Determination of diabatic coupling potentials by the inversion of inelastic
atom–atom scattering data: Case studies for He++Ne and Li+I
Robert Boyd, TakSan Ho, Herschel Rabitz, D. A. Padmavathi, and Manoj K. Mishra
Citation: J. Chem. Phys. 101, 2023 (1994); doi: 10.1063/1.467711
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Determination of diabatic coupling potentials by the inversion of inelastic
atom-atom scattering data: Case studies for He+ +Ne and Li+1
Robert Boyd, Tak-San Ho, and Herschel Rabitz
Department of Chemistry, Princeton University, Princeton, New Jersey 08544
D. A. Padmavathi and Manoj K. Mishra
Department of Chemistry, Indian Institute of Technology, Powai, Bombay 400 076, India
(Received 7 March 1994; accepted 12 April 1994)
A general iterative inversion algorithm based on first-order functional sensitivity analysis and
Tikhonov regularization is extended for the determination of diabatic coupling potentials from
inelastic scattering data. For simplicity, the two-state case is presented here, and it is assumed that
the (diagonal) diabatic potentials are known. "Noisy" and "noise-free" numerically simulated data,
calculated from model potentials for He + + Ne and Li + I, are used to illustrate the method. Various
coupling potential trial functions are used, ranging from a constant multiple of the model to a step
function. For most cases, the important regions of the coupling potential (i.e., those regions which
are most sensitive to the inelastic scattering data, including the region of crossing) are recovered to
high precision within four to seven iterations. Those cases which show a small range for
convergence may indicate a limit of the present algorithm, based solely on first-order functional
derivatives, and the need to extend it to include higher-order terms.
I. INTRODUCTION
Nonadiabatic electronic transitions, which arise from a
breakdown of the Born-Oppenheimer approximation, play
an important role in a wide variety of chemical and physical
phenomena, including spectroscopic processes, charge exchange collisions, surface physics, and some processes in
biological systems. I The nonadiabatic couplings between
electronic and nuclear degrees of freedom naturally play a
central role in the dynamics of such transitions. Unfortunately, it is usually difficult to perform ab initio calculations
to determine the coupling strength between corresponding
potential energy surfaces. In addition to the usual difficulties
encountered when obtaining the electronic quantities and energies to acceptable accuracies, the matrix elements
(alataq il.B> and (alazlaq;l.B> must be evaluated, where qi is
the ith internal nuclear coordinate, for the nonadiabatic coupling between two different electronic states a and f3. The
evaluation of these matrix elements (e.g., by finite difference
methods) is normally computationally expensive, especially
for multichannel cases, and is often plagued by numerical
inaccuracies. 2 Hence, the development of an algorithm which
could construct the coupling potential if the other potential
surfaces were known would be of interest. In this paper, we
propose a method for determining diabatic coupling potentials from inelastic scattering data, using the inversion algorithm of Ho and Rabitz. 3 This method is based on the
Tikhonov regularization procedure for ill-posed problems,
and has recently proven to be quite successful for potential
determination using data from elastic atom-atom scattering, 3 gas-surface scattering, and molecular vibration
spectroscopy.4 It should be noted that a semiclassical inversion method for recovering the coupling potential from inelastic differential cross section data at a fixed energy exists
J. Chern. Phys. 101 (3), 1 August 1994
for the two-state case. However, the algorithm proposed by
Child and Gerbef is limited in its application, e.g., to scattering by potentials which are purely repulsive at distances
less than the curve crossing point.
Here, we consider as case studies the following two-state
curve crossing problems. The first is the He + + Ne system,
which was also chosen by Child and Gerber to illustrate their
inversion algorithm, and for which the (forward) functional
sensitivity densities of the total cross sections were recently
calculated for various energies. 6 The second is the Li + I system, which is representative of charge exchange collisions,
and, hence, requires special attention due to the presence of
the Coulombic interaction. In both cases, the energy is fixed,
and a finite number of simulated, inelastic differential cross
section data points is used to construct the potential. Our
ultimate goal is not only to go beyond the semiclassical treatment in the two-state case, but also to make the inversion
method feasible for general multilevel curve crossing problems. In a separate paper, we plan to extend the method
described here to the recovery of the full diabatic potential
matrix.
The rest of this paper is organized as follows: In Sec.
II A, some of the relevant theory for inelastic scattering is
summarized. It is presented for completeness and clarity, and
serves to introduce the notation used throughout this paper,
since notations and boundary conditions were found to vary
widely in the references cited. In Sec. II B, the forward sensitivity analysis for differential cross sections is given, and in
Sec. II C, the inversion algorithm is presented. In Sec. III,
the results for the He + + Ne system are given and compared
with those of the semiclassical method. In Sec. IV, the results
from the Li + I system are presented, and finally, in Sec. V, a
brief summary with relevant conclusions is given.
0021-9606/94/101 (3)/2023/1 0/$6.00
© 1994 American Institute of Physics
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2023
Boyd et al.: Inversion of atom-atom scattering data
2024
I ][I-iKI ]-1
SI=[I+iK00
00'
II. THEORY
A. Coupled-channel Schrodinger equation
The standard multichannel model, which can be used to
describe the two-state systems of interest here, combined
with a partial wave expansion of the scattering functions,
leads to the following set of close-coupled equations: 7 ,8
(1)
*'
[
2
1=0
J
I
- 8iJPI(cos e),
0,
(12)
0
with =arg r(l + 1 + i 71/ k j)' Thus, for the two-state case of
interest here, accounting for the possibility of a Coulomb
interaction present in the exit channel, 0'12(e) is given by
2
(13)
(4)
B. Forward sensitivity analysis
where I is the identity matrix and
I
1
'"
2 ·(k.k.)112 ~ (21+ 1)[exp(i~;)S;j exp(i~j)
I
/L is the reduced mass of the system, E is the total energy,
and Zie and Z; e are the charges of the species in the ith
channel. In matrix form, the close-coupled equations become
Wij(r) = 8ij k i
fij(e)
(2)
(3)
+ WI(r) ]vI(r) =
(11)
I
where l2
where VICr) is the solution matrix, VCr) is the diabatic potential matrix with coupling potentials VijCr) = VjiCr) (i j),
[I;;
where K~ is the open-open submatrix of KI.
Finally, the differential cross section O'ijCE, e)
=dQijCE)ldO [where Qi/E) denotes the total cross section] from state i to state j is given byll
_ kj
2
O'ij( e) - k. Ifij( e) I ,
1)]
d2
2 271i 1(1+
I
2/L"
I
+ki ------=z- Uil="'il' £oJ VW(r)Vi,/r),
[ -;pI
u.,
r
r
r
I
2 2/L
k i =h'l [E- Vii(oo)],
(10)
0]
271i 1(1+
--;:------;:z-
The first-order response of the differential cross section
2/L
-h'l Vij(r).
(5)
In practice, Eq. (4) is solved, e.g., by virtue of the renormalized Numerov method9 to obtain standing wave solutions.
The asymptotic condition imposed on VICr) is given by
VI(r)- i(r) + NI(r)KI,
(6)
where KI is the reaction matrix, and iCr) and NICr) are
diagonal matrices with elements
Il/r)=8ijkiI/2Fl(71i,kir),
(7a)
l(
Nl/r) = 8ijki 1I2 G TJi ,kir).
(7b)
For open channels, F;C 71i ,kir) and Glc 71i ,kir) are the regular and irregular Coulomb functions,1O respectively. If 71i =0
(non-Coulomb system), F;C 71i ,kir) and G;C 71i ,kir) reduce
to the Ricatti-Bessel and Ricatti-Neumann functions, respectively.
The outgoing (+) and incoming (-) solutions to the
close-coupled radial equations are related to the standing
wave solutions by
(8)
80;/ e) to small functional variations in the potential
8Vi 'j'Cr) can be written as l3
OcTij( e) =
L, .~.,
J
I
fo'" K:~ j'( e,r) 8Vi ,j,(r)dr,
(14)
~I
where the kernel K:~ j' Ce, r), or functional sensitivity density,
is given by
[*
ij
_ 8Uij( e) _ k i
8fij( e) ]
Ki'j'(e,r)= 8V 'j,(r) -k j 2 Refij(e) 8V 'j'(r) ,
i
i
with Re denoting the real part of the argument and
senting the complex conjugate, and where
8fij( e)
8Vi 'j,(r)
(15)
* repre-
1
~
.1
2 ·(k.k-)1I2 £oJ (21+ Oexp(l~i)
I
J
I
8S;j
1=0
I
X 8V 'j,(r) exp(i~)PI(cos e).
(16)
i
The functional sensitivity density 8Si/8Vi ,j'Cr) can be derived by first considering the functional change of the wave
function 8U;!Cr), which obeys the equation
and obey the following asymptotic relation:
(9)
where S;I is an element of the scattering matrix Sl, related to
the reaction matrix KI by
=~
"''' ["
£oJ
Ii
i'
V,'",(r) 0"UI"+'I(r) +"
£oJ 8VW (r)U iI+]
,/r) ,
i'
(17)
and satisfies the boundary condition
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Boyd et at.: Inversion of atom-atom scattering data
2025
+ u~; (r) U\; (r) ]exp(i f,)P ,( 00'
-1/2
li7T
1Ji
)]}
8V 1+
8S 1a exp {.[
I kjr-T- k In(2k i r
.
a (r)- -k j
i
0) ] } ,
(20)
(18)
By left-multiplying the close-coupled equation for utter)
[Eq. (1)] by 8ult (r) and subtracting from it the closecoupled equation for 8ult (r) [Eq. (17)] right-multiplied by
Utt(r), summing over i and integrating over r, and by making use of the asymptotic relations for U;; (r) and 8U;t (r)
[Eqs. (9) and (I8), respectively], 8S~i 8Vij( r) is found to be
given by
8sL
oVji(r)
8S~J
8Vij(r)
ip.
1+
1+
=r;r Va (r)VjJ (r),
(I9a)
8S~J
ip.
1+
1+
8Vjj (r) =~ [U il (r)UjJ (r)
+ U~7(r)Ul;(r)];
ii= j.
(19b)
Hence, for the two-state case we are concerned with here,
with VII(r) and V 22 (r) fixed [8V 11 (r)=0 and 8V22 (r) =0],
the relevant kernel K(O,r)=KnUJ,r) is given by
c.
Regularized iterative inversion
The inversion algorithm described below assumes that
good experimental data for the inelastic differential cross
sections lTI2«(J) are available, and that the (diabatic) potential
functionalities V II (r) and V 22 (r) are known. The procedure
is new for curve crossing, and, in the examples that follow,
simulated data wiII be used as a test. The first step in the
inversion scheme is to choose a reference coupling potential
v?2(r). Next, the close-coupled equations are solved, as described above, to obtain approximate differential cross
sections a?Z<(J) and functional sensitivity densities
K«(J,r)=8u?2«(J)18v?2(r). At this point, it is often necessary to rescale the kernel K( (J,r) by introducing radial W r
and angular w (} weight functions as follows:
(2l+I)S~2 exp(i~~)PI(cos 0)]*
K(O,r)=2;;'k 2 Iml[i
I
where 1m denotes the imaginary part of the argument.
1=0
(21a)
K( (J,r) -+ K( (J,r)w ofw r
xl#' (2l+1)[U\;(r)U~;(r)
oud(J) =
(2Ib)
;
fo"" K«(J,r)8Vdr)w r
(2Ic)
dr.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1.5
I
I
I
I
I
\
I
I
\
\
\
\
I
\
\
\
\
1.0
\
\
\
\
\
'.,
'\"",
--------- -------------------------
..... ' ..............
-----------------
0.5
0.0L-____~_____ L_ _ _ _~_ _ _ _ _ _L_~==~=====k===_~
1.0
1.5
2.0
2.5
3.0
3.5
4.0
____
4.5
~
5.0
r (a.u.)
FIG. 1. The V lI (solid) and V 22 (dashed) potential curves from the Olson-Smith model for He++Ne.
J. Chern. Phys., Vol. 101, No.3, 1 August 1994
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Boyd at al.: Inversion of atom-atom scattering data
2026
0.20
C\I
----~
~
0.15
2I=::
'iii
2-C\I
.....
b
0.10
o
10
30
20
40
50
60
70
80
90
100
110
t? (degrees)
FIG. 2. Inelastic differential cross sections 0"12(8), weighted by sin(8), for the He+ +Ne system, at an energy of 0.919 a.u.
Typical choices for weight functions include W 0= sine 8) and
r2 or W r = e - r. If the asymptotic form of the coupling
potential is known, as is the case for some important classes
of nonadiabatic transitions, 1 the radial weight function W r
may be given that form to ensure proper behavior of the
potential in the asymptotic region. This is particularly desirable when the data being used is relatively insensitive to the
potential Vij(r) at large r.
To ensure that a smooth potential is recovered, a derivative constraint of order n may be imposed on Eq. (21c). This
has the effect of smoothing highly oscillatory kernels which,
when used for inversion, might otherwise give rise to potentials which are physically unreasonable. After integration by
parts n times, Eq. (21c) becomes3
Wr=
00"d8)=
fo'" K[n 1(8,r)D(n)(r)w
r
dr,
(22)
where 0(}"=0"12-d?2' and a is a regularization parameter
which may be optimally determined by minimizing the functional J(a)=110'12-a7~C(a)ll, as described in Ref. 3. An appropriate choice for the value of a is necessary to balance the
data residual IIKln] D(n) - ooW against the smoothness constraint IID(n)11 2•
The least-squares solution of Eq. (24) can be written
explicitly as 3
(25)
where ( I) denotes the inner product, and AT, ui' and v i are
the singular values, vectors, and functions, respectively, of
the Gram matrix M, defined as 3
where
(26)
(23)
K IO](8,r)=K(8,r),
D(n)(r)=rnoV(n)(r),
and
oV(n)(r)=d noV 12 (r)ldr n, and provided that 00 i )(r)-+0 as
r-+ OO for all i up to n. According to the Tikhonov regular-
Once 00 n )(r) has been obtained, oV 12 (r) can be calculated
by a series of n backward-folding integrations:
ization procedure, an approximate solution of Eq. (22) can be
obtained by minimizing the functional <I>(a,D(n» given by3
(27)
with
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Boyd et al.: Inversion of atom-atom scattering data
III. RESULTS AND DISCUSSION FOR He+ +Ne
>-bO
.,...
A. Potential functionalities and forward functional
sensitivity densities
.,c:
::1
IS
::::::
The Olson-Smith model for the He + + Ne system was
used to obtain differential cross sections <Td 0) and functional sensitivity densities K( O,r) in the manner described in
Sees. II A and II B for energies of 0.919 and 2.606 a.u. The
functional forms for the diabatic potentials are given by14
410
ti'
u
c:
205
.!!l
0.0
.!'l
"0
::1
..!!.
~
2027
-205
-410
<;>
..:
~
V 22 (r) = (21.1 r- 1 - 12.1 )e-rIO.678 + 0.617,
FIG. 3. Sensitivity densities &r12 (fJ)18Vdr), weighted by sin(e), for the
He+ +Ne system, at an energy of 0.919 a.u. &rdfJ)18V 12 (r)sin(fJ) is measured in atomic units.
noting that 8V<°)(r) = 8V I2 (r). Finally, a new, improved
coupling potential is obtained by adding 8V I2 (r) to the
original reference coupling potential 0l2(r), and the entire
procedure can be repeated using the sum as the new reference potential.
(28b)
where all values are expressed in atomic units. The diabatic
curves for VII (r) and V 22 (r) are shown in Fig. 1.
As a check for the differential cross sections <TdE,O)
obtained, an integration over 0 was performed to yield total
cross sections Qij(E) at the two energies of interest. The
results agreed with those given in Ref. 6. A plot of
sin(O)<TdE,O) at an energy 0[0.919 a.u. is given in Fig. 2.
Similarly,
the
functional
senSItIvIty
densities
8<TdE,0)18V12(r) were integrated over 0 to obtain
8Q 12 (E) I 8V 12 ( r), and the results were found to be in agreement
with
those
in
Ref.
6.
A
plot
of
sin(O)8<TdE,0)18Vdr) for an energy of 0.919 a.u. is given
in Fig. 3.
3.0
2.5
........
(\/
2.0
I
0
::i
1.5
«l
---tzl
1.0
0.5
---- ----- ----O.O~
1.1
____
-------
_ L_ _ _ _~L__ _ _ _~_ _ _ _~_ _ _ _ _ _~_ _ _ _~_ _ _ _ _ _L __ _ _ __ L_ _ _ _ _ _L_~
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
r (a.u.)
FIG. 4. Model (solid). initial reference (dashed), and recovered (dotted with x) coupling potentials for He+ + Ne, starting from Vi2(r) = 0.5 *V 12 (r).
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Boyd et al.: Inversion of atom-atom scattering data
2028
3.0
~.
2.5
\.
'II,
----
C\l
2.0
I
0
~
1.5
co
*.
---w
1.0
0,5
O,OL-____
1.1
~
____
1.3
~
______
1.5
~
1.7
____
""''''
_ L_ _ _ _ _ _L __ _ _ __ L_ _ _ _~_ _ _ _ _ _~_ _ _ _~_ _~
1.9
2.1
2.3
2,5
2.7
2,9
r (a.u,)
FIG. 5. Inversion results for He++Ne, starting from 012(r)=0.01 a.u. for r.;;5 a.u.; 012(r) =0 for r>5 a.u. Refer to Fig. 4 for fonts.
The sensitivity densities for both energies considered are
quite oscillatory, suggesting that a moderate to high derivative constraint (n =4 or higher) should be used to smooth the
kernel for inversion. The most prominent feature of K(r, e)
at both energies is a large peak centered roughly at e=40 deg
for £=0.919 a.u. and at e=15 deg for £=2.606 a.u. (corresponding approximately to the angular positions of the largest peaks in each respective differential cross section), and at
r=2.02 a.u. for both energies (the crossing point). For
£=0.919 a.u., the magnitude of the sensitivity densities becomes quite small for angles greater than 120 deg, and is
negligible for angles less than approximately 20 deg
[roughly in proportion to the magnitude of ude)]. Hence,
only differential cross sections from approximately fJ= 15 to
e= 120 deg are useful for purposes of inversion. In addition,
the sensitivity densities at all angles for distances less than
r= 1.6 a.u. are nearly zero. Thus, it is expected that the recovered coupling potential may not be accurate for r<1.6
a.u. Similar trends and conclusions apply for E=2.606 a.u.
B. The recovered potential
For the determination of the coupling potential, we focused on three variables: the radial weight function W r [in all
cases, the angular weight function was set to W 0= sin( fJ)], the
order of the derivative constraint n, as defined in Eq. (22),
and the choice of starting reference potential v? 2 ( r). Two
otherwise identical calculations were run at energies of 0.919
,and 2.606 a.u. Since the inversion results were nearly iden-
tical, and since the kernel at the lower energy appeared to
have a richer structure, all subsequent calculations were only
performed for the lower energy. In addition, noise was added
to the numerically simulated cross section data to approximate random errors in laboratory measurements, assuming a
normal distribution, for ±5% at the 95% confidence limit.
The choice of weight factor W r was investigated by numerically testing the two functions W r = r2 and W r = e - r.
With n=4, and v?2(r) = 0.8*V 12 (r), two inversions were
performed. The results were nearly identical, and both accurately reproduced the model potential from r= 1.1 a.u. to
around r=2.7 a.u., showing that the inversion is not very
dependent on choice of W r for this system. However, the
potential recovered using W r= r2 did not reproduce the
model potential as well in the asymptotic region, whereas the
choice of W r = e - r ensures correct asymptotic form. The radial weight function W r = e - r was used in all subsequent
calculations.
The order of the derivative constraint n should be chosen
such that a smooth potential is recovered. However, n should
not be so large as to destroy the geometric shape of the
original kernel K[Ol( O,r), or to unnecessarily add to the
overall computational expense; only the very high-frequency
oscillations in K[Ol( e,r) should be smoothed out. It was
found by numerical testing that a value of n =4 fits the above
criteria nicely. However, two other values of n were also
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2029
Boyd et al.: Inversion of atom-atom scattering data
3.0
2.5 \ \
\
,
'~.
\
\
\
\
\
---.
C\l
I
2.0
0
......
::i
«l
......
,
",,,"
''Ii'"
,,
'
,
1.5
.. ...
.
,
,
"
..
"""
C<l
"""""
""'...
1.0
.....
" " " " "- " ':.:,:.:~: : "' >"'"."-.-,-,~.' ' '".-
0.5
••
O,OL-____- L______L -____
l.1
l.3
l.5
~
____~______~____~______~____-L______L_~
l.7
1.9
2.1
2.3
2.5
2.7
2.9
r (a.u.)
FIG. 6. Inversion results for He++Ne, starting with v?2(r) = O.8*V 12 (r) and using data with up to ~% error for the inversion. Refer to Fig. 4 for fonts.
5%. The recovered differential cross sections (those obtained
by the method described in Sec. I A, using the recovered
potentials) converged within four iterations to within 0.05%
of the model ad8).
tested, n =2 and n =6, and both choices supported successful
inversions.
Four different starting reference potentials v?2(r) were
tested, using W r = e - rand n =4: three with exponential
forms, two smaller [v?2(r) = 0.8*Vdr) and v?2(r)
= 0.5* V I2 (r)] and one larger [v?2(r) = 1.5 *Vdr)] than
the model potential V 12 (r), and one with v?2(r) set equal to
a step function [v?2(r)=0.01 a.u. for r~5 a.u.; v?2(r)=0
for r>5 a.u.]. The results for the second and fourth cases are
shown in Figs. 4 and 5, respectively, and show that the successful recovery of the coupling potential in the region of
crossing is nearly independent of the choice of v?2(r). The
results for the first and third cases are similar to those for the
second, as shown in Fig. 4, and have been omitted for the
sake of brevity. For this system, all four choices of v?2(r)
yield recovered potentials whose absolute values agree well
with the model potential from a range of r = 1.6 to at least
r=2.3 a.u., with the poorest agreement coming, not surprisingly, from the choice v?2(r) set equal to a step function.
Noise was added to the cross sections to simulate experimental errors up to approximately ±5%, and an inversion
was carried out using v?2(r) = 0.8*Vdr), n=4, and
W r = e - r. The recovered potential is shown in Fig. 6, and
shows reasonably good agreement with the model potential,
as before, from around r= 1.6 to r=2.3 a.u.
In all cases described above, the initial guesses for
v?2(r) yielded cross sections dfz<e) that, on average, dif-
The inversion method of Child and Gerber was a semiclassical technique that involved two steps: In the first step,
S~2 was obtained from the differential cross sections O'de),
and in the second step the coupling potential V I2 (r) was
recovered from sb. Despite some severe limitations, the
method proved to be reasonably accurate in recovering the
coupling potential for He + + Ne, with results reported for the
range 2.3~r~1.55 a.u. (similar to the reliable range of the
inversion reported here); the maximum error at any given
point being 5.8%, with the average at 3.1 %.5
Many of the limitations inherent in the semiclassical
method are overcome in the regularized, iterative method
described in this paper: For the semiclassical approximation
made by Child and Gerber to be valid, the situation must
involve at least 60 significant partial waves;5 in the close
coupling approach used here, no such limitation exists. The
restriction to purely repulsive potentials likewise does not
apply to the method of Ho and Rabitz. In addition, the coupling potential recovered by the semiclassical method is only
fered from those produced using Vdr), adO) by 3% to
piecewise continuous, whereas that reported here is fully
C. Comparison with the semiclassical method of
Child and Gerber
J. Chern. Phys., Vol. 101, No.3, 1 August 1994
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Boyd et al.: Inversion of atom-atom scattering data
2030
I
I
I
I
I
I
I
I
I
0.4
I
I
I
,,
I
,
,
,,
,,
,,
,
0.2
,
,
I
0.3
I
I
I
I
I
::i
«i
.....
[Ll
I
I
,,
I
,
0.1
I
I
I
I
I
I
I
,,
,
0.0
--------
I
\
\
/,./..-~..---
,
-0.1
" ..
_
..........-
//
.....
..........
-0.2L-__~____~__~L-__- L____~__- i____~____L -_ _- L_ _ _ _~_ _~
2
4
6
8
10
12
14
16
18
20
22
24
r (a.u.)
FIG. 7. The V II (solid) and V 22 (dashed) potential curves for Li + l.
continuous, and there is no significant improvement in accuracy at the critical region around r=2.02 a.u., the region of
crossing, as is the case in our method.
One possible advantage the semiclassical method possesses is that no initial guess for the potential is used. However, as was demonstrated by the use of a step function,
recovery of the coupling potential is not strongly dependent
on this choice, so long as the functional response to changes
in the initial choice of (J"12(fJ) is largely first order. Also, the
fact that two steps are used in the semiclassical inversion
algorithm may increase computational costs and serve to
magnify any errors initially present or from the first inversion step, as one goes from O"de) to sb to V I2 (r).
In reality, both methods were highly successful in recovering the coupling potential from differential cross sections
for the He + + Ne system. However, the semiclassical method
could only be applied to a handful of other systems. The
ultimate measure of success for the method described here
will be determined by the number and variety of systems to
which it can be applied, and toward that end we now discuss
its application to the Li + I system.
IV. RESULTS AND DISCUSSION FOR Li+l
A. Potential functionalities and forward functional
sensitivity densities
The model potential of Faist and Levine was used to
determine O"12(e) and K( e,r) for the Li+ I system. The functional forms for the diabatic potentials are given bylS
V II (r)=[3100+ (2.996/r)12]e- rlO .44-1191.2/r6, (29a)
V 22 (r) = [1052 + (1. 839/r)8]e -rI0.3786 - 14.3977 / r
-46.5111r 4 - 0.823/r6- 5.3703/r7 + 2.326,
(29b)
Vdr)
= 17 .08e-0.8608r,
(29c)
where r is in A and E is in eV. To minimize the computational expense due to the number of significant partial waves
which must be included, all calculations were run at an energy of 2.336 eV, or just 0.01 eV above the threshold energy.
The diabatic curves for VII (r) and V 22 (r) are shown in
Fig. 7.
As before, O"12(e) and K( e,r) were evaluated in the manner described in Secs. II A and II B. The necessary Coulomb
functions were evaluated by Steed's method, based on
continued-fraction expansions of the quantities F' / F and
(G' + F')/( G + iF), as defined in Eqs. (7a) and (7b).16 The
kernel for this system is highly oscillatory, and shows a
rather large region of sensitivity, from at least r=6 to well
beyond r= 18 a.u. The large size of this region is expected
due to the strength of the coupling in the Li + I system, and is
typical for charge exchange collisions. This region is, as expected, roughly centered around r= 11. 75 a.u., the crossing
point.
J. Chem. Phys., Vol. 101, No.3, 1 August 1994
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Boyd et al.: Inversion of atom-atom scattering data
2031
7
6
5
,
(\)
I
0
,,
4
\~\
::i
ai
--ILl
,.
3
2
"''''',
'-"'-
'"''''
5
6
7
8
9
10
11
12
13
14
15
r (a.u.)
FIG. 8. Inversion results for Li+I, starting with v<i2(r) = 0.95*Vdr). See Fig. 4 for fonts.
B. The recovered potential
which supports many bound states. If the scattering data are
not sufficient in this case, additional information, such as
from spectroscopy, may need to be incorporated into the algorithm in order to obtain a. successful inversion.
Within the narrow window of convergence for choice of
v?z(r), the choices of the order of derivative constraint n
and angular weight function W (J were investigated and found,
as for the He + + Ne system, to be of minimal consequence
for obtaining a successful inversion. The results shown in
Fig. 8 used n=6, w(J=lIuu«(J), and wr=rz. In addition, the
choice W r = r2 was used throughout the above calculations,
in contrast with the choice wr=e- r used for the He++Ne
system.
In attempting to recover the coupling potential, starting
with initial guesses v?z(r) set equal to constants multiplied
by the model V I2 (r), it was found that the algorithm would
only converge if v?2(r) differed from V1z(r) by 5% or
less. The results from one inversion, using v?z(r)
= O.95*V I2 (r), are shown in Fig. 8. As could have been
anticipated, due to the large region of sensitivity for this
system, the recovered coupling potential was in excellent
agreement with the model V I2 (r) over a large range; well
before r=5 to well after r= 15 a.u. (essentially the entire
range where recovery was attempted).
The narrow range of v?z(r) that allowed for successful
recovery of the coupling potential can probably be explained
by considering a?2«(J). For v?2(r) differing from Vdr) by
just 5% [v?2(r) = 0.95*V I2 (r)], the corresponding a?2(fJ)
differed, on average, from ud(J) by more than 4200%! Even
so, the algorithm converged in seven iterations, such that the
recovered differential cross section was within 0.08% of
UI2«(J). Deviations of v?z(r) from V1z(r) which are more
than 5% probably bring the system too far beyond the regime
governed by first-order functional responses to allow for
convergence to an acceptable solution. An alternative explanation may be found by asking whether or not differential
cross section data are sufficient to recover a unique, physically acceptable coupling potential in a system where one or
In this paper, we have shown that the regularized inversion algorithm of Ho and Rabitz can be successfully applied
to recover coupling potentials from differential cross section
information. The results for the He + + Ne system indicate
that the algorithm is at least as successful as the semiclassical
method of Child and Gerber. However, it does not suffer
from the severe limitations of the older method. The successful recovery of V1Z(r) from use of "noisy" data is a positive
indication that this algorithm can be implemented to recover
coupling potentials from good laboratory data. The somewhat qualified success for the Li + I system indicates the ex-
more of the (diagonal) diabatic potentials possess a deep well
istence of a limit to which such an algorithm based solely on
V. SUMMARY
J. Chem. Phys., Vol. 101, No.3, 1 August 1994
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Boyd et al.: Inversion of atom-atom scattering data
2032
first-order functional sensitivity analysis and/or differential
cross section data can be applied. This study was the first
application of this algorithm for inversion from scattering
data to a system with a deep well, in addition to its first
application to a Coulombic system. Such systems often require the inclusion of thousands of partial waves (550 significant partial waves were needed for the Li + I system at
E=2.336 eV), and computational expense becomes an important consideration. Nevertheless, given the importance of
charge exchange collisions, further investigation, such as to
the inclusion of second-order corrections in the determination of 8Vdr), could prove to be worthwhile.
ACKNOWLEDGMENT
This research was supported by the Department of Energy.
I H. Nakamura, Int. Rev. Phys. Chern. 10, 123 (1991).
2F. A. Gianturco, A. Palma, and F. Schneider, Int. 1. Quantum Chern. 37,
729 (1990).
3T.-S. Ho and H. Rabitz, J. Chern. Phys. 89, 5614 (1988); 91, 7590 (1989),
and references therein.
4T._S. Ho and H. Rabitz, 1. Chern. Phys. 94, 2305 (1991); 96, 7092 (1992);
H. Heo, T.-S. Ho, K. K. Lehmann, and H. Rabitz, ibid. 97, 852 (1992).
SM. S. Child and R. B. Gerber, Mol. Phys. 38, 421 (1979).
6D. A. Padmavathi, M. K. Mishra, and H. Rabitz, Phys. Rev. A 48, 279
(1993).
7 A. C. Allison, Adv. At. Mol. Phys. 25, 323 (1988), and references therein.
8See, for example, L. I. Schiff, Quantum Mechanics (MCGraw-Hill, New
York, 1968).
9S. Shi and H. Rabitz, Comput. Phys. Rep. 10, 1 (1989), and references
therein.
lOSee, for example, Handbook of Mathematical Functions, edited by M.
Abramowitz and I. A. Stegun (Dover, New York, 1972).
II See, for example, M. S. Child, Molecular Collision Theory (Academic,
London, 1974).
12M. B. Faist, Ph.D. thesis, Ohio State University, Columbus, OH (1975).
13R. Guzman and H. Rabitz, 1. Chern. Phys. 86, 1395 (1987).
14R. E. Olson and F. T. Smith, Phys. Rev. A 3, 1607 (1971).
ISM. B. Faist and R. D. Levine, J. Chern. Phys. 64, 2953 (1976). The
parameter value of "3100" that appears in Eq. (29a) is a corrected, although arbitrary, number. In the original paper by Faist and Levine, the
parameter reads "lloo," which is obviously in error, since it gives rise to
a rather deep well in V II (r). The value 3100 reproduces the curve as
shown in Fig. 1 of Faist and Levine's paper.
16 A. R. Barnett, D. H. Feng, 1. W. Steed, and L. J. B. Goldfarb, Comput.
Phys. Commun. 8, 377 (1974).
J. Chem. Phys., Vol. 101, No.3, 1 August 1994
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