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Unification of “post” and “prior” approximations in sudden theories of scattering
Volume
117, number
6
CHEMICAL
PHYSICS
UNIFICATION OF “POST’ AND “PRIOR” IN SUDDEN THEORIES OF SCATTERING E. FICOCELLI Drporrimenro and Cenrro
Received
Chrmrca
Plarmi,
10 March 1985;
The “post” “complelely exact quanlnl
and “prior” energy orf-shell”
Unrversitb CNR,
dr Bari,
Vto Amendala di Barr.
Universird
in linal Iorm 17
Apnl
IL is pointed
T-mati.
Applicalions
calculations_
III sxttering
T-matrix
of Lhe formalism
= ~~,(YI-‘I
(la) (lb)
In (1) Ik,a(*)) represents tion, with the upper, (+) post or prior description For the present problem is given by (in au)
the full scattering wavefuno or <-), signs identifying the of the process, respectively. the full system Hamiltonian
+I&
+ v,
in Lhe hghl of sudden mcchamsms approximatmns
LO a rolatlonal
excilalion
may
problem
be
of atom-molecule
“unified”
are in good
in a single
agreemen
tilh
&lcr,> = onIan)
(3)
and Ik) a plane-wave state, corresponding to an energy
the
Vlki,Lyi) -
lY= -(1/2m)v2
Irafy
jumps
“post” and ‘prior” forms of the T-matrix lead to identical numerical results, when both are applied “exactly” to the collisional problem of interest [l] _ It follows from this that, Xhen approximations to the dynamics are used, it is very likely that imposing the equality of the two forms may lead to computational schemes that will behave more satisfactorily, from a numerical point _ _ - of __view. We wish to apply. the above general considerations to vii-rotational processes in atom-molecule collisions, for which the post and prior T-matrices can be wntten as Tfi = (kf, afl Wkj*(Y$+‘) 9
Ban,
Iraly
are analyzed
to scattering that
70126
OUL the1 the corresponding
also for multiquanlum
theory
I73.
70126 Barr.
1985
rorms ol Ihe scatlering
transfer.
I_ Post and prior sudden approximations It is well known
APPROXlMATIONS
VARRACCHIO
di Chimrca.
vib-rotauonal
5 July 1985
LETTERS
(2)
with the Iru>target states obeying the eigenvalue equation 0 009-2614/85/$03.30 0 JZlsevier Science Publishers B.V. (North-Holland Physiin Publishing Division)
E = k2/2m. A transition to a “sudden” regime can be made by
suitably approximating either of the scattertiE wave functions in (1). Such collisional schemes have been extensively investigated in the last few years; we refer the rtader to review articles [Z] for a list of papers on the SubJect. In particular, it has been pointed out by us [3] that, starting from the post scattering form (1 a), a GIOS (generalized infinite order sudden) approximation to the dynamics can be recovered in the form
Te E (IYfleXp(Ci)P”“(kfiki)l ai) 3
(4)
where the pararnetic dependence of the @(+)-matrix on molecular coordinates has been omitted for simplicity. The sudden amplitude in (4) obeys the “fied
molecule” @(+I
= v+
scattering equation [l] 1
v
Ei + (1/2m)V2
@(+I + isl
-
(5)
It should be stressed that TS(+)(k, ki) (=(krlTs(+)yCi)) is, characteristically, an off-shell T-matrbr element, invohin the energies of-the atom in both the entrance, P 3 = kJ2m, and exit, ef = kt/2m, channels, respeG
tively. Furthermore,
the C’i operator, argument of the 609
Volume
CHEMICAL
117, number 6
exponential Ci = -(Hog
in (4), -
PHYSICS
is defmed by [3 ]
Wi) d/dp
(6)
and it tends to reintroduce the effects of the nuclear motions into the dynamics. While we refer to the originaI papers for more details of the theory, here we point out that the derivation of(4) isessentially based on the trivial identity (a -x)-l
= exp(--x
d/da) a-1
(7)
that can be conveniently applied to the resolvent operator of the coIIision process considered. The approximation (4) has been obtained starting from the post colhsional form of the full T-matrix (la). if a sinular procedure is applied to the prior form, (lb), it can be readily shown that the following alternative approximation to the dynamics will be obtained Tc 2: ((YflT’(-‘(kf,
ki) exp (&I
I (Y,) _
(8)
Ts(-) amplitude represents the prior form to the sudden T-matrix and it is explicitly defined through the integral equation [l]
The
Ts(-I
= y+
1
Ts(-)
et + ( 1/2m)V2
+ ig
I/.
(9)
Once agam the sudden matrix element needed in (8) is in thz form of an off-shell amplitude, TS(-I(kr,ki), while C, is defined, similarly to (6). as zf = -&
-
of)
alder,
(10)
where the arrow denotes that this operator is acting to the left. Eqs (4) and (8) represent, in the GIOS picture, the two approxunations to the dynamics that we must now consider in a unified scheme.
2. The unified
formalism
From a comparison of (4) and (8) it is evident that the essential aspects of the post and prior sudden forms are contained into the TS(+) and Ts(-) scattering matrices that, as(5) and (9) indicate, wiII generally exhibit different numerical behaviour. In this regard, it is interesting to note that the initial projectile energy, E,, appears in the propagator of(S), whereas the final scattering energy, lf, is present inthe correspond610
5 July 1985
LEITEFG
ing quantity (9). This may cause widely differing behaviour of the amplitudes, particularly for multiquantum jumps, as is already well known within the context of the conventlanaI (on-shell) infinite order sudden (10s) approximation [2] _ Up to this point there is no reason to prefer one of the two forms, (4) or (8), over the other. We can refer back to scattering theory [l] end consider that, along with the post and prior T-matrices, one can also introduce a so called “completely off-shell” transition amplitude, that we shall denote as p(e), integral equation TS(e) = v+
v
I E + (1/2m)V72
+
Ts(c)
obeying the
_
(11)
1p
characteristic of (11) is that l, the energy in the denominator, is completely arbitrary and then generally different from both ci and lf_ We can then use such arbitrariness to reduce both the post and prior approximations to the same anaIytica.l form In other words, we can determine “that particular” energy of the projectile, say ?, such that The
exp(CJ
T’(+) = T’(-)
exp(Ef)
c Ts(T)
_
(12)
The determination of E in (12) is tantamount to saying that nuclear motion effects have been incorporated, in a highly non-perturbative fashion, into the structure of the sudden TS(iZ) amplitude. It is now very simple to show that the identrties in (12) can be established b,y approximating flM * $(wi + wf), in bothciand C,(see(6)and(lO)) andusingtheidentity (7) to absorb the ensuing exponentials into the structure of Ts(+) and Ts(-), respectively_ Performing the above simple manipulations reduces both collisional approximations to the same completely off-shell T-matrix, corresponding to an energy F = i(ei + Q)_ The final result of the present theory can then be summarized by saying that the expression TV ~(~fIT~(~f,ki;~(~i
+ Ef))l(Yi)
(13)
should represent, from a formal point of view, the most satisfactory approach to the description of a sudden dynamics. In fact, it respects intrinsic symAExtrieS of s~dtit%hg theCUy, SiIEe it iS that UIIiqUC form of the T-matrix, common to both the post and prior sudden approximations_ The Ts amplitude in (I 3) is a completely off-shell
CHEMICAL PHYSICS LEITERS
Volume 117. number 6
T-matrix, e-g, a quantity that cannot be obtained directly in terms of conventional scattetig wave functions_ In section 3 we outline a technique for its evaluatton and present preliminary numerical results for the He-H2 rotational excitation problem.
3. Numerical results and discussion We consider explicitly the He-H2 system, for which accurate quantal results are available m the literature [4--61. The interaction potential for this system is well represented by the two-term expansion Vr3-Y) = V,(r) + V,(r) &(a=
(14)
r) ,
with r the drstance of the atom from the molecular center of mass, 7 the angle between this and the molecular axis and the Vn and V2 components are exponential fits to the Krauss-Mies potential [7,8]. We have performed calculations for all three amplitudes below: Tfi 2: (or] TS(+‘(kf, ki) I Cui)I
(154
Tfi 21 (OZflTs’-‘(kf,
Cl=)
kl) I ai> 3
T~2:(~~lTS(kf,ki;~(Ei+
~f))lO!~)3
ow
where (15a) and (15b) derive from (4) a$ (S), respectively, by enforcing exp(CJ 2: exp(C,) z 1. Results based on (15a)-(1%) will be referred to, in the following, as the post, prior and unified sudden formalisms, respechvely_ The essential numerical step of the present calcuIations corresponds to solv-
5 July 1985
mg integral equations of the form T(Jrf, ki;
l) = V(kf> ki) V(kfa k’)T(k’,
+2m
s
ak’
ki; E)
k2-k2+iq
.
U6)
The str~ctture of this equation is common to all amplitudes appearing in (15). In fact, using k2/2m = 5 in the denominator of (16), leads to the Ts(+) amplitude (5) while k2/2m =$(cq + q) gives the Ts completely off-shell T-matrix (11). Finally, the choice k2/2m = eF, and interchange of the Tand V terms under the integral sign in (16) corresponds to the structure of Ts(-) in (9) AU the amplitudes of mterest, therefore, readily follow from (16) essentially by a specification of the k2/2m energy in the denom iuator. This equation can be solved using conventional partial-wave expansion techniques [l] and by suitably discretizing the ensuing dk integration, as we have recently done for similar e--molecule rotational excitation problems [9,10]. We refer to these papers for details of the computations and proceed directly to an analysis of the results based on (1 fS))_ We list, in table 1, 0 + 2 rotational excitation total cross sections, in different computational schemes The results of exact CC theory appear in column two, while those obtained from (15) are lrsted in columns three to five. Finally we also present, in the last column, for comparison, values of the 10s approximation from the literature [l 11. It can be seen from table 1, that while the post and prior schemes grossly overestimate and underestimate the total cross sections,
Table 1 Total cross sections (A’) for the 0 +Z rotational excitation in different approximations- close coupling (CC) results are taken from rafs 14.61; sudden values, rn o~hnnns three to five, are based on eqs. (UC). (Ma) and (i5b1, respectively (sea text); IOS are the infinite order sudden values
Energy
cc
unified
formalism
(ev) 0.1 0.9 15 2.1 3.0 4.2
2.05(-l) 3.33 4.13 4.48 4.61 4.5 1
al
6.29(-l) 3.89 4.20 3.97 3.86 3.25
Post
RiOI
10s
3.91 7.39 7.00 6.22 5.49 4.40
2 60(-2) l-87 2.42 261 2.64 2.38
1.27 406 4.35 4.24 3.89 3.47
=I 2.05 x 10-l. 611
Volume 117, number 6
CHEMICALPHYSICS
respectively, the results of the unified sudden formulation, in the third column, are in close agreement with the exact CC calculations. A comparison with IOS seems to indicate, besides, that the formalism based on (1%) is in better agreement with the exact values, particularly at threshold of the transition considered. To investigate Furtherthis point, we have calculated total cross sections for 0 +jf transitions, with jr up to 8, at the mcident energies tzr= 1,2 and 0.65 eV, respectively-.The correspondingresults aTeplotted in fii_ 1, where the values of the present theory are compared both to CS calculations and to IOS results
0
-I
\ .
-2
-3
\
b I-
O-
-l-
-2
LElI-EFtS
(which, incidentally, are automatically ob_tainedfrom a solution of the T-matrix equations (16)). Itshould be noted that both the IOS and the unified sudden formalism give identical Tesults for the 0 + 0 elastic process, as simple energy considerations readily show. The corresponding curves tend to separate, as soon as inelastic transitions are considered. In particular, while the unified formalism follows rather closely the CS values of fig. 1 (crosses), the IOS scheme shows deviations that may become rather large (up to two orders of magnitude) when higher values ofjf are considered To conclude such preliminary analysis of numerical data, we simply mention that, also for these multiquanta transitions, the results based on the Ts(* and Ts(-) amplitudes, eqs (15a) and (1 Sb), respectively, behave rather poorly_ These usually tend,to bracket the values of the unified formalism, based on (15~) and, therefore, have not been plotted in fig. 1. In summary, it would seem, from the present prehminary application% that restoring the basic symmetry of the post and prior approximations may lead to an improved description of the sudden dynamics of atom-molecule collisions. The corresponding unified formalism emphasizes the role of “completely off-shell” T-matrices and a more detailed study of such quantities may be performed, along the lines set out in this paper Off-shell T-amplitudes have started to make their appearance in sudden models of dynamical processes during the last few years [3,12] ; it is likely that increased attention will be paid to techmques of calculation for such quantities. In this regard it is worth pointing out that semiclassical approximations to off-shell T-matrix elements (in the spirit of WKB schemes for on&t% phase shifts) have recently appeared [ 131 and their usefulness for numerical implementation may be tested in the near future.
-
References
-at
[l]
[2] -
Fig 1 TotaI cross sections (A2) for 0 +ifrotational excita ticns at incident eneees l1 = I_2 eV, (a), and ei = 0.65 eV, (b). respectively: are the results of the umtied formal&n, eq. (15~); --are the values of the IOS (infimte order sudden) approxumation; X are the coupled state (CS) results of refs. 14.51.
612
5 July 1985
CJ. JoacU Quantum ~llision theory (North-Holland, Amsterdam, 1975). D.J. Kouri, in: Atom-molecule collisiontheory, ed. RB. Bernstein (Plenum Press, New York, 1979) p_ 301; T_ MuUoney and G-C SchaQ. Chern Phys 45 (1980) 213; E-F. Jendrek and M.H. Alexander, J. Chcm Phys 72 _ (1980) 6452; M. Baer, Ad& Chern Phys 49 (1982) 191; J. JeILinek and M Baer, J. Chem. Phys. 78 (1983) 4494.
Volume 117, number 6
CHEMICAL
[3] E. Ficncelli Varracchio. Chem Phys Letters I33 (1981) 324. [i] P_ Mffiuire and D.J_ Kouri, J. Chem Phys 60 (1974) 2488. [5] P. McGtie, Chem Phys 8 (1975) 231. [6] P. McGuire, J. Chem Phyr 62 (1975) 525. [7] h¶_Kzauss and F. Mies, J. Chem. Phys 42 (1965) 2703. [13] P. Mffiurre and D.A Micha, Intern. J. Quantum Chem 6 (1972) 111. [9] E. Ficocelh Vanacchio and UT_ Lamanna_ Chem Phys Letters lOl(1983) 38.
PHYSICS LEITERS
5 July 1965
[ 10 ] E. FicxxelIi Vanaccbia and UT_ Lamanna_ J. Phyr El7 (1984) 4395. [ll] G.PfefferandD.Secrest,J.ChemPhys67(1977)1394. [ 121 RB. Gerber, L.H. Beard and D.J. Kouri, J. Chem Phys 74 (1981) 4709, L-H. Beard and D J. Koui, J. Chem Phys 78 (1983) 220; L. Eno, J. Chem Pbys 80 (1984) 4196. [13] K Burnett and hl Bekley, Phyr Rev. A28 (1983) 3291.
613
117, number
6
CHEMICAL
PHYSICS
UNIFICATION OF “POST’ AND “PRIOR” IN SUDDEN THEORIES OF SCATTERING E. FICOCELLI Drporrimenro and Cenrro
Received
Chrmrca
Plarmi,
10 March 1985;
The “post” “complelely exact quanlnl
and “prior” energy orf-shell”
Unrversitb CNR,
dr Bari,
Vto Amendala di Barr.
Universird
in linal Iorm 17
Apnl
IL is pointed
T-mati.
Applicalions
calculations_
III sxttering
T-matrix
of Lhe formalism
= ~~,(YI-‘I
(la) (lb)
In (1) Ik,a(*)) represents tion, with the upper, (+) post or prior description For the present problem is given by (in au)
the full scattering wavefuno or <-), signs identifying the of the process, respectively. the full system Hamiltonian
+I&
+ v,
in Lhe hghl of sudden mcchamsms approximatmns
LO a rolatlonal
excilalion
may
problem
be
of atom-molecule
“unified”
are in good
in a single
agreemen
tilh
&lcr,> = onIan)
(3)
and Ik) a plane-wave state, corresponding to an energy
the
Vlki,Lyi) -
lY= -(1/2m)v2
Irafy
jumps
“post” and ‘prior” forms of the T-matrix lead to identical numerical results, when both are applied “exactly” to the collisional problem of interest [l] _ It follows from this that, Xhen approximations to the dynamics are used, it is very likely that imposing the equality of the two forms may lead to computational schemes that will behave more satisfactorily, from a numerical point _ _ - of __view. We wish to apply. the above general considerations to vii-rotational processes in atom-molecule collisions, for which the post and prior T-matrices can be wntten as Tfi = (kf, afl Wkj*(Y$+‘) 9
Ban,
Iraly
are analyzed
to scattering that
70126
OUL the1 the corresponding
also for multiquanlum
theory
I73.
70126 Barr.
1985
rorms ol Ihe scatlering
transfer.
I_ Post and prior sudden approximations It is well known
APPROXlMATIONS
VARRACCHIO
di Chimrca.
vib-rotauonal
5 July 1985
LETTERS
(2)
with the Iru>target states obeying the eigenvalue equation 0 009-2614/85/$03.30 0 JZlsevier Science Publishers B.V. (North-Holland Physiin Publishing Division)
E = k2/2m. A transition to a “sudden” regime can be made by
suitably approximating either of the scattertiE wave functions in (1). Such collisional schemes have been extensively investigated in the last few years; we refer the rtader to review articles [Z] for a list of papers on the SubJect. In particular, it has been pointed out by us [3] that, starting from the post scattering form (1 a), a GIOS (generalized infinite order sudden) approximation to the dynamics can be recovered in the form
Te E (IYfleXp(Ci)P”“(kfiki)l ai) 3
(4)
where the pararnetic dependence of the @(+)-matrix on molecular coordinates has been omitted for simplicity. The sudden amplitude in (4) obeys the “fied
molecule” @(+I
= v+
scattering equation [l] 1
v
Ei + (1/2m)V2
@(+I + isl
-
(5)
It should be stressed that TS(+)(k, ki) (=(krlTs(+)yCi)) is, characteristically, an off-shell T-matrbr element, invohin the energies of-the atom in both the entrance, P 3 = kJ2m, and exit, ef = kt/2m, channels, respeG
tively. Furthermore,
the C’i operator, argument of the 609
Volume
CHEMICAL
117, number 6
exponential Ci = -(Hog
in (4), -
PHYSICS
is defmed by [3 ]
Wi) d/dp
(6)
and it tends to reintroduce the effects of the nuclear motions into the dynamics. While we refer to the originaI papers for more details of the theory, here we point out that the derivation of(4) isessentially based on the trivial identity (a -x)-l
= exp(--x
d/da) a-1
(7)
that can be conveniently applied to the resolvent operator of the coIIision process considered. The approximation (4) has been obtained starting from the post colhsional form of the full T-matrix (la). if a sinular procedure is applied to the prior form, (lb), it can be readily shown that the following alternative approximation to the dynamics will be obtained Tc 2: ((YflT’(-‘(kf,
ki) exp (&I
I (Y,) _
(8)
Ts(-) amplitude represents the prior form to the sudden T-matrix and it is explicitly defined through the integral equation [l]
The
Ts(-I
= y+
1
Ts(-)
et + ( 1/2m)V2
+ ig
I/.
(9)
Once agam the sudden matrix element needed in (8) is in thz form of an off-shell amplitude, TS(-I(kr,ki), while C, is defined, similarly to (6). as zf = -&
-
of)
alder,
(10)
where the arrow denotes that this operator is acting to the left. Eqs (4) and (8) represent, in the GIOS picture, the two approxunations to the dynamics that we must now consider in a unified scheme.
2. The unified
formalism
From a comparison of (4) and (8) it is evident that the essential aspects of the post and prior sudden forms are contained into the TS(+) and Ts(-) scattering matrices that, as(5) and (9) indicate, wiII generally exhibit different numerical behaviour. In this regard, it is interesting to note that the initial projectile energy, E,, appears in the propagator of(S), whereas the final scattering energy, lf, is present inthe correspond610
5 July 1985
LEITEFG
ing quantity (9). This may cause widely differing behaviour of the amplitudes, particularly for multiquantum jumps, as is already well known within the context of the conventlanaI (on-shell) infinite order sudden (10s) approximation [2] _ Up to this point there is no reason to prefer one of the two forms, (4) or (8), over the other. We can refer back to scattering theory [l] end consider that, along with the post and prior T-matrices, one can also introduce a so called “completely off-shell” transition amplitude, that we shall denote as p(e), integral equation TS(e) = v+
v
I E + (1/2m)V72
+
Ts(c)
obeying the
_
(11)
1p
characteristic of (11) is that l, the energy in the denominator, is completely arbitrary and then generally different from both ci and lf_ We can then use such arbitrariness to reduce both the post and prior approximations to the same anaIytica.l form In other words, we can determine “that particular” energy of the projectile, say ?, such that The
exp(CJ
T’(+) = T’(-)
exp(Ef)
c Ts(T)
_
(12)
The determination of E in (12) is tantamount to saying that nuclear motion effects have been incorporated, in a highly non-perturbative fashion, into the structure of the sudden TS(iZ) amplitude. It is now very simple to show that the identrties in (12) can be established b,y approximating flM * $(wi + wf), in bothciand C,(see(6)and(lO)) andusingtheidentity (7) to absorb the ensuing exponentials into the structure of Ts(+) and Ts(-), respectively_ Performing the above simple manipulations reduces both collisional approximations to the same completely off-shell T-matrix, corresponding to an energy F = i(ei + Q)_ The final result of the present theory can then be summarized by saying that the expression TV ~(~fIT~(~f,ki;~(~i
+ Ef))l(Yi)
(13)
should represent, from a formal point of view, the most satisfactory approach to the description of a sudden dynamics. In fact, it respects intrinsic symAExtrieS of s~dtit%hg theCUy, SiIEe it iS that UIIiqUC form of the T-matrix, common to both the post and prior sudden approximations_ The Ts amplitude in (I 3) is a completely off-shell
CHEMICAL PHYSICS LEITERS
Volume 117. number 6
T-matrix, e-g, a quantity that cannot be obtained directly in terms of conventional scattetig wave functions_ In section 3 we outline a technique for its evaluatton and present preliminary numerical results for the He-H2 rotational excitation problem.
3. Numerical results and discussion We consider explicitly the He-H2 system, for which accurate quantal results are available m the literature [4--61. The interaction potential for this system is well represented by the two-term expansion Vr3-Y) = V,(r) + V,(r) &(a=
(14)
r) ,
with r the drstance of the atom from the molecular center of mass, 7 the angle between this and the molecular axis and the Vn and V2 components are exponential fits to the Krauss-Mies potential [7,8]. We have performed calculations for all three amplitudes below: Tfi 2: (or] TS(+‘(kf, ki) I Cui)I
(154
Tfi 21 (OZflTs’-‘(kf,
Cl=)
kl) I ai> 3
T~2:(~~lTS(kf,ki;~(Ei+
~f))lO!~)3
ow
where (15a) and (15b) derive from (4) a$ (S), respectively, by enforcing exp(CJ 2: exp(C,) z 1. Results based on (15a)-(1%) will be referred to, in the following, as the post, prior and unified sudden formalisms, respechvely_ The essential numerical step of the present calcuIations corresponds to solv-
5 July 1985
mg integral equations of the form T(Jrf, ki;
l) = V(kf> ki) V(kfa k’)T(k’,
+2m
s
ak’
ki; E)
k2-k2+iq
.
U6)
The str~ctture of this equation is common to all amplitudes appearing in (15). In fact, using k2/2m = 5 in the denominator of (16), leads to the Ts(+) amplitude (5) while k2/2m =$(cq + q) gives the Ts completely off-shell T-matrix (11). Finally, the choice k2/2m = eF, and interchange of the Tand V terms under the integral sign in (16) corresponds to the structure of Ts(-) in (9) AU the amplitudes of mterest, therefore, readily follow from (16) essentially by a specification of the k2/2m energy in the denom iuator. This equation can be solved using conventional partial-wave expansion techniques [l] and by suitably discretizing the ensuing dk integration, as we have recently done for similar e--molecule rotational excitation problems [9,10]. We refer to these papers for details of the computations and proceed directly to an analysis of the results based on (1 fS))_ We list, in table 1, 0 + 2 rotational excitation total cross sections, in different computational schemes The results of exact CC theory appear in column two, while those obtained from (15) are lrsted in columns three to five. Finally we also present, in the last column, for comparison, values of the 10s approximation from the literature [l 11. It can be seen from table 1, that while the post and prior schemes grossly overestimate and underestimate the total cross sections,
Table 1 Total cross sections (A’) for the 0 +Z rotational excitation in different approximations- close coupling (CC) results are taken from rafs 14.61; sudden values, rn o~hnnns three to five, are based on eqs. (UC). (Ma) and (i5b1, respectively (sea text); IOS are the infinite order sudden values
Energy
cc
unified
formalism
(ev) 0.1 0.9 15 2.1 3.0 4.2
2.05(-l) 3.33 4.13 4.48 4.61 4.5 1
al
6.29(-l) 3.89 4.20 3.97 3.86 3.25
Post
RiOI
10s
3.91 7.39 7.00 6.22 5.49 4.40
2 60(-2) l-87 2.42 261 2.64 2.38
1.27 406 4.35 4.24 3.89 3.47
=I 2.05 x 10-l. 611
Volume 117, number 6
CHEMICALPHYSICS
respectively, the results of the unified sudden formulation, in the third column, are in close agreement with the exact CC calculations. A comparison with IOS seems to indicate, besides, that the formalism based on (1%) is in better agreement with the exact values, particularly at threshold of the transition considered. To investigate Furtherthis point, we have calculated total cross sections for 0 +jf transitions, with jr up to 8, at the mcident energies tzr= 1,2 and 0.65 eV, respectively-.The correspondingresults aTeplotted in fii_ 1, where the values of the present theory are compared both to CS calculations and to IOS results
0
-I
\ .
-2
-3
\
b I-
O-
-l-
-2
LElI-EFtS
(which, incidentally, are automatically ob_tainedfrom a solution of the T-matrix equations (16)). Itshould be noted that both the IOS and the unified sudden formalism give identical Tesults for the 0 + 0 elastic process, as simple energy considerations readily show. The corresponding curves tend to separate, as soon as inelastic transitions are considered. In particular, while the unified formalism follows rather closely the CS values of fig. 1 (crosses), the IOS scheme shows deviations that may become rather large (up to two orders of magnitude) when higher values ofjf are considered To conclude such preliminary analysis of numerical data, we simply mention that, also for these multiquanta transitions, the results based on the Ts(* and Ts(-) amplitudes, eqs (15a) and (1 Sb), respectively, behave rather poorly_ These usually tend,to bracket the values of the unified formalism, based on (15~) and, therefore, have not been plotted in fig. 1. In summary, it would seem, from the present prehminary application% that restoring the basic symmetry of the post and prior approximations may lead to an improved description of the sudden dynamics of atom-molecule collisions. The corresponding unified formalism emphasizes the role of “completely off-shell” T-matrices and a more detailed study of such quantities may be performed, along the lines set out in this paper Off-shell T-amplitudes have started to make their appearance in sudden models of dynamical processes during the last few years [3,12] ; it is likely that increased attention will be paid to techmques of calculation for such quantities. In this regard it is worth pointing out that semiclassical approximations to off-shell T-matrix elements (in the spirit of WKB schemes for on&t% phase shifts) have recently appeared [ 131 and their usefulness for numerical implementation may be tested in the near future.
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References
-at
[l]
[2] -
Fig 1 TotaI cross sections (A2) for 0 +ifrotational excita ticns at incident eneees l1 = I_2 eV, (a), and ei = 0.65 eV, (b). respectively: are the results of the umtied formal&n, eq. (15~); --are the values of the IOS (infimte order sudden) approxumation; X are the coupled state (CS) results of refs. 14.51.
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CJ. JoacU Quantum ~llision theory (North-Holland, Amsterdam, 1975). D.J. Kouri, in: Atom-molecule collisiontheory, ed. RB. Bernstein (Plenum Press, New York, 1979) p_ 301; T_ MuUoney and G-C SchaQ. Chern Phys 45 (1980) 213; E-F. Jendrek and M.H. Alexander, J. Chcm Phys 72 _ (1980) 6452; M. Baer, Ad& Chern Phys 49 (1982) 191; J. JeILinek and M Baer, J. Chem. Phys. 78 (1983) 4494.
Volume 117, number 6
CHEMICAL
[3] E. Ficncelli Varracchio. Chem Phys Letters I33 (1981) 324. [i] P_ Mffiuire and D.J_ Kouri, J. Chem Phys 60 (1974) 2488. [5] P. McGtie, Chem Phys 8 (1975) 231. [6] P. McGuire, J. Chem Phyr 62 (1975) 525. [7] h¶_Kzauss and F. Mies, J. Chem. Phys 42 (1965) 2703. [13] P. Mffiurre and D.A Micha, Intern. J. Quantum Chem 6 (1972) 111. [9] E. Ficocelh Vanacchio and UT_ Lamanna_ Chem Phys Letters lOl(1983) 38.
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