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Resonances in the electronic quenching of O(1D) by N2. A numerical quantum mechanical study for the collinear collision
Volume
48, number
2
RESONANCES
CIICMICAL
IN THE ELECTRONIC
PIIYSICS
QUENCHING
LETITRS
OF O(lD)
1 June
1977
BY N,.
ANUMERlCALQUANTUMMECHANlCALSTUDYFORTHECOLLlNEARCOLLlSION G. DELGADO-SARRIO
dr Qo~crus, Dcpartamcnto dc Quim~a Cudtztiia. Unrvcrs~dad Autbnoma de ilfadrki. Madnd, Spam and Inrtituto dr Iktructura dr Ia hfatcria C.S.1.C , Madrid. Spaoz I%cul~ad
and J.A. BESWICK i: Laboratom Rccclvcd
de Phoroplrysiquc ~lfokularrc,
7 J.lnudry
UnwcrsltE de Pans 9:d, YI405 Orsay. France
1977
Rcvwzd rndnuscript rcccwed 7 March 1977
The closed coupled equations for the collincnr colhsion O(‘D) + N2 (‘Z$ +O(‘P) + N2(‘Ci) WVCbeen solved nunicw CAY for J model 0t t-0 CrOwIn;;potcnticd Lurvcs dasunlin~; n conSLant y3ln--orbat coupling Compdrnon bctwccn the rcwrlta of an nlom --ntom llkc model nnrl the converging result\ rcvc.11~ n cubst.mtial (fxtor of -40) cnhanccmcnt of the clcctroulc quenching .~t room tempcraturc topcthur with high vlhratromll excitation of Nz- Thcsc results. and tlw study of tbc peaks nppcaring in the qucnchg probabthtlcs ‘1su function of the incident energy, clearly confirm thrlt the high cffrcicncy of thr% rcact!on IS mainly due to rcsondnccb (qudsibound states), rls hrls been lately suggc\tcd.
. 1. lntrodmtion The collisional qucnchmg of the first excited state of the oxygen atom O(lD) + N,(’ Z’) -> O(3P) + N2(l 2:) has rcceivcd expcrimcntal F11and theorctical [2-S] attcntlon Idtcly. Thi5 interest has arisen not only from ihe importance of this mechanism in upper atmosphere reactions but also, from a purely thcoretical point of vlcw, because this “spin-forbidden” reactlon is known to occur with high efficiency [ I] . The firsr attempts [2,3] to explain the Hugh termal rate, (4-10) X IO-” cni3/molecule s, of this process were conducted using the semiclassical approximation to a model of curves crossing between the triplet and the singlet states. if a plausible spin-orbit couplmg is used (=80 cm-l [C,]) the rdte constant obtained in this way is 20-100 times smaller than the cxpcrlmental result. Moreover, the N, molecule is treated like an atom so there is no vibratlonal excitation of N2 after the colPresent address. Wcbmann Institute of Science, of Chcmlcal Phystcs, Rchovot, Isrxl.
358
Department
hsion. This is in contradiction with the cxpenmental result [I] which predicts that after the collision a substantlal amount of the 1.98 eV exothermicity js transformed m vibrational cxcltation of molecular nitrogen. Recently, Tully [4] has suggested that this violation of spin conservation rules must be associated with the formation of collision complexes whose hfetlme should be at least a few vibrational periods. Applying a statistIcal method closely related to the RRKM theory of unimolecular decomposition he was able to obtain good agreement with experimental results. A more elaborated statistical treatment [5] using classical trajcctory calculations has led to essentially the same resuits. If the resonance corresponds to quasibound vibrational excited states of the N,O molecule then, from a quantum mechanical point of view, three main results have to be expected: (a) The quenching probability should increase by one or two orders of magnitude going from the atomatom like model (Le., when the Nz molecule is treated like an atom) to the full consideration of the vlbration-
48. number 2
Volume
CIKMICAL
PHYSICS
al degrees of freedom because, if the complexes have a sufficiently long lifctimc (IO- * t --IO- t 3 s), the transition probability can be very large even for small couplings. (b) Asubstantial amount of the exothermicity should appear 3s vlbrationti cncrgy UT the nitrogen molecule. (c) TLC tmn4tion prohabdity should display a11 OScillatory behaviour as a function of the incident energy [7], not to be found in the atom-atom like calculation. Furthermore, if the resonances are well characterizcd, the peaks should appear at approximately the positron of the quasibound states calculated by simple pcrturbatlon theories [8,9] _ We present hcrc the numerical solution to the closed coupled equations for the collinear collision m a twocrossing electromc states mode! similar to that used in previous semiclassical calculations 12-41. Tl~c results arc in accord with points (a) through (c), confirming clearly that the high cfficlcncy of this reaction IS mamly due to complex fi>rmation.
LETTERS
I Iunc I977
Tahlc 1
-Rrameters - ---- of the - potential -- --- curves --_--
Rh2 = 1.128
A
= 0.293 WN2 a’ -_ 3 A-’
cv
RkO
calculations
- _‘_--
= L.184 A
E(‘)--.d3)-
1.98cV
a3 = 1.61 A-’
D’ = 3.8 CV --------
-.
D3 = 12.5 eV - _ _ _ _
__
(2)
,
and results
WC consider a simplified collmcar model with only two clcctronic states, a singlet state correlated with O(tD) + N,(tXs) and a triplet state correlated with 0(3 P) + N, (t C+). The potcntlal curves used are the same as the qualitative ones considered in refs. 12-41. We have represented them usmg the simple analytical forms,
where lf#‘)> represents the electronic part and XII’(K~~) ilrC thC ort~lollOrnla~ SOlIltiOflS to:
fi2 - d’ + $N~~ZN~(RNN -
- Rh2 1’
d&N
G2
=fi‘+&’
+ 1/2)X,I@~~)
1
2
eX+Z(i)(&,jo
vt3)(R~~,
RNO)
-
= ~CIQ
+ ~(3)eXp(-a(3)KNO)
R&J)]
~3,
) + k-(l)
(RNN
+ E(3),
-
x,~@NN)
-
(3)
After substitution of expansion (2) into the Schrodinger equation, we obtain the coupled
-
__
ly iepuhvc as a function of the N-O distance and h.ts been represented by an exponential function. On the other hand, the curve for the sir&t state has an attractive part which has been represented by a Morse pOtfXltii1~ With ITlUlllTlUITl - d’ ) at RN0 = Rho- &‘) and Ef3) are the asymptotic energies for the singtet and triplet states rcspectivcly. AJI the parameters used in the present calculation are listed in table t. The total wavefunction is expanded in the form:
+ l~‘3%#$v$Jz-o)l
2. Numerical
--
equations
,
= C v(i), ($1) + v!i.H 50 1111 N II ’
6~~)~ (1)
where the superscripts (1) and (3) correspond to the singlet and the triplet states respectively, RNN and RN0 arc the bond distances N-N and N-O respcctlvely, and pNz is the reduced mass of N2. WC have thus, in the two states, the same harmonic potential rcprescnting the N, bond, with frequency UN2 and equilibrium position Rk,. The potential for the triplet state IS pure-
(0 91,
(i+j=
I,3),
(4)
where v(‘k = nrz s ~RNNX;I(RNN)[V”‘(RNN,~~N~-“) -0D
-
$PN,d,(hN
and I$‘)
-- Rk2)I
XH@NN)
(5)
are the matrix clcments of the spin-orbit 359
Volume 48. nurnbcr 2
CIIEMICAL PHYSICS LETTERS
coupling yUv3) SO
= yi3,lF SO
= ((j”,l
~,,10’3,)
(6)
assumed to bc independent of the nuciear coordinates. In eqs. (Z), (4) and (S),f?N, -0 denotes the distance between the oxygen atom and the center-of-mass of the N2 molecule, i.e. R ~~ -0 =RN, +R&2,while I-CN*-0 is the reduced mass for tfle system N2-0, i.e. j,+* _. = ~UZO~N/(~VZN +7770). Thr: close coupling equations have been solved usmg a Fox integrator [IO] for a spin-orblt coupling of
80 cm-l nuclear
[GJ and assumed to be independent of the coordlnntcs.
For tlwrinnl
cncrgwz
only
one
channel tn the O(fD) G N2 tnitial ctatc is open but, 7 to 8 channeis arc open in the final state O(3P) + N2_ Denoting the partial quenching probability at total energy E by P,_,,,(E), for being after tile collision in the state 0(3P) + Nz(u = n) if imtiaffy tile system was in the state O(l D) + N2(u = II), we flavc calculated the mean quenching probability per cofhsion,
where fT-,- denotes
tfle rcfatlve mitlaf kinetic
E, = 1:’ - E(‘) - ;f“+jZ The total quenching thcrcfore:
energy
. proh.lbibty
@I per collision
is
(f)totA )= c;;l(po_,J. -V
69
In tabfc 2 we present tflc qucnchmg probablfltics at room tcnlpcrature. The first row and Prot.rl corresponds to the atom-atom like model (i.e., only the IJ= 0 vibrational channel of N2 is consldered). As the number of channels is Increazcd, the total qwznch-
PO--+,,
ing probablfitles increase and the converging results (for 18 ck11111e1s III the O(‘U) + N2 state and 8 channcls in the 0(31’) + N2 state) give a factor which is
36.6 times larger than tfle atom-atom like result. One interestmg way to look for tflc probability flow in this modcf IS to perform a set of calculations by artificially setting some of the couplings between the channels equal to zero. As an example, let us consider the case of only two channels in each electronic state. In fig. 1, WChi~ve represented this case. There arc three types of couplings between these states: (a) V-T coupfings,denoted by V{j\) and Vi:) whicfl are couplings between different channels of the same electronic 360
1 June 1977
Volume 48, number 2
CIKiWCAL
-I-_
-Il.
I
2
N
4
LT. I-TLRS
I Sune 1971
_J_
_-IL
3
PllYSICS
5
6
7
214
213
2lC
?I5
PIR
317
Zlrr
-0 Dlslance 6)
Pig. 1. Potcntinl CU~VCFused m the calculation ;md couplmg schcmc for the 2 X 2 chdnnel cw2.
l-l& 2 Prob.lbility
_
for the clcctroruc transition O(’ 0)
+ Nz (u = 0) + O(> P) f N2 (u = 0) for the 2
X I? chnnne[
wlcula-
tlon. (I) Spin-orbit couphng = 80 cm-t. ([I) spun -orbit coupling = 27 cm-‘, (111) 1$dnidcd by 10.
state; (b) E-V couplings, denoted by V{y3) which in our case only couples channels corresponding to different electronic states but with the same vibrational number. In table 3 we present the results for four dif-
incrcascs the quenching probability by a factor of two. Finally, in fig. 2 we show the quenching probability Po_,o as a function of the relative kinetic cncrgy for two wbrational channels of N, with two different
ferent coupling schemes: coupling scheme A where, having set v/j:) = V$ = Vi>“’ = 0 (so that we consider only V&“), is equal to the atom-atom hke model; coupling scheme B, with V&i) = V,(i13) = 0, which amounts to completely dccoupic the channel O(l D) + N2 (u = 1) and therefore resonances are not allowed; coupling scheme C, with V$ = Vdk3) = 0, corresponds to the decoupling of channel O(3P) + N, (u = 0); coupling scheme D whcrc all the couplings arc considered. The results of coupling schemes A and
D correspond, of course, to the two first rows of table 2 dnd arc presented here only for comparison. Comparing (a) and (b) we set that increaTing the
number of open channels for the repulsive potential dots not clringc signficantly the probabilitlcs. other hand, including only one closed channel
On the of the
attractive curve (case C) (and thus allowing resonances) Table 3 Mean quenching probabilities per collision for diffcrcnt coupling rchcmc~ in the 2 --
Couplmg scheme
--_~---
A
--.
---.
_--
--
.--
B
0.153 x 10-z
--
--.
__------ pa-+0 . _-_--__0.129 x 10-Z
C D
-
-_- ----
0.392 x 10-Z ----- ---__
_-
PO-1 ---
_--
-
X 2 channel cast
_ -.
_-
----
--
-
.-..
-
Obrcrv~iions
.---
atom--atom
0.135 x IO4
VA:’
0.759 x 10-Z
V$' and V$’
0.811 x IO-* ---
-.--
full 2 _ ---
and
_-_
__
__
--
--
.-
like coupiing
V&") - 0 no rcs‘,n‘lcIccs = 0 rcwr~~c~ccs
X 2 calculation
- ---
---
----
_-361
Volume 48. number 2 where u denotes oscillator
the quantum
and II the vibrational
1 June 1977
CIIEhllCAL PHYSICS LETTERS number of the Morse quantum number of
the N2 nioIecule. 111this case WC have necessarily n= I, and c~cu]~mn of this cricrgy for u in the range 3 I to 33 gives 2.12,2.165 and 2.21 eV. Without any doubt WC can sssocratc the peak observed in fig. 2 with the TCSOnancc ~32,~. The shift of 0.005 eV is probably simply due to the fact that we arc calculating here the position of the resonances using RN~ dnd RN~ -0 asif they wcrc identical to the normal modes of the iinear triatomic molecule. 1n this case of a well-defined peak WC can estimate the hfctimc of the cornplc~. Taking I‘ = 5 X 10d4 eV as an estnnate of the linewidth we have r =fi/r = IO-” <.
3. ConcIusiolts From the results prcscnted above, it is clear that the high efficiency of the quenching rcactron O(l D) + Nl('Ci)+ 0(3P) + N,(‘Zi) at room temperature, is mainly due to vibratronal resonances (quasibound states of N;O), as has been suggested before [4,5]. 77trs in turn can give a justification of the use of the collinear model. The ground state of N20 IS known to be linear and with symmetry lZi. Therefore, in order to conserve spin, rt dissociates into N,(X ‘Zf) + O(lD). Thus the collinear arrangement for the colhsion O(lD) with Nz(X ‘El) correlates with the ground state and so the depth of the well decrcasrs for a non-collinear collision configuration. The influence of the quasibound states in this case should bc consequently fess important [ 11-l 31 and only the collisions with 3 quasi-collinear arrarigcment should give high quenching probabilities. It should be stressed at tfris pomt that the potential curves used hcrc r?rc only quditntivc and they h.~vc been token directly from the earlier wmiclaWcalstudies [L-4] _It
parison between our calculated mean quenching probabilities per collision and the experimental rate constants is not possrblc (see however, ref. [IS] for the relation between collinear and three dimensional collision rates). However, relative quenching probabilities for different temperatures, if experimentally available could be subject to direct comparison with these colhnear mean quenching probabilities.
Acknowledgement WC arc indcbtcd to Dr. J.M. Launay for providing us wrth his version of the Fox integrator and to P. Vrllareal for computational assistance. We would also like to acknowledge the bcncfit of helpful discussions with Dr. hl. Bwr. This rcscnrch was supported, in part. by the “ServiceCulture1 de 1’Arnbassade de France en I!spagnc”.
References [ 11 T.G. Shger and G. Black, J. Chcm. Phys. 60 (1974) [2j
[ 31 (41 [S] (61
[ 7]
468. E.R. FIFIIC~ md
C. Bnucr, J. Chcm. Phys. 57 (1972) 1966. J.B. Dclos, J. Chcm. Phys. 59 (1973) 2365. J.C. Iuliy, J. Cbcm. Phys. 61 (1974) 61; 62 (1975) 1893. GX. Znhr. R. Preston and W-Ii. Mdlcr, J. Chcm. Phys. 62 (1975) 1127. T. Jdmanouchi and Il. IIoric, J. Pbys. Sot. Japan 7 (1952) 52. I’.M. Chapn1.m Jr. and C.1:. IIayec. J. Chcm. Phys. 62
(I 975) 4400. IS] W. Eastcs and R.A. Marcus, I Chcm. Phys. 59 (1973) 4757. [9] 0. Atabek, J A Bcswck and R. Lefcbvrc, Chem. Phys. Letters 33 (1975) 228.
[ 101 L. Tou, The numerical solutum of two-point boundary v.tluc problems in ordinary differential equations (Oxford Umv. Prcs\, London, 1957).
[IIf
S.D.
I’cyamrr~twrC~nci
I< J
B:wnku.
J. Chcm
Phy\.
should also be noted that only one repulsiveelectronic
49 (1968) 2473. ( 12 J A. Cl~utaan and G. Scgal, J. Chcm. Pbys. 57 (1972)
state was used in this calculation. In the collinear configuration two electronic states (3Z- and 3R) correlate with 0(3P) + Nz(* 2;) Ii I- 131. A study of the sensitivity of the final results as a function of small changes in the curve parameter as we11 as thc’influcnce of a second repulsive electromc state should be of importance. Pinally we should note that a direct com-
D.G. Iloppcr. P.J. Fortune, AC. Wzrhland T.O. Tlcman, Potential Cncrgy Surfaces for Triatomics II. Results of SCF and Preliminary OVC CJ~CuhtionS. ARL TR 750202, Vol. 2, Au Force Systems Command, June, 1975. [14] U. Tdno. Phys. Rev. 124 (1961) 1866. [ 151 W.A. Wasam Jr. and R.D. Levine, J. Chcm. Phys. 64 (1976) 3118.
362
3lX9. [ 131 Pd. Kr?u\,r.
48, number
2
RESONANCES
CIICMICAL
IN THE ELECTRONIC
PIIYSICS
QUENCHING
LETITRS
OF O(lD)
1 June
1977
BY N,.
ANUMERlCALQUANTUMMECHANlCALSTUDYFORTHECOLLlNEARCOLLlSION G. DELGADO-SARRIO
dr Qo~crus, Dcpartamcnto dc Quim~a Cudtztiia. Unrvcrs~dad Autbnoma de ilfadrki. Madnd, Spam and Inrtituto dr Iktructura dr Ia hfatcria C.S.1.C , Madrid. Spaoz I%cul~ad
and J.A. BESWICK i: Laboratom Rccclvcd
de Phoroplrysiquc ~lfokularrc,
7 J.lnudry
UnwcrsltE de Pans 9:d, YI405 Orsay. France
1977
Rcvwzd rndnuscript rcccwed 7 March 1977
The closed coupled equations for the collincnr colhsion O(‘D) + N2 (‘Z$ +O(‘P) + N2(‘Ci) WVCbeen solved nunicw CAY for J model 0t t-0 CrOwIn;;potcnticd Lurvcs dasunlin~; n conSLant y3ln--orbat coupling Compdrnon bctwccn the rcwrlta of an nlom --ntom llkc model nnrl the converging result\ rcvc.11~ n cubst.mtial (fxtor of -40) cnhanccmcnt of the clcctroulc quenching .~t room tempcraturc topcthur with high vlhratromll excitation of Nz- Thcsc results. and tlw study of tbc peaks nppcaring in the qucnchg probabthtlcs ‘1su function of the incident energy, clearly confirm thrlt the high cffrcicncy of thr% rcact!on IS mainly due to rcsondnccb (qudsibound states), rls hrls been lately suggc\tcd.
. 1. lntrodmtion The collisional qucnchmg of the first excited state of the oxygen atom O(lD) + N,(’ Z’) -> O(3P) + N2(l 2:) has rcceivcd expcrimcntal F11and theorctical [2-S] attcntlon Idtcly. Thi5 interest has arisen not only from ihe importance of this mechanism in upper atmosphere reactions but also, from a purely thcoretical point of vlcw, because this “spin-forbidden” reactlon is known to occur with high efficiency [ I] . The firsr attempts [2,3] to explain the Hugh termal rate, (4-10) X IO-” cni3/molecule s, of this process were conducted using the semiclassical approximation to a model of curves crossing between the triplet and the singlet states. if a plausible spin-orbit couplmg is used (=80 cm-l [C,]) the rdte constant obtained in this way is 20-100 times smaller than the cxpcrlmental result. Moreover, the N, molecule is treated like an atom so there is no vibratlonal excitation of N2 after the colPresent address. Wcbmann Institute of Science, of Chcmlcal Phystcs, Rchovot, Isrxl.
358
Department
hsion. This is in contradiction with the cxpenmental result [I] which predicts that after the collision a substantlal amount of the 1.98 eV exothermicity js transformed m vibrational cxcltation of molecular nitrogen. Recently, Tully [4] has suggested that this violation of spin conservation rules must be associated with the formation of collision complexes whose hfetlme should be at least a few vibrational periods. Applying a statistIcal method closely related to the RRKM theory of unimolecular decomposition he was able to obtain good agreement with experimental results. A more elaborated statistical treatment [5] using classical trajcctory calculations has led to essentially the same resuits. If the resonance corresponds to quasibound vibrational excited states of the N,O molecule then, from a quantum mechanical point of view, three main results have to be expected: (a) The quenching probability should increase by one or two orders of magnitude going from the atomatom like model (Le., when the Nz molecule is treated like an atom) to the full consideration of the vlbration-
48. number 2
Volume
CIKMICAL
PHYSICS
al degrees of freedom because, if the complexes have a sufficiently long lifctimc (IO- * t --IO- t 3 s), the transition probability can be very large even for small couplings. (b) Asubstantial amount of the exothermicity should appear 3s vlbrationti cncrgy UT the nitrogen molecule. (c) TLC tmn4tion prohabdity should display a11 OScillatory behaviour as a function of the incident energy [7], not to be found in the atom-atom like calculation. Furthermore, if the resonances are well characterizcd, the peaks should appear at approximately the positron of the quasibound states calculated by simple pcrturbatlon theories [8,9] _ We present hcrc the numerical solution to the closed coupled equations for the collinear collision m a twocrossing electromc states mode! similar to that used in previous semiclassical calculations 12-41. Tl~c results arc in accord with points (a) through (c), confirming clearly that the high cfficlcncy of this reaction IS mamly due to complex fi>rmation.
LETTERS
I Iunc I977
Tahlc 1
-Rrameters - ---- of the - potential -- --- curves --_--
Rh2 = 1.128
A
= 0.293 WN2 a’ -_ 3 A-’
cv
RkO
calculations
- _‘_--
= L.184 A
E(‘)--.d3)-
1.98cV
a3 = 1.61 A-’
D’ = 3.8 CV --------
-.
D3 = 12.5 eV - _ _ _ _
__
(2)
,
and results
WC consider a simplified collmcar model with only two clcctronic states, a singlet state correlated with O(tD) + N,(tXs) and a triplet state correlated with 0(3 P) + N, (t C+). The potcntlal curves used are the same as the qualitative ones considered in refs. 12-41. We have represented them usmg the simple analytical forms,
where lf#‘)> represents the electronic part and XII’(K~~) ilrC thC ort~lollOrnla~ SOlIltiOflS to:
fi2 - d’ + $N~~ZN~(RNN -
- Rh2 1’
d&N
G2
=fi‘+&’
+ 1/2)X,I@~~)
1
2
eX+Z(i)(&,jo
vt3)(R~~,
RNO)
-
= ~CIQ
+ ~(3)eXp(-a(3)KNO)
R&J)]
~3,
) + k-(l)
(RNN
+ E(3),
-
x,~@NN)
-
(3)
After substitution of expansion (2) into the Schrodinger equation, we obtain the coupled
-
__
ly iepuhvc as a function of the N-O distance and h.ts been represented by an exponential function. On the other hand, the curve for the sir&t state has an attractive part which has been represented by a Morse pOtfXltii1~ With ITlUlllTlUITl - d’ ) at RN0 = Rho- &‘) and Ef3) are the asymptotic energies for the singtet and triplet states rcspectivcly. AJI the parameters used in the present calculation are listed in table t. The total wavefunction is expanded in the form:
+ l~‘3%#$v$Jz-o)l
2. Numerical
--
equations
,
= C v(i), ($1) + v!i.H 50 1111 N II ’
6~~)~ (1)
where the superscripts (1) and (3) correspond to the singlet and the triplet states respectively, RNN and RN0 arc the bond distances N-N and N-O respcctlvely, and pNz is the reduced mass of N2. WC have thus, in the two states, the same harmonic potential rcprescnting the N, bond, with frequency UN2 and equilibrium position Rk,. The potential for the triplet state IS pure-
(0 91,
(i+j=
I,3),
(4)
where v(‘k = nrz s ~RNNX;I(RNN)[V”‘(RNN,~~N~-“) -0D
-
$PN,d,(hN
and I$‘)
-- Rk2)I
XH@NN)
(5)
are the matrix clcments of the spin-orbit 359
Volume 48. nurnbcr 2
CIIEMICAL PHYSICS LETTERS
coupling yUv3) SO
= yi3,lF SO
= ((j”,l
~,,10’3,)
(6)
assumed to bc independent of the nuciear coordinates. In eqs. (Z), (4) and (S),f?N, -0 denotes the distance between the oxygen atom and the center-of-mass of the N2 molecule, i.e. R ~~ -0 =RN, +R&2,while I-CN*-0 is the reduced mass for tfle system N2-0, i.e. j,+* _. = ~UZO~N/(~VZN +7770). Thr: close coupling equations have been solved usmg a Fox integrator [IO] for a spin-orblt coupling of
80 cm-l nuclear
[GJ and assumed to be independent of the coordlnntcs.
For tlwrinnl
cncrgwz
only
one
channel tn the O(fD) G N2 tnitial ctatc is open but, 7 to 8 channeis arc open in the final state O(3P) + N2_ Denoting the partial quenching probability at total energy E by P,_,,,(E), for being after tile collision in the state 0(3P) + Nz(u = n) if imtiaffy tile system was in the state O(l D) + N2(u = II), we flavc calculated the mean quenching probability per cofhsion,
where fT-,- denotes
tfle rcfatlve mitlaf kinetic
E, = 1:’ - E(‘) - ;f“+jZ The total quenching thcrcfore:
energy
. proh.lbibty
@I per collision
is
(f)totA )= c;;l(po_,J. -V
69
In tabfc 2 we present tflc qucnchmg probablfltics at room tcnlpcrature. The first row and Prot.rl corresponds to the atom-atom like model (i.e., only the IJ= 0 vibrational channel of N2 is consldered). As the number of channels is Increazcd, the total qwznch-
PO--+,,
ing probablfitles increase and the converging results (for 18 ck11111e1s III the O(‘U) + N2 state and 8 channcls in the 0(31’) + N2 state) give a factor which is
36.6 times larger than tfle atom-atom like result. One interestmg way to look for tflc probability flow in this modcf IS to perform a set of calculations by artificially setting some of the couplings between the channels equal to zero. As an example, let us consider the case of only two channels in each electronic state. In fig. 1, WChi~ve represented this case. There arc three types of couplings between these states: (a) V-T coupfings,denoted by V{j\) and Vi:) whicfl are couplings between different channels of the same electronic 360
1 June 1977
Volume 48, number 2
CIKiWCAL
-I-_
-Il.
I
2
N
4
LT. I-TLRS
I Sune 1971
_J_
_-IL
3
PllYSICS
5
6
7
214
213
2lC
?I5
PIR
317
Zlrr
-0 Dlslance 6)
Pig. 1. Potcntinl CU~VCFused m the calculation ;md couplmg schcmc for the 2 X 2 chdnnel cw2.
l-l& 2 Prob.lbility
_
for the clcctroruc transition O(’ 0)
+ Nz (u = 0) + O(> P) f N2 (u = 0) for the 2
X I? chnnne[
wlcula-
tlon. (I) Spin-orbit couphng = 80 cm-t. ([I) spun -orbit coupling = 27 cm-‘, (111) 1$dnidcd by 10.
state; (b) E-V couplings, denoted by V{y3) which in our case only couples channels corresponding to different electronic states but with the same vibrational number. In table 3 we present the results for four dif-
incrcascs the quenching probability by a factor of two. Finally, in fig. 2 we show the quenching probability Po_,o as a function of the relative kinetic cncrgy for two wbrational channels of N, with two different
ferent coupling schemes: coupling scheme A where, having set v/j:) = V$ = Vi>“’ = 0 (so that we consider only V&“), is equal to the atom-atom hke model; coupling scheme B, with V&i) = V,(i13) = 0, which amounts to completely dccoupic the channel O(l D) + N2 (u = 1) and therefore resonances are not allowed; coupling scheme C, with V$ = Vdk3) = 0, corresponds to the decoupling of channel O(3P) + N, (u = 0); coupling scheme D whcrc all the couplings arc considered. The results of coupling schemes A and
D correspond, of course, to the two first rows of table 2 dnd arc presented here only for comparison. Comparing (a) and (b) we set that increaTing the
number of open channels for the repulsive potential dots not clringc signficantly the probabilitlcs. other hand, including only one closed channel
On the of the
attractive curve (case C) (and thus allowing resonances) Table 3 Mean quenching probabilities per collision for diffcrcnt coupling rchcmc~ in the 2 --
Couplmg scheme
--_~---
A
--.
---.
_--
--
.--
B
0.153 x 10-z
--
--.
__------ pa-+0 . _-_--__0.129 x 10-Z
C D
-
-_- ----
0.392 x 10-Z ----- ---__
_-
PO-1 ---
_--
-
X 2 channel cast
_ -.
_-
----
--
-
.-..
-
Obrcrv~iions
.---
atom--atom
0.135 x IO4
VA:’
0.759 x 10-Z
V$' and V$’
0.811 x IO-* ---
-.--
full 2 _ ---
and
_-_
__
__
--
--
.-
like coupiing
V&") - 0 no rcs‘,n‘lcIccs = 0 rcwr~~c~ccs
X 2 calculation
- ---
---
----
_-361
Volume 48. number 2 where u denotes oscillator
the quantum
and II the vibrational
1 June 1977
CIIEhllCAL PHYSICS LETTERS number of the Morse quantum number of
the N2 nioIecule. 111this case WC have necessarily n= I, and c~cu]~mn of this cricrgy for u in the range 3 I to 33 gives 2.12,2.165 and 2.21 eV. Without any doubt WC can sssocratc the peak observed in fig. 2 with the TCSOnancc ~32,~. The shift of 0.005 eV is probably simply due to the fact that we arc calculating here the position of the resonances using RN~ dnd RN~ -0 asif they wcrc identical to the normal modes of the iinear triatomic molecule. 1n this case of a well-defined peak WC can estimate the hfctimc of the cornplc~. Taking I‘ = 5 X 10d4 eV as an estnnate of the linewidth we have r =fi/r = IO-” <.
3. ConcIusiolts From the results prcscnted above, it is clear that the high efficiency of the quenching rcactron O(l D) + Nl('Ci)+ 0(3P) + N,(‘Zi) at room temperature, is mainly due to vibratronal resonances (quasibound states of N;O), as has been suggested before [4,5]. 77trs in turn can give a justification of the use of the collinear model. The ground state of N20 IS known to be linear and with symmetry lZi. Therefore, in order to conserve spin, rt dissociates into N,(X ‘Zf) + O(lD). Thus the collinear arrangement for the colhsion O(lD) with Nz(X ‘El) correlates with the ground state and so the depth of the well decrcasrs for a non-collinear collision configuration. The influence of the quasibound states in this case should bc consequently fess important [ 11-l 31 and only the collisions with 3 quasi-collinear arrarigcment should give high quenching probabilities. It should be stressed at tfris pomt that the potential curves used hcrc r?rc only quditntivc and they h.~vc been token directly from the earlier wmiclaWcalstudies [L-4] _It
parison between our calculated mean quenching probabilities per collision and the experimental rate constants is not possrblc (see however, ref. [IS] for the relation between collinear and three dimensional collision rates). However, relative quenching probabilities for different temperatures, if experimentally available could be subject to direct comparison with these colhnear mean quenching probabilities.
Acknowledgement WC arc indcbtcd to Dr. J.M. Launay for providing us wrth his version of the Fox integrator and to P. Vrllareal for computational assistance. We would also like to acknowledge the bcncfit of helpful discussions with Dr. hl. Bwr. This rcscnrch was supported, in part. by the “ServiceCulture1 de 1’Arnbassade de France en I!spagnc”.
References [ 11 T.G. Shger and G. Black, J. Chcm. Phys. 60 (1974) [2j
[ 31 (41 [S] (61
[ 7]
468. E.R. FIFIIC~ md
C. Bnucr, J. Chcm. Phys. 57 (1972) 1966. J.B. Dclos, J. Chcm. Phys. 59 (1973) 2365. J.C. Iuliy, J. Cbcm. Phys. 61 (1974) 61; 62 (1975) 1893. GX. Znhr. R. Preston and W-Ii. Mdlcr, J. Chcm. Phys. 62 (1975) 1127. T. Jdmanouchi and Il. IIoric, J. Pbys. Sot. Japan 7 (1952) 52. I’.M. Chapn1.m Jr. and C.1:. IIayec. J. Chcm. Phys. 62
(I 975) 4400. IS] W. Eastcs and R.A. Marcus, I Chcm. Phys. 59 (1973) 4757. [9] 0. Atabek, J A Bcswck and R. Lefcbvrc, Chem. Phys. Letters 33 (1975) 228.
[ 101 L. Tou, The numerical solutum of two-point boundary v.tluc problems in ordinary differential equations (Oxford Umv. Prcs\, London, 1957).
[IIf
S.D.
I’cyamrr~twrC~nci
I< J
B:wnku.
J. Chcm
Phy\.
should also be noted that only one repulsiveelectronic
49 (1968) 2473. ( 12 J A. Cl~utaan and G. Scgal, J. Chcm. Pbys. 57 (1972)
state was used in this calculation. In the collinear configuration two electronic states (3Z- and 3R) correlate with 0(3P) + Nz(* 2;) Ii I- 131. A study of the sensitivity of the final results as a function of small changes in the curve parameter as we11 as thc’influcnce of a second repulsive electromc state should be of importance. Pinally we should note that a direct com-
D.G. Iloppcr. P.J. Fortune, AC. Wzrhland T.O. Tlcman, Potential Cncrgy Surfaces for Triatomics II. Results of SCF and Preliminary OVC CJ~CuhtionS. ARL TR 750202, Vol. 2, Au Force Systems Command, June, 1975. [14] U. Tdno. Phys. Rev. 124 (1961) 1866. [ 151 W.A. Wasam Jr. and R.D. Levine, J. Chcm. Phys. 64 (1976) 3118.
362
3lX9. [ 131 Pd. Kr?u\,r.