# Automating the Enterprise with Foundation Models

Michael Wornow*  Stanford University<EMAIL_ADDRESS>,  Avanika Narayan
Stanford University<EMAIL_ADDRESS>,  Krista Opsahl-Ong  Stanford
University<EMAIL_ADDRESS>,  Quinn McIntyre  Stanford University
<EMAIL_ADDRESS>,  Nigam H. Shah  Stanford University<EMAIL_ADDRESS>and
Christopher Ré  Stanford University<EMAIL_ADDRESS>
###### Abstract.

Automating enterprise workflows could unlock $4 trillion/year in productivity
gains. Despite being of interest to the data management community for decades,
the ultimate vision of end-to-end workflow automation has remained elusive.
Current solutions rely on process mining and robotic process automation (RPA),
in which a bot is hard-coded to follow a set of predefined rules for
completing a workflow. Through case studies of a hospital and large B2B
enterprise, we find that the adoption of RPA has been inhibited by high set-up
costs (12-18 months), unreliable execution (60% initial accuracy), and
burdensome maintenance (requiring multiple FTEs). Multimodal foundation models
(FMs) such as GPT-4 offer a promising new approach for end-to-end workflow
automation given their generalized reasoning and planning abilities. To study
these capabilities we propose ECLAIR, a system to automate enterprise
workflows with minimal human supervision. We conduct initial experiments
showing that multimodal FMs can address the limitations of traditional RPA
with (1) near-human-level understanding of workflows (93% accuracy on a
workflow understanding task) and (2) instant set-up with minimal technical
barrier (based solely on a natural language description of a workflow, ECLAIR
achieves end-to-end completion rates of 40%). We identify human-AI
collaboration, validation, and self-improvement as open challenges, and
suggest ways they can be solved with data management techniques. Code is
available at: https://github.com/HazyResearch/eclair-agents

**footnotetext: Denotes equal contribution

##  1\. Introduction

Figure 1. Differences between ECLAIR and traditional RPA. ECLAIR uses FMs to
learn expertise via video demonstrations (left), navigate GUIs given written
documentation (center), and audit completed workflows (right).

Digital workflows are the core of our modern economy, with 92% of jobs now
requiring digital skills (Bergson-Shilcock and Taylor, 2023). Many workflows
can and should be automated — across industries, workers average 3 hrs/day
doing repetitive digital workflows tangential to their core jobs (Anywhere,
2020), a phenomenon referred to as ”death by 1,000 clicks” (Schulte and Fry,
2019).

For example, consider a large B2B enterprise with tens of thousands of
customers (full case study in Section 3.2). A key accounting workflow is
ingesting customer contracts into a centralized database. After receiving a
contract via email, an analyst must manually extract relevant data and enter
it into an enterprise resource planning system (e.g. NetSuite, SAP, etc.)
before they can conduct downstream analyses. Doing this at enterprise scale
requires hundreds of individuals, each potentially taking 40 minutes per
contract.

As illustrated above, most enterprise workflows involve data integration,
ingestion, and transformation, and the market for process management is
projected to reach $65 billion by 2032 (Fahland et al., 2024). Decades of data
management work has gone into understanding and managing these workflows
(Casati and Shan, 2000; Sayal et al., 2002; Zeng et al., 2001; Hull et al.,
2013; Georgakopoulos et al., 1995; Jennings et al., 1998), yet the ultimate
vision of end-to-end automation — from understanding to execution to
monitoring — has remained elusive.

The current best-in-class solution is Robotic Process Automation (RPA), in
which workflows identified via process mining get manually encoded into a
fixed set of rules for a program to follow (Leno et al., 2021; Moreira et al.,
2023; Muthusamy et al., 2023). Narrowly scoped RPA deployments can have ROIs
of 30-200% and 2x the speed of workflows (Lhuer, 2016; da Costa et al., 2023).
However, more widespread adoption of RPA has been limited by three key failure
modes (Sarilo-Kankaanranta and Frank, 2021) which surfaced in interviews with
technology leaders at a hospital and B2B enterprise (full case studies in
Section 3):

  * •

High set-up costs: The desired workflow must be demonstrated to the RPA bot.
This is costly, as it requires a trained specialist to map workflows, write
automation scripts, and integrate with IT infrastructure (Moreira et al.,
2023; Perdana et al., 2023; Hull et al., 2013). A leading RPA vendor estimates
that 3-6 months of experience is needed to become proficient in RPA (UIPath,
2022). In our B2B case study, it took 12 months to go from project kickoff to
deployment.

  * •

Brittle execution: The RPA bot must execute the workflow. Since RPA relies on
hard-coded rules, bots cannot adapt to slight variations in input (e.g., a
button changing location on a screen, or a form field being renamed) (Ye et
al., 2023; Wewerka and Reichert, 2020; Chakraborti et al., 2020). This leads
to a “death by a thousand cuts” as the space of possibilities is essentially
unbounded. In our B2B case study, the RPA bot was initially only 60% accurate
and took 6 months of improvement to reach 95%.

  * •

Burdensome maintenance: RPA deployments often require human oversight to
validate outputs and fix edge cases. In our B2B case study, the bot required
continual monitoring by 2 full-time equivalents (FTEs).

The common cause behind these shortcomings is the difficulty of encoding
”tacit” (i.e. difficult to define) human workflow expertise into a rule-based
system like RPA (Muthusamy et al., 2023; Autor, 2014; Brynjolfsson et al.,
2023). Automating most workflows requires planning a sequence of actions and
executing them on a graphical user interface (GUI). This requires (a) visual
understanding, e.g. identifying a button on a screen; (b) real-time decision
making, e.g. knowing to scroll to locate a form field that has shifted
location; and (c) common sense to error correct, e.g. hitting escape when an
irrelevant pop-up appears.

Multimodal foundation models (FMs) such as GPT-4 (OpenAI, 2023) have
demonstrated visual understanding (Bavishi et al., 2023; Wang et al., 2023b;
Zhang et al., 2023) and generalized reasoning abilities (Yao et al., 2022; Wei
et al., 2022; Ahn et al., 2022; Wang et al., 2023c) for automating simple
digital workflows (Zheng et al., 2024; Gur et al., 2023; Yang et al., 2023a;
Yan et al., 2023; Zhang et al., 2024; Wu et al., 2024; Hong et al., 2023a).
This offers the possibility of sidestepping the failure modes of traditional
RPA, just as deep learning eclipsed rule-based approaches over the past decade
in machine learning. We thus ask the question: Can multimodal foundation
models automate enterprise workflows?

We take a first natural step in studying the opportunities and challenges of
applying multimodal FMs across all three stages of traditional RPA by
proposing ECLAIR – “Enterprise sCaLe AI for woRkflows”. As shown in Figure 1,
our system is defined as follows:

  1. (1)

Demonstrate: ECLAIR uses multimodal FMs to learn from human workflow expertise
by watching video demonstrations and reading written documentation. This
lowers set-up costs and technical barriers to entry. Initial experiments show
that ECLAIR can identify every step of a workflow based on screenshots from a
demonstration with 93% accuracy.

  2. (2)

Execute: ECLAIR observes the state of the GUI and plans actions by leveraging
the reasoning and visual understanding abilities of FMs (Yao et al., 2022;
Bavishi et al., 2023; OpenAI, 2023). Based solely on written documentation of
a workflow, ECLAIR improves end-to-end completion rates over an existing GPT-4
baseline from 0% to 40% on a sample of 30 web navigation tasks (Zhou et al.,
2023). However, this is still far from the accuracy needed for enterprise
settings, and we identify opportunities to close this gap.

  3. (3)

Validate: ECLAIR utilizes FMs to self-monitor and error correct. This reduces
the need for human oversight. When classifying whether a workflow was
successfully completed, ECLAIR achieves a precision of 90% and recall of 84%.

Our initial evaluations also identify several patterned failure modes for
future research. In Execute, ECLAIR has difficulty decomposing higher-level
steps into discrete actions (e.g. breaking ”Search XXX” into the sequence of
”click”, ”type XXX”, ”press enter”) and grounding actions to specific GUI
elements (e.g. differentiating two buttons with the same label). In Validate,
ECLAIR’s lack of heuristics for navigating GUIs make step-level validation
challenging (e.g. checking that a text field is first focused before typing).

While there is still progress to make, we are excited by the potential of
ECLAIR to automate entirely new categories of workflows that require real-time
decision-making, interaction with GUIs, and ”tacit” domain knowledge (Autor,
2014), as outlined in Figure 2. McKinsey estimates this could double the
amount of knowledge work that can be automated (Chui et al., 2023).

The rest of the paper is structured as follows. In Section 2, we discuss
related work on process mining, RPA, and applying FMs to workflow automation.
In Section 3, we provide case studies of a hospital and large B2B enterprise
which highlight the limitations of RPA. In Section 4, we outline how ECLAIR
can address these shortcomings. We conclude in Section 5 with a discussion of
future work and opportunities for the data management community.

Our contributions are: (1) two case studies highlighting the limitations of
process mining / RPA, (2) a framework, ECLAIR, for achieving end-to-end
enterprise workflow automation with multimodal FMs, (3) evaluations of ECLAIR
on 30 workflows involving enterprise web applications, and (4) proposals for
applying data management techniques to workflow automation.

##  2\. Background

We survey related work on RPA, process mining, data management tools for
workflow automation, and foundation models (FMs), then detail the problem
setting that ECLAIR seeks to solve.

Figure 2. ECLAIR can automate entirely new categories of workflows, such as
those that contain hard-to-describe steps, require complex decision making, or
are knowledge intensive. Listed examples are real-world hospital workflows
(see Section 3.1).

###  2.1. Related Work

Significant effort has gone into developing tools for understanding and
automating workflows.

Foundation Models (FMs) are deep learning models trained on large datasets
which can be adapted to a broad range of downstream tasks (Bommasani et al.,
2021). They have demonstrated robust world knowledge (Safavi and Koutra, 2021;
Jiang et al., 2020), reasoning (Yao et al., 2022; Wei et al., 2022), and
planning abilities (Ahn et al., 2022; Shen et al., 2023; Di Palo et al., 2023;
Wang et al., 2023c), and have achieved state-of-the-art results on data
processing tasks such as integration and cleaning (Narayan et al., 2022;
Kayali et al., 2023).

Process Mining is the identification and improvement of workflows based on
observational data (Van der Aalst, 2014; Reinkemeyer, 2020; Augusto et al.,
2018). Recent works applied FMs to process mining tasks such as Petri net
generation, workflow understanding, and process improvement (Fahland et al.,
2024; Rizk et al., 2023; Dumas et al., 2023; Vidgof et al., 2023; Berti and
Qafari, 2023; Grohs et al., 2023; Muthusamy et al., 2023), but were limited to
small case studies and unimodal models.

Robotic Process Automation (RPA) is the leading approach for automating
enterprise workflows (Dahabiyeh and Mowafi, 2023; Ivančić et al., 2019). In
RPA, a human manually defines a set of rules that a bot then follows to
accomplish a specific workflow (Ivančić et al., 2019; Chakraborti et al.,
2020; Wewerka and Reichert, 2020). These fixed rulesets make RPA brittle (e.g.
failing if a form changes the ordering of its fields) and difficult to
maintain (Fernandez and Aman, 2021; Moreira et al., 2023). FMs offer a
compelling alternative due to their robust reasoning capabilities (Yao et al.,
2022; Ahn et al., 2022) and ability to navigate GUIs (Hong et al., 2023a).
Initial FM-based approaches (Deng et al., 2023; Gur et al., 2023; Liu et al.,
2023) utilized Large Language Models (LLMs) to act on websites. Since LLMs can
only understand text, these works relied on scraping a webpage’s HTML as input
to the model. This prevented their application to native desktop and
virtualized software. Multimodal FMs address this limitation by attaching a
vision model to the base LLM (Zhang et al., 2023), which enables them to
directly reason over screenshots of a GUI (Humphreys et al., 2022; Furuta et
al., 2023; Shaw et al., 2023; Hong et al., 2023a; Bavishi et al., 2023).
Multimodal FMs have already shown promise in navigating websites (Zheng et
al., 2024; Yang et al., 2023a; He et al., 2024), mobile apps (Yan et al.,
2023), and desktop applications (Wu et al., 2024; Zhang et al., 2024). We aim
to design a system that helps bridge the gap between these proof-of-concepts
and enterprise-level solutions.

Data Management for Workflow Automation has been studied for nearly two
decades, with works ranging from business process management (Zeng et al.,
2001; Sayal et al., 2002; Georgakopoulos et al., 1995; Casati and Shan, 2000;
Rahman et al., 2015) to workflow automation and understanding (Hull et al.,
2013; Jennings et al., 1998). All of this work, however, pre-dates multimodal
FMs. As a result, the challenges around which these systems were designed
differ substantially from modern systems. Our work aims to build on these
prior efforts from the data management community by developing a system that
integrates FMs.

###  2.2. Problem Formulation

We aim to achieve end-to-end automation of enterprise workflows at minimal
cost. We have a workflow $w\in\mathcal{W}$ which consists of a sequence of
alternating states $s\in\mathcal{S}$ and actions $a\in\mathcal{A}$, such that
$w=(s,a,s^{\prime},a^{\prime\prime},...)$. Each workflow $w$ is done by a set
of workers during the course of business operations. These workers follow a
standard operating procedure (”SOP”), a form of written documentation which
outlines all of the steps and actions of the workflow. Once the workflows are
executed, an auditing process validates (either manually or programmatically)
whether each workflow $w$ was completed successfully. The goal is to automate
workflow $w$ by learning from human demonstrations and written documentation
(e.g. SOPs). We aim to directly operate on the GUI, as many workflows do not
have APIs or must be executed in native desktop/virtualized applications that
can only be observed visually (Hong et al., 2023a). This necessitates a
vision-based approach (Zheng et al., 2024; Assouel et al., 2023; Zhang et al.,
2024).

##  3\. Case Studies

We conducted interviews with business leaders across a number of industries
who led RPA projects. We select two organizations — a hospital and large B2B
enterprise — to serve as case studies. Both took over a year to deploy RPA
pilot projects, and both declined to expand RPA to additional workflows due to
its high set-up costs ($100k’s and months of development), unreliable
execution (accuracies started around 60% and peaked at 95%), and maintenance
requirements. We detail these challenges below.

###  3.1. Hospital Revenue Cycle Management (RCM)

Given the complexity of healthcare reimbursement, most hospitals have a
dedicated Revenue Cycle Management (RCM) department to ensure that timely
payment is collected for services delivered (Kilanko, 2023). Example RCM
workflows include verifying a patient’s insurance eligibility, obtaining prior
authorization, and processing claims (Kilanko, 2023). It is estimated that RCM
processes cost roughly 15% of every dollar of revenue gained (Bayley and
Levine, 2013). Additionally, 90% of health systems face RCM staffing shortages
(R1, 2022). Despite increased interest in automating RCM workflows, most
remain highly manual: roughly 94% of claims submissions and 76% of eligibility
verifications involve manual labor (Holloway et al., 2018). In conversations
with hospital IT leaders who considered automating certain RCM workflows, but
only managed a limited deployment, the following weaknesses of RPA surfaced:

(1) High Set-Up Costs. Integration with a hospital’s IT infrastructure is
costly, as is retraining staff (Kilanko, 2023). Hospital leadership estimated
that it took about 18 months and $10k’s to develop and deploy their RPA bot.
Significant back-and-forth with the vendor occurred as each workflow had to be
manually mapped and coded into a set of well-defined, ”always true” actions.

(2) Brittle Execution. It was estimated that $\sim$0.2 FTEs worth of effort
was saved with the RPA bot, as it could only handle two narrowly-scoped
workflows involving one payer and one department. Quarterly updates to the
hospital’s electronic health record and constant changes to payers’ websites
would break the bot. Eventually, this required the hospital to develop a
custom API that the bot could use to increase its reliability.

(3) Burdensome Maintenance. Given staffing constraints, the hospital chose to
outsource continued human oversight of the bot to the vendor as a managed
service for a fee. Despite this outsourced management, RCM managers still had
to manually review outputs to ensure compliance.

###  3.2. B2B Enterprise Invoice Processing

Large B2B enterprises must ingest a wide range of complex contracts and
process the information into systems of record such as NetSuite or SAP. These
workflows can vary widely depending on the specific product being invoiced,
the end purchaser, and the time the contract was written (Sahu et al., 2020).
Given its importance to enterprise financial planning, invoice processing has
also attracted attention from RPA vendors. We interviewed the head of revenue
operations at a large B2B enterprise which tried to implement RPA for a single
invoice processing workflow. After a year of development, 5 FTEs working with
the RPA bot were able to successfully accomplish a workflow that previously
took 20 FTEs. Despite this apparent success, however, the enterprise chose not
to expand RPA to other workflows, as the implementation proved to be too
painful:

(1) High Set-Up Costs. Going from initial contract with the RPA vendor ($150k)
to production deployment took over 12 months. External consultants (another
$100k) and 3 FTEs were required to integrate the bot with existing IT systems.
Major challenges were (a) accurately defining the workflow; and (b) the steep
learning curve for programming RPA bots, which required learning a proprietary
low-code development toolkit.

(2) Brittle Execution. The RPA bot was initially only 60% accurate post-
deployment and took 6 months of iteration to reach a final accuracy of 95%.
Beyond invoice processing, only $\sim$50% of desired workflows seemed feasible
to automate with RPA.

(3) Burdensome Maintenance. The cost of a wrongly processed invoice is high
($10k’s), so 2 FTEs were allocated to continuously monitor the bot to debug
issues, add new input formats, and inspect outputs. An outside firm also
conducted manual batch reviews.

##  4\. Preliminary Evaluations

In this section, we describe how ECLAIR can leverage multimodal FMs across all
three stages of the traditional RPA pipeline.

We provide preliminary experiments assessing the feasibility of this approach.
For our evaluations, we subsample 30 workflows from the WebArena benchmark
(Zhou et al., 2023). WebArena provides a set of interactive websites in which
an AI agent must complete complex workflows specified via natural language.
Specifically, we choose 30 workflows from the Gitlab and Adobe Magento
environments which their GPT-4 baseline model failed to complete (Gur et al.,
2023). We have human annotators record themselves completing each workflow and
write a step-by-step guide (”SOP”) on the steps they took.

###  4.1. Demonstrate

ECLAIR aims to learn from passively collected human demonstrations, with no
updates to the underlying FM’s weights. This limits the cost of labeling data,
simplifies deployment, and avoids known biases that arise when humans try to
articulate their work processes (Li et al., 2019). Our experiments show that
GPT-4 can accurately identify the steps of a workflow based on visual
observation of a human demonstration, with step-level precision of 0.94 and
recall of 0.95.

####  4.1.1. Can ECLAIR determine the steps of a workflow by viewing raw video
demonstrations?

This would enable ECLAIR to substantially improve the effectiveness and
scalability of process mining.

Hypothesis: A mulitmodal FM can generate accurate SOPs based on screenshots
taken at key frames from a video recording.

Set-Up: We prompt GPT-4 to generate an SOP for a workflow given a human
demonstration. We ablate various ways to provide the demonstration: just the
workflow description (WD); the workflow description and screenshots of key
frames from a video recording of the demonstration (WD+KF); or the workflow
description, key frames, and a textual action log of each click and keystroke
(WD+KF+ACT). Using a manually-written SOP as reference, a human annotator
calculates GPT-4’s precision (”What percent of steps in the GPT-4 SOP are in
the true SOP?”), recall (”What percent of steps in the true SOP are in the
GPT-4 SOP?”), and correctness (”By following the GPT-4 SOP, can I complete the
workflow?”).

Table 1. (Demonstrate) GPT-4 generation of SOPs. Metrics averaged across all 30 workflows. Method | # of Steps in SOP | Accuracy of SOP
---|---|---
Missing | Incorrect | Total | Precision | Recall | Correctness
WD | 1.57 | 3.58 | 13.67 | 0.75 | 0.81 | 0.60
WD+KF | 0.67 | 1.05 | 10.17 | 0.89 | 0.92 | 0.90
WD+KF+ACT | 0.63 | 0.57 | 9.63 | 0.94 | 0.95 | 0.93
Ground truth | 0 | 0 | 8.70 | 1 | 1 | 1

Takeaways: As shown in Table 1, the SOPs generated by GPT-4 using the
WD+KF+ACT strategy are judged as sufficiently ”correct” to complete 93% of
workflows. Even providing GPT-4 with screenshots alone (WD+KF) achieves 90%
correctness. However, WD+KF experiences almost twice as many hallucinations
(1.05 incorrect steps per SOP versus only 0.57 when the action trace is
included). Note that we do not do any workflow-specific prompt engineering or
data labeling to achieve these results. Additionally, we preprocess our video
demonstrations into a sequence of key frames using imperfect heuristics (i.e.
alignment with clicks and keystrokes). Future work may benefit from using a
model that can directly process video (rather than just images).

###  4.2. Execute

After defining the workflow, ECLAIR must execute a sequence of steps to
accomplish the workflow. We divide each step into two phases: (1) action
suggestion, i.e. planning what action to take; (2) action grounding, i.e.
translating the plan into actual clicks/keystrokes of GUI elements. We find
that providing domain knowledge to ECLAIR via SOPs doubles workflow completion
rates, while general-purpose models such as GPT-4 lag smaller models fine-
tuned on GUIs for action grounding.

####  4.2.1. Can ECLAIR accurately suggest the next action to take in a
workflow?

We investigate whether multimodal FMs can predict the next action in a
workflow based on the current state of the GUI and action history. We evaluate
if providing the FM with an SOP for the workflow increases overall workflow
completion rates.

Hypothesis: Providing high-level natural language guidance to the model via an
SOP improves workflow completion rates.

Set-Up: At each step of the workflow, the model takes as input the ground
truth history of actions, full SOP, and current GUI, and is expected to
generate the next action to take. We measure the accuracy of the model’s
suggested next action using a human annotator to evaluate whether it is
semantically equivalent to the corresponding ”ground truth” action for the
workflow.

Table 2. (Execute) GPT-4 average accuracy on next action suggestion with and without SOP guidance. SOP |  Next Action Suggestion Acc. |  Overall Workflow Completion Acc.
---|---|---
✘ | 0.83 | 0.17
✔ | 0.92 | 0.40

Takeaways: Our results in Table 2 demonstrate that SOPs improve overall
workflow completion rates by up to 23 points. While the SOP boosts the
accuracy of each step’s action suggestion to 0.92, the model struggles to
associate these suggested actions with the appropriate GUI elements, thereby
deflating its overall completion rate (recall that the model must ground its
actions to successfully complete the workflow). For example, when the action
is “Click on the profile button”, but the HTML element for the icon is
identified as ”svg” rather than ”button”, the model fails to select this
element.

####  4.2.2. Can ECLAIR accurately ”ground” its actions suggestions to GUI
elements?

Once ECLAIR suggests an action to take (i.e. “Click on the ”Submit” button”),
it must then map the action into mouse/keyboard commands which specify the
pixel location of the GUI element to interact with (Zheng et al., 2024).
Multimodal FMs are known to have difficulty with this, so we evaluate several
grounding strategies (Zheng et al., 2024).

Hypothesis: Providing bounding boxes for each element in a GUI improves a
multimodal FM’s ability to ground elements.

Set-Up: We sample a total of 120 and 302 webpages from the WebUI (Wu et al.,
2023b) and Mind2Web (Deng et al., 2023) datasets, respectively. We create a
natural language description for one element on each page, then prompt a model
to generate a bounding box (BB) for that element given a natural language
description of the BB and screenshot of the webpage. We measure ”accuracy” as
the percentage of predicted BBs whose center is within the element’s true BB
(i.e. If the model clicked on the center of its prediction, would it
successfully hit the target element?). We evaluate GPT-4 (OpenAI, 2023) and
CogAgent (Hong et al., 2023a) as two state-of-the-art closed/open source
multimodal FMs for GUI navigation. While CogAgent directly outputs BBs, GPT-4
does not. Thus, we use ”set-of-marks” prompting for GPT-4, in which we overlay
a unique numeric label on top of every element in the webpage screenshot
provided to GPT-4, and have it output the number of a labeled element (Yang et
al., 2023b). We generate these labels either directly from the webpage’s HTML
(”HTML”) or from a YOLONAS object detection model (”YOLO”) finetuned on 7k
WebUI webpages (Wu et al., 2023b). The latter simulates the setting where HTML
is not available.

Table 3. (Execute) Accuracy on grounding actions to GUI elements. ”S — M — L” is accuracy on small, medium, and large elements. Note: We found that HTML bounding boxes were not accurate in Mind2Web, so it is excluded. Model |  Bbox Source | Mind2Web | WebUI
---|---|---|---
S — M — L | Overall | S — M — L | Overall
GPT-4 | – | 0.01 — 0.03 — 0.16 | 0.07 | 0.00 — 0.03 — 0.14 | 0.05
GPT-4 | YOLO | 0.38 — 0.68 — 0.80 | 0.62 | 0.50 — 0.58 — 0.69 | 0.58
GPT-4 | HTML | – | – | 0.56 — 0.58 — 0.67 | 0.60
CogAgent | – |  0.55 — 0.71 — 0.87 | 0.71 |  0.71 — 0.64 — 0.75 | 0.70

Takeaways: The 18-billion parameter CogAgent outperforms GPT-4 on action
grounding. This suggests that smaller models purpose-built for GUI navigation
can be more effective than larger, general purpose multimodal FMs. However,
overall accuracy peaks at 70%, and interacting with smaller-sized elements
remains challenging. For GPT-4, using the bounding boxes generated by the
YOLONAS model performs similarly to the ”ground truth” HTML boxes. This
suggests that detecting elements on a GUI with a vision model is not the
bottleneck, but rather choosing which of those detected elements is the
desired element to interact with.

###  4.3. Validate

There are several levels of validation that ECLAIR must provide. At the
individual step-level, the agent should validate that its action suggestions
are (a) feasible to execute and (b) making progress towards accomplishing the
workflow. At the workflow-level, the agent should understand (a) whether the
workflow was successfully completed and (b) whether the steps taken to achieve
the workflow were sensible. We find that GPT-4 struggles with the former
lower-level details but performs well at the latter higher-level reasoning.

####  4.3.1. Can ECLAIR self-monitor at the individual step-level?

Identifying errors in individual actions would allow ECLAIR to perform error
correction in real-time. This would improve the reliability of the Execution
stage by enabling the model to avoid undesirable states and backtrack where
appropriate.

Hypothesis: A multimodal FM can detect if an action will succeed or fail based
on visual observation of changes in screen state.

Set-Up: First, we test the model’s ability to detect if an action failed (e.g.
typing had no effect because no text field was first focused). We sample
$(s,a,s^{\prime})$ traces from our dataset (positive examples) and generate
tuples where $s^{\prime}=s$ (negatives). We prompt GPT-4 to identify if the
action $a$ in $(s,a,s^{\prime})$ was successfully executed, given screen shots
for $s$ and $s^{\prime}$. We sample three negatives for each positive example.
Second, inspired by prior work on data cleaning (Benedikt et al., 2015;
Baltopoulos et al., 2011), we create a set of ”integrity constraints” defining
whether an action is viable at a particular state. For example, an ”integrity
constraint” for clicking a button is that the button is visible and not
disabled. We annotate constraints for all actions in our dataset, then prompt
GPT-4 with $(c,s)$ pairs where $c$ is the constraint for the action directly
after state $s$ (positive examples) and $(c,s^{\prime})$ pairs where
$s^{\prime}$ is a random state occurring before $s$ (negatives).

Table 4. (Validate) Performance of GPT-4 on self-validation tasks. Metrics averaged across all 30 workflows. Eval Type | Precision | Recall | F1
---|---|---|---
Actuation | 0.95 | 0.85 | 0.90
Integrity Constraint | 0.67 | 0.36 | 0.47
Workflow Completion | 0.90 | 0.84 | 0.87
Workflow Trajectory | 0.88 | 0.83 | 0.85

Takeaways: The results are shown in rows ”Actuation” and ”Integrity
Constraint” in Table 4. GPT-4 has a high precision (0.95) and recall (0.85)
when assessing whether an action was successfully executed. However, the low
integrity constraint scores indicate that it struggles to identify which
actions are viable given the state of the GUI. This could be due to only
observing static screenshots, which makes it difficult to discern animations
such as a blinking cursor. These results suggest that current multimodal FMs
cannot adequately self-monitor for enterprise use cases.

####  4.3.2. Can multimodal FMs self-monitor at the overall workflow-level?

Understanding whether a workflow was successfully completed or not can help
the agent (a) at runtime to know when to stop execution, and (b) post-
deployment for self-auditing.

Hypothesis: A multimodal FM can determine if a demonstration correctly
achieved a workflow based on visual observation of changes in screen state.

Set-Up: First, we test if the model can determine if it successfully completed
a workflow. To evaluate this, we sample full traces of
$(s^{1},a^{1},...,a^{n-1},s^{n})$ from our dataset (positive examples) and
truncate some by a random number of frames to get
$(s^{1},a^{1},...,a^{k-1},s^{k})$ (negatives) where $k<n$. Given the trace and
workflow description, we prompt GPT-4 to provide a binary assessment of
whether the workflow was successfully completed. Second, we investigate
whether the model understands the proper trajectory of actions in a workflow —
i.e. it is not sufficient to merely complete the workflow, but the steps taken
to complete it must align with its SOP. We sample full traces
$(s^{1},a^{1},...,a^{n-1},s^{n})$ from our dataset (positives) and either (a)
randomly shuffle or (b) randomly delete frames from this trace (negatives). We
provide GPT-4 with the SOP for the workflow, the workflow description, and the
trace, and have it output a binary assessment of whether the trace exactly
followed the SOP.

Takeaways: The F1 scores of 0.87 and 0.85 in rows ”Workflow Completion” and
”Workflow Trajectory” of Table 4 suggest that GPT-4 can self-monitor higher-
level properties of a workflow, but still has significant room for
improvement.

##  5\. Discussion

To be deployed, ECLAIR must meet a minimum level of performance. While this is
highly workflow-dependent, ECLAIR must be accurate enough such that the cost
of correcting its errors is outweighed by the efficiency gains of using it
(which might not require 100% accuracy). In our Section 3.2 case study, for
example, an accuracy of 95% was sufficient for the RPA bot to be deployed.
Defining such success criteria can be done via interviews with stakeholders,
financial modeling, and other enterprise planning methods (Leno et al., 2021)
We discuss considerations for deploying ECLAIR below.

Error Handling and Monitoring. We envision a multi-tiered system which
combines (1) self-validation, (2) programmatic heuristics, and (3) limited
human intervention to provide error correction. (1) Per our Section 4.3
experiments, a repository of integrity constraints — which have successfully
enhanced the quality of database schemas (Benedikt et al., 2015; Baltopoulos
et al., 2011) — could improve the accuracy of FM self-monitoring. Though FMs
can exhibit non-deterministic behavior, their reliability can be improved by
setting their temperature to 0, repeatedly querying (Shinn et al., 2023) and
ensembling predictions (Li et al., 2024), or eliciting confidence scores to
surface cases where intervention is necessary (Tian et al., 2023). (2)
Programmatic heuristics can also be used to detect failures, e.g. deviation
from the average time to execute a workflow. (3) Existing methods for auditing
human workflows can be re-applied, such as random spot checks or screenshots
of confirmation screens. However, we envision such monitoring to be more
limited – namely, to close the knowledge gap between ECLAIR and a particular
domain – as the outputs from monitoring can be repurposed into a dataset that
can be used to improve ECLAIR (Ouyang et al., 2022).

Human-ECLAIR Collaboration. While the long-term vision for ECLAIR is to
require minimal human interaction — i.e. only the Demonstrate phase (Section
4.1) needing human involvement — we acknowledge that human supervision may be
necessary for certain workflows. For example, a physician sign-off before
prescribing medications or tasks involving user authentication. To accomplish
this, the SOP could mark steps where the model transfers control to a human.
Alternatively, a whitelist of sensitive actions can be compiled to
automatically force transfer of control to a human when triggered, similar to
how kernels use interrupts to handle control flow (Mejia-Alvarez et al.,
2018). Finally, as mentioned in the prior section, generated human-ECLAIR
execution traces can be used to improve ECLAIR performance via fine-tuning or
few-shot prompting (Ouyang et al., 2022).

Self-Improvement. As ECLAIR repeatedly executes a workflow, it can observe the
effects of its actions on the environment. By documenting these observations,
ECLAIR can compile a database of common ”skills” that can later be transferred
to different workflows (Fu et al., 2024; Yang et al., 2023a; Wu et al., 2024;
Wang et al., 2023a; Park et al., 2023). Applying principles from self-driving
databases, which aim to continuously improve performance by implementing
sequences of ”actions” (i.e. changes to their configurations) based on
utilization patterns (Pavlo et al., 2021, 2017; Ma et al., 2018), could
provide a principled approach for such a self-improving workflow automation
system.

Multi-Agent Collaboration. Applying multiple agents to the same task can
improve accuracy (Li et al., 2024; Liang et al., 2023), as seen in recent work
on multi-agent software engineering (Hong et al., 2023b) and chatbot
applications(Wu et al., 2023a). Such an approach could be utilized within the
ECLAIR framework to create specialized agents for distinct subtasks or digital
environments. Prior work on collaborative data processing tools offers a
reference for how ECLAIR can be scaled to multi-agent (and multi-human)
workflows with shared resources. (Bhardwaj et al., 2015; Liu et al., 2022).

##  6\. Conclusion

We are excited for the potential of multimodal FMs to reimagine how work gets
done. By addressing the three main shortcomings of traditional process mining
and RPA (high set-up costs, brittle execution, and burdensome maintenance),
the realization of ECLAIR can help achieve the promise of enterprise workflow
automation.

###### Acknowledgements.

We thank Dan Fu, Ben Spector, Vishnu Sarrukai, Jon Saad-Falcon, Simran Arora,
Laurel Orr, and Silas Alberti for providing helpful feedback on this
manuscript. We thank Ishan Khare, Krrish Chawla, and Miguel Hernandez for
their assistance in collecting the dataset of workflow demonstrations. We
gratefully acknowledge the support of NIH under No. U54EB020405 (Mobilize),
NSF under Nos. CCF2247015 (Hardware-Aware), CCF1763315 (Beyond Sparsity),
CCF1563078 (Volume to Velocity), and 1937301 (RTML); US DEVCOM ARL under Nos.
W911NF-23-2-0184 (Long-context) and W911NF-21-2-0251 (Interactive Human-AI
Teaming); ONR under Nos. N000142312633 (Deep Signal Processing), N000141712266
(Unifying Weak Supervision), N000142012480 (Non-Euclidean Geometry), and
N000142012275 (NEPTUNE); Stanford HAI under No. 247183; NXP, Xilinx, LETI-CEA,
Intel, IBM, Microsoft, NEC, Toshiba, TSMC, ARM, Hitachi, BASF, Accenture,
Ericsson, Qualcomm, Analog Devices, Google Cloud, Salesforce, Total, the HAI-
GCP Cloud Credits for Research program, the Stanford Data Science Initiative
(SDSI), and members of the Stanford DAWN project: Facebook, Google, and
VMWare. The U.S. Government is authorized to reproduce and distribute reprints
for Governmental purposes notwithstanding any copyright notation thereon. Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views,
policies, or endorsements, either expressed or implied, of NIH, ONR, or the
U.S. Government. MW is supported by a Stanford HAI Graduate Fellowship and
Stanford Healthcare. AN is supported by the Knight-Hennessy Fellowship and the
NSF fellowship.

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# Landscape of nuclear deformation softness with spherical quasi-particle
random phase approximation

Nguyen Le Anh<EMAIL_ADDRESS>Department of Physics, Ho Chi Minh City
University of Education, 280 An Duong Vuong, District 5, Ho Chi Minh City,
Vietnam  Department of Theoretical Physics, Faculty of Physics and Engineering
Physics, University of Science, Ho Chi Minh City, Vietnam  Vietnam National
University, Ho Chi Minh City, Vietnam  Bui Minh Loc<EMAIL_ADDRESS>(present
address) Center for Exotic Nuclear Studies, Institute for Basic Science (IBS),
Daejeon 34126, Korea  San Diego State University, 5500 Campanile Drive, San
Diego, CA 92182  Panagiota Papakonstantinou<EMAIL_ADDRESS>Rare Isotope
Science Project, Institute for Basic Science, Daejeon 34047, Korea  Naftali
Auerbach<EMAIL_ADDRESS>School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 69978, Israel

###### Abstract

We investigate the stability and softness of nuclei against quadrupole,
octupole, and hexadecapole deformation. By applying the spherical Skyrme-force
Hartree-Fock Bardeen-Cooper-Schrieffer quasi-particle random phase
approximation, we diagnose ground-state deformation when imaginary solutions
are obtained, i.e., the spherical ground state collapses. We also calculate
the multipole polarizability in spherical nuclei with no collapse, as a
measure of softness. This numerically light and theoretically sound method is
found able to capture deformation patterns across the nuclide chart. The
connection between the intrinsic shape of nuclei and the dynamics of their
low-lying collective states is established and the role of shell structure is
discussed.

##  I Introduction

New experimental techniques and theoretical methods continue to advance our
understanding of nuclear deformation, in stable and exotic nuclei. Notably,
the hydrodynamic model description of relativistic heavy-ion collisions is
currently being used to infer deformation in nuclei. Specifically, the
elliptic flow $v_{2}$ and the triangular flow $v_{3}$, which are patterns of
particle movement observed in high-energy nuclear collisions, are used in the
study of quadrupole and octupole deformation, respectively Abdallah  _et al._
(2022); Jia (2022). The flow $v_{4}$ can also be measured, as has been done at
the Relativistic Heavy Ion Collider (RHIC), and be used to study hexadecapole
deformation.

In their ground state, most of the nuclei are not spherical but quadrupole
deformed Bohr and Mottelson (1998). A handful of octupole nuclei were
confirmed in the experiment Butler (2020). Very recently, evidence of
hexadecapole deformation in Uranium-238 at the RHIC was reported in Ref.
Ryssens  _et al._ (2023). Observables related to the $J^{\pi}=4^{+}_{1}$
states of the Krypton isotopes and, in particular, on the hexadecapole degree
of freedom were discussed in Ref. Spieker  _et al._ (2023). One question is
whether deformation values extracted through low-energy observables are
consistent with the new data obtained at high energy. A more general question
is whether we can understand in a systematic way which nuclei tend to develop
hexadecapole deformation.

A number of global theoretical models of nuclear deformation are available.
The finite-range-droplet model (FRDM) Möller  _et al._ (2016) is a
microscopic-macroscopic model, while the fully microscopic Skyrme Hartree-Fock
(HF) Bardeen-Cooper-Schrieffer (BCS) Goriely  _et al._ (2001), Gogny Hartree-
Fock-Bogolyubov Hilaire and Girod (2007) and deformed relativistic Hartree-
Bogoliubov theory in continuum (DRHBc) Zhang  _et al._ (2022) are based on a
mean-field approach and take as input an effective interaction or Lagrangian.
In mean-field approaches, the onset of deformation can be interpreted to some
extent in terms of shell structure that plays an important role in the
phenomenon of deformation: for example, magic nuclei are always spherical.
From the shell model, it is understood that the accumulating $p-n$ interaction
strength leads to additional configuration mixing and deviations from
spherical symmetry in the ground state Talmi (1962); Federman and Pittel
(1979). The Nilsson model has been very useful for relating nuclear
deformation and shell structure Nilsson (1955); Nilsson and Ragnarsson (1995).

From a theoretical point of view, to predict whether a nucleus is likely
deformed in its ground state, one typically performs precise calculations of
the density distributions and deformation parameters. In this work, we follow
a different computationally light approach to examine deformation patterns
throughout the nuclide chart. Specifically, we solve the self-consistent
spherical Skyrme HFBCS quasi-particle random phase approximation (QRPA) to
obtain the excited states and response function in the quadrupole, octupole,
and hexadecapole channels. One can then quantify the softness against
deformation or diagnose static deformation by using the formal properties of
the response function and the random phase approximation, respectively.

First, when no imaginary solutions are obtained, i.e., the RPA stability
matrix is positive-definite (all eigenvalues are real), the static
polarizability, which we might also call deformability, can be used as a
measure of stiffness. The static polarizability is formally determined by the
real part of the response function at vanishing excitation energy, which in
turn is related to the inverse energy-weighted sum rule of the imaginary part
of the response function via the Kramers–Kronig relation (see, e.g., Nozières
(1997)). Thus, by calculating the inverse-energy sum rule in a given channel
for a spherical nucleus, we are in effect calculating the deformability of the
nucleus.

Second, if an imaginary solution is obtained, that means that the spherical
HFBCS state assumed to be the ground state is not actually the lowest-energy
Skyrme HFBCS state Thouless (1960, 1961); Rowe (1968); Nakada (2016): If we
were to perform HFBCS calculations with the same Skyrme functional but
allowing for deformation, the ground state solution would show intrinsic
deformation. The point here is that there is no need to perform such deformed
HFBCS calculations: the fact that the (spherical) HFBCS-QRPA stability matrix
is not positive-definite already indicates that the spherical HFBCS state is
unstable against deformation. In addition, the present approach allows us to
study parity-odd shapes, such as octupole, and identify regions of the nuclear
chart where octupole and quadrupole deformation may coexist, perhaps leading
to triaxiality. The octupole deformation was studied in this way in Ref. Minh
Loc  _et al._ (2023).

We note that the excitation spectrum of deformed nuclei can be examined with
formulations of QRPA which allow for deformation, for example, the self-
consistent QRPA for use in axially symmetric nuclei as discussed Ref. Terasaki
and Engel (2010). Triaxially deformed nuclei are much more difficult to study
microscopically, owing mainly to computational limitations. However, we stress
that the objective of the present work is not to describe the excitation
spectrum but to assess static deformability.

The manuscript is organized as follows. In Sec. II, we describe briefly the
HFBCS-QRPA formalism and how it is employed in the present work. In Sec. III,
we present and discuss our results in the octupole, quadrupole, and
hexadecapole channels. We conclude in Sec. IV.

##  II Method

The HFBCS-QRPA method is well known and has been long in use Rowe (2010); Ring
and Schuck (1980). Self-consistent implementations, such as the one used here,
employ the same functional in calculating the HFBCS ground state and the QRPA
residual interaction. First, the HFBCS equations are solved for even-even
atomic nuclei. The resulting HFBCS mean fields and nuclear densities are
inputs to the QRPA calculation. Then in the RPA or QRPA, the excited states
$|\nu\rangle$ are the result of the operator $O^{+}_{\nu}$ acting on the
correlated ground state $|\tilde{0}\rangle$ with $J^{\pi}=0^{+}$

$|\nu\rangle=O^{+}_{\nu}|\tilde{0}\rangle.$ (1)

In general, the operator $O^{+}_{\nu}$ is expressed in terms of the creation
and annihilation of a pair of particle-hole states ($\alpha,\beta$) that are
obtained from the HF solution. The total angular momentum of the pair is
$(\lambda M)$. The RPA equation Ring and Schuck (1980) is written as

$\displaystyle\sum_{\beta}\left[A_{\alpha\beta}X_{\beta}^{(\nu)}+B_{\alpha\beta}Y_{\beta}^{(\nu)}\right]$
$\displaystyle=$ $\displaystyle\omega_{\nu}X_{\alpha}^{(\nu)},$
$\displaystyle\sum_{\beta}\left[B^{*}_{\alpha\beta}X_{\beta}^{(\nu)}+A_{\alpha\beta}^{*}Y_{\beta}^{(\nu)}\right]$
$\displaystyle=$ $\displaystyle-\omega_{\nu}Y_{\beta}^{(\nu)}.$ (2)

The matrices $A$ and $B$ are obtained from the HFBCS single-particle state
($\alpha,\beta$) and the residual interaction which is the antisymmetrized
particle-hole interaction Ring and Schuck (1980); Colò  _et al._ (2013). The
operator $O^{+}_{\nu}$ that excites the states $|\nu\rangle$ can be an
electromagnetic field $\hat{F}(\bm{r})$ with the $\lambda$-multipole component
being defined by

$\hat{F}_{\lambda M}(\bm{r})=e\sum_{i}^{A}r_{i}^{\lambda}Y_{\lambda
M}(\hat{r}_{i})\frac{1}{2}[1-\tau_{z}(i)],$ (3)

in which the sum runs over all nucleons. $Y_{\lambda M}(\hat{r}_{i})$ is the
spherical harmonic function; $e$ is the proton charge; $\tau_{z}$ is the third
component of the isospin operator. As the ground state is $0^{+}$, the final
total angular momentum $J$ is equal to the $\lambda$-multipole component of
the operator which is $\lambda=2,3,$ or $4$. They are connected to the
quadrupole, octupole, and hexadecapole moments of the matter distribution that
are used to quantify the intrinsic shape of the nucleus.

The degree of stability or softness of the atomic nucleus against variations
is expressed via the polarizability or deformability $\mathcal{C}_{\lambda}$
(curvature), which is obtained from the inverse energy-weighted sum rule of
the response function Ring and Schuck (1980); Abbas  _et al._ (1981). In the
(Q)RPA framework, it is calculated from the $m_{-1}^{\lambda}$ moment

$\displaystyle\mathcal{C}_{\lambda}$ $\displaystyle=$ $\displaystyle
2m_{-1}(\lambda)/A,$ (4) $\displaystyle m_{-1}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{n}|\langle\lambda_{n}||\hat{F}_{\lambda}||\tilde{0}\rangle|^{2}E_{n}^{-1}.$
(5)

The larger $\mathcal{C}_{\lambda}$ (W.u./MeV) is, the less stiff the system is
against deformations. For example, in the case of octupole deformation
$\mathcal{C}_{3}$ is around $1$ W.u./MeV for 96Ru, which is not considered
especially soft. In contrast, it is $4$ W.u./MeV (even larger) for 96Zr which
is an example of octupole softness nuclei Minh Loc  _et al._ (2023).

Our method in the study of nuclear deformation softness is described as
follows. Solving the QRPA equation (II) gives us the energies $\omega_{\nu}$
and the wave functions ($X^{(\nu)},Y^{(\nu)}$) of the excited states, where
$\nu$ enumerates those states. When the ground state is stable against the
particle-hole perturbations, the QRPA equation (II) has only real solutions
$\pm\omega_{\nu}$ Thouless (1960, 1961). If, however, a deformed HFBCS state
is energetically preferable to the spherical one, the QRPA solutions in the
relevant multipolarity will include imaginary values. We call the occurrence
of such an instability a “collapse” following Ref. Abbas  _et al._ (1981),
where the octupole-deformed softness of 96Zr was first recognized.

In Ref. Minh Loc  _et al._ (2023) we presented results of spherical QRPA
calculations in the octupole channel using various functionals. For certain
nuclei, we found that the results showed special sensitivity to the functional
used, namely the values for $\mathcal{C}_{3}$ varied greatly, with the low-
lying collective state at very low energy and in some cases “collapsed”. Such
results indicate that the nucleus is likely soft or even unstable against
specific deformation. 96Zr is an example of a soft nucleus, while 96Ru is an
example of a stiff nucleus against octupole deformation. Doubly magic nuclei
such as 208Pb are examples of stiff nuclei to all kinds of deformation. In the
next section, we highlight soft as well as stiff nuclei in the octupole,
quadrupole, and hexadecapole channels.

We applied the self-consistent HFBCS-QRPA calculation using the computer
program presented in Refs. Colò  _et al._ (2013); Colò and Roca-Maza (2021).
Skyrme forces SLy5 Chabanat  _et al._ (1998) and SkM∗ Bartel  _et al._ (1982)
are used. We computed the energies of first excited states
$E(2^{+}_{1}),E(3^{-}_{1})$, and $E(4^{+}_{1})$ for all even-even nuclei with
experimental data available in the NuDat database National Nuclear Data
Center, information extracted from the NuDat database . All used parameters
are described in detail in Ref. Colò  _et al._ (2013); Colò and Roca-Maza
(2021). When the spherical calculation for a given atomic nucleus has an
imaginary solution, we conclude that the spherical ground state is predicted
unstable against the QRPA operator (quadrupole, octupole, or hexadecapole), in
other words, the nucleus is predicted static deformed. In the case of the
quadrupole operator, as expected, we find the majority of nuclei to be
deformed. When all QRPA solutions are real, we use the multipole
polarizability as a comparative measure of softness. Softness against
deformation can be concluded also when the first excited state is found at
real but very low energy and highly collective. When, for example, the
transition strength in W.u. exceeds half the mass number, i.e., enhancement),
we can quite confidently classify the nucleus as soft.

##  III Results and Discussion

###  III.1 Octupole deformation softness

Figure 1: Octupole deformation landscape with the SLy5 force according to the
presence (collapse) or not (no collapse) of imaginary solutions. Lanthanide
and actinide regions are shaded. The dotted lines indicate the octupole-
driving numbers. This figure has been essentially adopted from Ref. Minh Loc
_et al._ (2023).

We begin with the octupole deformation, which is of negative parity and much
less studied than even-parity shapes. Generally, negative-parity nuclear
shapes are of interest in searches for physics beyond the standard model of
particle physics Auerbach  _et al._ (1996); Spevak  _et al._ (1997) (see
Pospelov and Ritz (2005); Butler (2020) for reviews). Motivated by the results
of recent heavy-ion experiments, which indicate octupole softness in 96Zr
Abdallah  _et al._ (2022); Zhang and Jia (2022), we examined potential static
octupole deformation softness across the nuclear chart in Ref. Minh Loc  _et
al._ (2023).

The results were consistent with the concept of octupole magic numbers: The
development of $3^{-}$ octupole deformation softness requires the presence of
a particle and a hole state of opposite parity but in close proximity and with
angular momenta such that they can couple to total angular momentum number
$3$. Such particle-hole pairs are very specific and owe their existence to the
spin-orbit splitting Minh Loc  _et al._ (2023); Abbas  _et al._ (1981). For
example, octupole deformation areas are found in the lanthanide and actinide
regions where the $1h_{11/2}$ or $1i_{13/2}$ hole orbitals come energetically
close to opposite-parity particle states $2d_{5/2}$ and $2f_{7/2}$,
respectively.

In Figure 1 that essentially adopted from Ref. Minh Loc  _et al._ (2023), we
recall the resulting landscape for octupole deformation softness. The
calculation was done only for nuclei for which experimental data are available
for the energy and strength of the $3^{-}_{1}$ transition to the ground state.
The blue circles in Figure 1 indicate nuclei where we find “collapse” in the
SLy5 spherical QRPA calculation, i.e., the presence of an imaginary solution,
indicating instability of the spherical shape against octupole deformation in
this model. More generally, it indicates softness against octupole deformation
Minh Loc  _et al._ (2023).

Figure 2: Light nuclei with strong enhanced octupole transition strength,
$B(E3)>0.5~{}A$ W.u., calculated with the SLy5 force. The dotted lines
indicate the octupole-driving numbers.

Also in Ref. Minh Loc  _et al._ (2023), it was pointed out that no “collapse”
was obtained in light nuclei. However, there were cases where enhanced $B(E3)$
transition strength was found at low energy and correspondingly high
polarizability. A high polarizability indicates strong octupole correlations
and a degree of softness. According to Ref. Minh Loc  _et al._ (2023), nuclei
with enhanced transition strength are highlighted in Figure 2. Not only the
“collapse” but also the enhancement of the transition strength are signatures
of nuclear deformation softness.

In addition, the transition from octupole deformation to octupole vibrations
has been suggested within two different models (see Refs. Bizzeti and Bizzeti-
Sona (2004, 2008) and Bonatsos  _et al._ (2005); Lenis and Bonatsos (2006)),
with 226Ra and 226Th near the actinide region appearing to be close to the
transition point. The experimental identification of stable octupole
deformation in 224Ra Gaffney  _et al._ (2013) supports these predictions.

In the following, we analyze the quadrupole and hexadecapole deformation
analogously.

###  III.2 Quadrupole deformation softness

Next, we demonstrate how the present approach works in the case of the well-
established quadrupole deformation. Quadrupole deformation is very common
throughout the nuclear chart and has been the most extensively studied shape
Möller  _et al._ (2016); Zhang  _et al._ (2022). Experimental data for the
first excited quadrupole state, $2^{+}_{1}$, have also been systematically
documented Pritychenko  _et al._ (2014). Its properties are closely related to
shell structure, for example, a high energy is associated with magicity.
Systematic QRPA calculations using SLy4 and SkM∗ force for the $2^{+}_{1}$
state in spherical nuclei were carefully discussed in Ref. Terasaki  _et al._
(2008). In that work, $155$ (SLy4) and $178$ (SkM∗) spherical nuclei were
selected and studied.

As already explained, in this work we solve the spherical QRPA equations for
all nuclei for which relevant data exist, regardless of sphericity: If no
imaginary solution is obtained, which means that the model does not predict
quadrupole deformation, the energy and strength of the $2^{+}_{1}$ state and
the quadrupole polarizability can be used as measures of the nucleus’s
softness against quadrupole deformation; if we obtain an imaginary solution,
then the nucleus is predicted to be deformed in its ground state.

The difference from the octupole case is that because of the positive parity
and low angular momentum of the $2^{+}$ state, many particle-hole components
are available for forming a $2^{+}$ quadrupole state and many particle-hole
pairs may lie close energetically in open-shell nuclei leading to softness or
to the instability of the spherical shape. As a result, the shell-structure
origin of quadrupole deformation is not as clearly identified as in the case
of octupole deformation. It is also unsurprising that quadrupole deformation
is so common across the nuclear chart. Figure 3 shows the quadrupole
deformation landscape with the mid-shell deformed regions highlighted by
circle or circular segments.

Figure 3: Quadrupole deformation landscape with the SLy5 force according to
the presence (collapse) or not (no collapse) of imaginary solutions. The mid-
shell deformed regions are highlighted by circle or circular segments. The
magic numbers are shown in dotted lines.

The most strongly deformed nuclei that are well-known in the literature,
namely 20-24Ne, 22-26Mg, 26,28Si, 98-102Sr, 102Zr, and 152Ce, can be
identified in Figure 3, too, as deformed.

Regarding regions in which quadrupole deformation appears, it would be
interesting to mention the parameter-independent predictions of the
$N_{p}N_{n}$ scheme Casten (1985) and the so-called $P$-factor Casten  _et
al._ (1987), which is determined from the number of valence protons and
valence neutrons counted from the nearest closed shell. Nuclei with $P>5$ are
expected to exhibit quadrupole deformation (see, for example, Figure 21 in
Ref. Casten and McCutchan (2007)).

In conclusion, our approach based on self-consistent QRPA captures the same
region of quadrupole deformation as well-known in literature such as in Ref.
Möller  _et al._ (2016) using the microscopic-macroscopic model.

###  III.3 Hexadecapole deformation softness

Hexadecapole deformation is not as thoroughly discussed in the literature as
quadrupole deformation. Although the parity in both cases is positive, the
number of single-particle pairs that can form a $J=4$ state is more limited.
In addition, most of those pairs can couple to $2^{+}$ states as well, so that
hexadecapole softness or deformation is expected to be favored in nuclei with
strong quadrupole softness or deformation. Figure 4 shows that it is around
neodymium ($Z=60$) and polonium ($Z=84$) that QRPA collapses in the $4^{+}$
channel. Observing the single-particle shell structure, the hexadecapole-soft
nuclei appear after closing the major shells. Regions of softness, however,
are more extended.

Figure 4: Hexadecapole deformation landscape with the SLy5 force according to
the presence (collapse) or not (no collapse) of imaginary solutions. The
dotted lines indicate the magic numbers. Lanthanide and actinide regions are
shaded.

The area where there are enhancements of $B(E4)$ transition strength is shown
in Figure 5. The extension of this area to the neutron-rich nuclei can provide
us with more information on the single-particle structure of the whole region.
The $Z,N=59-70,96-112$ regions of the nuclear chart are interesting also in
the context of shape coexistence, as they are candidates for the dual-shell
mechanism proposed in Ref. Martinou  _et al._ (2021).

Figure 5: Nuclei with especially enhanced hexadecapole transition strength,
$B(E3)>0.5~{}A$ W.u. in the lanthanide region ($Z=57-71$), as calculated with
SLy5 HFBCS-QRPA. These are certain Ce, Nd, Sm, and Gd isotopes with $N\geq
90$.

In Figure 6 we show a broader range of $\mathcal{C}_{4}$ and compare visually
the resulting landscape with predictions for the absolute value of the
hexadecapole deformation parameter based on the FRDM model Möller  _et al._
(2016). The regions of deformation or softness correspond almost one-to-one to
our results. Regarding the division of the Lanthanide region into two islands
instead of one, we note that one of the islands corresponds to positive and
one to negative values of $\beta_{4}$, a distinction that we cannot make with
our approach.

Figure 6: (a) Landscape of the polarizability $\mathcal{C}_{\lambda}$ with the
SLy5 force. The yellow color indicates nuclei for which we have not obtained a
result for the polarizability because static deformation is predicted
(collapse of spherical QRPA). (b) The absolute value of the hexadecapole
deformation parameter as predicted by the FRDM model Möller  _et al._ (2016)
for the nuclei studied in this work.

###  III.4 Nuclear deformation landscape

Our approach is applicable to the deformation softness and stiffness of any
multipolarity in atomic nuclei including quadrupole, octupole, and
hexadecapole modes from light to heavy nuclei, as we have seen. Figure 7 shows
the overview result of the “collapse” throughout the nuclide chart with two
different Skyrme forces, SkM∗ in Figure 7(a) and SLy5 in Figure 7(b). One can
see that the octupole or hexadecapole “collapse” coincides with the quadrupole
“collapse”. This means that the octupole and the hexadecapole deformation are
always on top of the quadrupole-deformation background. As mentioned in Ref.
Terasaki  _et al._ (2008), it was surprising to find that the two Skyrme
forces disagreed significantly on which nuclei are spherical. However, looking
at the landscape in Figure 7, we observe that the predicted areas of
deformation are similar, even though they are not identical. The important
conclusion is that changes predicted by different models in the single-
particle shell structure may differ somewhat, leading to differences in the
deformation landscape as predicted with different Skyrme forces Terasaki  _et
al._ (2008) and thus reaffirming the role of the shell-driving mechanism in
the evolution of nuclear deformation.

Figure 7: Nuclear deformation landscape with different SkM∗ (a) and SLy5 (b)
forces. The collapses are in color. The cross symbol presents nuclei without
any “collapse” in the completed diagnostic. The dotted lines indicate the
magic numbers.

Note that the original magic numbers, $2,8,20,28,50,82$, and $126$ Mayer
(1948); Haxel  _et al._ (1949) enhance stability and make nuclei spherical and
stiff against nuclear deformations. In the context of nuclear deformations,
the original magic numbers are interpreted as the stiff-multipole numbers. The
presence of strong multiple correlations decreases the stiffness of the atomic
nucleus, making them soft to the corresponding deformation. Therefore, the
octupole-driving particle numbers or octupole-magic numbers ($16,34,56,88$,
and $134$) represent soft-octupole numbers. The evolution of octupole magic
numbers in neutron-rich nuclei could be an interesting subject for future
work.

While the stiff-quadrupole-deformed nuclei corresponding to the original magic
number are trivial (Figure 8(a)), the stiff-octupole and the stiff-
hexadecapole magic numbers are less recognized and paid attention to so far.
In Figures 8(b) and (c), therefore, we show also the value of
$\mathcal{C}_{\lambda=3,4}$ (Eq. (5)) throughout the nuclide chart. The
quadrupole, octupole, and hexadecapole magic nuclei are in the colored
regions.

As octupole and hexadecapole softness coincide with quadrupole softness,
different types of magic numbers compete. As an example, in the case of
${}^{84}_{34}$Se50, our result shows that the effect of octupole deformation
softness is surpassed by the stiffness of the closed shell structure. Our
method based on the fully self-consistent HFBCS-QRPA can diagnose deformation
softness against one multipolarity at a time. For example, we cannot conclude
whether a specific nucleus will show, at the same time, quadrupole and
octupole deformation or quadrupole and hexadecapole. However, in qualitative
terms at least, our approach can reliably diagnose softness in atomic nuclei
and deformation patterns in the nuclide chart. Our results can serve as a
guide to more quantitative explorations.

Figure 8: Landscape for nuclear stiffness against multipole deformation
through the polarizability $\mathcal{C}_{\lambda}$ with the SLy5 force.

In soft nuclei, low-lying states exhibit greater collectivity, making them
more effectively described within the RPA framework. The low-lying collective
energy-level spectra below a few MeV reveal the shape of the nuclear ground
state. Our analysis with the Skyrme HFBCS-QRPA emphasizes the role of shell
driving mechanism in the evolution of nuclear deformation. It is reminded that
the spin-orbit interaction plays a crucial role in determining the single-
particle shell structure, magic numbers, and therefore nuclear deformations.
The strength of the spin-orbit interaction is determined empirically and is
not fully understood quantitatively, especially for weakly bound nuclei Kay
_et al._ (2017).

From the shell-model perspective, it has been shown that interactions between
neutrons and protons are primarily responsible for causing nuclear deformation
and that the same physical explanation holds across the nuclear chart Talmi
(1962); Federman and Pittel (1979). The review of the microscopic origin of
nuclear deformation from different approaches was discussed in Ref. Nazarewicz
(1994). The spin-orbit-like shells drive also the mechanism for shape
coexistence discussed in Ref. Martinou  _et al._ (2021). In general, the
degeneracy of eigenvalues of a single-particle Hamiltonian around the Fermi
level leads to instability concerning shape vibrations.

For completeness, in the Supplementary Material, we show the full range of
$\mathcal{C}_{\lambda}$ values and indicate nuclei with collapse for all three
multipolarities, as calculated with the SLy5 functional. We also show the
overview of the nuclear deformation softness landscape with three different
Skyrme forces based on the energy of the lowest excited state in each channel.
Finally, we provide tables with the numerical results obtained and used in the
figures.

##  IV Conclusion and future perspectives

The spherical QRPA is a numerically light and theoretically sound method for
diagnosing potential softness or deformation in nuclei, especially octupole,
hexadecapole, and higher-multipolarity shapes, which are much less studied
than quadrupole shapes and typically require special microscopic approaches.
It can capture the patterns of deformation in the nuclide chart and can help
identify easily areas of potential interest. Candidates for static quadrupole-
octupole deformed nuclei, as judged from the simultaneous softness or collapse
in both channels, were reported in Ref. Minh Loc  _et al._ (2023). In this
work, 122Te, 152Ce, 152-158Nd, 210Pb, and 206-218Po are found to be possible
candidates for static quadrupole-hexadecapole deformation.

Nuclear deformation is an ongoing research. Nowadays, new exotic beam
facilities have been built around the world. The impact of the nuclear shape
on the drip lines was recently discussed in Ref. Tsunoda  _et al._ (2020).
Theoretically, the determination of the drip lines has been updated up to
heavy nuclei from nuclear density functional theory Neufcourt  _et al._ (2020)
and up to Fe through the valence-space formulation of the in-medium similarity
renormalization group Stroberg  _et al._ (2021). Moreover, the difficulty in
assigning meaningful effective values to the shape parameters beta and gamma
because of large fluctuations, questioning the spherical shape of doubly magic
nuclei has been recently raised Poves  _et al._ (2020). Our work can serve as
a starting point for more detailed investigations in the future.

## Acknowledgements

N. L. A. was funded by the Master, PhD Scholarship Programme of Vingroup
Innovation Foundation (VINIF), code VINIF.2023.TS.003. B. M. L. was supported
by the Institute for Basic Science (IBS-R031-D1) and by SSAA under DOE NNSA
Contract No. DE-NA0004075. P. P. was supported by the Rare Isotope Science
Project of the Institute for Basic Science funded by the Ministry of Science,
ICT and Future Planning and the National Research Foundation (NRF) of Korea
(2013M7A1A1075764).

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