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

: A detailed study of elastic scattering of electrons and positrons from a hydrogen sulphide (H 2 S) 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 H 2 S 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 H 2 S. Therefore, the results for electron and positron impact di ﬀ erential, integral, momentum-transfer, absorption and total cross sections are obtained for the incident energies in the range of 10–500 eV. Comparisons of these di ﬀ erent types of cross section results with the available measurements and other calculations show good agreement, which suggests the applicability of our present approach.


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.
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 H 2 S, there are no earlier experimental or theoretical studies reported. However, such studies can be equally important to probe the nature of positron-H 2 S 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 H 2 S. 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-H 2 S 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 H 2 S 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.

Scattering Amplitude and Cross Sections
The scattering of electrons and positrons from a small molecule like H 2 S can be described in a non-relativistic manner by the Schrödinger equation as given below [30] (atomic units are used throughout) Here, E refers to the incident electron or positron energy, Ψ (r) is the wavefunction and V opt (r) is the total complex optical potential, which includes static V st (r), exchange V ex (r), polarisation V pol (r) and absorption V abs (r) potentials for electron, i.e., V opt (r) = V st (r) + V ex (r) + V pol (r) + iV abs (r). (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 H 2 S is a quasi-spherically symmetric molecule with heavier atom S at the center, the interaction potential V opt (r) is approximated as spherically symmetric. Further, to study the elastic scattering of an electron or a positron from H 2 S 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 Ψ (r) 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 where k 2 = 2E, k is the momentum of electron or positron and u l (r) is the radial wavefunction associated with partial wave l satisfying Equation (3) and can be solved subjecting to the following asymptotic boundary conditions, Here, δ l = δ Real + iδ 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, where P l (cos θ) is the l th order Legendre polynomial. f (θ) 2 gives differential cross section, while total elastic integral (σ el ), momentum transfer cross section (σ mtcs ) and absorption cross sections (σ abs ) are finally obtained by using the following expressions [30,31,42] and, Further, the total cross section (σ total ) is obtained by adding the total elastic integrated cross section (σ el ) with the absorption cross section (σ abs ).

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 H 2 S molecule by taking the squared modulus of the molecular wavefunction at a point r as seen by the projectile. 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 H 2 S. The molecular orbitals X i (r ) are further expressed as linear combination of atomic orbitals φ k (r ), i.e., here a ik is the molecular orbital co-efficient and k extends on the total number of atomic orbitals, which are 11 for H 2 S. Each atomic orbital is represented as where N k j and c k j are the normalization constants and coefficients of the Gaussian basis functions χ k j (r ), 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, Here, the unnormalized form of the Gaussian basis function χ k j (r ), which is centered on its nucleus A with coordinates A x , A y and A z can be represented as Here, m x , m y and m z are non-negative integers and α k j 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 where r P is the position vector of P with respect to the molecular frame can be given by and α and E k j,k j can be expressed as, . Further, the electronic static interaction potential between a projectile electron and molecule at a point r can be represented by 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., here, the expression r > is greater of r and r . 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 (1s 2 2s 2 2p 6 3s 2 3p 4 )), 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 (r), which is expressed as, Here, Z H n (n= 1-2 refers to two different hydrogen atoms of H 2 S molecule) and Z S are the atomic numbers of hydrogen and S atoms, respectively. Additionally, r H n and r S are respective position vectors of the hydrogen atoms and S atom from the centre of mass of H 2 S molecule. Thus, the total spherically-averaged static potential including the electronic as well as the nuclear charge distribution [46] for the H 2 S molecules is Here, r > is the maximum of (r, r H ) and r > is the maximum of (r, r S ). The total static potential for the electron-H 2 S elastic scattering given by Equation (18) can be further simplified and expressed as [46,47] Along with the static potential V st (r), we also include the exchange, polarisation and absorption potentials. We have chosen the modified exchange potential V ex (r) given by Gianturco and Scialla [36], The polarization potential V pol (r) is taken as [37] V pol (r) = where r cp is the crossing point of V co (r) and V p (r). For the long range, we have used polarisation potential given by, V p (r) = −α d /2r 4 , where α d is the dipole polarizability of H 2 S molecule and for the short range, we have chosen the following form given by Padial and Norcross [38] V co (r) = where r s = [3/4πρ(r)] 1/3 . To consider the different inelastic channels during the scattering process, we have used the semi-empirical absorption potential V abs (r), expressed as [39], Here T loc is the total local kinetic energy of the electron and can be represented as is unity which represents the Pauli-allowed final states of electron-H 2 S collision. Otherwise, θ(x) is zero referring to the Pauli-blocked collision. A 1 , A 2 , A 3 are the dynamic parameters that depend upon θ(x), ρ(r), E and ∆. Here ∆ is the threshold parameter from where the inelastic channels open. We have taken ∆ as the ionization energy of H 2 S. In the case of the positron scattering from the H 2 S 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 We have used the absorption potential given by Reid and Wadehra [41]. Its form is given as, Here, a 0 is the Bohr radius, which is taken unity and R is the 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., where, ∆ P = I − 6.8 eV, ∆ is taken as the first electronic excitation energy of H 2 S molecule. For H 2 S, 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].

Results and Discussion
Through the GAUSSIAN-16 software, we have obtained the analytical form of the Gaussian wavefunction of H 2 S 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 k j and exponents α k j as in Equations (10), (11) and (13) using the GAUSSIAN-16 package. The values of a ik are provided in Table 1, while α k j and c k j are given in Table 2.  Table 2. Calculated values of exponents α k j and coefficients c k j of STO-6G orbitals of H 2 S. Further, we have evaluated the integral given in Equation (16) in a molecular frame of H 2 S 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 • .  Figure 1. By using these coordinates, we can obtain the values of = 2.39296 and = 0.19464 (in atomic unit) as required in Equation (19). The angle between the two H atoms is 92.11°. is the product vector of the wavefunctions for H with S and two H atoms. In order that the vector coincides with the molecular frame, as given in Figure 2, one must rotate frame through Euler angles (α, β, γ). Here, we take vector to be along the zP axis in the rP 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 yp is decided by the cross product ×z and the remaining axis xp is specified to complete the standard coordinate axes. If we consider the angle between and the original z-axis to be β, then, at first, the yp axis is rotated by an angle β (0 ≤ β ≤ π) to bring into the z-axis. Subsequently, the z-axis is rotated through angle γ so that yp 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, − , /2), (0, − , 3 /2) and (0, , / 2), respectively, where, = cos (0.694). is the product vector, as defined in Equation (14), while xp is an arbitrary vector perpendicular to and yp is perpendicular to both and xp. (α, β, γ) are the standard Euler angles through which (xp, yp, ) are transformed to (x, y, z).

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  r P is the product vector of the wavefunctions for H with S and two H atoms. In order that the r P vector coincides with the molecular frame, as given in Figure 2, one must rotate r P frame through Euler angles (α, β, γ). Here, we take 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 r P ×z and the remaining axis x p is specified to complete the standard coordinate axes. If we consider the angle between 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 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, π − A, π/2), (0, π − A, 3π/2) and (0, π, π/2), respectively, where, A = cos −1 (0.694).  Figure 1. By using these coordinates, we can obtain the values of = 2.39296 and = 0.19464 (in atomic unit) as required in Equation (19). The angle between the two H atoms is 92.11°. is the product vector of the wavefunctions for H with S and two H atoms. In order that the vector coincides with the molecular frame, as given in Figure 2, one must rotate frame through Euler angles (α, β, γ). Here, we take vector to be along the zP axis in the rP 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 yp is decided by the cross product ×z and the remaining axis xp is specified to complete the standard coordinate axes. If we consider the angle between and the original z-axis to be β, then, at first, the yp axis is rotated by an angle β (0 ≤ β ≤ π) to bring into the z-axis. Subsequently, the z-axis is rotated through angle γ so that yp 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, − , /2), (0, − , 3 /2) and (0, , / 2), respectively, where, = cos (0.694). is the product vector, as defined in Equation (14), while xp is an arbitrary vector perpendicular to and yp is perpendicular to both and xp. (α, β, γ) are the standard Euler angles through which (xp, yp, ) are transformed to (x, y, z).

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,   Diagram showing Euler angles (α, β, γ) in H 2 S molecule where, (x, y, z) is the set of orthogonal vector for the fixed molecular frame. r P is the product vector, as defined in Equation (14), while x p is an arbitrary vector perpendicular to r P and y p is perpendicular to both r P and x p . (α, β, γ) are the standard Euler angles through which (x p , y p , r P ) are transformed to (x, y, z).

Electron-H 2 S Differential Cross Sections
We have obtained the electron impact differential cross section for H 2 S 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. 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.

Rawat et al. [14]
Brescansin et al. [16] Rawat et al. [14] Brescansin et al. [16] Jain et al. [19] (d) 100 eV Scattering angle (θ)  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.  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.

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. Rawat et al. [14] Brescansin et al. [16] Jain et al. [19] Scattering angle (θ)

Electron-H 2 S 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. 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.

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)).

Electron-H 2 S 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]. 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.

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].  Rawat et al. [14] Cho et al. [15] Brescansin et al. [16] Rawat et al. [14] Brescansin et al. [16] Machado et al. [18] Jain et al. [19]  Rawat et al. [14] Cho et al. [15] Brescansin et al. [16] Rawat et al. [14] Brescansin et al. [16] Machado et al. [18] Jain et al. [19]  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 so that these are readily available for various possible applications.

Positron-H 2 S 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-H 2 S 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.

Positron-H 2 S 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.

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 for positron-H2S elastic scattering.  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 for positron-H 2 S elastic scattering.

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 H 2 S 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 H 2 S and the same can be further extended to study bigger and more complex molecules as well.