Electron impact total cross section for acetylene over an extensive range of
impact energies (1 eV–5000 eV)
Minaxi Vinodkumar, Avani Barot, and Bobby Antony
Citation: J. Chem. Phys. 136, 184308 (2012); doi: 10.1063/1.4711922
View online: http://dx.doi.org/10.1063/1.4711922
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THE JOURNAL OF CHEMICAL PHYSICS 136, 184308 (2012)
Electron impact total cross section for acetylene over an extensive range
of impact energies (1 eV–5000 eV)
Minaxi Vinodkumar,1 Avani Barot,1 and Bobby Antony2
1
2
V. P. & R. P. T. P. Science College, Vallabh Vidyanagar 388120, Gujarat, India
Department of Applied Physics, Indian School of Mines, Dhanbad 826004, Jharkhand, India
(Received 12 January 2012; accepted 20 April 2012; published online 11 May 2012)
Comprehensive study on electron impact for acetylene molecule is performed in terms of eigenphase
diagram, electronic excitation cross sections as well as total cross section calculations from 1 eV to
5000 eV in this article. Computation of cross section over such a wide range of energy is reported
for the first time. We have employed two distinct formalisms to derive cross sections in these impact
energies. From 1 eV to ionization threshold of the target we have used the ab initio R-matrix method
and then spherical complex optical potential method beyond that. At the crossing point of energy,
both theories matched quite well and hence prove that they are consistent with each other. The results presented here expectedly give excellent agreement with other experimental values and theories
available. The techniques employed here are well established and can be used to predict cross sections
for other targets where data are scarce or not available. Also, this methodology may be integrated
to online database such as Virtual Atomic and Molecular Data Centre to provide cross section data
required by any user. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4711922]
I. INTRODUCTION
Acetylene is an industrial chemical found especially with
benzene, vinyl chloride, acrylic acid, esters, etc.1 and is used
as fuel in the high-temperature welding industry. It is a colourless gas with an unpleasant odour, and can be easily liquefied.
On account of its endothermic nature great care is required
in its use as it can decompose easily under pressure with an
explosive violence. Acetylene forms an explosive compound
with copper and hence the combination is to be avoided. Also,
it is shown that the range of explicability of acetylene mixed
with air is greater than that of any other gas and hence the
leakage of this gas must be strictly avoided. All these features
make it quite difficult to handle in the laboratories.
The reaction of acetylene with hydroxyl radicals is very
significant in the combustion of other atmospheric and astrophysical processes.2–4 Ethynyl radicals are also involved
in the formation of polyacetylenes and polycyclic aromatic
hydrocarbons5, 6 in the combustion of fossil fuels. The rate
and mechanism of the reaction between acetylene and OH
can be of an important degradation pathway,7 since they are
the major intermediates of rich hydrocarbon flames. The NO
chemistry of the flame is highly affected by the acetylene
compound since, they influence the post-flame behavior of
small hydrocarbons and hence on the resulting radicals. For
this reason this significant reaction has been the subject of
numerous experimental studies.8, 9 Detection of ethynyl radicals in the interstellar medium and planetary atmospheres10
has generated tremendous interest in the gas phase reactions
of these radicals. Also, they are of fundamental chemical interest due to their presence in a wide variety of natural and
artificial environments.
The justification of the role of C2 H radicals in combustion processes requires the knowledge of the cross sections
and rate constants of the principal formation and destruction
0021-9606/2012/136(18)/184308/8/$30.00
reactions. However, the methods of C2 H formation in combustion is not comprehensively studied.11 Knowledge of the
total cross sections of hydrocarbon molecules such as ethylene, acetylene, and methane, one can acquire information
about the carbon-to-carbon bonds and the physics of such
hydrocarbons.11
With the advent of highly sophisticated instruments and
high performing computers, accurate electron collision data
can be obtained now. However, such methods are quite tedious and take toll on human manpower and computational
resources. Also, these techniques can produce electron scattering cross sections only for limited targets out of the vast
number of molecular systems still to be studied. This has
prompted the researchers to look for faster methods to derive
electron impact collision data on the time scales required by
the industry.
Presently, we are focused on the calculation of electron
collision data for acetylene molecule in the energy range from
1 eV to 5 keV. For the low energy (below vertical ionization potential), we make use of the UK molecular R-Matrix
code through the Quantemol-N software package.12 In the intermediate and high energies (above vertical ionization potential), we employ the spherical complex optical potential
(SCOP) formalism.13 Formation of short-lived anions (resonances) and thus the possibility of their decay to produce
neutral and anionic fragments occur at low impact energies
below 10 eV. Such processes are very important in the understanding of local chemistry of electron target interaction upon
impact. Electron induced scattering cross sections at intermediate and high energies are essential especially in areas such
as astrophysics, atmospheric physics, and radiation physics.
However, it is obvious that such an approach needs to be consistent with each other at the transition energy (near ionization potential). The present results are in good agreement with
previous experimental and theoretical results. A review of the
136, 184308-1
© 2012 American Institute of Physics
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184308-2
Vinodkumar, Barot, and Antony
J. Chem. Phys. 136, 184308 (2012)
TABLE I. Literature survey for e -C2 H2 scattering.
Energy range
(eV)
Reference
0–5
0–10
0.05–5
0.01–20
1–10
1–40
1–400
10–500
Tossell14
Franz et al.15
Dressler and Allan16
Jain17
Gianturco and Stoecklin18
Brüche19
Sueoka and Mori20
Iga et al.21
10–1000
10–5000
50–500
200–1400
400–2600
Jiang et al.22
Jain and Baluja23
Iga et al.21
Ariyasinghe and Powers24
Xing et al.25
previous studies carried out on e -C2 H2 scattering is presented
in Table I.
From Table I we can see that there are substantial low energy scattering studies on this target both
theoretically14, 15, 17, 18 as well as experimentally.16, 19 The case
of intermediate to high energy regime is also not different.
There are significant contributions from the theoretical21–23
and the experimental groups.19–21, 24, 25 It is clear that e -C2 H2
is a well-studied system by theoreticians and experimentalist
alike. However, different groups have worked in different energy regimes with specific motive. No single work is carried
out over a wide range of impact energy starting from thermal
energy of the target (meV) to high energy (keV). In Sec. II, we
briefly discuss the salient features of the theoretical methodologies used for the present work.
II. THEORETICAL METHODOLOGY
The present calculations rest on two distinct methodologies that are well established in the two regimes of impact
energies, one below the ionization threshold of the target and
the other above it. Below ionization threshold of the target we
employ ab initio calculations using the R-matrix formalism26
through Quantemol-N27 software. Above this incident energy
we have used the SCOP formalism23, 28–31 to compute the total cross section. Before going to the details about these two
methodologies we discuss the target model employed for low
energy calculations.
A. Target model used for low energy calculations
The accuracy of collision data depends on the accuracy with which we can represent the target wave function.
Hence, it is imperative to have an appropriate target model.
Acetylene is a linear molecule with triple carbon-carbon bond
(2.27 ao ) and single carbon-hydrogen bond (2.00 ao ). We have
employed 6-31G* and DZP basis set for the target wave function representation and D2h point group symmetry of the order eight. The occupied and virtual molecular orbitals are obtained using Hartree-Fock self-consistent field optimization
Method
(Exp. - Experimental; Th. - Theoretical)
Multiple-scattering Xα (Th.)
Density functional theory and R-matrix method (Th.)
Exp.
A close-coupling formalism (Th.)
Ab initio calculations (Th.)
Exp.
Time of flight (Exp.)
Schwinger variational method + distorted-wave
approximation (Th.)
SCOP (Th.)
SCOP (Th.)
Relative flow technique (Exp.)
Linear transmission technique (Exp.)
Linear transmission technique (Exp.)
and were used to set up the C2 H2 electronic target states. The
ground state Hartree-Fock electronic configuration is 1Ag 2 ,
1B1u 2 , 2Ag 2 , 2B1u 2 , 3Ag 2 , 1B3u 2 , and 1B2u 2 . For establishing
a balance between the amounts of correlation incorporated in
the target wave function, out of 14 electrons, we have frozen
4 electrons in two molecular orbitals 1Ag and 1B1u . The rest
of the electrons are allowed to move freely in the active space
of 8 target occupied and virtual molecular orbitals 2Ag , 3Ag ,
1B2u , 1B3u , 2B1u , 3B1u , 1B3g , and 1B2g . A total of 20 target
states were represented by 426 configuration state functions
(CSFs) for 6-31G* and 24 target states were represented by
748 CSFs for DZP and the number of channels included in the
calculation were 52 using 6-31G* and 78 using DZP basis set.
The Quantemol-N modules GAUSPROP and DENPROP
generate target properties32 and constructs the transition
density matrix from the target eigenvectors obtained from
configuration interaction (CI) calculation. The multipole
transition moments obtained are then used to solve the outer
region coupled equations and the dipole polarizability α 0 .
These are computed using second-order perturbation theory
and the property integrals are evaluated by GUASPROP.
The CI calculation yields the ground state energy of
C2 H2 as −76.86 Hartree using both 6-31G* and DZP basis
set which is in excellent agreement with theoretical value
of −76.90 Hartree reported by Cui and Morokuma.33 The
present computed rotational constant of C2 H2 is 1.182 cm−1
which is in very good agreement with theoretical value
1.174 cm−1 reported by Jain and Baluja23 and experimental
data of 1.177 cm−1 reported in Ref. 36. The first electronic
excitation energy of C2 H2 is found to be 6.074 eV using
6-31G* basis set and 6.209 eV using DZP basis set. The first
excitation energy obtained using 6-31G* basis is in excellent
agreement with experimental value predicted by Dressler and
Allan16 and the value obtained using DZP basis is in excellent
agreement with theoretical value obtained by Malsch et al.34
and lower than the theoretical value of Ventura et al.35
The target properties along with available comparisons are
listed in Table II. The 10 electronic excitation thresholds
for acetylene are listed in Table III. It is to be noted that
3B1u and 3Au ; 1Au and 1B1u ; 3B3g and 3B2g are degenerate
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184308-3
Vinodkumar, Barot, and Antony
J. Chem. Phys. 136, 184308 (2012)
TABLE II. Target properties obtained for the e -C2 H2 molecule.
Ground-state energy (Hartree)
Present
−76.87(6-31G*)
−76.87 (DZP)
Rotational constant (cm−1 )
First excitation energy E1 (eV)
Theory
Present
Th./Exp.
Present
Theory
Exp.
−76.90 (Ref. 33)
6.074 (6-31G*)
6.209 (DZP)
6.00 (Ref. 16; Exp.)
6.20 (Ref. 34; Th.)
6.53 (Ref. 35; Th.)
1.182 (6-31G*)
1.182 (DZP)
1.1740 (Ref. 23)
1.1766 (Ref. 36)
states having energy 7.067 eV, 8.326 eV and 10.992 eV
respectively using 6-31G* basis set. Also using DZP basis
set 3B1u and 3Au ; 1Au and 1B1u ; 3B3g and 1B2g ; 3B3g and
1B2g are degenerate states having energy 6.97 eV, 7.91 eV,
10.66 eV and 10.94 eV respectively.
total wave function for the system is written as
ψIN (x1 , · · · , xN )
ζj (xN+1 )aIj k
ψkN+1 = A
I
+
j
χm (x1 , · · · , xN+1 )bmk ,
(1)
m
B. Low energy formalism (1 eV to ∼15 eV)
The most popular methodologies employed for low energy electron collision calculations are the Kohn variational
method, the Schwinger variational method, and the R-matrix
method, out of which the R-matrix is the most widely used
method. The ab initio R-matrix method relies on the division of configuration space into two spatial regions, namely,
inner region and the outer region. This division is considered to make sure that the calculations are viable computationally. This spatial distribution is the consequence of electronic charge distribution around the center of mass of the
system. The inner region is so chosen that it accommodates
the total wave function of the target molecule. Thus, all the
N+1 electrons are contained in the inner region, which makes
the problem numerically complex but very precise. The interaction potential consists of short-range potentials which are
dominant in this region, e.g., static, exchange, absorption, and
correlation polarization potentials. Moreover, the inner region
problem is solved independent of the energy of the scattering electron. In the outer region, exchange and correlation are
assumed to be negligible and the only long-range multipolar interactions between the scattering electron and the target
are included. A single-center approximation is assumed here
and this suffices quick, simple, and fast solutions in the outer
region. For the present system, the inner R-matrix radius is
taken as 10 ao while the outer region calculations are extended
up to 100 ao .
The inner region wave function is constructed using
close-coupling (CC) approximation.27 In the inner region, the
TABLE III. e -C2 H2 vertical excitation energies for all states below the
ionization threshold.
State
1A
g
3B
1u
3B , 3A
1u
u
3A
u
Energy (eV) Energy (eV)
(6-31G*)
(DZP)
0.00
6.07
7.07
7.82
0.00
6.21
6.97
7.57
State
1A
u
1A , 1B
u
1u
3B , 3B
3g
2g
1B3g , 1B2g
Energy (eV) Energy (eV)
(6-31G*)
(DZP)
7.99
8.33
10.99
...
7.68
7.91
10.66
10.94
where A is the anti-symmetrization operator, xN is the spatial and spin coordinate of the nth electron, ξ j is a continuum orbital spin-coupled with the scattering electron, and
aIjk and bmk are variational coefficients determined in the calculation. The accuracy of the calculation depends solely on
the accurate construction of this wave function given vide
Eq. (1). The first summation runs over the target states used
in the close-coupled expansion and a static exchange calculation has a single Hartree-Fock target state in the first sum.
The second summation runs over configurations χ m , where
all electrons are placed in target molecular orbitals. This sum
runs over the minimal number of configurations, usually 3 or
fewer, which is required to relax orthogonality constraints between the target molecular orbitals and the functions used to
represent the configuration. Our fully close-coupled calculation uses the lowest number of target states, represented by
a CI expansion in the first term and over a hundred configurations in the second. These configurations allow for both
orthogonality relaxation and short-range polarization effects.
We have employed fixed nuclei approximation and it has
been established that such an approximation yields very accurate cross sections for electron scattering by molecules at
low energy.37 However, the processes involving the nuclear
motion such as rotational and vibrational excitation, as well
as dissociative processes, are represented in the theory by
adiabatic-nuclei approximation. This approximation is valid
since the collision time is very short compared to the vibration and rotation times.37 The complete molecular orbital representation in terms of occupied and virtual target molecular
orbitals are constructed using the Hartree-Fock self-consistent
field method with Gaussian-type orbitals and the continuum
orbitals of Faure et al.38 and include up to g (l = 4) orbitals. The benefit of employing partial wave expansion for
low energy electron molecule interaction is its rapid convergence. In the case of dipole-forbidden excitations (J = 1),
the convergence of the partial waves is rapid. But in case of
dipole-allowed excitations (J = 1) the partial wave expansion converges slowly due to the long-range nature of the
dipole interaction. In order to account for the higher partial
waves not included in the fixed nuclei T matrices, the Born
correction is applied. For low partial waves (l ≤ 4) T matrices computed from the R-matrix calculations are employed to
compute the cross sections. The low partial wave contribution
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184308-4
Vinodkumar, Barot, and Antony
J. Chem. Phys. 136, 184308 (2012)
arising from the Born contribution is subtracted in order that
the final rotational cross section set only contains those partial waves due to R-matrix calculation. We have performed
the calculations with and without the dipole Born correction.
The R-matrix will provide the link between the inner region and the outer region. For this purpose the inner region
is propagated to the outer region potential until its solutions
matches with the asymptotic functions given by the Gailitis
expansion.26 Thus, by generating the wave functions, using
Eq. (1), and their eigenvalues are determined. These coupled
single-center equations describing the scattering in the outer
region are integrated to identify the K-matrix elements. Consequently, the resonance positions, widths, and various cross
sections can be evaluated using the T-matrix obtained from Smatrix which is in turn obtained from the K-matrix element.
C. High energy formalism
High energy electron scattering is modeled using the
well-established SCOP formalism23, 28–31, 39, 40 which employs
a partial wave analysis to solve the Schrödinger equation with
various model potentials as its input. The interaction of incoming electron with the target molecule can be represented
by a complex potential comprising of real and imaginary
parts as
Vopt (Ei , r) = VR (Ei , r) + iVI (Ei , r) ,
(2)
such that
VR (r, Ei ) = Vst (r) + Vex (r, Ei ) + Vp (r, Ei ),
(3)
where Ei is the incident energy. Equation (3) corresponds
to various real potentials to account for the electron target
interaction, namely, static, exchange, and the polarization
potentials, respectively. These potentials are obtained using
the molecular charge density of the target, the ionization
potential and the polarizability as inputs. We describe the
scattering within the fixed-nuclei approximation which neglects any dynamics involving the nuclear motion (rotational
and vibrational), whereas the bound electrons are taken to
be in the ground electronic state of the target at its optimized
equilibrium geometry. The molecular charge density may
be derived from the atomic charge density by expanding it
from the center of mass of the system. The charge density of
lighter hydrogen atom is expanded at the center of heavier
atom (carbon) by employing the Bessel function expansion as
in Gradashteyn and Ryzhik.41 This is a good approximation
since hydrogen atoms does not significantly act as scattering
centers and the cross sections are dominated by the central
atom size. Thus, the single-center molecular charge density is
obtained by a linear combination of constituent atomic charge
densities, renormalized to account for covalent molecular
bonding.
The atomic charge densities and static potentials (Vst ) are
formulated from the parameterized Hartree-Fock wave functions given by Cox and Bonham.42 The parameter free Hara’s
“free electron gas exchange model”43 is used to generate the
exchange potential (Vex ). The polarization potential (Vp ) is
constructed from the parameter free model of correlationpolarization potential given by Zhang et al.44 Here, various
multipole non-adiabatic corrections are incorporated in the
intermediate region which will approach the correct asymptotic form at large “r” smoothly. In the low energy region, the
small “r” region is not important due to the fact that higherorder partial waves are unable to penetrate the scattering region. However, in the present energy region, a large number
of partial waves contribute to the scattering parameters and
correct short-range behavior of the potential is essential.
The imaginary part in Vopt is called the absorption potential Vabs and accounts for the total loss of flux scattered
into the allowed electronic excitation or ionization channels.
The Vabs is not a long-range effect and its penetration towards
the origin increases with an increase in the energy. This implies that at high energies, the absorption potential accounts
the inner-shell excitations or ionization processes that may be
closed at low energies.
The well-known quasi-free model form of Staszewska
et al.45 is employed for the absorption part given by
8π
Tloc
Vabs (r, Ei ) = −ρ(r)
θ p2 − kF2 − 2
3
2
10kF Ei
× (A1 + A2 + A3 ),
(4)
where the local kinetic energy of the incident electron is
Tloc = Ei − (Vst + Vex + Vp )
(5)
and where p = 2Ei , kF = [3π ρ(r)] is the Fermi wave vector and further θ (x) is the Heaviside unit step-function such
that θ (x) = 1 for x ≥ 0, and is zero otherwise. A1 , A2 , and
A3 are dynamic functions that depend differently on θ (x), I,
, and Ei . Here, is the principal factor that decides the
values of total inelastic cross section, since below this value
ionization or excitation is not permissible. This is one of the
main characteristics of the Staszewska model.45 In the original Staszewska model45 , = I is considered and hence it
ignores the contributions coming from discrete excitations
at lower incident energies. This has been realized earlier by
Blanco and Garcia46, 47 and has elaborately discussed the need
to modify value. This has been attempted by us by considering as a slowly varying function of Ei around I. Such
an approximation is meaningful since fixed at I would not
allow excitation at energies Ei ≤ I. However, if is much
less than the ionization threshold, then Vabs, becomes unexpectedly high near the peak position. The amendment introduced by us is to give a reasonable minimum value 0.8 I to (Ref. 48) and also to express the parameter as a function of Ei
around I, i.e.,
2
2
1/3
(Ei ) = 0.8I + β(Ei − 1).
(6)
Here, the value of the parameter β is obtained by requiring that = I (eV) at Ei = Ep , the value of incident energy at
which Qinel reaches its peak. Ep can be found by calculating
Qinel by keeping = I. Beyond Ep , is kept constant and is
equal to I. The theoretical basis for variable is discussed in
more detail by Vinodkumar et al.49 Expression (6) is meaningful since if is fixed at the ionization potential it would
not allow any inelastic channel to open below I. Also, if it is
much less than I, then Vabs become significantly high close to
the peak position of Qinel .
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184308-5
Vinodkumar, Barot, and Antony
J. Chem. Phys. 136, 184308 (2012)
The complex potential thus formulated is used to solve
the Schrödinger equation numerically through partial wave
analysis. This calculation will produce complex phase shifts
for each partial wave which carries the signature of interaction of the incoming projectile with the target. At low impact
energies only a few partial waves are significant, but as the
incident energy increases more partial waves are needed for
convergence. The phase shifts (δ l ) thus obtained are employed
to find the relevant cross sections, total elastic (Qel ), and the
total inelastic cross sections (Qinel ) using the scattering matrix
Sl (k) = exp(2iδ l ).48 The total cross sections like the total elastic (Qel ) and the total inelastic cross sections (Qinel ) can be derived from the scattering matrix.39 The sum of these cross sections will give the total scattering cross section (TCS), QT .40
III. RESULTS AND DISCUSSION
In the present work, we have carried out a comprehensive computation for total cross section produced by collision of electrons with acetylene molecules in gas phase using two basis set viz. 6-31G* and DZP. We have presented
here the cross sections over a wide range of impact energies
from 1 eV to 5000 eV. The theoretical formalisms have its
own limits over the range of impact energies. More elaborately, the ab initio calculations are computationally viable
only up to around 20 eV, while SCOP formalism could be employed successfully from threshold of the target to 5000 eV.
Our attempt in the present work is to compute the total cross
section below the ionization threshold using close-coupling
formalism employing the R-matrix method and beyond ionization threshold using the SCOP formalism. The results so
obtained are consistent and there is a smooth transition at
the overlap of two formalisms. Thus, it is possible to provide
the total cross section over a wide range of impact energies
from meV to keV. We have presented our results in graphical
form (Figs. 1–4) and numerical values are tabulated separately
for low energy calculations and high energy calculations vide
Tables IV and V.
Figure 1 shows the eigenphase diagram for various doublet scattering states (2Ag , 2B2u , 2B3u , 2B1g , 2B1u , 2B3g , and
FIG. 1. C2 H2 eigenphase sums for a 20-state CC calculation.
FIG. 2. C2 H2 excitation cross section for a 20-state CC calculation from
initial state 1Ag .
2B2g ) of C2 H2 system. It is to be noted here that 2B2u and 2B3u
and 2B2g and 2B3g are degenerate states and hence a single
curve is shown for them in Fig. 1. The study of eigenphase diagrams is significant as they reflect the position of resonances
which are important features in the low energy regime. 2B3u
state shows a prominent structure between 3 and 4 eV which
predicts resonance in this energy range. This is reflected in
the TCS curves. Similarly, another structure is seen in 2B1u
state around 7 eV which can be visualized as a hump around
7 eV in the TCS curve. Moreover, they also show the important channels which are included in the calculations. It is to
be noted that as more states are included in the CC expansion
and retained in the outer region calculation, the eigenphase
sum increases reflecting the improved modeling of polarization interaction.
FIG. 3. TCS for e -C2 H2 scattering. Solid line: Present results (Q - mol DZP
with Born correction); Dashed line: Present results (Q -mol 6-31G* with
Born correction); short dashed line: Tossell;14 short dashed-dotted line: Franz
et al.;15 dashed-dotted-dotted line: Jain;17 dashed-dotted line: Gianturco and
Stoecklin;18 spheres: Brüche;19 stars: Dressler and Allan;16 open circles:
Sueoka and Mori.20
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184308-6
Vinodkumar, Barot, and Antony
J. Chem. Phys. 136, 184308 (2012)
TABLE V. Total cross sections (TCS) for e -C2 H2 scattering in Å2 using
SCOP formalism.
Energy
(eV)
16
20
30
40
60
80
100
150
200
300
400
FIG. 4. TCS for e -C2 H2 scattering. Solid line: Present results (Q-mol
DZP with Born correction); short Dashed line: Present results (Q-mol 631G* with Born correction); dashed line: Present (SCOP); short Dashed line:
Tossell;14 dashed line at low energy: Franz et al.;15 dotted line: Jain and
Baluja;23 dashed-dotted-dotted line: Jain;17 dashed-dotted line: Gianturco
and Stoecklin;18 short dotted line: Jiang et al.;22 short dashed-dotted: Iga
et al.;21 spheres: Brüche;19 stars: Dressler and Allan;16 open circles: Sueoka
and Mori;20 open diamonds: Xing et al.;25 open hexagons: Ariyasinghe and
Powers;24 open triangles: Iga et al.21
Figure 2 depicts the electronic excitation cross sections
for ground state 1Ag to low lying 8 excited states (3B1u , 3B1u ,
3Au , 3Au , 1Au , 1Au , 1B1u , and 3B3g ) for the 20-states CC calculation. 3B1u and 3Au and 1Au and 1Bu are degenerate states
are shown as single curve in Fig. 2. It can be seen from the
graph that the threshold of the first electronic excitation energy for C2 H2 is 6.21 eV. The electronic excitations to 3B1u
and 3B3g shows sharp increase near their respective thresholds which show the dominance of these energy levels in
the present calculation. The notable structure in 3B1u around
7 eV is reflected as hump around 7 eV in the TCS. These cross
TABLE IV. Total cross sections (TCS) for e -C2 H2 scattering in Å2 using
the R-matrix formalism.
Energy
(eV)
1.00
1.50
2.00
2.50
3.00
3.20
3.44
3.71
3.8
4.0
4.5
5.0
5.5
6.0
6.5
7.0
QT
(6-31G*)
QT
(DZP)
Energy
(eV)
QT
(6-31G*)
QT
(DZP)
24.68
23.53
23.83
26.76
34.61
38.75
42.26
44.41
44.22
42.93
37.87
33.61
30.72
28.82
27.44
26.69
24.07
23.68
25.47
31.25
41.26
44.43
46.00
45.07
44.36
42.46
37.62
33.92
31.35
29.62
28.36
27.58
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
26.10
25.66
25.23
24.76
24.21
23.62
23.02
22.41
21.76
21.26
20.83
20.42
20.04
19.55
19.07
19.05
26.99
26.56
26.17
25.67
25.11
24.50
23.89
23.24
22.67
22.17
21.50
20.89
20.45
19.98
19.57
19.18
QT
(SCOP)
Energy
(eV)
QT
(SCOP)
18.68
17.53
15.09
13.09
10.47
9.08
8.13
6.44
5.32
3.95
3.16
500
600
700
800
900
1000
1500
2000
3000
4000
5000
2.65
2.28
2.00
1.79
1.62
1.49
1.05
0.81
0.38
0.40
0.20
sections show the probability of excitation to various energy
levels of the target.
For the brevity of the figure we have presented the total
cross sections vide Figs. 3 and 4. Figure 3 shows the comparison of the present total cross section (using 6-31G* and
DZP) for low impact energies up to 10 eV obtained using the
R-matrix with all available results. The lower cross section at
the low impact energies clearly reflects the non-polar nature
of the molecule. The two salient features of this graph are a
strong peak around 2–3 eV which is attributed to 2 g shape
resonance and a broad hump around 7 eV. The present results
using static-exchange plus polarization can very distinctly
show both the features. The present results show a strong
peak of about 44 Å2 positioned at about 3.8 eV using 6-31G*
and 46.0 Å2 positioned at 3.4 eV using DZP. This feature
is also seen in experimental results of Brüche19 (39.5 Å2 at
2.75 eV), Dressler and Allan16 (38.0 Å2 at 2.95 eV), Sueoka
and Mori20 (35.8 Å2 at 2.5 eV), and theoretical data of
Tossell14 (87.0 Å2 at 2.63 eV), Jain17 (54.3 Å2 at 2.5 eV),
Gianturco and Stoecklin18 (51.94 Å2 at 2.75 eV), and Franz
et al.15 (50.8 Å2 at 2.6 eV). Present theoretical data yields
lowest peak compared to all theoretical data14, 15, 17, 18 and data
closer to experimental investigations.16, 19, 20 Thus, it could
be seen that present results of TCS are in good agreement
with other experimental results both in terms of peak value
of TCS as well as position of energy except measurements of
Tossell14 which show a very high peak compared to all other
data presented here although the peak position is 2.7 eV as
expected. The second feature of broad hump around 7 eV is
clearly evident from the Fig. 3 and it is in accordance with
the experimental results of Brüche19 and Sueoka and Mori20
and theoretical data of Jain.17 Beyond 4 eV the present
results are in good agreement with the theoretical results
of Gianturco and Stoeckelin,18 Jain16 and the experimental
results of Brüche.19
Figure 4 shows the comparison of total cross section for
e -C2 H2 scattering over a wide range of incident energy from
1 eV to 5000 eV. Beyond 10 eV the present data find excellent agreement with all the data presented here17, 19–25 except
theoretical results of Jiang et al.22 and Jain and Baluja.23 The
reason for this discrepancy is the fact that Jain and Baluja23
have added the contribution from rotational excitation
Downloaded 06 Jul 2012 to 144.82.174.90. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
184308-7
Vinodkumar, Barot, and Antony
using first order Born approximation incoherently while Jiang
et al.22 have used modified additivity rule which is basically
an additivity approach modified by geometrical screening and
this overestimates the results. The agreement of present results with all experimental groups 16, 19–21, 24, 25 clearly reflects
the success and consistency of SCOP formalism. Other important feature which can be clearly visualized from Fig. 4 is
a smooth transition of total cross section data at the overlap
energy (∼15 eV). Thus, we are able to present data for wide
energy range which is done probably for the first time for this
target.
J. Chem. Phys. 136, 184308 (2012)
cross sections over a wide range of impact energies in other
molecular systems where experiments are difficult or not possible to perform. Total cross section data find importance in
a variety of applications from aeronomy to plasma modeling.
Accordingly, such a methodology may be built into the design of online database to provide “data user” with the opportunity to request their own set of cross sections for use
in their own research. Such a prospect will be explored by
the emerging Virtual Atomic and Molecular Data Centre52
(http://batz.lpma.jussieu.fr/www_VAMDC).
ACKNOWLEDGMENTS
IV. CONCLUSIONS
A detailed study of e -C2 H2 system in terms of eigenphase, electronic excitations, and total cross sections is performed and reported in the article. We demonstrate here with
the help of eigenphase diagram (Fig. 2) that a CC calculation can give much more information than a simple staticexchange calculation at low energies. We can readily obtain
the position of resonances that may arise due to electron interaction from these graphs. In the present study, we can see
that the 2B3u state shows a prominent structure between 3 and
4 eV which is due to 2 g shape resonance and is clearly reflected as a strong peak of 44 Å2 at 3.8 eV using 6-31G*
basis set and 46 Å2 at 3.4 eV using DZP basis set around
the same energy in the TCS curve vide Fig. 3. We have performed close-coupling calculations employing UK molecular
R-matrix code below ionization threshold of the target while
SCOP formalism is used beyond it. We have demonstrated
through Fig. 3 that the results using these two formalisms are
consistent and show a smooth transition at the overlap energy
(∼15 eV). This feature confirms the validity of our theories
and hence enables us to predict the total cross sections from
meV to keV.50, 51
Acetylene is an extensively studied target both by theoreticians and experimentalist as discussed earlier. The values reported earlier differ significantly at most of the energy range, especially between theoretical and experimental
results. At about 2.5 eV (near the peak) the difference between the experimental and theoretical values is about 20%,
while between the measurements the variation is about 10%.
Beyond 20 eV, all the theoretical results except that of Iga
et al.,21 overestimate the experimental values by more than
25%. These discrepancies will have serious consequences if it
is to be used for further modeling. Present theoretical efforts
are to validate the cross sections for acetylene and come out
with a recommended value which may be of applied interest.
Hence, we have chosen this target so that present methodology may be put to test and benchmark our results by comparing with previous works. Even though there is a noticeable
deviation in the low energy region, present results are in excellent agreement with measurements at high energies. Also, at
low energies all the theoretical values overestimate the measurements to a large extend. Here, present results at midway
between the experimental values and other theories. In general, present results agree quite well with other measurements
throughout the energy range. Therefore, we are confident that
this methodology may be employed further to calculate total
M.V.K. thanks Department of Science and Technology, New Delhi, for financial support through Major Research Project Grant No. SR/S2/LOP-26/2008 and B.K.A.
thanks University Grants Commission, New Delhi, for financial support through Major Research Project Grant No. 39119/2010(SR) under which part of this work is carried out.
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