# One-Shot Weakly Supervised Video Object Segmentation

Mennatullah Siam
University of Alberta
<EMAIL_ADDRESS>&Naren Doraiswamy*
Indian Institute of Science (IISc)
<EMAIL_ADDRESS>&Boris N. Oreshkin*
ElementAI
<EMAIL_ADDRESS>&Hengshuai Yao
HiSilicon, Huawei Research
<EMAIL_ADDRESS>&Martin Jagersand
University of Alberta
<EMAIL_ADDRESS>equally contributing

###### Abstract

Conventional few-shot object segmentation methods learn object segmentation
from a few labelled support images with strongly labelled segmentation masks.
Recent work has shown to perform on par with weaker levels of supervision in
terms of scribbles and bounding boxes. However, there has been limited
attention given to the problem of few-shot object segmentation with image-
level supervision. We propose a novel multi-modal interaction module for few-
shot object segmentation that utilizes a co-attention mechanism using both
visual and word embeddings. It enables our model to achieve 5.1% improvement
over previously proposed image-level few-shot object segmentation. Our method
compares relatively close to the state of the art methods that use strong
supervision, while ours use the least possible supervision. We further propose
a novel setup for few-shot weakly supervised video object segmentation(VOS)
that relies on image-level labels for the first frame. The proposed setup uses
weak annotation unlike semi-supervised VOS setting that utilizes strongly
labelled segmentation masks. The setup evaluates the effectiveness of
generalizing to novel classes in the VOS setting. The setup splits the VOS
data into multiple folds with different categories per fold. It provides a
potential setup to evaluate how few-shot object segmentation methods can
benefit from additional object poses, or object interactions that is not
available in static frames as in PASCAL-$5^{i}$ benchmark.

##  1 Introduction

The few-shot learning literature has mainly focused on the classification task
such as Koch et al. (2015)Vinyals et al. (2016)Snell et al. (2017)Qi et al.
(2017)Finn et al. (2017)Ravi & Larochelle (2016)Sung et al. (2018)Qiao et al.
(2018). Recently, solutions for few-shot object segmentation which learns
pixel-wise classification based on few labelled samples have emerged Shaban et
al. (2017)Rakelly et al. (2018)Dong & Xing (2018)Wang et al. (2019)Zhang et
al. (2019a)Zhang et al. (2019b)Siam et al. (2019). Previous literature in few-
shot object segmentation relied on manually labelled segmentation masks. A few
recent works Rakelly et al. (2018)Zhang et al. (2019b)Wang et al. (2019)
started to conduct experiments using weak annotations in terms of scribbles,
bounding boxes. However, limited research was conducted on using image-level
supervision for few-shot object segmentation with one sole recent work Raza et
al. (2019).

In order to improve image-level few-shot object segmentation we propose a
multi-modal interaction module that leverages the interaction between support
and query visual features and word embeddings. The use of neural word
embeddings pretrained on GoogleNews Mikolov et al. (2013) and visual
embeddings from pretrained networks on ImageNet Deng et al. (2009) allows to
build such a model. We propose a novel approach that uses Stacked Co-Attention
to leverage the interaction among visual and word embeddings. We mainly
inspire from Lu et al. (2019) which proposed a co-attention siamese network
for unsupervised video object segmentation. However, our setup mainly focuses
on the few-shot object segmentation aspect to assess its ability to generalize
to novel classes. This aspect motivates why we meta-learned a multi-modal
interaction module that leverages the interaction between support and query
with image-level supervision.

Concurrent to few-shot object segmentation, video object segmentation(VOS) has
been extensively researched Pont-Tuset et al. (2017)Perazzi et al. (2016)Lu et
al. (2019)Tokmakov et al. (2016)Jain et al. (2017)Luiten et al. (2018). Two
main categories for video object segmentation are studied which are semi-
supervised VOS and unsupervised VOS. The semi-supervised VOS also named as
few-shot requires an initial strongly labelled segmentation mask to be
provided. In a recent work Khoreva et al. (2018) video object segmentation
using language expression has been studied. However, their work does not focus
on the aspect of segmenting novel categories which we try to address since it
can be of potential benefit to the few-shot learning community. The additional
temporal information in VOS can provide extra unlabelled data about the object
poses or its interactions with other objects to the few-shot object
segmentation method. It has the potential to be closer to human-like learning
rather than learning from a single frame. To the best of our knowledge we are
the first to propose a novel setup for video object segmentation that focuses
on the ability to generalize to novel classes by splitting the categories to
different folds which would better assess the generalization ability. The
setup as well only requires image-level labels for the first frame to segment
the corresponding objects unlike conventional semi-supervised VOS. Although
Youtube-VOS Xu et al. (2018) provided a way to evaluate on certain unseen
categories but it does not utilize any of the category label information in
the segmentation model. However, in order to ensure the evaluation for the
few-shot method is not biased to a certain category, it is best to split into
multiple folds and evaluate on different ones. To this end we use Youtube-VOS
dataset Xu et al. (2018) which is a large-scale VOS dataset with 65 different
object categories, the originally provided training data is further split into
5 different folds with 13 novel classes per fold. The novel setup opens the
door towards studying how few-shot object segmentation can benefit from
temporal information for the different object poses or its interactions. The
main contributions in this paper are summarized as follows:

  * •

A novel formulation for learning image-level labelled few-shot segmentation
based on a multi-modal interaction module is presented. It utilizes neural
word embeddings and attention mechanisms that relates the support and query
images. It enables attention to the most relevant spatial locations in the
query feature maps.

  * •

We propose a novel setup called few-shot weakly supervised video object
segmentation that can be of potential benefit to few-shot object segmentation.

##  2 Method

###  2.1 Stacked Co-Attention

A simple conditioning on the support set image would be insufficient where
neither sparse nor dense annotations is provided. In order to leverage the
interaction between the support set images and query set image a conditioned
support-to-query attention module is used to learn the correlation between
them. Initially a base network is used to extract features from support set
image $I_{s}$ and query image $I_{q}$ which we denote as $V_{s}\in R^{W\times
H\times C}$ and $V_{q}\in R^{W\times H\times C}$. Where $H$ and $W$ denote the
feature maps height and width respectively, while $C$ denote the feature
channels.

It is important to initially capture the class label semantic representation
through using semantic word embeddings Mikolov et al. (2013) before performing
the co-attention. The main reason is that both the support set and query set
can contain multiple common objects from different classes, so depending
solely on the support-to-query attention will fail in this case. A projection
layer is used on the semantic word embeddings to construct $z\in R^{d}$ where
$d$ is 256. It is then spatially tiled and concatenated with the visual
features resulting in $\tilde{V_{s}}$ and $\tilde{V_{q}}$. An affinity matrix
$S$ is then computed to capture the similarity between them using equation 1.

$S=\tilde{V_{s}}^{T}W_{co}\tilde{V_{q}}$ (1)

The feature maps are flattened into matrix representations where
$\tilde{V_{q}}\in R^{C\times WH}$ and $\tilde{V_{s}}\in R^{C\times WH}$, while
$W_{co}\in R^{C\times C}$ learns the correlation between feature channels. We
use a vanilla co-attention similar to Lu et al. (2019) where $W_{co}$ is
learned using a fully connected layer. The resulting affinity matrix $S\in
R^{WH\times WH}$ relates each column from $\tilde{V_{q}}$ and $\tilde{V_{s}}$.
A softmax operation is performed on the $S$ row-wise and column-wise depending
on what relation direction we are interested in following equations 2a and 2b.

$S^{c}=\textup{softmax}(S)$ (2a) $S^{r}=\textup{softmax}(S^{T})$ (2b)

$S^{c}_{*,j}$ has the relevance of the $j^{th}$ spatial location in $V_{q}$
with all spatial locations of $V_{s}$, where $j=1,...,WH$. The normalized
affinity matrix can be used to compute $U_{q}$ using equation 3 and $U_{s}$
similarly. $U_{q}$ and $U_{s}$ act as the attention summaries.

$U_{q}=\tilde{V_{s}}S^{c}$ (3)

The attention summaries are further gated using a gating function $f_{g}$ with
learnable weights $W_{g}$ and bias $b_{g}$ following equations 4a and 4b. The
gating function ensures the output to be in the interval [0, 1] in order to
mask the attention summaries using a sigmoid activation function $\sigma$. The
$\circ$ operator denotes the hadamard product or element-wise multiplication.

$f_{g}(U_{q})=\sigma{(W_{g}U_{q}+b_{g})}\in[0,1]$ (4a)
$U_{q}=f_{g}(U_{q})\circ U_{q}$ (4b)

The gated attention summaries $U_{q}$ are concatenated with the original
visual features $V_{q}$ and reshaped back to construct the final output from
the attention module. Figure 1 demonstrates our proposed method. We utilize a
ResNet-50 He et al. (2016) encoder pre-trained on ImageNet Deng et al. (2009)
to extract visual features. The segmentation decoder is comprised of an
iterative optimization module (IOM) Zhang et al. (2019b) and an atrous spatial
pyramid pooling (ASPP) Chen et al. (2017a)Chen et al. (2017b). In order to
improve our model we use two stacked co-attention modules. It allows the model
to learn a better representation by letting the support set guide the
attention on the query image and the reverse with respect to the support set
through multiple iterations.

Figure 1: Architecture of Few-Shot Object segmentation model with co-
attention. The $\oplus$ operator denotes concatenation, $\otimes$ denotes
element-wise multiplication, $\circ$ denotes matrix multiplication.

###  2.2 Few-Shot Weakly Supervised Video Object Segmentation

We propose a novel setup for few-shot video object segmentation where we
utilize image-level labelled first frame to learn object segmentation in the
sequence rather than using manual segmentation masks. More importantly the
setup is devised in a way to split the categories to multiple folds to assess
the generalization ability to novel classes. We utilize Youtube-VOS dataset
training data which has 65 categories, and we split into 5 folds. Each fold
has 13 classes that are used as novel classes, while the rest are used in the
meta-training phase. A randomly sampled category $Y_{s}$ and sequence
$V=\\{I_{1},I_{2},...,I_{N}\\}$ is used to construct support set
$S_{p}=\\{(I_{1},Y_{s})\\}$ and query images $I_{i}$.

##  3 Experimental Results

In this section we demonstrate results from experiments conducted on the
PASCAL-$5^{i}$ dataset Shaban et al. (2017) which is the most widely used
dataset for evaluating few-shot segmentation. We also conduct experiments on
our novel few-shot weakly supervised video object segmentation.

###  3.1 Experimental Setup

We report two evaluation metrics, the mean-IoU computes the intersection over
union for all 5 classes within the fold and averages them neglecting the
background Shaban et al. (2017). Whereas the binary-IoU metric proposed in
Rakelly et al. (2018)Dong & Xing (2018) computes the mean of foreground and
background IoU in a class agnostic manner. Both metrics are reported as an
average of 5 different runs to ensure a stable result following Wang et al.
(2019). We have also noticed some deviation in the validation schemes used in
previous works. Zhang et al. (2019b) follows a procedure where the validation
is performed on the $L_{test}$ classes to save the best model whereas Wang et
al. (2019) does not perform validation and rather trains for a fixed number of
iterations. We choose the approach followed in Wang et al. (2019) since we
consider that the $L_{test}$ classes are not available during the initial
meta-training phase.

###  3.2 Few-shot Weakly Supervised Object Segmentation

We compare the result of our proposed method with stacked co-attention against
the other state of the art methods for 1-way 1-shot segmentation on
pascal-$5^{i}$ in Table 1 using mean-IoU and binary-IoU metrics. We report the
results for both the two validation schemes where V1 is following Zhang et al.
(2019b) and V2 is following Wang et al. (2019) validation scheme. Without the
utilization of segmentation mask or even sparse annotations, our method with
the least supervision of image level labels performs relatively on-par 53.5%
compared to the current state of the art methods 56% showing the efficacy of
our proposed algorithm. It outperforms the previous one-shot weakly supervised
segmentation Raza et al. (2019) with 5.1%. Our proposed model outperforms the
baseline method which utilizes a co-attention module without using word
embeddings. Figure 3 shows the qualitative results for our proposed approach.

Table 1: Quantitative results for 1-way, 1-shot segmentation on the PASCAL-$5^{i}$ dataset showing mean-Iou Shaban et al. (2017) and binary-IoU Rakelly et al. (2018)Dong & Xing (2018). W: stands for using weak supervision from Image-Level labels. Method | W | fold 0 | fold 1 | fold 2 | fold 3 | mean-IoU | binary-IoU
---|---|---|---|---|---|---|---
FG-BG | ✗ | - | - | - | - | - | 55.1
Shaban et al. (2017) | ✗ | 33.6 | 55.3 | 40.9 | 33.5 | 40.8 | -
Rakelly et al. (2018) | ✗ | 36.7 | 50.6 | 44.9 | 32.4 | 41.1 | 60.1
Dong & Xing (2018) | ✗ | - | - | - | - | - | 61.2
Siam et al. (2019) | ✗ | 41.9 | 50.2 | 46.7 | 34.7 | 43.4 | 62.2
Wang et al. (2019) | ✗ | 42.3 | 58.0 | 51.1 | 41.2 | 48.1 | 66.5
Zhang et al. (2019b) | ✗ | 52.5 | 65.9 | 51.3 | 51.9 | 55.4 | 66.2
Zhang et al. (2019a) | ✗ | 56.0 | 66.9 | 50.6 | 50.4 | 56.0 | 69.9
Raza et al. (2019) | ✓ | - | - | - | - | - | 58.7
Baseline | ✓ | 38.6 | 56.6 | 43.8 | 38.2 | 44.3 | 60.2
Ours-V1 | ✓ | 49.4 | 65.5 | 50.0 | 49.2 | 53.5 | 65.6
Ours-V2 | ✓ | 42.1 | 65.1 | 47.9 | 43.8 | 49.7 | 63.8

(a) ’bicycle’

(b) ’bottle’

(c) ’bird’

Figure 3: Qualitative evaluation on PASCAL-$5^{i}$ 1-way 1-shot. The support
set and prediction on the query image are shown in pairs.

###  3.3 Few-shot Weakly Supervised Video Object Segmentation

Table 2 shows results on our proposed novel setup and comparing our method
with the baseline of using co-attention module without utilizing word
embeddings similar to Lu et al. (2019). It shows the potential benefit from
utilizing neural word embeddings to guide the attention module.

Table 2: Quantitative Results on Youtube-VOS One-shot weakly supervised setup. Method | fold 0 | fold 1 | fold 2 | fold 3 | fold 4 | Mean
---|---|---|---|---|---|---
Baseline | 40.1 | 33.7 | 47.1 | 36.4 | 36.6 | 38.8
Ours | 41.6 | 40.8 | 51.4 | 41.5 | 39.1 | 42.9

##  4 Conclusions

Our proposed method demonstrates great promise toward performing few-shot
object segmentation based on gated co-attention that leverages the interaction
between the support set and query image. Our model utilizes neural word
embeddings to guide the attention mechanism which improves the segmentation
accuracy compared to the baseline. We demonstrate promising results on
Pascal-$5^{i}$ where we outperform the previously proposed image-level
labelled one-shot segmentation method by 5.1% and perform closer to methods
that use strongly labelled masks. Our novel setup provides a mean to
experiment with the effect of capturing different object viewpoints, and
appearance changes in few-shot object segmentation. It closely mimics human
learning for novel objects from few labelled samples by aggregating
information from different viewpoints and capturing different object
interactions.

## References

  * Chen et al. (2017a) Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille.  Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs.  _IEEE transactions on pattern analysis and machine intelligence_ , 40(4):834–848, 2017a.
  * Chen et al. (2017b) Liang-Chieh Chen, George Papandreou, Florian Schroff, and Hartwig Adam.  Rethinking atrous convolution for semantic image segmentation.  _arXiv preprint arXiv:1706.05587_ , 2017b.
  * Deng et al. (2009) Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In _2009 IEEE conference on computer vision and pattern recognition_ , pp. 248–255. Ieee, 2009.
  * Dong & Xing (2018) Nanqing Dong and Eric P. Xing.  Few-shot semantic segmentation with prototype learning.  In _BMVC_ , volume 3, pp. 4, 2018.
  * Finn et al. (2017) Chelsea Finn, Pieter Abbeel, and Sergey Levine.  Model-agnostic meta-learning for fast adaptation of deep networks.  In _Proceedings of the 34th International Conference on Machine Learning-Volume 70_ , pp. 1126–1135. JMLR. org, 2017.
  * He et al. (2016) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pp. 770–778, 2016.
  * Jain et al. (2017) Suyog Dutt Jain, Bo Xiong, and Kristen Grauman.  Fusionseg: Learning to combine motion and appearance for fully automatic segmention of generic objects in videos.  _arXiv preprint arXiv:1701.05384_ , 2017.
  * Khoreva et al. (2018) Anna Khoreva, Anna Rohrbach, and Bernt Schiele.  Video object segmentation with language referring expressions.  In _Asian Conference on Computer Vision_ , pp. 123–141. Springer, 2018.
  * Koch et al. (2015) Gregory Koch, Richard Zemel, and Ruslan Salakhutdinov.  Siamese neural networks for one-shot image recognition.  In _ICML Deep Learning Workshop_ , volume 2, 2015.
  * Lu et al. (2019) Xiankai Lu, Wenguan Wang, Chao Ma, Jianbing Shen, Ling Shao, and Fatih Porikli.  See more, know more: Unsupervised video object segmentation with co-attention siamese networks.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 3623–3632, 2019.
  * Luiten et al. (2018) Jonathon Luiten, Paul Voigtlaender, and Bastian Leibe.  Premvos: Proposal-generation, refinement and merging for video object segmentation.  In _Asian Conference on Computer Vision_ , pp. 565–580. Springer, 2018.
  * Mikolov et al. (2013) Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean.  Distributed representations of words and phrases and their compositionality.  In _Advances in neural information processing systems_ , pp. 3111–3119, 2013.
  * Perazzi et al. (2016) F. Perazzi, J. Pont-Tuset, B. McWilliams, L. Van Gool, M. Gross, and A. Sorkine-Hornung.  A benchmark dataset and evaluation methodology for video object segmentation.  In _Computer Vision and Pattern Recognition_ , 2016.
  * Pont-Tuset et al. (2017) Jordi Pont-Tuset, Federico Perazzi, Sergi Caelles, Pablo Arbeláez, Alexander Sorkine-Hornung, and Luc Van Gool.  The 2017 davis challenge on video object segmentation.  _arXiv:1704.00675_ , 2017.
  * Qi et al. (2017) Hang Qi, Matthew Brown, and David G Lowe.  Learning with imprinted weights.  _arXiv preprint arXiv:1712.07136_ , 2017.
  * Qiao et al. (2018) Siyuan Qiao, Chenxi Liu, Wei Shen, and Alan L Yuille.  Few-shot image recognition by predicting parameters from activations.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 7229–7238, 2018.
  * Rakelly et al. (2018) Kate Rakelly, Evan Shelhamer, Trevor Darrell, Alyosha Efros, and Sergey Levine.  Conditional networks for few-shot semantic segmentation.  2018\.
  * Ravi & Larochelle (2016) Sachin Ravi and Hugo Larochelle.  Optimization as a model for few-shot learning.  2016\.
  * Raza et al. (2019) Hasnain Raza, Mahdyar Ravanbakhsh, Tassilo Klein, and Moin Nabi.  Weakly supervised one shot segmentation.  In _Proceedings of the IEEE International Conference on Computer Vision Workshops_ , pp. 0–0, 2019.
  * Shaban et al. (2017) Amirreza Shaban, Shray Bansal, Zhen Liu, Irfan Essa, and Byron Boots.  One-shot learning for semantic segmentation.  _arXiv preprint arXiv:1709.03410_ , 2017.
  * Siam et al. (2019) Mennatullah Siam, Boris N Oreshkin, and Martin Jagersand.  Amp: Adaptive masked proxies for few-shot segmentation.  In _Proceedings of the IEEE International Conference on Computer Vision_ , pp. 5249–5258, 2019.
  * Snell et al. (2017) Jake Snell, Kevin Swersky, and Richard Zemel.  Prototypical networks for few-shot learning.  In _Advances in Neural Information Processing Systems_ , pp. 4077–4087, 2017.
  * Sung et al. (2018) Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip HS Torr, and Timothy M Hospedales.  Learning to compare: Relation network for few-shot learning.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 1199–1208, 2018.
  * Tokmakov et al. (2016) Pavel Tokmakov, Karteek Alahari, and Cordelia Schmid.  Learning motion patterns in videos.  _arXiv preprint arXiv:1612.07217_ , 2016.
  * Vinyals et al. (2016) Oriol Vinyals, Charles Blundell, Tim Lillicrap, and Daan Wierstra.  Matching networks for one shot learning.  In _Advances in Neural Information Processing Systems_ , pp. 3630–3638, 2016.
  * Wang et al. (2019) Kaixin Wang, Jun Hao Liew, Yingtian Zou, Daquan Zhou, and Jiashi Feng.  Panet: Few-shot image semantic segmentation with prototype alignment.  In _Proceedings of the IEEE International Conference on Computer Vision_ , pp. 9197–9206, 2019.
  * Xu et al. (2018) Ning Xu, Linjie Yang, Yuchen Fan, Dingcheng Yue, Yuchen Liang, Jianchao Yang, and Thomas Huang.  Youtube-vos: A large-scale video object segmentation benchmark.  _arXiv preprint arXiv:1809.03327_ , 2018.
  * Zhang et al. (2019a) Chi Zhang, Guosheng Lin, Fayao Liu, Jiushuang Guo, Qingyao Wu, and Rui Yao.  Pyramid graph networks with connection attentions for region-based one-shot semantic segmentation.  In _Proceedings of the IEEE International Conference on Computer Vision_ , pp. 9587–9595, 2019a.
  * Zhang et al. (2019b) Chi Zhang, Guosheng Lin, Fayao Liu, Rui Yao, and Chunhua Shen.  Canet: Class-agnostic segmentation networks with iterative refinement and attentive few-shot learning.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 5217–5226, 2019b.


# Adhesive Transfer operates during Galling

Samuel R. Rogers<EMAIL_ADDRESS>Jaimie Daure  Philip Shipway  David Stewart
David Dye  Imperial College, South Kensington, London SW7 2AZ, UK  University
of Nottingham, University Park, Nottingham NG7 2RD, UK  Rolls-Royce plc,
Raynesway, Derby DE21 7WA, UK

###### Abstract

In order to reduce cobalt within the primary circuit of pressurised water
reactors (PWR’s), wear-resistant steels are being researched and developed. In
particular interest is the understanding of galling mechanisms, an adhesive
wear mechanism which is particularly prevalent in PWR valves. Here we show
that large shear stresses and adhesive transfer occur during galling by
exploiting the 2 wt.% manganese difference between 304L and 316L stainless
steels, even at relatively low compressive stresses of
$50\text{\,}\mathrm{MPa}$. Through these findings, the galling mechanisms of
stainless steels can be better understood, which may help with the development
of galling resistant stainless steels.

There has been renewed interest in nuclear power generation in recent years,
in an effort to reduce carbon emissions and reliance on fossil fuels. With new
regulations and a desire to reduce cobalt in pressurised water reactor (PWR)
primary circuits [1] alternative materials are required. Of particular
interest is the replacement of Stellite 6 (a cobalt hardfacing alloy) in valve
seatings with a galling-resistant stainless steel alloy.

Galling is an adhesive wear mechanism which is known to result in severe
surface degradation, and may result in sliding surfaces seizing [2, 3, 4].
Whilst self-mated stainless steel is well documented to show poor galling
resistance, particularly at elevated temperature, considerable efforts have
been undertaken to develop a galling resistant stainless steel or iron-based
hardfacing alloy [2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

Work has been produced which elucidates the galling mechanisms which may
occur, many of these have been on self-mated test pairs, making adhesive
transfer difficult to discern. Whilst a number of non self-mated tests have
also been performed, the two materials are often found to differ quite
considerably, be it through differing hardness, yield strengths, phases,
microstructures or widely differing chemistries [5, 18, 9, 19]. This work
seeks to address these issues, by performing non self-mated tests using two
very mechanically similar stainless steels but which differ sufficiently in
chemistry (namely molybdenum content) in order to observe any adhesive
transfer which may have occurred.

Figure 1: (a) ASTM G196 galling rig, redrawn from [4]; (b) ASTM G196 galling
sample, with a section removed to enable a view of the radial cross-section;
(c) top view of an ASTM G196 galling sample.

Galling tests were performed using the ASTM G196 method, Figure 1, under
atmospheric conditions at the University of Nottingham. An applied compressive
load of $50\text{\,}\mathrm{MPa}$ was used. $50\text{\,}\mathrm{MPa}$ was
chosen since this would ensure that galling took place between the surfaces.
Tests used non self-mated pairs of 314L/316L stainless steel. Virgin surfaces
were used for each test. Surfaces had been ground to a finish of
$\pm$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

Upon the completion of testing, the samples’ surfaces were scanned using white
light interferometry (WLI) before being processed using specially written
Python code to process the surfaces (ittilt correct, remove edge effects and
interpolate). Once processed, surfaces were quantified, using the maximum &
minimum surface heights, Rt (max. + min.), galled area and volume change.
These quantities are described in [2, 20].

Samples were sectioned and prepared for metallographic examination using SEM
in both secondary electron (SE) and backscattered electron (BSE) imaging
modes. Energy dispersive X-ray spectroscopy (EDX) was also used to form
elemental maps of cross-sections.

Sample | Sample Height / $\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | Galled Area / % | $\Delta$V / $\text{\,}{\mathrm{mm}}^{3}$
---|---|---|---
Max | Min | Rt
(a) | 257 | -233 | 490 | 12.04 | -0.2440
(b) | 162 | -182 | 344 | 18.61 | -0.0916
(c) | 262 | -126 | 388 | 19.75 | +0.3068
(d) | 171 | -219 | 390 | 18.87 | -0.3005
(e) | 198 | -237 | 435 | 21.69 | -0.2533
(f) | 270 | -175 | 445 | 16.82 | +0.0121

Figure 2: White light interferometry height maps of 316L (a), (c) & (e) vs
304L (b), (d) & (f) stainless steel samples tested at
$50\text{\,}\mathrm{MPa}$ in the non-oxidised conditions, with their
corresponding height scales and quantification measures. Tests were paired in
the following way: (a) & (b), (c) & (d) and (e) & (f). The red dashed line on
sample (c) denotes the cross-section which was used to image Figures 3 and 4.

When the galling damage of non self-mated tests, Figure 2, are compared with
that of self-mated 316L galling tests at the same applied stress, it can be
seen that the extent of the galling damage is very similar [2], confirming the
validity of using 304L as a mating pair with 316L stainless steel.

Since the damage morphology is consistent with that seen in self-mated 316L
galling tests [2], it can be concluded that the wedge formation and flow
mechanism has also occurred for 304L/316L non self-mated tests, Figure 2. A
full description for the wedge formation and flow mechanism can be found in
[2]. In each non self-mated test, the damage morphology is consistent with a
single surface providing most of the material for the galling peak. In the
case of Figure 2 surface (a) and (b), two instances of this have occurred,
with each surface being responsible for the growth of a galling peak. In
addition to material transfer, material has also been lost from the system in
this test, as given by the negative $\Delta$V for both surfaces. The test
between surface (c) and (d) of Figure 2 shows a single instance of galling to
have occurred, with surface (d) primarily giving material for adhesion
junction growth. It can be seen that the all of the material displaced from
surface (d) was adhesively transferred to surface (c), as shown by their
almost identical changes in volume, Figure 2. The final test, between surfaces
(e) and (f) show a primary galling scar, however, there are a number of
smaller scars which are also seen. Although adhesive transfer is seen to occur
from surface (e) to (f), material has been lost from the system.

Figure 3: A circumferential cross-section of a galling peak of 316L stainless
steel, Figure 2(c), from a non self-mated test of 304 vs 316L stainless steel
sample tested at a normal load of $50\text{\,}\mathrm{MPa}$. (a) The whole
peak as viewed using SE imaging; (b) Mo EDX map of the peak front; (c) peak
rear as viewed using SE imaging; (d) Mo EDX map of the peak rear.

In order to verify if adhesive transfer occurred between the non self-mated
samples, they needed to be sectioned in order to view the samples sub-surface.
Figure 3 shows a portion of a circumferential cross-section of the surface
shown in Figure 2(c), which is a 316L stainless steel sample, and shows the
main galling peak on the sample. As was seen for self-mated 316L mated pairs,
the underlying microstructure is clearly distinguished from the peak and the
tribologically affected zone (TAZ) [2].

It is interesting to note that the the galling peak is very clearly adhered to
the underlying material at the peak rear, Figure 3(a) & (c), whilst being
seemingly unattached at the peak front. This suggests that the adhesion
junction formed at the peak rear, and although adhesion may have also occurred
at the peak front, the adhesion bond in this region was not as strong as at
the original adhesion junction.

By performing an EDX scan over the front portion of the galling peak, it can
very clearly be seen that the predominant material of the peak is 304L
stainless steel, as evidenced by its low Mo concentration, Figure 3(b),
meaning that adhesive transfer of a significant volume has occurred. In
addition, a portion of 316L appears to be mechanically mixed within the 304L
peak, Figure 3(b). Whilst from this section alone, it is difficult to know how
this material got there, by observing the mechanical mixing on the right of
Figure 3(b), it can be seen that a small amount of the 316L substrate is being
mixed into the 304L peak. It may therefore be that the larger portion of 316L
within the peak has been mechanically mixed into the 304L. It is difficult to
give certainty on this, however, since it could be that the 316L first
transferred over to the 304L mating surface, before being mechanically mixed
and transferred back to the the 316L mating surface.

When observing the rear of the galling peak, Figure 3(c) & (d), further
portions of mechanically mixed 316L within the 304L peak can be seen. These
are finer in width and longer, suggesting that they have undergone a greater
extent of shear, and potentially mechanical mixing, than those towards the
front of the galling peak, Figure 3(b). Flow lines are much more visible
within both the galling peak and the substrate material at the peak rear.

Figure 4: A refinement of the wedge formation and growth mechanism, as
described in [2], with the addition of material flow, as described using
colour co-ordinated arrows.

With this information, we can refine the mechanism put forward in [2] in order
to include mechanical mixing within and between tribosurfaces, Figure 4. Two
asperities come into contact and form an adhesive junction, Figure 4(a),
subsequent shearing of this junction results in wedge formation (this can also
occur through the shearing of two flat surfaces that have adhered), Figure
4(b). At this point, the mechanism can continue in one of two ways: Figure
4(c) material from both tribosurfaces causes the wedge grows to such an extent
that excess material ahead of the prows folds over, whilst shear failure
occurs behind the prow, resulting in the formation of lips, or; Figure 4(d)
material from one tribosurface preferentially supplies material for wedge
growth, resulting in a very large wedge on one tribosurface and a considerably
smaller wedge on the opposing surface. The mechanism as shown in Figure 4(d)
was found to primarily occur in this work.

Figure 5: (a) A backscattered electron image and EDX maps of (b) Mo and (c) Ni
of a small galling peak on the same circumferential section as viewed in in
Figure 3. The scale bar is common across (a) - (c). (d) A sketch of the
galling peak shown in (a), showing the layers of 316L and 304L stainless steel
that have adhesively transferred through successive galling events to form the
final galling peak which is observed. Red portions of the sketch are 304L
stainless steel, with uncoloured regions being 316L stainless steel.

Across the circumferential cross-section viewed in Figure 3, an additional
smaller galling peak can be also be seen, Figure 5. If this small peak is
mapped using EDX, it can be seen that unlike in Figures 3, the entirety of the
peak appears to be made up of a number of mechanically mixed regions of 304L
and 316L stainless steels, Figure 5. Given that the regions of 304L seem to be
isolated within the 316L material within the peak, that all regions of 304L
appear relatively thin and long, and that the volume of 316L material within
the peak does not appear to have all come from this galling trough, it appears
an alternative explanation for the galling peak morphology is required. The
most likely possibility is that multiple instances of galling and adhesive
transfer have occurred, resulting in a galling peak with a layered internal
structure, Figure 5(d), and when considering the volume of 316L, a peak which
is considerably larger than the associated trough size would suggest.

In summary, as with self-mated 316L stainless steel, non self-mated 304L vs
316L galling pairs were found to gall via the wedge formation and growth
mechanism. Due to chemical variations between the two alloys (namely the
incorporation of molybdenum in 304L), adhesive transfer was observed between
surfaces. Mechanical mixing between surfaces was also observed, in addition to
the previously observed mechanical mixing in a single surface. A series of
smaller galling peaks were observed to have formed, likely through successive
galling and adhesive transfer events.

## Acknowledgements

We gratefully acknowledge support from Rolls-Royce plc, from EPSRC
(EP/N509486/1) and from the Royal Society (D. Dye Industry Fellowship). The
authors gratefully acknowledge funding from Rolls-Royce plc to the University
of Nottingham in support of this work.

## References

## References

  * [1] M. Vannerem.  Chemistry of operating civil nuclear reactors.  ONR Guide NS-TAST-GD-088 Revision 2, Office for Nuclear Regulation, January 2019.
  * [2] S.R. Rogers, D. Bowden, R. Unnikrishnan, F. Scenini, M. Preuss, D. Stewart, D. Dini, and D. Dye.  The interaction of galling and oxidation in 316L stainless steel.  Wear, 450-451:203234, 2020.
  * [3] ASTM International.  ASTM G98-17. Standard Test Method for Galling Resistance of Materials. ASTM International; 2017.
  * [4] ASTM International.  ASTM G196-08(2016) Standard Test Method for Galling Resistance of Material Couples. ASTM International; 2016.  2016\.
  * [5] J.R. Davis, editor.  Stainless Steels.  ASM Specialty Handbook. ASM International, December 1994.
  * [6] R. Tyagi A.P. Harsha, P.K. Limaye and A. Gupta.  Development of tribological test equipment and measurement of galling resistance of various grades of stainless steel.  Journal of Tribology, 138(2):024501–6, September 2015.
  * [7] W.J. Schumacher.  Wear compatibility of unlubricated stainles steel and other metal couples.  In W.A. Glaeser, K.C. Ludema, and S.K. Rhee, editors, Wear of Materials 1977, pages 134–139. American Society of Mechanical Engineers, 1977\.
  * [8] Australian Stainless Steel Development Association.  Galling and its Control. www.assda.asn.au/images/pdfs/faq%20no5%20galling.pdf [Accessed 13th January 2020].
  * [9] R.G. Wensel.  Galling of Unlubricated Close-Fitting Metal Parts. Atomic Energy of Canada Limited. Report number: AECL-6547, 1979.
  * [10] A. Oudriss Y. Chen M. Risbet P.-E. Mazeran T. Lesage, S. Bouvier.  Galling categories investigations in stainless steels.  Wear, 460-461, November 2020.
  * [11] H. Ocken.  The galling wear resistance of new iron-base hardfacing alloys: a comparison with established cobalt- and nickel-base alloys.  Surface and Coatings Technology, 76-77:456–461, 1995.
  * [12] J.-K. Kim and S.-J. Kim.  The temperature dependence of the wear resistance of iron-base NOREM02 hardfacing alloy.  Wear, 237(2):217–222, February 2000.
  * [13] J. H. Magee.  Two Galling Resistant Stainless Steels used for Bridge Hinge Pins. Carpenter Technology Corporation. 2018.
  * [14] R.T. Smith.  Development of a Nitrogen-Modified Stainless-Steel Hardfacing Alloy.  PhD thesis, The Ohio State University, 2015.
  * [15] M.C. Doran.  Nanoindentation as a characterization tool for wear resistance in stainless steels.  Master’s thesis, Ohio State University, 2016.
  * [16] D. Bowden.  Assessment of cobalt-free hardfacing stainless steel alloys for nuclear applications.  PhD thesis, University of Manchester, 2016.
  * [17] J.L. Daure, M.J. Carrington, D.G. McCartney, D.A. Stewart, and P.H. Shipway.  Measurement of friction in galling testing – an example of its use in characterising the galling behaviour of hardfacings at ambient and elevated temperature.  Wear, 476:203736, July 2021.
  * [18] J.V. Reid.  Adhesion of technological alloys.  PhD thesis, University of Waterloo, 1984.
  * [19] Committee of Stainless Steel Producers.  Review of the wear and galling characteristics of stainless steels.  Handbook.
  * [20] E.P. Whitenton L.K. Ives, M.B. Peterson.  The Mechanism, Measurement, and Influence of Properties on the Galling of Metals. National Institute of Standards and Technology. Report number: NISTIR 89-4064, 1989.


# Elastic scattering of electrons by water: an ab initio study

Francesca Triggiani 1,2, Tommaso Morresi 3,4, Simone Taioli 3,4,5∗ and Stefano
Simonucci 1,2∗

###### Abstract

In this work we devise a theoretical and computational method to compute the
elastic scattering of electrons from a non-spherical potential, such as in the
case of molecules and molecular aggregates. Its main feature is represented by
the ability of calculating accurate wave functions for continuum states of
polycentric systems via the solution of the Lippmann-Schwinger equation,
including both the correlation effects and multi-scattering interference
terms, typically neglected in widely used approaches, such as the Mott theory.
Within this framework, we calculate the purely elastic scattering matrix
elements. As a test case, we apply our scheme to the modelling of electron-
water elastic scattering. The Dirac-Hartree-Fock self-consistent field method
is used to determine the non-spherical molecular potential projected on a
functional space spanned by Gaussian basis set. By adding a number of multi-
centric radially-arranged $s$-type Gaussian functions, whose exponents are
system-dependent and optimized to reproduce the properties of the continuum
electron wave function in different energy regions, we are able to achieve
unprecedented access to the description of the low energy range of the
spectrum ($0.001<E<10$ eV) up to keV, finding a good agreement with
experimental data and previous theoretical results. To show the potential of
our approach, we also compute the total elastic scattering cross section of
electrons impinging on clusters of water molecules and zundel cation. Our
method can be extended to deal with inelastic scattering events and heavy-
charged particles.

##  1 Introduction

A complete understanding of the collision processes of charged particles with
atoms, molecules, and solids is of paramount importance in several areas of
physics, chemistry, and medicine. On the one hand, electron and ion beams are
used for manufacturing devices, such as in lithography techniques to fabricate
integrated circuits or other nanostructures (Dapor, 2020). On the other hand,
scattering experiments involving charged projectiles may represent a tool to
“extend our vision”. Indeed, by tuning the kinetic energy of the impinging
particles, we can characterise the spectrum of materials electronic and
optical properties and probe different time (or frequency) and length (or
momentum) scales, such as in energy loss (Azzolini et al., 2017, 2018b, 2020;
Pedrielli et al., 2021, 2022), secondary electron (Azzolini et al., 2018a;
Taioli et al., 2022a), and Auger (Taioli et al., 2009, 2010a; Colle et al.,
2004a, b; Taioli and Simonucci, 2015, 2021b; Taioli et al., 2015)
spectroscopies. Moreover, the modelling of charge transport and flow inside
materials (Dapor, 2020) and different media, such as stellar and interstellar
plasma (Maciel, 2013) or biological matter (Nikjoo et al., 2016), is
fundamental for the interpretation of phenomena ranging from the
nucleosynthesis of the chemical elements in evolved stars (Simonucci et al.,
2013; Taioli et al., 2022b; Palmerini et al., 2021; Mascali et al., 2022;
Vescovi et al., 2019; Morresi et al., 2018) to hadrontherapy treatments for
cancer cure (de Vera et al., 2022b; Taioli et al., 2020; Pedrielli et al.,
2021).
In all these processes the charged particles undergo scattering events that
can be classified as either elastic or inelastic. In inelastic scattering
energy is exchanged between the projectile and the target constituents
resulting in energy loss of the incident particle. The loss pattern, which can
be caused by several mechanisms, can be used to analyse the target properties.
For example, in materials science electron beams are employed as a probe of
the electronic and optical properties, such as in analytical techniques like
scanning (SEM) and transmission (TEM) electron microscopy. The energy
transferred to the specimen produces an energy loss of the primary beam, which
leads to a range of useful signals that can be used to characterize the
material. At variance, elastic scattering occurs with no energy loss
experienced by the incident primary electron. Elastically scattered electrons
change their direction of motion without changing their wavelength. For
example, coherent elastic scattering is a common mechanism exploited in
electron diffraction to analyze the structure of bulk crystals and surfaces,
such as in the low energy electron diffraction (LEED) technique.
Despite the recognized maturity of the experimental research field in electron
scattering and charge transport, the quality (or nature) and the quantity of
the experimental results available, the theoretical assessment of the cross
sections of scattering events involving electrons is still essential for
enabling us to resolve the combined effects on the spectral lineshape due to
the intrinsic features of the intraband and interband transitions, the
vibrational and rotational details, and the characteristics of the incident
beams and of the electron spectrometer apparatus. To deliver such an accurate
theoretical description of the irradiation and of its role to initiate
physical-chemical processes, one must develop a framework capable of dealing
with three main issues. First, a treatment of the electron repulsion that
includes the correlation effects within the system and between the projectile
and the (ionic) system, particularly at low energy where these effects are
more significant; second, the calculation of the continuum orbital at a given
positive energy, as the scattering matrix is rather sensitive to the quality
of such wave function; finally, a computational effort that should be of the
same order of magnitude as that of standard bound state calculations.
In this work we focus on these three aspects by developing a theoretical and
computational method for modelling the elastic collision of electrons from
non-spherical potentials, which we apply as a test case to electron-water
molecules scattering, a well-known and much studied problem. Water is indeed
the main constituent of biological tissues, and electron-water collisions are
crucial events e.g. in the hadrontherapy techniques for cancer cure that aim
to induce unrecoverable bio-damage to tumor cells. Surprisingly, the radiation
damage is mostly due to low-energy electrons, generated upon collisions of
impinging hadrons, such as protons or charged ions, with biological tissue
rich in water (Gorfinkiel et al., 2005; Taioli and Tennyson, 2006b, a; de Vera
et al., 2022b; Taioli et al., 2020). However, the mechanisms underlying the
radiation-based killing of tumor cells in cancer therapies have not yet been
fully understood, especially at low energy where quantum effects are dominant
(Song et al., 2021; Signorell, 2020). In fact, in our analysis we were driven
by the lack of reliable experimental and theoretical data in low-energy (below
10 eV) electron-water elastic scattering, despite the study of electron
scattering in liquid water being a very active field of research (Signorell,
2020; Gianturco et al., 1998; Gianturco and Scialla, 1987; Itikawa and Mason,
2005; Gorfinkiel et al., 2002; Zhang et al., 2009; Faure et al., 2004;
Kadokura et al., 2019; Gorfinkiel et al., 2005; Taioli and Tennyson, 2006b;
Song et al., 2021). Theoretical analysis is in particular hindered by the
intrinsic difficulty to fully account for the physical properties of water
assemblies from first principles, such as the existence of a permanent dipole
moment, the electron-electron exchange-correlation interaction between the
incoming electron and the scattering center, the numerous electronic excited
configurations coupled with roto-vibrational states, and the inter- and intra-
molecular multiple scattering effects.
Indeed, it was immediately understood that the elastic scattering cross
section for a polar molecule diverges at all impinging electron energies
within the Born approximation, where nuclei are fixed (Altshuler, 1957;
Garrett, 1972) (as in the case of a bare Coulomb potential generated by a
point-particle, see e.g. the Rutherford elastic cross section), and that
rotations should be taken into account to achieve a finite value. Moreover, it
was also realised that the electron elastic scattering is dominated by small
scattering angles (Fabrikant, 2016; Fedus and Karwasz, 2017). In fact, also in
currently and widely used approaches based on the Mott’s theory (Salvat et
al., 2005) these effects are included semi-empirically, while the inter-atomic
potential is treated as a mere superposition of atomic contributions
neglecting the overlapping tails of the atomic potentials. More recently,
Tennyson and coworkers used the R-matrix approach to study the electron and
positron collisions with polar molecules, such as water, at low energy
(Gorfinkiel et al., 2002; Zhang et al., 2009; Faure et al., 2004; Kadokura et
al., 2019; Gorfinkiel et al., 2005; Taioli and Tennyson, 2006b; Song et al.,
2021). The R-matrix approach to scattering consists in dividing the system
into an asymptotic region, where the non-interacting wave function is
analytically known, and a scattering region, where the electron-molecule
interaction is accounted for by high-accuracy quantum-mechanical methods, such
as configuration interaction or coupled cluster. The matching of the inner and
outer solutions at the boundary leads to important scattering information,
such as phase shift and transmission probability. Nevertheless, these analyses
were related to a single water molecule for impinging electron kinetic
energies larger than 10 eV, also due to the lack of experimental data,
typically carried out in water vapour, in this energy range.
With this in mind, our primary goal is to develop a method that does not
suffer the limitations of the previous approaches and allows one to reproduce
accurately the experimental results; moreover it is sufficiently general to be
applied not only to the analysis and interpretation of the elastic scattering
cross sections of electrons colliding with (polar) molecular aggregates but
also to be extended to the study of other inelastic scattering processes, such
as excitation and ionization, since its central feature is the ability of
calculating accurate wave functions for continuum states of polycentric
systems.
Our approach to assess the elastic scattering cross section (ESCS) of
electrons (in general applicable to charged particles) is fully ab initio and
consists in the following steps. First, we solve self-consistently the Dirac-
Hartree-Fock (DHF) equations for the single water molecule or the water
clusters, thus taking into account relativistic effects that may include spin-
flip and polarization of the beam spin. We expand the electronic wave
functions and the Coulomb potential for fixed nuclear positions using an
Hermite Gaussian function (HGF) basis set, which is enlarged and optimized
with more or less diffuse functions to reproduce the properties of the
continuum electron orbital in different energy regions. We then build the
scattering cross section by computing the solution of the Lippmann-Schwinger
(LS) equation for the scattering state (Taioli et al., 2010a) with the Coulomb
potential term projected into the functional space spanned by the Gaussian
basis set.
At first, we benchmark our method by comparing the results obtained using
finite differences with one- and two-dimensional model potentials, where exact
solutions for the scattering phases can be computed by accurately integrating
the Schroedinger (or Dirac) equations. Moreover, we discuss elastic scattering
on a single water molecule, and we further extend the investigation to
clusters of water molecules in liquid phase and water aggregates, such as the
zundel cation, to mimic the presence of surrounding molecular environment in
order to rigorously determine the effect of multiple scattering on the elastic
scattering cross section.
This article is organised as follows: in Section 2 we will discuss the basics
of our theoretical and computational method, such as the projected potential
method using HGF basis set in connection with the LS equation, and its
benchmark with a toy model potential in one and two dimensions; to show its
capabilities and accuracy, in Section 3 we apply our framework to the
calculation of the total elastic cross section of electrons colliding with a
single, a cluster, and an aggregate of water molecules. In Section 4 we
present final remarks and conclusions.

##  2 Theoretical and computational method

###  2.1 Differential and total elastic scattering cross sections

The ESCS, which we aim to compute in this work, is defined as follows:

$\sigma_{el}(T)=\int_{\Omega}\frac{d\sigma_{el}(T,\Omega)}{d\Omega}d\Omega,$
(1)

where $T$ is the kinetic energy of the primary beam, and
$d\Omega=2\pi\sin\theta d\theta d\phi$ defines the differential solid angle.
The differential elastic scattering cross section (DESCS) appearing in Eq. 1
can be obtained by following two different routes.
On the one hand one can apply the relativistic Mott theory (Mott, 1929),
whereby the DESCS of electrons impinging on a single molecule can be computed
as (Dapor, 2020; Salvat et al., 2005):

$\frac{d\sigma_{\mathrm{el}}(T,\theta)}{d\Omega}\,={\mathcal{N}}\,\sum_{m,n}\,e^{(i{\bf
q}\cdot{\bf
r}_{mn})}\,[f_{m}(\theta)f^{*}_{n}(\theta)\,+\,g_{m}(\theta)g^{*}_{n}(\theta)]\,,$
(2)

where ${\bf r}_{mn}\,=\,{\bf r}_{m}\,-\,{\bf r}_{n}$ is the distance between
the atoms $m$ and $n$ within the molecule, ${\mathcal{N}}$ is the target
atomic number density, and ${\bf q}$ and $\theta$ are the momentum transfer
and the scattering angle; finally $f_{m,n}(\theta)$ and $g_{m,n}(\theta)$ are
the direct and spin-flip scattering amplitudes of the atomic species $m,n$,
respectively. $f_{m,n}(\theta)$ and $g_{m,n}(\theta)$ can be obtained by
solving the Dirac equation in a central atomic potential for the single atomic
constituent.
To take into account the presence of bonded interactions between different
chemical species in Eq. 2 the sum is carried out using amplitudes weighted by
an exponential displacement rather than probabilities, allowing for
interference and intramolecular scattering. Nevertheless, within this approach
the calculation of the molecular scattering potential and of the continuum
electron wave function is carried out as a mere superposition of independent
atomic centers, thus neglecting the overlapping tails of the long-range part
of the Coulomb interaction and the relevant multiple scattering effects. In
this framework, consequently, the presence of the electrical polarity due to
charge separation, such as the permanent dipole moment of liquid water that
has a dramatic effect on the DESCS, is superimposed and treated semi-
empirically.
The presence of surrounding water molecules in the liquid phase, which are
randomly oriented can be taken into account by averaging the Eq. 2 over all
the possible spatial orientations, which leads to the following relation:

$\frac{d\sigma_{el}}{d\Omega}=\sum_{m,n}\frac{\text{sin}\left(kr_{m,n}\right)}{kr_{m,n}}(f^{*}_{m}(\theta)f_{n}(\theta)+g^{*}_{m}(\theta)g_{n}(\theta)).$
(3)

Again, Eq. 3 does neglect both multiple scattering effects, as atoms are
treated as independent scattering centers, and the overlap between the long-
range parts of the intermolecular potential.
To overcome these issues, we have developed an approach based on the full ab
initio treatment of the electron-electron (intermolecular and intramolecular)
interaction. Within our approach, the DESCS is assessed by:

$\frac{d\sigma_{el}}{d\Omega}=\frac{m^{2}}{4\pi^{2}}|\langle{\phi_{k\hat{n}}}|T^{+}(E)|\phi_{\mathbf{k}}\rangle|^{2}=\frac{m^{2}}{4\pi^{2}}|\langle{\phi_{k\hat{n}}}|V|{\psi^{+}_{\mathbf{k}}}\rangle|^{2},$
(4)

where $m$ is the mass of the impinging particle, $\phi_{k\hat{n}}$ and
$\phi_{\mathbf{k}}$ are the asymptotic electron wave-function long after and
long before the scattering, respectively, characterised by the modulus of the
momentum $k$, which is scattering invariant (elastic scattering), the
scattering angle identified by the unit vector $\hat{n}$, and incident
orientation defined by $\hat{\mathbf{k}}$; $T$ is the on-shell $T$-operator,
$V$ is the self-consistent potential and $\psi^{+}_{\mathbf{k}}(E)$ is the
scattering wave function.
The latter quantity can be obtained by following two different, while
equivalent routes. The first approach is the direct solution of the many-body
Schroedinger (or Dirac in a fully relativistic treatment of the electron
motion) hamiltonian for positive energy eigenvalues $E$:

$H\psi^{+}_{\mathbf{k}}(E)=(H_{0}+V)\psi^{+}_{\mathbf{k}}(E)=E\psi^{+}_{\mathbf{k}}(E),$
(5)

where $H_{0}$ and $V$ are the kinetic energy and the electron-electron Coulomb
repulsion operators, respectively. Furthermore, it has been proved (Taioli et
al., 2010a) that such many-body problem can be reduced to an effective single-
particle problem, in which the scattering states can be obtained from the
resolution of the following LS equation for the continuum orbital that can be
solved exactly using a Green’s function approach with the proper boundary
condition:

$\psi^{+}_{\mathbf{k}}(E)=\phi^{+}_{\mathbf{k}}(E)+G_{0}^{+}(E)V\psi^{+}_{\mathbf{k}},$
(6)

where

$G_{0}^{+}(E)=\lim_{\epsilon\rightarrow 0}(E-H_{0}+i\epsilon)^{-1}$ (7)

is the Green’s function of the free particle at the same energy $E$ of the
interacting hamiltonian in Eq. 5, and $\phi^{+}_{\mathbf{k}}(E)$ is the
eigenvector of the free hamiltonian $H_{0}$. This expression holds in both
non-relativistic and relativistic theories, notwithstanding wave functions are
4-components spinors in the latter framework (Reiher and Wolf, 2014). In this
work we adopt the Green’s function method of Eq. 6 by projecting both the
Coulomb potential and the continuum orbital using HGFs (for details on the use
of HGFs with the LS equation see the following section 2.3).
The DESCS in Eq. 4 is computed by solving either the Schroedinger or the Dirac
equation within a numerical scheme where the electron-electron Coulomb
repulsion is approximated. In our approach, we use a mean-field approximation
based on the Hartree-Fock (or Dirac-Hartree-Fock (DHF) in a fully relativistic
treatment) method, where the orbital wave function and the scattering
potential are obtained by the self-consistent solution of the relevant
equations (see the Appendix for further details).

###  2.2 Benchmark system: 1D multi channel scattering from toy model
potentials

Prior moving to the discussion of the results on water and water aggregates,
as a test we benchmark our method against a model system. In particular, we
consider the following two-channels problem ($\hbar=1$):

$-\frac{1}{2m}\nabla^{2}\psi_{j}\left(\mathbf{r}\right)+\sum_{k}V_{jk}\left(\mathbf{r}\right)\psi_{k}\left(\mathbf{r}\right)=E\psi_{j}\left(\mathbf{r}\right),$
(8)

where $j,k$ label the indices of the two channels. The hamiltonian in Eq. 8 is
intentionally non-relativistic avoiding thus the description of the electrons
in terms of spinors and total angular momentum, as our goal is to resort to
the simplest model in order to test the validity of the LS approach in
connection with HGFs as a basis set to expand continuum scattering states. In
fact, in a non-relativistic approximation to scattering we rely on the use of
wave functions and a good quantum number is represented by the orbital angular
momentum. We also limit our test for the sake of simplicity to the modelling
of two $s$-orbitals channels, as the additional basis functions that we
include in the simulation of the electron-water scattering contain only
$s$-type HGFs.
By considering a spherically symmetric potential $V(r)$, the wave function can
be conveniently factorised in radial and angular parts, i.e.
$\psi_{j}\left(\bf{r}\right)=\mathcal{R}_{jl}\left(r\right)Y_{lm}\left(\vartheta,\varphi\right)$,
which specifically for the $s$-channels case reads
$\psi_{j}\left(\bf{r}\right)=\mathcal{R}_{j0}/\sqrt{4\pi}$. From Eq. 8 one
writes the following system of coupled differential equations:

$\begin{split}&-\frac{1}{2m}\frac{d^{2}}{dr^{2}}u_{1}\left(r\right)+V_{11}\left(r\right)u_{1}\left(r\right)+V_{12}\left(r\right)u_{2}\left(r\right)=Eu_{1}\left(r\right)\\\
&-\frac{1}{2m}\frac{d^{2}}{dr^{2}}u_{2}\left(r\right)+V_{21}\left(r\right)u_{1}\left(r\right)+V_{22}\left(r\right)u_{2}\left(r\right)=Eu_{2}\left(r\right),\end{split}$
(9)

where $u_{j}\left(r\right)=r\mathcal{R}_{j}\left(r\right)$, and
$u:\mathbb{R}_{+}\rightarrow\mathbb{C}^{2}$ fulfills the condition
$u_{j}\left(0\right)=0$. We solve these equations by further assuming $m=1$
for the following model potentials:

$\begin{split}&V_{11}\left(r\right)=-V_{22}\left(r\right)=-14\frac{e^{-0.5r}}{r}+7\frac{e^{-0.2r}}{r}\\\
&V_{12}\left(r\right)=V_{21}\left(r\right)=2\frac{e^{-0.3r}}{r}.\end{split}$
(10)

Figure 1: Results of the toy model potential problem of Eq. (9). (a) Phase
shifts for the auto-channels obtained through the integration of the
differential equation (ODE) and through the projection on a Gaussian basis set
by solving the corresponding Lippmann-Schwinger equations. The Gaussian basis
set is composed by 40 well tempered $s$-Gaussian functions. (b) Comparison of
the $T$-matrix elements for the auto-channels using the LS and ODE approaches.

In Figure 1 we show the phase shifts (panel a) and the $T$-matrix elements
(panel b) calculated in the auto-channels, identified as those channels in
which both the incoming and outgoing wave funtions are in the same channel.
In particular, we compare the numerical solution obtained by integrating the
Eqs. 9 via a fourth-order Runge-Kutta method (ODE in Figure 1 represented by
the colored continuous lines), which is almost exact, with the results
obtained using the LS equation (points in Figure 1) in which the potential is
projected using 40 well-tempered $s$-type Gaussian functions (Huzinaga and
Klobukowski, 1993), which deliver the best convergence with the ODE
calculations (for details on the use of Gaussian functions with the LS
equation see the following section 2.3). We observe that the phase shifts in
Figure 1a) in the auto-channels are well reproduced apart from a small region
around 10 Hartree where the LS solutions deviate from the exact ones. This is
reflected in the $T$-matrix elements of Figure 1b), where the resonance in the
channel 2-2 around the energy of 10 Hartree is shifted towards larger energies
in the LS solution. This can be rationalized by recognizing that the basis set
is not well suited to describe accurately this energy region, while it works
very well in the low energy part. It is worth to notice that, of course, the
resolution achievable by a given basis set depends upon the number and type of
Gaussian functions. We will solve this issue in the calculation of the ESCS of
water.

###  2.3 Our method: Gaussian projection of the Lippmann-Schwinger equation

The numerical solution of the DHF equation (or Schroedinger equation in non-
relativistic approximation) can be found by expanding the wave functions and
the potential using HGFs, which represent one of the most suitable basis set
to study non-spherical polyatomic systems, such as water, water cluster and
water molecular aggregates (for more details on this procedure see Ref.
(Taioli and Simonucci, 2021a)). The HGFs are defined as follows:

$g(\mathbf{r})=g(u,v,w;a,\mathbf{R};\mathbf{r})=N{\partial^{u+v+w}\over\partial
X^{u}\partial Y^{v}\partial
Z^{w}}({2a\over\pi})^{3/4}\exp[-a(\mathbf{r}-\mathbf{R})^{2}],$ (11)

where $\mathbf{R}\equiv(X,Y,Z)$ is the center of the HGF (which can or cannot
coincide with the nuclei positions), $a$ determines the HGF width, ($u,v,w$)
represent the order of derivation that determines the symmetry of the HGF
(e.g. first derivative represents a $p$-type orbital, the second derivative a
$d$-type orbital…); finally, $N$ is a coefficient set to normalize the HGFs.
The mono- and bi- electronic integrals of the fully relativistic many-body
hamiltonian are evaluated by including several HGFs characterised by different
derivation orders and centers (Taioli et al., 2010b; Taioli and Simonucci,
2021a).
Our approach relies on the semi-projected potential method (Taioli et al.,
2010a; Taioli and Simonucci, 2021a), where the Coulomb potential is projected
in HGFs, while the kinetic energy term $H_{0}=c{\bf\alpha}\cdot{\bf p}+\beta
mc^{2}$ ($\alpha$ and $\beta$ are the Dirac matrices, see Eq. 15 of the
Appendix) is unprojected so as to recover the asymptotic continuum after the
Coulomb potential dies off. Within this method the LS equation 6 for the
scattering wave function identified by the quantum number $\gamma$ reads:

$\psi_{\gamma{\bf k}}^{+}({\bf
r})=e^{i\bf{k}\cdot\bf{r}}+G_{0}^{+}(\varepsilon)V_{\gamma}^{t}(\bf{r})\psi_{\gamma\bf{k}}^{+}(\bf{r}),$
(12)

where
$V_{\gamma}^{t}=\sum_{\lambda\mu\nu\tau}|\lambda>S^{-1}_{\lambda\mu}<\mu|\hat{V}_{\gamma}|\nu>S^{-1}_{\nu\tau}<\tau|,\quad
S_{\lambda\mu}=<\lambda|\mu>$ is the approximate DHF self-consistent potential
represented by a set of HGFs, which includes both the Coulomb and static
dipole components of the potential. By replacing $V_{\gamma}$ with
$V^{t}_{\gamma}$ the LS equation 12 can be solved as follows:

$\psi_{\gamma\bf{k}}^{+}(\vec{r})=e^{i\bf{k}\cdot\bf{r}}+G_{0}^{-}(\varepsilon)T_{\gamma}(\varepsilon)e^{i\bf{k}\cdot\bf{r}},$
(13)

where $T_{\gamma}$ is the transition operator defined by the equation
$T_{\gamma}=V^{t}_{\gamma}+V^{t}_{\gamma}G_{0}^{-}(\varepsilon)T_{\gamma}$.
Within this approach we make use of the Static Exchange Approximation, in
which one neglects the effects of the continuum orbital on the bound orbitals.
The elements of the HGF basis set are chosen to minimize the difference
$(V_{\gamma}-V^{t}_{\gamma})\psi_{\gamma\bf{k}}^{+}(\bf{r})$ inside the region
of interest.
As in this work we aim to assess the total elastic scattering cross section
for a range of electron kinetic energies, which can span several orders of
magnitudes (from 0.001 to 103 eV), one can guess that it is cumbersome to find
a unique HGF basis set able to reproduce the behaviour of the scattering wave
function in such a large energy region. In particular, we have found that the
basis set for different energy ranges cannot be one and the same, owing to the
difficulty of reproducing the continuum orbital at different regimes with the
same basis set. In our simulations of the liquid water we use thus two sets of
HGFs. The first basis set is centered on the oxygen and hydrogen atoms and is
common to all the test cases, independent of the number of water molecules
included in the calculation of the elastic scattering cross section. In
particular, the basis sets used to expand the bound and continuum orbitals are
made of totally uncontracted HGFs, that is of $(12s,8p,6d,4f)$ Gaussian
functions centered on each hydrogen nucleus and of $(20s,12p,8d,6f,4g)$
Gaussian functions centered on the oxygen nuclei. These uncontracted basis
functions must be optimized so as to reproduce orbital eigenfunctions of the
DHF hamiltonian at the energies of interest.
Depending on the electron kinetic energy range, we have found that the
inclusion of a second set of HGFs is critical to reproduce the elastic
scattering experimental data. This observation is in agreement with the toy
model potential calculations previously presented, where a small number of
HGFs was not sufficient to reproduce the accuracy of the exact solution at all
energies (see Figure 1 around 10 Hartree). This finding can be rationalized by
noticing that the insertion of this further $s$-symmetry basis set delivers
enough flexibility to reproduce the scattering wave function behaviour in
different energy regions: narrower HGFs (that is larger exponential
coefficients $a$ in Eq. 11) can more accurately represent the highly
oscillating behaviour of the scattering wave function at high energy, while
more diffuse HGF (that is smaller exponential coefficients $a$ in Eq. 11) can
better reproduce the slowly varying behaviour of the scattering wave function
at low energy. This further HGF basis set is uniformly distributed within a
sphere, roughly centered on the center of mass of the water molecule, and
characterised by a variable radius $R=1.25/\sqrt{a}$ (in a.u.), where $a$ is
the exponent of the HGF.

##  3 Results and discussion

###  3.1 Single water molecule

Figure 2: Theoretical elastic scattering cross section (continuous multi-
colored line) of electrons impinging on a single water molecule with different
kinetic energies compared to mixed experimental/theoretical results from Ref.
(Khakoo et al., 2013) (full blue squares with error bars), with the
recommended elastic (full purple circles) and total (green triangles) cross
sections from Ref. (Song et al., 2021). The recommended elastic cross section
of Ref. (Song et al., 2021) is based on the results given in Ref. (Gorfinkiel
et al., 2002) up to 6 eV and by interpolating the cross sections obtained with
different methods (Itikawa and Mason, 2005). The recommended total cross
section is obtained at energies between 0.1 and 7 eV via R-matrix calculations
(Zhang et al., 2009; Faure et al., 2004), at 7–50 eV is based on experiments
from Refs. (Szmytkowski, 1987; Szmytkowski and Mozejko, 2006; Kadokura et al.,
2019) and above 50 eV on experimental measurements from Ref. (Muñoz et al.,
2007). The cyan circles report the results obtained by using the Elsepa code
(Salvat et al., 2005). Data in the $x$ and $y$ axes are reported in $\log$
scale, $a$ is in bohr-2.

Using the approach presented in the methodological section, we carried out the
calculation of the total elastic scattering cross section of electrons
impinging on i) a single water molecule; ii) a cluster of 2 and 3 water
molecules; iii) the zundel cation.
In Figure 2 we show the results obtained for the single water molecule, which
is well studied both experimentally and theoretically. Our ab initio results
(continuous multi-colored line) are compared with other theoretical and
experimental approaches summarized in Ref. (Song et al., 2021) (see the figure
caption for details on the different points). We notice that our calculations
are performed in the fixed-nuclei approximation, i.e. we do not account for
rotational and vibrational degrees of freedom. We also stress that
unfortunately a single set of HGFs cannot accurately describe the elastic
cross section in the full spectrum, that is to say that one needs to include
more or less diffuse HGFs in order to be able to describe the oscillations of
the continuum wave function (and, ultimately, of the Coulomb potential tail)
in different energy ranges. Thus, we decided to add further $s-$symmetry HGFs
to the bound state basis set of water, whose width $a$ depends on the energy
region. Each colour of the line represents thus a different Gaussian width
(see the legend in Figure 2, reporting in bohr-2 the seven $a$ coefficients
used to accurately reproduce the experimental data). We notice that while
these coefficients where optimized by comparison with experimental results,
their physical meaning is clear: at low energies the variance of the Gaussian
functions should capture the long wavelengths related to low energies; on the
other hand at higher energies the dependence on the $a$ coefficient is less
stringent; in general, larger values of $a$ describe better the higher
oscillations. Interestingly, above 4 eV our results are in excellent agreement
with those by Khakoo et al. (Khakoo et al., 2013), while below 4 eV our data,
although following the same trend, are slightly outside the error bars. This
is in accordance with the fact that the results of Ref. (Khakoo et al., 2013)
were obtained using the Schwinger multichannel method (Takatsuka and McKoy,
1981), employing extensive basis sets and considering both polarization and
dipole-scattering effects. In our DHF framework, of course we also take into
account them, and our LS solutions must be stationary points of the Schwinger
variational principle. We also notice that at energies $>$ 10 eV, both our and
Ref. (Khakoo et al., 2013) results fit better the total cross section trend
(green triangles in Figure 2a)), than the recommended elastic one (full purple
circles (Song et al., 2021)). This point can be understood by noticing that
our approach is based on the inclusion in our model of the elastic channel
only, while the total cross section contains the contributions of both elastic
and inelastic scattering events. However, we have previously demonstrated (see
Refs. (Taioli et al., 2010a; Taioli and Simonucci, 2015)) that the total
probability is not significantly modified by the opening of multiple channels;
rather such total transition probability is redistributed among several
channels, while keeping constant the total value. Thus, it is not surprising
that our ESCS actually follows closely the experimental data.
Finally, in Figure 2 we also compare our ab initio calculations with the
Elsepa (Salvat et al., 2005) model (cyan points), which is fully relativistic
but the atomic rather than the molecular interaction potential is considered,
while the dipole is included semi-empirically. The discrepancy between our and
Elsepa results can be thus attributed to the different treatment of the
interaction potential.

###  3.2 Water clusters

To show the potential of our method to deal with polycentric systems, we have
studied the electron scattering with small clusters of water molecules, which
are shown in Figure 3.

Figure 3: Water clusters employed in this work as scattering centers. In
panels (a) and (b) we report two different configurations of water molecules
in the liquid phase. In panel (c) we show the zundel cation minimum
configuration as reported in (Huang et al., 2005).

In particular, in the panels a) and b) of Figure 3 we report two different
clusters made of three molecules, which were obtained by trimming a large
cubic cell of liquid water (i.e. density is 1 g / cm3), whose configuration
was optimized using Density Functional Theory implemented with a PBE
functional (Giannozzi et al., 2017). The two clusters are meant to describe
the different environment in which electrons travel inside liquid water,
having the dipole moments pointing towards different directions.

Figure 4: ESCS of electrons colliding at different kinetic energies with the
two water clusters (3 [H2O]1,2) reported in Figure 3, in comparison to the
results for the single water molecule (1 H2O) shown in Figure 2. Multi-colored
lines report our calculations; the cyan points report the values computed with
the Elsepa code (Salvat et al., 2005) on the water cluster 1 of Figure 3. The
total cross section of the water clusters has been divided by three, i.e. to
normalize it with the number of molecules within the clusters. Data in the $x$
and $y$ axes are reported in $\log$ scale, $a$ is in bohr-2.

In Figure 4 we show the ESCS of these two water clusters resulting from our
simulations (upper multi-colored lines) in comparison to both experimental
data (symbols), single water molecule calculations (lower multi-colored line),
and Elsepa (cyan points). Each colour of the line represents a different
Gaussian width (see the legend in Figure 4, reporting in bohr-2 the $a$
coefficients used in different energy ranges). Of course, the ESCSs of the
water clusters have been normalized by the number of molecules in the
simulation cell. We notice that the ESCS of the two water clusters
characterised by different water molecule configurations are similar. However,
by comparing our simulations of the single water molecule with water clusters,
we find a very good match in the low ($E<$ 1 eV) and high ($E>$ 30 eV) energy
range; at odds, we obtain a diverging behaviour in the intermediate regime
($1<E<30$ eV) with ESCS differing by almost 100% at around 3 eV. As the basis
set we use is comparable in both size and diffuseness for the three cases
investigated, we attribute such a discrepancy in the ESCS at intermediate
energy due to multiple scattering and a more accurate representation of the
inter- and intra-molecular potential. Indeed, Elsepa simulations of ESCS on
cluster 1 of Figure 3 follow closely the single water molecule behaviour. We
stress that experimental measurements are carried out on water vapour in
different conditions of relative humidity, while our simulations are carried
out using the standard density of liquid water. This finding highlights the
importance of an accurate treatment of the interatomic potential for dealing
with liquid phases as compared to vapour phase, where molecules are far apart
and, thus, more closely resembling the single molecule calculations.

###  3.3 Elastic scattering from the zundel cation

Finally, to show the flexibility of our approach to deal with charged system
we calculate the ESCS of electrons impinging on protonated water (zundel
cation), in which hydrogen ions bond with water molecules and form loosely
bound structures (see panel c) of Figure 3).

Figure 5: ESCS of electrons colliding at different kinetic energies with the
zundel cation in comparison to the water clusters (3 [H2O]1,2) reported in
Figure 3. Multi-colored lines report our calculations; the cyan points report
the values computed with the Elsepa code (Salvat et al., 2005) on the water
cluster 1 of Figure 3. The total cross section of the water clusters has been
divided by three, i.e. to normalize it with the number of molecules within the
clusters, and by two for the zundel cation. Data in the $x$ and $y$ axes are
reported in $\log$ scale, $a$ is in bohr-2.

In particular, in Figure 5 we compare the results for the zundel cation (upper
multi-colored line) obtained using our relativistic ab initio method with
those of the water clusters (of course all ESCSs have been normalized with
respect to the number of water molecules within the systems). Each colour of
the line represents a different Gaussian width (see the legend in Figure 5,
reporting in bohr-2 the $a$ coefficients used in different energy ranges). We
notice that the presence of different environments dramatically changes the
ESCS, particularly in the intermediate to low energy region ($<100$ eV). This
again shows up the importance of accurately modelling the intra- and inter-
molecular potential, as the different behaviour of the curves in Figure 5 can
be attributed to the different environment and multiple scattering effects
more than to a deficiency of the HGF basis set. Finally, we also report the
Elsepa simulations carried out for the zundel cation configuration, even
though we stress that Elsepa (cyan points) can deal only with neutral charge
state.

##  4 Conclusions

In this work we propose a first-principles method to compute elastic
scattering cross sections of electrons impinging on polycentric systems, such
as water and water aggregates. Our approach does not require approximations,
beyond the chosen many-body method (that is systematically improvable), on the
channels involved in the process. Indeed, the latter are inherently included
in the formalism by solving the LS equations for the scattering wave function.
Furthermore, the projection of the potential onto a functional space spanned
by HGFs allows one to deal with polycentric systems of increasing complexity
and different charge states, and to achieve a computational cost similar to
that of bound state calculations.
The modelling of the electron elastic scattering in (polar) molecules
typically is carried out either by ab initio methods, typically based on the
R-matrix approach (Faure et al., 2004; Gorfinkiel et al., 2005) or by
following the Mott theory. The main difference with the Mott theory is that in
our scheme the scattering potential is not a mere sum of atomic contributions,
and thus fully accounts for the polycentric nature of the target system,
including dipole moments and multiple scattering arising from the accurate
treatment of the inter- and intra-molecular potential. This is at odds with
the Mott theory, where the elastic scattering by different atoms belonging to
the same molecule is considered independently and the total cross section is
obtained as a weighted sum of the atomic cross sections. We have demonstrated
that both the inclusion of re-scattering and the accurate calculation of the
inter- and intra-molecular potentials are crucial to the assessment of the
elastic cross section.
On the other hand, ab initio methods so far used to calculate the ESCS of
electrons do not account for relativistic effects and are typically limited to
the modelling of a single water molecule. At variance, our scheme clearly
shows how the consideration of the liquid phase nature of water emerges as a
significant deviation from the single molecule case; moreover, it is fully
relativistic, so important effects present in the elastic scattering of
unpolarized electrons, such as spin polarization and flip, are taken fully
into account. While we recognize that relativistic effects are not of primary
importance in water systems, our framework paves the way towards a full
understanding and accurate treatment of the electron scattering also from
heavy-element nanoparticles (NPs), such as gold clusters, where the inclusion
of relativistic effects is of paramount importance. For example, gold NPs are
particularly interesting in hadrontherapy for cancer cure to increase the
relative biological effectiveness of ion and electron beams. Moreover, the use
of polarized electron beams is becoming increasingly important in quantum
information theory, which must deal with the decoherence of qubits from
quantum noise to protect information. Only a quantum-relativistic approach to
the elastic scattering of electron beams by atoms and molecules makes it
possible to determine the polarization degree upon scattering and to deliver
an accurate account of experimentally produced polarized ensembles of
electrons.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

Conceptualization, S.T., S.S. and M.T.; methodology, S.T., S.S., F.T., and
M.T.; software, S.S., M.T.; validation, S.S, F.T, and M.T.; investigation,
S.S., M.T, and F.T.; resources, S.T., and S.S.; writing–original draft
preparation, S.T, T.M., F.T.; writing–review and editing, S.T., S.S., F.T.,
and M.T.; visualization, M.T., F.T., and S.S.; supervision, S.T, S.S.; project
administration, S.T, S.S.; funding acquisition, S.T, S.S. All authors have
read and agreed to the published version of the manuscript.

## Funding

This action has received funding from the European Union under grant agreement
n. 101046651.

## Acknowledgments

We acknowledge fruitful discussions with Drs. M. Dapor, P. De Vera, and Profs.
I. Abril and R. Garcia-Molina.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Appendix

The many-particle Dirac equation for particles of mass $m$, interacting via a
generic scalar/vector potential is the following:

$\bigg{\\{}\sum_{i}(c\alpha_{i}\mathbf{p}_{i}+\beta_{i}mc^{2}+V_{i})+\sum_{i<j}[\beta_{i}\beta_{j}g_{S,ij}+(1-\alpha_{i}\cdot\alpha_{j})g_{V,ij}]\bigg{\\}}\psi(\mathbf{r}1,...,\mathbf{r}_{N})=E\psi(\mathbf{r}_{1},...,\mathbf{r}_{N}),$
(14)

where $V_{i}$ is the external potential, $\alpha_{i}$ and $\beta_{i}$ are the
following $4\times 4$ Hermitian matrices

$\alpha=\begin{pmatrix}0&\sigma\\\ \sigma&0\\\ \end{pmatrix},\hskip
28.45274pt\beta=\begin{pmatrix}I&0\\\ 0&-I\\\ \end{pmatrix},$ (15)

with $I$ the $2\times 2$ identity matrix and $\sigma$ the Pauli matrices;
$g_{V}$ and $g_{S}$ are respectively the vector and scalar potentials. In
second quantization, we can use the Hartree-Fock (HF) approximation to the
electron exchange-correlation interaction by assuming:

$\biggl{\langle}\hat{\Psi}^{\dagger}_{s_{1}}(\mathbf{r})\hat{\Psi}^{\dagger}_{s^{\prime}_{1}}(\mathbf{r^{\prime}})\hat{\Psi}_{s^{\prime}_{2}}(\mathbf{r^{\prime}})\hat{\Psi}_{s_{2}}(\mathbf{r})\biggr{\rangle}=\biggl{\langle}\hat{\Psi}^{\dagger}_{s_{1}}(\mathbf{r})\hat{\Psi}_{s_{2}}(\mathbf{r})\biggr{\rangle}\biggl{\langle}\hat{\Psi}^{\dagger}_{s^{\prime}_{1}}(\mathbf{r^{\prime}})\hat{\Psi}_{s^{\prime}_{2}}(\mathbf{r^{\prime}})\biggr{\rangle}-$
$\biggl{\langle}\hat{\Psi}^{\dagger}_{s_{1}}(\mathbf{r})\hat{\Psi}_{s^{\prime}_{2}}(\mathbf{r^{\prime}})\biggr{\rangle}\biggl{\langle}\hat{\Psi}^{\dagger}_{s^{\prime}_{1}}(\mathbf{r^{\prime}})\hat{\Psi}_{s_{2}}(\mathbf{r})\biggr{\rangle},$
(16)

where the density matrix

$\rho_{s^{\prime}_{2},s_{1}}=\biggl{\langle}\hat{\Psi}^{\dagger}_{s_{1}}(\mathbf{r})\hat{\Psi}_{s^{\prime}_{2}}(\mathbf{r^{\prime}})\biggr{\rangle}$
(17)

is the bloc matrix

$\rho(\mathbf{r^{\prime}},\mathbf{r})=\begin{pmatrix}\rho_{LL}(\mathbf{r^{\prime}},\mathbf{r})&\rho_{LS}(\mathbf{r^{\prime}},\mathbf{r})\\\
\rho_{SL}(\mathbf{r^{\prime}},\mathbf{r})&\rho_{SS}(\mathbf{r^{\prime}},\mathbf{r})\\\
\end{pmatrix},$ (18)

in which LS denote the large and small component of the Dirac bispinor, while
$s_{1},s_{2},s^{\prime}_{1}$ and $s^{\prime}_{2}$ label the upper and lower
bispinor components. For a many-body systems of electrons interacting only via
Coulomb repulsion $V$, the DHF equation can be rewritten as (Taioli and
Simonucci, 2021a; de Vera et al., 2022a):

$\begin{pmatrix}mc^{2}+V-E&-c\mathbf{\sigma}\cdot i\nabla\\\
c\mathbf{\sigma}\cdot
i\nabla&-mc^{2}-V-E\end{pmatrix}\begin{pmatrix}\Psi_{L}\\\
\Psi_{S}\end{pmatrix}=0.$ (19)

where $V=V_{H}+V_{exc}$ is written in HF approximation as the sum of the
Hartree potential and the exchange interaction. Please notice the different
sign in front of the interaction term acting on the upper and lower parts of
the spinor. This matrix equation was used to calculate the electronic
structure, and in particular to obtain the relativistic interaction potential
of water aggregates.

## References

  * Altshuler (1957) Altshuler, S. (1957).  Theory of low-energy electron scattering by polar molecules.  _Phys. Rev._ 107, 114–117.  10.1103/PhysRev.107.114
  * Azzolini et al. (2018a) Azzolini, M., Angelucci, M., Cimino, R., Larciprete, R., Pugno, N. M., Taioli, S., et al. (2018a).  Secondary electron emission and yield spectra of metals from monte carlo simulations and experiments.  _Journal of Physics: Condensed Matter_ 31, 055901
  * Azzolini et al. (2018b) Azzolini, M., Morresi, T., Abrams, K., Masters, R., Stehling, N., Rodenburg, C., et al. (2018b).  Anisotropic approach for simulating electron transport in layered materials: computational and experimental study of highly oriented pyrolitic graphite.  _The Journal of Physical Chemistry C_ 122, 10159–10166
  * Azzolini et al. (2017) Azzolini, M., Morresi, T., Garberoglio, G., Calliari, L., Pugno, N. M., Taioli, S., et al. (2017).  Monte carlo simulations of measured electron energy-loss spectra of diamond and graphite: role of dielectric-response models.  _Carbon_ 118, 299
  * Azzolini et al. (2020) Azzolini, M., Ridzel, O. Y., Kaplya, P. S., Afanas’ev, V., Pugno, N. M., Taioli, S., et al. (2020).  A comparison between monte carlo method and the numerical solution of the ambartsumian-chandrasekhar equations to unravel the dielectric response of metals.  _Computational Materials Science_ 173, 109420
  * Colle et al. (2004a) Colle, R., Embriaco, D., Massini, M., Simonucci, S., and Taioli, S. (2004a).  Ab initio calculation of the normal auger spectrum of c2h2.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ 37, 1237
  * Colle et al. (2004b) Colle, R., Embriaco, D., Massini, M., Simonucci, S., and Taioli, S. (2004b).  Auger-electron angular distributions calculated without the two-step approximation: Calculation of angle-resolved resonant auger spectra of c2h2.  _Physical Review A_ 70, 042708
  * Dapor (2020) Dapor, M. (2020).  _Transport of Energetic Electrons in Solids Computer Simulation with Applications to Materials Analysis and Characterization_.  Springer Tracts in Modern Physics, 271 (Cham: Springer International Publishing), 3rd ed. 2020. edn.
  * de Vera et al. (2022a) de Vera, P., Simonucci, S., Trevisanutto, P. E., Abril, I., Dapor, M., Taioli, S., et al. (2022a).  Simulating the nanometric track-structure of carbon ion beams in liquid water at energies relevant for hadrontherapy.  In _J. Phys.: Conf. Ser._ (IOP), vol. 2326, 012017
  * de Vera et al. (2022b) de Vera, P., Taioli, S., Trevisanutto, P. E., Dapor, M., Abril, I., Simonucci, S., et al. (2022b).  Energy deposition around swift carbon-ion tracks in liquid water.  _International Journal of Molecular Sciences_ 23, 6121
  * Fabrikant (2016) Fabrikant, I. I. (2016).  Long-range effects in electron scattering by polar molecules.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ 49, 222005.  10.1088/0953-4075/49/22/222005
  * Faure et al. (2004) Faure, A., Gorfinkiel, J. D., and Tennyson, J. (2004).  Low-energy electron collisions with water: elastic and rotationally inelastic scattering.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ 37, 801.  10.1088/0953-4075/37/4/007
  * Fedus and Karwasz (2017) Fedus, K. and Karwasz, G. P. (2017).  Electron scattering on molecules: search for semi-empirical indications.  _The European Physical Journal D_ 71, 138.  10.1140/epjd/e2017-80065-2
  * Garrett (1972) Garrett, W. (1972).  Low-energy electron scattering by polar molecules.  _Molecular Physics_ 24, 465–487.  10.1080/00268977200101621
  * Giannozzi et al. (2017) Giannozzi, P., Andreussi, O., Brumme, T., Bunau, O., Nardelli, M. B., Calandra, M., et al. (2017).  Advanced capabilities for materials modelling with quantum ESPRESSO.  _Journal of Physics: Condensed Matter_ 29, 465901.  10.1088/1361-648x/aa8f79
  * Gianturco et al. (1998) Gianturco, F. A., Meloni, S., Paioletti, P., Lucchese, R. R., and Sanna, N. (1998).  Low-energy electron scattering from the water molecule: Angular distributions and rotational excitation.  _The Journal of Chemical Physics_ 108, 4002–4012.  10.1063/1.475349
  * Gianturco and Scialla (1987) Gianturco, F. A. and Scialla, S. (1987).  Low‐energy electron scattering from water molecules: A study of angular distributions.  _The Journal of Chemical Physics_ 87, 6468–6473.  10.1063/1.453428
  * Gorfinkiel et al. (2005) Gorfinkiel, J., Faure, A., Taioli, S., Piccarreta, C., Halmova, G., and Tennyson, J. (2005).  Electron-molecule collisions at low and intermediate energies using the r-matrix method.  _The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics_ 35, 231–237
  * Gorfinkiel et al. (2002) Gorfinkiel, J. D., Morgan, L. A., and Tennyson, J. (2002).  Electron impact dissociative excitation of water within the adiabatic nuclei approximation.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ 35, 543.  10.1088/0953-4075/35/3/309
  * Huang et al. (2005) Huang, X., Braams, B. J., and Bowman, J. M. (2005).  Ab initio potential energy and dipole moment surfaces for h5o2+.  _The Journal of Chemical Physics_ 122, 044308.  10.1063/1.1834500
  * Huzinaga and Klobukowski (1993) Huzinaga, S. and Klobukowski, M. (1993).  Well-tempered gaussian basis sets for the calculation of matrix hartree—fock wavefunctions.  _Chemical Physics Letters_ 212, 260–264.  https://doi.org/10.1016/0009-2614(93)89323-A
  * Itikawa and Mason (2005) Itikawa, Y. and Mason, N. (2005).  Cross sections for electron collisions with water molecules.  _Journal of Physical and Chemical Reference Data_ 34, 1–22.  10.1063/1.1799251
  * Kadokura et al. (2019) Kadokura, R., Loreti, A., Kövér, A., Faure, A., Tennyson, J., and Laricchia, G. (2019).  Angle-resolved electron scattering from ${\mathrm{h}}_{2}\mathrm{O}$ near 0.  _Phys. Rev. Lett._ 123, 033401.  10.1103/PhysRevLett.123.033401
  * Khakoo et al. (2013) Khakoo, M. A., Silva, H., Muse, J., Lopes, M. C. A., Winstead, C., and McKoy, V. (2013).  Erratum: Electron scattering from h2o: Elastic scattering [phys. rev. a 78, 052710 (2008)].  _Phys. Rev. A_ 87, 049902.  10.1103/PhysRevA.87.049902
  * Maciel (2013) Maciel, W. (2013).  _Astrophysics of the Interstellar Medium_.  10.1007/978-1-4614-3767-3
  * Mascali et al. (2022) Mascali, D., Santonocito, D., Amaducci, S., Andò, L., Antonuccio, V., Biri, S., et al. (2022).  A novel approach to $\beta$-decay: Pandora, a new experimental setup for future in-plasma measurements.  _Universe_ 8, 80.  https://doi.org/10.3390/universe8020080
  * Morresi et al. (2018) Morresi, T., Taioli, S., and Simonucci, S. (2018).  Nuclear beta decay: Relativistic theory and ab initio simulations of electroweak decay spectra in medium-heavy nuclei and of atomic and molecular electronic structure (adv. theory simul. 11/2018).  _Advanced Theory and Simulations_ 1, 1870030
  * Mott (1929) Mott, N. (1929).  The scattering of fast electrons by atomic nuclei.  _Proc. R. Soc. London, Ser. A_ 124, 425
  * Muñoz et al. (2007) Muñoz, A., Oller, J. C., Blanco, F., Gorfinkiel, J. D., Limão Vieira, P., and García, G. (2007).  Electron-scattering cross sections and stopping powers in ${\mathrm{h}}_{2}\mathrm{O}$.  _Phys. Rev. A_ 76, 052707.  10.1103/PhysRevA.76.052707
  * Nikjoo et al. (2016) Nikjoo, H., Emfietzoglou, D., Liamsuwan, T., Taleei, R., Liljequist, D., and Uehara, S. (2016).  Radiation track, dna damage and response-a review.  _Reports on progress in physics. Physical Society (Great Britain)_ 79, 116601.  10.1088/0034-4885/79/11/116601
  * Palmerini et al. (2021) Palmerini, S., Busso, M., Vescovi, D., Naselli, E., Pidatella, A., Mucciola, R., et al. (2021).  Presolar grain isotopic ratios as constraints to nuclear and stellar parameters of asymptotic giant branch star nucleosynthesis.  _The Astrophysical Journal_ 921, 7
  * Pedrielli et al. (2021) Pedrielli, A., de Vera, P., Trevisanutto, P. E., Pugno, N. M., Garcia-Molina, R., Abril, I., et al. (2021).  Electronic excitation spectra of cerium oxides: from ab initio dielectric response functions to monte carlo electron transport simulations.  _Physical Chemistry Chemical Physics_ 23, 19173–19187
  * Pedrielli et al. (2022) Pedrielli, A., Pugno, N. M., Dapor, M., and Taioli, S. (2022).  In search of the ground state crystal structure of ta $\\_2$ o $\\_5$ from ab-initio and monte carlo simulations.  _Computational Materials Science_ 216, 111828
  * Reiher and Wolf (2014) Reiher, M. and Wolf, A. (2014).  _Relativistic Theory of the Electron_ (John Wiley & Sons, Ltd), chap. 5.  159–191.  https://doi.org/10.1002/9783527667550.ch5
  * Salvat et al. (2005) Salvat, F., Jablonski, A., and Powell, C. J. (2005).  elsepa—dirac partial-wave calculation of elastic scattering of electrons and positrons by atoms, positive ions and molecules.  _Computer Physics Communications_ 165, 157–190.  https://doi.org/10.1016/j.cpc.2004.09.006
  * Signorell (2020) Signorell, R. (2020).  Electron scattering in liquid water and amorphous ice: A striking resemblance.  _Phys. Rev. Lett._ 124, 205501.  10.1103/PhysRevLett.124.205501
  * Simonucci et al. (2013) Simonucci, S., Taioli, S., Palmerini, S., and Busso, M. (2013).  Theoretical estimates of stellar e- captures. i. the half-life of 7be in evolved stars.  _The Astrophysical Journal_ 764, 118
  * Song et al. (2021) Song, M.-Y., Cho, H., Karwasz, G. P., Kokoouline, V., Nakamura, Y., Tennyson, J., et al. (2021).  Cross sections for electron collisions with h2o.  _Journal of Physical and Chemical Reference Data_ 50, 023103.  10.1063/5.0035315
  * Szmytkowski (1987) Szmytkowski, C. (1987).  Absolute total cross sections for electron-water vapour scattering.  _Chemical Physics Letters_ 136, 363–367.  https://doi.org/10.1016/0009-2614(87)80267-1
  * Szmytkowski and Mozejko (2006) Szmytkowski, C. and Mozejko, P. (2006).  Electron-scattering total cross sections for triatomic molecules: No2 and h2o.  _Optica Applicata_
  * Taioli et al. (2022a) Taioli, S., Dapor, M., Dimiccoli, F., Fabi, M., Ferroni, V., Grimani, C., et al. (2022a).  The role of low-energy electrons in the charging process of lisa test masses.  _arXiv preprint arXiv:2209.12791_
  * Taioli and Simonucci (2015) Taioli, S. and Simonucci, S. (2015).  A computational perspective on multichannel scattering theory with applications to physical and nuclear chemistry.  In _Annual Reports in Computational Chemistry_ (Elsevier), vol. 11. 191–310
  * Taioli and Simonucci (2021a) Taioli, S. and Simonucci, S. (2021a).  Chapter three - relativistic quantum theory and algorithms: A toolbox for modeling many-fermion systems in different scenarios (Elsevier), vol. 17 of _Annual Reports in Computational Chemistry_. 55–111.  https://doi.org/10.1016/bs.arcc.2021.08.003
  * Taioli and Simonucci (2021b) Taioli, S. and Simonucci, S. (2021b).  The resonant and normal auger spectra of ozone.  _Symmetry_ 13, 516
  * Taioli et al. (2015) Taioli, S., Simonucci, S., a Beccara, S., and Garavelli, M. (2015).  Tetrapeptide unfolding dynamics followed by core-level spectroscopy: a first-principles approach.  _Physical Chemistry Chemical Physics_ 17, 11269–11276
  * Taioli et al. (2010a) Taioli, S., Simonucci, S., Calliari, L., and Dapor, M. (2010a).  Electron spectroscopies and inelastic processes in nanoclusters and solids: Theory and experiment.  _Physics reports_ 493, 237–319
  * Taioli et al. (2010b) Taioli, S., Simonucci, S., Calliari, L., and Dapor, M. (2010b).  Electron spectroscopies and inelastic processes in nanoclusters and solids: Theory and experiment.  _Physics Reports_ 493, 237–319.  https://doi.org/10.1016/j.physrep.2010.04.003
  * Taioli et al. (2009) Taioli, S., Simonucci, S., Calliari, L., Filippi, M., and Dapor, M. (2009).  Mixed ab initio quantum mechanical and monte carlo calculations of secondary emission from sio 2 nanoclusters.  _Physical Review B_ 79, 085432
  * Taioli and Tennyson (2006a) Taioli, S. and Tennyson, J. (2006a).  Waterwaves: wave particles dynamics on a complex triatomic potential.  _Computer physics communications_ 175, 41–51
  * Taioli and Tennyson (2006b) Taioli, S. and Tennyson, J. (2006b).  A wave packet method for treating nuclear dynamics on complex potentials.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ 39, 4379
  * Taioli et al. (2020) Taioli, S., Trevisanutto, P. E., de Vera, P., Simonucci, S., Abril, I., Garcia-Molina, R., et al. (2020).  Relative role of physical mechanisms on complex biodamage induced by carbon irradiation.  _The Journal of Physical Chemistry Letters_ 12, 487–493
  * Taioli et al. (2022b) Taioli, S., Vescovi, D., Busso, M., Palmerini, S., Cristallo, S., Mengoni, A., et al. (2022b).  Theoretical estimate of the half-life for the radioactive 134cs and 135cs in astrophysical scenarios.  _The Astrophysical Journal_ 933, 158
  * Takatsuka and McKoy (1981) Takatsuka, K. and McKoy, V. (1981).  Extension of the schwinger variational principle beyond the static-exchange approximation.  _Phys. Rev. A_ 24, 2473–2480.  10.1103/PhysRevA.24.2473
  * Vescovi et al. (2019) Vescovi, D., Piersanti, L., Cristallo, S., Busso, M., Vissani, F., Palmerini, S., et al. (2019).  The effects of a revised 7be e-capture rate on solar neutrino fluxes.  _Astronomy & Astrophysics_ 623, 7
  * Zhang et al. (2009) Zhang, R., Faure, A., and Tennyson, J. (2009).  Electron and positron collisions with polar molecules: studies with the benchmark water molecule.  _Physica Scripta_ 80, 015301.  10.1088/0031-8949/80/01/015301