The stability and diffusion of protons and deuterons in rare gas matrices are reexamined. These are known to be stabilized in rare gas matrices in the form of linear, centrosymmetric Rg₂H⁺ cations. The elementary step in their diffusion, displacement from one Rg–Rg bond to a neighboring one, can be modelled as an isomerization of the triangular Rg₃H⁺ cation. Using an analytic approximation for thermally averaged transmission coefficients for tunnelling through and reflection by a truncated parabolic potential barrier (R.T. Skodje and D.G. Truhlar,J. Phys. Chem. 85, 624 (1981)) we calculate the rate constants for this elementary diffusion step. The calculated rate constants are consistent with all experimental observations, and confirm that tunnelling makes the dominant contribution to the diffusion of protons and deuterons in rare gas solids. Deuteration reduces the tunnelling rates by 5 to 8 orders of magnitude, which agrees with the observation that D⁺ in rare gas solids is signficantly more stable than H⁺

Beyer, M.K.

Bondybey, V.E.

Savchenko, E.V.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10, p. 1045–1048The contribution of tunnelling to the diffusion of protonsand deuterons in rare gas solidsMartin K. Beyer1, Elena V. Savchenko2, and Vladimir E. Bondybey11Institut für Physikalische und Theoretische Chemie, Technische Universität München,4 Lichtenbergstraße, Garching 85747, GermanyE-mail: beyer@ch.tum.de; bondybey@ch.tum.de2 B. Verkin Institute for Low Temperature Physics and Engineering of the National Academyof Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, UkraineE-mail: savchenko@ilt.kharkov.uaThe stability and diffusion of protons and deuterons in rare gas matrices are reexamined. Theseare known to be stabilized in rare gas matrices in the form of linear, centrosymmetric Rg2H+ ca-tions. The elementary step in their diffusion, displacement from one Rg–Rg bond to a neighboringone, can be modelled as an isomerization of the triangular Rg3H+ cation. Using an analytic appro-ximation for thermally averaged transmission coefficients for tunnelling through and reflection bya truncated parabolic potential barrier (R.T. Skodje and D.G. Truhlar, J. Phys. Chem. 85, 624(1981)) we calculate the rate constants for this elementary diffusion step. The calculated rate con-stants are consistent with all experimental observations, and confirm that tunnelling makes thedominant contribution to the diffusion of protons and deuterons in rare gas solids. Deuteration re-duces the tunnelling rates by 5 to 8 orders of magnitude, which agrees with the observation thatD+ in rare gas solids is signficantly more stable than H+.PACS: 47.45.–n, 63.50.+x, 66.30.DnIntroductionInfrared absorptions associated with deuterons andprotons in solid rare gases have been known for morethan thirty years, and in fact some of the puzzling as-pects of their earliest observations were linked withtheir diffusion behavior. Initially, a strong infraredabsorption near 644 cm–1 was detected, whenever ar-gon with a small amount of deuterium was dischargedand deposited on a cold substrate held at 14 K. How-ever, no comparable absorption appeared when normalhydrogen was used instead [1,2]. This problem turnedout to lie in the fact that grating infrared spectrome-ters used at the time required typically some 60 min-utes to scan the usual IR-range from 4000–400 cm–1.When, on the other hand, the spectral scan wasstarted directly in the expected region, the corre-sponding absorption due to normal hydrogen was de-tected near 905 cm–1, shifted almost exactly by theappropriate 2 ratio. While the hydrogen absorptiondecreases in intensity with a half-life of a few minutes,the deuteron band is much more stable, with its rate ofdecrease being slower by several orders of magnitude.Initially there also was some doubt as to the chargestate of the carrier of the absorption bands, but subse-quent evidence pointed increasingly towards posi-tively charged protons [3–7]. Similar absorptionswere also found in solid krypton [2], and more re-cently in xenon [5], and it is now well established thatthe transitions are due to the 3 asymmetric stretchand 3 + n1 combination bands with the low fre-quency symmetric stretching mode of the linear,centrosymmetric Rg2H+ cations. Already the first in-vestigation of these absorptions pointed out the iso-electronic nature of Rg2H+ species with the well-known HX2– bihalide anions. From the chemical pointof view, protons — or deuterons — are essentiallydimensionless charges, and show therefore a clearpreference for a coordination number of two, that isthey are strongly solvated by two and only twostrongly bound rare gas ligands [8].As noted above, both H+ and D+ are observable insolid argon, krypton and xenon. H+ is quite short-lived in argon, and disappears on a much slower, 103 stime-scale in solid Kr and Xe, whereas D+ is muchmore long-lived in argon, and essentially stable in the© Martin K. Beyer, Elena V. Savchenko, and Vladimir E. Bondybey, 2003heavier rare gases, with the orders of magnitude dif-ference in the stability between protons and deuteronsbeing suggestive of a tunnelling process. Interest-ingly, neither protons, nor deuterons have thus farbeen observed in solid neon. We have recently dis-cussed these phenomena in this journal [9], and sug-gested that diffusion might be activated by room tem-perature black-body radiation, which exhibits a strongoverlap with the 3 band of Rg2H+. At that time, wedid not quantify the effect of tunnelling, but sug-gested that it might contribute to the observed differ-ences.A proton or a deuteron in a rare gas solid is insertedbetween two neighboring atoms to form the linear,centrosymmetric Rg2H+ entitity. Interestingly, thedistance between the two atoms with the proton in-serted is in fact considerably shorter than the distancebetween two nearest neighbour atoms in an unper-turbed rare gas lattice. An elementary step of diffusionthrough the solid then requires displacement of theproton from one Rg–Rg bond to a neighboring one,and can be modelled as an isomerization of a triangu-lar Rg3H+ cation, as shown schematically in Fig. 1. Inthe T-shaped, C2v cation, the proton is displacedstretching the Rgb–H+ and shortening the Rgc–H+distance to reach the equally C2v transition state. Wehave computed the barrier heights and frequencies ofthis transition state in our recent investigation [8,9].In the present work, we use these results to estimatethe transmission rate constants for H+ and D+ in solidneon, argon, krypton and xenon through the diffusionbarrier, accounting for thermal activation and tunnel-ling.Computational detailsThermally averaged transmission coefficients fortruncated parabolic potential barriers are calculatedwith the analytic approximation by Skodje and Truh-lar [10], which resolves some problems associatedwith the earlier approaches by Wigner [11] and Bell[12]. As an input into this formalism, the barrierheight and the imaginary frequency of the transitionstate are needed. The transition states of Rg3H+ havebeen calculated in our previous work [8,9] with theB3LYP functional and the 6-311++G(3df, 3pd) basisset on hydrogen, neon and argon, and the Stutt-gart/Dresden relativistic effective core potential ba-sic set [13] on argon, krypton, and xenon, employingthe Gaussian94 [14] program suite. In all cases, fre-quencies are scaled by a factor of 0.8 to account for theinfluence of the rare gas solid [9].Results and discussionAs argued previously [8,9], the reaction sequencedepicted in Fig. 1 for Rg3H+ represents one step in thediffusion of protons or deuterons in rare gas solids.Zero-point corrected barrier heights V0 and imaginaryfrequencies  are given in the Table. The key parame-ter to describe tunneling is the transmission coeffi-cient (T) at temperature T [10], which is more in-structively called the enhancement of the transfer ratedue to tunneling by Lill and Helms [15]. With = –2/hi (1)and = 1/kT (2)it is easily shown that the conditions  <  and V0 > 1are fulfilled for the species discussed in this work. Ac-cording to Skodje and Truhlar [10], (T) is then ap-proximated by the equation:(T) = [/( – )] exp [( – )V0] . (3)At T = 10 K this yields the exceedingly large num-bers which are listed in the Table. They indicate thatat these low temperatures, tunnelling is the only con-tribution to the diffusion of protons as well as deuter-ons.In order to get a quantitative estimate of the rateconstants involved in the diffusion process, classicalArrhenius rate constants are calculated. Since thebending mode 2 of the protonated rare gas dimer cor-responds to the imaginary mode in the transition state,its frequency is chosen as the Arrhenius A-factor, i.e.the frequency with which the barrier is probed. TheArrhenius rate constant kA then amounts to:kA(T) = c2 exp (–V0) . (4)Calculated values for kA are listed in the Table.They illustrate that thermal activation is in no casesufficient to overcome the diffusion barrier, and sig-nificant energy input e.g. from room temperature1046 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10Martin K. Beyer, Elena V. Savchenko, and Vladimir E. BondybeyMinimum C Minimum CTransition state CRga Rga RgaRgb Rgb RgbRgc RgcRgc2v 2v 2vFig. 1. Cluster model for proton diffusion in rare gas ma-trices. The proton changes its connectivity when it goesthrough the transition state of the Rg3H+ ion. A largenumber of these elementary steps lead to a random motionof the proton through the matrix. Reproduced from ref.[8] by permission of the PCCP owner societies.black-body radiation [9] would be required to explainthe diffusion. However, the enhancement of transferrate due to tunnelling described by (T) is sufficientlylarge to balance these low Arrhenius rate constants.With the assumption that the quantum mechanicalbarrier is probed at the same rate as the classical bar-rier, the semi-classical thermally averaged rate con-stant kSC which accounts for thermal activation andtunnelling amounts to:kSC= (T) kA(T) = c2[/( – )] exp (–V0).(5)These values are again given in the Table, and areshown in Fig. 2. Since most parameters, like V0 and ,are in the argument of the exponential function, theaccuracy of these results cannot be better than a feworders of magnitude. Nevertheless, the numbers clear-ly reflect the experimental observations: In solidneon, the deuteron changes place on a timescale of s,and the proton on a timescale of ns, both consistentwith the fact that neither H+ nor D+ have been ob-served in solid neon. In either case, the charged parti-cle is rapidly diffusing through the matrix and mayfind species like electrons, X– impurities, neutral hy-drogen atoms, H2O or H2 [16] to recombine.The barrier V0 is much higher for argon, kryptonand xenon than for neon. Consequently, diffusion rateconstants are reduced by 10 orders of magnitude ormore. Within this group of heavier, more polarizablerare gases, there is still a significant increase in thelifetime of the charge carrier for argon via krypton toxenon, since the barriers slightly increase, and moreimportantly, the frequencies in the transition stateconsiderably decrease. This is consistent with the ex-perimental observation that H+ is relatively short-lived in argon, and much easier to observe in kryptonand xenon. In the simultaneous presence of hydrogenThe contribution of tunnelling to the diffusion of protons and deuterons in rare gas solidsFizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 1047TableMatrix, dopant H+ or D+, barrier height V0 in kJ·mol–1, scaled imaginary frequency  in cm–1, scaled bending frequency 2in cm–1 (scaling factor 0.8 in both cases), enhancement of the transfer rate due to tunnelling (10 K), classical Arrheniusrate constant kA(10 K) in s–1, semiclassical rate constant which accounts for thermal activation and tunnelling kSC(10 K)in s–1.Matrix Dopant V0/ kJ· mol–1 -i/cm–1 2/cm–1  (10 K) kA(10 K)/s–1 kSC(10 K)/s–1Ne H+ 11.5 a 609 620 a 6.25·1055 1.98·10–47 1.24·109Ne D+ 13.8 b 441 444 b 9.66·1064 1.37·10–59 1.33·106Ar H+ 29.7 a 506 549 a 6.03·10141 1.50·10–142 9.06·10–1Ar D+ 29.9 b 366 390 b 3.97·10137 9.64·10–144 3.83·10–6SECP cAr H+ 30.7 a 506 518 a 3.59·10146 8.48·10–148 3.05·10–1Ar D+ 31.6 b 364 369 b 1.97·10145 1.20·10–152 2.37·10–7Kr H+ 32.3 a 471 486 a 1.32·10153 3.49·10–156 4.60·10–3Kr D+ 32.8 b 335 345 b 1.10·10149 6.06·10–159 6.63·10–10Xe H+ 35.5 a 441 444 a 1.36·10167 6.15·10–173 8.35·10–6Xe D+ 35.9 b 313 314 b 2.70·10161 3.54·10–175 9.57·10–14Comment: a From Ref. [8]; b From Ref. [9]; c The values above the horizontal line were computed considering all electrons explicitly.The computations below the divider were carried out using the quasi-relativistic effective core potential basis sets of the Stutt-gart/Dresden group [13].HH    SECPDD    SECP++++Ne Ar Kr Xe1050–5–10–15log (k    )scMatrix gasFig. 2. Semi-classical rate constant of the elementary dif-fusion step for Ne, Ar, Kr, and Xe. While both H+ and D+have a high mobility in a neon matrix, diffusion is consid-erably slowed down in argon, krypton, and xenon.Deuteration reduces the rate constants by 5 to 8 orders ofmagnitude, which stabilizes D+ relative to H+ in matricesof heavier rare gases.and deuterium, also the rate of deuteron disappear-ance is increased, suggesting that also the neutralsinks for the charge carrier are diffusing in part bytunnelling.As expected, however, deuteration reduces the dif-fusion rate constants kSC by 5 to 8 orders of magni-tude, which means that in cases where H+ is short-lived, D+ becomes long-lived. If H+ is long-lived, D+becomes essentially stable, as observed experimentallyfor xenon by Räsänen and coworkers [5].In the presence of two different rare gases, that isin terms of Fig. 1, if Rgb is different from Rgc, thenthe two minima on the right and left hand sides willno longer be equivalent, but will have substantiallydifferent energies. Under these circumstances the dif-fusion should stop, and the positive charge would re-main localized near the heavier, more polarizable raregas atom. Species like XeH+ and Xe2H+ should there-fore be observable in solid neon.ConclusionsThe calculations of rate constants which accountfor thermal activation as well as for tunnelling showthat the latter process, tunnelling, dominates the dif-fusion of protons and deuterons in rare gas solids. At atemperature of 10 K, tunnelling is faster than thermalactivation by 55 orders of magnitude for H+ in neon,up to 167 orders of magnitude for H+ in xenon. Thevalues are consistent with all experimental observa-tions. Deuteration reduces the tunnelling rates by 5 to8 orders of magnitude, which is consistent with theobserved, significantly higher stability of D+ in com-parison with H+ in rare gas solids. The results also in-dicate that the transition states proposed earlier areindeed relevant for the diffusion of protons and deu-terons in rare gas solids.Financial support from the Deutsche Forschungs-gemeinschaft through the program «Forderung derwissenschaftlichen Beziehungen deutscher Wissen-schaftler zu Wissenschaftlern in Ländern Mittel- undOsteuropas sowie Ländern der vormaligen UdSSR» isgratefully acknowledged.1. V.E. Bondybey, G.C. Pimentel, and P.N. Noble, J.Chem. Phys. 55, 540 (1971).2. V.E. Bondybey and G.C. Pimentel, J. Chem. Phys.56, 3832 (1972).3. D.E. Milligan and M.E. Jacox, J. Mol. Spectrosc. 46,460 (1973).4. C.A. Wight, B.S. Ault, and L.E. Andrews, J. Chem.Phys. 65, 1244 (1976).5. H. Kunttu, J. Seetula, M. Räsänen, and V.A. Apka-rian, J. Chem. Phys. 96, 5630 (1992).6. H.M. Kunttu and J.A. Seetula, Chem. Phys. 189, 273(1994).7. T.D. Fridgen and J.M. Parnis, J. Chem. Phys. 109,2155 (1998).8. M. Beyer, A. Lammers, E.V. Savchenko, G. Niedner-Schatteburg, and V.E. Bondybey, Phys. Chem. Chem.Phys. 1, 2213 (1999).9. M. Beyer, E.V. Savchenko, G. Niedner-Schatteburg,and V.E. Bondybey, Fiz. Nizk. Temp. 25, 1087 (1999)[Low Temp. Phys. 25, 814 (1999)].10. R.T. Skodja and D.G. Truhlar, J. Phys. Chem. 85,624 (1981).11. E. Wigner, Z. Phys. Chem. B19, 203 (1932).12. R.P. Bell, Trans. Faraday Soc. 55, 1 (1959).13. A. Nicklass, M. Dolg, H. Stoll, and H. Preuss, J.Chem. Phys. 102, 8942 (1995).14. Gaussian 94, Revision D.4, M.J. Frisch, G.W.Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson,M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Peters-son, J.A. Montgomery, K. Raghavachari, M.A. Al-La-ham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman,J. Cioslowski, B.B. Stefanov, A. Nanayakkara,M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen,M.W. Wong, J.L. Andres, E.S. Replogle, R. Gom-perts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. De-frees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gon-zalez and J.A. Pople, Gaussian, Inc.Pittsburgh PA(1995).15. M.A. Lill and V. Helms, J. Chem. Phys. 115, 7985(2001).16. M. Beyer, E.V. Savchenko, G. Niedner-Schatteburg,and V.E. Bondybey, J. Chem. Phys. 110, 11950(1999).1048 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10Martin K. Beyer, Elena V. Savchenko, and Vladimir E. Bondybey

The contribution of tunnelling to the diffusion of protons and deuterons in rare gas solids

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The contribution of tunnelling to the diffusion of protons and deuterons in rare gas solids

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