1. Phys. B: Al. Mol. Opt. Phys. 27 (1994) 1881-1888. printed in the UK
Electronic excitation of HZby positron impact: an application
of the Schwinger multichannel method
J L S Lino?, J S E Germanot and M A P Limaf
t Institute Tecnol6gico da AernnBytica CTA. 12728 S I Campos. SP. Brazil
lnstituto de Fisica 'Gleb Wataghin', UNICAMP, 13081 Camphas, SP,Brazil
Received 28 lune 1993. in final form I I February 1994
Abstract. In this paper we report integral.and differential cross sections for positron-impact
--t B'E? transition in H2 for collision energies of 13.5. 15. 17.5,
exciwion of the X'E:
20, 25 and 30 eV. These cross sections were derived from fixed-nuclei scanering amplitudes
in such a way thal low angular-momentum components are obrained using the Schwinger
multichannel method and the large ones are Ireated by the firs1 Born appmximation. To our
knowledge, up to date no experimental data are available for comparison. Our results differ
from previous theoretical results of Mukhejee and co-workers. We discuss possible reasons for
this disagreement
1. Introduction
In the last few years, there has been increasing theoretical activity concerning positronmolecule scattering (Armour er al 1990) Most of these calculations are for elastic processes
and are made with model polarization potentials. Good examples of these initiatives are
found in recent publications by Jain and Gianturco (1991) for e+<& and Gibson (1992) for
et-Hz scattering. In both cases low-energy elastic integral and differential cross section are
calculated. The first ab initio calculations were presented by Armour (1985) and Tennyson
(1986) for elastic e+-Hz using the Kohn variational and R-matrix methods, repectively.
More recently, German0 and Lima (1993) adapted the Schwinger multichannel method
(SMC) for positron scattering and also presented ab initio results for Hz and C& molecules.
In contrast to the theoretical case, most of the experimental data are from attenuatedbeam experiments and consequently restricted to total cross sections. Recent technological
developments allowing steady production of more intensive positron beams are triggering
more challenging experiments that may furnish integral and differential cross sections for
elastic and electronically inelastic positron-molecule scattering processes. Currently, for
electronic excitation cross sections by positron impact, the only theoretical results are those
of Mukhejee eta1 (1991). They have studied the X'Z; -+ BIZ: transition of Hz using
a close-coupling approximation. In this work we also present results for thii excitation.
Our calculations cover the impact energy range of 13.5 to 30 eV and were obtained with a
theory which combines the SMC and the first Born approximation (FBA). We will refer to this
combined method as the BSMC. The SMC was introduced some time ago to study electronmolecule scattering problems (Takatsuka and McKoy 1981) and has been extensively applied
to several molecular systems (see for example Winstead et al 1992, and references therein).
This first application for electronic excitation of molecules by positron impact has
several objectives. First, to establish benchmark calculations at the two-state level of
0953-4075/94/091881+08$19.50 @ 1994 IOP Publishing Ltd
1881
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J L S Lino et a1
approximation (benchmark here does not necessarily mean exact results or good agreement
with experimental data-just what should be expected at this level of approximation). This
sort of calculation is indispensable in the development of new methodologies in this area
As a matter of fact, this is a natural fust step towards a multichannel study involving
other 'competing' channels (any other open electronic state, including the ionization
and positronium-formation channels). Furthermore, in order to assess the importance of
polarization effects, which, in principle, are not expected to substantially affect the electronic
excitation cross sections at high energies, a two-state calculation is also necessw. It is
important to note that we cannot say beforehand which channel will have a bigger influence
on the excitation cross sections. In addition to the dipole allowed transitions and a few other
hints given hy pertubation theory, there are no simple rules to predict such an influence.
As a result it is important to know what effect each of these competing channels will
cause on the cross sections. To be convinced that we are not putting in all effects at the
same time and fortuitously getting the correct results we must start from the two-state level
of approximation. Electronic multichannel coupling (competition among bound electronic
states) which is difficult to establish in electron-molecule collisions is easier to implement
in the positron-molecule case (a good example of a possible simplification is that only
molecular singlet states can be coupled in scattering against a closed-shell target initially
in its ground state). So, as a second objective, it is important to discuss differences and
similarities with the electron-molecule excitation process in which exchange interactions
must be accounted for and are not needed in positron scattering. Finally, our results can
motivate experimentalists to look into these electronic excitation cross sections to better
establish the importance of multichannel effects as well as ionization and positroniumformation Channel couplings in these processes.
The essence of the BSMC rests in the fact that for dipoleallowed transitions such as
the X'Z: + B 'E$ in Ha, the long-range character of the dipolar coupling requires
a large number of partial waves for a proper convergence of the cross sections in the
forward direction. If a full multichannel treatment is used to calculate these excitation cross
sections, the need to include a large number of partial waves can add considerably to the
computational effort. Such an effort can be avoided if above a certain angular momentum
'e', the remaining partial waves, which are weakly scattered, are obtained from a weakcollision theory such as the FBA. In the following section we present a short review of the
SMC for positron scattering and the present scheme to complete the scattering amplitude
with FBA. Finally, in the last two sections we present and discuss the results of the present
application.
2. Theory
The method is essentialy an extension of the Schwinger multichannel formulation electronmolecule vakatsuka and McKoy 1984, Lima and McKoy 1988, and Lima et a1 1990). In the
SMC for positron-molecule scattering (German0 and Lima 1993) the variational expression
for the scattering amplitude is
Electronic excitation of H2
1883
and
+
A") = [QBQ P V P - VG$?V],
(3)
In these equations g,is a product of a target state and a plane wave, V is the interaction
potential, fi is the total energy minus the full Hamiltonian of the system, P(Q) is a
projection onto the open (closed) electronic target states, and GF) is the Green function
projected onto this P-space.
The final scattering amplitude which combines the SMC and the FBA is given in the body
frame as
where
and
Good techniques of completing the scattering amplitude have been discussed recently by
several authors. These procedures were recently used by Rescigno and Lengsfield (1992)
for elastic scattering and by Winstead et al(1993) in inelastic elecbon-molecule collisions.
In both cases the authors completed the first-order theory amplitude with a FBA evaluated
with the strongest piece of the potential (dipole approximation) in the laboratory frame.
Here we are employing the full-potential to evaluate the first Born scattering amplitude and
since we are dealing with electronic excitation cross section we are able to converge the
scattering amplitude with respect to a finite number of partial waves in the body frame. This
is possible because, although the potential for a dipole transition falls as l / r Zthe differential
cross sections (DCS) never diverge in the foward direction (the momentum transfer, which
causes the divergence for the l / r 2 potential to never be zero for an electronic excitation).
Therefore one can evaluate the first Bom amplitude in the body frame, expand it in partial
waves up to some maximum value l ~transform
~ , it into the lab frame and then combine it
with the SMC amplitude. In evaluating the body frame amplitude some care must be taken.
For example, in the first Born approximation the contribution from the nuclei is in about the
same magnitude as from the target electrons for Ikf- kil << 1. Indeed they differ by a sign
and since they can be large numbers one can run into numerical problems. To avoid this
we can either choose quadratures such that lkf- kil is never too small or use an analytical
expression for the strongest piece of the potential for cases where Ikf kil << 1. We used
two different quadratures for Ici and kf to avoid situations where l
k - ki I were too small.
-
3. Resulb and discussion
As a first application of our formulation we have obtained the differential and integral cmss
sections for excitation of the X 'E: --f B 'E: transition in Hz for positron-impact energies
J L S Lino et al
1884
\
i
Impact Energy (eV)
Figure 1.
X'E:
Integral inelastic cross sections for h e
+ BIZ: transition in e+-H* scaltering. The
full c w e is ule present results for basis A (same as
used by Gibson er nl (1987). for ulis same electronic
excitation of H2 by electron impact). The broken curve
is the present results for basis set B (Gibson cc al1984).
6.1
0
Quadrature(x100)
Figure 2. Convergence of the integral cross section
with respct Lo he number of quadrature points
(directions of k, and kd for the FBA. hnmpact energy
of 20 eV.
of 13.5, 15, 17.5, 20, 25 and 30 eV. Our calculations are canied out at the two-state level
of approximation. The ground state of Hz is represented by a Hartree-Fock wavefunction
and the excited state is obtained by moving an electron from the lug orbital to a lo;
improved virtual orbital-NO (Hunt and Goddard 1969). Both the target wavefunction and
the scattering wavefunctions were represented by a basis of Cartesian Gaussian functions.
In our studies of Hz we have employed two basis sets. For the basis set A (Gibson et af
1987), the internuclear distance is 1 . 4 0 0 7 ~the
. SCF energy was found to be -1.133 au, and
12.78 eV for the Ivo excitation energy. For the basis set B (Gibson et af 1984). we used
the same internuclear distance 1.4~0to obtain a SCF energy of -1.132 au , and an excitation
energy of 12.92 eV. In figure 1 we show a good agreement between the total cross sections
obtained with these two basis sets using only the SMC method (no completion with FBA). In
evaluating the FBA scattering amplitude we tested the convergence of the integral and the
differential cross section with respect to the number of quadrature points (directions of kj
and kr) which are needed to accomplish the frame transformation from the body to the lab
frame. A typical convergence of the integral cross section (20 eV) can be seen in figure 2.
In figure 3 we show OUT total excitation cross section of the SMC and BSMC methods in
comparison with the previous theoretical results of Mukherjee et al (1991). The results of
Mukherjee et a[ clearly reveal a disagreement with our BSMC. Mukherjee and co-workers
(1991) have calculated these integral cross sections using the close-coupling approximation
for the same excitation process in the energy range 15-75 eV. Their results are considerably
larger in magnitude. For a better judgement of the nature of the discrepancies, in table 1
we compare the most important partial cross sections obtained with the two methods. The
results show clearly (at least for 20 and 25 eV) that the relative contributions are indeed very
similar. This is an indication that the cross sections obtained with these two methods differ
roughly by a constant factor (it may be energy dependent). Although not shown, all partial
cross sections (with projected angular momentum greater than 2, up to 6) obtained with the
FBA also differ from their results by about this same factor. This is not expected, since, even
for low-energy scattering, the FBA results should be reasonably good for high partial waves.
1885
Electronic exciiaioii O ~ H Z
A possible reason for the discrepancies could be found in the description of the excited
state. Mukherjee et nl have employed regular virtual orbitals (and not WO) to describe the
excited state. Virtual orbitals are r;ot made to represent excited states of the molecule (see,
for instance, a discussion made b]; Shavitt (1977)). Although not shown in the figure, the
results may change substantially if we use regular virtual orbitals to describe the excited
state (in our case the cross section became smaller by a factor of two). It is difficult to
estimate if our virtual orbitals have any similiarity with those of Mukherjee er a1 (1991)
since they have used Slater-type functions and we are using Cartesian Gaussian functions
(and also due to the fact that we have different basis sizes). Except for the representation
of the target states, in principle, there are no differences between the closecoupling and the
Schwinger multichannel methods. IThis is true as long as both are applied at the two-state
level of approximation and are coilverged with respect to the description of the continuum
wavefunctions.
Table 1. Panial cmss +ions for the X
Enerev (eV)
-f
B transition in e+-H2 scattering
x*
x,,
n.
n,,
Am
A>?
,0.105
‘~0,239
#0.304
0.0533
0.328
0.00381
0.0506
0.0211
0.421
0.0882
0.147
0.250
16 (29)
29(28)
26(25)
26 (8)
13 (13)
15 (14)
Higher
svmmetries
Absolute Values (0;)
15
O.OM19
20
0,188
25
0,200
0.102
0.115
0.173
0.0026
0.0643
0.1792
1 (I)
4(4)
6(4)
2(1)
lO(8)
6(3)
66)
11 (7)
11 (12)
Relative Values‘ (percentage)
15
20
I8 (19)
17(16)
25
12(14)
~31(38)
i21 (28)
1 9 (23)
‘Defined as a hundred times the ratio betwer!n partial and integral cross sections for this transition. For comparison.
we include between parentheses lhe resulesof Mukhejee et a1 (1991).
Figure 3. Integral inelastic cross section ifor e+-H2.
Present calculation for the SMC (- - &); present
calculation using the FBACSMC (BSMC) (-.);
FBA only
(- - -); close-coupling result of Mukhejeei et a1 1991
(-
- .-).
Figure 4. Differential inelastic cross section by positron
impact at 13.5 eV. The full curve presents BSMC results
and the b m h curve the SMC results.
1886
J L S Lino et a1
20
Figure 5. Differential inelastic CIOSS section by positron
impact at IS eV. Present FBA results (- - -); present
BSMC results (-);
results obtained With SMC by itself
(-
Figure 6. Same as in figure 5 for 17.5 eV
- -),
The present DCS for the respective transition by positron impact are shown in figures
4-9. The importance of using the combined theory (BSMC) can be seen in all figures by
comparing these results with those obtained with the SMC method by itself. As expected,
the presence of the FBA amplitude reveals its importance in the small angular scattering
region.
Scattering Angle (deg)
Figure 7. Differential inelastic cross seaion at 20 eV.
Results obrained using FBA + SMC fM electron scattering
(Gibson er 41 1987) (- - -1; OUT results for positron
results for positron
s c a r i n g using the BSMC (-);
scattering using the SMC by itself; full curfe with
triangles. theoretical results of Branchett el ol (1991)
using the R-matrix technique for electon scattering;
stars. e x p r i m e n d data for electron scattering of
Khakoo and Trajmar (1986).
Scattering Angle (deg)
Figure 8. Same BS in figure 4 for 25 eV.
1887
10
Scattering Angle (deg)
Figure 9. Same as in figure 7 for 30 eV, except that the R-&x
available).
results are not included (not
In figures 7 and 9 we compare our results with those obtained for the same transition
in electron-Hz scattering at 20 and 30 eV. For 20 eV we compare OUI results with those
obtained by Branchett et al(l991) using the R-matrix technique. We also show in these
figures the experimental data of Khakoo and Trajmar (1986) for this same excitation but
obtained with electron scattering. We found that, although they present different shapes,
their magnitudes are quite similar. This is expected since in both cases (electron and positron
scattering) this electronic excitation is governed mainly by a dipole transition.
4. Conclusions
We have shown that our version of the Schwinger multichannel method, adapted for lowenergy positron-molecule scattering, can be used to calculate dipoleallowed excitation cross
sections. The Schwinger multichannel method for positrons was carried out with a discrete
basis expansion of the trial scattering wavefunction and was used to obtain the contribution
to the cross section for the strongly scattered low-angular-momentum partial waves. The
FBA was used to correct the contribution of the high-l partial waves. Our results obtained
at the two-state level of approximation seem to be very reliable. Multichannel studies
including other transitions in Hz by positron impact are presently under way.
Acknowledgments
The computation work was done at the Instituto de F'rote@o ao VBoKTA, SBo Jose dos
Camps-SP, and at Instituto de Ffsica Gleh Wataghin, Unicamp, Campinas, SP, Brazil.
JLSL acknowledges financial support from FundaqBo de Amparo I? Pesquisa do Estado de
SBo Paulo-FAPESP, Brazil. MAF'L acknowledges financial support from Conselho Nacional
de Pesquisa e Desenvolvimento Cientffico e Tecnol6gico-CNF'q, Brazil.
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J L S Lino et a1
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