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Importance of complex-mode collisions in the vibrational relaxation of strongly attracting molecules. The H2OHCl(ν=1) system
Volume 216, number 1,2
CHEMICAL PHYSICS LETTERS
24 December 1993
Importance of complex-mode collisions in the vibrational relaxation of strongly attracting molecules. The H,O-HCl( U= 1) system H.K. Shin Department of Chemistry, University ofNevada, Reno, NV89557, USA Received 18 August 1993; in final form 13 October 1993
The deexcitation of HCl( u= 1) in the HIO-HCl interaction is studied at thermal collision energies using classical trajectory techniques. In the presence of strong attraction, the collision partners form a complex, and the initial vibrational energy flows freely among various internal modes in the complex. When the complex redissociates, the major portion of the energy is found to localize in the rotational motion of HCl. This study establishes the occurrence of vibrational relaxation in complex-mode collisions and the main energy transfer pathway to be vibration-to-rotation in the complex. Energy transfer is found to be inefficient in direct-mode collisions.
1. Intruduction
Strong attractive forces operate between hydrogen chloride and water molecules, and these forces can significantly affect dynamics of collision-induced molecular processes. In the presence of strong attraction, a collision between Hz0 and HCl molecules can result in the formation of a complex, in which the H atom of HCl is in close proximity to the oxygen atom. In this complex, where the H-O interaction is of major importance in describing intramolecular dynamics, the rotational motions of HCl become hindered, and the relative motion of the collision partners transforms into a large-amplitude oscillatory motion. The H-O interaction causes a major disturbance in the HCl bond but not in Hz0 because of the heavy mass of the 0 atom compared with the H atom. When a vibrationally excited HCl is involved, its relaxation can then proceed either in a complex-mode collision or in a direct-mode collision. Recent studies show that the vibrational relaxation of HCl( tr= 1) is efficient at room temperature, and its negative temperature suggests the importance of complex-mode collisions [ 11. The importance of collision complexes in vibrational relaxation has already been noted in earlier studies [ 2-
41. If complex-mode collisions produce efficient energy transfer, the result can be of fundamental importance in studying not only vibrational relaxation of hydrogen-bond interaction systems but also vibrational relaxation of polar molecules or molecular ions which produce strong dipole-dipole or ion-dipole attractions. The purpose of this study is to establish the occurrence of complex-mode collisions in the H,O-HCl system and that the vibrational relaxation of HCl( V= 1) can efficiently proceed in such complexes. We consider the interaction at thermal collision energies and use classical trajectories techniques.
2. Complex-mode collisions Hydrogen-bond attraction between HCl and Hz0 molecules is known to be as strong as 5.74 kcal/mol [ 5 1. In the presence of such strong attraction, the interacting molecules can form a collision complex, which can survive for a duration greatly exceeding the vibrational period of HCl. When an H,O-HCl complex is formed, the center-of-mass separation between the colliding molecules transforms into a lowfrequency oscillatory motion of the c.m. separation
0009-2614/93/S 06.00 Q 1993 Elsevier Science Publishers B.V. All rights reserved. SSDI 0009-2614(93)El265-I
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denoted by R in fig. 1. A considerable number of theoretical studies have been reported in recent years for the structure of this complex indicating the equilibrium configuration of O-H-Cl is essentially linear. The calculations show that the O-Cl-H angle varies from 0 to 3.4” [6-lo]. The results are consistent with Legon and Willoughby’s microwave spectroscopic observations of the linear O-H-Cl configuration [ 111. In H,O-HCl, we take the molecular units to undergo internal motions about the equilibrium structure of HCl in the XY plane and Hz0 in the X2 plane with CIV symmetry. The experimentally determined value of 3.22 A will be used for the O-Cl equilibrium distance [ 111; the calculated values of the O-Cl distance lie between 3.09 and 3.28 A [ 5,9,10,12]. When the complex is formed, H of HCl is in strong interaction with 0 of Hz0 through the hydrogen-bond attraction, and we consider R to represent the distance between the c.m. of HCl and the 0 atom. The vibrational displacement of the HCl unit from its equilibrium bond length d will be denoted by x. In the complex, the HCl unit can undergo complete internal rotations when the OCl separation is large. When the separation is small, on the other hand, HCl undergoes hindered rotation about the O-H-Cl equilibrium configuration in the XY plane. This rotation is represented by 8 in fig. 1. In addition, the HCl bond undergoes the out-of-XY plane rotational motion and the angle # represents this motion. Although there is strong attraction between the colliding molecules, we do not expect the concentration of hydrogen-bond dimers in the gas phase to be large. No concentration data are available for the present system, but we note that in H20-H20, the
24 December 1993
attractive energy is 6 kcal/mol, which is quite close to 5.74 kcal/mol of H,O-HCl, and the dimer concentration is known to be only about 1% at 373 K [ 131. On the other hand, the concentration of collision complexes surviving one to several picoseconds, which is much longer than the duration of direct-mode collisions, can be significant. The overall interaction of two molecular units in the nonrigid H,O-HCl complex is dominated by the interaction between the 0 atom of Hz0 and the H atom of HCI. Other interatomic distances are significantly larger so the corresponding interaction is much weaker than the H-O interaction. The H-O interaction is assumed to contain exponentially repulsive and attractive terms acting between the atoms. The first step in writing analytical forms of interaction potentials is to express the intermolecular atom-atom distances in terms of the 0-HCl internal coordinates as Zig= [R2+y&(d+~)Z2ycR(d+x) c0s(tY*+~*)“*]“* and Zig= [R* +y& (d + x)* + Zy,R(d + x) cos(6* + @2)1’2]1’2, where yHcl = rnr& (mn+ ma). We use these two distances in two pairwise interactions (see, for example, ref. [ 14 ] ) between the 0 atom and each atom of HCl. The equilibrium configuration of the complex is described by the coordinates R=R,, x=0, 8~0 and $=O. The equilibrium atom-atom distances are then simply zc,H_o= R,- d and ~,,~_o = R, + d. The potential function representing the HCl and 0 interaction can then be represented by adding the terms of these two atom-atom distances as V(R, X, 6, #) =&&exp{(R,-d)/a-2yaR(d+x)
cos(tl*+#*) “*I “*/u}
-2 exp{ (R,-d)/2a-2y,R(d+x)
HCI Fig. 1. Complex-mode collision model. 28
[R*+&(d+x)*
cos(8*+#*) “*I
-2 exp{ (R,+d)/2u+ZhR(d+x)
[R*+y&(d+x)*
cos(e*+#*) “*I “*/2u})
+Dcl_o(exp{ (R,+d)/a+2%R(d+x)
[R*+&(d+x)*
"'/a}
[R*+&(d+x)’
cos(8*+@*) “*I “*/2u}) .
(1)
Both observed and calculated H20-HCl interaction energies reported in the literature represent the H-
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0 hydrogen-bond attraction Dna. Therefore, the first part of eq. (1) is of major importance, but the second part becomes significant when HCl rotates in the 8 direction. We take Bacskay’s recent data of 5.74 kcal/mol for Dn_o [ 5 1. The calculated values vary over a wide range of 4.74 to 8.93 kcal/mol [S12,15,16] but the majority of the calculations give a value close to 6 kcal/mol. While the value of DHa is large due to the hydrogen-bond interaction, Da_o for the Cl-O interaction is expected to be significantly smaller. We estimate it from the attractive well depths of Hz0 and Cl2 using combining rule as Da_p (DHzoDa2) ‘12, where DHto = 775k and D,,=357k [ 171 #l. This procedure gives Da* = 1.04 kcal/mol. Because of its strong polar character, we use DW instead of Dot here. From ~~0.25 A for HCl-HCl [ 181 and 0.20 A for H20H20 [ 19,201, we estimate a=0.22 A for H20-HCl also using the combining rule. The HCl molecule is considered to be a Morse oscillator; V(x) = Do x ( 1 - e-x’b)2, where Do and b are the Morse parameters.
3. Solution of the equations of motion The equations of motion are
(2) where p is the reduced mass of the collision system, M is the reduced mass of HCI, Z is the moment of inertia of HCl, and V= V(R, x, 8, @)+ V(x). Here D0=Dg+&=4.6192eV,withD~=4.4336eV,and b= (2D,,/M~o~)‘/~=O.535 A, where we use the fundamental frequency 2990.9463 cm-’ for o #2. The anharmonicity constants are o;c,= 52.8186 cm-‘, oeyecO.22437 cm-‘, and o&~= -0.01218 cm-’ [21].ThemomentofinertiaisZ=2.64~10-40gcm2, where the equilibrium bond distance d is 1.27455 A
1211. Standard numerical routines are used to solve the s’ Seepp.1111and12000fref.[17]forDcl,and~oandp. 168 for the combining laws. 12 Molecular constants are taken from ref. [ 2 11.
equations of motion [ 22,23 1. To determine the distribution of lifetimes, we sample 1080 trajectories for various initial phases of the HCl vibration, 8 rotation and @rotation at a given initial collision energy Eu. The initial conditions ( t = to) needed to solve the equations are R(E,
to)=aln[(DIER)+(DIER)‘l
+ 2a ln{cosh [ (E,/2p) lj2( to/u)
x(6,, t,)=bln
]
1+ (E~/D,-,)1/2 sin(Qt,,+&)
1- (Et/D,)
>’
t9(& 6,) = (2Eo/Z) 1’2t,,+6e , @CS,,to) = WqJ01’2h3
+a,,
(3)
where&!= [2(D,-Ez)/M]1/2/b, EE istheinitialvibrational energy of HCl, and 6s are the initial phases. We consider HCl to be in the excited state corresponding to the v= 1 energy of anharmonic oscillator determined by the term value expression E$/hc =o,(v+f)-w~=(v+1)2+wy,(u+1)3+o~= x (v+ f )‘. The solutions at time t are then R(6, E, t), x(6, E, t), C3(S,E, 2) and $(S, E, t), respectively. Here E z {ER, Ee, E+,}and 6 = (8, &, 6,). We take the collision energy corresponding to 300 K (i.e. E,=$RTand E,=E,=RTwith T7300 K). Trajectories are begun at the initial time to corresponding to a distance of 13.0 A, and the equations are integrated until trajectories reach time t. Because ma =BmH, we consider R ( t ) to describe essentially the O-Cl motion.
4. Results and discussion In fig. 2, we show the distribution of lifetimes of collision complexes. The distribution closely follows the first-order kinetics represented by ln(N,/N,) = -k(E,)t,, particularly in the range of 1 to 5 ps. Here N, is the number of trajectories with the lifetime tc and No is the total pumber of trajectories in the ensemble. About 30% of collisions lead to complexes surviving for 1 ps; the fraction for complexes surviving for 5 ps decreases to about 2%. Some collisions lead to lifetimes much longer than 5 ps. In the present calculations the upper limit of time is set at 29
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Fig. 2. Distribution of lifetimes of collision complexes.
13 ps, and there are some complexes which do not redissociate even at such a large time. The rate coefficient k(E,) obtained from the linear part of the plot is 0.70 ps-’ or the average lifetime of r= l/ k(&) = 1.43 ps. Since the vibrational period of the H20-HCl complex is estimated to be about 0.25 ps [ I], the lifetime of long-lived complexes surviving for 5 ps corresponds to 20 vibrational periods. Note that this time scale corresponds to about 450 vibrational periods of the HCI unit. To make a brief comment on the deexcitation probability, we note that the phase-averaged vibrational energy transfer AZ?from HCl( v= 1) is 0.122 eV or AE/hw=0.328. In the forced harmonic oscillator model for direct-mode collisions, the l-+0 deexcitation probability can be given in the form PIO= (&??/fiw) exp( -AE/fio) [24]. Since the duration of a direct-mode collision rd is 0.36 ps as shown below, on average, there are r/rd= 3.9 ( = 4) turning points; i.e. the complex-mode collision can be viewed as a series of four direct-mode collisions. Thus, the phase-averaged energy transfer after each turning point in complex-mode collisions can be estimated as ( r/rd ) (AZ?@0) . By replacing (AE/fio ) in the direct-mode expression by the latter quantity, we can estimate the complex-mode probability as PI,,= [(7/7dd)(~/fiw)]exP[-(7/7d)(~/~~)l
=0.077,
which is in fair agreement with the observed value of 0.1 reported as the upper limit of measurements at room temperature [ 25 1. We now consider several cases, each representing 30
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a special type of collision in order to establish the effectiveness of complex-mode collision in relaxing HCl (u= 1). All complex-mode collisions belong to one of these cases. We first take a case representing complexes with a lifetime very close to 5 ps. In figs. 3a-3e, we plot the collision trajectory R(t) and the time evolution of the kinetic energy of the O-Cl motion &(t) = f&t)‘, the HCl vibrational energy E,(t)=D,[1-e-x(‘)‘b]2+fM~(t)2, the HCl rotational energy in the f3direction E@(t) = iZ& t)2 and the HCl rotational energy in the @ direction Z&(t) =iZ&t)2, respectively. The plots show that there are four turning points and the O-Cl motion undergoes large-amplitude oscillations. There is efficient energy flow between the HCl vibration and 8 rotation as well as the O-Cl oscillation, but not the $ rotation, at and near each turning point. The HCl ( u= 1) vibration in the complex even gains some energy, and the entire energy is localized in the vi-
TIME (ps) Fig. 3. Dynamics of the collisions representing a complex with lifetime 5.1 ps in 3a-3e, lifetime 1.2 ps in 3a’-3e’, l&time 12.1 ps in 3a”-3e”. (a, a’, a”) Collision trajectory R(t); (b, b’, b”) kinetic energy of the O-Cl motion Ek( t); (c, c’, c” ) vibrational energy&(t); (d, d’, d”) rotationalenergyEo(t); and (e, e’, e”) rotational energy E,(t).
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bration when the O-Cl bond undergoes large-amplitude oscillations. However, at the final turning point, the hindered $ rotation strongly couples with the vibration and gains a large fraction of the vibrational energy as the complex redissociates. The vibrationalenergy lostby HCl(v= I) is (AE)J= 0.481 eV which is nearly 90% of the initial energy, and 0.326 eV of this energy is transferredto the out-of-planerotational motion of HCl, i.e. (AE&. (Here the subscript 6 simply means the energy transfer is phase dependent. ) The energy gained by the 19rotation in the XY plane, (A&&, is 0.140 eV. The rest, which is only 0.015 eV, is transferred to translation, indicating that the O-Cl motion is not efficient in removing the vibrational energy from HCl( v= 1). Note that A(t) initially determines the translational energy of the pre-collision state, but when the complex is formed, it describes the kinetic energy of the O-Cl oscillation. However, when the complex redissociates, k(t) again determines the translational energy but now of the post-collision state. Therefore, 67.8, 29.1, and 3.1% of AE from HCl( VI 1) are transferred to the $ rotation, 8 rotation, and translation, respectively. There are a number of collision complexes with a relatively short lifetime of about 1 ps as seen in fig. 2. A case representing such complex-mode collisions is shown in figs. 3a’-3e’. This collision, involving only two turning points, is also efficient in relaxing HCl (v= 1) . The amount of energy lost by HCl(v= 1) isOS16eK whichisaslargeas97%ofE?. Themajor portion of this energy is transferred to the 6 rotation, see fig. 3e’. The final distributions of AE in the @rotation, translation, and 8 rotation are 74.2, 18.1 and 7.7%, respectively. Note that the vibration to $ rotation energy transfer path is again the principal relaxation pathway, and translation is now more efficient than the 8 rotation in removing energy from HCl( v= 1) in this complex with a shorter lifetime. In this collision complex, the 8 rotation takes essentially all of the vibrational and translational energies, but just before redissociation it loses most of the energy to the # rotation. While most of the long-lived collisions follow a trajectory similar to the one shown in fig. 3a, some take a form plotted in fig. 3a”, where the amplitude of the O-Cl oscillation between the first and the final turning points is extremely large. In this representative case, the rebounding molecular units after the
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first impact tend to fly away from each other and then spend a long time near the apex, where the kinetic energy vanishes, of the barely bound trajectory. The O-Cl bond expands to a value as large as 7 A, but the complex fails to redissociate. At each turning point, energy efficiently flows between the HCl vibration, rotations, and O-Cl oscillation. The major portion of Ez is transferred to the rotational motions as the O-Cl bond begins the large amplitude motion, and this energy remains in the rotations (particularly in the # rotation) over the time period during which the O-Cl bond executes the large-amplitude oscillation; this is clearly seen in figs. 3d” and 3e”. At the final turning point, the vibrational motion regains about half of Et from the rotations. In this complex-mode collision with a lifetime as long as 12 ps, HCl(v= 1) loses (AQ~0.179 eV or one third of Ez. Although (dE)b is not as large as in the above cases where the lifetime is in the range of l-5 ps, the deexcitationis stillfairly eficient. Thus we notice that when the lifetime is exceptionally long, the vibration has the opportunity to get some energy transfer back from the rotations. The above trajectories are radically different from a direct-mode trajectory. The trajectory and time evolution of vibrational energy for a representative case of the direct-mode collision are shown in fig. 4a. The figure clearly shows the ineficiency of vibrational relaxation in a direct-mode collision. In this case, the initial vibrational energy of 0.542 eV decreases to 0.5 19 eV, which is only about a 4% change, on collision. The initial values of the & and $-rotational energies, both being the same with 0.0258 eV, decrease to 0.00245 and 0.0 10 1 eV, respectively. In this collision of a short duration, therefore, the rotational motions are inefficient to gain energy from the vibration. They actually lose some energy. The total energy lost by the vibration and rotations is only 0.06 16 eV, which is taken up by translation. We find other such direct-mode collisions to be similarly inefficient in promoting vibrational relaxation. The vibrational energy of HCl ( v= 1) in the directmode collision plotted in fig. 4a violently changes during collision. To check the time scale for this direct-mode collision, we replot R(t) on an expanded scale along with the conjugated momentum It)(t) in fig. 4b. In this figure we actually plot the reduced momentum defined as (p/ER) lf2R( t), which is 31
Volume 216, number 1,2
‘f2
0
CHEMICAL PHYSICS LETTERS
2
-.8 -.6 -4 52
4
b
0
.2 A
8
.6
IO
.6
TIME (ps) Fig. 4. Dynamics of a direct-mode collision. (a) The collision trajectory R(f)and the time evolution of the vibrational energy of the O-Cl motion EJ t). (b) The collision trajectory R(t)and the reduced momentum of the relative motion (p/Q 112&t),
evolved from the pre-collision value of -2’12. An accurate determination of the duration of a collision is not easy, since it is difficult to decide when the collision actually begins and when it ends. However, in the present study, the momentum starts to change from - 2”’ when the collision begins and then eventually _levels off to the constant value (p/ ER) 1/2R(oo) as the collision completes, so we can find the duration from these changes. In fig. 4b the momentum decreases from - 2”’ at t = - 0.58 ps and levels off to the constant final value of 2.28 at t= -0.22 ps, so the duration of collision is 0.36 ps. Throughout the present calculation, we find the duration of other direct-mode collisions is very close to this value. It is interesting to note a conventional estimate of the duration of collision as r=a/v, where u is the initial collision velocity (2ER/p)1/2, giving only 0.028 ps at E,=300 K. This value is an order of magnitude smaller than the numerical calculation shown above. The quantity a/v actually determines a short range around the classical turning point at the repulsive wall of the interaction potential; it rep 32
24 December 1993
resents the time between the minimum and the maximum of (,u/E,J ‘/‘k (t) in fig. 4b. However, fg 4b clearly shows that the collision begins long before the time where the momentum takes the minimum value and continues to beyond the time where the maximum appears. It is well known in the field of vibrational energy transfer that the analytical solutions of the equations of motion for exponential interaction potentials are hyperbolic functions giving a single turning point trajectory [26-291. When the initial collision energy is appropriately symmetrized, such single turning point solutions describe direct-mode collisions quite satisfactorily, but they are not useful in describing complex-mode collisions. Finally, we compare the power spectrum of the HCl vibration in the above direct collision case with that of the complex-mode case considered in fg 3a. The spectrum shown in fig. 5a is the Fourier transform of x( t) over the entire time range of the pm-collision to the post-collision through the complex state and it is much more complicated than the direct-mode case shown in fig. Sb. The frequency 2818 cm-’ in figs. 5a and 5b represents the HCl(v= 1) vibration. A large blue-shifted peak at 2969 cm-’ in the complex-mode collision stands for the vibration of the deexcited HCl, and this frequency is very close to the fundamental 2990 cm-i. Fig. 5a also shows signif-
I
a: Complex-Mode
b: Direct-Mode
Collision
Collision
2818 cm-1 #’
2600
2800 2700 , FREQUENCY
2900
3000
(cm-‘)
Fig. 5. Power spectra of (a) the complex-mode case considered in fg 3a and (b) the direct-mode case considered in fig. 4a
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red- and blue-shifts around the HCl ( v= 1) peak. These shifts represent the excited and deexcited states of HCl in the collision complex from its v= 1 initial state; see fig. 3c. The frequencies 2776 and 2792 cm-’ represent the HCl vibration between the second and third and between the third and fourth turning points, respectively. icant
Acknowledgement The computational part of this research was supported by an NSF Advanced Computing Resources grant at the Pittsburgh Supercomputing Center and by the National Supercomputing Center for Energy and the Environment at the University of Nevada at Las Vegas.
References 111 H.K. Shin, J. Chem. Phys. 98 ( 1993) 1964. [2] E.E. Ferguson, J. Phys. Chem. 90 (1986) 731. [ 3 ] P. Tosi, M. Ronchetti and A. Lagana, J. Chem. Phys. 88 (1988) 4814. [ 41 G. Ramachandran and G.S. Ezra, J. Chem. Phys. 97 ( 1992) 6322. [5]G.B.Bacskay,Mol.Phys. 77 (1992) 61. [6] M.M. Szczesniak, S. Schemer and Y. Bouteiller, J. Chem. Phys. 81 (1984) 5024. [7] Z. Latajka and S. Scheiner, J. Chem. Phys. 87 (1987) 5928. [S] J.E. Del Bene, J. Phys. Chem. 92 (1988) 5998.
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[9] W.B. De Almeida and A. HinchliiTe, Chem. Phys. 137 (1989) 143. [ lo] G.B. Bacskay, D.I. Kerdraon and N.S. Hush, Chem. Phys. 144 (1990) 53. [ 111 A.C. Legon and L.C. Willoughby, Chem. Phys. Letters 95 (1983) 449. [ 121 M.M. Szcze&& Z Iatajka and S. Scheiner, J. Mol. Struct. THEOCHEM 135 (1986) 179. [ 13 ] L.A. Curtiss, D.J. FNI-$ and M. Blander, J. Chem. Phys. 71 (1979) 2703. [ 141 J.A. Beswick and J. Jortner, Advan. Chem. Phys. 47 ( 1981) 363. [ 151 A. HinchlitTe, J. Mol. Struct. THEGCHEM 106 (1984) 361. [ 161 Y. Hannachi and B. Silvi, J. Mol. Struct. THEOCHEM 288 (1989) 483. [ 171 J.O. Hirschfelder, C.F. Ctutiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1964). [18]H.K.ShinandY.H.Kim,J.Chem.Phys.73(1980)3186. [ 191 R.O. Watts, Chem. Phys. 26 (1977) 367. [20] H. Hopkie, H. Kistenmacher and E. Clementi, J. Chem. Phys. 59 (1973) 1325. [2 1 ] K.P. Huber and G. Henberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979) p. 286. [ 221 C.W. Gear, Numerical initial values problems in ordinary differential equations (Prentice-Hall, New York, 197 1) . [23] MATH/LIBRARY (IMSL, Houston, 1989) p. 640. [ 241 H.K. Shin, in Molecular collisions, Part A, ed. W.H. Miller (PlenumPress,NewYork, 1976) pp. 131-210. [25] H.L. ChenandC.B.Moore, J. Chem.Phys. 54 (1971) 4072. [26] D. Rapp, J. Chem. Phys. 32 (1960) 735. [ 271 T.L. Cottrell and J.C. McCoubmy, Molecular energy @an&r in gases (Butterworths, London, 196 1) ch. 6. [28] J.D. Kelley and M. Wolfsberg, J. Chem. Phys. 44 (1966) 324. [29] J.T. Yardley, Introduction to molecular energy transfer (Academic Press, New York, 1980), ch. 4.
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Importance of complex-mode collisions in the vibrational relaxation of strongly attracting molecules. The H,O-HCl( U= 1) system H.K. Shin Department of Chemistry, University ofNevada, Reno, NV89557, USA Received 18 August 1993; in final form 13 October 1993
The deexcitation of HCl( u= 1) in the HIO-HCl interaction is studied at thermal collision energies using classical trajectory techniques. In the presence of strong attraction, the collision partners form a complex, and the initial vibrational energy flows freely among various internal modes in the complex. When the complex redissociates, the major portion of the energy is found to localize in the rotational motion of HCl. This study establishes the occurrence of vibrational relaxation in complex-mode collisions and the main energy transfer pathway to be vibration-to-rotation in the complex. Energy transfer is found to be inefficient in direct-mode collisions.
1. Intruduction
Strong attractive forces operate between hydrogen chloride and water molecules, and these forces can significantly affect dynamics of collision-induced molecular processes. In the presence of strong attraction, a collision between Hz0 and HCl molecules can result in the formation of a complex, in which the H atom of HCl is in close proximity to the oxygen atom. In this complex, where the H-O interaction is of major importance in describing intramolecular dynamics, the rotational motions of HCl become hindered, and the relative motion of the collision partners transforms into a large-amplitude oscillatory motion. The H-O interaction causes a major disturbance in the HCl bond but not in Hz0 because of the heavy mass of the 0 atom compared with the H atom. When a vibrationally excited HCl is involved, its relaxation can then proceed either in a complex-mode collision or in a direct-mode collision. Recent studies show that the vibrational relaxation of HCl( tr= 1) is efficient at room temperature, and its negative temperature suggests the importance of complex-mode collisions [ 11. The importance of collision complexes in vibrational relaxation has already been noted in earlier studies [ 2-
41. If complex-mode collisions produce efficient energy transfer, the result can be of fundamental importance in studying not only vibrational relaxation of hydrogen-bond interaction systems but also vibrational relaxation of polar molecules or molecular ions which produce strong dipole-dipole or ion-dipole attractions. The purpose of this study is to establish the occurrence of complex-mode collisions in the H,O-HCl system and that the vibrational relaxation of HCl( V= 1) can efficiently proceed in such complexes. We consider the interaction at thermal collision energies and use classical trajectories techniques.
2. Complex-mode collisions Hydrogen-bond attraction between HCl and Hz0 molecules is known to be as strong as 5.74 kcal/mol [ 5 1. In the presence of such strong attraction, the interacting molecules can form a collision complex, which can survive for a duration greatly exceeding the vibrational period of HCl. When an H,O-HCl complex is formed, the center-of-mass separation between the colliding molecules transforms into a lowfrequency oscillatory motion of the c.m. separation
0009-2614/93/S 06.00 Q 1993 Elsevier Science Publishers B.V. All rights reserved. SSDI 0009-2614(93)El265-I
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denoted by R in fig. 1. A considerable number of theoretical studies have been reported in recent years for the structure of this complex indicating the equilibrium configuration of O-H-Cl is essentially linear. The calculations show that the O-Cl-H angle varies from 0 to 3.4” [6-lo]. The results are consistent with Legon and Willoughby’s microwave spectroscopic observations of the linear O-H-Cl configuration [ 111. In H,O-HCl, we take the molecular units to undergo internal motions about the equilibrium structure of HCl in the XY plane and Hz0 in the X2 plane with CIV symmetry. The experimentally determined value of 3.22 A will be used for the O-Cl equilibrium distance [ 111; the calculated values of the O-Cl distance lie between 3.09 and 3.28 A [ 5,9,10,12]. When the complex is formed, H of HCl is in strong interaction with 0 of Hz0 through the hydrogen-bond attraction, and we consider R to represent the distance between the c.m. of HCl and the 0 atom. The vibrational displacement of the HCl unit from its equilibrium bond length d will be denoted by x. In the complex, the HCl unit can undergo complete internal rotations when the OCl separation is large. When the separation is small, on the other hand, HCl undergoes hindered rotation about the O-H-Cl equilibrium configuration in the XY plane. This rotation is represented by 8 in fig. 1. In addition, the HCl bond undergoes the out-of-XY plane rotational motion and the angle # represents this motion. Although there is strong attraction between the colliding molecules, we do not expect the concentration of hydrogen-bond dimers in the gas phase to be large. No concentration data are available for the present system, but we note that in H20-H20, the
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attractive energy is 6 kcal/mol, which is quite close to 5.74 kcal/mol of H,O-HCl, and the dimer concentration is known to be only about 1% at 373 K [ 131. On the other hand, the concentration of collision complexes surviving one to several picoseconds, which is much longer than the duration of direct-mode collisions, can be significant. The overall interaction of two molecular units in the nonrigid H,O-HCl complex is dominated by the interaction between the 0 atom of Hz0 and the H atom of HCI. Other interatomic distances are significantly larger so the corresponding interaction is much weaker than the H-O interaction. The H-O interaction is assumed to contain exponentially repulsive and attractive terms acting between the atoms. The first step in writing analytical forms of interaction potentials is to express the intermolecular atom-atom distances in terms of the 0-HCl internal coordinates as Zig= [R2+y&(d+~)Z2ycR(d+x) c0s(tY*+~*)“*]“* and Zig= [R* +y& (d + x)* + Zy,R(d + x) cos(6* + @2)1’2]1’2, where yHcl = rnr& (mn+ ma). We use these two distances in two pairwise interactions (see, for example, ref. [ 14 ] ) between the 0 atom and each atom of HCl. The equilibrium configuration of the complex is described by the coordinates R=R,, x=0, 8~0 and $=O. The equilibrium atom-atom distances are then simply zc,H_o= R,- d and ~,,~_o = R, + d. The potential function representing the HCl and 0 interaction can then be represented by adding the terms of these two atom-atom distances as V(R, X, 6, #) =&&exp{(R,-d)/a-2yaR(d+x)
cos(tl*+#*) “*I “*/u}
-2 exp{ (R,-d)/2a-2y,R(d+x)
HCI Fig. 1. Complex-mode collision model. 28
[R*+&(d+x)*
cos(8*+#*) “*I
-2 exp{ (R,+d)/2u+ZhR(d+x)
[R*+y&(d+x)*
cos(e*+#*) “*I “*/2u})
+Dcl_o(exp{ (R,+d)/a+2%R(d+x)
[R*+&(d+x)*
"'/a}
[R*+&(d+x)’
cos(8*+@*) “*I “*/2u}) .
(1)
Both observed and calculated H20-HCl interaction energies reported in the literature represent the H-
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0 hydrogen-bond attraction Dna. Therefore, the first part of eq. (1) is of major importance, but the second part becomes significant when HCl rotates in the 8 direction. We take Bacskay’s recent data of 5.74 kcal/mol for Dn_o [ 5 1. The calculated values vary over a wide range of 4.74 to 8.93 kcal/mol [S12,15,16] but the majority of the calculations give a value close to 6 kcal/mol. While the value of DHa is large due to the hydrogen-bond interaction, Da_o for the Cl-O interaction is expected to be significantly smaller. We estimate it from the attractive well depths of Hz0 and Cl2 using combining rule as Da_p (DHzoDa2) ‘12, where DHto = 775k and D,,=357k [ 171 #l. This procedure gives Da* = 1.04 kcal/mol. Because of its strong polar character, we use DW instead of Dot here. From ~~0.25 A for HCl-HCl [ 181 and 0.20 A for H20H20 [ 19,201, we estimate a=0.22 A for H20-HCl also using the combining rule. The HCl molecule is considered to be a Morse oscillator; V(x) = Do x ( 1 - e-x’b)2, where Do and b are the Morse parameters.
3. Solution of the equations of motion The equations of motion are
(2) where p is the reduced mass of the collision system, M is the reduced mass of HCI, Z is the moment of inertia of HCl, and V= V(R, x, 8, @)+ V(x). Here D0=Dg+&=4.6192eV,withD~=4.4336eV,and b= (2D,,/M~o~)‘/~=O.535 A, where we use the fundamental frequency 2990.9463 cm-’ for o #2. The anharmonicity constants are o;c,= 52.8186 cm-‘, oeyecO.22437 cm-‘, and o&~= -0.01218 cm-’ [21].ThemomentofinertiaisZ=2.64~10-40gcm2, where the equilibrium bond distance d is 1.27455 A
1211. Standard numerical routines are used to solve the s’ Seepp.1111and12000fref.[17]forDcl,and~oandp. 168 for the combining laws. 12 Molecular constants are taken from ref. [ 2 11.
equations of motion [ 22,23 1. To determine the distribution of lifetimes, we sample 1080 trajectories for various initial phases of the HCl vibration, 8 rotation and @rotation at a given initial collision energy Eu. The initial conditions ( t = to) needed to solve the equations are R(E,
to)=aln[(DIER)+(DIER)‘l
+ 2a ln{cosh [ (E,/2p) lj2( to/u)
x(6,, t,)=bln
]
1+ (E~/D,-,)1/2 sin(Qt,,+&)
1- (Et/D,)
>’
t9(& 6,) = (2Eo/Z) 1’2t,,+6e , @CS,,to) = WqJ01’2h3
+a,,
(3)
where&!= [2(D,-Ez)/M]1/2/b, EE istheinitialvibrational energy of HCl, and 6s are the initial phases. We consider HCl to be in the excited state corresponding to the v= 1 energy of anharmonic oscillator determined by the term value expression E$/hc =o,(v+f)-w~=(v+1)2+wy,(u+1)3+o~= x (v+ f )‘. The solutions at time t are then R(6, E, t), x(6, E, t), C3(S,E, 2) and $(S, E, t), respectively. Here E z {ER, Ee, E+,}and 6 = (8, &, 6,). We take the collision energy corresponding to 300 K (i.e. E,=$RTand E,=E,=RTwith T7300 K). Trajectories are begun at the initial time to corresponding to a distance of 13.0 A, and the equations are integrated until trajectories reach time t. Because ma =BmH, we consider R ( t ) to describe essentially the O-Cl motion.
4. Results and discussion In fig. 2, we show the distribution of lifetimes of collision complexes. The distribution closely follows the first-order kinetics represented by ln(N,/N,) = -k(E,)t,, particularly in the range of 1 to 5 ps. Here N, is the number of trajectories with the lifetime tc and No is the total pumber of trajectories in the ensemble. About 30% of collisions lead to complexes surviving for 1 ps; the fraction for complexes surviving for 5 ps decreases to about 2%. Some collisions lead to lifetimes much longer than 5 ps. In the present calculations the upper limit of time is set at 29
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Fig. 2. Distribution of lifetimes of collision complexes.
13 ps, and there are some complexes which do not redissociate even at such a large time. The rate coefficient k(E,) obtained from the linear part of the plot is 0.70 ps-’ or the average lifetime of r= l/ k(&) = 1.43 ps. Since the vibrational period of the H20-HCl complex is estimated to be about 0.25 ps [ I], the lifetime of long-lived complexes surviving for 5 ps corresponds to 20 vibrational periods. Note that this time scale corresponds to about 450 vibrational periods of the HCI unit. To make a brief comment on the deexcitation probability, we note that the phase-averaged vibrational energy transfer AZ?from HCl( v= 1) is 0.122 eV or AE/hw=0.328. In the forced harmonic oscillator model for direct-mode collisions, the l-+0 deexcitation probability can be given in the form PIO= (&??/fiw) exp( -AE/fio) [24]. Since the duration of a direct-mode collision rd is 0.36 ps as shown below, on average, there are r/rd= 3.9 ( = 4) turning points; i.e. the complex-mode collision can be viewed as a series of four direct-mode collisions. Thus, the phase-averaged energy transfer after each turning point in complex-mode collisions can be estimated as ( r/rd ) (AZ?@0) . By replacing (AE/fio ) in the direct-mode expression by the latter quantity, we can estimate the complex-mode probability as PI,,= [(7/7dd)(~/fiw)]exP[-(7/7d)(~/~~)l
=0.077,
which is in fair agreement with the observed value of 0.1 reported as the upper limit of measurements at room temperature [ 25 1. We now consider several cases, each representing 30
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a special type of collision in order to establish the effectiveness of complex-mode collision in relaxing HCl (u= 1). All complex-mode collisions belong to one of these cases. We first take a case representing complexes with a lifetime very close to 5 ps. In figs. 3a-3e, we plot the collision trajectory R(t) and the time evolution of the kinetic energy of the O-Cl motion &(t) = f&t)‘, the HCl vibrational energy E,(t)=D,[1-e-x(‘)‘b]2+fM~(t)2, the HCl rotational energy in the f3direction E@(t) = iZ& t)2 and the HCl rotational energy in the @ direction Z&(t) =iZ&t)2, respectively. The plots show that there are four turning points and the O-Cl motion undergoes large-amplitude oscillations. There is efficient energy flow between the HCl vibration and 8 rotation as well as the O-Cl oscillation, but not the $ rotation, at and near each turning point. The HCl ( u= 1) vibration in the complex even gains some energy, and the entire energy is localized in the vi-
TIME (ps) Fig. 3. Dynamics of the collisions representing a complex with lifetime 5.1 ps in 3a-3e, lifetime 1.2 ps in 3a’-3e’, l&time 12.1 ps in 3a”-3e”. (a, a’, a”) Collision trajectory R(t); (b, b’, b”) kinetic energy of the O-Cl motion Ek( t); (c, c’, c” ) vibrational energy&(t); (d, d’, d”) rotationalenergyEo(t); and (e, e’, e”) rotational energy E,(t).
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bration when the O-Cl bond undergoes large-amplitude oscillations. However, at the final turning point, the hindered $ rotation strongly couples with the vibration and gains a large fraction of the vibrational energy as the complex redissociates. The vibrationalenergy lostby HCl(v= I) is (AE)J= 0.481 eV which is nearly 90% of the initial energy, and 0.326 eV of this energy is transferredto the out-of-planerotational motion of HCl, i.e. (AE&. (Here the subscript 6 simply means the energy transfer is phase dependent. ) The energy gained by the 19rotation in the XY plane, (A&&, is 0.140 eV. The rest, which is only 0.015 eV, is transferred to translation, indicating that the O-Cl motion is not efficient in removing the vibrational energy from HCl( v= 1). Note that A(t) initially determines the translational energy of the pre-collision state, but when the complex is formed, it describes the kinetic energy of the O-Cl oscillation. However, when the complex redissociates, k(t) again determines the translational energy but now of the post-collision state. Therefore, 67.8, 29.1, and 3.1% of AE from HCl( VI 1) are transferred to the $ rotation, 8 rotation, and translation, respectively. There are a number of collision complexes with a relatively short lifetime of about 1 ps as seen in fig. 2. A case representing such complex-mode collisions is shown in figs. 3a’-3e’. This collision, involving only two turning points, is also efficient in relaxing HCl (v= 1) . The amount of energy lost by HCl(v= 1) isOS16eK whichisaslargeas97%ofE?. Themajor portion of this energy is transferred to the 6 rotation, see fig. 3e’. The final distributions of AE in the @rotation, translation, and 8 rotation are 74.2, 18.1 and 7.7%, respectively. Note that the vibration to $ rotation energy transfer path is again the principal relaxation pathway, and translation is now more efficient than the 8 rotation in removing energy from HCl( v= 1) in this complex with a shorter lifetime. In this collision complex, the 8 rotation takes essentially all of the vibrational and translational energies, but just before redissociation it loses most of the energy to the # rotation. While most of the long-lived collisions follow a trajectory similar to the one shown in fig. 3a, some take a form plotted in fig. 3a”, where the amplitude of the O-Cl oscillation between the first and the final turning points is extremely large. In this representative case, the rebounding molecular units after the
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first impact tend to fly away from each other and then spend a long time near the apex, where the kinetic energy vanishes, of the barely bound trajectory. The O-Cl bond expands to a value as large as 7 A, but the complex fails to redissociate. At each turning point, energy efficiently flows between the HCl vibration, rotations, and O-Cl oscillation. The major portion of Ez is transferred to the rotational motions as the O-Cl bond begins the large amplitude motion, and this energy remains in the rotations (particularly in the # rotation) over the time period during which the O-Cl bond executes the large-amplitude oscillation; this is clearly seen in figs. 3d” and 3e”. At the final turning point, the vibrational motion regains about half of Et from the rotations. In this complex-mode collision with a lifetime as long as 12 ps, HCl(v= 1) loses (AQ~0.179 eV or one third of Ez. Although (dE)b is not as large as in the above cases where the lifetime is in the range of l-5 ps, the deexcitationis stillfairly eficient. Thus we notice that when the lifetime is exceptionally long, the vibration has the opportunity to get some energy transfer back from the rotations. The above trajectories are radically different from a direct-mode trajectory. The trajectory and time evolution of vibrational energy for a representative case of the direct-mode collision are shown in fig. 4a. The figure clearly shows the ineficiency of vibrational relaxation in a direct-mode collision. In this case, the initial vibrational energy of 0.542 eV decreases to 0.5 19 eV, which is only about a 4% change, on collision. The initial values of the & and $-rotational energies, both being the same with 0.0258 eV, decrease to 0.00245 and 0.0 10 1 eV, respectively. In this collision of a short duration, therefore, the rotational motions are inefficient to gain energy from the vibration. They actually lose some energy. The total energy lost by the vibration and rotations is only 0.06 16 eV, which is taken up by translation. We find other such direct-mode collisions to be similarly inefficient in promoting vibrational relaxation. The vibrational energy of HCl ( v= 1) in the directmode collision plotted in fig. 4a violently changes during collision. To check the time scale for this direct-mode collision, we replot R(t) on an expanded scale along with the conjugated momentum It)(t) in fig. 4b. In this figure we actually plot the reduced momentum defined as (p/ER) lf2R( t), which is 31
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‘f2
0
CHEMICAL PHYSICS LETTERS
2
-.8 -.6 -4 52
4
b
0
.2 A
8
.6
IO
.6
TIME (ps) Fig. 4. Dynamics of a direct-mode collision. (a) The collision trajectory R(f)and the time evolution of the vibrational energy of the O-Cl motion EJ t). (b) The collision trajectory R(t)and the reduced momentum of the relative motion (p/Q 112&t),
evolved from the pre-collision value of -2’12. An accurate determination of the duration of a collision is not easy, since it is difficult to decide when the collision actually begins and when it ends. However, in the present study, the momentum starts to change from - 2”’ when the collision begins and then eventually _levels off to the constant value (p/ ER) 1/2R(oo) as the collision completes, so we can find the duration from these changes. In fig. 4b the momentum decreases from - 2”’ at t = - 0.58 ps and levels off to the constant final value of 2.28 at t= -0.22 ps, so the duration of collision is 0.36 ps. Throughout the present calculation, we find the duration of other direct-mode collisions is very close to this value. It is interesting to note a conventional estimate of the duration of collision as r=a/v, where u is the initial collision velocity (2ER/p)1/2, giving only 0.028 ps at E,=300 K. This value is an order of magnitude smaller than the numerical calculation shown above. The quantity a/v actually determines a short range around the classical turning point at the repulsive wall of the interaction potential; it rep 32
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resents the time between the minimum and the maximum of (,u/E,J ‘/‘k (t) in fig. 4b. However, fg 4b clearly shows that the collision begins long before the time where the momentum takes the minimum value and continues to beyond the time where the maximum appears. It is well known in the field of vibrational energy transfer that the analytical solutions of the equations of motion for exponential interaction potentials are hyperbolic functions giving a single turning point trajectory [26-291. When the initial collision energy is appropriately symmetrized, such single turning point solutions describe direct-mode collisions quite satisfactorily, but they are not useful in describing complex-mode collisions. Finally, we compare the power spectrum of the HCl vibration in the above direct collision case with that of the complex-mode case considered in fg 3a. The spectrum shown in fig. 5a is the Fourier transform of x( t) over the entire time range of the pm-collision to the post-collision through the complex state and it is much more complicated than the direct-mode case shown in fig. Sb. The frequency 2818 cm-’ in figs. 5a and 5b represents the HCl(v= 1) vibration. A large blue-shifted peak at 2969 cm-’ in the complex-mode collision stands for the vibration of the deexcited HCl, and this frequency is very close to the fundamental 2990 cm-i. Fig. 5a also shows signif-
I
a: Complex-Mode
b: Direct-Mode
Collision
Collision
2818 cm-1 #’
2600
2800 2700 , FREQUENCY
2900
3000
(cm-‘)
Fig. 5. Power spectra of (a) the complex-mode case considered in fg 3a and (b) the direct-mode case considered in fig. 4a
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red- and blue-shifts around the HCl ( v= 1) peak. These shifts represent the excited and deexcited states of HCl in the collision complex from its v= 1 initial state; see fig. 3c. The frequencies 2776 and 2792 cm-’ represent the HCl vibration between the second and third and between the third and fourth turning points, respectively. icant
Acknowledgement The computational part of this research was supported by an NSF Advanced Computing Resources grant at the Pittsburgh Supercomputing Center and by the National Supercomputing Center for Energy and the Environment at the University of Nevada at Las Vegas.
References 111 H.K. Shin, J. Chem. Phys. 98 ( 1993) 1964. [2] E.E. Ferguson, J. Phys. Chem. 90 (1986) 731. [ 3 ] P. Tosi, M. Ronchetti and A. Lagana, J. Chem. Phys. 88 (1988) 4814. [ 41 G. Ramachandran and G.S. Ezra, J. Chem. Phys. 97 ( 1992) 6322. [5]G.B.Bacskay,Mol.Phys. 77 (1992) 61. [6] M.M. Szczesniak, S. Schemer and Y. Bouteiller, J. Chem. Phys. 81 (1984) 5024. [7] Z. Latajka and S. Scheiner, J. Chem. Phys. 87 (1987) 5928. [S] J.E. Del Bene, J. Phys. Chem. 92 (1988) 5998.
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[9] W.B. De Almeida and A. HinchliiTe, Chem. Phys. 137 (1989) 143. [ lo] G.B. Bacskay, D.I. Kerdraon and N.S. Hush, Chem. Phys. 144 (1990) 53. [ 111 A.C. Legon and L.C. Willoughby, Chem. Phys. Letters 95 (1983) 449. [ 121 M.M. Szcze&& Z Iatajka and S. Scheiner, J. Mol. Struct. THEOCHEM 135 (1986) 179. [ 13 ] L.A. Curtiss, D.J. FNI-$ and M. Blander, J. Chem. Phys. 71 (1979) 2703. [ 141 J.A. Beswick and J. Jortner, Advan. Chem. Phys. 47 ( 1981) 363. [ 151 A. HinchlitTe, J. Mol. Struct. THEGCHEM 106 (1984) 361. [ 161 Y. Hannachi and B. Silvi, J. Mol. Struct. THEOCHEM 288 (1989) 483. [ 171 J.O. Hirschfelder, C.F. Ctutiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1964). [18]H.K.ShinandY.H.Kim,J.Chem.Phys.73(1980)3186. [ 191 R.O. Watts, Chem. Phys. 26 (1977) 367. [20] H. Hopkie, H. Kistenmacher and E. Clementi, J. Chem. Phys. 59 (1973) 1325. [2 1 ] K.P. Huber and G. Henberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979) p. 286. [ 221 C.W. Gear, Numerical initial values problems in ordinary differential equations (Prentice-Hall, New York, 197 1) . [23] MATH/LIBRARY (IMSL, Houston, 1989) p. 640. [ 241 H.K. Shin, in Molecular collisions, Part A, ed. W.H. Miller (PlenumPress,NewYork, 1976) pp. 131-210. [25] H.L. ChenandC.B.Moore, J. Chem.Phys. 54 (1971) 4072. [26] D. Rapp, J. Chem. Phys. 32 (1960) 735. [ 271 T.L. Cottrell and J.C. McCoubmy, Molecular energy @an&r in gases (Butterworths, London, 196 1) ch. 6. [28] J.D. Kelley and M. Wolfsberg, J. Chem. Phys. 44 (1966) 324. [29] J.T. Yardley, Introduction to molecular energy transfer (Academic Press, New York, 1980), ch. 4.
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