Wealth distribution is a complex and critical aspect of any society. Information exchange is considered to have played a role in shaping wealth distribution patterns, but the specific dynamic mechanism is still unclear. In this research, we used simulation-based methods to investigate the impact of different modes of information exchange on wealth distribution. We compared different combinations of information exchange strategies and moving strategies, analyzed their impact on wealth distribution using classic wealth distribution indicators such as the Gini coefficient. Our findings suggest that information exchange strategies have significant impact on wealth distribution and that promoting more equitable access to information and resources is crucial in building a just and equitable society for all.

§ INTRODUCTION

Wealth distribution is a crucial aspect of any society, and understanding how it forms is a complex and multifaceted issue. The importance of studying social wealth distribution lies not only in the fact that wealth is a crucial social resource, but also in the observation that statistical patterns similar to those found in wealth distribution can be identified in many other fields, such as the citation counts of research papers and the level of attention received by celebrities [Vermeulen, 2018]. To study the distribution of wealth is essentially to study the distribution of a group of similar social successes represented by the distribution of wealth. One important factor to be considered in wealth distribution is the role of information exchange [Coelho et al., 2005, Hu et al., 2007]. Information exchange plays a role in shaping wealth distribution patterns by allowing individuals and groups to acquire knowledge and resources that are necessary for building wealth. In recent years, advances in technology have made information exchange more accessible and efficient than ever before. The internet and social media platforms, in particular, have transformed the way people communicate and share information. These developments have created new opportunities for people to access information and resources and have made it easier for individuals and groups to collaborate and work together towards common goals. However, not all individuals and groups have equal access to information and resources [Nishi et al., 2015]. In many societies, there are significant disparities in access to education, technology, and other resources that are necessary for participating in information exchange. These disparities can create barriers that prevent some individuals and groups from fully participating in the economy and building wealth. Moreover, information exchange can also have unintended consequences that can exacerbate existing inequalities. For example, individuals and groups that are already wealthy may have more resources to invest in acquiring and sharing information, which can further consolidate their wealth and power. Despite these challenges, information exchange remains a crucial factor in shaping wealth distribution patterns. In this report, we will explore the ways in which information exchange affects wealth distribution, and the various ways in which society can work to promote more equitable access to information and resources. By understanding the role of information exchange in wealth distribution, we can work towards building a more just and equitable society for all.
In this research, we use simulation based method to focus on the investigation of the impact of various modes of information exchange on wealth distribution.

§ RELATED WORKS

Research methods for wealth distribution and information exchange mainly include theoretical analysis, empirical research, and numerical simulation. Theoretical analysis and empirical research mainly focus on the construction of theoretical models and analysis of empirical data, while numerical simulation can more intuitively demonstrate the impact of different information exchange strategies on wealth distribution. Therefore, in this research, we use numerical simulation to construct different information exchange strategies and analyze their impact on wealth distribution. In these numerical simulations, we use some classic models, such as the famous Barabási-Albert Model [Albert and Barabási, 2002] and Small-World Network Model [Watts and Strogatz, 1998], and make some improvements and extensions based on actual conditions. In each model, we adopt different information exchange strategies and compare their performance in wealth distribution. In addition, in this research, we also use some classic wealth distribution indicators, such as the Gini coefficient [Dorfman, 1979] and Lorenz curve [Gastwirth, 1971], to measure the distribution of wealth under different information exchange strategies. At the same time, we also consider some possible influencing factors, such as the initial distribution of node wealth and network structure, to more comprehensively evaluate the impact of different information exchange strategies on wealth distribution.

Basic Principles of Wealth Distribution

Wealth distribution is an important field of study in economics and sociology. Economists mainly focus on the disparity of wealth and inequality in the distribution of wealth, while sociologists are more concerned with issues of distribution fairness and social justice. Early research mainly focused on individual economic behavior and the role of market mechanisms, such as Adam Smith's theory of the "invisible hand" and David Ricardo's labor theory of value. However, in recent years, people have increasingly recognized the important role of social networks and information exchange in wealth distribution [Nishi et al., 2015].

Social Networks and Wealth Distribution

Social network theory holds that social networks have a significant impact on economic activity and the formation of wealth distribution. In these networks, behaviors such as information exchange, resource sharing, and cooperative transactions between individuals can promote the flow and distribution of wealth. For example, research by Albert-László Barabási and Réka Albert [Barabási and Albert, 1999] shows that social network structure plays an important role in the stability and equilibrium of wealth distribution. At the same time, numerous empirical studies have shown that behaviors such as information exchange and resource sharing between nodes in social networks are crucial for the flow and distribution of wealth.

Information Exchange and Wealth Distribution

Information exchange is a crucial factor in wealth distribution [Peress, 2004]. Information asymmetry often leads to inequality in resource allocation and wealth distribution. For example, in financial markets, those with more information can typically earn higher returns, exacerbating wealth inequality. However, transparency and fairness in information exchange can also promote balanced and stable wealth distribution. In recent years, with the continuous development and application of information technology, various new information exchange strategies and platforms have emerged, providing new opportunities to promote the flow and distribution of wealth.

Simulation Based Research

With the development of modern economics, more and more researchers begin to pay attention to the use of simulation methods to study economic phenomena [Herz and Merz, 1998, Moiseev and Akhmadeev, 2017]. The simulation method is a computer program-based simulation technique that helps researchers conduct experiments and analyzes in a virtual economic environment to explore possible economic changes and policy impacts. Compared with traditional statistical and empirical research methods, simulation methods have stronger theoretical and experimental control capabilities. They can provide researchers with a more realistic picture of the economic environment, allowing them to better simulate the effects of economic changes and policy implementation. In addition, simulation methods can help researchers better understand the nature of economic phenomena and provide more empirical evidence for decision-making. Simulation methods are widely used in economics. For example, they can be used to study the impact of tax policies on different groups [Altig et al., 2001] or to explore the stability of market regulatory mechanisms [Teufel et al., 2013].
Pluchino et al., 2018 used a simulation-based method to show that in Western culture, the dominant elite paradigm, which is characterized by high competition, overlooks the influence of external factors in personal success stories. Success often depends not only on personal qualities, such as talent and intelligence, but also on random factors. The research suggests that it is not reasonable to allocate too much honor or resources to lucky people, and recommends policy measures to improve elite management, diversity of thought, and innovation. However, their research assumes that everyone is a completely independent individual and does not take into account the influence of social networks on opportunity creation (reflected in luck). Their research also did not take into account that in different social cultures, changes in social networks will bring different luck distributions to individuals in different positions. Referring to the aforementioned influence of social network and information exchange on wealth distribution [Nishi et al., 2015, Barabási and Albert, 1999, Peress, 2004], we add the influence of social network and information exchange strategy to the simulation, so as to observe and analyze the influence of different information exchange strategies of individuals and different information exchange cultures of groups on the wealth distribution.

§ SIMULATION METHODOLOGY

§.§ Simulation Setups

Our simulations use the TvL model [Pluchino et al., 2018], an agent model based on a small number of simple assumptions that aims to describe the career evolution of a group of people under the influence of lucky or unlucky random events. We consider $N$ individuals (denoted by blue dots in Fig.<ref>), each of whom has a talent value of $T_i$ (intelligence, skill, ability, etc.), which obeys a normal distribution around a given mean $m_T$, with a standard deviation of $\sigma_T$, randomly distributed in a square world. The world has a periodic boundary condition (i.e., has a ring topology) surrounded by a certain number of "motion" events (denoted by green and red circles in Fig.<ref>), some of which are lucky and others unlucky (neutrals are not considered in the model events because they have no significant impact on the individual's life). In Fig.<ref>, we represent these events as colored circles: lucky events are green and represent a relative percentage of the total number of events, $pL$, and unlucky events are red and represent a percentage of the total number of events $100-pL$. The total number of events $N_E$ is uniformly distributed, but the distribution tends to be completely uniform only when $N_E$ is infinity. In our simulation, $N_E$ is proportional to $N/2$. Thus, at the start of each simulation, different regions of the world randomly distribute more lucky or unlucky events, while other regions are more neutral. Moving circles further randomly within the square lattice (i.e., the world) does not change the fundamental feature of the model, which is that different individuals, regardless of their talents, face different numbers of lucky or unlucky events during their lifetime.

An example of initial setup for simulations ($N=1000$ individuals (agents), with different degrees of talent (intelligence, skills, etc.), are randomly located in their positions within a square world of $200*200$ patches with periodic boundary conditions. During each simulation, which covers several dozens of years, they are exposed to a certain number $N_E$ of lucky (green circles) and unlucky (red circles) events, which move across the world following random trajectories (random walks). In this example, $N_E=500$.)

For a single simulation run, consider a 50-year working life period (from 20 to 70 years) with time steps of half a year for a total of 100 steps. At the start of the simulation, all agents have the same amount of capital of the order $C_i(0) = C(0) (i = 1, ..., N)$, denoting their starting wealth level. The purpose of this selection is to provide no initial advantage to anyone. While the talent of an agent is time independent, the capital of an agent varies over time. During the temporal evolution of the model, i.e., during the lifetime of the agent under consideration, all event circles are randomly moved around the world, and in doing so may intersect with some agent's positions. In each time step, each event circle is moved by a distance of length $v$ in a random direction. The radius of event circles is $r$. When the event circle and the individual change from disjoint to intersecting, we say that the event happened to this agent. After the intersection, the event circle will not disappear.

According to this, at a given time step (i.e., every half year), there are 3 different possible actions for a given agent $A_i$:

1. There is no event circle intercepting the location of agent $A_i$: this means that no important events have occurred in the past half a year. Agent $A_i$ does nothing.

2. The lucky event intercepts the position of agent $A_i$: this means that a lucky event has occurred in the past six months (note that the generation of innovative ideas is considered to be a lucky event that occurs in the brain of the agent). Therefore, agent $A_i$ will increase its capital by an order of magnitude with probability proportional to its talent $T_i$. Only when $rand [0,1] < T_i$, i.e., when the agent is smart enough to benefit from its luck, its capital order will become $C_i(t) = C_i(t-1) + dC$. Here, $dC$ is the average impact of each event on the magnitude of wealth.

3. Unfortunate event intercepts the position of agent $A_i$: this means that an unfortunate event has occurred within the past half year; thus, agent $A_i$ will reduce its capital by an order of magnitude, i.e., $C_i(t) = C_i(t-1) - dC$.

The above rules (including changing capital by orders of magnitude in the event of misfortune or luck, the probability of change being proportional to the talent of the agent, etc.) are simple and widely agreed because they are based on commonsense evidence that wealth in every person's life is usually characterized by very rapid growth or decline. In addition, these rules give highly talented people a significant advantage, because they can better exploit the opportunities that luck brings (including the ability to use the good ideas born in their brains). On the other hand, a car accident or sudden illness, for example, is always an unfortunate event, and individual talents play no role in avoiding such events. Thus, we can more effectively generalize the concept of "talent" as "any personal quality that increases the chances of being seized." In other words, the term "talent" broadly refers to intelligence, skill, cleverness, tenacity, determination, hard work, risk-taking, etc.

Research by Pluchino et al., 2018 showed that having the advantage of great talent is a necessary but not sufficient condition for attaining very high wealth. However, the matter of building a social network and obtaining information from it can neither be included in individual talent nor simply abstracted into luck. Because of changes in social networks themselves and obtaining information in social networks, both of these things depend not only on the information exchange strategies of individuals, but also on the information exchange strategies of the other party (or more generally, on the information exchange strategies of the group culture). This is a process of mutual influence, and this process cannot be decoupled into two independent variables of talent and luck. So, in order to study the influence of social network and information exchange on wealth distribution, we added the following settings.

1. The action of agent exchanging information in social network is added.

2. The action that the agent moves in the space according to the obtained information is added.

3. The action of the agent to update its social network based on the information obtained is added.

Therefore, compared with the original research [Pluchino et al., 2018], our research adds a social network through which agents can share their own information and get information from other agents, so that they can move to richer places (places with more lucky events). Each agent has the following 6 attributes: geographic location, talent value, wealth magnitude, social network links, social network update strategy, and geographic location movement strategy. Among them, the latter 3 items are newly added in our research.

§.§ Experiment Designs

As a preliminary research, we limit the update rules of social networks to the following 4 types:

1. Random: There is a random social relationship between agents

2. Location: There is a social relationship between agents whose distance between geographic locations is within a radius $R$.

3. Wealth: There is a social relationship between agents whose distance between wealth magnitudes is within $nC$ times $dC$.

4. Talent: There is a social relationship between agents whose distance between talent value is within $nT$ times standard deviation of talent value $T$.

We limit mobile strategies to the following 3 types:

1. Random: Random movement

2. Highest: Follow the agent with the highest order of wealth in the social network

3. Average: Follow the weighted average position of the agent position in the social network according to the order of wealth

We use mean wealth, wealth variance, and Gini coefficient 3 indicators to assess the macro-level results of the simulations.

§ RESULTS AND ANALYSIS

The frequency distributions of numbers of lucky (left) and unlucky (right) events are reported
separately on a log-linear scale. It can be seen that both distributions are well fitted by an exponential
distribution with similar negative exponents. (Random-Random)

In the left figure, talent is plotted as a function of wealth magnitude: the wealthiest people are not
the most talented. In the right figure, wealth magnitude is plotted as a function of talent: the wealthiest agents have talents only around average, while the most talented only have assets around their starting assets.

The final distribution of wealth among the population (log-lin scale). Despite the normal
distribution of talent, the distribution of wealth shows a strong double-exponential effect. (Random-Random)

We set population $N = 1000$, number of events $N_E = 500$, percent ratio of lucky events $pL = 50$, mean of talents $m_T = 0.6$, deviation of talents $\sigma_T = 0.1$, radius of events $r = 1$, moving speed of events $v = 1$, initial magnitude of wealth $C (0) = 5$, altitude of wealth magnitude changing $dC = 0.5$, radius of location neighborhood $R = 5$, radius of wealth neighborhood $nC = 3$, radius of talent neighborhood $nT = 1$ in our simulations. Fig.<ref> to Fig.<ref> show the simulation results with Random social network and Random mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Random social network and Highest mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Random social network and Average mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Location social network and Random mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Location social network and Highest mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Location social network and Average mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Wealth social network and Random mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Wealth social network and Highest mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Wealth social network and Average mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Talent social network and Random mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Talent social network and Highest mobile strategy. Fig.<ref> to Fig.<ref> show the simulation results with Talent social network and Average mobile strategy. Table <ref> shows the mean of wealth magnitude, standard deviation of wealth magnitude and Gini coefficient under different combinations of social network and mobile strategies.

The experimental data from the strategy combination revealed the impact of different strategies on the mean and standard deviation of wealth magnitude as well as the Gini coefficient. Looking at the mean of wealth magnitude, the wealth social network with random or average mobile strategy showed the highest mean wealth at 4.82, whereas the location social network and highest mobile strategy had a lower average wealth at 4.70. In terms of standard deviation, the random social network with highest mobile strategy had the highest standard deviation at 1.33, while the location social network with highest mobile strategy had the lowest at only 0.83. The random social network and random or highest mobile strategy also had the highest Gini coefficient at 0.13, while the location social network with highest mobile strategy had the lowest at 0.09. These data demonstrate that the random and wealth social network perform better in achieving high mean wealth, but tend to with high standard deviation and Gini coefficient. On the other hand, the location social network performs better in standard deviation and Gini coefficient, but limits the increase in mean wealth. The result of talent social network is somewhere in between. Also, highest mobile strategy tends to suppress the mean and standard deviation of wealth and the Gini coefficient at the same time, while average mobile strategy tends to increase the mean of wealth magnitude while increasing the standard deviation and Gini coefficient except for random social network scenarios.

Mean of wealth magnitude, standard deviation of wealth magnitude and Gini coefficient under
different combinations of social network and mobile strategies.

Policies     Mean of wealth     Standard deviation of wealth     Gini coefficient

Random-Random     4.82     1.24     0.13

Random-Highest     4.75     1.33     0.13

Random-Average     4.75     1.10     0.12

Location-Random     4.79     0.94     0.10

Location-Highest     4.70     0.83     0.09

Location-Average     4.78     1.04     0.11

Wealth-Random     4.82     1.32     0.13

Wealth-Highest     4.76     1.09     0.12

Wealth-Average     4.82     1.15     0.12

Talent-Random     4.78     1.18     0.12

Talent-Highest     4.74     0.99     0.11

Talent-Average     4.74     1.02     0.11

§ CONCLUSION

In this report, we focused on investigating the influence of various information exchange strategies on wealth distribution. We incorporated the effects of non-decouplable social networks and information exchange strategies into simulations of talent and luck, and analyze the effects of individual and group information exchange strategies and cultures on wealth distribution. As a preliminary research, we restricted the update rules of social networks to 4 types: random, location, wealth, and talent, and limit the mobile strategies to 3 types: random, highest, and average. We used 3 indicators, namely, mean wealth magnitude, standard deviation of wealth magnitude, and Gini coefficient to evaluate the macroscopic results of the simulation. The qualitative results of the research suggested that different social networks and information exchange strategies have varying impacts on wealth distribution. Specifically, the research found that the location social network tent to perform better in terms of standard deviation and Gini coefficient, but limits the increase in mean wealth. In contrast, the wealth social network performed better in achieving high mean wealth, but tent to have higher standard deviation and Gini coefficient. The talent social network falled somewhere in between. The choice of mobile strategy also had a significant impact on wealth distribution, with the highest mobile strategy suppressing both the mean and standard deviation of wealth and the Gini coefficient, while the average mobile strategy tent to increase the mean of wealth magnitude but also increase the standard deviation and Gini coefficient.

[Albert and Barabási, 2002]
Réka Albert and Albert-László Barabási.
Statistical mechanics of complex networks.
Reviews of modern physics, 740 (1):0 47, 2002.

[Altig et al., 2001]
David Altig, Alan J Auerbach, Laurence J Kotlikoff, Kent A Smetters, and Jan
Simulating fundamental tax reform in the united states.
American Economic Review, 910 (3):0 574–595,

[Barabási and Albert, 1999]
Albert-László Barabási and Réka Albert.
Emergence of scaling in random networks.
science, 2860 (5439):0 509–512, 1999.

[Coelho et al., 2005]
Ricardo Coelho, Zoltán Néda, José J Ramasco, and Maria Augusta
A family-network model for wealth distribution in societies.
Physica A: Statistical Mechanics and its Applications,
353:0 515–528, 2005.

[Dorfman, 1979]
Robert Dorfman.
A formula for the gini coefficient.
The review of economics and statistics, pages 146–149, 1979.

[Gastwirth, 1971]
Joseph L Gastwirth.
A general definition of the lorenz curve.
Econometrica: Journal of the Econometric Society, pages
1037–1039, 1971.

[Herz and Merz, 1998]
Bernhard Herz and Wolfgang Merz.
Experiential learning and the effectiveness of economic simulation
Simulation & Gaming, 290 (2):0 238–250,

[Hu et al., 2007]
Mao-Bin Hu, Rui Jiang, Qing-Song Wu, and Yong-Hong Wu.
Simulating the wealth distribution with a richest-following strategy
on scale-free network.
Physica A: Statistical Mechanics and its Applications,
381:0 467–472, 2007.

[Moiseev and Akhmadeev, 2017]
Nikita A Moiseev and Bulat A Akhmadeev.
Agent-based simulation of wealth, capital and asset distribution on
stock markets.
Journal of Interdisciplinary Economics, 290
(2):0 176–196, 2017.

[Nishi et al., 2015]
Akihiro Nishi, Hirokazu Shirado, David G Rand, and Nicholas A Christakis.
Inequality and visibility of wealth in experimental social networks.
Nature, 5260 (7573):0 426–429, 2015.

[Peress, 2004]
Joel Peress.
Wealth, information acquisition, and portfolio choice.
The Review of Financial Studies, 170 (3):0
879–914, 2004.

[Pluchino et al., 2018]
Alessandro Pluchino, Alessio Emanuele Biondo, and Andrea Rapisarda.
Talent versus luck: The role of randomness in success and failure.
Advances in Complex systems, 210 (03n04):0
1850014, 2018.

[Teufel et al., 2013]
Felix Teufel, Michael Miller, Massimo Genoese, and Wolf Fichtner.
Review of system dynamics models for electricity market simulations.
Working paper series in production and energy, 2013.

[Vermeulen, 2018]
Philip Vermeulen.
How fat is the top tail of the wealth distribution?
Review of Income and Wealth, 640 (2):0
357–387, 2018.

[Watts and Strogatz, 1998]
Duncan J Watts and Steven H Strogatz.
Collective dynamics of ‘small-world’networks.
nature, 3930 (6684):0 440–442, 1998.

The frequency distributions of numbers of lucky and unlucky events (Random-Highest)

Talent-Wealth plots (Random-Highest)

The final distribution of wealth among the population (Random-Highest)

The frequency distributions of numbers of lucky and unlucky events (Random-Average)

Talent-Wealth plots (Random-Average)

The final distribution of wealth among the population (Random-Average)

The frequency distributions of numbers of unlucky and lucky events (Location-Random)

Talent-Wealth plots(Location-Random)

The final distribution of wealth among the population (Location-Random)

The frequency distributions of numbers of unlucky and lucky events (Location-Highest)

Talent-Wealth plots(Location-Highest)

The final distribution of wealth among the population (Location-Highest)

The frequency distributions of numbers of unlucky and lucky events (Location-Average)

Talent-Wealth plots(Location-Average)

The final distribution of wealth among the population (Location-Average)

The frequency distributions of numbers of unlucky and lucky events (Wealth-Random)

Talent-Wealth plots(Wealth-Random)

The final distribution of wealth among the population (Wealth-Random)

The frequency distributions of numbers of unlucky and lucky events (Wealth-Highest)

Talent-Wealth plots(Wealth-Highest)

The final distribution of wealth among the population (Wealth-Highest)

The frequency distributions of numbers of unlucky and lucky events (Wealth-Average)

Talent-Wealth plots(Wealth-Average)

The final distribution of wealth among the population (Wealth-Average)

The frequency distributions of numbers of unlucky and lucky events (Talent-Random)

Talent-Wealth plots(Talent-Random)

The final distribution of wealth among the population (Talent-Random)

The frequency distributions of numbers of unlucky and lucky events (Talent-Highest)

Talent-Wealth plots(Talent-Highest)

The final distribution of wealth among the population (Talent-Highest)

The frequency distributions of numbers of unlucky and lucky events (Talent-Average)

Talent-Wealth plots(Talent-Average)

The final distribution of wealth among the population (Talent-Average)


# Relativistic calculations of molecular electric dipole moments of singly
charged aluminium monohalides

R. Bala<EMAIL_ADDRESS>V. S. Prasannaa<EMAIL_ADDRESS>B.
P. Das  Centre for Quantum Engineering, Research and Education, TCG CREST,
Salt Lake, Kolkata 700091, India

###### Abstract

In this work, we have studied the permanent electric dipole moments of singly
charged aluminium monohalides in their electronic ground state, $X^{2}\Sigma$,
using Kramers-restricted relativistic configuration interaction method. We
report our results from this method in the singles and doubles approximation
with those of Dirac-Fock calculations. For our finite field computations,
quadruple zeta basis sets were employed. We discuss the electron correlation
trends that we find in our calculated properties and have compared our results
with those from literature, wherever available.

Keywords

ab initio calculations, permanent electric dipole moments

††preprint: APS/123-QED

##  I Introduction

Molecular electric dipole moment is an important property in the field of cold
and ultracold molecules, due to its occurrence in a wide range of
applications. Molecular electric dipole moments (also known as permanent
electric dipole moments, and abbreviated hereafter as PDMs) are useful in the
study of long range dipole - dipole interactions [1], which in turn find
applications in the fields of quantum phase transitions [2, 3, 4], quantum
information processing [5], and quantum computing [6, 7]. PDMs are also
crucial inputs in searches for physics beyond the Standard Model of elementary
particles via the electric dipole moment of the electron (eEDM) [8, 9, 10,
11]. PDMs also become significant in studies that strive to improve
understanding of electrostatic bonding, for example, see Refs. [12].

In particular, the PDMs of aluminium monohalide systems may be interesting for
many applications in the coming years, given the recent interest in laser
cooling of molecular ions and works that have theoretically shown that AlF+
and AlCl+ can be laser cooled [13]. A few studies have been carried out on the
two systems prior to identifying that they are amenable to laser cooling [14,
15, 16, 17, 18, 19]. However, to the best of our knowledge, we could not find
works that calculated PDMs of the heavier counterparts, AlBr+, AlI+, AlAt+,
and the superheavy AlTs+. In this work, keeping in mind the near-future scope
of singly ionized molecules and the scarcity of literature for aluminium
monofluoride ions, we carry out accurate relativistic calculations of PDMs of
these systems with high-quality basis sets. We carry out relativistic
calculations as we expect relativistic effects to become increasingly
important as we progress from the lighter to the heavier molecules in the
series, especially in the case of AlTs+. We also study the correlation effects
and their trends in these systems.

This paper is organized into three sections. The theory and methodology used
to calculate the results of the aforementioned molecular properties are given
in section II. Section III includes the results and discussion on the computed
properties, while section IV presents a summary of this work.

##  II Theory and Methodology

###  II.1 Theory

We have employed the Dirac-Hartree-Fock (DF) as well as Kramers-restricted
Configuration Interaction (KRCI) approaches to obtain the PDMs ($\mu$). We use
the Dirac-Coulomb Hamiltonian throughout this work. We obtain the property of
interest to us using the finite field method (two point central difference) as
[20]

$\displaystyle{}\mu_{i}\,=\,-\left(\frac{dE}{d\varepsilon_{i}}\right)_{0}.$
(1)

In the above expression, $E$ refers to the energy, while $\varepsilon$ is the
perturbative parameter. Throughout, we consider the $z$ \- axis as the
internuclear axis of the considered diatomic molecule. Further details of the
approaches employed in this work can be found in Ref. [21].

###  II.2 Details of calculation

We first perform geometry optimization to determine the equilibrium bond
lengths of the considered molecules, using relativistic density functional
theory (DFT) and with Dyall’s v3z basis sets (of triple zeta quality) [22,
23]. The functional used in the DFT calculations is B3LYP. The equilibrium
bond lengths are found to be 1.623 Å for AlF+, 2.058 Å for AlCl+, 2.223 Å for
AlBr+, 2.512 Å for AlI+, 2.768 Å for AlAt+, and 2.928 Å for AlTs+. All of our
computations were carried out in the DIRAC22 [24] program.

The field dependent energies are computed at DF and KRCI levels of theory for
all aluminium halides at their optimized values of internuclear distances. For
the KRCI calculations, we have considered single - and double - excitations
with respect to the reference DF state (KRCI method in singles and doubles
approximation, abbreviated as KRCISD hereafter). In particular, the KRCISD
energies are evaluated using generalised active space (GAS) technique. In this
approach, the active orbitals are divided into three sub spaces: filled paired
(GAS1), filled unpaired (GAS2) and unfilled virtual orbitals (GAS3). We have
considered 9 active electrons for all the molecules and imposed an energy cut-
off of $10\,E_{h}$ for the virtuals, in order to avoid the associated steep
computational cost, particularly for the heavier systems.

The reported calculations have been performed using Gaussian charge
distribution for nuclei. Further, the aluminium atom is chosen to be at the
origin. The perturbative electric field strengths are chosen in the range
$-1\times 10^{-4}$ to $1\times 10^{-4}$ au. Relativistic uncontracted Dyall
basis sets of quadruple zeta quality (v4z) have been used for all calculations
[22, 23]. The details of the number of basis functions are:

  * •

Al: ($24s,\,14p,\,3d,\,2f,\,1g$),

  * •

F: ($18s,\,10p,\,3d,\,2f,\,1g$),

  * •

Cl: ($24s,\,14p,\,3d,\,2f,\,1g$),

  * •

Br: ($30s,\,21p,\,13d,\,2f,\,1g$),

  * •

I: ($33s,\,27p,\,18d,\,2f,\,1g$),

  * •

At: ($34s,\,31p,\,21d,\,14f,\,1g$), and

  * •

Ts: ($35s,\,35p,\,24d,\,16f,\,1g$).

We use atomic units throughout this work, unless mentioned otherwise.

##  III Results and Discussion

###  III.1 Equilibrium bond lengths

|
---|---
(a) | (b)
Figure 1:  a. Calculated bond lengths (in Angstrom) for AlF+ and b. AlCl+
systems. We have compared it with the available literature values (for AlF+
and AlCl+). In the legends, ‘MRCI’ refers to multireference configuration
interaction, ‘RHF’ to restricted Hartree-Fock, ‘PNO’ is pseudo-natural
orbitals coupled electron pair approximation, and ‘Expt’ is the abbreviated
notation for experiment.
---
Figure 2:  Molecular PDMs for the ground states of AlX+ (X: F, Cl, Br, I, and
At) ions in Debye (D), at DF (blue) and KRCI (green) levels of theory.

From the geometry optimization procedure using relativistic DFT, we find the
equilibrium bond lengths to be 1.623, 2.058, 2.223, 2.512, 2.768, and 2.928 Å
for AlF+, AlCl+, AlBr+, AlI+, AlAt+, and AlTs+, respectively. The numerical
values of all our obtained bond lengths along with literature values
(available only for AlF+ and AlCl+) are collected in Table (A1). Figure (1)
shows our results and how they compare with the available ab initio and
experimental results for AlF+ and AlCl+.

The molecular PDMs of aluminium monohalides discussed in the subsequent
subsections have been calculated at the corresponding equilibrium bond lengths
mentioned above. Our values of equilibrium bond lengths for AlF+ and AlCl+
show good agreement with the existing results. The maximum difference between
the values reported in the current work and those from literature using many-
body methods is 1.9% (3.0%) for AlF+ (AlCl+). Experimental data for
equilibrium bond length is available only for AlF+, among all the systems that
we considered, and our computed value differs from this result by 2.1%. Thus,
it is reasonable to assume that relativistic DFT with triple zeta quality
basis set provides sufficiently accurate results for equilibrium bond lengths
of the considered molecules.

###  III.2 Permanent dipole moments

The magnitude of PDMs for all the considered molecules along with the
available data from literature are collected in Table (A2). Figure (2)
contains the values of PDMs both at DF as well as KRCISD levels of theory.
From the table, it can be seen that with the choice of Al as origin, while
AlCl+ and AlBr+ possess the smallest values of PDM, the superheavy AlTs+ has
the largest PDM among the aluminium monohalide ions at 6.91 D. It can be seen
from the figure that the PDMs increase monotonically as one goes from lighter
to heavier systems, with the PDMs for AlF+ and AlCl+ being negative and with
those of the heavier systems being positive. The correlation trend itself is
found to be oscillatory, with the correlation effects being found to be very
large for AlCl+, AlBr+, and AlI+ (about 62, 86, and 31 percent respectively).
On the other hand, we find that electron correlation contributes to the PDMs
of AlAt+ and AlTs+ by less than 2.2 percent. The comparison of magnitudes of
our values of PDMs for AlF+ and AlCl+ with the existing calculations is
discussed in the following paragraphs.

Klein and Rosmus [18] have reported the molecular PDM for AlF+. They have used
pseudonatural orbital configuration interaction (PNO CI) method together with
the Gaussian type orbitals. The basis sets used in their work include
($12s,9p,2d$) and ($10s,6p,2d$) functions for Al and F atoms, respectively.
Further, the deepest five $s$ and four $p$ functions for Al and four $s$ and
three $p$ functions for F have been used in their contracted form. At the CI
level, they have considered single and double exciations with respect to the
reference Hartree-Fock (HF) state. To evaluate the effects of excitations
higher than single and double excitations, the coupled electron pair
approximation (CEPA) has been utilized in their work. The PNO CISD method and
CEPA approximation together is abbreviated as PNO - CEPA. Since they have
computed their PDM values in centre-of-mass system, we have translated the PDM
by considering Al atom as origin. With this transformation, their PDM for AlF+
ionic system is 2.29 D. Our PDM value of 2.56 D at DF level is larger than the
HF value given in their work by 7.1%. The correlated value of PDM in Ref. [18]
differs from our value at KRCISD level by approximately 1.7%.

Kang et al. [13] have computed the PDM for AlCl+ molecule using the MRCI
method with Davidson correction (MRCI+Q). They obtain a PDM of $\approx$ 0.27
D, which is smaller by 0.2 D with respect to our work (0.47 D). We note that
we use the KRCI approach, while they employ MRCI method. Besides this
difference, we consider the effects of relativity, bond length, choice of
active space, and choice of basis, all using KRCI, to possibly explain the
dependence of PDM on these factors.

_Relativity–_ The authors in Ref. [13] had incorporated spin-orbit coupling
effects in their calculations. We indirectly checked for its significance by
calculating the PDM using 4-component spinfree (0.43 D) as well as non-
relativistic (0.45 D) approaches, with the former including mass-velocity-
Darwin effects. Noting that our calculation with the Dirac-Coulomb Hamiltonian
had yielded 0.47 D, we expect that relativity contributes negligibly to the
PDM.

_Bond length–_ Further, the difference in bond length between our work and
theirs is 0.06 Å. We carried out KRCI calculations with the bond length used
in Kang et al. [13] (= 1.998 Å) and with 10 $E_{h}$ cut-off for virtuals, and
found the PDM to be 0.30 D (versus Kang et al.’s 0.27 D). Thus, difference in
bond length between our work and theirs was one source of discrepancy.

_Active space and single-particle basis–_ We note that their active space is
much smaller than ours. They had considered 9 active electrons and 3 virtual
molecular orbitals. On the other hand, our active space for relativistic
correlation calculation includes 9 active electrons and 115 virtual orbitals.
Moreover, Kang et al. [13] employ the aug-cc-pV5Z basis, while we employ
Dyall’s quadruple zeta quality basis sets. It is worth noting that within a
given virtual cut-off energy, the number of virtuals can vary between the two
bases. Therefore, we carried out KRCI calculations (all with 9 active
electrons) taking these possibilities into account, which we summarize below:

  * •

Our main result: Dyall v4z basis, 115 virtual orbitals (10 $E_{h}$ cut-off)
gives 0.47 D.

  * •

Dyall v4z basis, 43 virtual orbitals (1 $E_{h}$ cut-off) gives 0.17 D.

  * •

cc-pVQZ basis, 111 virtual orbitals (10 $E_{h}$ cut-off) gives 0.47 D.

  * •

cc-pVQZ basis, 37 virtual orbitals (1 $E_{h}$ cut-off) gives 0.15 D.

We emphasize at this point that a space of 3 virtual orbitals would correspond
to a cut-off much lesser than 1 $E_{h}$, and we carried out calculations with
1 $E_{h}$ cut-off due to practical limitations with the employed code. A
comparison of results 1 and 2, as well as 3 and 4 indicates that reduction in
important low-lying virtual orbitals can alter the result, while comparing
results 1 and 3, and 2 and 4, point at the result not being very sensitive to
choice of basis (in this case Dyall v4z vs cc-pVQZ).

We now compare our result with that obtained in Ref. [16]. The authors employ
MRCI, and obtain a PDM of 0.19 D. The size of their basis for both the atoms
is ($17s,\,12p)\,\rightarrow\,[6s,\,5p$], and is augmented with three $d$ and
one $f$ functions. Our basis size is substantially larger. Their choice of
active space is also significantly smaller than ours. Further, we do not
expect much difference from choice of bond lengths, as they chose 2.037 Å,
which is not very different from our 2.058 Å.

We now consider the sources of error in our calculations. We consider AlF+ as
a representative system for the error budget, and consider the choice of basis
(quality and role of diffuse functions), excitations (in particular, triples),
number of active electrons, the number of virtual orbitals, and finally the
choice of bond length. All these computations have been done at the double
zeta level, in view of the involved computational cost with a higher quality
basis. We expect that the error budget estimated with the double zeta basis
would not be very different from that for a higher quality basis. We have
found the error due to basis set size is at most -2.1 percent. We estimated
this value based on our expectation that the difference in PDM between a
pentuple zeta basis and the quadruple zeta one would not exceed that between
the quadruple zeta and a triple zeta basis. Further, the inclusion of
augmented functions to the basis was found to change the PDM by -5.1 percent.
The difference in the value of PDM obtained at the equilibrium bond length
reported in the current work and that obtained at the experimental value for
AlF+ is found to be -3.8 percent. This accounts for the possible error due to
difference in bond length. The inclusion of the triple excitations along with
singles and doubles changes the PDM by -3.8 percent. To understand the effect
of high lying virtuals, we have performed the PDM calculation with 15 $E_{h}$
virtual cut-off. The PDM with 15 $E_{h}$ virtual cut-off is found to be 2.49 D
that contributes to a change of 0.4 percent. Further, the inclusion of 15
electrons instead of 9 electrons in the active space changes the PDM by 1.2
percent. Finally, we also considered the effect of redistributing active
orbitals between GAS1 and GAS2, and found that when we flip the distribution,
that is, 1 orbital in GAS1 and 4 in GAS2 in place of 4 in GAS1 and 1 in GAS2,
the results change by -0.8 percent.

By considering all the factors discussed above, we expect the maximum error in
our computed results to be about 12 percent.

##  IV Summary

In summary, we report results for equilibrium bond lengths and the PDMs of
aluminium monohalides using high quality basis sets and accurate many-body
theory, in view of scarcity in literature for the heavier systems beyond AlF+
and AlCl+. Our findings of equilibrium bond lengths and PDMs for the two
lighter systems match well with the available results. We also report the
trends in PDMs across the considered systems. We find that correlation effects
can be very significant, for example, in AlCl+, AlBr+, and AlI+. In order to
understand the difference in the PDM of AlCl+ between our result and a
literature value, we carry out a detailed analysis to account for the effects
of relativity, bond length, choice of active space, and basis sets. We find
that for AlCl+, the size of the virtual orbital space as well as relatively
small changes in equilibrium bond length could affect the results
significantly, while relativistic effects may not play such a vital role.
Finally, we also choose AlF+ as a representative system for estimating the
error budget, and find that the net effect due to errors on the PDM is about
12 percent. The accuracy of the results of the PDM calculations in the present
work can be assessed when the PDMs of these laser coolable molecular ions
become available in the future by accurate spectroscopic measurements.

###### Acknowledgements.

All calculations reported in the current work were carried out on the Rudra
cluster, Sankhyasutra Labs (Bangalore, India).

## References

  * Tohme and Korek [2016] S. N. Tohme and M. Korek, Electronic structure calculation of the KYb molecule with dipole moments, polarizabilities, and ro-vibrational studies, Computational and Theoretical Chemistry 1078, 65 (2016).
  * Trefzger  _et al._ [2011] C. Trefzger, C. Menotti, B. Capogrosso-Sansone, and M. Lewenstein, Ultracold dipolar gases in optical lattices, Journal of Physics B: Atomic, Molecular and Optical Physics 44, 193001 (2011).
  * Astrakharchik  _et al._ [2007] G. E. Astrakharchik, J. Boronat, I. L. Kurbakov, and Y. E. Lozovik, Quantum phase transition in a two-dimensional system of dipoles, Phys. Rev. Lett. 98, 060405 (2007).
  * Tan  _et al._ [2011] L. Tan, Y.-Q. Zhang, and W.-M. Liu, Quantum phase transitions for two coupled cavities with dipole-interaction atoms, Phys. Rev. A 84, 063816 (2011).
  * Carr  _et al._ [2009] L. D. Carr, D. DeMille, R. V. Krems, and J. Ye, Cold and ultracold molecules: science, technology and applications, New Journal of Physics 11, 055049 (2009).
  * DeMille [2002] D. DeMille, Quantum computation with trapped polar molecules, Phys. Rev. Lett. 88, 067901 (2002).
  * Yelin  _et al._ [2006] S. F. Yelin, K. Kirby, and R. Côté, Schemes for robust quantum computation with polar molecules, Phys. Rev. A 74, 050301 (2006).
  * Abe  _et al._ [2018] M. Abe, V. S. Prasannaa, and B. P. Das, Application of the finite-field coupled-cluster method to calculate molecular properties relevant to electron electric-dipole-moment searches, Phys. Rev. A 97, 032515 (2018).
  * Fazil  _et al._ [2019] N. M. Fazil, V. S. Prasannaa, K. V. P. Latha, M. Abe, and B. P. Das, RaH as a potential candidate for electron electric-dipole-moment searches, Phys. Rev. A 99, 052502 (2019).
  * Sunaga  _et al._ [2019] A. Sunaga, M. Abe, M. Hada, and B. P. Das, Merits of heavy-heavy diatomic molecules for electron electric-dipole-moment searches, Phys. Rev. A 99, 062506 (2019).
  * Prasannaa  _et al._ [2020] V. S. Prasannaa, B. K. Sahoo, M. Abe, and B. P. Das, Significance of non-linear terms in the relativistic coupled-cluster theory in the determination of molecular properties, Symmetry 12, 10.3390/sym12050811 (2020).
  * Buckingham [1959] A. D. Buckingham, Molecular quadrupole moments, Q. Rev. Chem. Soc. 13, 183 (1959).
  * Kang  _et al._ [2017] S.-Y. Kang, F.-G. Kuang, G. Jiang, D.-B. Li, Y. Luo, P. Feng-Hui, W. Li-Ping, W.-Q. Hu, and Y.-C. Shao, Ab initio study of laser cooling of AlF+and AlCl+ molecular ions, Journal of Physics B: Atomic, Molecular and Optical Physics 50, 105103 (2017).
  * Dyke  _et al._ [1984] J. Dyke, C. Kirby, A. Morris, B. Gravenor, R. Klein, and P. Rosmus, A study of aluminium monofluoride and aluminium trifluoride by high-temperature photoelectron spectroscopy, Chemical Physics 88, 289 (1984).
  * Glenewinkel‐Meyer  _et al._ [1988] T. Glenewinkel‐Meyer, A. Kowalski, B. Müller, C. Ottinger, and W. H. Breckenridge, Emission spectra and electronic structure of group IIIa monohalide cations, The Journal of Chemical Physics 89, 7112 (1988).
  * Glenewinkel‐Meyer  _et al._ [1991] T. Glenewinkel‐Meyer, B. Müller, C. Ottinger, P. Rosmus, P. J. Knowles, and H. Werner, Ab initio calculations on the four lowest electronic states of AlF+ and AlCl+, The Journal of Chemical Physics 95, 5133 (1991).
  * Brites  _et al._ [2008] V. Brites, D. Hammoutène, and M. Hochlaf, Spectroscopy, metastability, and single and double ionization of AlCl, The Journal of Physical Chemistry A 112, 13419 (2008).
  * Klein and Rosmus [1984] R. Klein and P. Rosmus, Ab initio calculations of infrared transition probabilities in the electronic ground states of AlF and AlF+, Theoretica chimica acta 66, 21 (1984).
  * Bruna and Grein [2001] P. J. Bruna and F. Grein, Electron-spin magnetic moment (g factor) of X${}^{2}\Sigma^{+}$ diatomic radicals MX$(\pm)$ with nine valence electrons (M = Be, B, Mg, Al; X = N, O, F, P, S, Cl). An ab initio study, The Journal of Physical Chemistry A 105, 3328 (2001).
  * Buckingham [1967] A. D. Buckingham, Permanent and induced molecular moments and long-range intermolecular forces, in  _Advances in Chemical Physics_ (John Wiley & Sons, Ltd, 1967) pp. 107–142.
  * Bala  _et al._ [2019] R. Bala, H. S. Nataraj, and M. K. Nayak, Ab initio calculations of permanent dipole moments and dipole polarizabilities of alkaline-earth monofluorides, Journal of Physics B: Atomic, Molecular and Optical Physics 52, 085101 (2019).
  * Dyall [2006] K. G. Dyall, Relativistic quadruple-zeta and revised triple-zeta and double-zeta basis sets for the 4p, 5p, and 6p elements, Theoretical Chemistry Accounts 115, 441 (2006).
  * Dyall [2016] K. G. Dyall, Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the light elements H-Ar, Theoretical Chemistry Accounts 135, 128 (2016).
  * Jensen  _et al._ [morg] H. J. A. Jensen, A. S. P. Gomes, R. Bast, T. Saue, L. V. with contributions from I. A. Aucar, V. Bakken, C. Chibueze, J. Creutzberg, K. G. Dyall, U. E. S. Dubillard, E. Eliav, T. Enevoldsen, E. Fasshauer, T. Fleig, O. Fossgaard, L. Halbert, E. D. Hedegård, T. Helgaker, B. Helmich-Paris, J. Henriksson, M. van Horn, M. Ilias, C. R. Jacob, S. Knecht, S. Komorovsky, O. Kullie, J. K. Laerdahl, C. V. Larsen, Y. S. Lee, N. H. List, H. S. Nataraj, M. K. Nayak, P. Norman, G. Olejniczak, J. Olsen, J. M. H. Olsen, A. Papadopoulos, Y. C. Park, J. K. Pedersen, M. Pernpointner, J. V. Pototschnig, R. D. Remigio, M. Repisky, K. Ruud, P. Salek, B. Schimmelpfennig, B. Senjean, A. Shee, J. Sikkema, A. Sunaga, A. J. Thorvaldsen, J. Thyssen, J. van Stralen, M. L. Vidal, S. Villaume, O. Visser, T. Winther, S. Yamamoto, and X. Yuan, DIRAC, a relativistic ab initio electronic structure program, Release DIRAC22,  (2022 (http://www.diracprogram.org)), available at: http://dx.doi.org/10.5281/zenodo.6010450.

## Appendix

The numerical values of all the computed quantities in this work along with
the existing results are given in appendix.

Table A1: Equilibrium bond lengths (in Å) for AlX+ (X: F, Cl, Br, I , At, and Ts) molecular systems using relativistic DFT with triple zeta basis sets. We also compare our obtained values with those available in literature.  Molecule | Method |  $R_{e}$ (Å) | Ref.
---|---|---|---
AlF+ | DFT | 1.623 | This work
| MRCI | 1.593 | [13]
| RHF | 1.576 | [14]
| PNO-CEPA | 1.601 | [14]
| MRCI | 1.604 | [16]
| Expt | 1.59 $\pm$ 0.01 | [14]
AlCl+ | DFT | 2.058 | This work
| MRCI | 1.998 | [13]
| MRCI | 2.037 | [16]
| MRCI | 2.023 | [17]
AlBr+ | DFT | 2.223 | This work
AlI+ | DFT | 2.512 | This work
AlAt+ | DFT | 2.768 | This work
AlTs+ | DFT | 2.928 | This work
Table A2: Magnitude of permanent dipole moments (in Debye) for AlX+ (X: F, Cl,
Br, I, At, and Ts) molecular systems at DF and KRCISD level using quadruple
zeta (dyall.v4z) basis sets.

Molecule | Method | PDM# | Ref.
---|---|---|---
AlF+ | DF | 2.56 | This work
| KRCISD | 2.33 | This work
| MRCI† | 2.40 | [16]
| HF† | 2.39 | [18]
| PNO-CEPA† | 2.29 | [18]
AlCl+ | DF | 0.76 | This work
| KRCISD | 0.47 | This work
| MRCI†,a | 0.27 | [13]
| MRCI† | 0.19 | [16]
AlBr+ | DF | 0.07 | This work
| KRCISD | 0.50 | This work
AlI+ | DF | 1.49 | This work
| KRCISD | 2.16 | This work
AlAt+ | DF | 4.31 | This work
| KRCISD | 4.24 | This work
AlTs+ | DF | 6.76 | This work
| KRCISD | 6.91 | This work

#Results for PDM are rounded off at second decimal place.

†In Ref. [16, 18], the authors have reported the values of PDM in centre-of-
mass system. We have suitably translated them with Al chosen to be the origin
to make the comparison consistent with our values.

aThe PDM has been extracted from the graph given in Ref. [13].


∎ LHAASO short = LHAASO , long = Large High Altitude Air Shower Observatory ,
short-plural = , GRB short = GRB , long = gamma-ray burst , short-plural = s ,
CMB short = CMB , long = cosmic microwave background , short-plural = , EBL
short = EBL , long = extragalactic background light , short-plural = , PDF
short = PDF , long = probability distribution function , short-plural = s ,

e1Correspondence author<EMAIL_ADDRESS>
11institutetext: Department of Applied Physics, College of Science, China
Agricultural University, Qinghua East Road, Beijing 100083, People’s Republic
of China 22institutetext: CAS Key Laboratory of Theoretical Physics, Institute
of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, People’s
Republic of China 33institutetext: Theoretical Physics Division, Institute of
High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China 44institutetext: School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

# Multi-TeV photons from GRB 221009A: uncertainty of optical depth considered

Zhi-Chao Zhaoaddr1  Yong Zhouaddr2  Sai Wange1,addr3,addr4

(Received: date / Accepted: date)

###### Abstract

It is reported that the Large High Altitude Air Shower Observatory (LHAASO)
observed thousands of very-high-energy photons up to $\sim$18 TeV from GRB
221009A. We study the survival rate of these photons via considering the fact
that they are absorbed by the extragalactic background light. By performing a
set of $10^{6}$ Monte-Carlo simulations, we explore the parameter space
allowed by current observations and estimate the probability of predicting
that LHAASO detects at least one photon of 18 TeV from GRB 221009A. We find
that the standard physics is compatible with the observations of 18 TeV
photons within $3.5\sigma$ confidence interval. Our research method can be
straightforwardly generalized to study more data sets of LHAASO and other
experiments in the future.

††journal: Eur. Phys. J. C

##  1 Introduction

More than 5000 photons above 0.5 TeV emitted from GRB 221009A at redshift
$z\simeq 0.151$ 111https://gcn.gsfc.nasa.gov/gcn3/32648.gcn3 were observed by
the LHAASO 222https://gcn.gsfc.nasa.gov/gcn3/32677.gcn3 within 2000 seconds
after the first detection by Swift, Fermi-GBM, Fermi-LAT, and so on
333https://gcn.gsfc.nasa.gov/gcn3_archive.html. The highest energetic photons
were reported to reach $\sim 18$ TeV, representing the first observation of
photons above 10 TeV from GRB.  Such observations intrigued studies on photon
mixing with axion-like particles Galanti:2022pbg ; Galanti:2022xok ;
Baktash:2022gnf ; Carenza:2022kjt ; Gonzalez:2022opy ; Lin:2022ocj ;
Troitsky:2022xso ; Nakagawa:2022wwm ; Zhang:2022zbm , Lorentz symmetry
violation Baktash:2022gnf ; Li:2022vgq ; Finke:2022swf ; Zhu:2022usw ;
He:2022jdl ; Huang:2022xto ; Vardanyan:2022ujc ; Li:2022wxc , ultra-high-
energy cosmic rays Das:2022gon , dark photon Gonzalez:2022opy , sterile
neutrinos Cheung:2022luv ; Smirnov:2022suv ; Brdar:2022rhc , and
misidentification of the showers Baktash:2022gnf . In our work, we will
investigate the survival probability of multi-TeV photons from GRB 221009A by
considering the fact that they are significantly absorbed by the extragalactic
background light intervening between the GRB and the Earth. We will further
show whether the standard physics is still capable to interpret current
observations.

Very-high-energy photons can dissipate their energies via annihilation with
photons in CMB and EBL, producing electron-positron pairs. The threshold of
this channel to happen is $E_{\mathrm{th}}=m_{e}^{2}/E_{\mathrm{b}}$, where
$m_{e}$ and $E_{b}$ are the mass of electrons and the averaged energy of
background light, respectively. Therefore, for CMB photons, this threshold is
hundreds of TeV, implying that we can safely disregard the effect of CMB
photons. However, the energy of EBL photons can be higher by several orders of
magnitude than the CMB photons, changing the threshold to be lower by the same
magnitude. For example, the threshold is $\sim 2.6$ TeV if we consider
$E_{b}\sim 0.1$eV. Therefore, we should take into account the suppression
effect of EBL photons on the detected flux of $\sim 18$ TeV photons by LHAASO.

##  2 Flux of TeV photons and EBL attenuation

The EBL-suppressed flux $F_{o}$ depends on the intrinsic flux $F_{i}$ and the
optical depth $e^{-\tau}$ due to absorption by EBL. Therefore, we have

$F_{o}(E)=F_{i}(E)e^{-\tau(E,z)}\ ,$ (1)

where $E$ is the observed energy of photons and $z$ is the redshift of GRB
221009A. We use the tabulated data of the EBL and optical depth measured by
Ref. Dominguez:2010bv . The intrinsic flux of photons is approximated to be a
power-law MAGIC:2019lau

$F_{i}(E)=A_{i}\left(\frac{E}{0.5\mathrm{TeV}}\right)^{\alpha_{\mathrm{i}}}e^{-\frac{E}{E_{\rm
cut}}}\ ,$ (2)

where $A_{i}$ is an intrinsic flux at a pivot energy scale 0.5 TeV,
$\alpha_{\mathrm{i}}$ denotes a spectral index, and $E_{\mathrm{cut}}$ is a
cutoff energy scale. Based on the reports of Fermi-LAT, we have two measured
values of $\alpha_{i}$, namely, $-1.87\pm 0.04$
444https://gcn.gsfc.nasa.gov/gcn3/32658.gcn3 and $-2.12\pm 0.11$
555https://gcn.gsfc.nasa.gov/gcn3/32637.gcn3. However, they were obtained at
0.1–1 GeV, which is an energy range beyond the capability of LHAASO.
Meanwhile, there is not a report of spectral index from LHAASO at present.
During our parameter inference processes, we assume that the spectral index
$\alpha_{i}$ is $-2$ and $-3$, respectively. Our results can be adjusted to
fit any value of $\alpha_{i}$ between $-2$ and $-3$ if it is reported by
LHAASO in the future. Due to the same reason, we fix $E_{\mathrm{cut}}$ to 2
TeV, 5 TeV, and 10 TeV, respectively. Therefore, we leave $A_{i}$ to be
determined by the data sets of LHAASO.

By considering the performance of LHAASO LHAASO:2019qtb , we predict the
number of events within energy range $0.5-10$ TeV to be

$N_{>0.5\mathrm{TeV}}=T\int_{0.5\mathrm{TeV}}^{10\mathrm{TeV}}F_{o}(E)S_{\mathrm{eff}}(E,\theta)dE\
,$ (3)

where $S_{\mathrm{eff}}$ is the effective area of LHAASO-WCDA, as provided in
Fig. 6 of Chapter 3 in Ref. LHAASO:2019qtb , $\theta\simeq 28^{\circ}$ is the
zenith angle of GRB 221009A, and $T=2000$ seconds is the duration of LHAASO
observing run.

To explore the parameter space, we perform Bayesian analysis by considering
the fact that the number of photons above 0.5TeV is more than 5000, as
reported by LHAASO 666https://gcn.gsfc.nasa.gov/gcn3/32677.gcn3. We assume
that the event number follows Poisson distribution with PDF, i.e.

$p(k)=\lambda^{k}e^{-\lambda}/k!$ (4)

with $\lambda=5000$ being the expectation and $k=N_{>0.5\mathrm{TeV}}$. Our
results and conclusion are robust when choosing other value of $\lambda$
within a range $[5000,6000)$. If we consider smaller values of $\lambda$,
e.g., $\lambda=4000$, our results would not be altered significantly, i.e., a
difference of around 15%, which would not destroy our conclusions. We
implement the Bayesian inference by using the affine-invariant Markov chain
Monte Carlo (MCMC) ensemble sampler in emcee 2013PASP..125..306F . We assume
that $A_{i}$ has a uniform prior in the range $[10^{-5},10^{-2.5}]$ in units
of $\mathrm{TeV}^{-1}\mathrm{m}^{-2}\mathrm{s^{-1}}$. The optical depth is
sampled by following the tabulated data of median value and uncertainties of
$\tau$ 777http://side.iaa.es/EBL/, as described in Ref. Dominguez:2010bv .

(a) $\alpha_{i}=-2$

(b) $\alpha_{i}=-3$

Figure 1: Posterior probability distribution functions of $A_{i}$ estimated in
the case of $E_{\rm cut}=10,\ 5,\ 2$ TeV for $\alpha_{i}=-2$ and
$\alpha_{i}=-3$, respectively.

The results of Bayesian parameter inferences are shown as the one-dimensional
posterior PDF of $A_{i}$ in Fig. 1. The left panel shows the results in the
case of $\alpha_{i}=-2$ while the right one shows those in the case of
$\alpha_{i}=-3$. For any case, we find $A_{i}\simeq\mathrm{few}\times 10^{-4}$
$\mathrm{TeV}^{-1}\mathrm{m}^{-2}\mathrm{s^{-1}}$. In the following, we do
Monte Carlo simulations via sampling $A_{i}$ following its posterior PDF and
$\tau$ following the aforementioned tabulated data.

##  3 Probability of detecting TeV photons

Based on the above results, we will estimate the probability of predicting
that LHAASO observes at least one photon $\sim$18 TeV from GRB 221009A. During
an observation of $T=2000$ seconds, the event number of photons with energy
centered at $E$ is given by

$N(E)=T\int_{10\mathrm{TeV}}F_{o}(E^{\prime})S_{\mathrm{eff}}(E^{\prime},\theta)P(E,E^{\prime})dE^{\prime}\
,$ (5)

where
$P(E,E^{\prime})\propto\mathrm{exp}[-(E^{\prime}/E-1)^{2}/(2\sigma^{2})]$
stands for a Gaussian PDF with $\sigma=\Delta E/E$ being the energy resolution
of LHAASO-KM2A Cui:2014bda . We consider the energy range above 10 TeV, with
an emphasis on 18 TeV. For each given energy $E$, we perform a set of $10^{6}$
Monte-Carlo simulations. We count the number of models that predict $N(E)\geq
1$ and compute the corresponding probability via dividing this number by
$10^{6}$.

(a) $\alpha_{i}=-2$

(b) $\alpha_{i}=-3$

Figure 2: Probability of predicting that LHAASO observes at least one photon of multi-TeV from GRB 221009A within 2000 seconds for $E_{\rm cut}=10,\ 5,\ 2$ TeV respectively. The left panel shows the case of $\alpha_{i}=-2$ and the right panel shows the case of $\alpha_{i}=-3$. The dotted vertical lines denote 18 TeV photons while the dotted horizontal lines denote probabilities of observing at least one photon of around 18 TeV. — |  $E_{\rm cut}=10$ TeV |  $E_{\rm cut}=5$ TeV |  $E_{\rm cut}=2$ TeV
---|---|---|---
$\alpha_{i}=-2$ | 24.5 % | 12.4 % | 0.6 %
$\alpha_{i}=-3$ | 3.3 % | 1.2 % | 0.1 %
Table 1: List for probabilities of predicting that LHAASO observes at least
one photon of $\sim 18$ TeV from GRB 221009A within 2000 seconds.

Our results of Monte Carlo simulations are shown in Fig. 2. For a given
energy, we estimate a probability of predicting that LHAASO detects at least
one photon. The left (right) panel shows the results in the case of
$\alpha_{i}=-2$ ($\alpha_{i}=-3$). The labeling of $E_{\mathrm{cut}}$ is the
same as in Fig. 1. In particular, the dotted vertical lines denote 18 TeV in
the two panels, while the dotted horizontal lines denote the probabilities of
predicting that LHAASO is capable to detect at least one photon of 18 TeV from
GRB 221009A. Correspondingly, we also list these probabilities in Tab. 1. We
find that in either case the standard physics is compatible with the
observations of 18 TeV photons from GRB 221009A within $3.5\sigma$ confidence
interval. This prediction could be further tested with the observational data
sets of LHAASO in the future.

##  4 Summary

In this work, we have investigated the survival rate of very-high-energy gamma
rays within the energy range of LHAASO, by taking into account the effect of
EBL attenuation. In the framework of standard physics, we simulated the
probability of detecting the multi-TeV events from GRB 221009A. When
considering the energy resolution of LHAASO, we found that the standard
physics is still compatible with the observations of 18 TeV photons from GRB
221009A within $3.5\sigma$ confidence interval. The above conclusions might be
altered if we consider other measurements of EBL and optical depth Saldana-
Lopez:2020qzx ; Gilmore:2011ks ; Inoue:2012bk ; Kneiske:2010pt ; Finke:2009xi
; Franceschini:2017iwq ; Franceschini:2008tp , that are beyond the scope of
this paper. If the report of LHAASO can be confirmed in the future, we may
derive a novel constraint on the models of EBL or even discriminate different
models of EBL. We would leave such detailed studies to future works. Our
research method can also be straightforwardly generalized to study more data
sets of LHAASO and other experiments in the future. In addition, we did not
consider other sources of astrophysical uncertainties, e.g., the effect of
intergalactic magnetic fields. This effect may be considerable for GRB
221009A, especially in the Fermi-LAT energy band. In fact, the intergalactic
magnetic field strength had been estimated to be $\sim 10^{-16}$ Gauss, if
considering the delayed cascade photons observed by Fermi-LAT, as shown in
Ref. Xia:2022uua . Meanwhile, it was shown that LHAASO might have observed the
cascade photons from this GRB. However, it is challenging to discriminate them
from photons due to other astrophysical processes in practice.

We also noticed that other experiments detected photons from GRB 221009A, but
at energy bands different from LHAASO. For example, the Carpet-2 experiment
888ATel $\\#$15669 reported a single event $\sim 251$ TeV in coincidence with
GRB 221009A with statistical significance of $\sim 3.8\sigma$. This event
might imply an evidence of new physics due to such a high energy scale.
However, another possibility was also proposed to be a cosmic-ray origin due
to secondary emission from ultra-high-energy cosmic rays Das:2022gon ;
AlvesBatista:2022kpg ; Mirabal:2022ipw . Therefore, further observations in
future are necessary to remove these debates.

###### Acknowledgements.

We acknowledge Prof. Xiao-Jun Bi, Prof. Hai-Nan Lin and Dr. Liang-Duan Liu for
helpful discussions. ZCZ is supported by the National Natural Science
Foundation of China (Grant NO. 12005016). YZ is supported by the National
Natural Science Foundation of China (Grants No. 12047558 and No. 12075249),
the China Postdoctoral Science Foundation (Grant No. 2021M693238), and the
Special Research Assistant Funding Project of CAS. SW is supported by the
National Natural Science Foundation of China (Grant No. 12175243), the Key
Research Program of the Chinese Academy of Sciences (Grant No. XDPB15) and the
science research grants from the China Manned Space Project with No. CMS-
CSST-2021-B01.

## References

  * (1) G. Galanti, M. Roncadelli, and F. Tavecchio,  (2022), 2210.05659.
  * (2) G. Galanti, M. Roncadelli, and F. Tavecchio,  (2022), 2211.06935.
  * (3) A. Baktash, D. Horns, and M. Meyer,  (2022), 2210.07172.
  * (4) P. Carenza and M. C. D. Marsh,  (2022), 2211.02010.
  * (5) M. M. Gonzalez et al.,  (2022), 2210.15857.
  * (6) W. Lin and T. T. Yanagida,  (2022), 2210.08841.
  * (7) S. V. Troitsky,  Pisma Zh. Eksp. Teor. Fiz. 116, 745 (2022), 2210.09250.
  * (8) S. Nakagawa, F. Takahashi, M. Yamada, and W. Yin,  (2022), 2210.10022.
  * (9) G. Zhang and B.-Q. Ma,  (2022), 2210.13120.
  * (10) H. Li and B.-Q. Ma,  (2022), 2210.05563.
  * (11) J. D. Finke and S. Razzaque,  (2022), 2210.11261.
  * (12) J. Zhu and B.-Q. Ma,  (2022), 2210.11376.
  * (13) P. He and B.-Q. Ma,  Phys. Lett. B 835, 137536 (2022), 2210.14817.
  * (14) Y. Huang and B.-Q. Ma,  (2022), 2211.00231.
  * (15) V. Vardanyan, V. Takhistov, M. Ata, and K. Murase,  (2022), 2212.02436.
  * (16) H. Li and B.-Q. Ma,  (2022), 2210.06338.
  * (17) S. Das and S. Razzaque,  (2022), 2210.13349.
  * (18) K. Cheung,  (2022), 2210.14178.
  * (19) A. Y. Smirnov and A. Trautner,  (2022), 2211.00634.
  * (20) V. Brdar and Y.-Y. Li,  (2022), 2211.02028.
  * (21) A. Dominguez et al.,  Mon. Not. Roy. Astron. Soc. 410, 2556 (2011), 1007.1459.
  * (22) MAGIC, V. A. Acciari et al.,  Nature 575, 455 (2019), 2006.07249.
  * (23) LHAASO, A. Addazi et al.,  Chin. Phys. C 46, 035001 (2022), 1905.02773.
  * (24) D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman,  Publ. Astron. Soc. Pac. 125, 306 (2013), 1202.3665.
  * (25) LHAASO, S. Cui, Y. Liu, Y. Liu, and X. Ma,  Astropart. Phys. 54, 86 (2014).
  * (26) A. Saldana-Lopez et al.,  Mon. Not. Roy. Astron. Soc. 507, 5144 (2021), 2012.03035.
  * (27) R. C. Gilmore, R. S. Somerville, J. R. Primack, and A. Dominguez,  Mon. Not. Roy. Astron. Soc. 422, 3189 (2012), 1104.0671.
  * (28) Y. Inoue et al.,  Astrophys. J. 768, 197 (2013), 1212.1683.
  * (29) T. M. Kneiske and H. Dole,  Astron. Astrophys. 515, A19 (2010), 1001.2132.
  * (30) J. D. Finke, S. Razzaque, and C. D. Dermer,  Astrophys. J. 712, 238 (2010), 0905.1115.
  * (31) A. Franceschini and G. Rodighiero,  Astron. Astrophys. 603, A34 (2017), 1705.10256.
  * (32) A. Franceschini, G. Rodighiero, and M. Vaccari,  Astron. Astrophys. 487, 837 (2008), 0805.1841.
  * (33) Z.-Q. Xia, Y. Wang, Q. Yuan, and Y.-Z. Fan,  (2022), 2210.13052.
  * (34) R. Alves Batista,  (2022), 2210.12855.
  * (35) N. Mirabal,  (2022), 2210.14243.