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Temperature dependence of vibration—vibration energy exchange probabilities among highly excited levels: CO(υ-1) + CO(1) → CO(υ) + CO(0)
Volume 64, number 1
TEMPEMTURE
CHEMICAL
PHYSICS
LETTERS
DEPENDENCE OF VIBRATION-VIBRATION
PROBXBILITIES AMONG HIGHLY EXCITED LEVELS: CO(u-1)
15
June 1979
ENERGY EXCHANGE + CO(I) --f CO(u) + CO(O)
Received 5 March 1979
The temperature dependence of onequantum V - V mcr~) exchange probxbdiries imohing highly excited CO molrI CO(l) - CO(u) f CO(O) has been investigated. I‘or u = 8, the probabdity increases by an order of magnitude orer the temper&ture r.mse 100-3000 I;; for a = 15, the increase is nearly tno crders of mzgnitude. The dependence of energy ewh.mge probabilities on u is not stron,.0 The contribution of moleculx attraction dominates that of recules m CO(u-1)
pulsion in the process particularly
at Ioxxer temperatures.
l_ Introduction in this letter we shall report the temperature dependence of vibration-vibration (VV) energy exchange prob3bilities smong highly excited Iekels of CO molecules, which is an import3nt problem in underst3nding the performmce of CO 13sers. Many papers have been published on CO chemical lasers [ !] , but the temperature dependence of wbratiomd energy exchange probabilities involving highly excited levels of CO h3ve received little sttention. We shall 3ppIy the model of short-range interaction, which h3s 3lready been shown to produce srttisfxtory results for the energy exchange processes involving highly excited vibntional levels [2] , to the one-quantum process CO(u-I) + CO(l) -+ CO(u) + CO(O). where u = S-15 over the temperature rangeof 100 to3000 K_
2. Interaction potential energies We shall express the overall interaction the Morse type:
potentid
energy as the sum of four atom--atom
interaction
temx
of
(1) where rl ,z ~r-q~(d~s1)cosOl~~~,l(d+“1)Cos8~,r;,4 =:r~l(d+x1)cosOl r9~~~(J~“~)COSe~,8lte~) is the angle between the direction of the intenolecul~r distance I- nnd the axis of the first (second) molecule, D and a are the depth of the attractive potential well and the range parameter for the collision system, 41-Z = ?r~~,~/ (“rc f ?rzo), and d is the equilibrium bond distance of CO. The deviation of the bond distance from its equilibrium value d is denoted by xi_ The distances Ti~'Sare the equiIibrium values of ri’s_ which will have to be dekermined in the calculation of energy exchange probabilities_ Here, note that I-~ is the oxygen-oxygen distance, ~4 distances. When the atom-atom disis the carbon-carbon distance, and r3 4 are the remaining oxygen-carbon tances are introduced, eq. (1) become; 3n orientation dependent function_ V(I(~,8, ,e2, x1, x2), which can be *Theoretical Chemistry Group Contribution
No. 1092.
21
VoIume
1.5June 1979
PHYSICS LE-J-JERS
CHEWSTRY
64. number I
avenged -as
When ihe integration is performed, the contribution of each atom--atom interaction term contains the factor exp[(rfe - r)/~] or exp[(ri, - r)/%] _ To estimate the atom-atom equihbrium distance rr-ein this factor, we first equate the equilibrium distance re between the centers of mass of the colliding molecules to that of the LennardJones potential_ Since the Lennard-Jones parameter o is 3-70 a [3], re = 4.16 A_ The orientation avemge introduced in the formulation of Y(r, x1. x2) is based on the model of the rapid rotation of colliding molecules. Then, each moIecuIe can be viewed as two concentric spheres, the outer formed by the rotation of the carbon atom around the center of mass of CO and the inner by the oxygen atom. Thus, the sum of the interactions at the shortest distance between two spheres will determine the overall intemction energy_ When r = re, rIe is the nearest distsnce between two inner spheres, re - ?qzd = 3-19 a, r4e is between two outer spheres, i.e., re - 3q1 d = 2.87 A, and the distance between the outer sphere of molecule I and the inner of moIecuIe 2 is r2e = rse = re q1 d - qzd = 3-03 A_ Therefore. the factors exp [(rfe - re)/a] appearing in exp [(ri, - r)/a] = exp ](F, - re)/n ] - r&z] = exp[& - r&r] = exp(-1.13/a) X exp [(fe - Mr] are exp ](rJe - r-,)/a] = esp (-0.964/a), exp[&, and exp [(r4e - r&z] = exp (- l-29/ CI), w here CIis in ii_ Similar reMions will be obtained for the attractive part, i-e_, exp[(ri, - re)/%] _ With the introduction of these relations, the orientation Etvemged function crm be given as Y(G xJ, x2) = D (A exp [(re - r)/~] - 2B exp [(re - r)/2n] ] +D {A’exp
[Ire - r)la]
- B’exp [fre - r)/%]
)x1 x2 E V(r) + If’@. x1.x,),
(3)
\ihere -4 = 42;
1
exp[(rJ,
; reMI sinh’(Q,)
+ (Q, Q2)-’
iexp I(rze -
rePal+ e.up[(rze- r,)/al hinh RI1) sin!1(Q,>
f 42;” w.W.., - r,)/a] sinh2(Ql), A = Q;'
exp[(rJe - r,)la]
+XQ, Q,)-’
eNk,e
f (it cr?) cash (Q,) i- Q,’
[-2(rlla/d)
-
sinh(Q2) cosh(Q2) + (a/d)’ si&Q,)
r,)/al l-k1 a/d) sinh(Q?)
cash (Q2) + (Q/d)’ sinh (0,)
exp](r4c - re)!o] [-2(qta/~)sinIl(Ql)cosh
cash (QJ) - (9&d)
f s; cosh2(Q2)1 sinh (Q,)
cash (Q,)
sinh (Q2)1 (QJ) f (a/d)‘sir&Qt)
tqf
cosb’(Qt)]
_
Here B is the =me ;is A with Q replaced by ?a, and S’ is the same as 34’ with EIalso repIaced by 2~; QJ = qt&rand Q2 = q,d/tz. With the introduction ofexp[(ri, - r,)/a] and exp[(ri, - r,)/&] in the ith term of V(r) and V’(r.xt. _K?). we have included the contribution of each atom--atom interaction to the over& interaction_ If these factors are not included, the result wouid impIy the equal contribution of each atom-atom interaction. in which case the mtroduction of a factor of I/4 may be needed [4] _
3_ Energy exchange
probabifity
The wave equation for the oscillator state vector 19) describing the system of two interacting oscillators can be written in the standard form
22
Volume
64, number
I
CHEXICAL
PHYSICS
LETTERS
15June1979
where Ho is the hamiltonian of the unperturbed system and V’(r) represents V’[r(~),xI,x~] defined in eq. (3), where the relative coordinate r is parametrized in time through the solution of the equation of motion_ Eq_ (4) can be converted into iti
aim/at=ii(r)w>,
(3
where Z(f) = exp(iHOr/li)V’(r)
exp (-iHof/ti)
(6)
and the new state vector I+>>,which provides a complete description of the dynamics throughout the collision, is defined as IW = exp (-if$‘r/fi)lW_ In terms of the ladder operators u! and uj, the perturbation potential gnen in eq. (3) can be written as V’(r) = (fi/zQ’) (MI M,OI w7_)- U’ {A’ exp ( [re - r(t)] /a) - B’ exp ( [re - r(r)] /%)}(a; =F(f)(a;
++a1
+ a1 ) (a: f a2)
+a&
(7)
where M and o are the reduced mass and the angular frequency of the molecule CO, respectively. Note that while AZ1 and M2 are identical here, the frequencies w1 and o2 are different; OI is the transition angular frequency for i -1” of molecule 1 and w ? is for Ii --LX-’ of molecule 2. We now note that of the operators appearing in the product (aI -WI) (u: +a?) = aii< tnja2 + nl aI+ QI Q?, we need to include only =i a, and nl u; in the present onequantum exchangeprocess. Therefore, the solution of eq. (5) can be written in the f&m I@(r))=Texp
-+J C
F(r’){exp[i(wl
-00
The disentang!ing
- wz) r’] ui a2 + exp [-i(o
of this expression yields the following approximate of VV processes in the present system I@(E)> = exp
X =
[
-i
exp [--$j-i
j
F(8)
exp [i(wt --w,)t’]
s-0D
F(r’)F(t”)
exp[i(wl
df’n t d i]expb)
-~~)f’]
I - 0’)
1
expression,
_a
fir’)
exp[-i(wl--o?)t”]
(8)
t’] a1 ~$1 dt’ IcD(--)>.
which is suitable for the study
dt’dt”(ula~
1 1_
i dt*u;a?
exp [-i(wr-m2)t’]
-L&)
I@(---)>
U(t)1 jk),
(9)
where I+(-+ represents the initial state of the colhsion, I/X-)_and U(t) contains ah possible information about the dynamical processes going on between t = -00 and t. Hence, when @(---) is specified, we can obtain a unique solution for g(r) at all subsequent times, and in particular for t -+ +oo_Then, the VV energy exchange probability forjk +I%’ can be expressed in the simple form P/i?
= lim [<~%‘lU(r)ijk)] ‘, r--co
(10)
which becomes
(11) for/
>j
and~2 > k. k’ being the least of the four quantum numbers, where
~(+=q=
[~Q~M(w,w,)‘~‘]-’ $ _B
{A’exp
( [re--tit)]
/a) -
B’exp
( [r,-r(r)]
/3lr)hp
[i(wl--02)fl
df-
(12) 23
Volume 6-2,number i
15 June 1979
CHEhIICAL PHYSICS LETTERS
For the specific process under consideration, CO(u - 1) + CO( 1) * CO(u) + CO(O), k’ = 0, thus the wsum reduces to the single-term expression,j!= IO!_ When &-kllG(-)i’ is significantly small compared with unity, an approximate expression of the energy exchange probability is P,“r9 t =r ulG(-)12. For V(r) given above the classical path of the relative tr&IationaI motion can be determined from the equntion (13 where b is the impact par.uneter and r* is the largest root of the denominator_ In introducing the impact parameter, we assumed &at energy exchange is very smtt!i for r > T,, which is an appropriate treatment for the collisions with short-range interaction_ This equation can be readily integrated to obtain the trajectory exp[(rc - r)/Za] = (E*/~0)I’3cosQ~cosI~[(E*/2~)I’2(t/n)]
- sin +}-I,
(14)
where E” = E( 1 - b’/rz) and 0 = arc tan[B(D/AE)*IZ] _ Then the quantity G(m) contains the integrals of the w2) (2p/.Ea)*lz_ In evaluating the integral, type 12 (cash .$- sin o))-” ee’rc d$. where II = 1 or 2 rind JJZ = a(wl the p&s will have to be dispked in the upper-half circle because of the presence of the term sin 0, which represents the contribution of molecular attraction to the collision trajectory_ The contour integration is stmightforward I fG(-N;.,
= [%&J~
- wz) A’/MUA]
+ (B/A - B’/.4’)[a(w1 wiicref(k*)
- aa)]
z
~COSIl fj1E”)
-‘(AD/2~)‘&inh
= 7ia(wl - ~1) (p/2l?)1/1 and&E*) exchange probability can then be defined as
[
1 + g(P)]
{j&k*)
1
[ I i-g(b*)]
= 4~ arc tan [B(D/Ak -* )
)] ’ csch’
[Zf(p)]
,
(13
Ii?-1 _The thermal average oi the energy
(16) where the energy E appearing in the probability I~.xsbeen replaced by the “symmetrized” energy [5] E = a f[f? + fi(Wl - wz)] IE + E4r/‘}‘7 to account for the inelasticity of the collision process_ Note that the upper sign here is for exothermic processes and the lower for endothermic processes_ The integrand in eq. (16) is a complicated fun&ion of E and 6, but it can be readily handled on a computer_ The result for the exothermic VV process CO(v - I) + CO( 1) - CO(u) + CO(O), i.e_,i = u - 1, k = 1,I” = u and k’ = 0, is presented in the following section_
-Z_CaIcuIation and discussion Potential constants needed in the calculation we o, = 2170.21, w~_Y~ = 13.46 and wgc = 0.0308 cm-t [6] _ The energy Ievets are then aladated from G(u) = w,(u + $) - arxe(u f $)z + a,_-‘;(~ + 4)3_ Other values needed are d = 1.I25 z%[6] md the Lennrtrd-Jones constant u = 3.70 A and D = I IOk 131. The range parameter Q obtained from vibration-trzmslation rehxation data [7] is taken to be O.lSS ~1 [‘?I _ The determination of atomatom equiIibrium dist,mces has already been discussed above. Kumericai results of the thermal-avenge probability zre compared with experimental data IS] zt 100 M in fig_ l_ It is clear from the comp;lr%on that the use of the exponential potential is not satisfactory for the present system beIow u = 7, where the energy mismatch becomes smail; e-g_, for u = 7, AE = wI - wz = 157 cm-t_ For u > 7, on the other hand, the agreement appears to be satisfzctorv, although rhe extrapolation of experimental data to u > 10 appears to predict a somewhat stronger dependence of energy exchange probabilities on u than the calculated result_ In fact, in an earlier work Jeffers and
Volume 64, number 1
CHEMICAL
PHYSICS
15 June 1979
LE?TERS
id
------T I------
1 i
T”K Fig_ I_ Dcpendencc of energy e\Lhangc probabilities of CO(u - 1) + CO(l) -CO(v) + CO(O) on the vibrational qu.mfwn number u. Exprrinxntd data are taken from ref. [ 8 I_
rig. 1. Temperature dependence of energy c\chaxSc probabilities of CO(u - 1) + CO(l) - CO(v) + CO(O) fcr u = S-15_
Kelley [2] have noted that for large u the VV process can be explained in terms of short-range interactions, while for small u by long-ranSe dipole-dipole interactions [91_ The result for u > 7 shown in fig. 1 supports this earlier report. The present approach is based on short-rsnSe interaction 3s shown in eq. (3) and it can produce reliable vaIues of energy exchange probabikties for VV processes involving higher eilergy Ievels_ We shall now discuss the temperature dependence of energy eschange probabilities for u = S-15, for which the present approach appears to give satisfktory results. The temperature dependence of exothermic VV energy e.uchange probabdities determined over the range of IO0 to 3000 K is displayed in fig_ 2_ The figure shows that the dependence is in general signifkmr particularly for smaller values of u; for u = 8 it chsnges from 2.2 X IO-’ at 100 K to 3.1 X 10m7 Jt 3000 M, while the corresponding change for u = 15 is from 4.5 X IOm4 to 3.7 X 1O-1. However, an important resuit is that at a @en temperature nU probabilities of different u appear to merge above 1000 K. Based on this result we crtn predict energy exchange probabilities for u > I5 at such temperatures to be comparable to the corresponding values shown in fig_ 2; the estimation can be extended even to u < S at such temperatures. The probabilities at 1500, 2000 and 3000 K are appro.xi.mateiy 0_015,0_0?2 and O-035, respectively_ Note that above 2000 K the u = S probability becomes the smallest and the u = I5 probability the largest. but the difference is nor signitkmt. For the pureIy repulsive interaction, i.e., E’(r) = DA exp[(r, - r)la] and V’(r.xl.x,) = 0.4’ exp[(r,-r)fz].ulxu?. the quantity iG(-)I’ is simply [2rr&ol - ~2) A’/MwA ] 2 co& ff(E*l] cscha iZf(E*y] _Comparing thermal werage nrobabiiities obtained for this expression wit!1 the above result, we can detemline the effect of molecular attraction on VV energy exchange processes_ The appearance of g(E*) in eq. (I 5) is due to the contribution of molecular attraction to the collision trajectory, and it makes a large contribution to the overall value of PjF(7) prtrticularIy for Inrge u, where the productf(E*)g(E*) is large_ For example, for u = 10, the contribution of molecular attraction increases the energy exchange probability by factors of 34,7-I, 4.3,2_7, I .9 and 1.7 at 100,300, 500, 1000,200O and 3000 K, respectiveIy. For u = IS, the corresponding factors are 5 I, S.S,4.7.2.S, 1.9 and I-7. This comparison shows the major roIe of molecular attraction in the present energy exchange process; in particuisr the role of molecular attraction completely dominates the contribution of repulsive interaction at iow 25
VoIume
64, number
1
CHEMICAL
PHYSICS
15 June 1979
LETTERS
temperatures (< 300 K) such that the overall process can be describgd in terms of the attmctive interaction alone_ Although the expression derived above does not app!y to the collisions involving lower energy levels, the extension of the mode! shows that for u= 2, the attractive interaction increases tire probability by 12,5-I, 3.6,2.5, 19 and l-7 at !00,300,500, !000,2000 and 3000 K, respectively_ Another interesting result is that for different u, t!re contribution of molecular attraction is to increase the probability by the same factor at Irig!1 temperatures; e-g_, the factor is ! 9 at 2000 K and is 1.7 at 3000 The present approach uses t!re intemction potentia! which is obtained averaging over molecular orientations, zrndintroduces the contribution of nonzero-impact parameter collisions. Therefore, the comparison of this resu!t with that of collinear co!hsions wi!! show the importrmce of molecular orientations and noncollinear collisions_ For the co!Iinear arrsngement C-O + C-O, assuming the nearest 0-C intemction is of on!y importance, we obtain WY.X!, x2) = DCexF[(r,-r)/al whkh
exp((ql
xl
* q2x2)/a]
-
2exp
f(~,--W~~l
exp kl-~1
f 9&/~11,
gives
V(r) = D Cexp [(re -
r)/~]
-
2
exp [(r, - I-)/2~] )
and
!% When
“! Px2) = D {exP [(r, - r)/Q] - $ exp [(r, - r)/2Q] } (‘I! q,/Q’) X! X2.
the probability P/z’ obtained for t!ris interaction is averaged as 00
exp
(-E/kT)d(E/,W),
0 the result is found to be significantly larger than the above result obtained from eq_ (!6)_ For example, for u = IO, the probability of the collinear mode! takes the values of 3.5 X !Om3, 1-O X !O-3- and ! .8 X IO-? at 100, 300 ad 500 K, respectively, w!rich are larger than the corresponding va!ues obtained from eq_ (I 6) by :! factor of 2.7.3.3 and 3_6_ (A!thoug!r the present approach does not apply to small u, t!re calculation reveals the factors of 25,29 and 32 for u = 2.) Therefore, the present approach predicts that the “steric” factor due to the inclusion of the avemge effect of molecuiar orientations and nonzero impact parameter collisions is somewllat close
to I/S for the diatom-diatom
system rather t!xm I/9 which has been estimated in this type of collisions [3-JO] _
EZeferences [I ! I_W_St_ Smith, in: MoIecuhr energy transfer, eds_ R-D_ Levine and J_ Jortner (Wiley. New York. 1976) pp_ 91-98. [7] W-Q. Jeffers and J.D. Relley. J. Chem. Thys. 55 (1971) 4433. [3] J-O_ Hirschf&der, C-C. Curtiss and R-B_ Bird. Molecular theory of gases and liquids (Wiley. New York, 1964) pp_ 11 IO[4]
H.K. Shin, J. Chem. Phys. 60 (1974)
1064.
T-L. CottrelI and J.C.McCoubrey. Molecular energy transfer in gases (Butterworths. London, 161 G_ Herzberg_ Spectra of diatomic moIecuIes (Van Nostrand. Princeton, 1967) table 39. [S]
[7] R-C_ WIIikan, J_ Chem_ Phys_ 38 (1963) 2855_ 181 C_ Wittig and I_W_M_ Smith, Chem_ Phys. Letters 16 (1972)
f9j R-D. Shxma, [IO] K-F_ Heafeld
26
Phys. Rev. 177 (1969)
1961) p_ 134.
292_
2855.
and T-A. Litovitz, Absorption
and dispersion of ultrasonic waves (Academic
Press, New Ycrk,
1959) p- 278.
TEMPEMTURE
CHEMICAL
PHYSICS
LETTERS
DEPENDENCE OF VIBRATION-VIBRATION
PROBXBILITIES AMONG HIGHLY EXCITED LEVELS: CO(u-1)
15
June 1979
ENERGY EXCHANGE + CO(I) --f CO(u) + CO(O)
Received 5 March 1979
The temperature dependence of onequantum V - V mcr~) exchange probxbdiries imohing highly excited CO molrI CO(l) - CO(u) f CO(O) has been investigated. I‘or u = 8, the probabdity increases by an order of magnitude orer the temper&ture r.mse 100-3000 I;; for a = 15, the increase is nearly tno crders of mzgnitude. The dependence of energy ewh.mge probabilities on u is not stron,.0 The contribution of moleculx attraction dominates that of recules m CO(u-1)
pulsion in the process particularly
at Ioxxer temperatures.
l_ Introduction in this letter we shall report the temperature dependence of vibration-vibration (VV) energy exchange prob3bilities smong highly excited Iekels of CO molecules, which is an import3nt problem in underst3nding the performmce of CO 13sers. Many papers have been published on CO chemical lasers [ !] , but the temperature dependence of wbratiomd energy exchange probabilities involving highly excited levels of CO h3ve received little sttention. We shall 3ppIy the model of short-range interaction, which h3s 3lready been shown to produce srttisfxtory results for the energy exchange processes involving highly excited vibntional levels [2] , to the one-quantum process CO(u-I) + CO(l) -+ CO(u) + CO(O). where u = S-15 over the temperature rangeof 100 to3000 K_
2. Interaction potential energies We shall express the overall interaction the Morse type:
potentid
energy as the sum of four atom--atom
interaction
temx
of
(1) where rl ,z ~r-q~(d~s1)cosOl~~~,l(d+“1)Cos8~,r;,4 =:r~l(d+x1)cosOl r9~~~(J~“~)COSe~,8lte~) is the angle between the direction of the intenolecul~r distance I- nnd the axis of the first (second) molecule, D and a are the depth of the attractive potential well and the range parameter for the collision system, 41-Z = ?r~~,~/ (“rc f ?rzo), and d is the equilibrium bond distance of CO. The deviation of the bond distance from its equilibrium value d is denoted by xi_ The distances Ti~'Sare the equiIibrium values of ri’s_ which will have to be dekermined in the calculation of energy exchange probabilities_ Here, note that I-~ is the oxygen-oxygen distance, ~4 distances. When the atom-atom disis the carbon-carbon distance, and r3 4 are the remaining oxygen-carbon tances are introduced, eq. (1) become; 3n orientation dependent function_ V(I(~,8, ,e2, x1, x2), which can be *Theoretical Chemistry Group Contribution
No. 1092.
21
VoIume
1.5June 1979
PHYSICS LE-J-JERS
CHEWSTRY
64. number I
avenged -as
When ihe integration is performed, the contribution of each atom--atom interaction term contains the factor exp[(rfe - r)/~] or exp[(ri, - r)/%] _ To estimate the atom-atom equihbrium distance rr-ein this factor, we first equate the equilibrium distance re between the centers of mass of the colliding molecules to that of the LennardJones potential_ Since the Lennard-Jones parameter o is 3-70 a [3], re = 4.16 A_ The orientation avemge introduced in the formulation of Y(r, x1. x2) is based on the model of the rapid rotation of colliding molecules. Then, each moIecuIe can be viewed as two concentric spheres, the outer formed by the rotation of the carbon atom around the center of mass of CO and the inner by the oxygen atom. Thus, the sum of the interactions at the shortest distance between two spheres will determine the overall intemction energy_ When r = re, rIe is the nearest distsnce between two inner spheres, re - ?qzd = 3-19 a, r4e is between two outer spheres, i.e., re - 3q1 d = 2.87 A, and the distance between the outer sphere of molecule I and the inner of moIecuIe 2 is r2e = rse = re q1 d - qzd = 3-03 A_ Therefore. the factors exp [(rfe - re)/a] appearing in exp [(ri, - r)/a] = exp ](F, - re)/n ] - r&z] = exp[& - r&r] = exp(-1.13/a) X exp [(fe - Mr] are exp ](rJe - r-,)/a] = esp (-0.964/a), exp[&, and exp [(r4e - r&z] = exp (- l-29/ CI), w here CIis in ii_ Similar reMions will be obtained for the attractive part, i-e_, exp[(ri, - re)/%] _ With the introduction of these relations, the orientation Etvemged function crm be given as Y(G xJ, x2) = D (A exp [(re - r)/~] - 2B exp [(re - r)/2n] ] +D {A’exp
[Ire - r)la]
- B’exp [fre - r)/%]
)x1 x2 E V(r) + If’@. x1.x,),
(3)
\ihere -4 = 42;
1
exp[(rJ,
; reMI sinh’(Q,)
+ (Q, Q2)-’
iexp I(rze -
rePal+ e.up[(rze- r,)/al hinh RI1) sin!1(Q,>
f 42;” w.W.., - r,)/a] sinh2(Ql), A = Q;'
exp[(rJe - r,)la]
+XQ, Q,)-’
eNk,e
f (it cr?) cash (Q,) i- Q,’
[-2(rlla/d)
-
sinh(Q2) cosh(Q2) + (a/d)’ si&Q,)
r,)/al l-k1 a/d) sinh(Q?)
cash (Q2) + (Q/d)’ sinh (0,)
exp](r4c - re)!o] [-2(qta/~)sinIl(Ql)cosh
cash (QJ) - (9&d)
f s; cosh2(Q2)1 sinh (Q,)
cash (Q,)
sinh (Q2)1 (QJ) f (a/d)‘sir&Qt)
tqf
cosb’(Qt)]
_
Here B is the =me ;is A with Q replaced by ?a, and S’ is the same as 34’ with EIalso repIaced by 2~; QJ = qt&rand Q2 = q,d/tz. With the introduction ofexp[(ri, - r,)/a] and exp[(ri, - r,)/&] in the ith term of V(r) and V’(r.xt. _K?). we have included the contribution of each atom--atom interaction to the over& interaction_ If these factors are not included, the result wouid impIy the equal contribution of each atom-atom interaction. in which case the mtroduction of a factor of I/4 may be needed [4] _
3_ Energy exchange
probabifity
The wave equation for the oscillator state vector 19) describing the system of two interacting oscillators can be written in the standard form
22
Volume
64, number
I
CHEXICAL
PHYSICS
LETTERS
15June1979
where Ho is the hamiltonian of the unperturbed system and V’(r) represents V’[r(~),xI,x~] defined in eq. (3), where the relative coordinate r is parametrized in time through the solution of the equation of motion_ Eq_ (4) can be converted into iti
aim/at=ii(r)w>,
(3
where Z(f) = exp(iHOr/li)V’(r)
exp (-iHof/ti)
(6)
and the new state vector I+>>,which provides a complete description of the dynamics throughout the collision, is defined as IW = exp (-if$‘r/fi)lW_ In terms of the ladder operators u! and uj, the perturbation potential gnen in eq. (3) can be written as V’(r) = (fi/zQ’) (MI M,OI w7_)- U’ {A’ exp ( [re - r(t)] /a) - B’ exp ( [re - r(r)] /%)}(a; =F(f)(a;
++a1
+ a1 ) (a: f a2)
+a&
(7)
where M and o are the reduced mass and the angular frequency of the molecule CO, respectively. Note that while AZ1 and M2 are identical here, the frequencies w1 and o2 are different; OI is the transition angular frequency for i -1” of molecule 1 and w ? is for Ii --LX-’ of molecule 2. We now note that of the operators appearing in the product (aI -WI) (u: +a?) = aii< tnja2 + nl aI+ QI Q?, we need to include only =i a, and nl u; in the present onequantum exchangeprocess. Therefore, the solution of eq. (5) can be written in the f&m I@(r))=Texp
-+J C
F(r’){exp[i(wl
-00
The disentang!ing
- wz) r’] ui a2 + exp [-i(o
of this expression yields the following approximate of VV processes in the present system I@(E)> = exp
X =
[
-i
exp [--$j-i
j
F(8)
exp [i(wt --w,)t’]
s-0D
F(r’)F(t”)
exp[i(wl
df’n t d i]expb)
-~~)f’]
I - 0’)
1
expression,
_a
fir’)
exp[-i(wl--o?)t”]
(8)
t’] a1 ~$1 dt’ IcD(--)>.
which is suitable for the study
dt’dt”(ula~
1 1_
i dt*u;a?
exp [-i(wr-m2)t’]
-L&)
I@(---)>
U(t)1 jk),
(9)
where I+(-+ represents the initial state of the colhsion, I/X-)_and U(t) contains ah possible information about the dynamical processes going on between t = -00 and t. Hence, when @(---) is specified, we can obtain a unique solution for g(r) at all subsequent times, and in particular for t -+ +oo_Then, the VV energy exchange probability forjk +I%’ can be expressed in the simple form P/i?
= lim [<~%‘lU(r)ijk)] ‘, r--co
(10)
which becomes
(11) for/
>j
and~2 > k. k’ being the least of the four quantum numbers, where
~(+=q=
[~Q~M(w,w,)‘~‘]-’ $ _B
{A’exp
( [re--tit)]
/a) -
B’exp
( [r,-r(r)]
/3lr)hp
[i(wl--02)fl
df-
(12) 23
Volume 6-2,number i
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CHEhIICAL PHYSICS LETTERS
For the specific process under consideration, CO(u - 1) + CO( 1) * CO(u) + CO(O), k’ = 0, thus the wsum reduces to the single-term expression,j!= IO!_ When &-kllG(-)i’ is significantly small compared with unity, an approximate expression of the energy exchange probability is P,“r9 t =r ulG(-)12. For V(r) given above the classical path of the relative tr&IationaI motion can be determined from the equntion (13 where b is the impact par.uneter and r* is the largest root of the denominator_ In introducing the impact parameter, we assumed &at energy exchange is very smtt!i for r > T,, which is an appropriate treatment for the collisions with short-range interaction_ This equation can be readily integrated to obtain the trajectory exp[(rc - r)/Za] = (E*/~0)I’3cosQ~cosI~[(E*/2~)I’2(t/n)]
- sin +}-I,
(14)
where E” = E( 1 - b’/rz) and 0 = arc tan[B(D/AE)*IZ] _ Then the quantity G(m) contains the integrals of the w2) (2p/.Ea)*lz_ In evaluating the integral, type 12 (cash .$- sin o))-” ee’rc d$. where II = 1 or 2 rind JJZ = a(wl the p&s will have to be dispked in the upper-half circle because of the presence of the term sin 0, which represents the contribution of molecular attraction to the collision trajectory_ The contour integration is stmightforward I fG(-N;.,
= [%&J~
- wz) A’/MUA]
+ (B/A - B’/.4’)[a(w1 wiicref(k*)
- aa)]
z
~COSIl fj1E”)
-‘(AD/2~)‘&inh
= 7ia(wl - ~1) (p/2l?)1/1 and&E*) exchange probability can then be defined as
[
1 + g(P)]
{j&k*)
1
[ I i-g(b*)]
= 4~ arc tan [B(D/Ak -* )
)] ’ csch’
[Zf(p)]
,
(13
Ii?-1 _The thermal average oi the energy
(16) where the energy E appearing in the probability I~.xsbeen replaced by the “symmetrized” energy [5] E = a f[f? + fi(Wl - wz)] IE + E4r/‘}‘7 to account for the inelasticity of the collision process_ Note that the upper sign here is for exothermic processes and the lower for endothermic processes_ The integrand in eq. (16) is a complicated fun&ion of E and 6, but it can be readily handled on a computer_ The result for the exothermic VV process CO(v - I) + CO( 1) - CO(u) + CO(O), i.e_,i = u - 1, k = 1,I” = u and k’ = 0, is presented in the following section_
-Z_CaIcuIation and discussion Potential constants needed in the calculation we o, = 2170.21, w~_Y~ = 13.46 and wgc = 0.0308 cm-t [6] _ The energy Ievets are then aladated from G(u) = w,(u + $) - arxe(u f $)z + a,_-‘;(~ + 4)3_ Other values needed are d = 1.I25 z%[6] md the Lennrtrd-Jones constant u = 3.70 A and D = I IOk 131. The range parameter Q obtained from vibration-trzmslation rehxation data [7] is taken to be O.lSS ~1 [‘?I _ The determination of atomatom equiIibrium dist,mces has already been discussed above. Kumericai results of the thermal-avenge probability zre compared with experimental data IS] zt 100 M in fig_ l_ It is clear from the comp;lr%on that the use of the exponential potential is not satisfactory for the present system beIow u = 7, where the energy mismatch becomes smail; e-g_, for u = 7, AE = wI - wz = 157 cm-t_ For u > 7, on the other hand, the agreement appears to be satisfzctorv, although rhe extrapolation of experimental data to u > 10 appears to predict a somewhat stronger dependence of energy exchange probabilities on u than the calculated result_ In fact, in an earlier work Jeffers and
Volume 64, number 1
CHEMICAL
PHYSICS
15 June 1979
LE?TERS
id
------T I------
1 i
T”K Fig_ I_ Dcpendencc of energy e\Lhangc probabilities of CO(u - 1) + CO(l) -CO(v) + CO(O) on the vibrational qu.mfwn number u. Exprrinxntd data are taken from ref. [ 8 I_
rig. 1. Temperature dependence of energy c\chaxSc probabilities of CO(u - 1) + CO(l) - CO(v) + CO(O) fcr u = S-15_
Kelley [2] have noted that for large u the VV process can be explained in terms of short-range interactions, while for small u by long-ranSe dipole-dipole interactions [91_ The result for u > 7 shown in fig. 1 supports this earlier report. The present approach is based on short-rsnSe interaction 3s shown in eq. (3) and it can produce reliable vaIues of energy exchange probabikties for VV processes involving higher eilergy Ievels_ We shall now discuss the temperature dependence of energy eschange probabilities for u = S-15, for which the present approach appears to give satisfktory results. The temperature dependence of exothermic VV energy e.uchange probabdities determined over the range of IO0 to 3000 K is displayed in fig_ 2_ The figure shows that the dependence is in general signifkmr particularly for smaller values of u; for u = 8 it chsnges from 2.2 X IO-’ at 100 K to 3.1 X 10m7 Jt 3000 M, while the corresponding change for u = 15 is from 4.5 X IOm4 to 3.7 X 1O-1. However, an important resuit is that at a @en temperature nU probabilities of different u appear to merge above 1000 K. Based on this result we crtn predict energy exchange probabilities for u > I5 at such temperatures to be comparable to the corresponding values shown in fig_ 2; the estimation can be extended even to u < S at such temperatures. The probabilities at 1500, 2000 and 3000 K are appro.xi.mateiy 0_015,0_0?2 and O-035, respectively_ Note that above 2000 K the u = S probability becomes the smallest and the u = I5 probability the largest. but the difference is nor signitkmt. For the pureIy repulsive interaction, i.e., E’(r) = DA exp[(r, - r)la] and V’(r.xl.x,) = 0.4’ exp[(r,-r)fz].ulxu?. the quantity iG(-)I’ is simply [2rr&ol - ~2) A’/MwA ] 2 co& ff(E*l] cscha iZf(E*y] _Comparing thermal werage nrobabiiities obtained for this expression wit!1 the above result, we can detemline the effect of molecular attraction on VV energy exchange processes_ The appearance of g(E*) in eq. (I 5) is due to the contribution of molecular attraction to the collision trajectory, and it makes a large contribution to the overall value of PjF(7) prtrticularIy for Inrge u, where the productf(E*)g(E*) is large_ For example, for u = 10, the contribution of molecular attraction increases the energy exchange probability by factors of 34,7-I, 4.3,2_7, I .9 and 1.7 at 100,300, 500, 1000,200O and 3000 K, respectiveIy. For u = IS, the corresponding factors are 5 I, S.S,4.7.2.S, 1.9 and I-7. This comparison shows the major roIe of molecular attraction in the present energy exchange process; in particuisr the role of molecular attraction completely dominates the contribution of repulsive interaction at iow 25
VoIume
64, number
1
CHEMICAL
PHYSICS
15 June 1979
LETTERS
temperatures (< 300 K) such that the overall process can be describgd in terms of the attmctive interaction alone_ Although the expression derived above does not app!y to the collisions involving lower energy levels, the extension of the mode! shows that for u= 2, the attractive interaction increases tire probability by 12,5-I, 3.6,2.5, 19 and l-7 at !00,300,500, !000,2000 and 3000 K, respectively_ Another interesting result is that for different u, t!re contribution of molecular attraction is to increase the probability by the same factor at Irig!1 temperatures; e-g_, the factor is ! 9 at 2000 K and is 1.7 at 3000 The present approach uses t!re intemction potentia! which is obtained averaging over molecular orientations, zrndintroduces the contribution of nonzero-impact parameter collisions. Therefore, the comparison of this resu!t with that of collinear co!hsions wi!! show the importrmce of molecular orientations and noncollinear collisions_ For the co!Iinear arrsngement C-O + C-O, assuming the nearest 0-C intemction is of on!y importance, we obtain WY.X!, x2) = DCexF[(r,-r)/al whkh
exp((ql
xl
* q2x2)/a]
-
2exp
f(~,--W~~l
exp kl-~1
f 9&/~11,
gives
V(r) = D Cexp [(re -
r)/~]
-
2
exp [(r, - I-)/2~] )
and
!% When
“! Px2) = D {exP [(r, - r)/Q] - $ exp [(r, - r)/2Q] } (‘I! q,/Q’) X! X2.
the probability P/z’ obtained for t!ris interaction is averaged as 00
exp
(-E/kT)d(E/,W),
0 the result is found to be significantly larger than the above result obtained from eq_ (!6)_ For example, for u = IO, the probability of the collinear mode! takes the values of 3.5 X !Om3, 1-O X !O-3- and ! .8 X IO-? at 100, 300 ad 500 K, respectively, w!rich are larger than the corresponding va!ues obtained from eq_ (I 6) by :! factor of 2.7.3.3 and 3_6_ (A!thoug!r the present approach does not apply to small u, t!re calculation reveals the factors of 25,29 and 32 for u = 2.) Therefore, the present approach predicts that the “steric” factor due to the inclusion of the avemge effect of molecuiar orientations and nonzero impact parameter collisions is somewllat close
to I/S for the diatom-diatom
system rather t!xm I/9 which has been estimated in this type of collisions [3-JO] _
EZeferences [I ! I_W_St_ Smith, in: MoIecuhr energy transfer, eds_ R-D_ Levine and J_ Jortner (Wiley. New York. 1976) pp_ 91-98. [7] W-Q. Jeffers and J.D. Relley. J. Chem. Thys. 55 (1971) 4433. [3] J-O_ Hirschf&der, C-C. Curtiss and R-B_ Bird. Molecular theory of gases and liquids (Wiley. New York, 1964) pp_ 11 IO[4]
H.K. Shin, J. Chem. Phys. 60 (1974)
1064.
T-L. CottrelI and J.C.McCoubrey. Molecular energy transfer in gases (Butterworths. London, 161 G_ Herzberg_ Spectra of diatomic moIecuIes (Van Nostrand. Princeton, 1967) table 39. [S]
[7] R-C_ WIIikan, J_ Chem_ Phys_ 38 (1963) 2855_ 181 C_ Wittig and I_W_M_ Smith, Chem_ Phys. Letters 16 (1972)
f9j R-D. Shxma, [IO] K-F_ Heafeld
26
Phys. Rev. 177 (1969)
1961) p_ 134.
292_
2855.
and T-A. Litovitz, Absorption
and dispersion of ultrasonic waves (Academic
Press, New Ycrk,
1959) p- 278.