New mechanism for the vibrational mode-specific proton-transfer reaction NH3+ (ν) + NH3 → NH2 + NH4+

Chemical Physics 156 ( 199 1) 79-84 North-Holland

New mechanism for the vibrational mode-specific proton-transfer reaction NH: ( v) + NH3 -NH2 + NH: Akitomo Tachibana Tokio Yamabe 1,2

‘, Tetsuo

Suzuki

‘, Naoto

Yoshida

‘, Yasuhiro

Teramoto

and

Department of Hydrocarbon Chemistry, Faculty OfEngineering, Kyoto University, Kyoto 606. Japan

Received 2 January 199 1

A new mechanism has been presented for the vibrational mode-specific depression of the proton-transfer reaction NH,+ ( Y) + NHs-+NHz +NH: We adopt the standpoint of the ADO (average dipole orientation) theory, proposed by Su and Bowers (J. Chem. Phys. 58 ( 1973) 3027), and append a new long-range interaction created by a vibration-induced dipole to the AD0 theory. A “reaction zone” concept is crucial in this approach; outside of the zone the long-range intermolecular interactions play a key role. A qualitatively good agreement is observed for the experimental result of the title reaction obtained by Chupka and Russel (J. Chem. Phys. 48 (1968) 1527)

1. Introduction Study of ion-molecule reactions is one of the active fields of research in chemistry [ l-51. It is well known for ion-molecule reactions that in many cases one can estimate the reaction rate based on the classical mechanical theory, in which one does not need a microscopic reaction mechanism. Especially for ion-polar molecule reactions, Bowers and co-workers have reported studies based on a classical mechanical capture reaction theory, which is called the ADO (average dipole orientation) theory [ 31. Their calculated results are in good agreement with the results of corresponding experiments. In the ADO theory one considers the reaction rate to be the capture rate, i.e. the rate at which the reactants approach each other into the region of a certain intermolecular distance (which may be called “reaction zone”) in which the reaction would absolutely occur. In the ADO theory, one does not need to discuss the reaction zone in detail, because reaction rates can be estimated by only considering long-range in’ Also: Division of Molecular Engineering, Kyoto University, Kyoto 606, Japan. * Also: Institute for Fundamental Chemistry, 34-4 Nishihimkicho, Takano, Sakyo-ku, Kyoto 606, Japan. 0301-0104/91/$03.50

teractions which work outside the reaction zone. This is an interesting and important point. In the ADO theory, a reactant molecular ion is treated like a point charge; so that the modification of the theory taking into account finite size of the molecular ion is an interesting theme. For example, if the molecular ion is vibrationally excited, the reaction rate may change; it may be an important and interesting problem to discuss this effect. Actually such experimental studies, for example a proton transfer reaction NH: (v) +NH3-+NH2+ NH:, have been reported [ 4,5 1. We have been studying the modification of the ADO theory in this regard [ 61. In this connection, we are now interested in the estimation of the deviation of intra- and intermolecular potentials caused by molecular vibration. In the present paper we would like to discuss this point. In general, in order to study dynamical processes of molecules such as chemical reactions, intra- and intermolecular energy transfer, and so on, one must estimate the intra- and intermolecular potentials of the considered molecular systems. In theoretical studies this is usually done as follows: owing to rapid development of ab initio methods, we can calculate the intra- and intermolecular potential non-empirically, in case they are not too big. Then we assume an

0 1991 Elsevier Science Publishers B.V. All rights reserved.

appropriate functional form of the potential, with some fitting parameters, which is formally described as U(r)=U(r,c,,c,

,...),

(1)

where r and c, represent coordinates of configuration space and fitting parameters, respectively. Famous examples of such model potentials are given in extensive studies by Clementi et al. [ 71 and Jorgensen et al. [ 81. Atom-atom type functional forms, such as the Lennard-Jones type and the Coulombic type, are used in these studies. In other studies intermolecular potentials are decomposed into their contributions, such as electrostatic, exchange, polarization, and dispersion interactions, in order to clarify the physical and chemical origin of the interactions [9-l 11. By using the thus obtained U(r), thermodynamical properties and chemical reactions for various kinds of molecules have been studied [ 711 1. For the estimation of the deviation of the potential caused by molecular vibration, there seems to be an approach based on the Hellmann-Feynman theorem [ 12 1. One can express the force caused by molecular vibration as follows:

for example, to the problem of vibrational predissociation processes of van der Waals molecules [2,18,19]. In the following sections we will discuss the application of the present method to the interaction between the ion and the polar molecule, in particular to the vibrational mode-specific proton transfer reaction NH: (Y) +NH3-+NH2+NH:. It should be noted here that other reaction channels also exist for the reaction system composed of NH: (v) and NH3, in which charge-transfer and hydrogen abstraction processes within the reaction zone attached to the other reaction channels play a key role [ 5,201.

2. Interaction between the ion and the polar molecule We approximately decompose the total Hamiltonian into three parts, i.e. the Hamiltonian for the isolated ion, H,, that for the isolated polar molecule, HI, and that for the interaction between the ion and the polar molecule, H’. The interaction Hamiltonian H can be described by Buckingham theory [ 2 1,22 ] as H’ =

T2q,

q2 +

~2,(41P2,

-q2hx)

+TzuS(f41gzuP+~428,,8--,la~2S)+...

(2) where Qk is the kth vibrational coordinate, and p( r) is the electron density of the system. Here, if one can estimate the density p(r) in eq. (2), one can determine the force. This can be done not only by using the ab initio MO method but also by density functional theory [ 131, for example by using the local scaling method [ 14,15 1. The deviation of the potential AU can be obtained by integrating eq. (2 ), AU(Q,=U(Q)-U(O)=

c j u,dQkr

(3)

where Q denotes the sum of all Qk. This method is advantageous when one treats long-range forces, where molecules are mutually separated enough, and then the density p(r) can be approximated well by the sum of the densities for isolated molecules. Actually such approaches, that one fits p( r), have been used in order to obtain the interaction potentials [ 16,171. Because our present method treats the interactions caused by vibration, it can be applicable,

1

(4)

where ql, pla, and @i,, are total charge, dipole moment, and quadrupole moment for the ionic moleare those for the polar molcule, and q2, Pan, and @22aS ecule. T,, T201,and TzaB are given by T, =R-’

,

T2,=V,R-‘=-RJR’, Tw = V,V,R-‘=

(3R,R,-R26,,)/R5,

(5)

where R is the vector from the center of mass of the ion to that of the polar molecule and R = IR I.In the original ADO theory, the ion was assumed to have no permanent dipole, and eq. (4) reduces to Kno

= Tzaqip2a .

(6)

The interaction between the charge of the ion and the induced dipole of the polar molecule is given as the second-order perturbation term caused by H’ [ 2 11. Let us consider the force caused by the vibrational motion of the ion. In the present treatment uk in eq. (2) consists of two contributions; one from the de-

A. Tachibana et al. / The proton transfer reaction NH:(v) + NH, -NH,

rivative of Hi to Qk, and the other from the derivative of H’ to Qk. We denote the two contributions by L$lra and Up, respectively. U?‘, is given by

where V, is the potential term of Hi and pi is the electron density of the ion. Because coordinates of electrons do not depend on Qk,

Substituting

+ NH:

81

eq. ( 12) into ( lo), we obtain ( ~2s

> T2aa

(14)

.

Once we have determined Uptra and Up, we can obtain the deviation of the potential, AU(Qk), by integration (see eq. (3 ) ):

AU(Q,) = - t/dQk)

> (~2s) Tzap+ t&Q:

3 (15)

where wp is the vibrational terms which do not depend for example, the interaction and the induced dipole of constant terms.

frequency of Qk. Other on vibrations of the ion, between the ionic charge the molecule, appear as

(8) where Riand r, are position vectors of the ith nucleus and the jth electron of the ion, respectively, and nij and Nii are the unit vectors in the directions of Ri-rj and Ri-Rj, respectively. Up is given by (9) In the present case aH’ /aQk in eq. (9) can be well approximated by considering only its first term, i.e. the dipole-dipole interaction: < ~2~3 > T2as

,

(10)

where ,ula is the dipole induced by vibration of the kth vibrational mode. Hence pia is a function of Qk,, and in the equilibrium configuration it vanishes: ~,a

=~,rr(Qk) ; P,,(O) =O .

In eq. ( lo), ap,,/aQk

(11)

is given by

(12) where Rin is the position vector of the ith nucleus centered to the center of mass of the ion, and R, is that of the charge center for the positive charge of the ionic molecule: & = 1 Z,RilZ,,

.

(13)

3. Application to the proton transfer reaction NH;(v)+NH,+NH,+NH; 3.1. Potential In this section as an application we shall discuss the vibrational mode-specific depression of the proton transfer reaction NH: (v) + NHj+NH2 +NH: [ 4,5 1. This reaction was treated by Chesnavich and Bowers by phase space theory, but their result overestimated the reaction cross section compared to the experimental result of Chupka and Russell [4]. In order to explain this discrepancy, the transition state hypothesis has been introduced [ 41. According to this hypothesis, the vibrational mode which depresses the reaction is considered to be orthogonal to the reaction coordinate at the transition state. However, the NH: (v) +NHJ+NH2+NH: reaction is known to have no activation energy barrier, and hence no transition state [ 5 1. So that we can expect it to be proper that one analyzes this reaction from the standpoint of the ADO theory. We should examine the deviation of the long-range forces by vibration of the ion, and its effect on the capture rate determined by the ADO theory. It is needless to say that our standpoint is not appropriate for the reactions in which a rate-determining step exists in the reaction zone. For example, the vibrational energy dependence of the reaction rate is also examined for the reaction of NH; (v) and HZ0

[ 41. According to the result of the theoretical study, the reaction of NH: (v) and Hz0 has a potential energy barrier; the result of the ab initio HF/4-3 1G calculation is that the height of the potential barrier is 17 kcallmol [ 23 1. By taking this result into account, the rate-determining step seems to be the process involving the barrier in the reaction zone. Therefore, it does not seem proper that one analyzes the reaction of NH: (v) and HZ0 from the standpoint of the ADO theory. Actually, the reaction rate obtained by the ADO theory is very different from that obtained by experiment [ 5 1. We now treat the three configurations shown in fig. I. First of all, for a long intermolecular distance, the

dipole moment of NH3 faces the opposite direction of NH:. This configuration maximizes the attractive interaction between the total charge of NH: and the dipole moment of NH,, therefore, in this point the probability of appearance of this con~guration is expected to be large. As for vibrationally excited NH: (v), we consider three cases: ( 1) NH: (v) has no dipole moment at the equilibrium point, (2 ) NH: (v) has a dipole moment which faces the opposite direction of that of NH3, and (3) NH: (v) has a dipole moment which faces the same direction as that of NH,. For these cases we can write the interaction potentials as

+-

(1)

(16)

_.__._________.._.___ .._........ N

where pl ( Qk) takes the values of 0, - pl, and ,u, according to the cases ( 1)) (2 ), and (3 ), respectively.

P2

\

H

3.2. Reaction rate First we consider the vibrational energy dependence of the deviation of the interaction potential. It can be expected that the more the vibration of NH: (n/) is excited, the bigger the vibrationally induced dipole moment of NH; is. Hence, as shown in fig. 2, the barrier of the effective potential for configuration (2) in fig. 1 becomes higher as the vibrational energy of NH: (v) becomes bigger. In order to add this effect in the phase space theory, let us consider the following. The probability of appearance of the two con~~~tions, (2) and (3) in fig. 1, is assumed to be same. Then we describe the new cross section 0 as

H

o= -I-IPbS(~)+fe,s(v)f(~) , (3)

._.._.__._...__.___._.. v2

\

H

Fig. 1. The three configurations discussed in this paper. ( 1) The case that NH; has no dipole moment. (2) The case that the vibrationally induced dipole moment faces the opposite direction of the dipole moment of NH>. (3) The case that the GbrationaIty induced dipole moment faces the same direction as the dipole moment of NH>.

(17)

where o,,(v) is the cross section derived from the phase space theory [4]. This means that configuration (3) does not affect opafV) but configuration (2 ) does; the degree is taken into account by the factor f(u). The functional form off(v) is not known, but we can expect thatfl0) = 1 andf( v) -+Oas v becomes bigger. This mechanism of depression is originated from the detailed treatment of the interaction potential, which is left intact in the phase space treatment but is explicitly taken into account in the ADO the-

A. Tachibana et al. / The proton transfer reaction NH:(v) + NH, -+NH, + NH:

83

V Fig. 4. Schematic plot of the vibrational tion cross section u( 0).

dependence

of the reac-

4. Conclusion

Fig. 2. Schematic plot of the potential deviation induced by vibration of NH: as a function of the intermolecular distance R. The curve “v= Cl” corresponds to configuration ( 1) of fig. 1.The potential curves which have humps correspond to configuration (2) of fig. 1. The potential curves under the “v= 0” curve cormspond to configuration (3 ) of fig. I.

1.0

f

0.5

0.0

L V

Fig. 3. Schematic

plot ofS( v).

ory. Then, if we assume that f( v) is a monotone decreasing function of v, we can obtain a schematic plot off(v) versus v as shown in fig. 3. Using this and eq. ( 17), we can obtain the schematic fig. 4 for the vibrational energy dependence of the reaction cross section a(v). This plot reproduces the experimental result qualitatively well; see fig. 1 of Chesnavich and Bowers [ 4 1.

Our present method seems useful when one treats the case that long-range intermolecular forces play the dominant role in the considered problem, for example, ion-molecule reactions with no activation energy barrier. In particular, we have discussed the proton transfer reaction NH: (v) +NHs+NH2+NH$ from this point of view. We have found that there is a possibility that the vibration-induced dipole of the ion plays an important role and one can explain the mode-specific depression of this reaction without considering the reaction zone in detail. A more sophisticated treatment of this problem is now undertaken and will be reported elsewhere [ 6 1.

This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan, for which we express our gratitude. We are also grateful to the computer Center of the Institute for Molecular Science and the Data Processing Center of Kyoto University for their generous permission to use HITAC M-680H and S820 computers, and FACOM M-780130, VP400E, and VP-200 computers, respectively.

References [ I ] P. Ausloos, ed., Kinetics of Ion-Molecule Reactions (Plenum, New York, 1979). [2] R.D. Levine and R.B. Bernstein, Molecular Reaction Dynamics and Chemical Reactivity (Oxford Univ. Press, Oxford, 1987).

84

/i. Tuchrhana et al. /

Theproton transfer reaction :VH~(~))+XH~ -NH2 +NH:

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