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Quantum-mechanical normal-mode approach to collinear collisions of identical diatomics. Distorted wave formulation
vrihlme-29,
number I /
A purely quantum mschanical description of the .collincar c$ision.of two identical diaGngicsispres+3&titbiri a narmdmode dscompnsitiantechniquepreviously used kmiclassiciilly by Zelecbbw et aLThe mod&k .jhow to extend considerably the range of validity of the distort&wave apprdximation. Tht results are dkusse&and compared to the previous semiclassical results.
1. fnrfoduction
A few years ago, Zelechow et al. [I ] pubfished a semicIassical study of the T-V-V transfers in the coE.near identical diatomics. The main features of this treatment involved a separation of the variables by usln~ normal modes for the system throughout the colkion and use of the Kemer-Treanor solution for the:linGar’ fordng of n harmonic oscillator 121.The aim of the present communication is to-use&e same nom@-mode tech111r~uc ~PIu purely quantum mechanical first order treatment of the problem, then to Mike cIear tIie’&seconnei tiw of our results with those in ref. [ 11. A few recent articles of interest for the diatom-diatom colEdons within tire scruiclnssicul[3,4] and quantum mechanical [S--8] models are given in the list of references.
collidor~ of two
1lru clulksior~systani is pictured in iig. 1. The interaction between the two diatomics is assumed to be a func. don of X and ul+y3 only. Following ref. [ 1J, the two normal symmetric and antisymmetric coordinates are defined fls Y = 2-yYp fl
Y2)
(
Yil = 2-“2(Y,-
Y2),
(2) 111ttlrsc coardinates, aildin using the notationsp = mAmg/(mA+mg)andm ~S(m~+mg),theSchr~nger mol0cq10
1
molmub
2
Fig. 1. Model and notation for the faca to face collinear coltision of two identical xgqlecules.
‘Volkii
29, puinbcr 1
‘t
-_ equatidn fo; the problem
In:
CHEMICALPHYSICS~LkTTERS
:
.-
_’
1 November
1974
.’
is
use of an exponentially
repulsive potentjal:
yin;?
ex&-ur),
it is useful to redefine
cer-
Hence:
iI2 .~y,=C~~w/fi) Y,, y,=(pcd/fi)'i2~a, .y=mA/(mA+mg),
(Y=&fi/~W)‘/2
)
M =nzAj2mBJ
X =~&+&$2)‘Dx,
(4)
; where‘w is the free o&la& frequency (K/g) lD. La&r, Al energies ire measured in units of Aw; in particular .I~us: $ =E/fi~ and Cllint= Y~,,#u. ‘. The assumption that $(X, Ys,Ya) = $(s)(~,~s)$(a~Cy,) results’in the following d,ecoupled equations:
~(j)(x,yg)=8(s)~(s)(x,ys),
-a2 +52 $O)b,j= alla) .-zay,2 ,a. 1
I
‘1
,
(5)
where E$) + Z(y) = E.We observe that the y,-oscillator is not affected by the collision and the equation in )I~ is identical to that describing an atom--diatom collision. However, even though the SchrGdinger equation is most easily solved in the b,.~,) representation, the asymptotic conditions crf the physical problem must be stated in terms of states of the individual 1 and 2 molecular osciLlators. By th:.same conventions as in ref. [l], we denote’ by @,w a state with (k-j+l) quanta in molecule 1 and (j-1) quaka in molecule 2, and by *,?a state with (N-k+l) quanta in the symmetric mode ys and (k-l) quanta k the antisymmetric mode ya. The transformation matrices from one representation to the. other have.been studied extensively and tabulated for different values of Nin ref. [l]. The result is IV+1
.N+l
(6) If the initial state of the oscillators
is +,m, the total collision
wavefunction
should have the’asymptotic
form:
(7) where K:,, is the incidert 1, Q =~c2K~piw
relative wave number
+lv t. i -=-P?{Ko”,(N’)}2/2M
in the diatom-diato?
t Iv’ + 1 .
representation
and (8)
L7’p+&lar, Kin = KoUt(N). IThe quantities KoUt(N’) denote the’ emergent relative wave numbers associat.ed with the open channels. Fdr a .-given ii,, they depend only on the tota! number of quanta exch.a.Qed between translation and vibrations. If all the reduced atom-diatqi problems [&qssl(5)] have been solved for a given collision energy 8,_oll 5 CY’K$/ ‘...&, with the reduced iyrknetric oscillator being initiaJly in state j = 0,1;2, .:.fl,successively, then the solutions. .- .. fulfrl’n&essarily the asymptotic co’xiditions: :. ; ._ ‘.. .: _’ ‘. ” .. _’ _; ,,: : ._ .. .._,. .,.
V.&me
29, number 1
1974
..-
..
open
:. - e~p(~~i~~~~~~)
liwi +9(x;;,) x+1.
1 November
CHEhiICAL PHYSICS LETTERS
.)
+ F
ex~~~~;,CN+il-i)~B;i-f~~CYs)
(91
, :
..
where ~~s~=p2K~~~ulrtj~~~2=cr2{Koo,(N+f-j)}2/~+f~1/2
,.
.’
_,,
(10)
and+& =0,-l, 2... des&nates the set of the internal states of a harmonic &ci.Uator. ‘, The linear combination N+1WJ,.‘i,)
= g
: ‘.
c~?~~~~,Cx,u,)~~-r01,‘)
a
,;
(11)
is a solution of the Schr~~ger equation for the diatom-diatom problem, and it is readily vetified that it behaves correctly in the asymptotic region for the system initially in state Qjm. Thus the knowledge of the set of solutions of eqs. (5j allows one to obtain readily any solution of the diatom-diatom problem with a total number of quanta in both molecules less than or equal to N, whatever the initial dist~bution of the vibrational energ over tie two molecules. Reformulating the gsymptotic expression (9) ti terms of the coordinates CyI,y2) is the last step in obtaining explicitly the transition probabilities. The results are as follows. (a) Vibration to vibration (V-V) energy transfer (resonant processes):
(12) (b) Transiation to vibration (T-V) energy trhnsfer (n&resonant
processes):
Iv+1 P(sPJ~‘@]PT))
=Ko,t(N’>Ki~~
g
F_ q&p&*~+i~‘_~++i
2:.
All the B’s appearing in eqs. (12) and (13) are “‘he same as in eq. (?) which describes ffie fictitious at&-ciiatom process. If there appears in eq. (13) a coefficient cp which lies out.Gde the limits of the corresponding matrices in ref. [l J -for instance C’& forN= 2 andN’ = 1 - it should be takea equal to zero. : :
3. Distorted wave approximation (DWA)
The basic technical features of this Bjproximation-as applied in eqs. (5) ( I.e.; to treat the problem as an-atomdiatom syitem) are as follows. The equation involved is
“‘(X)
=
u,
(Isa)
79
v&ine29,numb‘er ,._ .I...
1
‘-
‘.
CHEMICAiPiIYSICS
LETT’EXi
‘.
.
.,
-: -
.-
1 No&be;
1974.
‘.
eq. (15a),, the &rnptotic behavx + +m. R. stands for order-ap&oximation for .pbtentialVjj(~) obtained by is then used in the rhs of eq.
.’ .Ai-‘/I_(K/T,!Ji&) CF,‘o’(x)l~,(x)l~~~(x)~ , : “a) ::I._.., where F] (x):Is~,@ homogeneous solution of eq. (15b). Finally
‘.
‘.
(16)
the CIW-wavefunction
is (17)
Its asymptotic
behavior
forx + fm is
.:. (18) ‘. In order to get the asymptotic form to conform with that’of eq. (9), t-ie function ismultiplied by -2i exp(ibj) s,o that ei_,f may be written as -2i kj+f exdi(6j + 8$}. Th e insertion of the above results for,,the description of each : fictiti6us~atomTdiatom collision into the formalism for the true diatom-diatom collision results in the following expressions; -. .: (a) VTV probabilities:.
_.’
‘.(b) T-V
(19)
probabilities:
~&]W i &,P” :
.,
N+l X E. x r&k=1
Cky C$’
CF$~~+l((x)l 617~_~+~~~_~~(x)l F$?,jffl(x>i
: where $9 jjT-k+ldviV’-k+l(x) = “L,~_k+lCV,)r~int(X,YS)I
exp{i(S$_k+l + 6$Lk+l))
’,
(20)
u~~yk+lCY~)’ .
The lower indexin the F(O)‘s and G’s denotes the number of quanta in the reduced symmetric oscillator; the upper index in or out identifies the associated collision wave number 2s Kin Ior Ko,,(N’), respectively. .. -The&formulae arc easily proved to satisfy the principle df detailed balance. It should be emphasized,that all :..transition probabilities (V-Vas well as T-V) are obt&ed.by solving purely elastic Schrijdinger equations, and ‘, computing integr+nvolving these functions. I :.,. The V-V transfer, in particular, arises from the occurrence of the .$ous zeroth order elastic atom-diatom ~,cqlLisions in different effective potentials’causing the cbrrespondingw~vefun&ns to’haye different asymptotic .’ .‘, ;.-p@s.‘~ ., ._ : ‘.. :8d’ _; -/ :.. .: : -.. ‘; .. :. ._ ‘, ,.. : ’ ., ., .: ,’/.? :: ,. .! P_, ‘. .‘.
Volume 29, number 1
;.
.4. C&e
of
expo~entja~y
Then a &ential
..
1 November 1974
CHEMICAL PHYSICS LET’liRS
:
ene&
= Gp(-x)
vin&,ys)
repulsive function
interaction
”
of +-he f&n
’ ,
” ,‘, :
’
{I +21~cI*yS +c2(YzJ$j
:
I .,
.:
_ ,.
‘.
,’
(21)
‘is used. This p~atential originates’from the usu& form: ?’int a exp(-ar) (see fI& I) by expanding the exponential to the second ,order my,_ The parameters CI and C,, which should mathematically equal one, are iqtroduced In order to study,the
physi&l
effects of their variatidn
on the transition
probabilities.
The ,first order term in Cl COP-
responds to the force acting on the reduced symmetric oscillatbr; physically it descritjes the direct T&V tra-tsfers. The second crder term in Cz allows the oscillator to vary its force constant ,thiouglr me colbsion, i.e., to adjust itself progressively to the external force, and essentially induces the V-V transfers. Higher order ierms are neglected.
~. Insertion of the potential in eq. (21) into the expression of the effective potential ckY$r) results in
vjjCx> = expf-x1 70) = expf- b - WY)11 C(O) ,
(221
where y(j)=
hx(j)‘=ln[y(j)/y(O)j
1 +C,c~~(j+1/2),
1 .’
4.1. v- ~~r~~~~~~~ties
‘:
‘After some sk~plification, the resuhing equation is 2
(231
;
when ci!ii small, a first order treatment is satisfactory and leads to results which are’more simple and easier to analyse physically. Thus the appro,ximation: ln[l’(j)/y(@)] = C20r2j leads to the formula: (24i
.
4.2. T- V probobilkies
.’ Some intermediate results for the distorted wave treatment of the reduced atom-‘~a~orn system wXrin an ex.ponentid interaction were previ&sly given by Mies [9]. In the case N! = N+l , their uss results in: Sinh(2~Ki,)Sinh[2nKo,t(N+1)] F(q?
’
‘,
= (4?7iFiCr/a)2
-+Y)
..
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29, number 1.
CiIEhfICALPHYSICS
.’
.
LETTkR:;
‘1
November 1974
. ..: ., .:‘- Wheri a is smail, the vaAab!e of the hypergeometric fijndtion.2F1, wluch depen.ds on LY,can be expanded and truncated to first order: I - 7(N- k + l)/y(N-,k +2) k @..In addifdn [7(N- k +2)1-l is taken equal to one ti-d the, G’s are approximated in the same way as in section 4.1. ThUS the following approximate formula is ob‘.t&ed, ,. Sinh(2?TKi,)
f&y
Sillh[22TK,,~(N+l)]
j’ +.a@-+‘)) = (4?NC,/ar)2 {Cosh(27TKi,) - COSh[2nK~,,(N+l)]}2 IV+1 x
iF,(l
2’
+j.K(? ,.l +iEi(-);2;Czor2)~~~-C~~C~~~1)(N-k+2)1’z~xp{iK’t)C~~2(~-k+l)}
.
(27)
This approximate exp&sion has proved quite reasonable for a G 0.15. Moreover its simp!e form makes the physical int’erpretation-easier. The formulas for N’ =N+2 are not given here since, within the distorted wave approxima-tion, the .corresponding results are known to be unrealistic.
5:Discussion 5.1. Gct&~l considerctims
It shquld first be emphasized that the -method is by no means equivalent to a distorted-wave (DW) treatment & dk@&) representation. The latter would be based on, the assumption that all probabilities are small. In contrast, the present treatment only a&.umes that the T-V probabilities are small; more precisely it is assumed that the .DW approximation correctly descriges each of the implied reduced atom-diatom collisipns. Since, for many realistic collision systems, there exists a range bf.collision energies within which the V-V probabilities are substantial wl@e &se for T-y are stitl small, a serious extension of the range of validity of the perturbational theory for this special problem by use of the present, technique is forseeable. Any fundamental drawback of the distorted-wave approximation is however reflected in our results. In particular, the sum of aU probabilities exceeds unity, since tile purely resonant V-V probabilities add to one. Morebver, multiquanta ,?-V transitions are very poorly described by a DW method. These two shortcomings could be, at .leziii in part, .corrected by using a K-matrix formalism for each of the contributing atom-diatom processes [lo]. .’ It is worthwhile to remark that transitions occurring essentially via a sequence of elementary processes involving a global exchange of no more thm one quantum between translation and vibration should be correctly described. ,Heie we call an elementary process any transition (ni, r22)+ (n;;tzi) for w&h the matrix element ‘OZ~~Y~)~n~C~~)l_CL”~n*~x~Ys)l~~~y~)~~z~~Y~)~ is ncn-zero,
vhere
-y&j,)
= i-y1
according
to eqs. (21) and (la):
+C,~7(Y~‘-Y2)+~c~~272(Y,fY2+.2Y~
’ Y,),..
For instance, the tfan,sition (2,O) --t (1,2) occurs iia elementary processes whose amplitudes interfere in the overall ‘:amplitud& The elementary processes that are possible in the present model are denoted by bars in the diagram within theN = 2 or 3 subspace of states 021, nz)_ ’ The l?W.cpproximation neglects,the back coupling for T.-V processes. The arrowsin .+hd diagram indicate that ‘. ,’ the ttisition is from a 2-quanta state td a 3-quanta state. ‘. 1’ .The ac_cuiAy ‘of our results can: be expected to be of the same order as. that of the: DW approximation.for ,‘:_ .; +:
2
.,.. .; ‘.. ._... i .:
‘:
I
: ;.
;: -I .,.I.
-,.
‘.,
,,
_,‘, .,: ‘,_ ,. ... .. . ,_ .’ ._
.. ._
Volume 29,numbc1
CflEMlCALPHYSICS.L~~ERS
I’
1 November
,:
1974
‘.atom-diatom problems. For these problems, it is known that the DW probabilities at low collision energy exhibit a systematic error depending only on the parameters M and cr of the system ([Z] and references therein). For many systems of interest, this error is typically qf the order of 2.5% and it becomes more important for large Y& ues of M, i-e,, for strongly adiabatic collisions. This is esse&aliy due to the failure of the DW approdmation to account for the compression of a molecule in the course of a collision; this could be corrected for bf treating each reduced atom-diatom collision within a perturbed-stationa~state (PSS) approximation. In the range of high collision energies, the method proposed obviously breaks down because of the on&t of the strong coupling regime for each atom-diatom collision.
.Since the approximate expression of the resonant transition probability in eq. (24) provides rather valid results in most cases (smah Q, see tables I and Z), it is clear that the V-V transition probabi~fies are essenti;aIIy given by s~p~e’t~gonomet~~~ functions of the argument 5~. An array of the explicit expressions of these functions for different values of N, j and j’ is not given here; the reader can find it as table 2 of ref. [I]. But he must keep in mind that P does not have quite the same significance as in the present article although the formal expressions Gf p are the same in the particular case of an exponential interaction (see section 5.4). In fig.,& the curves representing the variations of these functions versus jp are drawn for N 5 3. p is a func~on of 8C0ll as well as M and cy.Moreover the range of p in which the DW expressions lead to rea sonabIy good results corresponds to the values of p below those for the strong coupling regime being set; for instance, iFM= 0.5 and Q = 0.112, the upper limit of this range is p = 0.8; this is typical of a case where the atoms A and B have’masses close in magnitude. Consequently the range of validity of the formulas proposed varies from one set of parametersM and Q to another. This is.clearly observable in tables 1 and 2.
In the same way as in section 5.2 and within the same range of p, the appro~mate expression of the non-’ resonant transition probability in eq. (271, whose validity is extensive (see tables 1 and T), shows that the T-V transition probab~ties are approximately products of weighting factors depending on G,,, &f, (IIand the number of quanta exchanged between translation and vibrati,on~multiplied by trigonometrical functions of ~‘7 d&K(~) which depend on the trans.itions actually studied. Some .of these t~~onomet~~~ factors am given explicitly in table 3 for N varying fr&m 0 to, 3 and N’ IN-1 ; for’N=2 and N’=3, the variations of different T-V trigonometric~ fabtors corresponding to different couples 6.i’) are plotted against cpin fig. 3. It appears that for ., (. .-
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‘,..
:
T&fc’l ., : ,. .‘. .~. ,: ., _‘,’ .‘,;.,_I -z , ,.:.._(:@$la&d transitionprobtibilitiesat different& T_.’ coll of severalcoIlid& fysteiil~~(characteriicd by M and a), for.Ci = Cz = 1.0:The two:numbers’aljpearingin TV 8. ”x0., ” .’ .: ach’tcrni of the nrr?y.reprcsentin the left the “cxatit” DW result [cf. eq. (23) or (26)], on the right the.app&imnte ok [c&c+ (24),0~,!27)]. 9.28-l ;e,re-.z ‘.,,,,.,_ ‘..a: I ,. : -2 &IlS9.8~X lo-‘.., . . ._ L ;‘., “: . ‘. ,’ /,.‘. .*
--,
03
,‘a
i,,, .*
k”
..I.
;.
,:‘.
_.;.
Volume 29, n&ber
1
1 November 1974
CHEMICAL PHYSICS LETTERS
Tadle 2 Results of the study of the isotope effect in-the collision of hydrogen-type molecules driven by the =me interaction potentiat The four nuqnlxrs in each term (which are exact DW results) correspond, respectively, to H-H + H-H, M= 0.5, a= 0.315; H-D+D-H,M=0.25,rr~0.195;D~H+H-D,-M=1.0,~’=0.391;D-D+D-D,~~=0.5,u=0~265
., %oll
Transition
0.5
1.1
1.7
2.3
2.9
3.5
‘9.92-l 9.91-l 8.89-l 9.70-l
9.12-l 9.81-1 7.67-l 9.34-l
8.67-l 9.70-l 6.56-I 8.99-l
8.22-l 9.60-I 5.57-I 8.65-l
7.80-l 9.50-i 4.67-l 8.32-l
7.3471 9.39-l 3.88-l
8.78-2 1.93-2 2.33-l 6.60-Z
1.33-l. 2.97-2 3.44- 1 1.01-l
1.78-l 4.00-2 4.43 -1 1.35-1
2.20-l 5.03-2 5.33-l 1.68-I
2.6 L-1 6.05-2 6.12-I 2.00-l
5.00-S 7.31-6 4.87-6 4.71-6
9.28-3 3.29-3 1.64-3 2.42-3
5.31-2 2.57-2 1.06-2 1.98-2
1.40-l 8.26-2 2.78-2 6.43-2
2.63-l 1.ac-L 4.95-2 1.40-t
(1,0~(1,1)
2.49-5 3.88-6 2.80-6 2.36-6
4.88-3 1.63-3 1.10-3 1.26 -3
2.94-2 1.29-Z 8.25-3 1.07-2
8.15-2 4.19-2 2.54 -2 3.61-2
1.61-I 9.25-2 5.33-2 8.L 5-2
(l,W(O,2)
1.91-6 6.57-8 5.32-7 1.34-7
8.73-4 6.52-S 4.70-4 1.701
7.89-3 7,96-4 5.07-3 2.18-3
2.88-2 3.46-3 1.99-2 9.75 -3
7.01-2 9.53-3 4.93-2 2.72-2
(1,0)-+(1,0)
u,ol-a
(1,0)+&O)
1)
8.20-3 8.80-3 l.li-1 3.04-2 ‘.
c
6.00-L
Fig. 2. Variation oFthe approximate resonmt transition probabilities versus 4~ [cf. qs. (24) and (25)] for N= 3. Note the two transitions (3,0)+(1,2) and :3,0)4(0,3) would have a zero probability of occurrence for a DW. treziment in the bbyz) representation with tie trunuted potential in eq. (21).
reasonable values ofM and 01,at high 8coll, some transitions that would not be taken into account by 2 DW treatreach probabilities of Occurrence of the same o:der of magnitude as those allowed by this treatment. In that respect the method proposed extends considerably the range of validity of perturbation methods. In the harticular case of a highly asymmetrical system (e.g.; either CM +.HCl or HCl f Cl.H) the T-V transition probabilities remain quite small and insignificant up to much higher v&es of p, so that several cstiations in the variations of the‘V-V transition probabilitiej versus p occu’r H;iUrin the range of validity oftbe formillas. However, the collinear model clearly is p&ticularly unrealistic in such situations. ment in the (JJ~? y2) representation
8.5 :
~olume’29;number
CHEMICAL-PHYSICS
1
._ :: ‘. . .. : ,. .’ : : f’ ~.k&i& factors appearing in the approsimaie
LE?ERS,
1 November
:
:.
1974
.
Tible 3 ‘. T-y transition probabiiitiesP(*m -+ @lfc’f’) .>.[cr. eq. (i?jj for N= 1, &3;The.‘_ c, factors for F= n-tN’G+l with $coll =, e.are the same as those forN= n+l-if ._ I = n ‘cvlt Zcoll = e-l @ticiple ofdctafied bal-. ..:, .i :.’ ?nCe). The abbreriatioas C =. GO&Y,S = sin& and ,Y= $‘p are used .’
..
..
:
:
-Quanta--
,’
0
:-
1
:
‘2 :
‘.’ ..
. . ..
2’..
3.
” :
(0,0)-r
(1.0)
‘. (l,O)-,
u.0;
.‘.
..
‘:
[l,d)-w
eq. (27)
‘.
14.
,, .(2,0)-r
.’
._,
.. ‘. :
”
.,,
.
(c),2)?)
.
(3,O)
‘.
lip ‘. .-
.:.
:
‘.
.:
.. ,‘,
2r6 C4(1-.-8S”)iZ 3C%$_
.’
.:’
are’not taken into account bj, a DW treatment
..
:
3P 2pb 9p4l2 6P2.
3C2[ l&{3’8C2)j/2, l-3C2S2 3S2[1+C2(3-BS2)]/2 6S2C’L
in the (y 1,.;12>representation
:
~~1~~~~~~
,.
6C4‘$“‘
:
‘,
3P2 l-3P2
‘;;k+BC*)/2 ;‘._
32 3@j2
1-3c2s2
(3,0)*(4.0) : (3.0)~(3,13 (3.O)-l,(2,2)‘) (3,0)-t(1,3)a) !3,0) -c f0,4)“)
foetors
(l+p2)/2
.’
‘.3cW
(3,X) (231) -+ (2,2j (2,lI-F (1,3)“) f2,lj-c (0,4j3)
ET_Vhigcmmtrical
”
S2(1i,3C2)/2 3S4j2
.,
l/2 p? 3(1 -2P2j/2
3c4/:1.
‘, @(li-3s*)/2
c&1)-
:
transitions
,for small q
;
l/2
s2
(2,0)* (2,l) (2,0)-+(1,2$ f2.0)-, co,3F)
iZ,lj-(4,0)@,
,’ .“? The
.,
: l/2 ..’ cz .‘, ‘I;/2 .- ‘1
.,.
(l,l)-+ (3,01a) (l,l)-c(2,1) 3:
leading term
: ., ‘.’ -. ‘_
.’
.,; .
:
Angul& factor.’
U,Ol+(~,l)
.
‘.
.’
&I*p(n’p7;)
.:
:
:
.:
Tr&sition
NN’
N.,.. :
.’
:
3(1-6p2)/2 1-3p2
:,
w/2 :
6p2 for the truncated
potcntiai
in eq. (21)
‘.
,’
‘.,.’ .,
‘_
..
. ..
. .
,“,,
_..
.,‘,
., ,,: : .,”
.: ‘.
‘,
.-;I ,. .tiote
...,, ,_
..I’) &&three
t in $6 iy$y2)
.. : :‘,
1 November 1974
CHEMICAL PHYSICS LETTERS
“Volume 29, number 1
Table 4 Ipfluencz of C2 on the transition probabilities. The V-V transfers clearly increase for increasing C2 whereas the-T-V tramfen that do not require an intermediate V-V elementary pro&s decrease (anh&monic factor). On the other hand T-V transfers that rcquiie non-zero c2 tend to increase for jrjcreasing Cz until about C, = 1.3 H-H+H-H,M=b.5,~=0.315,C1=1.0,&coll=2.0 Transition
*’ c2
(2,OH2,0) (Z,Ofiz(l,l) (2,Ofi(O, 2) (l,lti(l.l) GGOI-GO) (2,0)-+(2,1) (2,Ofi*(1,2) (2, W(O, 3) (l,l)-r(3,0) (I,l+G,l)
0.0
0.3
0.6
0.9
‘1.2
I.5
1.00+0 0.00 0.00 1.00+0
9.68-l 3.20-2 2.68-4 9.36-l
8.85-1 1.11-l 3.54-3 7.77-l
7.73-l 2.12-1 1.46-2 5.76-l
6.51-1 3.12-L 3.77-2 3.76-l
5.29-L 3.96-l 7.47-2 2.07-l
1.00-l 3.33-2 1.55-3 1.49-5 2.38-3 6.43-2
6.35-2 2.33-2 3.57-3 1.27-l 5.54-3 3.90-2
3.97-2 1.72-2 4.66-3 3.51-4 7.27-3 2.31-2
2.46-2 1.29-2 4.88-3 6.144 7.56-3 1.55-2
1.5 L-2 9.74-3 4.52-3 8.44-4 6.90-3 3.93-3
1.54-1 5.14-2 0.00 0.00 0.00 .1.03-l
The effect of varying C2 from 0 to 1.5 on the transition probabilities (see table 4) is shov+n partly in the followi&remark: in addition to the fact that the terms in C, and C2 in the potential are essentially responsible for the T-V and V-V transfers respectively, the term in C, reduces the amount of T-:V transfer by means of the hypergeometric function 2F1 - cf. eq. (27) - it&modulus decreases drastically for increasing C2. This is known in literature for atom-djatom collisions as the alzhanlzqnic factor of Mies [9]. 5.4. Comparison with the semiclassical formdatiox The results presented in this article are quantum mechanical and of the first order. in perturbation ‘&ey are closely related to the semiclassical results of ref. [l]. In particular:
theory.
- Our formula (24) for resonant transition probabilities is exactly the same as formula (75) of ref. [l]. The respective derivations are, however, quite different. In ref. [l], p is associated with a.phase shift in time c&the intramolecular wavefunction; in the present work, the counterpart of it is a phase shift in spat-3 of the relative motion wavefunction (compare the defmition of p in eq. (25) above with eq. (55) of ref. [l] for 2 generaI semiclassical definition of p and with eq. (57) for the case of exponential iepulsion, respectively). - Our formula (27) for non-resonant transition probabilities is very similar to formula (61) of ref. [I] as applied to the case where m = N+l ; this was expected since formula (6 1) is a low collision energy approximation. Mareover, in order to make both formulas as similar as possible, the relative velocity used in the semiclassical treatment .- should be the average of the initial and final velocities. This procedure is well known in atom-diatom models [l I]; it appears to remain valid for diatom-diatom problems and is consistent with a first order perturbation treatment in qutitum m,echznics. Formula (61), however, contains nothing corresponding to our function 2Fl;. this stems probably from ‘their approximation for the treatment of the quadratic term in the potential.
6. Conclusions
A quantum mechanical ,version of the semiclassical treatment in ref.. cl] lias been developed and simpIe formulas have been proposed for the T-V-V transition probabili&s..The formul& ) both treatments have b.ee,nshown to ..’
:
. ‘, :.
87
.
‘. ‘.
..’
_
:.
,.
_;,,‘.V&me. .
2?,‘L&er
1. .
-:,, :: ,.:‘- 1 &vembel: 1974 . ‘;
,‘:1-
CHEMICAL ~~~YSIC&-?:El$
_:_.. . ‘. :,: ,: ._ ,’ .. :. i ._.. .,*, ,.,’ ,’ ,I,,be ilosel$ re$afkd t\o each, other. However, the qua&urn m,ochtical results’do not suffer tlie usuhl ‘shortcomi&s ..,’ pf ~~c.~s~c~ theory, i.e.., n~n-c~nse~ati~~ df energy and.&o&dt ‘behavior ndar t,& energetic threshold. More... oyFi,.;Lhe piksent:appidacfi is Sow being improved in a Straightforward iay by &i$a’bette; de~c~pti~n’of each ..
‘reduced atom-d&&m cofl+m: the work for s&h a description wiih& an adiabatic PSS framkwork is nearly’finis&d. Ai exact quantum mechanical tre&m&nt is’also possible bg IXXII~standard prq&ms which solve numerid ...ly-problemsofthisnature. ’ .,; ‘. :‘,, ..’ .’ : ,‘. ,;, : ,
-
,> ,,., ‘. .. .. . ‘, -. _’ ..: : .’ “,_:~~~~~r~d~~~~~ : : . . .- ;; ‘, ,. ” :,, ‘,., ,’ .‘, .._,,?‘he a&.&i aie indebted to Mr. R. ~7et~o~e.for m&h helpful assi&nce, Dr, J. Manz and Mr. C. Leforestier’ .. are gratefully ‘.
acl&owledged
for frxitful
discussions.
-
._
:.
:
. ...Refeten& ._
., .‘-
c
..’
1
‘.
‘,
.‘y .’
::
‘. .’ . . [ 11‘A. Zeliih&v, 13.Rapp.~&d ‘T.E. Sharp. 3. tiei.
.’
‘.
Phys. 49.(1968) 286. ,,:: .[2]‘D. Rapp and T,Kasul, Chem. Rev. 69 (1969) 61. ;‘._ [3] !.G. Kelley,J. CXem. Phys: 56 (1972)6108.’ ,,, [+I ,,H.K.~Shin,Chem. Phys Letters 3 (1969) 56.0. ..” .. .’ Is] M,E:Riley,and 4. Kuppermann, Chem. Phys. Letters 1 (1968) 537. .. .f6] V. Gutsc%ck, ~.‘McKoy and D.‘Diestler, J. C&em. Phys. !&?(1370) 4801. f7),‘q.J. Wilson, J. Chem. Phys 53 (1970) 2075.’ : ‘. IS]. M-H. Alexander; 3. Chem. Phys. 59 (1973) 6254. 19]‘F.H:Mies, J. Clxm. Phys. 40 (1964) $23:. .. [iO]‘?. Secres$ Ann. Rav; Phys Chem. 24 (1973) 37% [lf], F. Htidrich, K. Wilson a$d R. Rapp, J. Chcm. Phys. 54 (1971) 3885.
,.
: _.
: ‘.
;.. .. ;:.
; ,. “, ‘,
:
_(. .-
,,
:
‘.’ : ,.
‘.’
: :
..
“‘_ ”
.:
,.’
.-.
”
‘. ‘:
:
I
.‘
:
;
..
.. ,:
‘, ‘.
:
number I /
A purely quantum mschanical description of the .collincar c$ision.of two identical diaGngicsispres+3&titbiri a narmdmode dscompnsitiantechniquepreviously used kmiclassiciilly by Zelecbbw et aLThe mod&k .jhow to extend considerably the range of validity of the distort&wave apprdximation. Tht results are dkusse&and compared to the previous semiclassical results.
1. fnrfoduction
A few years ago, Zelechow et al. [I ] pubfished a semicIassical study of the T-V-V transfers in the coE.near identical diatomics. The main features of this treatment involved a separation of the variables by usln~ normal modes for the system throughout the colkion and use of the Kemer-Treanor solution for the:linGar’ fordng of n harmonic oscillator 121.The aim of the present communication is to-use&e same nom@-mode tech111r~uc ~PIu purely quantum mechanical first order treatment of the problem, then to Mike cIear tIie’&seconnei tiw of our results with those in ref. [ 11. A few recent articles of interest for the diatom-diatom colEdons within tire scruiclnssicul[3,4] and quantum mechanical [S--8] models are given in the list of references.
collidor~ of two
1lru clulksior~systani is pictured in iig. 1. The interaction between the two diatomics is assumed to be a func. don of X and ul+y3 only. Following ref. [ 1J, the two normal symmetric and antisymmetric coordinates are defined fls Y = 2-yYp fl
Y2)
(
Yil = 2-“2(Y,-
Y2),
(2) 111ttlrsc coardinates, aildin using the notationsp = mAmg/(mA+mg)andm ~S(m~+mg),theSchr~nger mol0cq10
1
molmub
2
Fig. 1. Model and notation for the faca to face collinear coltision of two identical xgqlecules.
‘Volkii
29, puinbcr 1
‘t
-_ equatidn fo; the problem
In:
CHEMICALPHYSICS~LkTTERS
:
.-
_’
1 November
1974
.’
is
use of an exponentially
repulsive potentjal:
yin;?
ex&-ur),
it is useful to redefine
cer-
Hence:
iI2 .~y,=C~~w/fi) Y,, y,=(pcd/fi)'i2~a, .y=mA/(mA+mg),
(Y=&fi/~W)‘/2
)
M =nzAj2mBJ
X =~&+&$2)‘Dx,
(4)
; where‘w is the free o&la& frequency (K/g) lD. La&r, Al energies ire measured in units of Aw; in particular .I~us: $ =E/fi~ and Cllint= Y~,,#u. ‘. The assumption that $(X, Ys,Ya) = $(s)(~,~s)$(a~Cy,) results’in the following d,ecoupled equations:
~(j)(x,yg)=8(s)~(s)(x,ys),
-a2 +52 $O)b,j= alla) .-zay,2 ,a. 1
I
‘1
,
(5)
where E$) + Z(y) = E.We observe that the y,-oscillator is not affected by the collision and the equation in )I~ is identical to that describing an atom--diatom collision. However, even though the SchrGdinger equation is most easily solved in the b,.~,) representation, the asymptotic conditions crf the physical problem must be stated in terms of states of the individual 1 and 2 molecular osciLlators. By th:.same conventions as in ref. [l], we denote’ by @,w a state with (k-j+l) quanta in molecule 1 and (j-1) quaka in molecule 2, and by *,?a state with (N-k+l) quanta in the symmetric mode ys and (k-l) quanta k the antisymmetric mode ya. The transformation matrices from one representation to the. other have.been studied extensively and tabulated for different values of Nin ref. [l]. The result is IV+1
.N+l
(6) If the initial state of the oscillators
is +,m, the total collision
wavefunction
should have the’asymptotic
form:
(7) where K:,, is the incidert 1, Q =~c2K~piw
relative wave number
+lv t. i -=-P?{Ko”,(N’)}2/2M
in the diatom-diato?
t Iv’ + 1 .
representation
and (8)
L7’p+&lar, Kin = KoUt(N). IThe quantities KoUt(N’) denote the’ emergent relative wave numbers associat.ed with the open channels. Fdr a .-given ii,, they depend only on the tota! number of quanta exch.a.Qed between translation and vibrations. If all the reduced atom-diatqi problems [&qssl(5)] have been solved for a given collision energy 8,_oll 5 CY’K$/ ‘...&, with the reduced iyrknetric oscillator being initiaJly in state j = 0,1;2, .:.fl,successively, then the solutions. .- .. fulfrl’n&essarily the asymptotic co’xiditions: :. ; ._ ‘.. .: _’ ‘. ” .. _’ _; ,,: : ._ .. .._,. .,.
V.&me
29, number 1
1974
..-
..
open
:. - e~p(~~i~~~~~~)
liwi +9(x;;,) x+1.
1 November
CHEhiICAL PHYSICS LETTERS
.)
+ F
ex~~~~;,CN+il-i)~B;i-f~~CYs)
(91
, :
..
where ~~s~=p2K~~~ulrtj~~~2=cr2{Koo,(N+f-j)}2/~+f~1/2
,.
.’
_,,
(10)
and+& =0,-l, 2... des&nates the set of the internal states of a harmonic &ci.Uator. ‘, The linear combination N+1WJ,.‘i,)
= g
: ‘.
c~?~~~~,Cx,u,)~~-r01,‘)
a
,;
(11)
is a solution of the Schr~~ger equation for the diatom-diatom problem, and it is readily vetified that it behaves correctly in the asymptotic region for the system initially in state Qjm. Thus the knowledge of the set of solutions of eqs. (5j allows one to obtain readily any solution of the diatom-diatom problem with a total number of quanta in both molecules less than or equal to N, whatever the initial dist~bution of the vibrational energ over tie two molecules. Reformulating the gsymptotic expression (9) ti terms of the coordinates CyI,y2) is the last step in obtaining explicitly the transition probabilities. The results are as follows. (a) Vibration to vibration (V-V) energy transfer (resonant processes):
(12) (b) Transiation to vibration (T-V) energy trhnsfer (n&resonant
processes):
Iv+1 P(sPJ~‘@]PT))
=Ko,t(N’>Ki~~
g
F_ q&p&*~+i~‘_~++i
2:.
All the B’s appearing in eqs. (12) and (13) are “‘he same as in eq. (?) which describes ffie fictitious at&-ciiatom process. If there appears in eq. (13) a coefficient cp which lies out.Gde the limits of the corresponding matrices in ref. [l J -for instance C’& forN= 2 andN’ = 1 - it should be takea equal to zero. : :
3. Distorted wave approximation (DWA)
The basic technical features of this Bjproximation-as applied in eqs. (5) ( I.e.; to treat the problem as an-atomdiatom syitem) are as follows. The equation involved is
“‘(X)
=
u,
(Isa)
79
v&ine29,numb‘er ,._ .I...
1
‘-
‘.
CHEMICAiPiIYSICS
LETT’EXi
‘.
.
.,
-: -
.-
1 No&be;
1974.
‘.
eq. (15a),, the &rnptotic behavx + +m. R. stands for order-ap&oximation for .pbtentialVjj(~) obtained by is then used in the rhs of eq.
.’ .Ai-‘/I_(K/T,!Ji&) CF,‘o’(x)l~,(x)l~~~(x)~ , : “a) ::I._.., where F] (x):Is~,@ homogeneous solution of eq. (15b). Finally
‘.
‘.
(16)
the CIW-wavefunction
is (17)
Its asymptotic
behavior
forx + fm is
.:. (18) ‘. In order to get the asymptotic form to conform with that’of eq. (9), t-ie function ismultiplied by -2i exp(ibj) s,o that ei_,f may be written as -2i kj+f exdi(6j + 8$}. Th e insertion of the above results for,,the description of each : fictiti6us~atomTdiatom collision into the formalism for the true diatom-diatom collision results in the following expressions; -. .: (a) VTV probabilities:.
_.’
‘.(b) T-V
(19)
probabilities:
~&]W i &,P” :
.,
N+l X E. x r&k=1
Cky C$’
CF$~~+l((x)l 617~_~+~~~_~~(x)l F$?,jffl(x>i
: where $9 jjT-k+ldviV’-k+l(x) = “L,~_k+lCV,)r~int(X,YS)I
exp{i(S$_k+l + 6$Lk+l))
’,
(20)
u~~yk+lCY~)’ .
The lower indexin the F(O)‘s and G’s denotes the number of quanta in the reduced symmetric oscillator; the upper index in or out identifies the associated collision wave number 2s Kin Ior Ko,,(N’), respectively. .. -The&formulae arc easily proved to satisfy the principle df detailed balance. It should be emphasized,that all :..transition probabilities (V-Vas well as T-V) are obt&ed.by solving purely elastic Schrijdinger equations, and ‘, computing integr+nvolving these functions. I :.,. The V-V transfer, in particular, arises from the occurrence of the .$ous zeroth order elastic atom-diatom ~,cqlLisions in different effective potentials’causing the cbrrespondingw~vefun&ns to’haye different asymptotic .’ .‘, ;.-p@s.‘~ ., ._ : ‘.. :8d’ _; -/ :.. .: : -.. ‘; .. :. ._ ‘, ,.. : ’ ., ., .: ,’/.? :: ,. .! P_, ‘. .‘.
Volume 29, number 1
;.
.4. C&e
of
expo~entja~y
Then a &ential
..
1 November 1974
CHEMICAL PHYSICS LET’liRS
:
ene&
= Gp(-x)
vin&,ys)
repulsive function
interaction
”
of +-he f&n
’ ,
” ,‘, :
’
{I +21~cI*yS +c2(YzJ$j
:
I .,
.:
_ ,.
‘.
,’
(21)
‘is used. This p~atential originates’from the usu& form: ?’int a exp(-ar) (see fI& I) by expanding the exponential to the second ,order my,_ The parameters CI and C,, which should mathematically equal one, are iqtroduced In order to study,the
physi&l
effects of their variatidn
on the transition
probabilities.
The ,first order term in Cl COP-
responds to the force acting on the reduced symmetric oscillatbr; physically it descritjes the direct T&V tra-tsfers. The second crder term in Cz allows the oscillator to vary its force constant ,thiouglr me colbsion, i.e., to adjust itself progressively to the external force, and essentially induces the V-V transfers. Higher order ierms are neglected.
~. Insertion of the potential in eq. (21) into the expression of the effective potential ckY$r) results in
vjjCx> = expf-x1 70) = expf- b - WY)11 C(O) ,
(221
where y(j)=
hx(j)‘=ln[y(j)/y(O)j
1 +C,c~~(j+1/2),
1 .’
4.1. v- ~~r~~~~~~~ties
‘:
‘After some sk~plification, the resuhing equation is 2
(231
;
when ci!ii small, a first order treatment is satisfactory and leads to results which are’more simple and easier to analyse physically. Thus the appro,ximation: ln[l’(j)/y(@)] = C20r2j leads to the formula: (24i
.
4.2. T- V probobilkies
.’ Some intermediate results for the distorted wave treatment of the reduced atom-‘~a~orn system wXrin an ex.ponentid interaction were previ&sly given by Mies [9]. In the case N! = N+l , their uss results in: Sinh(2~Ki,)Sinh[2nKo,t(N+1)] F(q?
’
‘,
= (4?7iFiCr/a)2
-+Y)
..
: ,,
\ :‘,
.’
-‘.. . .
,’ .,‘.
‘,
.’
., ‘.
1:‘.
:
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: ._.. ‘.
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81 ‘...’ ,.
’ Vdlumc
29, number 1.
CiIEhfICALPHYSICS
.’
.
LETTkR:;
‘1
November 1974
. ..: ., .:‘- Wheri a is smail, the vaAab!e of the hypergeometric fijndtion.2F1, wluch depen.ds on LY,can be expanded and truncated to first order: I - 7(N- k + l)/y(N-,k +2) k @..In addifdn [7(N- k +2)1-l is taken equal to one ti-d the, G’s are approximated in the same way as in section 4.1. ThUS the following approximate formula is ob‘.t&ed, ,. Sinh(2?TKi,)
f&y
Sillh[22TK,,~(N+l)]
j’ +.a@-+‘)) = (4?NC,/ar)2 {Cosh(27TKi,) - COSh[2nK~,,(N+l)]}2 IV+1 x
iF,(l
2’
+j.K(? ,.l +iEi(-);2;Czor2)~~~-C~~C~~~1)(N-k+2)1’z~xp{iK’t)C~~2(~-k+l)}
.
(27)
This approximate exp&sion has proved quite reasonable for a G 0.15. Moreover its simp!e form makes the physical int’erpretation-easier. The formulas for N’ =N+2 are not given here since, within the distorted wave approxima-tion, the .corresponding results are known to be unrealistic.
5:Discussion 5.1. Gct&~l considerctims
It shquld first be emphasized that the -method is by no means equivalent to a distorted-wave (DW) treatment & dk@&) representation. The latter would be based on, the assumption that all probabilities are small. In contrast, the present treatment only a&.umes that the T-V probabilities are small; more precisely it is assumed that the .DW approximation correctly descriges each of the implied reduced atom-diatom collisipns. Since, for many realistic collision systems, there exists a range bf.collision energies within which the V-V probabilities are substantial wl@e &se for T-y are stitl small, a serious extension of the range of validity of the perturbational theory for this special problem by use of the present, technique is forseeable. Any fundamental drawback of the distorted-wave approximation is however reflected in our results. In particular, the sum of aU probabilities exceeds unity, since tile purely resonant V-V probabilities add to one. Morebver, multiquanta ,?-V transitions are very poorly described by a DW method. These two shortcomings could be, at .leziii in part, .corrected by using a K-matrix formalism for each of the contributing atom-diatom processes [lo]. .’ It is worthwhile to remark that transitions occurring essentially via a sequence of elementary processes involving a global exchange of no more thm one quantum between translation and vibration should be correctly described. ,Heie we call an elementary process any transition (ni, r22)+ (n;;tzi) for w&h the matrix element ‘OZ~~Y~)~n~C~~)l_CL”~n*~x~Ys)l~~~y~)~~z~~Y~)~ is ncn-zero,
vhere
-y&j,)
= i-y1
according
to eqs. (21) and (la):
+C,~7(Y~‘-Y2)+~c~~272(Y,fY2+.2Y~
’ Y,),..
For instance, the tfan,sition (2,O) --t (1,2) occurs iia elementary processes whose amplitudes interfere in the overall ‘:amplitud& The elementary processes that are possible in the present model are denoted by bars in the diagram within theN = 2 or 3 subspace of states 021, nz)_ ’ The l?W.cpproximation neglects,the back coupling for T.-V processes. The arrowsin .+hd diagram indicate that ‘. ,’ the ttisition is from a 2-quanta state td a 3-quanta state. ‘. 1’ .The ac_cuiAy ‘of our results can: be expected to be of the same order as. that of the: DW approximation.for ,‘:_ .; +:
2
.,.. .; ‘.. ._... i .:
‘:
I
: ;.
;: -I .,.I.
-,.
‘.,
,,
_,‘, .,: ‘,_ ,. ... .. . ,_ .’ ._
.. ._
Volume 29,numbc1
CflEMlCALPHYSICS.L~~ERS
I’
1 November
,:
1974
‘.atom-diatom problems. For these problems, it is known that the DW probabilities at low collision energy exhibit a systematic error depending only on the parameters M and cr of the system ([Z] and references therein). For many systems of interest, this error is typically qf the order of 2.5% and it becomes more important for large Y& ues of M, i-e,, for strongly adiabatic collisions. This is esse&aliy due to the failure of the DW approdmation to account for the compression of a molecule in the course of a collision; this could be corrected for bf treating each reduced atom-diatom collision within a perturbed-stationa~state (PSS) approximation. In the range of high collision energies, the method proposed obviously breaks down because of the on&t of the strong coupling regime for each atom-diatom collision.
.Since the approximate expression of the resonant transition probability in eq. (24) provides rather valid results in most cases (smah Q, see tables I and Z), it is clear that the V-V transition probabi~fies are essenti;aIIy given by s~p~e’t~gonomet~~~ functions of the argument 5~. An array of the explicit expressions of these functions for different values of N, j and j’ is not given here; the reader can find it as table 2 of ref. [I]. But he must keep in mind that P does not have quite the same significance as in the present article although the formal expressions Gf p are the same in the particular case of an exponential interaction (see section 5.4). In fig.,& the curves representing the variations of these functions versus jp are drawn for N 5 3. p is a func~on of 8C0ll as well as M and cy.Moreover the range of p in which the DW expressions lead to rea sonabIy good results corresponds to the values of p below those for the strong coupling regime being set; for instance, iFM= 0.5 and Q = 0.112, the upper limit of this range is p = 0.8; this is typical of a case where the atoms A and B have’masses close in magnitude. Consequently the range of validity of the formulas proposed varies from one set of parametersM and Q to another. This is.clearly observable in tables 1 and 2.
In the same way as in section 5.2 and within the same range of p, the appro~mate expression of the non-’ resonant transition probability in eq. (271, whose validity is extensive (see tables 1 and T), shows that the T-V transition probab~ties are approximately products of weighting factors depending on G,,, &f, (IIand the number of quanta exchanged between translation and vibrati,on~multiplied by trigonometrical functions of ~‘7 d&K(~) which depend on the trans.itions actually studied. Some .of these t~~onomet~~~ factors am given explicitly in table 3 for N varying fr&m 0 to, 3 and N’ IN-1 ; for’N=2 and N’=3, the variations of different T-V trigonometric~ fabtors corresponding to different couples 6.i’) are plotted against cpin fig. 3. It appears that for ., (. .-
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: .:‘.,
(2,0)3(3>d) (2.0)+(2,1)’ :-- ;. ,. ., : ‘- (2;0)+(1,2) (2&&(0,3) ‘_ . ..’ (1,1)-+(3,0) (l,I)+c&1) :
,..,. ” .“(3,0)-+(3,0) : (3,0&(2,1) ,. .,:.‘. .(?,O)i(l,2) ~(pi.,. (3,0~(0,3) :. (2,1)_&, 1) .:,,. : .(2;Wc1,2) : ‘, ;-’
: .,: ‘. ,- ..,_ : ,.
: .,‘.
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”
,,
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‘,
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3.Ok3 3.85-2 4.65-2 2.26-l 4,.11-2.4.44-l .9.09-1’2.91YI 4.94-2 3.29-l 8.63-l 1.09-4
“.
;
‘-1.44-l 2.01-l 1.73-1 : 4.81-l .4.24-l 2.02-I 1.84-7 1.92-7 7.43-8 1.11-6 2.56-7 7.71-7
.4.5
.,
7.75--l 2.06-l 1.82-2 5.38-4 5.25-l 2.51-l 3.86-6 1.34-6 5.07-8 3.73-10 7.59-8, 2,55-6
Cl-Hi- H-CI,M= 1,7.7&u = 0.52.
‘%I~ .: . ..o.s. ._
‘. ,’ .‘-
‘,
i
:,.
‘..
,’
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9.h5-1 5.40-2 1.0’3-3 ,6.54-6 8.74-l 7 07-2 3.75_9,
.-..
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- 9.4%1 5.15-2 9.36-4 5.70-6 8.80-l 6.75-2 3 86_9
1.5 -I-
,..N-N+E~-N,~M=O.S,L~=O,~~~
(3;0)-+(3,0) .’ .‘. 9.82-l 9,81-l ,,, I. (3,dj-t(2,1) 1.77-2 l.HS-2 ,I’ ,. ” (3,Ot+(1,2) 1.07-4 1.17-4 _. .,I. ‘(3,0)+(0,3) ..’ .2:18-i 2.45-j ‘a ,. ‘.:, (2&+(2,1) 9.59-l 9.57-1 .’ ‘, (2,1)-+(1;2) .“’ 2.34-2 2.46-2 I ‘( GOt((3,Ob mo-+~2,1) .’ .. ‘.(2;0>+(1;2) ” :
.I.-
.’ .... ..:: .’ .. ..
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8.$6-l’ 1.19-1 5.37-3 &IO-$. 7.24-l. 1.51-l 1.26-4 4.85-5 6.50-6 1.85-7 9.66-6 8.10-5.
3.86-1.9.54-l 1.09-2 4.50-2 6.02-l 7.07-4 1.81-3. 3.70-6 3.70-l 8.95-l 1.76-2 5.90-2 3.40-4 1.07-3. 1.80-4 3.84-4 ‘1.16-4 2.60-S 3.16-4 3.50-7 3.37-5 3.88-5 6.69-5 7.03-4
9.5
8.81-l 1.14-l 4.89-3 7.03-S 7.36-1 1.45-1 1.24-4 4.68-S 6.01-6 l.G5-7 9.02-C) ., 7.91-5
.;
1
. .
5.5
‘.
2.00-3 4.20-4 2.03-S 6.21-4 2.98-3
4.15-3
S.li-1 1.76-I 1:27-2 3.06-4 5.93-l 2.18-r’
:. ..
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..
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1.00+0 1.00 to 4.50-6 9.51-6 3.02-11 3.01-14 .3.24-17 3.18-17 l.OO+O l.OO+O 1.27-5 1.27-5
0.5
%,I1
,,
7.61-‘1 2.17-1 X07-2 6.58-4 4.99-i 2.63-1 3.10-2 1.39-2 3.71-3 2,43-4 5.49-3 1.89-2
.7,5.
“ ‘.
.,, : ”
:
,. ‘:.
,‘.
IO~rl
_.:. 6.94-l
::,
Q/j-’
’ ’
3.19-Z MO-3 4.03-l 3.05-l i.o3-i 4.98-2 1.61-2, 1,.37--3 2837-2 GAO-2
v1
‘;.‘, ..’
.;,
‘..
:.
:
,,
.- ‘.‘
._L-1 ” z, ‘.
3.48,2. 1.50-3~. 3.82-l ‘;. ,g, 3.14-l 1.04-i .’ E: 5.14.-2.:’ 1.74.12 1.54-3 2.53-2.. -:’ -2 6.i7:i o
: 2.60-l ‘2.69-l
,.
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j.oocd ‘I, -l.ooib: 1.00+0 ‘,’ 8.56-5 1.80:4 ‘1.8174, ,‘, 1.09-8 .l.d9G 2.44-9 2.32-14.:. : .:2.i8-13 2.18-13 1 1.ooto. ,1.00+0 ; l.OO?O 1.14-4 2.41-4 241-4 4.57-6 ‘. 1.65-3 ’ 1.65:?* 5.49-4 5.49-4, : 1.52-6 1.54-10 I .. 1.2$-7.a 1,25-7: 2.91-15 5.34-12 5.35-12 ; 2.31-10. 1.88-7 1.88-7’ o 1.10-3 1.10-3 < 3.04-6 -13
1. :
1.5-i”,
7.51-l 2.26-1 2.27-2 7.58-4 4.80-l’. 2.72-1, 3.15-2 1.44-2 3.99-3 2.73-4 5.86-3’ 1.92-2
:g*s:
,.;: 2
. . ;, : ;
‘. i.oo+o 8.55-5 2.44-9 2.32.-14 1.00+0 1.14-4 4.56-6 1.52-6. 1.53-10 2.91-15 2.30-10 3.04-6
4.5’
H-CI+CI-H,M=i.40R4-2,a=
‘I
.
8.19-l 1.69--l 1.16-2 2.66-4 6.10-I 2.10-l 4.68-3 1193-3 ‘. 3.89-4 1.80-5 5.80-4 2.92-3
‘,..
:
T&fc’l ., : ,. .‘. .~. ,: ., _‘,’ .‘,;.,_I -z , ,.:.._(:@$la&d transitionprobtibilitiesat different& T_.’ coll of severalcoIlid& fysteiil~~(characteriicd by M and a), for.Ci = Cz = 1.0:The two:numbers’aljpearingin TV 8. ”x0., ” .’ .: ach’tcrni of the nrr?y.reprcsentin the left the “cxatit” DW result [cf. eq. (23) or (26)], on the right the.app&imnte ok [c&c+ (24),0~,!27)]. 9.28-l ;e,re-.z ‘.,,,,.,_ ‘..a: I ,. : -2 &IlS9.8~X lo-‘.., . . ._ L ;‘., “: . ‘. ,’ /,.‘. .*
--,
03
,‘a
i,,, .*
k”
..I.
;.
,:‘.
_.;.
Volume 29, n&ber
1
1 November 1974
CHEMICAL PHYSICS LETTERS
Tadle 2 Results of the study of the isotope effect in-the collision of hydrogen-type molecules driven by the =me interaction potentiat The four nuqnlxrs in each term (which are exact DW results) correspond, respectively, to H-H + H-H, M= 0.5, a= 0.315; H-D+D-H,M=0.25,rr~0.195;D~H+H-D,-M=1.0,~’=0.391;D-D+D-D,~~=0.5,u=0~265
., %oll
Transition
0.5
1.1
1.7
2.3
2.9
3.5
‘9.92-l 9.91-l 8.89-l 9.70-l
9.12-l 9.81-1 7.67-l 9.34-l
8.67-l 9.70-l 6.56-I 8.99-l
8.22-l 9.60-I 5.57-I 8.65-l
7.80-l 9.50-i 4.67-l 8.32-l
7.3471 9.39-l 3.88-l
8.78-2 1.93-2 2.33-l 6.60-Z
1.33-l. 2.97-2 3.44- 1 1.01-l
1.78-l 4.00-2 4.43 -1 1.35-1
2.20-l 5.03-2 5.33-l 1.68-I
2.6 L-1 6.05-2 6.12-I 2.00-l
5.00-S 7.31-6 4.87-6 4.71-6
9.28-3 3.29-3 1.64-3 2.42-3
5.31-2 2.57-2 1.06-2 1.98-2
1.40-l 8.26-2 2.78-2 6.43-2
2.63-l 1.ac-L 4.95-2 1.40-t
(1,0~(1,1)
2.49-5 3.88-6 2.80-6 2.36-6
4.88-3 1.63-3 1.10-3 1.26 -3
2.94-2 1.29-Z 8.25-3 1.07-2
8.15-2 4.19-2 2.54 -2 3.61-2
1.61-I 9.25-2 5.33-2 8.L 5-2
(l,W(O,2)
1.91-6 6.57-8 5.32-7 1.34-7
8.73-4 6.52-S 4.70-4 1.701
7.89-3 7,96-4 5.07-3 2.18-3
2.88-2 3.46-3 1.99-2 9.75 -3
7.01-2 9.53-3 4.93-2 2.72-2
(1,0)-+(1,0)
u,ol-a
(1,0)+&O)
1)
8.20-3 8.80-3 l.li-1 3.04-2 ‘.
c
6.00-L
Fig. 2. Variation oFthe approximate resonmt transition probabilities versus 4~ [cf. qs. (24) and (25)] for N= 3. Note the two transitions (3,0)+(1,2) and :3,0)4(0,3) would have a zero probability of occurrence for a DW. treziment in the bbyz) representation with tie trunuted potential in eq. (21).
reasonable values ofM and 01,at high 8coll, some transitions that would not be taken into account by 2 DW treatreach probabilities of Occurrence of the same o:der of magnitude as those allowed by this treatment. In that respect the method proposed extends considerably the range of validity of perturbation methods. In the harticular case of a highly asymmetrical system (e.g.; either CM +.HCl or HCl f Cl.H) the T-V transition probabilities remain quite small and insignificant up to much higher v&es of p, so that several cstiations in the variations of the‘V-V transition probabilitiej versus p occu’r H;iUrin the range of validity oftbe formillas. However, the collinear model clearly is p&ticularly unrealistic in such situations. ment in the (JJ~? y2) representation
8.5 :
~olume’29;number
CHEMICAL-PHYSICS
1
._ :: ‘. . .. : ,. .’ : : f’ ~.k&i& factors appearing in the approsimaie
LE?ERS,
1 November
:
:.
1974
.
Tible 3 ‘. T-y transition probabiiitiesP(*m -+ @lfc’f’) .>.[cr. eq. (i?jj for N= 1, &3;The.‘_ c, factors for F= n-tN’G+l with $coll =, e.are the same as those forN= n+l-if ._ I = n ‘cvlt Zcoll = e-l @ticiple ofdctafied bal-. ..:, .i :.’ ?nCe). The abbreriatioas C =. GO&Y,S = sin& and ,Y= $‘p are used .’
..
..
:
:
-Quanta--
,’
0
:-
1
:
‘2 :
‘.’ ..
. . ..
2’..
3.
” :
(0,0)-r
(1.0)
‘. (l,O)-,
u.0;
.‘.
..
‘:
[l,d)-w
eq. (27)
‘.
14.
,, .(2,0)-r
.’
._,
.. ‘. :
”
.,,
.
(c),2)?)
.
(3,O)
‘.
lip ‘. .-
.:.
:
‘.
.:
.. ,‘,
2r6 C4(1-.-8S”)iZ 3C%$_
.’
.:’
are’not taken into account bj, a DW treatment
..
:
3P 2pb 9p4l2 6P2.
3C2[ l&{3’8C2)j/2, l-3C2S2 3S2[1+C2(3-BS2)]/2 6S2C’L
in the (y 1,.;12>representation
:
~~1~~~~~~
,.
6C4‘$“‘
:
‘,
3P2 l-3P2
‘;;k+BC*)/2 ;‘._
32 3@j2
1-3c2s2
(3,0)*(4.0) : (3.0)~(3,13 (3.O)-l,(2,2)‘) (3,0)-t(1,3)a) !3,0) -c f0,4)“)
foetors
(l+p2)/2
.’
‘.3cW
(3,X) (231) -+ (2,2j (2,lI-F (1,3)“) f2,lj-c (0,4j3)
ET_Vhigcmmtrical
”
S2(1i,3C2)/2 3S4j2
.,
l/2 p? 3(1 -2P2j/2
3c4/:1.
‘, @(li-3s*)/2
c&1)-
:
transitions
,for small q
;
l/2
s2
(2,0)* (2,l) (2,0)-+(1,2$ f2.0)-, co,3F)
iZ,lj-(4,0)@,
,’ .“? The
.,
: l/2 ..’ cz .‘, ‘I;/2 .- ‘1
.,.
(l,l)-+ (3,01a) (l,l)-c(2,1) 3:
leading term
: ., ‘.’ -. ‘_
.’
.,; .
:
Angul& factor.’
U,Ol+(~,l)
.
‘.
.’
&I*p(n’p7;)
.:
:
:
.:
Tr&sition
NN’
N.,.. :
.’
:
3(1-6p2)/2 1-3p2
:,
w/2 :
6p2 for the truncated
potcntiai
in eq. (21)
‘.
,’
‘.,.’ .,
‘_
..
. ..
. .
,“,,
_..
.,‘,
., ,,: : .,”
.: ‘.
‘,
.-;I ,. .tiote
...,, ,_
..I’) &&three
t in $6 iy$y2)
.. : :‘,
1 November 1974
CHEMICAL PHYSICS LETTERS
“Volume 29, number 1
Table 4 Ipfluencz of C2 on the transition probabilities. The V-V transfers clearly increase for increasing C2 whereas the-T-V tramfen that do not require an intermediate V-V elementary pro&s decrease (anh&monic factor). On the other hand T-V transfers that rcquiie non-zero c2 tend to increase for jrjcreasing Cz until about C, = 1.3 H-H+H-H,M=b.5,~=0.315,C1=1.0,&coll=2.0 Transition
*’ c2
(2,OH2,0) (Z,Ofiz(l,l) (2,Ofi(O, 2) (l,lti(l.l) GGOI-GO) (2,0)-+(2,1) (2,Ofi*(1,2) (2, W(O, 3) (l,l)-r(3,0) (I,l+G,l)
0.0
0.3
0.6
0.9
‘1.2
I.5
1.00+0 0.00 0.00 1.00+0
9.68-l 3.20-2 2.68-4 9.36-l
8.85-1 1.11-l 3.54-3 7.77-l
7.73-l 2.12-1 1.46-2 5.76-l
6.51-1 3.12-L 3.77-2 3.76-l
5.29-L 3.96-l 7.47-2 2.07-l
1.00-l 3.33-2 1.55-3 1.49-5 2.38-3 6.43-2
6.35-2 2.33-2 3.57-3 1.27-l 5.54-3 3.90-2
3.97-2 1.72-2 4.66-3 3.51-4 7.27-3 2.31-2
2.46-2 1.29-2 4.88-3 6.144 7.56-3 1.55-2
1.5 L-2 9.74-3 4.52-3 8.44-4 6.90-3 3.93-3
1.54-1 5.14-2 0.00 0.00 0.00 .1.03-l
The effect of varying C2 from 0 to 1.5 on the transition probabilities (see table 4) is shov+n partly in the followi&remark: in addition to the fact that the terms in C, and C2 in the potential are essentially responsible for the T-V and V-V transfers respectively, the term in C, reduces the amount of T-:V transfer by means of the hypergeometric function 2F1 - cf. eq. (27) - it&modulus decreases drastically for increasing C2. This is known in literature for atom-djatom collisions as the alzhanlzqnic factor of Mies [9]. 5.4. Comparison with the semiclassical formdatiox The results presented in this article are quantum mechanical and of the first order. in perturbation ‘&ey are closely related to the semiclassical results of ref. [l]. In particular:
theory.
- Our formula (24) for resonant transition probabilities is exactly the same as formula (75) of ref. [l]. The respective derivations are, however, quite different. In ref. [l], p is associated with a.phase shift in time c&the intramolecular wavefunction; in the present work, the counterpart of it is a phase shift in spat-3 of the relative motion wavefunction (compare the defmition of p in eq. (25) above with eq. (55) of ref. [l] for 2 generaI semiclassical definition of p and with eq. (57) for the case of exponential iepulsion, respectively). - Our formula (27) for non-resonant transition probabilities is very similar to formula (61) of ref. [I] as applied to the case where m = N+l ; this was expected since formula (6 1) is a low collision energy approximation. Mareover, in order to make both formulas as similar as possible, the relative velocity used in the semiclassical treatment .- should be the average of the initial and final velocities. This procedure is well known in atom-diatom models [l I]; it appears to remain valid for diatom-diatom problems and is consistent with a first order perturbation treatment in qutitum m,echznics. Formula (61), however, contains nothing corresponding to our function 2Fl;. this stems probably from ‘their approximation for the treatment of the quadratic term in the potential.
6. Conclusions
A quantum mechanical ,version of the semiclassical treatment in ref.. cl] lias been developed and simpIe formulas have been proposed for the T-V-V transition probabili&s..The formul& ) both treatments have b.ee,nshown to ..’
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_:_.. . ‘. :,: ,: ._ ,’ .. :. i ._.. .,*, ,.,’ ,’ ,I,,be ilosel$ re$afkd t\o each, other. However, the qua&urn m,ochtical results’do not suffer tlie usuhl ‘shortcomi&s ..,’ pf ~~c.~s~c~ theory, i.e.., n~n-c~nse~ati~~ df energy and.&o&dt ‘behavior ndar t,& energetic threshold. More... oyFi,.;Lhe piksent:appidacfi is Sow being improved in a Straightforward iay by &i$a’bette; de~c~pti~n’of each ..
‘reduced atom-d&&m cofl+m: the work for s&h a description wiih& an adiabatic PSS framkwork is nearly’finis&d. Ai exact quantum mechanical tre&m&nt is’also possible bg IXXII~standard prq&ms which solve numerid ...ly-problemsofthisnature. ’ .,; ‘. :‘,, ..’ .’ : ,‘. ,;, : ,
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,> ,,., ‘. .. .. . ‘, -. _’ ..: : .’ “,_:~~~~~r~d~~~~~ : : . . .- ;; ‘, ,. ” :,, ‘,., ,’ .‘, .._,,?‘he a&.&i aie indebted to Mr. R. ~7et~o~e.for m&h helpful assi&nce, Dr, J. Manz and Mr. C. Leforestier’ .. are gratefully ‘.
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Phys. 49.(1968) 286. ,,:: .[2]‘D. Rapp and T,Kasul, Chem. Rev. 69 (1969) 61. ;‘._ [3] !.G. Kelley,J. CXem. Phys: 56 (1972)6108.’ ,,, [+I ,,H.K.~Shin,Chem. Phys Letters 3 (1969) 56.0. ..” .. .’ Is] M,E:Riley,and 4. Kuppermann, Chem. Phys. Letters 1 (1968) 537. .. .f6] V. Gutsc%ck, ~.‘McKoy and D.‘Diestler, J. C&em. Phys. !&?(1370) 4801. f7),‘q.J. Wilson, J. Chem. Phys 53 (1970) 2075.’ : ‘. IS]. M-H. Alexander; 3. Chem. Phys. 59 (1973) 6254. 19]‘F.H:Mies, J. Clxm. Phys. 40 (1964) $23:. .. [iO]‘?. Secres$ Ann. Rav; Phys Chem. 24 (1973) 37% [lf], F. Htidrich, K. Wilson a$d R. Rapp, J. Chcm. Phys. 54 (1971) 3885.
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