J. Phys. B: At. Mol. Opt. Phys. 31 (1998) 2091–2099. Printed in the UK
PII: S0953-4075(98)89874-X
Low-energy electron scattering by N2 , P2 , As2 and Sb2
M H F Bettega†, M A P Lima‡ and L G Ferreira‡
† Departamento de Fı́sica, Universidade Federal do Paraná, UFPR, Caixa Postal 19081, 81531990 Curitiba, Paraná, Brazil
‡ Instituto de Fı́sica Gleb Wataghin, Universidade Estadual de Campinas, UNICAMP, Caixa
Postal 6165, 13083-970, Campinas, São Paulo, Brazil
Received 9 December 1997
Abstract. We report elastic integral, momentum transfer and differential cross sections from
10–30 eV for electron scattering by X2 (X = N, P, As, Sb). These results were obtained at the
static-exchange approximation with the Schwinger Multichannel Method with Pseudopotentials
[M H F Bettega, L G Ferreira and M A P Lima 1993 Phys. Rev. A 47 1111]. Our results for
N2 are in good agreement with experimental data. We also compare our results with previous
calculations on XH3 (X = P, As, Sb) [M H F Bettega, M A P Lima and L G Ferreira 1996
J. Chem. Phys. 105 1029] and found, as expected, that the X2 cross sections are larger than the
corresponding XH3 cross sections.
1. Introduction
The calculation of low-energy scattering cross sections for molecules formed by heavy atoms
involves a lot of computational effort [1]. Even at the static-exchange approximation,
where the target is considered frozen in its ground state, this calculation is not simple.
Aiming at simplifying the electron-molecule collision calculations, we implemented soft
norm-conserving pseudopotentials [2] in the Schwinger multichannel (SMC) [3] method
[4]. The basic idea of the wedding between the SMC method and the pseudopotential
(SMCPP) is to replace the potential due to the core electrons and the nucleus of each
atom of the molecule by the pseudopotential, the valence electrons being described in a
many-body framework (Hartree–Fock approximation in the present implementation). As
a consequence, a considerable computational effort is saved and molecules with many
electrons may be investigated. Such a saving has allowed numerous applications of the
method for several molecular systems [5]. More recently norm-conserving pseudopotentials
were also implemented in the complex Kohn method [6, 7].
Molecular nitrogen is probably one of the most investigated molecules due to its
importance in several processes. For example, N2 is used in swarm and discharge plasmas
and plays a fundamental role in processes involving the atmosphere [8–9]. Its elastic cross
section presents a shape resonance in the 2 5g channel near 2 eV, and its correct description
has been the subject of several studies [9–11]. However, to our knowledge, the other
dimers formed by the atoms below nitrogen in the periodic table, and therefore with the
same valence electronic configuration, namely P2 , As2 and Sb2 , have been not investigated
even at the static-exchange approximation. In this paper we present results from calculations
of elastic integral, momentum transfer and differential cross sections in the energy range
from 10 to 30 eV for electron scattering by the dimers X2 (X = N, P, As, Sb). All the
c 1998 IOP Publishing Ltd
0953-4075/98/092091+09$19.50 2091
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M H F Bettega et al
calculations were performed with the SMCPP method at the static-exchange approximation.
In this energy range, polarization effects are not important and this approximation is expected
to give reliable cross sections [5].
2. Theoretical formulation
The SMC [3, 12] and SMCPP [4] methods have been discussed in earlier works and we
will review here only some key steps of these methods. The SMC method is a multichannel
extension of the Schwinger variational principle. Actually it is a variational approximation
for the scattering amplitude, where the scattering wavefunction is expanded in a basis of
(N+1)-particle Slater determinants,
X
± E
am
(k)|χm i,
(1)
|9kE i =
m
± E
(k) of this expansion are then variationally determined. The resulting
and the coefficients am
expression for the scattering amplitude in the body frame is
1 X
fkEi ,kEf = −
hSkEf |V |χm i d −1 mn hχn |V |SkEi i
(2)
2π m,n
where
dmn = hχm |A(+) |χn i
(3)
and
Ĥ P + P Ĥ
(V P + P V )
Ĥ
−
+
− V G(+)
(4)
A =
P V.
N +1
2
2
In the above equations SkEi , the solution of the unperturbed Hamiltonian H0 , is the product of
a target state and a plane wave, V is the interaction potential between the incident electron
and the target, |χm i is an (N+1)-electron Slater determinant used in the expansion of the
trial scattering wavefunction, Ĥ = E − H is the total energy of the collision minus the
full Hamiltonian of the system, with H = H0 + V , P is a projection operator onto the
open-channel space defined by target eigenfunctions |8l i,
(+)
P =
open
X
|8l ih8l |,
(5)
l
and G(+)
P is the free-particle Green’s function projected on the P -space.
For elastic scattering at the static-exchange approximation, the P operator is composed
only by the ground state of the target |81 i
P = |81 ih81 |
(6)
and the configuration space |χm i is
{|χm i} = A|81 i|ϕi i
(7)
where |ϕi i is a one-particle function represented by one molecular orbital.
With the choice of Cartesian Gaussian functions to represent the molecular and scattering
orbitals, all the matrix elements arising in equation (2) can be computed analytically, except
those from hχm |V G(+)
P V |χn i (VGV) that are evaluated by numerical quadrature [12].
The numerical calculation of the matrix elements from VGV represents the most
expensive step in the SMC code and demands almost the entire computational time of the
Low-energy electron scattering by N2 , P2 , As2 and Sb2
2093
scattering calculation. These matrix elements are reduced to a sum of primitive two-electron
integrals involving a plane wave and three Cartesian Gaussians
Z Z
1
E
E =
hαβ|V |γ ki
dEr1 dEr2 α(Er1 )β(Er1 ) γ (Er2 )eik.Er2
(8)
r12
and must be evaluated for all possible combinations of α, β and γ and for several directions
E We must also evaluate the one-electron integrals of the type
and moduli of k.
Z
Er
E = dEr α(Er )V̂ P P eik.E
hα|V̂ P P |ki
.
(9)
These one-electron integrals are more complex than those involving the nuclei, but they can
be calculated analytically, and their number is also reduced due to the smaller basis set. In
the above equation, V̂ P P is the nonlocal pseudopotential operator given by
with
V̂ P P (r) = V̂core (r) + V̂ion (r)
(10)
"
#
2
h
i
Zv X
core
core 1/2
V̂core (r) = −
c erf αi
r ,
r i=1 i
(11)
and
V̂ion (r) =
3 X
2
1 X
X
n=0 j =1 l=0
Anj l r 2n e−αj l r ×
2
+l
X
|lmihlm|,
(12)
m=−l
where Zv is the valence charge of the atom and in this application it is equal to 5 for N, P,
As and Sb. The coefficients cicore , Anj l , and the decay constants αicore and αj l are tabulated
in [2].
Even for small molecules, a large number of the two-electron integrals must be
evaluated. This limits the size of molecules in scattering calculations. In the SMCPP
method we need shorter basis set to describe the target and scattering and consequently the
number of two electron integrals is smaller than in the all-electron case. The reduction in
the number of these integrals allows the study of larger molecules than those reachable by
all-electron techniques.
3. Computational procedures
The ground state of the X2 (X = N, P, As, Sb) molecules is described by the valence
electronic configuration 2σg2 2σu2 3σg2 1πu4 (1 6g+ ). The basis functions we used in the
description of the valence part of the target state |81 i and to describe the scattering orbitals
|ϕi i are shown in table 1. These functions were generated by a variational procedure [13],
being suitable for these pseudopotential calculations. We have tested basis sets with different
sizes, ranging from 72 uncontracted functions (67 scattering orbitals) up to 96 uncontracted
functions (91 scattering orbitals). The results presented here were obtained with the smaller
set for N2 and with the larger set for P2 , As2 and Sb2 . All calculations were performed in
a fixed-nuclei static-exchange approximation at the experimental equilibrium geometries.
4. Results and discussion
In figures 1, 2 and 3 we compare our calculated cross sections for N2 with experimental
data from [14–16] and find a general good agreement. Figure 4 shows the integral elastic
2094
M H F Bettega et al
Table 1. Cartesian Gaussiana functions for the X atoms.
Type
N
Exponent
P
Exponent
As
Exponent
Sb
Exponent
Type
Coefficient
s
s
s
s
s
s
17.569870
3.423613
0.884301
0.259045
0.053066
0.022991
12.524060
3.745793
1.446075
0.594791
0.240327
0.073944
14.375400
5.479020
0.865623
0.524922
0.137104
0.023814
10.013550
3.825364
0.849370
0.564441
0.175107
0.016717
1.0
1.0
1.0
1.0
1.0
1.0
p
p
p
p
p
p
8.075437
2.923137
1.252239
0.511098
0.205334
0.079059
2.643781
1.064158
0.410957
0.201602
0.119230
0.027344
4.929918
1.266223
0.441445
0.156902
0.096300
0.054391
3.047568
0.838121
0.675204
0.267415
0.084885
0.018684
1.0
1.0
1.0
1.0
1.0
1.0
d
d
d
d
0.403039
0.091192
1.440483
0.470875
0.188833
0.079087
2.255294
0.478967
0.182590
0.072354
0.991919
0.282477
0.129957
0.025140
1.0
1.0
1.0
1.0
a Cartesian Gaussian functions are defined by
φlmn = Nlmn (x − ax )l (y − ay )m (z − az )n exp(−α|Er − aE |2 ).
cross section (10
-16
cm 2)
20
15
10
5
10
15
20
25
30
energy (eV)
Figure 1. Integral elastic cross section for N2 . Solid curve: our results; squares: experimental
results from [14]; circles: experimental results from [15]; diamonds: experimental results from
[16].
cross sections for the X2 dimers. For N2 , the cross section is a constant in the energy range
considered and presents a distinct behaviour from the cross sections for the other molecules.
In figure 5 we compare the integral cross sections for P2 , As2 and Sb2 with those for
PH3 , AsH3 and SbH3 [17]. Although the X2 cross sections are larger than the corresponding
XH3 cross sections in the entire energy range, as expected, they present the same pattern
Low-energy electron scattering by N2 , P2 , As2 and Sb2
2095
cross section (10
-16
cm 2)
15.0
12.5
10.0
7.5
5.0
10
15
20
25
30
energy (eV)
Figure 2. Momentum transfer cross section for N2 . Solid curve: our results; squares:
experimental results from [14]; circles: experimental results from [15].
cross section (10 -16 cm 2/sr)
10 eV
15 eV
10
20 eV
30 eV
1
0.1
0 30 60 90 120150180
angle (degrees)
Figure 3. Differential cross section for N2 at 10, 15, 20 and 30 eV. Solid curve: our results;
squares: experimental results from [14]; circles: experimental results from [15].
shown for the XH3 cross sections. We therefore repeat the procedure used in the case
of XH3 molecules: we divide the cross sections by the squared interatomic distance and
multiply the energy (k 2 ) by the same number. The results of this procedure are presented in
figure 6 that shows the similarity between As2 and Sb2 and that P2 has slightly larger cross
sections. The inset in figure 6 shows the results for the XH3 . Thus the figure suggests that
the scattering is mainly determined by the heavy atoms.
Figure 7 presents our results for the momentum transfer cross section (MTCS). As in
2096
M H F Bettega et al
60
cm 2)
P2
40
As 2
cross section (10
50
-16
N2
Sb 2
30
20
10
0
10
15
20
25
30
energy (eV)
Figure 4. Integral elastic cross section for X2 .
P2
PH 3
50
As 2
cross section (10
-16
cm 2)
60
40
AsH 3
Sb 2
30
SbH 3
20
10
10
15
20
25
30
energy (eV)
Figure 5. Integral elastic cross section for X2 and for XH3 .
the case of the integral cross sections, the MTCS for N2 has a distinct behaviour from the
MTCS for the other dimers. In figures 8 and 9 we present differential cross sections (DCS)
for X2 molecules at 10, 15, 20, 25 and 30 eV. The DCS for As2 and Sb2 are very similar,
specially at higher energies, and show a strong contribution from ` = 2. For P2 we also
found contribution from ` = 2 partial wave, and for N2 the DCS show only one minimum
for all energies. Finally, we have also investigated the existence of shape resonances at low
impact energies. As mentioned above, the nitrogen molecule supports a shape resonance at
about 2 eV (≈ 4 eV at the static-exchange level of approximation). Although not shown,
we have found no shape resonances for the other molecules of the family.
Low-energy electron scattering by N2 , P2 , As2 and Sb2
P2
As 2
15
Sb 2
10
20
PH3
normalized cross section (arb. units)
normalized cross section (arb. units)
20
2097
AsH3
SbH3
15
10
5
0
5
10
15
normalized wave vector (arb. units)
5
0
8
10
12
14
16
18
20
normalized wave vector (arb. units)
Figure 6. Normalized integral cross section for P2 , As2 and Sb2 .
N2
P2
20
As 2
cross section (10
-16
cm 2)
25
15
Sb 2
10
5
0
10
15
20
25
30
energy (eV)
Figure 7. Momentum transfer cross section for X2 .
5. Conclusions
In this paper we reported calculated integral, differential and momentum transfer cross
sections for scattering of low-energy electrons by X2 (X = N, P, As, Sb). These results
were obtained with the Schwinger multichannel method with pseudopotentials at the staticexchange level of approximation. Although these molecules scatter in different ways, we
found similarities in the cross sections for As2 and Sb2 . For P2 , As2 and Sb2 we found
a strong d-wave behaviour in the differential cross section plots and also found in their
2098
M H F Bettega et al
100
cm 2/sr)
N2
P2
As 2
Sb 2
cross section (10
-16
10
1
10 eV
0.1
0
30
60
90
120 150 180
angle (degrees)
Figure 8. Differential cross sections for X2 at 10 eV.
N2
P2
cross section (10 -16 cm 2/sr)
As 2
Sb 2
15 eV
20 eV
25 eV
30 eV
100
10
1
0.1
0 30 60 90 120150180
angle (degrees)
Figure 9. Differential cross sections for X2 at 15, 20, 25 and 30 eV.
integral cross sections the same pattern shown by the integral cross sections for PH3 , AsH3
and SbH3 . The calculated cross sections for X2 are larger than the calculated cross sections
for XH3 .
Acknowledgments
MHFB acknowledges partial support from Fundação da Universidade Federal do Paraná para
o Desenvolvimento da Ciência, da Tecnologia e da Cultura (FUNPAR). LGF and MAPL
Low-energy electron scattering by N2 , P2 , As2 and Sb2
2099
acknowledge partial support from Brazilian agency Conselho Nacional de Desenvolvimento
Cientı́fico e Tecnológico (CNPq). Our calculations were performed at CENAPAD-SP, at
CENAPAD-NE and at CCE-UFPR.
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