The mechanism of the interstellar isomerization reaction HOC+\HCO+

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THE JOURNAL OF CHEMICAL PHYSICS 130, 244308 共2009兲

The mechanism of the interstellar isomerization reaction HOC+ \ HCO+ catalyzed by H2: New Insights from the reaction electronic flux Stefan Vogt-Geisse and Alejandro Toro-Labbéa兲 QTC, Facultad de Química, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago 6094411, Chile

共Received 31 December 2008; accepted 12 May 2009; published online 26 June 2009兲 A theoretical study of the mechanism of the isomerization reaction HOC+ → HCO+ is presented. The mechanism was studied in terms of reaction force, chemical potential, reaction electronic flux 共REF兲, and bond orders. It has been found that the evolution of changes in REF along the intrinsic reaction coordinate can be explained in terms of bond orders. The energetic lowering of the hydrogen assisted 共catalyzed兲 reaction has been identified as being due to the stabilization of the H+3 transition state complex and the stepwise bond dissociation and formation of the H–O and H–C bonds, respectively. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3147702兴 I. INTRODUCTION

Since the first detection in 1970 共Ref. 1兲 the formyl ion 共HCO+兲 has been one of the most important molecules to astrophysics due to its characteristic rotational lines, which makes it, after CO, the most easily detectable molecule in the interstellar medium. Thanks to this peculiar feature astrophysics use HCO+ as a tracer molecule to get information about the physical properties of the interstellar medium.2 Additionally, the discovery of this ion confirmed chemical models incorporating ion-molecule reactions as the main synthetic reaction channel. The isomer of the formyl ion HOC+ was first detected experimentally in 1982 by Gudeman and Woods3 followed by its detection in the interstellar medium in 1983.4 To the date its detection together with HCO+ has been confirmed for several molecular clouds including DR 21ⴱ共OR兲, W51M W3共OR兲, Orion共3N,IE兲, Orion KL Sgr B2, and G34.3; the HCO+ / HOC+ ratios encountered in these objects range from 360 to 6000.5 Since the activation barrier of the unimolecular rearrangement was found to be too high for interstellar conditions 共40 kcal/mol兲 共Ref. 6兲, an alternative interconversion mechanism had to be proposed. First it was thought that the interconversion happened through reaction with atomic hydrogen, but it was found that this reaction as well possesses a considerable activation barrier.7 An alternative interconversion mechanism, the ion-molecule reaction, HOC+ + H2 → HCO+ + H2 ,

共1兲 8

was first proposed by Jarrold et al., but soon discarded because of a too large barrier height, which was found to be 10 kcal/mol relative to the separate reactants. Nevertheless a more recent ab initio calculation of the energy barrier, at higher level of theory 关CCSD共T兲兴, found that it is practically nonexistent taking the noninteracting reactants as the reference and of 11.9 kcal/mol with reference to the activated complex, making this reaction a viable depletion mechanism of HOC+.9 Furthermore kinetic studies indicate that this rea兲

Author to whom correspondence should be addressed. Electronic mail: [email protected].

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action is relatively fast at interstellar temperatures 共3.8⫾ 0.5⫻ 10−10 cm3 s−1 at 25 K兲, thus confirming the interstellar relevance of this reaction.10 Considering this state of the art, we analyze the reaction mechanisms of the unimolecular and H2-assisted isomerization reactions in order to get insights on the nature of the catalytic effect that is introduced by molecular hydrogen. Both reactions, the assisted and nonassisted isomerization, are studied along an intrinsic reaction coordinate 共IRC兲.11 Along this reaction coordinate, several global properties, such as reaction force,12–16 electronic chemical potential,17–19 reaction electronic flux 共REF兲,20,21 and electronic bond orders, are studied. Additionally a new useful qualitative descriptor is introduced, which is the derivative of the electronic bond order, which were obtained through natural bond orbital 共NBO兲 analysis.22 With the help of this tools, electron transfer processes are identified and the mechanism of both reactions in terms of electronic reordering can be well established. II. THEORETICAL BACKGROUND A. Energy and force profiles

Useful information about the mechanism of a chemical process can be obtained from the profile of reaction force. Along the IRC ␰, the reaction force is defined as12–16 F共␰兲 = −

dE , d␰

共2兲

where E共␰兲 is the potential energy. It has been shown in recent papers that the critical points of F共␰兲 define regions along ␰ in which different kinds of processes might be taking place.13 For a chemical reaction blue, where reactants and products are separated by an energy barrier, F共␰兲 presents two critical points, which are a minimum at ␰1 and a maximum at ␰2. These points define the following reaction regions:13,23 reactant region 共␰R ⱕ ␰ ⱕ ␰1兲, transition state region 共␰1 ⱕ ␰ ⱕ ␰2兲, products region 共␰2 ⱕ ␰ ⱕ ␰ P兲, and ␰R and ␰ P are the position of reactants and products, respectively. In

130, 244308-1

© 2009 American Institute of Physics

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the reactant region the reacting complex is prepared for chemical transformation mainly through structural rearrangements and preparative polarization processes. The TS region is characterized by actual charge transfer processes that take place to achieve the bond reordering from reactant to products. In the product region the molecular structure relaxes until reaching the reaction products. The energy involved as the reaction advances along the reaction coordinate can be characterized through the works determined at the different steps of the reaction, W1 = −

W3 = −



␰1

␰R



␰2

␰0

F共␰兲d␰ ;

F共␰兲d␰ ;

W2 = −

W4 = −



␰0

␰1



␰P

␰2

F共␰兲d␰; 共3兲 F共␰兲d␰ ,

with ␰0 as the position of the transition state. The activation energy 共⌬E⫽兲 can be therefore rationally partitioned as13–16,24,25 ⌬E⫽ = W1 + W2 .

共4兲

It is worth stressing the fact that the reaction force provides a rational way to partition the activation and reaction energies, ⌬Eⴰ = W1 + W2 + W3 + W4 .

共5兲

An important result that comes out from partition given by Eq. 共3兲 is that the physical nature of ⌬E⫽ can now be attributed through the relative weight of the components W1 and W2.24,25

B. Chemical potential and reaction electronic flux

The chemical potential is a global electronic property that describes the reactivity of molecular systems and has been defined within density functional theory as follows:17,18

␮=

冉 冊 ⳵E ⳵N

␷共r兲

= − ␹,

共6兲

where ␹ is the electronegativity.26,27 It represents the escaping tendency of the electronic cloud from equilibrium.19 The use of the finite difference approximation leads to the following working expressions for ␮:17,18

␮⬵−

共I + A兲 , 2

共7兲

where I and A are the first ionization potential and electron affinities, respectively. The above equation can be used to determine the chemical potential all along the reaction coordinate thus giving rise to ␮共␰兲. Having ␮共␰兲 at hand and in order to be able to rationalize the electronic transfer process when studying a chemical reaction along a reaction coordinate, a new useful descriptor has been recently introduced: the REF.20,21 The REF naturally arises from the chemical potential when considering it as a charge density gradient between two points of the reaction and is defined as

J共␰兲 = − Q

冉 冊

d␮ , d␰

共8兲

where Q is the transport coefficient that can be estimated from the reaction and activation energies and chemical potentials.12 Through this profile it will be possible to identify the regions along the reaction coordinate where polarization and charge transfer are taking place. Following the classic thermodynamic analogy, regions with J共␰兲 ⬎ 0 are associated with spontaneous arrangements of the electronic density, whereas J共␰兲 ⬍ 0 should be associated with nonspontaneous change of the electron density. A decomposition can be made in terms of polarization and transfer fluxes, such that20,21 J共␰兲 = J p共␰兲 + Jt共␰兲.

共9兲

For a bimolecular reaction the polarization flux is calculated when the reactive complex is separated into molecular fragments, A and B, which are treated separately through the counterpoise method,28,29 taking the sum of the fragments flux as polarization flux, J p共 ␰ 兲 = J A共 ␰ 兲 + J B共 ␰ 兲

and

Jt共␰兲 = J共␰兲 − J p共␰兲.

共10兲

The polarization flux takes into account the deformation of the electronic cloud of a given fragment in response to the external field created by the other fragment; of course it includes the intramolecular electronic transfer within the fragments. This effect takes place generally in the preparation or relaxation steps of the reaction. The intermolecular charge transfer, Jt共␰兲, includes the charge transfer between the fragments. With this decomposition at hand, a very detailed characterization of the charge transfer mechanism can be obtained and together with the rate of change of the bond order a complete picture of the bond formation, dissociation, and polarization effects can be given as shown in Sec. IV C. III. COMPUTATIONAL DETAILS

The profiles along the reaction coordinate for the various properties studied here were obtained using PBE functionals30 for exchange and correlation. The Dunning basis set aug-cc-pVTZ 共Ref. 31兲 was used, all calculations were preformed with the GAUSSIAN03 package.32 The transition states were localized using the quadratic synchronous transit33 approach and frequency calculations were preformed to check the nature of the stationary states. The minimum energy paths were obtained using the IRC 共IRC= ␰兲,11 which is arbitrarily zero at the transition state position. The profiles of energy, reaction force, chemical potential, and REF were obtained through single point calculations on the previously optimized geometries of the supermolecule obtained from the IRC procedure. Natural bond orbital34 analysis was carried out in which the Wiberg bond order was obtained along ␰ to monitor the evolution of the local electronic properties. The REF was obtained through numerical differentiation of the chemical potential profile ␮共␰兲, which, at each point ␰, was determined through Eq. 共7兲. For calculating the polarization flux following the partitioning of Eq. 共10兲, the Counterpoise routine28,29 of the GAUSSIAN03 package was used. With

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‡ ‡ TABLE I. Reaction energy 共⌬Eⴰ兲, forward 共⌬Efor 兲, reverse 共⌬Erev 兲, and reaction works associated with the different processes.

Reaction R1 R2

FIG. 1. 共Color online兲 Sketch of the geometry of reactants, TS, and products.

this routine the chemical potential for two independent fragments were calculated at each point of the reaction coordinate, using the corresponding fragment’s geometry already obtained at points ␰ and the basis set of the supermolecule. Having the chemical potentials ␮A共␰兲 and ␮B共␰兲 of the individual fragments at hand, the electronic flux J p共␰兲 and Jt共␰兲 were obtained according to Eq. 共10兲. IV. RESULTS AND DISCUSSION A. Energy and reaction force profiles

Figure 1 shows the reactions under study: reaction 1 共R1兲 is the nonassisted or free isomerization and reaction 2 共R2兲 represents the hydrogen-assisted protonic transfer isomerization. It can be seen that in R2 the hydrogen 共H1兲 leaving from the donor atom is the same as the one received by the acceptor atom, restricting the action of the molecular hydrogen to the assistance of the protonic transfer. The energy and reaction force profiles of R1 and R2 are displayed in Fig. 2 and Table I contains the complete energetic information of the isomerization in terms of reaction works. The global energy change show similar trends in reactions R1 and R2, since they are both of exothermic nature, indicating a greater thermodynamic stability for the HCO+ isomer. However R1 is 11 kcal/mol more exothermic than R2. Thus, the presence of the molecular hydrogen achieves a

FIG. 2. Relative energy 共in kcal/mol兲 and reaction force 共in kcal/ mol ␰兲 profiles of isomerization reactions R1 and R2.

⌬Eⴰ

‡ ⌬Efor

‡ ⌬Erev

W1

W2

W3

W4

⫺40.62 ⫺29.96

33.17 13.24

73.79 46.23

22.40 7.34

10.77 5.90

⫺32.65 ⫺24.27

⫺41.14 ⫺21.96

destabilization of the HCO+ product so that the assisted reaction gives a less stable product. This is a first indicator of an important ion-molecule interaction between H2 and the formyl ion to assist the protonic motion. The energetic barrier for the forward isomerization reac‡ tion in R1 is ⌬Efor = 33.17 kcal/ mol, while in R2 the barrier suffers a decrease of 19.93 kcal/mol to reach a value of ‡ ⌬Efor = 13.24 kcal/ mol, what once again shows the important role of H2 in assisting the hydrogenic motion. In the reverse reaction the lowering of the barrier is of 37%, which is considerably lower than for the forward reaction 共60%兲, thus suggesting a more important role of the hydrogen catalyst in the direct reaction. To get a more detailed vision on the energetic barrier, the characterization of the different contributions by the reaction force analysis can be of great utility. The reaction force divides the reaction in three characteristic regions; in the reactant region mostly structural rearrangements occur thus preparing the reactants to reach the second region, the TS region, which is characterized by reordering in the electronic cloud. In the last region there is structural relaxation to obtain the reaction products. To each of these processes there is associated a specific amount of work defined in Eq. 共3兲 and listed in Table I. Note that when going from R1 to R2 there is a lowering throughout all reaction works, suggesting that molecular hydrogen has a significant effect over all reaction regions. However the most significant decrease occurs in W1 at the reactant region. This can be explained considering the geometric restraints, which R1 has to overcome to bring the hydrogen near to the acceptor atom in order to reach the TS region and begin the electronic transfer. In R2 these constraints are relieved by the assisting hydrogen. The above observation can be confirmed by analyzing the H1–O–C angle along the reaction coordinate. In R1 this angle has a value of 82° at the force minimum, ␰1, thus indicating that most of the energy spent in the reactant region is used to bring the H1–O–C angle from 180° to 82°. At the TS the H1–O–C angle is 65°, just 23° below the value at ␰1, thus confirming the fact that W1 is mostly used in lowering this angle and bringing the hydrogen atom near the acceptor atom. Contrary to that, in R2 the H1–C–O angle is 109° in ␰1, consistent with a W1 value much smaller than that of R1. In the R2 isomerization a less sharp angle is necessary to begin the charge transfer. Consistent with the above observation are the relative contributions W1 and W2 to the forward reaction barrier, W1 in R1 represents 68% of the energy barrier, while in R2, it makes out only 55%, thus indicating that in R1 the structural rearrangement is much more relevant than in R2, in this latter the importance of both contributions W1 and W2 is essentially the same. When looking at the reaction works

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FIG. 4. 共Color online兲 Polarization and transfer flux 共in kcal/ mol ␰兲 for R2.

FIG. 3. Chemical potential 共in kcal/mol兲 and REF profiles 共in kcal/ mol ␰兲 for R1 and R2.

associated with the electronic transfer processes 共W2 , W3兲, a significant lowering is observed, emphasizing the important electronic effect of the assisting hydrogen. In summary, molecular hydrogen has a strong influence over all reaction regions, nevertheless it has a more significant effect over W1, which can be explained observing the reaction geometry, since in the nonassisted reaction there is a large unavoidable structural obstacle to activate the charge transfer.

B. Chemical potential and reaction electronic flux

The profiles of ␮ calculated form the ionization potential and electron affinities are shown in Fig. 3. In both reactions the chemical potential is lower in products than in reactants. However the way to reach this minimum in R1 is different than in R2, accounting for a different mechanism of charge transfer in both isomerizations. Notice that in both reactions the most dramatic change in chemical potential takes place in the TS region, confirming that it is in this stage of the reaction, where most electronic reordering occur. To get more insights on the nature of the electronic transfer processes the REF obtained from Eq. 共8兲 will be very useful. To obtain a qualitative trend and a proper comparison of both reactions, a transport coefficient Q = 1 was used.20,21 The REF of both reactions are also shown in Fig. 3. A similar trend in both reactions can be observed at first sight; the two profiles display a maximum at the reactant region, a minimum entering the TS region, and a large maximum also within this region. Then it follows a decrease in J共␰兲 to reach a minimum value at the product region to finally converge to a zero flux state at the product. A closer view shows that R1 presents a significant broad peak at the TS, the minimum spotted entering the TS region is quite small, whereas in R2 the minimum at ␰1 followed by the maximum just before ␰2 might be accounting for two main stages of electronic transfer, one before reaching the TS and a second more intense one leaving the TS.

To gain more insights on the REF, a decomposition was made for R2 considering H2 and HOC+ as the molecular fragments all along the reaction coordinate. The results for polarization and transfer fluxes are shown in Fig. 4. It can be noticed by inspection of Figs. 3 and 4 that the initial and final trend of J共␰兲 comes from the polarization flux, so that they can be associated with net polarization processes occurring during the activation and relaxation processes, these are deformations in the electronic cloud that subsequently activate the charge transfer. Note that in R1 this preparative polarization flux is larger than in R2 共Fig. 3兲, which confirms that it is in the reactant region where the catalytic effect of hydrogen is most important; this result is consistent with the reaction force analysis. The polarization flux profile of R2 also presents a quite constant flux at the TS region, which however cannot be considered as a preparative polarization flux like the one present at the begin and end of the reaction. The electronic transfer flux shows two peaks that coincide with the critical points found in the total flux profile indicating that the main changes in the electronic density are due to charge transfer between the two molecular fragments HOC+ and H2. C. Natural bond order analysis

It has been shown that the electronic reordering accompanying the hydrogenic motion presents a different form in the nonassisted than in the catalyzed reaction. This was displayed through the analysis of the REF, which had a different shape in both reactions. Now we go on to analyze the modifications in the local electronic density through the characterization of the bond order changes along the reaction coordinate. Figure 5 displays the bond orders along the reaction coordinate of all bonds in both reactions. The major changes in the bond orders of all bonds occur or are initiated within the TS region confirming that it is in this region where the biggest part of the electronic reordering takes place. In the transition state the bond of the transferred hydrogen to the carbon and oxygen atoms are remarkably weaker in R2 than in R1, which shows a higher affinity of this hydrogen atom to the molecular hydrogen than to the C–O fragment. Additionally, by analyzing the bonds between the three hydrogens, it can be noticed that at this point of the reaction the three bond orders take a similar value, indicating the existence of a H+3 cluster35 in the R2 transition state, which is only weakly bounded to the C–O fragment. Further sustain

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FIG. 5. 共Color online兲 Bond order evolution of all bonds for R1 and R2.

to this observation provides the natural population analysis, which indicates that most of the positive charge 共0.86兲 is shared between these three hydrogens, thus forming a H+3 ¯ CO transition state, in which the H+3 cluster is stabilized by ␴ resonance.36 On the other hand, the bond order displayed by the H2 molecule is different at the product from its value at the reactant, this is so because the reactant is defined by the complex H2 ¯ HOC, whereas in the product both fragments are practically separated, as shown in Fig. 1. To gain more insights on the electronic transfer mechanism, we analyzed the derivative of the bond order with respect to the reaction coordinate 共Fig. 6兲, which are the rates of dissociation and formation of the respective bonds. Naturally, within this definition a negative sign indicates bond weakening or dissociation, while a positive sign accounts for bond formation or strengthening. With these results at hand

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the difference in the charge transfer mechanism between the two isomerizations in study become evident. In R1, the charge transfer between the dissociating bond H1–O and the forming bond H1–C occur in a direct way, i.e., they both reach a maximum rate of change at the TS, where, from a configurational point of view, the direct charge transfer is most easy. Notice that both H1–C and C–O are being formed and strengthened 共positive sign in the derivative兲, respectively, while only the H1–O bond dissociates 共negative sign in the derivative兲, which means that in this reaction bond formation processes are dominant. On the other hand in R2 the minimum of the derivative of the H1–O bond order is located before the TS at ␰1, while the derivative of the H1–C bond order is close to zero. In addition, Fig. 6 shows at ␰1 the rates of the H1–H2 and H2–H3 bonds reach a quite symmetric maximum and a minimum, respectively, indicating the preparation of a H+3 cluster. Therefore, before reaching the TS, some electronic reordering takes place to form the stable H+3 cluster dissociating the H1–O and weakening H2–H3 bond while forming the H1–H2 and H1–H3 bonds. Note that at the TS the derivative of all hydrogen bonds 共H2–H3, H1–H2, and H1–H3兲 is zero, thus confirming the stability of the cluster at this point of the reaction. After passing the TS, a second electronic transfer step is observed, in which the H1–C bond is formed and the H+3 cluster dissociates to yield the products H2 + HCO+. This second charge transfer step is most significant just before leaving the TS region as the rate of change of all bonds being formed and cleaved reach their maximal expressions. The three maxima involved in this second electronic reordering step are higher, in absolute value, than in the first step, this is due to the formation of a stronger bond. Additionally throughout all the TS region the C–O bond is strengthened in a relative constant way 共see also Fig. 5兲, in contrast with R1, where the rate of change also reaches a maximum at the TS. The above analysis suggests that the lowering of the reaction barrier is due to a different charge transfer process. In R1 the charge transfer occurs in a direct way, which translates into the simultaneous dissociation and formation of the changing bonds, while in R2 the molecular hydrogen induces a charge transfer mechanism that takes place in a stepwise way: first the H1–O bond is dissociated entering the TS region then the H1–C bond is formed when leaving this region. D. Derivative of bond populations and reaction electronic flux

FIG. 6. 共Color online兲 The bond order derivatives of all bonds for R1 and R2.

Comparing the profile of the REF 共Fig. 3兲 and the derivative of the bond order 共Fig. 6兲 analyzed in the previous section, a clear qualitative agreement is observed. However this rate of change accounts only for charge transfer between bonds, thus the polarization effects, which are present in both reactions, do not appear. In R1 all three derivatives reach their maximum at the TS 共Fig. 6兲 with the difference that the H1–O derivative is negative whereas the H1–C and O–C derivatives have a positive sign. Since bond forming processes are dominant, the REF, which contains the global information about the charge transfer, has a positive sign. This result suggests that a positive sign in the REF can be attrib-

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FIG. 7. Sum of the derivatives of H1–C, H1–O, H1–H2, H1–H3, and H2–H3 bond orders for R2.

uted to the dominance of bond formation, which when the necessary conditions are set is an spontaneous process. In R2 the first minimum in the electronic transfer flux profile 共Fig. 4兲 coincides with the peaks of the derivative of the bonds involved in this first transfer step. Summing up the contribution of the derivative of all four changing bonds involved in this first charge transfer step 共H1–O, H1–H2, H1– H3, and H2–H3兲 a total negative derivative is obtained 共Fig. 7兲, hence indicating that a negative sign in the REF implies dominance of electronic flux to promote bond dissociation in a nonspontaneous process. In the second transfer step the flux has a positive sign, which again is in accordance with the sum of the bond order derivatives involved 共Fig. 7兲 in the second charge transfer step 共H1–C, H1–H2, H1–H3, and H2–H3兲, thus sustaining the fact that a positive sign in the REF corresponds to electronic motion promoting bond formation processes. Note also that, in both REF and the derivative of bond order, the intensity of the first peak is less than in the second one, which confirms that more electronic flux is necessary in the second transfer step to form the stronger H1–C bond. Through the rate of changes of the bond order the polarization flux of R2 displayed in Fig. 4 can be further decomposed. The polarization flux in the TS region 共Fig. 4兲 evidently corresponds to the strengthening of the C–O bond 共Fig. 5兲; it can be observed that the derivative of this bond 共Fig. 6兲 shows the same qualitative behavior in this region as J p共␰兲. Thus the polarization flux present in the TS region of Fig. 4 can be attributed to an intramolecular charge transfer to strengthen the C–O bond while the polarization flux in reactant and product region should be attributed to net polarization effect in the electronic cloud to promote the reaction. On the other hand, the transfer flux 共Fig. 4兲 is in good qualitative agreement with the sum of all bonds involving hydrogen atoms 共Fig. 7兲, which, as expected, are responsible for the intermolecular transfer flux. Note that this fact confirms that the electronic transfer goes through the molecular hydrogen, since, considering the way the molecular fragments were defined, the charge transfer from the dissociating to the forming bond would appear as an intramolecular polarization flux.

In summary, the analysis of REF and the derivative of the bond order suggest a completely different electronic transfer process for both reactions. R1 begins with structural motion to bring donor and acceptor atoms closer accompanied by a strong polarization of the electronic cloud to activate the charge transfer. Once donor and acceptor atoms are sufficiently close the charge transfer begins and reaches a maximum at the TS. Finally a relaxation to products occur, which once again implies a polarization flux. R2 also begins with preparative polarization flux, which however is less than in R1. This stage is followed by the first charge transfer step to form the stable H+3 ¯ CO transition state. Passed the TS the formation of the product begins and the second charge transfer step reaches it maximum just before leaving the TS region. Once again a strong polarization flux is observed throughout the end of the reaction on its way to relaxing to products. V. CONCLUDING REMARKS

In this work the mechanism of assisted and nonassisted interstellar isomerization reactions was fully characterized by the analysis of energy profiles, reaction force, chemical potential, REF, and population analysis with its derivative. The electronic transfer in the nonassisted reaction takes place in a direct way requiring a large structural activation, while in the hydrogen catalyzed reaction the charge transfer happens in two steps, hence facilitating the bond reordering and lowering the energy barrier. Additionally a qualitative relation between REF and the derivative of the bond population was found, thus assigning a positive sign in the REF to bond formation and a negative sign to bond dissociation processes. ACKNOWLEDGMENTS

The authors wish to thank financial support from project FONDECYT Grant No. 1090460 and FONDAP Grant No. 11980002 共CIMAT兲. D. Buhl and L. E. Snyder, Nature 共London兲 228, 267 共1970兲. P. Papadopoulos, Astrophys. J. 656, 792 共2007兲. 3 C. Gudeman and C. Woods, Phys. Rev. Lett. 48, 1344 共1982兲. 4 R. Woods, C. Gudeman, R. Dickman, P. Goldsmith, W. Huguenin, W. Irvine, A. Hjalmarson, L. Nyman, and H. Olofsson, Astrophys. J. 270, 583 共1983兲. 5 A. Apponi and L. Ziurys, Astrophys. J. 481, 800 共1997兲. 6 J. Martin, P. Taylor, and T. Lee, J. Chem. Phys. 99, 286 共1993兲. 7 S. Green, Astrophys. J. 277, 900 共1984兲. 8 M. Jarrold, M. Bowers, D. DeFrees, A. Mclean, and E. Herbst, Astrophys. J. 303, 392 共1986兲. 9 E. Herbst and D. Woon, Astrophys. J. 463, L113 共1996兲. 10 M. Smith, S. Schlemmer, J. von Richthofen, and D. Gerlich, Astrophys. J. 578, L87 共2002兲. 11 K. Fukui, Acc. Chem. Res. 14, 363 共1981兲. 12 A. Toro-Labbé, J. Phys. Chem. A 103, 4398 共1999兲. 13 S. Gutiérrez-Oliva, B. Herrera, A. Toro-Labbé, and H. Chermette, J. Phys. Chem. A 109, 1748 共2005兲. 14 B. Herrera and A. Toro-Labbé, J. Chem. Phys. 121, 7096 共2004兲. 15 J. Martinez and A. Toro-Labbé, Chem. Phys. Lett. 392, 132 共2004兲. 16 A. Toro-Labbé, S. Gutierrez-Oliva, M. Concha, J. Murray, and P. Politzer, J. Chem. Phys. 121, 4570 共2004兲. 17 R. Parr and W. Yang, Density Functional Theory of Atoms and Molecules 共Oxford University Press, New York, 1989兲. 18 P. Geerlings, F. De Proft, and W. Langenaeker, Chem. Rev. 共Washington, D.C.兲 103, 1793 共2003兲. 1 2

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