# OPV2V: An Open Benchmark Dataset and Fusion Pipeline for Perception with
Vehicle-to-Vehicle Communication

Runsheng Xu1∗, Hao Xiang1∗, Xin Xia1, Xu Han1, Jinlong Li2, Jiaqi Ma1 *Equal
contribution1University of California, Los Angeles, Mobility Lab. {rxx3386,
haxiang, x35xia, hanxu417<EMAIL_ADDRESS>State University,
Cleveland Vision and AI Lab<EMAIL_ADDRESS>
###### Abstract

Employing Vehicle-to-Vehicle communication to enhance perception performance
in self-driving technology has attracted considerable attention recently;
however, the absence of a suitable open dataset for benchmarking algorithms
has made it difficult to develop and assess cooperative perception
technologies. To this end, we present the first large-scale open simulated
dataset for Vehicle-to-Vehicle perception. It contains over 70 interesting
scenes, 11,464 frames, and 232,913 annotated 3D vehicle bounding boxes,
collected from 8 towns in CARLA and a digital town of Culver City, Los
Angeles. We then construct a comprehensive benchmark with a total of 16
implemented models to evaluate several information fusion strategies (i.e.
early, late, and intermediate fusion) with state-of-the-art LiDAR detection
algorithms. Moreover, we propose a new Attentive Intermediate Fusion pipeline
to aggregate information from multiple connected vehicles. Our experiments
show that the proposed pipeline can be easily integrated with existing 3D
LiDAR detectors and achieve outstanding performance even with large
compression rates. To encourage more researchers to investigate Vehicle-to-
Vehicle perception, we will release the dataset, benchmark methods, and all
related codes in https://mobility-lab.seas.ucla.edu/opv2v/.

##  I INTRODUCTION

Perceiving the dynamic environment accurately is critical for robust
intelligent driving. With recent advancements in robotic sensing and machine
learning, the reliability of perception has been significantly improved [1, 2,
3], and 3D object detection algorithms have achieved outstanding performance
either with LiDAR point clouds [4, 5, 6, 7] or multi-sensor data [8, 9].

Despite the recent breakthroughs in the perception field, challenges remain.
When the objects are heavily occluded or have small scales, the detection
performance will dramatically drop. Such problems can lead to catastrophic
accidents and are difficult to solve by any algorithms since the sensor
observations are too sparse. An example is revealed in Fig. 1(a). Such
circumstances are very common but dangerous in real-world scenarios, and these
blind spot issues are extremely tough to handle by a single self-driving car.

(a)

(b)

Figure 1: Two examples from our dataset. _Left_ : Screenshot of the
constructed scenarios in CARLA. _Middle_ : The LiDAR point cloud collected by
the ego vehicle. _Right_ : The aggregated point clouds from all surrounding
CAVs. The red circles represent the cars that are invisible to the ego vehicle
due to the occlusion but can be seen by other connected vehicles. (a): The ego
vehicle plans to turn left in a T-intersection and the roadside vehicles block
its sight to the incoming traffic. (b): Ego-vehicle’s LiDAR has no
measurements on several cars because of the occlusion caused by the dense
traffic.

To this end, researchers started recently investigating dynamic agent
detection in a cooperative fashion, such as USDOT CARMA [10] and Cooper [11].
By leveraging the Vehicle-to-Vehicle (V2V) communication technology, different
Connected Automated Vehicles (CAVs) can share their sensing information and
thus provide multiple viewpoints for the same obstacle to compensate each
other. The shared information could be raw data, intermediate features, single
CAV’s detection output, and metadata e.g., timestamps and poses. Despite the
big potential in this field, it is still in its infancy. One of the major
barriers is the lack of a large open-source dataset. Unlike the single
vehicle’s perception area where multiple large-scale public datasets exist
[12, 13, 14], most of the current V2V perception algorithms conduct
experiments based on their customized data [15, 16, 17]. These datasets are
either too small in scale and variance or they are not publicly available.
Consequently, there is no large-scale dataset suitable for benchmarking
distinct V2V perception algorithms, and such deficiency will preclude further
progress in this research field.

To address this gap, we present OPV2V, the first large-scale Open Dataset for
Perception with V2V communication. By utilizing a cooperative driving co-
simulation framework named OpenCDA [18] and CARLA simulator [19], we collect
73 divergent scenes with a various number of connected vehicles to cover
challenging driving situations like severe occlusions. To narrow down the gap
between the simulation and real-world traffic, we further build a digital town
of Culver City, Los Angeles with the same road topology and spawn dynamic
agents that mimic the realistic traffic flow on it. Data samples are shown in
Fig. 1 and Fig. 4. We benchmark several state-of-the-art 3D object detection
algorithms combined with different multi-vehicle fusion strategies. On top of
that, we propose an Attentive Intermediate Fusion pipeline to better capture
interactions between connected agents within the network. Our experiments show
that the proposed pipeline can efficiently reduce the bandwidth requirements
while achieving state-of-the-art performance.

##  II Related Work

Vehicle-to-Vehicle Perception: V2V perception methods can be divided into
three categories: early fusion, late fusion, and intermediate fusion. Early
fusion methods [11] share raw data with CAVs within the communication range,
and the ego vehicle will predict the objects based on the aggregated data.
These methods preserve the complete sensor measurements but require large
bandwidth and are hard to operate in real time [15]. In contrast, late fusion
methods transmit the detection outputs and fuse received proposals into a
consistent prediction. Following this idea, Rauch et al. [20] propose a
Car2X-based perception module to jointly align the shared bounding box
proposals spatially and temporally via an EKF. In [21], a machine learning-
based method is utilized to fuse proposals generated by different connected
agents. This stream of work requires less bandwidth, but the performance of
the model is highly dependent on each agent’s performance within the vehicular
network. To meet requirements of both bandwidth and detection accuracy,
intermediate fusion [22, 15] has been investigated, where intermediate
features are shared among connected vehicles and fused to infer the
surrounding objects. F-Cooper [22] utilizes max pooling to aggregate shared
Voxel features, and V2VNet [15] jointly reason the bounding boxes and
trajectories based on shared messages.

Sensors | Details
---|---
4x Camera | RGB, $800\times 600$ resolution, $110\degree$ FOV
1x LiDAR |  |  $64$ channels, $1.3~{}M$ points per second,
---
$120~{}m$ capturaing range, $-25\degree$ to $5\degree$
vertical FOV, $\pm 2~{}cm$ error
GPS & IMU |  $20~{}mm$ positional error, $2\degree$ heading error
TABLE I: Sensor specifications. Figure 2: Sensor setup for each CAV in OPV2V.
Figure 3: Examples of the front camera data and BEV map of two CAVs in OPV2V.
The yellow, green, red, and white lanes in the BEV map represent the lanes
without traffic light control, under green light control, under red light
control, and crosswalks.

Vehicle-to-Vehicle Dataset: To the best of our knowledge, there is no large-
scale open-source dataset for V2V perception in the literature. Some work [11,
22] adapts KITTI [14] to emulate V2V settings by regarding the ego vehicle at
different timestamps as multiple CAVs. Such synthetic procedure is unrealistic
and not appropriate for V2V tasks since the dynamic agents will appear at
different locations, leading to spatial and temporal inconsistency. [15]
utilizes a high-fidelity LiDAR simulator [23] to generate a large-scale V2V
dataset. However, neither the LiDAR simulator nor the dataset is publicly
available. Recently, several works [17, 24] manage to evaluate their V2V
perception algorithms on the CARLA simulator, but the collected data has a
limited size and is restricted to a small area with a fixed number of
connected vehicles. More importantly, their dataset is not released and
difficult to reproduce the identical data based on their generation approach.
T&J dataset [11, 22] utilizes two golf carts equipped with 16-channel LiDAR
for data collection. Nevertheless, the released version only has 100 frames
without ground truth labels and only covers a restricted number of road types.
A comparison to existing dataset is provided in Table II.

TABLE II: Dataset comparison. (†) The number is reported based on data used during their experiment. (††) Single LiDAR resolution’s data is counted. (‡) Ground truth data is not released in the T&J dataset and it only has 100 frames and LiDAR data. (-) means that the number is not reported in the paper and can’t be found in open dataset. (∗) means the data has the format mean$\pm$std. Dataset | frames | GT | Dataset | CAV | cities | Code | Open Dataset | Reproducibility&
---|---|---|---|---|---|---|---|---
3D boxes | Size | range |  |  |  | Extensibility
V2V-Sim [15] | 51,200 | - | - | 10 $\pm$ 7∗ | $>1$ |  |  |
[17] | 1,310† | - | - | 3, 5 | 1 |  |  |
[24] | 6,000†† | - | - | 2 | 1 | ✓ |  |
T&J [11, 22] | 100‡ | 0‡ | 183.7MB | 2 | 1 | ✓ | ✓ |
OPV2V | 11,464 | 232,913 | 249.4GB | 2.89 $\pm$ 1.06∗ | 9 | ✓ | ✓ | ✓

(a)

(b)

Figure 4: A caparison between the real Culver City and its digital town. (a)
The RGB image and LiDAR point cloud captured by our vehicle in Culver City.
(b) The corresponding frame in the digital town. The road topology, building
layout, and traffic distribution are similar to reality.

##  III OPV2V Dataset

###  III-A Data Collection

Simulator Selection. CARLA is selected as our simulator to collect the
dataset, but CARLA itself doesn’t have V2V communication and cooperative
driving functionalities by default. Hence, we employ OpenCDA [18], a co-
simulation tool integrated with CARLA and SUMO [25], to generate our
dataset111Codes for generating our dataset have been recently released in
https://github.com/ucla-mobility/OpenCDA/tree/feature/data_collection. It is
featured with easy control of multiple CAVs, embedded vehicle network
communication protocols, and more convenient and realistic traffic management.

Sensor Configuration. The majority of our data comes from eight default towns
provided by CARLA. Our dataset has on average approximately 3 connected
vehicles with a minimum of 2 and a maximum of 7 in each frame. As Fig. 2
shows, each CAV is equipped with 4 cameras that can cover 360° view together,
a 64-channel LiDAR, and GPS/IMU sensors. The sensor data is streamed at 20 Hz
and recorded at 10 Hz. A more detailed description of the sensor
configurations is depicted in Table I.

Culver City Digital Town. To incorporate scenarios that can better imitate
real-world challenging driving environments and evaluate models’ domain
adaptation capability, we further gather several scenes imitating realistic
configurations. An automated vehicle equipped with a 32-channel LiDAR and two
cameras is sent out to Culver City during rush hour to collect sensing data.
Then, we populate the road topology of digital town via RoadRunner [26],
select buildings based on agreement with collected data, and then spawn cars
mimicking the real-world traffic flow with the support of OpenCDA. We collect
4 scenes in Culver City with around 600 frames in total (See Fig. 4). These
scenes will be used for validation of models trained with simulated datasets
purely generated in CARLA. Future addition of data from real environments is
planned and can be added to the model training set.

Data Size. Overall, 11,464 frames (i.e. time steps) of LiDAR point clouds (see
Fig. 1) and RGB images (see Fig. 3) are collected with a total file size of
249.4 GB. Moreover, we also generate Bird Eye View (BEV) maps for each CAV in
each frame to facilitate the fundamental BEV semantic segmentation task.

Downstream Tasks. By default, OPV2V supports cooperative 3D object detection,
BEV semantic segmentation, tracking, and prediction either employing camera
rigs or LiDAR sensors. To enable users to extend the initial data, we also
provide a driving log replay tool222The tool can be found here.. along with
the dataset. By utilizing this tool, users can define their own tasks (e.g.,
depth estimation, sensor fusion) and set up additional sensors (e.g., depth
camera) without changing any original driving events. Note that in this paper,
we only report the benchmark results on 3D Lidar-based object detection.

Figure 5: Polar density map in log scale for ground truth bounding boxes. The polar and radial axes indicate the angle and distance (in meters) of the bounding boxes with respect to the ego vehicle. The color indicates the number of bounding boxes (log scale) in the bin. The darker color means a larger number of boxes in the bin. Figure 6: _Left_ : Number of points in log scale within the ground truth bounding boxes with respect to radial distance from ego vehicles. _Right_ : Bounding box size distributions.  TABLE III: Summary of OPV2V dataset statistics. Traffic density means the number of vehicles spawned around the ego vehicle within a 140m radius and aggressiveness represents the probability of a vehicle operating aggressive overtakes. Road Type | Percentage(%) |  | Length(s)
---
mean/std
| CAV number
---
mean/std
| Traffic density
---
mean/std
| Traffic Speed(km/h)
---
mean/std
| CAV speed(km/h)
---
mean/std
| Aggressiveness
---
mean/std
4-way Intersection | 24.5 | 12.5/4.2 | 2.69/0.67 | 29.6/26.1 | 19.3/8.8 | 21.3/10.2 | 0.09/0.30
T Intersection | 24.1 | 14.3/12.8 | 2.55/1.3 | 27.9/18.65 | 26.3/7.5 | 26.2/10.0 | 0.11/0.32
Straight Segment | 20.7 | 20.2/12.7 | 3.54/1.21 | 38.0/36.3 | 45.7/14.8 | 54.3/20.1 | 0.82/0.40
Curvy Segment | 23.3 | 17.8/6.8 | 2.86/0.95 | 19.1/9.2 | 45.8/15.1 | 51.6/19.2 | 0.50/0.51
Midblock | 4.7 | 10.0/1.3 | 3.00/1.22 | 21.8/8.2 | 45.1/8.3 | 50.7/11.5 | 0.20/0.44
Entrance Ramp | 2.7 | 9.3/0.9 | 2.67/0.57 | 20.3/2.8 | 54.8/1.7 | 66.7/4.8 | 0.67/0.57
Overall | 100 | 16.4/9.1 | 2.89/1.06 | 26.5/17.2 | 33.1/15.8 | 37.5/21.0 | 0.34/0.47
Figure 7: The architecture of Attentive Intermediate Fusion pipeline. Our
model consists of 6 parts: 1) Metadata Sharing: build connection graph and
broadcast locations among neighboring CAVs. 2) Feature Extraction: extract
features based on each detector’s backbone. 3) Compression (optional): use
Encoder-Decoder to compress/decompress features. 4) Feature sharing: share
(compressed) features with connected vehicles. 5) Attentive Fusion: leverage
self-attention to learn interactions among features in the same spatial
location. 6) Prediction Header: generate final object predictions.

###  III-B Data Analysis

As Table III depicts, six distinct categories of road types are included in
our dataset for simulating the most common driving scenarios in real life. To
minimize data redundancy, we attempt to avoid overlong clips and assign the
ego vehicles short travels with an average length of 16.4 seconds, dissimilar
locations, and divergent maneuvers for each scenario. We also allocate the
gathered 73 scenes with diverse traffic and CAV configurations to enlarge
dataset variance.

Fig. 5 and Fig. 6 reveal the statistics of the 3D bounding box annotations in
our dataset. Generally, the cars around the ego vehicle are well-distributed
with divergent orientations and bounding box sizes. This distribution is in
agreement with the data collection process where the object positions are
randomly selected around CAVs and vehicle models are also arbitrarily chosen.
As shown in Fig. 5, unlike the dataset for the single self-driving car, our
dataset still has a large portion of objects in view with distance $\geq$
100m, given that the ground truth boxes are defined with respect to the
aggregated lidar points from all CAVs. As displayed in Fig. 6, although a
single vehicle’s LiDAR points for distant objects are especially sparse, other
CAVs are able to provide compensations to remarkably boost the LiDAR points
density. This demonstrates the capability of V2V technology to drastically
increase perception range and provide compensation for occlusions.

##  IV Attentive Intermediate Fusion Pipeline

As sensor observations from different connected vehicles potentially carry
various noise levels (e.g., due to distance between vehicles), a method that
can pay attention to important observations while ignoring disrupted ones is
crucial for robust detection. Therefore, we propose an Attentive Intermediate
Fusion pipeline to capture the interactions between features of neighboring
connected vehicles, helping the network attend to key observations. The
proposed Attentive Intermediate Fusion pipeline consists of 6 modules:
Metadata sharing, Feature Extraction, Compression, Feature sharing, Attentive
Fusion, and Prediction. The overall architecture is shown in Fig. 7. The
proposed pipeline is flexible and can be easily integrated with existing Deep
Learning-based LiDAR detectors (see Table IV).

Metadata Sharing and Feature Extraction: We first broadcast each CAVs’
relative pose and extrinsics to build a spatial graph where each node is a CAV
within the communication range and each edge represents a communication
channel between a pair of nodes. After constructing the graph, an ego vehicle
will be selected within the group.333During training, a random CAV within the
group is selected as ego vehicle while in the inference, the ego vehicle is
fixed for a fair comparison. And all the neighboring CAVs will project their
own point clouds to the ego vehicle’s LiDAR frame and extract features based
on the projected point clouds. The feature extractor here can be the backbones
of existing 3D object detectors.

Compression and Feature sharing: An essential factor in V2V communication is
the hardware restriction on transmission bandwidth. The transmission of the
original high-dimensional feature maps usually requires large bandwidth and
hence compression is necessary. One key advantage of intermediate fusion over
sharing raw point clouds is the marginal accuracy loss after compression [15].
Here we deploy an Encoder-Decoder architecture to compress the shared message.
The Encoder is composed of a series of 2D convolutions and max pooling, and
the feature maps in the bottleneck will broadcast to the ego vehicle. The
Decoder that contains several deconvolution layers [27] on the ego-vehicles’
side will recover the compressed information and send it to the Attentive
Fusion module.

Attentive Fusion: Self-attention models [28] are adopted to fuse those
decompressed features. Each feature vector (green/blue circles shown in Fig.7)
within the same feature map corresponds to certain spatial areas in the
original point clouds. Thus, simply flattening the feature maps and
calculating the weighted sum of features will break spatial correlations.
Instead, we construct a local graph for each feature vector in the feature
map, where edges are built for feature vectors in the same spatial locations
from disparate connected vehicles. One such local graph is shown in Fig.7 and
self-attention will operate on the graph to reason the interactions for better
capturing the representative features.

Prediction Header: The fused features will be fed to the prediction header to
generate bounding box proposals and associated confidence scores.

Figure 8: The architecture of PIXOR with Attentive Fusion.

##  V Experiments

###  V-A Benchmark models

We implement four state-of-the-art LiDAR-based 3D object detectors on our
dataset and integrate these detectors with three different fusion strategies
i.e., early fusion, late fusion, and intermediate fusion. We also investigate
the model performance under a single-vehicle setting, named no fusion, which
neglects V2V communication. Therefore, in total 16 models will be evaluated in
the benchmark. All the models are implemented in a unified code framework, and
our code and develop tutorial can be found in the project website.

Selected 3D Object Detectors:  We pick SECOND [29], VoxelNet [4], PIXOR [30],
and PointPillar [5] as our 3D LiDAR detectors for benchmarking analysis.

Early fusion baseline: All the LiDAR point clouds will be projected into ego-
vehicles’ coordinate frame, based on the pose information shared among CAVs,
and then the ego vehicle will aggregate all received point clouds and feed
them to the detector.

Late fusion baseline:  Each CAV will predict the bounding boxes with
confidence scores independently and broadcast these outputs to the ego
vehicle. Non-maximum suppression (NMS) will be applied to these proposals
afterwards to generate the final object predictions.

Intermediate fusion: The Attentive Fusion pipeline is flexible and can be
easily generalized to other object detection networks. To evaluate the
proposed pipeline, we only need to add the Compression, Sharing, and Attention
(CSA) module to the existing network architecture. Since 4 different detectors
add CSA modules in a similar way, here we only show the architecture of
intermediate fusion with the PIXOR model as Fig. 8 displays. Three CSA modules
are added at the 2D backbone of PIXOR to aggregate multi-scale features while
all other parts of the network remain the same.

TABLE IV: Object detection results on Default CARLA Towns and digital Culver City. Method | Default | Culver
---|---|---
AP@IoU | AP@IoU
$0.5$ | $0.7$ | $0.5$ | $0.7$
PIXOR | No Fusion | 0.635 | 0.406 | 0.505 | 0.290
Late Fusion | 0.769 | 0.578 | 0.622 | 0.360
Early Fusion | 0.810 | 0.678 | 0.734 | 0.558
Intermediate Fusion | 0.815 | 0.687 | 0.716 | 0.549
PointPillar | No Fusion | 0.679 | 0.602 | 0.557 | 0.471
Late Fusion | 0.858 | 0.781 | 0.799 | 0.668
Early Fusion | 0.891 | 0.800 | 0.829 | 0.696
Intermediate Fusion | 0.908 | 0.815 | 0.854 | 0.735
SECOND | No Fusion | 0.713 | 0.604 | 0.646 | 0.517
Late Fusion | 0.846 | 0.775 | 0.808 | 0.682
Early Fusion | 0.877 | 0.813 | 0.821 | 0.738
Intermediate Fusion | 0.893 | 0.826 | 0.875 | 0.760
VoxelNet | No Fusion | 0.688 | 0.526 | 0.605 | 0.431
Late Fusion | 0.801 | 0.738 | 0.722 | 0.588
Early Fusion | 0.852 | 0.758 | 0.815 | 0.677
Intermediate Fusion | 0.906 | 0.864 | 0.854 | 0.775

###  V-B Metrics

We select a fixed vehicle as the ego vehicle among all spawned CAVs for each
scenario in the test and validation set. Detection performance is evaluated
near the ego vehicle in a range of $x\in[-140,140]m,y\in[-40,40]m$. Following
[15], we set the broadcast range among CAVs to be 70 meters. Sensing messages
outside of this communication range will be ignored by the ego vehicle.
Average Precisions (AP) at Intersection-over-Union (IoU) threshold of both 0.5
and 0.7 are adopted to assess different models. Since PIXOR ignores the $z$
coordinates of the bounding box, we compute IoU only on x-y plane to make the
comparison fair. For the evaluation targets, we include vehicles that are hit
by at least one LiDAR point from any connected vehicle.

Figure 9: Average Precision at IoU=0.7 with respect to CAV number. Figure 10:
Average Precision at IoU=0.7 with respect to data size in log scale based on
VoxelNet detector. The number$\times$ refers to the compression rate.

###  V-C Experiment Details

The train/validation/test splits are 6764/1981/2719 frames. The testing frames
contain all road types and are further split into two parts–CARLA default maps
and Culver City digital town. For each frame, we assure that the minimum and
maximum numbers of CAVs are 2 and 7 respectively. We use Adam Optizer [31] and
early stop to train all models, and it takes us 14 days to finish all training
on 4 RTX 3090 GPUs.

###  V-D Benchmark Analysis

Table IV depicts the performance of the selected four LiDAR detectors combined
with different fusion strategies. All fusion methods achieve $\geq$10% AP
gains at IoU 0.7 over no fusion counterparts for both default CARLA towns and
Culver City, showing the advantage of aggregating information from all CAVs
for V2V perception. Generally, because of the capability of preserving more
sensing measurements and visual cues, early fusion methods outperform late
fusion methods. Except for PIXOR at Culver City, intermediate fusion achieves
the best performance on both testing sets compared with all other methods. We
argue that the AP gains over early fusion originate from the mechanism of the
self-attention module, which can effectively capture the inherent correlation
between each CAV’s perception information. It is also worth noting that the
prediction results for Culver City are generally inferior to CARLA towns. Such
a phenomenon is expected as the traffic pattern in Culver City is more similar
to real life, which causes a domain gap with the training data. Furthermore,
we collect the Culver City data in a busy hour under a very congested driving
environment, which leads to vastly severe occlusions and makes the detection
task very challenging.

###  V-E Effect of CAV Quantity

We explore the detection performance as affected by the number of CAVs in a
complex intersection scenario where 150 vehicles are spawned in the
surrounding area. A portion of them will be transformed into CAVs that can
share information. We gradually increase the number of the CAVs up to 7 and
apply VoxelNet with different fusion methods for object detection. As shown in
Fig. 9, the AP has a positive correlation with the number of CAVs. However,
when the quantity reaches 4, the increasing rate becomes lower. This can be
due to the fact that the CAVs are distributed on different sides of the
intersection and four of them can already provide enough viewpoints to cover
most of the blind spots. Additional enhancements with 5 or more vehicles come
from denser measurements on the same object.

###  V-F Effect of Compression Rates

Fig. 10 exhibits the data size needed for a single transmission between a pair
of vehicles and corresponding AP for all fusion methods on the testing set in
CARLA towns. We pick VoxelNet for all fusion methods here and simulate
distinct compression rates by modifying the number of layers in Encoder-
Decoder. By applying a straightforward Encoder-Decoder architecture to squeeze
the data, the Attentive Intermediate Fusion obtains an outstanding trade-off
between the accuracy and bandwidth. Even with a 4096x compression rate, the
performance still just drop marginally (around 3%) and surpass the early
fusion and late fusion. Based on the V2V communication protocol [32], data
broadcasting can achieve 27 Mbps at the range of 300 m. This represents that
the time delay to deliver the message with a 4096x compression rate is only
about 5 ms.

##  VI CONCLUSIONS

In this paper, we present the first open dataset and benchmark fusion
strategies for V2V perception. We further come up with an Attentive
Intermediate Fusion pipeline, and the experiments show that the proposed
approach can outperform all other fusion methods and achieve state-of-the-art
performance even under large compression rates.

In the future, we plan to extend the dataset with more tasks as well as
sensors suites and investigate more multi-modal sensor fusion methods in the
V2V and Vehicle-to-infrastructure (V2I) setting. We hope our open-source
efforts can make a step forward for the standardizing process of the V2V
perception and encourage more researchers to investigate this new direction.

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# Capturing electronic correlations in electron-phonon interactions in
molecular systems with the $GW$ approximation

Antonios M. Alvertis  KBR, Inc., NASA Ames Research Center, Moffett Field,
California 94035, United States  Materials Sciences Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, United States
e-mail<EMAIL_ADDRESS>David B. Williams-Young  Applied Mathematics and
Computational Research Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, United States  Fabien Bruneval  Université Paris-
Saclay, CEA, Service de Corrosion et de Comportement des Matériaux, SRMP,
91191 Gif-sur-Yvette, France  Jeffrey B. Neaton  Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, United
States  Department of Physics, University of California Berkeley, Berkeley,
United States  Kavli Energy NanoScience Institute at Berkeley, Berkeley,
United States

## Abstract

Electron-phonon interactions are of great importance to a variety of physical
phenomena, and their accurate description is an important goal for first-
principles calculations. Isolated examples of materials and molecular systems
have emerged where electron-phonon coupling is enhanced over density
functional theory (DFT) when using the Green’s-function-based _ab initio_ $GW$
method, which provides a more accurate description of electronic correlations.
It is however unclear how general this enhancement is, and how employing high-
end quantum chemistry methods, which further improve the description of
electronic correlations, might further alter electron-phonon interactions over
$GW$ or DFT. Here, we address these questions by computing the renormalization
of the highest occupied molecular orbital energies of Thiel’s set of organic
molecules by harmonic vibrations using DFT, $GW$ and equation-of-motion
coupled-cluster calculations. We find that $GW$ can increase the magnitude of
the electron-phonon coupling across this set of molecules by an average factor
of $1.1-1.8$ compared to DFT, while equation-of-motion coupled-cluster leads
to an increase of $1.4-2$. The electron-phonon coupling predicted with the _ab
initio_ $GW$ method is generally in much closer agreement to coupled cluster
values compared to DFT, establishing $GW$ as an accurate way of computing
electron-phonon phenomena in molecules and beyond at a much lower
computational cost than higher-end quantum chemistry techniques.

##  1 Introduction

The interaction of electrons with atomic and lattice vibrations (phonons) in
solids is of central importance in chemistry and physics, defining carrier
transport properties [1, 2], superconducting critical temperatures [3, 4], the
rate of non-radiative recombination [5, 6], the reaction pathway of ultrafast
charge and energy transfer [7, 8], and more for materials and molecular
systems. In organic molecules and molecular crystals, atomic and lattice
vibrations can cause substantial renormalization of ground and excited state
properties [9, 10, 11, 12]. Therefore, accurate and predictive theories for
the interaction of electrons with vibrations are important in order to guide
the experimental search for systems with a wide range of functionality, as
well as to promote the fundamental understanding of a variety of physical
processes.

For decades, density functional theory (DFT) has arguably been the most widely
used framework for computing the properties of molecules and materials from
first principles, and it has been extensively used to calculate electronic and
vibrational degrees of freedom, as well as their interplay [13, 14, 15, 16,
17]. The choice of the DFT exchange-correlation functional has been known in
several systems to affect the predicted magnitude of these effects, with
examples demonstrating that inclusion of exact exchange to the DFT functional
tends to increase electron-phonon coupling [18] compared to cases where the
local density approximation (LDA) [19] and generalized gradient approximations
(GGA) [20] are used. Moreover, _ab initio_ many-body perturbation theory
within the $GW$ approximation [21, 22], which is known to yield highly
accurate properties for the electronic levels of molecular systems and solids
[23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45], has also been used to compute electron-phonon coupling,
showing significantly increased magnitude of these interactions over DFT in
some cases [46, 47, 48, 49, 50], but only small changes in others [49].
Further increases of electron-phonon coupling might also be expected with an
even more accurate description of electronic correlations, _e.g._ methods such
as coupled cluster (CC) [51]. Additionally, the challenge associated with
rigorously converging electron-phonon calculations in solids has led to the
suggestion that the discrepancies reported in the literature in the phonon-
induced fundamental gap renormalization obtained at different levels of
electronic structure theory could potentially be attributed to under-converged
calculations [52]. Given these practical challenges and inconclusive reports,
there remains no consensus on whether including exact exchange within DFT or
employing higher-level $GW$ calculations leads to a systematic increase of the
computed electron-phonon interactions, or whether examples in the literature
showing such an increase constitute isolated cases. To complicate matters
further, Kohn-Sham wavefunctions computed within DFT are most commonly used as
a starting point for $GW$ calculations, and the choice of starting point can
greatly affect the $GW$ results [53, 54]. However, the dependence of the
electron-phonon interactions computed with $GW$ on the functional employed for
the DFT starting point remains unknown.

Here we present a systematic study of electron-phonon coupling using Thiel’s
set of organic molecules [55], a set of 28 small molecules. As a proxy for the
magnitude of the coupling of molecular vibrations to electrons, we study the
renormalization of the highest occupied molecular orbital (HOMO) energy of
each molecule due to the zero-point motion of the atomic nuclei, which we
obtain within the harmonic approximation using a well-established finite-
displacements approach [56, 9]. We compute the zero-point renormalization
(ZPR) of the HOMO energies with DFT using several different functionals with
varying degrees of exact exchange, and we also perform $GW$ calculations with
various DFT starting points. We compare our DFT and $GW$ ZPR calculations to
the values obtained with accurate coupled-cluster (CC) methods. Our results
establish that the correction to DFT electronic correlations obtained within
the $GW$ and CC formalisms tends to systematically increase the magnitude of
electron-phonon coupling, particularly when employing DFT functionals with
semilocal or small amounts of exact exchange. That $GW$ and CC methods
generally predict stronger electron-phonon interactions compared to DFT can
have implications for accurate prediction of properties such as
superconductivity [47, 50] and finite-temperature excited state properties
[57] in other systems. The results with $GW$ are in overall good agreement
with CC, yet obtained at much lower computational cost ($N^{4}$ compared to
$N^{6}$ for coupled cluster), suggesting that the $GW$ method provides a
powerful and computationally efficient way of accurately modeling electron-
phonon interactions in molecular systems.

##  2 Theoretical background and computational methods

Figure 1: Thiel’s set of organic molecules.

In this paper we study the so-called Thiel’s set of organic molecules shown in
Fig. 1, which consists of $28$ diverse gas-phase molecular structures [55]. In
order to understand the impact of the level of the electronic structure theory
on the magnitude of the electron-phonon interactions, we focus on the
vibrationally-induced ZPR of the HOMO energy of the Thiel’s set molecules.
Section 2.1 provides an overview of the theoretical framework we employ here.
Then Section 2.2 goes into some detail on the electronic structure methods
used in this work in order to compute the HOMO energies of the Thiel’s set
molecules, and Section 2.3 summarizes the results of our vibrational and
electronic structure calculations to compute the HOMO ZPR at the various
levels of theory.

###  2.1 Vibrational averages of observables

In what follows, we review the formalism for computing vibrational averages
for quantum-mechanical operators, and some of the approximations relevant to
this work. Let us consider an operator $\mathcal{O}$ corresponding to an
observable of interest, in the presence of atomic motion at temperature $T$.
Within the Born-Oppenheimer approximation, the finite-temperature expectation
value of $\mathcal{O}$ may be written as

$\left\langle\mathcal{O}(T)\right\rangle_{\cal H}=\frac{1}{Z}\int
dX\mathcal{O}(X)e^{-\beta\cal H},$ (1)

where the canonical partition function $Z=\int dXe^{-\beta\cal H}$ involves
the configuration space integral $\int dX$ [58].

The Hamiltonian $\cal H$ of the system includes electronic and nuclear degrees
of freedom in general, and may be approximated at different levels. One
approach that is generally valid for small organic molecules is to assume
nuclear motion to be harmonic, reducing the vibrational contribution to the
Hamiltonian to the following form,

${\cal
H}^{\textrm{har}}\equiv\frac{1}{2}\sum_{n}(\nabla_{u_{n}}^{2}+\omega_{n}^{2}u_{n}^{2}),$
(2)

in atomic units. Here, $\omega_{n}$ is the frequency of the $n_{\text{th}}$
vibrational mode, and $u_{n}$ the respective eigendisplacement, which are
computed using a finite-displacements approach [59, 60] in conjunction with
DFT total energy and forces calculations. The ground state vibrational
properties of organic molecules are known to be fairly insensitive to the
amount of exact exchange in the employed DFT functional [18], and we therefore
compute the vibrational modes and frequencies of all structures considered
here with the global hybrid B3LYP [61], as implemented in the NWChem software
[62].

In the harmonic approximation, the integral of eq. 1 becomes [56]

$\left\langle\mathcal{O}(T)\right\rangle_{\cal H}=\int
d\mathbf{u}|\Phi(\mathbf{u};T)|^{2}\mathcal{O}(\mathbf{u}),$ (3)

where $\mathbf{u}$ a vector of atomic displacements associated with a given
vibrational mode in the system and

$|\Phi(\mathbf{u};T)|^{2}=\prod_{n}(2\pi\sigma^{2}_{n}(T))^{-1/2}\exp{\left(-\frac{u_{n}^{2}}{2\sigma^{2}_{n}(T)}\right)}$
(4)

is the harmonic density at temperature $T$, which in turn is a product of
Gaussian functions of width,

$\sigma^{2}_{n}(T)=\frac{1}{2\omega_{n}}\cdot\coth{\left(\frac{\omega_{n}}{2k_{B}T}\right)}.$
(5)

The expectation value of eq. 3 may be computed by a Monte Carlo sampling,
drawing $N$ random samples $\mathbf{u}_{i}$ from the harmonic density
distribution function of eq. 4, at which we compute the observable of interest
$\mathcal{O}(\mathbf{u}_{i})$, allowing us to finally obtain

$\left\langle\mathcal{O}(T)\right\rangle_{\text{MC}}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}\mathcal{O}(\mathbf{u}_{i}).$
(6)

This Monte Carlo (MC) sampling of the expectation value of observables has
strong parallels to the nuclear ensemble method [63].

Apart from the harmonic and adiabatic approximations, no further assumptions
have been made in deriving eq. 6. The Monte Carlo sampling method has the
advantage of including the effect of vibrations to all orders on the
observable of interest in a non-perturbative fashion [64]. We will focus here
on the effect of vibrations on electronic observables (see Section 2.2) and
refer to the interactions between electronic and vibrational degrees of
freedom as electron-phonon interactions, and use the term “phonon”
interchangeably with “molecular vibrations” and “normal modes”.

The Monte Carlo sampling method does not allow one to identify the
contribution of individual vibrational modes to the thermal average, as we
always consider collective displacements. In order to isolate the effect of
individual vibrations one can expand the observable of interest in a specific
vibrational coordinate as

$\mathcal{O}(\mathbf{u})=\mathcal{O}(\mathbf{0})+\sum_{n}\frac{\partial\mathcal{O}(\mathbf{0})}{\partial
u_{n}}u_{n}+\frac{1}{2}\sum_{n}\sum_{n^{\prime}}\frac{\partial^{2}\mathcal{O}(\mathbf{0})}{\partial
u_{n}\partial u_{n^{\prime}}}u_{n}u_{n^{\prime}}+...\hskip 2.84544pt,$ (7)

where $\mathbf{u}=\mathbf{0}$ represents the equilibrium geometry of the
molecule. In the above expansion of $\mathcal{O}(\mathbf{u})$ the third-order
term vanishes, hence by truncating eq. 3 to fourth order one arrives at the
so-called quadratic (Q) expectation value:

$\left\langle\mathcal{O}(T)\right\rangle_{\text{Q}}=\mathcal{O}(\mathbf{0})+\sum_{n}\frac{1}{2\omega_{n}}\cdot\frac{\partial^{2}\mathcal{O}}{\partial
u_{n}^{2}}[\frac{1}{2}+n_{B}(\omega_{n},T)],$ (8)

where $n_{B}(\omega_{n},T)$ is the Bose-Einstein distribution function for the
occupation of mode $n$ at temperature $T$. At a practical level, the
expectation value of eq. 8 may be obtained by approximating the second
derivative appearing for every vibrational mode in this equation by the
finite-difference formula

$\frac{\partial^{2}\mathcal{O}}{\partial
u_{n}^{2}}\approx\frac{\mathcal{O}(\delta u_{n})+\mathcal{O}(-\delta
u_{n})-2\mathcal{O}(\mathbf{0})}{\delta u_{n}^{2}}.$ (9)

The quadratic expectation value of eq. 8 is less accurate than the Monte Carlo
expectation value of eq. 6, and also has the disadvantage that the finite
difference formula employed to approximate the second derivative introduces
some dependence on the choice of $\delta u_{n}$ for every mode. While we will
not use this level of approximation to extract quantitative values of
observables in the presence of molecular vibrations, it offers the advantage
of separating the contribution of each normal mode, and we will employ it
below to gain further physical insights into the ZPR we compute with eq. 6.

###  2.2 Electronic structure calculation of molecular ionization potentials

Here we are interested in understanding the effect of vibrations on the HOMO
energies, $\epsilon_{\text{HOMO}}$, of the Thiel’s set molecules, therefore we
take $\mathcal{O}=\epsilon_{\text{HOMO}}$, using the notation of Section 2.1.
We compute all HOMO energies with DFT using a range of exchange-correlation
functionals. Specifically, we employ the GGA as formulated by Perdew and Burke
and Ernzerhof (PBE) [20]; and we employ global hybrid variants of PBE, namely
PBEh($\alpha$), that contain a finite amount of exact exchange that is
governed by the parameter $\alpha$ ranging from $0$ to $1$, namely

$E_{XC}=\alpha E^{EX}_{X}+(1-\alpha)E^{PBE}_{X}+E^{PBE}_{C},$ (10)

where $E_{XC}$ the PBEh exchange-correlation functional, $E^{EX}_{X}$ the
exact exchange energy, and $E^{PBE}_{X}$ and $E^{PBE}_{C}$ are the PBE
exchange and correlation energies respectively. We consider two cases,
$\alpha=0.25$ and $\alpha=0.5$, in order to gain insights into the impact of
the amount of exact exchange on the HOMO energies and their ZPR in this set of
molecules. The case with $\alpha=0.25$ is also referred to as the PBE0
functional, and we will refer to the case with $\alpha=0.5$ simply as the PBEh
functional. Moreover, we employ the BHLYP functional [65], which also contains
$50\%$ exact exchange. Functionals with about 50 % are known to provide an
excellent starting point for subsequent $GW$ calculations [31, 45] for
molecular systems.

Beyond DFT, we also perform $GW$ calculations using the so-called one-shot
$G_{o}W_{o}$ approach. Here the one-particle Green’s functions $G$ and
screened Coulomb interaction $W$ are constructed from the eigenvalues and
eigenvectors of a preceding DFT calculation ($W$ is obtained within the random
phase approximation [66]), and used to construct the self-energy $\Sigma=iGW$,
which in turn is used to generate quasiparticle energies via correction
$\Sigma-V_{XC}$ to generalized Kohn-Sham eigenvalues. Within this one-shot
approach, we do not iteratively update $\Sigma$ based on the computed $GW$
eigenvalues and eigenvectors, as the one-shot $G_{o}W_{o}$ approach has been
shown to produce highly accurate results for a wide range of molecular systems
[45]. While larger basis sets are often required to achieve convergence of the
orbital energies obtained within _ab initio_ many-body perturbation theory
[67], the ZPR is computed as a difference between orbital energies (see eq. 12
below), which converges faster. In the supplementary information Fig. 1 and
Fig. 2 we plot the convergence of the HOMO energy and HOMO ZPR respectively,
for the example case of pyrimidine, with respect to the basis set size. We
thus verify that the aug-cc-pVTZ basis produces converged results for the HOMO
energy ZPR, which are at most $1$ meV different from the values obtained using
aug-cc-pV5Z.

In order to assess the accuracy of the computed DFT and GW HOMO energy ZPR in
this work, we compare their values to the ZPR of the ionization potential (IP)
as computed by the equation-of-motion ionization potential coupled cluster
method with single and double excitations (EOM-IP-CCSD) [51]. For each of the
molecules in Thiel’s set, we compute the lowest IP within EOM-IP-CCSD,
corresponding to the negative of the HOMO energy according to the ionization
potential theorem. We perform all EOM calculations within the Massively
Parallel Quantum Chemistry (MPQC) code [68, 69], employing the aug-cc-pVTZ
basis set as in the case of our DFT and $GW$ calculations.

###  2.3 Extracting the HOMO energy ZPR

Figure 2: HOMO energies (black crosses) of furan (C4H4O) at different $0$ K
configurations, computed with one-shot $GW@\text{PBE0}$. The red dashed line
indicates the HOMO energy in the absence of molecular vibrations, the blue
dashed line is its converged $0$ K value when vibrations are accounted for,
and the blue solid line represents the cumulative average of the vibrational
average of the HOMO energy.

We now combine the vibrational and electronic structure calculations of
Sections 2.1 and 2.2 to compute the HOMO energy ZPR with different levels of
electronic structure theory. For each molecule within the Thiel’s set, we
generate $N=100$ displaced configurations $\mathbf{u}_{i}$, and following eq.
6, we obtain the HOMO energy at $0$ K as

$\epsilon_{\text{HOMO}}(0\,\mathrm{K})_{\text{MC}}=\frac{1}{100}\sum_{i=1}^{100}\epsilon_{\text{HOMO}}(\mathbf{u}_{i}).$
(11)

The values of $\epsilon_{\text{HOMO}}(\mathbf{u}_{i})$ are computed within the
different levels of electronic structure theory outlined in Section 2.2. We
find that $N=100$ is generally sufficient to converge these vibrational
averages, with values in the proximity of $N=50$ already being converged
within $1$ meV in most cases. In Fig. 2 we show an example of the convergence
of the HOMO energy of furan at $0$ K with $GW@\text{PBE0}$ (one-shot $GW$ with
PBE0 eigenenergies and eigenstates as starting point).

The ZPR for the HOMO energy of each molecule is then obtained as

$\text{ZPR}=\epsilon_{\text{HOMO}}(0\,\mathrm{K})-\epsilon_{\text{HOMO}}(\mathbf{u}=\mathbf{0}).$
(12)

We note that for certain limited cases of molecules, and depending on the
employed DFT functional, there can be accidental near-degeneracies between
HOMO and HOMO-1, and the displacements $\mathbf{u}_{i}$ can sometimes induce a
change in the ordering of these orbitals. In order to reliably apply eq. 12,
one needs to correctly identify the orbital that corresponds to the original
HOMO, which we achieve by monitoring the expectation value of the kinetic
energy operator of the near-degenerate orbitals. This case of accidental near-
degeneracies is different to the symmetry-imposed degeneracy of molecules such
as benzene, where the HOMO and HOMO-1 are degenerate. In these latter
degenerate cases, the ZPR of the HOMO and HOMO-1 is taken to be the average of
the two, which is a gauge-invariant quantity.

##  3 Results and discussion

In Fig. 3 we show the average (across the Thiel’s set) of the ratio of the
HOMO energy ZPR at the various DFT and $GW$ levels of theory to the ZPR
obtained within EOM-IP-CCSD ($\text{ZPR}_{\text{CCSD}}$). Table 1 gives the
mean absolute error (MAE) and mean signed error (MSE) of the ZPR at the
different levels of theory with respect to our EOM-IP-CCSD reference values.
All ZPR values for the HOMO energies of individual molecules are given in the
supplementary material. We have excluded butadiene, cyclopentadiene and
naphthalene from the averages of Fig. 3, due to the fact that their EOM-IP-
CCSD ZPR value is smaller than 2 meV, causing unphysically large oscillations
in the values of the ratio presented in Fig. 3.

Figure 3:  Average ratio ($\%$) of the HOMO energy ZPR obtained at the different levels of theory compared with that obtained with EOM-IP-CCSD, across the entire Thiel’s set with the exception of butadiene, cyclopentadiene and naphthalene (see main text). Blue bars indicate the average ratio of the DFT HOMO energy ZPR values to EOM-IP-CCSD, while orange bars correspond to the respective $GW$ values. The red bars are the standard error to the average values. Values near $100$% correspond to good agreement with EOM-IP-CCSD.  Level of theory | MSE | MAE
---|---|---
PBE | $-26.9$ | $32.8$
$GW@\text{PBE}$ | $2.9$ | $11.1$
PBE0 | $-18.7$ | $18.8$
$GW@\text{PBE0}$ | $-0.1$ | $6.9$
PBEh | $-5.5$ | $14.0$
$GW@\text{PBEh}$ | $-5.9$ | $8.9$
BHLYP | $-7.8$ | $20.3$
$GW@\text{BHLYP}$ | $-5.0$ | $8.6$
Table 1: Mean signed error (MSE) and mean absolute error (MAE) for the ZPR
values at the different DFT and $GW$ levels of electronic structure theory
with respect to the ZPR values of our EOM-IP-CCSD reference. All values are in
meV.

We first comment on the average ratio of the ZPR obtained using DFT with the
various functionals to $\text{ZPR}_{\text{CCSD}}$, indicated with blue bars in
Fig. 3. On average, as the content of exact exchange included in the
functional is increased, the magnitude of the ZPR increases, approaching the
EOM-IP-CCSD values. This is consistent with reports in the literature for the
fullerene $\text{C}_{60}$ [18], where employing a hybrid functional increases
the magnitude of electron-phonon interactions. Our result here shows that such
a trend seems to hold generally across diverse organic molecules. This trend
of increased ZPR with greater contents of exact exchange is consistent with
the well-known increase in the electronic localization in Hartree-Fock
compared to semi-local DFT [70]. The inclusion of exact exchange causes
increased localization between bonded atoms, consequently resulting in greater
variation of this density upon displacement of atoms, for example through
carbon-carbon stretching motions. Such high-frequency motions dominate in
small organic molecules such as the ones studied here, and are indeed known to
have a larger effect on electronic states with a more localized electronic
density [9, 10]. However, we note the inclusion of exact exchange is not
sufficient to yield good agreement with the EOM-IP-CCSD ZPR values on average.

We now comment on the effect of performing $GW$ calculations on top of the
different DFT starting points. The average $GW$ HOMO energy ZPR ratios to
$\text{ZPR}_{\text{CCSD}}$ in Fig. 3 (orange bars) show an increase compared
to their DFT counterparts in each case. This effect is particularly prominent
in the cases where functionals with a low exchange content (PBE and PBE0) are
employed, with the average ZPR ratio increasing by a factor of $1.7-1.8$ in
these two cases compared to DFT. This strong systematic increase of the
electron-phonon interactions predicted by $GW$ compared to that computed from
DFT functionals with low exchange content is consistent with multiple reports
in the literature for diverse systems [46, 47, 48, 49, 50]. In fact, Refs.
[46, 47, 48, 49, 50] all employ DFT starting points without any exact
exchange. We also see from Fig. 3 that the increase of the HOMO energy ZPR
from DFT to $GW$ persists even in cases with $50\%$ exact exchange (PBEh and
BHLYP), although it is a much weaker effect compared to using functionals with
a small content of exchange. The increase of electron-phonon interactions upon
inclusion of exact exchange can in part reproduce the increase computed with
$GW$. We note that the $GW$ HOMO energy ZPR values have significantly smaller
standard errors relative to DFT (with the exception of $GW@\text{PBE}$),
highlighting the overall better performance of $GW$, even over hybrid DFT
calculations with substantial exact exchange included. The additional
electronic correlations included in EOM-IP-CCSD also lead to a small increase
over $GW$. Notably, $GW$ calculations on top of DFT starting points with
no/low contents of exact exchange (PBE and PBE0 in this case) perform best in
terms of reproducing the EOM-IP-CCSD reference.

Figure 4: Comparison of the HOMO ZPR of the Thiel’s set molecules computed
with PBE/$GW@\text{PBE}$ (panel a) and PBE0/$GW@\text{PBE0}$ (panel b) to EOM-
IP-CCSD values. Perfect agreement is indicated by the diagonal line (black
dashes).

In order to gain a better understanding of the ZPR averages of Fig. 3 and the
large increase of the ZPR when performing $GW$ calculations using PBE and PBE0
DFT starting points, we plot in Fig. 4 the HOMO energy ZPR with PBE and
$GW@\text{PBE}$ (panel a) and with PBE0 and $GW@\text{PBE0}$ (panel b),
against the EOM-IP-CCSD ZPR for each of the Thiel’s set molecules. It is
evident that $GW$ HOMO energy ZPR values are in better agreement with EOM-IP-
CCSD, and are generally larger in magnitude compared DFT. Additionally, the
$GW$ ZPR values are more closely distributed around the black dashed line
along the diagonal, indicating full agreement with EOM-IP-CCSD values,
especially in the $GW@\text{PBE0}$ case.

Figure 5: Vibrational mode-resolved difference in the ZPR computed with DFT
PBE0 and $GW@\text{PBE0}$ for furan.

We also identify the effect of individual vibrational modes on the ZPR
differences at the different levels of theory. In Fig. 5 we show results for
furan, where by using the quadratic approximation of Section 2.1 to the ZPR,
we plot the difference between the PBE0 and $GW@\text{PBE0}$ HOMO energy ZPR
values decomposed into the different normal mode contributions. Furan is
representative of a more general phenomenon, where every individual mode tends
to contribute increased ZPR at the $GW$ level of theory compared to DFT. This
is in agreement with the previous finding in the case of $\text{C}_{60}$ [18],
where employing a hybrid functional caused an increase of the electron-phonon
coupling constants for the majority of the vibrational modes of the molecule.

## Conclusions

We have presented a systematic study of the zero-point renormalization of the
HOMO energy of the Thiel’s set of organic molecules computed at different
levels of theory, in particular using DFT functionals with varying degree of
exact exchange, $GW$ calculations using these DFT calculations as starting
points, and coupled-cluster calculations within the equation of motion
formalism. We find that DFT HOMO energy ZPR values are systematically
underestimated compared to coupled-cluster values, with the inclusion of exact
exchange somewhat improving agreement. HOMO energy ZPR values obtained within
the $GW$ formalism greatly increase the electron-phonon interactions predicted
by the DFT starting points and result in much better agreement with coupled-
cluster. The underestimation of the DFT ZPR is greatest when employing
functionals with low fractions of exact exchange, in agreement with examples
that have been reported in the literature over the past decade. Our study
establishes such an increase in the magnitude of electron-phonon coupling to
be a general feature of the $GW$ method for molecules, and through comparison
to coupled-cluster calculations emphasizes that indeed computing electron-
phonon interactions within methods that incorporate electronic correlations
beyond DFT is a necessary step towards achieving predictive accuracy for these
phenomena. Importantly, the favorable scaling of $GW$ calculations compared to
coupled cluster demonstrates that _ab initio_ $GW$ methods can be more
affordable and yet accurate for modeling electron-phonon interactions. Our
results may have implications beyond molecular physics and for a wide range of
systems in condensed matter, where electron-phonon interactions play an
important role, including, but not limited to, molecular crystals and
materials with potential for high-temperature superconductivity.

## Acknowledgments

This work was undertaken as part of the Photosynthetic Light Harvesting
program at LBNL and primarily supported by the U.S. Department of Energy,
Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences
Division, under Contract No. DE-AC02-05CH11231. DWY was supported by the
Center for Scalable Predictive methods for Excitations and Correlated
phenomena (SPEC), which is funded by the U.S. Department of Energy (DoE),
Office of Science, Office of Basic Energy Sciences, Division of Chemical
Sciences, Geosciences and Biosciences as part of the Computational Chemical
Sciences (CCS) program at Lawrence Berkeley National Laboratory under FWP
12553. This research used resources of the National Energy Research Scientific
Computing Center (NERSC), a Department of Energy Office of Science User
Facility using NERSC award BES-ERCAP0023385. Part of this work was performed
using HPC resources from GENCI–TGCC (Grant 2023-gen6018).

Figure 6: Table of contents figure.

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