# Science based AI model certification for new operational environments with
application in traffic state estimation

Daryl Mupupuni1, Anupama Guntu2, Liang Hong3, Kamrul Hasan4 and Leehyun Keel5
Department of Electrical and Computer Engineering, Tennessee State University
Nashville, TN 37209, USA
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###### Abstract

The expanding role of Artificial Intelligence (AI) in diverse engineering
domains highlights the challenges associated with deploying AI models in new
operational environments, involving substantial investments in data collection
and model training. Rapid application of AI necessitates evaluating the
feasibility of utilizing pre-trained models in unobserved operational settings
with minimal or no additional data. However, interpreting the opaque nature of
AI’s black-box models remains a persistent challenge. Addressing this issue,
this paper proposes a science-based certification methodology to assess the
viability of employing pre-trained data-driven models in new operational
environments. The methodology advocates a profound integration of domain
knowledge, leveraging theoretical and analytical models from physics and
related disciplines, with data-driven AI models. This novel approach
introduces tools to facilitate the development of secure engineering systems,
providing decision-makers with confidence in the trustworthiness and safety of
AI-based models across diverse environments characterized by limited training
data and dynamic, uncertain conditions. The paper demonstrates the efficacy of
this methodology in real-world safety-critical scenarios, particularly in the
context of traffic state estimation. Through simulation results, the study
illustrates how the proposed methodology efficiently quantifies physical
inconsistencies exhibited by pre-trained AI models. By utilizing analytical
models, the methodology offers a means to gauge the applicability of pre-
trained AI models in new operational environments. This research contributes
to advancing the understanding and deployment of AI models, offering a robust
certification framework that enhances confidence in their reliability and
safety across a spectrum of operational conditions.

###### Index Terms:

Artificial Intelligence (AI), Model Certification, Scientific Knowledge,
Physical Inconsistency, Traffic State Estimation (TSE)

##  I Introduction

Artificial intelligence (AI) has experienced tremendous growth and application
across several fields [1, 2, 3]. Application of AI has completely changed the
engineering sector over the last ten years by allowing engineers to design
creative ideas, solve complex problems, and expedite operations in several
engineering disciplines. For instance, various environmental engineering
problems were solved by AI based prediction models such as pollution control,
wastewater to improve the performance of several chemical and logical
treatment processes [4, 5]. AI is also useful in autonomous systems, proactive
maintenance in the aerospace and automobile industries [6]. AI is changing
engineering by automating tasks, improving designs, and enabling predictive
analysis. The integration of AI will continue driving innovation in the future
[7].

Although AI models like Artificial Neural Networks, Ensemble Approaches etc.,
continue to perform better than other models in many fields, but their
acceptance in delicate fields like finance and healthcare is questionable
because of the models’ difficulty in being understood and explained [8]. The
lack of interpretability or transparency also known as ”black-box models,” is
a key cause of concern in some AI models and it is challenging to comprehend
how they came to those conclusions [9]. The minimal amount of data available
for training AI models is another restriction. For AI models to learn
effectively, large and varied datasets are necessary to train the models.
Owing to the rarity of some events or the biases present in the data, it can
be difficult to obtain balanced datasets in some areas. As a result, the
model’s predictions may be biased or have skewed representation. This is also
reflected in the application of models developed from these datasets to be
employed in new operational situations [10].

Since AI models operate as black boxes, they must undergo certification to
ensure their effective performance across various environments. This
certification process can take various forms, including assessments that
verify adherence to minimum ethical standards and what we term as nuanced
evaluations [11]. Some important applications like robots in work spaces
shared with humans [12], social and environmental [13] need to be certified.
This type of certification study is also important in traffic state estimation
(TSE) which is not fully examined yet. The goal of the certification process
is to ascertain whether the TSE model developed under different traffic state
conditions can accurately anticipate in a new setting while upholding the
traffic conservation rule. By certifying the model, we hope to increase its
interpretability, strengthen its dependability, and demonstrate its
suitability for TSE example that occur in the real world [14]. The findings of
this study have implications for developing the TSE area and advance knowledge
of certifying deep learning models for safety-critical applications.

New operational environments refer to real-world settings where an AI system
is deployed that differ from those used to develop and test the system,
potentially exposing it to unfamiliar situations and inputs. In this
particular instance, the new environments are new traffic contexts. To fulfill
the urgent demand for safety, dependability, and transparency in AI
applications, deep learning models need certification [12]. The contributions
of this paper are the establishment of a thorough science-based AI model
certification methodology that ensures the resilience of deep learning models
in new operational situations by thorough examination and the incorporation of
known scientific concepts. This work enables one to ascertain the extent to
which AI models adapt to new complex and dynamic environments, reducing risks
and potential accidents by bridging the gap between conventional data-driven
approaches and science-based understanding. Additionally, the focus on
transparency and explainability encourages a better understanding of model
judgments, fostering public confidence and facilitating regulatory compliance
[15],[16]. We seek to increase the precision and dependability of TSE models,
resulting in safer and more effective transportation systems.

###  I-A Literature Review

Certifying AI models is pivotal when deploying them in unfamiliar operational
contexts, particularly where conventional data-driven methods may prove
inadequate. This is of paramount importance in applications like autonomous
vehicles, where certification ensures safety and dependability in diverse and
unpredictable conditions such as bad weather and unusual traffic patterns.
Similarly, in healthcare diagnostics, certification is necessary to handle
different patient populations, rare diseases, and unforeseen medical
situations, instilling confidence in the accuracy and clarity of AI-assisted
diagnoses especially in new operational environments [17], [18].

In natural disaster prediction and response artificial intelligence models are
used to anticipate natural disasters, such as earthquakes, hurricanes, or
wildfires, need to be certified to function well in disaster scenarios that
haven’t been encountered before [19]. These model certifications guarantees
their ability to adjust to shifting circumstances and deliver accurate early
warnings. In aerospace and aviation AI certification is needed to address the
difficulties of new operational contexts, such as air traffic changes, weather
disturbances, and emergency scenarios, certification is required for
autonomous drones and aircraft systems [15].

In the fields of industrial robotics and manufacturing, artificial
intelligence (AI) models must be certified to operate in dynamic production
environments with the possibility of varying raw material supply, equipment
breakdowns, or unforeseen production difficulties.AI models used for
environmental monitoring, such as those that predict weather, ocean currents,
or air quality, must be certified to handle a variety of geographic and
climatic conditions [20].

Certifying AI models is paramount in security and surveillance for operation
in uncontrolled environments, such as public spaces where unforeseen
occurrences and abnormalities may occur [21]. Financial risk assessment also
needs certification owing to financial markets being subject to abrupt
volatility and unforeseen events, AI models for risk assessment and prediction
in finance must be approved for new operational settings. For Power Grid
Management, AI models used to manage power grids and energy distribution
systems must be certified to react to unforeseen changes in energy supply,
demand, and potential grid failures [22]. For application in space
exploration, AI models used in space missions need to be certified to
withstand the harsh environments of uncharted space, where data may be
sporadic and unexpected [23].

Certification is necessary in order to check for several challenges that are
encountered when we are trying to employ pre-trained AI models in new
environments. Some of the challenges that may be encountered include models
that may be vulnerable to adversarial assaults, in which malicious inputs are
skillfully created to falsely estimate outcomes [21]. When pre-trained models
are used in new environments, privacy concerns may arise because vulnerable
data may unintentionally leak through model outputs. Ethics issues may also
arise if AI models are used in new environments without being properly vetted
or knowing their limitations, especially if important choices are based on
their outputs. Pre-trained models can be used in situations where they are not
in compliance with rules or legal requirements, which can result in legal
issues and liability issues [24]. Understanding how pre-trained models work
can be tricky because they are complex sometimes and difficult to interpret.

Another important application is traffic state estimation where this
certification is mandatory to test the models in different environments. It is
always difficult to acquire huge number of data in all environments since it
involves more cost to collect it. In that scenario, certification of AI model
in other environments is necessary. Thus, it is essential to carry out
thorough testing, validation, and fine-tuning of the pre-trained model in the
particular operating context in order to assure its safety, dependability, and
ethical use in order to reduce these risks. Additionally, keeping an eye on
the model’s performance and user input can aid in spotting and resolving
possible problems [21].

Pre-trained models can be quite effective and helpful in terms of time and
computational power, but they may not always function at their best in novel
and uncharted circumstances pausing risks of increased uncertainties. Pre-
trained models that are typically trained on a small number of datasets may
not have a thorough understanding of domains that are unrelated to their
training data. The availability of unbalanced datasets is a frequent problem
in traffic state estimation. A dataset that is uneven in the context of the
traffic state estimation example would be one where some traffic states or
situations are over represented while others are underrepresented or rare
[25]. The imbalance may be caused numerous things, including traffic patterns,
variable levels of congestion, or the scarcity of data from particular places
or times. For instance the traffic dataset that may be readily available is
mainly the NGSIM-CA data from California and the traffic state may not fully
represent the state of traffic in other states say Tennessee. The model may
generate inaccurate or deceptive results in new circumstances as a result of
its ignorance of the relevant domain. In some cases the input data in a
context for which the model has not yet been trained differs from that
context, the model may interpret the input incorrectly and produce false
results.

Pre-trained models may inherit biases from their training data, which is known
as bias amplification. These biases may intensify when applied in unknown
environments, producing unfair or discriminating outcomes. It is possible for
pre-trained models to over-fit to specific applications or datasets [26]. They
might not generalize when applied in novel and varied circumstances. Due to
domain variations, the model’s performance may suffer in new circumstances,
producing inconsistent results and decreasing user confidence.

It can be very beneficial to use pre-trained AI models in new environments,
but doing so requires careful preparation, ongoing monitoring, and the right
tweaks to make sure the model operates properly and safely. Striking a balance
between using the model’s pre-trained skills and tailoring it to the unique
demands of the operational setting is crucial.

###  I-B Methodology

This section of the paper provides background information on traffic flow
physics and its relevance to AI models, particularly in the context of traffic
data certification. It emphasizes the importance of understanding fundamental
traffic concepts, illustrated by the fundamental traffic diagram (Figure 1),
which depicts the linear relationship between traffic speed and density [27,
28, 29]. The conservation of vehicles principle, based on mass conservation,
is introduced as a fundamental physics concept governing traffic flow.
Additionally, mathematical models such as the Lighthill-Whitham-Richards (LWR)
model, based on fluid dynamics principles, are discussed for representing
traffic flow and analyzing congestion [30, 31].

Figure 1: Speed against density diagram in traffic flow

The section delves into the relationship between traffic density, flow rate,
and vehicle speed, highlighting the impact of these factors on traffic
conditions [32]. It also introduces equations describing traffic flow,
including the LWR equation. The Greenshields’ fundamental diagram is presented
to summarize the relationship between traffic density, flow rate, and free-
flow speed is mathematically defined by equations 1 and 2 [33, 34].

$q(x,t)=\frac{\partial N(x,t)}{\partial t}$ (1)

$\rho(x,t)=-\frac{\partial N(x,t)}{\partial x}$ (2)

The discussion transitions to the application of Artificial Intelligence (AI)
in traffic state estimation, categorizing it as a data-driven approach. The
potential of AI algorithms, particularly deep learning models, in forecasting
traffic conditions is emphasized, along with the importance of labeled
training data and computational resources [35, 36]. The structure of deep
learning neural networks and the role of the cost function in training are
explained. The paper provides equations for the mean squared error cost
function used in traffic state estimation for the assessment of the
metrics.The cost function, also known as the loss or objective function, is a
crucial mathematical measure in deep learning that quantifies the disparity
between a neural network’s expected output and the actual target values [31].

Accuracy in deep learning is explored, focusing on classification and
regression accuracy calculations and is defined by the equation.

$A_{D_{L}}=\frac{\sqrt{\sum_{x=1}^{X_{m}}\sum_{t=1}^{T_{n}}|\rho(x,t)-\hat{\rho}(x,t)|^{2}}}{\sqrt{\sum_{x=1}^{X_{m}}\sum_{t=1}^{T_{n}}|\rho(x,t)|^{2}}}$
(3)

Equation 3 is used to evaluate the performance of an AI model and the results
obtained from this equation are used to dteremine the errors associated with
the predictions.

Regression accuracy is specifically discussed in the context of traffic state
estimation, introducing the Root Mean Squared Error (RMSE) formula.

$C_{D_{L}}=\text{MSE}_{(\rho(x,t),\hat{\rho}(x,t))}=\frac{1}{N}\sum_{i=1}^{N}|\rho(x,t)-\hat{\rho}(x,t)|^{2}$
(4)

Equation 4 is used in the training process. N is the number of estimating
outputs in Mean Squared Error MSE. The estimated vehicle density
$\hat{\rho}(x,t)$ and the actual vehicle density, $\rho(x,t)$ at location x
and time t.

##  II Methodology Discussion: Model Assessment against Science Metrics Laws

To evaluate and certify AI models, the steps shown in figure 2 are followed.

Certifying the suitability of a Traffic State Estimation AI model for
deployment in new environments necessitates the integration of scientific
principles and traffic laws into the model assessment. This involves a
comprehensive examination of the model’s predictions in alignment with
established scientific principles, with a primary focus on the implementation
of the conservation of traffic. This approach serves to foster the development
of dependable models that strictly adhere to physical and mathematical
constraints, thereby establishing a foundational framework for the scientific
validation and acceptance of the AI model. Upon saving the model parameters, a
crucial step ensues, where the models undergo comparison with well-known
physical constraints and physics laws. This comparative analysis aims to
augment the reliability, robustness, and trustworthiness of deep learning
models designed for traffic state estimation.

The pivotal role played by the law of conservation of traffic in this process
is evident as predictions from the model are scrutinized for deviations from
ideality. Ensuring adherence to fundamental principles involves applying
traffic flow conservation rules or laws, validating that the predicted values
for density, velocity, and predicted flow rate align with the actual influx
and outflow of vehicles. The assessment extends to confirming the consistency
of predicted values for flow rate and density with the anticipated connections
suggested by the traffic flow model. An analysis of results is then conducted
to identify anomalies or deviations from the established rules of traffic
flow, scrutinizing expected density values and traffic variables. This
meticulous examination identifies areas, such as instances of congestion or
unexpected correlations between flow rate and density, facilitating a
comprehensive evaluation of the model’s adaptability to new environments.

##  III Experimental Configuration

Assessment of the performance of the AI system’s correctness and dependability
are required for science-based certification of AI models. [14]. This section
introduces a science-based certification framework for AI models, emphasizing
the evaluation of assessment in new environments. The certification process is
outlined in Figure 2.

Figure 2: Certification process

In this paper, the Lax-Hopf method is utilized for generating synthetic
traffic datasets under conditions where there is neither downstream flow nor
upstream flow. Moreover, it is specifically calibrated to conform to the
Greenshields’ model and traffic conservation laws.

Figure 3: Dataset generated with a $v_{f}$ =25

This experiment is done on a road segment defined by 1000 meters (x) observed
over a duration of 50 seconds (t). The traffic density values throughout the
domain were created by the Lax-Hopf method by setting up the following initial
density values $\rho(x,t)$ and initial flow rates.

Initial density: For interval at x in [0, 200] meters is 0.13 vehicle per
meter (veh/m), [200, 500] is 0.06 veh/m and [500, 1000] is 0.03 veh/m at t=0.

Free flow velocity ($v_{f}$) of 25 meters per second (55 miles per hour) and
jam density ($\rho_{m}$) of 0.15 veh/m were utilized. The dataset was created
at the time steps of 0.1 seconds and the distance step is 2 meters. The
generated synthetic dataset is displayed in figure 3.

The deep learning experimental set up uses a 10 hidden layer architecture with
40 neurons each. Two optimization algorithms were used that is the limited
Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) and Adam. The Tensorflow library for
machine learning was implemented in the development of our Traffic State
Estimation.The machine used to run the TSE was a 11th Gen Intel® Core™
i9-11900KF @ 3.50GHz × 16, with a Random Access Memory of 62.5GB.

##  IV Results and Discussions

This study conducted an insightful investigation into the conservation law of
traffic flow. It did so through a two-step modelling approach combined with
simulations of varied traffic conditions. First, a machine learning model was
trained on sensor data to estimate real-time traffic states like speed and
flow. Road geometry and counts then fed into a physics-based conservation
formula to calculate expected outflows. By generating new environments with
different $v_{f}$ values while keeping other parameters constant, we aimed to
assess the impact on the conservation law. While the conservation law provides
a fundamental framework for understanding traffic flow, it is crucial to
acknowledge potential deviations in real-world scenarios, influencing traffic
modeling and analysis.

The estimated results from the ML model were then directly compared against
the calculations derived from the established traffic flow physics model. The
differences or errors between the two results were analyzed. This section
further underscores the significance of certifying deep learning models with
physics laws to enhance predictions in Traffic State Estimation (TSE). The
deep learning model, trained in a specific environment and tested across
diverse settings, particularly aligns with scenarios featuring sensors
strategically positioned within the road network. This approach allows for the
observation and analysis of traffic variables at fixed locations over time,
mirroring real-world setups with fixed sensors.

The monitored traffic conditions and behaviors at predetermined positions
contribute to predicting traffic conditions in new or unobserved areas.
Through this comprehensive exploration, we emphasize the robustness of our
approach, bridging the gap between deep learning models and the underlying
physics governing traffic dynamics for more accurate TSE predictions. The
graphical representation in Figure 4 visually reinforces the trends observed,
validating the effectiveness of our methodology in real-world applications.

The deep learning (DL) model undergoes training using only 6% of the complete
dataset, specifically 15,000 samples, with a fixed $v_{f}$ of 25. The testing
phase employs the remaining dataset, encompassing data from various
environments. During training, we utilized 15,000 iterations with the Adam
optimizer and an additional 50,000 iterations with the L-BFGS optimizer.

The paper presented a systematic method to validate the ML model’s traffic
state estimations and identify any inconsistencies by benchmarking it against
well-defined traffic behavior according to fundamental physical conservation
laws. The conservation law provided a standardized numerical approach to
simulate ”ground truth” traffic states. The validation focused on deviations
from the Conservation Law by testing the model, which was trained on
$v_{f}$=25 meters/sec, across different free flow speed ($v_{f}$)
environments. Physics loss values were calculated by comparing the model’s
predictions to conservation law simulations for all test $v_{f}$ environments.
These physics loss values were then normalized to allow performance comparison
of the model across the various test $v_{f}$ conditions it was not directly
trained on. Figure 4 displays the normalized physics loss results from the
different $v_{f}$ environment tests, demonstrating the model’s generalized
performance when exposed to conditions beyond its training data.

Figure 4: Classification based on Normalized Physics Loss.

Figure 5 clearly shows the model’s Normalized Physics Loss (NPL) metric values
at varying $v_{f}$ test environments from 5 to 45. NPL effectively measures
the predictive error of the model in different conditions compared to its
training environment of 25.

$\displaystyle 0<\text{NPL}\leq 2$ ; Reuse $\displaystyle:$ $\displaystyle C$
$\displaystyle 2<\text{NPL}\leq 5$ ; Refine $\displaystyle:$ $\displaystyle R$
$\displaystyle\text{NPL}>5$ ; Discard $\displaystyle:$ $\displaystyle D$

Figure 5: Certification classification table

The certification category definitions utilize NPL to objectively classify a
model as reuse, refine or discard based on its robustness to domain shift.
Models with low NPL (0-2) show acceptable performance for direct reuse, while
those with intermediate NPL (2-5) may be refined with additional data or
introduction of physics informed correction. Models with high NPL ($>$5) face
significant domain shift and are discarded.

By applying these science-based metrics and definitions, the presented
approach offers a standardized, reproducible method to assess and certify AI
systems for safety and reliability when transferred to environments beyond
their training scope. This has tangible applications, for example in
certifying traffic prediction models across a range of real-world road
conditions.

The table and category definitions demonstrate how the proposed methodology
can guide careful model selection and development efforts to ensure models
perform robustly, benefiting deployment in critical domains like autonomous
vehicles. Overall, the paper presents an effective framework for certifying AI
according to scientific rather than just economic factors.

##  V Summary and Conclusions

This paper divulged a method of science-based certification of AI models to be
applied in new operational environments. We investigated the viability and
efficacy of certifying deep learning models for safety-critical traffic state
estimation using physics rules. Enhancing deep learning models’ dependability
and credibility in applications where precise traffic state estimation is
crucial for assuring safety was the goal and data is limited.

The findings of this work pave the door for additional approaches such as
physics-regulated deep learning model and transfer learning to better employ
the models in new operating conditions. Furthermore, in this paper we used
only synthetic data, future work should utilize the real world data for
betterment of certification process of the AI models.

## Acknowledgments

This work is supported in part by the National Science Foundation (NSF) under
Grant No. 2130990.

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# Towards CP Violation Studies on Superheavy Molecules: Theoretical and
Experimental Perspective

R. Mitraa,b, V. S. Prasannaac, R. F. Garcia Ruizd, T. K. Satoe, M. Abef, Y.
Sakemig, B. P. Das${}^{c}{{}^{,}}^{h}$ and B. K. Sahooa aAtomic, Molecular and
Optical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad
380009, India  bIndian Institute of Technology Gandhinagar, Palaj, Gandhinagar
382355, India  cCentre for Quantum Engineering, Research and Education, TCG
CREST, Salt Lake, Kolkata 700091, India  dMassachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  eAdvanced Science Research
Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan
fDepartment of Chemistry, Graduate School of Science, Hiroshima University,
1-3-2, Kagamiyama, Higashi-Hiroshima City, Hiroshima, Japan 739-8511  gCenter
for Nuclear Study, The University of Tokyo, Hongo, Bunkyo, Tokyo 113-0033,
Japan  hDepartment of Physics, Tokyo Institute of Technology, 2-12-1-H86
Ookayama, Meguro-ku, Tokyo 152-8550, Japan

###### Abstract

Molecules containing superheavy atoms can be artificially created to serve as
sensitive probes for study of symmetry-violating phenomena. Here, we provide a
detailed theoretical study for diatomic molecules containing the superheavy
lawrencium nuclei. The sensitivity to time-reversal violating properties was
studied for different neutral and ionic molecules. The effective electric
fields in these systems were found to be about 3-4 times larger than other
known molecules on which electron electric dipole moment experiments are being
performed. Similarly, these superheavy molecules exhibit an enhancement of
more than 5 times for parity- and time-reversal-violating scalar-pseudoscalar
nucleus-electron interactions. We also briefly comment on some experimental
aspects by discussing the production of these systems.

##  I Introduction

Recent advances in studies on heavy molecular systems are enabling new
opportunities in fundamental physics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13]. Symmetry-violating properties of nuclei and fundamental particles can be
highly enhanced in certain molecules. Precision experiments with ThO
molecules, for example, have provided the most stringent bound to electric
dipole moment (EDM) of an electron [1], constraining the existence of new
physics at the TeV scale [14]. The search for a permanent EDM of an elementary
particle has attracted a great deal of experimental and theoretical attention
as it is highly sensitive to new sources of time-reversal (T) violation
(equivalent to CP violation) [15], knowledge of which can be useful in
explaining the matter-antimatter asymmetry in the Universe [16, 17].

In molecules, the sensitivity to symmetry-violating phenomena scales with the
mass and charge of the nuclei [18]. Hence, molecules containing super-heavy
nuclei could serve as particularly sensitive laboratories. Future studies of
these systems can open up new opportunities in fundamental physics and
chemistry. There have been recent advances in the search for the electron EDM
using paramagnetic molecules, which would be a signature of CP violation
beyond the standard model of elementary particles. Moreover, spectroscopic
results from studies of superheavy elements have recently been reported, and
in view of these developments, it is desirable to investigate EDMs using
molecules with superheavy atoms. However, our experimental knowledge of such
rare molecules is in its infancy, and theoretical developments are critical to
motivate and guide the experimental progress [19, 20].

Within a molecule, the magnitude of the intrinsic electric field that an
electron with an EDM experiences, due to the other electrons and nuclei, can
be viewed as an effective electric field ($\mathcal{E}_{\mathrm{eff}}$) [18].
Because of their large atomic number, super-heavy radioactive elements are
expected to exhibit a substantial enhancement in their
$\mathcal{E}_{\mathrm{eff}}$, since one expects relativistic and electron
correlation effects to be prominent. Indeed, effective electric fields of CnH
[21, 22], CnF [23], LrO, NoF, RfN, E120F, and E121O [24] have been calculated,
and asserts the above statement. In this work, we focus our studies on
diatomic molecules containing Lr atoms, i.e. LrO, LrF+, and LrH+. Because of
short half-lives and the low production rates, superheavy elements, including
Lr, need to be handled on a single-atom scale. There are fourteen different
isotopes of Lr, of which we propose to employ 256Lr (half-life $\approx$ 27 s)
that has been used in several studies of atomic properties. A purified single
ion-beam of 256Lr, synthesized in the 249Cf (11B, $4n$) reaction, has been
successfully produced by using the ISOL (Isotope Separator On-Line) system at
the Tandem accelerator facility of Japan Atomic Energy Agency (JAEA) [25]. We
explore a possible alternative scheme to produce Lr atoms. Our work also aims
to extend the studies of molecular ions, which could enable precision
measurements with just a single molecular ion [28, 26, 27].

The manuscript is organized as follows: Sec. II presents the general theory of
molecular EDMs due to electron EDM and scalar-pseudoscalar (S-PS) nucleus-
electron interactions. The many-body method employed in the present work is
explained in Sec. III. Our theoretical results and discussions about the
sensitivity of these molecules to symmetry-violating properties are presented
in Sec. IV.

##  II Theory

Theoretical studies for the sensitivity to symmetry-violating properties have
been performed for the diatomic molecules LrO, LrF+, and LrH+. We employ a
4-component relativistic coupled-cluster (RCC) method for this purpose. The
fully relativistic aspect becomes especially relevant, as theoretical
calculations have predicted that the Lr atom would have a configuration
different from that expected by a non-relativistic treatment,
[Rn]$5f^{14}6d^{1}7s^{2}$, due to strong relativistic effects [29]. The
[Rn]$5f^{14}7s^{2}7p_{1/2}$ configuration would be most probable for the Lr
atom according to the measurement of the first ionization potential
measurement [25]. We employ the configuration for calculations in present
work. Since LrO, LrF+ and LrH+ have one unpaired electron each, these
molecules are sensitive to both the electron EDM and nucleus-electron S-PS
interactions [30]. Thus, these T-odd sources can induce an energy shift in the
molecular levels given by

$\displaystyle\Delta E\simeq-d_{e}\mathcal{E}_{\mathrm{eff}}+k_{s}W_{s},$ (1)

where the first term on the right hand side of Eq. (1) describes the
contribution from electron EDM and the second term shows that from the S-PS
interaction. In the above equation, $k_{s}$ is the coupling parameter for S-PS
interaction, with the accompanying quantity $W_{s}$ being the analogue of
$\mathcal{E}_{\mathrm{eff}}$, and like the effective electric field, it too
can only be calculated using many-body theory.

The interaction Hamiltonian due to electron EDM in a molecule is given by [31]

$\displaystyle H_{EDM}$ $\displaystyle=$
$\displaystyle-2icd_{e}\sum_{j=1}^{N_{e}}\beta\gamma_{5}p_{j}^{2}$ (2)
$\displaystyle=$ $\displaystyle\sum_{j=1}^{N_{e}}h_{EDM,j},$

where $c$ is the speed of light, $\beta$ is a Dirac matrix, $\gamma_{5}$ is
the product of Dirac matrices, $p_{j}$ is the momentum operator corresponding
to the $j^{th}$ electron, and $N_{e}$ is the total number of electrons in the
system. Assuming that the shift in energy is only due to the electron EDM, the
expression for $\mathcal{E}_{\mathrm{eff}}$ can be obtained as

$\displaystyle\mathcal{E}_{\mathrm{eff}}=-\frac{1}{d_{e}}\frac{\langle\Psi\arrowvert
H_{EDM}\arrowvert\Psi\rangle}{\langle\Psi\arrowvert\Psi\rangle},$ (3)

where $\arrowvert\Psi\rangle$ is the ground state wave function of the
molecule. Similarly, the S-PS interaction Hamiltonian manifested at molecular
level is given by [32, 31, 33] $H_{S-PS}=\sum_{A=1}^{N_{nuc}}H_{S-PS,A}$, with

$\displaystyle
H_{S-PS,A}=i\frac{G_{F}}{\sqrt{2}}k_{s,A}Z_{A}\sum_{j=1}^{N_{e}}\beta\gamma_{5}\rho_{A}(r_{A_{j}}),$
(4)

where $A$ denotes the $A^{th}$ nucleus, $Z_{A}$ is the nuclear charge of the
$A^{th}$ nucleus, $\rho_{A}$ the nuclear charge density, $G_{F}=2.219\times
10^{-14}$ (in atomic units (a.u.)) is the Fermi coupling constant, $N_{nuc}$
is the total number of nuclei in the molecule, and $k_{s,A}$ is the S-PS
coupling coefficient of the corresponding nucleus, $A$. We have accounted here
contribution only from the heaviest atom (since the contribution from the
heavier of the two atoms dominates) and denote $k_{s,A}$ as $k_{s}$ for
brevity. The quantity of theoretical interest in the S-PS interaction,
$W_{s}$, are given by

$\displaystyle W_{s}=\frac{1}{k_{s}}\frac{\langle\Psi\arrowvert
H_{S-PS}\arrowvert\Psi\rangle}{\langle\Psi\arrowvert\Psi\rangle}.$ (5)

We consider the Gaussian nuclear charge distribution to define nuclear charge
density, given by
$\rho_{A}(r)=\left(\frac{\eta_{A}}{\pi}\right)^{\frac{3}{2}}e^{-\eta_{A}r^{2}}$,
with $\eta_{A}=\frac{3}{2}(R_{A}^{rms})^{-2}$, where $R_{A}^{rms}$ is the root
mean squared nuclear charge radius of the $A^{th}$ nucleus.

Another property that plays an important role in determining the statistical
sensitivity of a molecule in an EDM experiment is the molecular permanent
electric dipole moment (PDM). The PDM can be evaluated as the expectation
value of the electric dipole operator, given as

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\sum_{A=1}^{N_{nuc}}Z_{A}R_{A}-\sum_{i=1}^{N_{e}}r_{i}$ (6)
$\displaystyle=$ $\displaystyle\mu_{nuc}-\mu_{e},$

where $R_{A}$ and $r_{i}$ are respectively the position of the $A^{th}$
nucleus and $i^{th}$ electron with respect to the origin. Since we employ the
Born-Oppenheimer approximation [34], where the nuclei are clamped, $R_{A}$ are
constants, and are determined from the minimum of the potential energy surface
for a given molecule. In the expression, $\mu_{nuc}$ and $\mu_{e}$ are the
nuclear and electronic contributions to PDM, respectively. Since the
investigated molecules are diatomic systems, each of the systems have only one
equilibrium bond length ($R_{e}$). We have set the Lr atom as the origin to
define $R_{e}$. Therefore, the PDMs of the charged systems are specified with
Lr as the origin. We add here that it is straightforward to obtain the PDM at
any other origin, including setting it at the centre of mass of the molecular
system, as the property simply scales with change in origin.

##  III Method Of Calculation

The molecular parameters were calculated by solving the many-body wave
function of the molecular system within a relativistic framework. To account
for the relativistic and electron correlation effects rigorously, we employ
here the RCC theory, expressing the wave function as [35]

$\displaystyle\arrowvert\Psi\rangle=e^{T}\arrowvert\Phi_{0}\rangle,$ (7)

where $\arrowvert\Phi_{0}\rangle$ is the Dirac-Hartree-Fock (DF) wave
function, and $T$ is the cluster operator. The latter is responsible for
generating particle-hole excitations which arise out of the DF state due to
the residual Coulomb interaction that is ignored in the evaluation of the DF
wave function. $T$ is expressed as

$\displaystyle T=T_{1}+T_{2}+T_{3}+\cdots+{T_{N}}_{e},$ (8)

where the operator $T_{i}$ accounts for all possible $i$ particle- $i$ hole
excitations. Due to the prohibitively expensive cost associated with including
all the excitations, we adopt the common practice where one restricts $T$ to
include only singles and doubles excitations (RCCSD method), in which the
excitation operators are defined using the second quantization operators as

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\sum_{i,a}t_{i}^{a}a_{a}^{\dagger}a_{i},$ (9)

and

$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{i,j,a,b}t_{ij}^{ab}a_{a}^{\dagger}a_{b}^{\dagger}a_{i}a_{j},$
(10)

where the indices $i$, $j$, $k$, $\cdots$ etc. denote the occupied orbitals,
and $a,b,c,\cdots$ identify the virtual orbitals, $t_{i}^{a}$ is the amplitude
for a single excitation from $i^{th}$ occupied orbital to $a^{th}$ virtual
orbital, whereas, $t_{ij}^{ab}$ is the amplitude of the doubles excitation
from the occupied $i$ and $j$ orbitals to the virtual $a$ and $b$ orbitals,
respectively.

Once the ground state wave function of a molecule is obtained, we calculate
the properties of interest by using the expectation value approach. The
expectation value of an operator $O$ in the RCC method is given by

$\displaystyle\langle O\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle\Phi_{0}\arrowvert
e^{T^{\dagger}}Oe^{T}\arrowvert\Phi_{0}\rangle}{\langle\Phi_{0}\arrowvert
e^{T^{\dagger}}e^{T}\arrowvert\Phi_{0}\rangle}$ (11) $\displaystyle=$
$\displaystyle\langle\Phi_{0}\arrowvert
e^{T^{\dagger}}Oe^{T}\arrowvert\Phi_{0}\rangle_{l},$

where the subscript $l$ means that only the linked terms contribute. In the
RCCSD method, since $T\simeq T_{1}+T_{2}$, we have

$\displaystyle\langle O\rangle$ $\displaystyle=$
$\displaystyle\langle\Phi_{0}\arrowvert
O\arrowvert\Phi_{0}\rangle_{l}+\langle\Phi_{0}\arrowvert(OT_{1}+h.c.)+T_{1}^{\dagger}OT_{1}+T_{2}^{\dagger}OT_{2}$
(12)
$\displaystyle+\frac{1}{2}(T_{1}^{\dagger}OT_{1}^{2}+h.c.)+\frac{1}{2}(T_{2}^{\dagger}OT_{1}^{2}+h.c.)$
$\displaystyle+(T_{2}^{\dagger}OT_{1}T_{2}+h.c.)+\cdots\arrowvert\Phi_{0}\rangle_{l},$

where h.c. stands for the hermitian conjugate term [36, 37]. It can be noted
that $OT_{2}$ and its h.c. term do not give us fully contracted terms owing to
the one-body form of the operators, which describe the properties that are
investigated here, i.e., $O=\mu$, $O=H_{EDM}$ and $O=H_{S-PS}$ for the
evaluation of $\mu$, $\mathcal{E}_{\mathrm{eff}}$ and $W_{s}$ respectively. In
the above expression, the first term gives the DF value and the $OT_{1}$ term
includes electron correlations arising through the core-polarization and pair-
correlation effects to all orders. Therefore, it contributes dominantly to the
electron correlation effects.

|  |
---|---|---
(a) | (b) | (c)
Figure I: Potential energy curves (PEC) of neutral LrO (sub-figure (a)), as
well as LrF+ (sub-figure (b)), and LrH+ (sub-figure (c)) molecular ions with a
range of typical molecular bond-lengths ($R$) using the RCCSD(T) method. The
plot shows that LrO can form a stable molecule around the equilibrium bond-
length $R_{e}\simeq 3.46$, while both the ions possess their minima around
$R_{e}\simeq 3.56$. All the values are given in a.u..

##  IV Results and Discussion

At the outset, it is critical to find if the systems of interest can form
bound states. The ground electronic state energies at several bond lengths
were calculated to construct the potential energy curve (PEC) of each
molecule. We then identify the minimum of the PEC, whose corresponding bond-
length provides the molecular $R_{e}$. We carry out this procedure for LrO,
LrF+, and LrH+ (Fig. I), while ensuring that for each molecule, we choose more
grid points around the minimum, to pin-point $R_{e}$ to an accuracy of two
decimal places in a.u.. We employ the RCCSD method with the inclusion of
partial triple excitations in a perturbational manner, using the Dirac 18
package [38, 39]. This approach is referred to as the RCCSD(T) method. We use
Dyall’s triple-zeta (TZ) v3z [40] basis sets. For lowering the computational
requirements while compromising little on accuracy, we have cut off the high-
lying virtual orbitals with energies above 2000 a.u.. The value of $R_{e}$ for
LrO was found to be 3.46 a.u., while that of LrF+, and LrH+ were found to be
3.56 a.u.. The PECs of each of the molecules exhibit a smooth trend with a
clear global minimum (see Figure I).

After finding the equilibrium bond-lengths of the investigated molecules, we
calculate the other properties of interest. For this purpose, we use Dyall’s
quadruple-zeta (QZ) v4z [40] basis sets. We employ the UTChem [41, 42] package
for DF calculations and for atomic orbital to molecular orbital integral
transformations, and in tandem, use the Dirac08 package for RCC calculations
[43]. We finally use our expectation value code to evaluate the values of the
properties [44]. We present the calculated values of
$\mathcal{E}_{\mathrm{eff}}$, $W_{s}$ and $\mu$ of LrO, LrF+, and LrH+ from
the DF and RCCSD methods in Table 1. We also compare our results for LrO with
the only available values in literature from Refs. [24, 45]. In Ref. [24], the
authors have used (37$s$, 34$p$, 14$d$, and 9$f$) uncontracted Gaussian type
functions for Lr, and a decontracted atomic natural orbital (ANO) basis set of
TZ quality for O. They performed their calculations by employing complex
generalised Hartree-Fock (cGHF) as well as complex generalised Kohn-Sham
(cGKS) theories. They obtained an $R_{e}$ of 3.51 a.u. and 3.53 a.u. with cGHF
and the cGKS approaches, respectively, which is in reasonably close proximity
to our estimated value of 3.46 a.u.. The value of $\mathcal{E}_{\mathrm{eff}}$
using cGHF method came out to be 322.58 GV/cm, while the cGKS method yielded
250.21 GV/cm. Our calculation, using the RCCSD method and with Dyall’s QZ
bases for Lr: (37$s$, 34$p$, 24$d$, and 14$f$) and O: (18$s$ and 10$p$), gives
an $\mathcal{E}_{\mathrm{eff}}$ of 258.92 GV/cm, and is in better agreement
with their results from the cGKS method than the cGHF approach. In Ref. [45],
the authors use analytic first derivatives for X2C (exact 2-component) CCSD
and CCSD(T) methods, and employed TZ quality basis sets. Further, they froze
several of the occupied orbitals in their computations, and obtain an
effective electric field of 263.9 GV/cm with CCSD and 246.5 GV/cm with the
CCSD(T) approaches. Although their CCSD results are in reasonable agreement
with our CCSD results obtained using all-electron fully relativistic CCSD
calculations and with a QZ basis, it could be fortuitous. This is evident from
the disagreement in the correlation trends in Ref. [45], where the DF result
is greater than that CCSD counterpart. In Table 1, we have also compared our
results with the corresponding values of ThO, where the most accurate EDM
measurement is available, and HgF, which possesses the largest
$\mathcal{E}_{\mathrm{eff}}$ estimated thus far for non-super-heavy systems.
The Lr molecules were found to have values of $\mathcal{E}_{\mathrm{eff}}$
that are 3-4 times larger than ThO and HgF. Similarly, the values of $W_{s}$
were found to be about 5 times larger.

Table 1: Calculated values of $\mathcal{E}_{\mathrm{eff}}$, $W_{s}$ and $\mu$
for LrO, LrF+, and LrH+ using the RCCSD method, and comparison with literature
values wherever they are available. We also compare these values with
calculations of the corresponding quantities for ThO and HgF (see the text for
further details).

Molecule |  $\mathcal{E}_{\mathrm{eff}}$ (GV/cm) |  $W_{s}$ (kHZ) |  $\mu$ (D) | Reference
---|---|---|---|---
LrO | 258.92 | 1032.23 | 4.58 | This work
| 250.21 | 938a |  | [24]
| 246.5 |  |  | [45]
LrF+ | 246.31 | 977.37 | 12.29 | This work
LrH+ | 343.38 | 1375.62 | 11.05 | This work
ThO | 87 | 232a | 4.27 | Ref. [2]
HgF | 115.42 | 264.7 | 2.61 | Ref. [8]

aIn Refs. [24] and [2], the authors give 469 and 116 kHz
---
for $W_{s}$, respectively, and we have multiplied their values
by a factor of two in the table for reasons pertaining to
consistency in notation.

Table 2: Contributions from the individual terms of the expectation value expression in the RCCSD method to $\mathcal{E}_{\mathrm{eff}}$, $W_{s}$ and $\mu$ for LrO, LrF+, and LrH+. The table shows the trends in different contributions between $\mathcal{E}_{\mathrm{eff}}$ and $W_{s}$.  Term |  $\mathcal{E}_{\mathrm{eff}}$ (GV/cm) |  $W_{s}$ (kHz) |  $\mu$ (D)
---|---|---|---
| LrO | LrF+ | LrH+ | LrO | LrF+ | LrH+ | LrO | LrF+ | LrH+
$O$ | 235.16 | 215.20 | 283.59 | 944.28 | 864.94 | 1135.23 |  $-$898.40 |  $-$828.32 |  $-$919.79
$OT_{1}+$h.c. | 65.02 | 35.52 | 76.06 | 254.36 | 140.13 | 300.45 |  $-$1.62 |  $-$0.56 |  $-$0.72
$T_{1}^{\dagger}OT_{1}$ | $-26.28$ |  $-$2.24 |  $-$9.56 |  $-$105.67 |  $-$9.02 | -38.91 | $-0.59$ |  $-$0.09 |  $-$0.14
$T_{1}^{\dagger}OT_{2}+$h.c. | $-8.11$ |  $-$2.87 | 0.95 |  $-$33.28 |  $-$11.82 |  $-$32.96 | 0.58 | 0.14 | 0.12
$T_{2}^{\dagger}OT_{2}$ |  $-$8.54 |  $-$1.70 |  $-$8.21 |  $-$34.43 |  $-$6.86 | 3.73 |  $-$0.48 |  $-$0.14 |  $-$0.14
Others | 1.67 | 2.4 |  $-$0.45 | 6.97 | 6.7 | 1.38 |  $-$0.44 | 0.08 | 0.02
Nuclear term | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 905.56 | 841.27 | 931.73
Total | 258.92 | 246.31 | 343.38 | 1032.23 | 977.37 | 1375.62 | 4.58 | 12.29 | 11.05

In Table 2, we present contributions to the values of
$\mathcal{E}_{\mathrm{eff}}$, $W_{s}$ and $\mu$ for LrO, LrF+, and LrH+ from
different terms of the RCCSD method, given in Eq. (12). The first term
corresponds to the DF value, while other terms represent correlation
contributions. As the table shows, $OT_{1}+$h.c. terms contribute the most to
the correlation effects. We note at this point that $OT_{1}$ primarily
contains in it correlation effects arising from interaction of pairs of
electrons. $T_{1}^{\dagger}OT_{1}$ also contains effects involving pairs of
electrons, but the interplay between the EDM and Coulomb interactions are more
complex. The next leading-order contributions arise from
$T_{1}^{\dagger}OT_{1}$. Albeit the terms related to $T_{2}$ operator give
comparatively small contributions, they are non-negligible. In fact, for LrO,
a sizeable amount of the contributions from $OT_{1}+$h.c. terms are cancelled
out by the other terms that are linear in $T$. The row denoted as ‘Others’
show that the non-linear terms are negligible even for the considered super-
heavy systems. Comparisons between the ratios of the magnitudes of
$OT_{1}+$h.c. and DF values of $\mathcal{E}_{\mathrm{eff}}$, $W_{s}$, and
$\mu$ in LrO, which come out to be $\approx 0.28$, $\approx 0.27$ and $\approx
0.23$, respectively, indicate that the electron correlation effects are almost
equally important in all these quantities. Similarly, for
$\mathcal{E}_{\mathrm{eff}}$, $W_{s}$, and $\mu$, we find these ratios as
$\approx 0.16$, $\approx 0.16$ and $\approx 0.27$, respectively, for LrF+, and
$\approx 0.27$, $\approx 0.17$ and $\approx 0.26$, respectively, for LrH+.
These results suggest that the electron correlation trends are almost similar
in both molecular ions.

We now briefly comment on the molecular orbital information in the chosen
systems. We expect that the singly occupied molecular orbital (SOMO) electron
is localized in the $s$ orbital of Lr from the following reasoning: as Lr has
$7s^{2}7p^{1}$ configuration, while O, H+, and F+ are expected to pull two
electrons towards themselves owing to their larger electronegativity, thus
leading to Lr $7s^{1}$. We verified this reasoning explicitly by examining the
atomic components of the SOMO for all three systems, and indeed found that the
atomic orbitals from Lr provide the dominant contributions. We also carried
out population analysis, and found that in LrF+, for example, the SOMO is
predominantly made out of the $s$ function of Lr (0.86), followed by its $d$
(0.09) and $p$ functions (0.0388).

As seen in Table 2, the major contribution to the properties of interest in
this work comes from the DF part. Therefore, we take a closer look at the DF
contribution, in order to understand the possible reasons for observing large
values of $\mathcal{E}_{\mathrm{eff}}$ and $W_{s}$ in the studied superheavy
molecules, as compared to other systems. Typically, in these molecules, the
heavier atom provides most of the contributions to
$\mathcal{E}_{\mathrm{eff}}$ and $W_{s}$. Due to the short-range and odd-
parity nature of the scalar interaction Hamiltonians, the $s_{1/2}$ and
$p_{1/2}$ orbitals generally contribute predominantly to
$\mathcal{E}_{\mathrm{eff}}$ and $W_{s}$. It is known that relativistic
effects deform the inner core orbitals, $s_{1/2}$ and $p_{1/2}$, strongly in
the heavier atomic system. Thus, it is anticipated that these orbitals can
strongly influence the $\mathcal{E}_{\mathrm{eff}}$ and $W_{s}$ values in LrO,
LrF+, and LrH+. We explicitly verify this argument by decomposing the DF
contribution to $\mathcal{E}_{\mathrm{eff}}$ (we do not repeat the analysis
for $W_{s}$, as we expect for it similar trends as the effective electric
field) for all the three systems as

$\displaystyle\mathcal{E}_{\mathrm{eff}}^{DF}$ $\displaystyle=$
$\displaystyle\frac{1}{d_{e}}\langle\Phi_{0}|H_{EDM}|\Phi_{0}\rangle$ (13)
$\displaystyle=$
$\displaystyle\frac{1}{d_{e}}\sum_{j}\langle\phi_{j}|h_{EDM}|\phi_{j}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{d_{e}}\langle\phi_{v}|h_{EDM}|\phi_{v}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{d_{e}}\sum_{k}\sum_{l}C_{k}C_{l}\langle\chi_{v,k}|h_{EDM}|\chi_{v,l}\rangle.$

In above expression, the sum over all MO contributions boils down to only the
valence molecular orbital (SOMO, denoted as $v$ in the above set of equations)
term due to the fact that contributions from the orbitals with opposite spin
components of the closed-shell configuration cancel out each other. In the
last line, we have expanded the valence molecular orbital as the sum of atomic
orbitals (AOs), $|\chi_{v,i}\rangle$, where $i$ can be $l$ or $k$. Further
details about this decomposition of SOMO to AOs can be found in, for example,
Ref. [46]. Note that the SOMO contains contributions from both the constituent
atoms of a molecule. Of all the terms in Eq. (13), the contributions from the
$s$ and $p_{1/2}$ orbitals of Lr dominates, and accounts for about 232, 214,
and 281 GV/cm, for LrO, LrF+, and LrH+, respectively, as shown in Fig. II.
Note that the DF values for these systems are 235, 215, and 284 GV/cm for LrO,
LrF+, and LrH+, respectively. The other contributions, such as those from
$p_{3/2}$ and $d_{3/2}$ of Lr, $s$ and $p_{1/2}$ of the lighter atom etc, are
less than 1 GV/cm. It is also worth noting that while the DF values of LrO and
LrH+ themselves are different only by about 50 GV/cm, the total effective
electric fields are apart by over 80 GV/cm. This is attributed to the
significant cancellation between the $OT_{1}$ \+ h.c. and the
$T_{1}^{\dagger}OT_{1}$ terms in LrO, as shown in Table 2.

Figure II: Bar plots showing effective electric fields of LrO, LrF+, and LrH+
at three levels: the $s-p_{1/2}$ mixing contributions from Lr at the DF level
of theory (as red bars), the DF values themselves (as green bars), and the
total values (in blue). All units are in GV/cm.

Using the values of PDM and bond-length of a molecule, we estimate the
polarizing electric field for that system, which is given by
$E_{pol}=\frac{2B}{\mu}$, with $B$ being the rotational constant. The
$E_{pol}$ of LrO is 18.38 kV/cm, while they are 5.79 kV/cm and 101.03 kV/cm in
LrF+ and LrH+, respectively. The $E_{pol}$ required to polarize LrO and LrF+
are practically achievable in the laboratory. The larger, and thus less
desirable, value of $E_{pol}$ in LrH+ can be attributed to its smaller reduced
mass. Therefore, LrH+ may not be as suitable as the other two considered
candidates, but it can also be considered in an experiment if any alternative
suitable technique to measure EDM in this ion can be found.

We now turn our attention to estimating the production rates of Lr molecules
for an EDM experiment. We propose to use the RIKEN heavy-ion linear
accelerator (RILAC) facility because a high intensity ion beam is readily
available to produce atoms of interest. As for 256Lr production, we propose
the 209Bi(48Ca, $1n$) reaction as a possible candidate, where a production
cross-section of 60 nanobarns (nb) has been reported [48], although it is
necessary to have a prospect of stable supply of 48Ca. As explained in Ref.
[49], the 249Cf(11B, $4n$) reaction was employed in the single ion beam
production of 256Lr at JAEA due to its relatively high cross-section of 122
nb. The 249Cf target material is, however, radioactive, and too rare to
prepare a sufficiently large target, which can be applied to a beam from the
RILAC. On the other hand, 209Bi is stable and easy to handle to make a target
with a large area. A typical target thickness is 300 $\mu gcm^{-2}$ [50]. The
RILAC facility can typically provide a $3-p\mu A$ 48Ca beam. Under the
situation, Lr atoms can be produced with a rate of one atom per second. In the
case that GARIS (GAs-filled Recoil Ion Separator) is applied to mass-
separation and single ion beam production, a transparent efficiency of 50% is
expected [51]. In addition, in preliminary experiments at JAEA, about 20% of
Lr can be converted to LrO. Thus, we estimate N $\approx$ 0.1 molecules per
second. It means only about one-atom-per-minute molecular beam could be
produced, which presents major challenges for experiments with neutral
molecules. On the other hand, molecular ions such as LrF+ and LrH+ can be
efficiently guided and trapped by electromagnetic fields, enabling experiments
even with just a single molecular ion.

##  V Conclusion

We have scrutinized the sensitivity of LrF+, LrH+, and LrO to symmetry-
violating properties. Firstly, we showed that bound states with a stable
minimum does occur in the chosen molecular systems. Thereafter, we reported
the computed values of $\mathcal{E}_{\mathrm{eff}}$, $\mu$, and $W_{s}$ for
the aforementioned molecules. We observed that the values of
$\mathcal{E}_{\mathrm{eff}}$ for the three superheavy molecules are about 3-4
times larger than other molecules on which EDM experiments are being
performed. Similarly, the $W_{s}$ values were found to be about 5 times
larger. We discuss a feasible pathway to produce Lr atoms and hence produce Lr
molecules. We also study the properties of LrF+ and LrH+ molecules and their
potential for future single ion experiments. Our precisely estimated bond
lengths and $\mu$ values of the LrF+, LrH+, and LrO molecules can also be
useful to guide other experimental setup using these superheavy molecules.

## Acknowledgement

Most of the calculations reported in this work were performed using the
Vikram-100 high-performance computing facility at PRL, Ahmedabad. The rest of
the calculations were carried out on National Supercomputing Mission’s (NSM)
computing resource, ‘PARAM Siddhi-AI’, at C-DAC Pune, which is implemented by
C-DAC and supported by the Ministry of Electronics and Information Technology
(MeitY) and Department of Science and Technology (DST), Government of India.
This work was partially supported by the Office of Nuclear Physics, U.S.
Department of Energy, under grants DE-SC0021176 and DE-SC0021179.

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UMPA, Ecole Normale Supérieure de Lyon
Email<EMAIL_ADDRESS>

# On some collinear configurations in the planar three-body problem

Alexei Tsygvintsev

###### Abstract.

In this paper, we further investigate the planar Newtonian three-body problem
with a focus on collinear configurations, where either the three bodies or
their velocities are aligned. We provide an independent proof of Montgomery’s
result [6], stating that apart from the Lagrange’s solution, all negative
energy solutions to the zero angular momentum case result in syzygies, i.e.,
collinear configurations of positions. The concept of generalised syzygies,
inclusive of velocity alignments, was previously explored by the author for
bounded solutions in [8]. In this study, we broaden our scope to encompass
negative energy cases and provide new bounds. Our methodology builds upon the
elementary Sturm-Liouville theory and the Wintner-Conley “linear” form of the
three-body problem, as previously explored in the works of Albouy and
Chenciner [1, 2, 3].

###### Key words and phrases:

dynamical systems, celestial mechanics, three-body problem, syzygies

##  1\. Introduction

We consider the motion of three points with masses $m_{1}$, $m_{2}$, and
$m_{3}$, denoted $P_{1}$, $P_{2}$, and $P_{3}$, respectively, in the plane.
These points have positions $(x_{i},y_{i})\in\mathbb{R}^{2}$, where $i=1,2,3$.

The Newtonian three-body problem [9] involves finding the motion of three
point masses under the influence of their mutual gravitational attraction.

The equations of motion can be written in the complex compact form as follows

$\ddot{z}_{1}=m_{2}\frac{z_{21}}{|z_{21}|^{3}}-m_{3}\frac{z_{13}}{|z_{13}|^{3}},\quad\ddot{z}_{2}=m_{3}\frac{z_{32}}{|z_{32}|^{3}}-m_{1}\frac{z_{21}}{|z_{21}|^{3}},\quad\ddot{z}_{3}=m_{1}\frac{z_{13}}{|z_{13}|^{3}}-m_{2}\frac{z_{32}}{|z_{32}|^{3}}\,,$
(1.1)

where $z_{k}=x_{k}+iy_{k}\in\mathbb{C}$, $k=1,2,3$ and $z_{kl}=z_{k}-z_{l}$.
We assume that the total linear momentum is zero:

$\sum\limits_{k}m_{k}\dot{z}_{k}=\sum\limits_{k}m_{k}z_{k}=0\,,$ (1.2)

by placing the centre of mass at the origin of the coordinate system.

The word syzygy ( from Late Latin s$\mathrm{\bar{y}}$zygia =“conjunction” )
historically been used by astronomers to describe the alignment of celestial
bodies, and in this context, it refers to a configuration where all three
points lie on a straight line. In general, a solution of the $N$-body problem
is said to have a syzygy at $t=t_{0}$ if at that moment all bodies belong to a
certain straight line.

Let $t\mapsto z_{i}(t)$, $i=1,2,3$ be any solution of the equations (1.1)
defined for $t\in I=[0,a)$, $a>0$. In our work [8] we proposed to study the
natural generalisation of syzygies by adding the supplementary condition of
collinearity of velocities.

######  Definition 1.1.

The three bodies $P_{1},P_{2},P_{3}$ form a generalised syzygy at the moment
$t_{0}\in I$ if at least one of the complex triplets
$(z_{1},z_{2},z_{3})(t_{0})$ (positions) or
$(\dot{z}_{1},\dot{z}_{2},\dot{z}_{3})(t_{0})$ (velocities) belongs to the
same straight line passing through the origin (see Figure 1).

Figure 1.  A generalised syzygy: (A) – the positions are collinear (eclipse),
(B) – the velocities (dashed arrows) are collinear

In his work [6], Montgomery demonstrated that aside from the Lagrange
solution, every solution to the Newtonian three-body problem with zero angular
momentum and negative energy inevitably encounters a syzygy, defined as a
collinear configuration of positions. Non-syzygy solutions with nonzero
angular momentum are still relatively unexplored. Diacu demonstrated in [5]
that the set of initial conditions leading to syzygy solutions in the planar
three-body problem is non-empty and open.

In particular, the orbit of one particle crosses the line of the other two and
can not be tangent to this line in the transition point. Generalised syzygies
may occur more frequently, as they include collinear configurations of
velocities as well.

The paper is organised as follows.

In Chapter 2, we provide a new independent proof of Montgomery’s result [6] on
the existence of syzygies in the zero angular momentum case. The general
approach involves formulating a second-order “inear” matrix equation
$\ddot{X}=AX$ (see [1, 2, 3] for more detail), where the matrix $X$
characterises the configuration. The term “linear” is used since the matrix A
also depends on the configuration. However, despite this dependence, we can
draw certain conclusions based on the general properties of $A$.

In Chapter 3, we impose a certain algebraic restriction on the mutual
distances between three bodies. In the case of periodic behaviour, this
restriction guarantees the existence of a syzygy.

Finally, in Chapter 4, we present a simple algebraic condition that defines an
open set of initial conditions leading to a generalised syzygy, which holds
for arbitrary angular momentum.

##  2\. Zero Angular Momentum Case

Let $\Gamma\,:\,t\mapsto(z_{1}(t),z_{2}(t),z_{3}(t))$ be a zero angular
momentum solution with negative energy of the three-body problem (1.1). In
this section we prove that if $\Gamma$ is collision free for a sufficiently
long period of time, it will always encounter a syzygy. A similar result was
first established by Montgomery in [6] using the shape sphere approach. Our
proof is essentially algebraic and is more elementary.

Before proceeding further, we need to establish some preliminary results.

After introducing the new variables $w_{i}=m_{i}z_{i}$, $i=1,2,3$ the
relations (1.2) yield

$\sum\limits_{i}w_{i}=\sum\limits_{i}\dot{w}_{i}=0.$ (2.1)

Writing

$w_{k}=X_{k}+iY_{k},\quad X_{k}=m_{k}x_{k},\quad Y_{k}=m_{k}y_{k},\quad
k=1,2,3\,,$ (2.2)

and using (2.1), one derives from the equations (1.1) the following $2\times
2$ matrix equation

$\ddot{X}=AX,\quad X=\left[\begin{array}[]{ll}X_{1}&Y_{1}\\\
X_{2}&Y_{2}\end{array}\right],\quad
A=\left[\begin{array}[]{lllll}-m_{2}\rho_{3}-m_{13}\rho_{2}&&m_{1}\rho_{32}\\\
m_{2}\rho_{31}&&-m_{1}\rho_{3}-m_{32}\rho_{1}\end{array}\right]\,,$ (2.3)

where

$\rho_{1}=1/|z_{32}|^{3},\quad\rho_{2}=1/|z_{13}|^{3},\quad\rho_{3}=1/|z_{21}|^{3},\quad
m_{ij}=m_{i}+m_{j},\quad\rho_{ij}=\rho_{i}-\rho_{j}\,.$ (2.4)

The matrix $A$ is related to the Wintner-Conley endomorphism encoding the
forces. Its higher dimension version was investigated in [1, 2, 3] in the
study of $n$-body configurations and the same equation (2.3) was used (up to
reduction, transposition and some scaling factor).

######  Theorem 2.1.

Let $t\mapsto(z_{1}(t),z_{2}(t),z_{3}(t))$, $t\in[0,T_{1}]$ be a zero angular
momentum collision-free solution to the three-body problem (1.1) with negative
energy $H=-\alpha$, $\alpha>0$ where

$T_{1}(\alpha)=\frac{\sqrt{2}\pi\Sigma}{\alpha^{3/2}}\,,$ (2.5)

and

$\Sigma=\frac{(m_{3}m_{2})^{3/2}}{m_{32}^{1/2}}+\frac{(m_{1}m_{3})^{3/2}}{m_{13}^{1/2}}+\frac{(m_{2}m_{1})^{3/2}}{m_{21}^{1/2}}\,.$
(2.6)

Then there exists $t_{0}\in[0,T_{1}]$ such that the three bodies form a syzygy
at the moment $t=t_{0}$.

###### Proof.

Introducing $x=[X_{1},X_{2}]^{T}$, $y=[Y_{1},Y_{2}]^{T}$ and using the
equation (2.3), we obtain $\ddot{x}=Ax$, $\ddot{y}=Ay$. Therefore,
$\ddot{\Delta}_{1}$ can be written as follows

$\ddot{\Delta}_{1}=\det([Ax,y])+\det([x,Ay])+2\det([\dot{x},\dot{y}])=\det([Ax,y])+\det([x,Ay])+2\Delta_{2}\,.$
(2.7)

A simple algebraic computation shows that
$\det([Ax,y])+\det([x,Ay])=\mathrm{Tr}(A)\Delta_{1}$.

Thus, as follows from (2.7),

$\ddot{\Delta}_{1}=\mathrm{Tr}(A)\Delta_{1}+2\Delta_{2}\,,$ (2.8)

where

$\mathrm{Tr}(A)=-(m_{32}\rho_{1}+m_{13}\rho_{2}+m_{21}\rho_{3})\,.$ (2.9)

######  Lemma 2.1.

Let $t\mapsto z_{i}(t)$, $i=1,2,3$ be any solution of the three-body problem
(1.1) with negative energy $H=-\alpha$, where $\alpha>0$. Then
$\mathrm{Tr}(A)\leq-\alpha^{3}/\Sigma^{2}$ along this solution with $\Sigma$
defined by (2.6).

###### Proof.

By introducing $r_{i}=\rho_{i}^{1/3}$, $i=1,2,3$ the total energy of the
three-body problem (1.1) can be written as

$H=K-U(r),\quad U(r)=m_{3}m_{2}r_{1}+m_{1}m_{3}r_{2}+m_{2}m_{1}r_{3},\quad
r=(r_{1},r_{2},r_{3})\,,$ (2.10)

where $K\geq 0$ is the kinetic energy.

Since $K-U=-\alpha$ and $K\geq 0$, one obtains $U\geq\alpha$.

Writing
$F(r)=-\mathrm{Tr}(A)(r)=m_{32}r_{1}^{3}+m_{13}r_{2}^{3}+m_{21}r_{3}^{3}$, we
shall calculate, for any $s>0$, the minimum of the function $r\mapsto F(r)$ on
the compact set

$K_{s}=\\{r\in\mathbb{R}^{3}\,|\,U(r)=s,r_{i}\geq 0,i=1,2,3\\}\,.$ (2.11)

Because $K_{s}$ is convex (triangle) and $F$ is a convex function on $K_{s}$
(since $d^{2}F\geq 0$ on $K$), it is sufficient to determine its local
minimum. One computes:

$\nabla U=(m_{3}m_{2},m_{1}m_{3},m_{2}m_{1}),\quad\nabla
F=3(m_{32}r_{1}^{2},m_{13}r_{2}^{2},m_{21}r_{3}^{2})\,.$ (2.12)

The Lagrange multiplier $\lambda$ found from the equations $\nabla
F=\lambda\,\nabla U$, $U=s$ is given by

$\lambda=\frac{3s^{2}}{\Sigma^{2}}\,,$ (2.13)

with $\Sigma$ defined in (2.6).

The corresponding extremum point is

$r^{*}=(r_{1}^{*},r_{2}^{*},r_{3}^{*})=\frac{s}{\Sigma}\left(\sqrt{\frac{m_{3}m_{2}}{m_{32}}},\sqrt{\frac{m_{1}m_{3}}{m_{13}}},\sqrt{\frac{m_{2}m_{1}}{m_{21}}}\right)\in
K_{s}\,,$ (2.14)

which is a local minimum of $F$ because $d^{2}F(r^{*})$ is positive definite
and $d^{2}U=0$. Thus, by substitution:

$\min_{r\in K_{s}}\,F(r)=F(r^{*})=\frac{s^{3}}{\Sigma^{2}}\,.$ (2.15)

Hence, considering $s\geq\alpha$, one shows that $U(r)\geq\alpha$ implies
$F(r)=-\mathrm{Tr}(A)(r)\geq\displaystyle\frac{\alpha^{3}}{\Sigma^{2}}$. The
proof of the Lemma 2.1 is finished. ∎

The function $t\mapsto C(t)=\dot{X}X^{-1}$ satisfies, using (2.3), the
following matrix Riccati equation

$\dot{C}+C^{2}=A\,.$ (2.16)

Combined with the Cayley-Hamilton identity

$C^{2}-\mathrm{Tr}(C)C+\det(C)I_{2}=0\,,$ (2.17)

the equation (2.16) yields

$\dot{C}+\mathrm{Tr}(C)C=A+\det(C)I_{2}\,.$ (2.18)

Applying Liouville’s formula to the equation $\dot{X}=CX$, we obtain:

$\dot{\Delta}_{1}=\mathrm{Tr}(C)\Delta_{1}\,.$ (2.19)

Multiplying both sides of equation (2.18) by $\Delta_{1}$, we get

$\frac{d}{dt}\left(\Delta_{1}C\right)=\Delta_{1}A+\Delta_{1}\det(C)I_{2}=\Delta_{1}A+\Delta_{2}I_{2}\,,$
(2.20)

where we have used $\det(C)=\Delta_{2}/\Delta_{1}$.

Introducing the adjugate matrix $\tilde{X}$ of $X$ and using
$C=\dot{X}X^{-1}$, we compute

$\Delta_{1}C=\dot{X}\tilde{X}=\left[\begin{array}[]{lll}\dot{X}_{1}&\dot{Y}_{1}\\\
\dot{X}_{2}&\dot{Y}_{2}\end{array}\right]\,\left[\begin{array}[]{rrr}Y_{2}&-Y_{1}\\\
-X_{2}&X_{1}\end{array}\right]\,.$ (2.21)

The matrix $A(\rho_{1},\rho_{2},\rho_{3})$, defined in (2.3) can be written as
a linear combination

$A=\rho_{1}A_{1}+\rho_{2}A_{2}+\rho_{3}A_{3}\,,$ (2.22)

where $A_{1}$, $A_{2}$, and $A_{3}$ are $2\times 2$ constant matrices that
depend on the masses and are defined by

$A_{1}=\left[\begin{array}[]{rrr}0&0\\\
-m_{2}&-m_{32}\end{array}\right],\,A_{2}=\left[\begin{array}[]{rrr}-m_{13}&-m_{1}\\\
0&0\end{array}\right],\,A_{3}=\left[\begin{array}[]{rr}-m_{2}&m_{1}\\\
m_{2}&-m_{1}\end{array}\right]\,.$ (2.23)

Let $E=M_{3}(\mathbb{R})$ be the Euclidean space of real $2\times 2$ matrices
equipped with the inner product $<\,,\,>$ defined by
$<A,B>=\mathrm{Tr}(A^{T}B)$ for $A,B\in E$.

We check that $A_{1},A_{2},A_{3}\in E$ are linearly independent and span a
$3$-dimensional vector subspace of $E$. Furthermore, we have the identity

$-\frac{A_{1}+A_{2}+A_{3}}{M}=I_{2},\quad M=\sum m_{i}\,.$ (2.24)

Combining equations (2.22), (2.24) and (2.20), we obtain

$\frac{d}{dt}(\dot{X}\tilde{X})=\left(\Delta_{1}\rho_{1}-\frac{\Delta_{2}}{M}\right)A_{1}+\left(\Delta_{1}\rho_{2}-\frac{\Delta_{2}}{M}\right)A_{2}+\left(\Delta_{1}\rho_{3}-\frac{\Delta_{2}}{M}\right)A_{3}\,.$
(2.25)

Let us check, with help of the equation (2.25), constancy of the angular
momentum. We introduce the matrix

$L=\left[\begin{array}[]{lrr}-m_{3}^{-1}&m_{1}^{-1}+m_{3}^{-1}\\\ \\\
-m_{2}^{-1}-m_{3}^{-1}&m_{3}^{-1}\end{array}\right]\,.$ (2.26)

It can be checked that $<L,A_{i}>=0$, $i=1,2,3$.

Taking the inner product of both sides of the equation (2.25) with $L$, we
find that

$\frac{d}{dt}\left<(\dot{X}\tilde{X}),L\right>=0\,,$ (2.27)

i.e. the function $I=\left<\dot{X}\tilde{X},L\right>=const$ is time
independent.

One can easily verify that $I$, written as a function of variables $X_{i}$,
$Y_{j}$, $1\leq i,j\leq 2$ (by excluding $(X_{3},Y_{3})$ with help of (2.1))
is the angular momentum of the three-body problem (1.1) :

$I=\sum_{i=1}^{3}m_{i}\,R_{i}\times\dot{R}_{i}=\sum_{i=1}^{3}\frac{S_{i}}{m_{i}}\,,$
(2.28)

where $R_{i}=(x_{i},y_{i})$, $S_{i}=X_{i}\dot{Y}_{i}-\dot{X}_{i}Y_{i}$, $1\leq
i\leq 3$.

The equation (2.20) can be written, by replacing $\Delta_{2}$ using equation
(2.8), in the following form

$\frac{d}{dt}\left(\dot{X}\tilde{X}-\frac{\dot{\Delta}_{1}}{2}I_{2}\right)=\Delta_{1}\left(A-\frac{\mathrm{Tr}(A)}{2}I_{2}\right)\,.$
(2.29)

Applying (2.22), we can transform it to

$\frac{d}{dt}\left(\dot{X}\tilde{X}-\frac{\dot{\Delta}_{1}}{2}I_{2}\right)=\Delta_{1}\sum_{i=1}^{3}\rho_{i}\tilde{A}_{i}\,,$
(2.30)

where $\tilde{A}_{i}=A_{i}-\displaystyle\frac{\mathrm{Tr}(A_{i})}{2}I_{2}$,
$\mathrm{Tr}(\tilde{A}_{i})=0$, $1\leq i\leq 3$.

It can be shown, using (2.23), (2.24), that
$\tilde{A}_{1}+\tilde{A}_{2}+\tilde{A}_{3}=0$. Substituting
$\tilde{A}_{3}=-\tilde{A}_{1}-\tilde{A}_{2}$ in (2.30) we obtain finally

$\frac{d}{dt}\left(\dot{X}\tilde{X}-\frac{\dot{\Delta}_{1}}{2}I_{2}\right)=\Delta_{1}(\rho_{1}-\rho_{3})\tilde{A}_{1}+\Delta_{1}(\rho_{2}-\rho_{3})\tilde{A}_{2}\,,$
(2.31)

where

$\tilde{A}_{1}=\left[\begin{array}[]{rrr}m_{32}/2&0\\\
-m_{2}&-m_{32}/2\end{array}\right],\,\tilde{A}_{2}=\left[\begin{array}[]{rrr}-m_{13}/2&-m_{1}\\\
0&m_{13}/2\end{array}\right],\quad m_{ij}=m_{i}+m_{j}\,.$ (2.32)

Moreover, as follows from (2.19), (2.21), one always has

$\mathrm{Tr}(\dot{X}\tilde{X})=\dot{\Delta}_{1},\quad\mathrm{det}(\dot{X}\tilde{X})=\Delta_{1}\Delta_{2}\,.$
(2.33)

Let $I=\left<\dot{X}\tilde{X},L\right>=k$ where $k\in\mathbb{R}$ is the
constant value of the angular momentum. Computing
$\left<\dot{X}\tilde{X},L\right>$ using (2.21), it is easy to derive the
following equality:

$\dot{X}\tilde{X}-\frac{\dot{\Delta}_{1}}{2}I_{2}=\frac{b}{m_{2}}\tilde{A}_{1}-\frac{a}{m_{1}}\tilde{A}_{2}-k\frac{m_{3}}{2}J\,,$
(2.34)

where

$a=\left|\begin{array}[]{lll}X_{1}&Y_{1}\\\
\dot{X}_{1}&\dot{Y}_{1}\end{array}\right|,\quad
b=\left|\begin{array}[]{lll}X_{2}&Y_{2}\\\
\dot{X}_{2}&\dot{Y}_{2}\end{array}\right|,\quad
J=\left[\begin{array}[]{rrr}1&0\\\ 0&-1\end{array}\right]\,.$ (2.35)

For $k=0$, the formula (2.34) is given by

$\dot{X}\tilde{X}-\frac{\dot{\Delta}_{1}}{2}I_{2}=\frac{b}{m_{2}}\tilde{A}_{1}-\frac{a}{m_{1}}\tilde{A}_{2}=\left[\begin{array}[]{rrr}\beta
a+\gamma b&a\\\ -b&-\beta a-\gamma b\end{array}\right]=R\,,$ (2.36)

where

$\beta=\frac{1}{2}\left(\frac{m_{3}}{m_{1}}+1\right),\quad\gamma=\frac{1}{2}\left(\frac{m_{3}}{m_{2}}+1\right)\,,$
(2.37)

and

$-\mathrm{det}(R)=\beta^{2}a^{2}+(2\beta\gamma-1)ab+\gamma^{2}b^{2}\,.$ (2.38)

As follows from (2.37), one always has $\beta,\gamma>1/2$. Using these
inequalities, it is easy to show that the quadratic form in (2.38), as a
function of $a$ and $b$, is positive and therefore

$\mathrm{det}(R)(a,b)\leq 0,\quad\forall\,a,b\in\mathbb{R}\,.$ (2.39)

Since $\mathrm{Tr}(R)=0$, the eigenvalues of $R$ are real. Hence, according to
(2.36), the eigenvalues $\lambda_{1}$, $\lambda_{2}$ of $\dot{X}\tilde{X}$ are
also real. Thus, using (2.33), we have the following inequality holding for
arbitrary $a$, $b\in\mathbb{R}$

$\mathrm{Tr}(\dot{X}\tilde{X})^{2}-4\,\mathrm{det}(\dot{X}\tilde{X})=\dot{\Delta}_{1}^{2}-4\Delta_{1}\Delta_{2}\geq
0\,.$ (2.40)

Suppose that the solution $t\mapsto(z_{1}(t),z_{2}(t),z_{3}(t))$ is syzygy
free in the interval $[0,T_{1}]$. In particular, we can suppose

$\Delta_{1}(t)>0,\quad\forall t\in[0,T_{1}]\,,$ (2.41)

so that the function $\delta:\,t\mapsto\sqrt{\Delta_{1}(t)}$ is differentiable
in this interval.

Differentiating $\delta$ twice using (2.8), we obtain

$\ddot{\delta}=\eta\delta,\quad\eta=\left(\frac{\mathrm{Tr}(A)}{2}-\frac{\dot{\Delta}_{1}^{2}-4\Delta_{1}\Delta_{2}}{4\Delta_{1}^{2}}\right),\quad\quad
t\in[0,T_{1}]\,.$ (2.42)

According to (2.40) and Lemma 2.1:

$\eta(t)\leq\frac{\mathrm{Tr}(A)}{2}\leq-\zeta^{2},\quad\zeta^{2}=\frac{\alpha^{3}}{2\Sigma^{2}},\quad
t\in[0,T_{1}]\,.$ (2.43)

As follows from the zero comparison theorem of the Sturm-Liouville theory [4],
the solution $\delta$ of (2.42) always has a zero between any two consecutive
zeros of any solution $y$ of the equation $\ddot{y}=-\zeta^{2}y$ whose general
solution is $y(t)=A\cos(\zeta t+\phi_{0})$, $A,\phi_{0}\in\mathbb{R}$. For a
nonzero $A$, every two consecutive zeros of $y$ are separated by an interval
of the length $\pi/\zeta$ . Hence, $\delta(t_{0})=\Delta_{1}(t_{0})=0$ for
some $0<t_{0}<T_{1}=\pi/\zeta$ with $T_{1}$ given by the formula (2.5). This
contradicts our hypothesis (2.41). The proof of Theorem 2.1 is finished.

∎

##  3\.  Existence of syzygies for periodic solutions

In this section we will show that any periodic solution of the three-body
problem will have a syzygy, i.e. exhibits a collinear configuration of the
bodies as soon as some geometrical constraints are imposed on the shape of the
triangle formed by the bodies. Not every periodic solution has a syzygy; for
example, the Lagrange periodic solution has the three bodies situated at the
vertices of an equilateral triangle and never aligned.

######  Definition 3.1.

A periodic solution of the three-body problem (1.1) is called $\theta$-rigid
if there exists a non-zero vector
$\theta=(\theta_{1},\theta_{2},\theta_{3})\in\mathbb{R}^{3}$ such that

$\theta_{1}(\rho_{3}(t)-\rho_{2}(t))+\theta_{2}(\rho_{1}(t)-\rho_{3}(t))+\theta_{3}(\rho_{2}(t)-\rho_{1}(t))\geq
0,\quad\forall\,t\in\mathbb{R}\,,$ (3.1)

where the sum is strictly positive for at least one $t_{0}\in\mathbb{R}$.

For instance, let us consider a periodic solution for which

$|z_{32}(t)|>|z_{13}(t)|\Leftrightarrow\rho_{1}(t)<\rho_{2}(t),\quad\forall
t\in\mathbb{R}\,.$ (3.2)

Thus, it is $\theta$-rigid with respect to the vector
$(\theta_{1},\theta_{2},\theta_{3})=(0,0,1)$.

An example of this is Euler’s collinear periodic solution, in which all bodies
are perpetually collinear and each body describes an elliptical orbit,
implying that a syzygy occurs at every moment.

We note that the Lagrange equilateral solution is not $\theta$-rigid for any
choice of $\theta$. Indeed, in this case $\rho_{1}=\rho_{2}=\rho_{3}$ and,
therefore, the sum in (3.1) is always zero.

One might ask whether it is possible to have a periodic solution without
syzygies, in which one side of the triangle is always smaller than the other
and thus satisfies one of the conditions of the form (3.2).

Our next theorem provides a negative response to this question.

######  Theorem 3.1.

Every $\theta$-rigid periodic solution to the three-body problem (1.1) admits
a syzygy.

###### Proof.

Let $t\mapsto z_{i}(t)$, $i=1,2,3$ be a periodic solution of the three-body
problem (1.1) with period $\tau>0$.

Let us suppose that it is $\theta$-rigid and has no syzygies i.e.
$\Delta_{1}(t)=\det(X(t))\neq 0$, $\forall t\in[0,\tau]$. Without loss of
generality, we can assume that

$\Delta_{1}(t)>0,\quad\forall t\in[0,\tau]\,.$ (3.3)

Let $S_{i}=X_{i}\dot{Y}_{i}-\dot{X}_{i}Y_{i}$, $i=1,2,3$ be three oriented
areas of parallelograms formed by the vectors $(X_{i},Y_{i})$ and
$(\dot{X}_{i},\dot{Y}_{i})$. Then, using (2.3) and (2.1), one derives the
following equations

$\dot{S}_{1}=m_{1}\Delta_{1}(\rho_{3}-\rho_{2}),\quad\dot{S}_{2}=m_{2}\Delta_{1}(\rho_{1}-\rho_{3}),\quad\dot{S}_{3}=m_{3}\Delta_{1}(\rho_{2}-\rho_{1})\,.$
(3.4)

Thus,

$\frac{d}{dt}\left(\sum_{i=1}^{3}\frac{\theta_{i}S_{i}}{m_{i}}\right)=\Delta_{1}S,\quad
S=\theta_{1}(\rho_{3}-\rho_{2})+\theta_{2}(\rho_{1}-\rho_{3})+\theta_{3}(\rho_{2}-\rho_{1}),\quad
t\in[0,\tau]\,.$ (3.5)

Integrating (3.5) and using the periodicity of $S_{i}$, $i=1,2,3$ we find

$\int_{0}^{\tau}\Delta_{1}S\,dt=0\,,$ (3.6)

which obviously contradicts (3.3) and that $S(t_{0})>0$ for some
$t_{0}\in[0,\tau]$. The proof is finished. ∎

As pointed out by Richard Montgomery to the author, in the case of the
$\theta=(0,0,1)$ rigid solution, the corresponding orbit, when mapped to the
shape sphere [7], belongs to the half-sphere bounded by the isosceles circle
$|z_{32}|=|z_{13}|$. According to Theorem 3.1, it should thus always intersect
the collinear plane of the shape space.

##  4\. Sufficient condition for the existence of generalised syzygies

In this section, we revisit a simple geometric condition – first introduced in
our previous study [8] – which, based on initial positions and velocities,
guarantees the occurrence of generalised syzygies (see Definition 1.1).

######  Definition 4.1.

The configuration of the bodies $P_{1}$, $P_{2}$ and $P_{3}$ at the moment
$t_{0}\in I$ is called antisymmetric, if the oriented areas of parallelograms
spanned by the vectors $(z_{j}(t_{0}),z_{k}(t_{0}))$ and
$(\dot{z}_{j}(t_{0}),\dot{z}_{k}(t_{0}))$ are nonzero and have opposite signs
for some $j\neq k$ (see Figure 2). This condition is equivalent algebraically
to

$\left|\begin{array}[]{lll}x_{j}&y_{j}\\\
x_{k}&y_{k}\end{array}\right|(t_{0})\cdot\left|\begin{array}[]{lll}\dot{x}_{j}&\dot{y}_{j}\\\
\dot{x}_{k}&\dot{y}_{k}\end{array}\right|(t_{0})=(x_{j}y_{k}-y_{j}x_{k})(t_{0})\cdot(\dot{x}_{j}\dot{y}_{k}-\dot{y}_{j}\dot{x}_{k})(t_{0})<0.$
(4.1)

Figure 2.  Antisymmetric configuration of three bodies

######  Proposition 4.1.

The property of being antisymmetric is independent of the choice of the pair
of the bodies: once verified for a particular pair of bodies $P_{j}$, $P_{k}$,
$j\neq k$, the condition (4.1) will be also satisfied for all other possible
choices of pairs $P_{n}$, $P_{m}$, $n\neq m$.

###### Proof.

We observe that the condition (4.1) is invariant under the permutation of $j$
and $k$. Without loss of generality, we can assume that (4.1) holds for
$(j,k)=(1,2)$ and show that it is true also for the case where $(j,k)=(2,3)$.
The proof for other cases is similar.

The product of two determinants in (4.1) for $j=2$, $k=3$ can be written as

$\delta=\underbrace{\left|\begin{array}[]{lll}x_{2}&y_{2}\\\
x_{3}&y_{3}\end{array}\right|\cdot\left|\begin{array}[]{lll}\dot{x}_{2}&\dot{y}_{2}\\\
\dot{x}_{3}&\dot{y}_{3}\end{array}\right|}_{a}=\displaystyle\frac{1}{m_{2}^{2}m_{3}^{2}}\left|\begin{array}[]{lll}m_{2}x_{2}&m_{2}y_{2}\\\
m_{3}x_{3}&m_{3}y_{3}\end{array}\right|\cdot\left|\begin{array}[]{lll}m_{2}\dot{x}_{2}&m_{2}\dot{y}_{2}\\\
m_{3}\dot{x}_{3}&m_{3}\dot{y}_{3}\end{array}\right|\,.$ (4.2)

As can be seen from the equalities (1.2):

$\begin{array}[]{llll}[m_{3}x_{3},m_{3}y_{3}]=-[m_{1}x_{1},m_{1}y_{1}]-[m_{2}x_{2},m_{2}y_{2}],\\\
[m_{3}\dot{x}_{3},m_{3}\dot{y}_{3}]=-[m_{1}\dot{x}_{1},m_{1}\dot{y}_{1}]-[m_{2}\dot{x}_{2},m_{2}\dot{y}_{2}]\,.\end{array}$
(4.3)

Therefore, by using (4.2), (4.3), and the elementary properties of
determinants, we have:

$\delta=\displaystyle\frac{1}{m_{2}^{2}m_{3}^{2}}\left|\begin{array}[]{lll}m_{2}x_{2}&m_{2}y_{2}\\\
-m_{1}x_{1}&-m_{1}y_{1}\end{array}\right|\cdot\left|\begin{array}[]{lll}m_{2}\dot{x}_{2}&m_{2}\dot{y}_{2}\\\
-m_{1}\dot{x}_{1}&-m_{1}\dot{y}_{1}\end{array}\right|=\displaystyle\frac{m_{1}^{2}}{m_{3}^{2}}\underbrace{\left|\begin{array}[]{lll}x_{1}&y_{1}\\\
x_{2}&y_{2}\end{array}\right|\cdot\left|\begin{array}[]{lll}\dot{x}_{1}&\dot{y}_{1}\\\
\dot{x}_{2}&\dot{y}_{2}\end{array}\right|}_{b}$ (4.4)

Obviously, the sign of $a$ in (4.2) and $b$ in (4.4) are the same. That
finishes the proof. ∎

######  Proposition 4.2.

Both definitions of a generalised syzygy and an antisymmetric configuration,
are invariant when replacing $(x_{i},y_{i})$ by $(X_{i},Y_{i})$ and
$(\dot{x}_{i},\dot{y}_{i})$ by $(\dot{X}_{i},\dot{Y}_{i})$ for $i=1,2,3$.

###### Proof.

It follows immediately from (2.2) and the identity below, verified for any
$j\neq k$:

$\left|\begin{array}[]{llll}x_{j}&y_{j}\\\
x_{k}&y_{k}\end{array}\right|\cdot\left|\begin{array}[]{llll}\dot{x}_{j}&\dot{y}_{j}\\\
\dot{x}_{k}&\dot{y}_{k}\end{array}\right|=\frac{1}{m_{j}^{2}m_{k}^{2}}\left|\begin{array}[]{llll}X_{j}&Y_{j}\\\
X_{k}&Y_{k}\end{array}\right|\cdot\left|\begin{array}[]{llll}\dot{X}_{j}&\dot{Y}_{j}\\\
\dot{X}_{k}&\dot{Y}_{k}\end{array}\right|\,.$ (4.5)

∎

The author has shown in [8] that the existence of a generalised syzygy can be
guaranteed by the antisymmetry condition (4.1), provided that the mutual
distances of the three bodies remain bounded. The following theorem
generalises this result to the case of negative energy, which includes
unbounded (escape) solutions in particular. Moreover, the proof presented here
is simpler and more straightforward.

######  Theorem 4.1.

Let $t\mapsto(z_{1}(t),z_{2}(t),z_{3}(t))$ be a solution of the three-body
problem (1.1) with negative energy $H=-\alpha$, $\alpha>0$. We assume that the
initial configuration of the bodies $P_{1}$, $P_{2}$, $P_{3}$ at $t=0$ is
antisymmetric and the solution is collision free for $t\in[0,T]$ where

$T(\alpha)=\frac{\pi\Sigma}{\alpha^{3/2}}\,,$ (4.6)

and $\Sigma$ is defined by (2.6). Then there exists $t\in[0,T]$ such that the
three bodies attain a generalised syzygy at time $t=t_{0}$.

###### Proof.

Writing $d=\Delta_{2}/\Delta_{1}$, the equation (2.8) can be transformed into
the second-order Hill’s linear differential equation:

$\ddot{\Delta}_{1}=(\mathrm{Tr}(A)+2d)\Delta_{1}\,.$ (4.7)

We suppose now that the solution of (1.1) $t\mapsto z_{i}(t)$, $i=1,2,3$ is
defined in the interval $[0,T]$, with $T$ given in (4.6). Assuming it starts
at $t=0$ from an antisymmetric configuration and that no generalised syzygy
happens for any $0<t\leq T$ we obtain

$d(t)=\frac{\Delta_{2}}{\Delta_{1}}(t)<0,\quad\Delta_{1}(t)\neq 0,\quad\forall
t\in[0,T]\,.$ (4.8)

Therefore, according to (4.7), (4.8) and the Lemma 2.1:

$\ddot{\Delta}_{1}=\phi\Delta_{1},\quad\phi=\mathrm{Tr}(A)+2d\leq-\theta^{2},\quad\theta^{2}=\alpha^{3}/\Sigma^{2},\quad
t\in[0,T]\,.$ (4.9)

Comparing (4.7) with $\ddot{y}=-\theta^{2}y$, and using the same Sturm-
Liouville argument as in the proof of Theorem 2.1, we conclude that
$\Delta_{1}$ admits at least one zero in the interval $[0,\pi/\theta]=[0,T]$.
This contradicts our hypothesis (4.8). The proof of Theorem 4.1 is finished. ∎

##  5\. Conclusion

Montgomery [6] showed the existence of syzygies in the three-body problem for
the case of zero angular momentum and negative energy, except for the Lagrange
homothetic solutions. Our Theorem 4.1 is free from the restriction on the
angular momentum and contains both an easy-to-check sufficient condition and
an upper bound on the time instant when the generalised syzygy occurs. Theorem
3.1 deals with the collinear configurations (syzygies) in the periodic case.
Under the assumption that the triangle formed by the bodies obeys a geometric
restriction called $\theta$-rigidity, we show that the bodies become aligned
at some instant, resulting in a syzygy in the corresponding solution. We also
conjecture that a similar result holds for bounded non-periodic solutions. A
more in-depth analysis of equations (2.25) and (2.16) may reveal additional
interesting properties regarding collinear configurations in the three-body
problem.

Akcnowledgments
I am deeply grateful to Alain Chenciner and Richard Montgomery for their
invaluable suggestions and insightful remarks.

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