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On the theory of collision-induced vibrational transitions in polyatomic molecules: The mode-matching model
Volume 44, number 2
1 December 1976
CHEMICAL PHYSICS LETTERS
ON THE TIiEORY OF COLLISION-INDUCED VIBRATIONAL TRANSITIONS IN POLYATOMIC MOLECULES: THE MODE-MATCHING MODEL
Adolf MIKLAVC and Sighart FISCHER Institut fziirTheoretische Physik. Technische UniversitZt Miinchen, 8046 Gurching, W-Ge&ny Received 19 August 1976
A modification of the SSH theory for collision-induced vibrational transitions in poiyatomic motecules is proposed. The breathing-sphere model assumptions are avoided by considering the angular relation between the direction of approach and the normal modes displacements of the atoms involved in the contract (mode-matching). The resuIts, as compared to the breathing-sphere model, indicate a considerable, mode speciiic reduction of the transition probabilities. MramoIecular transitions of CH3Cl are studied as an example.
1.
Introduction
q = 4ir2/.lVl(uh,
decades already, the SSH theory has been by far the most widely used theoretical tool for interpreting the large volume of experimental results obtained on collision-induced vibrational energy exchange in molecules [l-3] . The theory was first developed by Schwartz, Slawsky and Herzfeld [4,5] for collisions of diatomic molecules and later extended by Tanczos [6] and Stretton [7] so that it could be used also for polyatomic molecules. The probability Piy[(a, b) that during a binary collision vibrational mode QU in one molecule will change its quantum state from i to j, while simultaneously a second mode For two
QL, in the same, its state
or in the other
molecule,
= cpo(l)p,(2)[~‘-~(~)]2[yk-‘(~)]2}F,,
where Ft, = (8p12kT)(8n3pAE/h
with
will
change
from k to I, is given by [7]
2 2 -@olkT ) e
(1)
q’ = 4n21.tu’l&
and AL?= hu,(i-j)
+ hv&--1)
= $X(u2 - 8)
.
p is the reduced mass of the colliding pair which ap-
proach each other with an effective relative velocity u and recede with an effective velocity u’. The integration is over the thermal distribution (ZIPr(K)) of the molecular velocities which is assumed to be in&xwellian, h and k are Planck and BoItzmann constants respectively and OLis the exponential rep&ion parameter obtained by matching an exponential curve to a suitable intermolecular potent% whose mimimum value is @o. tc is the distance of closest approach and r. is the separation at zero potential enerm. The translational factor F,, defined by eq. (2j, is obtained on the basis of a three-dimensional version of first-order distorted wave theory. A clear derivation of this factor is given in ref. [2] _ Kere we will discuss somewhat more in detail the remaining factors in (I), i.e. the socalled geometrical or steric factors PO and the vibrational factors V related to vibrational transitions in molecules. For the later ones the so-called breathingsphere model has been employed [6,7]. It wiI1 then be clear that the breathing-sphere model cannot be adequate, particular!y for intramoIecuIar transitions in polyatomic mo’,ecules. We propose an improved 209
method for calculating vibrational factors and apply it to collision-induced intramole’cular transitions of CH3C1. In the case of the n6 -+ v3 transition of CHSCl we find a reduction of the transition probability by a factor of 4.6 which is in good agreement with experiments by Lee and Ronn [8 J . In the case of more complex transitions, a reduction by more than an order of magnitude is found.
2. The breathing-sphere
model
To obtain the vibrational factors V in the SSH formula (l), the breathing-sphere model is commonly adopted. One first assumes that the two molecules interact with each other via two surface atoms, atom s in the first and atom s’ in the second molecule, and that this interaction is of the following form: V= V, exp 01 --r+
[(
CAr-&)+ a
p
Ar,@)
)I*
(3)
Ars(;l) is simply the length of the Cartesian displacemcnt of the surface atom s from its equilibrium position due to the normal vibra?ion Q,. The sum overtz is over all normal vibrations of the first molecule and that over I, is the corresponding sum for the second molecule. Furthermore, we have for the Cartesian component Ax,,(a) of Ar&) the relatron A-x&)
= cs&dQa,
(4)
so that
Here Q, are mass-weighted normal modes of the molecules. The probability for e.g. a transition ji, + ia f I ) Inside the first molecule is then proportional to A&. Since one does not know which surface atom is involved in the contact, one calculates the transition probability for each surface atom and then takes an average. This leads to the following expressions for the vibrational factors in (1) [6,7] : V2&
42-J = 1 ,
P(ia-GQ
210
1)=
1 December 1976
CHEMICALPHYSICS LETI-JZRS
Volume 44, number 2
(6)
(i,+l*l)(i,~l) V2(& --f ia f 2) = 16Y2
4 % CA;=. s s
The coefficients A, are defined by eq. (5). Ns is the number of the surface atoms in the molecule and 3h = 4n2v /h. Note that in our expression the coefficients A,4. are not divided by masses, because we are using the mass-weighted normal coordinates. The breathing-sphere model contains the important assumption that during the collision the two molecules approach each other with orientation most favourable for all normal-mode amplitudes involved. In the case of single-mode changes within one molecule this is possible for one special direction of approach of the collision partners only. For intramolecular transitions involving two or more normal modes this is not possible. To correct for this assumption, steric factors PO are introduced for each molecule_ PO is usually taken to be l/3 for linear molecules while for nonlinear molecules it is takenas 2/3 following Tanczos [6] . There seems to be a considerable amount of arbitrariness in this choice, particularly for larger molecules, since it is not clear how these steric factors could be deduced in a reasonably well-based manner. The situation is particularly complex in the case of intramolecular mode-to-mode transitions. The displacements of an atom due to two normal vibrations can even be orthogonal to each other. The orientation most favourable for deactivation of one mode may therefore be most unfavourable for the activation of the other. In the cases of such “mode mismatching” one can expect therefore the orientational effects to be of considerable importance. It should be pointed out that these questions are of primary importance for the discussion of pathways of vibrational relaxation. Some experimental information on such pathways is available for the liquid phase as well as the gas phase (see e.g. refs. [8-121).
3. The mode-matching
model
In the following we wish to suggest a model in which we go beyond the breathing-sphere model assumptions. We want to consider in detail the angular relation between the direction of approach and the normalmode displacements of the atoms involved in the contact. Let us again imagine that the two molecules interact
Volume 44, number 2
prohibited on grounds of the geometry of the molecule. If one calculates the matrix elements in the same approximation as in the SSH theory and carries through the averaging over the surface atoms and the possible directions, one fmds e.g. in case of an intramoIecu1ar transition (ia + ia + 1) (ib * ib f 1). instead of(l), (2) and (6),
via two surface atoms s and s’ and that the interaction is of the form V= V. exp(-
arssn) .
(7)
The vector r,,~ between the atoms s and s’ can be written as rss* = r - (xs + x,‘), where xs and xso are the infinitesimal displacements of the two atoms from their equilibrium positions due to the normal vibrations. We can therefore expand
=r ‘ss
I
XpX$
l
1+-C 2
‘ss
=r-i
l
(xs
+ Xsl)
xs
- xs*
--
,
r2
(xs + $4
1
?
(8)
,
where ii = r/r is the direction vector cf the line connecting the interacting atoms s and s’ in equilibrium. Since the displacements xs are small, we neglected terms other than linear in xs in expression (8). The displacement xs can moreover be expressed as a sum xs = I&x&r) of th e “elementary displacements” x,(a), each due to a particular normal-mode and vibration QU The interaction (7) then assumes the form V=Voexp
[
-a
1 December 1976
CHEMICAL PHYSICS LETTERS
r-
Ci.x,(a)
(
-C;i*x&‘) a
a
)I .
I
(9)
for the corresponding transition probability. The factor Fu is defined by eq. (2) and 7 = 4n2v/fr. The summation over s runs over alI N, surface atoms. S20, is the solid angle in which the direction vector n can lie. $Zo, clearly depends on the geometry of the moIecule at each surface atom s. In case of a more comptex intramoIecular transition (ia -F ia +-n,)($
one obtains p(ia + ia * n=)(ib-cib = F(a;
If one replaces the scalar products ii x&a) with their maximal values Ix,(a)1 = b&z), i.e. if one assumes the collision to be in the most favourable direction for all normal modes simultaneously (which is clearly impossible for polyatomic molecules), one obtains the interaction which is assumed in the breathing-sphere model [eq. (3)] . The scalar products i xs can be expressed in terms of normal modes Qa
--f ib +- nb)(ic --r ic + n,} .._
ia, n,,
* nb)(i,+i
c f n,)......
va; ib, “b, vb; ic, ncuc;
.__)
l
(13)
l
fi
l
xs= 5 ~saGx?a ,
(10)
so that finally V= V. exp
-c
a
u,,($Q,
-c
us*& a’
Q,#
11 .
(11) The coeffickkts u,(G) reflect that coupling to a particular normal vibration depends on the direction of approach &. One can then take an average over all directions ii, pospibly excluding those ii which are
where the factorF(cu;i,,n,,v,;i6,nb,vb; i,,tt,,v,; . ..) is the sarre as in the breathing-sphere theory. The steric factors PO do not appear in our theory. The other quantities are defined as in eq. (I 2). It is cIear that for the vibrational transitions in the partner molecuIe one obtains an extra factor which is of the same form as (13). The coefticients u,(h) of the expansion (10) can be obtained using the standard normal-mode anaIysis [13,14]. Suppose: the molecule has N atoms and x represents the 3Afdimensioi2al vector of Cartesian displacements of these atoms (measured from their equilibrium position). Furthermore, Iet r be the internal coordinates, s the symmetry coordinates and Q the normal modes of the moIecuIe (there are 3iV - 6 of each of them). If one defines the matrices 6 and L 211
Table 1
by the rekttions r=L*Q,
~=B*x,
Ratios of the transition probabilities P calculated with the present method to the corresponding probabilities Pb_sph_ obtained by breathing-sphere calculations (with the steric factor 1) for intramolecular transitions in CH3CI
(14)
then one has the following relation between x and Q [14] : x=&j--’
o B=. [B o M-’
. $1-1
.
L .
Transition
Q.
(15)
Y3 -+ YgW
M-l is the inverse mass matrix of the molecule and the index T denotes the transpose of a matrix. Sometimes the matrix 1 is given in the literature which relates the symmetry coordinates to the normal modes, i.e. s = 3 Q. Since s = U r, where 11-l = UT, one can define a new matrix B through the relation l
UI -+2u2 y1 -Ys(4 +us.(b) Iq -c&J2 +2v&) Y4 (0) -, vg (a) + yj (b)
(16)
and it is easy to show then that eq. (12) can be rewritten in the form x=&.3=.@
.M--’
Furthermore, 141:
.g’T]-1
.‘L.Q_
(17)
the following reIation must hold [13,
&ET=& ET . These relations are very whereg=z*M”-’ helpful since they provide a consistency check. Eq. (13) can tJms be written as: l
.gT
.5--l
.z.
Q.
(18)
4. Application to CH3CJ We have carried through the calculations proposed above for intramolecular mode-mode transitions in CHjCf induced by collisions with inert gases. Some comparison with experiment is possible in this case.
In our calculations formula (17) was used. The molecular parameters, the symmetry coordinates and the matrix r were taken from ref. [15]. The 9 X 15 matrix B from eq. (14) was calculated by the method described in detail in ref. [13], and the similar matrix E [eq. (Id)] was then obtained for the symmetry coordinates as defined in ref. [IS]. In table 1 the ratios of our transition probabilities to those calculated with the breathing-sphere model (with the steric factor equal to 1) are given for a member of intiamolecular transitions. The standard notation is used for the modes [16]. We see that in most cases poor mode matching reduces the transition probability by more than an order of magnitude. As expected, these effects are stronger in m0ie complex transitions, especially when the displacements of the atoms undergoing the collision are perpendicular to each other for the normaI modes involved. In table 2 we compare the transition probabilities per collision,-calculated with the present and with the breathing-sphere model, to these obtained experimentally by Lee and Ronn [S] for the intramolecular process
CH&1(“6) + X + CH&)
+ X f 283 cm-‘.
Table 2 Measured and calculated transition probabilities
for the v6 + v3 transition in CH3CI-X collisions
Colhsion
Prob/coUisiona)
partner X
(experimen
ze Ar Kr Xe
0.0012 0.00091 0.00088 0.0010 0.00076
td)
a) From ref. [8]. b, From ref. [8], but with the steric factor 1 instead of 2!3. 212
pfpb.sph. 0.218 0.059 0.023 0.029 0.019
l
s=U.B*XI~=X
x=M-1
1 December 1976
CHEMICAL PHYSICS LETTERS
Volume 44, number 2
ProbfcoUision (present c&uIation) -
Prob/colIisionb) (b.sph. Axlation)
0.00135 0.00105 0.00088 0.00062 0.00050
0.0062 0.0048 0.00405 0.0028 0.0023
Volume 44, number 2
CHEMICAL PHYSICS LETTERS
For the collision partners X different rare gases were taken. The breathing-sphere calculations were already carried through by Lee and Ronn [8], using formulas developed by Tanczos and Stretton [6,7]. We give their results, without the steric factor of 2/3 assumed by Tanczos.
5. Conclusions In our model we assume that the contact between the colliding molecules occurs only via two atoms. Furthermore, we assume that a direction of approach can be defined during the collision. This means the time of collision defined as 7c = L/c where L is the interaction length and u the average velocity, should be shorter than the period of rotation rr. In the original papers by Schwartz, Slawsky and Herzfeld [4,5] these assumptions also entered. The breathing-sphere model, on the other hand, may be better motivated in the limit of fast rotation 7c > 7,, since than the interaction potential could be preaveraged over the angular part. But even such an average cannot be done independently for the different normal modes involved. That means, the breathing-sphere model may be applicable for transitions involving a single quantum change but it must fail for intramolecular transitions involving excitations or deactivations of differenr normal modes. In this case the concept of steric factors is also no longer helpful. Such intramolecular processes are the predominant processes involved in the fast energy equilibration observed by Laubereau, Kaiser et al. [9-l l] in the liquid phase, as well as by Flynn, Weitz, Ronn et al. [8,12] in the gas phase. The reduction factors, compared to the breathing-sphere model, run typically over one to two orders of magnitude and are very mode specific. Therefore we find it necessary to account for the mode matching in order to understand pathways for the rapid intramolecular relaxation. In a recent study [l l] on CH,I individual rates for the vi to v4 as well as those to combination and overtones of bending modes have been measured. We found it necessary to consider the energy release into translational as welI as rotational degrees of freedom. In this case it is no longer possible to factorize the expression for the rate into a vibrational and a translationai contribution since the participation of rota-
1 December LX%
tionaI transitions depends upon the direction of approach. We shalI expend the present model to include rotations within the semiclassical frame in a forthcoming paper. Acknowledgement Tile authors wish tqthank Professor W. Kaiser and Dr. A Laubereau for many stimulating discussions which initiated the present investigation. Referencesr11 K.F. HerzfeId and T.A. Litovitz. Absorption and dispersion of ultrasonic waves (Academic Press, New York, 1959). 121G.M. Burnett and A.M. North, eds.. Transfer and storage of energy by molecules, Vol. 2 (Wiley-Interscience, New York, 1969). r31 J.F. Clarke and MMcChcsney, Dynamics of reIasing gases (Buttexworths, London, 1976). 141 R.N. Schwartz, Z.I. Slawsky and K.F. Henfeld,J. Chem. Phys. 20 (1952) 1951. [51 R.N. Schwartz and K.F. Herzfcld, I. Chem. Phys. 22 (1954) 767. 161 F.I. Tannos, J. Chem. Phys. 25 (1956) 439. 10.53. 171 J.L. Strctton, Trans. Faraday Sot. 61(1965) 181 S.M. Lee and A.M. Ronn, Chem. Phys. Letters 22 (1973) 279. [91 A. Laubereau, L. Kirschner and W. Kaiser, Opt. Commun. 9 (1972) 182. r101 A. Laubereau, G. Kehl and W. Kaiser. Opt. Commun. 1 L (1974) 74. 1111 K. Spanner, A. Laubereau and W. Kaiser, Chem. Phys. Letters 44 (1976) 88. 1121 E. Weitz, G. Flynn and A.M. Ronn, J. Chem. Phys/ 56 (1972) 6060; E. Weitz and G. Flynn, J. Chem. Phys. 58 (1973) 2679; Ann. Rev. Phys. Chem. 25 (1974) 275; B.L. Earl and A.M. Ronn, Chem. Phys. 12 (I9761 113; Y. Langsarn, S.M. Lee and A.M. Ronn, Chcm. Phys. 14 (1976) 375; B.L. Earl, P.C. isolani and A.M. Ronn. Chem. Phys. Letters 39 (1976) 95. S.T. Lin, B.L. Earl and A.M. Ronn, Chem. Phys. 16 (1976) 117; L.A.Gamss, B.H. Kohn, A.M. Roan and G.W. Flynn, Chem. Phys. Letters41 (1976) 413. [ 13 ] E.B. Wilson Jr.. J.C. Decius and PC Cross. MoIecuIar vibrations (McGraw-Hill. New York, 195.5). [ 141 P. Cans, Vibrating molecules (Chapman and HaII, London, 1971). [ 151 W.T. King, I.M. Mills and B. Crawford. I. Chem. Phys 27 (i9.57) 455. [ 161 G. Her&erg, Molecular spectra and mofecuIar structure, Vol. 2 (Van Nostrand, Princeton, 1959).
213
1 December 1976
CHEMICAL PHYSICS LETTERS
ON THE TIiEORY OF COLLISION-INDUCED VIBRATIONAL TRANSITIONS IN POLYATOMIC MOLECULES: THE MODE-MATCHING MODEL
Adolf MIKLAVC and Sighart FISCHER Institut fziirTheoretische Physik. Technische UniversitZt Miinchen, 8046 Gurching, W-Ge&ny Received 19 August 1976
A modification of the SSH theory for collision-induced vibrational transitions in poiyatomic motecules is proposed. The breathing-sphere model assumptions are avoided by considering the angular relation between the direction of approach and the normal modes displacements of the atoms involved in the contract (mode-matching). The resuIts, as compared to the breathing-sphere model, indicate a considerable, mode speciiic reduction of the transition probabilities. MramoIecular transitions of CH3Cl are studied as an example.
1.
Introduction
q = 4ir2/.lVl(uh,
decades already, the SSH theory has been by far the most widely used theoretical tool for interpreting the large volume of experimental results obtained on collision-induced vibrational energy exchange in molecules [l-3] . The theory was first developed by Schwartz, Slawsky and Herzfeld [4,5] for collisions of diatomic molecules and later extended by Tanczos [6] and Stretton [7] so that it could be used also for polyatomic molecules. The probability Piy[(a, b) that during a binary collision vibrational mode QU in one molecule will change its quantum state from i to j, while simultaneously a second mode For two
QL, in the same, its state
or in the other
molecule,
= cpo(l)p,(2)[~‘-~(~)]2[yk-‘(~)]2}F,,
where Ft, = (8p12kT)(8n3pAE/h
with
will
change
from k to I, is given by [7]
2 2 -@olkT ) e
(1)
q’ = 4n21.tu’l&
and AL?= hu,(i-j)
+ hv&--1)
= $X(u2 - 8)
.
p is the reduced mass of the colliding pair which ap-
proach each other with an effective relative velocity u and recede with an effective velocity u’. The integration is over the thermal distribution (ZIPr(K)) of the molecular velocities which is assumed to be in&xwellian, h and k are Planck and BoItzmann constants respectively and OLis the exponential rep&ion parameter obtained by matching an exponential curve to a suitable intermolecular potent% whose mimimum value is @o. tc is the distance of closest approach and r. is the separation at zero potential enerm. The translational factor F,, defined by eq. (2j, is obtained on the basis of a three-dimensional version of first-order distorted wave theory. A clear derivation of this factor is given in ref. [2] _ Kere we will discuss somewhat more in detail the remaining factors in (I), i.e. the socalled geometrical or steric factors PO and the vibrational factors V related to vibrational transitions in molecules. For the later ones the so-called breathingsphere model has been employed [6,7]. It wiI1 then be clear that the breathing-sphere model cannot be adequate, particular!y for intramoIecuIar transitions in polyatomic mo’,ecules. We propose an improved 209
method for calculating vibrational factors and apply it to collision-induced intramole’cular transitions of CH3C1. In the case of the n6 -+ v3 transition of CHSCl we find a reduction of the transition probability by a factor of 4.6 which is in good agreement with experiments by Lee and Ronn [8 J . In the case of more complex transitions, a reduction by more than an order of magnitude is found.
2. The breathing-sphere
model
To obtain the vibrational factors V in the SSH formula (l), the breathing-sphere model is commonly adopted. One first assumes that the two molecules interact with each other via two surface atoms, atom s in the first and atom s’ in the second molecule, and that this interaction is of the following form: V= V, exp 01 --r+
[(
CAr-&)+ a
p
Ar,@)
)I*
(3)
Ars(;l) is simply the length of the Cartesian displacemcnt of the surface atom s from its equilibrium position due to the normal vibra?ion Q,. The sum overtz is over all normal vibrations of the first molecule and that over I, is the corresponding sum for the second molecule. Furthermore, we have for the Cartesian component Ax,,(a) of Ar&) the relatron A-x&)
= cs&dQa,
(4)
so that
Here Q, are mass-weighted normal modes of the molecules. The probability for e.g. a transition ji, + ia f I ) Inside the first molecule is then proportional to A&. Since one does not know which surface atom is involved in the contact, one calculates the transition probability for each surface atom and then takes an average. This leads to the following expressions for the vibrational factors in (1) [6,7] : V2&
42-J = 1 ,
P(ia-GQ
210
1)=
1 December 1976
CHEMICALPHYSICS LETI-JZRS
Volume 44, number 2
(6)
(i,+l*l)(i,~l) V2(& --f ia f 2) = 16Y2
4 % CA;=. s s
The coefficients A, are defined by eq. (5). Ns is the number of the surface atoms in the molecule and 3h = 4n2v /h. Note that in our expression the coefficients A,4. are not divided by masses, because we are using the mass-weighted normal coordinates. The breathing-sphere model contains the important assumption that during the collision the two molecules approach each other with orientation most favourable for all normal-mode amplitudes involved. In the case of single-mode changes within one molecule this is possible for one special direction of approach of the collision partners only. For intramolecular transitions involving two or more normal modes this is not possible. To correct for this assumption, steric factors PO are introduced for each molecule_ PO is usually taken to be l/3 for linear molecules while for nonlinear molecules it is takenas 2/3 following Tanczos [6] . There seems to be a considerable amount of arbitrariness in this choice, particularly for larger molecules, since it is not clear how these steric factors could be deduced in a reasonably well-based manner. The situation is particularly complex in the case of intramolecular mode-to-mode transitions. The displacements of an atom due to two normal vibrations can even be orthogonal to each other. The orientation most favourable for deactivation of one mode may therefore be most unfavourable for the activation of the other. In the cases of such “mode mismatching” one can expect therefore the orientational effects to be of considerable importance. It should be pointed out that these questions are of primary importance for the discussion of pathways of vibrational relaxation. Some experimental information on such pathways is available for the liquid phase as well as the gas phase (see e.g. refs. [8-121).
3. The mode-matching
model
In the following we wish to suggest a model in which we go beyond the breathing-sphere model assumptions. We want to consider in detail the angular relation between the direction of approach and the normalmode displacements of the atoms involved in the contact. Let us again imagine that the two molecules interact
Volume 44, number 2
prohibited on grounds of the geometry of the molecule. If one calculates the matrix elements in the same approximation as in the SSH theory and carries through the averaging over the surface atoms and the possible directions, one fmds e.g. in case of an intramoIecu1ar transition (ia + ia + 1) (ib * ib f 1). instead of(l), (2) and (6),
via two surface atoms s and s’ and that the interaction is of the form V= V. exp(-
arssn) .
(7)
The vector r,,~ between the atoms s and s’ can be written as rss* = r - (xs + x,‘), where xs and xso are the infinitesimal displacements of the two atoms from their equilibrium positions due to the normal vibrations. We can therefore expand
=r ‘ss
I
XpX$
l
1+-C 2
‘ss
=r-i
l
(xs
+ Xsl)
xs
- xs*
--
,
r2
(xs + $4
1
?
(8)
,
where ii = r/r is the direction vector cf the line connecting the interacting atoms s and s’ in equilibrium. Since the displacements xs are small, we neglected terms other than linear in xs in expression (8). The displacement xs can moreover be expressed as a sum xs = I&x&r) of th e “elementary displacements” x,(a), each due to a particular normal-mode and vibration QU The interaction (7) then assumes the form V=Voexp
[
-a
1 December 1976
CHEMICAL PHYSICS LETTERS
r-
Ci.x,(a)
(
-C;i*x&‘) a
a
)I .
I
(9)
for the corresponding transition probability. The factor Fu is defined by eq. (2) and 7 = 4n2v/fr. The summation over s runs over alI N, surface atoms. S20, is the solid angle in which the direction vector n can lie. $Zo, clearly depends on the geometry of the moIecule at each surface atom s. In case of a more comptex intramoIecular transition (ia -F ia +-n,)($
one obtains p(ia + ia * n=)(ib-cib = F(a;
If one replaces the scalar products ii x&a) with their maximal values Ix,(a)1 = b&z), i.e. if one assumes the collision to be in the most favourable direction for all normal modes simultaneously (which is clearly impossible for polyatomic molecules), one obtains the interaction which is assumed in the breathing-sphere model [eq. (3)] . The scalar products i xs can be expressed in terms of normal modes Qa
--f ib +- nb)(ic --r ic + n,} .._
ia, n,,
* nb)(i,+i
c f n,)......
va; ib, “b, vb; ic, ncuc;
.__)
l
(13)
l
fi
l
xs= 5 ~saGx?a ,
(10)
so that finally V= V. exp
-c
a
u,,($Q,
-c
us*& a’
Q,#
11 .
(11) The coeffickkts u,(G) reflect that coupling to a particular normal vibration depends on the direction of approach &. One can then take an average over all directions ii, pospibly excluding those ii which are
where the factorF(cu;i,,n,,v,;i6,nb,vb; i,,tt,,v,; . ..) is the sarre as in the breathing-sphere theory. The steric factors PO do not appear in our theory. The other quantities are defined as in eq. (I 2). It is cIear that for the vibrational transitions in the partner molecuIe one obtains an extra factor which is of the same form as (13). The coefticients u,(h) of the expansion (10) can be obtained using the standard normal-mode anaIysis [13,14]. Suppose: the molecule has N atoms and x represents the 3Afdimensioi2al vector of Cartesian displacements of these atoms (measured from their equilibrium position). Furthermore, Iet r be the internal coordinates, s the symmetry coordinates and Q the normal modes of the moIecuIe (there are 3iV - 6 of each of them). If one defines the matrices 6 and L 211
Table 1
by the rekttions r=L*Q,
~=B*x,
Ratios of the transition probabilities P calculated with the present method to the corresponding probabilities Pb_sph_ obtained by breathing-sphere calculations (with the steric factor 1) for intramolecular transitions in CH3CI
(14)
then one has the following relation between x and Q [14] : x=&j--’
o B=. [B o M-’
. $1-1
.
L .
Transition
Q.
(15)
Y3 -+ YgW
M-l is the inverse mass matrix of the molecule and the index T denotes the transpose of a matrix. Sometimes the matrix 1 is given in the literature which relates the symmetry coordinates to the normal modes, i.e. s = 3 Q. Since s = U r, where 11-l = UT, one can define a new matrix B through the relation l
UI -+2u2 y1 -Ys(4 +us.(b) Iq -c&J2 +2v&) Y4 (0) -, vg (a) + yj (b)
(16)
and it is easy to show then that eq. (12) can be rewritten in the form x=&.3=.@
.M--’
Furthermore, 141:
.g’T]-1
.‘L.Q_
(17)
the following reIation must hold [13,
&ET=& ET . These relations are very whereg=z*M”-’ helpful since they provide a consistency check. Eq. (13) can tJms be written as: l
.gT
.5--l
.z.
Q.
(18)
4. Application to CH3CJ We have carried through the calculations proposed above for intramolecular mode-mode transitions in CHjCf induced by collisions with inert gases. Some comparison with experiment is possible in this case.
In our calculations formula (17) was used. The molecular parameters, the symmetry coordinates and the matrix r were taken from ref. [15]. The 9 X 15 matrix B from eq. (14) was calculated by the method described in detail in ref. [13], and the similar matrix E [eq. (Id)] was then obtained for the symmetry coordinates as defined in ref. [IS]. In table 1 the ratios of our transition probabilities to those calculated with the breathing-sphere model (with the steric factor equal to 1) are given for a member of intiamolecular transitions. The standard notation is used for the modes [16]. We see that in most cases poor mode matching reduces the transition probability by more than an order of magnitude. As expected, these effects are stronger in m0ie complex transitions, especially when the displacements of the atoms undergoing the collision are perpendicular to each other for the normaI modes involved. In table 2 we compare the transition probabilities per collision,-calculated with the present and with the breathing-sphere model, to these obtained experimentally by Lee and Ronn [S] for the intramolecular process
CH&1(“6) + X + CH&)
+ X f 283 cm-‘.
Table 2 Measured and calculated transition probabilities
for the v6 + v3 transition in CH3CI-X collisions
Colhsion
Prob/coUisiona)
partner X
(experimen
ze Ar Kr Xe
0.0012 0.00091 0.00088 0.0010 0.00076
td)
a) From ref. [8]. b, From ref. [8], but with the steric factor 1 instead of 2!3. 212
pfpb.sph. 0.218 0.059 0.023 0.029 0.019
l
s=U.B*XI~=X
x=M-1
1 December 1976
CHEMICAL PHYSICS LETTERS
Volume 44, number 2
ProbfcoUision (present c&uIation) -
Prob/colIisionb) (b.sph. Axlation)
0.00135 0.00105 0.00088 0.00062 0.00050
0.0062 0.0048 0.00405 0.0028 0.0023
Volume 44, number 2
CHEMICAL PHYSICS LETTERS
For the collision partners X different rare gases were taken. The breathing-sphere calculations were already carried through by Lee and Ronn [8], using formulas developed by Tanczos and Stretton [6,7]. We give their results, without the steric factor of 2/3 assumed by Tanczos.
5. Conclusions In our model we assume that the contact between the colliding molecules occurs only via two atoms. Furthermore, we assume that a direction of approach can be defined during the collision. This means the time of collision defined as 7c = L/c where L is the interaction length and u the average velocity, should be shorter than the period of rotation rr. In the original papers by Schwartz, Slawsky and Herzfeld [4,5] these assumptions also entered. The breathing-sphere model, on the other hand, may be better motivated in the limit of fast rotation 7c > 7,, since than the interaction potential could be preaveraged over the angular part. But even such an average cannot be done independently for the different normal modes involved. That means, the breathing-sphere model may be applicable for transitions involving a single quantum change but it must fail for intramolecular transitions involving excitations or deactivations of differenr normal modes. In this case the concept of steric factors is also no longer helpful. Such intramolecular processes are the predominant processes involved in the fast energy equilibration observed by Laubereau, Kaiser et al. [9-l l] in the liquid phase, as well as by Flynn, Weitz, Ronn et al. [8,12] in the gas phase. The reduction factors, compared to the breathing-sphere model, run typically over one to two orders of magnitude and are very mode specific. Therefore we find it necessary to account for the mode matching in order to understand pathways for the rapid intramolecular relaxation. In a recent study [l l] on CH,I individual rates for the vi to v4 as well as those to combination and overtones of bending modes have been measured. We found it necessary to consider the energy release into translational as welI as rotational degrees of freedom. In this case it is no longer possible to factorize the expression for the rate into a vibrational and a translationai contribution since the participation of rota-
1 December LX%
tionaI transitions depends upon the direction of approach. We shalI expend the present model to include rotations within the semiclassical frame in a forthcoming paper. Acknowledgement Tile authors wish tqthank Professor W. Kaiser and Dr. A Laubereau for many stimulating discussions which initiated the present investigation. Referencesr11 K.F. HerzfeId and T.A. Litovitz. Absorption and dispersion of ultrasonic waves (Academic Press, New York, 1959). 121G.M. Burnett and A.M. North, eds.. Transfer and storage of energy by molecules, Vol. 2 (Wiley-Interscience, New York, 1969). r31 J.F. Clarke and MMcChcsney, Dynamics of reIasing gases (Buttexworths, London, 1976). 141 R.N. Schwartz, Z.I. Slawsky and K.F. Henfeld,J. Chem. Phys. 20 (1952) 1951. [51 R.N. Schwartz and K.F. Herzfcld, I. Chem. Phys. 22 (1954) 767. 161 F.I. Tannos, J. Chem. Phys. 25 (1956) 439. 10.53. 171 J.L. Strctton, Trans. Faraday Sot. 61(1965) 181 S.M. Lee and A.M. Ronn, Chem. Phys. Letters 22 (1973) 279. [91 A. Laubereau, L. Kirschner and W. Kaiser, Opt. Commun. 9 (1972) 182. r101 A. Laubereau, G. Kehl and W. Kaiser. Opt. Commun. 1 L (1974) 74. 1111 K. Spanner, A. Laubereau and W. Kaiser, Chem. Phys. Letters 44 (1976) 88. 1121 E. Weitz, G. Flynn and A.M. Ronn, J. Chem. Phys/ 56 (1972) 6060; E. Weitz and G. Flynn, J. Chem. Phys. 58 (1973) 2679; Ann. Rev. Phys. Chem. 25 (1974) 275; B.L. Earl and A.M. Ronn, Chem. Phys. 12 (I9761 113; Y. Langsarn, S.M. Lee and A.M. Ronn, Chcm. Phys. 14 (1976) 375; B.L. Earl, P.C. isolani and A.M. Ronn. Chem. Phys. Letters 39 (1976) 95. S.T. Lin, B.L. Earl and A.M. Ronn, Chem. Phys. 16 (1976) 117; L.A.Gamss, B.H. Kohn, A.M. Roan and G.W. Flynn, Chem. Phys. Letters41 (1976) 413. [ 13 ] E.B. Wilson Jr.. J.C. Decius and PC Cross. MoIecuIar vibrations (McGraw-Hill. New York, 195.5). [ 141 P. Cans, Vibrating molecules (Chapman and HaII, London, 1971). [ 151 W.T. King, I.M. Mills and B. Crawford. I. Chem. Phys 27 (i9.57) 455. [ 161 G. Her&erg, Molecular spectra and mofecuIar structure, Vol. 2 (Van Nostrand, Princeton, 1959).
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