Calculations of Ionization Cross-Sections of Acenes Based on Electron and Positron Impact

Abstract

We present calculations of the ionization cross sections for collisions of electrons and positrons with the acene molecules naphthalene, anthracene, tetracene, pentacene, and hexacene. The computations are performed using the binary-encounter Bethe (BEB) model and its modifications for positrons. The results show that all acenes exhibit maxima in their ionization cross sections at the same incident energy, regardless of molecular size. Furthermore, we find that the magnitude of the cross sections scales linearly with the number of rings in the acene molecules.

1. Introduction

Crystals and polycrystalline layers derived from organic materials have attracted sustained research attention for many years, motivated both by fundamental interest and by potential technological applications. Although the electro-optical properties of organic materials are generally inferior to those of inorganic semiconductors, which have been central to electronics for decades, they offer distinct advantages such as low production costs, ease of processing, and favorable mechanical characteristics. Key application areas for these materials include electronics (optical sensors, photodiodes, light-emitting diodes, field-effect transistors) and energy conversion (solar cells) [1,2].

Among organic materials, acenes deserve particular attention. They are composed of linear chains of fused benzene rings, with the general formula ${\mathrm{C}}_{2+4n}$${\mathrm{H}}_{4+2n}$, where n is the number of rings. The first members of the series are naphthalene ($n=2$), anthracene ($n=3$), tetracene ($n=4$), pentacene ($n=5$), and hexacene ($n=6$), as shown in Figure 1. The systematic relationship among these structures enables controlled studies of size-dependent trends.

In the crystalline state, acene molecules are held together by weak van der Waals forces [3,4]. As a result, their properties often resemble those of the isolated molecules [4,5]. Detailed investigations of the isolated species are therefore valuable for molecular electronics. Acenes also offer a practical advantage: high-quality polycrystalline layers can be fabricated via thermal vapor deposition on a variety of substrates (see, e.g., [2,6]).

Despite extensive research on crystalline and polycrystalline oligoacenes, the mechanisms governing charge-carrier generation, annihilation, and transport remain incompletely understood [1]. Electron transport studies provide insight into electronic conduction, while positron transport studies aid in modeling positron annihilation lifetime experiments [7] and in defect characterization [8].

Monte Carlo simulations are a powerful tool for probing charge-carrier transport in acenes [9,10]. Such simulations require molecular cross-section data as input [9,11]. Wiciak-Pawłowska et al. [12] showed that molecular structure has a strong impact on secondary electron yield. Cross sections should therefore be computed at the molecular level rather than approximated from atomic data.

The aim of this work is to present ionization cross sections for electron and positron impact on the acenes naphthalene, anthracene, tetracene, pentacene, and hexacene (see Figure 1). For naphthalene, Gupta et al. [13] reported electron-impact ionization cross sections. For anthracene, Singh et al. [14] and Krishnadas et al. [15] presented similar results. These studies employed different computational methods, and some of their results are discussed in Section 4. For the remaining systems, we are not aware of any experimental or theoretical cross-section data. By comparing the available and calculated results, we derive a simple scaling relation for the maxima of the ionization cross sections as a function of the number of benzene rings in the acene.

This paper is organized as follows. Section 2 introduces the theoretical models. Section 3 describes the computational details. Section 4 presents the calculated cross sections and discusses the scaling behavior. Finally, Section 5 summarizes the main findings.

Figure 1. Molecular geometries of the five acenes. The figures have been generated with the program Avogadro [16]. (a) Naphthalene, (b) anthracene, (c) tetracene, (d) pentacene, and (e) hexacene.

Figure 1. Molecular geometries of the five acenes. The figures have been generated with the program Avogadro [16]. (a) Naphthalene, (b) anthracene, (c) tetracene, (d) pentacene, and (e) hexacene.

2. Methods

2.1. Cross Section for Ionization by Electron Impact

The ionization cross sections for electron impact are calculated using the binary-encounter Bethe (BEB) model developed by Kim et al. [17,18]. In this model, the total ionization cross section of a molecule with incident electron kinetic energy E is expressed as the sum of the contributions from all occupied molecular orbitals:

$${\sigma }_{\mathrm{ion}}\left(E\right)=\sum _i^{n_{\mathrm{occ}}}{\sigma }_i^{\mathrm{BEB}}\left(E\right).$$

(1)

Here, ${\sigma }_i^{\mathrm{BEB}}\left(E\right)$ denotes the partial cross section for ionization of orbital i with binding energy $B_i$. For compact notation, the reduced kinetic energy of the incoming electron is defined as $t_i=E/B_i$, and the reduced kinetic energy of the bound electron in orbital i prior to ionization is $u_i=U_i/B_i$. With these definitions, the BEB model of Kim et al. [17,18] takes the following form:

$${\sigma }_i^{\mathrm{BEB}}\left(E\right)={\displaystyle \frac{S_i}{t_i+u_i+1}}\left[{\displaystyle \frac{ln\left(t_i\right)}{2}}\left(1-{\displaystyle \frac{1}{t_i^2}}\right)+\left(1-{\displaystyle \frac{1}{t_i}}\right)-{\displaystyle \frac{ln\left(t_i\right)}{t_i+1}}\right].$$

(2)

Here, the factor in the front of the parentheses is defined as

$$S_i=4\pi a_0^2{N}_i{\left({\displaystyle \frac{R}{B_i}}\right)}^2,$$

(3)

where the occupation of the orbital is given by $N_i$, R is the Rydberg constant, and $a_0$ is the Bohr radius.

2.2. Cross Section for Ionization by Positron Impact

In collisions between positrons and molecules, a molecular ion can be formed through three primary channels: annihilation, positronium formation, and direct ionization. In this work, we consider only direct ionization by positron impact. Four models are employed, all of which are modifications of the BEB model of Kim et al. [17,18]: the BEB-0 and BEB-W models of Fedus and Karwasz [19], and the BEB-A and BEB-B models of Franz et al. [20].

2.2.1. BEB-0 Model

Fedus and Karwasz [19] derived the BEB-0 model to describe direct ionization by positron impact. It is obtained from the electron-impact BEB expression in Equation (2) by removing the exchange term, which accounts for indistinguishability between the incident and target electrons in the continuum (the last term in the brackets):

$${\sigma }_i^{\mathrm{BEB}-0}\left(E\right)={\displaystyle \frac{S_i}{t_i+u_i+1}}\left[{\displaystyle \frac{ln\left(t_i\right)}{2}}\left(1-{\displaystyle \frac{1}{t_i^2}}\right)+\left(1-{\displaystyle \frac{1}{t_i}}\right)\right].$$

(4)

2.2.2. BEB-W Model

Klar [21] derived a Wannier-type threshold law [22], which describes positron-impact ionization cross sections close to the ionization threshold. As a modification of the BEB-0 model, Fedus and Karwasz [19] proposed the following:

$${\sigma }_i^{\mathrm{BEB}-\mathrm{W}}\left(E\right)={\displaystyle \frac{S_i}{t_i+u_i+1+f_i^{\mathrm{W}}}}\left[{\displaystyle \frac{ln\left(t_i\right)}{2}}\left(1-{\displaystyle \frac{1}{t_i^2}}\right)+\left(1-{\displaystyle \frac{1}{t_i}}\right)\right],$$

(5)

where an acceleration term

$$f_i^{\mathrm{W}}={\displaystyle \frac{1}{{(t_i-1)}^{1.65}}}$$

(6)

is introduced in the denominator to enforce Klar’s threshold law.

2.2.3. BEB-A Model

Rost and Heller [23] argued that Klar’s threshold law may only be valid up to ∼3 eV above the ionization threshold. Later, high-precision measurements by Ashley et al. [24] and theoretical analyses by Ihra et al. [25] indicated that close to threshold, the cross section increases exponentially. Jansen et al. [26] further extended these results, showing that the threshold law remains applicable for at least 10 eV above threshold. Franz et al. [20] incorporated this behavior by introducing a modified acceleration term into the BEB-W formulation,

$$f_i^{\mathrm{A}}={\displaystyle \frac{1}{{(t_i-1)}^{\alpha -1}{e}^{-{\beta }_i\sqrt{t_i-1}}}},$$

(7)

with $\alpha =2.64$ and ${\beta }_i=0.489\sqrt{B_i/\left(2R\right)}$. The resulting BEB-A model is written as follows:

$${\sigma }_i^{\mathrm{BEB}-\mathrm{A}}\left(E\right)={\displaystyle \frac{S_i}{t_i+u_i+1+f_i^{\mathrm{A}}}}\left[{\displaystyle \frac{ln\left(t_i\right)}{2}}\left(1-{\displaystyle \frac{1}{t_i^2}}\right)+\left(1-{\displaystyle \frac{1}{t_i}}\right)\right].$$

(8)

2.2.4. BEB-B Model

Franz et al. [20] also proposed the BEB-B model, which avoids the need for an explicit acceleration term in the denominator. Instead, threshold behavior consistent with the law of Jansen et al. [26] is achieved through a weighting function $g_i\left(t_i\right)$:

$${\sigma }_i^{\mathrm{BEB}-\mathrm{B}}\left(E\right)={\displaystyle \frac{S_i}{t_i+u_i+1}}\left[{\displaystyle \frac{ln\left(t_i\right)}{2}}\left(1-{\displaystyle \frac{1}{t_i^2}}\right)+\left(1-g_i\left(t_i\right)\right)\left(1-{\displaystyle \frac{1}{t_i}}\right)+g_i\left(t_i\right){\left(1-{\displaystyle \frac{1}{t_i}}\right)}^{\alpha }\right],$$

(9)

with

$$g_i\left(t_i\right)=e^{-{\beta }_i\sqrt{t_i-1}},$$

(10)

where $\alpha =2.64$ and ${\beta }_i=0.489\sqrt{B_i/\left(2R\right)}$, as in the BEB-A model.

3. Computational Details

The geometries of all five acenes were optimized using the B3LYP hybrid density functional [27,28] with the 6-311++G** basis set of Raghavachari et al. [29], implemented in the program package Gaussian 16 [30]. In all cases, the optimized structures were found to transform according to the $D_{2h}$ point group.

For the calculation of the ionization cross sections, the binding energies and the expectation values of the kinetic energy of the electrons prior to removal were obtained using the Hartree–Fock method, the Outer Valence Green’s Function (OVGF) method [31], and the 6-311++G** basis set [29]. The OVGF method was used for the binding energies of the valence orbitals, while Koopmans’ theorem [32] was applied to all remaining orbitals.

In

Table 1. First and second ionization energies of the acenes from experiments, EOM-CCSD calculations, and present OVGF calculations. All values are given in eV.

	${\tilde{\mathit{X}}}^{+}$			${\tilde{\mathit{A}}}^{+}$
Molecule	exp	EOM	OVGF	exp	EOM	OVGF
naphthalene	8.15 a	8.10 d	7.96	8.88 a	8.76 c	8.64
anthracene	7.41 b	7.35 d	7.19	8.57 a	8.48 c	8.37
tetracene	6.97 b	6.82 d	6.65	8.44 a	8.29 c	8.19
pentacene	6.61 b	6.45 d	6.27	8.03 a	8.01 c	7.69
hexacene	6.44 c	6.17 d	5.98	7.55 a	7.57 c	7.37

^a Experimental value from Clar and Schmidt [33]. ^b Experimental value from Schmidt [34]. ^c Experimental value from Boschi et al. [35]. ^d Vertical ionization energies computed with bt-PNO-IP-EOM-CCSD/def2-TZP//B3LYP/6-311++G** from Wagner et al. [36].

, we compare our OVGF values for the vertical ionization energies of the two lowest states of the acenes with experimental results from Clar and Schmidt [33], Schmidt [34], and Boschi et al. [35]. Also shown are calculations by Wagner et al. [36] performed using Equation-of-Motion Coupled-Cluster (EOM) theory. The OVGF values are about 0.1–0.2 eV lower than the vertical ionization energies obtained with the EOM method. Compared to the experiment, the OVGF values are underestimated by about 0.2 eV for naphthalene. This discrepancy increases with molecular size and reaches about 0.5 eV for hexacene. The experimental values correspond to adiabatic ionization energies, whereas both the OVGF and EOM results represent vertical ionization energies. Better agreement could be achieved by accounting for geometry relaxation of the ions, see, e.g., Deleuze et al. [37] and Andrzejak and Petelenz [38].

4. Results and Discussion

In Figure 2, we examine the influence of the ionization thresholds on the cross section. We investigate this effect for hexacene because the difference between the experimental ionization threshold and the theoretical value obtained with the OVGF method is nearly 0.5 eV for the first ionization threshold and about 0.2 eV for the second. We compare our computations with results in which the first two binding energies are shifted to match the experimental vertical ionization energies. The left panel shows the full range of the cross section from 5 eV to 5000 eV. On this scale, the curves appear identical. In the right panel, we show the region near the first few ionization thresholds. One can see that the difference between the two curves is less than $1\times {10}^{-20}$ m². The effect is so small because of the large density of states in this energy regime.

In Figure 3, we compare our results for electron-impact ionization of naphthalene and anthracene with selected data from the literature. For naphthalene, Gupta et al. [13] examined the performance of the spherical complex optical potential (SCOP) method and BEB models based on different sets of orbitals. They found the most accurate results using the BEB model with binding energies and orbital kinetic energies obtained from the $\omega$B97X density functional of Chai and Head-Gordon [39]. The $\omega$B97X functional is a long-range corrected variant of the Becke97 functional [40]. The resulting ionization energy of 8.15 eV is in excellent agreement with the experimental vertical ionization energy. In the left panel of Figure 3, we compare our results with the BEB cross section from Gupta et al. [13] based on the $\omega$B97X parameters. The difference is less than one percent.

For anthracene, Krishnadas et al. [15] computed the electron-impact ionization cross section using several approaches, including the additivity rule, SCOP, the pixel counting method (PCM), and BEB models with different orbital sets. In the right panel of Figure 3, we show our BEB results alongside those of Krishnadas et al. [15], which are based on binding energies and orbital kinetic energies obtained with the Hartree$2^{̆}$013Fock (HF) method, the B3LYP density functional, and the $\omega$B97XD density functional [41]. The $\omega$B97XD functional is a modification of $\omega$B97X that includes empirical atom–atom dispersion corrections. Singh et al. [14] also reported BEB cross sections based on HF orbitals, which are nearly identical to those of Krishnadas et al. [15]. We find that results obtained with B3LYP are too high, while those based on HF are about 20% too low. The $\omega$B97XD results agree very closely with ours, differing by less than one percent.

In Figure 4, the computed cross sections for direct ionization of the five acenes are shown for both electron and positron impact. All cross sections exhibit a similar shape, independent of projectile and target, as already noted by Rost and Pattard [42]. The cross sections rise from threshold, reach a maximum between 55 and 70 eV, and decrease at higher energies. For most collision energies, the cross sections for ionization by positron impact are larger than those for electron impact, in agreement with previous studies, e.g., Thomas and Gupta [43] for cyanopolyynes. This difference arises from the last term in Equation (2), which describes electron-exchange contributions and is absent in the positron case.

The largest cross sections are obtained for positron impact using the BEB-0 model. The results from the BEB-W model are slightly smaller. In calculations with the BEB-A and BEB-B models, the cross sections increase more slowly near threshold, consistent with the threshold law of Jansen et al. [26]. As shown by Franz et al. [20], the BEB-A and BEB-B models improve the accuracy of positron-impact ionization cross sections of nonpolar molecules from threshold up to the maximum, while at higher energies, they converge to the BEB-0 and BEB-W values.

For electron impact, the cross sections are the smallest in magnitude above 50 eV. Below 40 eV, however, they are slightly larger than those obtained for positron impact with the BEB-A and BEB-B models. For collision energies above 300 eV, all five models yield nearly identical cross sections. At still higher energies, positron- and electron-impact ionization cross sections show very similar behavior, consistent with earlier findings (see, e.g., Rost and Pattard [42]).

In

Table 2. Maxima of the cross section for ionization by electron impact using the BEB model. The energies $E_{max}$ at the maxima are given in eV, and the maxima of the cross sections are given in ${10}^{-20}$ m².

Molecule	${\mathit{E}}_{\mathit{max}}$	${\mathsf{\sigma }}_{\mathit{ion}}$
naphthalene	69.6	23.2
anthracene	69.0	32.3
tetracene	68.8	41.2
pentacene	68.4	50.5
hexacene	68.5	58.3

, the maxima of the cross sections for ionization by electron impact are presented for the five acenes, together with the corresponding energies at which the maxima occur. In

Table 3. Maxima of the cross section for direct ionization by positron impact using the models BEB-0, BEB-W, BEB-A, and BEB-B. The energies $E_{max}$ at the maxima are given in eV, and the maxima of the cross sections are given in ${10}^{-20}$ m².

	BEB-0		BEB-W		BEB-A		BEB-B
Molecule	${\mathit{E}}_{\mathit{max}}$	${\mathsf{\sigma }}_{\mathit{ion}}$	${\mathit{E}}_{\mathit{max}}$	${\mathsf{\sigma }}_{\mathit{ion}}$	${\mathit{E}}_{\mathit{max}}$	${\mathsf{\sigma }}_{\mathit{ion}}$	${\mathit{E}}_{\mathit{max}}$	${\mathsf{\sigma }}_{\mathit{ion}}$
naphthalene	57.7	28.0	62.4	27.4	71.4	25.8	69.6	25.6
anthracene	57.3	39.0	61.9	38.1	70.9	35.9	69.0	35.6
tetracene	57.1	49.8	61.8	48.6	70.7	45.8	68.8	45.4
pentacene	56.8	60.9	61.5	59.5	70.4	56.1	68.4	55.6
hexacene	56.8	70.3	61.6	68.6	70.5	64.8	68.6	64.2

, the corresponding data are shown for ionization by positron impact using the four different models: BEB-0, BEB-W, BEB-A, and BEB-B. The position of the maximum is nearly independent of the size of the acene. For ionization by electron impact, the maxima appear at roughly 69 eV. For ionization by positron impact, the position of the maxima depends on the model: about 57 eV for the BEB-0 model, around 62 eV for the BEB-W model, about 71 eV for the BEB-A model, and around 69 eV for the BEB-B model.

In all models, the value of the maximum cross section increases with the number of rings in the acenes. In Figure 5, the maxima of the cross sections are shown as a function of the number n of rings. The relationship between the maximum $M_n$ of the cross section and the number n of rings can be expressed as a linear function:

$$M_n=M_0+n·D,$$

(11)

where $M_0$ and D denote the intercept and the slope of the line, respectively. For each model, a straight line was fitted through the data points via linear regression using Scikit-learn [44]. In

Table 4. Results of the linear fitting of the relationship between the maxima $M_n$ of the ionization cross section and the number n of rings.

Molecule	Model	${\mathit{M}}_0$/${10}^{-20}$   ${\mathrm{m}}^2$	D/${10}^{-20}$   ${\mathrm{m}}^2$
electrons	BEB	5.77 ± 0.59	8.85 ± 0.14
positrons	BEB-0	7.00 ± 0.73	10.65 ± 0.17
	BEB-W	6.84 ± 0.72	10.39 ± 0.17
	BEB-A	6.41 ± 0.66	9.82 ± 0.15
	BEB-B	6.34 ± 0.66	9.73 ± 0.15

, the intercepts and slopes, together with their uncertainties, are given for the various models.

For ionization by electron impact, the maximum value of the cross section increases by roughly $9\times {10}^{-20}$ m² for each additional ring. For positron impact, the slopes in Equation (11) show only a weak dependence on the model, with differences well within the uncertainties of the linear regression. The maxima of the cross section increase by approximately $10\times {10}^{-20}$ m² per ring.

In Figure 6, we compare the normalized ionization cross sections for each BEB model. The normalization is performed by dividing each cross section by its maximum value. If the kinetic energy of the impinging particle is at least 10 eV above the ionization threshold, the shapes of the cross sections are independent of the acene, as already predicted by Rost and Pattard [42]. For impact energies close to threshold, small differences are observed between the acenes, as expected from the differences in their ionization thresholds.

5. Conclusions

In this paper, we presented cross sections for ionization by electron and positron impact for the five shortest acenes: naphthalene, anthracene, tetracene, pentacene, and hexacene. From previous studies, we assume that the cross sections for electron impact are accurate to within 10% of the experimental values [18]. For positron impact, we reported results obtained with four different models: BEB-0, BEB-W, BEB-A, and BEB-B. Based on earlier work, we assume that the BEB-A and BEB-B models provide the most reliable results for nonpolar molecular targets [20], with an accuracy of about 15% [20,45].

All acenes exhibit maxima of the ionization cross sections at nearly the same energy, independent of the number of rings: around 69 eV for electron impact. According to the BEB-A model, the maximum for positron impact is near 71 eV. We demonstrated that the maxima follow a simple linear relation of the form $M_n=M_0+n·D$, where n is the number of rings. For electron impact, the maxima are described by

$$M_n=(5.77+n·8.85)\times {10}^{-20}{\mathrm{m}}^2,$$

(12)

while for positron impact, the BEB-A model yields

$$M_n=(6.41+n·9.82)\times {10}^{-20}{\mathrm{m}}^2.$$

(13)

These relations allow straightforward extrapolation of the maxima for higher acenes. This is relevant because systematic theoretical investigations of higher acenes are hindered by the multi-reference character of their electronic ground states, as noted by Yang et al. [46] and Bettinger et al. [47].

In a related study on electron- and positron-impact ionization of cyanopolyynes (${\mathrm{HC}}_n$N), Thomas and Gupta [43] derived a linear relationship between the maxima of the cross section and the square root of the ratio of dipole polarizability to ionization potential. We argue that the extrapolation presented here is easier to apply, as it requires only the number of rings. These findings also suggest that theoretical models based on the additivity rule (see, e.g., Mahla and Antony [48]) should perform well for this class of compounds. However, Krishnadas et al. [15] showed that a direct application of the additivity rule may overestimate the total ionization cross section; therefore, some modifications or scaling may be necessary.

This study can be regarded as an entry point to the investigation of related organic compounds. Functionalized acenes and heteroacenes are being explored as materials for organic electronics [6]. Acenes also belong to the larger family of polycyclic aromatic hydrocarbons (PAHs) [49], some of which have been identified in diffuse interstellar bands [50].