IR,NMR,UV

# Twisted electron impact elastic cross sections of polyatomic molecules: All
active electron multicentered approach

Raul Sheldon Pinto  Department of Physics, Birla Institute of Technology and
Science, Pilani, Pilani Campus 333-031 (India<EMAIL_ADDRESS>pilani.ac.in Rakesh Choubisa  Department of Physics, Birla Institute of
Technology and Science, Pilani, Pilani Campus 333-031 (India).
<EMAIL_ADDRESS>
###### Abstract

The absolute elastic differential and Integral (total) cross-sections of
polyatomic molecules by plane and twisted electron beam (Bessel Beam) impact
are presented with $CO_{2}$ and $N_{2}O$ as examples. The cross sections are
computed theoretically in the First Born Approximation(FBA). The target
molecules are modeled using multicentered wavefunctions with Gaussian Type
Orbitals(GTOs) and optimized through Density Functional Theory. The electron
density obtained as a function of spatial coordinates is used to calculate the
elastic form factor while considering the active role of all electrons in the
molecule. Coulomb direct potentials are used for the interactions of the
incident beam with all electrons and nuclei. An orientation averaging of the
differential cross-section is performed to mimic experimental situations. The
Young’s interference effects due to the nuclei are considered. The cross
sections are calculated at intermediate energies ranging from 300eV to 1keV.
The proposed methodology can certainly be applied to any polyatomic molecule.
The interactions of Bessel beams (Twisted Electron Beam) with topological
charges $m_{l}$ = 1, 2, and 3 are analyzed. The average over impact parameters
of the differential cross-sections representing the beam’s interaction with a
large number of uniformly transversely distributed molecules is also studied
and presented. Finally, the Integral elastic cross-section (ICS) of the
molecules by plane wave and twisted beams is calculated and presented.

###### keywords:

American Chemical Society, LaTeX

##  1  Introduction

The quantum mechanical study of scattering and ionization processes of
molecular targets using diverse projectiles uncovers essential information on
the target’s spatial and electronic structure and the nature of interactions
in the quantum regime. The motive is to arrive at the cross-sections of
targets which reveal these essential information when plotted as a spectrum.
One must study the time-resolved scattering to obtain complete information on
the interactions and obtain the electronic structure of inelastic and
ionization processes. We focus only on the steady-state elastic scattering
processes where we draw important inferences on the internal structure of the
molecules and useful information on the fundamental interactions of the
targets with the projectiles.

Numerous articles have established the theoretical and experimental plane wave
scattering and ionization cross sections since the mid-1900s. Reported are
electron scattering cross-sections for almost every small atom or molecule.
Obtaining exact molecular orbital wavefunctions, electron-electron
correlations, and interaction potentials analytically can be formidable for
large molecules. Particularly in theoretical molecular physics, most orbitals
are seen to be modeled using Slater Type Orbitals (STOs) for scattering theory
calculations. The challenge with STOs is that their integrals are tedious and
can be daunting to compute 1. Little work is reported on the contribution of
core orbitals in addition to the valence in the cross-sections going beyond
the frozen core approximation. None of the work for twisted scattering and
ionization has included the effect of all the electrons and their nuclei in
their potentials to compute differential cross-sections. We use an all-active
electron model, where each electron in the molecule along with the nuclear
attractive potentials contribute to the overall electrostatic potential
through which the incident electron beam is scattered. We use density
functional theory (DFT) to estimate the electron density which is an integral
factor in the scattering computations. Quantum chemists have been using
density functional theory extensively to obtain the ground state of many
electron atoms, molecules, and solids. However, very little work using DFT in
quantum scattering calculations on molecular targets is found 2. Using DFT
with GTOs simplifies obtaining the molecular orbitals and the electron
densities required to compute the potentials hence simplifying the process of
obtaining the desired scattering amplitudes.

The study of twisted electron impact scattering and ionization processes has
recently been of augmented research interest as it has shown promising
applications in high-resolution imaging3, manipulation of nanoparticles by
tweezing 4, and quantum communication5. Twisted beams have additional degrees
of freedom due to their orbital angular momenta enabling them to act as higher
information containers (qubits, qutrits, qudits, etc.). The helical motion of
twisted electrons with time produces a magnetic field. This magnetic field can
interact with magnetic moments and thus can also be used to characterize the
magnetic properties of the targets under consideration. Owing to the control
of the size of the beam, one may generate very high magnetic fields at the
nanoscale 6. To realize these applications, it is essential to understand the
nature and dynamics of the interaction of twisted electron beams with matter.
Many types of twisted beams have been developed. The most natural is the
Bessel beam which is the solution of the Helmholtz equation in cylindrical
coordinates7.

The study of twisted electron beam ionization with atoms and small molecules
such as the Hydrogen atom, Hydrogen molecule, He atom, $H_{2}O$, $CH_{4}$ and
$NH_{3}$ molecules are found in the literature 8, 9, 10. Theoretical studies
on the elastic scattering of twisted beams have been reported for atomic
targets 11, 12 and for diatomic molecules with an example of $H_{2}$ molecule
13. The complexity in orbital wavefunctions for larger systems in which the
electron correlations play a significant role in the interactions is much more
difficult to handle and is not seen in the literature. Experimentally, no work
is found on the elastic scattering of molecules by twisted electrons. Elastic
scattering is an important naturally occurring phenomenon; Choosing to study
the interactions of electrons (plane and twisted) with two greenhouse gas
molecules i.e. $CO_{2}$ and $N_{2}O$, we also contribute to environmental
research. Both $CO_{2}$ and $N_{2}O$ are linear and isoelectronic, having 22
electrons each. Studying phenomena of such complex interactions occurring
between twisted electron beams and multielectron systems is detrimental to
understanding the fundamental quantum interactions and a stepping stone to
realizing the mentioned applications.

##  2 Theoretical Formalism

In this study, we follow the right-handed coordinate system. All calculations
are done in atomic units. Gauss Legendre Quadrature has been used to compute
the integrals. The molecules are constructed using GaussView, optimized using
GAUSSIAN (G09), and analyzed through Multiwfn. Density Functional Theory is
used to obtain the electron densities. The effect of the interactions of the
molecular electrons with the projectile is brought about by the electron
density.

$CO_{2}$ and $N_{2}O$ molecules are independently constructed using GaussView
specifying the initial coordinates, atoms, and bonds. A ground state geometry
optimization DFT calculation is performed in GAUSSIAN (G09) 14 requesting the
output of a wavefunction file (.wfn file). We use the density functional B3LYP
with the 6-311G* Pople basis set. We choose B3LYP as it is a common choice to
model the electron correlation of small molecules 15. The choice of functional
can vary from one molecule to the other based on the electron correlations.
The generated .wfn file is then analyzed through Multiwfn, a code developed by
Lu et. al. at MIT16. We use Multiwfn to extract the electron density as a
function of the spatial Cartesian coordinate points that we provide.
Initially, the points are specified in spherical polar coordinates each for r,
$\theta$, and $\phi$ and are obtained by pre-running the Gauss-Legendre
Quadrature loops with N=12. The number of such points obtained is $N^{3}$ =
1728. The points are then converted into Cartesian and fed to Multiwfn to get
the set of values for the electron density for those points. Multiwfn uses
only Cartesian coordinates16. The angle-resolved elastic differential cross-
section and Integral cross-section calculations are performed using the
target’s electron density using the elastic scattering form factor17.

Figure 1, depicts the elastic scattering of an electron beam with initial
wavevector $k_{i}$ and final wavevector $k_{s}$ by a carbon dioxide molecule
with scattering angle being $\theta_{s}$. The elastic scattering is
azimuthally symmetric.

Figure 1: Theoretical model with an example of scattering of plane waves by
$CO_{2}$ molecule

The origin of the coordinate system is taken to be at the molecule’s center of
mass (COM). The three nuclei are the three centers of each molecule. The
molecules $CO_{2}$ and $N_{2}O$ are triatomic, isoelectronic, linear, and
greenhouse gases. They have 22 electrons each. $CO_{2}$ doesn’t have a
permanent dipole moment while $N_{2}O$ does.

In this article, we describe our theoretical model in the following
subsections:

###  2.1 Plane Waves

The incident and scattered electron projectiles are described by plane waves
and are given by: $\frac{1}{({2\pi})^{3/2}}exp(i\bm{k_{i}}\cdot\bm{r})$ and
$\frac{1}{({2\pi})^{3/2}}exp(i\bm{k_{s}}\cdot\bm{r})$ respectively. The
expressions for the initial and final state composite wavefunctions of the
interactions are then given as:

$\braket{\bm{r}}{\psi_{i}}=\frac{1}{({2\pi})^{3/2}}exp(i\bm{k_{i}}.\bm{r})\psi_{m}(\bm{r})$
(1)
$\braket{\bm{r}}{\psi_{f}}=\frac{1}{({2\pi})^{3/2}}exp(i\bm{k_{s}}.\bm{r})\psi_{n}(\bm{r})$
(2)

$\psi_{m}$ and $\psi_{n}$ are the normalized initial and final molecular
wavefunctions while $\bm{k_{i}}$ and $\bm{k_{s}}$ are the initial and final
wavevectors of the projectile respectively. The initial and final wavevectors
differ only in direction and not in magnitude. The magnitude of the
wavevectors is $k_{i}=k_{s}=k$. The momentum transfer of this interaction is
given by $\bm{\Delta}=\textbf{k}_{i}-\textbf{k}_{s}$. Its magnitude in terms
of scattering angle $\theta_{s}$ can be deduced to have the value:

$\Delta=2k\sin(\theta_{s}/2)$ (3)

###  2.2 Monochromatic Bessel Beam

A twisted Bessel electron beam is a beam of electrons that possesses Orbital
Angular Momentum (OAM) with their wavefunction having a helical structure. The
individual contributions of the electron’s OAM contribute to the overall OAM
of the beam. The beam can be described as a superposition of plane waves each
having a longitudinal and transverse component of momentum. The transverse
momentum leads to a divergence of the beam from the propagation axis. The
beam’s axis of propagation houses the line of singularity, where the phase is
indeterminate. The expression for the Bessel beam propagating along the z-axis
in the positive z-direction in cylindrical coordinates and evolving in time is
given by:

$\psi_{\mathbf{k}_{i}m_{l}}^{(tw)}\left({r,\phi,z,t,b}\right)=J_{m_{l}}(k_{i_{\perp}}r)e^{ik_{i_{\parallel}}z}e^{im_{l}\phi}e^{-i\omega
t}e^{-i\mathbf{k_{i}}\cdot\mathbf{b}}$ (4)

where $J_{m_{l}}$ is the Bessel function of order $m_{l}$. It varies with the
transverse component wavevector and the radial coordinate.
$e^{ik_{i_{\parallel}}z}$ is the plane wave phase component with
$k_{i_{\parallel}}$ being the magnitude of the longitudinal component of the
wavevector. $e^{im_{l}\phi}$ is the azimuthal phase with $m_{l}$ being the
topological charge which is exactly equal to the Bessel function’s order.
$e^{-i\mathbf{k_{i}}\cdot\mathbf{b}}$ is the phase containing the effect of
impact parameter ($\bm{b}$) and $e^{-i\omega t}$ is the time evolutionary
phase. In this article we present only a steady state case with impact
parameter $\bm{b}$ = 0 and average over all impact parameters; thus ignoring
the time-dependence completely and impact-parameter-dependent phase partially.

We consider the impact of a monochromatic Beam of a wavevector
$\mathbf{k_{i}^{tw}}$. This incident wavevector in terms of its Cartesian
components is given as:

$\textbf{k}_{i}^{tw}=\textbf{k}_{i_{\parallel}}+\textbf{k}_{i_{\perp}}$ (5)

here, the $\mathbf{k}_{i_{\parallel}}$ is the longitudinal wavevector and
$\mathbf{k}_{i_{\perp}}$ is the transverse wavevector. They are given by:

$\mathbf{k}_{i_{\parallel}}=k_{z}\hat{z}=\left(k_{i}\cos\theta_{p}\right)\hat{z}$
(6)
$\mathbf{k}_{i_{\perp}}=\left(k_{i}\sin\theta_{p}\cos\phi_{p}\right)\hat{x}+\left(k_{i}\sin\theta_{p}\sin\phi_{p}\right)\hat{y}$
(7)

Here, $\theta_{p}$ and $\phi_{p}$ are the polar and azimuthal angles of the
twisted wavevector $\textbf{k}_{i}^{tw}$.

Now, the expressions for the initial and final state wavefunctions are given
by:

$\braket{\bm{r}}{\psi_{i}}=\psi_{\varkappa m}^{(tw)}(\bm{r})\psi_{m}(\bm{r})$
(8)
$\braket{\bm{r}}{\psi_{f}}=\frac{1}{({2\pi})^{3/2}}exp(i\bm{k_{s}}.\bm{r})\psi_{n}(\bm{r})$
(9)

$\psi_{\varkappa m}^{(tw)}(\bm{r})$ is the incident twisted beam wavefunction
(here, Bessel Beam), and $\psi_{m}(r)$ and $\psi_{n}(r)$ are the normalized
initial and final molecular wavefunctions. The scattered wave is considered to
be a plane wave. The orbital angular momentum is lost somewhere. In addition,
due to the limitations of the detector, it would be extremely difficult to
detect a twisted wave at a particular solid angle.

To subdue the effect of particular beam orientations, we consider a
generalized Bessel beam by working in the Fourier basis. The Fourier-transform
of (4) is given by:

$\psi_{\varkappa
m}^{(tw)}\left(\mathbf{k}_{i}\right)=\int_{0}^{\infty}\frac{dk_{i\perp}}{2\pi}k_{i\perp}\int_{0}^{2\pi}\frac{d\phi_{p}}{2\pi}a_{\varkappa
m}\left(\mathbf{k}_{i\perp}\right)\\\ e^{i\mathbf{k}_{i}\cdot\mathbf{r}}\\\ $
(10)

while the Fourier amplitude of the twisted wavefunction is given by:

$a_{\varkappa
m_{l}}\left(\mathbf{k}_{i\perp}\right)=\sqrt{\frac{2\pi}{\varkappa}}(-\mathrm{i})^{m_{l}}\mathrm{e}^{\mathrm{i}m_{l}\varphi_{k}}\delta\left(k_{i\perp}-\varkappa\right)$
(11)

The momentum transfer of the Bessel beam interaction is given by
$\bm{\Delta^{tw}}={\textbf{k}}_{i}^{tw}-{\textbf{k}}_{s}$. Its magnitude in
terms of scattering angle ($\theta_{s}$) can be deduced to have the value:

$\Delta^{tw}=2k\sin(\theta_{ps}/2)$ (12)

where $\theta_{ps}$ is the angle between $\bm{k_{i}^{tw}}$ and $\bm{k_{s}}$.
It is calculated as follows:

$\theta_{ps}=\cos^{-1}\left[\cos{\theta_{p}}\cos{\theta_{s}}+\sin{\theta_{p}}\sin{\theta_{s}}\cos{\phi_{p}}\right]$
(13)

. Here, $\phi_{s}$ = 0 since we consider coplanar asymmetric geometry.

###  2.3 Effect of the molecular electrons and the nuclear centers

The direct Coulomb interaction potential between the incident electron and
target molecule is:

$V_{d}=-\sum_{i=1}^{M}\frac{Z_{i}}{|\bm{r_{0}}-\bm{r_{i}|}}+\sum_{j=1}^{N}\frac{1}{|\bm{r_{0}}-\bm{r_{j}|}}$
(14)

here, i is summed over the M nuclear centers each having charge $Z_{i}$,
position vector $\bm{r_{i}}$, and producing an attractive potential. j is
summed over the N molecular electrons each having charge 1, position vector
$\bm{r_{j}}$, and producing a repulsive potential. $\bm{r_{0}}$ are the
coordinates of the incident projectile, Since we work with sufficiently high
energies the exchange interactions are negligible and hence we ignore their
contributions. Here, M = 3 and N = 22. M and N will vary with other molecules.

###  2.4 Orientation averaging

Experimentally, it is extremely challenging to orient a molecule in a
particular direction, hence to obtain experimentally realizable results an
averaging over orientations must be performed. Conventionally, it is performed
by changing the orientation of the molecule in particular directions18.
Existing work report summing over all spherical harmonics pertaining to a
particular orbital. Since this is tedious and can not be standardized we
propose a new way of orientation averaging by performing a passive rotation of
the plane of incidence over all angles to give an overall effect of
orientation averaging. We do so by incrementing the angles of the incident
wavevectors by a small constant angle (used here $10^{\circ}$) independently
from 0 to 180∘ measured down from the z-axis and 0 to 360∘ projection measured
from the x-axis on the XY plane, keeping the molecular orientation fixed. This
is similar to increments in the $\theta$ and $\phi$ angles respectively in
spherical coordinates; We denote these increments by $\theta_{1}$ and
$\phi_{1}$. The angular increments are done respecting the planar geometry of
the interactions and keeping all other angles fixed with respect to the
varying incident direction. The scattered vector and momentum transfer vector
are automatically rotated incrementally by this constant angle $\theta_{1}$
and $\phi_{1}$.

Figure 2 depicts the orientation averaging that we propose where $k_{i}$ is
incremented by $10^{\circ}$. The symbol near $\phi_{1}$ implies that the
rotation is into the plane of paper. The molecule could have any shape (shown
here as spherical as an example).

Figure 2: Varying incident electron beam direction keeping molecule’s
orientation fixed with respect to coordinate axes

The cross-sections are calculated for each increment of the incidence
direction, summed, and divided by the total number of increments. Since we use
an interval of $\theta_{1}$ = $\phi_{1}$ = $10^{\circ}$. Including
$0^{\circ}$, we obtain 703 distinct incident directions, implying 703 distinct
orientations and hence 703 cross-section computations. The electron densities
and molecular orbital coordinates remain fixed with reference to the static
frame. $\theta_{1}$ and $\phi_{1}$ are explicitly used in the formalism in the
following subsections.

###  2.5 Scattering Amplitudes

Here, we provide the formalism for the angle-resolved differential scattering
amplitude calculations for the orientation averaged system in the first Born
Approximation.

####  2.5.1 Plane-wave Scattering Amplitude

The elastic scattering transition matrix element/scattering amplitude in terms
of the initial state, final state, and the direct potential $V_{d}$ (equation
14) is given by:

$T_{fi}=\braket{\psi_{f}}{V_{d}}{\psi_{i}}$ (15)

$\psi_{i}$ and $\psi_{f}$ are the composite initial and final wavefunctions
(equations 1 and 2 or equations 8 and 9) respectively. Since we consider
elastic scattering the initial and final states change only in the
projectile’s direction. Hence we can represent them both by the state
$\ket{i}$. Therefore,

$\displaystyle
T_{ii}^{pw}=V_{ii}=\braket{i}{V_{d}}{i}=-\sum_{j=1}^{M}Z_{j}\int\frac{\mathrm{e}^{i\bm{\Delta}\cdot\bm{r_{0}}}d\bm{r_{0}}}{|\bm{r_{0}}-\bm{l_{j}}|}+\int\int\frac{\rho(\bm{r})\mathrm{e}^{i\bm{\Delta}\cdot\mathbf{r_{0}}}d\bm{r_{0}}d\bm{r}}{|\bm{r_{0}}-\bm{r}|}$
(16)

$\bm{r_{0}}$ are the coordinates of the projectile, $\bm{l_{j}}$ is the
position vector of each nucleus, and $\bm{r}$ are the coordinates of the
molecule. The density of the molecules as a function of spatial coordinates is
given in terms of inner products of the molecular wavefunctions $\psi_{n}$ and
$\psi_{m}$. Since we assume molecular wavefunctions to not change therefore
$m=n$ and,

$\rho(\bm{r})=|\psi_{m}(\bm{r})|^{2}$ (17)

This electron density is obtained by DFT calculations.

$\bm{\Delta}$ is the momentum transfer. To simplify equation (16) we solve the
integral analytically using Bethe’s Integral as given by 17, 19:

$\int\frac{e^{i\bm{\Delta}\cdot\mathbf{r}_{0}}}{\left|\mathbf{r}_{0}-\mathbf{r}_{1}\right|}d\mathbf{r_{0}}=\frac{4\pi}{\Delta^{2}}e^{i\bm{\Delta}\cdot\mathbf{r}_{1}}\\\
$ (18)

Using equation (18) and simplifying (16) we obtain the transition matrix
element / Scattering amplitude of the plane wave that we split into two parts:

$T_{ii}^{pw}=-\frac{2}{\Delta^{2}}(\alpha-\chi)$ (19)

$\alpha$ is the contribution to the nuclear centers. $\chi$ is the elastic
scattering form factor, also known as the X-ray form factor which introduces
the contribution of all the electrons into the scattering amplitude (pw stands
for plane waves).

$\alpha$ the contribution of the nuclear centers whose $j^{th}$ position
vector given by $\bm{l_{j}}$ is given by:

$\alpha=\sum_{j=1}^{3}Z_{j}exp(i\bm{\Delta\cdot l_{j}})$ (20)

To evaluate $\bm{\Delta\cdot l_{j}}$ the angle that $\bm{\Delta}$ and
$\bm{l_{j}}$ make with the fixed coordinate axes is required.
$\theta_{\Delta}$ is the angle that $\bm{\Delta}$ makes with the coordinate
axes and is given by:

$\theta_{\Delta}=\cos^{-1}\left({\frac{k}{\Delta}(1-\cos{\theta_{s}})}\right)$
(21)

and since the molecules lie on the z-axis, therefore, the angle made by the
nuclear center’s position vector with the z-axis is $\theta_{\bm{l_{j}}}$ = 0∘

For $CO_{2}$ equation (20) reads:

$\alpha=8\exp(il_{1}\Delta\cos(\theta_{\Delta}+\theta_{1}))+8\exp(-il_{1}\Delta\cos(\theta_{\Delta}+\theta_{1}))+6$
(22)

$l_{1}$ is the modulus of the magnitude of the position vectors of the Oxygen
nuclei. The carbon nucleus is at the origin of our coordinate system as it
houses the COM of the molecule. In addition, there is a $\theta_{1}$ added
which accounts for the angular increment in $\theta_{\Delta}$ as the system is
orientation averaged.

(20) simplifies to:

$\alpha=16\left[cos(l_{1}\Delta cos(\theta_{\Delta}+\theta_{1}))\right]+6$
(23)

The same for $N_{2}O$ equation (20) reads:

$\displaystyle\alpha=8exp\left(il_{1}\Delta\cos\left(\theta_{\Delta}+\theta_{1}\right)\right)+7\left[\exp\left(-il_{2}\Delta\cos\left(\theta_{\Delta}+\theta_{1}\right)\right)\right.\left.+exp(-il_{3}\Delta\cos(\theta_{\Delta}+\theta_{1}))\right]$
(24)

Here, $l_{1},l_{2},$ and $l_{3}$ are the magnitudes of the position vectors of
each nuclear center from the COM the origin of our coordinate system.

The target’s elastic scattering form factor is given by:

$\chi=\int exp(i\bm{\Delta\cdot r})\rho(\bm{r})\bm{dr}$ (25)

The electron density $\rho(\bm{r})$ as a function of spatial coordinates $r$,
$\theta$, and $\phi$ is introduced into the code from the output of Multiwfn.
The cosine of the angle between $\bm{\Delta}$ and $\bm{r}$ to be used in (25)
is given by:

$\cos(\theta_{\Delta
r})=\cos(\theta_{\Delta}+\theta_{1})\cos\theta+\sin(\theta_{\Delta}+\theta_{1})\sin\theta\cos(\phi-\phi_{1})$
(26)

Here, $\theta_{1}$ and $\phi_{1}$ account for the angular increment in
$\theta_{\Delta}$ as the system is orientation averaged.

####  2.5.2 Bessel scattering amplitude

The molecule’s twisted scattering amplitude / Transition Matrix Elements are
given in terms of plane wave scattering amplitude / Matrix elements as
follows:

$T_{ii}^{tw}(\varkappa,\bm{\Delta^{tw}})=(-i)^{m}\int^{2\pi}_{0}\frac{d\phi_{p}}{2\pi}e^{im\phi_{p}}T_{ii}^{pw}(\bm{\Delta^{tw}})\\\
$ (27)

Note that, $T_{ii}^{pw}$ is now a function of $\bm{\Delta^{tw}}$. The angles
$\bm{\Delta^{tw}}$ makes with the fixed coordinate axes are:

$\theta_{\Delta^{tw}}=\cos^{-1}\left({\frac{k}{\Delta^{tw}}(\cos\theta_{p}-cos\theta_{s})}\right)$
(28)

and,

$\phi_{\Delta^{tw}}=\cos^{-1}\left[\frac{k(\sin\theta_{p}\cos\phi_{p}-\sin\theta_{s})}{\Delta^{tw}\sin\theta_{\Delta^{tw}}}\right]$
(29)

For the twisted case, $\Delta$ in the equations (20)-(25) must be replaced by
${\Delta^{tw}}$.

The cosine of the angle between $\bm{\Delta^{tw}}$ and $\bm{r}$ to be used in
(25) is given by:

$\cos\theta_{\Delta^{tw}r}=\cos(\theta_{\Delta^{tw}}+\theta_{1})\cos\theta+\sin(\theta_{\Delta^{tw}}+\theta_{1})\sin\theta\cos(\phi-\phi_{\Delta^{tw}}-\phi_{1})$
(30)

With these modifications in the expressions of $\alpha$ and $\chi$ in plane
waves of (19). We compute the twisted Bessel scattering amplitude.

###  2.6 Cross-Sections

Once $T_{ii}$ for plane waves or twisted waves are obtained, the differential
and Integral cross-section can be obtained as follows:

####  2.6.1 Plane-wave differential Cross-section

The plane wave differential cross-section is given by:

$\frac{d\sigma}{d\Omega}^{pw}=|T_{ii}^{pw}|^{2}$ (31)

####  2.6.2 OAM-dependent differential cross-section

The twisted cross-section for particular values of m is given by:

$\left(\frac{d\sigma}{d\Omega}\right)^{tw}=|T_{ii}^{tw}|^{2}$ (32)

####  2.6.3 Impact parameter averaged Bessel differential cross-section

The averaging over impact parameters is performed to mimic experimental
situations and yield experimentally realizable results. The averaging
considers the number of target molecules to be tending to $\infty$. This
averaging is done in addition to the orientation averaging. The impact
parameter averaged differential cross section treats the target to be
macroscopic and doesn’t depend on the topological charge of the beam but
rather only on its opening angle $\theta_{p}$ 12, 20.

$\left({\frac{d\sigma(\bm{\Delta})}{d\Omega}}\right)_{av}^{tw}=\frac{1}{2\pi\cos\theta_{p}}\int_{0}^{2\pi}d\phi_{p}\left(\frac{d\sigma(\bm{\Delta})}{d\Omega}\right)^{pw}$
(33)

####  2.6.4 Integral Cross section (ICS)

The total (Integral) elastic cross sections for all the above cases of plane
and twisted waves are obtained by integrating over all solid angles of the
scattered electron.

$\sigma=\int\frac{d\sigma}{d\Omega}d\Omega=2\pi\int^{\pi}_{0}d\theta_{s}\sin{\theta_{s}}\frac{d\sigma}{d\Omega}$
(34)

##  3 Results and Discussions

###  3.1 Plane-wave Elastic Differential Cross Section

The results for plane wave absolute elastic electron differential cross
section from carbon dioxide and nitrous oxide are presented in this
subsection. The results are plotted for intermediate incident electron
energies ranging from 300eV to 1keV. We note that we haven’t evaluated the
differential cross sections for the singular point $\theta_{s}=$ 0∘ as the
cross sections blow up. The cross sections begin from $\theta_{s}=$ 1∘.

Figures 3 and 4 depict 4 subplots each of differential cross section plotted
against the scattering angle $\theta_{s}$ for plane wave electron impact.
Subfigures (a), (b), (c), and (d) depict the results for $E_{i}$ = 300eV,
500eV, 800eV, and 1keV respectively. The blue continuous curve represents our
model. The red points are the results obtained experimentally.

0306090120150180$10^{0}$$10^{2}$$10^{4}$present modelIga et.
al.(1999)0306090120150180$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$present
modelIga et. al.(1984)0306090120150180$10^{-1}$$10^{1}$$10^{3}$present
modelIga et.
al.(1984)0306090120150180$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$present
modelIga et. al.(1984)$\theta_{s}(^{\circ})$$\frac{d\sigma}{d\Omega}$
$\left(\frac{(a_{0})^{2}}{sr}\right)$(a)(b)(c)(d)

Figure 3: Absolute elastic differential cross sections for plane wave electron
impact on $CO_{2}$. Incident energy: (a) 300eV, (b) 500eV (c) 800eV (d) 1keV.
The points represented by red dots are experimental values obtained by Iga et.
al.21, 22

Figure 3 depicts plots for $CO_{2}$. The experimental data points are taken
from Iga et. al. (1999) 22 for $E_{i}$ = 300eV, while for $E_{i}$ = 500eV,
800eV, and 1keV by Iga et. al. (1984) 21. The results obtained by our
theoretical model match closely with the experimental data obtained by Iga et.
al.(1984) at $E_{i}$ = 500eV, 800eV, and 1keV. However, there is some
deviation for $E_{i}$ = 300eV from that of Iga et. al.(1999)

0306090120150180$10^{0}$$10^{2}$$10^{4}$present modelLee et.
al.0306090120150180$10^{-1}$$10^{1}$$10^{3}$$10^{5}$present modelLee et.
al.0306090120150180$10^{-1}$$10^{1}$$10^{3}$present modelLee et.
al.0306090120150180$10^{-1}$$10^{1}$$10^{3}$present modelBoechat et.
al.$\theta_{s}(^{\circ})$$\frac{d\sigma}{d\Omega}$
$\left(\frac{(a_{0})^{2}}{sr}\right)$(a)(b)(c)(d)

Figure 4: Absolute elastic differential cross sections for plane wave electron
impact on $N_{2}O$. Incident energy: (a) 300eV, (b) 500eV (c) 800eV (d) 1keV.
The points represented by red dots are experimental values obtained by Lee et.
al. and Boechat et. al. 23,24

Figure 4 depicts the plots for $N_{2}O$. The experimental data points
corresponding to $E_{i}$ = 300eV, 500eV, and 800eV are from Lee et. al. 23
while for 1keV from Boechat et. al. 24. We note that Boechat et. al. have
reported data only till the scattering angle $\theta_{s}$ = 15.25∘. Our
theoretical model for $N_{2}O$ can predict the electron elastic differential
cross section closely barring some deviations.

The results obtained by our model for both these molecules are in good
agreement with the experimental published data. The good agreement of the
results of our model with those of the experimental data establishes that our
model can determine with sufficient accuracy the elastic differential cross
sections for plane waves at the intermediate energies 300eV, 500eV, 800eV, and
1keV. We now use the benchmarked model to study the interactions of the
monochromatic Bessel beam with molecular targets in this model.

###  3.2 Twisted-wave elastic differential cross-section

Here, we present the results of the twisted Bessel electron beam impact
elastic differential cross section for $CO_{2}$ and $N_{2}O$ molecules at
intermediate energies ranging from 300eV to 1keV. These results are divided
into OAM-dependent and impact parameter averaged differential cross sections.

####  3.2.1 OAM Dependence

This subsection presents the results for twisted electron impact absolute
elastic differential cross sections for topological charges $m_{l}=$ 1, 2, and
3 at impact parameter $\bm{b}$ = 0. The opening angle for these calculations
is $\theta_{p}=$ 10∘. As a reference, the plane wave result is also shown.

Figures 5 and 6 depict 4 subplots each of differential cross sections plotted
against the scattering angle $\theta_{s}$ for Bessel wave electron impact on
carbon dioxide molecule and nitrous oxide molecules respectively. Subfigures
(a), (b), (c), and (d) in each figure depict the results for $E_{i}$ = 300eV,
500eV, 800eV, and 1keV respectively. The plane wave DCS is shown by a solid
dark blue curve. The twisted DCS for different $m_{l}$ are shown with lighter
colors.

0306090120150180$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$$10^{5}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$0306090120150180$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$0306090120150180$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$0306090120150180$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$$\theta_{s}(^{\circ})$$\frac{d\sigma}{d\Omega}$
$\left(\frac{(a_{0})^{2}}{sr}\right)$(a)(b)(c)(d)

Figure 5: OAM dependent differential cross section of Bessel beam impact on
$CO_{2}$ molecule at energies (a) 300eV, (b) 500eV, (c) 800ev and (d)1keV

For $CO_{2}$ we note that for incident energy $E_{i}$ = 300eV the twisted DCS
peaks at the opening angle of $\theta_{p}$ = 10∘ with the value of $10^{4}$.
The peak’s amplitude reduces for higher incident energy when the energy is
increased from 300eV to 1keV as shown in subplots (a) to (d). We also report
that the value of DCS approaching the singular point $\theta_{s}=$ 0∘ tends to
0 rather than $\infty$ as compared to the plane wave case. For plane waves,
the DCS at the singular point is of the order of 1010 ${a_{0}}^{2}$ while for
the twisted it is of the order of 10-9 ${a_{0}}^{2}$. We have not plotted the
DCS at the singular point $\theta_{s}=$ 0∘ as the range on the y-axis would be
too high suppressing important features in the graph. For higher topological
charges, the DCS beyond $\theta_{s}=$ 30∘ is lesser than those with lower
topological charges. For all cases, the DCS shows an oscillating behavior. The
peak’s amplitude of the differential cross section at 300eV is greater than
that of 1keV by one order of magnitude i.e. the peak’s amplitude is of the
order of $10^{4}$ ${a_{0}}^{2}$ at 300eV while the its just below $10^{3}$
${a_{0}}^{2}$ for 1keV. It is also seen that the peak becomes slightly
narrower as the energy is increased from 300eV to 1keV. However, the
differential cross-section of the target by twisted electron beam impact is
much lower than the plane wave results. The differential cross section
reported is the least for $m_{l}$ = 3 while the highest for $m_{l}$ = 1.

0306090120150180$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$$10^{5}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$0306090120150180$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$$10^{5}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$0306090120150180$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$0306090120150180$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$$\theta_{s}(^{\circ})$$\frac{d\sigma}{d\Omega}$
$\left(\frac{(a_{0})^{2}}{sr}\right)$(a)(b)(c)(d)

Figure 6: OAM dependent differential cross section of Bessel beam impact on
$N_{2}O$ molecule at energies (a) 300eV, (b) 500eV, (c) 800ev and (d)1keV

The results for the DCS of the $N_{2}O$ molecule are similar to those of the
$CO_{2}$ molecule, except for smoother curves with lesser oscillations. The
DCS peak’s amplitude at $\theta_{p}$ = 10∘ at 300eV is greater than that of
1keV by one order of magnitude i.e. the peak’s amplitude is just above
$10^{4}$ ${a_{0}}^{2}$ at 300eV while it is just above $10^{3}$ ${a_{0}}^{2}$
for 1keV. It is also seen that the peak becomes slightly narrower as the
energy is increased from 300eV to 1keV. However, the differential cross-
section of the target by twisted electron beam impact is much lower than that
of plane waves. The least differential cross section is reported for $m_{l}$ =
3 while the highest for $m_{l}$ = 1 similar to those of $CO_{2}$.

The peaks get sharper as the energy is increased. The peak’s amplitude at
$\theta_{s}=$ 10∘ is the same for all the topological charges in all the
graphs of Figures 5 and 6. The DCSs seem to merge as energy is increased from
300eV to 1keV. For large incident energies for example 1keV the DCS values of
the three topological charges merge, especially for $m_{l}=$ 1 and $m_{l}=$
2\. It is hard to differentiate between $m_{l}=$ 1 and $m_{l}=$ 2 at 1keV
energy for both $CO_{2}$ and $N_{2}O$ molecules. We assume that the similar
behavior between the two molecules is due to them being isoelectronic, linear,
and triatomic. A smooth decay of the differential cross-section is presented
by Koshelava et. al. 11 for atomic systems. This is contrary to what we see in
the differential cross-section i.e. we see a lot of fluctuations. We suspect
it is due to the all-active electron contribution to the differential cross-
section.

####  3.2.2 Impact Parameter Averaging

In this subsection, we present the macroscopic experimentally obtainable
differential cross sections also known as the impact parameter averaged
differential cross section. It represents the beam impacting an infinite
number of target molecules uniformly distributed transversely resembling the
experimental situations. Hence, called cross sections for macroscopic targets
of molecules. Here we present the above for $CO_{2}$ and $N_{2}O$ molecules
with the opening angle $\theta_{p}$ being 6∘, 20∘, and 45∘. The plane wave
results are also shown for comparison.

Figures 7 and 8 depict 4 subplots each of impact parameter averaged
differential cross sections plotted against the scattering angle $\theta_{s}$
for Bessel electron beam impact on $CO_{2}$ and $N_{2}O$ molecules
respectively. Subfigures (a), (b), (c), and (d) depict the results for $E_{i}$
= 300eV, 500eV, 800eV, and 1keV respectively. The plane wave DCS is shown by a
solid dark blue curve. The twisted DCS for different $\theta_{p}$ is shown
with magenta, light blue, and green colors.

0306090120150180$10^{-2}$$10^{0}$$10^{2}$$10^{4}$$10^{6}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$0306090120150180$10^{-2}$$10^{0}$$10^{2}$$10^{4}$$10^{6}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$0306090120150180$10^{-2}$$10^{0}$$10^{2}$$10^{4}$$10^{6}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$0306090120150180$10^{-3}$$10^{-1}$$10^{1}$$10^{3}$$10^{5}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$$\theta_{s}(^{\circ})$$\left(\frac{d\sigma}{d\Omega}\right)_{av}\left(\frac{(a_{0})^{2}}{sr}\right)$(a)(b)(c)(d)

Figure 7: Impact parameter averaged Bessel Differential Cross-Section for
$CO_{2}$ (a) 300eV, (b) 500eV, (c) 800ev and (d)1keV

For $CO_{2}$, we see peaks in the differential cross section at the given
opening angles of $\theta_{p}$ for energies $E_{i}$ = 300eV to 1keV. As
$\theta_{p}$ is raised from 6∘ to 45∘ the peak’s value reduces and the width
broadens. There are nearly two orders of magnitude difference between the
peak’s value of $\theta_{p}$ = 6∘ and 20∘ and nearly the same difference
between those of 20∘ to 45∘ for each incident energy. However, we note that as
the incident energy is increased from $E_{i}$ = 300eV to 1keV the amplitudes
of the peaks gradually decrease. The difference in the peak’s amplitude is
nearly one order of magnitude at energies $E_{i}$ = 300eV to 1keV i.e. from
$10^{6}$ to $10^{5}$ and $10^{4}$ to $10^{3}$ for those corresponding to
$\theta_{p}$ = 6∘ and 20∘. However, for $\theta_{p}$ = 45∘ it reduces in
magnitude of nearly two orders i.e. $10^{3}$ to $10^{1}$. We see that as the
opening angle increases the magnitude of the peak of the differential cross
section decreases and the FWHM increases. The twisted differential cross
sections nearly follow the plane wave differential cross section after
$\theta_{s}$ = 90∘.

0306090120150180$10^{0}$$10^{2}$$10^{4}$$10^{6}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$0306090120150180$10^{-1}$$10^{1}$$10^{3}$$10^{5}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$0306090120150180$10^{-1}$$10^{1}$$10^{3}$$10^{5}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$0306090120150180$10^{-1}$$10^{1}$$10^{3}$$10^{5}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$$\theta_{s}(^{\circ})$$\left(\frac{d\sigma}{d\Omega}\right)_{av}\left(\frac{(a_{0})^{2}}{sr}\right)$(a)(b)(c)(d)

Figure 8: Impact parameter averaged Bessel Differential Cross-Section for
$N_{2}O$ (a) 300eV, (b) 500eV, (c) 800eV and (d)1keV

Similar peak features are seen in the differential cross section for $N_{2}O$
except that $N_{2}O$ being dipolar shows a smoother curve as compared to
$CO_{2}$. The peak’s amplitude variation with the opening angle, $\theta_{p}$,
shows a sharp decrease from $\theta_{p}$ = 6∘ to $\theta_{p}$ = 45∘ as
compared to those of $CO_{2}$. The peak’s value for $E_{i}$ = 300eV at
$\theta_{p}$ = 6∘ is of the order of $10^{7}$ which reduces to $10^{6}$ for
$E_{i}$ = 1keV. The same decreasing trend is found for $\theta_{p}$ = 20∘ and
45∘. This is in contrast to the peak’s amplitude for $CO_{2}$ at $\theta_{p}$
= 45∘ where it is reduced by two orders of magnitude.

###  3.3 Integral Elastic Cross Section (ICS)

The integral cross sections corresponding to the electron elastic scattering
of $CO_{2}$ and $N_{2}O$ molecules for various twisted electron topological
charges and the beam’s interactions with macroscopic molecular targets
averaging over all impact parameters are presented in this section. Figures 9
and 10 depict 2 subplots each representing the integral cross-section plots of
the target molecules by twisted Bessel beam impact which depends on
topological charges and opening angle $\theta_{p}$ respectively. The plane
wave integral cross sections are also plotted for reference which are
represented by a continuous dark blue curve. The integral cross-sections are
plotted as a function of incident energies where the differential cross-
section is integrated over $\theta_{s}$ according to equation (34).

We note that the differential cross-section of the plane waves blows up in the
forward direction, leading to the blowing up of the integral cross-section,
hence, the differential cross-sections are integrated from 5∘ to 180∘ for both
the OAM-dependent and macroscopic target cases. The blowing up of the integral
cross section is due to the blowing up of the differential cross section as
$\theta_{s}\rightarrow 0^{\circ}$. Also, we have plotted the integral cross-
sections from $E_{i}=10$eV to $E_{i}=1k$eV. For $E_{i}\rightarrow$ 0eV, the
momentum transfer $\Delta\rightarrow$ 0 $a_{0}^{-1}$, which blows up the
scattering amplitude given by (18), hence we ignore results below $E_{i}=10$eV
resulting in our model not accurately predicting the features at lower
energies. This is also due to the Born approximation failing at lower energies
and plane waves representing the incident and scattered electrons not agreeing
at lower energies.

02004006008001000$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$02004006008001000$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$10^{5}$pw$m_{l}=1$$m_{l}=2$$m_{l}=3$$E_{i}$(eV)$\sigma((a_{0})^{2})$(a)(b)
Figure 9: Integral Elastic Cross-Section OAM dependent (a) $CO_{2}$ and (b)
$N_{2}O$

Figure 9 depicts the ICS results for the topological charges $m_{l}$ = 1, 2,
and 3. The impact parameter used here is $\bm{b}$ = 0. We note the integral
cross-sections obtained for all three twisted topological charges have
considerably lower values than those of plane waves. This can also be inferred
from the differential cross-section of the twisted beams being lower than
plane waves in Figures 7 and 8.

For $CO_{2}$ as shown in Figure 9 (a), the cross sections of the three
topological charges decrease with energy and intertwine beyond 250eV. The
integral cross sections reach about $10^{1}$ order of magnitude at $E_{i}$ =
1keV. For $N_{2}O$, as shown in Figure 9 (b) the integral cross sections show
deviations from smooth decay but do not intertwine much. As the energy
increases they attain the value of the order of $10^{1}$ at $E_{i}$ = 1keV.
Here it is seen that for $m_{l}$ = 1 the integral cross section is the highest
while for $m_{l}$ = 3 it’s the least.

02004006008001000$10^{1}$$10^{2}$$10^{3}$$10^{4}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$02004006008001000$10^{1}$$10^{2}$$10^{3}$$10^{4}$$10^{5}$pw$\theta_{p}=6^{\circ}$$\theta_{p}=20^{\circ}$$\theta_{p}=45^{\circ}$$E_{i}$(eV)$\sigma((a_{0})^{2})$(a)(b)

Figure 10: Integral Elastic Cross-Section macroscopic targets (a) $CO_{2}$ and
(b) $N_{2}O$

Figure 10 (a) and (b) depict the impact parameter averaged ICS results as a
function of incident energy $E_{i}$ for macroscopic $CO_{2}$ and $N_{2}O$
molecules respectively. This implies that the beam impacts a large number of
such molecules. We see that just as the plane waves they decrease as the
energy increases. The integral cross-sections of the twisted waves are nearly
equal or greater in magnitude than the plane wave cross-sections. This can
also be inferred from the differential cross-section plots in Figures 7 and 8
of impact parameter averaging where the twisted wave differential cross-
sections are greater in magnitude than those of the plane waves. As before,
the impact parameter averaged ICS for $CO_{2}$ for different $\theta_{p}$
intertwine while for $N_{2}O$ dont intertwine and fall smoothly.

###  3.4 Conclusions

We have presented the formalism and the methodology to obtain accurate elastic
differential and integral cross-sections of molecules by plane and Bessel
electron beams impact theoretically with $CO_{2}$ and $N_{2}O$ molecules as
examples. We have used DFT to obtain the electron density which brings about
the effects of all molecular electrons in the cross-section, thus formulating
the all-active electron approach. We have also used a multicentered approach
by considering three centers for each molecule. This methodology is a powerful
approach to determine cross-sections of complex molecular targets with ease.
We have benchmarked our model for plane wave experimental data and extended it
to the twisted wave case investigating the cross sections for OAM-dependence
and impact parameter averages. We have seen that the differential cross-
sections depend on the OAM and the opening angle, $\theta_{p}$ of twisted
beams. In the future one can extend the work to inelastic scattering and
ionization processes on other molecular targets to investigate electronic
structure using this methodology. One could also investigate the scattering at
low energies where our model fails to be accurate.

###  3.5 Acknowledgments

We wish to acknowledge the Department of Science and Technology(DST), Govt. of
India for the funding through the grant CRG/2021/003828.

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