Which Is Better at Predicting Quantum-Tunneling Rates: Quantum

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Letter

Which is better at predicting quantum-tunneling rates: quantum transition-state theory or free-energy instanton theory? Yanchuan Zhang, Thomas Stecher, Marko Cvitas, and Stuart C. Althorpe J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/jz501889v • Publication Date (Web): 27 Oct 2014 Downloaded from http://pubs.acs.org on November 2, 2014

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The Journal of Physical Chemistry Letters

Which is better at predicting quantum-tunneling rates: quantum transition-state theory or free-energy instanton theory? Yanchuan Zhang,† Thomas Stecher,† Marko T. Cvitaš,‡ and Stuart C. Althorpe∗,† Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom, and

b

Ruđer Bošković Institute, Department of Physical Chemistry,

Bijenička Cesta 54, 10000 Zagreb, Croatia. E-mail: [email protected]

Abstract Quantum transition-state theory (QTST) and free-energy instanton theory (FEIT) are two closely-related methods for estimating the quantum rate coefficient from the free-energy at the reaction barrier. In calculations on one-dimensional models, FEIT typically gives closer agreement than QTST with the exact quantum results at all temperatures below the crossover to deep tunneling, suggesting that FEIT is a better approximation than QTST in this regime. Here we show that this simple trend does not hold for systems of greater dimensionality. We report tests on several collinear and three-dimensional reactions, in which QTST outperforms FEIT over a range of temperatures below cross-over, which can extend down to half the cross-over temperature ∗

To whom correspondence should be addressed University of Cambridge ‡b Ruđer Bošković Institute †

1 ACS Paragon Plus Environment


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(below which FEIT outperforms QTST). This suggests that QTST-based methods such as ring-polymer molecular dynamics (RPMD) may often give closer agreement with the exact quantum results than FEIT.

Keywords: Quantum tunneling, dynamics


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Proton-transfer reactions involving tunneling through a barrier are widespread in chemistry, in fields ranging from gas-phase reactions 1–4 to enzyme catalysis. 5–7 Tunneling can accelerate the rates of such reactions by orders of magnitude and can also give rise to unusual isotope effects. Conventional (i.e. Wigner-Eyring-Polanyi 8 ) TST cannot treat tunneling, and an exact quantum treatment 9 is possible only for the simplest gas-phase reactions. There are therefore a variety of approximate methods, 10–27 which aim to treat quantum tunneling in various temperature regimes, an important class of which are those based on evaluating a path-integral free-energy in the region of the reaction barrier. 19–27 These methods scale much better with system size than exact quantum dynamics, since the simulations can be done using the ‘ring-polymer’ techniques first used to calculate static properties. 28,29 There are two main approaches behind such free-energy methods. The older approach is based on the so-called ‘Im F’ expression, 16–19 which is obtained by analytically continuing the partition function into the complex plane. To our knowledge, there is no clear derivation of this approach from first principles, although it has been used extensively in various branches of physics, 30 and its complete steepest-descent approximation gives the famous ‘instanton’ rate derived earlier by Miller from the quantum flux-side correlation function; 15 its partial steepest-descent implementation gives what we will refer to here as the free-energy instanton theory (FEIT), first used by Mills and coworkers. 27 The other approach is quantum transition-state theory (QTST), which was originally obtained heuristically, first in the special case of a centroid constraint, 20–22 then more generally as the TST limit of ring-polymer molecular dynamics (RPMD) rate-theory. 19,23,24 However QTST was shown recently to have a rigorous derivation as the short-time limit of the quantum flux through a path-integral dividing surface. 31–33 The similarities of the QTST and FEIT rate expressions will be explored later in this letter. Previous work, 19 published before the derivation of QTST, reported comparisons between FEIT and QTST for one-dimensional reaction barriers, which showed that FEIT gave closer agreement than QTST with the exact quantum rate-coefficients, at all temperatures below



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the cross-over temperature Tc , 17

Tc =

h ¯ ω‡ , 2πkB

(1)

where ω ‡ is the barrier frequency and kB is the Boltzmann constant. We will refer to this range of temperatures as the ‘deep-tunneling’ regime.∗ Above Tc , the Boltzmann matrix is dominated by fluctuations around a point on top of the barrier; below Tc , it is dominated by fluctuations around the delocalised ‘instanton’ path, which is a periodic orbit on the inverted potential well. These results suggested that QTST may be an approximation to FEIT. It was also found 19 that QTST consistently under-predicted the exact quantum rate for symmetric barriers, and over-predicted it for asymmetric barriers. Here we extend these comparisons to a set of collinear and three-dimensional reactions, to find out whether FEIT continues to give a better approximation than QTST to the exact quantum rate. As mentioned above, the QTST rate is identical to the TST limit of RPMD rate-theory, which corresponds to applying classical rate-theory to the ring-polymer Hamiltonian. 19,23,24 For a system with f degrees of freedom, whose classical Hamiltonian is

H=

f X p2i + V (x1 , ..., xf ) , 2m i i=1

(2)

the ring-polymer Hamiltonian is 23 f N X X p2i,j HN (x, p) = + UN (βN , x), 2mi i=1 j=1 ∗

Other definitions of this term are sometimes used in the literature.


(3)

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where

UN (βN , x) =

N X

V (x1,j , ..., xf,j )

(4)

j=1

+

f X i=1

N mi X (xi,j+1 − xi,j )2 . 2(βN h ¯ )2 j=1

is the ring-polymer potential energy surface, and βN = β/N with β = 1/kB T . Defining a dividing surface s(x) to be a function of the ring-polymer coordinates which divides reactants from products, one then takes the t → 0+ limit of the classical flux-side time-correlation function in ring-polymer space to obtain the QTST rate, 31 1 kQTST (β)Qr (β) = lim N →∞ (2π¯ h)N f

Z

Z dp

dx e−βN HN (p,x)

×δ[s(x)]vs (p, x)h[vs (p, x)].

(5)

where

vs (p, x) =

f N X X ∂s(x) pi,j i=1 j=1

∂xi,j mi

,

(6)

is the instantaneous flux through s(x). Quantum transition-state theory gives the exact quantum rate if there is no recrossing of s(x) by the quantum flux, 31,33 but it does not give a rigorous upper bound, since coherent recrossing may increase the rate by removing destructive interference. However, if realtime coherence effects are small (as is likely for single-surface chemical reactions at room temperatures), QTST gives a good approximation to an upper bound, in which case the optimal dividing surface s(x) is the surface that minimizes kQTST . At temperatures above the cross-over temperature Tc , the optimal dividing surface can usually be well-approximated



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by

s(x) = q 0 − s0 ,

where q 0 =

PN i

(7)

qi /N is the ring-polymer centroid along the classical reaction coordinate q,

and s0 is located at the top of the barrier. This important special case of QTST corresponds to the centroid-TST method. 20–22 For symmetric or weakly asymmetric barriers, the centroid dividing surface continues to work below Tc , typically down to Tc /2. For asymmetric barriers, however, the dividing surface mixes in other degrees of freedom at temperatures below Tc ; above Tc /2, it can be approximated by 19

s = q 0 cosφ − rsinφ − s0 ,

where r =

(8)

p 2 2 (q+1 + q−1 )/N , and q±1 are degenerate free ring-polymer normal modes with

frequency 2π/β¯ h in the N → ∞ limit. The FEIT rate 27 is obtained by analytically continuing the partition function into the complex plane, and is given by 2 kFEIT Qr (T ) = β¯ h

r

π Qs=0 (T ), 2β|F 00 (0)|

(9)

where Qs=u (T ) is the constrained partition function 1 Qs=u (T ) = lim N →∞ (2π¯ h)N f

Z

Z dp

dx e−βN HN (p,x)

×δ[s(x) − u].

(10)

and F (u) is the ring-polymer free-energy 1 F (u) = − ln[Qs=u (T )]. β


(11)

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The function s in the delta function is usually chosen to be the same as the ring-polymer dividing surface s of Eq. (5) (i.e. it is chosen to be close to the surface that maximizes the free-energy). Clearly the expressions for kQTST and kFEIT are very similar. In the special case that the mass m associated with the s coordinate is not configuration-dependent, we obtain the simple relation 2π kFEIT = kQTST β¯ h

r

m . |F 00 (0)|

(12)

At temperatures below the cross-over temperature, the steepest-descent limit of kFEIT yields the ‘instanton’ rate. 15–17,19,34 To our knowledge, this correspondence is the nearest thing to a rigorous justification for the use of kFEIT , namely that it provides an improved description of the thermal fluctuations around the instanton. It therefore makes sense only to apply FEIT at temperatures below Tc . Tests on one-dimensional models have found that FEIT gives a consistent improvement on the QTST rate in this regime. Examples of this trend are shown in Fig. 1, for a symmetric Eckart barrier potential, using parameters taken from ref. 19. Figures 2 and 3 show comparisons of kQTST and kFEIT with the exact quantum rates for some collinear and three-dimensional reactions using potential energy surfaces 35–39 taken from the POTLIB website. 40 The free-energy F (u) was calculated using umbrella integration 41 along the reaction coordinate s, with harmonic bias potentials placed at equally spaced windows to yield the mean force f (s), which was then fitted to a spline 42 to give the second derivative F 00 (0) needed to evaluate Eq. (12). For the symmetric collinear reactions, we used s = q 0 (with classical reaction coordinate q taken to be the unstable normal mode of the collinear transition-state); for the asymmetric Cl + H2 reaction, we used the conical reaction coordinate of Eq. (8), obtaining the values of φ (which varied from φ = 0.299 to φ = 0.422 radians from 300 K to 100 K) from instanton calculations at each temperature. For the fully-dimensional H + H2 and D + H2 reactions, we used the dividing surface of ref. 2,43 and



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44, which interpolates between the asymptotic dividing surface

s0 (q) = R∞ − |R|,

(13)

in which |R| is the vector between reactant atom A and reactant diatom BC center-ofmass (and R∞ is chosen large enough to ensure the A-BC interaction is negligible), and the transition-state dividing surface ‡ ‡ s1 (q) = rBC − min[rAB , rAC ] − (rBC − rAB ),

(14)

where rXY is the distance between atoms X and Y and the daggered quantities are the values at the transition-state. Note that this dividing surface is not optimal, as is reflected in the apparent shift in the cross-over temperature in Fig. 3. The collinear calculations used N = 256 (T > 200K) and 512 (T ≤ 200K) polymer beads; the three-dimensional calculations used N = 192 (T ≤ 300K), 128 (T = 350, 400K) and 64 (T ≥ 500K). The collinear Cl + HCl reaction was also tested, but found to give poor results for both kQTST and kFEIT , most probably because of the large amount of dividing-surface recrossing in this heavy-light-heavy reaction. Unlike the one-dimensional results in Fig. 1, the collinear and three-dimensional results in Figs. 2 and 3 do not show consistently better performance of FEIT over QTST at all temperatures below Tc . Instead, there is a range of temperatures below Tc across which QTST is closer to the exact quantum result than FEIT; in the H/D + H2 calculations, this range extends as far down as half the cross-over temperature Tc . Once the temperature is below this range, however, then FEIT does give better results than QTST, where trends first identified in ref. 19 (that QTST underestimates the rate for symmetric barriers and overestimates it for asymmetric barriers) also appear to hold. †

†

Note that small changes to

Additional calculations would be needed to confirm whether the upward trend in the QTST rate for Cl + H2 continues below 200 K.


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theses results would be obtained if the dividing surfaces were optimised perfectly, but this is unlikely to change the trends in Figs. 2 and 3. In summary, when applying QTST and FEIT to two- and three-dimensional collinear reactions, we have not been able to find a direct over-the-barrier reaction which replicates the one-dimensional behavior shown in Fig. 1. Instead, we find that QTST gives better agreement than FEIT with the exact quantum rate coefficient over a range of temperature below cross-over. In addition to our findings, many recent studies have shown that RPMD methods can produce results close to accurate quantum rates and experimental results for many gas-phase reactions. 45–48 Further systems need to be explored in the future, but the trend found here suggests that QTST-based methods give more accurate results than FEIT, except at very low temperatures.

Acknowledgments YZ acknowledges funding from the British Council in New Zealand and the University of Cambridge. SCA acknowledges funding from the UK Engineering and Physical Sciences Research Council.

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1.2

103

1.0 100 10-3 10-6 2

Ratio

k cms−1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.8

Quantum QTST FEIT Classial-TST

4

0.6 6

βħω‡

8

10

12

0.4 2

Quantum QTST FEIT

4

6

βħω‡

8

10

12

Figure 1: Comparison of exact quantum, QTST, FEIT and classical TST rate coefficients k for the symmetric Eckart barrier used in ref. 19. The panel on the right plots the ratios of the QTST and FEIT rate coefficients to the exact quantum rate. The blue dashed line indicates the cross-over temperature Tc .



104

2.1

100

1.8

Quantum QTST FEIT Classical-TST

10-12

k cms−1

10-16

Ratio

H +H2

10-8

2

4

6 1000/T

8

10

0.6 1.4

10-7

1.2

10-14

H +ClH

10-21

10-35


2

4

6 1000/T

8

10

0.6

H +BrH Quantum QTST FEIT Classical-TST

2

4

6 1000/T

8

10

10-2 10-4

1.3

Quantum QTST FEIT

H +ClH

2

4

6 1000/T

8

10 Quantum QTST FEIT

H +BrH

2

4

6 1000/T

2.7

8

Cl +H2 Quantum QTST FEIT Classical-TST

1.9

10 Quantum QTST FEIT

2.3

2

10

1.1

0.7

Ratio

100

8

0.9

104 102

6 1000/T

1.5

10-15 10-20

4

1.7

Ratio

10-10

1.0

2

0.8

100 10-5

H +H2

1.2

100

10-28

k cms−1

1.5

Quantum QTST FEIT

0.9

Ratio

k cms−1

10-4

k cms−1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Cl +H2

1.5 1.1

3

1000/T

4

5

0.7

2

3

1000/T

4

Figure 2: Same comparison as Fig. 1 for several collinear reactions.


5

Page 17 of 17

10-12

2.1

10-14

1.8

10-18 10-20 10-22

H +H2

Ratio

k cms−1

10-16


2

10-12

3 1000/T

4

10-18 10-20 10-22

H +H2

1.2

0.6

5

2

3 1000/T

4

2.1

D +H2 Quantum QTST FEIT Classical-TST

2

1.5

5 Quantum QTST FEIT

1.8 Ratio

10-16

1.5

Quantum QTST FEIT

0.9

10-14

k cms−1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


D +H2

1.2

0.9 3 1000/T

4

5

0.6

2

3 1000/T

4

5

Figure 3: Same comparison as Fig. 1 for fully-dimensional H + H2 and D + H2 .

Graphical TOC Entry


Which Is Better at Predicting Quantum-Tunneling Rates: Quantum

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