# Energy Efficiency Maximization in IRS-Aided Cell-Free Massive MIMO System

Si-Nian Jin, _Member, IEEE_ , Dian-Wu Yue, _Senior Member, IEEE_ , Yi-Ling
Chen, Qing Hu, _Member, IEEE_ S.-N. Jin, D.-W. Yue and Q. Hu are with the
College of Information Science and Technology, Dalian Maritime University,
Dalian 116026, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>hq0518@dlmu.edu.cn). Y.-L. Chen is with the School of Intelligence and
Electronic Engineering, Dalian Neusoft University of Information, Dalian
116023, China (e-mail: chenyiling@neusoft.edu.cn).

###### Abstract

In this paper, we consider an intelligent reflecting surface (IRS)-aided cell-
free massive multiple-input multiple-output system, where the beamforming at
access points and the phase shifts at IRSs are jointly optimized to maximize
energy efficiency (EE). To solve EE maximization problem, we propose an
iterative optimization algorithm by using quadratic transform and Lagrangian
dual transform to find the optimum beamforming and phase shifts. However, the
proposed algorithm suffers from high computational complexity, which hinders
its application in some practical scenarios. Responding to this, we further
propose a deep learning based approach for joint beamforming and phase shifts
design. Specifically, a two-stage deep neural network is trained offline using
the unsupervised learning manner, which is then deployed online for the
predictions of beamforming and phase shifts. Simulation results show that
compared with the iterative optimization algorithm and the genetic algorithm,
the unsupervised learning based approach has higher EE performance and lower
running time.

###### Index Terms:

Intelligent reflecting surface, cell-free, energy efficiency, iterative
optimization, unsupervised learning.

##  I Introduction

Recently, cell-free massive multiple-input multiple-output (MIMO) has emerged
as a promising technology to effectively alleviate inter-cell interference
[1]. In this system, a large number of distributed access points (APs) are
linked to the central processing unit (CPU) through the backhaul link and
provide services to all the users without cell boundaries. However, the large-
scale deployment of APs will bring some problems, such as high deployment cost
and power consumption. To address these difficulties, a promising technique
called intelligent reflecting surface (IRS) has been proposed as a cost-
efficient and energy-efficient solution [2, 3]. Specifically, IRS consists of
a large number of passive low-cost reflecting elements, each of which can
reflect the electromagnetic incident signals to any directions, constructing a
favorable propagation environment.

Inspired by IRS, IRS-aided cell-free massive MIMO system has received a lot of
attention, where spectral efficiency was extensively studied [4, 5]. Besides,
energy efficiency (EE) as another key performance metric, some existing works
also investigated the EE of this system [6, 7, 8, 9]. For instance, the
researchers in [6, 7, 8] proposed some iterative optimization algorithms to
solve total EE maximization problem. Furthermore, in order to maintain
fairness among users, an alternate optimization algorithm was proposed to
enhance the EE of the worst user [9]. However, all of the above-mentioned
iterative optimization algorithms have high computational complexity. This
makes these algorithms unsuitable for some practical scenarios, such as high
mobility communication scenarios. To overcome the drawback of conventional
algorithms, in the context of IRS-aided communication systems, deep learning
(DL) based on approach was recently proposed to address related issues, mainly
focusing on the spectral efficiency optimization [10, 11]. The significant
advantage of DL approach is that it can provide ideal predictions with low
computational complexity based on trained neural network. Although the DL
approach has many advantages, it is still a blank for research work using DL
approach to solve EE maximization problem in IRS-aided cell-free massive MIMO
system.

To fill this gap, this paper investigates the EE maximization problem by
jointly optimizing the beamforming at APs and the phase shifts at IRSs. First,
since the EE maximization problem in this paper is partially different from
the problem in [6, 7, 8, 9], we develop a novel iterative optimization
algorithm to solve the corresponding problem. Then, an unsupervised learning
based approach is proposed to tackle the EE maximization problem.
Specifically, we model a two-stage deep neural network (DNN) and design a
reasonable loss function, and train this DNN in an unsupervised manner to
learn the optimal beamforming and phase shifts. At last, simulation results
show that compared with the traditional genetic algorithm (GA) and the
proposed iterative optimization algorithm, the unsupervised learning based
approach can achieve better EE performance with extremely low running time.

The rest of this paper is organized as follows. Section II presents system
model and problem formulation. Section III and Section IV propose an iterative
optimization algorithm and an unsupervised learning based approach to solve EE
maximization problem, respectively. Numerical results are discussed in Section
V, and Section VI concludes this paper.

_Notations:_ ${\left({\cdot}\right)^{*}}$, ${\left({\cdot}\right)^{\rm T}}$,
${\left({\cdot}\right)^{\rm H}}$, ${\rm trace}\left({\cdot}\right)$,
$\left|{\cdot}\right|$, $\left\|{\cdot}\right\|$, ${\mathop{\rm
Re}\nolimits}\left\\{{\bf{\cdot}}\right\\}$ and
${\rm{diag}}\left\\{{\bf{\cdot}}\right\\}$ denote conjugate, transpose,
conjugate-transpose, trace, absolute value, Euclidean norm, real part and
diagonal operation, respectively. $\mathbb{C}$ and $\mathbb{R}$ denote the
sets of complex and real numbers, respectively.
$\mathcal{CN}\left({0,\sigma^{2}}\right)$ denotes a circularly symmetric
complex Gaussian random variable with mean 0 and covariance $\sigma^{2}$.
${\left[{\bf{X}}\right]_{mn}}$ denotes the $\left({m,n}\right){\mathop{\rm
th}\nolimits}$ element of matrix ${\bf{X}}$. ${\left[x\right]^{+}}$ denotes
the operation of $\max\left\\{{0,x}\right\\}$.

##  II SYSTEM MODEL AND PROBLEM FORMULATION

Figure 1: The downlink IRS-aided cell-free massive MIMO system.

This paper considers the downlink IRS-aided cell-free massive MIMO system, as
shown in Fig. 1. For this system, through the instructions of the CPU, $M$
single-antenna APs are deployed to serve $K$ single-antenna users with the
help of $L$ IRSs, where each IRS is equipped with $N$ reflecting elements and
the total number of reflecting elements at all IRSs is $I=LN$. Let
$\mathcal{M}=\left\\{{1,\cdots,M}\right\\}$,
$\mathcal{K}=\left\\{{1,\cdots,K}\right\\}$ and
$\mathcal{I}=\left\\{{1,\cdots,I}\right\\}$ as the index sets of the APs, the
users and the total elements of all IRSs, respectively.

###  II-A Transmission Model

The notation ${{g}_{{\rm AU},mk}}\in{\mathbb{C}^{1\times 1}}$,
${{\bf{g}}_{{\rm AI},ml}}\in{\mathbb{C}^{N\times 1}}$, and ${{\bf{g}}_{{\rm
IU},lk}^{\rm H}}\in{\mathbb{C}^{1\times N}}$ denote the channel from AP $m$ to
user $k$, the channel from AP $m$ to IRS $l$, and the channel from IRS $l$ to
user $k$, respectively. For convenience, we define ${{\bf{g}}_{{\mathop{\rm
AU}\nolimits},k}^{\rm H}}=\left[{{g_{{\mathop{\rm
AU}\nolimits},1k}},\cdots,{g_{{\mathop{\rm
AU}\nolimits},Mk}}}\right]\in{\mathbb{C}^{1\times M}}$ and
${{\bf{G}}_{{\mathop{\rm AI}\nolimits},l}}=\left[{{{\bf{g}}_{{\mathop{\rm
AI}\nolimits},1l}},\cdots,{{\bf{g}}_{{\mathop{\rm
AI}\nolimits},Ml}}}\right]\in{\mathbb{C}^{N\times M}}$ as the channel vector
from all APs to user $k$ and the channel matrix from all APs to IRS $l$,
respectively. In addition, by using the channel estimation methods [12], all
channel state information is assumed to be perfectly obtained by CPU. Due to
the existence of IRS, there are some cascaded channels between the APs and the
users, we define ${\bf{g}}_{{\mathop{\rm AIU}\nolimits},k}^{\rm
H}\in{\mathbb{C}^{1\times M}}$ as the composite AP-IRS-user channel from all
APs to user $k$ via all IRSs, which is given as

${{\bf{g}}_{{\mathop{\rm AIU}\nolimits},k}^{\rm
H}}=\sum\nolimits_{l=1}^{L}{{\bf{g}}_{{\mathop{\rm
IU}\nolimits},lk}^{\mathop{\rm
H}\nolimits}{{\bf{\Phi}}_{l}}{{\bf{G}}_{{\mathop{\rm A}\nolimits}{\mathop{\rm
I}\nolimits},l}}}={{\bf{v}}^{\mathop{\rm H}\nolimits}}{{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},k}},\;\;k\in\mathcal{K},$ (1)

where the phase shift matrix of IRS $l$ can be optimized through a diagonal
matrix ${{\bf{\Phi}}_{l}}={\mathop{\rm diag}\nolimits}\left\\{{{{\mathop{\rm
e}\nolimits}^{j{\theta_{l1}}}},\cdots,{{\mathop{\rm
e}\nolimits}^{j{\theta_{lN}}}}}\right\\}$ with
${\theta_{ln}}\in\left[{0,2\pi}\right]$, and other notations are defined as

${{\bf{v}}^{\mathop{\rm
H}\nolimits}}=\left[{{{\rm{e}}^{j{\theta_{11}}}},\cdots,{{\rm{e}}^{j{\theta_{1N}}}},\cdots,{{\rm{e}}^{j{\theta_{L1}}}},\cdots,{{\rm{e}}^{j{\theta_{LN}}}}}\right]\in{\mathbb{C}^{1\times
I}},$ (2) ${{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},lk}}={\rm{diag}}\left\\{{{\bf{g}}_{{\mathop{\rm
IU}\nolimits},lk}^{\mathop{\rm H}\nolimits}}\right\\}{{\bf{G}}_{{\mathop{\rm
A}\nolimits}{\mathop{\rm I}\nolimits},l}}\in{\mathbb{C}^{N\times M}},$ (3)
${{\bf{G}}_{{\mathop{\rm AIU}\nolimits},k}}={\left[{{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},1k}^{\mathop{\rm T}\nolimits},\cdots,{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},Lk}^{\mathop{\rm T}\nolimits}}\right]^{\mathop{\rm
T}\nolimits}}\in{\mathbb{C}^{I\times M}}.$ (4)

Based on the above system model, the received signal of user $k$ can be
expressed as

${r_{k}}=\left({{{\bf{v}}^{\mathop{\rm H}\nolimits}}{{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},k}}+{{\bf{g}}_{{\mathop{\rm AU}\nolimits},k}^{\rm
H}}}\right)\sum\nolimits_{j=1}^{K}{{{\bf{w}}_{j}}{x_{j}}}+{n_{k}},\;\;k\in\mathcal{K},$
(5)

where ${\bf{W}}\in{\mathbb{C}^{M\times K}}$ represents the beamforming matrix
at all APs, which its $m$th row and $k$th column element is
${\left[{\bf{W}}\right]_{mk}}={w_{mk}}$. Thus,
${{\bf{w}}_{j}}={\left[{{w_{1j}},\cdots,{w_{Mj}}}\right]^{\mathop{\rm
T}\nolimits}}\in{\mathbb{C}^{M\times 1}}$ is the $j$th column of ${\bf{W}}$,
representing the beamforming vector for user $j$. ${x_{j}}$ and
${n_{k}}\sim\mathcal{CN}\left({0,{{\sigma^{2}}}}\right)$ are the unit-power
information symbol for user $j$ and the noise at user $k$, respectively. Based
on (5), the achievable rate of user $k$ can be given by

${R_{k}}={\log_{2}}\left({1+\frac{{{{\left|{\left({{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},k}}+{{\bf{g}}_{{\mathop{\rm AU}\nolimits},k}^{\rm
H}}}\right){{\bf{w}}_{k}}}\right|}^{2}}}}{{\sum\limits_{j\neq
k}^{K}{{{\left|{\left({{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},k}}+{{\bf{g}}_{{\mathop{\rm AU}\nolimits},k}^{\rm
H}}}\right){{\bf{w}}_{j}}}\right|}^{2}}}+{\sigma^{2}}}}}\right).$ (6)

###  II-B Optimization Problem Formulation

To implement the design of EE optimization algorithm, we define the total
power consumption of system as

${P_{{\mathop{\rm
total}\nolimits}}}=\sum\limits_{k=1}^{K}{\alpha{\bf{w}}_{k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{k}}}+{P_{{\mathop{\rm fix}\nolimits}}},$ (7)

where $\sum\nolimits_{k=1}^{K}{\alpha{\bf{w}}_{k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{k}}}$ is the transmit power of all APs and
$\alpha={\upsilon^{-1}}$ with $\upsilon\in\left({0,1}\right]$ being the power
amplifier efficiency. Furthermore, ${P_{{\mathop{\rm
fix}\nolimits}}}=M{P_{{\mathop{\rm
AP}\nolimits}}}+K{P_{{\rm{User}}}}+I{P_{{\mathop{\rm IRS}\nolimits}}}$ denotes
the static power consumption, where ${P_{{\rm{AP}}}}$, ${P_{{\rm{User}}}}$ and
${P_{{\mathop{\rm IRS}\nolimits}}}$ are the circuit power consumption of each
AP, each user and each reflecting element of IRS, respectively. Thus, EE is
given by

${\mathop{\rm
EE}\nolimits}=\frac{{B\sum\nolimits_{k=1}^{K}{{R_{k}}}}}{{\sum\nolimits_{k=1}^{K}{\alpha{\bf{w}}_{k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{k}}}+{P_{{\mathop{\rm fix}\nolimits}}}}},$ (8)

where $B$ is the bandwidth of the channel.

Based on (8), the problem of maximizing EE can be formulated as

$\displaystyle\mathop{\max}\limits_{{\bf{W}},{\bf{v}}}\;\;$
$\displaystyle{\mathop{\rm EE}\nolimits}$ (9a) $\displaystyle\;{\mathop{\rm
s.t.}\nolimits}\;\;\;$
$\displaystyle{R_{k}}\geq{R_{\min}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;k\in\mathcal{K},$
(9b)
$\displaystyle{\left\|{{{{\bf{\bar{w}}}}_{m}}}\right\|^{2}}\leq{P_{\max}},\;\;\;\;\;\;\;m\in\mathcal{M},$
(9c)
$\displaystyle{\left|{{{\left[{\bf{v}}\right]}_{i}}}\right|^{2}}=1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;i\in\mathcal{I}.$
(9d)

The constraint in (9b) guarantees the user’s rate requirements of all users,
where $R_{\min}$ is the minimum data rate requirement. The constraint in (9c)
is a transmit power constraint, where
${{\bf{\bar{w}}}_{m}}=\left[{{w_{m1}},\cdots,{w_{mK}}}\right]\in{\mathbb{C}^{1\times
K}}$ and $P_{\max}$ are the beamforming vector at AP $m$ and the maximum
transmit power for each AP, respectively. The unit modulus constraint for each
reflecting element of IRS is provided in (9d). Due to the non-convexity of
this problem, it is difficult to solve.

##  III ITERATIVE OPTIMIZATION ALGORITHM

To solve the problem of maximizing EE, we first decompose the problem (9) into
two subproblems, i.e., the beamforming at APs and the phase shifts at IRSs,
respectively. Then, we design the corresponding algorithm to alternately
optimize them.

###  III-A Optimizing ${\bf{W}}$ for Given ${\bf{v}}$

For a fixed ${\bf{v}}$, problem (9) can be transformed into

$\displaystyle\mathop{\max}\limits_{\bf{W}}\;\;$
$\displaystyle\frac{{B\sum\nolimits_{k=1}^{K}{{{\log}_{2}}\left({1+\frac{{{{\left|{{{\bf{h}}_{{\mathop{\rm
AU}\nolimits},k}^{\rm H}}{{\bf{w}}_{k}}}\right|}^{2}}}}{{\sum\nolimits_{j\neq
k}^{K}{{{\left|{{{\bf{h}}_{{\mathop{\rm AU}\nolimits},k}^{\rm
H}}{{\bf{w}}_{j}}}\right|}^{2}}}+{\sigma^{2}}}}}\right)}}}{{\sum\nolimits_{k=1}^{K}{\alpha{\bf{w}}_{k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{k}}}+{P_{{\mathop{\rm fix}\nolimits}}}}}$ (10a)
$\displaystyle\;{\mathop{\rm s.t.}\nolimits}\;\;$
$\displaystyle\left\|{{\bf{h}}_{{\mathop{\rm AU}\nolimits},k}^{\mathop{\rm
H}\nolimits}{{\bf{W}}_{k}},{\sigma}}\right\|\leq\frac{{{\bf{h}}_{{\mathop{\rm
AU}\nolimits},k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{k}}}}{{\sqrt{{2^{{R_{\min}}}}-1}}},\;\;\;\;k\in\mathcal{K},$
(10b)
$\displaystyle{\left\|{{{{\bf{\bar{w}}}}_{m}}}\right\|}\leq{\sqrt{P_{\max}}},\;\;\;m\in\mathcal{M},$
(10c)

where
${{\bf{W}}_{k}}=\left[{{{\bf{w}}_{1}},\cdots,{{\bf{w}}_{k-1}},{{\bf{w}}_{k+1}},\cdots,{{\bf{w}}_{K}}}\right]\in{\mathbb{C}^{M\times\left({K-1}\right)}}$
and ${\bf{h}}_{{\mathop{\rm AU}\nolimits},k}^{\mathop{\rm
H}\nolimits}={{\bf{v}}^{\mathop{\rm H}\nolimits}}{{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},k}}+{{\bf{g}}_{{\mathop{\rm AU}\nolimits},k}^{\rm H}}$
represents an aggregated channel between all APs and user $k$.

In problem (10), the constraints of (10b) and (10c) are convex, but the
objective function of (10a) is non-convex. In order to convert problem (10)
into convex problem, we apply the quadratic transform [13] to the objective
function (10a), so problem (10) can be transformed into

$\displaystyle\mathop{\max}\limits_{{\bf{W}},{\bf{y}},z}\;\;$
$\displaystyle{f_{1}}\left({{\bf{W}},{\bf{y}},z}\right)$ (11a)
$\displaystyle\;{\mathop{\rm s.t.}\nolimits}\;\;\;$
$\displaystyle\left({10{\rm{b}}}\right),\left({10{\mathop{\rm
c}\nolimits}}\right),$ (11b)

where ${\bf{y}}$ refers to a collection of variables
$\left\\{{{y_{1}}{\rm{,}}\cdots{\rm{,}}{y_{K}}}\right\\}$ and

${y_{k}}=\frac{{{\bf{h}}_{{\mathop{\rm AU}\nolimits},k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{k}}}}{{\sum\nolimits_{j\neq
k}^{K}{{{\left|{{\bf{h}}_{{\mathop{\rm AU}\nolimits},k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{j}}}\right|}^{2}}}+{\sigma^{2}}}},\;k\in\mathcal{K},$
(12)
${z}=\frac{{\sqrt{\sum\nolimits_{k=1}^{K}{{{\log}_{2}}\left({1+\frac{{{{\left|{{{\bf{h}}_{{\mathop{\rm
AU}\nolimits},k}^{\rm H}}{{\bf{w}}_{k}}}\right|}^{2}}}}{{\sum\nolimits_{j\neq
k}^{K}{{{\left|{{{\bf{h}}_{{\mathop{\rm AU}\nolimits},k}^{\rm
H}}{{\bf{w}}_{j}}}\right|}^{2}}}+{\sigma^{2}}}}}\right)}}}}{{\alpha\sum\nolimits_{k=1}^{K}{{\bf{w}}_{k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{k}}}+{P_{{\mathop{\rm fix}\nolimits}}}}},$ (13)
$\begin{array}[]{l}{f_{1}}\left({{\bf{W}},{\bf{y}},z}\right)=-{z^{2}}\left({\alpha\sum\limits_{k=1}^{K}{{\bf{w}}_{k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{k}}}+{P_{{\mathop{\rm fix}\nolimits}}}}\right)+\\\
2z\sqrt{\sum\limits_{k=1}^{K}{{{\log}_{2}}\left(\begin{array}[]{l}1+2{\mathop{\rm
Re}\nolimits}\left\\{{y_{k}^{*}{\bf{h}}_{{\mathop{\rm
AU}\nolimits},k}^{\mathop{\rm H}\nolimits}{{\bf{w}}_{k}}}\right\\}-\\\
{\left|{{y_{k}}}\right|^{2}}\left({\sum\limits_{j\neq
k}^{K}{{{\left|{{\bf{h}}_{{\mathop{\rm AU}\nolimits},k}^{\mathop{\rm
H}\nolimits}{{\bf{w}}_{j}}}\right|}^{2}}}+{\sigma^{2}}}\right)\end{array}\right)}}\end{array}.$
(14)

When the auxiliary variables ${\bf{y}}$ and $z$ are fixed, problem (11) is a
convex problem with respect to ${\bf{W}}$, and therefore the optimal
${\bf{W}}$ can be found by using the modern optimization solver (e.g. Mosek
[14]). By solving problem (11), the suboptimal solution of problem (10) can
also be obtained correspondingly.

###  III-B Optimizing ${\bf{v}}$ for Given ${\bf{W}}$

For a fixed ${\bf{W}}$, problem (9) can be equivalent to the problem of
maximizing sum rate as follows

$\displaystyle\mathop{\max}\limits_{\bf{v}}\;$
$\displaystyle\sum\limits_{k=1}^{K}{{{\log}_{2}}\left({1+\frac{{{{\left|{{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{a}}_{kk}}+{b_{kk}}}\right|}^{2}}}}{{\sum\nolimits_{j\neq
k}^{K}{{{\left|{{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{a}}_{kj}}+{b_{kj}}}\right|}^{2}}}+{\sigma^{2}}}}}\right)}$
(15a) $\displaystyle\;{\mathop{\rm s.t.}\nolimits}\;$
$\displaystyle\sum\limits_{j\neq k}^{K}{{{\left|{{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{a}}_{kj}}+{b_{kj}}}\right|}^{2}}}+{\sigma^{2}}\leq\frac{{{{\left|{{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{a}}_{kk}}+{b_{kk}}}\right|}^{2}}}}{{{2^{{R_{\min}}}}-1}},\;k\in\mathcal{K},$
(15b)
$\displaystyle{\left|{{{\left[{\bf{v}}\right]}_{i}}}\right|^{2}}=1,\;\;\;\;\;\;\;i\in\mathcal{I},$
(15c)

where ${{\bf{a}}_{kj}}={{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},k}}{{\bf{w}}_{j}}$ and ${b_{kj}}={{\bf{g}}_{{\mathop{\rm
AU}\nolimits},k}^{\rm H}}{{\bf{w}}_{j}}$ for $k,j\in\mathcal{K}$.

Since the problem (15a) is non-convex, we apply the quadratic transform [13]
and the Lagrangian dual transform [15] to (15a), so problem (15) can be
transformed into problem (16) as follows

$\displaystyle\mathop{\max}\limits_{{\bf{v}},{\boldsymbol{\gamma}},{\boldsymbol{\varepsilon}}}\;\;$
$\displaystyle{f_{2}}\left({{\bf{v}},{\boldsymbol{\gamma}},{\boldsymbol{\varepsilon}}}\right)$
(16a) $\displaystyle\;{\mathop{\rm s.t.}\nolimits}\;\;\;$
$\displaystyle\left({15{\mathop{\rm b}\nolimits}}\right),\left({15{\mathop{\rm
c}\nolimits}}\right),$ (16b)

where ${\boldsymbol{\gamma}}=\left[{{\gamma_{1}},\cdots,{\gamma_{K}}}\right]$
and
${\boldsymbol{\varepsilon}}=\left[{{\varepsilon_{1}},\cdots,{\varepsilon_{K}}}\right]$
are two auxiliary vector variables introduced to make the problem more
tractable and

${\gamma_{k}}=\frac{{{{\left|{{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{a}}_{kk}}+{b_{kk}}}\right|}^{2}}}}{{\sum\nolimits_{j\neq
k}^{K}{{{\left|{{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{a}}_{kj}}+{b_{kj}}}\right|}^{2}}}+{\sigma^{2}}}},\;k\in\mathcal{K},$
(17)
${\varepsilon_{k}}=\frac{{\sqrt{1+{\gamma_{k}}}\left({{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{a}}_{kk}}+{b_{kk}}}\right)}}{{\sum\nolimits_{j=1}^{K}{{{\left|{{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{{\bf{a}}_{kj}}+{b_{kj}}}\right|}^{2}}}+{\sigma^{2}}}},\;k\in\mathcal{K},$
(18)
${\bf{\Theta}}=\sum\limits_{k=1}^{K}{\sum\limits_{j=1}^{K}{{{\left|{{\varepsilon_{k}}}\right|}^{2}}{{\bf{a}}_{kj}}{\bf{a}}_{kj}^{\mathop{\rm
H}\nolimits}}},$ (19)
${\bf{u}}=\sum\limits_{k=1}^{K}{\left({\sqrt{1+{\gamma_{k}}}\varepsilon_{k}^{*}{{\bf{a}}_{kk}}-{{\left|{{\varepsilon_{k}}}\right|}^{2}}\sum\limits_{j=1}^{K}{b_{kj}^{*}{{\bf{a}}_{kj}}}}\right)},$
(20)
${f_{2}}\left({{\bf{v}},{\boldsymbol{\gamma}},{\boldsymbol{\varepsilon}}}\right)=\sum\limits_{k=1}^{K}{\left({{{\log}_{2}}\left({1+{\gamma_{k}}}\right)-{\gamma_{k}}}\right)}-{{\bf{v}}^{\mathop{\rm
H}\nolimits}}{\bf{\Theta v}}+2{\mathop{\rm
Re}\nolimits}\left\\{{{{\bf{v}}^{\mathop{\rm H}\nolimits}}{\bf{u}}}\right\\}.$
(21)

By using semidefinite relaxation [3] and some matrix transformations for
problem (16), the problem (16) can be transformed into

$\displaystyle\mathop{\max}\limits_{\bf{Q},{\boldsymbol{\gamma}},{\boldsymbol{\varepsilon}}}\;\;$
$\displaystyle{\mathop{\rm
trace}\nolimits}\left({{\bf{\bar{\Theta}Q}}}\right)$ (22a)
$\displaystyle\;{\mathop{\rm s.t.}\nolimits}\;\;\;$
$\displaystyle\sum\limits_{j\neq
k}^{K}{\left({{\rm{trace}}\left({{{\bf{C}}_{kj}}{\bf{Q}}}\right)+{{\left|{{b_{kj}}}\right|}^{2}}}\right)}+{\sigma^{2}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\frac{{{\rm{trace}}\left({{{\bf{C}}_{kk}}{\bf{Q}}}\right)+{{\left|{{b_{kk}}}\right|}^{2}}}}{{{2^{{R_{\min}}}}-1}},\;\;\;k\in\mathcal{K},$
(22b) $\displaystyle\;{\left[{\bf{Q}}\right]_{ii}}=1,\;\;\;i=1,\cdots,I+1,$
(22c) $\displaystyle\;{\bf{Q}}\succeq 0,$ (22d)

where ${\bf{\bar{\Theta}}}=\left[\begin{array}[]{l}-{\bf{\Theta}},{\bf{u}}\\\
{\>{\bf{u}}^{\mathop{\rm H}\nolimits}},\;0\end{array}\right]$ and
${{\bf{C}}_{kj}}=\left[\begin{array}[]{l}{{\bf{a}}_{kj}}{\bf{a}}_{kj}^{\mathop{\rm
H}\nolimits},{{\bf{a}}_{kj}}b_{kj}^{*}\\\ {b_{kj}}{\bf{a}}_{kj}^{\mathop{\rm
H}\nolimits},\;\;\;0\end{array}\right]$.

When the auxiliary variables ${\boldsymbol{\gamma}}$ and
${\boldsymbol{\varepsilon}}$ are both fixed, problem (22) is a convex problem,
we can solve it via Mosek. After getting the optimized ${\bf Q}$, we can use
the Gaussian randomization on ${\bf Q}$ to get a feasible rank-one solution
${\bf v}$ for problem (15) [3]. Therefore, by alternately solving problem (11)
and problem (22), we can obtain a suboptimal solution ${\bf W}$ and ${\bf v}$
to the problem (9), the specific algorithm is presented in
${\bf{Algorithm\>1}}$. Based on [16], the complexity of ${\bf{Algorithm\>1}}$
is approximated as
$\mathcal{O}\left({{\rm{ite}}{{\rm{r}}_{1}}\sqrt{M+K}{M^{3}}{K^{4}}+{\rm{ite}}{{\rm{r}}_{2}}{K^{1.5}}{I^{6.5}}}\right)$,where
${\rm iter}_{1}$ and ${\rm iter}_{2}$ represent the number of iterations to
solve problem (11) and problem (22), respectively.

Algorithm 1 Beamforming and Phase shift for problem (9)

1: Initialize ${\bf{W}}^{0}$ and ${\bf{v}}^{0}$ to a feasible value and set
$t=0$.

2: For given ${\bf{v}}^{t}$, update ${\bf y}$ and $z$ by (12) and (13).

3: Update ${\bf{W}}$ by solving problem (11) for fixed ${\bf y}$ and $z$.

4: Repeat steps 2-3 until EE does not increase and output the optimized
beamforming ${\bf{W}}^{t+1}$.

5: For given ${\bf{W}}^{t+1}$, update ${\boldsymbol{\gamma}}$ and
${\boldsymbol{\varepsilon}}$ by (17) and (18).

6: Update ${\bf{v}}$ by solving the problem (22) together with Gaussian
randomization for fixed ${\boldsymbol{\gamma}}$ and
${\boldsymbol{\varepsilon}}$.

7: Repeat steps 5-6 until EE does not increase and output the optimized phase
shifts ${\bf{v}}^{t+1}$.

8: If EE does not increase or $t=T$, the iteration process is terminated.
Otherwise, set $t=t+1$ and go to step 2.

Figure 2: An overview of the proposed network.

##  IV UNSUPERVISED LEARNING BASED APPROACH

Unlike the time-consuming iterative optimization algorithm in section III,
this section propose an unsupervised learning based approach to tackle problem
(9), where this approach can quickly output the optimal $\bf W$ and $\bf v$ by
inputting the channel state information of the corresponding scenario to the
network trained by unsupervised learning manner.

###  IV-A Network Architecture and Training

The detailed network architecture and implementation are illustrated in Fig.
2. Based on problem (9), it is intuitive to use the real and imaginary parts
of channels ${{\bf{G}}_{{\mathop{\rm AIU}\nolimits},k}}$ and
${{\bf{g}}_{{\mathop{\rm AU}\nolimits},k}^{\rm H}}$, $\forall k\in\mathcal{K}$
as the input of the network, where the total number of features is $2MIK+2MK$.
After these features are fed into the network, they will be normalized and fed
into the phase shift prediction network. In the phase shift prediction
network, we use five fully-connected (FC) layers, where the first four FC
layers has $2MIK+2MK$ neurons and the last layer has $I$ neurons,
respectively. For the first four FC layers, each of them is first followed by
the batch normalization (BN) layer and then followed by the Leaky rectified
linear unit (LeakyRelu) activation layer. For FC5 layer, it uses a linear unit
to output the phase shift prediction ${{\bf{v}}_{{\mathop{\rm
pred}\nolimits}}}=\cos\left({{{{\boldsymbol{\theta}}}_{{\mathop{\rm
pred}\nolimits}}}}\right)+j\sin\left({{{\boldsymbol{\theta}}_{{\mathop{\rm
pred}\nolimits}}}}\right)$ based on the predicted real vector
${{\boldsymbol{\theta}}_{{\mathop{\rm pred}\nolimits}}}\in{\mathbb{R}^{I}}$.
Based on ${{\bf{v}}_{{\mathop{\rm pred}\nolimits}}}$, optimal effective
channel ${\bf{H}}_{{\mathop{\rm AU}\nolimits}}^{{\mathop{\rm
pred}\nolimits}}=\left[{{\bf{h}}_{{\mathop{\rm AU}\nolimits},1}^{\rm
pred};\cdots;{\bf{h}}_{{\mathop{\rm AU}\nolimits},K}^{\rm pred}}\right]$ with
${\bf{h}}_{{\mathop{\rm AU}\nolimits},k}^{\rm pred}={{\bf{v}}^{\mathop{\rm
H}\nolimits}_{\rm pred}}{{\bf{G}}_{{\mathop{\rm
AIU}\nolimits},k}}+{{\bf{g}}_{{\mathop{\rm AU}\nolimits},k}^{\rm H}}$ can be
formed and then the real and imaginary parts of ${\bf{H}}_{{\mathop{\rm
AU}\nolimits}}^{{\mathop{\rm pred}\nolimits}}$ will be normalized and used as
input to the beamforming prediction network. In the beamforming prediction
network, convolutional neural network is designed with four convolutional
(Conv) layers with 256, 256, 128, 64 filters of size $M\times K$ and one FC
layer with $2MK$ neurons, where FC layer is followed by a Linear unit to
output the real and imaginary parts of the beamforming prediction ${\bf
W}_{\rm pred}$. In addition, the usage of BN layer and activation function in
beamforming prediction network is similar to the phase shift prediction
network.

During training, we apply the Adam optimizer to update weights and set the
initial learning rate to $0.01$. Maximal epoch, batch size and early stopping
of patience are set as 500, 1024 and 20, respectively. To speed up the
convergence, when the validation loss does not decrease within 10 epochs, the
learning rate will be reduced by a factor of 0.5 until the minimum learning
rate threshold $10^{-6}$ is reached. Besides, for sufficient training and
accurate evaluation of model, we generate $1.6\times 10^{5}$, $4\times 10^{4}$
and $10^{3}$ samples for training, validation and testing, respectively.
Finally, it is worth noting that since unsupervised learning does not require
labels, it greatly reduces the cost of obtaining training samples.

###  IV-B Loss Function

Given that the goal of problem (9) is to maximize EE under some constraints of
(9b), (9c) and (9d), we apply the following loss function for model training
as

${\mathop{\rm
Loss}\nolimits}=\frac{1}{T}\sum\limits_{t=1}^{T}{\left(\begin{array}[]{l}-{\mathop{\rm
EE}\nolimits}+\sum\nolimits_{k=1}^{K}{{\beta_{\rm{1}}}{{\left[{{R_{\min}-{R_{k}}}}\right]}^{+}}}\\\
+\sum\nolimits_{m=1}^{M}{{\beta_{\rm{2}}}{{\left[{{\left\|{{{{\bf{\bar{w}}}}_{m}}}\right\|^{2}}-{P_{\max}}}\right]}^{+}}}\end{array}\right)},$
(23)

where $T$ is the mini-batch size. ${\beta_{1}}$ and ${\beta_{2}}$ are the
penalty coefficient for the rate constraint of (9b) and the power constraint
of (9c), where each penalty term will have a positive value only if the
corresponding constraints are violated, enforcing the training process towards
satisfying the given requirement. In addition, the constraint of (9d) is not
added to loss function because it is already implemented in the process of
converting the real vector ${{\boldsymbol{\theta}}_{{\mathop{\rm
pred}\nolimits}}}$ to ${{\bf{v}}_{{\mathop{\rm pred}\nolimits}}}$. Therefore,
by choosing appropriate ${\beta_{1}}$ and ${\beta_{2}}$, loss function can be
used to solve problem (9) efficiently. Finally, based on [17], the complexity
of unsupervised learning based approach is approximated as
$\mathcal{O}\left({{M^{2}}{K^{2}}\left({16{I^{2}}+{1.07*{10}^{5}}}\right)}\right)$.

##  V Numerical Results

In this section, simulation results are provided to validate the EE
performance of the iterative optimization algorithm and the unsupervised
learning based approach. First, we consider a simulation setup, where $M=13$
APs and $L=2$ IRSs configured with $N=50$ reflecting elements provide services
to $K=3$ single-antenna users. Besides, the $m$-th AP is located at
$\left({10\left({m-1}\right)\;{\mathop{\rm m}\nolimits},-40\;{\mathop{\rm
m}\nolimits},5\;{\mathop{\rm m}\nolimits}}\right),m\in\mathcal{M}$, and the
IRSs are located at $\left({40\;{\mathop{\rm m}\nolimits},10\;{\mathop{\rm
m}\nolimits},10\;{\mathop{\rm m}\nolimits}}\right)$ and
$\left({80\;{\mathop{\rm m}\nolimits},10\;{\mathop{\rm
m}\nolimits},10\;{\mathop{\rm m}\nolimits}}\right)$, respectively. Finally,
the users with a height of 1.65 m are located in a uniform circular area with
a radius of 2 m, and the center of the region can move along the line between
$\left({0\;{\mathop{\rm m}\nolimits},0\;{\mathop{\rm m}\nolimits}}\right)$ and
$\left({120\;{\mathop{\rm m}\nolimits},0\;{\mathop{\rm m}\nolimits}}\right)$.

For all channels in simulation setup, we apply a Rician fading channel
modeling approach similar to reference [4]. For example, we assume that the
path loss is 30 dB at the reference distance 1 m. The path loss exponent of
the AP-IRS link, the IRS-user link and the AP-user link are set as 2.2, 2.8
and 3.5, respectively. The Rician factor of the AP-IRS link, the IRS-user link
and the AP-user link are set as 10 dB, 5 dB and $-\infty$ dB, respectively.
Besides, other simulation parameters are given as the maximum transmit power
of each AP ${P_{\max}}=0\;{\mathop{\rm dB}\nolimits}$, the minimum rate
requirement of each user ${R_{\min}}=1\;{\rm bit/s/Hz}$, the circuit power of
each AP ${P_{{\mathop{\rm AP}\nolimits}}}=10\;{\mathop{\rm dBm}\nolimits}$,
the circuit power of each user ${P_{{\mathop{\rm
User}\nolimits}}}=10\;{\mathop{\rm dBm}\nolimits}$, the circuit power of each
IRS element ${P_{{\mathop{\rm IRS}\nolimits}}}=0\;{\mathop{\rm
dBm}\nolimits}$, the bandwidth $B=1\;{\mathop{\rm MHz}\nolimits}$, the noise
power ${\sigma^{2}}=-60\;{\mathop{\rm dBm}\nolimits}$, the power amplifier
efficiency $\upsilon=0.8$, and the penalty coefficient
${\beta_{1}}={\beta_{2}}=50$, respectively. Finally, all operations are
executed by software (Python 3.6, TensorFlow 2.6.0 and Mosek 9.3.20) and
hardware (two Xeon Platinum 8368 CPUs, a GeForce RTX 3090 GPU and 512 GB RAM).

Figure 3: Cumulative distribution of the energy efficiency. Figure 4: The loss
of unsupervised learning based approach on the epochs and the EE of iterative
optimization algorithm on the number of iterations.

To validate the efficiency of our proposed scheme, Fig. 3 compares the
cumulative distribution functions of EE for the iterative optimization
algorithm, the unsupervised learning based approach and the GA, where the
population size (PS) and the maximum number of generations play a key role in
the performance of GA. In our simulations, we set the maximum number of
generations to 20000 and set the PS to $\left\\{{300,400}\right\\}$,
respectively. In Fig. 3, we can observe that the $95\%-{\rm{likely}}$ EE by
the unsupervised learning based approach, the iterative optimization
algorithm, the GA with PS 300 and PS 400 are about 3.14 Mbit/Joule, 1.92
Mbit/Joule, 0.91 Mbit/Joule and 1.39 Mbit/Joule, respectively. Besides, the
average running time of these schemes are given as 0.001 s, 6.592 s, 18.281 s
and 21.977 s, respectively. Given the highest EE performance and the lowest
running time, the unsupervised learning based approach is regarded as an
optimal scheme compared to other schemes. In addition, we can also observe
that for the $95\%-{\rm{likely}}$ EE, the iterative optimization algorithm
only consumes about $30\%$ of the running time to reach 2.1 (or 1.38) times EE
performance of GA with PS 300 (or 400), respectively. Since the unsupervised
learning based approach takes a lot of time to train a perfect DNN, the
iterative optimization algorithm is also considered a competitive approach in
scenarios where training time is limited.

An exemplary training process is illustrated in Fig. 4(a). As can be readily
observed, our proposed DNN architecture can approach convergence after about
100 epochs. This verifies the feasibility of the DNN architecture. In
addition, the impact of iteration number on the EE of the iterative
optimization algorithm is illustrated in Fig. 4(b). It can be observed that as
the number of iterations increases, the EE can approach convergence after 20
iterations. This also shows that the iterative optimization algorithm has good
convergence.

##  VI Conclusions

For the IRS-aided cell-free massive MIMO system, we proposed an iterative
optimization algorithm and an unsupervised learning based approach to tackle
EE maximization problem, respectively. The simulation results show that,
benefiting from the universal approximation properties of the DNN, the
unsupervised learning based approach exceeds the iterative optimization
algorithm and GA in terms of EE and running time. This shows that the
unsupervised learning based approach is ideal for solving EE maximization
problem. In addition, when the training time is limited, the iterative
optimization algorithm performs relatively well in terms of EE, running time
and convergence, indicating that it is also an excellent scheme.

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# Photoelectron circular dichroism of O 1$s$-photoelectrons of uniaxially
oriented trifluoromethyloxirane: Energy dependence and sensitivity to
molecular configuration

G. Nalin<EMAIL_ADDRESS>Institut für Kernphysik, Goethe-
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany  K.
Fehre<EMAIL_ADDRESS>Institut für Kernphysik, Goethe-
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany  F.
Trinter  Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, 22607
Hamburg, Germany  Molecular Physics, Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  N. M. Novikovskiy
Institut für Physik und CINSaT, Universität Kassel, Heinrich-Plett-Strasse 40,
34132 Kassel, Germany  Institute of Physics, Southern Federal University,
344090 Rostov-on-Don, Russia  N. Anders  Institut für Kernphysik, Goethe-
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany  D.
Trabert  Institut für Kernphysik, Goethe-Universität, Max-von-Laue-Strasse 1,
60438 Frankfurt am Main, Germany  S. Grundmann  Institut für Kernphysik,
Goethe-Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
M. Kircher  Institut für Kernphysik, Goethe-Universität, Max-von-Laue-Strasse
1, 60438 Frankfurt am Main, Germany  A. Khan  Institut für Kernphysik, Goethe-
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany  R.
Tomar  Institut für Kernphysik, Goethe-Universität, Max-von-Laue-Strasse 1,
60438 Frankfurt am Main, Germany  M. Hofmann  Institut für Kernphysik, Goethe-
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany  M.
Waitz  Institut für Kernphysik, Goethe-Universität, Max-von-Laue-Strasse 1,
60438 Frankfurt am Main, Germany  I. Vela-Perez  Institut für Kernphysik,
Goethe-Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
H. Fukuzawa  Institute of Multidisciplinary Research for Advanced Materials,
Tohoku University, Sendai 980-8577, Japan  K. Ueda  Institute of
Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai
980-8577, Japan  J. Williams  Department of Physics, University of Nevada,
Reno, Nevada 89557, United States  D. Kargin  Institut für Chemie und CINSaT,
Universität Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel, Germany  M.
Maurer  Institut für Chemie und CINSaT, Universität Kassel, Heinrich-Plett-
Strasse 40, 34132 Kassel, Germany  C. Küstner-Wetekam  Institut für Physik und
CINSaT, Universität Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel, Germany
L. Marder  Institut für Physik und CINSaT, Universität Kassel, Heinrich-Plett-
Strasse 40, 34132 Kassel, Germany  J. Viehmann  Institut für Physik und
CINSaT, Universität Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel, Germany
A. Knie  Institut für Physik und CINSaT, Universität Kassel, Heinrich-Plett-
Strasse 40, 34132 Kassel, Germany  T. Jahnke  Institut für Kernphysik, Goethe-
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
European XFEL GmbH, Holzkoppel 4, 22869 Schenefeld, Germany  M. Ilchen
<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron (DESY), Notkestrasse
85, 22607 Hamburg, Germany  Institut für Physik und CINSaT, Universität
Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel, Germany  European XFEL GmbH,
Holzkoppel 4, 22869 Schenefeld, Germany  R. Dörner  Institut für Kernphysik,
Goethe-Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
R. Pietschnig<EMAIL_ADDRESS>Institut für Chemie und CINSaT,
Universität Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel, Germany  Ph. V.
Demekhin<EMAIL_ADDRESS>Institut für Physik und CINSaT,
Universität Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel, Germany  M. S.
Schöffler<EMAIL_ADDRESS>Institut für Kernphysik, Goethe-
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany

###### Abstract

The photoelectron circular dichroism (PECD) of the O 1$s$-photoelectrons of
trifluoromethyloxirane(TFMOx) is studied experimentally and theoretically for
different photoelectron kinetic energies. The experiments were performed
employing circularly polarized synchrotron radiation and coincidentelectron
and fragment ion detection using Cold Target Recoil Ion Momentum Spectroscopy.
The corresponding calculations were performed by means of the Single Center
method within the relaxed-core Hartree-Fock approximation. We concentrate on
the energy dependence of the differential PECD of uniaxially oriented TFMOx
molecules, which is accessible through the employed coincident detection. We
also compare results for differential PECD of TFMOx to those obtained for the
equivalent fragmentation channel and similar photoelectron kinetic energy of
methyloxirane (MOx), studied in our previous work. Thereby, we investigate the
influence of the substitution of the methyl-group by the trifluoromethyl-group
at the chiral center on the molecular chiral response. Finally, the presently
obtained angular distribution parameters are compared to those available in
literature.

##  I Introduction

The interaction of circularly polarized light with chiral molecules leads to a
range of chiroptical effects. One of such effects, namely the photoelectron
circular dichroism (PECD), has been demonstrated to have strong potential as a
sensitive probe of this chiral light-molecule interaction. The effect
manifests itself as a forward-backward asymmetry of the photoelectron angular
emission distribution with respect to the light propagation direction, which
inverts upon switching of the light’s helicity or upon exchanging molecular
enantiomers. After the first theoretical formulation provided by Ritchie in
1976 Ritchie76  and the first experimental observation by Böwering et al. in
2001 Bowering01 , PECD rapidly evolved into a well-established sensitive
chiral recognition technique Powis08a ; Nahon15 ; Turchini17 ; Hadidi18 .

PECD is a universal chiroptical effect occurring in all regimes Beaulieu16NJP
, from one-photon to multiphoton ionization Lux12AngChm ; Lehmann13jcp  and
strong-field ionization Beaulieu16NJP ; Fehre19 ; Fehre19jpca , with a
majority of experiments, so far, reporting on the ionization of valence
electrons Garcia03 ; Nahon06 ; Garcia08 ; Garcia10 ; Tia13 . PECD shows a
complex dependence on the electronic Garcia14  and vibronic Garcia13
configuration, as well as the conformation Turchini13 ; Nahon16 , dimerization
Nahon10 , clustering Daly11  of molecules, and the energy of the emitted
photoelectron Powis08b . The typical strength of PECD is on the order of a few
percent. Its unique sensitivity has been exploited to study ultrafast
molecular dynamics Beaulieu16FD ; Ilchen21tfmox , especially time-resolved
molecular relaxation Comby16 , and it has been employed for real-time
determination of the enantiomeric excess in racemic mixtures Comby18 .

Several experiments reported on PECD after core ionization of chiral molecules
Stener04 ; Nahon06 ; Powis08b ; Hergenhahn04 ; Alberti08 ; Ulrich08 ;
Pitzer16cpc ; Pitzer16jpb ; Tia17 ; Ilchen17 ; Hartmann19 . In our recent work
Tia17 , we performed a differential study of PECD after O $1s$-photoionization
of uniaxially oriented methyloxirane molecules (MOx, C3H6O). To this end, the
O $1s$-photoelectrons were detected in coincidence with two charged fragments.
The molecular fragmentation axis was assumed to coincide with the emission
direction of those ionic fragments relying on the validity of the so-called
axial-recoil approximation Zare72 . This assumption is often well justified
for cases of rapid Auger decay of an inner-shell vacancy and subsequent
Coulomb explosion of a molecular dication. It was found that a selection of
certain fragmentation directions of MOx with respect to the light propagation
direction yields an increase of the PECD by a factor of 10, as compared to
randomly oriented molecules.

Figure 1: Top: Structure of S- and R-methyloxirane (C3H6O, MOx). Bottom:
Structure of R- and S-trifluoromethyloxirane (C3H3F3O, TFMOx). Note that
substitution of the methyl-group (CH3) with the trifluoromethyl-group (CF3) at
the chiral center causes a renaming of the respective R- and S-enantiomers
according to the CIP rule Cahn66 .

In the present work, we extend our previous study Tia17  to a close relative
of MOx, the trifluoromethyloxirane (TFMOx, C3H3F3O) molecule, in which all
three hydrogen atoms of the methyl-group are replaced by much heavier fluorine
atoms. For the further discussion, it is important to stress that by
definition of the Cahn-Ingold-Prelog (CIP Cahn66 ) rules, this substitution
exchanges the assigned handedness (S- to R-enantiomer and vice versa) despite
the unchanged connectivity of the molecular backbone and the stereo-center in
MOx and TFMOx (see Fig. 1). We investigate here the O $1s$-photoelectrons of
TFMOx for different kinetic energies. One of these energies coincides with
that of the study on MOx presented in Ref. 32. We selected, furthermore, for
our investigation of TFMOx one of the fragmentation channels which was studied
previously for MOx.

Figure 2: Ratio of enantiomers present in the samples of racemic RS-TFMOx (a)
and its S- and R-enantiomers (b and c, respectively) as experimentally
determined using HPLC.

##  II Research Methods

###  II.1 Samples

An RS-mixture and samples of pure S- or R-TFMOx enantiomers were commercially
obtained (SynQuest Laboratories) and used as received. Their chemical identity
and purity were verified by nuclear magnetic resonance (NMR) spectroscopy. The
1H, 13C, and 19F NMR data were recorded on Varian VNMRS-500 MHz or MR-400 MHz
spectrometers at room temperature. Chemical shifts were referenced to residual
protic impurities in the solvent (1H) or the deuterio solvent itself (13C) and
reported relative to external Si(CH3)4 (1H, 13C) or C6H5CF3(19F). The chemical
purity after integration of the fluorine NMR signal has been estimated to be
higher than 96% for the RS-TFMOx mixture and higher than 96% or 98% for the
S-TFMOx or R-TFMOx enantiomers, respectively. The NMR shifts were assigned
based on 2D-NMR (HSQC) spectra. The enantiomeric purity of the samples was
investigated by high-performance liquid chromatography (HPLC) using a
CHIRALPAK IG (4.6 mm $\times$ 250 mm) chiral analytical column. According to
these measurements, the RS-TFMOx mixture is racemic within margins of error
(S-TFMOx : R-TFMOX = 49% : 51%), and the enantiopure S-TFMOx or R-TFMOx have
an enantiomeric excess of approximately 95% (see Fig. 2 for details).

###  II.2 Experiments

The experiment was conducted during 8-bunch mode (pulsed operation) at
synchrotron SOLEIL (Saint-Aubin, France) at beamline SEXTANTS using Cold
Target Recoil Ion Momentum Spectroscopy (COLTRIMS, Doerner00 ; Ullrich03 ;
Jahnke04 ). The light’s helicity was switched every two hours between circular
left (CL) and circular right (CR). The samples were expanded through a nozzle
of 60 $\mu$m diameter, and the expanding gas was skimmed two times, resulting
in a free molecular gas jet of about 1 mm diameter. The jet was intersected at
right angle with the synchrotron-radiation beam. A closed-loop recycling
system was installed and used to reduce sample consumption Fehre20rsi . After
photoionization, the ionic fragments and electrons were accelerated by the
electric field of the COLTRIMS spectrometer in opposite directions towards
time- and position-sensitive detectors. A mesh-free spectrometer with
electrostatic lenses and field-free drift regions was employed in order to
improve the analyzer’s angle and mass resolutions, and to increase the overall
detection efficiency. The electric field at the point of interaction was set
to $E=116~{}$V/cm. The time- and space-focusing geometry of the COLTRIMS
spectrometer doerner1997nimb ; Schoeffler2011njp ; Schoeffler2013prl  resulted
in an solid angle of acceptance of 4$\pi$ for both, photoelectrons up to 15 eV
and ions up to 10 eV kinetic energy.

The electron energy calibration was carried out using a series of measurements
on argon atoms, ionizing both, the $2p_{1/2}$ and $2p_{3/2}$ shells. From this
series, a calibration function was created which links the measured electron
positions-of-impact and times-of-flight with the photoelectron energy. The
five photon energies for the TFMOx measurement (541.5, 542.5, 544.5, 546.5,
and 550.1 eV) were chosen such that O $1s$-photoelectrons with kinetic
energies of 3.1, 4.1, 6.1, 8.1, and 11.7 eV are created. The energy-
calibration procedure covered this range of electron kinetic energies. The
absolute and relative spatial orientation of the electron and ion detectors
was checked comparing the molecular-frame photoelectron angular distributions
of N2 K-shell electrons measured during the same beamtime to those reported in
Ref. 46. The particles’ impact positions and times-of-flight were measured by
position- and time-sensitive microchannel plate (MCP) detectors, using delay-
line anodes for position read-out (Roentdek GmbH). A HEX90 anode
($\varnothing$ 90 mm) equipped with an 80 mm diameter MCP Jagutzki02  was used
for the electron, and a HEX125L anode ($\varnothing$ 125 mm) equipped with an
efficiency-enhanced 75 mm diameter funnel MCP (Hamamatsu) Fehre18  for the ion
detection. The momentum vector of each fragment at the instant of ionization
is derived from their time-of-flight and impact position on the detector. Data
were recorded with typical rates of 6 kHz on the ion and 10 kHz on the
electron detector. The combined CF${}_{3}^{+}$($m/z=69$) –
C2Hi=1,2,3O+($m/z=41,42,43$) breakup channel contains roughly $2.5\times
10^{5}$ valid events for each enantiomer, light helicity, and photon energy.

In order to cross-check the experimental results presented here, we performed
a second short experiment at the permanent COLTRIMS end-station of the PIPE
instrument, located at beamline P04 of the synchrotron PETRA III (DESY,
Hamburg, Germany) Viefhaus13 . In this case, hexagonal delay-line detectors
with an active diameter of 80 mm (HEX90 anodes) were used for both, ion and
electron detection. The ion arm of that spectrometer was a 17 cm long
acceleration region without any meshes, while the electron arm was built in a
time-focussing geometry Wiley1955rsi . The dichroic parameter $\beta_{1}$ was
retrieved from the measured data by integrating over all possible
fragmentation channels of TFMOx. Therefore, it was sufficient to record
datasets for each polarization direction and photon energy for only 20
minutes. The data were recorded at detection rates of 15 kHz for the electron
detector. The photoelectron-energy calibrations as well as detector-
orientation calibrations were similarly performed as described above for the
measurements at synchrotron SOLEIL.

Figure 3: Retrieval of the anisotropy and dichroic parameters from the
measured PECD of R-TFMOx recorded at the O 1s-photoelectron energy of 11.7 eV.
A fit (red line) of the experimental data (circles with error bars) using Eq.
(2) yielded $\beta_{1}=0.76\%\pm 0.06\%$ and $\beta_{2}=1.14\pm 0.15$.

###  II.3 Data analysis

Within the electric-dipole approximation, the laboratory-frame photoelectron
angular emission distribution can be expanded in terms of Legendre polynomials
$P_{L}$ via the dichroic parameter $\beta_{1}$ and the anisotropy parameter
$\beta_{2}$. For randomly oriented molecules and in case of an ionization
employing circularly polarized photons (with helicity $\pm 1$), the
corresponding expansion reads:

$I_{\pm 1}(\theta)\propto
1\pm\beta_{1}P_{1}(\cos\theta)-\frac{1}{2}\beta_{2}P_{2}(\cos\theta),$ (1)

where the emission angle $\theta$ is defined with respect to the light
propagation direction Cherepkov82 . We utilize one of the standard definitions
of PECD Hergenhahn04 ; Tia17 , which is given by the normalized difference of
the angular distributions (1) recorded for left- and right-handed circularly
polarized light (referred to as $+1$ and $-1$, respectively):

$PECD(\theta)=\frac{I_{+1}(\theta)-I_{-1}(\theta)}{I_{+1}(\theta)+I_{-1}(\theta)}=\frac{\beta_{1}P_{1}(\cos\theta)}{1-\frac{1}{2}\beta_{2}P_{2}(\cos\theta)}.$
(2)

We used this equation to extract the parameters $\beta_{1}$ and $\beta_{2}$
from the measured $\theta$-dependent PECD. Figure 3 illustrates the fitting
procedure for a photoelectron kinetic energy of 11.7 eV using R-TFMOx
enantiomers as a target. In this example, the fitting procedure yields
$\beta_{1}=0.76\%\pm 0.06\%$ and $\beta_{2}=1.14\pm 0.15$. The uncertainties
represent the standard error of the fitting.

In order to check for possible systematic experimental errors, various subsets
of the experimental data were considered separately for this analysis. For
example, the high electric fields of the spectrometer degraded the time-of-
flight resolution and thus the resolution of the momentum component parallel
to the spectrometer axis. As one cross-check, only the central part of the
photoelectron time-of-flight distribution was examined and analyzed as a
subset. Furthermore, in order to exclude possible influences of inhomogeneous
detection probabilities, the electron detector was divided into its left and
right half along the light propagation direction, and each subset was analyzed
separately. The influence of possible ion feedback from the ion detector was
investigated with the help of applying different conditions on the ion time-
of-flight and by a selection of different ranges of sums of momenta for
various fragmentation channels. All these subsets of the data yielded (within
statistical errors) the same angular distribution parameters $\beta_{1}$ and
$\beta_{2}$.

###  II.4 Theory

The O $1s$-photoionization of TFMOx was described by the ab initio theoretical
approach used in our previous angle-resolved studies on MOx Tia17 ; Hartmann19
and TFMOx Ilchen17  molecules. The electronic structure and dynamics
calculations were carried out by employing the Single Center (SC) method and
code Demekhin11 ; Galitskiy15 , which provide an accurate description of the
partial angular momentum photoelectron continuum waves of ionized molecules.
The calculations were performed for the equilibrium molecular geometry of the
neutral ground electronic state of TFMOx. Details of the calculations can be
found in our previous study of TFMOx Ilchen17 . Therefore, only essential
differences between the previous and present calculations are discussed below.
While the calculations in Ref. 33 were carried out within the frozen-core
Hartree-Fock (FCHF) approximation, in the present work, the photoionization
transition amplitudes were computed within the relaxed-core Hartree-Fock
(RCHF) approximation, which accounts for the major effect of the monopole
relaxation of molecular orbitals induced by the creation of the inner-shell
vacancy. In addition, we increased here the SC expansions of the occupied
orbitals by partial harmonics with $\ell_{c},|m_{c}|\leq 79$ (as compared to
$\ell_{c},|m_{c}|\leq 59$ in Ref. 33), while those for the partial
photoelectron continuum waves were kept unchanged with
$\ell_{\varepsilon},|m_{\varepsilon}|\leq 29$.

The calculations were performed for different orientations of the molecular
frame (MF) with respect to the laboratory frame (LF, $Z_{LF}$ is defined by
the propagation direction of the circularly polarized light), which is given
by the two Euler angles $\alpha$ and $\beta$. Note, that the third angle
$\gamma$ (which defines a rotation around the laboratory $Z_{LF}$ axis) is
irrelevant in case of using circularly polarized light. In the experiment, a
breakup into two molecular fragments was examined. Thus, only a single
fragmentation axis was determined. Therefore, the corresponding differential
$PECD(\theta,\beta)$ was obtained by numerical integration of the
computational results over the orientation angle $\alpha$. Here, the
orientation interval $\alpha\in[0,2\pi)$ was covered in steps of
$\Delta\alpha=0.025\pi$. Since the fragmentation dynamics can be rather
complicated (and not all of the particles are detected in coincidence), it is
not straightforward to relate the fragmentation axis obtained in the
experiment from the asymptotic momenta of the two detected ions to a given
molecular orientation at the instant of ionization. Therefore, just as in our
previous work on uniaxially oriented MOx Tia17 , we used the molecular
orientation axis at the instant of the photoionization process as a free
parameter in the calculations, and searched for the best visual correspondence
between the computed and measured differential PECD. Details on this procedure
can be found in the supplemental material document of our previous work Tia17
.

##  III Results and discussion

###  III.1 Differential PECDs: MOx vs. TFMOx

Figure 4: Measured differential PECD of the uniaxially oriented enantiomers of
MOx (left column, data from Ref. 32) and TFMOx (right column, present work) as
a function of the photoelectron emission angle $\theta$ and the molecular
orientation angle $\beta$. The PECD maps were obtained for O
$1s$-photoionization yielding similar electron kinetic energies (11.5 eV for
MOx and 11.7 eV for TFMOx). Conceptually the same fragmentation channels are
chosen, i.e., CH${}_{3}^{+}$($m/z=15$) – C2H2O+($m/z=42$) for MOx and
CF${}_{3}^{+}$($m/z=69$) – C2Hi=1,2,3O+($m/z=41,42,43$) for TFMOx. According
to the CIP rule, opposite enantiomers of these two chiral molecules with
identical geometrical structure are compared in each row (see Fig. 1 for
details).

In our previous work on uniaxially oriented MOx Tia17 , a differential PECD
was studied for a photoelectron kinetic energy of 11.5 eV and two
fragmentation channels. One of the channels represents a single-bond breakage
with the loss of the methyl-group, i.e., a breakup into
CH${}_{3}^{+}$($m/z=15$) – C2H2O+($m/z=42$). Therefore, we decided to focus in
the present work on a similar fragmentation channel of TFMOx, which represents
a single-bond breaking with the loss of the trifluoromethyl-group. We observed
that additional losses of hydrogen atoms on the oxirane ring do not alter the
measured electron emission distributions. Therefore, we examined here the
differential PECD combining the datasets of the breakup channels
CF${}_{3}^{+}$($m/z=69$) – C2Hi=1,2,3O+($m/z=41,42,43$), which we designated
as $\mathbf{p}(69~{}\mathrm{amu})-\mathbf{p}(43~{}\mathrm{amu})$ for brevity.
The relative momentum (i.e., the difference of the momentum vectors) of those
two fragments defines a fragmentation axis of the molecule, and its
orientation Euler angle $\beta$ in the laboratory frame (with respect to the
light propagation direction) is thus known. The molecule is that way, however,
only uniaxially oriented which leaves one degree of freedom for its rotation
around this axis.

We first consider the photoelectron kinetic energy of 11.7 eV, which is very
similar to that used in Ref. 32 for MOx. The measured differential PECDs of
uniaxially oriented enantiomers of TFMOx are depicted in the right column of
Fig. 4 in panels (b) and (d). For some molecular orientations (i.e., some
spatial orientations of the molecular fragmentation), the differential PECD
reaches about 20%. Our measurements thus confirm the conclusion of Ref. 32
that fixing already one axis of a chiral molecule in space enhances the chiral
asymmetry by about an order of magnitude (as compared to about 1% observed for
the randomly oriented molecules shown in Fig. 3). The asymmetry switches its
sign upon exchanging of the enantiomers, which confirms the chiral origin of
the observed effect. This normalized difference obeys also the analytically
derived asymmetry property Tia17 :
$PECD(\pi-\theta,\pi-\beta)=-PECD(\theta,\beta)$, which implies a flip of its
sign upon rotation of this two-dimensional map by $180^{\circ}$ around the
normal to the picture plane. Please note that this equivalence of changing the
sign of the light helicity and exchanging the enantiomer holds only for
enantiomeric pure samples and perfectly circularly polarized light (see Ref.
10 for more details).

For comparison, panels (a) and (c) in the left column of Fig. 4 show the
differential PECD of MOx from Ref. 32. One can see that—for equal
fragmentation channels and similar photoelectron kinetic energies—the
differential PECDs observed for these two molecules are very similar in the
strengths and structures. It is surprising that exchanging the three hydrogen
atoms in the methyl-group by the much heavier scatterers (fluorine atoms) does
not significantly change the observed PECD landscape. In each row of this
figure, we compare the data for MOx and TFMOx molecules of similar geometrical
structures (therefore, opposite R- and S-enantiomers). One can see that the
differential PECDs of the two molecules of similar geometrical structures have
opposite-sign landscapes, whereas for identical R- or S-enantiomers of both
molecules the measured sign landscapes are very similar. This fact confirms
that the inner-shell PECD effect emerges as a result of a complex multiple
scattering of the emitted photoelectron wave by the molecular potential.
Indeed, similar stereo-descriptors of enantiomers assume a similar circular
arrangement of the stronger-to-weaker scatterers, which couple rotational
motion of a photoelectron, driven by the electric field vector of circularly
polarized light, with its translational motion (like a nut on a bolt Powis08b
; Tia17 ).

###  III.2 Energy dependence of PECD

Figure 5: Experimental (left column) and theoretical (right column)
differential PECD of uniaxially oriented R-TFMOx as a function of the
photoelectron emission angle $\theta$ and the molecular orientation angle
$\beta$, obtained for different kinetic energies (indicated in each panel) of
the O $1s$-photoelectron and the combined fragmentation channels
CF${}_{3}^{+}$($m/z=69$) – C2Hi=1,2,3O+($m/z=41,42,43$).

A full account of the differential PECD of uniaxially oriented TFMOx, measured
for the five aforementioned photoelectron kinetic energies, is depicted in the
left column of Fig. 5. For all energies, there are molecular orientations at
which the measured differential PECD reaches maximal values of about $\pm 16$
to $\pm 20\%$ [panels (a), (c), (e), (g), and (i) in the left column], which
are at least larger by a factor of 10 than the respective orientation-averaged
values $\beta_{1}$ (see Sec. III.3 and Fig. 6). Qualitatively, all measured
differential PECDs depict similar landscapes including their signs.
Nevertheless, a clear trend in the development of the measured differential
PECDs with the increase of the photoelectron kinetic energy can be noticed.
For the two lower kinetic energies of 3.1 eV in panel (a) and 4.1 eV in panel
(c), as well as 6.1 eV in panel (e), we observe a sequence of about
45∘-inclined positive-negative-positive-negative stripes (from top to bottom).
For the two higher kinetic energies of 8.1 eV in panel (g) and 11.7 eV in
panel (i), it develops in a clear sequence of three horizontal positive-
negative-positive islands at the top and an alternated sequence at the bottom.

In order to calculate the differential PECD of uniaxially oriented TFMOx, we
first assumed validity of the axial-recoil approximation Zare72 , for which
the momentum difference of two fragments represents an orientation of the
molecular axis in space at the instant of ionization (i.e., the molecule does
not rotate significantly during the Auger decay and subsequent fragmentation).
This approximation worked well in our previous study Tia17 . There, we found
that—for the single-bond breaking with the loss of the methyl-group—the
optimal molecular axis, which provides the best description of the experiment,
connects the oxygen atom with the carbon of the methyl-group (see supplemental
material document of Ref. 32). Starting with this assumption, we searched for
the optimal molecular fragmentation axis, which provides the best match
between theory and experiment for the differential PECDs of uniaxially
oriented TFMOx for all five electron energies.

As a result of our search, we find that the best visual correspondence between
the measured and computed differential PECD is provided by a fragmentation
axis which connects the oxygen atom with the CH2-group on the oxirane ring. It
should be stressed that all five photoelectron energies yielded very similar
optimal fragmentation axes which coincide within a solid angle of about 0.2
sr. The fact, that the axis obtained this way does not connect two charged
fragments (the trifluoromethyl-group and the oxirane ring) but lies almost in
the plane of the oxirane ring, indicates a breakdown of the axial-recoil
approximation for the TFMOx molecule. Indeed, a considerably larger mass of
the trifluoromethyl-group (as compared to the methyl-group in MOx) and a much
richer electronic structure of fluorine atoms (as compared to hydrogens) may
lead to the fact, that the centers of charge of the fragments do not coincide
with their centers of mass. In this case, an intricate rotation of two
fragments with respect to each other takes place in the course of a
considerably slower dissociation of TFMOx (as compared to the same
fragmentation channel of MOx with the much lighter methyl-group), and, as a
consequence, the differences of their asymptotic momenta do not represent the
molecular orientation at the instant of photoionization.

The results of our calculations of the differential PECD of R-TFMOx, obtained
for the optimal fragmentation axis, are depicted in the right column of Fig.
5. The calculations provide a semi-quantitative theoretical description of the
measured differential PECD. Indeed, the overall landscapes and signs of the
experimental differential PECDs, including the arrangement of stripes and
islands discussed above, are reproduced by the theory. Detailed distributions
of the experimental and theoretical asymmetries in their forms and strengths
are, however, somewhat different. This leads to the fact that averaging the
theoretical and experimental signals over molecular orientations yields
different $\beta_{1}$ values. This fact can be seen, e.g., for the
photoelectron energies of 8.1 and 11.7 eV in the forward emission direction.
In particular in the experiment, molecular orientations yielding positive
chiral asymmetries dominate over those which yield a negative PECD, while this
situation is opposite in the theory [cf., in Fig. 5 panels (g) and (h), and
separately panels (i) and (j), for the emission angles around
$\theta=0^{\circ}$ and orientation angles around $\beta=90^{\circ}$ with
highest contribution weights]. This difference is reflected in opposite signs
of the computed and measured dichroic parameters at 8.1 and 11.7 eV [see panel
(a) of Fig. 6 and its discussion in Sec. III.3].

###  III.3 Anisotropy and dichroic parameters

Figure 6: Dichroic parameter $\beta_{1}$ (a) and the anisotropy parameter
$\beta_{2}$ (b) as given by Eq. (1) for the O $1s$-photoelectrons of R-TFMOx.
Circles: experimental results recorded at the BL13-2 beamline of SSRL using a
VMI setup, as reported in Ref. 33. Stars: $\beta_{1}$ obtained by the present
reanalysis of the raw data underlying the values shown by circles from Ref. 33
(see Sec. III.3). Triangles: results of the present experiment performed at
beamline SEXTANTS of SOLEIL using COLTRIMS. Squares: corresponding $\beta_{1}$
values measured at beamline P04 of PETRA III. Dashed curve: frozen-core
Hartree-Fock calculations performed with the SC method as reported in Ref. 33.
Solid curve: present relaxed-core Hartree-Fock calculations performed with the
SC method.

The angular distribution parameters $\beta_{1}$ and $\beta_{2}$ of the O
$1s$-photoelectrons of TFMOx, measured at SOLEIL and defined by Eq. (1), are
depicted in Fig. 6 by black triangles. Panel (b) reveals that our experimental
results for the anisotropy parameter $\beta_{2}$ are in a good agreement with
the experimental and theoretical results reported in Ref. 33. The latter were
computed by using the SC method within the FCHF approximation and are shown as
a blue dashed curve. Contrary to that, however, our measured results for the
dichroic parameter $\beta_{1}$ disagree considerably with the experimental and
theoretical results reported in Ref. 33 in an energy range between 3 and 6 eV.
As a cross-check, the dichroic parameter $\beta_{1}$ obtained from a second
experiment (see Sec. II.2 for details) with a different setup at a different
synchrotron-radiation facility is shown in panel (a) of Fig. 6 by green
squares. It confirms the results from our main experiment (black triangles).

In order to understand the considerable deviation between the presently and
previously measured dichroic parameters $\beta_{1}$ seen in panel (a) of Fig.
6 in the energy range of 3–6 eV, in a joint effort with the authors of Ref.
33, we reanalyzed the original Stanford Synchrotron Radiation Lightsource
(SSRL) raw data recorded at the BL13-2 beamline with a VMI setup. The
reanalysis of raw images was performed independently for every run of a
certain photon energy, helicity of the light, and enantiomer, using the
following steps. The intensity in the surrounding of the momentum spheres was
subtracted as background. Then, the momentum sphere had to be centered. It
turned out that the center varied from run to run on the level of a few
percent of the radius of the momentum sphere giving rise for systematic errors
if not corrected for. This shift had not been accurately corrected for in the
original analysis in Ref. 33. An Abel transformation was applied to the
centered distributions followed by a transformation into polar coordinates.
Finally only those electrons, which have the proper kinetic energy, are
selected for further analysis. This is done by gating on a certain thickness
of the momentum sphere of the Abel-inversed data. The resulting subset of data
(intensity as function of the emission angle with respect to the light
propagation axis) is treated analogue as the COLTRIMS data (see Sec. II.3).
Results of this reanalysis of the SSRL data from Ref. 33 are shown in panel
(a) of Fig. 6 by pink stars (note that the reanalysis was performed only in
the relevant energy range of the present experiment). The specified
uncertainties correspond to the fit error from the extraction of the
$\beta_{1}$ parameter. A subdivision of each run from the SSRL measurement
into two independent data sets confirmed both, the retrieved values of the
$\beta_{1}$ parameter and the respective uncertainties. As one can see from
this figure, the dichroic parameters determined in all three experiments are
now consistent.

In order to find a possible reason for the disagreement between the previously
computed and all measured dichroic parameters in panel (a) of Fig. 6, we
cross-checked the computational results of Ref. 33. First of all, we
reexamined the equilibrium geometry of the molecule obtained in Ref. 33 on the
(2,2)-CASSCF/6-31+G(d,p) level of theory and used in the subsequent
calculations. For this purpose, we now performed its optimization with DFT
methods using the B3LYP hybrid functional with a 6-311+G(d,p) basis set. The
minimal-energy structure was confirmed by the absence of imaginary
frequencies. It turned out that both equilibrium structures differ only
negligibly and the angular distribution parameters recomputed here in the FCHF
approximation basically confirm those computational results of Ref. 33 (not
shown here for brevity). As a next step, we recomputed the angular
distribution parameters of the O $1s$-photoelectrons of TFMOx using the RCHF
approximation. It is known to be more accurate for core ionization than the
FCHF approximation used in the previous work Ilchen17 . The results of these
calculations are shown in Fig. 6 by red solid curves. As one can see, the
effect of the core relaxation caused by the created inner-shell vacancy does
not significantly change the energy dependence of the computed angular
distribution parameters, and the presently computed $\beta_{1}$ and
$\beta_{2}$ parameters agree well with the previously computed and measured
data from Ref. 33 (blue dashed curve and yellow circles). For the dichroic
parameter $\beta_{1}$, which is of primary interest here, relaxation of the
core has just somewhat reduced its maximal and minimal values. A reason for
the remaining disagreement between the theoretical and experimental dichroic
parameters remains unknown (see also discussion in Sec. IV).

##  IV Conclusions and outlook

A joint experimental and theoretical study of the photoelectron circular
dichroism (PECD) of the O $1s$-photoelectrons of uniaxially oriented
trifluoromethyloxirane (TFMOx) has been presented. We report the differential
PECD for the combined single-bond breakup channels CF${}_{3}^{+}$($m/z=69$) –
C2Hi=1,2,3O+($m/z=41,42,43$) with the loss of the trifluoromethyl-group at
five photoelectron kinetic energies. The differential PECD of TFMOx, measured
for a photoelectron kinetic energy of 11.7 eV, is very similar to that
observed in our previous work Tia17  for an almost equal (11.5 eV)
photoelectron kinetic energy of the O $1s$-photoelectrons and the
(conceptually) same fragmentation channel (single-bond breaking with the loss
of the methyl-group) of methyloxirane (MOx). The measured PECD landscapes
exhibit equal signs if one compares identical S- or R-enantiomers of those two
molecules which possess similar hierarchic stereo arrangement of scatterers,
and opposite signs for different enantiomers which have similar geometrical
structures.

The molecular fragmentation axis, found theoretically for this breakup
channel, does not connect the CF${}_{3}^{+}$ and C2H3O+ fragments, which is a
clear indication that for larger molecules, the connection between the
orientation of the molecule at the instant of photoionization and the
asymptotic momenta of the detected fragments have a rather intricate
interconnection. The computed differential PECDs reproduce those experimental
results at all photoelectron kinetic energies on a semi-quantitative level.
Having rather similar landscapes, the computed and measured differential PECDs
differ in their forms and maximal asymmetry strengths. As a consequence,
averaging of those rather strong asymmetries over all molecular orientations
yields different small values of the theoretical and experimental dichroic
parameters $\beta_{1}$. We performed a reanalysis of the raw data from the
experiment published in Ref. 33 leading to a revision of some of the data
points previously reported. These corrected previous experimental results
agree with new results from two different experiments performed at two light
sources in the present work.

Being very similar in their geometrical structures, S/R-MOx and R/S-TFMOx
molecules are very different in their chemical and, thus, electronic
structures. This could be a reason for a very different degree of agreement
between theory and experiment in our earlier study of MOx Tia17  and in the
present and earlier studies of TFMOx Ilchen17 . Because of considerably more
electrons in TFMOx, the respective calculations seem to be much more sensitive
to minor details of the molecular potential, especially for slow
photoelectrons. Experimentally, very different masses of constituent parts in
TFMOx make an unambiguous assignment of a molecular orientation axis to a
certain fragmentation channel impossible. In order to clarify the remaining
disagreement between the theoretical and experimental dichroic parameters
$\beta_{1}$, the O $1s$-photoionization of TFMOx needs to be reinvestigated
with alternative theoretical methods (e.g., TDDFT Stener05  or CMS-X$\alpha$
Powis00 ). Reference 28 reports a comprehensive theoretical study of the
photoelectron angular distributions from chiral derivatives of oxirane by the
former theoretical approach (note that TFMOx was not studied there). As one
can learn from Fig. 8 of Ref. 28, dichroic parameters computed for O
1s-photoelectrons of all considered oxirane derivatives exhibit a clear
enhancement to about 5–8% within the photoelectron kinetic energy interval of
2–8 eV, supporting our theoretical findings on TFMOx. Taking also into account
the robustness of the present and previous Ilchen17  calculations performed by
the SC method, a reason for the remaining disagreement between the theoretical
and experimental dichroic parameters $\beta_{1}$ can lie beyond the (fixed-
nuclei electric-dipole approximation) electronic structure calculations
performed here.

## Authors contribution

The experiment was prepared and carried out by G.N., K.F., F.T.,N.A., D.T.,
S.G., M.K., A.K., R.T., M.H., M.W., I.V.-P., G.K., J.S.,D.T., H.F., K.U.,
J.B.W., C.K.-W., L.M., J.V., A.K., T.J., and M.S.S.Samples were analyzed by
D.K., M.M., and R.P. Data analysis wasperformed by G.N., K.F., M.I., and
M.S.S. Theoretical calculationswere performed by N.M.N. and P.V.D. Initial
draft was createdby G.N., K.F., R.D., P.V.D., and M.S.S. All authors discussed
theresults and commented on the manuscript.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

This work was funded by the Deutsche Forschungsgemeinschaft (DFG)–Project No.
328961117–SFB 1319 ELCH (Extreme light for sensing and driving molecular
chirality). This research was also undertaken as part of the ASPIRE Innovative
Training Network, which has received funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie Grant
Agreement No. 674960. H.F. and K.U. acknowledge the XFEL Priority Strategy
Program of MEXT, the Research Program of ‘Dynamic Alliance for Open
Innovations Bridging Human, Environment and Materials’ and IMRAM project for
support. K.F. acknowledges support by the German National Merit Foundation.
M.I. acknowledges funding of a Peter-Paul-Ewald Fellowship by the Volkswagen
foundation. M.S.S. thanks the Adolf-Messer Foundation for financial support.
We thank the staff of SOLEIL for running the facility and providing beamtime
under project 20180746 and beamline SEXTANTS for excellent support. We also
thank the staff of PETRA III for running the facility and providing beamtime
under project H-20010092 and beamline P04 for excellent support. The work at
PETRA III was supported by BMBF.

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