact QM probabilityare given as functions of the barrier width

representation the energies of the ground and excited statesare the generalized eigenvalues of the 2-by-2 HamiltonianmatrixHwith the overlapmatrix S For a symmetric potentialthe matrix elements are

ℎ11= ℎ22= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119903⟩ ℎ12= ℎ21= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119901⟩ (15)

11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)

H1205952119904119894= 119864

2119904

119894S1205952119904119894119894 = 0 1 (17)

The energy level splitting is Δ2119904 = 11986421199041minus 119864

2119904

0

Following [24] we use the quasiclassical relation betweenthe energy level splitting and the tunneling probability

1198641minus 1198640= Δ =

21198640

120587

radic119879 (1198640) (18)

with the ground state energy used instead of the frequency ofthe harmonic reactant well In the two-state representationthe energy level splitting Δ2119904 following from (17) yields anestimate for the tunneling 1198792119904 The same (18) gives estimatesof the quasiclassical energy level splitting ΔQC once thequasiclassical WKB tunneling probability 119879QC is computedfrom (3) Two pairs of states of the full potential 119899 = 0 119899 = 1and 119899 = 2 119899 = 3 are analyzed the higher energy states 119899 = 2and 119899 = 3 are described in the basis of the first excited statesof the reactant and product wells

Table 1 lists the ground and first excited energy levels ob-tained exactly using quasiclassical WKB approximation andwithin the two-state approximation labeled QM QC and2119904 respectively are listed in the upper half of Table 1The samequantities for the next highest pair of levels (Δ = 119864

3minus1198642) are

given in the lower half of Table 1 In the quasiclassical WKBtreatment the energy levels are defined by the width of the

1 2 3 4

Energy [Eh]

00001

1e minus 08

Tunn

elin

gT

(E)

L = 2 QML = 2 QC

E0

E1

Figure 4 Transmission through the symmetric rectangular barrierof Section 31 of the width 119871 = 2119886

0 Exact QM and QC WKB

probabilities are shown as a solid line and a dash respectively Theenergies of the ground and first excited states for the well width119889 = 2119886

0are indicated with dot-dashes

reactant well and therefore do not depend on the barrierwidth The ground state energy in the two-state approx-imation is remarkably accurate even for narrow barrierswhile the accuracy of 119899 = 2 level deteriorates for smallerbarrier widths The ratios of the quasiclassical and two-stateestimates of tunneling probability to the exact QM values asa function of the barrier width are plotted in Figure 3 for thelowest pair of states The agreement between 1198792119904 and 119879QM isexcellent for the barrier width 119871 gt 10 119886

0 The quasiclassical

WKB results underestimate the tunneling probability by afactor of 2This discrepancy can be understood by comparingquasiclassical WKB and QM tunneling at the energies ofthe asymptotic eigenstates shown in Figure 4 for the barrierwidth 119871 = 2119886

0 The transmission probabilities at these ener-

gies differ by a factor of 2 For a wider well of the width119889 = 4119886

0(not shown here) the ground state energy happens to

be near the intersection of the QM and quasiclassical WKBtunneling curves at 119864 = 02599119864

ℎ which coincidentally

yields much better agreement than the results shown inFigure 3 Therefore the accuracy of the quasiclassical resultsdepends on the reactant well width even if the accuracy ofthe quasiclassical tunneling probability does not At very lowenergies the discrepancy between the QM and QC tunnelingis in orders of magnitude but fortunately the energy regimebelow the zero-point energy of reactants does not contributeto the tunneling in a double-well system

To summarize the two-state tunneling probabilities forthe lowest pair of eigenstates are remarkably accurate exceptfor very narrow barriers and may provide useful estimatesif the rest of the states are of much higher energy Thebarrier can be considered sufficiently wide when the overlapof the reactant and product well eigenstates ⟨120601

119901|120601119903⟩ is less

than a few percent (5 percent in the current example)The QC expression (18) relating the energy level splitting


Table1Energylevelssplittin

gsand

tunn

elingp

robabilityfor

the119899=01and119899=23pairofstatesofthesym

metric

rectangu

lard

ouble-wellpotentia

lofSectio

n31Th

ereactantp

rodu

ctstates

overlap

ofthetwo-stated

escriptio

nisgivenin

thelastcolum

nTh

eenergylevelsandlevelsplittings

areg

iven

in119864ℎN

umbersin

parenthesesa

rethep

owerso

f10

119871[1198860]

119864

QM0

Δ

QM

119879

QM

119864

QC0

Δ

QC

119879

QC

119864

2s 0Δ

2119904

119879

2119904

⟨120601119901|120601119903⟩

Forthe119899=01pairof

energy

states

808800

1919

(minus9)

1205

(minus17)

08800

1174

(minus9)

439

(minus18)

08800

1913

(minus9)

117(minus17)

3554(minus9)

408800

4192(minus5)

5752(minus9)

08800

2564(minus5)

2095(minus9)

08800

4210(minus5)

564

6(minus9)

440

8(minus5)

208770

6196(minus3)

1257

(minus4)

08800

3790(minus3)

4577(minus5)

08769

6224(minus3)

1243

(minus4)

4035(minus3)

108422

00753

00185

08800

4608(minus3)

6765(minus3)

08425

7564

(minus2)

1989

(minus2)

3398(minus2)

05

07433

02618

02105

08800

01607

8225(minus2)

07483

02637

03064

00922

025

060

9804881

05900

08800

03000

02868

06289

04906

1502

01476


energy

states

833053

1099

(minus4)

1481

(minus8)

33053

1690

(minus4)

6450(minus9)

33053

0974(minus4)

2141(minus9)

4712(minus4)

432993

1227

(minus2)

1848

(minus4)

33053

1886

(minus2)

8031(minus5)

32999

1087

(minus2)

2676(minus5)

2790(minus2)

232477

01334

2144(minus2)

33053

01992

8962(minus3)

32472

0114

63071(minus3)

01644

131337

04856

02267

33053

064

7400947

30860

03487

00315

03225

05

300

6810

078

06166

33053

11672

03077

28262

05329

00877

03905

025

29086

14864

08529

33053

15672

05547

25604

06101

01401

03961


Table 2 Thermal rate constants 119896(119879) for the asymmetric rectangular double well of Section 32 The first column lists the temperature thatis 119896119861119879 in units of the barrier height119881

119887The thermal rate constants obtained using (9) using exact energy levels and QM and QC transmission

probabilities are given in the second and third columns respectively The last column contains the thermal rate constants obtained from QCtunneling probability and the energy levels of the reactant potential with infinitely high walls

119896119861119879119881119887

119879

QM119864

QM119899

119879

QC119864

QM119899

119879

QC119864

QC119899

001 691 (minus13) 265 (minus13) 738 (minus15)002 101 (minus8) 380 (minus9) 778 (minus10)004 123 (minus6) 464 (minus7) 252 (minus7)01 399 (minus5) 267 (minus5) 196 (minus5)02 131 (minus3) 197 (minus3) 241 (minus3)04 195 (minus2) 266 (minus2) 364 (minus2)10 0134 0159 019820 0286 0320 037140 0450 0483 0533

and tunneling is inaccurate for the higher energy pairs ofldquosplitrdquo levelsThe accuracy of the QC approximation dependson both the accuracy of QC tunneling probability anddiscrepancy between the energy levels of the full potentialand of the asymptotic reactant well For one-dimensionalsystems the use of exact energy eigenstates of the full potentialrather than those of an isolated reactant well in (9) ismore appropriate even with approximate calculation of thetunneling probability

32 Asymmetric Rectangular Barrier Now let us consider anasymmetric rectangular double-well potential given by (8)and sketched in Figure 1 There are 7 states with energiesbelow the barrier top The ground and first excited statesare localized on the reactant and product sides respectivelyThe eigenstates of the full potential with even quantum num-bers correlate with the eigenstates of the reactant well thedifference in energies of the full and asymptotic potentialsincreases with the quantum number The parameter valuesfor the potential are 119881

119887= 40119864

ℎ 1198810= 04119864

ℎ and 119871 = 10119886

0

The well width is 119889 = 401198860 The particle mass is119898 = 1 a u

Exact QM energy levels and eigenfunctions needed tocompute projections on the reactant well are obtained aseigenvalues and eigenvectors of the Hamiltonian matrix inthe Colbert-Miller discrete variable representation [27]

Exact QM transmission probability is given by (13) Qua-siclassical transmission probability is defined by (3) and set to(1) for energies above the barrier top The exact and approxi-mate rate constants computed from (9) are listed inTable 2 fortemperatures measured in units of the barrier top 119881

119887 Results

of the fully QM fully QC and mixed QCQM (QC trans-mission evaluated at exact QM energy levels) descriptions areshown in Figure 5 and in Table 2 At the lowest temperatureequivalent to 1 of the barrier height the QC rate constantis 93 times lower than the QM result while the mixedQCQM estimate is only 2 times lower At high temperatureswhen the high energy eigenstates significantly contribute tothe reaction rate constant the discrepancy between exactand approximate rate constants becomes smaller with themixed QCQM estimate being closer to the exact result at allenergies One concludes that generally the accuracy of 119896(119879)

0 005 01 015 02

Rate

cons

tant

Exact QM QM levelsQC T(E)Reactant levelsQC T(E)

k(T

)

0

02

04

06

08

Temperature [Vb]

0 10 20 30

00001

1e minus 08

1e minus 12

1e minus 16

Figure 5 Transmission through the asymmetric rectangular barrierof Section 32 The thermal reaction rate constant is given as afunction of temperature measured in units of the barrier height[119896119861119879119881119887] where119881

119887= 4119864ℎThe rate constants are obtained using the

exact QM transmission probability and energy levels (circlessolidline) the QC transmission probability evaluated at the exact QMenergy levels (trianglesdot-dash) and at the QC energy levels of theisolated reactant well (squaresdash) The insert shows 119896(119879) on thelinear scale for wide range of temperatures

depends both on the accuracy ofQC transmission probability119879

QC and on the difference between exact and asymptotic-wellenergy levels Using exact QM energy levels and functions inconjunction with quasiclassical transmission probabilities ismore accurate than the fully QC description

Note that for the rectangular potential considered here119879

QC(119864) = 1 for 119864 gt 119881

119887is inaccurate because QC approxi-

mation does not describe above-the-barrier reflection leadingto oscillations of 119879QM For general potentials the reactionprobability may be computed using for example the time-dependent QM wave packet approach [28] used in the next


example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values

33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission

119879

QMpar (119864) = (1 + exp(minus

2120587119864

119908119887

))

minus1

119908119887=radic

119896119887

119898

(19)

through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows

119881 (119909)

=

119896(119909 minus 1199091)

2

2

119909 lt 119909119871reactant region

minus

119896119887119909

2

2

+ 119881119887

119909119871le 119909 le 119909

119877barrier top region

119896(119909 minus 1199092)

2

2

+ 1198810119909 gt 119909

119877product region

(20)

with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909

119877mdash

the potential and its first derivative is continuous function at119909 = 119909

119871and 119909 = 119909

119877 Here a symmetric double well 119881

0= 0

is considered the remaining parameter values are 1199092= minus1199091

= 11198860 119896 = 119896

119887= 80119864

ℎ119886

minus2

0 119881119887= 20119864

ℎ and 119909

119877= minus119909119871= 05119886

0

The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881

119887would be equivalent to

03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature

The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878

119903119901(119864)|

2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878

119903119901(119864) describes trans-

mission from reactants to products

119878119903119901(119864) =

(2120587)

minus1

120578

lowast

119901(119864) 120578119903 (

119864)

int

infin

minusinfin

⟨120601

minus

119901

100381610038161003816100381610038161003816

119890

minus120580119905100381610038161003816100381610038161003816

120601

+

119903⟩ 119890

120580119864119905119889119905 (21)

The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+

119903⟩

taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus

119901⟩

is placed in the product region of the potential The time-dependent overlap of evolving |120601+

119903⟩ with the stationary ⟨120601minus

119901|

or correlation function 119862(119905) = ⟨120601

minus

119901(119909 0) | 120601

+

119903(119909 119905)⟩ is

computed and Fourier-transformed into the energy domain

The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy

120578119903119901(119864) = radic

119898

2120587119901

int

infin

minusinfin

119890

minus120580119901119909120601

plusmn

119903119901(119909) 119889119909 119901 =

radic2119898119864

(22)

A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures

The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881

1198875 = 4119864

ℎ(the

barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864

ℎ This energy range however

is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30

34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH

3system for a

constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘

One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape

We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels


00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)

Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896

119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)

Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO

distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896

119867119896119863(KIE) using exact QM formulation

in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top

The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896

119867119896119863should be largest at low

temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in


Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and

fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns

119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]

27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671

lowast mdash mdash


lowast mdash mdash


lowast mdash mdashAsterisk marks KIEQCKIEQM

the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude

When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864

119899rather than the

energy corresponding to 120588119899as we have defined it in (11) In

the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error

4 Conclusions

Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy

levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates

The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH

3system with

constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions

References

[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011

[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002

[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010

[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008

[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002

[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011

[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013

[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009

[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010

[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012

[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982

[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983

[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997

[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996

[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974

[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999

[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978

[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927

[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004

[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990

[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005

[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005

[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010

[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978

[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952

[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom

[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992

[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993

[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988

[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982

[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


Furthermore 119873(119864) determines the thermal reaction rateconstant as the Boltzmann average over the collision energy[12]

119896 (120573) =

1

2120587

int

infin

minusinfin119873(119864) 119890

minus120573119864119889119864

int

infin

minusinfin119890

minus120573119864119889119864

(2)

where 120573 is the inverse temperature 120573 = (119896119861119879)

minus1 (The atomicunits ℎ = 1 are used throughout) The cumulative reactionprobability can also be equivalently determined from expres-sions involving correlation functions of the flux operatoreigenstates evolved in time [12]These flux-based approachesnaturally connect to quantum and classical transition statetheories (beyond the scope of this paper)

Reactions in condensed phase are typically described bythe potential energy profile along the reaction coordinate of adouble-well character Therefore definitions of probabilitiesor rate constants related to scattering eigenstates are inappli-cable The flux-based approaches involving time-correlationfunctions are not formally guaranteed to reach a plateau (anddo not reach it in one-dimensional models) at long timeswithout inclusion of other degrees of freedom In the full-dimensional treatment the other ldquobathrdquo degrees of freedomare responsible for the wavefunction decoherence and forthe energy dissipation from the reactive degree of freedomThe plateau behavior of correlation functions is required fora rigorous definition of thermal reaction rate constants Ithas been argued though not formally proven that alreadyin two degrees of freedom oscillatory behavior of correlationfunctions is sufficiently damped with time to make the flux-based approaches work [13] Of course high-dimensionalmodels give more realistic representation of reactions incondensed phase but are often impractical If the couplingof the reaction coordinate to the bath degrees of freedom isweak or the bath degrees of freedom are not involved in 119905rearrangement of atoms then a reliable estimate of quantumtunneling probability as a characteristic of the barrier and ofthe potential energy profile along the reaction coordinate isimportant in itself Given that significance of QM tunnelingis the highest in the regime of just a few reactantproducteigenstates contributing to reactivity a simple expression forthe QM tunneling probabilities which can be implementedby solving the time-independent Schrodinger equation isuseful In the time domain treating a chemical reaction asan event proceeding from reactants to products rather thanthe transition state flux can be useful as well especially if thepotential depends on time for example mimicking the effectof substrate motion in reactions on surfaces [14]

For bound potentials the QM reaction probabilities andrate constants are usually estimated using the transition statetheory [15] or using the quasiclassicalWKB expression for thetunneling probability evaluated at the zero-point energy ofthe reactant well 119864 = 119864

0[16]

119879

QC(119864) = exp(minus2int

1199092

1199091

radic2119898 (119881 (119909) minus 119864)119889119909)

119881 (1199091) = 119881 (119909

2) = 119864

(3)

The transition state estimate is accurate for a single barrier atenergies comparable to its height while the WKB expressionis accurate at low energies when transmission from productsto reactants is justifiably neglected However the product re-gion and a possibility of a reverse reaction should be incor-porated for a more rigorous treatment the energy level split-tings measured spectroscopically are due to tunneling anddepend on all regions of the potential

Even though the approximate treatments of tunneling canbe surprisingly accurate the search for more conceptuallyrigorous expressions led to the development of less traditionalapproaches such as the stationary state decomposition forquadratic potentials [17] via the quantum Hamilton-Jacobiequation adapted for energy eigenstates [18 19] the exte-rior complex scaling method [20] and more recently thequantum instanton theory [21] ring polymer dynamics [22]and perturbative ldquobeyond instantonsrdquo correction to theWKBapproximation [23] The quantum tunneling is known to bevery sensitive to the barrier shape and energy as shownfor example in the context of enzyme catalysis by Hayand coworkers [9] Thus to assess the importance of QMtunneling we propose an adaptation of definitions of the QMtunneling probability and reaction rate constant based onexact energy levels of the bound system and exact QM tun-neling through the barrier of the corresponding scatteringsystem The expressions are obtained from the analysis ofa rectangular double-well potential (Section 2) and its unam-biguous mapping onto a scattering problem (For generalpotentials there is some ambiguity associated with the def-inition of the reactant and product regions) After modelapplications to rectangular and piecewise quadratic potentialwe use the approach to compute KIE on the reaction rateconstant of protondeuteron transfer in a model HOndashHndashCH3along a linear reaction path with realistic potential

(Section 3) Section 4 concludes

2 Formalism

The modified expression for the reaction rate constants isbased on the common functional form of the energy eigen-states for the open scattering system and the double-wellsystem (or any sequence of wells and barriers) consistingof rectangular potential steps Let us consider a one-dimen-sional open scattering system with a potential being constantin the asymptotic regions as 119909 rarr plusmninfin and arbitrary in theinteraction region referred to as the ldquobarrierrdquo region

119881 (119909) =

0 119909 lt 1199091reactant region

119881119887 (119909) 119909

1le 119909 le 119909

2barrier region

1198810

119909 gt 1199092product region

(4)

Describing scattering in the energy domain the energy eigen-states with appropriate boundary conditions such as a wave-function incoming from the left in the direction from




120595119864(119909)

=

exp (120580119896119903119909)


119905 (119864) exp (120580119896119901119909) 119909 gt 119909

2 119896119901= radic2119898 (119864 minus 119881

0)

(5)


120595119864(119909) = 119891

119864(119909) 119909

1le 119909 le 119909

2 (6)


119879 (119864) equiv 119873 (119864) = |119905 (119864)|

2119896119901

119896119903

(7)




119881 (119909) =

infin 119909 le 1199090 119909 ge 119909

3walls

0 1199090lt 119909 lt 119909

1reactant region

119881119887 1199091le 119909 le 119909

2barrier region

1198810 1199092lt 119909 lt 119909

3product region

(8)




119896 (120573) =

1

2120587

sum119899120588119899119879 (119864119899) 119890

minus120573119864119899

sum119899120588119899119890

minus120573119864119899

(9)

minus4 minus2 0 2 4

Coordinate x [a0]

Ener

gyE

[har

tree]

4

2

0

x0 x1 x2 x3


n = 0 rwn = 1 rw

Ψ1

Ψ0

Ψ2



bound system

119867120601119899= 119864119899120601119899 (10)



120588119899= ⟨120601119899

10038161003816100381610038161003816

119875119903

10038161003816100381610038161003816

120601119899⟩ = int

119909119903

minusinfin

1003816100381610038161003816

120601119899(119909)

1003816100381610038161003816

2119889119909 (11)


defined as 119909 lt 119909119903


120588119899= int

1199091

1199090

1003816100381610038161003816

120601119899 (119909)

1003816100381610038161003816

2119889119909 (12)





is a root of 119881(119909(119899)119903) = 119864

119899in the



0and 119909

3 respectively


minus2 minus1 0 1 2

0

05

x1 xL xR x2

Vb

V0

Pote

ntia

l [Eh

]

Coordinate [a0]


1and 119909

2are the


119871and 119909

119877are









119879 (119864)

=

4119896119903119896

2

119887119896119901

(119896119903+ 119896119901)

2

119896

2

119887+[119896

2

119903119896

2

119901+119896

2

119887(119896

2

119887minus 119896

2

119903minus 119896

2

119901)] sin2 (119896

119887119871)

for 119864 gt 119881119887

4119896119903119896

2

119887119896119901

(119896119903+ 119896119901)

2

119896

2

119887+[119896

2

119903119896

2

119901+119896

2

119887(119896

2

119887+ 119896

2

119903+ 119896

2

119901)] sinh2 (119896

119887119871)

for 1198810lt 119864 lt 119881

119887

(13)


119887minus 119864| and 119871 = 119909

2minus 1199091






1199092 is fixed at 119889 = 2119886


119887is set to 4119864

ℎ The




QM is defined as

119879

QM=

1205880119879 (1198640) + 1205881119879 (1198641)

1205880+ 1205881

(14)


119903and 120601




6 80

0

05

1

15

2

25


TQCTQM

T2sTQM



ℎ11= ℎ22= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119903⟩ ℎ12= ℎ21= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119901⟩ (15)

11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)

H1205952119904119894= 119864

2119904

119894S1205952119904119894119894 = 0 1 (17)


2119904

0


1198641minus 1198640= Δ =

21198640

120587

radic119879 (1198640) (18)



3minus1198642) are


1 2 3 4

Energy [Eh]

00001

1e minus 08

Tunn

elin

gT

(E)

L = 2 QML = 2 QC

E0

E1















119901|120601119903⟩ is less




gsand

tunn

elingp

robabilityfor


metric

rectangu

lard

ouble-wellpotentia

lofSectio

n31Th

ereactantp

rodu

ctstates

overlap

ofthetwo-stated

escriptio

nisgivenin

thelastcolum

nTh


areg

iven

in119864ℎN

umbersin

parenthesesa

rethep

owerso

f10

119871[1198860]

119864

QM0

Δ

QM

119879

QM

119864

QC0

Δ

QC

119879

QC

119864

2s 0Δ

2119904

119879

2119904

⟨120601119901|120601119903⟩


energy

states

808800

1919

(minus9)

1205

(minus17)

08800

1174

(minus9)

439

(minus18)

08800

1913

(minus9)

117(minus17)

3554(minus9)

408800

4192(minus5)

5752(minus9)

08800

2564(minus5)

2095(minus9)

08800

4210(minus5)

564

6(minus9)

440

8(minus5)

208770

6196(minus3)

1257

(minus4)

08800

3790(minus3)

4577(minus5)

08769

6224(minus3)

1243

(minus4)

4035(minus3)

108422

00753

00185

08800

4608(minus3)

6765(minus3)

08425

7564

(minus2)

1989

(minus2)

3398(minus2)

05

07433

02618

02105

08800

01607

8225(minus2)

07483

02637

03064

00922

025

060

9804881

05900

08800

03000

02868

06289

04906

1502

01476


energy

states

833053

1099

(minus4)

1481

(minus8)

33053

1690

(minus4)

6450(minus9)

33053

0974(minus4)

2141(minus9)

4712(minus4)

432993

1227

(minus2)

1848

(minus4)

33053

1886

(minus2)

8031(minus5)

32999

1087

(minus2)

2676(minus5)

2790(minus2)

232477

01334

2144(minus2)

33053

01992

8962(minus3)

32472

0114

63071(minus3)

01644

131337

04856

02267

33053

064

7400947

30860

03487

00315

03225

05

300

6810

078

06166

33053

11672

03077

28262

05329

00877

03905

025

29086

14864

08529

33053

15672

05547

25604

06101

01401

03961





119896119861119879119881119887

119879

QM119864

QM119899

119879

QC119864

QM119899

119879

QC119864

QC119899




119887= 40119864

ℎ 1198810= 04119864

ℎ and 119871 = 10119886

0




119887 Results


0 005 01 015 02

Rate

cons

tant


k(T

)

0

02

04

06

08

Temperature [Vb]

0 10 20 30

00001

1e minus 08

1e minus 12

1e minus 16







QC(119864) = 1 for 119864 gt 119881






119879


2120587119864

119908119887

))

minus1

119908119887=radic

119896119887

119898

(19)


119881 (119909)

=

119896(119909 minus 1199091)

2

2


minus

119896119887119909

2

2

+ 119881119887

119909119871le 119909 le 119909


119896(119909 minus 1199092)

2

2

+ 1198810119909 gt 119909


(20)


119877mdash


119871and 119909 = 119909


0= 0


= 11198860 119896 = 119896

119887= 80119864

ℎ119886

minus2

0 119881119887= 20119864

ℎ and 119909

119877= minus119909119871= 05119886

0





119903119901(119864)|


119903119901(119864) describes trans-


119878119903119901(119864) =

(2120587)

minus1

120578

lowast

119901(119864) 120578119903 (

119864)

int

infin

minusinfin

⟨120601

minus

119901

100381610038161003816100381610038161003816

119890

minus120580119905100381610038161003816100381610038161003816

120601

+

119903⟩ 119890

120580119864119905119889119905 (21)


119903⟩


119901⟩



119901|


minus

119901(119909 0) | 120601

+

119903(119909 119905)⟩ is



120578119903119901(119864) = radic

119898

2120587119901

int

infin

minusinfin

119890

minus120580119901119909120601

plusmn

119903119901(119909) 119889119909 119901 =

radic2119898119864

(22)



1198875 = 4119864

ℎ(the





3system for a





00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)


119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)











119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




120595119864(119909)

=

exp (120580119896119903119909)


119905 (119864) exp (120580119896119901119909) 119909 gt 119909

2 119896119901= radic2119898 (119864 minus 119881

0)

(5)


120595119864(119909) = 119891

119864(119909) 119909

1le 119909 le 119909

2 (6)


119879 (119864) equiv 119873 (119864) = |119905 (119864)|

2119896119901

119896119903

(7)




119881 (119909) =

infin 119909 le 1199090 119909 ge 119909

3walls

0 1199090lt 119909 lt 119909

1reactant region

119881119887 1199091le 119909 le 119909

2barrier region

1198810 1199092lt 119909 lt 119909

3product region

(8)




119896 (120573) =

1

2120587

sum119899120588119899119879 (119864119899) 119890

minus120573119864119899

sum119899120588119899119890

minus120573119864119899

(9)

minus4 minus2 0 2 4

Coordinate x [a0]

Ener

gyE

[har

tree]

4

2

0

x0 x1 x2 x3


n = 0 rwn = 1 rw

Ψ1

Ψ0

Ψ2



bound system

119867120601119899= 119864119899120601119899 (10)



120588119899= ⟨120601119899

10038161003816100381610038161003816

119875119903

10038161003816100381610038161003816

120601119899⟩ = int

119909119903

minusinfin

1003816100381610038161003816

120601119899(119909)

1003816100381610038161003816

2119889119909 (11)


defined as 119909 lt 119909119903


120588119899= int

1199091

1199090

1003816100381610038161003816

120601119899 (119909)

1003816100381610038161003816

2119889119909 (12)





is a root of 119881(119909(119899)119903) = 119864

119899in the



0and 119909

3 respectively


minus2 minus1 0 1 2

0

05

x1 xL xR x2

Vb

V0

Pote

ntia

l [Eh

]

Coordinate [a0]


1and 119909

2are the


119871and 119909

119877are









119879 (119864)

=

4119896119903119896

2

119887119896119901

(119896119903+ 119896119901)

2

119896

2

119887+[119896

2

119903119896

2

119901+119896

2

119887(119896

2

119887minus 119896

2

119903minus 119896

2

119901)] sin2 (119896

119887119871)

for 119864 gt 119881119887

4119896119903119896

2

119887119896119901

(119896119903+ 119896119901)

2

119896

2

119887+[119896

2

119903119896

2

119901+119896

2

119887(119896

2

119887+ 119896

2

119903+ 119896

2

119901)] sinh2 (119896

119887119871)

for 1198810lt 119864 lt 119881

119887

(13)


119887minus 119864| and 119871 = 119909

2minus 1199091






1199092 is fixed at 119889 = 2119886


119887is set to 4119864

ℎ The




QM is defined as

119879

QM=

1205880119879 (1198640) + 1205881119879 (1198641)

1205880+ 1205881

(14)


119903and 120601




6 80

0

05

1

15

2

25


TQCTQM

T2sTQM



ℎ11= ℎ22= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119903⟩ ℎ12= ℎ21= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119901⟩ (15)

11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)

H1205952119904119894= 119864

2119904

119894S1205952119904119894119894 = 0 1 (17)


2119904

0


1198641minus 1198640= Δ =

21198640

120587

radic119879 (1198640) (18)



3minus1198642) are


1 2 3 4

Energy [Eh]

00001

1e minus 08

Tunn

elin

gT

(E)

L = 2 QML = 2 QC

E0

E1















119901|120601119903⟩ is less




gsand

tunn

elingp

robabilityfor


metric

rectangu

lard

ouble-wellpotentia

lofSectio

n31Th

ereactantp

rodu

ctstates

overlap

ofthetwo-stated

escriptio

nisgivenin

thelastcolum

nTh


areg

iven

in119864ℎN

umbersin

parenthesesa

rethep

owerso

f10

119871[1198860]

119864

QM0

Δ

QM

119879

QM

119864

QC0

Δ

QC

119879

QC

119864

2s 0Δ

2119904

119879

2119904

⟨120601119901|120601119903⟩


energy

states

808800

1919

(minus9)

1205

(minus17)

08800

1174

(minus9)

439

(minus18)

08800

1913

(minus9)

117(minus17)

3554(minus9)

408800

4192(minus5)

5752(minus9)

08800

2564(minus5)

2095(minus9)

08800

4210(minus5)

564

6(minus9)

440

8(minus5)

208770

6196(minus3)

1257

(minus4)

08800

3790(minus3)

4577(minus5)

08769

6224(minus3)

1243

(minus4)

4035(minus3)

108422

00753

00185

08800

4608(minus3)

6765(minus3)

08425

7564

(minus2)

1989

(minus2)

3398(minus2)

05

07433

02618

02105

08800

01607

8225(minus2)

07483

02637

03064

00922

025

060

9804881

05900

08800

03000

02868

06289

04906

1502

01476


energy

states

833053

1099

(minus4)

1481

(minus8)

33053

1690

(minus4)

6450(minus9)

33053

0974(minus4)

2141(minus9)

4712(minus4)

432993

1227

(minus2)

1848

(minus4)

33053

1886

(minus2)

8031(minus5)

32999

1087

(minus2)

2676(minus5)

2790(minus2)

232477

01334

2144(minus2)

33053

01992

8962(minus3)

32472

0114

63071(minus3)

01644

131337

04856

02267

33053

064

7400947

30860

03487

00315

03225

05

300

6810

078

06166

33053

11672

03077

28262

05329

00877

03905

025

29086

14864

08529

33053

15672

05547

25604

06101

01401

03961





119896119861119879119881119887

119879

QM119864

QM119899

119879

QC119864

QM119899

119879

QC119864

QC119899




119887= 40119864

ℎ 1198810= 04119864

ℎ and 119871 = 10119886

0




119887 Results


0 005 01 015 02

Rate

cons

tant


k(T

)

0

02

04

06

08

Temperature [Vb]

0 10 20 30

00001

1e minus 08

1e minus 12

1e minus 16







QC(119864) = 1 for 119864 gt 119881






119879


2120587119864

119908119887

))

minus1

119908119887=radic

119896119887

119898

(19)


119881 (119909)

=

119896(119909 minus 1199091)

2

2


minus

119896119887119909

2

2

+ 119881119887

119909119871le 119909 le 119909


119896(119909 minus 1199092)

2

2

+ 1198810119909 gt 119909


(20)


119877mdash


119871and 119909 = 119909


0= 0


= 11198860 119896 = 119896

119887= 80119864

ℎ119886

minus2

0 119881119887= 20119864

ℎ and 119909

119877= minus119909119871= 05119886

0





119903119901(119864)|


119903119901(119864) describes trans-


119878119903119901(119864) =

(2120587)

minus1

120578

lowast

119901(119864) 120578119903 (

119864)

int

infin

minusinfin

⟨120601

minus

119901

100381610038161003816100381610038161003816

119890

minus120580119905100381610038161003816100381610038161003816

120601

+

119903⟩ 119890

120580119864119905119889119905 (21)


119903⟩


119901⟩



119901|


minus

119901(119909 0) | 120601

+

119903(119909 119905)⟩ is



120578119903119901(119864) = radic

119898

2120587119901

int

infin

minusinfin

119890

minus120580119901119909120601

plusmn

119903119901(119909) 119889119909 119901 =

radic2119898119864

(22)



1198875 = 4119864

ℎ(the





3system for a





00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)


119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)











119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


minus2 minus1 0 1 2

0

05

x1 xL xR x2

Vb

V0

Pote

ntia

l [Eh

]

Coordinate [a0]


1and 119909

2are the


119871and 119909

119877are









119879 (119864)

=

4119896119903119896

2

119887119896119901

(119896119903+ 119896119901)

2

119896

2

119887+[119896

2

119903119896

2

119901+119896

2

119887(119896

2

119887minus 119896

2

119903minus 119896

2

119901)] sin2 (119896

119887119871)

for 119864 gt 119881119887

4119896119903119896

2

119887119896119901

(119896119903+ 119896119901)

2

119896

2

119887+[119896

2

119903119896

2

119901+119896

2

119887(119896

2

119887+ 119896

2

119903+ 119896

2

119901)] sinh2 (119896

119887119871)

for 1198810lt 119864 lt 119881

119887

(13)


119887minus 119864| and 119871 = 119909

2minus 1199091






1199092 is fixed at 119889 = 2119886


119887is set to 4119864

ℎ The




QM is defined as

119879

QM=

1205880119879 (1198640) + 1205881119879 (1198641)

1205880+ 1205881

(14)


119903and 120601




6 80

0

05

1

15

2

25


TQCTQM

T2sTQM



ℎ11= ℎ22= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119903⟩ ℎ12= ℎ21= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119901⟩ (15)

11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)

H1205952119904119894= 119864

2119904

119894S1205952119904119894119894 = 0 1 (17)


2119904

0


1198641minus 1198640= Δ =

21198640

120587

radic119879 (1198640) (18)



3minus1198642) are


1 2 3 4

Energy [Eh]

00001

1e minus 08

Tunn

elin

gT

(E)

L = 2 QML = 2 QC

E0

E1















119901|120601119903⟩ is less




gsand

tunn

elingp

robabilityfor


metric

rectangu

lard

ouble-wellpotentia

lofSectio

n31Th

ereactantp

rodu

ctstates

overlap

ofthetwo-stated

escriptio

nisgivenin

thelastcolum

nTh


areg

iven

in119864ℎN

umbersin

parenthesesa

rethep

owerso

f10

119871[1198860]

119864

QM0

Δ

QM

119879

QM

119864

QC0

Δ

QC

119879

QC

119864

2s 0Δ

2119904

119879

2119904

⟨120601119901|120601119903⟩


energy

states

808800

1919

(minus9)

1205

(minus17)

08800

1174

(minus9)

439

(minus18)

08800

1913

(minus9)

117(minus17)

3554(minus9)

408800

4192(minus5)

5752(minus9)

08800

2564(minus5)

2095(minus9)

08800

4210(minus5)

564

6(minus9)

440

8(minus5)

208770

6196(minus3)

1257

(minus4)

08800

3790(minus3)

4577(minus5)

08769

6224(minus3)

1243

(minus4)

4035(minus3)

108422

00753

00185

08800

4608(minus3)

6765(minus3)

08425

7564

(minus2)

1989

(minus2)

3398(minus2)

05

07433

02618

02105

08800

01607

8225(minus2)

07483

02637

03064

00922

025

060

9804881

05900

08800

03000

02868

06289

04906

1502

01476


energy

states

833053

1099

(minus4)

1481

(minus8)

33053

1690

(minus4)

6450(minus9)

33053

0974(minus4)

2141(minus9)

4712(minus4)

432993

1227

(minus2)

1848

(minus4)

33053

1886

(minus2)

8031(minus5)

32999

1087

(minus2)

2676(minus5)

2790(minus2)

232477

01334

2144(minus2)

33053

01992

8962(minus3)

32472

0114

63071(minus3)

01644

131337

04856

02267

33053

064

7400947

30860

03487

00315

03225

05

300

6810

078

06166

33053

11672

03077

28262

05329

00877

03905

025

29086

14864

08529

33053

15672

05547

25604

06101

01401

03961





119896119861119879119881119887

119879

QM119864

QM119899

119879

QC119864

QM119899

119879

QC119864

QC119899




119887= 40119864

ℎ 1198810= 04119864

ℎ and 119871 = 10119886

0




119887 Results


0 005 01 015 02

Rate

cons

tant


k(T

)

0

02

04

06

08

Temperature [Vb]

0 10 20 30

00001

1e minus 08

1e minus 12

1e minus 16







QC(119864) = 1 for 119864 gt 119881






119879


2120587119864

119908119887

))

minus1

119908119887=radic

119896119887

119898

(19)


119881 (119909)

=

119896(119909 minus 1199091)

2

2


minus

119896119887119909

2

2

+ 119881119887

119909119871le 119909 le 119909


119896(119909 minus 1199092)

2

2

+ 1198810119909 gt 119909


(20)


119877mdash


119871and 119909 = 119909


0= 0


= 11198860 119896 = 119896

119887= 80119864

ℎ119886

minus2

0 119881119887= 20119864

ℎ and 119909

119877= minus119909119871= 05119886

0





119903119901(119864)|


119903119901(119864) describes trans-


119878119903119901(119864) =

(2120587)

minus1

120578

lowast

119901(119864) 120578119903 (

119864)

int

infin

minusinfin

⟨120601

minus

119901

100381610038161003816100381610038161003816

119890

minus120580119905100381610038161003816100381610038161003816

120601

+

119903⟩ 119890

120580119864119905119889119905 (21)


119903⟩


119901⟩



119901|


minus

119901(119909 0) | 120601

+

119903(119909 119905)⟩ is



120578119903119901(119864) = radic

119898

2120587119901

int

infin

minusinfin

119890

minus120580119901119909120601

plusmn

119903119901(119909) 119889119909 119901 =

radic2119898119864

(22)



1198875 = 4119864

ℎ(the





3system for a





00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)


119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)











119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


6 80

0

05

1

15

2

25


TQCTQM

T2sTQM



ℎ11= ℎ22= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119903⟩ ℎ12= ℎ21= ⟨120601119903

10038161003816100381610038161003816

119867

10038161003816100381610038161003816

120601119901⟩ (15)

11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)

H1205952119904119894= 119864

2119904

119894S1205952119904119894119894 = 0 1 (17)


2119904

0


1198641minus 1198640= Δ =

21198640

120587

radic119879 (1198640) (18)



3minus1198642) are


1 2 3 4

Energy [Eh]

00001

1e minus 08

Tunn

elin

gT

(E)

L = 2 QML = 2 QC

E0

E1















119901|120601119903⟩ is less




gsand

tunn

elingp

robabilityfor


metric

rectangu

lard

ouble-wellpotentia

lofSectio

n31Th

ereactantp

rodu

ctstates

overlap

ofthetwo-stated

escriptio

nisgivenin

thelastcolum

nTh


areg

iven

in119864ℎN

umbersin

parenthesesa

rethep

owerso

f10

119871[1198860]

119864

QM0

Δ

QM

119879

QM

119864

QC0

Δ

QC

119879

QC

119864

2s 0Δ

2119904

119879

2119904

⟨120601119901|120601119903⟩


energy

states

808800

1919

(minus9)

1205

(minus17)

08800

1174

(minus9)

439

(minus18)

08800

1913

(minus9)

117(minus17)

3554(minus9)

408800

4192(minus5)

5752(minus9)

08800

2564(minus5)

2095(minus9)

08800

4210(minus5)

564

6(minus9)

440

8(minus5)

208770

6196(minus3)

1257

(minus4)

08800

3790(minus3)

4577(minus5)

08769

6224(minus3)

1243

(minus4)

4035(minus3)

108422

00753

00185

08800

4608(minus3)

6765(minus3)

08425

7564

(minus2)

1989

(minus2)

3398(minus2)

05

07433

02618

02105

08800

01607

8225(minus2)

07483

02637

03064

00922

025

060

9804881

05900

08800

03000

02868

06289

04906

1502

01476


energy

states

833053

1099

(minus4)

1481

(minus8)

33053

1690

(minus4)

6450(minus9)

33053

0974(minus4)

2141(minus9)

4712(minus4)

432993

1227

(minus2)

1848

(minus4)

33053

1886

(minus2)

8031(minus5)

32999

1087

(minus2)

2676(minus5)

2790(minus2)

232477

01334

2144(minus2)

33053

01992

8962(minus3)

32472

0114

63071(minus3)

01644

131337

04856

02267

33053

064

7400947

30860

03487

00315

03225

05

300

6810

078

06166

33053

11672

03077

28262

05329

00877

03905

025

29086

14864

08529

33053

15672

05547

25604

06101

01401

03961





119896119861119879119881119887

119879

QM119864

QM119899

119879

QC119864

QM119899

119879

QC119864

QC119899




119887= 40119864

ℎ 1198810= 04119864

ℎ and 119871 = 10119886

0




119887 Results


0 005 01 015 02

Rate

cons

tant


k(T

)

0

02

04

06

08

Temperature [Vb]

0 10 20 30

00001

1e minus 08

1e minus 12

1e minus 16







QC(119864) = 1 for 119864 gt 119881






119879


2120587119864

119908119887

))

minus1

119908119887=radic

119896119887

119898

(19)


119881 (119909)

=

119896(119909 minus 1199091)

2

2


minus

119896119887119909

2

2

+ 119881119887

119909119871le 119909 le 119909


119896(119909 minus 1199092)

2

2

+ 1198810119909 gt 119909


(20)


119877mdash


119871and 119909 = 119909


0= 0


= 11198860 119896 = 119896

119887= 80119864

ℎ119886

minus2

0 119881119887= 20119864

ℎ and 119909

119877= minus119909119871= 05119886

0





119903119901(119864)|


119903119901(119864) describes trans-


119878119903119901(119864) =

(2120587)

minus1

120578

lowast

119901(119864) 120578119903 (

119864)

int

infin

minusinfin

⟨120601

minus

119901

100381610038161003816100381610038161003816

119890

minus120580119905100381610038161003816100381610038161003816

120601

+

119903⟩ 119890

120580119864119905119889119905 (21)


119903⟩


119901⟩



119901|


minus

119901(119909 0) | 120601

+

119903(119909 119905)⟩ is



120578119903119901(119864) = radic

119898

2120587119901

int

infin

minusinfin

119890

minus120580119901119909120601

plusmn

119903119901(119909) 119889119909 119901 =

radic2119898119864

(22)



1198875 = 4119864

ℎ(the





3system for a





00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)


119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)











119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



gsand

tunn

elingp

robabilityfor


metric

rectangu

lard

ouble-wellpotentia

lofSectio

n31Th

ereactantp

rodu

ctstates

overlap

ofthetwo-stated

escriptio

nisgivenin

thelastcolum

nTh


areg

iven

in119864ℎN

umbersin

parenthesesa

rethep

owerso

f10

119871[1198860]

119864

QM0

Δ

QM

119879

QM

119864

QC0

Δ

QC

119879

QC

119864

2s 0Δ

2119904

119879

2119904

⟨120601119901|120601119903⟩


energy

states

808800

1919

(minus9)

1205

(minus17)

08800

1174

(minus9)

439

(minus18)

08800

1913

(minus9)

117(minus17)

3554(minus9)

408800

4192(minus5)

5752(minus9)

08800

2564(minus5)

2095(minus9)

08800

4210(minus5)

564

6(minus9)

440

8(minus5)

208770

6196(minus3)

1257

(minus4)

08800

3790(minus3)

4577(minus5)

08769

6224(minus3)

1243

(minus4)

4035(minus3)

108422

00753

00185

08800

4608(minus3)

6765(minus3)

08425

7564

(minus2)

1989

(minus2)

3398(minus2)

05

07433

02618

02105

08800

01607

8225(minus2)

07483

02637

03064

00922

025

060

9804881

05900

08800

03000

02868

06289

04906

1502

01476


energy

states

833053

1099

(minus4)

1481

(minus8)

33053

1690

(minus4)

6450(minus9)

33053

0974(minus4)

2141(minus9)

4712(minus4)

432993

1227

(minus2)

1848

(minus4)

33053

1886

(minus2)

8031(minus5)

32999

1087

(minus2)

2676(minus5)

2790(minus2)

232477

01334

2144(minus2)

33053

01992

8962(minus3)

32472

0114

63071(minus3)

01644

131337

04856

02267

33053

064

7400947

30860

03487

00315

03225

05

300

6810

078

06166

33053

11672

03077

28262

05329

00877

03905

025

29086

14864

08529

33053

15672

05547

25604

06101

01401

03961





119896119861119879119881119887

119879

QM119864

QM119899

119879

QC119864

QM119899

119879

QC119864

QC119899




119887= 40119864

ℎ 1198810= 04119864

ℎ and 119871 = 10119886

0




119887 Results


0 005 01 015 02

Rate

cons

tant


k(T

)

0

02

04

06

08

Temperature [Vb]

0 10 20 30

00001

1e minus 08

1e minus 12

1e minus 16







QC(119864) = 1 for 119864 gt 119881






119879


2120587119864

119908119887

))

minus1

119908119887=radic

119896119887

119898

(19)


119881 (119909)

=

119896(119909 minus 1199091)

2

2


minus

119896119887119909

2

2

+ 119881119887

119909119871le 119909 le 119909


119896(119909 minus 1199092)

2

2

+ 1198810119909 gt 119909


(20)


119877mdash


119871and 119909 = 119909


0= 0


= 11198860 119896 = 119896

119887= 80119864

ℎ119886

minus2

0 119881119887= 20119864

ℎ and 119909

119877= minus119909119871= 05119886

0





119903119901(119864)|


119903119901(119864) describes trans-


119878119903119901(119864) =

(2120587)

minus1

120578

lowast

119901(119864) 120578119903 (

119864)

int

infin

minusinfin

⟨120601

minus

119901

100381610038161003816100381610038161003816

119890

minus120580119905100381610038161003816100381610038161003816

120601

+

119903⟩ 119890

120580119864119905119889119905 (21)


119903⟩


119901⟩



119901|


minus

119901(119909 0) | 120601

+

119903(119909 119905)⟩ is



120578119903119901(119864) = radic

119898

2120587119901

int

infin

minusinfin

119890

minus120580119901119909120601

plusmn

119903119901(119909) 119889119909 119901 =

radic2119898119864

(22)



1198875 = 4119864

ℎ(the





3system for a





00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)


119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)











119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





119896119861119879119881119887

119879

QM119864

QM119899

119879

QC119864

QM119899

119879

QC119864

QC119899




119887= 40119864

ℎ 1198810= 04119864

ℎ and 119871 = 10119886

0




119887 Results


0 005 01 015 02

Rate

cons

tant


k(T

)

0

02

04

06

08

Temperature [Vb]

0 10 20 30

00001

1e minus 08

1e minus 12

1e minus 16







QC(119864) = 1 for 119864 gt 119881






119879


2120587119864

119908119887

))

minus1

119908119887=radic

119896119887

119898

(19)


119881 (119909)

=

119896(119909 minus 1199091)

2

2


minus

119896119887119909

2

2

+ 119881119887

119909119871le 119909 le 119909


119896(119909 minus 1199092)

2

2

+ 1198810119909 gt 119909


(20)


119877mdash


119871and 119909 = 119909


0= 0


= 11198860 119896 = 119896

119887= 80119864

ℎ119886

minus2

0 119881119887= 20119864

ℎ and 119909

119877= minus119909119871= 05119886

0





119903119901(119864)|


119903119901(119864) describes trans-


119878119903119901(119864) =

(2120587)

minus1

120578

lowast

119901(119864) 120578119903 (

119864)

int

infin

minusinfin

⟨120601

minus

119901

100381610038161003816100381610038161003816

119890

minus120580119905100381610038161003816100381610038161003816

120601

+

119903⟩ 119890

120580119864119905119889119905 (21)


119903⟩


119901⟩



119901|


minus

119901(119909 0) | 120601

+

119903(119909 119905)⟩ is



120578119903119901(119864) = radic

119898

2120587119901

int

infin

minusinfin

119890

minus120580119901119909120601

plusmn

119903119901(119909) 119889119909 119901 =

radic2119898119864

(22)



1198875 = 4119864

ℎ(the





3system for a





00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)


119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)











119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




119879


2120587119864

119908119887

))

minus1

119908119887=radic

119896119887

119898

(19)


119881 (119909)

=

119896(119909 minus 1199091)

2

2


minus

119896119887119909

2

2

+ 119881119887

119909119871le 119909 le 119909


119896(119909 minus 1199092)

2

2

+ 1198810119909 gt 119909


(20)


119877mdash


119871and 119909 = 119909


0= 0


= 11198860 119896 = 119896

119887= 80119864

ℎ119886

minus2

0 119881119887= 20119864

ℎ and 119909

119877= minus119909119871= 05119886

0





119903119901(119864)|


119903119901(119864) describes trans-


119878119903119901(119864) =

(2120587)

minus1

120578

lowast

119901(119864) 120578119903 (

119864)

int

infin

minusinfin

⟨120601

minus

119901

100381610038161003816100381610038161003816

119890

minus120580119905100381610038161003816100381610038161003816

120601

+

119903⟩ 119890

120580119864119905119889119905 (21)


119903⟩


119901⟩



119901|


minus

119901(119909 0) | 120601

+

119903(119909 119905)⟩ is



120578119903119901(119864) = radic

119898

2120587119901

int

infin

minusinfin

119890

minus120580119901119909120601

plusmn

119903119901(119909) 119889119909 119901 =

radic2119898119864

(22)



1198875 = 4119864

ℎ(the





3system for a





00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)


119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)











119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


00001

001

1

QMQCParabolic

0 10 20 30

1

1e minus 08

1e minus 06

Energy [Eh]

T(E

)

(a)

0 05 1 15 20

05

Rate

cons

tant

0 01 02

00001

001

Temperature [Vb]

(b)


119861119879119881119887] where 119881

119887= 20119864

ℎ

0

10

20

30

40

50

08 1 12 14 16 18 2

272829

Ener

gy (m

E h)

ROH (A)

(a)

0 500 1000 1500

103

102

101

100

KIE

Temperature (K)

(b)











119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




119877CO 119896

QM119896

QC119896

QC119896

QM119864

QM0

[119898119864ℎ] 119864

QC0

[119898119864ℎ]


lowast mdash mdash


lowast mdash mdash








4 Conclusions




3system with





Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

research article calculation of the quantum-mechanical...

Documents