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Vibrational and rotational energy transfer upon molecular collisions
Chemical Physics 16 (1976) 269-279 0 North-Holland Publishing Company
VISRATIONAL
AND ROTATIONAL
ENERGY TRANSFER
UPON MOLECULAR
COLLISIONS*
Lue-Yung CHOW CHIU** Optical Physics Division, National Bureau of Standards, Washington. D.C. 20234. and Department of Chetnistiy. Howard University. Washington. D.C. 20059. USA Received 23 February 1976
The cross section of the rotational and vibrational energy transfer is derived by using the first Born approximation which quantizes the translational motion of the colliding particles. The theory developed here integrates the intermolecular potential V(R) over all regions of the internuclear distance R by obtaining a Fourier transform of I/(R). This differs from previous semiclassical (impact-parameter) treatments which either considered the short-range repulsive interaction or expanded V(R) into a long-range multipole expansion. The cross section obtained here is expressed in a very simple algebraic expression which can be readily calculated. This will be illustrated by examples of COa(OO1) + Na(u = 0) = COa(OO0) + Na(u= 1) + -t. Calculations have been made both for the exo18.6cm-tandCO(u=l)+CO(u=1)=CO(u=0)+CO(v=2)+27cm thermic and for the endothermic reactions. The comparison of the present results with esperimental results as well as with previously calculated results will be discussed.
1. Introduction With the advent of the powerful monochromatic beams of laser light, which can excite molecules into a specific vibrational and rotational energy state, there have been extensive experimenta! measurements [I] on the vibrational and rotational energy transfer processes. Theoretically the short-range repulsive forces between the molecules were believed to cause the vibrational energy transfer upon molecular collisions. Based on this premise, the theory of Schwartz, Slawsky and Herzfeld [2] predicts that the cross section increases linearly with temperature for exact or near resonance collisions. This has been successful in explaining a wide variety of vibrational relaxation rates, especially the earlier data obtained by shock wave [3] _However, recent laser measurements [4-71 show a negative temperature dependence which contradicts the theory based on short-range forces. Mahan [S] first suggested Rm3 interaction due to long-range dipole potential. Using the first Born approximation, Sharma and Brau [9] considered long-range dipole-quadrupble force between CO,(OOl) and N-,(u=O) to calculate the energy transfer cross section. Their temperature dependence agrees cIosely with that of experimental measurements [4]. The theory based on long-range forces has also been applied by Sharma [lo] to other systems. Dillon and Stephenson [ 1 l] have developed a multiquantum theory to calculate the vibrational energy transfer between excited HF (and also DF, HCl) and CO,. They have also obtained a good agreement with experimental results. Previous theoretical studies on vibrational energy transfer so far have been based on a classical path approximation [8-l 11: The probability of energy exchange from the first order time-dependent perturbation is then taken as P=ti-21jeinet/fi
(flu[R(;)]i)dt12,
where Ae is the energy discrepancy, and V[R(r)]
(L-1) is the intermolecular potential which depends on time exphcitly
* Work supported in part by a research grant from the U.S. Energy Research and Development Administration. ** Present and permanent address: Department of Chemistry. Howard University, Washington, DC. 20059.
270
L.-Y_ Chow Chief Vibrational
and rotational
energy transfers
through the trajectory R(r). This initial and final state wave functions, ii) and
2. General method We consider the following vibrational and rotational energy transfer process between two molecules A and B: A*(.@,)
+ B(vbJb)*
A(u;J;)
+ B*(ubJ;)
+ Ae*
(2.1)
where
is the internal energy discrepancy between the initial and the final molecular states. u’s and J’s are respectively the vibrational and rotational quantum numbers, B’s are-the rotational constants and Aeo is the vibrational energy difference for Ja= Jb = 0. The transition probability from the initial state ‘ki to final states af (between vf and vf + dvr> is given by [ 18) do=(25rlti)I(‘kfIV19i)126(Ef-Ei)dvf.
(2.3)
where ‘Pi = L-3~2exp(iki-R)@i and ‘Z!f = L-3/2exp(ikf -R)Gf are the unperturbed wave functions [ 191 describing the relative translational motion of A and J3 (which is taken to be a plane wave normalized in a box of dimension
..
L.-Y. Chow ChiulVibrational and rotational energy transfers
271
L , LB3f2 exp (ik.R)) , as well as their internal motion (0). ki and kf are respectively the propagation vector of the initial and the final plane waves, and R is the relative coordinate of B to A. Ei and Efare respectively the energy of states *i and *p The perturbation V=
(2.4)
Ceieif*..
ij
B’
which causes energy transfer, is the electrostatic interaction energy between A and B. ei and ej are the charges of the charged particles i andj belonging to molecules A and B respectively_ The operator V here does not depend on t explicitly due to quantization of the translational motion_ Integrating (2.3) over the final states dvf= (L/2r)3kf2dkfdCk and dividing it by the incident flux S =?iki/L3p, we obtain the differential cross section for the energy transfer as fo3ows: d~=(~/2~fi2)‘(~~/~~)I~exp(-i~~R)(~~IVl~~)dR12d~,
(2.5)
where 4 = kf - ki is the momentum transfer vector, and JJ is the reduced mass of A and B. Eq. (2.5) above is equivalent to the formula of the first Born approximation. Instead of expanding the intermolecular potential V by a two-center multipole-multipole expansion [16,i7], we integrate dR over all the region by the following procedure. According to the diagram in fig. 1, we have (2 -6) Ib3 where ria and rib are respectively the position vector from the charged particles i (which belongs to A) and j (which belongs to B) to the center of the mass of molecules A and B. Substituting eq. (2.6) into (2.4) and taking the Fourier transform, we have rii=R-ri,+r-
JVexp(-iq-R)dR=(4n/q*)
~eieiexp[-iQ-(ria-r~~)]_
(2.7).
The integration of V(R) over all the region R (including the overlap region) is therefore accomplished_ The transform shown on the right hand side of eq. (2.7) equals 4?r/q* times a product of two operators, namely Zi ei exp (-iq-ri,) (which operates on the wave functions centered on A only) and Zj ei exp (iq-rib) (which operates on the wave functions centered on B only). Substituting (2.7) into (2-S), and using q = kf-ki and qdq = (kfki/2?r) dG? 1201, we therefore have q3 (--$)* do(q) = 8n
1 ~ei4(0/.lexP[-ip.(ri,‘jb)li0,)Izdq-
e-8)
The above differential cross section depends on q, the momentum of transfer. To obtain the total cross section one has to perform integration over dq. The right hand side expression becomes vanishingly small [21] and can be
Fig. 1. Schematic diagramshowingthe relationshipof rij to ria, r1%and R. ria and r,-t,are the position vectors from the chaxged particles i andi to the center of mass of moleculesA and B respectively.
212
J.-Y_ Chow C?ziu/Vibrational and rotational energy
transfers
neglected when q is la’rge, i.e., q >-l/a, (a0 is the Bohr radius), the contribution to the cross section mainly comes from small q value. We therefore expand exp (iq-r) in terms of Bessel functionsjl(qr) and Pl(cos 0,) by Rayleigh expansion [22] along q (as z-axis), and then let jr(qr) = (qr)l/(2Z + l)!! Next, we rotate the coordinates twice such that the molecular figure axes of A and B may be used as separate quantization axes for each molecular system. We therefore have [22]
ix,[_iq-(ria_rjb)]
00
co
=,co[go(-#a
(i)‘b(2Za + 1)(2zb + 1) C(21a ’ 1)!!(2zb
‘l)!!]-’
a (2.9)
where C&(@) = [4~/(21+ l)] II2 Ylr,l(Ocp)is the normalized spherical harmonics. (Oi,l~ia) and (Oib~jt,) refer respfctively to the figure axis of rotating molecules A and B. The rotational matrices D~~~a(rra&y~) and D~bnlb(ab&,-y,,) rotate the quantization axis (along q) into those along the molecular figure axes of A and B respeciively. The internal state wave functions [ 191 r& = 99 @ and @f = @?+” of the initial and final states are taken as product wave functions for both molecules. Assuming both A and l5 are m _ the.. electronic C state, the wave function for each molecule is then by Born-_Oppenheimer approximation a product of an e$ctronie wave runction &(r, R), a vibrational wave function @v(R) and a rotational wave function [22] @=,&R) = (ZJ -I-l/&+/2 D&(@$ , where r represents all the electronic coordinates with respect to the figure axis and R is here the inter&clear distance. Hence we have
and similar expressions for @ and @I”_We assume that the electronic wave function remains the same for initial and final states. To obtain the energy transfer cross section for molecules initially at (uaJa) and (u&,) and finally at (u:Jl) and (ubJ{). we substitute (2.9) into (2.8) followed by summing-over the magnetic sublevels MA and MA of the final states and averaging over the magnetic sublevels Ma and Mb of the initial states. After integrating the rotational matrices separately for molecules A and B, and applying the orthogonal reiation [23] of ClebschGordan coefficients (which are generated through the integration of the rotational matrices), we obtained the following expression for the cross section C(J, I, J;; 000)2 C(J,, 1, J;; 000)2
where Qi,(R,)
=
(@e,(raR,) 1Feir;z Cla,O(eia%a)l@e(raRJ)
(2.12)
is the electric 2&-pole moment of the molecule A and (&I Q*a 1u,) is therefore the transition matrix element of the 2%pole moment between vibrational states I& and ua. Similar expressions hold for molecule B. The cross section expressed in (2.11) depends on the momentum transfer, and such q-dependent cross section might become experimentally measurable [24] in the near future. However to obtain estimates of the total cross section, we integrate eq. (2.ll)over the momentum transfer dq, and obtain (22.13) where
L.-Y.
Chow Chiu/Vibmtional
273
and rotational energy transfers
~r,*,=(2z,+1)(221~~1)[(2z,+1)!!(2z~~1)!!l-~(2z,~2~~-2)-~ x C(J,Z, J;; OOO)~c(J,,
z, J;;
~100)~ I@;
1Q,,1yJ2 &J;1Qlb1ub)12
(2.14)
and qmin =
k~-
ki ~ (it -
~r)ltl~i.
(2.15)
The above (2.15) is true when ki = kr (or ki - kr4 ki)_ Thr‘s implies the condition that the energy discrepancy Ae(= ‘i - ef) is much smaller than the kinetic energy of the reIative motion or the condition of near resonance. To estimate @max we recall that da(q) (as expressed in (2.8)) becomes vanishingly small [21] (and can be neglected) for large values of q, and the multipole expansion of the plane wave is valid when qr < 1 or q C 1 /I-_Since the electrons in the outer shell of the molecule are the ones to interact with another molecule causing energy transfer, we let r be the radius of the outer electron shell, namely [25] q,,,ax = 2/d,
(2.16)
where d is approximately the average moiecuI& diameter_ The dependence of the cross section on the d-values chosen will be illustrated in the examples given in the next section. Eq. (2.13) gives the energy transfer cross section for a given initial relative velocity ui_ To compare with the measurements over a distribution of vi, we average (2.13) over the Boltzmann distribution at temperature T_ Since almost all of the experimental cross sections reported are derived from the rate constant k, (cm3 s-l) of the reaction by u = k,/(v), we therefore carry out the corresponding averaging by -
o(T)=
(va(v))/~v)
(2.17)
instead of the straightfonvard averaging (a(u) ). The result from (2.17) will be smaller than’the result from (a(u) 1 by an exact factor l/2. Substituting (2.13) into (2.17), we have (2.18) where 7fazb is given by (2.14) (u)= (8kT/np)lj2 and AE = ei-- efin unit of cm-l is given by (2.2). Eq. (2.18) gives the energy transfer cross section for molecules initially at (u,J,) and (ubJb), and finally at (II!&) and (r&J& Since the measured cross section is often an averaged value over all of the rotational states, we average eq. (2.18) over the initial rotational states (JaJb) and sum it over the final rotational states (Ji Jb) as follows: (2.19) where n,(Ja) and “b(Jb) are respectively the rotational distribution functions of the initial rotational states (J,Jb) at temperature T- In the above summation, only terms which give positive vahre for UJ=J~+J;&( T) (i.e., 17 Iqmin 1) will contribute. If one is interested in the dependence of cross section on the initial rotational ‘%nax states, say J=, we perfom the average over the rotational states Jb only, i.e., (2.20) The cross section for energy transfer as given by (2.13) or (2.17) is a sum over terms of vibrational multipolemultipole transition moments of molecules A and B, 71,~~ as given by eq. (2.14). These transition moments also appear in Sharma and Brau’s [9] semi-classical (impact parameter) treatment. Their transition moments arise from expressing the intermolecular potential, V= ZV ei ei/rr7, in a long-range multipole-multipole expansion. Here the vibrational transition moments are derived through the multipole expansion of the plane waves, exp(-iq-rr,) and exp(iq-rib) of two separate molecular systems. For each multipole interaction the formalism presented here depends upon the minimum and maximum momentum transfer in eq. (2.13) and eq. (2_18). Since
274 Qmin
L.-Y_ Chow Chiu/Vibrationai and rotational energy transfers
z Ae/?iui (eq. 2.15), the second term in eq. (2.18) will be small for Ae = 0 (exact resonance) or for large*- In these cases the cross section will be inversely proportional
to temperature. The treatment of ref. [9] also yields a l/Tdependence at high temperature. While Qmin is completely specified by eq. (2.15), qrnax is not. Since the cross section due to each multipole-multipole transition moment will depend upon 4$$+(b-r)(2/d)*&+‘b-r), t h e cross sections will be extremely sensitive to the value of d used when I, and/or 1, are large. In this case we can treat d as an adjustable parameter whose value should be approximately the molecular diameter. This is different from the procedure used in ref. [9] . There, an impact parameter = b was chosen and straight line trajectories used for b >d and a parabolic interpolation procedure used for b
3. Numerical examples and discussions To illustrate the method, we calculate the cross section of energy transfer for the following reactions at near resonance:
C02(001)+NZ(~=0)~
CO,(OOO) •t Nz(u = 1) + Aeu = 18.6 cm-l
(3-l)
and CO@=1)+CO(u=1)~CO(u=0)+CO(u=2)+Aen=27cm-t
(3.2)
For the homonuclear diatomic molecule Nq, the transition dipole moment is zero and the leading term for CO, + N, reaction will be dipole-quadrupole interaction, namely Za(COa) = 1 and lb&) = 2. The next term, quadrupole-quadrupole interaction, is absent. This is because the vibrational states (001) and (000) of CO2 considered in eq. (3.1) are respectively odd and even upon inversion and the transition quadrupole moment over these states becomes zero. The dipole-quadrupoIe interaction will therefore be the onfy term in eq. (2.18) that survives. As for the CO + CO reaction, we consider both the leading dipole-dipole term, i.e., I, = 1, = 1 and the dipole-quadrupole term, i.e., 1, = 1, lb = 2 and la = 2, &, = 1_ Since the cross section calculated from the dipolequadrupole term is less than 0.6% of that from the dipole-dipole term we are justified in neglecting higher order terms. To compare with the measured cross section, the cross section for energy transfer has been averaged over the initial rotational states and summed over the fmal rotational states as given by eq. (2.19). The summations over the final states Jk and Ji are quickly terminated into a few terms due to the vanishiig of Clebsch-Gordan coefticients. To carry out the average over the initial states J, and Jb, Maxwell-Boltzmann distribution is assumed for the rotational distribution function n(J) as introduced in eq. (2.19). For a homonuclear diatomic molecule, the symmetry factor due to nuclear spin has to be included in n(J). In the case of 14N2, the states with total nuclear spin I = 2 and (f belong to the even rotational states and states of I= 1 belong to the odd rotational states [26] . The rotational distribution for b12 is therefore given by n (Jb) = z;’
3 OLJ~(2Jb + 1) exP I--Jb(Jb
+ l)Y] 9
(3.3)
where
y = h cBb/k T,
(3.5)
‘and e!_rb= 2 for even Jb and CXJb = 1 for odd Jb_ Siilarly for CO,(OOl) we have only odd rotational states and for CO,(OOO) we have only even rotational states [27] to contribute to the sum of the rotational levels. To cakulate the cross section uen fdr the reverse endothermic reaction, not only the initial and final states are reversed, but also the sign of Aeo_
L-Y.
Chow Cliiuf Vibmtional
and rotational
energy transfers
275
TabIe I Molecular parameters used in calculating the energy transfer cross sections Parameter
co2
co
N2
l(OIQ,11,12
1.0 x 10-37
-
1.17x
r(OlQ*l1)l2
-
1.1 X 10ms4 Stat C2 cm2 Cl
1.0
3.57 A
3.15 A
3.4 A
dce’
stit c2 .d
@Ref. 1291. b)Ref. [30]. ‘)Ref. [32]. and van der W&s constants [ 28]_
d)Ref.
a)
[31].
e)The average classical molecular diieterd,
10-3~statc~crr~b)
X lo-s5
Stat C* cm* d)
is derived from viscosity data
As indicated in (2.18), the cross section calculated depends on (q,,)2(fa+k1) = (2/d)2(za+Ib-1) whered is approximately the average molecular diameter. The average diameter considered here is the average of the classical values [Zg] derived from viscosity data and van ddr Waak constants. These average diameters are listed as d, in table 1, where the transition moments I (u’ I Ql I u) I* are also listed_ The transition dipole moments I(0 IQ, I 1) 12 for CO, and CO are, respectively, the measured values of Houghton [29] and of Young and Eachus [30]_ The transition quadmpole moment I(0 I Q2! 1) I2 for CO is the calculated value of Billiigsley and Krauss 1311, and I(0 I Qz I 1) I2 for N, is the calculated value of Cade [32] _ Using the d, and the transition moments listed in table 1, the cross section for both the exothermic and the endothermic reactions of CO + CO (given by eq. (3.2)), have been calculated over the temperature range IOOlOOOK. The experiments of Stephenson and Mosburg [6] on CO f CO were done for the endothermic reaction and their reported cross sections of the exothermic reaction, were obtained by the statistical relation, uex =
uen exp (Af,/kT).
(3.6)
Tf’Kl 2.0’0°0
300 co(v=II+co(v=l~
125 I cx. ~CO(v=01+CO(v=2)+27cm-’ 200
100
1.6
Fig. 2. The dashed line and the solid line are respectivelythe cross sections for the exothermic and endothermic reactions calculated independentlyfrom eq. (2 19). The experimentalexotheimic and endothermic cross sections of ref. [7] are represented here respectively by circles and triangles, and they are related by the statistical relation eq. (3.6).
276
L.-Y. Chow Chiu/lfibmtionaIand rotationalenergy transfers
In fig. 2, the theoretical and the experimental cross sections for both the exothermic and the endothermic reactions are plotted against l/T, and we have good agreement in both cases. Since the theoretical
_--
00
2
3 +x IC+K-’
)
4
5
Fig. 3:The dotted line is calculated by using the molecular parameters listed in table 1; the solid tine is calculated by usingthe adjusted d = 2.82 A and the same transition moments in table 1; the dashed line is calculated by using the experimental transition quadrupole moment for Nz (6 X lo- ss Stat C* cm* [37]) instead and the adjusted d = 2.50 A. The circles with error bars are the Bxperimental points of ref. [4] _
L.-Y.
Chow Chiu/VibrationaZ and rotationaZ energy
transfetS
277
= (2/d)4 by a single d, [36] by applying eq. (3.7) is not as straightforward as the case of approximation of &, two similar molecules (e.g., CO + CO). The same criterion in the choice of d (i.e., letting d = d,) perhaps Cannot be applied here. The transition quadrupole moment of N,, I(0 IQ21 1)d2 = 1.10 X 10ms4 Stat C2 cm4, used in these calculations is the calculated value of Cade [32] _This value differs by a factor of 6 from Nesbet’s [37] calculation (1.7 X 1O-55 Stat C2 crn4) and differs by a factor of 2 from Shapiro and Gush’s [38] experimental value (6 X lo-s5 Stat C2 cm4), which was measured indirectly from the collisionally induced infrared transition. While the dipole-quadrupole term contributes less than 0.6% to the total cross section for CO + CO reaction, the dipole-quadrupole is the leading term for CO, + N, reaction_ The uncertainty in I(0 1Q2 (1) I* is carried directly into the calculated cross section. For this reason we have also used Shapiro and Gush’s experimental i(0 IQ21 1) l2 to calculate the cross sections_ An adjusted value of d = 2.5 A is found in this case. Using this value, the calculated cross sections, shown as a dashed line in fig. 3, also agree closely with the experimental cross sections over the entire temperature range. The adjusted d = 2.5 a and 2.8 8, are all in the vicinity of the classical diameters. In fact, the molecular diameters derived recently from the molecular refraction [39] data are 2.86 A and 2.4 A respectively for CO2 and N2. Since the calculated cross section u due to each multipole-multipole transition moment depends upon q$h$zb-l) e (2/d) 2 ( ‘a+‘b-‘) where we have used (2/d) to approximate qrnax. This d has the meaning of a molecular diameter, and may be approximated by classical diameter, d,. Since qmax =2/d is an approximation to begin with, it is reasonable for us to treat d as an adjustable parameter (rather than identifying it with a fmed particular classical value) especially when D is very sensitive to the d-value. For CO + CO, the dipole--dipole is the leading term to contribute to the total cross section, the latter therefore depends only on &, -(2/d)2. The cross section in this case is not sensitive to the d-value and parametrization on d is not needed. However for the case of CO, + N,, where the cross section is very sensitive to the d-value (having a (2/d)4 dependence), treating d as an adjustable parameter is therefore needed. The adjusted d = 2.82 a found here is indeed in the vicinity of the classical diameter_ The recent distorted wave calculation of Sharma and Picard 1401 also showed that the cross section of CO, + N2 reaction is very sensitive to their diameter d chosen. The semiclassical impact parameter calculations [9,35] for CO2 f N2 and CO + CO also depend on the molecular diameters chosen. In that method, a distanced was assumed such that straight line trajectories used for b > d (b is the impact parameter) and parabolic interpolation procedures used for b
2-M
L.-Y.
Chow ChiulVibmtional
and rotational energy
transfers
Table 2 Cross sections for different values of Peg (1) Reaction: COs(OO1) + Ns(u= 0) = COs(OO0) + Ns(u= 1) + AEO AE~ (cm-*)
0.0 10.0 18.6 =) 20.0 30.0
(oex(T)) x 10” (cm*) 200 K
300 K
1000 K
1.32 1.26 1.23 1.22 0.99
0.89 0.85 0.82 0.82 0.75
0.27 0.26 0.25 0.25 0.24
(2) Reaction: CO(~=~)+CO(~=~)=CO(U=O)+CO(~=~)+ Aeo (cm-t)
AE~
( oex (T) 1 x 10’6 (cm2) 200
0.0 10.0 15.0 20.0 27.0 a) 35.0
K
1.07 0.97 0.91 0.85 0.77 0.72
300 K
1000 K
0.68 0.66 0.63 0.59 0.54 0.50
0.21 0.20 0.20 0.20 0.20 0.18
a) Pee = 18.6 cm-’ and Ae,, = 27.0 cm-r are the actual vibrational energy discrepancies for reactions CO2 + N2 and CO + CO respectively. These values have been used for cross section calculations presented in fig. 2 and fig. 3.
Acknowledgement The author wishes to thank Dr. Frederick Mies for his interest and comments and Drs. Karl Kessler, Richard Deslattes and John Cooper for their efforts to make possible her stay at the National Bureau of Standards (during her sabbatical leaves from Howard University). Dr. John Cooper’s valuable suggestions, interest and critical reading of the-manuscript are gratefully acknowledged.
References [l] [2] [3] [4] [5] [6] [7] [S]
For a review on recent experimental measunzments,please see C.B. Moore, (a) Advan. Chetn Phys. 23 (1973) 41: (b) of Chemical Research 2 (1969) 103. R.N. Schwartz, Z.I. Slawsky and-K-F. Herzfeld, J. Cbem. Phys. 20 (1952) 1591; K.F. Herzfeld, J. Chem. Phys. 47 (1967) 743. For a review on this. please see ref. [la] and also R.L. Taylor and S. Bitterman. Rev. Mod. Phys. 41 (1969) 26. W.A. Rosser, A.D. Wood and E.T. Gerry, J. Chem. Phys. 50 (1969) 4996. C-B. Moore, RE. Wood, B.-L. Hu and J-T. Yardiey, J. Chem. Phys. 46 (1967) 4222. J.C. Stephenson and E.R. Mosburg, Jr., J. Chem. Phys. 60 (1974) 3562; J.C. Stephenson, Appl. Phys. Lett. 22 (1973) 576. J.C. Stephenson, J. Chem. Phys 60 (1974) 4289; J.C. Stephenson and C.B. Moore, J. Chcm. Phys. 56 (1972) 1295. B.H. Mahan, J. Chem. Pbys. 46 (1967) 98. counts
AC-
L.-Y. Citow Chiu/Vibmtional [9] 1101 Ill1 1121 1131 [ 141 Cl51 [ 161 1171 [18] 1191
and rotational
energy transfers
219
R-D. Sharma and CA. Brau, J. Chem. Phys. 50 (1969) 924. R-D. Sharma, J. Chem. Phys. 50 (1969) 919; Phys. Rev. 177 (1969) 102; Phys. Rev. A 2 (1970) 173. T-A. Dillon and J.C. Stephenson, Phys. Rev. A 6 (1972) 1460; J. Chem. Phys. 58 (1973) 2056. R.J. Cross, Jr. and R-G. Gordon, J. Chem. Phys. 45 (1966) 3571. A.M. Arthurs, Proc. Cambridge Phil. Sot. 57 (1961) 904. R.G. Gordon and Y.N. Chiu. J. Chem. Phys. 55 (1971) 1469; Y-N. Chiu. J. Chem. Phys. 55 (1971) 5053. L.-Y. Chow Chiu, J. Chem. Phys. 60 (1974) 2079: Phys. Rev. A 5 (1972) 2055. M-E. Rose, J. Math. and Phys. 37 (1958) 215. C-G. Gray and J. Van Kranendonk, Can. J. Phys. 44 (1966) 2411. L.D. Landau and E.M. Lifshitz, Quantum Mechanics, 2nd Ed. (Addison-Wesley, Reading, Massachusetts, 1965) p. 147. *i and qfhere are the eigenfunctions of the unperturbed Hamiltonian Ho. when the interaction potential V between A and B is zero. After coordinates transformation, Ho can be written as Ho=Hi(a)
+ Hi(b)-
(fi*/2Ma)V%,-
(fi2/2Mb)Vfb,
where Hi(a) and Hi(b) describe respectively the (internal) motions of particles belonging to A and B with respect to the center of mass A and B. The remaining two kinetic energy terms are the kinetic energies of the center of ma= A and of that of B. These two terms can then be transformed into the kinetic energy of the relative motion between A and B, -V&,
and the kinetic energy of the center of mass of the whole system, - [7i2/2(Ma +Mb)]V& (which will be neglected). Both Wi and Wf are therefore a product of the internal state wave function and the wave function for relative motion. [ 201 Ref. [ 181, p. 573. 1211 The cross section is proportional to I(O,(r,R,)
1eXp(-iqq,)
1Qe(ra R,)) (@,(rbRb)
i exp(iq-r,%)
Ioe(rb Rb)) 1’.
where oe(r. R) is the molecular electronic wave function which can be represented by a linear combination of atomic orbitals such as r”-%+Y lm (09) (where c - l/ao)_ The above absolute value square therefore breaks into a linear combination of terms which are approximately proportional to a factor of (q2rm2 + I)-*p (where p > 4). This factor becomes vanishingly small (compared with unity) when 41-t > l(or q > l/no). [Xl M.E. Rose, Elementary Theory of Angular hlomentum Wiley. New York, 1961). 1231 The orthogonal relation used here is c C(J’IJ; MM’
-M’. m. -M)
C(J’I’J; -M’. m. -&f) = 611’(2J f 1)/(21+
I),
which can be easily derived from the relations given by ref. 1221. (241 For recent experimental observations on momentum transfer generated by molecular collisions, see T.W. Meyer and C.K. Rhodes, Phys. Rev. Lett. 32 (1974) 637. [25] See p_ 576 of ref. [ 181 for a parallel treatment in the atomic case. [26] Let P be an operator which interchanges two 14N nuclei and @I is the total nuclear spin wave function, then POI= Gl)‘@~ follows. When P applies to the total wave function @ = @epeo,@JeI, we therefore have P@ = (-l)J’f+ = Q and consequently (-l)J+I = 1. &cc the vibntional states (001)
and (000) of CO, are’respectivcly odd and even upon inversion of the nuclear coordinates we therefore have (-l)J= 1 for COz(OO0) and (-l)J+l = 1 forCOz(001). [28] (a) J.O. Hirschfelder, CF. Curtis and R.B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954) p. IllO1111; (b) W.J. Moore. Physical Chemistry, 4th Ed. (Prentice-Hall, New Jersey, 1972) p. 157. 1291 J.T. Houghton, Proc. Phys. Sot. (London) 91 (1967) 439. [30] L.A. Young and W.J. Eachus, J. Chem. Phys. 44 (1966) 4195. [31] F.P. Billing&y and hf. Krauss, J. Chem. Phys. 60 (1974) 2767. [32] P. Cad& value of I(0 IQz 11) 1’ for Nt is quoted in ref. [S] as the private communication. 1331 H.T. Powell, J. Chem. Phys. 59 (1973) 4937. [34] P.B. Sackett, A. Horduik and H. Schlossberg, Appl. Phys. Lert. 22 (1973) 367. [35] R.D. Sharma, Chem. Phys. Lett. 30 (1975) 261. [36] By (3.7), we have (2/~f~)~ = (2/d,) (2/db)3. Where da and db refer respectively to values of CO2 and Nz in table 1. [37] R.K- Nesbet, J. Chem. Phys. 40 (1964) 3619. 1381 MM. Shapiro and H.P. Gush, Can. J. Phys. 44 (1966) 949. (391 M. Itrplus and R.N. Porter, Atoms and Molecules 0V.G Benjamin, New York, 1970) p. 255 and ref. [28(b)]. [40] R.D. Sharma and R.H. Picard, J. Chem. Phys. 62 (1975) 3340. [27]
VISRATIONAL
AND ROTATIONAL
ENERGY TRANSFER
UPON MOLECULAR
COLLISIONS*
Lue-Yung CHOW CHIU** Optical Physics Division, National Bureau of Standards, Washington. D.C. 20234. and Department of Chetnistiy. Howard University. Washington. D.C. 20059. USA Received 23 February 1976
The cross section of the rotational and vibrational energy transfer is derived by using the first Born approximation which quantizes the translational motion of the colliding particles. The theory developed here integrates the intermolecular potential V(R) over all regions of the internuclear distance R by obtaining a Fourier transform of I/(R). This differs from previous semiclassical (impact-parameter) treatments which either considered the short-range repulsive interaction or expanded V(R) into a long-range multipole expansion. The cross section obtained here is expressed in a very simple algebraic expression which can be readily calculated. This will be illustrated by examples of COa(OO1) + Na(u = 0) = COa(OO0) + Na(u= 1) + -t. Calculations have been made both for the exo18.6cm-tandCO(u=l)+CO(u=1)=CO(u=0)+CO(v=2)+27cm thermic and for the endothermic reactions. The comparison of the present results with esperimental results as well as with previously calculated results will be discussed.
1. Introduction With the advent of the powerful monochromatic beams of laser light, which can excite molecules into a specific vibrational and rotational energy state, there have been extensive experimenta! measurements [I] on the vibrational and rotational energy transfer processes. Theoretically the short-range repulsive forces between the molecules were believed to cause the vibrational energy transfer upon molecular collisions. Based on this premise, the theory of Schwartz, Slawsky and Herzfeld [2] predicts that the cross section increases linearly with temperature for exact or near resonance collisions. This has been successful in explaining a wide variety of vibrational relaxation rates, especially the earlier data obtained by shock wave [3] _However, recent laser measurements [4-71 show a negative temperature dependence which contradicts the theory based on short-range forces. Mahan [S] first suggested Rm3 interaction due to long-range dipole potential. Using the first Born approximation, Sharma and Brau [9] considered long-range dipole-quadrupble force between CO,(OOl) and N-,(u=O) to calculate the energy transfer cross section. Their temperature dependence agrees cIosely with that of experimental measurements [4]. The theory based on long-range forces has also been applied by Sharma [lo] to other systems. Dillon and Stephenson [ 1 l] have developed a multiquantum theory to calculate the vibrational energy transfer between excited HF (and also DF, HCl) and CO,. They have also obtained a good agreement with experimental results. Previous theoretical studies on vibrational energy transfer so far have been based on a classical path approximation [8-l 11: The probability of energy exchange from the first order time-dependent perturbation is then taken as P=ti-21jeinet/fi
(flu[R(;)]i)dt12,
where Ae is the energy discrepancy, and V[R(r)]
(L-1) is the intermolecular potential which depends on time exphcitly
* Work supported in part by a research grant from the U.S. Energy Research and Development Administration. ** Present and permanent address: Department of Chemistry. Howard University, Washington, DC. 20059.
270
L.-Y_ Chow Chief Vibrational
and rotational
energy transfers
through the trajectory R(r). This initial and final state wave functions, ii) and
2. General method We consider the following vibrational and rotational energy transfer process between two molecules A and B: A*(.@,)
+ B(vbJb)*
A(u;J;)
+ B*(ubJ;)
+ Ae*
(2.1)
where
is the internal energy discrepancy between the initial and the final molecular states. u’s and J’s are respectively the vibrational and rotational quantum numbers, B’s are-the rotational constants and Aeo is the vibrational energy difference for Ja= Jb = 0. The transition probability from the initial state ‘ki to final states af (between vf and vf + dvr> is given by [ 18) do=(25rlti)I(‘kfIV19i)126(Ef-Ei)dvf.
(2.3)
where ‘Pi = L-3~2exp(iki-R)@i and ‘Z!f = L-3/2exp(ikf -R)Gf are the unperturbed wave functions [ 191 describing the relative translational motion of A and J3 (which is taken to be a plane wave normalized in a box of dimension
..
L.-Y. Chow ChiulVibrational and rotational energy transfers
271
L , LB3f2 exp (ik.R)) , as well as their internal motion (0). ki and kf are respectively the propagation vector of the initial and the final plane waves, and R is the relative coordinate of B to A. Ei and Efare respectively the energy of states *i and *p The perturbation V=
(2.4)
Ceieif*..
ij
B’
which causes energy transfer, is the electrostatic interaction energy between A and B. ei and ej are the charges of the charged particles i andj belonging to molecules A and B respectively_ The operator V here does not depend on t explicitly due to quantization of the translational motion_ Integrating (2.3) over the final states dvf= (L/2r)3kf2dkfdCk and dividing it by the incident flux S =?iki/L3p, we obtain the differential cross section for the energy transfer as fo3ows: d~=(~/2~fi2)‘(~~/~~)I~exp(-i~~R)(~~IVl~~)dR12d~,
(2.5)
where 4 = kf - ki is the momentum transfer vector, and JJ is the reduced mass of A and B. Eq. (2.5) above is equivalent to the formula of the first Born approximation. Instead of expanding the intermolecular potential V by a two-center multipole-multipole expansion [16,i7], we integrate dR over all the region by the following procedure. According to the diagram in fig. 1, we have (2 -6) Ib3 where ria and rib are respectively the position vector from the charged particles i (which belongs to A) and j (which belongs to B) to the center of the mass of molecules A and B. Substituting eq. (2.6) into (2.4) and taking the Fourier transform, we have rii=R-ri,+r-
JVexp(-iq-R)dR=(4n/q*)
~eieiexp[-iQ-(ria-r~~)]_
(2.7).
The integration of V(R) over all the region R (including the overlap region) is therefore accomplished_ The transform shown on the right hand side of eq. (2.7) equals 4?r/q* times a product of two operators, namely Zi ei exp (-iq-ri,) (which operates on the wave functions centered on A only) and Zj ei exp (iq-rib) (which operates on the wave functions centered on B only). Substituting (2.7) into (2-S), and using q = kf-ki and qdq = (kfki/2?r) dG? 1201, we therefore have q3 (--$)* do(q) = 8n
1 ~ei4(0/.lexP[-ip.(ri,‘jb)li0,)Izdq-
e-8)
The above differential cross section depends on q, the momentum of transfer. To obtain the total cross section one has to perform integration over dq. The right hand side expression becomes vanishingly small [21] and can be
Fig. 1. Schematic diagramshowingthe relationshipof rij to ria, r1%and R. ria and r,-t,are the position vectors from the chaxged particles i andi to the center of mass of moleculesA and B respectively.
212
J.-Y_ Chow C?ziu/Vibrational and rotational energy
transfers
neglected when q is la’rge, i.e., q >-l/a, (a0 is the Bohr radius), the contribution to the cross section mainly comes from small q value. We therefore expand exp (iq-r) in terms of Bessel functionsjl(qr) and Pl(cos 0,) by Rayleigh expansion [22] along q (as z-axis), and then let jr(qr) = (qr)l/(2Z + l)!! Next, we rotate the coordinates twice such that the molecular figure axes of A and B may be used as separate quantization axes for each molecular system. We therefore have [22]
ix,[_iq-(ria_rjb)]
00
co
=,co[go(-#a
(i)‘b(2Za + 1)(2zb + 1) C(21a ’ 1)!!(2zb
‘l)!!]-’
a (2.9)
where C&(@) = [4~/(21+ l)] II2 Ylr,l(Ocp)is the normalized spherical harmonics. (Oi,l~ia) and (Oib~jt,) refer respfctively to the figure axis of rotating molecules A and B. The rotational matrices D~~~a(rra&y~) and D~bnlb(ab&,-y,,) rotate the quantization axis (along q) into those along the molecular figure axes of A and B respeciively. The internal state wave functions [ 191 r& = 99 @ and @f = @?+” of the initial and final states are taken as product wave functions for both molecules. Assuming both A and l5 are m _ the.. electronic C state, the wave function for each molecule is then by Born-_Oppenheimer approximation a product of an e$ctronie wave runction &(r, R), a vibrational wave function @v(R) and a rotational wave function [22] @=,&R) = (ZJ -I-l/&+/2 D&(@$ , where r represents all the electronic coordinates with respect to the figure axis and R is here the inter&clear distance. Hence we have
and similar expressions for @ and @I”_We assume that the electronic wave function remains the same for initial and final states. To obtain the energy transfer cross section for molecules initially at (uaJa) and (u&,) and finally at (u:Jl) and (ubJ{). we substitute (2.9) into (2.8) followed by summing-over the magnetic sublevels MA and MA of the final states and averaging over the magnetic sublevels Ma and Mb of the initial states. After integrating the rotational matrices separately for molecules A and B, and applying the orthogonal reiation [23] of ClebschGordan coefficients (which are generated through the integration of the rotational matrices), we obtained the following expression for the cross section C(J, I, J;; 000)2 C(J,, 1, J;; 000)2
where Qi,(R,)
=
(@e,(raR,) 1Feir;z Cla,O(eia%a)l@e(raRJ)
(2.12)
is the electric 2&-pole moment of the molecule A and (&I Q*a 1u,) is therefore the transition matrix element of the 2%pole moment between vibrational states I& and ua. Similar expressions hold for molecule B. The cross section expressed in (2.11) depends on the momentum transfer, and such q-dependent cross section might become experimentally measurable [24] in the near future. However to obtain estimates of the total cross section, we integrate eq. (2.ll)over the momentum transfer dq, and obtain (22.13) where
L.-Y.
Chow Chiu/Vibmtional
273
and rotational energy transfers
~r,*,=(2z,+1)(221~~1)[(2z,+1)!!(2z~~1)!!l-~(2z,~2~~-2)-~ x C(J,Z, J;; OOO)~c(J,,
z, J;;
~100)~ I@;
1Q,,1yJ2 &J;1Qlb1ub)12
(2.14)
and qmin =
k~-
ki ~ (it -
~r)ltl~i.
(2.15)
The above (2.15) is true when ki = kr (or ki - kr4 ki)_ Thr‘s implies the condition that the energy discrepancy Ae(= ‘i - ef) is much smaller than the kinetic energy of the reIative motion or the condition of near resonance. To estimate @max we recall that da(q) (as expressed in (2.8)) becomes vanishingly small [21] (and can be neglected) for large values of q, and the multipole expansion of the plane wave is valid when qr < 1 or q C 1 /I-_Since the electrons in the outer shell of the molecule are the ones to interact with another molecule causing energy transfer, we let r be the radius of the outer electron shell, namely [25] q,,,ax = 2/d,
(2.16)
where d is approximately the average moiecuI& diameter_ The dependence of the cross section on the d-values chosen will be illustrated in the examples given in the next section. Eq. (2.13) gives the energy transfer cross section for a given initial relative velocity ui_ To compare with the measurements over a distribution of vi, we average (2.13) over the Boltzmann distribution at temperature T_ Since almost all of the experimental cross sections reported are derived from the rate constant k, (cm3 s-l) of the reaction by u = k,/(v), we therefore carry out the corresponding averaging by -
o(T)=
(va(v))/~v)
(2.17)
instead of the straightfonvard averaging (a(u) ). The result from (2.17) will be smaller than’the result from (a(u) 1 by an exact factor l/2. Substituting (2.13) into (2.17), we have (2.18) where 7fazb is given by (2.14) (u)= (8kT/np)lj2 and AE = ei-- efin unit of cm-l is given by (2.2). Eq. (2.18) gives the energy transfer cross section for molecules initially at (u,J,) and (ubJb), and finally at (II!&) and (r&J& Since the measured cross section is often an averaged value over all of the rotational states, we average eq. (2.18) over the initial rotational states (JaJb) and sum it over the final rotational states (Ji Jb) as follows: (2.19) where n,(Ja) and “b(Jb) are respectively the rotational distribution functions of the initial rotational states (J,Jb) at temperature T- In the above summation, only terms which give positive vahre for UJ=J~+J;&( T) (i.e., 17 Iqmin 1) will contribute. If one is interested in the dependence of cross section on the initial rotational ‘%nax states, say J=, we perfom the average over the rotational states Jb only, i.e., (2.20) The cross section for energy transfer as given by (2.13) or (2.17) is a sum over terms of vibrational multipolemultipole transition moments of molecules A and B, 71,~~ as given by eq. (2.14). These transition moments also appear in Sharma and Brau’s [9] semi-classical (impact parameter) treatment. Their transition moments arise from expressing the intermolecular potential, V= ZV ei ei/rr7, in a long-range multipole-multipole expansion. Here the vibrational transition moments are derived through the multipole expansion of the plane waves, exp(-iq-rr,) and exp(iq-rib) of two separate molecular systems. For each multipole interaction the formalism presented here depends upon the minimum and maximum momentum transfer in eq. (2.13) and eq. (2_18). Since
274 Qmin
L.-Y_ Chow Chiu/Vibrationai and rotational energy transfers
z Ae/?iui (eq. 2.15), the second term in eq. (2.18) will be small for Ae = 0 (exact resonance) or for large
to temperature. The treatment of ref. [9] also yields a l/Tdependence at high temperature. While Qmin is completely specified by eq. (2.15), qrnax is not. Since the cross section due to each multipole-multipole transition moment will depend upon 4$$+(b-r)(2/d)*&+‘b-r), t h e cross sections will be extremely sensitive to the value of d used when I, and/or 1, are large. In this case we can treat d as an adjustable parameter whose value should be approximately the molecular diameter. This is different from the procedure used in ref. [9] . There, an impact parameter = b was chosen and straight line trajectories used for b >d and a parabolic interpolation procedure used for b
3. Numerical examples and discussions To illustrate the method, we calculate the cross section of energy transfer for the following reactions at near resonance:
C02(001)+NZ(~=0)~
CO,(OOO) •t Nz(u = 1) + Aeu = 18.6 cm-l
(3-l)
and CO@=1)+CO(u=1)~CO(u=0)+CO(u=2)+Aen=27cm-t
(3.2)
For the homonuclear diatomic molecule Nq, the transition dipole moment is zero and the leading term for CO, + N, reaction will be dipole-quadrupole interaction, namely Za(COa) = 1 and lb&) = 2. The next term, quadrupole-quadrupole interaction, is absent. This is because the vibrational states (001) and (000) of CO2 considered in eq. (3.1) are respectively odd and even upon inversion and the transition quadrupole moment over these states becomes zero. The dipole-quadrupoIe interaction will therefore be the onfy term in eq. (2.18) that survives. As for the CO + CO reaction, we consider both the leading dipole-dipole term, i.e., I, = 1, = 1 and the dipole-quadrupole term, i.e., 1, = 1, lb = 2 and la = 2, &, = 1_ Since the cross section calculated from the dipolequadrupole term is less than 0.6% of that from the dipole-dipole term we are justified in neglecting higher order terms. To compare with the measured cross section, the cross section for energy transfer has been averaged over the initial rotational states and summed over the fmal rotational states as given by eq. (2.19). The summations over the final states Jk and Ji are quickly terminated into a few terms due to the vanishiig of Clebsch-Gordan coefticients. To carry out the average over the initial states J, and Jb, Maxwell-Boltzmann distribution is assumed for the rotational distribution function n(J) as introduced in eq. (2.19). For a homonuclear diatomic molecule, the symmetry factor due to nuclear spin has to be included in n(J). In the case of 14N2, the states with total nuclear spin I = 2 and (f belong to the even rotational states and states of I= 1 belong to the odd rotational states [26] . The rotational distribution for b12 is therefore given by n (Jb) = z;’
3 OLJ~(2Jb + 1) exP I--Jb(Jb
+ l)Y] 9
(3.3)
where
y = h cBb/k T,
(3.5)
‘and e!_rb= 2 for even Jb and CXJb = 1 for odd Jb_ Siilarly for CO,(OOl) we have only odd rotational states and for CO,(OOO) we have only even rotational states [27] to contribute to the sum of the rotational levels. To cakulate the cross section uen fdr the reverse endothermic reaction, not only the initial and final states are reversed, but also the sign of Aeo_
L-Y.
Chow Cliiuf Vibmtional
and rotational
energy transfers
275
TabIe I Molecular parameters used in calculating the energy transfer cross sections Parameter
co2
co
N2
l(OIQ,11,12
1.0 x 10-37
-
1.17x
r(OlQ*l1)l2
-
1.1 X 10ms4 Stat C2 cm2 Cl
1.0
3.57 A
3.15 A
3.4 A
dce’
stit c2 .d
@Ref. 1291. b)Ref. [30]. ‘)Ref. [32]. and van der W&s constants [ 28]_
d)Ref.
a)
[31].
e)The average classical molecular diieterd,
10-3~statc~crr~b)
X lo-s5
Stat C* cm* d)
is derived from viscosity data
As indicated in (2.18), the cross section calculated depends on (q,,)2(fa+k1) = (2/d)2(za+Ib-1) whered is approximately the average molecular diameter. The average diameter considered here is the average of the classical values [Zg] derived from viscosity data and van ddr Waak constants. These average diameters are listed as d, in table 1, where the transition moments I (u’ I Ql I u) I* are also listed_ The transition dipole moments I(0 IQ, I 1) 12 for CO, and CO are, respectively, the measured values of Houghton [29] and of Young and Eachus [30]_ The transition quadmpole moment I(0 I Q2! 1) I2 for CO is the calculated value of Billiigsley and Krauss 1311, and I(0 I Qz I 1) I2 for N, is the calculated value of Cade [32] _ Using the d, and the transition moments listed in table 1, the cross section for both the exothermic and the endothermic reactions of CO + CO (given by eq. (3.2)), have been calculated over the temperature range IOOlOOOK. The experiments of Stephenson and Mosburg [6] on CO f CO were done for the endothermic reaction and their reported cross sections of the exothermic reaction, were obtained by the statistical relation, uex =
uen exp (Af,/kT).
(3.6)
Tf’Kl 2.0’0°0
300 co(v=II+co(v=l~
125 I cx. ~CO(v=01+CO(v=2)+27cm-’ 200
100
1.6
Fig. 2. The dashed line and the solid line are respectivelythe cross sections for the exothermic and endothermic reactions calculated independentlyfrom eq. (2 19). The experimentalexotheimic and endothermic cross sections of ref. [7] are represented here respectively by circles and triangles, and they are related by the statistical relation eq. (3.6).
276
L.-Y. Chow Chiu/lfibmtionaIand rotationalenergy transfers
In fig. 2, the theoretical and the experimental cross sections for both the exothermic and the endothermic reactions are plotted against l/T, and we have good agreement in both cases. Since the theoretical
_--
00
2
3 +x IC+K-’
)
4
5
Fig. 3:The dotted line is calculated by using the molecular parameters listed in table 1; the solid tine is calculated by usingthe adjusted d = 2.82 A and the same transition moments in table 1; the dashed line is calculated by using the experimental transition quadrupole moment for Nz (6 X lo- ss Stat C* cm* [37]) instead and the adjusted d = 2.50 A. The circles with error bars are the Bxperimental points of ref. [4] _
L.-Y.
Chow Chiu/VibrationaZ and rotationaZ energy
transfetS
277
= (2/d)4 by a single d, [36] by applying eq. (3.7) is not as straightforward as the case of approximation of &, two similar molecules (e.g., CO + CO). The same criterion in the choice of d (i.e., letting d = d,) perhaps Cannot be applied here. The transition quadrupole moment of N,, I(0 IQ21 1)d2 = 1.10 X 10ms4 Stat C2 cm4, used in these calculations is the calculated value of Cade [32] _This value differs by a factor of 6 from Nesbet’s [37] calculation (1.7 X 1O-55 Stat C2 crn4) and differs by a factor of 2 from Shapiro and Gush’s [38] experimental value (6 X lo-s5 Stat C2 cm4), which was measured indirectly from the collisionally induced infrared transition. While the dipole-quadrupole term contributes less than 0.6% to the total cross section for CO + CO reaction, the dipole-quadrupole is the leading term for CO, + N, reaction_ The uncertainty in I(0 1Q2 (1) I* is carried directly into the calculated cross section. For this reason we have also used Shapiro and Gush’s experimental i(0 IQ21 1) l2 to calculate the cross sections_ An adjusted value of d = 2.5 A is found in this case. Using this value, the calculated cross sections, shown as a dashed line in fig. 3, also agree closely with the experimental cross sections over the entire temperature range. The adjusted d = 2.5 a and 2.8 8, are all in the vicinity of the classical diameters. In fact, the molecular diameters derived recently from the molecular refraction [39] data are 2.86 A and 2.4 A respectively for CO2 and N2. Since the calculated cross section u due to each multipole-multipole transition moment depends upon q$h$zb-l) e (2/d) 2 ( ‘a+‘b-‘) where we have used (2/d) to approximate qrnax. This d has the meaning of a molecular diameter, and may be approximated by classical diameter, d,. Since qmax =2/d is an approximation to begin with, it is reasonable for us to treat d as an adjustable parameter (rather than identifying it with a fmed particular classical value) especially when D is very sensitive to the d-value. For CO + CO, the dipole--dipole is the leading term to contribute to the total cross section, the latter therefore depends only on &, -(2/d)2. The cross section in this case is not sensitive to the d-value and parametrization on d is not needed. However for the case of CO, + N,, where the cross section is very sensitive to the d-value (having a (2/d)4 dependence), treating d as an adjustable parameter is therefore needed. The adjusted d = 2.82 a found here is indeed in the vicinity of the classical diameter_ The recent distorted wave calculation of Sharma and Picard 1401 also showed that the cross section of CO, + N2 reaction is very sensitive to their diameter d chosen. The semiclassical impact parameter calculations [9,35] for CO2 f N2 and CO + CO also depend on the molecular diameters chosen. In that method, a distanced was assumed such that straight line trajectories used for b > d (b is the impact parameter) and parabolic interpolation procedures used for b
2-M
L.-Y.
Chow ChiulVibmtional
and rotational energy
transfers
Table 2 Cross sections for different values of Peg (1) Reaction: COs(OO1) + Ns(u= 0) = COs(OO0) + Ns(u= 1) + AEO AE~ (cm-*)
0.0 10.0 18.6 =) 20.0 30.0
(oex(T)) x 10” (cm*) 200 K
300 K
1000 K
1.32 1.26 1.23 1.22 0.99
0.89 0.85 0.82 0.82 0.75
0.27 0.26 0.25 0.25 0.24
(2) Reaction: CO(~=~)+CO(~=~)=CO(U=O)+CO(~=~)+ Aeo (cm-t)
AE~
( oex (T) 1 x 10’6 (cm2) 200
0.0 10.0 15.0 20.0 27.0 a) 35.0
K
1.07 0.97 0.91 0.85 0.77 0.72
300 K
1000 K
0.68 0.66 0.63 0.59 0.54 0.50
0.21 0.20 0.20 0.20 0.20 0.18
a) Pee = 18.6 cm-’ and Ae,, = 27.0 cm-r are the actual vibrational energy discrepancies for reactions CO2 + N2 and CO + CO respectively. These values have been used for cross section calculations presented in fig. 2 and fig. 3.
Acknowledgement The author wishes to thank Dr. Frederick Mies for his interest and comments and Drs. Karl Kessler, Richard Deslattes and John Cooper for their efforts to make possible her stay at the National Bureau of Standards (during her sabbatical leaves from Howard University). Dr. John Cooper’s valuable suggestions, interest and critical reading of the-manuscript are gratefully acknowledged.
References [l] [2] [3] [4] [5] [6] [7] [S]
For a review on recent experimental measunzments,please see C.B. Moore, (a) Advan. Chetn Phys. 23 (1973) 41: (b) of Chemical Research 2 (1969) 103. R.N. Schwartz, Z.I. Slawsky and-K-F. Herzfeld, J. Cbem. Phys. 20 (1952) 1591; K.F. Herzfeld, J. Chem. Phys. 47 (1967) 743. For a review on this. please see ref. [la] and also R.L. Taylor and S. Bitterman. Rev. Mod. Phys. 41 (1969) 26. W.A. Rosser, A.D. Wood and E.T. Gerry, J. Chem. Phys. 50 (1969) 4996. C-B. Moore, RE. Wood, B.-L. Hu and J-T. Yardiey, J. Chem. Phys. 46 (1967) 4222. J.C. Stephenson and E.R. Mosburg, Jr., J. Chem. Phys. 60 (1974) 3562; J.C. Stephenson, Appl. Phys. Lett. 22 (1973) 576. J.C. Stephenson, J. Chem. Phys 60 (1974) 4289; J.C. Stephenson and C.B. Moore, J. Chcm. Phys. 56 (1972) 1295. B.H. Mahan, J. Chem. Pbys. 46 (1967) 98. counts
AC-
L.-Y. Citow Chiu/Vibmtional [9] 1101 Ill1 1121 1131 [ 141 Cl51 [ 161 1171 [18] 1191
and rotational
energy transfers
219
R-D. Sharma and CA. Brau, J. Chem. Phys. 50 (1969) 924. R-D. Sharma, J. Chem. Phys. 50 (1969) 919; Phys. Rev. 177 (1969) 102; Phys. Rev. A 2 (1970) 173. T-A. Dillon and J.C. Stephenson, Phys. Rev. A 6 (1972) 1460; J. Chem. Phys. 58 (1973) 2056. R.J. Cross, Jr. and R-G. Gordon, J. Chem. Phys. 45 (1966) 3571. A.M. Arthurs, Proc. Cambridge Phil. Sot. 57 (1961) 904. R.G. Gordon and Y.N. Chiu. J. Chem. Phys. 55 (1971) 1469; Y-N. Chiu. J. Chem. Phys. 55 (1971) 5053. L.-Y. Chow Chiu, J. Chem. Phys. 60 (1974) 2079: Phys. Rev. A 5 (1972) 2055. M-E. Rose, J. Math. and Phys. 37 (1958) 215. C-G. Gray and J. Van Kranendonk, Can. J. Phys. 44 (1966) 2411. L.D. Landau and E.M. Lifshitz, Quantum Mechanics, 2nd Ed. (Addison-Wesley, Reading, Massachusetts, 1965) p. 147. *i and qfhere are the eigenfunctions of the unperturbed Hamiltonian Ho. when the interaction potential V between A and B is zero. After coordinates transformation, Ho can be written as Ho=Hi(a)
+ Hi(b)-
(fi*/2Ma)V%,-
(fi2/2Mb)Vfb,
where Hi(a) and Hi(b) describe respectively the (internal) motions of particles belonging to A and B with respect to the center of mass A and B. The remaining two kinetic energy terms are the kinetic energies of the center of ma= A and of that of B. These two terms can then be transformed into the kinetic energy of the relative motion between A and B, -
and the kinetic energy of the center of mass of the whole system, - [7i2/2(Ma +Mb)]V& (which will be neglected). Both Wi and Wf are therefore a product of the internal state wave function and the wave function for relative motion. [ 201 Ref. [ 181, p. 573. 1211 The cross section is proportional to I(O,(r,R,)
1eXp(-iqq,)
1Qe(ra R,)) (@,(rbRb)
i exp(iq-r,%)
Ioe(rb Rb)) 1’.
where oe(r. R) is the molecular electronic wave function which can be represented by a linear combination of atomic orbitals such as r”-%+Y lm (09) (where c - l/ao)_ The above absolute value square therefore breaks into a linear combination of terms which are approximately proportional to a factor of (q2rm2 + I)-*p (where p > 4). This factor becomes vanishingly small (compared with unity) when 41-t > l(or q > l/no). [Xl M.E. Rose, Elementary Theory of Angular hlomentum Wiley. New York, 1961). 1231 The orthogonal relation used here is c C(J’IJ; MM’
-M’. m. -M)
C(J’I’J; -M’. m. -&f) = 611’(2J f 1)/(21+
I),
which can be easily derived from the relations given by ref. 1221. (241 For recent experimental observations on momentum transfer generated by molecular collisions, see T.W. Meyer and C.K. Rhodes, Phys. Rev. Lett. 32 (1974) 637. [25] See p_ 576 of ref. [ 181 for a parallel treatment in the atomic case. [26] Let P be an operator which interchanges two 14N nuclei and @I is the total nuclear spin wave function, then POI= Gl)‘@~ follows. When P applies to the total wave function @ = @epeo,@JeI, we therefore have P@ = (-l)J’f+ = Q and consequently (-l)J+I = 1. &cc the vibntional states (001)
and (000) of CO, are’respectivcly odd and even upon inversion of the nuclear coordinates we therefore have (-l)J= 1 for COz(OO0) and (-l)J+l = 1 forCOz(001). [28] (a) J.O. Hirschfelder, CF. Curtis and R.B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954) p. IllO1111; (b) W.J. Moore. Physical Chemistry, 4th Ed. (Prentice-Hall, New Jersey, 1972) p. 157. 1291 J.T. Houghton, Proc. Phys. Sot. (London) 91 (1967) 439. [30] L.A. Young and W.J. Eachus, J. Chem. Phys. 44 (1966) 4195. [31] F.P. Billing&y and hf. Krauss, J. Chem. Phys. 60 (1974) 2767. [32] P. Cad& value of I(0 IQz 11) 1’ for Nt is quoted in ref. [S] as the private communication. 1331 H.T. Powell, J. Chem. Phys. 59 (1973) 4937. [34] P.B. Sackett, A. Horduik and H. Schlossberg, Appl. Phys. Lert. 22 (1973) 367. [35] R.D. Sharma, Chem. Phys. Lett. 30 (1975) 261. [36] By (3.7), we have (2/~f~)~ = (2/d,) (2/db)3. Where da and db refer respectively to values of CO2 and Nz in table 1. [37] R.K- Nesbet, J. Chem. Phys. 40 (1964) 3619. 1381 MM. Shapiro and H.P. Gush, Can. J. Phys. 44 (1966) 949. (391 M. Itrplus and R.N. Porter, Atoms and Molecules 0V.G Benjamin, New York, 1970) p. 255 and ref. [28(b)]. [40] R.D. Sharma and R.H. Picard, J. Chem. Phys. 62 (1975) 3340. [27]