Study of Electron and Positron Elastic Scattering from Hydrogen Sulphide Using Analytically Obtained Static Potential

Abstract

A detailed study of elastic scattering of electrons and positrons from a hydrogen sulphide (H₂S) molecule is presented using the method of partial wave phase shift analysis with suitably chosen complex optical potentials. The important aspect of our present work is that we uniquely obtain static potential in an analytical form and use it along with exchange (only for electron), polarization and purely imaginary absorption potentials to define the complex optical potential. The static potential is evaluated by obtaining charge density from the H₂S molecule using the molecular wavefunction represented through an accurate analytical form of the Gaussian orbitals. The primary aim of our study is to test our present approach, as applied to the electron and positron scattering from H₂S. Therefore, the results for electron and positron impact differential, integral, momentum-transfer, absorption and total cross sections are obtained for the incident energies in the range of 10–500 eV. Comparisons of these different types of cross section results with the available measurements and other calculations show good agreement, which suggests the applicability of our present approach.

1. Introduction

Electron- and positron-molecule elastic scattering processes are important in understanding their transport and drift in any molecular medium as well as their interaction with electrons and nuclei present in the molecule. One common way to study such a process is using the method of partial wave phase shift analysis with complex optical potential which is usually expressed in terms of static, exchange (for electron), polarization and purely imaginary absorption potentials. The study of electron impact is different from that of positron impact, mainly in the sense that the static potentials seen by them with respect to molecular targets are, respectively, attractive and repulsive. Since a positron is distinguishable from the target electrons, the exchange potential must be dropped for positron scattering studies. However, the polarization potential is attractive for both the electron and positron scattering. Thus, it would be interesting for the current research to study both the electron and positron scattering from any important molecule. In the case of inelastic scattering of the positron, the positronium formation channel opens, in addition to excitation and ionization. Positronium is short-lived when it is produced and is annihilated by combining with an electron to form gamma rays, so the study of electron and positron scattering from astrophysical molecules is an interesting topic for the current research. In the present work, we focus on the analysis of elastic electron and positron scattering from hydrogen sulphide by taking a suitably defined complex optical potential, where we use our uniquely obtained analytical form of the static potential.

The study of H₂S is of great significance as it is a quasi-spherical symmetric molecule and commonly found in most of the astrophysical environments, such as in interstellar space and on the comet Austin (1989C1) [1]. Recent evidence of the clear detection of gaseous H₂S above the Uranus cloud deck has been noticed at a pressure level of between 1.2 bar and 3 bar [2] and it is expected that H₂S is the main sulphur hydride found in dense clouds. H₂S is also detected in the proto-planetary disks of GG Tau A [3] and hence, to characterize the properties and initial conditions of planet formation, we need to study the molecular species present in it. In the astronomical environment, positron annihilation gamma rays originate from solar flares [4] towards the Milky Way [5]. The detection of such rays gives us vital information about the solar system and the surrounding environment. Thus, the presence of H₂S has been clearly observed in different astrophysical environments. Besides the astrophysical environments, highly polar H₂S has been very active in chemical reactions where it has been used in a large number of chemical and physical processes [6]. H₂S is also an essential biological gas transmitter in health and diseases [7], as well as being used to develop drugs for cardiovascular disease [8]. The electron impact processes also play various roles in biomedical applications such as modeling software [9,10,11] to estimate radiation-induced damage in biological systems. In addition, H₂S is also used in plasma nitro-carburizing processes along with N₂ and H₂ [12] and hence, to understand and characterize such a plasma, the electron scattering cross sections from H₂S are required.

Before we discuss our method to study electron/positron scattering from H₂S, it is worth giving a brief review of the available experimental and theoretical work undertaken so far. Many studies have reported elastic scattering cross sections in the different ranges viz. low, intermediate, and high incident electron energies. Gulley et al. [13] measured the differential cross sections (DCS), integrated cross sections (ICS), and momentum transfer cross sections (MTCS) for incident electron energy from 1–30 eV using a crossed electronic-molecular beam apparatus with an angular variation of 20°–130°. Similarly, with the addition of the relative flow technique, Rawat et al. [14] measured the cross sections in the energy range of 100–500 eV through 10°–140° angular distributions. Further, the most recent experiments on e-H₂S cross sections have been reported by Cho et al. [15] and Brescansin et al. [16] in the energy range 3–30 eV and 30–100 eV, respectively, over the scattering angle 10°–180° with a crossed-beam electron spectrometer combined with a magnetic angle-changing device. Besides experimental work, several previous theoretical calculations have been completed, for instance, Rawat et al. [14] reported DCS, ICS, MTCS in the energy range from 0.5–500 eV using the Schwinger variational method combined with the distorted-wave approximation. A modified similar approach has been used by Brescansin et al. [16] to calculate the DCS, ICS, and MTCS in the energy range of 1 to 500 eV. Varella et al. [17] performed their calculation in between 5–30 eV energy, using a Schwinger multichannel method with pseudo-potentials, along with a Born closure procedure in order to account for the long-range potential of the targets. Machado et al. [18] calculated DCS in the range of 2–50 eV using fixed-nuclei and static-exchange approximations for the low partial-wave scattering amplitudes via the Schwinger variational iterative method, whereas, Born approximation with a point-dipole potential was used to account for higher partial waves. Jain et al. [19] performed calculations for 50–1000 eV by constructing an optical potential from a near Hartree–Fock one-center expansion. Gianturco et al. [20] calculated the DCSs up to 25 eV by using the single center expansion method. Yuan and Zhang [21] performed the DCS, TCS and MTCS calculations from 0.1–50 eV within the Born–Oppenhermer approximation. Nishimura and Itikawa [22] employed a close-coupling method to determine the ab initio electrostatic potential and reported DCSs from 3–30 eV. Gupta and Baluja [23] obtained DCS results from 3–15 eV by using the R-matrix method. Further, Aouchiche et al. [24] reported DCSs and ICSs in a wide range of energy from 10–1000 eV. They used partial wave formalism with a spherically symmetric complex optical potential, where the static contribution was deduced from a single center Hartree–Fock target description. For the total absorption cross section (ACS) only the theoretical calculations by Brescansin et al. [16] and Jain and Baluja [25] are available; both of them use a quasi-free spherical model. On the other hand, Szmytkowsky et al. [26] reported the total cross section (TCS) results from 6–370 eV using a modified electron spectrometer working in a linear transmission mode. Zecca et al. [27] performed TCS measurements on a Ramsauer-type apparatus for electron scattering in the energy range from 5–4000 eV. There are a couple of theoretical calculations available for TCS, such as from Brescansin et al. [16], Jain and Baluja [25] and Joshipura and Vinodkumar [28] and Limbachiya et al. [29]. For positron scattering from H₂S, there are no earlier experimental or theoretical studies reported. However, such studies can be equally important to probe the nature of positron–H₂S interactions, as well as to compare with similar electron impact results.

As mentioned earlier, most of the electron-molecule elastic scattering has been studied with the optical potential approach using the partial wave phase shift analysis with suitably chosen complex optical potential, which consists of the static potential along with the exchange, polarization and imaginary absorption potentials. The entire complex optical potential is then used in the Schrödinger equation [30,31,32] which is solved by partial wave phase shift analysis to obtain the scattering cross sections. As one can see from the review given above, this approach is quite successful and has been widely applied in studying the elastic scattering of electrons with different atoms and molecules, including H₂S. However, various optical potential choices make a real difference in the accuracy of the results obtained from such an approach. For example, evaluating the target molecule’s static potential as seen by the projectile electron or positron is an arduous and challenging task for a multicenter target [33,34], as one needs to correlate interactions between all the individual atoms and the projectile. There are many methods available that have been used in the past to obtain the multicenter molecular potentials. The single-center expansion (SCE) method [35] is one of the most widely implemented approaches in which wavefunction of the molecular orbitals are expanded with respect to the center of mass of a polyatomic molecule and numerical integration is used to obtain the potentials. Whereas, we have developed a method which obtains the spherically-averaged static potential by evaluating multicenter integrals using Gaussian orbitals centered on the various nuclei. It is worth mentioning here that our static potential is mostly analytic, except for the inclusion of the well-known error functions, which is advantageous due to available algorithms for their numerical evaluation. Thus, the accuracy of our potential is dependent only on the reliability of the Gaussian wavefunctions that we have used to represent the molecule. In fact, we have used STO-6G Gaussian wavefunctions in this work, which are quite accurate. For other potentials viz. exchange, polarization and absorption components of the complex optical potential, we have used standard and widely tested expressions [36,37,38,39,40,41]. Here, one of our main aims is to test our method for obtaining the static potential as applied to e-H₂S scattering by comparing our calculated DCS, ICS, MTCS, ACS and TCS results with the available measurements and other calculations. This will provide us the confidence to extend the applications of the present method to study scattering from other complex molecules in a similar way. We further utilize the method to report similar cross section results for the elastic collision of positrons with H₂S and compare these with electron impact results.

In Section 2, we briefly describe the theoretical method to obtain the static potential and different cross sections. Thereafter, our calculated results are presented and compared with other theoretical and experimental results in Section 3.

2. Theory

2.1. Scattering Amplitude and Cross Sections

The scattering of electrons and positrons from a small molecule like H₂S can be described in a non-relativistic manner by the Schrödinger equation as given below [30] (atomic units are used throughout)

$$\left[\frac{1}{2}{\nabla }^2+E-V_{opt}\right]\Psi \left(\mathit{r}\right)=0.$$

(1)

Here, E refers to the incident electron or positron energy, $\Psi \left(\mathit{r}\right)$ is the wavefunction and $V_{opt}\left(r\right)$ is the total complex optical potential, which includes static $V_{st}\left(r\right)$, exchange $V_{ex}\left(r\right)$, polarisation $V_{pol}\left(r\right)$ and absorption $V_{abs}\left(r\right)$ potentials for electron, i.e.,

$$V_{opt}\left(r\right)=V_{st}\left(r\right)+V_{ex}\left(r\right)+V_{pol}\left(r\right)+i{V}_{abs}\left(r\right).$$

(2)

In the case of positron scattering, the exchange potential needs to be dropped, as already explained earlier. The choices of these potentials have been discussed in detail in the following Section 2.2. Although H₂S is a quasi-spherically symmetric molecule with heavier atom S at the center, the interaction potential $V_{opt}\left(r\right)$ is approximated as spherically symmetric. Further, to study the elastic scattering of an electron or a positron from H₂S molecule, we consider it a rigid non-rotating system and exclude any rotational excitation contribution of the molecule in our present calculations.

By expanding the wavefunction $\Psi \left(\mathit{r}\right)$ in terms of its partial waves and phase shifts in the usual standard manner and then substituting in Equation (1), we obtain the following radial equation

$$\left(\frac{d^2}{d{r}^2}+k^2-2V_{opt}\left(r\right)-\frac{l\left(l+1\right)}{r^2}\right)u_l\left(r\right)=0,$$

(3)

where $k^2=2E$, $k$ is the momentum of electron or positron and $u_l\left(r\right)$ is the radial wavefunction associated with partial wave l satisfying Equation (3) and can be solved subjecting to the following asymptotic boundary conditions,

$$u_l\left(r\right)~\frac{1}{k}\sin \left(kr-\frac{l\pi }{2}+{\delta }_l\right), \mathrm{for} r\to \infty \mathrm{and} u_l\left(r\right)~ 0 \mathrm{for} r\to 0.$$

(4)

Here, ${\delta }_l={\delta }_{Real}+i{\delta }_{abs}$ is the complex phase shift of l^th outgoing partial wave with respect to the free spherical wave in the presence of the potential. Further, the solution of Equation (3) gives us the phase shift at each kinetic energy of the incident electron [30,31], in terms of which we can express the scattering amplitude by the following expression,

$$f\left(\theta \right)=\frac{1}{2ik}{\displaystyle \sum }_{l=0}^{\infty }\left(2l+1\right)\left(e^{2i{\delta }_l}-1\right)P_l\left(\cos \theta \right),$$

(5)

where $P_l\left(\cos \theta \right)$ is the l^th order Legendre polynomial. ${\left|f\left(\theta \right)\right|}^2$ gives differential cross section, while total elastic integral (${\sigma }_{el}$), momentum transfer cross section (${\sigma }_{mtcs}$) and absorption cross sections (${\sigma }_{abs})$ are finally obtained by using the following expressions [30,31,42]

$${\sigma }_{el}=\frac{\pi }{k^2}{\displaystyle \sum }_{l=0}^{\infty }\left(2l+1\right){\left|1-e^{2i{\delta }_l}\right|}^2,$$

(6)

$${\sigma }_{mtcs}=\frac{4\pi }{k^2}{\displaystyle \sum }_{l=0}^{\infty }\left(l+1\right){\sin }^2({\delta }_{l+1}-{\delta }_l)$$

(7)

and,

$${\sigma }_{abs}=\frac{\pi }{k^2}{\displaystyle \sum }_{l=0}^{\infty }\left(2l+1\right)\left[1-{\left|e^{-2Im{\delta }_l}\right|}^2\right].$$

(8)

Further, the total cross section $\left({\sigma }_{total}\right)$ is obtained by adding the total elastic integrated cross section (${\sigma }_{el}$) with the absorption cross section (${\sigma }_{abs}$).

2.2. Complex Optical Potential

To solve Equation (3), we first need to construct an appropriate complex optical potential by properly defining its different component potentials as given in Equation (2). Out of these component potentials the calculation of the static potential is quite nontrivial and of fundamental importance in the present work. We first calculate the charge density of H₂S molecule by taking the squared modulus of the molecular wavefunction at a point ${\mathit{r}}^{\prime }$ as seen by the projectile.

$$\rho \left({\mathit{r}}^{\prime }\right)={\displaystyle \sum }_i{\left|X_i\left({\mathit{r}}^{\prime }\right)\right|}^2,$$

(9)

where $X_i$ is the molecular wavefunction of the i^th molecular orbital and here $i$ sum over the number of molecular orbitals, which are 9 for H₂S. The molecular orbitals $X_i\left({\mathit{r}}^{\prime }\right)$ are further expressed as linear combination of atomic orbitals ${\mathsf{\varphi }}_k\left({\mathit{r}}^{\prime }\right)$, i.e.,

$$X_i\left({\mathit{r}}^{\prime }\right)={\displaystyle \sum }_k{a}_{ik}{\mathsf{\varphi }}_k\left({\mathit{r}}^{\prime }\right),$$

(10)

here $a_{ik}$ is the molecular orbital co-efficient and k extends on the total number of atomic orbitals, which are 11 for H₂S. Each atomic orbital is represented as

$${\mathsf{\varphi }}_k\left({\mathit{r}}^{\prime }\right)={\displaystyle \sum }_j{N}_{kj}{c}_{kj}{\chi }_{kj}\left({\mathit{r}}^{\prime }\right),$$

(11)

where $N_{kj}$ and $c_{kj}$ are the normalization constants and coefficients of the Gaussian basis functions ${\chi }_{kj}\left({\mathit{r}}^{\prime }\right)$, respectively. The sum over j denotes the number of Gaussian basis function. Thus, by combining Equations (10) and (11) with Equation (9), the total charge density can be expressed in the following general form as,

$$\rho \left({\mathit{r}}^{\prime }\right)={\displaystyle \sum }_i{\displaystyle \sum }_{k,k^{\prime }}{a}_{ik}{a}_{i{k}^{\prime }}{\displaystyle \sum }_{j,j^{\prime }}{N}_{kj}{N}_{k^{\prime }{j}^{\prime }}{c}_{kj}{c}_{k^{\prime }{j}^{\prime }}{\chi }_{kj}\left({\mathit{r}}^{\prime }\right){\chi }_{k^{\prime }{j}^{\prime }}\left({\mathit{r}}^{\prime }\right).$$

(12)

Here, the unnormalized form of the Gaussian basis function ${\chi }_{kj}\left({\mathit{r}}^{\prime }\right)$, which is centered on its nucleus A with coordinates $A_x, A_y$ and $A_z$ can be represented as

$${\chi }_{kj}\left({\mathit{r}}^{\prime }\right)=e^{-{\alpha }_{kj}{\left|{\mathit{r}}^{\prime }-\mathsf{A}\right|}^2}{\left(x^{\prime }-A_x\right)}^{m_x}{\left(y^{\prime }-A_y\right)}^{m_y}{\left(z^{\prime }-A_z\right)}^{m_z},$$

(13)

Here, $m_x$, $m_y$ and $m_z$ are non-negative integers and ${\alpha }_{kj}$ are the exponents of the Gaussian basis function. To expand Equation (12) further, we used the important property of Gaussian orbitals, i.e., the product of the exponential parts of two Gaussian orbitals with different centers A and B can be written as an exponential of the same form at a center P on the line joining A and B. Thus, the product of two Gaussian functions in the expression of Equation (12) becomes

$$\begin{array}{l}{\chi }_{kj}\left({\mathit{r}}^{\prime }\right){\chi }_{k^{\prime }{j}^{\prime }}\left({\mathit{r}}^{\prime }\right)=e^{-{\alpha }_{kj}{\left|r^{\prime }-\mathit{A}\right|}^2}{e}^{-{\alpha }_{k^{\prime }{j}^{\prime }}{\left|r^{\prime }-\mathit{B}\right|}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {(x^{\prime }-A_x)}^{m_x}{(y^{\prime }-A_y)}^{m_y}{(z^{\prime }-A_z)}^{m_z}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {(x^{\prime }-B_x)}^{m_x^{\prime }}{(y^{\prime }-B_y)}^{m_y^{\prime }}{(z^{\prime }-B_z)}^{m_z^{\prime }}\\ \hspace{1em}\hspace{1em}\hspace{1em}=E_{kj,k^{\prime }{j}^{\prime }}{e}^{-\alpha {\left|r^{\prime }-r_P\right|}^2}{(x^{\prime }-A_x)}^{m_x}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {(y^{\prime }-A_y)}^{m_y}{(z^{\prime }-A_z)}^{m_z}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {(x^{\prime }-B_x)}^{m_x^{\prime }}{(y^{\prime }-B_y)}^{m_y^{\prime }}{(z^{\prime }-B_z)}^{m_z^{\prime }},\end{array}$$

(14)

where ${\mathit{r}}_P$ is the position vector of P with respect to the molecular frame can be given by

$${\mathit{r}}_P=\frac{{\alpha }_{kj}A+{\alpha }_{k^{\prime }{j}^{\prime }}B}{{\alpha }_{kj}+{\alpha }_{k^{\prime }{j}^{\prime }}},$$

and $\alpha$ and $E_{kj,k^{\prime }{j}^{\prime }}$ can be expressed as,

$$\alpha =\left({\alpha }_{kj}+{\alpha }_{k^{\prime }{j}^{\prime }}\right)$$

and

$$E_{kj,k^{\prime }{j}^{\prime }}=e^{\left(-\frac{{\alpha }_{kj}{\alpha }_{k^{\prime }{j}^{\prime }}}{{\alpha }_{kj}+{\alpha }_{k^{\prime }{j}^{\prime }}}{\left|A-B\right|}^2\right)}.$$

Further, the electronic static interaction potential between a projectile electron and molecule at a point $\mathit{r}$ can be represented by

$$V\left(r\right)={\displaystyle \int }\frac{\rho \left({\mathit{r}}^{\prime }\right)}{\left|\mathit{r}-{\mathit{r}}^{\prime }\right|}d{\mathit{r}}^{\prime }.$$

(15)

Now, expanding the denominator and using the property of spherical harmonics, we get the spherically averaged static potential over all the orientations of the molecule, i.e.,

$$\begin{array}{l}V\left(r\right)={\displaystyle \sum _i}{\displaystyle \sum _{k,k^{\prime }}}{a}_{ik}{a}_{i{k}^{\prime }}{\displaystyle \sum }_{j,j^{\prime }}{N}_{kj}{N}_{k^{\prime }{j}^{\prime }}{c}_{kj}{c}_{k^{\prime }{j}^{\prime }}{E}_{kj,k^{\prime }{j}^{\prime }}\\ \hspace{1em}\times {\displaystyle \int }{e}^{-\alpha {\left|r^{\prime }-r_p\right|}^2}{(x^{\prime }-A_x)}^{m_x}{(y^{\prime }-A_y)}^{m_y}{(z^{\prime }-A_z)}^{m_z}\\ \hspace{1em}\times {(x^{\prime }-B_x)}^{m_x^{\prime }}{(y^{\prime }-B_y)}^{m_y^{\prime }}{(z^{\prime }-B_z)}^{m_z^{\prime }}\frac{1}{r_{>}}d{r}^{\prime },\end{array}$$

(16)

here, the expression $r_{>}$ is greater of $r$ and $r^{\prime }$. Evaluation of the integrals in the above equation is conducted by considering the multiplications of the two different atomic Gaussian orbitals i.e., two s-orbital of hydrogen atoms (i.e., H (1s)) or say with s-orbital of Sulphur atom (i.e., S (1s²2s²2p⁶3s²3p⁴)), s-orbital of H atom with the p-orbital of S atom and two p-orbitals of S atom. The integrals of Equation (16) for the interactions of s-s, s-p and p-p orbitals have been evaluated in terms of well-known error functions [43,44,45]. We have evaluated expression (Equation (16)) in a particular molecular frame of reference and this has been further discussed Section 3.

Further, to obtain the complete spherically averaged static potential, we have to take along with the electronic part of the static potential, the effective nuclear potential $V_{nucl}\left(r\right)$, which is expressed as,

$$V_{nucl}\left(r\right)=-{\displaystyle \sum }_{n=1}^2\frac{Z_{H_n}}{\left|\mathit{r}-{\mathit{r}}_{H_n}\right|}-\frac{Z_S}{\left|\mathit{r}-{\mathit{r}}_S\right|}.$$

(17)

Here, $Z_{H_n}$ (n= 1–2 refers to two different hydrogen atoms of H₂S molecule) and $Z_S$ are the atomic numbers of hydrogen and S atoms, respectively. Additionally, ${\mathit{r}}_{H_n}$ and ${\mathit{r}}_S$ are respective position vectors of the hydrogen atoms and S atom from the centre of mass of H₂S molecule. Thus, the total spherically-averaged static potential including the electronic as well as the nuclear charge distribution [46] for the H₂S molecules is

$$V_{st}\left(r\right)=V\left(r\right)-\frac{2Z_H}{r_{>}^{″}}-\frac{Z_S}{r_{>}^{‴}}.$$

(18)

Here, $r_{>}^{″}$ is the maximum of $\left(r,r_H\right)$ and $r_{>}^{‴}$ is the maximum of $\left(r,r_S\right)$. The total static potential for the electron-H₂S elastic scattering given by Equation (18) can be further simplified and expressed as [46,47]

$$V_{st}\left(r\right)=\{\begin{array}{c}V\left(r\right)-\frac{2Z_H}{r}- \frac{Z_S}{r} \mathrm{for} r\ge r_H. \\ V\left(r\right)-\frac{2Z_H}{r_H}-\frac{Z_S}{r} \mathrm{for} r_S<r<r_H. \\ V\left(r\right)-\frac{2Z_H}{r_H}-\frac{Z_S}{r_S} \mathrm{for} r\le r_S.\end{array}$$

(19)

Along with the static potential $V_{st}\left(r\right)$, we also include the exchange, polarisation and absorption potentials. We have chosen the modified exchange potential $V_{ex}\left(r\right)$ given by Gianturco and Scialla [36],

$$V_{ex}\left(r\right)=\frac{1}{2}\left[E-V_{st}+\frac{3}{10}{\left(3{\pi }^2\rho \left(r\right)\right)}^{2/3}\right]-\frac{1}{2}{\left\{{\left[E-V_{st}+\frac{3}{10}{\left(3{\pi }^2\rho \left(r\right)\right)}^{2/3}\right]}^2+4\pi \rho \left(r\right)\right\}}^{1/2}.$$

(20)

The polarization potential $V_{pol}\left(r\right)$ is taken as [37]

$$V_{pol}\left(r\right)=\{\begin{array}{c}{V}_{co}\left(r\right) if r<r_{cp}\\ \\ V_p\left(r\right) if r\ge r_{cp}\end{array},$$

(21)

where $r_{cp}$ is the crossing point of $V_{co}\left(r\right)$ and $V_p\left(r\right)$. For the long range, we have used polarisation potential given by, $V_p\left(r\right)=-{\alpha }_d/2r^4$, where ${\alpha }_d$ is the dipole polarizability of H₂S molecule and for the short range, we have chosen the following form given by Padial and Norcross [38]

$$V_{co}\left(r\right)=\{\begin{array}{c}0.0311\ln \left(r_s\right)-0.0584+0.006r_s\ln \left(r_s\right)-0.015r_s, r_s\le 0.7\\ -0.07356+0.02224 \ln \left(r_s\right), 0.7\le r_s<10.0\\ -0.584r_s^{-1}+1.988r_s^{-3/2}-2.450r_s^{-2}-0.733r_s^{-5/2}, r_s\ge 10.0\end{array}$$

(22)

where $r_s={\left[3/4\pi \rho \left(r\right)\right]}^{1/3}$.

To consider the different inelastic channels during the scattering process, we have used the semi-empirical absorption potential $V_{abs}\left(r\right)$, expressed as [39],

$$V_{abs}\left(r,E\right)=-\rho \left(r\right)\sqrt{\frac{T_{loc}}{2}}\left(\frac{8\pi }{10k_f^{3}E}\right)\theta \left(k^2-k_f^2-2\Delta \right)\times \left(A_1+A_2+A_3\right).$$

(23)

Here $T_{loc}$ is the total local kinetic energy of the electron and can be represented as

$$T_{loc}=E-\left(V_{st}+V_{ex}+V_{pol}\right),$$

and $k_f=\sqrt[3]{3{\pi }^2\rho \left(r\right)}$ is the Fermi wave vector. $\theta \left(x\right)$ is the Pauli-blocking function. For $x\ge 0$, the value of $\theta \left(x\right)$ is unity which represents the Pauli-allowed final states of electron–H₂S collision. Otherwise, $\theta \left(x\right)$ is zero referring to the Pauli-blocked collision.$A_1, A_2, A_3$ are the dynamic parameters that depend upon $\theta \left(x\right)$, $\rho \left(r\right)$, E and $\Delta$. Here $\Delta$ is the threshold parameter from where the inelastic channels open. We have taken $\Delta$ as the ionization energy of H₂S.

In the case of the positron scattering from the H₂S molecule, we have reversed the sign of the static potential as given in Equation (19) for the electron scattering case and added to it the following polarization potential given by Jain et al. [40] for the short range

$$V_{co}\left(r\right)=\{\begin{array}{c}-0.91r_s^{-1/2}+\left[0.00255 \ln (r_s)-0.0575\right]\ln \left(r_s\right)+0.5835, r_s<0.302\\ -0.461525-0.04549r_s^{-2}, 0.302\le r_s<0.56\\ -\frac{4.3637}{{\left(r_s+2.5\right)}^3}+\frac{-6.5755+0.4776r_s}{{\left(r_s+2.5\right)}^2}+\frac{1.43275}{r_s+2.5}-0.3149, 0.56\le r_s<8.0\end{array}$$

(24)

We have used the absorption potential given by Reid and Wadehra [41]. Its form is given as,

$$V_{ab}=4\pi {\left(\frac{a_{0}R}{ϵE_F}\right)}^2\{\begin{array}{c}f\left(1\right); {ϵ}^2-\delta \ge 1\\ f\left(\sqrt{{ϵ}^2-\delta }\right); 1\ge {ϵ}^2-\delta \ge 0\\ f\left(0\right); 0\ge {ϵ}^2-\delta \end{array}.$$

(25)

Here, $a_0$ is the Bohr radius, which is taken unity and $R$ is the Rydberg constant. $ϵ=\sqrt{\frac{E}{E_F}}, \delta =\frac{\Delta \left(E\right)}{E_F},$ where $E_F$ is the Fermi kinetic energy. $\Delta \left(E\right)$ is an important parameter for positron scattering and for absorption potential in the low to high impact energy, we chose it as proposed by Stevens et al. [48] i.e.,

$$\Delta \left(E\right)=\frac{\Delta -\left(\Delta -{\Delta }_P\right)}{\left[1+{(E/3I-1)}^2\right]},$$

(26)

where, ${\Delta }_P=I-6.8 \mathrm{eV}$, $\Delta$ is taken as the first electronic excitation energy of H₂S molecule.

For H₂S, the required values of the dipole polarizability, ionization potential and electronic excitation energy are taken, respectively, as 24.53 a.u. [49], 10.457 eV [50] and 6.3 eV [51].

3. Results and Discussion

Through the GAUSSIAN-16 software, we have obtained the analytical form of the Gaussian wavefunction of H₂S with the STO-6G Gaussian basis set. In fact, we tested our calculation by using three basis sets viz. STO-3G, STO-4G and STO-6G wavefunctions and achieved the optimization. We obtained coefficients $a_{ik}$ and $c_{kj}$ and exponents ${\alpha }_{kj}$ as in Equations (10), (11) and (13) using the GAUSSIAN-16 package. The values of $a_{ik}$ are provided in

Table 1. Energy eigenvalues of the molecular orbitals of H₂S with coefficients $a_{ik}$ as in Equation (10).

Energy Eigenvalues (eV)		−91.82436	−8.62558	−6.31578	−6.31354	−6.31008	−0.88948	−0.50648	−0.39461	−0.28071
	1s	0.99783	−0.35478	0.00000	−0.00731	0.00000	0.08458	0.00000	0.04775	0.00000
S	2s	0.00649	1.04753	0.00000	0.02175	0.00000	−0.28399	0.00000	−0.14345	0.00000
	2px	0.00000	0.00000	0.00000	0.00000	0.98740	0.00000	0.00000	0.00000	−0.30318
	2py	0.00000	0.00000	0.98494	0.00000	0.00000	0.00000	−0.18896	0.00000	0.00000
	2pz	−0.00017	−0.01711	0.00000	0.98503	0.00000	0.05106	0.00000	−0.22009	0.00000
	3s	−0.00087	0.04782	0.00000	−0.01091	0.00000	0.79115	0.00000	0.61998	0.00000
	3px	0.00000	0.00000	0.00000	0.00000	0.04632	0.00000	0.00000	0.00000	1.03186
	3py	0.00000	0.00000	0.05717	0.00000	0.00000	0.00000	0.54074	0.00000	0.00000
	3pz	0.00016	−0.00875	0.00000	0.05621	0.00000	−0.12548	0.00000	0.68194	0.00000
H1	1s	0.00021	−0.01041	−0.01140	0.01169	0.00000	0.21629	0.48298	−0.33510	0.00000
H2	1s	0.00021	−0.01041	0.01140	0.01169	0.00000	0.21629	−0.48298	−0.33510	0.00000

, while ${\alpha }_{kj}$ and $c_{kj}$ are given in

Table 2. Calculated values of exponents ${\alpha }_{kj}$ and coefficients $c_{kj}$ of STO-6G orbitals of H₂S.

Atom	Orbital	Exponent	Coefficient
S	1s	0.5529038289E + 04	0.9163596281E − 02
		0.1013743118E + 04	0.4936149294E − 01
		0.2836087927E + 03	0.1685383049E + 00
		0.9742727471E + 02	0.3705627997E + 00
		0.3783386178E + 02	0.4164915298E + 00
		0.1558207360E + 02	0.1303340841E + 00
	2s	0.3455896791E + 03	−0.1325278809E − 01
		0.6840121655E + 02	−0.4699171014E − 01
		0.2125904712E + 02	−0.3378537151E − 01
		0.8179121699E + 01	0.2502417861E + 00
		0.3552198128E + 01	0.5951172526E + 00
		0.1628232301E + 01	0.2407061763E + 00
	2p	0.3455896791E + 03	0.3759696623E − 02
		0.6840121655E + 02	0.3767936984E − 01
		0.2125904712E + 02	0.1738967435E + 00
		0.8179121699E + 01	0.4180364347E + 00
		0.3552198128E + 01	0.4258595477E + 00
		0.1628232301E + 01	0.1017082955E + 00
	3s	0.1294439442E + 02	−0.7943126362E − 02
		0.3466625105E + 01	−0.7100264172E − 01
		0.1300021248E + 01	−0.1785026925E + 00
		0.5819134077E + 00	0.1510635058E + 00
		0.2879592903E + 00	0.7354914767E + 00
		0.1484042983E + 00	0.2760593123E + 00
	3p	0.1294439442E + 02	−0.7139358907E − 02
		0.3466625105E + 01	−0.1829277070E − 01
		0.1300021248E + 01	0.7621621429E − 01
		0.5819134077E + 00	0.4145098597E + 00
		0.2879592903E + 00	0.4889621471E + 00
		0.1484042983E + 00	0.1058816521E + 00
H	1s	0.3552322122E + 02	0.9163596281E − 02
		0.6513143725E + 01	0.4936149294E − 01
		0.1822142904E + 01	0.1685383049E + 00
		0.6259552659E + 00	0.3705627997E + 00
		0.2430767471E + 00	0.4164915298E + 00
		0.1001124280E + 00	0.1303340841E + 00

.

Further, we have evaluated the integral given in Equation (16) in a molecular frame of H₂S which is fixed with the S atom at (0.0000, 0.0000, 0.1030) Å and the two H atoms at the positions (0.0000, 0.9616, −0.8239) Å and (0.0000, −0.9616, −0.8239) Å as shown in Figure 1. By using these coordinates, we can obtain the values of $r_H=2.39296$ and $r_S=0.19464$ (in atomic unit) as required in Equation (19). The angle between the two H atoms is 92.11°.

${\mathit{r}}_P$ is the product vector of the wavefunctions for H with S and two H atoms. In order that the ${\mathit{r}}_P$ vector coincides with the molecular frame, as given in Figure 2, one must rotate ${\mathit{r}}_P$ frame through Euler angles (α, β, γ). Here, we take ${\mathit{r}}_P$ vector to be along the z_P axis in the r_P frame. This choice makes it possible to have the angle α = 0, which leads to significant simplifications in our geometrical considerations. Further, the direction of y_p is decided by the cross product ${\mathit{r}}_P$×z and the remaining axis x_p is specified to complete the standard coordinate axes. If we consider the angle between ${\mathit{r}}_P$ and the original z-axis to be β, then, at first, the y_p axis is rotated by an angle β (0 ≤ β ≤ π) to bring ${\mathit{r}}_P$ into the z-axis. Subsequently, the z-axis is rotated through angle γ so that y_p coincides with the y-axis of the original molecular frame. Following the above procedure of rotation, we find the Euler angles (α, β, γ) for S-H1, S-H2 and H1-H2 to be ($0,\pi -A,\pi /2), \left(0,\pi -A,3\pi /2\right) \mathrm{and} \left(0,\pi ,\pi /2\right)$, respectively, where, $A={\cos }^{-1}\left(0.694\right)$.

3.1. Electron-H₂S Differential Cross Sections

We have obtained the electron impact differential cross section for H₂S in the range 15–500 eV and shown results only at a few selected energies. In Figure 3, we have shown our DCS results for 15, 20, 50, 100 eV and compared these with the available experimental results of Gulley et al. [13], Rawat et al. [14], Cho et al. [15] and Brescansin et al. [16] along with the theoretical calculations of Rawat et al. [14], Brescansin et al. [16], Varella et al. [17], Machado et al. [18] and Jain et al. [19].

For 15 eV, as shown in Figure 3a, our present calculation shows good agreement with the experimental measurement of Gulley et al. [13] up to 70°, thereafter, it slightly underestimates other results up to 120°. At the backward angles, our results show similar a nature and reasonable match with previous theoretical [17,18] calculations. It can be seen from Figure 3b that our results at 20 eV are in good agreement with both the measurements [13,15] up to 60° and then similar to other calculations [14,16,17,18] show two minima around 70° and 140°. Our DCSs are in better agreement with the measurements of Cho et al. [15] at the large scattering angles. Figure 3c shows that at 50 eV our results show an overall good qualitative agreement with the measured data of Brescansin et al. [16] and other theories [14,16,18,19] except for relatively sharper minima close to 55° and 135°. At 100 eV present DCSs show good agreement with measurements [14,16] till 50° and beyond that, these are slightly higher. Our results are in good agreement with the calculations of Jain et al. [19]. However, in comparison to the DCSs values from Rawat et al. [14] and Brescansin et al. [16], our results are somewhat larger, excluding the sharper minimum around 115° which is also observed by Jain et al. [19]. This is because both the calculations do not include the contribution from rotational excitation, which on the other hand, is considered in theoretical calculations of Rawat et al. [14] and Brescansin et al. [16], who showed that the sharp minimum at 100 eV disappears by including rotational excitation in calculating DCS. Since in the present calculation, we have taken the scattering from the target from a spherically averaged potential in the calculation, this might be the reason for the sharp minima which could disappear if we had considered the scattering first from the different molecular orientation and then taken the average of the results [52]. The same reason can be responsible for getting a sharper minimum around 55° for 50 eV.

Figure 4 shows our calculated DCS results at high incident electron energies from 150–500 eV. These results have been compared with the available experimental results of Rawat et al. [14], as well as the theoretical calculations of Brescansin et al. [16], Rawat et al. [14] and Jain et al. [19]. Our DCSs show an overall reasonable agreement with both the experimental [14] and theoretical [14,16,19] results. However, the agreement among various results is better at small scattering angles. We also notice that as the energy increases, the position of minimum shifts toward the forward scattering angle. Further, as expected the backscattering decreases with the increased incident electron energy due to lesser electron–target interaction and hence, scattering is dominated in the forward direction.

3.2. Electron-H₂S Integral and Momentum Transfer Cross Section

In Figure 5 we have compared our calculated integral and momentum transfer cross section with the experimental results of Gulley et al. [13], Rawat et al. [14], Cho et al. [15], Brescansin et al. [16] and as well as with the theoretical results of Rawat et al. [14], Brescansin et al. [16], Machado et al. [18] and Jain et al. [19]. Only measurements and calculations that have been used for comparison with our DCSs are included in Figure 5. Figure 5a shows that our ICS results are in better agreement with the measurements [13,15] in comparison to the other calculations [14,16,19], which are slightly higher up to 100 eV. The present ICSs also show good agreement with the theoretical results of Machado et al. [18]. The difference among various calculated ICSs tends to reduce beyond 100 eV. Figure 5b displays our MTCSs and their comparison with other results. As seen in the case of ICSs, the present MTCSs results also show better agreement with the measurements of Gulley et al. [13] and Cho et al. [15] as compared to the other theoretical results [14,16,19], which overestimate the experimental cross sections up to 100 eV. All theoretical curves merge at higher energies from 100–500 eV.

3.3. Electron-H₂S Absorption and Total Cross Section

In Figure 6, we have shown the absorption and total cross sections with respect to the incident electron energy up to 500 eV. The present ACSs have been compared with the calculations of Brescansin et al. [16] and Jain and Baluja [25] in Figure 6a. We see that our calculated ACSs follow a similar behavior as previously reported results [16,25] in the entire energy range from ionization threshold to 500 eV. We have used the same absorption potential as taken by Brescansin et al. [16] and Jain and Baluja [25] but differ in taking a combination of the other potential components (see Equation (19)).

In Figure 6b our calculated TCSs, i.e., the total elastic cross section plus total absorption cross section, has been compared with the previous experimental measurements of Szmytkowsky et al. [26] and Zecca et al. [27] along with the theoretical results of Brescansin et al. [16], Jain and Baluja [25] and Joshipura and Vinodkumar [28].

Our present TCS results are slightly lower than the experimental results of Szmytkowsky et al. [26] but lie within the error bars of the measurements reported by Zecca et al. [27]. Previously reported theoretical results [16,25,28] also show good agreement with our calculations in the entire energy range, whereas the calculations of Jain and Baluja [25] are higher at low energies but match well beyond 70 eV. Overall, our results obtained by using an analytical static potential can be said to be in good agreement with the previous measurements and calculations. In addition to the graphical display of the cross sections, we have also tabulated the present cross sections in

Table 3. Cross sections for electron-H₂S scattering. The DCS is in units of ${10}^{-16}{\mathrm{cm}}^2/\mathrm{sr}$ and ICS, MTCS, ACS and TCS are in units of ${10}^{-16}$ cm².

DCS
Angle (deg)	15 eV	20 eV	50 eV	100 eV	150 eV	200 eV	300 eV	500 eV
0	25.1610	26.6501	34.0321	17.0130	18.1060	19.0440	21.5230	23.6450
5	24.1281	25.2460	29.9162	13.9210	13.5510	13.2680	13.1140	11.9590
10	21.2822	21.6572	21.4790	8.6657	7.3693	6.7151	5.9198	4.7034
15	17.5680	17.2411	14.1612	5.1474	4.0113	3.4772	2.7449	1.8663
20	13.8621	13.1460	9.0720	3.0349	2.1488	1.7617	1.2962	0.8442
30	8.0617	7.2216	2.9852	0.9611	0.6514	0.5354	0.3810	0.2470
40	4.3292	3.5418	0.5673	0.3287	0.2632	0.2117	0.1607	0.1305
50	2.0106	1.3875	0.0237	0.1466	0.1517	0.1334	0.1095	0.0689
60	0.7741	0.3847	0.0513	0.1302	0.1403	0.1156	0.0743	0.0419
70	0.2951	0.1135	0.1489	0.1692	0.1316	0.0905	0.0512	0.0302
80	0.2221	0.2033	0.2148	0.1839	0.0994	0.0608	0.0364	0.0236
90	0.2801	0.3758	0.2378	0.1484	0.0563	0.0348	0.0236	0.0201
100	0.3250	0.4784	0.2183	0.0799	0.0207	0.0165	0.0171	0.0189
110	0.3303	0.4674	0.1603	0.0174	0.0085	0.0145	0.0204	0.0207
120	0.3491	0.3908	0.0879	0.0037	0.0321	0.0358	0.0347	0.0252
130	0.4512	0.3237	0.0362	0.0692	0.0953	0.0811	0.0596	0.0318
140	0.6762	0.3374	0.0357	0.2189	0.1894	0.1444	0.0925	0.0391
150	1.0033	0.4515	0.0898	0.2462	0.2968	0.2150	0.1279	0.0464
160	1.3501	0.6156	0.1789	0.6397	0.3949	0.2796	0.1596	0.0527
170	1.6156	0.7661	0.2539	0.7997	0.4634	0.3249	0.1814	0.0569
180	1.7216	0.8306	0.2913	0.8589	0.4881	0.3415	0.1889	0.0579
ICS	25.50	20.73	8.15	4.97	4.13	3.68	2.86	2.19
MTCS	8.91	5.83	1.51	1.15	1.07	1.07	0.88	0.47
ACS	0.55	1.44	5.12	5.02	4.18	3.59	2.83	1.99
TCS	28.61	24.76	15.33	10.41	8.19	6.94	5.42	3.89

so that these are readily available for various possible applications.

3.4. Positron-H₂S Differential Cross Section

In Figure 7, we have displayed positron impact differential cross sections for 20, 50, 100, 500 eV. So far, there are no experimental or theoretical results reported for positron—H₂S collisions with which we can compare our cross section results. From Figure 7, we notice that the position of the minima in the DCS curves move towards the forward angles as the energy increases. For the sake of comparison, in Figure 8 our DCS results for electron and positron scattering at selected energies are displayed. A clear difference in the shapes of their DCS curves can be noticed. In the case of the positron, the DCSs curves show monotonic behavior at large scattering angles, unlike electron scattering DCSs. This reveals the different nature of interactions of the two charged particles with a molecular target; for instance, exchange potential is absent in the case of the positron, while the static potential is of opposite signs for the two cases.

3.5. Positron-H₂S Elastic, Momentum, Absorption and Total Cross Section

Figure 9 depicts our positron impact results for elastic, momentum, absorption and total cross sections and compared these with the corresponding electron scattering cross sections.

In Figure 9a, we can see that the momentum transfer cross sections of the positron are lower than those for the electron. These differences become smaller for energies higher than 40 eV and can be traced back to the behavior of the DCS results, which are larger for electron backward scattering. A similar feature, as found for MTCSs, can be seen for the elastic cross section results in Figure 9b. However, in the case of the absorption cross sections for positron scattering we find the opposite, i.e., these are higher than the ACSs for electron collisions in the entire energy range up to 500 eV. For the positron scattering, the inelastic channels also include the positronium formation in the ACS results. Further, we observe that the total cross sections for electron scattering are higher than the corresponding results for positron scattering up to 30 eV and the difference continues to reduce as the energy increases. This is quite expected, as with increasing energy, the polarization and exchange potentials do not influence the cross sections significantly. Since the ACSs for the positron are somewhat higher than the electrons’ scattering results, we obtain slightly higher TCS results for positron scattering. The TCSs in most of the previous calculations for electron and positron scattering from other molecules [53] also show similar behaviors, i.e., the electron and positron cross sections approach each other for higher energies. Finally, all the cross sections are also provided in

Table 4. Cross sections for positron-H₂S scattering. The DCS is in units of ${10}^{-16}{\mathrm{cm}}^2/\mathrm{sr}$ and ICS, MTCS, ACS and TCS are in units of ${10}^{-16}$ cm².

DCS
Angle (deg)	20 eV	50 eV	100 eV	500 eV
0	16.7280	15.7601	15.0310	16.439
5	14.3860	10.8720	7.0699	1.0818
10	9.3056	4.2009	1.4995	2.3921
15	5.0222	1.3262	0.4113	2.2818
20	2.4881	0.3603	0.5887	1.1657
30	0.4157	0.3904	0.8018	0.2895
40	0.1007	0.5507	0.5577	0.1292
50	0.2471	0.4176	0.1975	0.0648
60	0.3645	0.2342	0.1214	0.0384
70	0.3504	0.1311	0.0982	0.0227
80	0.2744	0.0956	0.0778	0.0171
90	0.1868	0.0862	0.0584	0.0119
100	0.1259	0.0799	0.0447	0.0095
110	0.0927	0.0711	0.0364	0.0078
120	0.0796	0.0616	0.0311	0.0064
130	0.0785	0.0532	0.0275	0.0059
140	0.0817	0.0472	0.0248	0.0049
150	0.0852	0.0434	0.0229	0.0047
160	0.0870	0.0416	0.0219	0.0043
170	0.0889	0.0398	0.0208	0.0041
180	0.0893	0.0394	0.0206	0.0048
ICS	9.01	6.30	4.45	1.28
MTCS	1.86	1.15	0.72	0.17
ACS	14.03	10.09	7.24	2.66
TCS	23.04	16.39	11.69	3.94

for positron-H₂S elastic scattering.

4. Conclusions

In the present work, we have analytically calculated spherically averaged static potential to study the electron and positron impact elastic scattering from hydrogen sulfide, using the complex optical model potential approach. Along with the static potential, exchange, polarization and absorption potentials are also considered in our calculations. The target H₂S is represented by STO-6G basis set to calculate the charge density and subsequently, the static potential. We obtained Euler angles for calculating the overlap integrals involving the product of orbital wavefunctions of one atom with another in a particular molecular frame. Solving integrals in this manner makes our method unique in comparison to the other existing theoretical techniques. We have reported the results for DCS for electron and positron impact energies from 20–500 eV along with ICS, MTCS, ACS and TCS from 10–500 eV. All our reported results for electron scattering are compared with the available other theoretical and experimental results and we observed overall a good agreement. For positron scattering, we reported the cross section results for the first time and compared them with the electron scattering cross sections to study the difference between the two cases. We found that the total cross sections for electron and positron impact differ at lower energies. Thereafter, both the cross sections are very close to each other, as expected and noticed in most previous studies on other molecules. From our present study, we can say that we have presented a successful application of our technique of using the analytical static potential to describe electron and positron elastic collision with H₂S and the same can be further extended to study bigger and more complex molecules as well.