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A quantum-mechanical normal-mode approach to collinear collisions of identical diatomics. V-V resonant processes when the interaction potential includes an attractive part
Volume 40; number-2
1 June 1976
CHEMICAL PHYSLCS LETTERS .-
-
.-
1
..
-MECHANICAL NORMAL-MODE APPROACH TO COLLINEAR COLLISIONS A&iN-IUi OF IDENTICAL DIATOMICS. V-V RESONANT PROCESSES WHEN THE INTERAQJTION POTENTEAL INCLUDES AN A-iTaACTIVE G. BERGERON,
C. JLEFORESTIER
LuBomroire de Chimie lhebrique
PART
and X. CHAEUISAT
‘f Umiversir~ de Paris-Sud, 91405 Orsay. France
Received24 Februari 1976
Various collinear collision systems of two identical homonuclear diatomic molecules are investigated. The interaction potential includes an attractive we!1 (Morse-type potential). Some analytical formulas arc derived for the resonant V-V transition probabilities within the distorted wave (DW) framework. The resuhs are compared with those of an exact (close-coupling) treatment of the same model.
1. Introduction
In previous papers [I .2] a formalism for the collinear collision of two identical diatomic molecules was described. It was based on a normal-mode expansion [3] of the collision wavefunction and applied to a variety of model systems. However for all of them the interaction poteniial was assumed to be purely repulsive (exponential) 141. The present communication aims at applying the same formalism to resonant V-V processes in homonuclear diatomics when the interaction potential in-
cludes an attractive part (Morse potential). The distorted wave (DW) version of the formalism leads to analytical expresions of the transition probabilities. The numerical values of these are compared to the
results of an exact (close~co~pling)treatment of the same model problem.
over this potential is expanded to second order with respect to the displacements of the diatomics from their equilibrium positions. The following notation (expressed in reduced units) is used henceforth: M= ~A/L?LvB ,
OL= a y @/pcdy
K = (2M(D)‘/2/rY
,
A(n)=K[B,,(23’* N(n)
)
k, = (2M &,,)1’2/ly, cI)]r’*
,
= KBnn (21’2 or)/ [B,, (23’2 (Y)] 1’2 ,
where mA and mg are atomic masses, 2a is the steep ness parameter of the repulsive part of the Morse potential, 7 = mA/(mA + mB), p and w are, respectively, the reduced mass and frequency of the oscillators. CD and ecoI are, respectively the well depth and the collision energy expressed in F&J units. Last
of all: 2, Mcdel and DW fotialism
B,(.g)
= 1 + .g2 (n f 32
-
As in refs. [ I.21 an expansion of the collision
The model consists in two identical harmonic diatomic molecules lying on a_line (AB 4 BA). The interaction potential is of the Morse-type. It operates between the inner atoms (B) facing each other. More* The Laboratoire de Chimie Theorique is associatedwith the CNRS (E.R.A. no. 549).
wavefunction
in terms of both the overall symmetric
and antisymmetric normal modes of the two diatomic
molecules decouples the diatom-diatom collision problem and reduces it to a set of atom-diatom like problems. A subsequent use of the DW approximation for description of each reduced atom-diatom
collision results in the following
expression of the transition probability for a resonant vibration-tovibration o/-v) process [2] :
IV+1
= . gl
2
rs(y) C$!
exp[2i 6(N-k+
I)]
I
,
(1)
where S(n) is the asymptotic phase of the solution wavefunction F,,(x) of the elartic SchrGdinger equation:
I &(x1 = 0 -
a) exp(-x1]
(2)
The $-‘s are constant coupling matrix elements. They are studied in detail and tabulated in ref. [3]. @jw designates an intramolecular state in which there are N quanta in the two diatomic molecules and j- 1 in the right hand one. For the present problem S(n) is known analytically’ [S-7] :
exp[2i6(n)]
= [2A(~z)]-“ik~
vl +2ikti) ’ lT(l-2ikin)
I’(; - N(n) - ik. m ) I?(: --N(n) + ikti) *
(3?
Insertion of this result into eq. (1) provides the V-V transition probabilities. However the explicit expressions are rather complicated, in particular because of the occurrence of JTfunctions in eq. (3). In order to obtain an approximate but much simpler expression of the transition probabilities we replace the I’ function by the usual (asymptotic) expansion [S] :
lnJY(z)Z(z-!)ln(z)-ztiln(2n)+-...
mentary simplification is to expand*furthermore eq. (5) to first order with respect to a2. This supposes a2 to he small, a reasonable assumption for most coIlision systems (see results below). Finally i is neglected with respect to K in all,expressions whnre (K - 2) appears. Then the simplified expression of the transition probability is:
iv+1 =
kGlCkt)
$?
exp [i(k - I)p]
I
2 ,
(6)
where
{d2/dx2 +kk - K2 [i3~,~(2~~2~) exp(-k)
-2 B,&‘2
1 June i976
CHEMICAL PHYSICS LETTERS
Volume 40, fiumber-2 _
.
(4)
p = 2a(2M)“2
-(0”2
[2&Z
+ ,cg”2
ta&E,,/CD)“2]
.
(7)
The V-V transition probabilities thus have the same form as previously derived for an exponentially repuisive interaction potential (cf. ref. [2], eqs. (24) and (25)). The expression of p is remarkably simple. It is a sum of three terms, the first of which would remain alone if the Morse potential was replaced by its repulsive part alone. The second term is constant; it depends parametrically on the well depth. The third term depends on Cg as well as & col and tends to 0 when & col + 0. At this stage, on the strength ofeq. (7), it should be noticed that the interpretation of a collision driven l;y a potential including an attractive well as a purely repulsive collision by replacing C-1 by C c0l t Cg is dubious [9,10]. Finally, replacing the C,Q-‘sby their numerical values results in simple trigonometrical expressions of the transition probabsities. These expressions depend on p/2 only. For instance (see table 2, ref[3]): $rz$
= sina
,
If we take into account in the I”s involved in eq. (3); only those terms which depend explicitly on n (the other ones have no influence on the transition probabilities), there results a simplified expression of 6(n): 26(n) =- kin In [B/&P’=
a)] - ZN(Lz) x (n)
-2kjJy ~rr(n)l, where i - A$) - ikin = I-&) exp [i x(n)] . A supple-
(5) (8) 295
.-
Volume 40, number 2
.
1..June 1976
CHEMICAL PHYSICS LETTERS
3.SemicLssici33 fonnddion The semiclassical investigation of the problem within the framework proposed m sections 2 and 4 of ref. [3] is straightforward_ As an ingredient it makes use of the relative motion trajectory given analytically in refs. [l l-131. It is interesting to mention that the ver$ same expressions of the transition probabilities as in eq. (6) come out; along with the same p as in eq. (7)*. In fact, this is not surprising since the same happens to be true for a purely repulsive potential under
the small CY assumption. Let us note that neglecting l/2 with respect to K in the DW treatment is consistent with the fact that a semiclassical treatment of the relative motion neglects the quantization of the Morse weli (K > l/2 is the requirement that the Morse interaction potential must fulfill in order to support bound states).
In the exadt quantum-mechanical calculations, the same normal-mode decoupling scheme as in the DW treatment is used. The resulting close-coupling equations are solved by means of the very stable Fox integrator [18,19]_ Some results are pictured in figs. l-3. PO1_,li and P02<20 are plotted versus the collision energy for the three systems. On each plot, the DW, the DW + smalI (Yapproximation (DW,,,) and the exact
transition probabilities are represented. Although these results are rather limited, some conclusions clearly emerge from them: (i) For all of the three systems, the discrepancy between DW and DW,,, resultsis much smaller than that between exact and DW results. This is true whatever & co, within the range studied. Consequently there seems to be no reason to use the intricate DW formulas, instead of their simple DW,,, versions. (ii) The gap between exact and DW results increases strongly with collision energy. As expected, this gap
is much more pronounced for two-quanta transitions than for one-quantum
4. Results and discussion
In order to test the DW results by comparison with exact close-coupling results, DW and exact
A
I&-(”
computations were performed for three different model collision systems, namely: (l)M=OS, (2)M=
CY= 0.0645
(a =0.0508, 0.5 , co = 0.441
(3)M=OS,
(a = 1.106,
,
; 08-
a= 0.0715 ; ct =.0.0635
transitions.
.
These systems correspond roughly to 02 + 02, Cl, iCl2 and Br2 + BrZ, respectively **_ The values of 9 were derived from data found in ref. [IS]. The a’s were determined for the repulsive pz!rt of the Morse function to coincide with the repulsive interaction used by Secrest and Johnson [ 16]#. ‘ihe
calculations are not reproduced here. They are given expiicitly in ref. [ 141. **The study is restrioted to the CaSeof homonuclear diatomic molecules becatie a Morse potential is a (rough) representation cf a van der N’aalsinteraction. For heteronuclear molecules multipolar interactions could Se more
important as well. * Internal excitation (Feshbach) resonances are not looked for in the present study. Th&e resonances are very narrow and difficult to__lbcatew&in the range ot parameters . selected [ 171..
.296.
_,.._‘_
..
OLBS-
I
2
3
Fig. 1. Variation with ccllision energy of P~o+o~ and Pzn + 02 for 02 + 02 ((D F 0.0508, (L= 0.0645. The solid curws are exact results, the dashed curves are RW rest&. and the dotted lines are DWappresults.
VoIume40. number2
CH&lICAL
Fig. 2. Variation with collision energy of P10 + 01 and P204m for 0.441, a = 0.0715).
a2+cl2 0 =
(iii) This gap depends almost exclusively on the transition probability regardless of either the collision system or the collision knergy. This is illustrated in fig. 4, where the ratio I$:+ lo/~‘lo is plotted versus Po~~lo for the three systems. The points reported in this plot define remarkably weli a single curve, from which they depart sIightly only for very small values of ecol. Thus, from concl_usions,(i) aD_d (iii), it appears that a DWapp transition probability multiplied by a specific correction factor depending only on the probability itself results in a good estimate of the exact-transition probability. (iv) The exact quantum-mechanical transition probabilities are satisfactorily reproduced by means
PHYSICS LETTERS
1 June 1976
Fig. 3. Variation tith collision energy of Pro -01 Pzo+~~ for Br2 •i-Br2 (cb = 1.106, P = 0.0635).
and
of the analytical formulas derived within the DW,,
formalism [cf. eqs. (S)] provided that sin’(&) is replaced by the exact value of PO1~ 1o. For instance: ~20-4
= 2PlO+Ol(1
%-co2
= (pzo~ol)2
P30-+21
= 3P10+OI
P3O-42
= T &O+Od2
P3OLO3
= (qo+Ol)3
P21+12
=P,o,ol
-qO,Ol)
’
’ (’ -P10’01)2 (1 -P,,+,,)
’
’ 0 -
3Plo _&2
, etc.
C%
297
.Volume40, number 2,
CHEMICAL PHYSICS LETTERS
ln
-.
ul¶
.
1 J&e 1976
LIZ
versusPexaCt lo_,ol. The CTOSS~S are for 02 + 02, the dots for Qy + Cl2 and the triangles for Br2 + Br2. Regardless of either the collisionenergy or the nature of the collisionsystem, the points ye remarkably close to a
IGig.4. Hot of the ratio P~~ol/~~_lI sin&e curve.
t-
This is illustrated in table 1. By the way, we understand why vibration in the colliding diatom& prior to collision enhances the transition probabilities of resonant processes. For instance, for a one-quantum exchange between the $0 +ll> two diatomics: PZl Pso P 10401 within the range studied (see whatever EC01 table 1). Indeed, in fig. 5, the DWapp Plb+Ol, P20+11, P 30+21 and P21_.12 are plotted ve=usflO_,O1. -There we observe that the inequalities above remain trueup tOPol+io = 0.22. Similarly, for a two-quanta +12
>
+*1
>
-%-m
exchange: P 30+12’p20+02
Fig. 5. “Wapp plot of PI0 + ,,1, p20+11.~30-+21.~21+12. Sb 3 12 versus PIO + 01 within the range 0 to
p20 -+ 02 ad
’
0.5 [cf. eq. (9)]_
Table 1 Some transition probabilities of resonant processe! at various collision energies for the collision system Clp + Cl2 @= O-441 and a = 0.07 15). The numbers in parentheses are DW-app4ke transition probabilititk computed from the exact value of f’I+,ol by means of the approximate expressions in eq. (9). 0.917 - 1 means the same as 0.917 X lo-’ G01
PlO-+Ol
ho+
0.5
0.500 - i
0.917 - 1 (0.950 - 1)
1.5 2.5
_.$5 --
0.930 - 1
11
-0.163-
0.138
(0.169) 0.529 jb.23tf)
0.182
0.281 ‘: ._.(0.298)--
p30-+21
P 21412.
p2cJ402
0.127
0.161
0.234 - 2
(0.135) 0.217 (O-230)
(0.171) 0.262 (0.275)
(0.250 - 2) O%lL - 2
6.288 (0.308)
0.329 .. (0.347) :
‘.
--
1
(0.865 - 2) 0.179 - 1 ‘(0.190 - 1)
; (0.3310.31611 1)
p30-c
12
‘0.623 - 2 (0.713 - 2)
p30 -P a3
0.103 - 3 (0.125 - 3)
0.208 i 1
0.669 - 3
(0.235 - 1) 0.434-l’ (0.493 - 1)
(0.804 - 3) 0.220 - 2 (0.263. - 2)
Volumg $I, Gnber
Achowltidgment
CHEMICAL !?HYSICS LETTERS
2,.
.’ .
Dr. J.M. Launay is gratefully a&&edged for providing us with his version of the Fox integrator_
References [ l] J.H. CIaske and Et Thiele, Chem. Whys 4 (1974) 447. 121 G. Be&on and X. Chap&at, Chem. Phys. Letters 29 (1974) 77. [3] A. Zelechow, D. Rapp and T-E. Sharp, J. Chem._Phys_ 49 (1968) 286. [4] G. Bergeron, C. Leforestier and X. Chapuisat, Chem. .: Phys. titters 36 (1975) 152. [5] P.M. Morse and H. Fesbbch, Methods of theoretical physics (McGr2wHilL New York, 1953) p_ 492. [ 61 M. Abramowitz and LA. Stegun, NatI. Bur. Std. Appl. hi&h. Ser. 55 (1965).
1 June 1976
[7} D. storm and E. Thieb. J. &em. Phys, 59 (1973) 5102. 181 _ _ M. Abramowitz and I.A. Stegun, Handbook of mathmatical ftinctions (Dover, l&w York. 1968) p. 257. 191 D. Storm and E. Thiele, I. Cheti. Phys. 59 (1973) 3313. [lo] M.H. Alexander. J. Chem. Pbys. 59 (1973) 6254. [ ll] T-L- Cottrell and N. R&m, Trans. Faraday Sot. 51 (1955) 159. [12] E-A. And+%. Chem. Phys: Letters lL(l971) 429. [13] R.C. Amme, Advan. Chem. Phys. 28 (1975) 171. 1141 G. Bergeron, Thesis; Univ. Paris XI, Orsay (1976). [IS] A.F. Wagner and V. McKay, I. Chem. Phys. 58 (1973) 5561. [16] D. Secrest and B-W.Johnson. 5_Chem. Phys. 45 (1966) 4556. [ 171 W. Eastes and R.A. Marcus.J. Che?_ Phys. 59 (1973) 4757. [ 181 L. Fox, The numerinl solution of two-point boundary value problems in ordinary differential equations (Oxford Univ. Press, London, 1957). [19] G. Bergeron, X. Chapuisatand J.M. Launay, Chem. Phys. Letters 38 (1976) 349.
299
1 June 1976
CHEMICAL PHYSLCS LETTERS .-
-
.-
1
..
-MECHANICAL NORMAL-MODE APPROACH TO COLLINEAR COLLISIONS A&iN-IUi OF IDENTICAL DIATOMICS. V-V RESONANT PROCESSES WHEN THE INTERAQJTION POTENTEAL INCLUDES AN A-iTaACTIVE G. BERGERON,
C. JLEFORESTIER
LuBomroire de Chimie lhebrique
PART
and X. CHAEUISAT
‘f Umiversir~ de Paris-Sud, 91405 Orsay. France
Received24 Februari 1976
Various collinear collision systems of two identical homonuclear diatomic molecules are investigated. The interaction potential includes an attractive we!1 (Morse-type potential). Some analytical formulas arc derived for the resonant V-V transition probabilities within the distorted wave (DW) framework. The resuhs are compared with those of an exact (close-coupling) treatment of the same model.
1. Introduction
In previous papers [I .2] a formalism for the collinear collision of two identical diatomic molecules was described. It was based on a normal-mode expansion [3] of the collision wavefunction and applied to a variety of model systems. However for all of them the interaction poteniial was assumed to be purely repulsive (exponential) 141. The present communication aims at applying the same formalism to resonant V-V processes in homonuclear diatomics when the interaction potential in-
cludes an attractive part (Morse potential). The distorted wave (DW) version of the formalism leads to analytical expresions of the transition probabilities. The numerical values of these are compared to the
results of an exact (close~co~pling)treatment of the same model problem.
over this potential is expanded to second order with respect to the displacements of the diatomics from their equilibrium positions. The following notation (expressed in reduced units) is used henceforth: M= ~A/L?LvB ,
OL= a y @/pcdy
K = (2M(D)‘/2/rY
,
A(n)=K[B,,(23’* N(n)
)
k, = (2M &,,)1’2/ly, cI)]r’*
,
= KBnn (21’2 or)/ [B,, (23’2 (Y)] 1’2 ,
where mA and mg are atomic masses, 2a is the steep ness parameter of the repulsive part of the Morse potential, 7 = mA/(mA + mB), p and w are, respectively, the reduced mass and frequency of the oscillators. CD and ecoI are, respectively the well depth and the collision energy expressed in F&J units. Last
of all: 2, Mcdel and DW fotialism
B,(.g)
= 1 + .g2 (n f 32
-
As in refs. [ I.21 an expansion of the collision
The model consists in two identical harmonic diatomic molecules lying on a_line (AB 4 BA). The interaction potential is of the Morse-type. It operates between the inner atoms (B) facing each other. More* The Laboratoire de Chimie Theorique is associatedwith the CNRS (E.R.A. no. 549).
wavefunction
in terms of both the overall symmetric
and antisymmetric normal modes of the two diatomic
molecules decouples the diatom-diatom collision problem and reduces it to a set of atom-diatom like problems. A subsequent use of the DW approximation for description of each reduced atom-diatom
collision results in the following
expression of the transition probability for a resonant vibration-tovibration o/-v) process [2] :
IV+1
= . gl
2
rs(y) C$!
exp[2i 6(N-k+
I)]
I
,
(1)
where S(n) is the asymptotic phase of the solution wavefunction F,,(x) of the elartic SchrGdinger equation:
I &(x1 = 0 -
a) exp(-x1]
(2)
The $-‘s are constant coupling matrix elements. They are studied in detail and tabulated in ref. [3]. @jw designates an intramolecular state in which there are N quanta in the two diatomic molecules and j- 1 in the right hand one. For the present problem S(n) is known analytically’ [S-7] :
exp[2i6(n)]
= [2A(~z)]-“ik~
vl +2ikti) ’ lT(l-2ikin)
I’(; - N(n) - ik. m ) I?(: --N(n) + ikti) *
(3?
Insertion of this result into eq. (1) provides the V-V transition probabilities. However the explicit expressions are rather complicated, in particular because of the occurrence of JTfunctions in eq. (3). In order to obtain an approximate but much simpler expression of the transition probabilities we replace the I’ function by the usual (asymptotic) expansion [S] :
lnJY(z)Z(z-!)ln(z)-ztiln(2n)+-...
mentary simplification is to expand*furthermore eq. (5) to first order with respect to a2. This supposes a2 to he small, a reasonable assumption for most coIlision systems (see results below). Finally i is neglected with respect to K in all,expressions whnre (K - 2) appears. Then the simplified expression of the transition probability is:
iv+1 =
kGlCkt)
$?
exp [i(k - I)p]
I
2 ,
(6)
where
{d2/dx2 +kk - K2 [i3~,~(2~~2~) exp(-k)
-2 B,&‘2
1 June i976
CHEMICAL PHYSICS LETTERS
Volume 40, fiumber-2 _
.
(4)
p = 2a(2M)“2
-(0”2
[2&Z
+ ,cg”2
ta&E,,/CD)“2]
.
(7)
The V-V transition probabilities thus have the same form as previously derived for an exponentially repuisive interaction potential (cf. ref. [2], eqs. (24) and (25)). The expression of p is remarkably simple. It is a sum of three terms, the first of which would remain alone if the Morse potential was replaced by its repulsive part alone. The second term is constant; it depends parametrically on the well depth. The third term depends on Cg as well as & col and tends to 0 when & col + 0. At this stage, on the strength ofeq. (7), it should be noticed that the interpretation of a collision driven l;y a potential including an attractive well as a purely repulsive collision by replacing C-1 by C c0l t Cg is dubious [9,10]. Finally, replacing the C,Q-‘sby their numerical values results in simple trigonometrical expressions of the transition probabsities. These expressions depend on p/2 only. For instance (see table 2, ref[3]): $rz$
= sina
,
If we take into account in the I”s involved in eq. (3); only those terms which depend explicitly on n (the other ones have no influence on the transition probabilities), there results a simplified expression of 6(n): 26(n) =- kin In [B/&P’=
a)] - ZN(Lz) x (n)
-2kjJy ~rr(n)l, where i - A$) - ikin = I-&) exp [i x(n)] . A supple-
(5) (8) 295
.-
Volume 40, number 2
.
1..June 1976
CHEMICAL PHYSICS LETTERS
3.SemicLssici33 fonnddion The semiclassical investigation of the problem within the framework proposed m sections 2 and 4 of ref. [3] is straightforward_ As an ingredient it makes use of the relative motion trajectory given analytically in refs. [l l-131. It is interesting to mention that the ver$ same expressions of the transition probabilities as in eq. (6) come out; along with the same p as in eq. (7)*. In fact, this is not surprising since the same happens to be true for a purely repulsive potential under
the small CY assumption. Let us note that neglecting l/2 with respect to K in the DW treatment is consistent with the fact that a semiclassical treatment of the relative motion neglects the quantization of the Morse weli (K > l/2 is the requirement that the Morse interaction potential must fulfill in order to support bound states).
In the exadt quantum-mechanical calculations, the same normal-mode decoupling scheme as in the DW treatment is used. The resulting close-coupling equations are solved by means of the very stable Fox integrator [18,19]_ Some results are pictured in figs. l-3. PO1_,li and P02<20 are plotted versus the collision energy for the three systems. On each plot, the DW, the DW + smalI (Yapproximation (DW,,,) and the exact
transition probabilities are represented. Although these results are rather limited, some conclusions clearly emerge from them: (i) For all of the three systems, the discrepancy between DW and DW,,, resultsis much smaller than that between exact and DW results. This is true whatever & co, within the range studied. Consequently there seems to be no reason to use the intricate DW formulas, instead of their simple DW,,, versions. (ii) The gap between exact and DW results increases strongly with collision energy. As expected, this gap
is much more pronounced for two-quanta transitions than for one-quantum
4. Results and discussion
In order to test the DW results by comparison with exact close-coupling results, DW and exact
A
I&-(”
computations were performed for three different model collision systems, namely: (l)M=OS, (2)M=
CY= 0.0645
(a =0.0508, 0.5 , co = 0.441
(3)M=OS,
(a = 1.106,
,
; 08-
a= 0.0715 ; ct =.0.0635
transitions.
.
These systems correspond roughly to 02 + 02, Cl, iCl2 and Br2 + BrZ, respectively **_ The values of 9 were derived from data found in ref. [IS]. The a’s were determined for the repulsive pz!rt of the Morse function to coincide with the repulsive interaction used by Secrest and Johnson [ 16]#. ‘ihe
calculations are not reproduced here. They are given expiicitly in ref. [ 141. **The study is restrioted to the CaSeof homonuclear diatomic molecules becatie a Morse potential is a (rough) representation cf a van der N’aalsinteraction. For heteronuclear molecules multipolar interactions could Se more
important as well. * Internal excitation (Feshbach) resonances are not looked for in the present study. Th&e resonances are very narrow and difficult to__lbcatew&in the range ot parameters . selected [ 171..
.296.
_,.._‘_
..
OLBS-
I
2
3
Fig. 1. Variation with ccllision energy of P~o+o~ and Pzn + 02 for 02 + 02 ((D F 0.0508, (L= 0.0645. The solid curws are exact results, the dashed curves are RW rest&. and the dotted lines are DWappresults.
VoIume40. number2
CH&lICAL
Fig. 2. Variation with collision energy of P10 + 01 and P204m for 0.441, a = 0.0715).
a2+cl2 0 =
(iii) This gap depends almost exclusively on the transition probability regardless of either the collision system or the collision knergy. This is illustrated in fig. 4, where the ratio I$:+ lo/~‘lo is plotted versus Po~~lo for the three systems. The points reported in this plot define remarkably weli a single curve, from which they depart sIightly only for very small values of ecol. Thus, from concl_usions,(i) aD_d (iii), it appears that a DWapp transition probability multiplied by a specific correction factor depending only on the probability itself results in a good estimate of the exact-transition probability. (iv) The exact quantum-mechanical transition probabilities are satisfactorily reproduced by means
PHYSICS LETTERS
1 June 1976
Fig. 3. Variation tith collision energy of Pro -01 Pzo+~~ for Br2 •i-Br2 (cb = 1.106, P = 0.0635).
and
of the analytical formulas derived within the DW,,
formalism [cf. eqs. (S)] provided that sin’(&) is replaced by the exact value of PO1~ 1o. For instance: ~20-4
= 2PlO+Ol(1
%-co2
= (pzo~ol)2
P30-+21
= 3P10+OI
P3O-42
= T &O+Od2
P3OLO3
= (qo+Ol)3
P21+12
=P,o,ol
-qO,Ol)
’
’ (’ -P10’01)2 (1 -P,,+,,)
’
’ 0 -
3Plo _&2
, etc.
C%
297
.Volume40, number 2,
CHEMICAL PHYSICS LETTERS
ln
-.
ul¶
.
1 J&e 1976
LIZ
versusPexaCt lo_,ol. The CTOSS~S are for 02 + 02, the dots for Qy + Cl2 and the triangles for Br2 + Br2. Regardless of either the collisionenergy or the nature of the collisionsystem, the points ye remarkably close to a
IGig.4. Hot of the ratio P~~ol/~~_lI sin&e curve.
t-
This is illustrated in table 1. By the way, we understand why vibration in the colliding diatom& prior to collision enhances the transition probabilities of resonant processes. For instance, for a one-quantum exchange between the $0 +ll> two diatomics: PZl Pso P 10401 within the range studied (see whatever EC01 table 1). Indeed, in fig. 5, the DWapp Plb+Ol, P20+11, P 30+21 and P21_.12 are plotted ve=usflO_,O1. -There we observe that the inequalities above remain trueup tOPol+io = 0.22. Similarly, for a two-quanta +12
>
+*1
>
-%-m
exchange: P 30+12’p20+02
Fig. 5. “Wapp plot of PI0 + ,,1, p20+11.~30-+21.~21+12. Sb 3 12 versus PIO + 01 within the range 0 to
p20 -+ 02 ad
’
0.5 [cf. eq. (9)]_
Table 1 Some transition probabilities of resonant processe! at various collision energies for the collision system Clp + Cl2 @= O-441 and a = 0.07 15). The numbers in parentheses are DW-app4ke transition probabilititk computed from the exact value of f’I+,ol by means of the approximate expressions in eq. (9). 0.917 - 1 means the same as 0.917 X lo-’ G01
PlO-+Ol
ho+
0.5
0.500 - i
0.917 - 1 (0.950 - 1)
1.5 2.5
_.$5 --
0.930 - 1
11
-0.163-
0.138
(0.169) 0.529 jb.23tf)
0.182
0.281 ‘: ._.(0.298)--
p30-+21
P 21412.
p2cJ402
0.127
0.161
0.234 - 2
(0.135) 0.217 (O-230)
(0.171) 0.262 (0.275)
(0.250 - 2) O%lL - 2
6.288 (0.308)
0.329 .. (0.347) :
‘.
--
1
(0.865 - 2) 0.179 - 1 ‘(0.190 - 1)
; (0.3310.31611 1)
p30-c
12
‘0.623 - 2 (0.713 - 2)
p30 -P a3
0.103 - 3 (0.125 - 3)
0.208 i 1
0.669 - 3
(0.235 - 1) 0.434-l’ (0.493 - 1)
(0.804 - 3) 0.220 - 2 (0.263. - 2)
Volumg $I, Gnber
Achowltidgment
CHEMICAL !?HYSICS LETTERS
2,.
.’ .
Dr. J.M. Launay is gratefully a&&edged for providing us with his version of the Fox integrator_
References [ l] J.H. CIaske and Et Thiele, Chem. Whys 4 (1974) 447. 121 G. Be&on and X. Chap&at, Chem. Phys. Letters 29 (1974) 77. [3] A. Zelechow, D. Rapp and T-E. Sharp, J. Chem._Phys_ 49 (1968) 286. [4] G. Bergeron, C. Leforestier and X. Chapuisat, Chem. .: Phys. titters 36 (1975) 152. [5] P.M. Morse and H. Fesbbch, Methods of theoretical physics (McGr2wHilL New York, 1953) p_ 492. [ 61 M. Abramowitz and LA. Stegun, NatI. Bur. Std. Appl. hi&h. Ser. 55 (1965).
1 June 1976
[7} D. storm and E. Thieb. J. &em. Phys, 59 (1973) 5102. 181 _ _ M. Abramowitz and I.A. Stegun, Handbook of mathmatical ftinctions (Dover, l&w York. 1968) p. 257. 191 D. Storm and E. Thiele, I. Cheti. Phys. 59 (1973) 3313. [lo] M.H. Alexander. J. Chem. Pbys. 59 (1973) 6254. [ ll] T-L- Cottrell and N. R&m, Trans. Faraday Sot. 51 (1955) 159. [12] E-A. And+%. Chem. Phys: Letters lL(l971) 429. [13] R.C. Amme, Advan. Chem. Phys. 28 (1975) 171. 1141 G. Bergeron, Thesis; Univ. Paris XI, Orsay (1976). [IS] A.F. Wagner and V. McKay, I. Chem. Phys. 58 (1973) 5561. [16] D. Secrest and B-W.Johnson. 5_Chem. Phys. 45 (1966) 4556. [ 171 W. Eastes and R.A. Marcus.J. Che?_ Phys. 59 (1973) 4757. [ 181 L. Fox, The numerinl solution of two-point boundary value problems in ordinary differential equations (Oxford Univ. Press, London, 1957). [19] G. Bergeron, X. Chapuisatand J.M. Launay, Chem. Phys. Letters 38 (1976) 349.
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