- Email: [email protected]
Unreactive energy bands in atom—molecule collisions
‘.
,~U&.ic;ll Gajectory c&uIationr
(“
for exchkge
1. ~ntr~dust~~~
”
“1 it-l& been-shown in several studies of the H +I& .’ exchange reaction on a collinear potential-energy & .. face that Y&etota.! reaction probability is a co’mpli- .’ +ted function of relative tr~s~ation~ energy. Pro,nounced fluctuations in reaction prob;lbility were ob‘* tained in the quantum dynamical calculations of .!&esger [I], fuppermanri and co-workers [2--51, Wu _ 1, et al. [6,?] and Levine and 7;YuIS]. ~Ie~‘R~~tuat~ons ‘have been discus&d in terms of rnterference between resonant (coliision complex). z$d ndnresonant (direct) scattering. Such fluc~3tions also appear in classical :, trajectory calculations [5,9; lo& although apparently. ’ to’s’ lesser’extent IS], where,they have been de&bed as “multiple colli~ons~,’ [ll]. The effect-of the choice’. .i &f,vibrational pr~bab~it~e~ has bee’h shown by. ::Careless and Hvatt 112) +I$ amdyzed by Bbwman et al. ,[ 131. -Tk most, thoiough.discussion to.date, in: : ~clirdirig’acomparison betwken quantum, seml&la.&&.l jar.4 classic.~.~ilculations,‘was givenby Duff and;-. Truhlar,[f4], ., ;., . .._ .,. ‘,, ,’ ‘.,; ,.‘l Our oii;n approach to this prdblem owes much to : ~e~~~~wo~k~f We and c&workers [15;!$].:‘. : .. [.‘.‘: :..:-, .,) ._ ,’
_.,
.&‘:~-“.’ : ..“L :-. : : ‘. ,. . :_ .,:, .: ‘-1; ,.‘..
.
‘.,
.’
.,.
::.i’:. ., :. ,: ‘. ,.., ,:. “.
trajectory
times
the
exist-
is shown.
-
These au~hors_r~sed many of the, ~porta~t questions. which can be treated bQ molecular dynamics studies, ~clu~~~ fluctuations in reaction probability, but due to tel~~ol~~cal limitations were,unabfe to fulIy am.wer them.
.2. hlc:$rhad’of‘Glcofation.
‘. 1:
.I
’ En this study..we restric! ourselves to classical tra-
jectoly cal~ukitions o? the,,co&near SSl?fi$ [17] potentialenergy surface. ‘&hissurface, obtained quantum mech:micaUy, has a barrier height of 11 .O kc&i/mole ’ and very smopth contours. The formulation ofthe .. cl~ss~~~.equ~~i~~sof motion is that given by Wall and Porter, 1161. Mass combina~ons are of the,form ‘, H i.L;-H, where, H implies a heavy atom and L a light atom; L was, always chose% to be a hydrpgen’atom. ,’ A$calculations a&me #at the zer~~p~~~:~~r~~o~~ -;. e:rerm‘ is present iri the diatomic o&lla~or L-H at the startgf’a trajectory’. The magriitudgCtif the’ZPg is-ob- 1.‘. ta$& ; relative’jo Hz, from.‘the ratio of:@ square ,’., “, rdot o:l the reduced mar&s,;.AII tra)ectoiies &&at );p;B TAB =,:(!:_ti.0.nm ,. :and ,, emi.,, when ,, : _ 0; ‘Be__exceed .~ &SS., :’ ,‘,.,I
._
energy surface show and band positions
reactiorrs on the caUineq SSMK potential
ence of s&active and unreactive r&ions, or bonds. The r&&ion&ip ktween Th-, system T + HT exh%t; a strjkiq unreactive band. .’ ..
_.
.’
,: .‘. :
‘..::
..-
_,,_ -y _‘.. ‘,.’ ,( ,.. ‘.y _., _,,
Volume 30, number 2
CHEMICAL PHYSICS LETTERS
run: The initial distance ‘BC is allowed to vary between its classical limits; the initial choice of rgC is
molqand
15 January 1975
begins again above 24 kcal/mole.
The solid
line on the figure indicates the relative time per tra,jectqry. Except for the apparently ar~omalous time peak near 10 kcal/mole, al1 such peaks are related to the onset of a band separation between R and LJ. Closer examination of any of the band boundaries invariably reveals a more complicated “fine structure” or separation into Rand U trajectories. Thiz behavior, termed statistical, has been noted previoudy [14, and references therein], and can be related to the transfer of energy into the Hg symmetric stretching mode [8] and the subsequent “hesitation” of the systern near the activated complex. Fig. 2 shows the behaviour of the trajectory times and bands as a function of E,, for different phase angles. To distinguish between Rand U regions at each Q we have used solid lines (R) and dotted Lines (U).
equivalent to choosing a “phase angle” to describe the initial conditions. Owing to the signiricant anharmonicity,.use of the simple harmonic oscillator model to describe the vibrational chase is insufficient. Instead; a harmonic function having one force constant for compression and another for extension gives a good empirical fit to-the true dynamic behavior and allows definition of the phase angle to be in accord with the following extrema: 0” (equilibrium distance), 90” (maximum compression), 180’ (equilibrium), 270’ (maximum extension). The equations of motion were integrated by a Rurlge-Kutta startup and an AdamsBashforth 4th order predictor corrector routine using an integration step size of 2.0 X lo-l6 s.
3. Results ‘For the mass combination characteristic of H + H,, H = L = 1.008 amu, ZPE = 6.20 k&/mole, we first
dium grid of energies {ca. 0.2 kcal/mole)
fig. 1 shows a very clear separation into regions in which all tra-. jectories are reactive (R) or unreactive (U). The reac-
tive threshold occurs at relative translational energy Et= 8.0 kcal/mole (total energy E = 14.2 kcal/mole). An unreactive
e
10
band occurs from 14.2 to 15.7 kcall
I2
20
22
24
32
.26
Fig. 1. Reaction time plotted.aginst transtitional energy, for H + Hz. Renctiv6 bxnds (R) znd unrextive bands (IJ) are shown by dashed lines Trajectory times are i&oted solid line (maximum lime = 1100 X JO_‘6 s).
‘.
38
44
50
Ei (kcol/mo!e)
Et(kcallmolel
:
by the
Fig. 2. Reactive (sobi) &id unreactive (dotted) bands at different phase angles, as a function&f relative translational en‘. ergy, for H +.Hz. The dashed lines connect the bands at dif-ferent 6.’ ‘:
‘-
‘. :
,.
.. .,
201
15 JaillJXY
CIiEh~ICAL PHYSICS LE’lTERs
Volume 30, number 2
1975
‘.
Dashed lines an the rigure are used to connect time peaks at different J$[ 15JCIt is seen that there’is a con-
tinuous shift of the band position with increas;nk 9, ‘and that the time peaks are closely related to the band boundaries. Tltis very useful property means $!at ihe. outcome of each individual trajectory contributes to the.precise location of the energy bands; From the ovarall behavicr, one can construct the sort of dia; gramshown in fig. 33; in which the bands have been connected so that R and U regions are shown:The apparently anomalous time peak at @= 0, E,=,lO.O is clearly related to the presence.of a nearby unreactive band. ,. Given.the behavior of the R.and U bands,‘and an assumed vibrational distribution, the reaction probabiJity can be obtained. Assuming a classical enharmonic distribution appropriate to the SSMK surface with : zero-point energy, the classical result (solid line) in fig. 3b was obtained by taking a weighted average over the different phase angles. The dip in probability near 15 kcaI/moIe is related to the position of the fust unreactive band. The gradual decline in reaction probability at increasingEt is seen to be due to the increasing contribution of unreactive bands at hi&r energy. For comparison, the quantum dynamical re-
suit of Bowman and Kupperma& [S] on‘a similar surface is rather tiell reprodu$.d *Although the band behavior in the H + Hz exchange reaction is an interesting result, the overall effect on reactidn probability. is relatively slight, the most not,able feature being a small dip. The question now arises: Is there a potential-ener,gy surface, or mass combination,-for which the re.action probability drops to zero.at an energy close to threshold, and remains zero over a significant range? There are indications in the work of Diestler [I] that probability dips are related to the barrier height and the amount of vibrational energy present. It was also suggested by Wall and Porter [16] that mass ef- : fects are important, through their influence on the skew angle for the energy surface. We began to study the question by doing a systematic variation of mass combinations H + LH (expected to show the most extreme effe.cts) on the same SSMK surface, where H was allowed to take on values 2(D), 3(T), 10,20, and 40. Ail mass combinations showed band behavior as .in H + H-H, and some regularities were evident, but the most striking results were obtained for T +HT. These results are summarized in fig. 4. Fig. 4a shows
” These authors also obtained a classical result on the PK SW face which lacked the probability
dip present in the quan-
turn cdculaiion and our own cl&&l results It now ap pears that this !vas due to their choice of too cmme an energy
mesh.
E,tkcal/mole;
Fig. 3: is) Reaciivi +nd unreactive regions, for H.+ Hz,:at different d and Et &&vc are shaded). (b) Solid line: reaction ” probability based on the.$%XK vibrational distzibtition func tioy: (classical anhan~~oriic oscillator). Da&ed line: reaction ‘pr,obability based on ‘&e.quaxtwn dynamica! calcuiations Of : Bownan and Kuppemxii [S]. _,
._
::.
Fig. ?. (a) React+ (shaded) and‘unreacke (unshaded) re&ns for’r ?HT. (b) Reaction probability, based on the,SSMK distrjbutionfunction, showing thelarge unreactive band
VoIumi: 30, ni.imbei 2
:
a rapidly o&Ilatin~ series of bands near threshold {again ,S kcallmolej, followed by ari e&mow umq” active band which lasts essentitiy from 3_0to 40 kcd/ -.
Over t@s range the reaction probability is zero (fig. 4b), e&q~ for a sniall X region near 35. kcali
mole,
._ mole. It therefore appears Likely that given an energy sur-
face similar to the one discussed, a suitable mass combination; and a moleculbr beam apparatus with velociiy selection, such effects should be experimentally. observable, The extension of thiis study to coplanar
trajectories is in progress.
References, [l] D.J. DiestIer, I. C&em.Phys 56 (1972) 2092. [Z] G.C Scbatz qnd A. Kuppc~a~, J. atem. Phys. 59 (1973) 964. t31 D.G. Truhlar and ;4. Xupper&~,
(1972) 2232.
I.5 Januxy 1975
CHEMICAL PHYSICS LEA-lXRS
-. “.
J. Che.m. Phys. 56
” 141 I.hf.
Bowman and A. Kup~ermann, Chem. Fhys Letters
1?(19'?3) 166. and A. Kuppermann,Gem. Phyz tetters I2 (1971) 1. .,’ [61 S.F. Wu and RX%.ZwIne, hfol. Pi~~js.22 (1971) 881. .[?I SF. Wu, B.R. Johnson and R.D. Levine, MoL Fnys. 25 (1973) 509. 181 R.D. Levine and S,F. [Vu; Cheti. I&s Letters I1 Et9711
Is1
J.&f.Bovnnm
557.
PI K.P. gong and D.j. DiestIer, f. Chem.. khys 56 (1972) 3200. D. Russell &.ndI.C. r;i&t, 1. CI+nk P~J’E-51fl969) IWJ. CC Rankhand W.H. Miller, E, Chem. Phyr 55 (1971) 3150. P.N. Careleskand D. Hyatt, Chem. Phys. Letters 14 (1972) 3.58, J.M. Bowmsn. A. Ruppermzn~ and G.C. Schatz, C&em. Phys. Letters 19 (1973) 21. J.W. Duff and D.G. T~hlz=, Chem. I&%. 4 (1974) 1.
F.T. Wall, L..k.H.iUer Jr. and I. Bk?ur, I. Chem. I?hy% 29 (1958,255. F.T. WJ.I and RN. Porter, J. Chem. IX&. 39 (1963)
3112. I. Shavitt, Rhf. Stevens, $.I_.. ?&nn znd bi. I&plus, J.
C&em. Phys. 48 (1968) 2700.
:
~.
:
(
‘(
’ ‘. ..
.:’ _.‘.
‘.
:
.~
,.
.
.’
.‘., ,,’
/:
.
‘,
.’
. ;
‘.
.
.I
:
‘_
:.
-
203
,~U&.ic;ll Gajectory c&uIationr
(“
for exchkge
1. ~ntr~dust~~~
”
“1 it-l& been-shown in several studies of the H +I& .’ exchange reaction on a collinear potential-energy & .. face that Y&etota.! reaction probability is a co’mpli- .’ +ted function of relative tr~s~ation~ energy. Pro,nounced fluctuations in reaction prob;lbility were ob‘* tained in the quantum dynamical calculations of .!&esger [I], fuppermanri and co-workers [2--51, Wu _ 1, et al. [6,?] and Levine and 7;YuIS]. ~Ie~‘R~~tuat~ons ‘have been discus&d in terms of rnterference between resonant (coliision complex). z$d ndnresonant (direct) scattering. Such fluc~3tions also appear in classical :, trajectory calculations [5,9; lo& although apparently. ’ to’s’ lesser’extent IS], where,they have been de&bed as “multiple colli~ons~,’ [ll]. The effect-of the choice’. .i &f,vibrational pr~bab~it~e~ has bee’h shown by. ::Careless and Hvatt 112) +I$ amdyzed by Bbwman et al. ,[ 131. -Tk most, thoiough.discussion to.date, in: : ~clirdirig’acomparison betwken quantum, seml&la.&&.l jar.4 classic.~.~ilculations,‘was givenby Duff and;-. Truhlar,[f4], ., ;., . .._ .,. ‘,, ,’ ‘.,; ,.‘l Our oii;n approach to this prdblem owes much to : ~e~~~~wo~k~f We and c&workers [15;!$].:‘. : .. [.‘.‘: :..:-, .,) ._ ,’
_.,
.&‘:~-“.’ : ..“L :-. : : ‘. ,. . :_ .,:, .: ‘-1; ,.‘..
.
‘.,
.’
.,.
::.i’:. ., :. ,: ‘. ,.., ,:. “.
trajectory
times
the
exist-
is shown.
-
These au~hors_r~sed many of the, ~porta~t questions. which can be treated bQ molecular dynamics studies, ~clu~~~ fluctuations in reaction probability, but due to tel~~ol~~cal limitations were,unabfe to fulIy am.wer them.
.2. hlc:$rhad’of‘Glcofation.
‘. 1:
.I
’ En this study..we restric! ourselves to classical tra-
jectoly cal~ukitions o? the,,co&near SSl?fi$ [17] potentialenergy surface. ‘&hissurface, obtained quantum mech:micaUy, has a barrier height of 11 .O kc&i/mole ’ and very smopth contours. The formulation ofthe .. cl~ss~~~.equ~~i~~sof motion is that given by Wall and Porter, 1161. Mass combina~ons are of the,form ‘, H i.L;-H, where, H implies a heavy atom and L a light atom; L was, always chose% to be a hydrpgen’atom. ,’ A$calculations a&me #at the zer~~p~~~:~~r~~o~~ -;. e:rerm‘ is present iri the diatomic o&lla~or L-H at the startgf’a trajectory’. The magriitudgCtif the’ZPg is-ob- 1.‘. ta$& ; relative’jo Hz, from.‘the ratio of:@ square ,’., “, rdot o:l the reduced mar&s,;.AII tra)ectoiies &&at );p;B TAB =,:(!:_ti.0.nm ,. :and ,, emi.,, when ,, : _ 0; ‘Be__exceed .~ &SS., :’ ,‘,.,I
._
energy surface show and band positions
reactiorrs on the caUineq SSMK potential
ence of s&active and unreactive r&ions, or bonds. The r&&ion&ip ktween Th-, system T + HT exh%t; a strjkiq unreactive band. .’ ..
_.
.’
,: .‘. :
‘..::
..-
_,,_ -y _‘.. ‘,.’ ,( ,.. ‘.y _., _,,
Volume 30, number 2
CHEMICAL PHYSICS LETTERS
run: The initial distance ‘BC is allowed to vary between its classical limits; the initial choice of rgC is
molqand
15 January 1975
begins again above 24 kcal/mole.
The solid
line on the figure indicates the relative time per tra,jectqry. Except for the apparently ar~omalous time peak near 10 kcal/mole, al1 such peaks are related to the onset of a band separation between R and LJ. Closer examination of any of the band boundaries invariably reveals a more complicated “fine structure” or separation into Rand U trajectories. Thiz behavior, termed statistical, has been noted previoudy [14, and references therein], and can be related to the transfer of energy into the Hg symmetric stretching mode [8] and the subsequent “hesitation” of the systern near the activated complex. Fig. 2 shows the behaviour of the trajectory times and bands as a function of E,, for different phase angles. To distinguish between Rand U regions at each Q we have used solid lines (R) and dotted Lines (U).
equivalent to choosing a “phase angle” to describe the initial conditions. Owing to the signiricant anharmonicity,.use of the simple harmonic oscillator model to describe the vibrational chase is insufficient. Instead; a harmonic function having one force constant for compression and another for extension gives a good empirical fit to-the true dynamic behavior and allows definition of the phase angle to be in accord with the following extrema: 0” (equilibrium distance), 90” (maximum compression), 180’ (equilibrium), 270’ (maximum extension). The equations of motion were integrated by a Rurlge-Kutta startup and an AdamsBashforth 4th order predictor corrector routine using an integration step size of 2.0 X lo-l6 s.
3. Results ‘For the mass combination characteristic of H + H,, H = L = 1.008 amu, ZPE = 6.20 k&/mole, we first
dium grid of energies {ca. 0.2 kcal/mole)
fig. 1 shows a very clear separation into regions in which all tra-. jectories are reactive (R) or unreactive (U). The reac-
tive threshold occurs at relative translational energy Et= 8.0 kcal/mole (total energy E = 14.2 kcal/mole). An unreactive
e
10
band occurs from 14.2 to 15.7 kcall
I2
20
22
24
32
.26
Fig. 1. Reaction time plotted.aginst transtitional energy, for H + Hz. Renctiv6 bxnds (R) znd unrextive bands (IJ) are shown by dashed lines Trajectory times are i&oted solid line (maximum lime = 1100 X JO_‘6 s).
‘.
38
44
50
Ei (kcol/mo!e)
Et(kcallmolel
:
by the
Fig. 2. Reactive (sobi) &id unreactive (dotted) bands at different phase angles, as a function&f relative translational en‘. ergy, for H +.Hz. The dashed lines connect the bands at dif-ferent 6.’ ‘:
‘-
‘. :
,.
.. .,
201
15 JaillJXY
CIiEh~ICAL PHYSICS LE’lTERs
Volume 30, number 2
1975
‘.
Dashed lines an the rigure are used to connect time peaks at different J$[ 15JCIt is seen that there’is a con-
tinuous shift of the band position with increas;nk 9, ‘and that the time peaks are closely related to the band boundaries. Tltis very useful property means $!at ihe. outcome of each individual trajectory contributes to the.precise location of the energy bands; From the ovarall behavicr, one can construct the sort of dia; gramshown in fig. 33; in which the bands have been connected so that R and U regions are shown:The apparently anomalous time peak at @= 0, E,=,lO.O is clearly related to the presence.of a nearby unreactive band. ,. Given.the behavior of the R.and U bands,‘and an assumed vibrational distribution, the reaction probabiJity can be obtained. Assuming a classical enharmonic distribution appropriate to the SSMK surface with : zero-point energy, the classical result (solid line) in fig. 3b was obtained by taking a weighted average over the different phase angles. The dip in probability near 15 kcaI/moIe is related to the position of the fust unreactive band. The gradual decline in reaction probability at increasingEt is seen to be due to the increasing contribution of unreactive bands at hi&r energy. For comparison, the quantum dynamical re-
suit of Bowman and Kupperma& [S] on‘a similar surface is rather tiell reprodu$.d *Although the band behavior in the H + Hz exchange reaction is an interesting result, the overall effect on reactidn probability. is relatively slight, the most not,able feature being a small dip. The question now arises: Is there a potential-ener,gy surface, or mass combination,-for which the re.action probability drops to zero.at an energy close to threshold, and remains zero over a significant range? There are indications in the work of Diestler [I] that probability dips are related to the barrier height and the amount of vibrational energy present. It was also suggested by Wall and Porter [16] that mass ef- : fects are important, through their influence on the skew angle for the energy surface. We began to study the question by doing a systematic variation of mass combinations H + LH (expected to show the most extreme effe.cts) on the same SSMK surface, where H was allowed to take on values 2(D), 3(T), 10,20, and 40. Ail mass combinations showed band behavior as .in H + H-H, and some regularities were evident, but the most striking results were obtained for T +HT. These results are summarized in fig. 4. Fig. 4a shows
” These authors also obtained a classical result on the PK SW face which lacked the probability
dip present in the quan-
turn cdculaiion and our own cl&&l results It now ap pears that this !vas due to their choice of too cmme an energy
mesh.
E,tkcal/mole;
Fig. 3: is) Reaciivi +nd unreactive regions, for H.+ Hz,:at different d and Et &&vc are shaded). (b) Solid line: reaction ” probability based on the.$%XK vibrational distzibtition func tioy: (classical anhan~~oriic oscillator). Da&ed line: reaction ‘pr,obability based on ‘&e.quaxtwn dynamica! calcuiations Of : Bownan and Kuppemxii [S]. _,
._
::.
Fig. ?. (a) React+ (shaded) and‘unreacke (unshaded) re&ns for’r ?HT. (b) Reaction probability, based on the,SSMK distrjbutionfunction, showing thelarge unreactive band
VoIumi: 30, ni.imbei 2
:
a rapidly o&Ilatin~ series of bands near threshold {again ,S kcallmolej, followed by ari e&mow umq” active band which lasts essentitiy from 3_0to 40 kcd/ -.
Over t@s range the reaction probability is zero (fig. 4b), e&q~ for a sniall X region near 35. kcali
mole,
._ mole. It therefore appears Likely that given an energy sur-
face similar to the one discussed, a suitable mass combination; and a moleculbr beam apparatus with velociiy selection, such effects should be experimentally. observable, The extension of thiis study to coplanar
trajectories is in progress.
References, [l] D.J. DiestIer, I. C&em.Phys 56 (1972) 2092. [Z] G.C Scbatz qnd A. Kuppc~a~, J. atem. Phys. 59 (1973) 964. t31 D.G. Truhlar and ;4. Xupper&~,
(1972) 2232.
I.5 Januxy 1975
CHEMICAL PHYSICS LEA-lXRS
-. “.
J. Che.m. Phys. 56
” 141 I.hf.
Bowman and A. Kup~ermann, Chem. Fhys Letters
1?(19'?3) 166. and A. Kuppermann,Gem. Phyz tetters I2 (1971) 1. .,’ [61 S.F. Wu and RX%.ZwIne, hfol. Pi~~js.22 (1971) 881. .[?I SF. Wu, B.R. Johnson and R.D. Levine, MoL Fnys. 25 (1973) 509. 181 R.D. Levine and S,F. [Vu; Cheti. I&s Letters I1 Et9711
Is1
J.&f.Bovnnm
557.
PI K.P. gong and D.j. DiestIer, f. Chem.. khys 56 (1972) 3200. D. Russell &.ndI.C. r;i&t, 1. CI+nk P~J’E-51fl969) IWJ. CC Rankhand W.H. Miller, E, Chem. Phyr 55 (1971) 3150. P.N. Careleskand D. Hyatt, Chem. Phys. Letters 14 (1972) 3.58, J.M. Bowmsn. A. Ruppermzn~ and G.C. Schatz, C&em. Phys. Letters 19 (1973) 21. J.W. Duff and D.G. T~hlz=, Chem. I&%. 4 (1974) 1.
F.T. Wall, L..k.H.iUer Jr. and I. Bk?ur, I. Chem. I?hy% 29 (1958,255. F.T. WJ.I and RN. Porter, J. Chem. IX&. 39 (1963)
3112. I. Shavitt, Rhf. Stevens, $.I_.. ?&nn znd bi. I&plus, J.
C&em. Phys. 48 (1968) 2700.
:
~.
:
(
‘(
’ ‘. ..
.:’ _.‘.
‘.
:
.~
,.
.
.’
.‘., ,,’
/:
.
‘,
.’
. ;
‘.
.
.I
:
‘_
:.
-
203