We have performed accurate quantum mechanical calculations

for the coplanar H + H_2 exchange reaction, using sufficient rotational and vibrational basis functions in the close-coupling expansion to ensure convergence. We repeated these calculations with a converged rotational basis set but with only one vibrational basis function, in analogy to what Saxon and Light and Wolken and Karplus, respectively, did for the similar coplanar and three dimensional reaction. The vibrationally converged and one-vibration results differ substantially for the coplanar as well as the collinear reaction, indicating the crucial role played by virtual vibrational channels

Baer, Michael

Kuppermann, Aron

Schatz, George C.

English

Caltech Authors - Main

Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Coplanar and collinear quantum mechanical reactive scattering: 
The importance of virtual vibrational channels in the H + H2 
exchange reaction* 
Aron Kuppermann, George C. Schatzt, and Michael Baer+ 
Arthur Amos Noyes Laboratory of Chemical Physics, § California Institute of Technology, Pasadena, California 91109 
(Received 22 August 1974) 
We have performed accurate quantum mechanical cal-
culations for the coplanar H + H2 exchange reaction, 
using sufficient rotational and vibrational basis func-
tions in the close-coupling expansion to ensure con-
vergence. We repeated these calculations with a con-
verged rotational basis set but with only one vibra-
tional basis function, in analogy to what Saxon and 
Light1 and Wolken and Karplus, 2 respectively, did for 
the similar coplanar and three dimensional reaction. 
The vibrationally converged and one-vibration results 
differ substantially for the coplanar as well as the col-
linear reaction, indicating the crucial role played by 
virtual vibrational channels. 
To solve the Schrodinger equation for the coplanar 
reaction, we first integrated the appropriate coupled 
equations into the interaction region from each of the 
three arrangement channel regions, using an extension 
of the method developed by Kuppermann. 3 The result-
ing solutions were then smoothly matched on three con-
veniently chosen surfaces in configuration S{>ace. The 
R matrix and other asymptotic quantities were then ob-
tained. 
Calculations for the Porter-Karplus surface4 using 4 
or 5 vibrations and 10 or 12 rotations per vibration for 
a total of 40 to 60 channels yielded reaction probabili-
ties that change by less than 2%-5% as additional vibra-
tional or rotational basis functions are added, over the 
total energy range 0. 30-0.60 eV. Without forcing 
orthogonalization at any time, the results satisfy conser-
vation of flux to 0. 5% or better and time reversal invari-
ance to 6% or better. The calculations were repeated 
using the same number of rotations but only one vibra-
tion, and introducing an appropriate vibrational orthog-
onalization. 
The resulting total reaction cross sections a~ are 
plotted in Figs. 1(a) and 1(b) and show differences be-
tween the vibrationally converged and one-vibration 
results greater than 3 orders of magnitude at low ener-
gies. The ratio of the one-vibration to vibrationally 
converged ortho- para rate constants is 3. 15 at 300 oK 
and 532 at 100°K. 
Using the method developed previously, 3 we calculated 
the collinear converged5 and one-vibration reaction prob-
abilities for the same potential energy surface. The 
ratios of the coplanar to collinear cross sections are 
plotted in Fig. l(c). Although these cross sections vary 
individually by about 12 orders of magnitude over the 
energy range considered, their ratios vary by less than 
2 orders of magnitude, indicating a remarkably similar 
energy dependence. Virtual vibrational channels are 
furthermore about equally important in the collinear6 
and coplanar H + H2 reaction. This will probably still be 
the case for this system in three dimensions as well as 
for other reactions. 
We have also calculated the reactive, inelastic, and 
antisymmetrized differential cross sections for coplanar 
0.1 
0.5 CONVERGED 
--.:: 0.4 
.c 
0 
~0.3 
crt? 
0.2 
0.1 
1 VIBRATION 
Eo (eV) 
0.2 
.P 
/ 
'/ 
0.3 
/ 
/ 
I' 
/ 
I 
(a) 
(b) 
FIG. 1. Total reactive cross section a"f versus the total energy 
E and translational energy E 0 for the coplanar exchange reac-
tion H + H2 (v= O,j= 0) -H2 (v= O,j') + H (summed over all j') 
where the atoms are considered distinguishable. Arrows in 
abscissa indicate energies at which the vibration-rotation states 
(v = O,j') of H2 become accessible. (a) Linear plot; (b) semi-
logarithmic plot. In (c) we plot the ratio rrr /P~ versus energy, 
where Pf is the collinear total reaction probability (the collin-
ear total reaction cross section) for reagents H2 in"= 0 initially. 
In all cases, a solid line indicates vibrationally converged re-
sults, while a dashed line indicates one-vibration results as de-
fined in text. 
4362 The Journal of Chemical Physics, Vol. 61, No. 10, 15 November 1974 Copyright© 1974 American Institute of Physics 
Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Letters to the Editor 4363 
H + H2• Contrary to the results of Saxon and Light
1 our 
reactive differential cross sections show only backward 
peaking in both the one-vibration and vibrationally con-
verged results. In addition, we find no evidence for a 
quasiselection rule governing the reaction probabilities 
similar to the ortho- ortho and para- para nonreactive 
selection rule. We were able to artificially produce 
such a reactive quasiselection rule using matching pro-
cedures in which incorrect symmetry restrictions were 
placed on the matched wavefunctions. 7 
The inelastic cross sections are generally backward 
or sideward peaked and exhibit no fine oscillatory 
structure. Quantum symmetry oscillations resulting 
from interference between the direct and exchange 
amplitudes characterize some ortho- ortho and 
para- para antisymmetrized differential cross sections 
at total energies greater than 0. 5e V. 
In conclusion, virtual vibrational channels play a 
central role in the collinear and coplanar H + H2 reac-
tion, and probably in the three dimensional one and in 
many other reactions. Therefore, the results of cal-
culations using only open vibrational channels must be 
used with great caution. Once those virtual channels 
are included in a close-coupling reactive scattering 
calculation and convergence ascertained, a wealth of 
detailed physically meaningful information is obtained 
against which a variety of dynamical models can be 
tested. Work along these lines is proceeding in this 
laboratory. 
We thank Ambassador College for the use of their 
computational facilities in most of the calculations 
reported here. 
*Research supported in part by the U. S. Air Force Office of 
Scientific Research. 
tNational Science Foundation predoctoral fellow. Work per-
formed in partial fulfillment of the requirements for the Ph. D. 
degree in Chemistry at the California Institute of Technology. 
tPresent address: Soreq Nuclear Research Center, Yavne, 
Israel; and Department of Chemical Physics, The Weizmann 
.Institute of Science, P. 0. Box 26, Rehovot, Israel. 
§Contribution No. 4945. 
1R. P. Saxon and J. C. Light, J. Chern. Phys. 56, 3874 (1972); 
56, 3885 (1972). 
2G. Wolken and M. Karplus, J. Chern. Phys. 60, 351 (1974). 
3A. Kuppermann, Potential Energy Surfaces in Chemistry, 
edited by W. Lester (University of California at Santa Cruz, 
Santa Cruz, 1970), pp. 121-129; VII International Conference 
on the Physics of Electronic and Atomic Collisions, Abstracts 
of Papers (North-Holland, Amsterdam, 1971), p. 3; G. C. 
Schatz, J. M. Bowman, and A. Kuppermann, J. Chern. Phys. 
58, 4023 (1973); G. C. Schatz and A. Kuppermann, ibid. 59, 
964 (1973); J. M. Bowman, G. C. Schatz, andA. Kuppermann, 
Chern. Phys. Lett. 24, 378 (1974). 
4R. N. Porter and M. Karplus, J. Chern. Phys. 40, 1105 (1964). 
5These results are identical to those of D. J. Diestler, J. Chern. 
Phys. 54, 4547 (1971). 
6Virtual channels have been found to be important in many col-
linear collisions, such as in M. E. Riley and A. Kuppermann, 
Chern. Phys. Lett. 1, 537 (1968); D. G .. Truhlar and A. Kup-
permann, J. Chern. Phys. 52, 3841 (1970); 56, 2232 (1972); 
s. F. Wu and R. D. Levine, Mol. Phys. 22, 881 (1971); and 
Ref. 3 above. 
7Recently J. C. Light has obtained similar results in a modified 
version of his one-vibration method (private communication). 
Laser spectroscopy of supersonic molecular beams: Application 
to the N02 spectrum 
R. E. Smalley, B. L. Ramakrishna, D. H. Levy*, and L. Wharton 
The James Franck Institute and The Department of Chemistry. The University of Chicago, Chicago, Illinois 60637 
(Received 6 September 1974) 
Traditional high resolution optical spectroscopy of the 
type discussed in Herzberg's cla.ssic works1 has been a 
definitive source of information about the excited elec-
tronic states of small molecules. Because the compli-
cation caused by rotational structure increases rapidly 
with increasing molecular weight, highresolution opti-
cal studies have been limited to diatomic and reasonably 
small polyatomic molecules as may be seen in Tables 
61-82 of Volume III, R.ef. 1. The technique of matrix 
isolation2_. eliminates rotational structure and thus al-
lows work on larger molecules, and on the whole this 
technique has been highly successful. Unfortunately, 
matrix isolation spectroscopy suffers from two funda-
mental problems. First, the matrix completely re-
moves the rotational structure and therefore all infor-
mation carried by it is lost. Second, the spectrum is 
perturbed by the matrix, and it is sometimes not clear 
whether a structural feature deduced from the spectrum 
is characteristic of the free molecule or is induced by 
the matrix. 
In this paper we describe an experiment using a super-
sonic molecular beam5- 7 which provides a great reduc-
tion in rotational structure without completely removing 
it, and which allows high resolution study of free mole-
cules unperturbed even by gas phase collisions. In 
this experiment, a supersonic beam of 5% N02 in Ar 
was crossed with a tunable dye laser beam and the fluo-
rescence excitation spectrum was observed. Because of 
rotational cooling during the supersonic expansion, the 
highly complicated and largely unassigned absorption 
spectrum of N02 has been simplified to the point where 
individual vibronic bands are well separated and a com-
plete rovibronic assignment now appears to be within 
reach. 
The molecular beam source8- 10 had a 0. 05 mm nozzle 
The Journal of Chemical Physics, Vol. 61, No. 10, 15 November 1974 Copyright© 1974 American Institute of Physics 


Coplanar and collinear quantum mechanical reactive scattering: The importance of virtual vibrational channels in the H + H_2 exchange reaction

Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Coplanar and collinear quantum mechanical reactive scattering: 
The importance of virtual vibrational channels in the H + H2 
exchange reaction* 
Aron Kuppermann, George C. Schatzt, and Michael Baer+ 
Arthur Amos Noyes Laboratory of Chemical Physics, § California Institute of Technology, Pasadena, California 91109 
(Received 22 August 1974) 
We have performed accurate quantum mechanical cal-
culations for the coplanar H + H2 exchange reaction, 
using sufficient rotational and vibrational basis func-
tions in the close-coupling expansion to ensure con-
vergence. We repeated these calculations with a con-
verged rotational basis set but with only one vibra-
tional basis function, in analogy to what Saxon and 
Light1 and Wolken and Karplus, 2 respectively, did for 
the similar coplanar and three dimensional reaction. 
The vibrationally converged and one-vibration results 
differ substantially for the coplanar as well as the col-
linear reaction, indicating the crucial role played by 
virtual vibrational channels. 
To solve the Schrodinger equation for the coplanar 
reaction, we first integrated the appropriate coupled 
equations into the interaction region from each of the 
three arrangement channel regions, using an extension 
of the method developed by Kuppermann. 3 The result-
ing solutions were then smoothly matched on three con-
veniently chosen surfaces in configuration S{>ace. The 
R matrix and other asymptotic quantities were then ob-
tained. 
Calculations for the Porter-Karplus surface4 using 4 
or 5 vibrations and 10 or 12 rotations per vibration for 
a total of 40 to 60 channels yielded reaction probabili-
ties that change by less than 2%-5% as additional vibra-
tional or rotational basis functions are added, over the 
total energy range 0. 30-0.60 eV. Without forcing 
orthogonalization at any time, the results satisfy conser-
vation of flux to 0. 5% or better and time reversal invari-
ance to 6% or better. The calculations were repeated 
using the same number of rotations but only one vibra-
tion, and introducing an appropriate vibrational orthog-
onalization. 
The resulting total reaction cross sections a~ are 
plotted in Figs. 1(a) and 1(b) and show differences be-
tween the vibrationally converged and one-vibration 
results greater than 3 orders of magnitude at low ener-
gies. The ratio of the one-vibration to vibrationally 
converged ortho- para rate constants is 3. 15 at 300 oK 
and 532 at 100°K. 
Using the method developed previously, 3 we calculated 
the collinear converged5 and one-vibration reaction prob-
abilities for the same potential energy surface. The 
ratios of the coplanar to collinear cross sections are 
plotted in Fig. l(c). Although these cross sections vary 
individually by about 12 orders of magnitude over the 
energy range considered, their ratios vary by less than 
2 orders of magnitude, indicating a remarkably similar 
energy dependence. Virtual vibrational channels are 
furthermore about equally important in the collinear6 
and coplanar H + H2 reaction. This will probably still be 
the case for this system in three dimensions as well as 
for other reactions. 
We have also calculated the reactive, inelastic, and 
antisymmetrized differential cross sections for coplanar 
0.1 
0.5 CONVERGED 
--.:: 0.4 
.c 
0 
~0.3 
crt? 
0.2 
0.1 
1 VIBRATION 
Eo (eV) 
0.2 
.P 
/ 
'/ 
0.3 
/ 
/ 
I' 
/ 
I 
(a) 
(b) 
FIG. 1. Total reactive cross section a"f versus the total energy 
E and translational energy E 0 for the coplanar exchange reac-
tion H + H2 (v= O,j= 0) -H2 (v= O,j') + H (summed over all j') 
where the atoms are considered distinguishable. Arrows in 
abscissa indicate energies at which the vibration-rotation states 
(v = O,j') of H2 become accessible. (a) Linear plot; (b) semi-
logarithmic plot. In (c) we plot the ratio rrr /P~ versus energy, 
where Pf is the collinear total reaction probability (the collin-
ear total reaction cross section) for reagents H2 in"= 0 initially. 
In all cases, a solid line indicates vibrationally converged re-
sults, while a dashed line indicates one-vibration results as de-
fined in text. 
4362 The Journal of Chemical Physics, Vol. 61, No. 10, 15 November 1974 Copyright© 1974 American Institute of Physics 
Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Letters to the Editor 4363 
H + H2• Contrary to the results of Saxon and Light
1 
our 
reactive differential cross sections show only backward 
peaking in both the one-vibration and vibrationally con-
verged results. In addition, we find no evidence for a 
quasiselection rule governing the reaction probabilities 
similar to the ortho- ortho and para- para nonreactive 
selection rule. We were able to artificially produce 
such a reactive quasiselection rule using matching pro-
cedures in which incorrect symmetry restrictions were 
placed on the matched wavefunctions. 7 
The inelastic cross sections are generally backward 
or sideward peaked and exhibit no fine oscillatory 
structure. Quantum symmetry oscillations resulting 
from interference between the direct and exchange 
amplitudes characterize some ortho- ortho and 
para- para antisymmetrized differential cross sections 
at total energies greater than 0. 5e V. 
In conclusion, virtual vibrational channels play a 
central role in the collinear and coplanar H + H2 reac-
tion, and probably in the three dimensional one and in 
many other reactions. Therefore, the results of cal-
culations using only open vibrational channels must be 
used with great caution. Once those virtual channels 
are included in a close-coupling reactive scattering 
calculation and convergence ascertained, a wealth of 
detailed physically meaningful information is obtained 
against which a variety of dynamical models can be 
tested. Work along these lines is proceeding in this 
laboratory. 
We thank Ambassador College for the use of their 
computational facilities in most of the calculations 
reported here. 
*Research supported in part by the U. S. Air Force Office of 
Scientific Research. 
tNational Science Foundation predoctoral fellow. Work per-
formed in partial fulfillment of the requirements for the Ph. D. 
degree in Chemistry at the California Institute of Technology. 
tPresent address: Soreq Nuclear Research Center, Yavne, 
Israel; and Department of Chemical Physics, The Weizmann 
.Institute of Science, P. 0. Box 26, Rehovot, Israel. 
§Contribution No. 4945. 
1R. P. Saxon and J. C. Light, J. Chern. Phys. 56, 3874 (1972); 
56, 3885 (1972). 
2G. Wolken and M. Karplus, J. Chern. Phys. 60, 351 (1974). 
3A. Kuppermann, Potential Energy Surfaces in Chemistry, 
edited by W. Lester (University of California at Santa Cruz, 
Santa Cruz, 1970), pp. 121-129; VII International Conference 
on the Physics of Electronic and Atomic Collisions, Abstracts 
of Papers (North-Holland, Amsterdam, 1971), p. 3; G. C. 
Schatz, J. M. Bowman, and A. Kuppermann, J. Chern. Phys. 
58, 4023 (1973); G. C. Schatz and A. Kuppermann, ibid. 59, 
964 (1973); J. M. Bowman, G. C. Schatz, andA. Kuppermann, 
Chern. Phys. Lett. 24, 378 (1974). 
4R. N. Porter and M. Karplus, J. Chern. Phys. 40, 1105 (1964). 
5These results are identical to those of D. J. Diestler, J. Chern. 
Phys. 54, 4547 (1971). 
6Virtual channels have been found to be important in many col-
linear collisions, such as in M. E. Riley and A. Kuppermann, 
Chern. Phys. Lett. 1, 537 (1968); D. G .. Truhlar and A. Kup-
permann, J. Chern. Phys. 52, 3841 (1970); 56, 2232 (1972); 
s. F. Wu and R. D. Levine, Mol. Phys. 22, 881 (1971); and 
Ref. 3 above. 
7Recently J. C. Light has obtained similar results in a modified 
version of his one-vibration method (private communication). 
Laser spectroscopy of supersonic molecular beams: Application 
to the N02 spectrum 
R. E. Smalley, B. L. Ramakrishna, D. H. Levy*, and L. Wharton 
The James Franck Institute and The Department of Chemistry. The University of Chicago, Chicago, Illinois 60637 
(Received 6 September 1974) 
Traditional high resolution optical spectroscopy of the 
type discussed in Herzberg's cla.ssic works1 has been a 
definitive source of information about the excited elec-
tronic states of small molecules. Because the compli-
cation caused by rotational structure increases rapidly 
with increasing molecular weight, highresolution opti-
cal studies have been limited to diatomic and reasonably 
small polyatomic molecules as may be seen in Tables 
61-82 of Volume III, R.ef. 1. The technique of matrix 
isolation2_. eliminates rotational structure and thus al-
lows work on larger molecules, and on the whole this 
technique has been highly successful. Unfortunately, 
matrix isolation spectroscopy suffers from two funda-
mental problems. First, the matrix completely re-
moves the rotational structure and therefore all infor-
mation carried by it is lost. Second, the spectrum is 
perturbed by the matrix, and it is sometimes not clear 
whether a structural feature deduced from the spectrum 
is characteristic of the free molecule or is induced by 
the matrix. 
In this paper we describe an experiment using a super-
sonic molecular beam5- 7 which provides a great reduc-
tion in rotational structure without completely removing 
it, and which allows high resolution study of free mole-
cules unperturbed even by gas phase collisions. In 
this experiment, a supersonic beam of 5% N02 in Ar 
was crossed with a tunable dye laser beam and the fluo-
rescence excitation spectrum was observed. Because of 
rotational cooling during the supersonic expansion, the 
highly complicated and largely unassigned absorption 
spectrum of N02 has been simplified to the point where 
individual vibronic bands are well separated and a com-
plete rovibronic assignment now appears to be within 
reach. 
The molecular beam source8- 10 had a 0. 05 mm nozzle 
The Journal of Chemical Physics, Vol. 61, No. 10, 15 November 1974 Copyright© 1974 American Institute of Physics 


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Coplanar and collinear quantum mechanical reactive scattering: The importance of virtual vibrational channels in the H + H_2 exchange reaction

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