Restricted intramolecular vibrational relaxation in polyatomics and laser selective effects

Volume

69, number

1

CHEhIICAL

RESTRICTED AND

LASER

IN-I-RAMOLECULAR SELECTIVE

Everett THIELE,

Myron

1 January

PHYSICS LETTERS

VIBRATIONAL

RELAXATION

IN POLYATOMICS

EFFECTS

F. GOODMAN

and Jamss STONE

Deparmiort of Biologrcal Sc12rrc2s and Center for Laser Srtrdres Lit~rrers~t_~ of Southern Los Ingeles, CahJbmta 9OOOT. (iSA Recewed

I October

purporting

to

Cahfornra,

1979

A throrl for ummoleculx yields. and product irxgnent ment

1980

dlssocl3tlon translattonal

- demonstrate”

raptd

apphcable to sclecttre laser ewltatton 1s presented. E\pressIons for Metlmes, energy dlstrlbutlons are given 3s ;I iunctton oi IVR rate. We analyze an e\pertIVR.

In RRK [l] and RRKM [Z] theory It IS assumed that IVR (mtramolecular vtbrattonal rzlaxatton) is rapid compared to the rat2 of untmolecular decay [3,

41. As a consequence, detarled knowledge of the molecular dynamics IS not needed and relattvely su-nple statistical factors determtne the mean hfetlme of an excited molecule. The advantage to thrs approach IS evrdent m the successful apphcat1ons of these statIsticaI theones. especrally RRKhl theory. rn many areas [3-61 mcludmg multrphoton drssocratton [7- 1 11. We, however. do not accept the v12w that the statrstrcal theories of urnmolecular decay should always apply m multrphoton dtssocrattons for the followmg two Important reasons (1) Th2 extremely mterestmg possrbrhtres of selective chemistry would be thsorettcally derued, a pnorr. Any potential for selectrvrty that might be built m by the excrtatron process is washed out before reactton can occur if IVR IS too raprd. Conversely, genuine examples of laser selectrvlty * could not be described by RI3.K or RRKM theory as they presently srand, (1) It IS posstble to show that, m czrtam cases, agreement between RRK and RRKM theory and exper* In ref [ 131 UC hare argued that there IS no reason to doubt that the Hall and Kaldor expertment on the laser mduced reaction of cl clopropane 15 an example oi laser selectivity.

rment does not mdrcate the vahdtty of the raprd IVR assumptton, but rather the insensrttvrty of th2 experiment to the actual rate of the IVR process [ 121. Because the RRK and RRKM theones do not exphcrtly include IVR 111therr formulatron, it IS not possrble, wrthout further theory, to 2xtract from such experrments numerical bounds on the IVR rates. Usmg a srmphfied descnptron of IVR, called the restncted IVR model, we [ 12.141 have -made some progress m formulatmg a theory of urnmolecular decay that does exphcrtly mclude the competition between IVR and decay. In Its srmplest form thrs theory mtroduces mto the quantum RRK theory a parameter Q which can take values mtermedlate between cr = 1 for extremely rapid IVR, and o = 0 for negligrbly slow IVR. We have already used this theory to show that the success of RRK and RRKM theones m dsscnbmg pressure fall-off curves from conventional thermal experiments [I?] puts a lower bound on the IVR rat2 (=loIo s-t for a molecule wrth twenty vrbrattonal degrees of freedom) which 1s not so restrictive as to rule out the posstbtlity of laser selective chemistry wtth currently available techmques (see fig. 1). In thrs apphcatron our theory qualitatively resembles an earlier restilt due to Sole [ 153 in which he mterpolated between the classical versrons of RRK theory [I] (extremely rapid IVR) and the no-IVR theory of Slater [161-

Volume

69, number

CHEMICAL

1

PHYSICS

FIN. 1. Theoretical fall-oii tunes as a funcuon of the dunenslonless

for rhe resmcred IVR model pressure rm/~c. where l/r, 1s the colhslon frequency per molecule. These cumes are calculated for the various values of the parameter o and with an equal number of oscdlators III both groups WA = Fg = LO). Other parameters have values that are representatwe for the thermal cxperlment, m = 20. I&h 7 = 1.5. For pi < 1 O-’ the ewstence of the “false” high pressure hmlt transluon region is clearly mdlcated. Notxr that 011s tranntlon regon ends at a pressure ior ahlch l/~c - a/rm and also that for Q > IO”’ the transItIon repon ISunobservable bemg bounded by the colhnear tigh pressure hmns of the u = 0 and u = 1 CULLS a= lOA corresponds to an IVR rate o/~m of about IO” s-l.

In this paper we use the resmcted

IVR

model to

denve an expressron for the mean hfetrme of an excited molecule that, unhke RRK and RRKM theory, depends on botIr the total mternai energy arid the dlsmbzition of that energy among the molecule’s Internal

degrees of freedom, and IS thus well surted for the descnptron of laser selectrve effects. The mean lifetime also depends exp!icttly on the IVR rate through the adjustable parameter cr. With a specral choice of statrstrcal factors the Q = 1 case corresponds exactly to the quantum RRK theory wrth all the molecule’s Wbrdtional degrees of freedom “active”, whrle (Y= 0 corresponds to RRK theory wrth only a subset of the degrees of freedom acttve. in presenting a theory for the mean lifetime of a selectrvely excited molecule we are antrcrpatmg future experimental evrdence confirming laser selective effects. Although we have strong reservations

concernmg

the thermal assump-

LETTERS

L January

L98Cl

tion [17] gomg rnto the recent calculation by Shtdtz and Yablonovitch [ 181 for the multiphoton dissociatton efficrency of SF,, we do basically agree with the conclusion that, under conditions where collisions are ummportant (pressure below 1 Torr) and the laser intensity and pulse length are such that there is essentiafly no chemrcal reaction dunng the laser p&e, then the drssocratron efficiency is relatively insensitive to the partrcular statrstrcal model used to calculate bfetunes. Since thrs conclusion IS based on the observation that product yreld depends, under these experimental conditrons, primarrly on the fraction of moIecules exerted above the dissocratron threshold it- apphes equally well to variations in the lifetimes that mght be caused by less than rapid IVR. But this represents only one narrow set of experimental conditions. More generally, the evidence suggesting laser selectivity because of less than rapid n/R has come from an experiment [ 131 in which the pressure of an mert buffer gas is vaned over a range O-500 Torr, and from an expenment [ 191 in which the intensity was varied over a range sufficieni

to include a regime

where cornpetItIon between chermcal reaction and radrattve pumping IS Important. Ln this paper we also show that the center-of-mass translatronal energy distnbution for fragments produced by multiphoton dissocration in snnple bondruptures, as measured m the crossed laser molecular beam experiments of Lee and co-workers [I 11, is trot sensrtive to the rate of WR relaxation; therefore, contrary to recent suggestions (1 i ,201, Lee’s measurements cannot demonstrate that MC relaxation is rapid. In fact, the theoretical predictions (when computed for the same average translational energy) for the two extremes. 01= 0 and Q = 1, are too ciose to be drstmgurshed experimentally. The resmcted IVR model IS based on a master equation * w(n,

t)/dtl,,

= (l/r,)

- (l/Tm)p(n,

r, 3

c L(‘Cn’)P(‘r’, #I’

t) (11

in which the matnx L(Iz, n’) for the probabrlity of intramolecular transitrons from one microstate n’ to * For a recent &scussion of tie apphcabibty of a master equatxon description of IVR see ref. [?I].

I9

CHEMICAL

Volume 69, number 1 another

nucrostate

PHYSICS

II on the same energy shell IS given

by I!@, 12’) = (1 - a)/& + Q/S, L(n, II’) = cY/g.

nEJand?l’EI.

II E J and ~1’E I’ f J ;

(3

O
311I

gl

(3)

to be the total number of microstates belongmg to the energy shell. Clearly the L(tt, tt’) satisfy the probablhty requirement L(tt, tz’) = 1 .

F

(1)

In eq. (1) T, IS mrerprered as a mean time betlteeen those mtramolecular “colhslon” events which scramble a molecule’s vibrational energ). 1/TV can be identlfied with the Hugh pressure. pre-exponential -i factor and hence l/r, = 101-1 s-1 [I?]. The basic features of the restricted IVR model are contained m the solution to eq. (1) for the transItIon probablhtles detlned by eq. (Z)_ Thus solution [12.1-l] describes a relaxation to the mlcrocanomcal dlstrlbution that proceeds on t\\o rime scales. Devlatlons from a uniform dlstrlbutlon of states wlthm any set J disappear at a rate l/r,, , while devlatlons of the fotal set populations. C,IEI P(tz. t), from their mlcrocanomcal ensemble values, g[/g, disappear at the potentially much slower rate a/r,. TO mclude the posslbdlty of ummolecular reaction m the formahsm, one supposes the mlcrostates dlvtded mto two classes, R for reactant states and P for product states. A vahd model for reactlon then results from rhe assumption that the IVR process can * The follo\\mg IS one possible defmluon of the sets I suppose that the molecule’s wbratlonal degrees of freedom are dtwded into two groups A and B. lfn quanta of lxer eaergy hate been absorbed then the set I ml&t consist of all mvzrostates for which the total energy 1s partItIoned so that I quanta belong to the group-ii modes and n - I quanta belong to the groupB modec

20

1 January

LETTERS

1980

mduce transitions from reactant states to product states but not from product states to reactant states. In other words, one takes L(tz. tz’) = 0,

n’ EP,

(5)

thereby allowing the product states to act as a sink. More specifically, one mcorporates unimolecular reaction mto the restrzcted IVR model by using the expresslgns gven m eq. (2) for all transition probablhties which are not zero because of eq. (5). In ths context it IS convenient to define IR = I n R and If = I n P. where I = JRU Jp. In other words, fR IS the subset of all reactant states belonging to I, and JP IS the subset of all product states belonging to f. We take & for the number of states belonging to JR and 8 for the number of states belongmg to JP. where g/=$+&-

(6)

The master equation, reduced equations

eq. (l), yields m this case the

(8) where

e(t) =t,2R w. .=

0

and

6(t)

= c P(tz, t) . PIEIP

Eq (7) IS a closed set of equations descrlbmg the competition between the slower Q/T, JYR process and urnmolecular decay. One can find relattvely simple expressions for the Laplace transforms Pr(s) = j

e-=

Pf (t)dt

0

and

QJ(s) = J

e-sr [dl$(t)/dt]

0

Usmg standard

manipulations

dt .

(9)

Volume

69, number

1

CHEMICAL

PHYSICS

(11)

1 himary

LEITERS

l$

[1

XC

auf o+(l

1980

-1 (17)

-

1

-a),&

where

The mean lifetune, (0, for arbitrary initial conditions

yr =s+l_-_(Y _

IS eastly expressed as a linear combination of the WJ_ The overall yield of product states be!onging to a partrcular set I is another useful quantity that can be computed from Q,(S)_ One has

rm

Summmg

rm

(12)

gI

eq. (10) over I yields (13)

Eq. (13), when substrtuted mto eqs. (10) and (1 l), completes the solution for the quantrtres PAS) and

QIWLn the specral case, (Y= 1,

one easriy computes

YI = fl(t

= =)

- fl(t

= 0) = Q,(s)l,=,

Defining Y/ to be the yield, given the initial condition one computes from eqs. (i 1). (12). (13)and (18)

#(r=0)=61,J,

&

from

%‘I

eqs. (12) and (13)

Z,,Py(t=O)

~w~la=L=S+(l,s )8,g

Inverting

m the Laplace transform

g’=$$-

’

(14)

gives

~~(‘)IU=‘=~~(r=O)exp[-~~t].

(15)

Eq. (15) descrrbes an exponential loss of total reactant state populatron m a way that IS independent of the mitial drstnbutron of reactant states on the energy shell, and IS characterrzed by a smgle decay ccnstant, (1 /rm)gp/g, that depends only on the total mternal energy. These features of the (Y= 1 decay process, shared by RRK and RRKM theory, make such theones incapable of descrrbmg any laser selectivrty effects [ 14]_ In the general case (CY# 1) the decay behavior 1s more comphcated One can still. however, define a tseful quantity, the mean lifettme [ 161, that 1ssomewhat analogous to the smgle decay constant of eq. (15) (f)=

irzedr=-g

%I,=,

.

(16)

0

For d * l,(t) depends on the uutral reactant states, @(C = 0). Defining lifetime. given the mitial condrtron one computes from eqs. (1 l), (12),

-

drstnbutron of (0’ to be the mean @((t = 0) = sl, J, (13) and (16)

,

J

1

In another paper [ 121 we have used an expression equrvalent to eq. (19) to analyze for those factors favoring the possibility of selective bond breakmg in reactions in whrch two distinct product channels are

open at a given mternal energy. Before going on to a new application for the Yl, which is the computation of the translation energy distnbutton of fragments produced m srmple bond rupture, it will be helpful tc specrfy more closely the degeneracy factors that appear in eqs. (17), (19) etc. One parttcularly simple choice for the relevant degeneracy factors follows from modelling the mokcule’s vrbrattonal degrees of freedom after the energy level scheme of an F-fold degenerate oscillator. Then the energy (above the ground level) is always some integral number of quanta E, =nhX,

n=0,1,....

The total degeneracy of a level with energy, &,, , is equal to the number of wak; m which the n indistinguishable quanta can be distributed among the F dis-

tinguulshableoscillators: g=g(n,F)=(n+FDefiig

I)!/n!(F-

l)!.

(23)

the set I to be that collection of microstates

for whrch the energy is distributed such that I quanta belong to the group-A oscillators and rz - I quanta 21

Volume 69. belong

number

1

to the group-B

CHEMlCAL

oscillators

g~=g(l,F~)g(~l--.~~).

PHYSICS

one has

I=O.

Y,

l,..JZ,

g(l-ttt-I.F4-l)g(tt-I,F~)

=g(l-

m. F&(tz

- I. FB) .

mGI
(23)

Each term m the sum III eq. (22) accounts for Just those product states which result when exactly HI + I quanta hate accumulated In rhe breakable oscillator. Now. m a simple bond rupture one can assume thar these I excess quanta , YICI no zddltlonal energy. wind up as the rranslarlonal energy of ihe product fragments =_ Therefore. definmg PI 3s the probnblhty of finding exactly I quanta of translational energy, and remembermg that all sets 12 HZ+ I \\111 form some product states with 1 excess quanta m the breakable oscillator. one has 1, P, * ,_z+, Y,gU - MI- 1. F, - 1Mlf F,) Fmally. applymg tfle normahzat1on Z$=-O”‘PI = I. one obtams

=\Vc Icier to that Ass of rcxnons acnons. See ref. 141.

Table 1 Translanonal

energy

dlsrrlbuuon

rc) = 0.6

rhat

condltlon

rorsl

LIB 3 iuncuon

g(I-m-&FAg(I-

(31)

where F, and F’, are rhe number of group-A and group-B oscillators respectively and F = F, + FB If product forms wherever IPI or more quanta accumulate m one particular “ breakable” group-A oscillator (the Kasse: crlterla ior reactlon) then 1-W &=,z

1

LETTERS

calls I) pe 1 re-

of

ProbabfllI)

1)

m.FA)

January 1980

I = 0, 1, . ..n-m.

’

(23) for the average transla-

Using eq. (23) one compures rlonal energy

(24) The probablhtles, Pr, and the average translatlorxl energy, k), depend on the IVR rate through the Y, dependence on cr. In the special case CY= 1, eqs. (23) and (24) reduce to the simple predlctlons of RRK theory g(n-m-I,FA+FB-1) P&=1

=

g(n - m, F,+

w,= , =

n F,

+

)?I r;-,

FB)

’

(W

-

For a: < 1 effects due to the form of the lnltlal drstnbutlon of reactant states on the energy shell, and the less than raprd IVR rate begm to play a role. To assess the slgmficance of these factors m determrnmg the translatlonal energy dlstrlbutlon we have wrltten a short computer program to evaluate eqs. (23) and (24) for the maximally non-random mltlal condltlon corresponding to all N vrbratronal @(t=0)=6(,z‘ quanta mltlall$ locahzed m the group-A oscillators. The results of one such calculation are compared to the purely starlstical results, of eq. (25), m table 1. Because W IS m practice [ II] an experunentally deterrruned quanrlrl the reported values of n and (Y were selected to be consistent wlrh the constraint that k) be constant.

IX’R r3rc oil quanta

II

cl

I=D

I=

32

l.ooo

30

b) I.64 x 1O-3

0.613 0619

0 245 0.339

‘8 26

1.77 x 10-3 0

I

of iranskrlonal

energ)

a)

(r)/rm

I= z

I=3

1=4

0.093 0.090

0.033 0.033

0.011 0.012 0.012

3.43 x 10s 7.70 x 10s

0.617 O.~-lO 0.092 0.034 1.72 X lo6 0.600 0.757 0.099 0.033 0.009 3.37 x 10s __ ___3) This calculation HIS cawed OUI assuming ?O degrees of rlbranond irecdom (10 group-A oscfflators and 10 group-B oscfflators) and an ac1wation enerm correspondmg IO MI = 20 quanra 7112 values offi for rhe low probabdny, high enerp tA of the dlsrrlbutlon (5 < 1 C n - ttz) have been omltted. b)a = 1 IS the purei) sratlstlcal result cf RRK theor).

Volume 69. number

f

CHEMICAL PHYSICS LEZTERS

The results reported in table 1 clearly illustrate that translational energy drstnbutrons are far too insensitrve to the IVR rate (cu/T,) to support the conclusion that IVR rates are rapid- The values of Pt for varrous Q begin to diverge only in the ummportant, low probabrbty, hrgh energy tail of the drstnbutton . This msensrtivrty can actually be mferred m a rather general way from the form of eq. (25). On expandmg the relevant factor& (cf. eq. (20)) one obtams for the probability of I quanta of tr~slational energy

where for (Y= 1. F = F, + F, is the total number of degrees of freedom. One can denve an ldentrcal result for Q = 0, except that here F = FA IS reduced to the number of actrve degrees of freedom. Thus, the Pf, when expressed as a function of (~1, drffcr m the WO extreme cases cy= 1 and cv= 0, only m the srze of a small (for small Z) correctron term of order (I+ 1)/F. in conclusron, the vrew that energy m a molecule, excited above threshold by a laser Geld, IS more or less randomfy drstnbuted in all vrbracronal degrees of freedom before reaction can occur IS, m fact, not weti estabhshed. We gratefuily acknowledge support from NSF grant CHE76-g4I80. we dso wash to thank Ms. Sarah Wrrghr for sktied prepararton of the mamscrrpt.

[ 1 j L-S. Rassel, J. Phys Chem. 32 (1918) 12.5, 1065, Kmettcs of homogeneous New Yorb, 1932).

reacttons

(Chemtcaf

Catalog,

(I?] R.k Marcus end OK Rtce. J. Phys. Coffoid Cbem 55 (1951) 894. R.A. Marcus, J. Chem Phyf 20 (1952) 359. P-J. Robfson and K-A_ Holbrook, fkti~mofectdar reactions ~Vdey-Sntersctence, New York, 1972). W- Forst, Theory of unimoferular reactions (Academic Press, New York, 1973). B.S. Rabmorttch and D.W. Setser, Advan. Photochcm 3 (196-t) 1. RA_ hfarcus, J. Chem. Phys. 62 (1975) 1372. J-G. Black, P. Kolcdner, M.J. Shuitz, E. Yabfonovltch and N. Bloembeqen, Phyr Rev. A19 (i979) 70-k J-L. Lyman, J. Chem. Phys. 67 (1977) 1868. J-C- Srephcnson, D.S. King, M.F. Goodman and I. Stone, J. Chen Phys. 70 (1979) -M96. J.R Acherhaft and H.W. Galbnuth. in Laser Specuostop) IV, Proceedmgs of the 4th fntematronaf Conference, Rotlach-Egem (1979), eds. H. Wafther and K.W. Rorhe (Spnn8er, Berbn), to be publtshed. XaS- Sudbo, P-A- Schulz, E-R Grant, Y-R Shen and Y-T- Lee. J. Chcm. Phys. 70 (1979) 912. E- Thtele, hf.F. Goodman and J. Stone, in Laser Appftuttons to Chemtstry tssue of Opttcaf Engmecrutg, ed. T.F. George (Jan./Feb. 1980), to be publahed. RB. Hall and A_ Eialdor, J. Chrm. Phys. 70 (1979) 1027. h¶ F. Goodman, J Stone and E. Thfele, m: Toptcs in current phystcs, ed CD. Cantreff (Sprtnger. Bertin. 1979), to be pubbshed. XI. Sole, MoL Phys. 12. (1967) 101, 11 (1966) 379. N.B. Slater, Theory of ummolecular reacnons (Comc!l kuv_ Press, Ithaca, 1959). E Thtele, J. Stone and hl F. Goodman, C&m. Phys. Letters 66 (1979) 457. M J_ Shultt and E- Yabforovrtch. J. Cbem Phya 68 (1978) 3007. [ 191 A Kaldor, RB. Ha& D_hf. Co\, J.A. Horsfey, P_ Rabmowttz and G hf Kramer, J. Am. Chem. Sot. 101 (LY79) 4165, D-WI. Co\, RB. Half, J.k Horslcy, G.hL Kramer. PRabmovxz and A. Kafdor, Science 705 (1979) 390. [7-O] f- Oref and B.S. Rabmovrtcb, Accounts Chen Res. I2 (1979) 166. [ 1-11 R.G. Kay, J. Cbem. Phys. 60 (f974) 5205.

23