J . Phys. B At. Mol. Opt. Phys. 28 (1995) 467475. Printed in the UK
Elastic electron scattering by water molecules
Luiz E Machadot, Lee Mu-Taot, Luiz M Brescansins, Marco A P Lima5
and V McKoyll
t Departamento de Flsica, UFSCx. 13565-905 Siio Carlos, Siio Poulo, Brazil
1 Departamento de Qulmica. LIFSCar,
13565-905 SSo Culos, Siio Paulo, Brazil
$ lnstituto de Fisica Gleb Wataghin, UNICAMP, 13083-970 Campinas, SHo Paulo, Brazil
ll Arthur Ames Noyes Laboratory of Chemical Physics, California Institute of Technology.
Pasadena. CA 91125, USA
Received 4 January 1994, in final form 25 October 1994
Abstract. Elastic differential and momentum transfer cross sections ae reported for electron
scattering by HzO ai impact energies ranging from 4 to 50 eV. The iterative Schwinger vwiational
merhod in the fixed-nuclei. static-exchange approximation is used to calculate the low partial
wave scattering amplitudes and the higher partial wave contributions are included via closure
using the Born approximation for a point-dipole, Comparison of our calculated cross sections
with recent experimental and other theoretical results is encouraging.
1. Introduction
Collisions of electrons with water molecules play an important role in several fields (Trajmar
et al 1983) such as atmospheric physics, radiation biology and chemistry and plasma
physics. However, only in the last decade have these collision processes been studied
more extensively. In addition to the pioneering work of Briiche (1929), measurements of
total cross sections (Tcs) have been reported by several groups (Sokolov and Sokolova 1981,
Sueoka et a1 1986, Szmytkovski 1987, Zecca et nl 1987 and Nishimura and Yano 1988).
Differential cross sections (DCS) for elastic scattering and vibrational excitation were reported
by Jung etal (1982), Danjo and Nishimura (1985), Katase eta! (1986), Shyn and Cho (1987)
and Johnstone and Newel1 (1991). Vibrationally and rotationally inelastic processes have
also been studied by Jung et al(1982) and Shyn et al (1988). On the theoretical side, there
are several earlier Born-based calculations (Crawford 1967, Itikawa 1972, Jung et al 1982).
These collision cross sections have also been studied using more sophisticated treatments,
including the static-exchange-polarizationmodel potential approach (Jain and Thompson
1982, 1983), the static-exchange Schwinger variational multichannel (SMC) (Brescansin et
al 1986), the local modified semi-classical exchange (MSCE) (Gianturco and Scialla 1987,
Gianturco 1991) methods and, very recently, the complex Kohn variational (Rescigno and
Lengsfield 1992), the combined free-gas plus correlation-polarization (Okamoto et al 1993)
methods, and that of Brescansin et al (1992) applying the iterative Schwinger variational
method (ISVM) (Lucchese et al 1982). In this latter article we reported calculations of
some preliminary DCS at 15, 20 and 30 eV in the static-exchange level of approximation.
where the partial-wave expansions were truncated at 1, =IO and no further correction was
made in the scattering amplitudes. These non-fully-converged expansions led to qualitative
discrepancies between our DCS and the experimental data (Danjo and Nishimura 1985,
0953475/95/030467+Ll9$19.50 @ 1995 IOP Publishing Ltd
467
468
L E Machodo et a1
Shyn and Cho 1987, Johnston and Newell 1991) in the forward direction. Also, for incident
energies below 15 eV, some unphysical oscillations were found in the DCS.
In this paper we report the results of calculations of the differential and momentumtransfer cross sections for elastic scattering of electrons by H20 for incident energies
EO ranging from 4 to 50 eV. In these studies the low-order partial-wave components of
the T-matrix are obtained in the fixed-nuclei and static-exchange approximations using
ISVM (Lucchese et al 1982) and the higher-order terms are included via a Born-closure
approximation based on a point-dipole potential. As stressed recently by Rescigno and
Lengsfield (1992). the use of Born closure is essential in such body-fixed, adiabatic-nuclei
calculations for polar molecules since the associated partial-wave expansions of the Tmatrix are essentially divergent. The ISVM, an iterative procedure based on the Schwinger
variational method, is capable of providing highly converged estimates of these elastic
partial-wave T-matrix elements and has been used to study elastic scattering of electrons by
linear molecules (Lucchese and McKoy 1980, 1982, Lee et al 1990, 1992a). Extension of
this method to non-linear molecules includes applications to studies of the photoionization
cross sections of CH4 (Braunstein et al 1988). H20 (Machado et al 1990) and C&
(Brescansin et al 1993), ion rotational distributions of H20 (Lee et al 1992b, 1992~)as
well as of the elastic scattering of electrons by H20 (Brescansin et a1 1992).
The organization of the paper is as follows. In section 2 we outline the method and
procedures used in these studies. In section 3 some aspects of the calculations are discussed,
while our results along with their comparison with other available data are presented in
section 3.
2. Theory
In this section we will briefly discuss the method and procedures used in these studies.
Details can be found elsewhere (Brescansin et a1 1992). The differential cross section for
elastic electron-molecule scattering is given by
where f ( & , S ’ ) is the laboratory-frame (LF) scattering amplitude,
and k lie in the
directions of the momenta of the incident and scattered electron, respectively, and (a,@, y )
are the Euler angles orienting the principal axes of the molecule.
As in our previous work (Brescansin et nl 1992). the body-frame (BF) T-matrix can be
conveniently expanded in partial waves as
where
and i lie along the momenta of the incident and scattered electron in the BF,
respectively, and X;’(i) are symmetry-adapted functions (Burke et al 1972) which are
expanded in the usual spherical harmonics as follows:
Xg(3 = x b L ; y i m ( ? ) .
(3)
m
Here p is an irreducible representation (IR) of the molecular point group, p is a component of
this representation and h distinguishes between different bases of the same IR corresponding
Elastic electroil scattering by water molecules
469
to the same value of 1 , The coefficients bFmsatisfy important orthogonality relations and
are tabulated for the C2. and oh point groups by Burke er al (1972).
A BF-Born-closure formula for the T-matrix can be written as
pplS"M
where Tk,lh;,,h,
are the partial-wave T-matrix elements calculated via the lSVM and truncated
at some appropriate'values of the pairs ( I , h ) and ( P , h'), T$l,h, are the corresponding
partial-wave Born T-matrix elements for electron scattering by a point dipole, and T B is
the complete Born T-matrix. In evaluating the differential cross sections, TB is calculated
directly in the laboratory frame. The summation in (4) is also written in the LF via the usual
frame transformations (Edmonds 1960). Details of the calculation of the differential cross
sections are given in the appendix. The wavefunctions and the reactance K-matrices, which
are related to the T-matrices, are obtained from the numerical solution of the one-particle
Lippmann-Schwinger equation using the ISVM.
3. Numerical procedure
The SCF wavefunction used in these static-exchange calculations was obtained with the
standard [3s2p,2s] contracted Gaussian basis of Dunning and Hay (1977) augmented with
one d function (exponent of 0.34) on the oxygen and one p function (exponent of 0.13) on
the hydrogens. At the experimental equilibrium geometry of R(0-H) = l.8lao and B(H0-H) = 104.5" (Snyder and Basch 1972) this basis gives an SCF energy of -76.0199 au
and an electric dipole moment of 0.762 au, compared with the HartreeFock (m) limit of
-76.0632 au (Dunning et al 1972) and the measured dipole moment of 0.724 au (Meyer
1977). respectively. The l a , , 2al, 3a1, Ibz and Ib, orbital energies are -20.563, -1.358,
-0.582, -0.717 and -0.512 au, respectively. In the numerical solution of the LippmannSchwinger equation using the ISVM all single-centre expansions were truncated at 1, = 10 and
all possible values of h I were retained for a given 1. The resulting orbital normalizations
were better than 0.999 for all bound orbitals. We verified that this expansion was sufficient
to provide cross sections accurate to within 5%. The initial Lz basis set used in the ISVM
is shown in table 1. Four iterations (Lucchese et al 1982) of the ISVM provided converged
solutions of the LippmannSchwinger equation for the static-exchange potential.
<
'4
'
'
3b
'
'
Bb
'
'
$0
'
'Id0
'
'
Scattering Angle (deg)
150'
',I
Figure 1. DCS for e--H*O scattering tu E,, = 6 eV.
Full curve: present Born-closure ISVM results: longdashed c u m present results with a pure L2 basis:
broken curve: present ISVM resulls wilh B finite
number of partial waves.
470
L E Machado ei al
Table 1. Bacis sei used for the initial scnttering functions.
Type of Caxtesian
scattering
symmeiry
Centre
kat
0
Gaussian function'
z
XI.
H
Y?. 22
s
I.2
kaz
kbi
0
YZ
H
Y
0
Y
YZ
kh
a
H
Y
0
XE
x
H
2
Exponents
2.0. 0 5 . 0.1, 0.025
2.0, 0.5, 0.05
0.2 0.05
0.8, 0.2
0.5. 0.1
4.0. 2.0, i.0,0.5,0.1
1.2 0.4, 0.1
8.0, 4.0, 2.0, 1.0, 0.5
2.0, 1.0, 0.5, 0.1
2.0, 1.0, 0.5
8.0.4.0, 2.0. 1.0. 0.5
2.0. 1.0, 0.5.0.1
2.0, 1.0, 0.5
Cartesian Gaussian basis Fu,nctions are defined as
,pl.m.n,A
(r) = N ( x
- A,)% - Ay)"%- A,)"exp(-ulr - AI')
wilh N a normalization constant.
4. Results and discussion
In order to illustrate the influence of the iterative procedure and of the need for Born
closure on our calculated DCS, figure 1 shows the differential cross sections obtained from
(i) the Schwinger variational principle (no iterations) with the L2 basis of table 1, (ii)
the lSVM without Born closure and (iii) the lsvM with Born closure for a specific low
incident energy (6 eV). As noted recently by Rescigno and Lengsfield (1992) the smooth
behaviour and the absence of unphysical oscillations in the DCS obtained from a purely
L2 Schwinger variational calculation is due to the inability of the L2 basis to adequately
represent the long-range character of the dipole potential. The resulting cross sections
consequently do not reveal the underlying divergence of the partial-wave expansion of the
body-frame, fixed-nuclei T-matrix for electron scattering by polar molecules. On the other
hand, the cross section with a limited number of partial waves obtained with the ISVM,
which properly accounts for the long range of the dipole potential, is oscillatory due to the
lack of convergence of such partial-wave expansions. As expected, when this partial-wave
expansion is properly supplemented with Born closure, the resulting cross section shows
physically reasonable behaviour with angle.
Figures 2-7 show our fixed-nuclei Born-closure differential cross sections for collision
energies of 4.6, 10, 15,30 and 50 eV, along with some selected experimental data of Danjo
and Nishimura (1985). Shyn and Cho ( I 987) and Johnstone and Newell (1991). Theoretical
cross sections of Gianturco (1991), Rescigno and Lengsfield (1992) and Okamoto er al
(1993) are also included for comparison where available. As expected for a polar molecule,
our cross sections show very strong forward-peaking. Our results also show the backward
enhancement in the DCS which was predicted by Brescansin eta[ (1986) and later confirmed
by measurements of Shyn and Cho (1987). Agreement between our calculated cross sections
and available experimental data is generally good. Our results also agree quite well with
those of Rescigno and Lengsfield (1992) and Okamoto er al (1993) over the angular and
energy ranges covered here. Particularly at 4 eV our static-exchange DCS agree very well
with the recent p a h i z e d SCF results of Rescigno and Lengsfield (1992), reflecting the minor
Elustic electron scattering by water molecules
60
30
150
1 0
90
160
Scattering Angle (deg)
6 eV
Cl0
8
5:
-1
"P
VI
Scattering Angle (deg)
'
' 3b '
'
6b '
'
do
'
' 150'
' 150'
Scattering Angle (deg)
Figure 2. DCS for e--HzO collisions for 4 eV. Full
curve, present Born-closure ISVM results; long-broken
cutve, calculated Bom-closure Kohn variational
results of Rescigno and Lengsfield (1992); shortbroken curve. theoretical results of Gimturco (1991);
open squares. measurements of Danjo and Nishimura
(1985); full circles. experimenhl results of Shyn nnd
Cho (1987).
I
2
10 - 2
41 1
o
Figure 3. DCS for ec-UzO collisions for Eo =
6 eV. Full cun'e. present Born-closure ISVM results;
long-dashed curve. calculated Bom-closure Kohn
variational results of Rescigno and Lengslield (1992);
broken c w e , calculations of Okamoto er a1 (1993):
rhon-dashed c w e . theoretical results of Gianturca
(1991); full circles, experimental results of Shyn
and Cho (1987); open triangle. measurements of
Johnstone and Newell (1991).
',io
Figure 4. Same as figure 3 for IO eV.
role of polarization for collision of electrons with strongly polar molecules. On the other
hand, although the theoretical results of Gianturco (1991) reproduce the experimental data
at 4 and 6 eV, at higher energies his calculations clearly underestimate the measured DCS
at large scattering angles. It should be noted that, although the method used by Gianturco
(1991) is essentially equivalent to that of Okamoto et a[ (1993),this latter work correctly
describes the DCS in the intermediate- and large-angle region. Presumably, the discrepancies
between the results of these two calculations indicate the sensitivity of their results to adopted
exchange model potential.
In figure 8 we show our calculated momentum-transfer cross sections (MTCS) along
with the experimental data of Danjo and Nishimura (1985), Shyn and Cho (1987) and
412
L E Machado er a1
Figure 5. Same 1%figure 3 for 15 eV
10''
,
d
,
50
"50'
,
'
d0 ' ' $0
'
'120
Scattering Angle (deg)
'do' 'do'
150
' 1 0 ' '150'
Scattering Angle '[deg)
1kO
Figure 6. ncs for e'-HzO scattering EO =
30 eV. Full curve: present Bom-closure ISVM rcsults;
broken curve: calculated results of Okamoto et al
(1993). open Uingles: masuremen& of Johnstone
and NeweU (1991).
',IFfgure
7. Same as figure 6 for 50 eV
Johnstone and Newell (1991) and the calculated results of Rescigno and Lengsfield (1992)
and Okamoto er a1 (1993). Large discrepancies are seen in the experimental results. Above
10 eV our calculated MTCS agree well with the most recent measurements of Johnstone
and Newell (1991). Also, our results are in good agreement with those of Rescigno
and Lengsfield (1992) and of Okamoto et al (1993). The experimental results of Danjo
and Nishimura (1985) are systematically lower than our calculated values above 8 eV.
Since the backward enhancement of the DCS contributes significantly to the calculated MTCS
(Brescansin et al 1986), Danjo and Nishimura probably underestimated their MTCS derived
from the extrapolation of their measured DCS to larger scattering angles.
In summary, we have used the ISVM and a Born-closure to study elastic scattering of
Elastic electron scattering by water molecules
413
1
20,
..
U 5-
5 :
Figure S. m s for elastic-elecvon-H~O. Full curve:
present Bom-closure ISVM results. longdashed curve:
calculotions of Rescigno and Lengstield (1992);broken
curve: theoreticd results of Oknmoto el a1 (1993);
full circles: experimental results of Shvn and Cho
(1987); open triangles: measured values of Johnstone
and Newell (1991); open squares: m e i u r e m e n t ~of
Danjo and Nishimum (1985).
1
0
00
10
30
20
Energy (eV)
40
electrons by H 2 0 in the 4-50 eV energy range. Comparison between our calculated DCS and
MTCS with selected experimental and other theoretical results is encouraging and shows the
adequacy of the procedures used here for studying elastic electron-polar-molecule collisions.
Application of this method to other non-linear systems is under way.
Acknowledgments
This research was supported by Brazilian Agencies Conselho Nacional de Desenvolvimento
Cientifico e Tecnol6gico (CNF'q), Fundaq3.o de Amparo h. Pesquisa do Estado de S5.o Paula
(FAPESP), FINEP-PADCT and the US National Science Foundation.
Appendix
In the laboratory frame equation (4) can be written as
T = TB + Tdff
where
du
duB
do"'
+-do'
L E Machado et a1
474
where d o B / d Q is the Born expression for the DCS for a point-dipole,
d_o B
dz
=_2
dS2
3k2 (1 - cos 8 )
and
do'
dS2
1
1
-= 8n2
+ cc
dru sin 9, d p dyTB'TdiR
In (A5) and (A6), d is the target dipole moment, k = lk'l, 8 the scattering angle, and the
coefficients A r ( k ) are given by
PI&,
bl;h',m;
bPI"1' b'"'
l l h l m l I'h"'
bP'
PIP7
PI1
Ihmal,h,.I~h;(k)ai,l'h.(k)
x (I,OlO I LO)(l{Ol'O I LO)(ll
- mlIm I L - M)(l;m;l'm' I L M )
(A81
where ( j t m j jzm?lj,m,) are the usual Clebsch-Gordan coeflicients and the auxiliary
amplitudes a&h,(k) are defined as
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