It is shown that the phase factor i^(J+j+I) omitted in a theory of atom‐diatom monreactive scattering formulated by schatz and kupperman but included in the Choi, Poe, and Tang theory does not lead to errors in the analysis of te (H,H_2) system

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
Letters to the Editor 
This was taken as evidence that the observation tube in 
Ref. 2 might "have harbored incipient or diffuse pink 
afterglow. " It is not surprising to find these emissions 
in the discharge as these emission bands are usually 
present in auroral afterglows. In order to obtain some 
information concerning the origin of these bands, it is 
important to determine their spectral intensity distri-
bution. Unfortunately, this was not done in their work. 
Furthermore, there is no simple logical reason why 
the presence of these bands in the discharge may ex-
clude their presence in the afterglow, especially in ex-
periments under conditions of high N-atom concentra-
tions where the contribution due to excited species is 
important. 
Brennen and Shuman also suggested that several fea-
tures present in Fig. 1b of Ref. 1 and Fig. 1 of Ref. 2 
might be identified as bands in the N2 second positive 
(2P) and N; first negative (1N) systems, both of which 
were considered "telltale diagnostics" of the pink after-
glow. They have apparently not considered studies by 
various workers who have detected these emissions in 
a variety of N2 afterglows. 
3
-
5 In fact, we have recently 
observed emissions due to N2(2P), N;(1N) and Gold-
stein-Kaplan bands as well as the 5577 A and 3466 A 
lines in the yellow N2 afterglow. The intensity distribu-
tions of these band systems were found to be quite dif-
ferent from those reported in other afterglows. Kinetic 
studies on the emission properties indicate that these 
emission bands, somewhat resembled the highly vibra-
tional bands of N2(1P) system, develop principally in 
the observation vessel as a consequence of complex in-
teractions between energetic species. 
The pink afterglow was normally observed in our sys-
tem at pressures between 5 and 15 Torr and at a flow 
time of a few msec after the discharge. The glow was 
well isolated from the discharge and the subsequent 
yellow afterglow, and had a lifetime of a few msec, all 
of which are very similar to the pink afterglow original-
ly described by Beale and Broida. 6 We are quite con-
fused by Brennen and Shuman's statement that "We have 
sometimes seen the pink afterglow diffusely distributed 
throughout the entire volume of our observation tube. " 
If it is implied that the pink glow they observe simply 
survived from the discharge, how do they explain the 
mean halflife of the order of 10-3 sec estimated by Beale 
and Broida, and others? On the other hand, if the pink 
afterglow is created by active species in the observa-
tion vessel, how shall we define the pink afterglow? 
Interactions involving energetic species under condi-
tions of high atom concentrations 'are· undoubtedly com-
plicated. Argument over classical characterization of 
afterglows of this nature may serve little purpose. Our 
present interest is simply to try to understand the gen-
eral nature of energy transfer processes involving en-
ergetic species in the long-lived yellow afterglow of 
nitrogen. 
1W. Brennen and M. E. Shuman, J. Chern. Phys. 66, 4248 
(1977). 
2A. Y. -M. Ung, J. Chern. Phys. 65, 2987 (1976). 
3Y. Tanaka and A. S. Jursa, J. Opt. Soc. Am. 51, 1239 
(1961). 
4G. N. Hays, C. J. Tracy, A. R. de Monchy, and H. J. 
Oskam, Chern. Phys. Lett. 14, 352 (1972). 
5A. Y. -M. Ung, Chern. Phys. Lett. 32, 193 (1974). 
6G. E. Beale, Jr. and H. P. Broida, J. Chern. Phys. 31, 
1030 (1959). 
Comment on "Theory of collisions between an atom and a 
diatomic molecule in the body-fixed coordinate system''a) 
George C. Schatz 
Department of Chemistry, Northwestern University, Evanston, Illinois 60201 
Aron Kuppermann 
Arthur Amos Noyes Laboratory of Chemical Physics, b) California Institute of Technology, Pasadena, 
California 91125 
(Received II October 1978) 
In two recent papers, 1• 2 Choi, Poe, and Tang (CPT) 
have presented a theory of atom-diatom nonr.eactive 
scattering using body fixed (BF) coordinates which is 
quite similar to a theory previously developed3 and 
used' by us in applications to the H + H2 reaction. In 
comparing their and our asymptotic scattering matrix 
solutions, CPT noted that our lacked a phase factor 
iJ•J•l
5 and further stated1 that "erroneous numerical 
results could be introduced by omitting this phase fac-
tor." We show here that, contrary to this statement, 
this phase factor difference does not cause our results 
to be in error. In addition, we point out that CPT mis-
quoted our conclusions concerning the 0 dependence of 
the H + H2 rotationally inelastic nonreactive transition 
probabilities. 
CPT's argument concerning phase factors is based on 
the following asymptotic form for the BF S-matrix so-
J. Chem. Phys. 70(06), 15 Mar. 1979 0021-9606/79/063151-03$01.00 © 1979 American Institute of Physics 3151 
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3152 Letters to the Editor 
lution, which they derived by a unitary transformation 
on the corresponding space fixed (SF) solution1 (ex-
pressed in the quantum number notation of Eq. (5. 3) of 
Ref. 3, 5 and ignoring closed channels which are not 
essential here): 
and s"::;~· is CPT' s scattering matrix. 6 Except for an 
unimportant 27T(V, J' ) 112 factor, Eq. (2) differs only by 
the phase factor i"•J•l from an analogous expression used 
by us3 [Eq. (5. 5) of Ref. 3), by Pack, 7 and by Walker 
and Light8 to define a scattering matrix (which we de-
note here as s'J.f;rf). CPT argued1 that because the 
quantum number j is summed over in Eq. (1), a wave 
function >It'{J~0 '[s] obtained from an expression without 
the i 1 •J•l factor will be different than one with it, pre-
suming that the same S matrix is used in both solutions. 
They further argued that this "phase factor has nothing 
to do with the definition of the S matrix, " implying that 
the same S matrix would be obtained with either solu-
tion. 
We will now show that this last statement is, in fact, 
incorrect, and that the S matrix obtained using our 
asymptotic solution differs from theirs by a phase fac-
tor iJ-J'. To prove this, we examine the incoming 
spherical wave parts of both solutions. In Eq. (2) that 
is 
X27Te(~' Y'\.t•J'+l exp( -i(k,J'R- (J+ j)7T/2)], (3) 
while ours is identical to Eq. (3) except for the i1 •J'•l 
factor (and the unimportant 27T(V.,. 1.)
112). This means 
that the CPT incident solution equals i.T+J'•l times our 
incident solution. Since both solutions solve the same 
Schrodinger equation, 9 this same factor must likewise 
where 
relate the outgoing wave solutions, and this implies 
(equating coefficients of corresponding terms) that, 
·l+J'•lsv'J'O' .J+J+ISv'J'O' 
t JvJO =t JvJO • 
or 
Sv• J' n• ·J-J'sv' J' n• JvJO = t JvJO • 
(1) 
(2) 
(4} 
(5} 
It is now clear that CPT's S-matrix solution [Eq. (1}] 
and ours differ only by the multiplicative phase factor 
i"•J'•l (note the presence of j' rather than j in this fac-
tor), and this cannot have any measurable consequence. 
We should note that the factor iJ-J' in Eq. (5) can also 
be derived by comparing scattering amplitudes [using 
CPT's Eq. (28) and our Eq. (5. 31)101, or by comparing 
the relationship between BF and space fixed (SF) scat-
tering matrices [CPT's Eq. (21) and our Eq. (5. 14)11 ]. 
CPT do correctly point out that their BF R matrix 
boundary conditions {and wave functions) are, «com-
pletely different, " from ours. This is, of course, not 
surprising since their BF R and S matrices are not re-
lated by the same equation (Eq. (30) of Ref. 1) as ours 
[Eq. (5. 7) of Ref. 3). This difference does not affect 
any observable properties such as transition probabil-
ities or cross sections, since these are defined in terms 
of the S matrix, (i.e., plane wave) solution. 
Finally, we note that in examining transition probabil-
ities for ro1ationally inelastic nonreactive H + H2 colli-
sions, CPT. fmproperly attributed to us12 the conclu-
sion that (quoting from Ref. 2), "for low J, the K val-
ues are conserved approximately in closed-coupling 
TABLE I. Comparison of rotationally inelastic transition probabilities P1~~~· for nonreactive 
H+H2, aJ=15. 
CPTb atE= O. 673 eV 
- SJ' 
0 0 1 2 
-4 2.397(-3) 6.423(-4) 5. 551(- 5) 
-3 2.434(-3) B. 660(-4) 1. 782(-4} 
-2 5. 343(-3) 9. 676(-4) 2.115(-4) 
-1 7.949(-3) 2.876(-3) 3. 834(-4) 
0 1.060(-2) 4. 992(-3) 1. 019(- 3) 
1 7. 949(-3) B. 916(-3) 2.255(-3) 
2 5.343(-3) 8. 838(-3) 5. 851(-3) 
3 2.434(-3) 4. 950(-3) 9. 891(-3) 
4 2. 397(-3) 6.128(-3) 6.046(-3) 
a2.397(-3) should be read as 2.397X10-3, etc. 
l>aeference 2. 
SK" at E=O. 70 eV 
0 1 2 
2. 900(-3) 6. 614(-4) 1.326(-5) 
2. 805(-3) 7.493(-4) 3. 039(-4) 
7. 225(-3) 9.577(-4) 1. 595(-4) 
9. 813(-3) 3. 671(-3) 4. 236(-4) 
1. 399(- 2) 6. 562(-3) 1. 203(-3) 
9. 813(-3) 1.167(-2) 2.925(-3) 
7. 225(-3) 1. 093(-2) 7. 992(-3) 
2. 805(-3) 6.255(-3) 1. 296(-2) 
2. 900(-3) 7.798(-3) 7. 333(-3) 
"These results were generated as part of the calculations described in Ref. 4. although they have 
not previously been published. 
J. Chern. Phys., Vol. 70, No.6, 15 March 1979 
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 
calculations and dominant contributions of transition 
probabilities are from K6 =K,.=0," (i.e., il=O' =0). 
They proceded to challenge this staiement as if it had 
been made for nonreactive scattering. In Ref. 12 we 
did make a statement similar to this, but it refers 
specifically to reactive scattering (in H + H2). Our con-
clusion for nonreactive scattering was discussed 
thoroughly in Ref. 4, where we stated, "An examina-
tion of other nonreactive transition probabilities indi-
cates no strong tendency for a m1 = -m1.(0=0') "i.-
conserving" selection rule as has been assumed in ro-
tationally inelastic scattering." In fact, there is no 
essential difference between the 0, O' dependence of 
CPT's transition probabilities and ours. We demon-
strate this in Table 113 for the j = 2 to j' = 4 transition. 
a>Research supported in part by the United States Air Force 
Office of Scientific Research (Grant No. AFOSR-77-3394). 
b>contribution No. 5890. 
1B. H. Choi, R. T. Poe, and K. T. Tang, J. Chern. Phys. 69, 
411 (1978). 
2B. H. Choi, R. T. Poe, and K. T. Tang, J. Chern. Phys. 69, 
422 (1978). 
3G. C. Schatz and A. Kuppermann, J. Chern. Phys. 65, 4642 
(1976). 
4G. C. Schatz and A. Kuppermann, J. Chern. Phys. 65, 4668 
(1976). 
5We use the same notation as in Ref. 3 with regard to definition 
of the vibrational, orbital, rotational, total angular momen-
tum and body fixed projection quantum numbers (v,l,j,J, and 
Q, respectively). The arrangement channel index A will, 
however, be suppressed, and we let primed quantum numbers 
denote initial states and unprimed denote final states. This 
notation is similar to CPT, 1 but the quantum number 0' is 
denoted by them as -K,. and 0 as K8• 
6Note that ~~/ff here would have been labelled as S"j(j1,_ 0 , in 
Eq. (5. 5) of Ref. 3. 
7R. T Pack, J. Chern. Phys. 60, 633 (1974). 
8R. B. Walker and J. C. Light, Chern. Phys. 7, 84 (1975). 
9CPT argued1 that iirJ;}'0 ' [S) in Eqs. (1) and (2) does not solve 
the BF Schrodinger equation if the i 1 • J • 1 is omitted because 
the BF Hamiltonian does not commute with a diagonal matrix 
having diagonal elements i 1 • J • 1• Asymptotically, however, 
the Hamiltonian is diagonal in j so their argument does not 
apply, and both wave functions solve the. BF Schrodinger equa-
tion in that region. Either solution may furthermore be 
matched to solutions which solve the Schrodinger equation at 
small R as shown in detail in Ref. 3 for our solution. 
1'C PT used the same plane wave solution as we, and it is easy 
to see that the scattering amplitudes are simply proportional. 
By relating these quantities. and using Eq. (28) of Ref. 1 and 
Eq. (5. 31) of Ref. 3, we again derive Eq. (5). Note that the 
meaning of primed and unprimed quantum numbers in Eq. 
(5. 31) is reversed from that used here. 
11m this comparison, a phase factor difference i 1-" arises in 
the SF scattering matrices because of differences in the 
definition ofthe SF scattering solutions. 
12A. Kuppermann and G. C. Schatz, J. Chern. Phys. 62, 2502 
(1975). 
13The actual differences between their and our transition prob-
abilities in Table I are probably due to a combination of 
effects, including the slightly different energies at which the 
calculations were done, the different vibration-rotation basis 
sets used, and the different representations of the potential 
used (CPT truncated the expansion of the potential in Legendre 
polynomials at the P 2 term while we included the P4 term.) 
On "Theory of collisions between an atom and a diatomic 
molecule in the body-fixed coordinate system"a> 
B. H. Choi and A. T. Poe 
Department of Physics, University of California, Riverside, California 92521 
K. T. Tang 
Department of Physics, Pacific Lutheran University Tacoma, Washington 98447 
(Received 9 November 1978) 
In recent papers with the above titles, 1• 2 we presented 
a detailed formulation for the treatment of parities in a 
body-fixed (BF) close-coupling approximation. This is 
directly amenable to numerical computation of elastic 
and inelastic atom-diatom scattering cross sections. 
The main thrust of our paper is the parity boundary 
conditions. However, in the preceding comment by 
Schatz and Kuppermann, they raised some questions re-
garding the relationship between the nonparity boundary 
conditions for S matrix developed by us and by them. It 
seems necessary for us to clarify the implications of 
some statements which we made about nonparity asymp-
totic boundary conditions for S matrix, not to be misun-
derstood by other colleagues in this field who have em-
ployed different forms of the boundary condition from 
ours. (Hereafter, the "asymptotic" will be deleted.) 
We stated that if AP(R) is another boundary condition 
different from our A(R) defined in Ref. 1 as 
A1.,K.,,J,.x ,.(R) == 27Ti
1•J.,•t 
X (o1.,1.,oK.,-K"' exp{- i(k.,R- (J + j 7 )7T/2]} 
- ~,.K.,,JcxKa exp{i[k.,R- (J + j 7 )7T/2]}), (1) 
then 
(2) 
for some nonsingular matrices P and Q.3 Here, AP(R) 
is an expression in terms of corresponding S matrix 
J. Chem. Phys. 70(06), 15 Mar. 1979 0021·9606/79/063153-02$0 1. 00 © 1979 American Institute of Physics 3153 


Comment on "Theory of collisions between an atom and a

diatomic molecule in the body-fixed coordinate system"

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 
This was taken as evidence that the observation tube in 
Ref. 2 might "have harbored incipient or diffuse pink 
afterglow. " It is not surprising to find these emissions 
in the discharge as these emission bands are usually 
present in auroral afterglows. In order to obtain some 
information concerning the origin of these bands, it is 
important to determine their spectral intensity distri-
bution. Unfortunately, this was not done in their work. 
Furthermore, there is no simple logical reason why 
the presence of these bands in the discharge may ex-
clude their presence in the afterglow, especially in ex-
periments under conditions of high N-atom concentra-
tions where the contribution due to excited species is 
important. 
Brennen and Shuman also suggested that several fea-
tures present in Fig. 1b of Ref. 1 and Fig. 1 of Ref. 2 
might be identified as bands in the N2 second positive 
(2P) and N; first negative (1N) systems, both of which 
were considered "telltale diagnostics" of the pink after-
glow. They have apparently not considered studies by 
various workers who have detected these emissions in 
a variety of N2 afterglows. 3- 5 In fact, we have recently 
observed emissions due to N2(2P), N;(1N) and Gold-
stein-Kaplan bands as well as the 5577 A and 3466 A 
lines in the yellow N2 afterglow. The intensity distribu-
tions of these band systems were found to be quite dif-
ferent from those reported in other afterglows. Kinetic 
studies on the emission properties indicate that these 
emission bands, somewhat resembled the highly vibra-
tional bands of N2(1P) system, develop principally in 
the observation vessel as a consequence of complex in-
teractions between energetic species. 
The pink afterglow was normally observed in our sys-
tem at pressures between 5 and 15 Torr and at a flow 
time of a few msec after the discharge. The glow was 
well isolated from the discharge and the subsequent 
yellow afterglow, and had a lifetime of a few msec, all 
of which are very similar to the pink afterglow original-
ly described by Beale and Broida. 6 We are quite con-
fused by Brennen and Shuman's statement that "We have 
sometimes seen the pink afterglow diffusely distributed 
throughout the entire volume of our observation tube. " 
If it is implied that the pink glow they observe simply 
survived from the discharge, how do they explain the 
mean halflife of the order of 10-3 sec estimated by Beale 
and Broida, and others? On the other hand, if the pink 
afterglow is created by active species in the observa-
tion vessel, how shall we define the pink afterglow? 
Interactions involving energetic species under condi-
tions of high atom concentrations 'are· undoubtedly com-
plicated. Argument over classical characterization of 
afterglows of this nature may serve little purpose. Our 
present interest is simply to try to understand the gen-
eral nature of energy transfer processes involving en-
ergetic species in the long-lived yellow afterglow of 
nitrogen. 
1W. Brennen and M. E. Shuman, J. Chern. Phys. 66, 4248 
(1977). 
2A. Y. -M. Ung, J. Chern. Phys. 65, 2987 (1976). 
3Y. Tanaka and A. S. Jursa, J. Opt. Soc. Am. 51, 1239 
(1961). 
4G. N. Hays, C. J. Tracy, A. R. de Monchy, and H. J. 
Oskam, Chern. Phys. Lett. 14, 352 (1972). 
5A. Y. -M. Ung, Chern. Phys. Lett. 32, 193 (1974). 
6G. E. Beale, Jr. and H. P. Broida, J. Chern. Phys. 31, 
1030 (1959). 
Comment on "Theory of collisions between an atom and a 
diatomic molecule in the body-fixed coordinate system''a) 
George C. Schatz 
Department of Chemistry, Northwestern University, Evanston, Illinois 60201 
Aron Kuppermann 
Arthur Amos Noyes Laboratory of Chemical Physics, b) California Institute of Technology, Pasadena, 
California 91125 
(Received II October 1978) 
In two recent papers, 1• 2 Choi, Poe, and Tang (CPT) 
have presented a theory of atom-diatom nonr.eactive 
scattering using body fixed (BF) coordinates which is 
quite similar to a theory previously developed3 and 
used' by us in applications to the H + H2 reaction. In 
comparing their and our asymptotic scattering matrix 
solutions, CPT noted that our lacked a phase factor 
iJ•J•l5 and further stated1 that "erroneous numerical 
results could be introduced by omitting this phase fac-
tor." We show here that, contrary to this statement, 
this phase factor difference does not cause our results 
to be in error. In addition, we point out that CPT mis-
quoted our conclusions concerning the 0 dependence of 
the H + H2 rotationally inelastic nonreactive transition 
probabilities. 
CPT's argument concerning phase factors is based on 
the following asymptotic form for the BF S-matrix so-
J. Chem. Phys. 70(06), 15 Mar. 1979 0021-9606/79/063151-03$01.00 © 1979 American Institute of Physics 3151 
Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
3152 Letters to the Editor 
lution, which they derived by a unitary transformation 
on the corresponding space fixed (SF) solution1 (ex-
pressed in the quantum number notation of Eq. (5. 3) of 
Ref. 3, 5 and ignoring closed channels which are not 
essential here): 
and s"::;~· is CPT' s scattering matrix. 6 Except for an 
unimportant 27T(V, J' ) 112 factor, Eq. (2) differs only by 
the phase factor i"•J•l from an analogous expression used 
by us3 [Eq. (5. 5) of Ref. 3), by Pack, 7 and by Walker 
and Light8 to define a scattering matrix (which we de-
note here as s'J.f;rf). CPT argued1 that because the 
quantum number j is summed over in Eq. (1), a wave 
function >It'{J~0 '[s] obtained from an expression without 
the i 1 •J•l factor will be different than one with it, pre-
suming that the same S matrix is used in both solutions. 
They further argued that this "phase factor has nothing 
to do with the definition of the S matrix, " implying that 
the same S matrix would be obtained with either solu-
tion. 
We will now show that this last statement is, in fact, 
incorrect, and that the S matrix obtained using our 
asymptotic solution differs from theirs by a phase fac-
tor iJ-J'. To prove this, we examine the incoming 
spherical wave parts of both solutions. In Eq. (2) that 
is 
X27Te(~' Y'\.t•J'+l exp( -i(k,J'R- (J+ j)7T/2)], (3) 
while ours is identical to Eq. (3) except for the i1 •J'•l 
factor (and the unimportant 27T(V.,. 1.)112). This means 
that the CPT incident solution equals i.T+J'•l times our 
incident solution. Since both solutions solve the same 
Schrodinger equation, 9 this same factor must likewise 
where 
relate the outgoing wave solutions, and this implies 
(equating coefficients of corresponding terms) that, 
·l+J'•lsv'J'O' .J+J+ISv'J'O' 
t JvJO =t JvJO • 
or 
Sv• J' n• ·J-J'sv' J' n• JvJO = t JvJO • 
(1) 
(2) 
(4} 
(5} 
It is now clear that CPT's S-matrix solution [Eq. (1}] 
and ours differ only by the multiplicative phase factor 
i"•J'•l (note the presence of j' rather than j in this fac-
tor), and this cannot have any measurable consequence. 
We should note that the factor iJ-J' in Eq. (5) can also 
be derived by comparing scattering amplitudes [using 
CPT's Eq. (28) and our Eq. (5. 31)101, or by comparing 
the relationship between BF and space fixed (SF) scat-
tering matrices [CPT's Eq. (21) and our Eq. (5. 14)11 ]. 
CPT do correctly point out that their BF R matrix 
boundary conditions {and wave functions) are, «com-
pletely different, " from ours. This is, of course, not 
surprising since their BF R and S matrices are not re-
lated by the same equation (Eq. (30) of Ref. 1) as ours 
[Eq. (5. 7) of Ref. 3). This difference does not affect 
any observable properties such as transition probabil-
ities or cross sections, since these are defined in terms 
of the S matrix, (i.e., plane wave) solution. 
Finally, we note that in examining transition probabil-
ities for ro1ationally inelastic nonreactive H + H2 colli-
sions, CPT. fmproperly attributed to us12 the conclu-
sion that (quoting from Ref. 2), "for low J, the K val-
ues are conserved approximately in closed-coupling 
TABLE I. Comparison of rotationally inelastic transition probabilities P1~~~· for nonreactive 
H+H2, aJ=15. 
CPTb atE= O. 673 eV 
- SJ' 
0 0 1 2 
-4 2.397(-3) 6.423(-4) 5. 551(- 5) 
-3 2.434(-3) B. 660(-4) 1. 782(-4} 
-2 5. 343(-3) 9. 676(-4) 2.115(-4) 
-1 7.949(-3) 2.876(-3) 3. 834(-4) 
0 1.060(-2) 4. 992(-3) 1. 019(- 3) 
1 7. 949(-3) B. 916(-3) 2.255(-3) 
2 5.343(-3) 8. 838(-3) 5. 851(-3) 
3 2.434(-3) 4. 950(-3) 9. 891(-3) 
4 2. 397(-3) 6.128(-3) 6.046(-3) 
a2.397(-3) should be read as 2.397X10-3, etc. 
l>aeference 2. 
SK" at E=O. 70 eV 
0 1 2 
2. 900(-3) 6. 614(-4) 1.326(-5) 
2. 805(-3) 7.493(-4) 3. 039(-4) 
7. 225(-3) 9.577(-4) 1. 595(-4) 
9. 813(-3) 3. 671(-3) 4. 236(-4) 
1. 399(- 2) 6. 562(-3) 1. 203(-3) 
9. 813(-3) 1.167(-2) 2.925(-3) 
7. 225(-3) 1. 093(-2) 7. 992(-3) 
2. 805(-3) 6.255(-3) 1. 296(-2) 
2. 900(-3) 7.798(-3) 7. 333(-3) 
"These results were generated as part of the calculations described in Ref. 4. although they have 
not previously been published. 
J. Chern. Phys., Vol. 70, No.6, 15 March 1979 
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 
calculations and dominant contributions of transition 
probabilities are from K6 =K,.=0," (i.e., il=O' =0). 
They proceded to challenge this staiement as if it had 
been made for nonreactive scattering. In Ref. 12 we 
did make a statement similar to this, but it refers 
specifically to reactive scattering (in H + H2). Our con-
clusion for nonreactive scattering was discussed 
thoroughly in Ref. 4, where we stated, "An examina-
tion of other nonreactive transition probabilities indi-
cates no strong tendency for a m1 = -m1.(0=0') "i.-
conserving" selection rule as has been assumed in ro-
tationally inelastic scattering." In fact, there is no 
essential difference between the 0, O' dependence of 
CPT's transition probabilities and ours. We demon-
strate this in Table 113 for the j = 2 to j' = 4 transition. 
a>Research supported in part by the United States Air Force 
Office of Scientific Research (Grant No. AFOSR-77-3394). 
b>contribution No. 5890. 
1B. H. Choi, R. T. Poe, and K. T. Tang, J. Chern. Phys. 69, 
411 (1978). 
2B. H. Choi, R. T. Poe, and K. T. Tang, J. Chern. Phys. 69, 
422 (1978). 
3G. C. Schatz and A. Kuppermann, J. Chern. Phys. 65, 4642 
(1976). 
4G. C. Schatz and A. Kuppermann, J. Chern. Phys. 65, 4668 
(1976). 
5We use the same notation as in Ref. 3 with regard to definition 
of the vibrational, orbital, rotational, total angular momen-
tum and body fixed projection quantum numbers (v,l,j,J, and 
Q, respectively). The arrangement channel index A will, 
however, be suppressed, and we let primed quantum numbers 
denote initial states and unprimed denote final states. This 
notation is similar to CPT, 1 but the quantum number 0' is 
denoted by them as -K,. and 0 as K8• 
6Note that ~~/ff here would have been labelled as S"j(j1,_ 0 , in 
Eq. (5. 5) of Ref. 3. 
7R. T Pack, J. Chern. Phys. 60, 633 (1974). 
8R. B. Walker and J. C. Light, Chern. Phys. 7, 84 (1975). 
9CPT argued1 that iirJ;}'0 ' [S) in Eqs. (1) and (2) does not solve 
the BF Schrodinger equation if the i 1 • J • 1 is omitted because 
the BF Hamiltonian does not commute with a diagonal matrix 
having diagonal elements i 1 • J • 1• Asymptotically, however, 
the Hamiltonian is diagonal in j so their argument does not 
apply, and both wave functions solve the. BF Schrodinger equa-
tion in that region. Either solution may furthermore be 
matched to solutions which solve the Schrodinger equation at 
small R as shown in detail in Ref. 3 for our solution. 
1
'C PT used the same plane wave solution as we, and it is easy 
to see that the scattering amplitudes are simply proportional. 
By relating these quantities. and using Eq. (28) of Ref. 1 and 
Eq. (5. 31) of Ref. 3, we again derive Eq. (5). Note that the 
meaning of primed and unprimed quantum numbers in Eq. 
(5. 31) is reversed from that used here. 
11m this comparison, a phase factor difference i 1-" arises in 
the SF scattering matrices because of differences in the 
definition ofthe SF scattering solutions. 
12A. Kuppermann and G. C. Schatz, J. Chern. Phys. 62, 2502 
(1975). 
13The actual differences between their and our transition prob-
abilities in Table I are probably due to a combination of 
effects, including the slightly different energies at which the 
calculations were done, the different vibration-rotation basis 
sets used, and the different representations of the potential 
used (CPT truncated the expansion of the potential in Legendre 
polynomials at the P 2 term while we included the P4 term.) 
On "Theory of collisions between an atom and a diatomic 
molecule in the body-fixed coordinate system"a> 
B. H. Choi and A. T. Poe 
Department of Physics, University of California, Riverside, California 92521 
K. T. Tang 
Department of Physics, Pacific Lutheran University Tacoma, Washington 98447 
(Received 9 November 1978) 
In recent papers with the above titles, 1• 2 we presented 
a detailed formulation for the treatment of parities in a 
body-fixed (BF) close-coupling approximation. This is 
directly amenable to numerical computation of elastic 
and inelastic atom-diatom scattering cross sections. 
The main thrust of our paper is the parity boundary 
conditions. However, in the preceding comment by 
Schatz and Kuppermann, they raised some questions re-
garding the relationship between the nonparity boundary 
conditions for S matrix developed by us and by them. It 
seems necessary for us to clarify the implications of 
some statements which we made about nonparity asymp-
totic boundary conditions for S matrix, not to be misun-
derstood by other colleagues in this field who have em-
ployed different forms of the boundary condition from 
ours. (Hereafter, the "asymptotic" will be deleted.) 
We stated that if AP(R) is another boundary condition 
different from our A(R) defined in Ref. 1 as 
A1.,K.,,J,.x ,.(R) == 27Ti1•J.,•t 
X (o1.,1.,oK.,-K"' exp{- i(k.,R- (J + j 7 )7T/2]} 
- ~,.K.,,JcxKa exp{i[k.,R- (J + j 7 )7T/2]}), (1) 
then 
(2) 
for some nonsingular matrices P and Q.3 Here, AP(R) 
is an expression in terms of corresponding S matrix 
J. Chem. Phys. 70(06), 15 Mar. 1979 0021·9606/79/063153-02$0 1. 00 © 1979 American Institute of Physics 3153 


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