Importance of rotational motion in the vibrational relaxation of H2O molecules. Relaxation of the bending level in H2O+Ar collisions

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CHEMICAL PHYSICS LETTERS

IMPORTANCE OF ROTATIONAL MOTION IN THE VIBRATIONAL OF Hz0 MOLECULES. RELAXATION OF THE BENDING LEVEL IN H,O+Ar COLLISIONS

23 March 1990

RELAXATION

J. REE * and H.K. SHIN Department of Chemistry, University of Nevada, Rena, NV 8952 7, USA

Received 26 December 1989

A model of vibration-to-rotation energy transfer is developed for the relaxation of the bending mode of Hz0 in the HIO+Ar collision. In the model the energy release AE of the v,= 2 --LI transition is effkiently removed by rotation. Transition probabilities are calculated by a semiclassical procedure. The model correctly predicts both the temperature dependence and magnitude of the (020-O 10) probability over the temperature range of 200-1000 K. The importance of the coupling of rotation with bending is discussed. A model which assumes the removal of the energy release by translation seriously underestimates the probability especially at lower temperatures andpredictsa temperaturedependence whichistoo steep.

1. Introduction

The vibrational relaxation of Hz0 molecules is involved in many chemical and physical processes. Determination of relaxation rates in the gas phase provides critical information for modeling of Hz0 lasing processes near room temperature and atmospheric processes at low temperatures and understanding high temperature dynamics of combustion product gases, of which Hz0 is a major component [ l-71. In a near-resonant VV process of H20, recent studies show that translational motion is efficient in transferring the energy mismatch AE [ 71. For example, for energy exchange between the stretching levels 00 1 and 100 in the Hz0 + Ar collision, AE is only 105 cm- ’ and is efficiently transferred to translation. If AE is large, however, translational motion is no longer capable of removing the energy release and the model of VT energy transfer is not useful in explaining the vibrational relaxation of these hydrogen containing molecules. Vibrational relaxation processes which occur in H,O involve a large value of A& and the VT energy transfer model is not useful in predicting the magnitude and temperature dependence of relaxation rates. In such cases, it may be necessary to invoke the participation of the rotational motion of H20. One such system, which has been the subject of continuing research, is the relaxation of bending levels v2= 1 or 2. The dominant path for the relaxation of the stretching levels is relaxation down to the v2= 2 level followed by relaxation of single bending quantum to the bending fundamental level, which finally returns to the ground level. Therefore, the bending motion plays a crucial role in the relaxation pathway of stretching levels. Here, AE is 1595 cm-’ for v2= I +O and 1556 cm-’ for v*=2+ 1. The purpose of this work is to develop a VR energy transfer model and use it to calculate the semiclassical u2= 24 1 transition probability in H,O+Ar over the temperature range of 2001000 K. For this process, accurate experimental data are available [ 6 1, a critical test of such a VR model can be made. The result will be compared with that of a VT model.

’ Permanent address: Department of Chemistry, College of Education, Chonnam National University, Kwangju, Korea.

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2. Interaction potential energies The intramolecular and collision coordinates are defined in fig. 1. The molecule is assumed to rotate in the xy plane as indicated in the figure; the motion of this rotation relative to the incident atom is expressed by 8+ 4Ay. The angle IAy represents the effect of the bending motion on rotation. The instantaneous displacements from the two OH bonds are represented by Ar, and Ar, and the bending displacement from the equilibrium H-O-H angle by Ay. These internal coordinates are related to the normal coordinates as [ 8,9] Ari= GilQI +cizQz +GQS @=cxQ~

+c,,Q,+c~Qs

i=l,2,

I

(la) (lb)

,

where transformation coeffkients are cl1 =cz, =5.608x 10” g-‘j2, -+3=5.698X 10” g- “2, c31= -1.075x 10’8 g-‘/2 cm-‘, ~32~ -1.192x fig. 1, approximate expressions of the three atom-atom distances are R,=R-(&,+A~,)cos(&~A~)+[(&~+A~,)~/~R]

cl2 ~~~~~2.692 x 10” g-1/2, cl3 = 10” g-l/’ cm-‘, and c~~=O.From

sin2(8+jAy)+6cos(8-fy),

Pa) ,

R2=R-(d,,+Ar2)cos(y-8+fAy)+[(d,,+Ar2)2/2R]sin2(y-8+~Ay)f~cos(8-fy)

(2b)

R3=R+6cos(8-fy).

(2c)

The sine-square terms containing 1/R in the first two expressions are second-order contributions and are found to be significant in the present VR model, although they are unimportant in near-resonant processes [ 7 ] _ In eq. (2c), to the lirst order in the normal coordinates we find S=&+&Q, + 62Q2, where So is the equilibrium distance between the center of mass of Hz0 and the 0 atom (0.0655 A), where [ 81 6, = (Mif2/mo)a2, dz=(Mf’*/mo)al, al=0.8024, a,=O.5967, and Mr=2m,mo/(2m,+mo). Thus, from eqs. (1) and (2), all atom-diatom distances can be expressed in terms of the normal coordinates, i.e. Rig Ri( Ql ,Qz, Q3). Following ref. [ 71, we set up the potential energy as a sum of two H-Ar and the 0-Ar atom-pair interactions, each of which is assumed to be the Morse type; lJ= U,_,,(R, ) + U,_,,(R,) t U,_,,(R,). When the normalcoordinate-dependent atom-atom distances are introduced in these interactions and exponential parts containing Qts are expanded, we obtain the interaction energy for the one-quantum bending relaxation in the form U(R, 0,Q2)=[Aexp(l-R/a)-Bexp(il-R/2a)]+

[A’ exp(l-R/a)-B’exp(jl-R/2a)]Qz

=Uo(K&O)fU’W,&Q,),

(3)

A=Dexp[-&cos(f?-fy)]{exp[Lcos&f(a/R)L2sin2f?] texp[Lcos(y-8)-t(a/R)L*sin’(y-0)]+1},

i

Fig. 1. Collision coordinates ter of mass of HZO.

for H*O+Ar.

* represents

the cen-

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A’=Dexp[-6bcos(8-fy)]{[~;~cos8-6;cos(8-fy)-tLc,,sine - (a/2R) ( iL2cs2 sin 20+2L~‘,~ sin*@)] exp[L cos f3- (aL2/2R)

sin281

+ (~;~~0~(~-~)--8;cos(~-~f~)-_:L~~~sin(y-8)-(a/2R){~L~c~~sin[2(y-8)] + 2Lc;,sin2(y-8)))

exp[Lcos(y-f3)-(uL2/2R)

sin2(y-8)]-6zcos(t9-fy)).

S&=&/a, and D, I, a are potential constants to be determined. In A’, ch=cij/a and Here, L=d&a, J;=&/a, so these primed quantities and c3*are in units of g-‘/l cm-‘. The quantity B is equal to 2A except that a is now replaced by 2~; B’ and 2A’ are similarly related. The first term U,, describes the translational and rotational motion of Hz0 during its approach to Ar, while the second term u’ determines one-quantum transitions in the bending mode.

3. Trajectory expressions

The appearance of 1/R in eq. (3) greatly hinders the determination of an analytical expression for the collision trajectory R(t). The effect of the terms containing l/R on the interaction energy is important in the neighborhood of the most probable distance R* for energy transfer, and it rapidly becomes insignificant as R moves away from R*. This distance can be determined from the expression [ IO] R*=al2a ln[ (2,~/D)‘/*uAw], where Ao=AE/fi. In determining the trajectory, we thus replace l/R in U, by l/R*. With this replacement, the collision trajectory obtained from the solution of the equation of motion for a given rotation angle e is exp(fl-R/2a)=2E/{(4AE+B2)“Zcosh[

(E/2fl)‘/2(t/a)]

-p} ,

(4)

where E is the initial collision energy andp is the reduced mass of the collision system. We now give the rotation angle as a function of time [ 11,121: e(t)=eo+nt

,

(5)

where D is the angular velocity (rad/s) about the z axis and 0, is the initial phase. Since the classical expression for the energy of a Hz0 molecule rotating about the z axis is ER= jZ,52*, we express s2= (2Ea/Zz)1’2. With these time-dependent quantities, the overall interaction given by eq. (3) takes the form U(t, Q2) = U,(t) +U’(t, Q2).

4. Transition probability expression The Hamiltonian responsible for vibrational transitions involving the bending mode only can be written in terms of ladder operators (al, a2) as H(t)=hw,(afa,+f)+~(t~(af+a2),

(6)

where F(t)={A’(t)

exp[l-R(t)/a]-B’(t)

exp[fl-R(t)/2a]}(h/2w2)“*.

The time evolution of the quantum state ) y(t)) pendent Schrijdinger equation ifiIrfXt))=H(t)lcy(t)) in the form 222

after collision can be determined by solving the time-de(7)

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CHEMICAL PHYSICS LETTERS

Iy/(t))=ex~k~(tMl

exh5(Oa21

ewk(O44

wk4(~Vllvth)~~

lY(~0))=10~20)

3

(8)

where Z is the identity operator and g, is a complex-valued function of time. The time evolution of the quantum state can be determined by substituting eq. (8 ) in eq. ( 7). For an arbitrary bending state v,, the general form of the perturbed wave function can be obtained in the form

The probability of 02O-tOlO vibrational deexcitation can be written as

To determine g’s, we derive a linear relation between the coefficients of a$, az, a$u2, and Zin the Hamiltonian and those in I q(t) > by substituting eqs. (6 ) and (8) in eq. (7 ) [ 131:

-:[;;I=[;

_;,

z-z

;]I]*

(11)

Thus, g,‘s can be determined by solving four first-order differential equations subject to the initial conditions gl(tO)=O. Here, g3(t) is simply exp( -iiw?t). These equations will be solved for F(I) numerically using the fourth-order Runge-Kutta method. It should be noted that if the rotational motion is neglected (i.e. either a rotationally frozen or rotation-averaged model), F(t) appears in a simple form containing hyperbolic functions and the solution can be obtained in explicit forms [ 141.

5. Results and discussion Potential constants used in the calculation are [ 15,161: D(II,O) = 370 K, D(Ar) = 124 K, a(H*O) ~2.525 A, and o(Ar) = 3.418 A. The combining laws are used to calculate 0=214 K and 0~2.970 A for H20-Ar. The value of 1 is taken to be equal to a/a. The interaction range parameter a is calculated using the expressions [ 171 a= (a/x’/‘) [r( 7/12)/r( l/12)] (~D/x)“‘~, where x= [ ( f~)“2xAwak~]2’3. This value is weakly temperature dependent; it changes from 0.202 A at 300 IS to 0.189 A at 1000 K. (Note that a value of 0.2 8, for a is often assumed in energy transfer calculations [ 181.) The most probable distance R* is found to be 1.72 and 1.82 A at 300 and 1000 K, respectively. The normal frequencies [8,19] II’are v,=3825.32, u2= 1653.91, and u3=3935.50 cm-‘. The energy release AE calculated from the vibrational term expression [ 191 is 1556 cm-’ for v2=2+ 1. The bond distance doH is taken to be 0.956 8, [20]. The calculated value for the moment of inertia is 1,=2.93x 10C4’ g cm*. The transition probability given by eq. (10) is a function of rotational energy ER, translational energy E, impact parameter b, and rotational phase &: Po20-010=P020_0,0 (ER, E, b, $). The impact parameter is introduced in the expression by replacing E with E( 1 - b2/R**) in the range 0 R*, which is an appropriate assumption for short-range interaction. The distance R* will not be very far from the closest distance of approach in the classical head-on collision. In the model, the molecule is assumed to undergo rotation in the xy plane (i.e. about the z axis ) Therefore, in calculating the thermal-average transition probability, it is necessary to average P020_0,0(ER,E, 6, 0,) over #’ See for the normal frequencies ref. [ 19, p. 2071, for the vibrational term expression ref. [ 19, p. 2051, and for the anbarmonicity constants ref. [ 19, p. 2821.

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the z-direction energy. In the classical limit, the average can be made over the distribution function written in the form JK(E

,

R

dE

=

R

dp=.I% dp, S%.,dpx Ii?’dt SFd@18de sin 8exp[- (d/2L+p~12Zy+d/XUkTl f.~~d’~(T/@x)“2( T/6$)“‘(

T/8z)1’2

,

(12)

where 8,=fi2/2Z&, etc. When the relation p:=2Z& function reduces to

is used for z-axis rotation, the normalized distribution

fR(ER) d&= (kT/tiR)L’2exp(-ER/kT)

.

d(E,/kT)

(13)

We then average the transition probability over translational energy and impact parameter using the distribution function [ 71 f,(E,

b) dEdb=2(E/kT)(b/R*)

exp( -E/kT)

d(E/kT)

d(b/Z?*) .

(14)

Finally, the phase dependence will be removed by averaging the probability over 0, between 0 and 271.With these four integration steps, the final form of the transition probability averaged over Z&, E, 6, and 0, is

In the present VR model, where the energy release AE is removed by rotation, we replace the rotational energy ER appearing in the probability expression by the symmetrized energy Z$ = t [ ( ER+ AE) ‘I2 + Eg2] 2, which is equivalent to taking the average velocity f (QiSL&). When this is done, we obtain Pg,,,(Ec,, E, b, 0,) and the average is the VR probability. On the other hand, when we invoke the VT model, AE is then removed by translation, so E will have to be replaced by E? = a{ [E( 1 - b2/R*2) +AE]lf2+ [E( 149/R*‘)] ‘12j2. The result will be denoted by P&& T). Note the appearance of impact parameter b in the latter symmetrization. In fig. 2 we present calculated values of both the VR and VT probabilities for H20(020) +Ar-+ H,O(OlO) +Ar+AE= 1556 cm-t from 200 to 1000 K and compare them with recent experimental data [6]. Note that four additional points of earlier measurements by Moore and coworkers are for HziBO [ 5,2 1 ] and are plotted for comparison. It is important to compare the results of ref. [6] with the VT model. As shown in the figure, the latter model completely fails to predict the temperature dependence. At temperatures below 300 K, where the VR mechanism appears to be of major importance, the VR and VT probabilities differ by a factor as large as an order of magnitude. For example, at 200 and 300 K, the VR probability is larger than the VT probability by factors of 19 and 7.8, respectively. As temperature increases, however, translational motion becomes effective in removing the energy release, and it can eventually catch up to the efficiency of rotational motion. At 1000 K, the VR and VT probabilities are 2.06 x 10 -2 and 1.45 x 1O-‘, respectively. Thus, we may state that at low temperatures near or below 300 K, the relaxation is essentially a pure VR process but at higher temperatures both VR and VT mechanisms are effective in removing AE. At sufficiently high temperatures, the VT process can even be of major importance. The comparison of VR and VT models suggests that it is physically more realistic to consider a VRT mechanism. It is tempting to set up PvRT T), where SvRand fvTdetermine the 020-OlO( T) =fvR~&Wl( T, +f VT PVT02w10( fractional contributions of VR and VT pathways and are temperature dependent. For example, if we takef&= 1.O at 200 K and fVR= 0.10 at 1000 K, with some intermediate fractions at temperatures between them, a temperature dependence which is in near perfect agreement can result. However, it does not appear possible to

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Fig. 2. Temperature dependence of the VR and VT probabilities for 020-010. The VT probability at 200 K is 8.8~ 10W5(not shown). Experimentaldata: l ,ref. [6]; x,ref. [5]; + ref. [21].

23 March 1990

Fig. 3. Phase (0,) dependence of VR and VT energy transfer probabilities at 300 and 1000 K. The values at 300 K are multiplied by two.

determine these fractions in the present model other than fitting to the experimental data, but the results displayed in fig. 2 clearly suggest the importance of such a VRT model. To discuss the source of rotation-vibration coupling, we consider the term (c&+Ar,) cos( 0+ $Ay) of RI given by eq. (2a). With eqs. (la), i= 1, and (lb), we can convert this term into

When the constants are evaluated, the contribution of this term to the bending-rotation coupling which enters in U’ (R, 0, QZ) is exp[ -S& cos(& iy)] exp[L cos 0- (a/2R)L2 sin%] ( -5.70x

10” sin 19-2.69x 10” cos @)Q*g-‘j2,

where the first term is the result of a direct coupling between rotation and bending and is of major importance in the VR mechanism. The second term is for the coupling of the Qz mode of bond stretching Ar, with rotation and is much less important. A similar analysis can be presented for the corresponding terms of Rz. During the calculation, we found the integrand of eq. ( 16) to vary strongly with the initial phase $. In fig. 3, we plot the ~3,dependence of both Pz&O,O(T, $) and P$_,,, (T, $) at 300 and 1000 K. Note that %LoL0(7? = (2x)- ’ soZr p T&,lo( T, 0,) de,. The figure shows a strong phase dependence of these quantities: they take a maximum value at $= 50”, 170”, and 290”, an interval of 120”. The probability at phase zero is very small compared with these extreme values, indicating that phase-zero calculations of energy transfer probabilities are totally unsatisfactory. Even in the VT model, the phase dependence is important. We now check the approximation used in deriving the atom-atom distances R,. At 300 K, the most probable distance R* is 1.72 A. For this R*, the distance R, given by eq. (2~) is essentially identical to the exact value, 225

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e.g. the largest deviation is about 0.11 which occurs at 8= 320”. The deviations of distances R, and R2 from the exact values are somewhat larger. An error of about 5W is found between 8= 30” and 40’ for R, and between 60” and 80” for R2,At other angles, errors are typically about lo/o.Thus, the use of eq. (2) for the present formulation appears to be quite satisfactory.

Acknowledgemen This research was supported by the University of Nevada Foundation and NSF Advanced Computing Resources grant (CHE-890039P) at the Pittsburgh Supercomputing Center. One of the authors (JR) wishes to acknowledge the Korea Science and Engineering Foundation for a postdoctoral fellowship.

References [ I ] W.S. Benedict, M.A. Pollack and W.J. Tomlinson, IEEE J. Quantum Electron. QE-5 ( f969) 108. (21 W.J. Sajeant, Z. Kucerovsky and E. Brannen, Appl. Opt. I 1 (1972) 735. [3] R.T.V. KungandR.E. Center, J. Chem. Phys. 62 (1975) 2187. [4] H.E. Bass, R.G. Keeton and D. Williams, J. Acoust. Sot. Am. 60 (1976) 74. [ 51J. Finzi, F.E. Hovis, V.N. Panfilov, P. Hess and C.B. Moore, J. Chem. Phys. 67 (1977) 4053. [6] P.F. Zittel and D.E. Masturzo, J. Chem. Phys. 90 (1989) 977. [ 7] J. Ree and H.K. Shin, Chem. Phys. Letters 163 (1989) 308. [ 81 W.H. Shaffer and R.R. Newton, J. Chem. Phys. 10 (1942) 405. [9]F.DormanandC.C.Lin, J.Mol.Spectry. 12 (1964) 119. ] lo] H.K. Shin, 3. Am. Chem. Sot. 90 (1968) 3025. [It] G.D.B. Sorensen, J. Chem. Phys. 57 (1972) 5241. [12]J.Stricker,J.Chem.Phys.64(1976) 1261. [13]H.K. Shin, J. Chem. Phys. 81 (1984) 1725. [ 141 H.K. Shin, Chem. Phys. Letters 97 (1983) 41. [ 151J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1964) p. 1110. [ 161F.J. Zeleznikand R.A. Svehla, J. Chem. Phys. 53 (1970) 632. [17]H.K.Shin, J.Chem.Phys.41 (1964) 2864. [ 181J.T. Yardley, Introduction to molecular energy transfer (Academic Press, New York, 1980) ch. 4. [ 191 G. Henberg, Infrared and Raman spectra (Van Nostrand, Princeton, 1968). [20] G. Herzberg, Electronic spectra of polyatomic molecules (Van Nostrand, Princeton, 1968) p. 585. [21] FE. Hovis and C.B. Moore, J. Chem. Phys. 72 (1980) 2397.

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