Isotope Effects and Quantum-Mechanical Tunneling - American

3 Isotope Effects and Quantum-Mechanical Tunneling R A L P H E. W E S T O N ,

JR.

Downloaded by MICHIGAN STATE UNIV on November 27, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/bk-1975-0011.ch003

Department of Chemistry, Brookhaven National Laboratory, Upton, L. I., Ν. Y.

11973

Introduction The phenomenon of quantum-mechanical tunneling is not observ able i n the macroscopic world which we experience directly. Sup pose that a b a l l i s rolled along the ground towards a small ridge. If i t s kinetic energy i s greater than the potential energy of the b a l l at the top of the ridge it will surmount the barrier; if the kinetic energy i s less than this, the b a l l w i l l not get over the hill. However, a moving particle of atomic or electronic mass does not obey Newtonian mechanics. Instead, it behaves as a wave packet with a wavelength given by the de Broglie expression λ = h/mv

(1)

where h i s Planck's constant, m i s the mass of the particle, and v i t s velocity. For purposes of calibration, it is convenient to remember that a hydrogen atom, moving with thermal velocity at 300 K, has a wavelength of about 1 Å(0.1 nm). When such a particle encounters a barrier, represented by an increase i n potential energy, it does not behave l i k e a macroscopic particle. Instead, there is a f i n i t e probability of "leakage" or "tunneling" through the barrier even if the kinetic energy is less than the potential energy at the barrier summit; conversely, there is a f i n i t e probability of reflection even if the kinetic energy i s greater than this. The extent of tunneling, the transmission probability қ, i s defined by қ= ( A / A ) , where A and A are the wave function amplitudes for the incident and transmitted waves (Fig. 1). 2

t

i

i

t

As one would expect, κ depends on the mass of the particle, its velocity, and the shape and height of the barrier. Two con venient parameters are 44

In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.

3.

WESTON

Quantum-Mechanical

Tunneling

α = 2frV*/hv*

and

45

ε = E/V*

(2)

where V* is the barrier height and v* i s a "vibrational frequency" defined by 1

1

2

2


v* - (2w)"" {-y" [d V(x)/dx ]}

lâ

(3)

where μ i s the reduced mass appropriate to motion i n the χ direc tion. For a particular form of the potential barrier, V(x), known as the Eckart barrier (see below), the dependence of κ on ε and α is shown i n Fig. 2. The classical limit i s simply a step function at ε = 1, i . e . when the kinetic energy of the particle i s equal to the potential energy at the barrier summit. Large values of a, corresponding to high, or nearly f l a t , barriers lead to a similar form of κ. Low values of a, on the other hand, lead to f i n i t e values of κ with ε much less than unity, and to values of κ less than unity even when Ε i s greater than V . The physical significance of quantum-mechanical tunneling was recognized very early i n the development of wave mechanics, and there are many examples of physical phenomena i n which tunneling i s important. Here i s a very incomplete l i s t of examples, chosen principally on the basis of h i s t o r i c a l interest: 1. The cold emission of electrons from a metal cathode at a high negative voltage (1). 2. The emission of α-particles from an atomic nucleus (2,3). 3. The effect of the double minimum i n the potential energy for nh3 on vibrational energy levels (Λ). 4. The possibility of tunneling i n a chemical reaction i n volving motion of a proton or a hydrogen atom, which seems to have been f i r s t recognized by R. P. Bell (5). 5. The tunnel diode, a semiconductor device of considerable practical importance. The discovery of this by L. Esaki i n 195658 (6) led to his sharing i n the 1973 Nobel prize for physics (2)· In fact, i t is interesting to note that the other two recipients of the Nobel prize i n physics for that year, B. D. Josephson and I. Giaever, were also honored for discoveries involving tunneling in solids (8,9). The Nobel prize awards should, i n themselves, provide ample proof of the reality and the practical significance of tunnelingI Tunneling i n Chemical Reactions Now that i t has been made evident that tunneling i s predicted by quantum mechanics, and that there are a number of physical manifestations of i t , what about the particular area of chemical reactions?



isotopes a n d c h e m i c a l

principles

VU)

Theory of Elementary Gas Reaction Rates Figure 1. Permeability of a barrier to a particle with kinetic energy less than barrier height. The dashed line represents the wave function for a particle of energy, E, moving from the left and interacting with an Eckart barrier (solid line) (29).

0

0.2 0.4

0.6 0.8

1.0

1.2

1.4

1.6

1.8 2.0 2.2

i - E/V* Gas Phase Reaction Rate Theory Figure 2. Transmission probability κ(ε) as a function of reduced energy e(= E/V*) and a(= 2*V*/hv*) for a symmetrical Eckart barrier (30)


3

Quantum-Mechanical

WESTON

Tunneling

47

The potential energy of interaction for the reactants i n a hypothetical reaction such as


A + BC —> AB + C can be represented by a contour map (Fig. 3). Along the reaction path, the potential energy has the form of a one-dimensional barrier, similar to that i n Fig. 1. If the mass of the atom being transferred (B i n this example) i s sufficiently small, and i f the barrier has suitable dimensions, tunneling may become important. I am deliberately evading here some subtle questions, such as: 1. What i s the precise reaction path that should be used to obtain the one-dimensional potential? 2. Is i t correct to treat the problem as one-dimensional, when the potential energy i s a function of two dimensions? These and related questions have been discussed i n d e t a i l by others (10,11). For a one-dimensional barrier of arbitrary shape, numerical methods can be used to calculate the transmission probability κ (12). Before the availability of large computers, this problem was often circumvented by approximating the barrier with an Eckart function (13). The symmetric form of this, which I shall use later, i s 2

V(x) = V*/cosh [(2 V*)\v*x] U

.

(4)

This potential has the great advantage of leading to analytical solutions of the resulting Schrbdinger equation, so that much less computational effort i s required to calculate κ than i s required for an arbitrary barrier shape. It i s ideally suited for testing the qualitative effects of changing properties of the barrier. With an Eckart barrier, the transmission probability for a parti cle with energy Ε is found to be K ( E ) = [cosh(2αε^)-1]/[cosh(2ae^) + δ]

(5)

2

where δ = cosh(4a -π^)^ (a > π/2) or δ = cos|4a -w2p (a < π/2) , and the other quantities have already been defined. 2

In a chemical reaction system at thermal equilibrium, the Boltzmann distribution of molecular energies must be taken into account i n obtaining the average transmission probability. The tunneling factor i s usually defined as the ratio of this averaged transmission probability to that obtained with the classical values K(E)

=0,

Ε < V * ;

κ(Ε)

= 1,

Ε ^ V

American Chemical Society Library

16thPrinciples; St, N.W. In Isotopes and1155 Chemical Rock, P.; Washington, 20036 ACS Symposium Series; American ChemicalD.C. Society: Washington, DC, 1975.


48

ISOTOPES

ANDCHEMICAL

PRINCIPLES

Chemical Kinetics Figure 3. Hypothetical potential energy surface for the collinear reaction A + BC —» AB + C. The solid lines are equipotentials and the dashed line is the reaction path ( 31 ).


3.

WESTON

Quantum-Mechanical

Tunneling

49

That i s ,

Γ*(Τ)

(E)exp(-E/RT)dE

IS

=

(6)


exp(-E/RT)dE

Γ*(Τ)

exp(V*/RT) J K(E)exp(-E/RT)d(E/RT)

-

.

(7)

ο This expression must be numerically integrated over the appro priate energy range. Tunneling and Kinetic Isotope Effects The Magnitude of Primary Hydrogen Isotope Effects. As Eq.(l) i l l u s t r a t e s , the wavelength associated with a moving particle i s inversely proportional to the mass. In a comparison of two isotopic species i n a chemical reaction, this mass dependence leads to different values of v* and a. Since a is larger (v* smaller) for the heavier species, the values of κ(Ε) are closer to the classical values, and this shows up i n Γ*(Τ). Since relative differences i n mass are greatest for the isotopes of hydrogen, one might expect important differences i n the tunneling corrections for reactions involving the motion of H, D, or Τ atoms or ions. In fact, almost since the discovery of deuterium, such reactions have been studied i n an attempt to find evidence for tunneling i n a chemical reaction (14). To find evidence for tunneling, one must f i r s t account for kinetic isotope effects that do not depend on tunneling. The most direct method of doing this is by means of activated complex theory, which leads to a formulation of the rate constant i n terms of partition functions of the reactants and the activated complex. Arguments concerning the validity of activated complex theory are easy to provoke and d i f f i c u l t to settle, and I shall not consider this question here. It can then be shown (15,16) that the ratio of rate constants for two isotopic forms of the same reactant (AX| and AX2) is given by k

k

l 2

ν

Γ

ΐ* ΐ* v

2

Γ

2

3 r

"

7

r

i

(

A

X

i*

)

3 r

i-1 Γ (ΑΧ *) ±

2

"

6

r

j-1

j
t u n

t u n

H

t u n

D

H

D

When the tunneling correction i s combined with the rest of the rate constant ratio, (^H^D^class* resultant form of I ^ A Q is almost identical with that for tunneling only. This i s because ln(k /k ) has almost exactly the Arrhenius form, with ln(AQ class/ v i r t u a l l y independent of temperature (see the curves labelled^NT" i n Figs. 7a and 7a") . t h e

H

D

c l

f

Figs. 7a

1

and 7a" i l l u s t r a t e the following important features:

1. Regardless of the barrier height, there is a f a i r l y small temperature region i n which I I I A Q i s below the limit of -0.69 predicted by Bell's model i n the absence of tunneling (indicated by the lower dashed l i n e ) . The extent of the excursion below this limit increases with increasing barrier height. It is i n this temperature region that most experiments have been done, and earlier calculations of tunneling were made. 2. At lower temperatures, InAQ exceeds the upper limit of 0.35 predicted by B e l l i n the absence of tunneling (indicated by the upper dashed l i n e ) . This i s contrary to what had been expected before our calculations were made, because earlier calculations concentrated on the region to the l e f t of the inflection point i n Fig. 6. 3. The value of AQ does not correlate simply with theêxtent of tunneling, which i s indicated by values of ΙηΓ^* for Vg = 10 on the top of Fig. 7. In our work, we also examined other model reactions, and a few other barrier shapes, with similar results.



56

ISOTOPES

AND

CHEMICAL

PRINCIPLES

Thus, we conclude that experimental A Q values can be used as quantitative indicators of quantum-mechanical tunneling only i n conjunction with model calculations. Although AQ values outside of Bell's range of 0.5-/2 probably cannot be found i n the absence of tunneling, A Q values inside this range do not necessarily indi cate the absence of large tunneling factors. In spite of this somewhat uncertain situation, i t i s interesting to see i f there are experimental values of A Q outside the "non-tunneling" range. Data from reactions i n which tunneling has been invoked are given in Table I. (This does not mean that tunneling i s unimportant i n other reactions, but only that i t has not been specifically searched f o r ) . Without c r i t i c a l l y evaluating the experimental data of Table I, I simply point out that there are several reac tions i n which abnormally low A Q values have been observed. Relative Tritium-Deuterium Isotope Effects. Because some isotope experiments are best done with D substitution and others with Τ substitution, there has been for some time an interest i n , and a necessity for, the correlation of the rate constant ratios k / k and k /k . This correlation may be defined as H

D

H

T

ln(k /k ) H

T

Several theoretical investigations of the allowable range i n in the absence of tunneling have been made. Using a simplified model for a hydrogen-transfer reaction, Swain and coworkers (23) ob tained a value of 1.442, considering only zero-point-energy effects. Bigeleisen (24),using a more complete model, proposed a range of 1.33-1.58 for the relative isotope effects, including Wigner tunneling (valid i f κ is close to unity). He stated that exten sive tunneling should lead to abnormally low values of r. More O'Ferrall and Kouba (25) calculated isotope effects for some model proton-transfer reactions involving linear four- and fiveatom transition states. They found deviations of only 2% from the Swain value of r, (1.442), even when tunneling corrections for a parabolic barrier were included. It has also been shown by Lewis and Robinson (26) that tunneling corrections do not necessarily produce significant changes i n the relative isotope effect. Recently, Stern and Vogel (27) investigated relative tritiumdeuterium isotope effects for 180 model reaction systems. They found that, within the temperature range 20-1000°K, was re stricted to the range 1.33 < r^^ 1.58 provided that: 1. ^ H / ^ D f l c t significant force-constant changes between reactant and transition state at the isotopic position(s). A N D

r e

e


WESTON

Quantum-Mechanical

Tunneling

Table I. Arrhenius Preexponential Factors and Relative Tritium-Deuterium Isotope Effects Reaction

fa


0.032 1. 4-Nitrophenylnitromethane + tetramethylguanidine (in toluene)

Ref,

-

a

-

a

2. Same reaction i n dich1oromethane

0.45

3. Leucocrystal violet + chloranil

0.04

1.31

b

4. 2-Carbethoxycyclopentanone + F

0.042

1.32

c,d

5. 2-Ni tropropane + 2,4,6-1rime thy1pyridine

0.15

1.39

b

6. Oxidation of 1-phenyl-2,2,2-trifluoroethanol

0.25

1.46

b

7. 2-Carb ethoxycyclopentanone + chloracetate ion

0.35

1.72

c,d

8. Acetophenone + OH

0.38

1.38

e

9. l-Bromo-2-phenylpropane + ethoxide

0.40

1.48

10. 2-Carb ethoxycy clopentanone + Ό^)

0.44

1.48

f,g c,d

11. Pyridinediphenylborane + HÔ

0.94

1.38

h

12. 2,2-Diphenylethylbenzenesulfonate + methoxide

1.1

1.48

i

*Ref. 19 b

Ref. 26

C

B e l l , R. P., Fendley, J . Α., and Hulett, J . R., Proc. Roy. Soc. Ser. A (1956), 235, 453.

d

Jones, J . R., Trans. Faraday Soc. (1969), 65, 2430.

e

Jones, J . R., i b i d . (1969), 65, 2138.

f

Shiner, V. J . , J r . and Smith, M. L., J . Amer. Chem. Soc. (1961) , 83, 593. Shiner, V. J . , J r . and Martin, B., Pure Appl. Chem. (1964), 8, 371.

g

^Lewis, E. S. and Grinstein, R. H., J . Amer. Chem. Soc. (1962) , 84, 1158. S f i l l i , Α. V., J . Phys. Chem. (1966), 70, 2705.


58

ISOTOPES

AND

CHEMICAL

PRINCIPLES

2. kjj/kj. and kjj/kj} are both greater than unity. 11


3. kjj/kj. and k^/k^ exhibit "regular temperature dependences at a l l temperatures, "regular" being defined i n their paper. Stern and I (28) re-examined the model reactions meeting the above three c r i t e r i a , to see what effect the inclusion of tunnel ing would have on the relative isotope effect. The tunneling cor rections were calculated as described i n the preceding sub-section. Although this was done for a large number of model reactions, I s h a l l discuss here only the model for hydrogen-atom abstraction that has already been mentioned i n the preceding sub-section. Again, two variations were considered: i n one, V J J * = = Vj.*; i n the other, V * = V * + 1, V * = V * + 1.45. Figs. 8a and 8b show the ratios of tunneling corrections, Γ */Γβ* Ξ T y^ and Η*/ Τ* H/T> two cases. In addition, the relative tritium-deuterium tunneling correction D

R

T

R

Η

Γ

Γ

Ξ

T

f

o

r t

n

R

e

L~ =

lnT

H/T

/lnT

H/D

is indicated. The slightly different shapes of l n T ^ and l n T ^ vs. logT result i n a temperature dependence of t. At high tempera tures , Γ* is given by the Wigner expression, and i t can be shown (24) that R

as Τ l-(v */v *) D

H

T

H

D

—> 0

2

If the barrier height depends on isotopic substitution, there i s a temperature at which is unity and Jt, is zero. At a slightly higher temperature, T ^ becomes unity and t^has a pole (±°°) . R

The relative isotope effect including the tunneling correc tion, which we designate as r , can be expressed i n terms of t and f

LnB

r

=

H/D

st

Γ

n

e r

a

t

lnR

H/D]

e

In this expression, Rg/D ^ constant ratio with no tun neling correction, and R^/n the same ratio including tunneling. It is apparent that the weighting factors multiplying £ and târe complementary; as a result, plots of the weighted and weighted £ contributions vs. logT w i l l be, approximately, displaced "mirror images" (cf. Figs. 8a and 8b ). Because the two contributions are not exactly out of phase, a slight temperature dependence of r^ remains. Fig. 9 illustrates the effect on tând on of f

f


3.

WESTON

Quantum-Mechanical

Tunneling

* Η( Η η Γ


147.6

32.6

4.40

0.39

147.6

32.6

ν

59

= Ι 0 )

4.40

Q39

147.6

32.6

4.40

0.39

β

Τ„ν, (Κ) Journal of Chemical Physics Figure 7. Arrhenius pre-exponential factors as functions of temperature for a model hydrogen-atom abstraction with various barrier heights: (a), tunneling cor rection only, V * = V *; (α'), complete isotope effect, V * = V *; (a")> com plete isotope effect, V^* = V * + 1. The curves are labeled with V * in kcal/ mole. (NT indicates that no tunneling correction is included.) Numbers at the top of each graph are ΙηΓ * for V * =10, as a function of T . Dashed lines are at A = Vz andV~2 (22). D

H

D

H

H

H

Η

H

ar!l

Q



60

ISOTOPES

147.6 32.6 1 1 1 1 11II

2.0

4.40

1

A N DCHEMICAL

PRINCIPLES

0.39

! 1 1 1 1 III ία)

1.8 - 5

1

1.6

10

1.4 _

2C ' * 3 0

1.2

1 ! 1 1 11II

1

1 1 llllll

Journal of Chemical Physics Figure 9. Relative tritium-deuterium kinetic isotope effects as functions of tem perature with various barrier heights: (a),(b), tunneling only; (a'),(b') complete isotope effect. Unprimed letters are for isotope-independent barrier heights, and primed letters are for V * = V,/* + 1, V * = W + ΙΛ5 kcal/mole. The curves are labeled with V * in kcal/mole. Numbers at the top of the graphs are ΙηΓ * for V * = 10 kcal/mole, as a function of log T. Dashed horizontal lines are the nontunneling limits," 1.33 and 1.58. The infinite-temperature limit of the relative isotope effect is indicated by "n" at right border (28). f

D

T

H

Η

H

* T * S » the conclusions for the case when the barrier height i s isotope-dependent are very similar. Note that the abnormal behavior of i n Fig. 8b and 9b i s removed by the weighting factor to give the smooth curves of Figs. 8b and 9b'. The only cases for which the discontinuity i n r j i s retained are physically unrealistic models i n which secondary isotope effects are combined with an isotope-dependent barrier height. V

V

an

Isotope Effects and Quantum-Mechanical Tunneling - American

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