# Imaging charge migration in chiral molecules using time-resolved x-ray
diffraction

Sucharita Giri<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Bombay, Powai, Mumbai 400076 India  Jean Christophe
Tremblay<EMAIL_ADDRESS>Laboratoire de Physique et
Chimie Théoriques, CNRS-Université de Lorraine, UMR 7019, ICPM, 1 Bd Arago,
57070 Metz, France  Gopal Dixit<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Bombay, Powai, Mumbai 400076 India

###### Abstract

Four-dimensional imaging of charge migration is crucial to the understanding
of several ubiquitous processes in nature. The present work focuses on imaging
of charge migration in an oriented epoxypropane: a chiral molecule. A linearly
polarized pulse is used to induce the charge migration, which is imaged by
time-resolved x-ray diffraction. It is found that the total time-resolved
diffraction signals are significantly different for both enantiomers.
Furthermore, a connection between time-resolved x-ray diffraction and the
electronic continuity equation is discussed by analyzing the time-dependent
diffraction signal and the time derivative of the total electron density in
the momentum space.

##  I Introduction

X-ray diffraction is the method of choice when it comes to extracting the
real-space structure of solids and molecules with atomic-scale spatial
resolution Nielsen . Thanks to technological advances that have made it
possible to generate intense ultrashort pulses from novel x-ray source: x-ray
free-electron lasers (XFELs) ishikawa2012 ; pellegrini2016physics ; emma2 .
X-ray pulses with pulse duration of a few femtoseconds are routinely employed
to perform various kinds of experiments at different XFEL sources. Moreover,
generation of attosecond x-ray pulses was demonstrated experimentally in
recent years hartmann2018attosecond ; bucksbaum2019attoXR . These ultrashort
x-ray pulses have enabled extension of static x-ray diffraction into the time
domain with exceptional temporal resolution lindroth2019challenges . Time-
resolved x-ray diffraction (TRXD) within a pump-probe configuration is an
emerging method to obtain snapshots of a temporarily evolving electronic
charge distribution in matter with spatiotemporal resolution on atomic and
electronic scales peplow2017next . An electronic movie of ultrafast charge
migration can be made by putting the recorded snapshots together as a function
of pump-probe delay time dixit2012 ; vrakking2012 .

In this work, we investigate imaging of pump-induced charge migration in a
chiral molecule using TRXD. Oriented epoxypropane (1,2-propylene oxide) is
used as a prototypical system. The notion of chirality is omnipresent in
nature barrett2014spontaneous ; gal2012discovery . Understanding the chirality
is essential for a broad range of sciences including origin of homochirality
on earth. A molecule without inversion symmetry or a symmetry plane which is
not superimposable with its own mirror image is known as a chiral molecule
cahn1966specification . A pair of such chiral molecules is known as an
enantiomers. Enantiomers exhibit identical physical properties, but show a
strong enantiomeric preference during chemical and biological reactions.
Discerning the enantiomeric excess and handedness of chiral molecules is
crucial in chemistry, biology and pharmaceuticals.

Confirming the handedness of chiral molecules by determining their spatial
arrangement is known as absolute configuration determination Nielsen ;
bijvoet1951determination ; santoro2020absolute . X-ray crystallography is one
of the most reliable methods to determine the absolute configuration
flack1999absolute . In recent years, other methods such as microwave
spectroscopy patterson2013enantiomer ; eibenberger2017enantiomer , Coulomb-
explosion imaging pitzer2013direct ; herwig2013imaging , high-harmonic
generation cireasa2015probing ; wang2017high ; neufeld2019ultrasensitive ;
baykusheva2019real , vibrational circular dichroism fuse2019vibrational ;
magyarfalvi2011vibrational , and photoelectron circular dichroism
bowering2001asymmetry ; lux2012circular ; dreissigacker2014photoelectron ;
beaulieu2017attosecond ; beaulieu2018photoexcitation ; rozen2019controlling ;
comby2016relaxation  have been employed to discern enantiomers. Here we
demonstrate that TRXD within the pump-probe configuration offeres an
alternative method to probe the handedness of chiral molecules.

In previous work hermann2020probing ; tremblay2020time , we demonstrated that
the diffraction signal, obtained from TRXD, is related to the time derivative
of the momentum-space density. When a pump pulse induces a charge migration in
matter, the time-dependent charge dynamics is associated with the electronic
flux densities by obeying the quantum continuity equation sakurai1967advanced
. Time-resolved x-ray diffraction thereby encodes indirectly the information
about electronic flux densities accompanying charge migration. The flux
densities contains crucial mechanistic information about non-equilibrium
electron dynamics in matter okuyama2012dynamical ; diestler2013computation ;
takatsuka2014chemical ; hermann2014electronic ; yamamoto2015electron ;
bredtmann2014x ; okuyama2009electron . In recent years, several proposals have
been put forward to image the flux densities and ring currents associated with
charge migration using TRXD, ultrafast x-ray absorption and high-harmonic
generation tremblay2020time ; hermann2020probing ; bredtmann2014x ;
popova2015imaging ; carrascosa2020mapping ; nam2020monitoring ; cohen19prl .
The present work shows that TRXD and electronic flux density analysis provide
complementary information on the charge migration in oriented chiral molecules
driven by linearly polarized light. The intrinsic dynamics and its detection
are drastically different, and the complementary information can thus be used
to discriminate the enantiomers.

##  II Models and Methods

We simulate ultrafast electron dynamics using the hybrid time-dependent
density functional theory-configuration interaction (TDDFT-CI) methodology
klinkusch2016 ; hermann2016tddft . This combination of methods provides a good
balance between accuracy and computational efficiency. A generic $N$-electron
wave packet $\Psi(\mathbf{r}^{N},t)$ is represented at any given time $t$ as a
linear combination of the ground state Slater determinant and singly excited
many-body excited states. In spin-free representation, the wave packet takes
the form

$\Psi(\mathbf{r}^{N},t)=\sum\limits_{k=0}^{N_{\rm
states}}C_{k}(t)\Phi_{k}(\mathbf{r}^{N}).$ (1)

In the case of field-free temporal evolution,
$C_{k}(t)=c_{k}~{}\textrm{exp}({-iE_{k}t})$. This wave packet evolves
according to the many-electron time-dependent Schrödinger equation

$i\partial_{t}\Psi(\mathbf{r}^{N},t)=\left[\hat{H}_{0}-\hat{\mu}\cdot\mathcal{F}(t)\right]\Psi(\mathbf{r}^{N},t).$
(2)

Here, $\hat{H}_{0}$ is the field-free Hamiltonian within the Born-Oppenheimer
approximation. The field-molecule interaction is treated in the semi-classical
dipole approximation, with $\hat{\mu}$ the molecular dipole operator and
$\mathcal{F}(t)$ an applied external electric field.

The lowest-lying excited eigenstates $\Phi_{k}(\mathbf{r}^{N})$ of the many-
body electronic Hamiltonian are calculated as linear combinations of Slater
determinants. Each excited $N$-electron state is expressed as a linear
combination of singly excited configuration state functions

$\Phi_{k}(\mathbf{r}^{N})=\sum\limits_{ar}D_{a,k}^{r}\Phi_{a}^{r}(\mathbf{r}^{N})$
(3)

The configuration state functions $\Phi_{a}^{r}(\mathbf{r}^{N})$ are spin
symmetrized combinations of excited Slater determinants. They are defined with
respect to the Kohn-Sham ground state $\Phi_{0}(\mathbf{r}^{N})$, by taking an
electron from an occupied molecular orbital $a$ to a virtual molecular orbital
$r$.

In the hybrid TDDFT-CI method, all expansion coefficients
$\\{D_{0,k},D_{a,k}^{r}\\}$ are obtained from linear-response TDDFT. This step
is performed using standard quantum chemistry programs. The information from
the quantum chemistry package is postprocessed using the open-source toolbox
detCI@ORBKIT. The contributions of the different Slater determinants extracted
from the quantum chemistry programs are first pruned by removing all
contributions below some numerical threshold (here chosen as $10^{-5}$) and
then renormalized. In the basis of pseudo-CI eigenfunctions chosen for the
propagation, the Hamiltonian is considered diagonal. The matrix elements of
the dipole moment operator in Eq. (2) are computed by numerical integration.
Finally, the time-evolution of the coefficients $C_{k}(t)$ in Eq. (1) is
performed by direct numerical integration of Eq. (2) using a preconditioned
adaptive step size Runge-Kutta algorithm 04:TC:passrk .

In this work, all $N_{\textrm{states}}=31$ lowest-lying excited states below
the ionization threshold are used to obtain convergence, checked by repeating
the laser-driven dynamics simulations with
$N_{\textrm{states}}=\\{16,21,26,31\\}$. The states are computed using the
CAM-B3LYP functional 04:YTH:cam  and aug-cc-pVTZ basis sets dunning1989ccpvxz
on all atoms, as implemented in Gaussian16 frisch2016gaussian . The motion of
nuclei are typically much slower in comparison to the electronic motion and
therefore nuclei are treated as frozen in this work.

Integrating the time-dependent many-electron wave packet in Eq. (2) leads to
the one-electron quantum continuity equation

$\partial_{t}\rho(\mathbf{r},t)=-\vec{\nabla}\cdot\mathbf{j}(\mathbf{r},t).$
(4)

Here, $\rho(\mathbf{r},t)$ is the one-electron density and
$\mathbf{j}(\mathbf{r},t)$ the electronic flux density barth2009concerted .
From the many-body wavefunction, the expectation value of the one-electron
density operator

$\displaystyle\hat{\rho}\left(\mathbf{r}\right)=\sum_{k}^{N}\delta\left(\mathbf{r}-\mathbf{r}_{k}\right)$
(5)

yields the one-electron density. The Dirac delta distribution
$\delta\left(\mathbf{r}-\mathbf{r}_{k}\right)$ evaluates the probability of
finding electron $k$ with position $\mathbf{r}_{k}$ at a point of observation,
$\mathbf{r}$. The expectation value of the operator

$\displaystyle\hat{j}\left(\mathbf{r}\right)=\frac{1}{2}\sum_{k}^{N}\left(\delta_{k}(\mathbf{r})\hat{p}_{k}+\hat{p}^{\dagger}_{k}\delta_{k}(\mathbf{r})\right)$
(6)

yields the flux density, where the momentum operator
$\hat{p}_{k}=-i\vec{\nabla}_{k}$ of electron $k$, is applied either to the
right or to the left. The molecular orbitals to compute the integrals are
expressed as linear combinations of atom-centered Gaussian functions, and all
integrals are computed analytically The charge density and flux density thus
obtained satisfy very accurately the continuity equation as given in Eq. (4).
Details on the computation of the one-electron integrals from the many-body
wavefunction using the detCI@ORBKIT toolbox hermann2016orbkit ; pohl2017open ;
hermann2017open  are given elsewhere.

When the probe pulse duration is much shorter than the timescale of the charge
migration, the TRXD signal can be simulated using the expression for the
differential scattering probability (DSP) dixit2014theory

$\frac{dP}{d\Omega}=\frac{dP_{e}}{d\Omega}\sum_{j}\left|\int\mathrm{d}\mathbf{r}~{}\langle\Phi_{j}|\hat{n}(\mathbf{r})|\Psi({\textrm{T}})\rangle~{}e^{-i\mathbf{Q}\cdot\mathbf{r}}\right|^{2},$
(7)

where $\frac{dP_{e}}{d\Omega}$ is the Thomson scattering cross section of a
free electron. The signal probes the coherences between eigenstates of the
unperturbed Hamiltonian, $|\Phi_{k}\rangle$, contained in an electronic wave
packet $|\Psi({\textrm{T}})\rangle$ with T the pump-probe delay time. In
addition, $\mathbf{Q}$ is the photon momentum transfer and
$\hat{n}(\mathbf{r})$ is the electron density operator. The summation over $j$
includes all zeroth-order states included in the dynamics, which amounts to
detecting all the scattered photons within a given solid angle, irrespective
of the mean photon energy.

After the pump pulse and in the absence of an external field, the time-
dependent coefficients in Eq. (1) simplify to the analytical form containing
an amplitude and energy-dependent phase. At ${\textrm{T}}=0$, Eq. (7) reduces
to the time-independent diffraction signal as

$\frac{dP}{d\Omega}=\frac{dP_{e}}{d\Omega}\sum_{j}\left|\sum_{k}c_{k}\int
d\mathbf{r}~{}\langle\Phi_{j}|\hat{n}(\mathbf{r})|\Phi_{k}\rangle~{}e^{-i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}.$
(8)

As evident from this equation, the time-independent diffraction signal is
emerging from the joint contribution of the elastic and time-independent
inelastic parts of the total diffraction signal. This term serves as a
constant background in TRXD. To image the charge migration, we will present
time-dependent difference diffraction signals where the total diffraction
signal at zero delay time is subtracted from the subsequent delay times.

References kowalewski2017monitoring ; simmermacher2019theory  discuss the
different roles of time-independent and time-dependent diffraction signals
from the total diffraction signal. Moreover, detailed discussions about
probing electronic coherences in the wavepacket have been documented in Refs.
dixit2012 ; simmermacher2019electronic . Furthermore, the time resolution in
TRXD depends on several factors such as temporal duration on x-ray pulse,
jitter between pump and probe pulses, and pump-probe time delay.

Figure 1: Time-dependent electronic flux densities for (a) R- and, (b)
S-enantiomers of epoxypropane in the $xz$-plane at $2$ fs, $6$ fs, $10$ fs,
and $14$ fs after the pump pulse. Violet, maroon and gray colors are
representing carbon, oxygen and hydrogen atoms, respectively. Here, the
$z$-axis is defined as the chiral axis in the molecular frame of reference.

All many-electron dynamical simulations are performed using in-house codes
klinkusch2011 ; 04:TC:passrk ; 08:TKS ; 08:TS:gloct ; 10:TKKS . The integrals
required to compute the TRXD signals and electronic flux densities are
calculated using ORBKIT OKgit . Mayavi mayavi  and Matplotlib
hunter2007matplotlib  are used to visualize the electronic flux densities and
the time-resolved x-ray signals, respectively.

##  III Results and Discussion

Our starting point is a pair of oriented R- and S-enantiomers of epoxypropane.
In recent years, it has become possible to orient chiral molecules in space.
This has been demonstrated both theoretically and experimentally in Refs.
tutunikov2018selective ; tutunnikov2020observation ; saribal2020detecting .
Orientation can be achieved using either skewed-polarized pulses, a
centrifuge, or a combination of a static field and linearly polarized pulse.
The resulting orientation has the R- and S-enantiomers pointing in opposite
directions along the chiral axis. As the molecules are fixed in their own
frame of reference, everything in this work is described in the molecular
frame, where $z$ defines the chiral axis.

As in previous work, oriented epoxypropane is exposed to a linearly polarized
pump pulse of 10 fs pulse duration with 9.1$\times$1014 W/cm2 peak intensity
to initiate the charge migration dynamics. The pulse is polarized at 45∘ in
the $yz$-plane of the molecular frame of reference and has a 160 nm wavelength
with sine-squared envelope. As a result of the interaction between the pump
pulse and epoxypropane, ultrafast charge migration is set into motion.

For the R-enantiomer, $32\%$, $7\%$, and $2\%$ electronic populations are
transferred by the end of the pump pulse from ground state to the
3${}^{\text{rd}}$, 4${}^{\text{th}}$ and 5${}^{\text{th}}$ excited electronic
states, respectively. The rest of the population remains in the ground state,
which is responsible for a dynamics on the attosecond timescale. On the other
hand, the 3${}^{\text{rd}}$, 4${}^{\text{th}}$ and 5${}^{\text{th}}$ excited
states are populated at $37\%$, $10\%$ and $13\%$, respectively, while nearly
30$\%$ remains in the ground state for the S-enantiomer giri2020time . The
reason behind different electronic populations in R- and S-enantiomer can be
attributed to different temporal evolution of the magnitude and the phase of
the Cartesian components of the total dipole moment, as discussed in Ref.
giri2020time . Since the populations at the end of the pump pulse are
different for both enantiomers, the field-free charge migration looks
significantly different in both cases. This can be revealed by investigation
of the transient flux densities.

The electronic flux densities accompanying the charge migration for both
enantiomers are presented in Fig. 1. To avoid the fast oscillating
contribution stemming from the coherent interaction with the ground state in
the charge migration, only contributions from the excited states are
considered in the reconstructed signal. Snapshots are presented for $2$ fs,
$6$ fs, $10$ fs, and $14$ fs, and the 2D representation in the $xz$-plane is
obtained by integrating over the $y$ coordinate. The arrows of the vector
field are color-coded according to their magnitude. In both enantiomers,
almost all the flux density is concentrated around the oxygen atom (in
maroon). It is important to mention that, maybe surprisingly, the chiral
carbon atom is not involved in the dynamics, as revealed by the transient flux
densities giri2020time . The most striking difference lies in the magnitude of
the currents. The strength of the flux densities for the R-enantiomer is weak
in comparison to that for the S-enantiomer during the charge migration. At all
times, they both exhibit a similar circular pattern around the oxygen atom.
The direction of the circular charge migration changes as the dynamics
progresses.

Since the direction of the electron flow is more prominent for the
S-enantiomer, as reflected in Fig. 1(b), it can therefore be more readily
described. Arrows corresponding to the flux densities are emerging from the
oxygen atom in all directions, corresponding to an approximately circular
current density. Yet, the longest arrows in red represent the most probable
direction for the flow of electrons, and these are changing with time. At $2$
fs, arrows are mostly emerging in the positive $x$-direction and it is almost
reversed in the next snapshot, i.e., at $6$ fs. The flux densities for the
S-enantiomer are along the same direction at $2$ and $14$ fs, which indicate
the partial recurrence. Similar observations can be made for the circular
component of the current density in the R-enantiomer, where the direction of
the circular current is opposite to that of the S-enantiomer. In contrast, the
current component with the largest magnitude points in the same direction for
both enantiomers at all times. A tentative explanation could be that the
circular component stems from the phase transferred to the molecule by the
laser field. This leads to opposite relative phases among the three excited
states. On the other hand, the dominant component of the flux density could be
associated with the response of the electron cloud in the electric field
during the preparation phase. This electronic response behavior would then be
inherited by the subsequent charge migration dynamics, explaining that both
enantiomers react with the same phase.

The different timescale of the charge migration can be understood by analyzing
the characteristic timescales involved in the system, i.e.,
$\tau=\textit{h}/\Delta E$, where $\Delta E$ is the energy difference between
two eigenstates. The characteristic timescales corresponding to $E_{3}$ = 7.56
eV, $E_{4}$ = 7.73 eV, and $E_{5}$ = 7.84 eV are $\tau_{43}=24.32$ fs,
$\tau_{53}=14.77$ fs and $\tau_{54}=37.59$ fs. Different contributions of the
excited states and their respective interference including these timescales
are the reason for non-identical charge migration in the two excited
enantiomers.

Figure 2: Time-independent diffraction patterns for (a) R- and (b) S-
enantiomers of epoxypropane. The diffraction patterns are presented in
$Q_{z}-Q_{x}$ plane in units of ${dP_{e}}/{d\Omega}$. An ultrashort x-ray
pulse of 8 keV mean energy is aimed to obtain 1.55 Å spatial resolution in
these simulations. All diffracted photons up to 60∘ are collected in the
detector.

Time-resolved x-ray diffraction is a technique that can be used to image the
dynamics of charge migration with atomic-scale spatiotemporal resolutions. In
recent years, different formalisms of TRXD have been developed to image an
increasing number of processes henriksen ; dixit2013jcp ;
bennett2018monitoring ; simmermacher2019electronic ; dixit2013prl ;
dixit2014theory ; santra2014comment ; bredtmann2014x ; dixit2017time ;
kowalewski2017monitoring . To simulate time-resolved diffraction signal, the
DSP is computed according to Eq.(8). The six lowest-lying excited eigenstates
(i.e., $j=[0,6]$) were sufficient to obtain the converged diffraction signals
under the assumption that the sum-over-states expression was truncated due to
the detector response.

Figure 3: Time-resolved diffraction patterns for (a) R- and, (b) S-enantiomers
of epoxypropane in $Q_{z}-Q_{x}$ plane at 2 fs, 6 fs, 10 fs, and 14 fs.
Intensity of the diffraction patterns is presented in units of
${dP_{e}}/{d\Omega}$. Here, time-dependent diffraction signal at T = 0 fs is
subtracted to the subsequent delay time. An ultrashort x-ray pulse of 8 keV
mean energy is aimed to obtain 1.55 Å spatial resolution in these simulations.
All diffracted photons up to 60∘ are collected in the detector.

The time-independent contributions to the total diffraction signal for both
enantiomers after the excitation are presented in Fig. 2. These signals are
simulated using Eq. (8). From the figure, one can see that the time-
independent diffraction signals are different for both enantiomers. First, the
intensity of the time-independent signal is stronger for the R-enantiomer in
comparison to the S-enantiomer. This is especially true in the low-momentum
region. Also, the signals are not mirror images of each other, which could
allow one to discriminate between the enantiomers by simple inspection of the
two diffraction signals. Since these diffraction signals are static in nature,
the dynamical information about the charge migration is lost.

Time-resolved diffraction signals at selected delay times for both enantiomers
are presented in Fig 3. The total diffraction signal at T = 0 fs is subtracted
from the signal at subsequent delay times. It is observed that the time-
independent signal is three orders of magnitude stronger than this time-
dependent signal. Similar observations have been reported earlier in the case
of TRXD for different systems carrascosa2020mapping ; keefer2021 . In spite of
having low magnitude, the information about the ultrafast dynamics of charge
migration is imprinted in the different diffraction signals as they are
significantly different for both enantiomers at different delay times. The
central parts as well as the higher-momentum signals are changing in a very
distinct way for both enantiomers. Note that $\langle\Phi_{i}|\Phi_{j}\rangle$
is not exactly zero in our case due to numerical limitations, which impose
using a small but finite value with a threshold value of $10^{-5}$ on the
linear-response TDDFT coefficients. This, together with the numerical
coefficient renormalization, explains the small but non zero value of the
diffraction signal at the center in Fig. 3.

Figure 4: Time derivative of electron density in momentum space,
$|\partial_{t}\rho({\mathbf{Q}})|$, for (a) R- and, b) S-enantiomers of
epoxypropane in $Q_{z}-Q_{x}$ plane at 2 fs, 6 fs, 10 fs, and 14 fs.

By comparing the time-dependent diffraction pattern of the enantiomers at 2
fs, it can be seen that the overall signals are not very different for both
enantiomers. There are two lobes of positive signal around the center and they
are of almost the same intensity with different orientations for the
respective enantiomers. The negative part of the signal is scattered at higher
momenta. There is a mismatch in the intensity at the central part of the lobes
for the two enantiomers.

At the next time step, i.e., at 6 fs, the diffraction signal is reduced
significantly for both enantiomers. The signal is just reversed around the
center, dominated by a negative value and spreading to higher momenta for the
S-enantiomer. For the R-enantiomer, the signal at 10 and 14 fs are similar to
the one at 2 fs and reflect slower dynamics. In contrast, for the
S-enantiomers, the signal at 10 fs is comparable to the signal at 2 fs but not
identical due to the presence of four dips with a minimum value around the
center. There is overall depletion of the signal at 14 fs compared to the
previous time step. The oscillation period of the time-dependent signal
correlates well with the timescale of the charge migration found in the flux
density for the enantiomers as shown in Fig. 1.

In order to understand how the time-dependent diffraction signals are related
to the flux density, the time derivative of the electron density in momentum
space, $|\partial_{t}\rho({\mathbf{Q}})|$, at selected delay times is
presented in Fig. 4. At T = 0 fs, $|\partial_{t}\rho({\mathbf{Q}})|$ is zero
(see Eq. (S7) in Ref. hermann2020probing ). Therefore, subtracting
$|\partial_{t}\rho({\mathbf{Q}})|$ at T = 0 fs is not required. The
$|\partial_{t}\rho({\mathbf{Q}})|$ is related to the flux densities, and it
can be obtained by a Fourier transform of the electronic continuity equation,
Eq. (4).

For both enantiomers, the overall signal of $|\partial_{t}\rho({\mathbf{Q}})|$
decreases drastically from 2 fs to 6 fs. There is a partial revival of the
signal at 10 fs for the R-enantiomer as observed in Figs. 1 and 3. The
intensity of $|\partial_{t}\rho({\mathbf{Q}})|$ at 14 fs increases compared to
that in 10 fs for the R-enantiomer. In contrast,
$|\partial_{t}\rho({\mathbf{Q}})|$ for the S-enantiomer at 14 fs is
approximately similar to the one at 2 fs.

As evident from Figs 3 and 4, it is not straightforward to establish visual
one-to-one correspondence between the DSP and
$|\partial_{t}\rho({\mathbf{Q}})|$, which is in contrast to the earlier work
discussed in Ref. hermann2020probing . There are two simple reasons behind
this. The first one is that more than two electronic states are contributing
to the electronic wavepacket in epoxypropane. As a result, time-dependent
terms yield more than one sine term in the expression of
$|\partial_{t}\rho({\mathbf{Q}})|$ (see Eq. (S7) in Ref. hermann2020probing ).
The other reason can be attributed to a significant contribution stemming from
the sine square term in comparison to the sine term in the expression of
$\Delta{dP}/{d\Omega}$ (see Eq. (S12) in Ref. hermann2020probing ). These
findings are in contrast to an earlier case, where the wavepacket was composed
of only two electronic states, and the sine square term was negligibly small
in comparison to the sin term hermann2020probing . Therefore, the comparison
between the DSP and $|\partial_{t}\rho({\mathbf{Q}})|$ can only be
qualitative.

Figure 5: Asymmetry parameter $Y$ along the (a) $Q_{x}$ and, (b) $Q_{z}$ axes
as a function of the delay time. The 1D TRXD signal is obtained from Eq. (7)
and integrated along the other axes.

As the time-resolved diffraction signals for the two enantiomers are
different, we can define the asymmetry parameter as

$Y=\frac{\sigma_{R}-\sigma_{S}}{\sigma_{R}+\sigma_{S}},$ (9)

where $\sigma_{R}$ and $\sigma_{S}$ are the DSPs of the R- and S-enantiomers,
respectively. Here, the DSP is represented by $\sigma$ as an abbreviation.
Figures 5(a) and 5(b) present, respectively, the asymmetry parameter along
$Q_{x}$ and $Q_{z}$ axes, respectively, as a function of time. The domination
of the red part over the blue is an indication that the signal for the
S-enantiomer is stronger than that for the R-enantiomer along the $Q_{x}$
axis. If we look only around the $Q_{x}=0$ line, the signal for the
R-enantiomer is dominating over the S-enantiomer for all the time steps. The
case is almost reversed for $Q_{x}\neq 0$.

The signals are more interesting along the chiral axis, i.e., along the
$Q_{z}$ axis. For a specific point in momentum space, the signal is changing
in a periodic fashion. The same is true if we fix the time-delay and walk
along the $Q_{z}$ axis. These signals encode important information about the
dynamical evolution of the system, which is unique for a given choice of laser
tagging parameters. The asymmetry parameter can be used to determine the
enantiomeric ratio in a sample of unknown concentrations $C_{R}$ and $C_{S}$
in the R and S enantiomers, respectively. In this case, the measured signal of
the unknown mixture is given as the weighted sum of the two signals-
$\sigma_{\textrm{sample}}=C_{R}\sigma_{R}+C_{S}\sigma_{S}$. By subtracting
from the TRXD signal of a pure enantiomer, say, $\sigma_{R}$, this difference

$\displaystyle\Delta\sigma$
$\displaystyle=\sigma_{R}-(C_{R}\sigma_{R}+C_{S}\sigma_{S})$ (10)
$\displaystyle=(1-C_{R})\sigma_{R}-C_{S}\sigma_{S}=C_{S}(\sigma_{R}-\sigma_{S})$

can be simply normalized and related to the asymmetry parameter

$\frac{\Delta\sigma}{\sigma_{R}+\sigma_{S}}=C_{S}\left(\frac{\sigma_{R}-\sigma_{S}}{\sigma_{R}+\sigma_{S}}\right)=C_{S}Y,$
(11)

Hence, from the knowledge of the TRXD signals of both enantiomers, the
concentration of an enantiomer can be directly obtained by comparing the
intensity of the difference signal with that of the asymmetry parameter. As
the dynamics becomes more intricate at longer times, the time-dependent
monitoring of the different signals would potentially lead to a more precise
determination of the concentrations.

##  IV Conclusion

The present work aimed at imaging the charge migration in epoxypropane using
TRXD. The charge migration in oriented epoxypropane was triggered by a
linearly polarized pulse at 45∘ in the $yz$-plane in the molecular frame of
reference. For each enantiomer, the induced charge migration was different,
and is imaged by TRXD. The dominating time-independent and different TRXD
signals were analyzed separately. It was found that time-dependent diffraction
signals are significantly different for both enantiomers. We believe that the
present proof-of-principle results should motivate further studies of laser-
induced charge migration in chiral molecules imaged by TRXD. In particular,
this could pave the way for a new, sensitive technique to determine the
enantiomeric concentrations in unknown samples.

## Acknowledgements

G. D. acknowledges support from Science and Engineering Research Board of
India (Project No. ECR/2017/001460) and the Ramanujan Fellowship (Grant No.
SB/S2/ RJN-152/2015). S. G. acknowledges CSIR for a Senior Reasearch
Fellowship. We are grateful to the anonymous referee for constructive
suggestions.

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