Conference Proceedings Paper Information Entropy of Molecular Tunneling

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1 Conference Proceedngs Paper Inforaton Entropy of Molecular Tunnelng A. V. Stepanov, * and M. A. Stepanov Natonal Ozone Montorng Research and Educatonal Centre, Belarusan State Unversty, Kurchatov str. 7 86, Mnsk, Belarus Departent of Techncal Physcs, Inforaton Technologes and Robotcs Faculty, Belarusan Natonal Techncal Unversty, B. Khelntsky str. 9, Buldng, 003 Mnsk, Belarus; sst@al.ru * Correspondence: stepav@bsu.by; Tel.: Abstract: Molecular tunnelng process has been consdered by eans of radaton theory. The forula for nforaton entropy calculaton has been derved by eans of nteracton odel of theral equlbru radaton wth a olecule at low teperatures. The physcal eanng of nforaton entropy for low-teperature plateau of unolecular checal reacton has been deterned. It s a easure of converson of theral radaton energy to echancal energy that oves atos n a olecule durng eleentary actvaton act. It s also a easure of uncertanty of ths energy converson. The converson takes place at a teperature when the average energy of the eleentary actvaton act s equal to a part of zero energy of the transforng olecule. Two unolecular reactons have been nvestgated. These are Fe-CO bond recobnaton n β-heoglobn and double proton transfer n benzoc acd der for sequental deuteraton of hydrogen bond and varous hydrostatc pressures. Usng the nforaton entropy forula t s possble to calculate ts value n energy unts of easureents for low-teperature plateau. Probabltes of occurrence of the reactons under consderatons, ther effcency and ean-square fluctuatons of the dstrbuton functon paraeters have also been deterned. Keywords: Shannon entropy; nforaton entropy of olecular tunnelng; theral equlbru radaton; nteracton odel of theral radaton wth a olecule at low teperatures; Fe-CO bond recobnaton n β-heoglobn; double proton transfer n benzoc acd der. Introducton The actvaton process odel was proposed n 990 [], ts physcal nature beng alternatve to the well-known Boltzann-Arrhenus odel. As a result of ts developent, the nteracton odel of theral equlbru radaton wth a olecule at low teperatures was created n 00 to explan such a puzzlng process as olecular tunnelng []. The nteracton process was caused by subsequent realzaton of the olecule s sub-conforatons. Each sub-conforaton state was approxated by eans of a haronc quantu oscllator. The oscllator was consdered non-deal,.e., ts degeneracy and excluson of soe excted vbraton levels was taken nto account. The olecular tunnelng tself was due to a part of the oscllator s zero energy, the energy beng the result of the transton fro the prevous sub-conforaton state to the next one. All the process features were allowed for whle choosng the dstrbuton functon paraeters. Ths ade possble a nuercal sulaton of four olecular tunnelng reactons. These were photonduced converson of rhodopsn and deuterated rhodopsn nto prelurhodopsn, low-teperature polyerzaton of foraldehyde, and Fe-CO bond recobnaton n β-heoglobn. Inforaton entropy of actvaton process has been derved recently [3]. Its physcal eanng has also been deterned. It would be useful to do the sae for olecular tunnelng process, snce several researchers beleve that nforaton entropy s conceptually ore fundaental even by coparson wth energy [4]. So n ths paper we derve the forula for nforaton entropy of olecular tunnelng. Then two The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07

2 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 reactons are nvestgated nuercally. These are Fe-CO bond recobnaton n β-heoglobn and double proton transfer n benzoc acd der. The second reacton s consdered for sequental deuteraton of hydrogen bond and for varous hydrostatc pressures. For all these reactons the values of nforaton entropy are calculated. Probabltes of occurrence of the processes under consderaton, ther effcency and ean-square fluctuatons of the dstrbuton functon paraeters are also deterned.. Inforaton Entropy of Molecular Tunnelng Accordng to the actvaton process odel [5], the actvaton barrer for olecular tunnelng process can be represented as a set of dentcal quantu oscllators (Fgure ). These oscllators denote nteredate sub-conforatons of a olecule as the latter transfors. The barrer s n therodynac equlbru wth the theral radaton. Any par of adjacent oscllators has ther energy levels shfted by μhν. Here h s Planck constant, μ s the paraeter descrbng the levels shft, and ν s the effectve frequency of the quantu oscllator averaged over the generalzed reacton coordnate. The value of ν equals to the Ensten spontaneous esson coeffcent A [6]. The probablty of a transton over ths actvaton barrer can be wrtten as [] P nμ ( n+ μ) hν/ nhν/ e Ω [( n+ μ) hν ] e Ω [( n+ μ) hν ] = = ( n+ μ) hν/ nhν/ e Ω [( n+ μ) hν ] e Ω [( n+ μ) hν ] 0 0 where Ώ s the degree of degeneracy for (n + μ) hν vbraton level. Consequently, nforaton entropy for the process under consderaton can be expressed by eans of energy unts of easureent [3] nhν nhν, μ = nhν nhν 0 e Ω [( n+ μ) hν ] e Ω [( n+ μ) hν ] S k n e Ω [( n+ μ) hν ] e Ω [( n+ μ) hν ] 0 0 () Now, let us take nto account the fact that dfferent levels (or groups of levels) n the quantu oscllators can have the dfferent degrees of degeneracy, e.g., Ω, Ω, Ω3 (Fgure ),.e., the vbraton levels up to quantu nuber are non-degenerate, the ones fro quantu nubers to η have the degeneracy Ω, and those fro η to ζ, Ω, and fro ζ to, Ω3 [5,7]. The transton s forbdden f Ω = 0. Fgure. Arrangeent of the quantu oscllators averaged over the generalzed coordnate of checal reacton on the potental barrer.

3 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 Fgure. Partton of the energy levels of the averaged quantu oscllator takng nto account of the degree of degeneracy ( < η < ζ). Consderng the levels partton, the denonator of () ay be presented as nhν nhν η nhν e [( n μ) hν] e [( n μ) hν] e 0 0 ς nhν nhν +Ω [( n+ μ) hν] e +Ω 3[( n+ μ) hν] e. η ς Ω + = +Ω + + On the other hand, ths denonator can be splfed analytcally [,7] 0 nhν + A Q + B Q + CQ e Ω [( n+ μ) hν] =, Q hν / Q= e, A=Ω where, B=Ω Ω, So Equaton () can be transferred nto C =Ω and 3 Ω. nhν η nhν, μηζ,,, Q S =k e [( n ) h ] e +Ω + μ ν + + AQ + BQ + CQ 0 ζ nhν nhν Q +Ω [( n+ μ) hν] e +Ω 3 [( n+ μ) hν] e n η ζ + AQ + BQ + CQ nhν e +Ω n+ h e 0 +Ω ( n+ ) h e +Ω ( n+ ) h e. η nhν ζ nhν nhν [( μ) ν] [ μ ν] 3 [ μ ν] η ζ () where Entropy s addtve, so () can be separated nto four ters: S = S + S + S + S,,,,,,0,, η,, ζ,,, 3

4 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 S k e n e nhν nhν,,0 Q Q μ = + AQ + BQ + CQ 0 + AQ + BQ + CQ η nhν ( ) ( ) ( Q) Ω n+ μ hν ( Q) Ω n+ μ hν,, S k e n e nhν μ η = + AQ + BQ + CQ + AQ + BQ + CQ ζ nhν ( ) ( ) ( Q) Ω n+ μ hν ( Q) Ω n+ μ hν,, S k e n e nhν μη ζ = + AQ + BQ + CQ η + AQ + BQ + CQ ( ) nhν ( ) ( Q) Ω n+ μ hν ( Q) Ω n+ μ hν,, S k e n e nhν 3 3 μζ = + AQ + BQ + CQ ζ + AQ + BQ + CQ These four ters can be calculated usng the technque descrbed n [3], so we only wrte the fnal expresson for nforaton entropy:, μηζ,,, hν Q A Q + ηb Q + ζ C Q Q S = + k n T Q + AQ + BQ + CQ + AQ + BQ + CQ Ω nω ( Q Q ) +Ω nω ( Q Q ) +Ω nω Q k + AQ + BQ + CQ η η ζ ζ 3 3 (3) hν / Q= e, A=Ω wth, B=Ω Ω, C =Ω and 3 Ω. It should be noted there s no paraeter μ responsble for olecular tunnelng n (3). Let us try to overcoe ths apparent sunderstandng n the next secton. 3. Inforaton Entropy of Low-Teperature Plateau for Fe-CO Bond Recobnaton n β-heoglobn Such a process can be sulated wth a set of quantu oscllators wth non-degenerate vbraton levels [], that s Ω = Ω = Ω3 =. It s the dstnctve feature of ths reacton that allows one to splfy (3) as hν Q S = k n( Q) (4) T Q get Equaton (4) was derved n the prevous paper [3]. By ultplyng both parts of (4) nto T/, we hν S T hν = n( e ) hν (5) e The frst ter of (5) defnes the average energy of eleentary actvaton act. The second one descrbes a easure of energy uncertanty per quantu subsyste,.e., a olecule s sub-conforaton. It s known fro experents there always exsts a teperature of transton fro a low-teperature plateau to an actvaton part for any Arrhenus dependence. Solely fro the above reasonng we wrte the followng equalty: hν = μhν hν e Hence the teperature of transton T can be found as (6) 4

5 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 T hν = k n( + ) μ (7) Inforaton entropy can be expressed n ters of μ only. To do ths, we substtute (7) nto (5):, μ [ μ μ μ μ] S = k ( + ) n( + ) n. (8) The followng values of the paraeters for the reacton under consderaton can be used: ν = Hz, = , μ =.8 0 6, h = J s, and k = J/K []. Inserton of these values nto (7) gves T = K. Now t s possble to calculate the value of nforaton entropy for the begnnng of actvaton part of experental Arrhenus dependence. To do ths, we substtute the value of T nto (4) to get S = J/K. It should be noted that entropy S s easured n energy unts. On the other hand, (8) gves the,μ value of nforaton entropy for the low-teperature plateau S = J/K. Coparson of the two values shows that S s the requred paraeter correspondng to the low-teperature plateau. Notce that T s approxately equal to the teperature of actvaton barrer creaton T = K [8]. Besdes, Laura Esensten found experentally that a checal reacton rate constant of Fe-CO bond recobnaton n β-heoglobn s teperature ndependent for teperatures lower than 0 K [9]. All thngs consdered, the nteracton odel of theral equlbru radaton wth a olecule at low teperatures s realstc for olecular tunnelng representaton. So we have exaned the deal exaple of olecular tunnelng reacton,.e., when an actvaton barrer can be represented as a set of oscllators wth non-degenerate vbraton levels. Let us consder another reacton where the degeneracy and excluson of soe excted vbraton levels wll be taken nto account. 4. Inforaton Entropy of the Double Proton Transfer n the Benzoc Acd Der Ths process was nuercally sulated n [5]. The followng forula for the average energy of the eleentary actvaton act was used: ηζ,,, μ Q A Q + ηb Q + ζ CQ ε = hν + + μ Q + A Q + B Q + CQ (9) By analogy wth (6) we can wrte a transcendental equaton: Q A Q + ηb Q + ζ CQ hν + μhν = Q + A Q + B Q + CQ (0) One has to solve (0) nuercally to get the teperature of transton T fro the low-teperature plateau to the actvaton part of Arrhenus dependence. Substtuton of T nto (3) gves the value of nforaton entropy for the low-teperature plateau of the process under consderaton. Experental studes of the teperature dependence of the checal reacton rate constant were ade for onocrystallne sples wth sequental deuteraton of hydrogen bonds [0] and polycrystallne ones under varous hydrostatc pressures []. 5

6 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, Sequental Deuteraton of the Hydrogen Bond 4... Inforaton Entropy of the Low-Teperature Plateau Nuercal sulaton of ths process showed the paraeters Ω3 and ζ are excess [5]. So (0) can be rewrtten as η Q A Q + ηbq hν + μhν η = Q + AQ + BQ () The calculated values of T can be found n Table. Equaton (3) ay be splfed here because nω = 0 : η, η, hν Q A Q + ηb Q Q S = + k n η η T Q + AQ + BQ + AQ + BQ η Ω nω( Q Q ) k η + AQ + BQ () The derved values of nforaton entropy are shown n Table. Table. Inforaton entropy for partal and coplete deuteraton of the hydrogen bond n the benzoc acd der. Bond H-H D-H D-D 9 ν 0 Hz 8,4 5,0, Ω 0,75 0,80 0,95 Ω,0,0,0,0,0,0 η μ 4, ,0 0 3, T, K 56,37 68,587 5,757 S,,η, J/K 8, , , S,μ, J/K,73 0 4,8 0 5, Now let us return to the deal set of oscllators wth non-degenerate vbraton levels. Ths set s a,μ splfcaton for our deuteraton process, but we can use (8) to get the values of S and copare,,η,,η,μ the wth those of S. As seen fro Table, S are 3 0 tes saller than S. Ths result can be reasonably expected Assuptons Needed for Calculaton of the Requred Paraeters To calculate probabltes of occurrence of the reactons under consderaton as well as ther effcency and ean-square fluctuatons of the entropy paraeters, one has to know two teperatures [3]. These are the teperature of actvaton barrer creaton T and the teperature T when the barrer s overcoe by absorbng only two quanta of theral equlbru radaton (TER). These teperatures were derved analytcally for the set of deal quantu oscllators [8]. The correspondng actvaton barrer was n quas-equlbru wth TER. Teperature ndependent Ensten coeffcents A and B were used to get the values of T and T. However, these coeffcents depend on teperature n our case, so t s not possble to wrte down a set of equatons that satsfy the Ensten detaled balancng prncple for a olecule nteractng wth TER. One has to nvoke soe other assuptons to calculate T and T. Such assuptons ust not be contradctory to a physcal eanng of the actvaton process odel. 6

7 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 It has just been found n Secton 3, that T s approxately equal to T (T = 0,39 K vs. T = 0,776 K), where T s the teperature of transton fro the low-teperature plateau to the actvaton part for the Arrhenus dependence of Fe-CO bond-recobnaton n β-heoglobn. Let us assue that T s the teperature of actvaton barrer creaton f the barrer conssts of non-deal quantu oscllators. Of course, ths assupton requres a ore rgorous proof. Nevertheless, we can state that at T the actvaton barrer begns to anfest tself by defnng the checal reacton rate constant of the process under consderaton. To get the value of T, the followng reasonngs ust be taken nto account. Frst, as the olecule under transforaton s n therodynac quas-equlbru wth TER, ts probablty of transforaton should be slghtly less or greater than 0.5 [3]. Second, ean-square fluctuatons of the dstrbuton functon paraeters ust be less than paraeter values theselves, for the dstrbuton functon has no physcal eanng otherwse. So the change of nforaton entropy of actvaton that defnes the experental Arrhenus dependence can be derved usng (): Δ S = S S (3),, η,, η,, η T T Suppose that a unolecular reacton s a large therodynac equlbru fluctuaton. Then we can calculate ts probablty of occurrence usng Ensten Forula [,3] W = e,, η ΔS k (4) To get a unolecular reacton effcency, both left-hand and rght-hand sdes of () should be ultpled by T/ η, η, T Q A Q η B Q Q S = hν + + n η η Q + AQ + BQ + AQ + BQ η Ω nω( Q Q ) kt η + AQ + BQ (5) The frst ter of the rght-hand sde of (5) s the average energy of the eleentary actvaton act [3,5]. The two other ters descrbe uncertanty of ths energy. The uncertanty s due to unevenness and fuzzness of the dstrbuton functon representng the actvaton process. If the left-hand sde of (5) s taken to be 00%, a unolecular reacton effcency can be easly calculated. So f a unolecular reacton can be represented as a large fluctuaton, then nforaton entropy change,,η Δ S s a consequence of ths fluctuaton. Knowng,,η Δ S one can get ean-square relatve fluctuatons of the entropy paraeters. The followng forula proposed by Ensten [] can be used Δ = k,, S η Δ ( ) Δ (6) where Δ s one of the paraeters that deternes the entropy change as a result of the fluctuaton Calculaton Results The results are suarzed n Tables and 3. In Table ν a = ν s the effectve actvaton frequency. As seen fro the results, there exsts a correlaton between Δ ν a and Δ ν. The correlaton can be wrtten as ( Δ ν a ) = ( Δ ν ). When calculatng T and W, the results turn out to be very senstve to. Ths paraeter descrbes the lowest vbraton level of haronc oscllator where a quantu sub-conforaton of the transforng olecule can be called non-deal. 7

8 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 Table. Paraeters for sequental deuteraton of the hydrogen bond n the benzoc acd der. Bond H-H D-H D-D ν (Hz) 8, ,0 0 9, νа (Hz),93 0,5 0 3,95 0 Ώ 0,75 0,8 0,95 Ώ,0,0,0,0,0,0 η T 56,37 68,587 5,757 T 58 97,4 34 W 0,546 0,546 0,546 Δ ν (Hz),74 0 9, , Δ 5,47 0,04 66,09 Δ ν a (Hz) 5, ,75 0 0,058 0 Δν (Hz) 5, ,75 0 0,058 0 ν Δ (Hz) 4, ,0 0 0,057 0 ΔΩ 0,09 0,7 0, ΔΩ 0,079 0,034 0,006 Δ 0,994 0,985 0,995 Δ η 57,593 49,94 30,33 Table 3 deonstrates the effcency of the reactons under consderaton. Table 3. Coparson of the average energy of the eleentary actvaton act and ts uncertanty for sequental deuteraton of the hydrogen bond. Bond H-H D-H D-D Teperature, K T T T T T T 56, ,587 97,4 5, ε (%),458 54,55 9,8 84,6 35,85 85,575 ( ε (%) 87,54 45,484 90,77 5,784 64,49 4,45 S T ) / As teperature ncreases, so does the reacton effcency. Ths result s soewhat unexpected, snce the teperature dependence of Fe-CO bond recobnaton n β-heoglobn was shown to be just the opposte [3]. When the results of Table 3 are copared wth those of Table 4, t s apparent that a unolecular reacton effcency ncreases wth ncreasng teperature f the reacton s represented as a set of non-deal quantu oscllators. Table 4. Reacton effcency of Fe-CO bond recobnaton n β-heoglobn for varous teperatures. Process Teperature, K T Fe-CO Recobnaton T 0,397 80,86 ε (%) 93,0 63,98 S T ) / ( ε (%) 6,798 34,09 8

9 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, Inforaton Entropy of the Double Proton Transfer n the Polycrystallne Benzoc Acd Saple under Varous Hydrostatc Pressures As n the prevous secton, the actvaton barrer can be represented as a set of deal or non-deal quantu oscllators. The correspondng values of nforaton entropy for low-teperature plateaus η are gven n Table 5. It can be easly seen that S,,, μ are 3 tes saller than S. Table 5. Inforaton entropy of the double proton transfer n the polycrystallne benzoc acd saple under varous hydrostatc pressures. Pressure, kbar 0,0,3,0,4,9 3,3 9 ν 0 Hz 8,4 8,9 9,4 9,9 0,4 0, Ω,86 0,86 0,88 0,89 0,88 0,87 Ω,0,0,0,0,0,0,0,0,0,0,0,0 η μ 3,45 0 5,955 0,3 0,66 0,5 0,99 0 T, K 9,7 03,808 8,55 67,95 84,384 06,6 S,,η, J/K 7, , , , , , S,μ, J/K, , , , , , Square roots of ean-square relatve fluctuatons of the dstrbuton functon paraeters are shown n Table 6. Once agan, there s a correlaton between the effectve actvaton frequency ν a and oscllator frequency ν. It can be wrtten as Δ ν = Δ ν a The value of s stll extreely portant when calculatng T and W. The reacton probablty decreases wth ncreasng hydrostatc pressure appled to the saple. The pressure dependence s attrbutable to the departure of the therodynac syste under consderaton fro the equlbru state, so that the probablty s less than 0.5. Table 6. Paraeters for the double proton transfer n the polycrystallne benzoc acd der under varous hydrostatc pressures. Pressure, kbar 0,0,3,0,4,9 3,3 ν 0 9. Hz 8,4 8,9 9,4 9,9 0,4 0, νа 0, Hz 3,696 3,738 3,76 3,76 3,744 3,706 Ω 0,86 0,86 0,88 0,89 0,88 0,87 Ω,0,0,0,0,0,0,0,0,0,0,0,0 η T 9,7 03,808 8,55 67,95 84,384 06,6 T ,9 5,7 39,9 5,65 W 0,539 0,5 0,5 0,476 0,47 0,46 Δ ν 0 9, Hz Δ,4,563 3,07 3,848 3,657 3,586,6,08,39,99,95 0,837 Δ ν a 0 (Hz),06,076,8,463,37,9 Δν 0 (Hz) ν Δ 0 (Hz),06,076,8,463,37,9 0,976 0,998,057,87,64,8 ΔΩ 0,08 0,08 0,059 0,057 0,066 0,08 9

10 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 ΔΩ 0,036 0,037 0,048 0,066 0,058 0,053 Δ 0,999 0,983 0,999 0,999 0,999 0,999 Δ η 7,67 7,788 30,947 36,83 33,58 3,085 Once agan, as teperature ncreases, so does the reacton effcency (Table 7). Table 7. Coparson of the average energy of the eleentary actvaton act and ts uncertanty for varous hydrostatc pressures appled to the saple. Pressure, kbar Teperature, K ε (%) ( S T ) / ε (%) 0,0,3,0,4,9 3,3 T 9,7,636 87,364 T 60 6,46 30,54 T 03,808 3,08 86,99 T 65 66,353 33,647 T 8,55 5,38 84,68 T 46,9 43,447 56,553 T 67,95 7,9 8,7 T 5,7 9,58 70,47 T 84,384 6,49 83,509 T 39,9 35, 64,78 T 06,6 5,94 84,076 T 5,65 40,836 59, Calculaton of Ratos between Mean Square Relatve Fluctuatons of the Dstrbuton Functon Paraeters and the Paraeters Theselves The calculated rato values for the two reactons of nterest are shown n Tables 8 and 9. It was shown earler [3] that f the actvaton barrer s represented as a set of deal quantu oscllators then all the three paraeters ν, and νa have equal effect on the large therodynac fluctuaton, snce ther approprate ratos are all about 93%. As the data of Tables 8 and 9 suggest, the stuaton s dfferent for non-deal quantu oscllators set. The odel paraeters do not nfluence the therodynac fluctuaton to the sae extent. Partcular attenton should be gven to the degeneraces Ω and Ω. Ther nor change of several percent wll gve rse to the large therodynac fluctuaton responsble for unolecular reacton progress. Table 8. Square roots of ean-square relatve fluctuatons of the dstrbuton functon paraeters dvded by the accordng paraeters (n percent). Bond H-H D-H D-D Δ ν /ν (%) Δ / (%) 7,07 4,068 6,78,379 3,349 6,73 a ν a Δ ν / (%) 7,069 4,068 6,78 ΔΩ / Ω (%),976,59,74 ΔΩ / Ω (%) 7,873 3,436 0,59 Δ / (%) 99,49 98,53 99,56 Δ η /η (%) 3,539 9,377 33,59 0

11 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 Table 9. Square roots of ean-square relatve fluctuatons of the dstrbuton functon paraeters dvded by the accordng paraeters (n percent). Pressure, kbar Δ ν /ν (%) Δ / (%) 8,74 8,796 3,654 38,878 35,6 3,895 6,4 6,686 8,098 3,554 3,096 3,874 a ν a Δ ν / (%) 8,74 8,796 3,654 38,878 35,6 3,895 ΔΩ / Ω (%) 9,554 9,467 6,75 6,593 7,536 9,64 ΔΩ / Ω (%) 3,57 3,7 4,756 6,597 5,799 5,344 Δ / (%) 99,88 98,3 99,973 99,99 99,986 99,995 Δ η /η (%) 35,935 36,089 40,9 47,83 43,063 40,37 5. Conclusons Molecular tunnelng s one of the ost nterestng and yet lttle-known low-teperature checal processes. Ths s anly because a tunnelng olecule ass s three orders of agntude greater than that of electron, so ts de Brogle wavelength s sall copared to a checal bond length. Snce ths process occurs wdely n nature, ts nvestgaton s a fundaental proble. The proposed nteracton odel of theral equlbru radaton wth a olecule at low teperatures descrbes the process by usng a dstrbuton functon. The functon represents an actvaton barrer as the olecule s sub-conforaton states set. All the states share a coon trat, that of provdng the tunnelng process by eans of a part of zero-pont vbratons energy. To study the olecular tunnelng process n ore detal a concept of nforaton entropy for actvaton process has been used. Its physcal eanng was deterned earler []. The results of ths nvestgaton can be suarzed as follows:. The forula for nforaton entropy of olecular tunnelng has been derved.. Arrhenus dependence for olecular tunnelng process s caused by nforaton entropy change. In addton, nforaton entropes for low-teperature plateau and for the begnnng of the actvaton part of Arrhenus dependence are equal. 3. Probablty of occurrence of the double proton transfer n the benzoc acd der has been calculated for sequental deuteraton of the hydrogen bond and varous hydrostatc pressures. 4. Effcency coparson has been ade for Fe-CO bond recobnaton n β-heoglobn and double proton transfer n benzoc acd der. The frst reacton s ore effectve at low teperatures, whle the teperature dependence of the second one s the opposte. 5. There exsts a correlaton between ean-square fluctuatons of the effectve actvaton frequency and quantu oscllator frequency. 6. The calculated ean-square fluctuatons of the dstrbuton functon paraeters ndcate they do not nfluence a large entropy fluctuaton to the sae extent. Here the actvaton barrer s represented as a set of non-deal quantu oscllators where the degeneracy and excluson of soe exted vbraton levels are taken nto account. Acknowledgents: We thank N.D. Knyazhevch and A.G. Svetashev for any helpful suggestons. Author Contrbutons: The actvaton process odel was created and developed by A. V. Stepanov. He also dd the nuercal sulatons. M. A. Stepanov derved the equatons and wrote the paper. Conflcts of Interest: The authors declare no conflct of nterest.

12 The 4th Internatonal Electronc Conference on Entropy and Its Applcatons (ECEA 07), Noveber st Deceber 07; Scforu Electronc Conference Seres, Vol. 4, 07 References. Tavgn, V.L.; Stepanov, A.V. Actvaton Process Model. Phys. Status Sold B 990, 6, Stepanov, A.V. Interacton odel of theral radaton wth olecule at low teperatures: olecular tunnellng. J. Mol. Struct. 00, 578, Stepanov, A.V. Inforaton entropy of actvaton process: Applcaton for low-teperature fluctuatons of a yoglobn olecule. Int. J. Mod. Phys. B 05, 9, Jaynes, E.T. Inforaton theory and statstcal echancs. Phys. Rev. 957, 06, Stepanov, A.V.; Tsvrko, M.P. Sulaton of the double proton transfer dynacs n the benzoc acd der. J. Appl. Spectrosc. 004, 7, Stepanov, A.V. Actvaton process odel: Ensten coeffcents for actvaton barrer. J. Mol. Struct. 007, 805, Stepanov, A.V.; Tavgn, V.L. Developent of the actvaton process odel: Copensaton effect. Int. J. Quantu Che. 996, 59, Stepanov, A.V. Modelng of etaaterals: A globular proten as a etaateral prototype for electroagnetc-acoustc converson at low teperatures. Proc. SPIE 0, 8070, Esensten, L. Molecular tunnelng n hee protens. Int. J. Quantu Che. 976, 0, Stöckl, A.; Meer, B.H.; Kres, R.; Meyer, R.; Ernst, R.R. Hydrogen bond dynacs n sotopcally substtuted benzoc acd ders. J. Che. Phys. 990, 93, Horsewll, A.J.; McDonald, P.J.; Vjayaraghavan, D. Hydrogen bond dynacs n benzoc acd ders as a functon of hydrostatc pressure easured by nuclear agnetc resonance. J. Che. Phys. 994, 00, Ensten, A. Quantentheore des enatogen dealen Gases. Zwete Abhandlung. Stzung Phys.-Math. Kl. 95, 8, Lesne, A. Shannon entropy: a rgorous noton at the crossroads between probablty, nforaton theory, dynacal systes and statstcal physcs. Math. Struct. Coput. Sc. 04, 4, e by the authors. Lcensee MDPI, Basel, Swtzerland. Ths artcle s an open access artcle dstrbuted under the ters and condtons of the Creatve Coons Attrbuton (CC BY) lcense (