- Email: [email protected]
On the applicability of the classical trajectory equations in inelastic scattering theory
1 Februxy
CHEMICAL PHYSICS LEl-lXRS
Volume 30, numb& 3
1975
ON THE APPLICABILITY OF THE CLASSICAL TRAJECTORY EQUATIONS IN INELASTIC SCATTEPSNG THEORY
Gert Due BILLING Chemistry Loboratory Received 27 September
III, H.C. @ted
Institute, 2100 Copertiragen 4, Denmark
1974
Ey using the classical trajectory equations and an average trajectory we have been able to predict the single quantum transitions for the one-dimensional atom-diatom collision problem to within lo-30%. The double quantum transiiions POZ ye correct to within at least a factor of two.
procedures. In ref. [2] we suggested the use of the following two methods:
1. Introduction
In a semi-classical description of inelastic collisions one can in principle use two different methods, namely the semi-classical S-matrix method developed by Miller and Marcus [l] and the classical trajectory method which uses a quantum mechanical description of one or several degrees of freedom, whereas the remaining are treated classically [2]. In the first method one has the complicating feature that complex trajectories have to be introduced in order to get non-zero amplitudes for classically forbidden transitions. In principle there should be no problems in t-his extension of ordinary classical mechanics. In praitice howcqer some difficulties such as root searching in complex space and stability problems will have to be overcome. The classical trajectory
method
is foknally
(b)
V,,=
Here V is the intermolecular potential, V,,, is an average potential, i.e., a potential which is independent of the quantum coordinate, q is the total wavefunction, qn an eigenstate to the internal hamiltonian and un the initial velocity in channel n. From either. the method (a) or (b) and the classical trajectory equations:
a more
correct description because of the use of a quantum mechanical treatment of a,part of the scattering : problem. Here however it has turned out that the coupling between the classical and the qu.antum degrees of kkedom is rather involved [3] _ Thus we have to solve a boundary value problem in order to :, determine “the classic.al trajecto,ry”. This can apparently not always be done because the iterative procedure used to obtain the trajectory’will fail to converge in some cases [2,4]. . ,I We have therefore been interested in alternative -.,’ .’
the transition probabiIitiesP,, can be obtained as P nm = lc,(~)l* wherkc,,,(--)=6,&. In eq.(l) En is the internal energy of state tz and the total wavc-
function has been expanded as 9 = zrr cn’p,.,. Note in mCthpd (a) we have to use a different trajector$ for every P,m . In ref. [2] ‘we suggested that method (b):&ould be used at high energies. In this
,that
limit the average potential (Ik(Vle> should be useful. ‘Ibis~follows from the use of Ehrenfest’s theorem and ,-.
..
‘.
CHEMICAL
V+rn~ 30, number 3
PHYSICS LElTERS
1 February 1975
.,
.A
[2.,7]. In refs. [2,8] w.e have mainly,investigated the results of ;I first-order treatment of the quantum transition (her: vibrational transitions). kthis paper we give some .falues qbtained by the methods (a) and (c) using an nth order expansion .of the tot.4 Gavefunction, i.e., L?e probabilities listed here are the converged semi-classical probabilities. Since the onedimensionill atom-diatom collision problem is thoroughly investigated we have chosen dthissystem for the numerical kalculations. Secrest and Johnson [9] ahaveperformed exact quantum mechanical calculations on eight different systems. These systems can roughly be divided !nto tw,o subgroups, consisting of those having either small 0; large reduced mass. In ref. [S] we investigated a system from each group, namely no. I (m z(3.5, %r = 0.114) and 7,(m = 514, aySJ 7 O-2973), wllere
has been inves$gat&d @ somk detail by:I’horson et & [5]. The reason for suggesting method (a) at lower energies is that’this choice of average trajectory has turned out to give excellent results for curve crossing. pr?,bletis with a linear potential &Td exponential interaction 161. At energies close to threshold this symtietrization will furthermore give better results tbti the one nom-idly used in semi-classical theories, namelyd=j(vi +.fJ&. It is now natural and cqnvenient to combine the methods (a) and (b) to method(c): V, = (*I v.1G4 (aveiage potential) , and u= [u, urn)!‘* .(effective velocity for the tramition
.
n +m arm -+nj.
This method will correspond to (b) at high energies (un = urn) a+ to (aj at low energies (because we bve Vav -_”Ic,12(1p,I~,iQ x (~~\vI~pn>in both cases). Note also that method (c) obeys the principle of detailed palance and comerves enerm along the aver*e trajectory. -The reason for dealing with this semi-classical approach @that an extension of this theory to threedirnension~ calculatidns of atom-diatom and diatomdiatom collision problems is rather Straightforward
M = mB
772~ ,I(mg
+ nzc) ,
k, is the force constant for BC and ~1is the reduced mass of the system A + BC.
Tible'l Probabilities for single quantum transitions obtained by different theories. The last column gives the power, i-k., the numbers in the first row should be multiplied by lo-‘. The total enero E is given ti units of the ground state energy $hr~ System
Trarhition
E
.l
7.6 8.8
-;
o-1 O-l 1-2 .O,l l-2 o-1 ’ 2 .x1 l-2 o-1 O-l l-2
lO.@ 12.0
‘:
16.0 ‘.
. 4.9Js.5 6.9455
7
8;?455
..’ : .’
‘.;6.7882 _‘.. : 7 .;-‘. ,392 ; :
4.30 2.03 2.23 6.58 1.52
4.90 2.36
. ,,.
: ,.
.‘.-
.: :.
.~..
ref. [12]
3.38 2.26
1.86
6.04
_’ : ,.
,. ..
,,, :-
:
Miller ref. [IO]
ITFITS
(c)
7.85
1:53 5.97. : 1’ 1.97 .. 2.37
._
-:‘: ‘..‘._
(a)
?.SS 1.32 1.92 1.66 1.12 2.93 2.30
0,-l l-2 .G’-1 l-2
-
Exact ref. [9]
3.36. ‘, 3.40 :
-.. ..’
Power
LMXCUS
ref. [Ill
4.01 1:YJ 2.02 6.39 1.44
4.45 2.09 2.30 6.81 1.57
4.69
-5
2.52
-5 .- 4
2.91 l.‘iS 1.89 1.65 0.87 2.64 1.70
2.94 1.36 1.97 1.71 2.35 ‘. 6.30’. 4.69 ”
3.03
3.32 125 3.30 3.40
1.36 ‘5.24 ,- 1.80 2.66
-4 -4
1.22..
-:
: 2.51 ..
.. ;
:--
‘:
-3 -3 -2 -2 -4 -3. -4
:
,:
6.50
:.‘:_ :
;
-2 :-3 -1 -j.
-.: .. .
:’ .,
: /
1 ..
.,
::. :
,‘, ..,
; -,
,,,, . .
i ..
,, .:
;.
... .;
..., ..,,
Volume 30, rwnbor
3
CHEhiICAL
Table 2. The vibrational transition &obabilities System
E
Exact ref.’ [9]
PHYSICS LETTERS
Pm o’,cained by different theories
(a)
(cl
2. Results In tables 1 and 2 we have compared the numerical results obtained by the methods (a).and (c) for the systems 1 and 7 with the exact results of Secrest and Johnson [9], the semi-classical values obtained by Miller and George [IO], Marcus and Stine [l I] and ~ues’obt~ed by the ITFITS method introduced by Heidrich et aL[12]. We see that the resuultsobtained by method (c) are somewhat better than both method (a) and the ITFITS theory and almost as good as the values obtained by lMiIler and Marcus (exc+,foT some of the PO2 values), Especially for system 7 we get an improvement by using method (c) instead of (a). The reason for this that for this system we’get a large distortion of the trajectory 1 during the collision by using the time-dependent average potential instead of the “static” potential used in method (a), We also note t&t it is only for’ the very small probabilities that we get a relatively large devialion from the exact results (lo-30% for single qua&m tran&ons’and as much as a factor of two for double quaturn transitions).
.-
The r~la~iveiy’go~d agriement &ith the exact re.sults bbtained by using method (c) is rather encouraging. We think that this method is able to @edi& t&e iritegr;al’cross tiections to within at least a factor qf two: Be&use of this and because of its simplicity .:
1 February 1975
ITFITS ref. [12]
‘.
Miller
hI2Icus
ref. [IO]
ref. [i l.1
Power
-
we beliecthat &is method is w&h while investigating further. Thus dre plan. to use it to determine bkh the integral and the differential cross sections for combined rotations-~brational transitions in the He-H2 system.
References [i] WH. Miller,
(21 131 141 ES]
J. Chem. Kyis. 53 (1970) 1949,3578; Chem. Phys. Letters 7 (1970) 431; I. (Them. Phys. 54 (1971) 5386; RA. Marcus, Chem. Phys. Letters 7 (1970) 52.5; I. Chem. Phys. 54 (1971) 3965; J.N.L. Connor and R.A. Marcus, I. Cbem. Pirys. 5.5 (1971) 5636; W.H. Wang and R.A. hfucus, I. CTnem. Fhys, 55 (1971) 5663. G.D. Billing, J. C’hem.Phys., to be published. P. Pechukis, Phys. Rev. lSl(1969) 166,174. A.P. Penner and R. Wallace,Phys. Rev. A9 (19741 1136. J-B; Delos, W.B. Thorson and S.K. Knudson, Phys. Rev. A6 (1972).709;
J.B. Defosand
W.B. Thorson.
??hys.
Rev. A6 (1972)
720. [6] A.M. \VooUey and S.E. Nielsen, C&m. whys. titters .21(1973) 491. [7] G.D. Billing,
5658. [ii 1.3. Stine and R.A.
.-
Ma.rcw, Chem. Whys. Letters 15
(1972)536. 1121 FE. H&rich, K.R. W&on znd D. Phys. 54 (1971) 3885.
;
:.
..
,’
Rapp, I:
., .
..
;..
Chem:
CHEMICAL PHYSICS LEl-lXRS
Volume 30, numb& 3
1975
ON THE APPLICABILITY OF THE CLASSICAL TRAJECTORY EQUATIONS IN INELASTIC SCATTEPSNG THEORY
Gert Due BILLING Chemistry Loboratory Received 27 September
III, H.C. @ted
Institute, 2100 Copertiragen 4, Denmark
1974
Ey using the classical trajectory equations and an average trajectory we have been able to predict the single quantum transitions for the one-dimensional atom-diatom collision problem to within lo-30%. The double quantum transiiions POZ ye correct to within at least a factor of two.
procedures. In ref. [2] we suggested the use of the following two methods:
1. Introduction
In a semi-classical description of inelastic collisions one can in principle use two different methods, namely the semi-classical S-matrix method developed by Miller and Marcus [l] and the classical trajectory method which uses a quantum mechanical description of one or several degrees of freedom, whereas the remaining are treated classically [2]. In the first method one has the complicating feature that complex trajectories have to be introduced in order to get non-zero amplitudes for classically forbidden transitions. In principle there should be no problems in t-his extension of ordinary classical mechanics. In praitice howcqer some difficulties such as root searching in complex space and stability problems will have to be overcome. The classical trajectory
method
is foknally
(b)
V,,=
Here V is the intermolecular potential, V,,, is an average potential, i.e., a potential which is independent of the quantum coordinate, q is the total wavefunction, qn an eigenstate to the internal hamiltonian and un the initial velocity in channel n. From either. the method (a) or (b) and the classical trajectory equations:
a more
correct description because of the use of a quantum mechanical treatment of a,part of the scattering : problem. Here however it has turned out that the coupling between the classical and the qu.antum degrees of kkedom is rather involved [3] _ Thus we have to solve a boundary value problem in order to :, determine “the classic.al trajecto,ry”. This can apparently not always be done because the iterative procedure used to obtain the trajectory’will fail to converge in some cases [2,4]. . ,I We have therefore been interested in alternative -.,’ .’
the transition probabiIitiesP,, can be obtained as P nm = lc,(~)l* wherkc,,,(--)=6,&. In eq.(l) En is the internal energy of state tz and the total wavc-
function has been expanded as 9 = zrr cn’p,.,. Note in mCthpd (a) we have to use a different trajector$ for every P,m . In ref. [2] ‘we suggested that method (b):&ould be used at high energies. In this
,that
limit the average potential (Ik(Vle> should be useful. ‘Ibis~follows from the use of Ehrenfest’s theorem and ,-.
..
‘.
CHEMICAL
V+rn~ 30, number 3
PHYSICS LElTERS
1 February 1975
.,
.A
[2.,7]. In refs. [2,8] w.e have mainly,investigated the results of ;I first-order treatment of the quantum transition (her: vibrational transitions). kthis paper we give some .falues qbtained by the methods (a) and (c) using an nth order expansion .of the tot.4 Gavefunction, i.e., L?e probabilities listed here are the converged semi-classical probabilities. Since the onedimensionill atom-diatom collision problem is thoroughly investigated we have chosen dthissystem for the numerical kalculations. Secrest and Johnson [9] ahaveperformed exact quantum mechanical calculations on eight different systems. These systems can roughly be divided !nto tw,o subgroups, consisting of those having either small 0; large reduced mass. In ref. [S] we investigated a system from each group, namely no. I (m z(3.5, %r = 0.114) and 7,(m = 514, aySJ 7 O-2973), wllere
has been inves$gat&d @ somk detail by:I’horson et & [5]. The reason for suggesting method (a) at lower energies is that’this choice of average trajectory has turned out to give excellent results for curve crossing. pr?,bletis with a linear potential &Td exponential interaction 161. At energies close to threshold this symtietrization will furthermore give better results tbti the one nom-idly used in semi-classical theories, namelyd=j(vi +.fJ&. It is now natural and cqnvenient to combine the methods (a) and (b) to method(c): V, = (*I v.1G4 (aveiage potential) , and u= [u, urn)!‘* .(effective velocity for the tramition
.
n +m arm -+nj.
This method will correspond to (b) at high energies (un = urn) a+ to (aj at low energies (because we bve Vav -_”Ic,12(1p,I~,iQ x (~~\vI~pn>in both cases). Note also that method (c) obeys the principle of detailed palance and comerves enerm along the aver*e trajectory. -The reason for dealing with this semi-classical approach @that an extension of this theory to threedirnension~ calculatidns of atom-diatom and diatomdiatom collision problems is rather Straightforward
M = mB
772~ ,I(mg
+ nzc) ,
k, is the force constant for BC and ~1is the reduced mass of the system A + BC.
Tible'l Probabilities for single quantum transitions obtained by different theories. The last column gives the power, i-k., the numbers in the first row should be multiplied by lo-‘. The total enero E is given ti units of the ground state energy $hr~ System
Trarhition
E
.l
7.6 8.8
-;
o-1 O-l 1-2 .O,l l-2 o-1 ’ 2 .x1 l-2 o-1 O-l l-2
lO.@ 12.0
‘:
16.0 ‘.
. 4.9Js.5 6.9455
7
8;?455
..’ : .’
‘.;6.7882 _‘.. : 7 .;-‘. ,392 ; :
4.30 2.03 2.23 6.58 1.52
4.90 2.36
. ,,.
: ,.
.‘.-
.: :.
.~..
ref. [12]
3.38 2.26
1.86
6.04
_’ : ,.
,. ..
,,, :-
:
Miller ref. [IO]
ITFITS
(c)
7.85
1:53 5.97. : 1’ 1.97 .. 2.37
._
-:‘: ‘..‘._
(a)
?.SS 1.32 1.92 1.66 1.12 2.93 2.30
0,-l l-2 .G’-1 l-2
-
Exact ref. [9]
3.36. ‘, 3.40 :
-.. ..’
Power
LMXCUS
ref. [Ill
4.01 1:YJ 2.02 6.39 1.44
4.45 2.09 2.30 6.81 1.57
4.69
-5
2.52
-5 .- 4
2.91 l.‘iS 1.89 1.65 0.87 2.64 1.70
2.94 1.36 1.97 1.71 2.35 ‘. 6.30’. 4.69 ”
3.03
3.32 125 3.30 3.40
1.36 ‘5.24 ,- 1.80 2.66
-4 -4
1.22..
-:
: 2.51 ..
.. ;
:--
‘:
-3 -3 -2 -2 -4 -3. -4
:
,:
6.50
:.‘:_ :
;
-2 :-3 -1 -j.
-.: .. .
:’ .,
: /
1 ..
.,
::. :
,‘, ..,
; -,
,,,, . .
i ..
,, .:
;.
... .;
..., ..,,
Volume 30, rwnbor
3
CHEhiICAL
Table 2. The vibrational transition &obabilities System
E
Exact ref.’ [9]
PHYSICS LETTERS
Pm o’,cained by different theories
(a)
(cl
2. Results In tables 1 and 2 we have compared the numerical results obtained by the methods (a).and (c) for the systems 1 and 7 with the exact results of Secrest and Johnson [9], the semi-classical values obtained by Miller and George [IO], Marcus and Stine [l I] and ~ues’obt~ed by the ITFITS method introduced by Heidrich et aL[12]. We see that the resuultsobtained by method (c) are somewhat better than both method (a) and the ITFITS theory and almost as good as the values obtained by lMiIler and Marcus (exc+,foT some of the PO2 values), Especially for system 7 we get an improvement by using method (c) instead of (a). The reason for this that for this system we’get a large distortion of the trajectory 1 during the collision by using the time-dependent average potential instead of the “static” potential used in method (a), We also note t&t it is only for’ the very small probabilities that we get a relatively large devialion from the exact results (lo-30% for single qua&m tran&ons’and as much as a factor of two for double quaturn transitions).
.-
The r~la~iveiy’go~d agriement &ith the exact re.sults bbtained by using method (c) is rather encouraging. We think that this method is able to @edi& t&e iritegr;al’cross tiections to within at least a factor qf two: Be&use of this and because of its simplicity .:
1 February 1975
ITFITS ref. [12]
‘.
Miller
hI2Icus
ref. [IO]
ref. [i l.1
Power
-
we beliecthat &is method is w&h while investigating further. Thus dre plan. to use it to determine bkh the integral and the differential cross sections for combined rotations-~brational transitions in the He-H2 system.
References [i] WH. Miller,
(21 131 141 ES]
J. Chem. Kyis. 53 (1970) 1949,3578; Chem. Phys. Letters 7 (1970) 431; I. (Them. Phys. 54 (1971) 5386; RA. Marcus, Chem. Phys. Letters 7 (1970) 52.5; I. Chem. Phys. 54 (1971) 3965; J.N.L. Connor and R.A. Marcus, I. Cbem. Pirys. 5.5 (1971) 5636; W.H. Wang and R.A. hfucus, I. CTnem. Fhys, 55 (1971) 5663. G.D. Billing, J. C’hem.Phys., to be published. P. Pechukis, Phys. Rev. lSl(1969) 166,174. A.P. Penner and R. Wallace,Phys. Rev. A9 (19741 1136. J-B; Delos, W.B. Thorson and S.K. Knudson, Phys. Rev. A6 (1972).709;
J.B. Defosand
W.B. Thorson.
??hys.
Rev. A6 (1972)
720. [6] A.M. \VooUey and S.E. Nielsen, C&m. whys. titters .21(1973) 491. [7] G.D. Billing,
5658. [ii 1.3. Stine and R.A.
.-
Ma.rcw, Chem. Whys. Letters 15
(1972)536. 1121 FE. H&rich, K.R. W&on znd D. Phys. 54 (1971) 3885.
;
:.
..
,’
Rapp, I:
., .
..
;..
Chem: