# LitLLM: A Toolkit for Scientific Literature Review

Shubham Agarwal1,2,3, Issam H. Laradji1,4, Laurent Charlin2,3,5, Christopher
Pal1,2,5
1ServiceNow Research, 2Mila - Quebec AI Institute, 3HEC Montreal, Canada
4UBC, Vancouver, Canada, 5Canada CIFAR AI Chair
Correspondence<EMAIL_ADDRESS>
###### Abstract

Conducting literature reviews for scientific papers is essential for
understanding research, its limitations, and building on existing work. It is
a tedious task which makes an automatic literature review generator appealing.
Unfortunately, many existing works that generate such reviews using Large
Language Models (LLMs) have significant limitations. They tend to
hallucinate—generate non-factual information—and ignore the latest research
they have not been trained on. To address these limitations, we propose a
toolkit that operates on Retrieval Augmented Generation (RAG) principles,
specialized prompting and instructing techniques with the help of LLMs. Our
system first initiates a web search to retrieve relevant papers by summarizing
user-provided abstracts into keywords using an off-the-shelf LLM. Authors can
enhance the search by supplementing it with relevant papers or keywords,
contributing to a tailored retrieval process. Second, the system re-ranks the
retrieved papers based on the user-provided abstract. Finally, the related
work section is generated based on the re-ranked results and the abstract.
There is a substantial reduction in time and effort for literature review
compared to traditional methods, establishing our toolkit as an efficient
alternative. Our open-source toolkit is accessible at
https://github.com/shubhamagarwal92/LitLLM and Huggingface space
(https://huggingface.co/spaces/shubhamagarwal92/LitLLM) with the video demo at
https://youtu.be/E2ggOZBAFw0

LitLLM: A Toolkit for Scientific Literature Review

Shubham Agarwal1,2,3, Issam H. Laradji1,4, Laurent Charlin2,3,5, Christopher
Pal1,2,5 1ServiceNow Research, 2Mila - Quebec AI Institute, 3HEC Montreal,
Canada 4UBC, Vancouver, Canada, 5Canada CIFAR AI Chair Correspondence:
<EMAIL_ADDRESS>

##  1 Introduction

Scientists have long used NLP systems like search engines to find and retrieve
relevant papers. Scholarly engines, including Google Scholar, Microsoft
Academic Graph, and Semantic Scholar, provide additional tools and structure
to help researchers further. Following recent advances in large language
models (LLMs), a new set of systems provides even more advanced features. For
example, Explainpaper111https://www.explainpaper.com/ helps explain the
contents of papers, and Writefull222https://x.writefull.com/ helps with
several writing tasks, including abstract and title generation. There are, of
course, many other tasks where similar technologies could be helpful.

Systems that help researchers with literature reviews hold promising
prospects. The literature review is a difficult task that can be decomposed
into several sub-tasks, including retrieving relevant papers and generating a
related works section that contextualizes the proposed work compared to the
existing literature. It is also a task where factual correctness is essential.
In that sense, it is a challenging task for current LLMs, which are known to
hallucinate. Overall, creating tools to help researchers more rapidly
identify, summarize and contextualize relevant prior work could significantly
help the research community.

Recent works explore the task of literature review in parts or in full. For
example, Lu et al. (2020) proposes generating the related works section of a
paper using its abstract and a list of (relevant) references. Researchers also
look at the whole task and build systems using LLMs like ChatGPT for
literature review Haman and Školník (2023); Huang and Tan (2023). While these
LLMs tend to generate high-quality text, they are prone to hallucinations
Athaluri et al. (2023). For example, the Galactica system was developed to
reason about scientific knowledge (Taylor et al., 2022). While it outperforms
contemporary models on various scientific tasks, it generates made-up content
like inaccurate citations and imaginary papers.333see e.g., What Meta Learned
from Galactica

Figure 1: LitLLM on Huggingface Space. Our system works on the Retrieval
Augmented Generation (RAG) principle to generate the literature review
grounded in retrieved relevant papers. User needs to provide the abstract in
the textbox (in purple) and press send to get the generated related work (in
red). First, the abstract is summarized into keywords (Section 3.1), which are
used to query a search engine. Retrieved results are re-ranked (in blue) using
the Paper Re-Ranking module (Section 3.2), which is then used as context to
generate the related work (Section 3.3). Users could also provide a sentence
plan (in green) according to their preference to generate a concise, readily
usable literature review (See Section 3.3.2).

As a step forward, we explore retrieval-augmented-generation (RAG) to improve
factual correctness Lewis et al. (2020). The idea is to use the retrieval
mechanism to obtain a relevant list of existing papers to be cited which
provides relevant contextual knowledge for LLM based generation.

LitLLM is an interactive tool to help scientists write the literature review
or related work section of a scientific paper starting from a user-provided
abstract (see Figure 1). The specific objectives of this work are to create a
system to help users navigate through research papers and write a literature
review for a given paper or project. Our main contributions are:

  * •

We provide a system based on a modular pipeline that conducts a literature
review based on a user-proposed abstract.

  * •

We use Retrieval Augmented Generation (RAG) techniques to condition the
generated related work on factual content and avoid hallucinations using
multiple search techniques.

  * •

We incorporate sentence-based planning to promote controllable generation.

Figure 2: Schematic diagram of the modular pipeline used in our system. In the
default setup, we summarize the research abstract into a keyword query, which
is used to retrieve relevant papers from an academic search engine. We use an
LLM-based reranker to select the most relevant paper relative to the provided
abstract. Based on the re-ranked results and the user-provided summary of
their work, we use an LLM-based generative model to generate the literature
review, optionally controlled by a sentence plan.

##  2 Related Work

LLMs have demonstrated significant capabilities in storing factual knowledge
and achieving state-of-the-art results when fine-tuned on downstream Natural
Language Processing (NLP) tasks Lewis et al. (2020).

However, they also face challenges such as hallucination, outdated knowledge,
and non-transparent, untraceable reasoning processes Huang et al. (2023); Gao
et al. (2023); Li et al. (2024). These limitations have motivated the
development of RAG (Retrieval Augmented Generation), which incorporates
knowledge from external databases to enhance the accuracy and credibility of
the models, particularly for knowledge-intensive tasks Gao et al. (2023). RAG
has emerged as a promising solution to the challenges faced by LLMs. It
synergistically merges LLMs’ intrinsic knowledge with the vast, dynamic
repositories of external databases Gao et al. (2023). This approach allows for
continuous knowledge updates and integration of domain-specific information in
an attempt to limit the effect of outdated knowledge. The proposed work builds
upon the advancements around RAG to provide a more efficient solution for
academic writing.

On the other hand, there has been a notable emphasis on utilizing Large
Language Models (LLMs) for tasks related to information retrieval and ranking
Zhu et al. (2023). The work by Sun et al. (2023) leverages generative LLMs
such as ChatGPT and GPT-4 for relevance ranking in information retrieval,
demonstrating that these models can deliver competitive results to state-of-
the-art supervised methods. Pradeep et al. (2023b, a) introduce different
open-source LLM for listwise zero-shot reranking, further motivating the
proposed approach of using LLMs for reranking in our work.

The exploration of large language models (LLMs) and their zero-shot abilities
has been a significant focus in recent research. For instance, one study
investigated using LLMs in recommender systems, demonstrating their promising
zero-shot ranking abilities, although they struggled with the order of
historical interactions and position bias Hou et al. (2023). Another study
improved the zero-shot learning abilities of LLMs through instruction tuning,
which led to substantial improvements in performance on unseen tasks Wei et
al. (2021). A similar approach was taken to enhance the zero-shot reasoning
abilities of LLMs, with the introduction of an autonomous agent to instruct
the reasoning process, resulting in significant performance boosts Crispino et
al. (2023). The application of LLMs has also been explored in the context of
natural language generation (NLG) assessment, with comparative assessment
found to be superior to prompt scoring Liusie et al. (2023). In the domain of
Open-Domain Question Answering (ODQA), a Self-Prompting framework was proposed
to utilize the massive knowledge stored in LLMs, leading to significant
improvements over previous methods Li et al. (2022). Prompt engineering has
been identified as a key technique for enhancing the abilities of LLMs, with
various strategies being explored Shi et al. (2023).444This paragraph was
generated using our platform with some minor modifications based on a slightly
different version of our abstract.

##  3 Pipeline

Figure 2 provides an overview of the pipeline. The user provides a draft of
the abstract or a research idea. We use LLM to first summarize the abstract in
keywords that can be used as a query for search engines. Optionally, the users
could provide relevant keywords to improve search results. This query is
passed to the search engine, which retrieves relevant papers with the
corresponding information, such as abstracts and open-access PDF URLs. These
retrieved abstracts with the original query abstract are used as input to the
other LLM Re-ranker, which provides a listwise ranking of the papers based on
the relevance to the query abstract. These re-ranked abstracts with the
original query are finally passed to the LLM generator, which generates the
related work section of the paper. Recently, Agarwal et al. (2024) showed that
prompting the LLMs with the sentence plans results in reduced hallucinations
in the generation outputs. These plans contain information about the number of
sentences and the citation description on each line, providing control to meet
author preferences. We include this sentence-based planning in the LLM
generator as part of this system. In the following, we provide more details
about each of the modules.

Figure 3: Different retrieval strategies as discussed in Section 3.1

###  3.1 Paper Retrieval Module

In our toolkit, we retrieve relevant papers using the Semantic Scholar API.
Other platforms could be used, but the S2 Platform is well-adapted to this use
case. It is a large-scale academic corpus comprising 200M+ metadata records
across multiple research areas, providing information about papers’ metadata,
authors, paper embedding, etc. The Recommendations API also provides relevant
papers similar to any seed paper. Figure 3 shows our system’s different
strategies. We describe these three settings that we use to search for
references:

  * •

User provides an abstract or a research idea (roughly the length of the
abstract). We prompt an LLM (see Figure 4) to summarize this abstract in
keywords which can be used as a search query with most APIs.

  * •

Users can optionally also provide keywords that can improve search results.
This is similar (in spirit) to how researchers search for related work with a
search engine. This is particularly useful in interdisciplinary research, and
authors would like to include the latest research from a particular domain,
which could not be captured much in the abstract.

  * •

Lastly, any seed paper the user finds relevant enough to their idea could be
used with the Recommendations API from search engines to provide other closely
related papers.

Figure 4: Prompt used to summarize the research idea by LLM to search an
academic engine

###  3.2 Paper Re-Ranking Module

Recent efforts have explored the application of proprietary LLMs for ranking
Sun et al. (2023); Ma et al. (2023) as well as open-source models like Pradeep
et al. (2023a, b). These approaches provide a combined list of passages
directly as input to the model and retrieve the re-ordered ranking list Zhang
et al. (2023). Typically, a retriever first filters top-k potential
candidates, which are then re-ranked by an LLM to provide the final output
list. In our work, we use the instructional _permutation generation_ approach
Sun et al. (2023) where the model is prompted to generate a permutation of the
different papers in descending order based on the relevance to the user-
provided abstract, thus producing an ordered list of preferences against
providing intermediate scores. Figure 5 showcases the prompt we used for LLM-
based re-ranking.

Figure 5: Ranking prompt based on the permutation generation method Figure 6:
Prompt for Retrieval Augmented Generation

###  3.3 Summary Generation Module

We explore two strategies for generation: (1) Zero-shot generation and (2)
Plan-based generation, which relies on sentence plans for controllable
generation, described in the following

####  3.3.1 Zero-shot generation

While LLMs can potentially search and generate relevant papers from their
parametric memory and trained data, they are prone to hallucinating and
generating non-factual content. Retrieval augmented generation, first
introduced in Parvez et al. (2021) for knowledge tasks, addresses this by
augmenting the generation model with an information retrieval module. The RAG
principles have been subsequently used for dialogue generation in task-
oriented settings Thulke et al. (2021), code generation Liu et al. (2020);
Parvez et al. (2021) and product review generation Kim et al. (2020). RAG
drastically reduces hallucinations in the generated output Gao et al. (2023);
Tonmoy et al. (2024).

Our work builds upon the principles of RAG, where we retrieve the relevant
papers based on the query and augment them as context for generating the
literature review. This also allows the system to be grounded in the retrieved
information and be updated with the latest research where the training data
limits the parametric knowledge of the LLM. Figure 6 shows our system’s prompt
for effective Retrieval Augmented Generation (RAG).

####  3.3.2 Plan based generation

To get the best results from LLM, recent research shifts focus on designing
better prompts (Prompt Engineering) including 0-shot chain-of-thought
prompting Kojima et al. (2022); Zhou et al. (2022), few-shot prompting Brown
et al. (2020) techniques, few-shot Chain-of-thought prompting Wei et al.
(2022) and in-context prompting Li and Liang (2021); Qin and Eisner (2021).
However, the longer context of our problem statement (query paper and multiple
relevant papers) hinders the application of these techniques for response
generation.

We utilized sentence plan-based prompting techniques drawing upon insights
from the literature of traditional modular Natural Language Generation (NLG)
pipelines with intermediary steps of sentence planning and surface realization
Reiter and Dale (1997); Stent et al. (2004). These plans provide a sentence
structure of the expected output, which efficiently guides the LLM in
generating the literature review in a controllable fashion as demonstrated in
concurrent work (Agarwal et al., 2024). Figure 7 (in Appendix) shows the
prompt for plan-based generation with an example template as:

Please generate {num_sentences} sentences in {num_words} words. Cite {cite_x}
at line {line_x}. Cite {cite_y} at line {line_y}.

##  4 Implementation Details

We build our system using Gradio Abid et al. (2019), which provides a nice
interface to quickly and efficiently build system demos. Our user interface is
also available at HuggingFace
Space555https://huggingface.co/spaces/shubhamagarwal92/LitLLM. We query the
Semantic Scholar API available through the Semantic Scholar Open Data Platform
(Lo et al., 2020; Kinney et al., 2023) to search for the relevant papers.
Specifically, we use the Academic Graph666https://api.semanticscholar.org/api-
docs/graph and Recommendations777https://api.semanticscholar.org/api-
docs/recommendations API endpoint. In this work, we use OpenAI
API888https://platform.openai.com/docs/guides/gpt to generate results for LLM
using GPT-3.5-turbo and GPT-4 model. At the same time, our modular pipeline
allows using any LLM (proprietary or open-sourced) for different components.
We also allow the end-user to sort the retrieved papers by relevance (default
S2 results), citation count, or year.

##  5 User Experience

As a preliminary study, we provided access to our user interface to 5
different researchers who worked through the demo to write literature reviews
and validate the system’s efficacy. We also provide an example in the demo
with an abstract for a quick start. Particularly, the users found the 0-shot
generation to be more informative about the literature in general while the
plan-based generation to be more accessible and tailored for their research
paper, as also evident in our demo video.999https://youtu.be/E2ggOZBAFw0.
Table 1 (in Appendix) shows the output-related work for a recent paper Li et
al. (2023) that was randomly chosen with a number of cited papers as 4. Our
system generated an informative query Multimodal Research: Image-Text Model
Interaction and retrieved relevant papers where the top recommended paper was
also cited in the original paper. While zero-shot generation provides valuable
insights into existing literature, plan-based generation produces a more
succinct and readily usable literature review.

##  6 Conclusion and Future Work

In this work, we introduce and describe LitLLM, a system which can generate
literature reviews in a few clicks from an abstract using off-the-shelf LLMs.
This LLM-powered toolkit relies on the RAG with a re-ranking strategy to
generate a literature review with attribution. Our auxiliary tool allows
researchers to actively search for related work based on a preliminary
research idea, research proposal or even a full abstract. We present a modular
pipeline that can be easily adapted to include the next generation of LLMs and
other domains, such as news, by changing the source of retrieval information.

Given the growing impact of different LLM-based writing assistants, we are
optimistic that our system may aid researchers in searching relevant papers
and improve the quality of automatically generated related work sections of a
paper. While our system shows promise as a helpful research assistant, we
believe that their usage should be disclosed to the readers, and authors
should also observe caution in eliminating any possible hallucinations.

In the future, we would also like to explore academic search through multiple
APIs, such as Google Scholar. This work only considered abstracts of the query
paper and the retrieved papers, which creates a bottleneck in effective
literature review generation. With the advent of longer context LLMs, we
envision our system ingesting the whole paper (potentially leveraging an
efficient LLM-based PDF parser) to provide a more relevant background of the
related research. We consider our approach as an initial step for building
intelligent research assistants which could help academicians through an
interactive setting (Dwivedi-Yu et al., 2022).

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  * Zhao et al. (2022) Zijia Zhao, Longteng Guo, Xingjian He, Shuai Shao, Zehuan Yuan, and Jing Liu. 2022.  Mamo: Fine-grained vision-language representations learning with masked multimodal modeling.  _Proceedings of the 46th International ACM SIGIR Conference on Research and Development in Information Retrieval_.
  * Zhou et al. (2022) Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. 2022.  Large language models are human-level prompt engineers.  _arXiv preprint arXiv:2211.01910_.
  * Zhu et al. (2023) Yutao Zhu, Huaying Yuan, Shuting Wang, Jiongnan Liu, Wenhan Liu, Chenlong Deng, Zhicheng Dou, and Ji-Rong Wen. 2023.  Large language models for information retrieval: A survey.  _arXiv preprint arXiv:2308.07107_.

## Appendix

Figure 7: Prompt for sentence plan-based generation

In the following, we provide snippets of code to retrieve results from the
Semantic Scholar API for both recommendation and query-based search:

⬇

# QUERY BASED SEARCH

def query_search_s2(query: str,

num_papers_api: int,

fields: str):

rsp = requests.get("https://api.semanticscholar.org/graph/v1/paper/search",

headers={"X-API-KEY": S2_API_KEY},

params={"query": query, "limit": num_papers_api, "fields": fields})

rsp.raise_for_status()

results = rsp.json()

# Total papers found

total = results["total"]

papers = results["data"]

return papers

⬇

# PAPER SEARCH

def get_paper_data(paper_url: str,

fields: str):

"""

Retrieves data of one paper based on URL

"""

rsp =
requests.get(f"https://api.semanticscholar.org/graph/v1/paper/URL:{paper_url}",

headers={"X-API-KEY": S2_API_KEY},

params={"fields": fields})

results = rsp.json()

return rewsults

⬇

# RECOMMENDATION API

def get_recommendations_from_s2(

arxiv_id: str, num_papers_api: int,

fields: str):

"""

Get recommendations from S2 API

"""

query_id = f"ArXiv:{arxiv_id}"

rsp =
requests.post("https://api.semanticscholar.org/recommendations/v1/papers/",

json={

"positivePaperIds": [query_id]},

params={"fields": fields,

"limit": num_papers_api})

results = rsp.json()

papers = results["recommendedPapers"]

return papers

Abstract for paper: Do DALL-E and Flamingo Understand Each Other? Li et al.
(2023)
---
The field of multimodal research focusing on the comprehension and creation of
both images and text has witnessed significant strides. This progress is
exemplified by the emergence of sophisticated models dedicated to image
captioning at scale, such as the notable Flamingo model and text-to-image
generative models, with DALL-E serving as a prominent example. An interesting
question worth exploring in this domain is whether Flamingo and DALL-E
understand each other. To study this question, we propose a reconstruction
task where Flamingo generates a description for a given image and DALL-E uses
this description as input to synthesize a new image. We argue that these
models understand each other if the generated image is similar to the given
image. Specifically, we study the relationship between the quality of the
image reconstruction and that of the text generation. We find that an optimal
description of an image is one that gives rise to a generated image similar to
the original one. The finding motivates us to propose a unified framework to
finetune the text-to-image and image-to-text models. Concretely, the
reconstruction part forms a regularization loss to guide the tuning of the
models. Extensive experiments on multiple datasets with different image
captioning and image generation models validate our findings and demonstrate
the effectiveness of our proposed unified framework. As DALL-E and Flamingo
are not publicly available, we use Stable Diffusion and BLIP in the remaining
work. Project website: https://dalleflamingo.github.io.
LLM summarized query: Multimodal Research: Image-Text Model Interaction
Retrieved papers: (User Input: 4)
$[1]$ CoCa: Contrastive Captioners are Image-Text Foundation Models (2022)
Cited by 702 (Yu et al., 2022)
$[2]$ MAMO: Fine-Grained Vision-Language Representations Learning with Masked
Multimodal Modeling (2022) Cited by 0 (Zhao et al., 2022)
$[3]$ Dynamic Modality Interaction Modeling for Image-Text Retrieval (2021)
Cited by 88 (Qu et al., 2021)
$[4]$ WIT: Wikipedia-based Image Text Dataset for Multimodal Multilingual
Machine Learning (2021) Cited by 185 (Srinivasan et al., 2021)
Zero shot Generation (Section 3.3.1)
The field of multimodal research has seen significant advancements in recent
years, with the development of models such as Flamingo and DALL-E that focus
on image captioning and text-to-image generation respectively. However, the
question of whether these models can understand each other and work in harmony
is a topic of interest. In this context, the work of [1] presents the
Contrastive Captioner (CoCa), a model that combines contrastive loss and
captioning loss to pretrain an image-text encoder-decoder foundation model.
This model, while efficient, does not fully address the interaction between
image and text modalities at a fine-grained level. The work of [2] addresses
this gap by proposing a jointly masked multimodal modeling method that focuses
on fine-grained multimodal representations. This method performs joint masking
on image-text input and integrates both implicit and explicit targets for the
masked signals to recover. However, this approach, while effective, does not
fully address the challenges of intra-modal reasoning and cross-modal
alignment that are inherent in image-text retrieval. The work of [3] presents
a novel modality interaction modeling network based on the routing mechanism,
which is the first unified and dynamic multimodal interaction framework
towards image-text retrieval. This model can dynamically learn different
activated paths for different data, providing a more flexible approach to
modality interaction. However, the design of interaction patterns in this
model still relies heavily on expert experience and empirical feedback, which
may limit its applicability in different contexts. The work of [4] introduces
the Wikipedia-based Image Text (WIT) Dataset, a large-scale dataset for
multimodal, multilingual learning. This dataset, while providing a rich
resource for multimodal learning, does not directly address the question of
how different models can understand each other and work together. In light of
these previous works, our study proposes a reconstruction task where Flamingo
generates a description for a given image and DALL-E uses this description as
input to synthesize a new image. We argue that these models understand each
other if the generated image is similar to the given image. This approach
allows us to study the relationship between the quality of the image
reconstruction and that of the text generation, and to propose a unified
framework to finetune the text-to-image and image-to-text models. Our
extensive experiments validate our findings and demonstrate the effectiveness
of our proposed unified framework.
Plan based generation (Section 3.3.2): Generate the output using 5 sentences.
Cite [1] on line 2. Cite [2], [3] on line 3. Cite [4] on line 5.
The field of multimodal research has seen significant advancements in the
comprehension and creation of both images and text, with models like Flamingo
and DALL-E leading the way. In a similar vein, the Contrastive Captioner
(CoCa) model presented in [1] pretrains an image-text encoder-decoder
foundation model with contrastive loss and captioning loss, achieving state-
of-the-art performance on a broad range of downstream tasks. Other works have
also focused on improving the fine-grained image-text interaction, with [2]
proposing a jointly masked multimodal modeling method and [3] developing a
dynamic multimodal interaction framework for image-text retrieval. These
models, while effective, often rely on expert experience and empirical
feedback, which may limit their flexibility. The introduction of the
Wikipedia-based Image Text (WIT) Dataset in [4] provides a large, multilingual
dataset for pretraining multimodal models, offering a more diverse set of
concepts and real-world entities for model training. This diversity and scale
could potentially enhance the performance of models like Flamingo and DALL-E,
further advancing the field of multimodal research.

Table 1: We show an example generated related work for a randomly chosen
recent paper Li et al. (2023) with LLM summarized query and retrieved papers.
We show the generated related work from our system using both zero-shot and
plan-based generation, producing a more succinct and readily usable literature
review. Note: The number of citations is retrieved by Semantic Scholar at the
date of submission of this work.


#  Ab initio calculations of the hyperfine structure of 109Cd, 109Cd+ and
reevaluation of the nuclear quadrupole moment $Q$(109Cd)

Benquan Lu  National Time Service Center, 710600 Lintong, China  The
University of Chinese Academy of Sciences, 100088 Beijing, China  Xiaotong Lu
National Time Service Center, 710600 Lintong, China  The University of Chinese
Academy of Sciences, 100088 Beijing, China  Tao Wang<EMAIL_ADDRESS>Department of Physics, and Center of Quantum Materials and Devices, Chongqing
University, Chongqing, 401331, China  Hong Chang<EMAIL_ADDRESS>National Time Service Center, 710600 Lintong, China  The University of Chinese
Academy of Sciences, 100088 Beijing, China

###### Abstract

Large scale ab initio calculations of the electric contribution (i.e. the
electric field gradient) to the electric quadrupole hyperfine interaction
constants $B$ for the $5p~{}^{2}P_{3/2}$ state in 109Cd+ ion and the
$5s5p~{}^{3}P_{1,2}$ states in neutral 109Cd atom were performed. To probe the
sensitivity of $B$ to different electron correlation effects, six sets of
variational multi-configuration Dirac-Hartree-Fock and relativistic
configuration interaction calculations employing different strategies were
carried out. The calculated electric field gradients, together with
experimental values of $B$, allow us to extract a new value of the nuclear
electric quadrupole moment $Q$(109Cd) = 0.634(22) b which is about 5% larger
than the recommended value [Stone, At. Data Nucl. Data Tables 111-112, 1
(2016); Pyykkö, Mol. Phys. 116, 1328 (2018)]. Efforts were made to provide a
realistic theoretical uncertainty for the final $Q$ value based on statistical
principles and correlation with the magnetic dipole hyperfine interaction
constant $A$.

###### pacs:

32.10.Fn, 31.15.vj, 21.10.Ky

##  I INTRODUCTION

Cadmium (Z = 48), which has an open shell and two protons less than “magic”
tin, is of broad interest since its nucleus is closer to the shell and has
more characteristics. Furthermore, the knowledge of nuclear properties in the
vicinity of the tin isotopes is helpful for understanding stellar
nucleosynthesis. In comparison to the magnetic dipole moment $\mu_{I}$, the
nuclear electric quadrupole moment $Q$ can give a more rigorous test of the
nuclear shell model since it has a very typical linear behavior with regard to
the number of valence nucleons Mayer and Jensen (1955); Horie and Arima
(1955). In addition, the quadrupole moment is a unique and excellent tool to
study nuclear deformation and shape coexistence Neyens (2003), especially for
exotic nuclei in the vicinity of the proton or neutron drip line Campbell  _et
al._ (2016). The nuclear electric quadrupole moment of cadmium has been proved
to be the most revealing case to study the above phenomenons Yordanov  _et
al._ (2013, 2016). Additionally, specific questions concerning the nuclear
structure of the cadmium isotopes, such as shell quenching Kautzsch  _et al._
(2000); Dillmann  _et al._ (2003), sphericity Dworschak  _et al._ (2008),
deformation Jungclaus and Egido (2006); Rodríguez  _et al._ (2008), or whether
vibrational nuclei exist at all Garrett and Wood (2010), require the nuclear
electric quadrupole moment.

The nuclear trends of the nuclear quadrupole moments $Q$ among 12 neutron-rich
isotopes in the range 107-129Cd were studied Yordanov  _et al._ (2013). The
$Q$(Cd) values given in Ref. Yordanov  _et al._ (2013) were based on measured
atomic hyperfine structures for odd neutron number $N$ isotopes from the
collinear laser spectroscopy at ISOLDE-CERN. They were derived by combining
the measured hyperfine parameter $B$ of $5p~{}^{2}P_{3/2}$ state in cadmium
ion with the calculated electronic contribution $B/Q$ for each isotope. The
$B/Q$ values were resulted from the Dirac-Hartree-Fock calculation.
Afterwards, the nuclear quadrupole moments of 107,109,121,123Cd and
111m-123mCd were obtained from the ratio of the hyperfine interaction constant
$B$, ${}^{A}Q$/${}^{109}Q$ = ${}^{A}B$/${}^{109}B$, by using the quadrupole
moment of the long-lived 109Cd isotope as a reference Frömmgen  _et al._
(2015). The nuclear quadrupole moment of 109Cd was derived from the
combination of the measured $B$ value of $5s5p~{}^{3}P_{2}$ state in neutral
109Cd atom and the value of $B/Q$. The $B/Q$ was calculated, with an
uncertainty of 6%, using the multi-configuration Dirac-Hartree-Fock (MCDHF)
theory, in which the main first-order electron correlations were considered.
However, it has been found that it is essential to take into account the
higher-order electron correlations for achieving satisfactory accuracy of
physical quantities concerned, for example, the hyperfine structure constants
for the $5s5p~{}^{1}P_{1}$ and ${}^{3}P_{1,2}$ states in 87Sr Lu  _et al._
(2019). In addition, due to the sensitivity of the hyperfine parameters to
different electron correlation effects, it is sometimes necessary to do
multiple sets of computations, each using a different computing strategy and
correlation model, in order to assess the quality of the calculations Bieroń
_et al._ (2018). As a result, to obtain high accurate value of $B/Q$,
different computational models should be established to systematically
consider the electron correlation effects.

In this work, we performed six different computational approaches to obtain
accurate values on the $B/Q$s for the $5p~{}^{2}P_{3/2}$ state in 109Cd+ ion
and the $5s5p~{}^{3}P_{1,2}$ states in neutral 109Cd atom by using the MCDHF
theory. In our calculation, the electron correlation effects, especially the
core-core and higher-order correlations, on the $B/Q$s were investigated in
detail. Combining the calculated $B/Q$s with the available experimental $B$, a
value of the nuclear quadrupole moment of 109Cd was evaluated with an
uncertainty of 3%. The uncertainty was estimated based on statistical
principles and the error bar of the magnetic dipole hyperfine interaction
constant $A$.

##  II Thoery

In the MCDHF method, the wave function $\Psi(\Gamma PJM_{J})$ for an atomic
state is expressed as a linear combination of configuration state functions
(CSFs) $\Phi_{j}(\gamma_{j}PJM_{J})$ with the same parity $P$, total angular
momentum $J$ and its projection quantum numbers $M_{J}$, i.e.,

$\Psi(\Gamma PJM_{J})=\sum_{j}^{N}c_{j}\Phi_{j}(\gamma_{j}PJM_{J}),$ (1)

where, $c_{j}$ represents the mixing coefficient corresponding to the $j$th
configuration state function, and $\gamma$ stands for the other quantum
numbers which can define the state uniquely. The configuration state functions
$\Phi_{j}(\gamma_{j}PJM_{J})$ are built from sums of products of the one-
electron Dirac orbitals

$\phi(r,\theta,\varphi,\sigma)=\frac{1}{r}\binom{P(r)\chi_{\kappa
m}(\theta,\varphi,\sigma)}{iQ(r)\chi_{-\kappa m}(\theta,\varphi,\sigma)},$ (2)

where $P(r)$ and $Q(r)$ are the radial wavefunctions. The coefficients $c_{j}$
and the radial parts of the one-electron orbitals are optimized simultaneously
in the self-consistent field (SCF) procedure. The SCF radial equations to be
iteratively solved are derived from the application of the variational
principle on a weighted Dirac-Coulomb energy functional of the targeted atomic
states according to the extended optimal level (EOL) scheme Grant  _et al._
(1980). Once a radial orbital set has been determined, the relativistic
configuration interaction (RCI) computation in which only the expansion
coefficients are varied. The Breit interaction and quantum electrodynamic
(QED) corrections can be included in the RCI computation.

The hyperfine structure of atomic energy levels is caused by the interaction
between the electrons and the electromagnetic multipole moments of the
nucleus. The interaction Hamiltonian can be represented by a multipole
expansion

$H_{hfs}=\sum_{k\geq 1}\mathbf{T}^{(k)}\cdot\mathbf{M}^{(k)},$ (3)

where $\mathbf{T}^{(k)}$ and $\mathbf{M}^{(k)}$ are spherical tensor operators
of rank $k$ in the electronic and nuclear space, respectively. The $k$ = 1 and
$k$ = 2 terms represent the magnetic dipole and the electric quadrupole
interactions. The higher-order hyperfine interactions, including the nuclear
magnetic octupole interaction, are negligible Beloy  _et al._ (2008); Singh
_et al._ (2013). The magnetic dipole and the electric quadrupole hyperfine
interaction constants ($A$ and $B$) are defined as

$A=\frac{\mu_{I}\left\langle
PJ\left\|\mathbf{T}^{(1)}\right\|{PJ}\right\rangle}{I\sqrt{J(J+1)(2J+1)}}$ (4)

and

$B=2Q\left\\{\frac{J(2J-1)}{(J+1)(2J+3)}\right\\}^{1/2}\left\langle
PJ\left\|\mathbf{T}^{(2)}\right\|{PJ}\right\rangle.$ (5)

Here, $\mu_{I}$ and $Q$ are the magnetic dipole moment and the electric
quadrupole moment of the nucleus. The electric tensor operators
$\textbf{T}^{(1)}$ and $\textbf{T}^{(2)}$ are given by

$\textbf{T}^{(1)}=\sum_{j}-i\alpha
r_{j}^{-2}(\bm{\alpha}_{j}\cdot\mathbf{l}_{j}\mathbf{C}^{(1)}(j)),$ (6)
$\textbf{T}^{(2)}=\sum_{j}-r_{j}^{-3}\mathbf{C}^{(2)}(j),$ (7)

where $i$ is the imaginary unit, $r_{j}$ is the radial coordinate of the $j$th
electron, $\mathbf{l}$ is the orbital angular momentum operator,
$\mathbf{C}^{(k)}$ is a spherical tensor of rank $k$, $\alpha$ is the fine
structure constant and $\bm{\alpha}_{j}$ is the Dirac matrix.

According to Eq. (5), the nuclear electric quadrupole moment (in mb) can be
extracted by

$Q=\frac{4.2559579}{\mathrm{EFG}}B,$ (8)

where the hyperfine interaction constant $B$ and the electric field gradient
(EFG) which represents the electric charge distribution in the vicinity of the
nucleus are expressed in MHz and a.u., respectively. In the present work, we
will monitor the related $B/Q\propto$ EFG ratio value (in MHz/b) instead of
the EFG.

##  III Calculations of $A$ and $B/Q$

In this work, we adopted the active space approach to capture electron
correlations Roos  _et al._ (1980); Olsen  _et al._ (1988). According to the
perturbation theory, the electron correlations can be divided into the first-
order and the higher-order correlations Fischer  _et al._ (1997); Li  _et al._
(2012). The first-order electron correlation effect, composed of valence-
valence (VV), core-valence (CV) and core-core (CC) correlations, is captured
by the CSFs generated through the single (S) and double (D) excitations from
the occupied orbitals in the single reference configuration set. The higher-
order electron correlations are usually considered in two ways Zhang  _et al._
(2017). One is to add CSFs generated by the triple (T) and quadruple (Q)
excitations from the mono-reference configuration and the other is to include
those produced through the SD excitations from the multi-reference (MR)
configurations. By using the TQ-excitation method, the higher-order electron
correlations are more difficult to deal with as the number of CSFs grows
rapidly and easily goes beyond the current computer resources. Alternatively,
the SD-excitation MR approach was adopted to estimate the higher-order
electron correlations. In this method, the CSFs with large mixing coefficients
in the first-order correlation were selected to form the MR configuration set.
The CSFs generated by SD excitations from the MR configuration set can capture
the main higher-order electron correlations. In this calculation, the
outermost $5s$ and $5p$ electrons in the reference configuration were treated
as the valence electrons and the others the core.

###  III.1 Cd+ ion

For calculating the hyperfine interaction constants of the
$5p~{}^{2}P_{1/2,3/2}$ states in Cd+ ion, two different approaches were used.
In the first approach, called optimal level 1 (OL1), the wave function for the
${}^{2}P_{3/2}$ state was optimized independently. In the second approach,
called optimal level 2 (OL2), the wave function was generated with the EOL
form. In the OL1 scheme, the energy of the ${}^{2}P_{3/2}$ state is optimized
to minimize in variational calculations, but the OL2 scheme can expand the
calculations to cover the ${}^{2}P_{1/2,3/2}$ states in the $5p$
configuration.

Our calculation was started in the Dirac-Hartree-Fock (DHF) approximation. The
occupied orbitals in the reference configuration, or called spectroscopic
orbitals, were optimized and kept frozen in the following computations. The VV
and CV correlations between the electrons in the core and valence electron,
labeled as CV model, were considered in the SCF procedure through the
configuration space expanded by SD-excitation CSFs from the reference
configuration $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}4d^{10}5p$.
The SD-excitation is restricted that at most one electron is excited from the
core shells. The occupied shells were successively opened for substitutions
into the virtual set, starting with $4spd$, followed by $3spd$, and then by
$2sp$ and $1s$. Meanwhile, the virtual orbital (VO) is added layer by layer,
and only the added virtual orbitals were variable. In order to fully consider
the VV and CV correlations, seven layers of VOs were augmented to make sure
the convergence of parameters under investigation. In Table 1, the
progressively opened core shells and the expanded VOs are presented. The
numbers of configuration state functions (NCSFs) for states in the $5p$
configuration generated in various computational models are also shown in this
table. To raise computational efficiency, the CSFs which do not interact with
the reference configurations were removed Jönsson  _et al._ (2007); Fischer
_et al._ (1997). The orbital set obtained in the CV model was used for the
subsequent RCI calculation.

Table 1: The numbers of configuration state functions (NCSF) in various computational models. |  |  |  | NCSF |
---|---|---|---|---|---
Reference configurations | VO | Model |  $J$=1/2 |  $J$=3/2 |
{$5p$} |  | DHF | 1 | 1 |
{$4s^{2}4p^{6}4d^{10}5p$} | {5s, 6p, 5d, 4f} |  | 117 | 207 |
{$3s^{2}\cdots 5p$} | {6s, 7p, 6d, 5f, 5g} |  | 877 | 1607 |
{$2s^{2}\cdots 5p$} | {7s, 8p, 7d, 6f, 6g, 6h} |  | 2615 | 4849 |
{$1s^{2}\cdots 5p$} | {8s, 9p, 8d, 7f, 7g, 6h} |  | 4816 | 8871 |
{$1s^{2}\cdots 5p$} | {9s,10p, 9d, 8f, 8g, 6h} |  | 7428 | 13695 |
{$1s^{2}\cdots 5p$} | {10s,11p,10d, 9f, 8g, 6h} |  | 9973 | 18305 |
{$1s^{2}\cdots 5p$} | {11s,12p,11d, 9f, 8g, 6h} | CV | 11998 | 21853 |
{$4s^{2}4p^{6}4d^{10}5p$} | {11s,12p,11d, 9f, 8g, 6h} | CC | 67050 | 171725 |
{$4s^{2}4p^{6}4d^{10}5p$} | {9s, 8p, 6d, 5f, 5g} | Tri | 2442229 | 4427350 |
|  |  |  $J$=0 |  $J$=1 |  $J$=2
{$5s5p$} |  | DHF | 1 | 2 | 1
{$4s^{2}4p^{6}4d^{10}5s5p$} | {6s, 6p, 5d, 4f} |  | 246 | 871 | 787
{$3s^{2}\cdots 5s5p$} | {7s, 7p, 6d, 5f, 5g} |  | 1793 | 7440 | 7229
{$2s^{2}\cdots 5s5p$} | {8s, 8p, 7d, 6f, 6g, 6h} |  | 5285 | 23196 | 23024
{$1s^{2}\cdots 5s5p$} | {9s, 9p, 8d, 7f, 7g, 6h} |  | 9677 | 43143 | 42946
{$1s^{2}\cdots 5s5p$} | {10s,10p, 9d, 8f, 8g, 6h} |  | 14934 | 67678 | 67571
{$1s^{2}\cdots 5s5p$} | {11s,11p,10d, 9f, 8g, 6h} |  | 20078 | 91403 | 90816
{$1s^{2}\cdots 5s5p$} | {12s,12p,11d, 9f, 8g, 6h} | CV | 24178 | 109719 | 107940
{$4s^{2}4p^{6}4d^{10}5s5p$} | {12s,12p,11d, 9f, 8g, 6h} | CC | 58058 | 446978 | 433156
+{$4s^{2}4p^{6}4d^{8}5s5p5d^{2}$; | {8s, 7p, 6d, 4f} | MR | 2021707 | 6405941 | 9432299
$4s^{2}4p^{6}4d^{10}5p6s$; |  |  |  |  |
$4s^{2}4p^{6}4d^{10}5p5d$; |  |  |  |  |
$4s^{2}4p^{6}4d^{10}5s6d$} |  |  |  |  |

The CC electron correlation related to the $n=4$ shell was estimated with the
CSFs generated by SD excitations from the $n=4$ core shell to the full layers
of the virtual orbitals. This computational model is marked as CC in Table 1.
As analyzed in Ref. Porsev and Derevianko (2006); Bieroń  _et al._ (2008,
2009), the three-particle effects are often very important for the hyperfine
interactions. Therefore, the contributions of the higher-order electron
correlation effects were considered by triple excitations from the $n=4,5$
core shell in this work. The CSF space is further expanded by triple
excitations from the single reference configuration to the first three layers
of the virtual orbitals, which is labeled as ‘Tri’. In practice, the GRASP2018
package Fischer  _et al._ (2019) was employed to perform calculations.

In Table 2, the values of $A$ and $B/Q$ for ${}^{2}P_{1/2}$ and
${}^{2}P_{3/2}$ states in 109Cd+ ion in various computational models are
shown. Previous experimental measurement of the hyperfine interaction constant
$A$ and theoretical calculation of the $B/Q$ have also been displayed. From
the table, we can see that the calculated results from the OL1 and OL2
approaches are almost the same. It also can be seen that the VV and CV
correlations are the dominant correlation effects for the hyperfine
parameters. In Fig. 1, the values of $A$ constants and $B/Q$ for the
${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ states in the OL2 approach as functions of
the virtual layers are presented. From the figures, it can be seen that the VV
and CV correlation effects on the hyperfine interaction constants are
converged when seven virtual layers were added. The CC correlation in the
$n=4$ core shell is also important. It contributes about 24% for the hyperfine
constants $A$ of the ${}^{2}P_{1/2}$ state and 27% for the ${}^{2}P_{3/2}$
state. For $B/Q$, its contribution is about 26% for the ${}^{2}P_{3/2}$ state.

Figure 1: The values of $A$ constants and $B/Q$ for ${}^{2}P_{1/2}$ and
${}^{2}P_{3/2}$ states as functions of the virtual layers in the CV model (OL2
approach).

As is well known that three-particle effects are important for the hyperfine
structure Bieroń  _et al._ (2018); Jönsson  _et al._ (1996). To consider the
higher-order electron correlations, we adopted the triple substitutions from
the $n=4,5$ shells. It can be observed that the contribution of the triple
substitutions on the hyperfine interaction constants are about 6%, opposite to
the contribution of the CC correlation. The calculated $A$ value for the
${}^{2}P_{3/2}$ state obtained in the ‘Tri’ model is closer to the
experimental measurement. The four-particle effects on the values of $A$ and
$B/Q$ were estimated to be about 0.2% and 0.1%, which determined that the
higher-order electron correlations considered by the triple substitutions are
sufficient. Therefore, the CC and higher-order electron correlations cannot be
neglected for accurately calculating the hyperfine parameters in Cd+. In
addition, it was found that the Breit interaction and QED corrections have
negligible effects on the hyperfine structures.

Our calculated value of $A$ for the ${}^{2}P_{3/2}$ state agrees to about the
6% level the experimental measurement Yordanov  _et al._ (2013). This rough
agreement determined that our computational models are reasonable. The error
bar of the magnetic dipole hyperfine interaction constant $A$ can be used as a
source for estimating the uncertainty of the calculated $B/Q$, since they both
have similar $r^{-3}$ dependence on the radial part of the electronic wave
function Bieroń  _et al._ (2005); Bieroń and Pyykkö (2001). Additionally, our
calculated value of $B/Q$ for the ${}^{2}P_{3/2}$ state lies in the range of
the result with error bar given by Yordanov et al. Yordanov  _et al._ (2013)
using the Dirac-Hartree-Fock calculation.

Table 2: $A$ (MHz), $B/Q$ (MHz/b) and $Q$ (b) values for the ${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ states in 109Cd+ ($I^{\pi}$ = 5/2+ and $\mu$ = $-$0.8278461(15)$\mu_{N}$ Frömmgen  _et al._ (2015)) calculated with OL2 and OL1 (in parentheses) approaches in various computational models. The $Q$ values are extracted from the relation $Q=B_{expt}/(B/Q)$, where the experimental value is $B_{expt}(^{2}P_{3/2})$ = 403(1) Yordanov  _et al._ (2013). | ${}^{2}P_{1/2}$ | ${}^{2}P_{3/2}$
---|---|---
Models |  $A$ (MHz) |  $A$ (MHz) |  $B/Q$ (MHz/b) |  $Q$ (b)
DHF |  $-$506.23 |  $-$78.63($-$78.54) | 454.85(454.33) | 0.886(0.887)
CV |  $-$728.72 |  $-$124.96($-$124.58) | 755.72(754.57) | 0.533(0.534)
CC |  $-$586.82 |  $-$98.08($-$97.71) | 599.81(598.40) | 0.672(0.673)
Tri |  $-$613.76 |  $-$104.52($-$104.28) | 640.94(640.76) | 0.629(0.629)
Others |  |  $-$111.4(2)111Measured by Yordanov et al. Yordanov  _et al._ (2013) | 666(27)222Calculated by the Dirac-Hartree-Fock calculation Yordanov  _et al._ (2013) |

###  III.2 Cd atom

Similar to the calculations in Cd+, two different approaches were also used
for the hyperfine structure calculations for the $5s5p~{}^{3}P_{1,2}$ states
in neutral Cd atom. In the first approach (OL1), the wave functions for the
${}^{3}P_{1}$ and ${}^{3}P_{2}$ states were optimized independently. In the
second approach (OL4), the wave functions were optimized for all four states
of the $5s5p$ configuration ($5s5p~{}^{3}P_{0},^{3}P_{1},^{3}P_{2},^{1}P_{1}$)
in the EOL approach, with standard weights.

After the calculation in the DHF approximation, the VV and CV correlations
were considered in the SCF procedure through the configuration space expanded
by restricted SD-excitation CSFs from the $5s5p$ configuration. The restricted
double excitations were described as the same as the calculations in Cd+. And
the occupied shells in the reference configuration were opened in succession
while the virtual orbital is added layer by layer. The reference
configuration(s), the VOs and the NCSFs for states in the $5s5p$ configuration
generated in different computational models are also presented in Table 1.

Furthermore, the CC correlation in the $n=4$ core shell was included as well
in the RCI calculation. In this calculation, the configuration space was
expanded with SD excitations from the occupied $n$ = 4 core shell to the full
layers of virtual orbitals. The MR-SD model was applied to estimate the
higher-order electron correlation effects among the $n$ = 4, 5 shells. As
mentioned earlier, the MR configuration set was formed by selecting the
dominant CSFs in the CC model, i.e., those CSFs with mixing coefficients
$c_{j}$ larger than 0.04. In this work, the MR-configuration is composed of
{$4s^{2}4p^{6}4d^{8}5s5p5d^{2}$; $4s^{2}4p^{6}4d^{10}5p6s$;
$4s^{2}4p^{6}4d^{10}5p5d$; $4s^{2}4p^{6}4d^{10}5p6d$}. The SD excitations were
allowed from the MR configurations to three layers of virtual orbitals. This
model is marked as MR.

The values of $A$ and $B/Q$ for the $5s5p~{}^{3}P_{1,2}$ states in 109Cd atom
in various computational models are shown in Table 3. The previous
experimental measurement of the hyperfine interaction constant $A$ and
theoretical calculation of the $B/Q$ have been displayed as well. From the
table, we can see that there is a difference between the calculated results
from the two approaches (OL1 and OL4). For constant $A$, the difference is
less than 1%, but about 6% for $B/Q$. The VV and CV correlations make the main
contribution to the hyperfine interaction constants of both states. In Fig. 2,
we present the values of $A$ constants and $B/Q$s for the ${}^{3}P_{1}$ and
${}^{3}P_{2}$ states in the OL4 approach as functions of the virtual layers.
After optimizing seven layers of correlation orbitals, the VV and CV
correlation effects on the hyperfine interaction constants were converged. The
CC correlation in the $n=4$ core shell changes the $A$ constants by a factor
of 19% for the ${}^{3}P_{1}$ and 14% for the ${}^{3}P_{2}$ states, and 37% for
the $B/Q$ values of the two states. We can also observe that the higher-order
electron correlations, estimated in the MR-SD model, make an opposite
contribution to the CC correlation effects, which was also observed in Ref. Lu
_et al._ (2019) for Sr I, Ref. Bieroń  _et al._ (2009) for Au I and Ref. Zhang
_et al._ (2017) for the Al+ ion. Therefore, the CC correlation and the higher-
order electron correlations cannot be neglected for estimating the
uncertainty. To the best of our knowledge, there is no experimental value for
the ${}^{1}P_{1}$ state, so we exclude this state as a reference to extract
the $Q$ value.

Figure 2: The values of $A$ constants and $B/Q$s for ${}^{3}P_{1}$ and
${}^{3}P_{2}$ states as functions of the virtual layers in the CV model (OL4
approach).

Fr$\rm{\ddot{o}}$mmgen et al. Frömmgen  _et al._ (2015) also calculated the
values of $A$ and $B/Q$ for the ${}^{3}P_{2}$ state by using the MCDHF method.
By following the calculation model presented in Ref. Frömmgen  _et al._
(2015), our calculated value of $A$ constant obtained in the CV model agrees
with theirs, but the $B/Q$ value is about 9% larger. The reason for this
difference is unclear at present. Our values of $A$ for ${}^{3}P_{1}$ and
${}^{3}P_{2}$ states in the final model are in close agreement with the
experimental measurements. This determined that our computational model is
reasonable.

Table 3: $A$ (MHz), $B/Q$ (MHz/b) and $Q$ (b) values for the $5s5p~{}^{3}P_{1,2}$ states in 109Cd ($I^{\pi}$ = 5/2+ and $\mu$ = $-$0.8278461(15)$\mu_{N}$ Frömmgen  _et al._ (2015)) calculated with OL4 and OL1 (in parentheses) approaches in various computational models. The $Q$ values are extracted from the relation $Q=B_{expt}/(B/Q)$, where the experimental values are $B_{expt}(^{3}P_{1})$ = $-$165.143(5) Thaddeus and McDermott (1963) and $B_{expt}(^{3}P_{2})$ = 296(2) Frömmgen  _et al._ (2015). Models | ${}^{3}P_{1}$ | ${}^{3}P_{2}$
---|---|---
|  $A$ (MHz) |  $B/Q$ (MHz/b) |  $Q$ (b) |  $A$ (MHz) |  $B/Q$ (MHz/b) |  $Q$ (b)
DHF |  $-$884.04($-$882.04) |  $-$156.64($-$191.53) | 1.054(0.862) |  $-$761.48($-$735.71) | 272.28(334.86) | 1.087(0.884)
CV |  $-$1255.68($-$1258.46) |  $-$299.52($-$306.47) | 0.551(0.539) |  $-$998.23($-$998.56) | 530.31(545.83) | 0.558(0.542)
CC |  $-$1057.17($-$1056.36) |  $-$218.90($-$251.13) | 0.754(0.658) |  $-$876.28($-$860.49) | 385.86(446.97) | 0.767(0.662)
MR |  $-$1118.03($-$1120.47) |  $-$254.77($-$266.15) | 0.648(0.620) |  $-$910.06($-$904.97) | 452.06(474.03) | 0.655(0.624)
Others |  $-$1148.784(7)111Measured by Thaddeus and Mcdermott Thaddeus and McDermott (1963) |  |  |  $-$917.5(1)222Measured by Fr$\rm{\ddot{o}}$mmgen et al. Frömmgen  _et al._ (2015) |  500(30)333Calculated by the MCDHF method Frömmgen  _et al._ (2015) |

##  IV The nuclear electric quadrupole moment of 109Cd

Combining the calculated $B/Q$ of the ${}^{2}P_{3/2}$ state in Cd+ ion and the
${}^{3}P_{1}$, ${}^{3}P_{2}$ states in Cd atom with the measured hyperfine
interaction constants $B$(${}^{2}P_{3/2}$) = $-$403(1) MHz Yordanov  _et al._
(2013), $B$(${}^{3}P_{1}$) = $-$165.143(5) MHz Thaddeus and McDermott (1963)
and $B$(${}^{3}P_{2}$) = 296(2) MHz Frömmgen  _et al._ (2015), six values of
the nuclear electric quadrupole moment $Q$(109Cd) were obtained. The estimated
results are also presented in Tables 2 and 3.

The scattering of these six values gives us an information about the
dependence of the calculated values of $B/Q$ on the choice of our
computational methods and provides a basis for estimating the error bar of the
nuclear quadrupole moment $Q$(109Cd). The estimated values of $Q$ are
distributed around their median value ($Q$ = 0.629 b), with a rather flat
distribution. For an estimate of the accuracy, we obtained a value of $Q$ =
0.634 b on average of the six evaluations with one standard deviation of 14
mb. The difference between the median and average results is negligible
compare to the error bar.

As mentioned above, the difference between the calculated and experimental
values of the magnetic dipole hyperfine constant
$\left|A_{Cal.}-A_{Exp.}\right|$ may be used as another tool to estimate the
error bar for the determination of $Q$ Bieroń  _et al._ (2009, 2018); Lu  _et
al._ (2019); Schiffmann  _et al._ (2020). Considering the uncertainties of $A$
for the ${}^{2}P_{3/2}$, and ${}^{3}P_{1,2}$ states, an average error bar of
22 mb is obtained. We chose the larger of the above two error bars and finally
proposed that $Q$(109Cd) = 0.634(22) b.

Table 4 shows the available nuclear electric quadrupole moments of the 109Cd
nucleus for comparison. To our knowledge, Laulainen and McDermott Laulainen
and McDermott (1969) first reported a nuclear electric quadrupole moment of
109Cd, based on their measurement on the hyperfine-structure splitting of the
$5s5p~{}^{3}P_{1}$ state by the optical double-resonance technique and a
simple calculation on the EFG in the semi-empirical approximation. This value
has been used as a reference for almost 50 years Stone (2005). Yordanov et al.
Yordanov  _et al._ (2013) calculated the value of $B/Q$ for the
$5p~{}^{2}P_{3/2}$ state in 109Cd+ ion by using the Dirac-Hartree-Fock
calculation. The $Q$ value was extracted from the combination of the
measurement of $B$ using the collinear laser spectroscopy and the calculation
of $B/Q$. This value was mentioned in the review from Pyykkö Pyykkö (2018) and
Stone Stone (2016). Most recently, Fr$\rm{\ddot{o}}$mmgen et al. Frömmgen  _et
al._ (2015) also reported a value $Q$ = 0.592(36) based on the $B/Q$ value of
the $5s5p~{}^{3}P_{2}$ state in the 109Cd atom given by the MCDHF theory. This
value matches the quoted value of $Q$ = 0.604(25) b. Only the main first-order
electron correlations were included in their calculations, and the corrections
resulting from unrestricted double and triple substitutions were assessed to
be smaller than the uncertainty of $B/Q$. Compared to the two previous values,
our value is about 5% larger and the error bar is relatively smaller. It
should be stressed that the nuclear quadrupole moment of 109Cd nucleus was
extracted from the hyperfine structures of atomic and ionic cadmium. The
values of $B/Q$ were calculated in different theoretical frameworks with
detailed consideration of the electron correlation effects, especially the CC
and the higher-order electron correlations.

Table 4: $Q$ value from various works in barn (b). |  $Q$(b) | Atomic systems
---|---|---
Laulainen and McDermott Laulainen and McDermott (1969) | 0.69(7) |  $5s5p~{}^{3}P_{1}$ 109Cd atom
Yordanov et al. Yordanov  _et al._ (2013) | 0.604(25) |  $5p~{}^{2}P_{3/2}$ 109Cd+ ion
Fr$\rm{\ddot{o}}$mmgen et al. Frömmgen  _et al._ (2015) | 0.592(36) |  $5s5p~{}^{3}P_{2}$ 109Cd atom
This work | 0.634(22) |  $5p~{}^{2}P_{3/2}$ 109Cd+ ion
|  |  $5s5p~{}^{3}P_{1,2}$ 109Cd atom

##  V Conclusion

By using the MCDHF theory, six different computational approaches were
employed to determine the expectation values of the $B/Q$s for the
$5p~{}^{2}P_{3/2}$ state in 109Cd+ ion and the $5s5p~{}^{3}P_{1,2}$ states in
neutral 109Cd atom. Different electron correlation effects on the hyperfine
interaction constants $A$ and the $B/Q$s, especially for the correlations
related to the core shells and higher-order electron correlations, were
considered systematically. It was found that the contributions of the CC and
the higher-order electron correlations on the $B/Q$s are remarkable and offset
each other partly. Therefore, it is essential to take into account both of
them in order to achieve high accuracy of hyperfine parameters. Combining the
calculated $B/Q$s with the precise experimental values of the electric
quadrupole hyperfine interaction constant $B$, we determined the nuclear
quadrupole moment of 109Cd, $Q$ = 0.634(22) b. Our result was obtained based
on a combination of neutral atomic and single ionic systems. We obtained a
reliable uncertainty for the $Q$ value which was controlled at 3% level by
accounting for statistical principles and the correlation with the magnetic
dipole hyperfine interaction constant $A$. The nuclear electric quadrupole
moment of 109Cd isotope can be used as a reference for determining the
quadrupole moment of the other isotopes of Cd, especially the Cd isomers.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
under Grant No. 61775220, the Strategic Priority Research Program of the
Chinese Academy of Sciences under Grant No. XDB21030100, the Key Research
Project of Frontier Science of the Chinese Academy of Sciences under Grant No.
QYZDB-SSW-JSC004, and the West Light Foundation of the Chinese Academy of
Sciences under Grant No. XAB2018B17.

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