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Collision-induced predissociation by van der Waals' interaction
Volume 4, number 4
COLLISION-INDUCED
CHEMICAL PHYSICS LETTERS
PREDISSOCIATION
BY
VAN
I November L969
DER
WAALS’
INTERACTION
J. E. SELWYN* and J. I. STEINFELD*’ Department
of Ckenlistry, Massachusetts Institute of Technology, Canlbridge, Massachusetts 02139, USA Received 21 August 1969
We have deveIoped a theory that satisfactorily predicts the relative magnitudes of the observed fluorescence quenching cross sections for B(%‘IO+)IZ, based on van der Waals perturbation. Applications to other U sgstems are also discussed.
1. INTRODUCTION Coincident with the measurement of rotational and vibrational energy-transfer cross sections for iodine have come values for the B - X fluorescence quenching cross section in the absence of a magnetic field [1,2]. Data is available for self- and foreign-gas quenching; both 7~’ = 15 and v’ = 25 have been studied in detail. Various attempts have been made to correlate these cross sections with molecular parameters [Z-4]. None of these, however, has been based on a systematic quantum mechanical approach to the problem. Early observations [5-‘71 of 1(2~3,2) atoms following irradiation of 12 vapor below the dissociation limit suggested that the quenching results from a collision-induced adiabatic transition from the bound B3nO+ : state of 12 to a repulsive electronic state th% correlates with ground-state atomic iodine. The original contention of RUssler [3] was that the ratio of the collision cross section to the gas kinetic cross section (hereinafter called the quenching efficiency) would be proportional to the product of the polarizability u of the quenching molecule and the duration of the collision, which is proportional to the square root of the reduced mass (pln). Steinfeld [2] has tested this correlation for his data on the v’ = 25 state and has replotted Brown’s data [l] for the LI’ = 15 state using a better value for the excited state radiative lifetime. Although the correlation obtained with up’/2 is fairly good, certain gases (notably CO2 and NO) behave anomalously.
2. DERIVATION OF THE QUENCHING CRCSS SECTION The collision between the excited molecule and the quencher, which results in predissociation, may be regarded as a time dependent perturbation on the stationary state Hamiltonian for the system. Thus we must first consider wave functions appropriate to the stationary state in which the perturbation is zero. The initial state of the system can be represented by a product wave function as follows:
where $1; is the wave function for the excited 3”O& bound state of I2 an< I&Q is the wave function for the quencher molecule. ‘L’he electronic coordinates are represented by r, the interatomic distance in 12 by R2, and RI is the distance between the quencher and center of mass of 12. The Born-Oppenheimer approximation when applied to the I2 wave function gives q
r
eo$
r, RI )xv (4 1 (2)
where initial
the effect of the quencher molecule on the excited state wave function has been ignored.
This is valid for large RI, i.e. before the collision. After the collision the I2 wave function will be represented by
91, = 7 cj$(r,R~)x~(R%)
(3)
= CC.$J.WXEW2) j
* Nationti Science Foundation Predoctoral ** Alfred P.Sloan Research Fellow.
Fellow.
33
where the repulsive states $j are characterized by the total electronic angular momentum, a, parity P, and the inversion symmetry (i = + or 217
Volume 4, number 4
CHEMICAL
HYSICS ‘
1 November
LETTERS
1969
t IOL-
2
1
cq I-
If
F-
F, ,,,,,,, l
0 II 01
g Re3 Fig.
Iap
(scaled)
1. Relative quenching efficiencies for K;(B%J,+) in (A: data of ref. (11) and v = 25 (0: data of u ref. [2]).
for a = 0 states) and x&32) is the free particle wave function for the dissociating pair. The total is the product wave function after the collision -
Fig.
kq =~P(EL)I(J/iIV(r,R1)I~f}12.
2. Relative
G where X1 = EiXli, potential that gives forces. Proceeding ment in eq. (4), we
quenching
l
IO
( , , , ,,., ma
CbCXd%l,
efficiencies ref. [lo]).
and Y1 in spherical
Xl a R sin6 cosb, (4)
for SO;
(data of
coordinates
Yl a Rsin0
gives
sin@ .
Assuming axial symmetry of the molecular Hamiltonian (i.e. unperturbed -wave functions during the collision),
Thus, (X1X2 + YlY2 - 22122)
(51
This potential is the same rise to the van der Waals to calculate the matrix elefind
etc.
(6)
218
, , ,,,,
Note that no cross terms appear in this expression (81. To see why they vanish, consider term of the form
WritingXl
An appropriate form for Vcan be obtained [8] by considering the classical expression for the interaction between the two charge distributions in the Y2 and the quencher. The potential for this interaction is =E?
,
rl/gf.
The quenching rate can be predicted by the “Fermi Golden Rule” for constant transition rates, which gives the quenching rate
V(r,R1)
“I
7
U= 15
of this expression and
,
I
.:
Q R sinQ[6(SZtl)
+ 6(5L -I)]
and
The complete product will thus be zero when the sum over 52 is performed. The form of the expression for the polarizability of the quencher molecule suggests a simplification to the rate expression, since
a
Fig. 3.
Relative
1 Xavernher
CHEMICAL PHYSICS LETTERS
Volume 4. number 4
quenching
efficiencies
ref. [121).
for
1969
NOi (data of
It has been found that the denominator of this expression is very nearly equal to the ionization energy, 1. Substituting eq. (7) into eq. (6) we find that
where the u’s are the Lennard-Jones Substituting we find o;
= i””
27R1
[;’
KS
(&Cb?*]
parameters.
dR1
=
(11) (8)
=($&)1’2
In terms
of these
matrix
elements,
the
rate
may
be written
$
A . C
From
A.
~~~~~~~(f),(xElxv.:/Z
equation (11) we may obtain the parametric
dependence, o2 Q:‘L%Y_ 3 q RC
(12)
3. RESULTS FOR IODINE
(9) This expression has units of reciprocal time. Recall that this rate is derived for a fixed contiguration, i.e. a given value of Rl. What we wish to know, however, is whether a particular collision will result in quenching. This probability is simply the integral of the rate over time. We may write dt as (dt/dRI)dRl where (df/dRI) is approximated as the inverse of the mean thermal velocity, hg/BkT) 'I2. The probability as a function of RI isthen integrated over the distribution of approach distances, weighted by the appropriate differential area to give the quenching cross section. The upper limit on RI is -0; the lower limit is just the hard-sphere collision radius which will be taken as
The results of our theory for 13 are illustrated in fig. 1. The data for the two vibrational states that have been studied extensively are shown. This Log-log plot shows better agreement. with eq. (12) than previous attempts to correlate the cross section with cY$/~. Again. as in the previous theories, the cross section for NO is Iatger than predicted. The only plausible explanation Ear this particular divergence has been presented by Steinfeld {4]. Whereas totally symmetric collision partners interact with the B state to give A’ complexes which correlate with the 0; repulsive state of I2, NO which has a 2R ground state can correlate with other final states since it may aIso produce A" complexes. Thus, the enhanced quenching ability of NO has been attributed to the opening up of additional channels for quenching. Recent preliminary values of v’ = 43 quenching 219
‘Volume 4, number 4
CtiMlCAL
sections also indicate eq. (12) [?I.
crqss
4. APPLICATIONS
-pod agreement
PtiYsICS LETTERS
with
TO OTHER SYSTEMS
RecentIy! data has become available on the cross sections for the quenching of the A(IBI)--t X(lAl) fluorescence in SO2 [lo]. In this case, tine quenching has been ascribed to callisioninduced Internal conversion. Although the final state’is discrete in this instance, a “continuum” of vibrational states is available after “non resonant” emission. Thus, a treatment analogous to that for I2 would seem applicable. The results obtained from eq. (12) are scaled and plotted against the experimental cross sections in fig. 2. Although there is still some degree of scatter, the. results seem to lie clearly along the expected straight line of unit slope. The present considerations give a better fit to the measured results than do previous attempts to fit the data to u or ~gl/~. Again. it should be noted that the cross section for NO is somewhat larger than predicted. Finally, we shall consider instances of fluorescence quenching in three polar molecules, viz, NO, NO2 and CH30H 111-13). Eq. (12) can be shown to be equally applicable in the case of polar molecules; the basic Rl-dependence remains the same. The results of this theory for NO2 are presented in fig. 3. For NO2 1121 good agreement is again observed although the cross sections for NO and CO2 are still underpredicted by eq. (12).
1 November 1969
In the case of methanol [13], the experimental cross sections show far greater variability than do those predicted (see fig. 4). The collisioninduced predissociation in methanol not only shows a much more complicated chemistry than the previous examples, but also shows wide scatter on an Lyj.fij2 plot. Finally brief mention must be made regarding the.observed fluorescence in NO [ll]. The theory predicts the correct order for the relative size of the quenching cross sections of He, Ar, H2 and N2 but it fails once more in predictions for NO itself and CO2.
REFERENCES [I] R. L. Brown and W. Klemperer, J. Chem. Phys. 41 (1964) 3072. [Z] J. 1. Steinfeld and W.Klemperer. J. Chcm. Phys.42 (1965) 3475. [3] F.Rossler. Z.Physik 96 (1933) 251. [4] J. I. Steinfeld. J. Chem. Phys. 44 (1966) 2740. [S] L. A. Turner and E.W. Samson, Phys. Rev. 37 (1931) 1684. [S] L.A. Turner, Phys. Rev.41 (1932) 627. (71 E. Rabinowitch and W. C. Wood, J. Chem. Phys.4 (1936) 35.9. (81 H. Fdargenau, Rev. Mod. Phys. 11 (1939) 1. [9] J. I. Steinfeld and R. B. Kurzel, to be published. [I?] H. D. AIettee, to be published. [ll] H. P. Broida and T. Carrington, J. Chem. Phys. 35 (19ti3) 136. [IZ] G. f-I. Meyers, D. M. Silver and F. Kaufman, J. Chem. Phys.44 (1966) 718. 1131 J.Hagege, P.C. Roberge and C.\rermeil, Trans Faraday Sot. 64 (1968) 3289.
COLLISION-INDUCED
CHEMICAL PHYSICS LETTERS
PREDISSOCIATION
BY
VAN
I November L969
DER
WAALS’
INTERACTION
J. E. SELWYN* and J. I. STEINFELD*’ Department
of Ckenlistry, Massachusetts Institute of Technology, Canlbridge, Massachusetts 02139, USA Received 21 August 1969
We have deveIoped a theory that satisfactorily predicts the relative magnitudes of the observed fluorescence quenching cross sections for B(%‘IO+)IZ, based on van der Waals perturbation. Applications to other U sgstems are also discussed.
1. INTRODUCTION Coincident with the measurement of rotational and vibrational energy-transfer cross sections for iodine have come values for the B - X fluorescence quenching cross section in the absence of a magnetic field [1,2]. Data is available for self- and foreign-gas quenching; both 7~’ = 15 and v’ = 25 have been studied in detail. Various attempts have been made to correlate these cross sections with molecular parameters [Z-4]. None of these, however, has been based on a systematic quantum mechanical approach to the problem. Early observations [5-‘71 of 1(2~3,2) atoms following irradiation of 12 vapor below the dissociation limit suggested that the quenching results from a collision-induced adiabatic transition from the bound B3nO+ : state of 12 to a repulsive electronic state th% correlates with ground-state atomic iodine. The original contention of RUssler [3] was that the ratio of the collision cross section to the gas kinetic cross section (hereinafter called the quenching efficiency) would be proportional to the product of the polarizability u of the quenching molecule and the duration of the collision, which is proportional to the square root of the reduced mass (pln). Steinfeld [2] has tested this correlation for his data on the v’ = 25 state and has replotted Brown’s data [l] for the LI’ = 15 state using a better value for the excited state radiative lifetime. Although the correlation obtained with up’/2 is fairly good, certain gases (notably CO2 and NO) behave anomalously.
2. DERIVATION OF THE QUENCHING CRCSS SECTION The collision between the excited molecule and the quencher, which results in predissociation, may be regarded as a time dependent perturbation on the stationary state Hamiltonian for the system. Thus we must first consider wave functions appropriate to the stationary state in which the perturbation is zero. The initial state of the system can be represented by a product wave function as follows:
where $1; is the wave function for the excited 3”O& bound state of I2 an< I&Q is the wave function for the quencher molecule. ‘L’he electronic coordinates are represented by r, the interatomic distance in 12 by R2, and RI is the distance between the quencher and center of mass of 12. The Born-Oppenheimer approximation when applied to the I2 wave function gives q
r
eo$
r, RI )xv (4 1 (2)
where initial
the effect of the quencher molecule on the excited state wave function has been ignored.
This is valid for large RI, i.e. before the collision. After the collision the I2 wave function will be represented by
91, = 7 cj$(r,R~)x~(R%)
(3)
= CC.$J.WXEW2) j
* Nationti Science Foundation Predoctoral ** Alfred P.Sloan Research Fellow.
Fellow.
33
where the repulsive states $j are characterized by the total electronic angular momentum, a, parity P, and the inversion symmetry (i = + or 217
Volume 4, number 4
CHEMICAL
HYSICS ‘
1 November
LETTERS
1969
t IOL-
2
1
cq I-
If
F-
F, ,,,,,,, l
0 II 01
g Re3 Fig.
Iap
(scaled)
1. Relative quenching efficiencies for K;(B%J,+) in (A: data of ref. (11) and v = 25 (0: data of u ref. [2]).
for a = 0 states) and x&32) is the free particle wave function for the dissociating pair. The total is the product wave function after the collision -
Fig.
kq =~P(EL)I(J/iIV(r,R1)I~f}12.
2. Relative
G where X1 = EiXli, potential that gives forces. Proceeding ment in eq. (4), we
quenching
l
IO
( , , , ,,., ma
CbCXd%l,
efficiencies ref. [lo]).
and Y1 in spherical
Xl a R sin6 cosb, (4)
for SO;
(data of
coordinates
Yl a Rsin0
gives
sin@ .
Assuming axial symmetry of the molecular Hamiltonian (i.e. unperturbed -wave functions during the collision),
Thus, (X1X2 + YlY2 - 22122)
(51
This potential is the same rise to the van der Waals to calculate the matrix elefind
etc.
(6)
218
, , ,,,,
Note that no cross terms appear in this expression (81. To see why they vanish, consider term of the form
WritingXl
An appropriate form for Vcan be obtained [8] by considering the classical expression for the interaction between the two charge distributions in the Y2 and the quencher. The potential for this interaction is =E?
,
rl/gf.
The quenching rate can be predicted by the “Fermi Golden Rule” for constant transition rates, which gives the quenching rate
V(r,R1)
“I
7
U= 15
of this expression and
,
I
.:
Q R sinQ[6(SZtl)
+ 6(5L -I)]
and
The complete product will thus be zero when the sum over 52 is performed. The form of the expression for the polarizability of the quencher molecule suggests a simplification to the rate expression, since
a
Fig. 3.
Relative
1 Xavernher
CHEMICAL PHYSICS LETTERS
Volume 4. number 4
quenching
efficiencies
ref. [121).
for
1969
NOi (data of
It has been found that the denominator of this expression is very nearly equal to the ionization energy, 1. Substituting eq. (7) into eq. (6) we find that
where the u’s are the Lennard-Jones Substituting we find o;
= i””
27R1
[;’
KS
(&Cb?*]
parameters.
dR1
=
(11) (8)
=($&)1’2
In terms
of these
matrix
elements,
the
rate
may
be written
$
A . C
From
A.
~~~~~~~(f),(xElxv.:/Z
equation (11) we may obtain the parametric
dependence, o2 Q:‘L%Y_ 3 q RC
(12)
3. RESULTS FOR IODINE
(9) This expression has units of reciprocal time. Recall that this rate is derived for a fixed contiguration, i.e. a given value of Rl. What we wish to know, however, is whether a particular collision will result in quenching. This probability is simply the integral of the rate over time. We may write dt as (dt/dRI)dRl where (df/dRI) is approximated as the inverse of the mean thermal velocity, hg/BkT) 'I2. The probability as a function of RI isthen integrated over the distribution of approach distances, weighted by the appropriate differential area to give the quenching cross section. The upper limit on RI is -0; the lower limit is just the hard-sphere collision radius which will be taken as
The results of our theory for 13 are illustrated in fig. 1. The data for the two vibrational states that have been studied extensively are shown. This Log-log plot shows better agreement. with eq. (12) than previous attempts to correlate the cross section with cY$/~. Again. as in the previous theories, the cross section for NO is Iatger than predicted. The only plausible explanation Ear this particular divergence has been presented by Steinfeld {4]. Whereas totally symmetric collision partners interact with the B state to give A’ complexes which correlate with the 0; repulsive state of I2, NO which has a 2R ground state can correlate with other final states since it may aIso produce A" complexes. Thus, the enhanced quenching ability of NO has been attributed to the opening up of additional channels for quenching. Recent preliminary values of v’ = 43 quenching 219
‘Volume 4, number 4
CtiMlCAL
sections also indicate eq. (12) [?I.
crqss
4. APPLICATIONS
-pod agreement
PtiYsICS LETTERS
with
TO OTHER SYSTEMS
RecentIy! data has become available on the cross sections for the quenching of the A(IBI)--t X(lAl) fluorescence in SO2 [lo]. In this case, tine quenching has been ascribed to callisioninduced Internal conversion. Although the final state’is discrete in this instance, a “continuum” of vibrational states is available after “non resonant” emission. Thus, a treatment analogous to that for I2 would seem applicable. The results obtained from eq. (12) are scaled and plotted against the experimental cross sections in fig. 2. Although there is still some degree of scatter, the. results seem to lie clearly along the expected straight line of unit slope. The present considerations give a better fit to the measured results than do previous attempts to fit the data to u or ~gl/~. Again. it should be noted that the cross section for NO is somewhat larger than predicted. Finally, we shall consider instances of fluorescence quenching in three polar molecules, viz, NO, NO2 and CH30H 111-13). Eq. (12) can be shown to be equally applicable in the case of polar molecules; the basic Rl-dependence remains the same. The results of this theory for NO2 are presented in fig. 3. For NO2 1121 good agreement is again observed although the cross sections for NO and CO2 are still underpredicted by eq. (12).
1 November 1969
In the case of methanol [13], the experimental cross sections show far greater variability than do those predicted (see fig. 4). The collisioninduced predissociation in methanol not only shows a much more complicated chemistry than the previous examples, but also shows wide scatter on an Lyj.fij2 plot. Finally brief mention must be made regarding the.observed fluorescence in NO [ll]. The theory predicts the correct order for the relative size of the quenching cross sections of He, Ar, H2 and N2 but it fails once more in predictions for NO itself and CO2.
REFERENCES [I] R. L. Brown and W. Klemperer, J. Chem. Phys. 41 (1964) 3072. [Z] J. 1. Steinfeld and W.Klemperer. J. Chcm. Phys.42 (1965) 3475. [3] F.Rossler. Z.Physik 96 (1933) 251. [4] J. I. Steinfeld. J. Chem. Phys. 44 (1966) 2740. [S] L. A. Turner and E.W. Samson, Phys. Rev. 37 (1931) 1684. [S] L.A. Turner, Phys. Rev.41 (1932) 627. (71 E. Rabinowitch and W. C. Wood, J. Chem. Phys.4 (1936) 35.9. (81 H. Fdargenau, Rev. Mod. Phys. 11 (1939) 1. [9] J. I. Steinfeld and R. B. Kurzel, to be published. [I?] H. D. AIettee, to be published. [ll] H. P. Broida and T. Carrington, J. Chem. Phys. 35 (19ti3) 136. [IZ] G. f-I. Meyers, D. M. Silver and F. Kaufman, J. Chem. Phys.44 (1966) 718. 1131 J.Hagege, P.C. Roberge and C.\rermeil, Trans Faraday Sot. 64 (1968) 3289.