capbtabboxtable[][]  11institutetext: IIIT-Hyderabad, India
11email<EMAIL_ADDRESS>ajoy.mondal@, and<EMAIL_ADDRESS>
# LEANet for Enhancing Classroom Slide Narration System

Jobin K.V  Ajoy Mondal  Jawahar C.V

###### Abstract

For the last three decades, teachers or the teaching community have been
successfully using slide presentations as an efficient and effective tool for
classroom communication. However, blind and visually impaired (vi) students
find it challenging to adjust to this teaching model. This shortcoming
motivates us to design a solution that generates audio descriptions
corresponding to the slide content. This problem poses as an image-to-markup
language generation task. The initial step is to extract logical labels such
as title, text, equation, figure, and table from the slide image. In the
classroom slide images, the logical labels are distributed based on the
location of the image. Most semantic segmentation networks do not reflect such
unique attributes in the architecture. In this paper, we propose a slide image
segmentation architecture called Location Encoding Attention Network (leanet).
This architecture utilizes the location of the logical regions to segment the
slide image efficiently. Publicly available benchmark datasets such as WiSe
and SPaSe validate the performance of our segmentation architecture. We
extract the slide content (information) using four well-established modules
such as ocr, figure classification, equation description, and table structure
recognizer. With this information, we build a Classroom Slide Narration System
(csns) for helping vi students to understand the slide content. The users give
better feedback on the quality output of the proposed csns in comparison to
existing systems like Facebook’s Automatic Alt-Text (aat) and Tesseract.

###### Keywords:

Slide image segmentation Logical regions Location encoding Classroom slide
narration

Figure 1: Presents sample slide images with considerable diversity with
respect to theme and content. First Row: a wide variety in slide themes.
Second Row: a large scale diversity in the slides’ content.

##  1 Introduction

Slide presentations play an essential role in classroom teaching, technical
lectures, massive open online courses (mooc), and many formal
meetings.Presentations are an efficient and effective tool for the students to
understand the subject better. However, the visually impaired (vi) students do
not benefit from this type of delivery due to their limitation in reading the
slides. Human assistants could help vi students to access and interpret the
slide’s content. However, at every time, it is impossible and impractical for
the vi students to have a human reader; because of the cost and the limited
availability of the trained personnel. From this perspective, this would be an
important issue in modern classrooms.

The automatic extraction of logical regions from these slide images is the
initial step for understanding the slide’s content. We define automatic
extraction of logical parts of slide images as segmentation of semantically
similar regions, closely related to natural image semantic segmentation in
computer vision [long2015fully, chen2017deeplab, zhao2017pyramid]. The goal is
to assign logical labels like, title, paragraph, list, equation, table, figure
to each pixel in the slide images. Due to the large variability in the theme
(i.e., the layout of the slide), style, and content of the slide shown in
Figure 1, the extraction of meaningful regions is a challenging task.

The text extraction from these regions is useful to the vi student only if the
text is tagged with its logical function. The left image of Figure 2, the
plain text “Discovering Attention Patterns” is more meaningful to the vi
student only if he knows the text is the heading of the slide. Similarly, the
detection of figures, equations, and table regions are also essential for
understanding the slides. Our goal is to develop a system that meaningfully
describes the classroom slides with a markup text, as shown in Figure 2.

Figure 2: Illustrate qualitative comparison between outputs obtained by the
proposed csns and the existing assistive systems such as Automatic Alt-Text
(aat) [wu2017automatic] and Tesseract ocr [smith2007overview]. The proposed
system generates a markup text, where each logical content is tagged with its
appropriate logical function. The audio file generated using this markup text
gives a clear picture of the classroom slide’s content to the blind or vi
students.

The classroom slides have a complex design, as shown in Figure 1. The same
text could appear in various labels such as “heading”, “paragraph”, and
“list”. Hence, only the visual features learned by the existing semantic
segmentation networks (e.g., fcn [long2015fully], Deeplab [chen2017deeplab],
pspnet [zhao2017pyramid], etc.) are not sufficient to distinguish various
logical regions. Limited works [haurilet2019spase, haurilet2019wise] on
classroom slide segmentation utilize $(x,\ y)$ coordinate values to enhance
the quality of segmentation results. However, all these works
[haurilet2019spase, haurilet2019wise] do not utilize location information
efficiently for segmentation. In contrast, Choi _et al._ [Choi_2020_CVPR]
utilize the position encoding [vaswani2017attention] to impose the height
information in hanet for semantic segmentation of urban-scene images. In the
case of slide images, height and width information of each logical region are
equally important for segmenting regions accurately. In this work, we propose
a Classroom Slide Segmentation Network, called Location Encoding Attention
Network (leanet) to segment slide images accurately. The proposed network
consists of (i) Attention Module which utilizes the location encoding of the
logical region (height and width of a region) and (ii) Atrous Spatial Pyramid
Pooling (aspp) module which extracts multi-scale contextual features. The
experiments on publicly available benchmark datasets WiSe [haurilet2019wise]
and SpaSe [haurilet2019spase] establish the effectiveness of the proposed
leanet over state-of-the-art techniques — Haurilet _et al._
[haurilet2019wise], Haurilet _et al._ [haurilet2019spase], hanet
[Choi_2020_CVPR], danet [fu2019dual], and dranet [fu2020scene].

We propose a Classroom Slide Narration System (csns) for the proposed slide
segmentation for helping vi students for reading and understanding the slide’s
content. The proposed csns creates audio corresponding to the segmented
region’s content in reading order. We use four existing recognizers —ocr
[smith2007overview], equation descriptor [mondal2019textual], table recognizer
[sachin2020table], and figure classifier [jobin2019docfigure] to recognize
content corresponds to segmented region. The proposed csns system has two
modes to present the slide’s content in audio format — (i) Interactive Mode:
the user can select a specific region on the slide for reading the slide’s
content, and (ii) Non-interactive Mode: the system narrates all extracted
information in reading order. We evaluate the effectiveness of the developed
system on WiSe dataset [haurilet2019wise] and compare the user experiences
with the existing assistive systems aat [wu2017automatic] and Tesseract ocr
[smith2007overview]. The proposed system attains a better user experience than
the existing assistive systems.

The contributions of this work are as follows:

  * •

We proposed a leanet segmentation architecture to efficiently segment the
classroom slide images with utilizing the location encoding.

  * •

We present a csns as use of slide segmentation to narrate slide’s content in
reading order. It helps the vi students to read and understand the slide’s
content.

  * •

Experiments on publicly available benchmark datasets WiSe and SPaSe establish
the effectiveness of the proposed leanet and csns over existing slide
segmentation and assistive techniques.

##  2 Related Work

##### Visual Assistance System:

According to the World Health Organization (who), at least $2.2$ billion
people have a vision impairment or blindness worldwide. The computer vision
community has undertaken significant efforts to deploy automated assistive
systems for such people. Some of the unique assistive systems based on
computer vision are — (i) voice111https://www.seeingwithsound.com
[auvray2007learning] vision technology is a sensory substitution system for
blind people which gives visual experience through live camera views by image-
to-sound renderings, (ii) Electro-Neural Vision System (envs)
[meers2004vision] provides a virtual perception of the three-dimensional
profile and color of the surroundings through electrical pulses, (iii) a
wearable travel aid for environment perception and navigation
[bai2019wearable], (iv) a smartphone-based image captioning for visually and
hearing impaired person [makav2019smartphone]. (v) Currency Recognition using
Mobile Phones [singh2014currency] (v) Facebook’s Automatic Alt-Text (aat)
[wu2017automatic] system is one of the most widely used assistive features,
which automatically recognizes a set of $97$ objects present in Facebook
photos, (vi) FingerReader [shilkrot2015fingerreader] is a wearable text
reading device to assist blind people and dyslexic readers. All these existing
assistive systems fail to properly narrate the classroom slides for a vi
student due to various logical regions present in the slides.

##### Document Image Segmentation:

Document images are dominated by textual regions and graphics regions.
Graphics include natural images, various types of graphs, plots, and tables.
Historically, the document image segmentation task is considered as dividing a
document into various regions. Existing techniques [amin2001page,
ha1995recursive] are either bottom-up or top-down. In the case of bottom-up
[amin2001page], individual words are first detected based on hand-crafted
features. Therefore, detected words are grouped to form lines and paragraphs.
Segmenting textual regions (like paragraphs) into lines and words helps to
find semantic regions present in the document in the case of top-down
[ha1995recursive]. These methods mostly use feature engineering and do not
necessitate any training data while applying clustering or thresholding.

With the recent advancement in deep convolutional neural networks (cnns),
several neural-based models [chen2015page, vo2016dense, renton2017handwritten,
wick2018fully, yang2017learning] have been proposed to segment document
images. Chen et al. [chen2015page] considered a convolutional auto-encoder to
learn features from cropped document image patches. Thereafter, those features
are used to train an svm classifier [cortes1995support] to segment document
patches. Vo et al. [vo2016dense] and Renton et al. [renton2017handwritten]
proposed lines detection algorithm in the handwritten document using fcn. In
[yang2017learning], Yang et al. proposed an end-to-end, multi-modal, fully
convolutional network for extracting semantic structures from document images.
Inspired by document image segmentation and natural image segmentation using
deep cnns, Haurilet _et al._ [haurilet2019wise, haurilet2019spase] explore
DeepLab [chen2017deeplab] architecture on classroom slide segmentation task.
More recently, danet [fu2019dual], hanet [Choi_2020_CVPR], and dranet
[fu2020scene] capture long-range dependency to improve performance by
extending self-attention mechanism. Our proposed leanet is strongly influenced
by danet [fu2019dual], hanet [Choi_2020_CVPR], and dranet [fu2020scene].

##  3 Classroom Slide Narration System

Classroom Slide Narration System (csns) is developed as a use case of slide
segmentation to help vi students to read and understand slide’s content
without human assistance. The proposed csns consists of three modules — (i)
identification of logical regions present in the slide (i.e, slide
segmentation), (ii) extraction of meaningful information from each segmented
region, and (iii) create an audio narration of extracted information to help
vi students read the slides in a proper order. We discuss each of these
modules in detail.

###  3.1 Slide Segmentation

We propose a Classroom Slide Segmentation Network, called leanet, to segment
classroom slide images accurately. The proposed network aims to learn more
discriminating features for semantic associations through the location
encoding attention mechanism. Fig. 3 shows the proposed network with its three
modules — (i) an attention module that utilizes the location encoding of
logical regions of slide image, (ii) a multi-scale feature extraction module
to extract multi-scale features through an Atrous Spatial Pyramid Pooling
(aspp) layer [chen2017deeplab], and (iii) feature concatenation module which
combines low-level and high-level features with the guidance of encoding
location information. Finally, a simple decoder network predicts the high-
resolution, multi-label, dense segmentation output. We follow the existing
works [chen2017deeplab, zhao2017pyramid, fu2020scene] and employ a pre-trained
residual network (e.g., ResNet-$101$) with dilated strategy [zhang2018context]
as the backbone. Note that, we remove down-sampling operations and employ
dilated convolutions in the last two ResNet blocks, thus enlarging the final
feature map size to $1/8$ of the input image. It retains more detailed
features without adding extra parameters. The features from the dilated
residual network are fed into the aspp layer to extract multi-scale features.
While in the attention module, we build a relationship between any two pixels
with and without location encoding, and this results in an attention map
scaled with location encoding. Low-level and high-level features are
concatenated in the feature concatenation module to enrich the object
representation. Finally, a decoder takes enrich features and produces dense
segmentation output.

Figure 3: Illustrate the architecture of the proposed leanet for classroom
slide segmentation. The network consists of three modules (i) attention module
(upper dotted region), (ii) multi-scale feature extraction module (Atrous
Spatial Pyramid Pooling layer, lower dotted region), (iii) feature
concatenation module. Here, $\oplus$ and $\otimes$ represent the element-wise
sum and multiplication of features, respectively.

####  3.1.1 Attention Module

The previous attempts [chen2017deeplab, zhao2017pyramid] suggest that the
local features generated by traditional fully Convolutional Networks (fcns)
could not capture rich context, leading to misclassification of objects and
stuff. While Choi _et al._ [Choi_2020_CVPR] discussed that urban-scene images
have common structural priors depending on a spatial position. The authors
also discuss using height-wise contextual information to estimate channels
weights during pixel-level classification for urban-scene segmentation. Fu _et
al._ [fu2019dual] also discuss the self-attention mechanism to capture the
spatial dependencies between any two positions of the feature maps. Inspired
by these works [Choi_2020_CVPR, fu2019dual], we introduce an attention module
that utilizes location encoding to enhance the contextual information and
relative position to the local feature. It enriches the contextual
relationship between local features and location encoding. The attention
module selectively encodes a broader range of contextual information and
location into each pixel according to semantic-correlated relations, thus
enhancing discrimination power for dense predictions.

As illustrated in Fig. 3 (inside the dotted box), we use a convolution layer
to obtain features of dimension reduction $\mathbf{A}\in$ $\mathbb{R}^{C\times
H\times W}$. We first feed it into a convolution layer to generate two new
feature maps $\mathbf{B}$ and $\mathbf{C}$, respectively, where
$\\{\mathbf{B},\mathbf{C}\\}\in\mathbb{R}^{C\times H\times W}$. We add the
location encoding $\mathbf{L}$ (refer sec 3.1.2) to the $\mathbf{B}$. Then we
reshape them to $\mathbb{R}^{C\times N}$, where $N=H\times W$ is the number of
pixels. After that, we perform a matrix multiplication between the transpose
of $\mathbf{C}$ and $\mathbf{B}+\mathbf{L}$, and apply a soft-max layer to
calculate the spatial attention map $\mathbf{S}\in\mathbb{R}^{N\times N}$ :

$s_{ji}=\frac{\exp\left((B_{i}+L_{i})\cdot
C_{j}\right)}{\sum_{i=1}^{N}\exp\left((B_{i}+L_{i})\cdot C_{j}\right)},$ (1)

where $s_{ji}$ measures the $i^{th}$ position’s impact on $j^{th}$ position.
The more similar feature representations of the two position contributes to
greater correlation between them. Meanwhile, we feed feature $\mathbf{A}$ into
a convolution layer to generate a new feature map
$\mathbf{D}\in\mathbb{R}^{C\times H\times W}$ and reshape it to
$\mathbb{R}^{C\times N}$. Then we perform a matrix multiplication between
$\mathbf{D}$ and the transpose of $\mathbf{S}$ and reshape the result to
$\mathbb{R}^{C\times H\times W}$. Finally, we multiply it by a scale parameter
$\alpha$ and perform a element-wise sum operation with the features
$\mathbf{A}$ to obtain the final output $\mathbf{E}\in\mathbb{R}^{C\times
H\times W}$ as follows:

$E_{j}=\alpha\sum_{i=1}^{N}\left(s_{ji}D_{i}\right)+(1-\alpha)\times A_{j},$
(2)

where $\alpha$ is initialized as 0 and leaned gradually.

####  3.1.2 Location Encoding

We use the sinusoidal position encoding [vaswani2017attention] as $\mathbf{L}$
matrix. For injecting positional encoding, we follow the strategy proposed in
Transformer [vaswani2017attention, Choi_2020_CVPR]. In our approach, we give
importance to both height and width location encoding. The dimension of the
positional encoding is the same as the channel dimension $C$ of the
intermediate feature map that combines with $\mathbf{L}$. The positional
encoding is defined as

$\displaystyle PE_{v}(v,2i)$ $\displaystyle=\sin\left(v/100^{2i/C}\right);\ \
PE_{v}(v,2i+1)$ $\displaystyle=\cos\left(v/100^{2i/C}\right),$ $\displaystyle
PE_{h}(h,2i)$ $\displaystyle=\sin\left(h/100^{2i/C}\right);\ \ PE_{h}(h,2i+1)$
$\displaystyle=\cos\left(h/100^{2i/C}\right),$

where $v$ and $h$ denote the vertical and horizontal location index in the
entire image ranging from $0$ to ${H/r}-1$ and $0$ to ${W/r}-1$, and $i$ is
the channel location index ranging from $0$ to ${C}-1$. The $r$ is called the
reduction ratio, which determines the frequency of $\sin$ or $\cos$ wave.
Here, we apply $\sin$ and $\cos$ location encoding for intermediate layers of
the features. The new representation $\mathbf{L}$ incorporating positional
encoding is formulated as

$\mathbf{L}=\beta PE_{h}\oplus(1-\beta)PE_{v},$

where $\oplus$ is an element-wise sum and $\beta$ is initialized as $0$ and
learned gradually. Adding both the horizontal and vertical position encoding
enables the attention module to learn the location information in both
direction. The $\beta$ value is learned based on the importance of width and
height locations. Figure 4 shows the location encoding value of a layer in the
$\mathbf{L}$ matrix. Finally, we add random jitters to the location encoding
$PE_{v}$ and $PE_{h}$ for the better generalization of location encoding. The
height and width of the location encoding matrix $\mathbf{L}$ adjusted to the
dimension of the feature vector.

Figure 4: Presents visualization of various location encoding with frequency
$2$, $1$, and $0.5$ and the $\beta$ value $0$, $0.5$, and $1$.

###  3.2 Information Extraction

We group the classroom slide image into four logical regions - text, figure,
equation, and table and explore existing recognizers to recognize the content
of these four regions.

##### OCR:

Optical Character Recognition (ocr) module is used to recognize the content of
text regions present in the slide image. The logical regions such as heading,
paragraph, and enumeration are grouped into a text region. The curved or
skewed text does not commonly appear in the slide images. The only
complexities are text with various fonts and colors. Tesseract ocr
[smith2007overview] recognizes text with various fonts and colors of the slide
images.

##### Figure Classification Module:

Most figures appearing on the slides are belongs to document figures such as
different types of graphs, sketches, block diagram , maps, etc. The WiSe
dataset has limited figure labels include Diagram, Legend, Logos, Maps,
Realistic, and other images. To overcome this drawback, we use ResNet
[he2016deep] trained on DocFigure [jobin2019docfigure] to recognize the
category label of each figure present in slides. In addition to it, we further
classify the realistic image using ResNet [he2016deep] trained on ImageNet
[krizhevsky2012imagenet].

##### Equation Description:

The equation frequently occurs in the classroom slides. Describing the
mathematical equation to a vi student is a challenging task due to the complex
mathematical notations. Various approaches [mondal2019textual,
medjkoune2017combining] exist to describe equations. We follow med model
[mondal2019textual] for describing equation in natural language to interpret
the logical meaning of equation to vi students.

##### Table Structure Recognition:

Recognition of the table present in the slide includes detecting cells,
identifying cells’ associations, and recognizing its content. We use
TabStruct-Net [sachin2020table] to recognize the physical structure of the
table and then Tesseract ocr [smith2007overview] to recognize content.

Figure 5: Illustrate the information extraction pipeline for the proposed csns
solution from the classroom slide image. The four modules extracts slide
information and save as individual Json object. The x, y, w, and h represents
the bounding rectangle of the region.

The classroom slide image information extraction pipeline is visualy described
in the figure 5. All the information from the four module are combined and
save to a json file. All these processes are done on the server and then send
the json file to a mobile app, and the vi student understand the slide’s
content by interacting with the app.

###  3.3 Application

We develop a mobile application for vi student to read and understand the
classroom slide’s content inspired by the study on an empirical investigation
into difficulties experienced by vi internet users [murphy2008empirical]. For
this purpose, we place a camera in the classroom, which is connected to the
server for capturing slide images. Each blind student has a mobile app, which
is connected to the server through a WiFi connection. Using the mobile
application (CAPTURE button), the student can request the server to get
information on the current slide. The server captures the current slide image
through the camera, extracts the information as json file, and sends it back
to the mobile application. Figure 6(b) shows a screenshot of the developed
mobile app with one example. The app has two modes — (i) interactive and (ii)
non-interactive. In the interactive mode, the user can navigate various
logical regions of slides by themselves. The app shows the captured slide
image with all extracted information. When the user touches one particular
area, it automatically plays the audio sound using android tts library of that
region’s content along with category label. In the non-interactive mode, the
app automatically plays the audio corresponding to the slide’s content. In
this case, the screen contains the READ ALL button; when the user touches this
particular area, the app automatically plays the audio sound of the full
slide’s content. These features are demonstrated in the demo
video222https://youtu.be/PnPYrA8ykF0

##  4 Experiments

In this section, we first describe the implementation details of proposed
leanet architecture. Then, we demonstrate the effectiveness of the leanet
architecture by extensive quantitative analysis including ablation studies. We
evaluate the approach with two different classroom slide segmentation datasets
including WiSe [haurilet2019wise] and SpaSe [haurilet2019spase]. For all the
quantitative experiments, we measure the segmentation performance in terms of
mean Intersection over Union (mIoU) and pixel accuracy matrix. Finally, we
demonstrate the csns and compare with Facebook’s Automatic Alt-Text (aat)
[wu2017automatic] and Tesseract ocr [smith2007overview].

###  4.1 Experimental Setup

We use the residual network (e.g., ResNet-101) pre-trained on ImageNet
[krizhevsky2012imagenet] as the backbone network for all the experiments. We
replace a single $7\times 7$ convolution with three of $3\times 3$
convolutions in the first layer of ResNet-101. We also adopt an auxiliary
cross-entropy loss in the intermediate feature map. Since for the multi-label
segmentation task we need to predict multiple classes per pixel, we replace
the softmax output layer in the previous deep models with a sigmoid activation
function and train these models using binary cross entropy loss. We train the
network using sgd optimizer with an initial learning rate of $2e-2$ and
momentum of $0.9$. We set the weight decays of $5e-4$ and $1e-4$ for backbone
networks and segmentation networks, respectively. The learning rate schedule
follows the polynomial learning rate policy [liu2015parsenet]. The initial
learning rate is multiplied by$(1-\frac{iteration}{max\ iteration})^{0.9}$. We
use typical data augmentations such as random scaling in the range of
$[0.5,2]$, Gaussian blur, color jittering, and random cropping to avoid over-
fitting. We set crop size and batch size of $540\times 720$ and $2$,
respectively. We use mean Intersection over Union (mIoU) [chen2017deeplab] and
Pixel Accuracy (pa) [haurilet2019wise, haurilet2019spase] for evaluation
purpose.

###  4.2 Dataset

##### WiSe:

The WiSe [haurilet2019wise] dataset consists of $1300$ slide images. It is
divided into a training set, validation set, and test set consisting of $900$,
$100$, and $300$ slide images annotated with $25$ overlapping region
categories.

##### SPaSe:

The SpaSe [haurilet2019spase] dataset contains dense, pixel-wise annotations
of $25$ classes for $2000$ slide images [haurilet2019spase]. It is split into
training, validation, and test sets containing $1400$, $100$, and $500$
images, respectively.

###  4.3 Ablation Study

In the ablation study, we use the ResNet-101 backbone with an output stride of
$16$ and evaluate on the WiSe dataset. First, we examine the best feature
vector for adding location encoding in the leanet architecture. Based on Fig.
3 of leanet architecture, the location encoding can be added to any of the
feature vectors including, $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$,
$\mathbf{D}$. We empirically find out the best feature for adding location
encoding. In addition to it, we also conduct experiments by changing the
frequency of the $\sin$ wave encoding in the $\mathbf{L}$ matrix. Finally, we
also compare with CoordConv [liu2018intriguing] approach.

| Features | Frequency | mIoU
---|---|---
$\mathbf{A}$ |  $\mathbf{B}$ |  $\mathbf{C}$ |  $\mathbf{D}$
|  |  |  | 1 | 51.07
✓ |  |  |  | 1 | 51.44
|  ✓ |  |  | 1 | 52.26
|  |  ✓ |  | 1 | 52.12
|  |  |  ✓ | 1 | 50.04
|  ✓ |  |  | 2 | 51.14
|  ✓ |  |  | 0.5 | 50.32
$\mathbf{A}$ \+ CoordConv [liu2018intriguing](width+ height) | 51.12
(a)
| $\beta$ | mIoU
---|---
0 | 51.73
1 | 50.25
0.5 | 51.83
learning | 52.26
(b)
Table 1: It shows ablation studies and the impact of hyper-parameters on a
validation set of the WiSe dataset. Table (a) shows the result of segmentation
by adding location encoding to the features $\mathbf{A}$, $\mathbf{B}$,
$\mathbf{C}$, and $\mathbf{D}$. Table (b) shows the effect of manual $\beta$
values and as a learnable parameter.

Table 1 shows the ablation studies conducted in the proposed network
architecture by changing the various hyper-parameters. The Table 1(a) shows
that the location encoding on the feature vector outperformed the traditional
CoordConv [liu2018intriguing] approach. The experiment also shows that adding
location encoding to the intermediate feature $\mathbf{B}$ and $\mathbf{C}$ is
comparatively better than adding it to $\mathbf{A}$ and $\mathbf{D}$. The best
frequency of $\sin$ wave in the $\mathbf{L}$ is found as 1. While Table 1 (b)
shows the learnable $\beta$ value outperforms the manually assigned $\beta$
value.

###  4.4 Comparison with State-of-the-Art Techniques

Comparison results of the proposed network with state-of-the-art techniques
are presented in Table 2. From the table, we observe that none of the
attention mechanism (e.g., hanet [Choi_2020_CVPR], danet [fu2019dual], and
dranet [fu2020scene]) is better than Haurilet _et al._ [haurilet2019spase] on
SPaSe dataset. While in case of WiSe dataset, all attention architectures
hanet [Choi_2020_CVPR], danet [fu2019dual], and dranet [fu2020scene] are
better than than Haurilet _et al._ [haurilet2019wise]. dranet [fu2020scene]
obtains the best performance ($35.77$% mean IoU and $78.64$% PA) among all
existing techniques on SPaSe dataset. While danet [fu2019dual] obtains the
best performance ($44.85$% mean IoU and $88.80$% PA) among all existing
techniques on WiSe dataset. The proposed network obtains the best results
($46.74$% mean IoU and $89.80$% PA) on Wise and ($36.17$% mean IoU and
$79.11$% PA) on SPaSe dataset as compared to the existing techniques. The Fig.
7 shows the qualitative performance of the proposed approach with other
attention architecture.

Methods | Datasets
---|---
| SPaSe | WiSe
| mean IoU | PA | mean IoU | PA
Haurilet _et al._ [haurilet2019spase] | 35.80 | 77.40 | - | -
Haurilet _et al._ [haurilet2019wise] | - | - | 37.20 | 88.50
HANet [Choi_2020_CVPR] | 32.37 | 76.54 | 39.35 | 88.72
DANet [fu2019dual] | 33.12 | 77.39 | 44.85 | 88.80
RANet [fu2020scene] | 35.77 | 78.64 | 43.31 | 88.77
Ours | 36.17 | 79.11 | 46.74 | 89.80
Table 2: illustrates the comparison of the proposed method with state-of-the-
art techniques on benchmark datasets WiSe and SPaSe datasets.

(a) Shows the user ranking for various approaches. I and N indicates
interactive and non-interactive modes, respectively.

(b) Illustrates the user interface of the developed mobile application. The
mobile screen displays the camera-captured slide image. Various colors
highlight the slide’s logical regions for better visualization. The Read All
button plays generated audio sound corresponding to the slide’s content.

###  4.5 Narration System Evaluation

We establish the effectiveness of the developed slide narration system by
comparing the performance with Facebook’s Automatic Alt-Text (aat)
[wu2017automatic] and Tesseract ocr [smith2007overview]. We use $300$ test
images of the WiSe dataset for evaluating the system. We select $30$
volunteers who can read and write English. We provide list of $300$ images
along with narrating audio files produced by various techniques: the proposed
slide narration system, Facebook’s Automatic Alt-Text, and Tesseract. Each
volunteer can randomly choose $5$ slide images from the given list and
evaluate corresponding audio outputs generated by various techniques. The
volunteer give a grade (value in the range $[0,\ 10]$) to the generated audio
files by various techniques by comparing these audio files with the original
slide content. We collect all these grades and plot them as a bar chart (shown
in Figure 6(a)). The user gives the best grade for the audio file
corresponding to the ground truth segmentation of the slide. However, the
proposed csns also performed comparatively better as the ground truth
segmentation.

Figure 7: The comparison of the output images of proposed leannet architecture
and other attention architectures. To visualize the overlapped label region,
we use separate image for each label.

##  5 Summary

This paper presents a classroom slide segmentation network (leanet) to segment
classroom slide images more accurately. The attention module in the proposed
network enriches the learned contextual features by utilizing the location
encoding of the logical region (height and width of a region). We develop a
Classroom Slide Narration System as an application of classroom slide
segmentation for helping vi students to read and understand the slide’s
content. Experiments on benchmark datasets WiSe and SpaSe conclude that the
proposed leanet achieves the best slide segmentation results as compared to
the state-of-the-art methods. We also demonstrate the effectiveness of our
developed Classroom Slide Narration System over existing assistive tools in
terms of user experiences.


# Revisiting  the Divergent Multipole  Expansion of Atom–Surface Interactions
with an Emphasis on $\bm{\alpha}$–Quartz, Hydrogen and Positronium

Ulrich D. Jentschura  Department of Physics and LAMOR, Missouri University of
Science and Technology, Rolla, Missouri 65409, USA

###### Abstract

We revisit the derivation of multipole contributions to the atom-wall
interaction previously presented in [G. Łach et al., Phys. Rev. A 81, 052507
(2010)]. A careful reconsideration of the angular-momentum decomposition of
the second-, third- and fourth-rank tensors composed of the derivatives of the
electric-field modes leads to a modification for the results for the
quadrupole, octupole and hexadecupole contributions to the atom-wall
interaction. Asymptotic results are given for the asymptotic long-range forms
of the multipole terms, in both the short-range and long-range limits.
Calculations are carried out for hydrogen and positronium in contact with
$\alpha$-quartz; a reanalysis of analytic models of the dielectric function of
$\alpha$-quartz is performed. Analytic results are provided for the multipole
polarizabilities of hydrogen and positronium. The quadrupole correction is
shown to be numerically significant for the atom-surface interaction of
positronium, both for a perfect conductor as well as for interactions with
$\alpha$-quartz. The expansion into multipoles is shown to constitute a
divergent, asymptotic series.

###### Contents

  1. I Introduction
  2. II Multipole Interactions
    1. II.1 General Formulation
    2. II.2 Perfect Conductor
  3. III Dielectric Surface
    1. III.1 Dipole Term
    2. III.2 Quadrupole Term
    3. III.3 Octupole Term
    4. III.4 Hexadecupole Term
    5. III.5 Some General Short–Range Results
  4. IV Interactions for Hydrogen and Positronium
    1. IV.1 $\maybebm{2^{\ell}}$–Pole Polarizabilities ($\maybebm{\ell=1,2,3,4}$)
    2. IV.2 Multipole Corrections
  5. V Conclusions

##  I Introduction

Multipole corrections to the atom-surface interactions have been considered in
Refs. Łach  _et al._ (2010, ), with an application to helium interacting with
$\alpha$-quartz. The theory of atom-surface interactions is well known, for
both perfect conductors as well as realistic dielectric materials (see Refs.
Lifshitz (1955); Landau and Lifshitz (1960); Dzyaloshinskii  _et al._ (1961a,
b); Parsegian (1974); Spruch and Tikochinsky (1993); Tikochinsky and Spruch
(1993a, b), pp. 261–263 of Ref. Milonni (1994), and Ref. Łach  _et al._
(2010)). In Ref. Łach  _et al._ (2010, ), the quadrupole, octupole and
hexadecupole contributions to the atom-wall interaction were evaluated. Here,
we update the analysis presented in Ref. Łach  _et al._ (2010) with an
emphasis on the isolation of angular-momentum components of the quantized
electric field near the wall.

Let us discuss the basis for the update, in terms of the initial expressions
used for the evaluations, both for the cases discussed in Ref. Łach  _et al._
(2010) as well as for the modifications discussed here. Specifically,
according to Eq. (22) of Ref. Łach  _et al._ (2010) (in accordance with Ref.
Milonni (1994)), the dipole part of the atom-wall interaction can be expressed
as follows,

$\mathcal{E}_{\ell=1}=-\frac{1}{2}\sum_{\vec{k}\lambda}\sum_{i}\alpha_{\ell=1}(\omega)\big{|}E^{i}_{\vec{k}\lambda}(\vec{r})\big{|}^{2}\,,$
(1)

where $\vec{E}_{\vec{k}\lambda}(\vec{r})$ is the (in general complex) mode
function of the electric field near the wall, $\vec{k}$ is the wave vector,
and $\lambda$ is the polarization. The index $i$ enumerates the Cartesian
components. The dipole polarizability of the atom is
$\alpha_{\ell=1}(\omega)=\alpha_{1}(\omega)$, and it is evaluated at the
angular frequency $\omega=\omega_{k}=c\,|\vec{k}|$, where $c$ is the speed of
light. The mode function $\vec{E}_{\vec{k}\lambda}(z)$ of the electric field
is related to the mode function $\vec{A}_{\vec{k}\lambda}(z)$ of the vector
potential by the relation

$\vec{E}_{\vec{k}\lambda}(\vec{r})=\mathrm{i}\omega_{k}\,\vec{A}_{\vec{k}\lambda}(\vec{r})\,,$
(2)

where the mode functions in Eqs. (18), (19) and (20) of Ref. Łach  _et al._
(2010). The formalism for the treatment of the dipole contribution is well
established.

The starting point for the analysis of the quadrupole, octupole, and
hexadecupole corrections to the atom-wall interaction has been given by Eq.
(8) of Ref. Łach  _et al._ (2010) as follows,

$\mathcal{E}_{\ell=2}=-\frac{1}{12}\,\sum_{\vec{k}\lambda}\sum_{ij}\alpha_{2}(\omega)\,\big{|}\nabla^{j}E^{i}_{\vec{k}\lambda}(\vec{r})\big{|}^{2}\,,$
(3)

where $\nabla^{j}\equiv\partial/\partial x^{j}$ and the indices $i$ and $j$
stand for the Cartesian components, and $\alpha_{2}$ is the quadrupole
polarizability of the atom. We reserve the index $i$ for the component of the
electric field. The octupole energy shift has been analyzed in Ref. Łach  _et
al._ (2010) based on the expression

$\mathcal{E}_{\ell=3}=-\frac{1}{180}\sum_{\vec{k}\lambda}\sum_{ijk}\alpha_{3}(\omega)\,\big{|}\nabla^{j}\nabla^{k}E^{i}_{\vec{k}\lambda}(\vec{r})\big{|}^{2}\,,$
(4)

where $\alpha_{3}=\alpha_{\ell=3}$ is the octupole polarizability of the atom.
The expressions (3) and (4) need to be substantiated by the specification of
the angular-momentum component relevant to the analysis. Namely, the virtual
atomic transitions contributing to the quadrupole, octupole and hexadecupole
polarizabilities of the atom connect the atomic ground state to virtual states
with angular momenta $\ell=2$ and $\ell=3$. However, when one calculates sums
such as

$\sum_{ij}\big{|}\nabla^{j}E^{i}_{\vec{k}\lambda}(\vec{r})\big{|}^{2}\,,$ (5)

without separating the ($\ell=2$)-component of the second-rank tensor
$\nabla^{j}E^{i}$ first, then one sums over the ($\ell=0$)- and
($\ell=1$)-components of the second-rank tensor $T^{ji}=\nabla^{j}E^{i}$ as
well. While the proper isolation of the angular-momentum components does not
change the functional form of the results reported in Ref. Łach  _et al._
(2010), some prefactors receive modifications. For the quadrupole term, the
proper isolation of the ($\ell=2$)-component leads to the modified expression

$\mathcal{E}_{\ell=2}=-\frac{1}{12}\,\sum_{\vec{k}\lambda}\sum_{ij}\alpha_{2}(\omega)\,\big{|}\left.\\{\nabla^{j}E^{i}_{\vec{k}\lambda}(\vec{r})\\}\right|_{\ell=2}\big{|}^{2}\,.$
(6)

where the ($\ell=2$)-component of a tensor is given by the traceless,
symmetric component as follows,

$\left.T^{ij}\right|_{\ell=2}=\frac{1}{2}\,\left(T^{ij}+T^{ji}\right)-\frac{1}{3}\,\delta^{ij}\,{\text{Tr}}(T)\,,$
(7)

and ${\text{Tr}}(T)=\sum_{i}T^{ii}$ is the trace. The octupole term is
substantiated by the expression

$\mathcal{E}_{3}=-\frac{1}{180}\sum_{\vec{k}\lambda}\sum_{ijk}\alpha_{3}(\omega)\,\big{|}\left.\\{\nabla^{j}\nabla^{k}E^{i}_{\vec{k}\lambda}(\vec{r})\\}\right|_{\ell=3}\big{|}^{2}\,,$
(8)

where the isolation of the octupole ($\ell=3$)-component of a third-rank
tensor has recently been discussed by Itin and Reches in Ref. Itin and Reches
(2022). The isolation of the ($\ell=3$)-component of a third-rank tensor is
not completely trivial. The modified expressions given in Eqs. (6) and (8)
have phenomenological consequences which are discussed in the following.

Finally, the hexadecupole energy shift is calculated as follows,

$\mathcal{E}_{4}=-\frac{1}{5040}\sum_{\vec{k}\lambda}\sum_{ijk}\alpha_{4}(\omega)\,\big{|}\left.\\{\nabla^{j}\nabla^{k}\nabla^{l}E^{i}_{\vec{k}\lambda}(\vec{r})\\}\right|_{\ell=4}\big{|}^{2}\,,$
(9)

The isolation of the ($\ell=4$)-component from a Cartesian tensor is discussed
in Ref. Andrews and Ghoul (1982).

We use Systéme International (SI mksA) units in the following, before we
switch to atomic units in Sec. IV. The basis for the treatment of multipole
interactions is discussed in Sec. II, the case of a perfect conductor in Sec.
II.2, and dielectric materials in Sec. III. Interactions with hydrogen and
positronium atoms are discussed in Sec. IV. Conclusions are drawn in Sec. V.

##  II Multipole Interactions

###  II.1 General Formulation

In order to put things into perspective, and in terms of some orientation, we
briefly review the concept of the multipole oscillator strength, inspired by
Chap. 5 of Ref. Jentschura and Adkins (2022). The $2^{\ell}$-pole
polarizability is written as

$\alpha_{\ell}(\omega)=\frac{1}{2\ell+1}\sum_{m}\left<1S\left|Q_{\ell
m}\left(\frac{1}{H-E_{1S}+\hbar\omega}\right.\right.\right.\\\
+\left.\left.\left.\frac{1}{H-E_{1S}-\hbar\omega}\right)Q^{*}_{\ell
m}\right|1S\right>\,,$ (10)

where the $Q_{\ell m}$ tensor is given as

$Q_{\ell m}=\sum_{a}e\,\sqrt{\frac{4\pi}{2\ell+1}}\,r_{a}\,Y_{\ell
m}(\hat{r}_{a})\,,$ (11)

and the sum over $a$ runs over the atomic electrons. The spherical harmonic is
$Y_{\ell m}$, and the shorthand notation $\hat{r}_{a}$ summarizes the polar
and azimuth angles $\theta_{a}$ and $\varphi_{a}$ which define $\hat{r}_{a}$
uniquely, and vice versa. Here,

$\left.T^{ij}\right|_{\ell=2}=\frac{1}{2}\,\left(T^{ij}+T^{ji}\right)-\frac{1}{3}\,\delta^{ij}\,\mathrm{trc}(T)$
(12)

is the traceless quadrupole component of a general tensor $T^{ij}$. The
octupole energy shift ($\ell=3$) is evaluated according to Eq. (8),

$\mathcal{E}_{3}=-\frac{1}{180}\sum_{\vec{k}\lambda}\sum_{ijk}\alpha_{3}(\omega)\,\big{|}\left.\\{\nabla^{j}\nabla^{k}E^{i}_{\vec{k}\lambda}(\vec{r})\\}\right|_{\ell=3}\big{|}^{2}\,.$
(13)

The ($\ell=3$)-component can be extracted from a third-rank tensor $T^{ijk}$
as follows. One first decomposes $T^{ijk}$ into a totally symmetric part
$S^{ijk}$, a totally skew-symmetric part $A^{ijk}$, and a remainder term
$N^{ijk}$,

$\displaystyle S^{ijk}=$ $\displaystyle\;T^{(ijk)}$ (14a) $\displaystyle=$
$\displaystyle\;\frac{1}{6}\left(T^{ijk}+T^{jki}+T^{kij}+T^{jik}+T^{kji}+T^{ikj}\right)\,,$
$\displaystyle A^{ijk}=$ $\displaystyle\;T^{[ijk]}$ (14b) $\displaystyle=$
$\displaystyle\;\frac{1}{6}\left(T^{ijk}+T^{jki}+T^{kij}-T^{jik}-T^{kji}-T^{ikj}\right)\,,$
$\displaystyle N^{ijk}=$ $\displaystyle\;T^{[ijk]}-S^{ijk}-A^{ijk}\,.$ (14c)

The matrix $S^{ijk}$ contains both components with $\ell=1$ and $\ell=3$. One
then forms a vector $\alpha^{k}$ which ones promotes to a matrix $K^{ijk}$,

$\displaystyle\alpha^{k}=$
$\displaystyle\;\delta^{ij}\,S^{ijk}=\frac{1}{3}\,\left(\delta^{ij}\,T^{ijk}+\delta^{ij}\,T^{ikj}+\delta^{ij}\,T^{kij}\right)\,,$
$\displaystyle K^{ijk}=$
$\displaystyle\;\frac{1}{5}\,\left(\alpha^{i}\,\delta^{jk}+\alpha^{j}\,\delta^{ik}+\alpha^{k}\,\delta^{ij}\right)\,.$
(15)

Finally, the ($\ell=3$)-component which we need for our considerations is
obtained as

$\left.T^{ijk}\right|_{\ell=3}=S^{ijk}-K^{ijk}\,.$ (16)

For the hexadecupole energy shift, we need the ($\ell=4$)-component of the
Cartesian tensor

$T^{ijk\ell}=\nabla^{i}\nabla^{j}\nabla^{k}E^{\ell}_{\vec{k}\lambda}(\vec{r})\,.$
(17)

It can be calculated as follows Andrews and Ghoul (1982),

$\left.T^{ijk\ell}\right|_{\ell=4}=T^{(ijk\ell)}-\frac{1}{7}\left(\delta^{ij}T^{(\rho\rho
k\ell)}+\delta^{ik}T^{(\rho\rho j\ell)}\right.\\\
\left.+\delta^{i\ell}T^{(\rho\rho jk)}+\delta^{jk}T^{(\rho\rho
i\ell)}+\delta^{j\ell}T^{(\rho\rho ik)}+\delta^{k\ell}T^{(\rho\rho
ij)}\right)\\\
+\frac{1}{35}\left(\delta^{ij}\delta^{k\ell}+\delta^{ik}\delta^{j\ell}+\delta^{i\ell}\delta^{jk}\right)T^{(\rho\rho\sigma\sigma)}\,.$
(18)

Here, the Einstein summation is understood, with dummy indices ($\rho$ and
$\sigma$) being summed over. Furthermore, the symmetrization of a tensor is
defined as follows,

$T^{(ijk\ell)}=\frac{1}{24}\,\bigl{[}T^{ijk\ell}+T^{ij\ell
k}+T^{ikj\ell}+T^{ik\ell j}+T^{i\ell jk}\\\ +T^{i\ell
kj}+T^{jik\ell}+T^{ji\ell k}+T^{jki\ell}+T^{jk\ell i}+T^{j\ell ik}+T^{j\ell
ki}\\\ +T^{kij\ell}+T^{ki\ell j}+T^{kji\ell}+T^{kj\ell i}+T^{k\ell
ij}+T^{k\ell ji}+T^{\ell ijk}\\\ +T^{\ell ikj}+T^{\ell jik}+T^{\ell
jki}+T^{\ell kij}+T^{\ell kji}\bigr{]}$ (19)

For the case $\ell=4$, just as is the case for $\ell=3$, the isolation of the
component of highest angular momentum is a nontrivial exercise Andrews and
Ghoul (1982).

###  II.2 Perfect Conductor

Based on the formalism outlined in Secs. I and (II), and in Chap. 5 of Ref.
Jentschura and Adkins (2022), it is relatively straightforward to evaluate the
atom-wall interaction for a perfect conductor. We first recall the known
result for the atomic-dipole contribution, e.g., from Eq. (27) of Ref. Łach
_et al._ (2010),

$\mathcal{E}_{1}(z)=-\frac{\hbar}{16\pi^{2}\epsilon_{0}z^{3}}\int\limits_{0}^{\infty}\\!{\rm
d}\omega\,\alpha_{1}(\mathrm{i}\omega)\\\\[0.4878pt]
\times\left[1\\!+\\!\frac{2\omega z}{c}+2\left(\frac{\omega
z}{c}\right)^{2}\right]\\!\exp\left(-\frac{2\omega z}{c}\right)\,.$ (20)

The $(\ell=2)$-contribution to the energy shift is as follows,

$\mathcal{E}_{2}(z)=-\frac{\hbar}{16\pi^{2}\epsilon_{0}z^{5}}\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{2}(\mathrm{i}\,\omega)\,\exp\left(-\frac{2\omega
z}{c}\right)\\\\[0.4878pt] \times\left[\frac{3}{4}+\frac{3}{2}\,\frac{\omega
z}{c}+\frac{4}{3}\,\left(\frac{\omega
z}{c}\right)^{2}+\frac{2}{3}\,\left(\frac{\omega
z}{c}\right)^{3}+\frac{1}{6}\left(\frac{\omega z}{c}\right)^{4}\right]\,.$
(21)

This result constitutes a correction to a result previous derived in Eq. (32)
of Ref. Łach  _et al._ (2010). For the octupole term, we have

$\mathcal{E}_{3}(z)=-\frac{\hbar}{16\pi^{2}\epsilon_{0}z^{7}}\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{3}(\mathrm{i}\,\omega)\mathrm{e}^{-2\omega
z/c}\,\\\\[0.4878pt] \times\left[\frac{2}{3}+\frac{4}{3}\,\frac{\omega
z}{c}+\frac{92}{75}\,\left(\frac{\omega
z}{c}\right)^{2}+\frac{152}{225}\,\left(\frac{\omega
z}{c}\right)^{3}\right.\\\\[0.4878pt] \left.+\frac{6}{25}\,\left(\frac{\omega
z}{c}\right)^{3}+\frac{4}{75}\left(\frac{\omega
z}{c}\right)^{4}+\frac{4}{675}\left(\frac{\omega z}{c}\right)^{6}\right]\,.$
(22)

For large distances, the multipole polarizabilities are suppressed by higher
powers of $z$. Specifically, the $2^{\ell}$-pole contribution scales as
$z^{-2\ell+1}$ for $z\to 0$ and as $z^{-2\ell+2}$ for $z\to\infty$. The
general structure of the short-range asymptotic limit is given in Eq. (52),
which has the same functional dependence but a different prefactor than Eq.
(49) of Ref. Łach  _et al._ (2010). In the context of the current
investigations, it is useful to clarify that we understand by short range (see
also Ref. Łach  _et al._ (2010)) the distance regime $a_{0}\ll z\ll
a_{0}/\alpha$, where $a_{0}$ is the Bohr radius and $\alpha$ is the fine-
structure constant. As explained in Ref. Zaremba and Kohn (1976), the first
inequality $a_{0}\ll z$ should be taken with a grain of salt in this context;
the short-range expressions derived below are actually valid down to distance
regions of a few angstroms away from the surface Zaremba and Kohn (1976); Tao
and Rappe (2014)). From the point of view of physisorption, what we refer to
as the short-range regime rather constitutes a long-range distance Zaremba and
Kohn (1976); Tao and Rappe (2014)). By contrast, the long-range regime as
considered in the current investigation refers to atom-surface distances $z\gg
a_{0}/\alpha$.

Finally, on the basis of Eqs. (9) and (18), while employing, otherwise, the
formalism of Ref. Łach  _et al._ (2010), the hexadecupole energy shift is
obtained as follows,

$\mathcal{E}_{4}(z)=-\frac{\hbar}{16\pi^{2}\epsilon_{0}z^{9}}\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{4}(\mathrm{i}\,\omega)\mathrm{e}^{-2\omega
z/c}\,\\\\[0.4878pt] \left[\frac{5}{8}+\frac{5}{4}\,\frac{\omega
z}{c}+\frac{115}{98}\,\left(\frac{\omega
z}{c}\right)^{2}+\frac{100}{147}\,\left(\frac{\omega
z}{c}\right)^{3}\right.\\\\[0.4878pt] +\frac{79}{294}\,\left(\frac{\omega
z}{c}\right)^{4}+\frac{11}{147}\left(\frac{\omega
z}{c}\right)^{5}+\frac{32}{2205}\left(\frac{\omega
z}{c}\right)^{6}\\\\[0.4878pt] \left.+\frac{4}{2205}\left(\frac{\omega
z}{c}\right)^{7}+\frac{1}{8820}\left(\frac{\omega z}{c}\right)^{8}\right]\,.$
(23)

This concludes the discussion of interactions with a perfect conductor.

##  III Dielectric Surface

###  III.1 Dipole Term

In order to analyze the multipole contributions for a dielectric, one consults
the transverse electric (TE) and transverse magnetic (TM) modes given in Eqs.
(18), (19) and (20) of Ref. Łach  _et al._ (2010). In this case, TE stands for
incident waves whose electric field is transverse to the plane of incidence,
whereas TM stands for waves whose electric field is in the plane of incidence.
The calculation is described in detail in Ref. Łach  _et al._ (2010). For
given wave vector $\vec{k}$, one has two polarization vectors, one for the the
TE mode, and another one for the TM mode. With these ideas in mind, it is
relatively easy to rederive the following result, originally derived in Ref.
Lifshitz (1955) for a dipole polarizable particle in contact with a dielectric
surface,

$\displaystyle\mathcal{E}_{1}(z)=$
$\displaystyle\;-\frac{\hbar}{8\pi^{2}\epsilon_{0}\,c^{3}}\,\int_{0}^{\infty}\mathrm{d}\omega\,\omega^{3}\,\alpha_{1}(\mathrm{i}\omega)\,$
$\displaystyle\;\times\int_{1}^{\infty}\mathrm{d}p\,\exp\left(-\frac{2\,p\,\omega\,z}{c}\right)\,H(\epsilon(\mathrm{i}\omega),p)\,,$
(24)

where

$H(\epsilon,p)=\frac{\sqrt{\epsilon-1+p^{2}}-p}{\sqrt{\epsilon-1+p^{2}}+p}+(1-2p^{2})\frac{\sqrt{\epsilon-1+p^{2}}-p\,\epsilon}{\sqrt{\epsilon-1+p^{2}}+p\,\epsilon}.$
(25)

For the convenience of the reader, a remark might be in order. Namely, in
order to obtain Eq. (III.1) from Ref. Lifshitz (1955), one sets, in Ref.
Lifshitz (1955),
$\epsilon_{2}(\omega)=1+N_{V}\,\alpha_{1}(\omega)/\epsilon_{0}$, for material
number 2, one expands to first order in the volume density $N_{V}=N/V$, where
$N$ is the number of atoms and $V$ is the volume, and one applies the
principle of virtual work. This leads to the potential given in Eq. (III.1).
The fact that Ref. Lifshitz (1955) implicitly contains the atom-surface
interaction, was also pointed out in Ref. Zaremba and Kohn (1976). It is
useful to investigate the function

$K(\epsilon,z)=\int_{1}^{\infty}\mathrm{d}p\,\exp\left(-\frac{2\,p\,\omega\,z}{c}\right)\,H(\epsilon,p)\,.$
(26)

For $\epsilon\to\infty$, we have $H(\epsilon,p)\approx 2\,p^{2}$, and

$K(\epsilon,z)\mathop{\approx}^{\epsilon\to\infty}\frac{1}{2}\,\left(\frac{c}{z\omega}\right)^{3}\,\left[1+2\,\frac{z\omega}{c}+2\,\left(\frac{z\omega}{c}\right)^{2}\right]\,,$
(27)

which shows that the formula (34) is consistent with the result for the dipole
interaction with a perfect conductor. For $z\to 0$, we have, on the other
hand,

$K(\epsilon,z)=\frac{1}{2}\,\left(\frac{c}{z\,\omega}\right)^{3}\,\frac{\epsilon-1}{\epsilon+1}+{\mathcal{O}}(z^{-1})\,.$
(28)

Inserting (28) into (34), we have the short-range limit as

$\mathcal{E}_{1}(z)\mathop{=}^{z\to
0}-\frac{\hbar}{16\pi^{2}\,\epsilon_{0}\,z^{3}}\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{1}(\mathrm{i}\omega)\,\frac{\epsilon(\mathrm{i}\omega)-1}{\epsilon(\mathrm{i}\omega)+1}\,.$
(29)

The long-range limit is obtained as a function of the static polarizability
$\alpha(0)$ and the static dielectric function $\epsilon(0)$ as follows,

$\mathcal{E}_{1}(z)\mathop{=}^{z\to\infty}-\frac{3c\hbar\,\alpha_{1}(0)}{32\pi^{2}\,\epsilon_{0}\,z^{4}}\,\Psi_{1}(\epsilon(0))\,.$
(30)

Here, $\Psi_{1}(\epsilon)$ is a function which is normalized to unity in the
limit $\epsilon(0)\to\infty$ (limit of a perfect conductor) and which can
otherwise be expressed as follows,

$\Psi_{1}(\epsilon)=A_{1}(\epsilon)+B_{1}(\epsilon)\,\ln\left(\frac{\sqrt{\epsilon-1}-\sqrt{\epsilon}+1}{\sqrt{\epsilon-1}+\sqrt{\epsilon}-1}\right)\\\
+C_{1}(\epsilon)\,\ln\left(\frac{\sqrt{\epsilon+1}-\sqrt{\epsilon}+1}{\sqrt{\epsilon+1}+\sqrt{\epsilon}-1}\right)\,.$
(31)

The coefficients involve both fractional and integer powers of $\epsilon$,

$\displaystyle A_{1}(\epsilon)=$
$\displaystyle\;\frac{6\epsilon^{2}-3\epsilon^{3/2}-4\epsilon-3\sqrt{\epsilon}+10}{6(\epsilon-1)}\,,$
(32a) $\displaystyle B_{1}(\epsilon)=$
$\displaystyle\;\frac{2\epsilon^{3}-4\epsilon^{2}+3\epsilon+1}{2(\epsilon-1)^{3/2}}\,,$
(32b) $\displaystyle C_{1}(\epsilon)=$
$\displaystyle\;-\frac{\epsilon^{2}}{\sqrt{\epsilon+1}}\,.$ (32c)

The expansion about the perfect-conductor limit is

$\Psi_{1}(\epsilon)=1-\frac{5}{4\sqrt{\epsilon}}+\frac{22}{15\epsilon}+\mathcal{O}\left(\frac{1}{\epsilon^{3/2}}\right)\,,$
(33)

which is tantamount to an expansion about a large static dielectric function
$\epsilon\equiv\epsilon(0)$.

###  III.2 Quadrupole Term

The generalization to the quadrupole polarizability reads as follows,

$\mathcal{E}_{2}(z)=-\frac{\hbar}{16\pi^{2}\,\epsilon_{0}\,c^{5}}\int_{0}^{\infty}\mathrm{d}\omega\,\omega^{5}\,\alpha_{2}(\mathrm{i}\omega)\\\
\times\int_{1}^{\infty}\mathrm{d}p\,\mathrm{e}^{-2p\omega
z/c}\left(\frac{p^{2}}{2}-\frac{1}{3}\right)\,H(\epsilon(\mathrm{i}\omega),p)\,.$
(34)

In the perfect conductor limit ($\epsilon\to\infty$), this result is in
agreement with the previously derived result given in Eq. (21). The short-
range limit reads as follows,

$\mathcal{E}_{2}(z)\mathop{=}^{z\to
0}-\frac{3\hbar}{64\pi^{2}\,\epsilon_{0}\,z^{5}}\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{2}(\mathrm{i}\omega)\,\frac{\epsilon(\mathrm{i}\omega)-1}{\epsilon(\mathrm{i}\omega)+1}\,.$
(35)

The long-range limit is obtained as follows,

$\mathcal{E}_{2}(z)\mathop{=}^{z\to\infty}-\frac{35c\hbar\,\alpha_{2}(0)}{384\pi^{2}\,\epsilon_{0}\,z^{6}}\,\Psi_{2}(\epsilon(0))\,,$
(36)

where $\Psi_{2}$ has the same structure as $\Psi_{1}$,

$\Psi_{2}(\epsilon)=A_{2}(\epsilon)+B_{2}(\epsilon)\,\ln\left(\frac{\sqrt{\epsilon-1}-\sqrt{\epsilon}+1}{\sqrt{\epsilon-1}+\sqrt{\epsilon}-1}\right)\\\
+C_{2}(\epsilon)\,\ln\left(\frac{\sqrt{\epsilon+1}-\sqrt{\epsilon}+1}{\sqrt{\epsilon+1}+\sqrt{\epsilon}-1}\right)\,.$
(37)

The coefficients are given as follows,

$\displaystyle A_{2}(\epsilon)$
$\displaystyle=\frac{1}{140(\epsilon-1)^{2}}\left[-120\epsilon^{4}+60\epsilon^{7/2}+380\epsilon^{3}\right.$
$\displaystyle\left.-180\epsilon^{5/2}-364\epsilon^{2}+75\epsilon^{3/2}+348\epsilon+75\sqrt{\epsilon}-224\right]\,,$
(38a) $\displaystyle B_{2}(\epsilon)$
$\displaystyle=\frac{3}{28}\frac{8\epsilon^{5}-28\epsilon^{4}+40\epsilon^{3}-34\,\epsilon^{2}+7\epsilon+5}{(\epsilon-1)^{5/2}}\,,$
(38b) $\displaystyle C_{2}(\epsilon)$
$\displaystyle=\frac{3}{7}\,\frac{\epsilon^{2}(2\epsilon-1)}{\sqrt{\epsilon+1}}\,.$
(38c)

The expansion about the perfect-conductor limit is

$\Psi_{2}(\epsilon)=1-\frac{31}{28\sqrt{\epsilon}}+\frac{338}{245\epsilon}+\mathcal{O}\left(\frac{1}{\epsilon^{3/2}}\right)\,.$
(39)

###  III.3 Octupole Term

For the interaction with a dielectric surface, the octupole energy shift reads
as follows,

$\mathcal{E}_{3}(z)=-\frac{\hbar}{16\pi^{2}\,\epsilon_{0}\,c^{7}}\int_{0}^{\infty}\mathrm{d}\omega\,\omega^{7}\alpha_{3}(\mathrm{i}\omega)\int_{1}^{\infty}\mathrm{d}p\mathrm{e}^{-2p\omega
z/c}\\\
\times\left(\frac{8}{135}\,p^{4}-\frac{16}{225}\,p^{2}+\frac{4}{225}\right)\,H(\epsilon(\mathrm{i}\omega),p)\,.$
(40)

The calculation of the dielectric response function for imaginary input
frequencies is a nontrivial problem. The short-range limit is given by the
expression

$\mathcal{E}_{3}(z)\mathop{=}^{z\to
0}-\frac{\hbar}{24\pi^{2}\,\epsilon_{0}\,z^{7}}\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{3}(\mathrm{i}\omega)\,\frac{\epsilon(\mathrm{i}\omega)-1}{\epsilon(\mathrm{i}\omega)+1}\,.$
(41)

The long-range limit is

$\mathcal{E}_{3}(z)\mathop{=}^{z\to\infty}-\frac{77c\hbar\,\alpha_{3}(0)}{800\pi^{2}\,\epsilon_{0}\,z^{8}}\,\Psi_{3}(\epsilon(0))\,,$
(42)

where $\Psi_{3}$ reads as follows,

$\Psi_{3}(\epsilon)=A_{3}(\epsilon)+B_{3}(\epsilon)\,\ln\left(\frac{\sqrt{\epsilon-1}-\sqrt{\epsilon}+1}{\sqrt{\epsilon-1}+\sqrt{\epsilon}-1}\right)\\\
+C_{3}(\epsilon)\,\ln\left(\frac{\sqrt{\epsilon+1}-\sqrt{\epsilon}+1}{\sqrt{\epsilon+1}+\sqrt{\epsilon}-1}\right)\,.$
(43)

The coefficients are given as follows,

$\displaystyle A_{3}(\epsilon)$
$\displaystyle=\frac{1}{11088(\epsilon-1)^{3}}\biggl{[}-2520\epsilon^{11/2}-28560\epsilon^{5}$
$\displaystyle\;+13860\epsilon^{9/2}+66528\epsilon^{4}-31290\epsilon^{7/2}$
$\displaystyle\;-76336\epsilon^{3}+29120\epsilon^{5/2}-2415\epsilon^{3/2}$
$\displaystyle\;+62016\epsilon^{2}-48675\epsilon-6195\sqrt{\epsilon}+19888\biggr{]}\,,$
(44a) $\displaystyle B_{3}(\epsilon)$
$\displaystyle=\frac{5}{528(\epsilon-1)^{7/2}}\bigl{[}48\epsilon^{7}-288\epsilon^{6}+712\epsilon^{5}$
$\displaystyle\;-976\epsilon^{4}+890\epsilon^{3}-444\epsilon^{2}+5\epsilon+59\bigr{]}\,,$
(44b) $\displaystyle C_{3}(\epsilon)$
$\displaystyle=-\frac{5}{33}\,\frac{3\epsilon^{2}-6\epsilon+1}{\sqrt{\epsilon+1}}\,.$
(44c)

The expansion about the perfect-conductor limit is

$\Psi_{3}(\epsilon)=1-\frac{265}{264\sqrt{\epsilon}}+\frac{914}{693\epsilon}+\mathcal{O}\left(\frac{1}{\epsilon^{3/2}}\right)\,.$
(45)

###  III.4 Hexadecupole Term

For the interaction with a dielectric surface, the octupole energy shift reads
as follows,

$\mathcal{E}_{4}(z)=-\frac{\hbar}{16\pi^{2}\,\epsilon_{0}\,c^{9}}\int_{0}^{\infty}\mathrm{d}\omega\,\omega^{9}\alpha_{4}(\mathrm{i}\omega)\int_{1}^{\infty}\mathrm{d}p\mathrm{e}^{-2p\omega
z/c}\\\
\times\left(\frac{1}{252}\,p^{6}-\frac{1}{147}\,p^{4}+\frac{1}{294}\,p^{2}-\frac{1}{2205}\right)\,H(\epsilon(\mathrm{i}\omega),p)\,.$
(46)

The short-range limit is

$\mathcal{E}_{4}(z)\mathop{=}^{z\to
0}-\frac{5\hbar}{128\pi^{2}\,\epsilon_{0}\,z^{9}}\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{4}(\mathrm{i}\omega)\,\frac{\epsilon(\mathrm{i}\omega)-1}{\epsilon(\mathrm{i}\omega)+1}\,.$
(47)

The long-range limit is obtained as follows,

$\mathcal{E}_{4}(z)\mathop{=}^{z\to\infty}-\frac{1287c\hbar\,\alpha_{4}(0)}{12544\pi^{2}\,\epsilon_{0}\,z^{10}}\,\Psi_{4}(\epsilon(0))\,,$
(48)

where $\Psi_{4}$ reads as follows,

$\Psi_{4}(\epsilon)=A_{4}(\epsilon)+B_{4}(\epsilon)\,\ln\left(\frac{\sqrt{\epsilon-1}-\sqrt{\epsilon}+1}{\sqrt{\epsilon-1}+\sqrt{\epsilon}-1}\right)\\\
+C_{4}(\epsilon)\,\ln\left(\frac{\sqrt{\epsilon+1}-\sqrt{\epsilon}+1}{\sqrt{\epsilon+1}+\sqrt{\epsilon}-1}\right)\,.$
(49)

The coefficients are given as follows,

$\displaystyle A_{4}(\epsilon)$
$\displaystyle=\frac{1}{205920(\epsilon-1)^{4}}\biggl{[}-40320\epsilon^{8}+20160\epsilon^{15/2}$
$\displaystyle\;+369600\epsilon^{7}-181440\epsilon^{13/2}-1338624\epsilon^{6}$
$\displaystyle\;+641760\epsilon^{11/2}+2670816\epsilon^{5}-1252440\epsilon^{9/2}$
$\displaystyle\;-3242464\epsilon^{4}+1409520\epsilon^{7/2}+2767936\epsilon^{3}$
$\displaystyle\;-690270\epsilon^{5/2}-2110464\epsilon^{2}-62265\epsilon^{3/2}$
$\displaystyle\;+1300000\epsilon+118125\epsilon^{1/2}-376480\biggr{]}\,,$
(50a) $\displaystyle B_{4}(\epsilon)$
$\displaystyle=-\frac{7}{4576(\epsilon-1)^{9/2}}\bigl{[}128\epsilon^{9}-1216\epsilon^{8}+4608\epsilon^{7}$
$\displaystyle\;-9600\epsilon^{6}+12720\epsilon^{5}-12120\epsilon^{4}$
$\displaystyle\;+8080\epsilon^{3}-2580\epsilon^{2}-405\epsilon+375\bigr{]}\,,$
(50b) $\displaystyle C_{4}(\epsilon)$
$\displaystyle=\frac{7}{143}\,\frac{4\epsilon^{3}-18\epsilon^{2}+12\epsilon-1}{\sqrt{\epsilon+1}}\,.$
(50c)

The expansion about the perfect-conductor limit is

$\Psi_{4}(\epsilon)=1-\frac{1323}{1430\sqrt{\epsilon}}+\frac{18050}{14157\epsilon}+\mathcal{O}\left(\frac{1}{\epsilon^{3/2}}\right)\,.$
(51)

###  III.5 Some General Short–Range Results

In the short-range limit, a few simplifications and generalizations are
possible, especially in regard to two-body bound systems like hydrogen and
positronium. First, we may point out the generalization of the short-range
expressions given in Eqs. (29), (35), (41), and (47), to arbitrary multipole
orders. Indeed, the general result for the $2^{\ell}$-pole effect reads as
follows,

$\mathcal{E}_{\ell}(z)\mathop{=}^{z\to
0}-\frac{\hbar}{16\pi^{2}\,\epsilon_{0}\,z^{2\ell+1}}\,\\\
\times\frac{\ell+1}{2\ell}\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{\ell}(\mathrm{i}\omega)\,\frac{\epsilon(\mathrm{i}\omega)-1}{\epsilon(\mathrm{i}\omega)+1}\,.$
(52)

This result has the same functional form as Eq. (49) of Ref. Łach  _et al._
(2010), but a different, updated prefactor.

##  IV Interactions for Hydrogen and Positronium

###  IV.1 $\maybebm{2^{\ell}}$–Pole Polarizabilities
($\maybebm{\ell=1,2,3,4}$)

We aim to give an update on multipole corrections to atom-surface
interactions, beyond the discussion in Ref. Łach  _et al._ (2010). It turns
out that closed-form results for the multipole polarizabilities, up to the
hexadecupole term, can be given for hydrogen and positronium. In order to
ensure concise formulas, we now switch to atomic units (see Chap. 2 of Ref.
Jentschura and Adkins (2022)) with $a_{0}=1$, $E_{h}=1$,
$\epsilon_{0}=1/(4\pi)$, and $e=1$ (unit elementary charge). The atoms of
interest are hydrogen and positronium. With techniques outlined in Ref.
Gavrila and Costescu (1970) and in Chap. 4 of Ref. Jentschura and Adkins
(2022), it is possible to derive closed-form expressions for hydrogen and
positronium which we use in the nonrelativistic, and non-recoil limit. We
consider the specialization of the tensor (11) to a one-electron atom and work
with the matrix element

$Q_{\ell}^{({\rm H})}(\omega)=\frac{1}{2\ell+1}\sum_{m}\left<Q_{\ell
m}\frac{1}{H-E^{({\rm H})}_{1S}+\omega}Q^{*}_{\ell m}\right>_{1S}\,,$ (53)

where the superscript indicates the atom (H stands for hydrogen). From
$Q_{\ell}^{({\rm H})}(\omega)$, one obtains the $2^{\ell}$-pole polarizability
as follows,

$\alpha^{({\rm H})}_{\ell}(\omega)=Q_{\ell}^{({\rm
H})}(\omega)+Q_{\ell}^{({\rm H})}(-\omega)\,.$ (54)

The polarizability of positronium can be obtained from the hydrogen result as
follows,

$\alpha^{({\rm Ps})}_{\ell}(\omega)=2^{2\ell+1}\left[Q_{\ell}^{({\rm
H})}(2\omega)+Q_{\ell}^{({\rm H})}(-2\omega)\right]\,.$ (55)

According to Eq. (4.154) of Ref. Jentschura and Adkins (2022), the dipole term
can be expressed in terms of a variable $t$,

$Q_{1}^{({\rm
H})}(\omega)=\frac{2t^{2}\,p^{(1)}(t)}{3\,(1-t)^{5}\,(1+t)^{4}}+\frac{256\,t^{9}\,f(t)}{3\,(1+t)^{5}\,(1-t)^{5}}\,,$
(56)

where the photon energy is parameterized by the $t$ variable,

$t=t(\omega)=\frac{1}{\sqrt{1+2\omega}}\,.$ (57)

The polynomial $p^{(1)}(t)$ incurred in Eq. (56) is

$p^{(1)}(t)=3-3t-12t^{2}+12t^{3}+19t^{4}-19t^{5}-26t^{6}-38t^{7}\,.$ (58)

Within the representation in terms of the $t$ variable, the transition to
positronium amounts to the replacement $t\to t^{\prime}$ where
$t^{\prime}=t^{\prime}(\omega)=(1+4\omega)^{-1/2}$. The quadrupole term reads
as follows,

$Q_{2}^{({\rm
H})}(\omega)=\frac{t^{2}\,p^{(2)}(t)}{5\,(1-t)^{7}\,(1+t)^{6}}-\frac{4096\,t^{11}\,(t^{2}-4)\,f(t)}{5\,(t^{2}-1)^{7}}\,.$
(59)

The polynomial $p^{(2)}(t)$ reads as follows,

$p^{(2)}(t)=45-45t-285t^{2}+285t^{3}+786t^{4}-786t^{5}-1322t^{6}\\\
+1322t^{7}+2865t^{8}+5327t^{9}-553t^{10}-1495t^{11}\,.$ (60)

The octupole term is

$\displaystyle Q_{3}^{({\rm H})}(\omega)=$
$\displaystyle\;\frac{9\,t^{2}\,p^{(3)}(t)}{7(1-t)^{9}(1+t)^{8}}$
$\displaystyle\;-\frac{36864\,t^{13}\,(t^{2}-4)\,(t^{2}-9)f(t)}{7\,(t^{2}-1)^{9}}\,.$
(61)

It contains the polynomial $p^{(3)}(t)$ which reads as follows,

$p^{(3)}(t)=70-70t-595t^{2}+595t^{3}+2280t^{4}-2280t^{5}-5309t^{6}\\\
+5309t^{7}+9134t^{8}-9134t^{9}-24077t^{1}0-49651t^{11}\\\
+6532t^{12}+20092t^{13}-323t^{14}-1725t^{15}\,.$ (62)

Finally, the hexadecupole term reads as follows,

$\displaystyle Q_{4}^{({\rm H})}(\omega)=$
$\displaystyle\;\frac{t^{2}\,p^{(4)}(t)}{9(1-t)^{11}(1+t)^{11}}$
$\displaystyle\;-\frac{262144\,t^{15}\,(t^{2}-4)\,(t^{2}-9)\,(t^{2}-16)\,f(t)}{9\,(t^{2}-1)^{11}}\,.$
(63)

The polynomial $p^{(4)}(t)$ is given by

$p^{(4)}(t)=14175-14175t-150255t^{2}+150255t^{3}\\\
+732060t^{4}-732060t^{5}-2191500t^{6}+2191500t^{7}\\\
+4624386t^{8}-4624386t^{9}-8049042t^{10}+8049042t^{11}\\\
+23686124t^{12}+51811348t^{13}-7367996t^{14}\\\
-24613572t^{15}+484375t^{16}+3316713t^{17}+14153t^{18}\\\ -145225t^{19}\,.$
(64)

The asymptotic limits are as follows. For high photon energy, one obtains the
result

$Q_{\ell}^{({\rm
H})}(\omega)=\frac{2^{2\ell-1}\,\Gamma(2\ell+3)}{2\ell+1}\,\frac{1}{\omega}+\mathcal{O}\left(\frac{1}{\omega^{2}}\right)\,.$
(65)

By contrast, the result in the static limit is

$Q_{\ell}^{({\rm
H})}(\omega=0)=\frac{2\,(\ell+2)}{\ell(\ell+1)}\,\Gamma(2\ell+3)\,,$ (66)

for the general multipole order $2^{\ell}$. For the static value of the
polarizability, one multiplies the result given in Eq. (66) by a factor $2$.
The result (66) can be used in Eqs. (30), (36), (42) and (48), in order to
obtain long-range asymptotics for a perfect conductor. For positronium, one
multiplies the result given in Eq. (66) by $2^{2\ell+1}$ in order to obtain
the static limit of the $2^{\ell}$-pole polarizability of positronium.

Figure 1:  A reanalysis of the dielectric function of $\alpha$-quartz
(ordinary axis) is performed based on Eq. (72), with fitting parameters given
in Table 1. The panels refer to (a) lattice resonance region, real part of
$\rho(\omega)$, (b) lattice resonance region, imaginary part of
$\rho(\omega)$, (c) interband region, real part of $\rho(\omega)$, (d)
interband region, imaginary part of $\rho(\omega)$. The angular frequency is
given in atomic units [a.u.]. The data points are taken from Ref. Phillipp
while solid curve is given in Eq. (72). Figure 2:  Same as Fig. 2, but for the
extraordinary axis. Table 1:  We indicate the coefficients for the first few
resonances for $\alpha$-quartz according to the fitting formula (72), for the
ordinary and the extraordinary optical axis. The values for $\omega_{k}$,
$\gamma_{k}$ and $\gamma^{\prime}_{k}$ are measured in atomic units, i.e., in
units of $E_{h}/\hbar$. The subscript $k$ numbers the resonances. Ordinary
Axis: Vibrational Excitations
---
$k$ | $\alpha_{k}$ | $\omega_{k}$ | $\gamma_{k}$ | $\gamma^{\prime}_{k}$
1 | $1.045\times 10^{-2}$ | $1.827\times 10^{-3}$ | $1.301\times 10^{-5}$ | $1.631\times 10^{-5}$
2 | $8.541\times 10^{-2}$ | $2.218\times 10^{-3}$ | $1.835\times 10^{-5}$ | $-1.528\times 10^{-6}$
3 | $2.022\times 10^{-3}$ | $3.180\times 10^{-3}$ | $4.026\times 10^{-5}$ | $1.019\times 10^{-4}$
4 | $1.111\times 10^{-2}$ | $3.668\times 10^{-3}$ | $3.381\times 10^{-5}$ | $3.423\times 10^{-5}$
5 | $5.850\times 10^{-2}$ | $5.234\times 10^{-3}$ | $3.959\times 10^{-5}$ | $-3.881\times 10^{-5}$
6 | $4.472\times 10^{-2}$ | $5.339\times 10^{-3}$ | $3.776\times 10^{-5}$ | $4.041\times 10^{-4}$
Ordinary Axis: Interband Excitations
$k$ | $\alpha_{k}$ | $\omega_{k}$ | $\gamma_{k}$ | $\gamma^{\prime}_{k}$
7 | $1.341\times 10^{-2}$ | $3.899\times 10^{-1}$ | $1.423\times 10^{-2}$ | $-1.195\times 10^{-1}$
8 | $8.086\times 10^{-2}$ | $4.287\times 10^{-1}$ | $9.996\times 10^{-2}$ | $4.432\times 10^{-1}$
9 | $1.568\times 10^{-2}$ | $5.245\times 10^{-1}$ | $5.322\times 10^{-2}$ | $7.333\times 10^{-1}$
10 | $2.520\times 10^{-1}$ | $6.270\times 10^{-1}$ | $9.280\times 10^{-2}$ | $8.041\times 10^{-1}$
11 | $1.772\times 10^{-1}$ | $8.237\times 10^{-1}$ | $4.981\times 10^{-1}$ | $-4.361\times 10^{-1}$
Extraordinary Axis: Vibrational Excitations
$k$ | $\alpha_{k}$ | $\omega_{k}$ | $\gamma_{k}$ | $\gamma^{\prime}_{k}$
1 | $3.628\times 10^{-2}$ | $1.736\times 10^{-3}$ | $2.316\times 10^{-5}$ | $2.316\times 10^{-5}$
2 | $6.734\times 10^{-2}$ | $2.418\times 10^{-3}$ | $2.811\times 10^{-5}$ | $4.267\times 10^{-5}$
3 | $1.023\times 10^{-2}$ | $2.430\times 10^{-3}$ | $9.981\times 10^{-5}$ | $-3.154\times 10^{-3}$
4 | $1.114\times 10^{-2}$ | $3.578\times 10^{-3}$ | $3.594\times 10^{-5}$ | $2.118\times 10^{-3}$
5 | $1.026\times 10^{-1}$ | $5.307\times 10^{-3}$ | $4.383\times 10^{-5}$ | $2.533\times 10^{-4}$
Extraordinary Axis: Interband Excitations
$k$ | $\alpha_{k}$ | $\omega_{k}$ | $\gamma_{k}$ | $\gamma^{\prime}_{k}$
6 | $1.351\times 10^{-2}$ | $3.899\times 10^{-1}$ | $1.428\times 10^{-2}$ | $-1.172\times 10^{-1}$
7 | $7.775\times 10^{-2}$ | $4.283\times 10^{-1}$ | $9.861\times 10^{-2}$ | $4.422\times 10^{-1}$
8 | $1.512\times 10^{-2}$ | $5.245\times 10^{-1}$ | $5.229\times 10^{-2}$ | $7.323\times 10^{-1}$
9 | $2.430\times 10^{-1}$ | $6.272\times 10^{-1}$ | $9.108\times 10^{-2}$ | $7.971\times 10^{-1}$
10 | $1.859\times 10^{-1}$ | $8.209\times 10^{-1}$ | $5.049\times 10^{-1}$ | $-4.160\times 10^{-1}$

We are now in the position to derive some more closed-form expressions for
interactions with a perfect conductor, i.e., in the limit
$\epsilon(\mathrm{i}\omega)\to\infty$. Specifically, for multipole
interactions with atomic hydrogen, one obtains the following result for the
integral over the $2^{\ell}$-pole polarizability,

$\int_{0}^{\infty}\mathrm{d}\omega\,\alpha_{\ell}(\mathrm{i}\omega)=\frac{\pi\,\Gamma(2\ell+3)}{2^{2\ell+1}\,(2\ell+1)}\frac{a_{0}^{2\ell}}{\hbar}\,.$
(67)

The general result for the $2^{\ell}$-pole energy shift for hydrogen (H)
interacting with a perfect conductor thus is as follows,

$\mathcal{E}^{({\rm H})}_{\ell}(z)\mathop{=}^{z\to
0,\epsilon\to\infty}-\frac{(\ell+1)\,\Gamma(2\ell+3)}{2^{2\ell+4}\,\ell\,(2\ell+1)}\,\frac{E_{h}}{(z/a_{0})^{2\ell+1}}\,.$
(68)

Here, $E_{h}$ is the Hartree energy, and $a_{0}$ is the Bohr radius. The ratio
$z/a_{0}$ is equal to the atom-wall distance, expressed in atomic units (see
Chap. 2 of Ref. Jentschura and Adkins (2022)). For positronium (Ps), one
obtains the result

$\mathcal{E}^{({\rm Ps})}_{\ell}(z)\mathop{=}^{z\to
0,\epsilon\to\infty}-\frac{(\ell+1)\,\Gamma(2\ell+3)}{16\,\,\ell\,(2\ell+1)}\,\frac{E_{h}}{(z/a_{0})^{2\ell+1}}\,.$
(69)

In the short-range regime, the expansion into multipoles constitutes an
expansion in powers of $a_{0}/z$, where $a_{0}$ is the Bohr radius and $z$ is
the atom-wall distance.

The general results given in Eqs. (68) and (69) demonstrate that the sum over
the multipole potentials (at least for hydrogen and positronium interacting
with a perfect conductor) constitutes a divergent, asymptotic series. The
divergence, for any distance $z$, happens due to the factorial growth of the
prefactor $\Gamma(2\ell+3)$. Optimal truncation of the multipole expansion at
the smallest term of the series then constitutes a valid procedure for
obtaining theoretical predictions Caliceti  _et al._ (2007), while Borel
summation can be used in order to sum the divergent series Jentschura (2000).

Table 2:  Coefficient $C_{(2\ell+1)0}$ multiplying the leading term for the
$2^{\ell}$-pole contribution to the atom-surface interaction are given in
atomic units, for hydrogen interacting with a (perfect) conductor, and with
$\alpha$-quartz. $C_{(2\ell+1)0}$ for Hydrogen
---
| $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$
Conductor | 0.250 | 0.844 | 7.50 | 123
$\alpha$–Quartz | 0.0599 | 0.178 | 1.48 | 23.4
Table 3:  Same as Table 2 for positronium. $C_{(2\ell+1)0}$ for Positronium
---
| $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$
Conductor | 1.00 | 13.5 | 480 | 31500
$\alpha$–Quartz | 0.302 | 3.73 | 127 | 8130

###  IV.2 Multipole Corrections

We aim to combine the analysis of the multipole polarizabilities given in Sec.
IV.1 with an update on $\alpha$-quartz (Ref. Łach  _et al._ (2010)). For the
data presented in Phillipp , we used the following fit formula in Ref. Łach
_et al._ (2010),

$\displaystyle\rho(\omega)\equiv$
$\displaystyle\;\frac{\epsilon(\omega)-1}{\epsilon(\omega)+2}=\frac{[n(\omega)+\mathrm{i}\,k(\omega)]^{2}-1}{[n(\omega)+\mathrm{i}\,k(\omega)]^{2}+2}$
$\displaystyle\simeq$
$\displaystyle\;\sum_{k=1}^{n}\frac{\alpha_{k}\,\omega_{k}^{2}}{\omega_{k}^{2}-\mathrm{i}\,\gamma_{k}\,\omega-\omega^{2}}\,.$
(70)

Here, $n(\omega)$ are $k(\omega)$ are the dispersive and absorptive parts of
the index of refraction, while the functional form is inspired by the
Clausius–Mossotti equation. We take the opportunity to point out the missing
factor $\omega_{k}^{2}$ in the numerator of the fitting function given in Eq.
(70) of Ref. Łach  _et al._ (2010) (see also Ref. Łach  _et al._ ). The
missing prefactor had previously been supplemented in Eq. (21) of Ref.
Jentschura  _et al._ (2016). For completeness, it might be useful to point out
that the expression $(\epsilon(\omega)-1)/(\epsilon(\omega)+1)$, which appears
in the integrand of the short-range expressions (29), (35), (41), and (47),
can be obtained from $\rho(\omega)$ as follows,

$\frac{\epsilon(\omega)-1}{\epsilon(\omega)+1}=\frac{3\rho(\omega)}{\rho(\omega)+2}\,.$
(71)

For intrinsic silicon Moore  _et al._ (2022), we have recently found that a
better analytic representation can be obtained based on the following fit
formula,

$\displaystyle\rho(\omega)\simeq$
$\displaystyle\;\sum_{k=1}^{n}\frac{\alpha_{k}\,(\omega_{k}^{2}-\mathrm{i}\,\gamma^{\prime}_{k}\,\omega)}{\omega_{k}^{2}-\mathrm{i}\,\gamma_{k}\,\omega-\omega^{2}}\,.$
(72)

where the expression
$[\alpha_{k}\,(\omega_{k}^{2}-\mathrm{i}\,\gamma^{\prime}_{k}\,\omega)]$ can
be regarded as a complex oscillator strength that includes a phenomenological
model of radiative reaction Moore  _et al._ (2022). We find that such a model
represents the data from Ref. Phillipp very well (see Figs. 1 and 2, as well
as Table 1). This finding is nontrivial in view of the necessity for any fit
function to fulfill the Kramers–Kronig relationships (see, e.g., Chap. 6 of
Ref. Jentschura (2017)), which relate the real and imaginary parts of $\rho$
and $\epsilon$. The functional form of our model (72) fulfills the
Kramers–Kronig relationships automatically. While the oscillator strengths and
resonance frequencies for vibrational excitations differ in between the
ordinary and the extraordinary axis of $\alpha$-quartz (see Table 1), we find
that the influence of the low-frequency vibrational excitations on the leading
short-range expansion coefficients reported below is numerically negligible.

We are now in the position to analyze the multipole corrections for
$\alpha$-quartz. In general, we can say that these intriguing corrections to
atom-surface interactions have generated considerable interest [see, e.g.,
Ref. Łach  _et al._ (2011), Eq. (2) of Ref. Tao and Rappe (2014), and Eq. (8)
of Ref. Zheng  _et al._ (2017)]. In order to put this finding into
perspective, we should point out that it has recently become possible to
calculate the multipole corrections to polarizabilities more accurately from
first principle, for general multi-electron atoms Kaur  _et al._ (2022). In
this article, in order to facilitate the analysis of the multipole
corrections, we have evaluated exact expressions for hydrogen and positronium
up to the hexadecupole order (see Sec. IV.1).

It is indicated to include a brief discussion on the magnitude of the
multipole corrections. We continue to use atomic units. A priori, the short-
range approximations are valid for $z\leq 1/\alpha\approx 137.036$ in atomic
units, i.e., for distances smaller than about 100 Bohr radii Łach  _et al._
(2010); Jentschura and Moore (2023). The leading term for the $2^{\ell}$-pole
multipole term is proportional to $1/z^{2\ell+1}$. It is well known that
Lifshitz theory cannot be used for arbitrarily close approach to the surface.
The probability density of atomic wave functions (for ground-state hydrogen
atoms) decrease with a probability $|\psi|^{2}\sim\exp(-2r)$, where $r$ is the
distance from the nucleus in atomic units. For a distance of $z=0.5\,{\rm
nm}$, which is roughly equal to $z=10$ in atomic units, one has
$|\psi|^{2}\sim 10^{-9}$, eliminating the overlap as a possible limiting
factor. It has been stressed in Ref. Zaremba and Kohn (1976) that, for
vanishing overlap of the atomic wave function with the surface, the exchange
of electrons between the atom and the substrate can be neglected, which, in
turn, makes it possible to treat the electrons (and nuclei) of the atom and
the surface as distinguishable. It has explicitly been shown in Refs. Manson
and Ritchie (1984); Paranjape and Pathak (1985) that, for distances of the
order of a few atomic units, the interaction with the surface approaches a
regime where Lifshitz theory can be used (see also Fig. 1 of Ref. Manson and
Ritchie (1984)).

We can thus use an exemplary distance of $z=10$ atomic units, in order to
analyze the magnitude of the multipole corrections. It has recently been shown
in Ref. Jentschura and Moore (2023) that higher-order terms in the atom-
surface potential contain logarithms of the atom-wall distance, $\ln(z)$,
leading to a semi-analytic expansion of the atom-surface potential in powers
of $z$ and $\ln(z)$, described by coefficients with two indices. Hence, we
will refer to the leading short-range coefficient multiplying for the atom-
surface dipole and quadrupole terms as $C_{30}$ and $C_{50}$ (rather than
$C_{3}$ and $C_{5}$), where the first index counts the power of $z$ and the
second index (equal to zero) indicates the absence of a logarithm Jentschura
and Moore (2023). For a perfect conductor, we thus write the relations

$\mathcal{E}_{1}(z)\approx-\frac{C_{30}}{z^{3}}\,,\quad\mathcal{E}_{2}(z)\approx-\frac{C_{50}}{z^{5}}\,,\quad
1\ll z\ll\alpha^{-1}\,,$ (73)

with the following exemplary results (for the dipole and quadrupole
coefficients) for perfect conductors [see Eq. (68) and (69)]

$\displaystyle C_{30}^{({\rm H})}\;\mathop{=}^{\epsilon\to\infty}\;$
$\displaystyle\;\frac{1}{4}\,,\qquad$ $\displaystyle C_{50}^{({\rm
H})}\;\mathop{=}^{\epsilon\to\infty}\;\frac{27}{32}\,,$ (74) $\displaystyle
C_{30}^{({\rm Ps})}\;\mathop{=}^{\epsilon\to\infty}\;$
$\displaystyle\;1\,,\qquad$ $\displaystyle C_{50}^{({\rm
Ps})}\;\mathop{=}^{\epsilon\to\infty}\;\frac{27}{2}\,.$ (75)

For $\alpha$-quartz, we refer to Tables 2 and 3 for the multipole
coefficients. For positronium, at $z=10$ atomic units, one has the following
ratio of the quadrupole corrections to the leading dipole term,

$\displaystyle\frac{\mathcal{E}^{({\rm Ps})}_{2}(z=10)}{\mathcal{E}^{({\rm
Ps})}_{1}(z=10)}=$ $\displaystyle\;\frac{C^{({\rm Ps})}_{50}}{C^{({\rm
Ps})}_{30}\times(10)^{2}}$ $\displaystyle=$
$\displaystyle\;\left\\{\begin{array}[]{cc}0.135&\;\;\mbox{(conductor)}\\\
0.124&\;\;\mbox{($\alpha$--quartz)}\\\ \end{array}\right.\,.$ (78)

A deviation by 13.5% is larger than the uncertainty of many current
measurements of the dielectric function of materials Palik (1985). For
hydrogen, the quadrupole correction is

$\displaystyle\frac{\mathcal{E}^{({\rm H})}_{2}(z=10)}{\mathcal{E}^{({\rm
H})}_{1}(z=10)}=$ $\displaystyle\;\frac{C^{({\rm H})}_{50}}{C^{({\rm
H})}_{30}\times(10)^{2}}$ $\displaystyle=$
$\displaystyle\;\left\\{\begin{array}[]{cc}0.0336&\;\;\mbox{(conductor)}\\\
0.0296&\;\;\mbox{($\alpha$--quartz)}\\\ \end{array}\right.\,,$ (81)

which is of the order of a few percent. These observations are consistent with
the literature; for example, the importance of an accurate understanding of
quadrupole corrections has been stressed in the last-but-one paragraph of Ref.
Tao and Rappe (2014).

##  V Conclusions

We have (re-)derived the quadrupole, octupole and hexadecupole corrections to
the atom-wall interaction, with a special emphasis on the isolation of the
relevant angular-momentum components from the derivative tensors of the
electric field. The functional form of our results is the same as the one
obtained in in Ref. Łach  _et al._ (2010), but important differences are
obtained for the prefactors.

The general results for the dipole, quadrupole, octupole and hexadecupole
contributions to the atom-wall interactions have been given in Eqs. (III.1),
(34), (40) and (46), respectively. Short-range limits have been given for
either term in Eqs. (29), (35), (41), and (47), and the long-range limits have
been analyzed in Eqs. (30), (36), (42) and (48). For the long-range limit, we
have found rather concise formulas for the dependence of the coefficient
multiplying the term proportional to $z^{-2\ell-2}$, in terms of logarithms
[see Eqs. (31), (37), (43) and (49)]. The analytic results for the long-range
limit constitute an important addition to the results originally reported in
Refs. Łach  _et al._ (2010, ).

Furthermore, we find that the expansion into multipole terms constitutes an
asymptotic, divergent series [see Eqs. (68) and (69)]. Series with factorially
divergent coefficients can, in many cases, be summed using generalizations of
the Borel method Jentschura (2000); Caliceti  _et al._ (2007). Furthermore, we
find that it is possible to express the quadrupole, octupole, and hexadecupole
polarizabilities of the hydrogen and positronium in closed analytic form. This
enables us to reanalyze multipole corrections, with a special emphasis on
hydrogen and positronium. We find that the quadrupole correction is
phenomenologically relevant [see Eqs. (IV.2) and (IV.2)]. A reanalysis of the
dielectric function of $\alpha$-quartz using the functional form given in Eq.
(72) reveals very good agreement with numerical data from Ref. Phillipp . For
the short-range coefficients of the multipole corrections to the atom-surface
interactions (perfect conductor and $\alpha$-quartz), we present results in
Tables 2 and 3. These data confirm the rapid growth of the coefficients
multiplying the multipole corrections, both for interactions with perfect
conductor and $\alpha$-quartz, consistent with the eventual factorial
divergence of the series.

## Acknowledgments

The authors acknowledge insightful conversations with Carsten Ullrich,
Christopher Moore, Yakov Itin and István Nándori. This research was supported
by NSF grant PHY–2110264.

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