Electron induced chemistry of thiophene

Abstract

A comprehensive theoretical study of electron scattering with thiophene over a wide impact energy range is reported in this article. Total, elastic, differential and momentum transfer cross sections were computed at low energy using an ab initio R-matrix method through QUANTEMOL-N. The R-matrix calculations were carried out using the Complete Active Space–Configuration Integration (CAS–CI) method employing Static Exchange (SE) and Static Exchange plus Polarization (SEP) models. Beyond the ionization threshold, from intermediate to high energy the calculations were carried out using the Spherical Complex Optical Potential (SCOP) formalism. There is a smooth crossover of the two formalisms at the overlap energy, and hence we are able to predict the cross sections over a wide energy range. Apart from the scattering cross section calculations, the other focus was to obtain resonances which are important features at low energy. We observed three prominent structures in the total cross section (TCS) curve. The first peak at 2.5 eV corresponds to the formation of a σ* resonance which is attributed to a Feshbach resonance, in good agreement with earlier predicted experimental and theoretical values of 2.65 eV and 2.82 eV, respectively. The second peak observed at 4.77 eV corresponds to the shape resonance that resembles earlier predicted experimental values of 5 eV and 5.1 eV, which is attributed to ring rupture. The third peak at 8.06 eV is attributed to a core excited shape resonance. There is a lone previous theoretical dataset for the total cross section by da Costa et al. [R. F. da Costa, M. T. do N. Varella, M. Lima, and M. Bettega, 2013, J. Chem. Phys., 138, 194306] from 0 to 6 eV, and no other theoretical or experimental work is reported at low energy to the best of our knowledge. Hence the present work is important to fill the void of scattering data as the earlier work is fragmentary. The differential, momentum transfer and excitation cross sections beyond 6 eV are reported for the first time.

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1 Introduction

Thiophene (Fig. 1) is one of the most important and largely studied heterocyclic aromatic compounds,^1–3 yet it remains a focal point of on-going research owing to the central role it plays in the challenges facing modern science and technology. Specifically, the materials that include thiophene units, such as thiophene polymers and oligomers, possess various important properties,^4–6 which make them promising as photochromatic molecular switches,^7–9 organic semiconductors,^10,11 solar cells,^12,13 light-emitting diodes and field-effect transistors.^14–16 For many of these applications, understanding the fundamental electronic structure, spectroscopy, photophysics and scattering data of thiophene is of primary importance. Hence, considerable efforts are constantly being made to gain further knowledge in these fields. Apart from this industrial importance, low energy electron impact studies for bio-molecules and related systems have gained prominence after the pioneering work reported by Boudaiffa et al.¹⁷ Now it is well accepted that bond cleavages in DNA produced by the impact of slow secondary electrons occur through the formation of transient negative ion formation (anions). It is worth mentioning that the formation of such metastable states (transient negative ion formation) is a very efficient way to release energy to the nuclear degrees of freedom, especially through the vibrational channel. As a result, the process may eventually give rise to one or more separated fragments, which is also known as a dissociative electron attachment (DEA). Therefore, the understanding and the characterization of resonances represent important steps towards a deeper insight into the damage induced to DNA through DEA mechanisms.¹⁸ Looking to the importance of such studies of thiophene, it is surprising that not much work has been carried out on electron impact studies. To date, a low energy electron impact study was carried out recently by da Costa et al.,²⁰ where they reported electron impact integral, momentum transfer and differential cross sections for thiophene molecules using the Schwinger multichannel method for very low impact energies from 0.5 eV to 6 eV. Mozejko et al.¹⁹ reported integral elastic cross sections from 40 eV to 3000 eV and ionization cross sections from threshold to 3000 eV using the Binary Encounter Bethe (BEB) method. Apart from these, Modelli and Burrow²¹ reported electron transmission spectra for thiophene. Very recently Holland et al.²² resolved the photoabsorption spectrum between 5 and 12.5 eV and Hedhili et al.²³ reported measurements for electron impact on multilayer thiophene condensed on a polycrystalline platinum substrate. Wehlitz and Hartman²⁴ studied double ionization of thiophene using monochromatized synchrotron radiation over a wide range of photon energies, in combination with the ion time of flight technique. Zhang et al.²⁵ measured the binding energy spectrum and momentum distributions of the valence orbitals of thiophene using electron momentum spectroscopy. Haberkern et al.²⁶ measured high-resolution electron-energy-loss spectra of thiophene in the range of the low-lying singlet–triplet excitations. Muftakhov et al.⁴⁴ studied the dissociative attachment of electrons to thiophene in the gas phase in the energy range 0–12 eV. Looking to the literature survey, it is quite evident that electron impact scattering studies are very scarce and in the absence of any experimental data and fragmentary theoretical data, the present study will be quite meaningful and important. In the present work we study electronic excitation cross sections, differential cross sections, momentum transfer cross sections and total cross sections for e-C₄H₄S scattering.

Fig. 1 Schematic diagram of the thiophene molecule.

Electron–molecule collision cross sections from very low energy up to threshold play an important role in determining electron transport properties and the electron energy distribution of a swarm of electrons drifting through various gases. They also play a significant role in modeling low temperature plasmas. In addition to practical interest, electron scattering data are of fundamental theoretical importance towards the understanding of various electron assisted molecular chemistry.²⁷ The electron bombardment of a molecule may result in the formation of positive and negative ions. The latter may be produced by resonance attachment, dissociative resonance capture and ion-pair formation. These mechanisms provide us with vast understanding about various chemical changes that will take place with the target at different electron energies. The resonance processes usually occur in the 0–10 eV energy region and the ion-pair processes at energies above 10 eV.

This paper is organised as follows: in Section 2 we describe first the target model, and then describe the salient features of the theoretical methodologies employed for low energy as well as high energy calculations. Section 3 is devoted to results and discussions of the results obtained, and finally we end up with the conclusions of the present study.

2 Theoretical methodology

Considerable progress has been made in the experimental and theoretical study of electron–molecule collision processes in the past decades. By utilizing better electron spectrometers and adopting position-sensitive detectors, experimentalists are capable of producing accurate cross-sectional data on electron collisions with larger molecules and even exploring free radical species. However, given the vast number of molecular systems and the requirement for an ever-increasing amount of data, the experimental community are unable to meet the demands of the myriad of data users. In this respect one must look to theory to provide much of the required electron-scattering data. On the theoretical front, with the advent of high-performing computers and the development of very accurate theories, computation of reliable cross-section data is now possible, at least for smaller targets. These theoretical methods are computationally taxing and consume longer computing time. Thus, there is a demand for more generic and faster calculations to provide reliable data to the user community.

For low energy (0.01 eV to about 20 eV), we employed ab initio calculations using QUANTEMOL-N²⁸ which utilizes the UK molecular R-matrix code,²⁹ while the SCOP method is employed for calculating total (elastic plus inelastic) cross sections beyond the ionization threshold up to 5 keV.³⁰ The salient features of these two formalisms are briefly discussed in the following subsections. Before going to the details of the theoretical methods we also discuss the target model employed for the present system.

2.1 Target model

The accuracy of scattering data depends on the accuracy of the target wave function, hence, it is advisable to have an appropriate target model. For many-electron targets like C₄H₄S, the relative energy between the N-target electrons and the N + 1 target plus scattering electron becomes important since neither the target nor the scattering wave functions have energies close to the exact value for the given system. This requires careful choice of the configurations in terms of a complete active space (CAS) and the valence configuration interaction (CI) representation of the target system.²⁸ This is realized by characterizing the low lying electronic states of the target and by generating a suitable set of orbitals. The molecular orbitals are generated by performing a self-consistent field (SCF) calculation of the ground state of the molecule (X1A₁). Since the SCF procedure is inadequate to provide a good representation of the target states, we improve the energies of these states by invoking the variational method of configuration interaction (CI) in which we take a linear combination of configuration state functions (CSFs) of a particular overall symmetry. This lowers the energies and the correlation introduced provides a better description of the charge cloud and the energies. For all the states included here, we employ a CI wave function to represent the target states.

The Hartree–Fock electronic configuration for the ground state of C₄H₄S at its equilibrium geometry in C_2v point group symmetry is 1a₁², 1b₂², 2a₁², 3a₁², 2b₂², 4a₁², 3b₂², 5a₁², 1b₁², 6a₁², 4b₂², 7a₁², 8a₁², 5b₂², 9a₁², 6b₂², 10a₁², 7b₂², 2b₁², 11a₁², 3b₁², 1a₂². Out of a total of 44 electrons, 36 electrons are frozen in 1a₁, 2a₁, 3a₁, 4a₁, 5a₁, 6a₁, 7a₁, 8a₁, 9a₁, 10a₁, 11a₁, 1b₁, 2b₁, 1b₂, 2b₂, 3b₂, 4b₂, 5b₂, 6b₂, 7b₂ molecular orbitals and 8 electrons are kept free to move in the active space of the 3b₁, 4b₁, 8b₂, 1a₂ molecular orbitals. We employed 6-311G* Gaussian type orbitals (GTO) and Double Zeta plus Polarization (DZP) basis sets.

The target wave functions are computed using the complete active space–configuration integration (CAS–CI) method. They are subsequently improved using a pseudo-natural orbital calculation. The Born correction for this polar molecule is employed to account for higher partial waves, l > 4. In the static-exchange-polarization (SEP) model, the ground state of the molecule is perturbed by single and double excitations of the electrons, thus leading to the inclusion of polarization effects. The SEP model augments the static exchange (SE) model by including polarization effects. Thus polarization effects are accounted for by including closed channels in a CI expansion of the wave function of the entire scattering system. These electronic and angular momentum channels altogether generated 864 configuration state functions (CSFs) and 138 channels in the calculation.

The QUANTEMOL-N modules GAUSPROP and DENPROP³¹ are employed to construct the transition density matrix from the target eigenvectors obtained from the Configuration Integration (CI) expansion and generate the target properties. The multipole transition moments obtained are then used to solve the outer region coupled equations and the dipole polarizability α₀. These are computed using second-order perturbation theory and the property integrals are evaluated by GUASPROP.³¹ Our self-consistent field (SCF) calculations yielded target parameters such as the ground state energy, the first electronic excitation energy, rotational constant, dipole moment and ionization energy which are listed in Table 1.

Table 1 Target properties obtained for the C₄H₄S molecule using 6-311G* and DZP basis sets

Target property (Unit)	Present		Other
	6-311G*	DZP	Th./Exp.
Ground state energy (Hartree)	−551.35	−551.34	−551.37 (ref. 25)
First excitation energy (eV)	4.51	4.46	5.78 (ref. 22)
			5.64 (ref. 22)
			3.8 (ref. 32)
Rotational constant (cm^−1)	0.2683	0.2683	0.2683 (ref. 33)
Dipole moment (Debye)	0.79	0.68	0.55 (ref. 34)
			0.64 (ref. 25)
			0.55 (ref. 35)
			0.53 (ref. 36)
Ionization potential (eV)	8.91	8.93	8.87 (ref. 32)
			8.86 (ref. 33)

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The self-consistent field calculations yielded the ground state energies of −551.35 Hartree and −551.34 Hartree using 6-311G* and DZP basis sets, which are in very good agreement with −551.37 Hartree reported by Zhang et al.²⁵ We report 10 electronic excitation states below the ionization threshold of the target for thiophene with the first electronic excitation energy obtained at 4.51 eV using the 6-311G* and at 4.46 eV using the DZP basis sets, as listed in Table 1. The present first excitation energy is close to 4.7 eV as reported by Palmer et al.³² and slightly lower compared to 5.64 eV as reported by Holland et al.²² The present rotational constant of 0.2683 cm⁻¹ is in perfect agreement with the theoretical value of 0.2683 cm⁻¹ reported in the CCCBDB (Computational Chemistry Comparison and Benchmark DataBase).³³ The present computed dipole moments of 0.79 D obtained using the 6-311G* and 0.68 D obtained using the DZP basis sets are close to the measured value of 0.64 D reported by Zhang et al.,²⁵ and slightly higher compared to 0.55 D reported by CRC³⁷ and Ogata et al.³⁵ and 0.53 D reported by Pozdeev et al.³⁶ It can be easily seen that the dipole moment is very sensitive to the basis set chosen. The DZP basis set gives better target property calculations for thiophene as compared to the 6311G* basis set, as is evident from Table 1. The present calculated ionization threshold is 8.91 eV or 8.93 eV using the 6311G* and DZP basis sets, against 8.87 eV or 8.86 eV reported by Palmer et al.³² or in the CCCBDB,³³ respectively. Table 2 shows vertical excitation energies for thiophene obtained using the 6311G* and DZP basis sets.

Table 2 Vertical excitation energies for thiophene below the ionization threshold for the 6-311G* and DZP basis sets

6-311G*		DZP
State	Energy (eV)	State	Energy (eV)
1A1	0.00	1A1	0.00
3B2	4.51	3B2	4.46
3A1	5.71	3A1	5.52
3B1	7.41	1A1	7.24
1B1	7.59	3A1	7.54
3A2	7.68	3B1	7.57
1A2	7.89	1B2	7.67
1B2	7.98	3A2	7.68
1A1	8.28	1B1	7.88
3A2	8.44	1A2	7.98
1A2	8.52	3B2	8.01

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2.2 Low energy formalism (0.01–20 eV)

The low energy calculations use the close coupling method in which the total wave function of the system (e + molecule) is represented as a superposition of the ground state and all excited state wave functions of the target. The most popular methodologies employed for low-energy electron collision calculations are the Kohn variational method,^38,39 the Schwinger multichannel method,^40,41 and the R-matrix method²⁹ which make use of the close coupling (CC) method. Out of the three methods, the R matrix is the most widely used method. The basic idea behind the R-matrix method²⁹ is to split the configuration space describing the scattered electron and target into an inner region, which is a sphere of radius ‘a’ about the target center of mass, and an outer region. The boundary between these two regions is defined by the R-matrix radius. This radius is chosen large enough so that, in the external region, only known long-range forces are effective and anti-symmetrization effects such as exchange and electron–electron correlation can be neglected. In fact, in the internal region where the scattered electron has penetrated the charge distribution of the target, the interaction is strong. In the inner region, the full electron–molecule problem is solved using Quantum Chemistry codes. The inner region is usually chosen to have a radius of around 10 au and the outer region is extended to about 100 au. The choice of this value depends on the stability of the results obtained in the inner region and outer region calculations. We describe the scattering within the fixed-nuclei (FN) approximation that neglects any dynamics involving the nuclear motion (rotational as well as vibrational), whereas the bound electrons are taken to be in the ground electronic state of the target at its optimized nuclear geometry. This is an effect of the extent of the electronic charge density distribution around the center of mass of the target. In the present study we considered 13 au for the inner R-matrix radius.

In the inner region the total wave function for the system is written as

where A is the antisymmetrization operator, x_N is the spatial and spin coordinate of the n^th electron, ζ_j is a continuum orbital spin-coupled with the scattering electron and a_Ijk and b_mk are variational coefficients determined in the calculation. The summation in the first term runs over the target states used in the close-coupled expansion. The summation in the second term runs over configurations χ_m, where all electrons are placed in target molecular orbitals. The number of these configurations varies considerably with the model employed. With the wave function given by eqn (1), a static exchange calculation has a single Hartree–Fock target state in the first sum. The second term runs over the minimal number of configurations, usually 3 or fewer, required to relax orthogonality constraints between the target molecular orbitals and the functions used to represent the configuration. Our fully close-coupled system 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.

The target and the continuum orbitals are represented by Gaussian type orbitals and the molecular integrals are generated by the appropriate molecular package. The R-matrix will provide the link between the inner region and outer region. For this purpose the inner region is propagated to the outer region potential until its solutions match with the asymptotic functions given by the Gailitis expansion.²⁹ Thus, by generating the wave functions, using eqn (1), their eigenvalues are determined. These coupled single centre equations describing the scattering in the outer region are integrated to identify the K-matrix elements. The K-matrix is a symmetric matrix whose dimensions are the number of channels. All the observables are basically deduced from it, and further, it is used to deduce the T-matrix using the relation:

The T-matrices are in turn used to obtain various total cross sections. The K-matrix is diagonalized to obtain the eigenphase sum. The eigenphase sum is further used to obtain the position and width of the resonance by fitting them to the Breit Wigner profile⁴² using the program RESON.⁴² General diagrams for the ‘QUANTEMOL-N’ UK molecular R-matrix codes⁴³ for the target calculations, inner region calculations and outer region calculations are presented in Fig. 2, 3 and 4, respectively.

Fig. 2 Structure of a target calculation in the polyatomic UK molecular R-matrix codes.

Fig. 3 Structure of the inner region calculation in the polyatomic UK molecular R-matrix codes.

Fig. 4 The main modules used for the outer region in the polyatomic UK molecular R-matrix codes.

Fig. 2 shows the steps taken to generate the target wave functions and associated properties as a prelude to a full scattering run.³¹ The first three modules, SWMOL3, SWORD and SWFJK perform the necessary integral evaluation. SWTRMO converts integrals over atomic orbitals to integrals over molecular orbitals which are used in the (CI) calculation. SWEDMOS makes the orbital set orthonormal by Schmidt orthogonalization. The configuration generation for the Hamiltonian matrix is done by CONGEN. SCATCI is the workhorse of the inner region calculation that constructs the Hamiltonian matrix and diagonalizes it. Modules GAUSPROP and DENPROP calculate the target properties. It should be noted that the molecular R-matrix with pseudo states (RMPS) calculations involve running the target section twice. Fig. 3 shows the modules involved in the inner region calculations. The integrals over atomic orbitals are again computed using SWMOL3 and those involving a continuum orbital are adjusted for the finite dimension of the R-matrix radius using GAUSTAIL. Thus target properties, (N + 1) CI vectors and boundary amplitudes computed using the inner region calculation go as the input for the outer region calculation.

Unlike the inner region codes, the outer region is run as a single program. Fig. 4 shows the important modules used for the scattering calculations. SWINTERF acts as the boundary between the inner region and outer region codes. It reads in the target properties, boundary amplitudes, R-matrix pole positions and associated eigenvectors from the inner region. From these it constructs a list of asymptotic channels necessary to construct the R-matrix. RSOLVE is the main workhorse of the outer region code. The output from RSOLVE is a set of K-matrices, and TMATRX is used to turn K-matrices to T-matrices, which, in turn, through IXSEC, gives integral cross sections. Module EIGENP diagonalizes the K-matrix to give the eigenphase sum, which is analyzed for resonances by RESON. TIMEDEL fits these resonances and provides the position and width of resonances. POLYDCS⁴⁵ calculates differential and momentum transfer cross sections.

Differential cross section (DCS) study is very important as it provides a large amount of information about the interaction processes. Indeed, the evaluation of DCSs is a stringent test for any scattering theory as it is sensitive to effects which are averaged out in integral cross sections. The DCS for a polyatomic molecule is represented by:

where σ represents the cross-section, Ω represents the solid angle, represents the differential cross-section and P_L represents the Legendre polynomial of order L. The details about A_L have already been discussed by Gianturco and Jain.⁴⁶ For a polar molecule this expansion over L converges slowly due to the long range nature of the dipole potential. To overcome this problem we use the closure formula given by:Here the superscript B denotes the fact that the relevant term is calculated under the Born approximation with an electron point dipole interaction. It is clear that convergence of the series is faster as the contribution arising from the Born term is subtracted, as seen in eqn (4). The quantity for any initial rotor state is given by the sum over all final rotor states as

The calculated dipole moment (D) and rotational constants (A = 0.2683 cm⁻¹, B = 0.1804 cm⁻¹, C = 0.1079 cm⁻¹) for C₄H₄S are used in the calculation of the elastic DCS (J = 0 → J′ = 0) and rotationally inelastic (J = 0 → J′ = 1, 2, 3, 4 and 5) DCSs at different collision energies.

In fact, the Momentum Transfer Cross Section (MTCS) is obtained by integrating the differential cross sections (DCS) with a weight factor of (1 − cos θ).

where σ_m is a momentum transfer cross section.

2.3 Higher energy formalism (threshold to 5 keV)

The R-matrix calculations cannot be extended beyond 20 eV even with the latest modern computers due the complexities involved in the scattering calculations. Hence, the scattering calculations above the ionization threshold are studied using the SCOP formalism.^30,47 In this formalism, the electron–molecule system is represented by a complex optical potential comprising real and imaginary parts as

V_opt(r, E_i) = V_R(r) + iV_I(r, E_i) (7)

such that

V_R(r, E_i) = V_st(r) + V_ex(r, E_i) + V_p(r, E_i) (8)

where E_i is the incident energy. Eqn (8) corresponds to various real potentials to account for the electron target interaction, namely static, exchange and polarization potentials, respectively. These potentials are obtained employing the target geometry, molecular charge density of the target, the ionization potential and polarizability as inputs. The molecular charge density may be derived from the atomic charge density by expanding it from the center of mass of the system. The molecular charge density so obtained is renormalized to account for the total number of electrons present. The atomic charge densities and static potentials (V_st) are formulated from the parameterized Hartree–Fock wave functions given by Cox and Bonham.⁴⁸

The parameter free Hara’s free electron gas exchange model⁴⁹ is used to account for any exchange between the incoming electron and one of the target electrons through the exchange potential (V_ex). The polarization potential (V_p) arises from the transient redistribution of the target charge cloud due to the incoming electron which gives rise to dipole and quadrupole moments. This potential is formulated from the parameter free model of the correlation–polarization potential given by Zhang et al.⁵⁰ Here, various multipole non-adiabatic corrections are incorporated in the intermediate region which will approach the correct asymptotic form at large r smoothly. The target parameters like ionization potential (IP) and dipole polarizability (α₀) of the target used here are the best available from the literature.⁵¹

The imaginary part in V_opt, called the absorption potential V_abs accounts for the total loss of flux scattered into the allowed inelastic channels, namely discrete electronic excitation channels or excitations leading to continuum states i.e. ionization channels. The expression used here is vibrationally and rotationally elastic. This is due to the fact that the non-spherical terms do not contribute much to the total potential at the present high energy range.

The well-known quasi-free model of Staszewska et al.⁵² is employed for the absorption part, given by

where T_loc is the local kinetic energy of the incident electron which is given by

T_loc = E_i − (V_st + V_ex + V_p) (10)

Here p² = 2E_i, is the Fermi wave vector and A₁, A₂ and A₃ are dynamic functions that depend differently on θ(x), I, Δ and E_i. Here, I is the ionization threshold of the target, θ(x) is the Heaviside unit step-function and Δ is an energy parameter below which V_abs = 0. Hence, Δ is the principal factor which decides the values of the total inelastic cross section, since below this value ionization or excitation is not allowed. This is one of the main characteristics of the Staszewska model.⁵² This has been modified by us by considering Δ as a slowly varying function of E_i around I. Such an approximation is meaningful since Δ fixed at I would not allow excitation at energies E_i ≤ I. However, if Δ is much less than the ionization threshold, then V_abs becomes unexpectedly high near the peak position. The amendment introduced is to give a reasonable minimum value of 0.8I to Δ⁵³ and also to express the parameter as a function of E_i around I, i.e.

Δ(E_i) = 0.8I + β(E_i − I) (11)

Here the parameter β is obtained by requiring that Δ = I (eV) at E_i = E_p, the value of incident energy at which the present total inelastic cross section Q_inel reaches its peak. E_p can be found by calculating Q_inel by keeping Δ = I. Beyond E_p, Δ is kept constant and is equal to I. The expression given in eqn (11) is meaningful as Δ fixed at the ionization potential would not allow any inelastic channel to open below I. Also, if it is much less than I, then V_abs becomes significantly high close to the peak position of Q_inel. This has been elaborately discussed in our earlier paper by Vinodkumar et al.⁵³

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 carry the signature of the interaction of the incoming projectile with the target. Phase shifts are the key ingredients for all scattering calculations. At low energies only a few phase shifts (5–6 for absorption and 100 for polarization at the ionization threshold) are significant, but as the incident energy increases, more phase shifts (around 40 for absorption and 100 for polarization) are needed for convergence. The phase shifts δ_l thus obtained are employed to find the relevant total elastic (Q_el) and total inelastic (Q_inel) cross sections using the scattering matrix S_l(k) = exp(2iδ_l).⁵⁴ Then, the total scattering cross section (TCS) Q_T is obtained by adding these two cross sections.⁵⁴

3 Results and discussion

The present work reports total cross sections for e-C₄H₄S scattering. We have employed the ab initio R-matrix code below the ionization threshold of the target. In this energy range the total cross section is the sum of the total elastic and total electronic excitation cross sections. Above it, we have computed the total cross section as the sum of total elastic and total inelastic cross sections using the SCOP formalism. With the amalgamation of these two formalisms we are able to predict the total cross sections over a wide energy range.^55–58 The numerical results of the total cross sections for C₄H₄S are reported from 0.01 eV to 5000 eV as listed in Table 3 and are also plotted graphically (Fig. 10).

Table 3 Total Cross Sections (TCSs) for e-C₄H₄S scattering (energies (E) are in eV and TCSs are in Ų)

E	R-matrix^a	E	R-matrix^a	E	SCOP^b	E	SCOP^b
a TCS calculated using R-matrix formalism.b TCS calculated using SCOP formalism.
0.01	626.48	4.89	50.70	19	28.22	700	5.29
0.25	46.82	5	50.56	20	27.32	800	4.78
0.5	39.48	6	44.18	30	26.11	900	4.37
0.75	37.61	7	40.36	40	26.51	1000	4.03
1	36.89	8	39.36	50	24.80	2000	2.65
1.5	36.38	9	37.87	60	22.70	3000	2.25
2	36.88	10	36.42	70	20.91	4000	1.84
2.45	54.49	11	35.69	80	19.45	5000	1.44
2.47	54.80	12	34.97	90	18.24	—	—
2.5	54.57	13	33.13	100	17.22	—	—
3	40.08	14	31.71	200	12.72	—	—
3.5	39.65	15	31.26	300	9.59	—	—
4	42.13	16	31.38	400	7.88	—	—
4.5	47.78	17	30.65	500	6.75	—	—
4.75	50.24	18	29.51	600	5.92	—	—

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Fig. 5 e-C₄H₄S excitation cross sections from the ground state (X1A₁) to the 3B₂ – solid line, 3A₁ – dash line, 3B₁ – dot line, 1B₁ – dash dot, 3A₂ – dash dot dot, 1A₂ – short dash, 1B₂ – short dot, 1A₁ – short dash dot.

It is important to study eigenphase diagrams as they provide the positions of resonances which are important features of collision chemistry in the low energy regime. Resonances are a common characteristic of electron molecule scattering at low impact energies and lead to distinctive structure in pure vibrational excitation cross sections.⁵⁹ Resonances occur when the incident electron is temporarily captured by the target to form a negative ion (an anion) which subsequently decays, either by autodetachment (often leaving the target vibrationally/electronically excited) or by dissociating the molecule to produce a net product anion (a process known as Dissociative Electron Attachment (DEA)). In the last few decades of development of negative ion mass spectrometry of resonant dissociative electron attachment, considerable progress has been achieved in understanding the fragmentation process. Presently, a recursive procedure for detecting and performing Breit Wigner fits to the eigenphase diagram is done through the program RESON.⁴² This program generates new energy points and marks those points where the numerically computed values of the second derivative changes sign from positive to negative. Finer grids are constructed about each of these points which are used as inputs for the Briet Wigner fit⁴² and the two most important parameters (position and width) relating to resonances are obtained.

Table 4 gives the positions and widths of resonances obtained in the present case using R-matrix calculations.

Table 4 Position and width of resonance states for C₄H₄S

State	Present			Other’s position
	Basis state	Position	Width
2B1	DZP	2.51	0.33	2.82 (ref. 20)
2A2	DZP	4.35	1.28	5.3 (ref. 23)
	—	—		5.35 (ref. 44)
	—	—		5.48 (ref. 44)
2A2	6-311G*	7.77	0.01	8.5 (ref. 44)
2A1	6-311G*	10.70	0.56	9.5 (ref. 23)
	—	—		10.2 (ref. 44)
2B1	DZP	11.36	0.13	11 (ref. 23)
2A2	DZP	11.97	0.84	—
2A2	DZP	13.21	0.16	—
2A2	DZP	14.40	0.06	—
2A2	DZP	15.17	0.05	16 (ref. 23)
2B1	6-311G*	17.60	0.46	—
2B2	6-311G*	17.83	0.46	—
2A2	DZP	17.90	0.57	—
2A1	DZP	18.51	1.24	—
2A2	6-311G*	18.51	0.33	—
2B2	DZP	18.69	1.10	—
2B2	DZP	18.73	0.84	—
2A1	DZP	19.48	1.62	—

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Hedhili et al.²³ and Muftakhov et al.⁴⁴ have performed a detailed study on DEA. Muftakhov et al.⁴⁴ detected the formation of C₄H⁻, S⁻, SH⁻, C₃H₂⁻, SC₂H₂⁻, SC₂H⁻, C₂H₃⁻, C₄H⁻, SC₄H₂⁻ and SC₄H₃⁻ anions in the energy range 0 to 12 eV. The resonances were observed at 3.5 eV, 5.3 eV, 6.4 eV, 8.5 eV, 8.9 eV and 10.2 eV by Muftakhov et al.⁴⁴ The resonances detected by Hedhili et al.²³ were at 3.8 eV, 4.7 eV, 5 eV, 6.5 eV, 12.1 eV, 12.7 eV, 13.3 eV, 13.9 eV and 14.3 eV. The present position and width of the resonances calculated using the Briet Wigner profile are listed in Table 4. We observed the resonances at 2.51 eV, 4.35 eV, 7.77 eV, 10.7 eV, 11.36 eV, 11.97 eV, 13.21 eV, 14.40 eV, 15.17 eV, 17.6 eV, 17.83 eV, 17.90 eV, 18.51 eV, 18.69 eV, 18.73 eV and 19.48 eV. Muftakhov et al.⁴⁴ suggested that the resonance observed at 5.3 eV involved excitation of the molecule into the lowest singlet state and electron capture into a diffuse molecular orbital of a quasi-Rydberg state. We observed this resonance at 4.77 eV which is close to 5.3 eV as reported by Muftakhov et al.⁴⁴ and 5 eV as reported by Hedhili et al.²³ Ion yields seen above 6.5 eV were assigned to core-exited resonances in which the incident electron and a second electron, promoted from a inner orbital, reside in quasi-Rydberg states. Resonance like structures are readily apparent in the yield functions of C₂H⁻ and C₂HS⁻ at 11 eV and 16 eV. The resonances above 16 eV may be attributed due to a dissociative desorption process. The resonances reported in Table 4 are responsible for fragmentation of C₄H₄S into various anions C₂H⁻, S⁻, SH⁻, C₃H₂⁻, SC₂H₂⁻, SC₂H⁻, C₂H₃⁻, C₄H⁻, SC₄H₂⁻ and SC₄H₃⁻ at different energies.

Fig. 5 presents electron-impact excitation cross sections from the ground state (X1A₁) to the first eight excited states (1A₁, 1B₂, 1A₂, 3A₂, 1B₁, 3B₁, 3A₁ and 3B₂) obtained using the R-matrix calculation. Fig. 6 shows the sum of our rotationally resolved differential cross sections summed over all transitions (J = 0 to J′ = 0 to 5) for incident energies 1 eV, 2 eV, 3 eV, 4 eV, 5 eV, 6 eV, while Fig. 7 shows the same for incident energies 8 eV, 10 eV, 12 eV, 15 eV and 20 eV. da Costa et al.²⁰ have reported differential cross sections for 1 eV, 2 eV, 3 eV, 4 eV, 5 eV and 6 eV. The scattering is dominated by the elastic component 0 → 0 and dipole component 0 → 1. The elastic component shows two minima centred at 40° and 108° in the 1 eV curve, which indicates the dominance of a p-wave in the interference pattern arising due to various partial wave amplitudes. The results of da Costa et al.²⁰ are qualitatively in agreement and the position of the minima is also comparable. For the 2 eV curve the minima occur at around 40° and 120° in agreement with the results of da Costa et al.²⁰ For the 3 eV curve there is a single minimum observed at around 129°, both in the present case as well as in the results of da Costa et al.²⁰ At 4 eV the minimum is at around 98° for both the present and da Costa et al.²⁰ data. At 5 eV and 6 eV the present DCS results show two minima while the results of da Costa et al.²⁰ show a single minimum. This discrepancy may be attributed to the difference in the treatment of polarization effects and in the partial wave cut off for the Born corrections adopted in the two calculations.⁶⁰ We have compared present DCS results with those of da Costa et al.²⁰ for 1 eV to 6 eV, but for brevity of the figure we have shown comparison of our results for 1 eV and 2 eV with that of da Costa et al.²⁰ in Fig. 6. As the energy increases, the convergence with respect to J is rapid. The divergence at the forward angle is confirmed as being due to dipole allowed transitions 0 → 1 dominating the scattering. The differential cross sections decrease as the incident energy increases. The sharp enhancement in the forward direction is a result of the strong long-range dipole component of the interaction potential. In the absence of any comparisons, either theoretical or experimental, for 8 eV, 10 eV, 12 eV, 15 eV and 20 eV, we plotted all the DCS curves on the same figure (Fig. 7).

Fig. 6 Rotationally resolved Differential Cross Sections (DCS) for incident energies of 1 eV to 6 eV; present (thick lines): solid line – 1 eV, dash line – 2 eV, dot line – 3 eV, dash dot line – 4 eV, dash dot dot line – 5 eV, short dash line – 6 eV; da Costa et al.²⁰ (thin lines): short dash dot line – 1 eV, short dot line – 2 eV.

Fig. 7 Rotationally resolved Differential Cross Sections (DCS) for incident energies of 8 eV, 10 eV, 12 eV, 15 eV and 20 eV; present: solid – 8 eV, dash – 10 eV, dot – 12 eV, dash dot – 15 eV, dash dot dot – 20 eV.

The Momentum Transfer Cross Sections (MTCS) indicate the importance of backward scattering and are an important quantity that forms the input to solve the Boltzmann equation for the calculation of the electron distribution function for a swarm of electrons drifting through a molecular gas. In contrast to the divergent behavior of DCS in the forward direction, the MTCS does not diverge due to the multiplicative factor (1 − cos θ). A further test of the quality of our DCS is shown by the momentum transfer cross section (MTCS) in Fig. 8 from energies of 0.01 eV to 20 eV. The MTCS cross sections are computed for SEP and SE models. In thiophene there is long lived σ* shape resonance which is seen as a peak in the MTCS curve at 2.79 eV, which is in excellent agreement with the same resonance predicted at 2.78 eV by da Costa et al.²⁰ We also predict a strong peak at 4.86 eV which is not observed in the MTCS curve of da Costa et al.²⁰ The various peaks or structures observed in MTCS correspond to various resonance processes. The present MTCS curves for both SE and SEP approximations are in good agreement with the results of da Costa et al.²⁰ No other theoretical or experimental comparisons are reported for thiophene.

Fig. 8 e-C₄H₄S Momentum Transfer Cross Section (MTCS); present: SEP – solid line, SE – dash dot dot line; da costa et al.:²⁰ SEP – short dot line, SE – dot line.

Due to the presence of long range dipole interactions, the total cross section at low energy is diverging in the fixed nucleii approximation due to a singularity in the differential cross section in the forward direction. It is well known that the cross sections of dipole dominated processes only converge slowly with partial waves. To obtain converged cross sections, the effect of rotation must be included along with a large number of partial waves. The higher partial waves (l ≥ 4) are included using a Born correction as given in the work of Chu and Dalgarno.⁶¹ This is done by adjusting the T-matrices using the CC cross sections generated by the code POLYDCS.⁴⁵ In this procedure our low l T-matrices are added to analytic dipole Born T-matrices using adiabetic nuclear rotation (ANR).^62–64 The Born contribution for partial waves higher than l = 4 to the elastic cross section at energies below 0.5 eV is quite large as seen from Fig. 10.

For brevity, the total cross sections for the e-C₄H₄S scattering results are plotted in two figures. The target model employed for the present calculations is CAS (c) = 3, number of states per symmetry (n) = 3 and R-matrix radius (r) = 13, abbreviated as c3n3r13. In Fig. 9, we have compared low energy (0 to 20 eV) total cross section data for e-C₄H₄S scattering using 6-311G* and DZP basis sets with the lone theoretical data of da Costa et al.²⁰ They have done calculations for impact energies 0 to 6 eV only. The present calculations for total cross sections at low energy are carried out using both SE and SEP models. We observe three prominent structures for the SEP approximation employing the DZP basis set and two prominent structures using the 6311G* basis set. They are at 2.5 eV, 4.77 eV and 8.06 eV with peak values of total cross sections as 56.87 Ų, 51.91 Ų and 38.11 Ų, respectively. The first peak corresponds to the formation of a σ* resonance which can be attributed to a Feshbach resonance as predicted by da costa et al.²⁰ and Muftakhov et al.⁴⁴ The second peak observed at 4.77 eV corresponds to the resonance which was suggested to involve excitation of the molecule into the lowest singlet state and electron capture into a diffuse molecular orbital of a quasi-Rydberg state by Muftakhov et al.⁴⁴ They observed this peak at 5.3 eV, which is very close to the present value of 4.77 eV. The third peak at 8.06 eV is attributed to a core excited shape resonance. The present SE approximation yields three structures at 3.0 eV, 6.45 eV and 12.49 eV with cross section values of 54.93 Ų, 40.92 Ų and 37.68 Ų, respectively. It is quite evident that the SEP approximation gives more refined calculations and the resonance peaks shift to lower energy. The elastic cross sections of da Costa et al.²⁰ are qualitatively in good agreement with the present results with their second peak at 2.52 eV of 45.66 Ų very close to our peak at 2.50 eV of 56.87 Ų of the σ* shape resonance. The lower peak at 0.53 eV of 59.24 Ų predicted by da Costa et al.²⁰ is not observed in our calculations. There are no other theoretical or experimental results reported to the best of our knowledge.

Fig. 9 e-C₄H₄S total scattering cross sections; c3n3r13 using: dot lines – DZP basis set, dash dot dot – 6-311G* basis set; dash dot line – present Hartree–Fock calculations; dash line – da Costa et al.²⁰

Finally, in Fig. 10 we report the total cross section for e-C₄H₄S scattering over a wide range of impact energies starting from 0.01 eV to 5000 eV. We are able to report the total cross sections over such a wide range due to smooth cross over of the data obtained through two methodologies, viz. low energy data using R-matrix through QUANTEMOL-N and intermediate to high energy data through SCOP formalism. We find a smooth transition of the present data using the 6 311G* as well as the DZP basis sets with the SCOP data at 17.2 eV and 19.9 eV, respectively. The only high energy data for total cross section is reported by Mozejko et al.¹⁹ who have employed the additivity rule for their calculations. The total cross sections reported by Mozejko et al.¹⁹ are higher compared to the present data, as expected, and the discrepancy decreases with an increase in the energy. No structure is observed in the data of Mozejko et al.¹⁹ as they used the additivity rule which does not include any molecular properties in their calculation. No other theoretical or experimental data have been reported so far.

Fig. 10 e-C₄H₄S total scattering cross sections; c3n3r13 using: dot lines – DZP basis set, dash dot dot – 6-311G* basis set, solid lines – present SCOP, short dash – Mozejko et al.¹⁹

Beyond 40 eV no prominent structures are found in the total cross section curve. We would like to point out that in the present intermediate to high-energy region, the static term dominates over the exchange and polarisation contributions. In the SCOP approach we obtain information on the absorption cross sections due to all allowed inelastic channels. At intermediate energies (from threshold to up to about 200 eV) the electronic excitations diminish and the inelastic channel for electron–molecule collisions corresponding to direct and dissociative ionization dominates. As expected, the V_abs has no dominant long-range effect and it penetrates towards the inner shells with an increase in energy. We also note that the electronic and vibrational excitations are too small in this energy range to make any sizeable contributions to the total cross sections. Beyond 200 eV, the total cross sections follow Born Bethe decline. At intermediate to high energies (20 eV–5000 eV) the cross sections are of relevance in applied areas such as plasma deposition, etching processes in the semiconductor industry and electrostatic precipitators for the processing of atmospheric pollutants. The present target molecule, thiophene, is of special interest as it is an important molecule in aeromatics which is a fascinating and rapidly evolving field, in which the various cross sectional data from low energy to high energy are employed in the kinetic modelling of reaction rates and the understanding of reaction mechanisms. There are no theoretical or experimental results available for this target beyond 6 eV to the best of our knowledge. So the present work may inspire more thoreticians/experimentalists to take up this task, as thiophene is very important system from the point of view of its diverse applications as discussed in the introduction.

4 Conclusion

The elastic, differential, momentum transfer, excitation and total cross sections are reported for the first time for electron impact above 6 eV with C₄H₄S using the R-matrix method with an adequate target representation. The total electron scattering cross sections for thiophene over a wide electron energy range is computed by adopting an amalgamation of two distinct theoretical methods viz. the ab initio R-matrix (low energy – 0 ≤ E_i ≤ 20 eV) and the SCOP formalism for E_i > threshold. The data computed using the two formalisms was found to merge smoothly at 17.2 eV for results obtained using the 6-331G* and at 19.9 eV for results obtained using the DZP basis sets. The present data for total cross sections at low energy is in good agreement with the previous data reported by da Costa et al.²⁰ This composite formalism is therefore able to produce a robust set of total cross section data when used in tandem (Table 3).

Further, the computed target properties such as the ground state energy, first electronic excitation energy, dipole moment, ionization potential and rotational constant match very well with the earlier predicted theoretical and experimental results as evident from Table 1. In Table 2 we have reported ten electronic excitation energy states for thiophene using DZP and 6-311G* basis sets. We observed the formation of a σ* shape resonance at 2.51 eV, which is close to the σ* resonance experimentally predicted by Modelli and Burrow²¹ at 2.65 eV and the resonance theoretically predicted at 2.82 eV by da Costa et al.²⁰ The second peak observed at 4.77 eV corresponds to the shape resonance which is very close to 5.1 eV as reported by Muftakhov et al.,⁴⁴ and this resonance is responsible for ring rupture which is reported to be at 5 eV by Hedhili et al.²³ The third peak at 8.06 eV is attributed to a core excited shape resonance. The peaks at 11 eV and 16 eV are attributed to a dissociate desorption process and lead to the formation of C₂H⁻ and C₂HS⁻. The knowledge of the dissociation dynamics through these resonances for thiophene needs to be investigated further on the complex potential energy surfaces given by these resonant states. Such electron impact studies of thiophene have gained prominence due its many fold applications as discussed in the introduction. The present work is important as there is a paucity of electron impact studies on thiophene. Also, the scattering data are important in developing models to estimate the radiation damage of living cells due to radio therapy, as secondary electrons produced during irradiation (e.g. in cancer therapy) can cause large effects on the biomolecules.

Acknowledgements

Minaxi Vinodkumar acknowledges DAE-BRNS, Mumbai for the major research project [37(3)/14/44/BRNS-2014] under the financial support of which part of this work is carried out.

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