# Body Gesture Recognition to Control a Social Robot

Javier Laplaza, Joan Jaume Oliver, Ramón Romero, Alberto Sanfeliu and Anaís
Garrell All authors work in the Institut de Robòtica i Informàtica Industrial
de Barcelona (IRI), Catalonia, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, we propose a gesture based language to allow humans to interact
with robots using their body in a natural way. We have created a new gesture
detection model using neural networks and a custom dataset of humans
performing a set of body gestures to train our network. Furthermore, we
compare body gesture communication with other communication channels to
acknowledge the importance of adding this knowledge to robots. The presented
approach is extensively validated in diverse simulations and real-life
experiments with non-trained volunteers. This attains remarkable results and
shows that it is a valuable framework for social robotics applications, such
as human robot collaboration or human-robot interaction.

##  I Introduction

Finding natural and efficient communication channels is essential in Human-
Robot Interaction (HRI). If we take a look at the way humans communicate with
each other, we see that about 70% of the communication is non-verbal
communication [16]. Moreover, when humans want to communicate with other
agents with whom they do not share a common spoken language –foreigners, babys
or animals– most of the communication is non-verbal [2, 7].

When it comes to communication with robots, it is possible to establish a set
of gestures to communicate certain ideas in a similar way that gesture
language works between humans. But, similarly to gesture language, this
approach requires that both agents know which gestures compose the language
and what meaning has each gesture. Furthermore, previous attempts of creating
such body language with robots are generally designed by people used to work
with robots. We argue that such kind of languages cannot be expected to find
success when robots face humans with no previous experience with robots.

This inspired us to seek a natural way for humans to communicate with robots
using gestures. Our goal is to create a natural gesture dictionary and explore
how humans rate gesture based communication versus other communication
channels.

To achieve this, we first collected a dataset using human volunteers making a
predefined set of communication gestures. Then, we create a gesture detection
model to allow the robot identify human gestures. Finally, we wanted to study
how convenient was using gestures for communication. To do so, we carried out
several experiments using non-trained volunteers and the IVO robot [10], see
Fig.1, in order to compare gesture based communication to other communication
channels.

Figure 1: By raising the left arm, the human is able to tell the IVO robot to
turn to the left side.

The remainder of our study unfolds in the following manner. Section II
introduces the related work in human-robot gesture recognition. Section III
provides an overview of the contributions we describe in this article,
presenting the model architecture. Section IV describes the created dataset
the authors used to train and test the system. In Section V, we present the
setting of the experiments and our evaluation methodology, which is
subsequently employed in the results. Finally, conclusions are given in
sections VI.

##  II Related work

Most of the literature related to gesture detection focuses on hand gesture
detection. Great examples of this are [4], [1] and [8]. Touchless interaction
methods that will not use sound or speech for the communication need to
somehow sense the human body’s position.

Wearable sensors are widely used in gesture recognition. For instance,
incorporating IMUs in the wrist or some combination of them in the arms to
track the body motion that will be used to determine the robot commands, like
in [6], [13] and [14]. Moreover, users can use data-gloves that track the hand
and its finger’s position, [5] and [17].

Nevertheless, the fact of using specific wearables does not guarantee the
touchless methods any advantages. Using gadgets on the human body takes back
some of the touchless technologies strong points and also induces to a less
natural communication.

This explains the snowball in research on communication with the bare human
body, dressed as one would dress to interact with another human, without
adding extra instruments or devices on the user.

In this particular work, we are interested in hand gestures and body gestures
with the use of a camera, it provides a richer language to express more
complex concepts than full body gestures, and that might be the reason why not
so many works focus on full body gesture communication with robots.

Some work on full body gesture communication focus on identifying where the
human is pointing at to identify a specific object or region, one example of
this being [18]. Other works focus on identifying emotions expressed by the
body posture, such as [15]. Moreover, Some gestures are very restricted to
specific tasks, in [9] the approach is to detect whether a human is willing to
collaborate or not during a hand-over operation according to his gestural
expression.

The work proposed in [12] is similar to our approach, but they use the
relative position of hands and faces to define poses. More importantly, they
use UAV as the platform to interact, which has different dynamics than ground
robots and also a different perspective of humans when using a camera.

Another similar work is presented in [3], focusing in optical flow techniques
used in order to extract features of the human body.

In [19] a set of gestures are proposed for HRI operations. While very similar
to our approach, in our work we focus on allowing very general gestures
instead of defining them beforehand. Also, we study how human volunteers rate
the interaction with the robot using gestures.

##  III Model Architecture

In order to properly identify different body gestures, we have been working on
a model that consists on a linear classifier with some hidden layers that are
using different body positions as input paramaters while get gestures
probability as outputs.

Although we are feeding our model with different body positions information,
our inputs are videos with four dimensions $(x,y,z,t)$. In the work presented
in this paper temporal component has been omitted and only the last frame of
each video is used as input. This means that we only take into account those
gestures that are static.

From this frame, we used MediaPipe Pose machine learning solution [11] in
order to extract the full body coordinates (from head to feet). This ML model
predicts the location of 33 pose landmarks each following the
$(x,y,z,visibility)$ structure that will be used as our Classifier input.

Once we pre-process the video to obtain our $[33,4]$ matrix input containing
all body positions coordinates, we can set up our classifier.

Our model uses four Linear $y=xA^{T}+b$ data transformations fully connected
layers of size $256,128,64$ and $8$ combined with
${ReLU}(x)=(x)^{+}=\max(0,x)$ activation functions.

The outputs from our classifier consists in a one dimensional vector with
length equals to the number of gestures that could be predicted, being each
vector element the model score for each specific gesture.

From this vector, a ${Softmax}$ layer is applied in order to normalize the
model scores and the maximum gesture probability is then considered to be the
output gesture.

Figure 2: Schematic view of the neural network used for the gesture detection.

As for training and test strategies, and due to the low number of gestures and
data available, instead of using the common $70\%-30\%$ strategy, the one out
method was used. More precisely, from the $8$ persons available all where used
for train except for the $2$ one that was exclusively used for test purposes.

On one hand, it is interesting to say that the loss function used for training
is the Cross-Entropy Loss, and on the other hand, the chosen optimization
algorithm for this network is the Adaptive Moment Estimation (${Adam}$) with a
learning rate $\alpha=0.01$ and
$\beta_{1}=0.9,\beta_{2}=0.999,\epsilon=10^{-8}$ hyperparameters.

##  IV Dataset

Since we couldn’t find a suitable dataset for our goal, we decided to create a
dataset in our laboratory.

The main feature of our gesture based communication dictionary is naturalness.
We want that everyone can communicate with the robot, not only people who is
already familiar with robots.

We divide the defined gestures in two groups: static and dynamic gestures (see
Fig. 3):

  * •

Static gestures

    * –

Attention: Catch the robot’s attention to give him an order.

    * –

Right: Order the robot to turn right.

    * –

Left: Order the robot to turn left.

    * –

Stop: Order the robot to stop its trajectory.

    * –

Yes: Approve a robot’s information.

    * –

Shrug: Inform the robot that you don’t understand his information.

    * –

Random: Random gesture, not necessarily a communication gesture.

    * –

Static: Human is standing still.

  * •

Dynamic gestures

    * –

Greeting: Greet the robot.

    * –

Continue: Order the robot to continue its path after telling him to stop.

    * –

Turn-back: Order the robot to turn 180 degrees.

    * –

No: Deny a robot’s information.

    * –

Slowdown: Order the robot to reduce its speed.

    * –

Come: Order the robot to reach your position.

    * –

Back: Order the robot to move back.

As for data recording, each human volunteer was recorded using an RGB camera.
When human volunteers where asked to make a gesture they were provided with a
vague explanation of the gesture intention. This was done to collect data that
felt most natural to each volunteer. There was no restriction on which arms
should be moved in each gesture whatsoever. Thus, different volunteers could
make the same gesture in a very different way, using one arm or the other, or
even both of them.

Each gesture was repeated three times, first 1 meter away from the camera,
then 4 meters away and finally 6 meters away.Each video contains information
of only one gesture, and all the videos were recorded indoors.

Finally, we use MediaPipe [11] to extract the 3D joints of the human in the
video.

We also consider a wide range of users regarding age, gender, education level
and culture. Taking this into account, we created our dataset thanks to 10
human volunteers, 7 men and 3 women. Each volunteer was asked to perform

Figure 3: Some samples of static gestures recorded in the dataset.

##  V Experiments

###  V-A Model results

We train the model until we see that it starts overfitting in the test
dataset. Then, we stop the training and check the results on this test
dataset. These results can be seen in Table 4.

Figure 4: Precision, recall and F1-score for every model gesture class.

The results show that the model accuracy for certain gestures is remarkable.
Specifically, attention, right, left, shrug and static gestures rate an
F1-score above 0.8. This result was expected, since all those gestures are
very different from other gestures in the dataset.

On the other hand, there are two gestures that the model particularly
struggles classify: stop and yes. Again, this result was expected, since both
gestures present an overall body pose very similar, only the hand position
differs.

Finally, the random gesture has a F1-score of 0.64, which is understandable
for a class that include very different body poses.

In order to further study how do our defined gestures interfere with each
other, we create a confusion matrix (see Fig. 5). From this matrix, we confirm
that our model isn’t able to classify gestures yes and stop. Possible ways to
tackle this can be increasing model complexity or using a more complete
Mediapipe model that incorporates finger and face information.

Figure 5: Model confussion matrix. Figure 6: Samples of gestures used by the
user: turn left (top left), look upwards (top right), look downwards (bottom
left) and approach (bottom right).

###  V-B User Study

The results presented in the previous section demonstrate that the robot is
able to detect and recognize human natural gesture. A user study was also
conducted to determine whether the body gesture recognition to control our
robot enhances the usability and the comfort of the robot from the point of
view of the human. We compared our method with the use of a remote controller.

The hypothesis we endeavored to test was as follows: “Participants will feel
more comfortable and will perceive difference between the use of body gesture
recognition and the use of a remote controller.”

We asked humans to communicate different orders to the robot, specifically:

  * •

Order the robot to move closer.

  * •

Order the robot to move away.

  * •

Order the robot to turn to the right.

  * •

Order the robot to turn to the left.

  * •

Order the robot to look up.

  * •

Order the robot to look down.

In the first experiment, the human had to use the natural gestures to express
to the robot what action must perform. We conducted these experiments in a
Wizard-of-Oz way, since using the gesture detector may lead to missing some of
the gestures, and it can cause a negative impact to the user perception of
gestures as a channel of communication.

Then, we repeated the same experiment but this time we gave a remote
controller to the human, thus that he/she could tele-operate the robot after
some instruction.

In each case, we asked the volunteers to make all gestures/commands in a
random order. We also chose randomly between the gesture communication and the
controller as the first experiment for each volunteer in order to avoid
possible biases. Refer to Fig.6 to seen some samples of the gesture
communication.

Figure 7: Evaluation from 1 (low) to 7 (high) of the main aspects related to
the robot behavior in handover task.

For the experiments, we selected 15 people (8 men, 7 women) on the University
Campus. Participants ranged in age from 19 to 50 years (M=29.5, SD=9.2), and
represented a variety of University majors and occupations including computer
science, mathematics, biology, finance and chemistry. For each individual
selected, we randomly activated one of the two behaviors to deliver an object
to the volunteer, it is, the activation of the prediction module or the not
use of it. It should be mentioned that none of the participants had previous
experience working or interacting with robots.

Participants were asked to complete a variety of surveys. Our independent
variables considered whether participant make use of our gesture recongnition
or the remote control. The main dependent variables involved participants’
perceptions of the sociability, naturalness, security and comfort
characteristics. Each of these fields, was evaluated by every participant
using a questionnaire to fill out after the experiment, see Table I, based on
[kirby2010social].

Participants were asked to answer a questionnaire, following their encounter
with the robot in each mode of behavior. To analyze their responses, we
grouped the survey questions into four scales: the first measured sociability
robot behavior, while the second naturalness, and third and fourth evaluated
the comfort. Both scales surpassed the commonly used 0.7 level of reliability
(Cronbach’s alpha).

Each scale response was computed by averaging the results of the survey
questions comprising the scale. ANOVAs were run on each scale to highlight
differences between the three robot behaviors.

Below, we provide the results of comparing the two different methods. To
analyze the source of the difference, four scores were examined:
“sociability”, “naturalness”, “security” and “comfort”, plotted in Fig. 7. For
all four aspects, the evaluation score plotted in Fig. 7, pairwise comparison
with Bonferroni demonstrate there were difference between the two kind of
behavior approaches, $p<0.05$.

Therefore, after analyzing these four components, we may conclude that if the
robot is capable of understand people’s body gesture the acceptability of the
robots increases, and participants perceived the robot as a social entity.

Question
---
| How positive/negative did you feel
---
about your interaction with the robot?
How human-like did the robot behave?
How social was the robot’s behavior?
| How hopeful/despairing did you feel
---
about your interaction with the robot?
How natural was the robot’s behavior?
| How excited/calm did you feel
---
about your interaction with the robot?
How safe did you feel around the robot?
| How much control did you feel
---
about your interaction with the robot?
| How agitated/comfortable did you feel
---
about your interaction with the robot?
How comfortable did you feel near the robot?
TABLE I: Each block of questions correspond to the quality being measured.
From top to bottom: sociability, naturalness, security and comfort.

##  VI Conclusions

We created a dataset of natural gestures to communicate with a robot. We also
developed a new neural network architecture able to classify the static
gestures of the dataset. We reach high classification accuracy (¿80%)) in most
of the gestures, except for gestures that look very similar. Our future work
will try to add also the dynamic gestures to the model and decouple gestures
that can be easily confused.

The findings presented in the previous section reinforce the notion that the
robot’s ability to understand human body gestures is an important skill to
master in order to achieve natural interaction with people. Overall, people
interacting and communicating with the robot using natural gestures enhances
the engagement between the robot and the human.

The experiments we conducted yielded conclusive results. We found that people
felt their interaction with the robot was more natural when the robot
communicated through gestures. Detailed analysis showed that these capacities
improved the human’s perception of the robot’s security and sociability. We
also found that the amount of speech and comments made by the robot seems to
be appropriate for this type of scenario. Finally, volunteers perceived our
robot more sociable and closer when it recognized and understood their
gestures, and the interactions were longer.

## References

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kirby2010social


# Reference system approach within the white-dwarfs theory

M.V. Vavrukh, D.V. Dzikovskyi, N.L. Tyshko

(Received September 7, 2017, in final form October 5, 2017)

###### Abstract

Reference system approach of non-relativistic electron fluid theory was
adapted for calculation of characteristics of the electron-nuclear model at
the densities typical of degenerate dwarfs. Two- and three correlation
functions of degenerate relativistic electron gas have been calculated in the
momentum-frequency representation in the local field approximation. Main
contributions of the Coulomb interactions to the energy and equation of state
of the model at $T=0$ K have been calculated in the adiabatic approximation.

Key words: electron-nuclear model, correlation functions, local field
correction function, zero-temperature energy, equation of state

PACS: 05.30.Fk, 97.20Rp, 97.60Bw

###### Abstract

Áàçèñíèé ïäõä íåðåëÿòèâñòñüêî¿ òåîð¿ åëåêòðîííî¿ ðäèíè àäàïòîâàíî äî
ðîçðàõóíêó õàðàêòåðèñòèê åëåêòðîí-ÿäåðíî¿ ìîäåë ïðè ãóñòèíàõ, ùî âäïîâäàþòü
âèðîäæåíèì êàðëèêàì. Ðîçðàõîâàíî äâî- òà òðèàñòèíêîâ êîðåëÿöéí ôóíêö¿
âèðîäæåíîãî ðåëÿòèâñòñüêîãî åëåêòðîííîãî ãàçó â ìïóëüñíî-àñòîòíîìó
ïðåäñòàâëåíí ó íàáëèæåíí ëîêàëüíîãî ïîëÿ. Â àäàáàòèíîìó íàáëèæåíí îáèñëåíî
îñíîâí âíåñêè êóëîíâñüêèõ âçàìîäé â åíåðãþ òà ðâíÿííÿ ñòàíó ìîäåë ïðè $T=0$ K.

Ключов слова: åëåêòðîí-ÿäåðíà ìîäåëü, êîðåëÿöéí ôóíêö¿, ïîïðàâêà íà ëîêàëüíå
ïîëå, åíåðãÿ ìîäåë ïðè $T=0$ K, ðâíÿííÿ ñòàíó

##  1 Introduction

A hundred years have passed since the discovery of degenerate dwarfs [1]. The
theory of an internal structure of cold dwarfs was developed by Chandrasekhar
in 1940s, and it was based on the equation of state of ideal relativistic
electron gas at $T=0$ K [2, 3]. Generalization of this theory followed in the
next decades, when in the works of many authors there were investigated
effects of such important factors as axial rotation [4, 5], Coulomb
interactions [6], incomplete degeneration of an electron subsystem [7, 8],
effects of magnetic fields [9, 10], effects of general relativity [5, 11],
processes of neutronization [12], etc. Interpretation of the whole diversity
of properties of the dwarfs obtained from the observations of space [13, 14],
requires constructing a general theory that also takes into account the
effects of the above mentioned factors, among which there are the competing
ones.

The effect of interactions that play an important role in determining the
structure of dwarfs at different masses and luminosities, and especially for
the case of massive cold dwarfs, is the least studied. Basing on Wigner-Seitz,
Thomas-Fermi approximations and non-relativistic random phase approximation
corresponding to the approximate accounting of two-particle electron
correlations, Salpeter [6] showed that Coulomb interactions lead to a small
decrease of pressure of degenerate relativistic electron gas at $T=0$ K, which
is still considered to be the basis of Chandasekhar’s theory [10].

Due to the high density ($\sim 10^{5}$ g/cm3), the matter in the cores of
degenerate dwarfs has a metal electron structure with completely collectivized
electrons, and the Fermi momentum is of the order $m_{0}c$. Thus, the non-
relativistic approach is not applicable, and it complicates the calculations,
because the correlation functions of the reference system (of an interacting
relativistic electron gas) are less investigated than the functions of the
analogical non-relativistic model. On the other hand, the electron-nuclear
model at “dwarfs” densities is slightly non-ideal, which allows one to use a
random phase approximation in order to account for the interactions. A
consistent approach that corresponds to the modern metals many-electron theory
is proposed in [15]. A disadvantage of this work is the use of approximate
expressions for two- and three particle correlation functions of ideal
relativistic degenerate electron gas. In the present work we use an accurate
static two-electron correlation function and a three-particle correlation
function in the long-wavelength approximation, which improves the reliability
of the results for the energy of the ground state of the model and equation of
state.

##  2 General relations

We consider a more realistic, compared to [15], spatially homogeneous
electrically neutral model, which consists of $N_{\text{e}}$ electrons,
$N_{1}$ nuclei of charge $z_{1}$ and $N_{2}$ nuclei of charge $z_{2}$ in the
volume $V$ in thermodynamic limit $N_{\text{e}},\ V\rightarrow\infty,\
N_{\text{e}}/V=\text{const}$ at the temperatures much lower than the
temperature of the electron subsystem degeneration. Generalization for a
larger number of nuclei species is obvious.

For the Hamiltonian of the model

$\displaystyle\hat{H}=\hat{H}_{0}+\hat{V}_{\text{ee}}+\sum\limits_{i=1}^{2}\hat{V}_{\text{en}}^{i}+\hat{V}_{\text{nn}}\,,$
(2.1)

we use a secondary quantization representation for the electrons and a
coordinate one for the nuclear subsystem:

$\displaystyle\hat{H}_{0}=\sum\limits_{\mathbf{k},s}E_{k}\,a_{\mathbf{k},s}^{+}a_{\mathbf{k},s}$
(2.2)

is the Hamiltonian of free electrons
$(E_{k}=[(m_{0}c^{2})^{2}+\hbar^{2}k^{2}c^{2}]^{1/2}-m_{0}c^{2})$,

$\displaystyle\hat{V}_{\text{ee}}=\frac{1}{2V}\sum\limits_{\mathbf{q}\neq
0}V_{q}\,\hat{I}_{2}(\mathbf{q},-\mathbf{q})$ (2.3)

is the operator of electron-electron interactions,

$\displaystyle\hat{V}_{\text{en}}^{i}=-\frac{1}{V}z_{i}\sum\limits_{\mathbf{q}\neq
0}V_{q}\,S_{-\mathbf{q}}^{(i)}\,\hat{\rho}_{\mathbf{q}}$ (2.4)

is the operator of electron interactions with $i$-th nuclear subsystem,

$\displaystyle\hat{V}_{\text{nn}}=\frac{1}{2V}\sum\limits_{\mathbf{q}\neq
0}V_{q}\sum\limits_{i,j}z_{i}z_{j}\big{[}S_{\mathbf{q}}^{(i)}S_{-\mathbf{q}}^{(j)}-N_{i}\delta_{i,j}\big{]}$
(2.5)

is the sum of direct nuclear interactions. Here, $V_{q}=4\piup e^{2}/q^{2}$,
$S_{\mathbf{q}}^{(i)}=\sum\nolimits_{l=0}^{N_{i}}\exp{[\mathrm{i}(\mathbf{q,R}_{l}^{i})]}$
is the structure factor of $i$-th nuclear subsystem,

$\displaystyle\hat{I}_{2}(\mathbf{q},-\mathbf{q})$
$\displaystyle=\sum\limits_{\mathbf{k}_{1},\mathbf{k}_{2}}\sum\limits_{s_{1},s_{2}}a^{+}_{\mathbf{k}_{1}+\mathbf{q},s_{1}}a^{+}_{\mathbf{k}_{2}-\mathbf{q},s_{2}}a_{\mathbf{k}_{2},s_{2}}a_{\mathbf{k}_{1},s_{1}},$
$\displaystyle\hat{\rho}_{q}$
$\displaystyle=\sum\limits_{\mathbf{k},s}a^{+}_{\mathbf{k+q},s}a_{\mathbf{k},s}\,,$
(2.6)

$a^{+}_{\mathbf{k},s},a_{\mathbf{k},s}$ are the creation and annihilation
operators of electrons in quantum states with the given vector $\mathbf{k}$
and the spin variable $s$, $\mathbf{R}_{l}^{i}$ is the radius-vector of $l$-th
nucleus with the charge $z_{i}$.

For the calculation of a partition function of an electron subsystem in the
fixed nuclei field in the grand canonical ensemble

$\displaystyle
Z(\mu)=\operatorname{Sp}_{\text{e}}\exp{\left[-\beta\left(\hat{H}_{0}-\mu\hat{N}_{\text{e}}+\hat{V}_{\text{ee}}+\sum\limits_{i=1}^{2}\hat{V}_{\text{en}}^{i}\right)\right]},$
(2.7)

we summarize the reference system approach developed in [16, 17] for a
description of non-relativistic model of electron liquid. In the formula
(2.7), $N_{\text{e}}$ is the operator of the number of electrons, $\mu$ is the
variable of chemical potential. As in [16], let us move to the “frequency”
representation of electron operators

$\displaystyle
a_{\mathbf{k},s}(\nu_{*})=\int\limits_{0}^{\beta}a_{\mathbf{k},s}(\beta^{\prime})\psi_{\nu_{*}}(\beta^{\prime})\mathrm{d}\beta^{\prime},$
(2.8)

where
$\psi_{\nu_{*}}(\beta^{\prime})=\beta^{-1/2}\exp{(\mathrm{i}\nu_{*}\beta^{\prime})}$,
$\nu_{*}=(2n+1)\piup\beta^{-1}$, $n=0;\pm 1;\pm 2;\ldots.$ In the “frequency”
representation

$\displaystyle\exp{\left[-\beta\left(\hat{H}_{0}-\mu\hat{N}_{\text{e}}+\hat{V}_{\text{ee}}+\sum\limits_{i}\hat{V}_{\text{en}}^{i}\right)\right]}=\exp{(-\beta\mathcal{\hat{H}}_{\mu})T_{\beta}\\{\hat{S}_{\text{ee}}(\nu)\hat{S}_{\text{en}}(\nu)\\}},$
(2.9)

and $\mathcal{\hat{H}}_{\mu}=\hat{H}_{0}-\mu\hat{N}_{\text{e}}$. The operator
$T_{\beta}$ is a generalized ordering operator,

$\displaystyle\hat{S}_{\text{ee}}(\nu)$ $\displaystyle=\exp{\bigg{[}-(2\beta
V)^{-1}\sum\limits_{\mathbf{q}\neq
0}\sum\limits_{\nu}V_{q}\,\hat{\rho}_{\mathbf{q},\nu}\,\hat{\rho}_{-\mathbf{q},-\nu}\bigg{]}},$
$\displaystyle\hat{S}_{\text{en}}(\nu)$
$\displaystyle=\exp{\bigg{[}V^{-1}\sum\limits_{i=1}^{2}z_{i}\sum\limits_{\mathbf{q}\neq
0}V_{q}\,S_{-\mathbf{q}}^{(i)}\,\hat{\rho}_{\mathbf{q},0}\bigg{]}},$
$\displaystyle\hat{\rho}_{\mathbf{q},\nu}$
$\displaystyle=\sum\limits_{\mathbf{k},s}\sum\limits_{\nu_{*}}a^{+}_{\mathbf{k+q},s}(\nu_{*}+\nu)a_{\mathbf{k},s}(\nu_{*}),$
(2.10)

moreover, $\nu=2\piup n\beta^{-1}$; $n=0,\pm 1,\pm 2,\ldots,$ and the
calculation of the average of the product of operators
$a_{\mathbf{k},s}(\nu_{*})$ is performed according to the rule [18, 19]

$\displaystyle-\langle
T_{\beta}\\{a_{\mathbf{k}_{1},s_{1}}(\nu_{*})a^{+}_{\mathbf{k}_{2},s_{2}}(\nu_{*}^{\prime})\\}\rangle_{\mathcal{H}_{\mu}}=G_{\mathbf{k}_{1},s_{1}}(\nu_{*})\delta_{\nu_{*},\nu_{*}^{\prime}}\delta_{{s}_{1},s_{2}}\delta_{\mathbf{k}_{1},\mathbf{k}_{2}}\,,$
(2.11)

where
$G_{\mathbf{k},s}(\nu_{*})=(\mathrm{i}\nu_{*}-E_{k}+\mu)^{-1}\exp{(\mathrm{i}\delta\nu_{*})}$
is the spectral image of one-electron Green’s function of an ideal system
$(\delta\rightarrow+0)$, which is the reference system for a description of
the interacting electron gas model.

We use model with the Hamiltonian $H_{0}+\hat{V}_{\text{ee}}$ as a reference
system to calculate $Z(\mu)$:

$\displaystyle
Z(\mu)=\exp{[-\beta\Omega_{\text{e}}(\mu)]}\langle\hat{S}_{\text{en}}(\nu)\rangle_{\text{e}}\,,$
$\displaystyle\exp{[-\beta\Omega_{\text{e}}(\mu)]}=\exp{[-\beta\Omega_{0}(\mu)]}\langle
T_{\beta}\hat{S}_{\text{ee}}(\nu)\rangle_{0}\,,$
$\displaystyle\langle\hat{S}_{\text{en}}(\nu)\rangle_{\text{e}}=\langle
T_{\beta}\\{\hat{S}_{\text{ee}}(\nu)\hat{S}_{\text{en}}(\nu)\\}\rangle_{0}[\langle
T_{\beta}\hat{S}_{\text{ee}}(\nu)\rangle_{0}]^{-1}.$ (2.12)

The symbol $\langle\hat{A}\rangle_{0}$ denotes a statistical averaging over
the states, and $\langle\hat{A}\rangle_{\text{e}}$ for the states of the
reference system;

$\displaystyle\Omega_{0}(\mu)=-\beta^{-1}\sum\limits_{\mathbf{k},s}\ln{\\{1+\exp{[-\beta(E_{k}-\mu)]}\\}}$
(2.13)

is the grand potential of an ideal system of electrons, and
$\Omega_{\text{e}}(\mu)$ is the grand potential of the interacting electron
gas.

Expanding the operator $\hat{S}_{\text{en}}(\nu)$ in the formula (2.12) in a
power series, averaging by states for the reference system and presenting the
result in an exponential form, we have obtained contributions to the grand
potential of electron-nuclear interactions

$\displaystyle\Omega_{\text{en}}$ $\displaystyle=-\sum\limits_{n\geqslant
2}(n!V^{n})^{-1}\sum\limits_{i_{1},i_{2},\ldots,i_{n}}z_{i_{1}}z_{i_{2}}\ldots
z_{i_{n}}\sum\limits_{\mathbf{q}_{1},\ldots,\mathbf{q}_{n}\neq
0}V_{q_{1}}V_{q_{2}}\ldots
V_{q_{n}}S_{-\mathbf{q}_{1}}^{(i_{1})}S_{-\mathbf{q}_{2}}^{(i_{2})}\ldots
S_{-\mathbf{q}_{n}}^{(i_{n})}$
$\displaystyle\quad\times\tilde{\mu}_{n}(\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{n}|0)\delta_{\mathbf{q}_{1}+\ldots+\mathbf{q}_{n},0}\,,$
(2.14)

where

$\displaystyle\tilde{\mu}_{n}(\mathbf{q}_{1},\ldots,\mathbf{q}_{n}|0)=\beta^{-1}\langle
T_{\beta}\\{\hat{\rho}_{\mathbf{q}_{1},0}\
\hat{\rho}_{\mathbf{q}_{2},0}\ldots\hat{\rho}_{\mathbf{q}_{n},0}\\}\rangle_{\text{e}}^{\text{c}}$
(2.15)

is a cumulant part of the average of the product of $n$ operators
$\hat{\rho}_{\mathbf{q},0}$. This is a static limit of $n$-particles
correlation function of the reference system

$\displaystyle\tilde{\mu}_{n}(y_{1},\ldots,y_{n})=\tilde{\mu}_{n}(\mathbf{q}_{1},\ldots,\mathbf{q}_{n}|\nu_{1},\ldots,\nu_{n})=\beta^{-1}\langle
T_{\beta}\\{\hat{\rho}_{\mathbf{q}_{1},\nu_{1}}\hat{\rho}_{\mathbf{q}_{2},\nu_{2}}\ldots\hat{\rho}_{\mathbf{q}_{n},\nu_{n}}\\}\rangle_{\text{e}}^{\text{c}}.$
(2.16)

The functions (2.15), (2.16) are a generalization of “many-tails” of non-
relativistic theory of the electron-ion model of metals [16, 17] and have a
well-defined physical meaning. In particular, the functions (2.16) are a
spectral representation of $n$-particles correlation functions, which are
given in the coordinate space. For example, the binary distribution function
of the interacting electron gas $F_{2}(\mathbf{r})$ is associated with the
function $\tilde{\mu}_{2}(\mathbf{q},-\mathbf{q}|\nu,-\nu)$ of the expression

$\displaystyle F_{2}(\mathbf{r})=1+[\beta
N_{\text{e}}(N_{\text{e}}-1)]^{-1}\sum\limits_{\nu}\sum\limits_{\mathbf{q}\neq
0}\tilde{\mu}_{2}(\mathbf{q},-\mathbf{q}|\nu,-\nu)\exp{[\mathrm{i}(\mathbf{q},\mathbf{r})].}$
(2.17)

The dynamic correlation functions of $n$-particles (2.16) are determined by
the polarization operators $M_{n}(y_{1},\ldots,y_{n})$ of $n$-particles [17]

$\displaystyle\mu_{2}(y,-y)=M_{2}(y,-y)\left[1+\frac{V_{q}}{V}M_{2}(y,-y)\right]^{-1},$
$\displaystyle\mu_{n}(y_{1},\ldots,y_{n})=M_{n}(y_{1},\ldots,y_{n})\prod\limits_{i=1}^{n}\left[1+\frac{V_{q_{i}}}{V}M_{2}(y_{i},-y_{i})\right]^{-1},\quad
n\geqslant 3.$ (2.18)

These ratios generalize the well-known random phase approximation, in which
$M_{2}(y,-y)\Rightarrow\tilde{\mu}_{2}^{0}(y,-y),\\\
M_{n}(y_{1},\ldots,y_{n})\Rightarrow\tilde{\mu}_{n}^{0}(y_{1},\ldots,y_{n})$,
where

$\displaystyle\tilde{\mu}_{n}^{0}(y_{1},\ldots,y_{n})=\beta^{-1}\langle
T_{\beta}\\{\hat{\rho}_{\mathbf{q}_{1},\nu_{1}}\ldots\hat{\rho}_{\mathbf{q}_{n},\nu_{n}}\\}\rangle_{0}^{\text{c}}$
(2.19)

is the spectral image of correlation functions of an ideal electron gas model,

$\displaystyle
M_{2}(y,-y)=\tilde{\mu}_{2}^{0}(y,-y)\left[1-\frac{V_{q}}{V}\tilde{\mu}_{2}^{0}(y,-y)\,G(y)\right]^{-1},$
$\displaystyle
M_{n}(y_{1},\ldots,y_{n})=\tilde{\mu}_{n}^{0}(y_{1},\ldots,y_{n})\prod\limits_{i=1}^{n}\left[1-\frac{V_{q_{i}}}{V}\tilde{\mu}_{2}^{0}(y_{i},-y_{i})\,G(y_{i})\right]^{-1}$
(2.20)

for the $n\geqslant 3$, so the problem of calculating the functions (2.16) is
reduced to the calculation of correlation functions of the ideal relativistic
gas model (2.19) and the local field correction function $G(y)$ for the
relativistic interacting electron gas.

##  3 Correlation functions of ideal degenerate relativistic electron gas

Static and dynamic correlation functions of the non-relativistic ideal
electron gas are well known. The analytical expression for
$\tilde{\mu}_{2}^{0}(y,-y)$ was obtained in [20]. The function
$\tilde{\mu}_{3}^{0}(\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{1}-\mathbf{q}_{2}|0,0,0)$
and the function
$\tilde{\mu}_{4}^{0}(\mathbf{q}_{1},-\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{2}|0,\ldots,0)$
were calculated in [18, 19, 21]. The dynamic functions
$\tilde{\mu}_{3}^{0}(y_{1},y_{2},-y_{1}-y_{2})$ and
$\tilde{\mu}_{4}^{0}(y_{1},-y_{1},y_{2},-y_{2})$ were first calculated in
[16]. The calculation of these functions for a relativistic model is a complex
problem, because the electron spectrum is not a quadratic function of the wave
vector.

According to the definition (2.19), the functions
$\tilde{\mu}_{n}^{0}(y_{1},\ldots,y_{n})$ are a convolution of one-particle
Green’s functions, the same as in a non-relativistic case:

$\displaystyle\mu_{2}^{0}(y,-y)=-\frac{1}{\beta}\sum\limits_{\mathbf{k},s}\sum\limits_{\nu_{*}}G_{\mathbf{k},s}(\nu_{*})G_{\mathbf{k+q},s}(\nu_{*}+\nu),$
$\displaystyle\mu_{3}^{0}(y_{1},y_{2},y_{3})=\frac{2}{\beta}\sum\limits_{\mathbf{k},s}\sum\limits_{\nu_{*}}G_{\mathbf{k},s}(\nu_{*})G_{\mathbf{k+q}_{1},s}(\nu_{*}+\nu_{1})G_{\mathbf{k-q}_{2},s}(\nu_{*}-\nu_{2})\delta_{\mathbf{q}_{1}+\mathbf{q}_{2}+\mathbf{q}_{3},0}\delta_{\nu_{1}+\nu_{2}+\nu_{3},0}\,,$
$\displaystyle\mu_{4}^{0}(y_{1},-y_{1},y_{2},-y_{2})=\frac{1}{\beta}\sum\limits_{\mathbf{k},s}\sum\limits_{\nu_{*}}G_{\mathbf{k},s}(\nu_{*})G_{\mathbf{k-q}_{1},s}(\nu_{*}-\nu_{1})\sum_{\sigma=\pm
1}G_{\mathbf{k}-\sigma\mathbf{q}_{2},s}(\nu_{*}-\sigma\nu_{2})$
$\displaystyle\qquad\qquad\qquad\qquad\times[2G_{\mathbf{k},s}(\nu_{*})+G_{\mathbf{k+q}_{1}+\sigma\mathbf{q}_{2},s}(\nu_{*}+\nu_{1}+\sigma\nu_{2})].$
(3.1)

Factorizing the products of Green’s functions and using the ratio

$\displaystyle\frac{1}{\beta}\sum\limits_{\nu_{*}}G_{\mathbf{k},s}{(\nu_{*})}=n_{\mathbf{k},s}=\\{1+\exp{[\beta(E_{k}-\mu)]}\\}^{-1},$
(3.2)

we obtain representation functions $\tilde{\mu}_{n}^{0}(y_{1},\ldots,y_{n})$:

$\displaystyle\mu_{2}^{0}(y,-y)=-2\mathop{\symoperators
Re}\nolimits\sum\limits_{\mathbf{k},s}\frac{n_{\mathbf{k},s}}{\mathrm{i}\nu+E_{\mathbf{k}}-E_{\mathbf{k+q}}}\,,$
$\displaystyle\mu_{3}^{0}(y_{1},y_{2},y_{3})=\delta_{y_{1}+y_{2}+y_{3},0}[\gamma_{3}(y_{1},-y_{2})+\gamma_{3}(y_{2},-y_{3})+\gamma_{3}(y_{3},-y_{1})],$
$\displaystyle\gamma_{3}(y_{1},y_{2})=2\mathop{\symoperators
Re}\nolimits\sum\limits_{\mathbf{k},s}n_{\mathbf{k},s}(\mathrm{i}\nu_{1}+E_{\mathbf{k}}-E_{\mathbf{k+q}_{1}})^{-1}(\mathrm{i}\nu_{2}+E_{\mathbf{k}}-E_{\mathbf{k+q}_{2}})^{-1}\ldots.$
(3.3)

###  3.1 Two-particle correlation function

Rewriting the sum over vector $\mathbf{k}$ via integral using a spherical
coordinate system and integrating by the angular variables, we obtain a
representation:

$\displaystyle\tilde{\mu}_{2}^{0}(y,-y)$
$\displaystyle=\frac{3N_{\text{e}}}{m_{0}c^{2}x^{2}}J_{2}(q_{*},\tilde{\nu}|x),$
$\displaystyle J_{2}(q_{*},\tilde{\nu}|x)$
$\displaystyle=\frac{1}{2xq_{*}}\sum\limits_{s}\int\limits_{0}^{\infty}\mathrm{d}k_{*}k_{*}n_{k_{*},s}A(k_{*}|q_{*},\tilde{\nu}),$
$\displaystyle A(k_{*}|q_{*},\tilde{\nu})$
$\displaystyle=\sum\limits_{\sigma=\pm
1}\sigma\biggl{\\{}\big{[}1+(k_{*}+\sigma
q_{*})^{2}\big{]}^{1/2}-\tilde{\nu}\arctan{\big{[}\tilde{\nu}\,^{-1}\eta_{\sigma}(k_{*},q_{*})\big{]}}$
$\displaystyle\quad+\frac{1}{2}(1+k_{*}^{2})^{1/2}\ln{[\tilde{\nu}^{2}+\eta_{\sigma}^{2}(k_{*},q_{*})]}\biggr{\\}},$
$\displaystyle\eta_{\sigma}(k_{*},q_{*})$
$\displaystyle=\big{[}1+(k_{*}+\sigma
q_{*})^{2}\big{]}^{1/2}-(1+k_{*}^{2})^{1/2}.$ (3.4)

Here, there appear dimensionless variables

$\displaystyle k_{*}=\frac{x|\mathbf{k}|}{k_{\text{F}}}\,,\qquad
q_{*}=\frac{x|\mathbf{q}|}{k_{\text{F}}}\,,\qquad\tilde{\nu}=\frac{\nu}{m_{0}c^{2}}\,,$
(3.5)

where $x=\hbar k_{\text{F}}(m_{0}c)^{-1}$ is a relativistic parameter
[$k_{\text{F}}=(3\piup^{2}N_{\text{e}}/V)^{1/3}$ is the Fermi wave number]. In
the static case,

$\displaystyle q_{*}x\,J_{2}(q_{*},0|x)$
$\displaystyle=\frac{2}{9}(R_{+}-R_{-})\left(1+\frac{7}{4}x^{2}-\frac{q_{*}^{2}}{8}\right)+\frac{5q_{*}x}{72}(R_{+}+R_{-})+\frac{q_{*}x}{12}R_{0}$
$\displaystyle+\frac{R_{0}^{3}}{3}\ln{\left|\frac{R_{+}-R_{0}}{R_{-}-R_{0}}\right|}+\frac{q_{*}}{8}\left(1+\frac{q_{*}^{2}}{6}\right)\left[2\ln{|x+R_{0}|}-\ln{|(R_{+}+x+q_{*})(R_{-}+x-q_{*})|}\right]$
$\displaystyle+\frac{S_{q}^{3}}{6}\Biggl{(}\ln{\left|\frac{1+\frac{1}{2}q_{*}^{2}+\frac{1}{2}xq_{*}+S_{q}R_{+}}{1+\frac{1}{2}q_{*}^{2}-\frac{1}{2}xq_{*}+S_{q}R_{-}}\right|}-\ln{\left|\frac{1+\frac{1}{2}q_{*}x+S_{q}R_{0}}{1-\frac{1}{2}q_{*}x+S_{q}R_{0}}\right|}-2\ln{\left|\frac{x+\frac{1}{2}q_{*}}{x-\frac{1}{2}q_{*}}\right|}\Biggr{)},$
$\displaystyle R_{0}$ $\displaystyle=(1+x^{2})^{1/2},\qquad
S_{q}=\left(1+\frac{1}{4}q_{*}^{2}\right)^{1/2},\qquad
R_{\pm}=\left[1+(q_{*}\pm x)^{2}\right]^{1/2}.$ (3.6)

As in the non-relativistic case, the function $J_{2}(q_{*},0)$ has a weak
logarithmic peculiarity of type
$(x-\frac{1}{2}q_{*})\ln{|x-\frac{1}{2}q_{*}|}$ in the vicinity of $q_{*}=2x$
($|\mathbf{q}|=2k_{\text{F}}$). In general, the correlation functions of a
relativistic model are similar to the corresponding functions of a non-
relativistic model, which is clearly visible from the asymptotic of the
function $\tilde{\mu}_{2}^{0}(\mathbf{q},-\mathbf{q}|0)$:

$\displaystyle\tilde{\mu}_{2}^{0}(\mathbf{q},-\mathbf{q}|0)\Rightarrow\begin{cases}\dfrac{3N_{\text{e}}(1+x^{2})^{1/2}}{m_{0}c^{2}x^{2}}+\ldots&\text{by}\,\,q\rightarrow
0;\vspace{0.8ex}\\\
\dfrac{2N_{\text{e}}}{m_{0}c^{2}q_{*}}+\ldots&\text{by}\,\,q\rightarrow\infty\end{cases}$
(3.7)

in the relativistic case,

$\displaystyle\tilde{\mu}_{2}^{0}(\mathbf{q},-\mathbf{q}|0)\Rightarrow\begin{cases}\dfrac{3N_{\text{e}}}{2\varepsilon_{\text{F}}}+\ldots&\text{by}\,\,q\rightarrow
0;\vspace{0.8ex}\\\
\dfrac{2N_{\text{e}}}{\hbar^{2}q^{2}/2m_{0}}+\ldots&\text{by}\,\,q\rightarrow\infty\end{cases}$
(3.8)

in the non-relativistic approximation $(2\varepsilon_{\text{F}}\equiv
m_{0}c^{2}x^{2})$. A peculiarity of the function
$\tilde{\mu}_{2}^{0}(\mathbf{q},-\mathbf{q}|0)$ is its strong dependence on
the relativistic parameter, as shown in figure 1.

Figure 1: Dependence of the function $J_{2}(q_{*},0|x)$ on the wave vector
$\mathbf{q}$ at different values of the relativistic parameter.

In the dynamic case, the $\tilde{\mu}_{2}^{0}(y,-y)$ can be represented as the
approximation

$\displaystyle\tilde{\mu}_{2}^{0}(y,-y)$
$\displaystyle\approx\frac{3N_{\text{e}}}{m_{0}c^{2}x^{2}}\tilde{J}_{2}(q,v|x),$
$\displaystyle\tilde{J}_{2}(q,v|x)$
$\displaystyle=\frac{1}{4}C(q_{*}|x)\biggl{[}1-v\left(\arctan\frac{1+q_{*}/2x}{v}-\arctan\frac{1-q_{*}/2x}{v}\right)$
$\displaystyle\quad+\frac{x}{2q_{*}}\left(1-\frac{q_{*}^{2}}{4x^{2}}+v^{2}\right)\ln\frac{v^{2}+(1+q_{*}/2x)^{2}}{v^{2}+(1-q_{*}/2x)^{2}}\biggr{]},$
$\displaystyle v$ $\displaystyle=\nu_{*}(2xq_{*})^{-1}C(q_{*}|x),\qquad
C(q_{*}|x)=(1+x^{2}+q_{*}^{2})^{1/2}+(1+x^{2})^{1/2},$ (3.9)

that only slightly deviates near the maximum from the numerical result,
calculated by formula (3.4). In the non-relativistic limit and the long-
wavelength approximation $C(q_{*}|x)\rightarrow 2$,
$v\rightarrow\nu(2\varepsilon_{\text{F}}q/k_{\text{F}})^{-1}$, and the
function (3.9) coincides with the known expression for a non-relativistic
function $\tilde{\mu}_{2}^{0}(y,-y)$ [15]. The expression (3.9) is here only
for completeness. We used the result of numerical calculations for the dynamic
function $\tilde{\mu}_{2}^{0}(y,-y)$ to obtain the correlation energy of the
reference system (the model of interacting electron gas).

###  3.2 Three-particle correlation function

The function $\tilde{\mu}_{3}^{0}(y,-y,0)$, which is a partial case of three-
particle dynamic functions when $\mathbf{q}_{2}=-\mathbf{q}_{1}$,
$\nu_{2}=-\nu_{1}$, at a full degeneration has an exact analytical image:

$\displaystyle\tilde{\mu}_{3}^{0}(y,-y,0)=\frac{3N_{\text{e}}}{(m_{0}c^{2})^{2}q_{*}x^{3}}(1+x^{2})^{1/2}A(x|q_{*},\tilde{\nu}).$
(3.10)

In the static limit,

$\displaystyle\tilde{\mu}_{3}^{0}(\mathbf{q},-\mathbf{q},0|0)=\frac{3N_{\text{e}}}{(m_{0}c^{2}x^{2})^{2}}J_{3}(\mathbf{q},-\mathbf{q},0|x),$
$\displaystyle
J_{3}(\mathbf{q},-\mathbf{q},0|x)=\frac{R_{0}}{\tilde{q}}\left(\tilde{R}_{+}-\tilde{R}_{-}+R_{0}\ln{\left|\frac{\tilde{R}_{+}-R_{0}}{\tilde{R}_{-}-R_{0}}\right|}\right),$
$\displaystyle\tilde{R}_{\pm}=[1+x^{2}(1\pm\tilde{q})^{2}]^{1/2},\qquad
R_{0}=(1+x^{2})^{1/2}.$ (3.11)

In the formula (3.11), a “non-relativistic” scale was used for the wave vector
$(\tilde{q}\equiv|\mathbf{q}|/k_{\text{F}})$. In the long-wavelength limit,

$\displaystyle\tilde{\mu}_{3}^{0}(\mathbf{q},-\mathbf{q},0|0)=\frac{3N_{\text{e}}}{(m_{0}c^{2}x^{2})^{2}}(1+2x^{2}).$
(3.12)

Dependence of the dimensionless factor $J_{3}(\mathbf{q},-\mathbf{q},0|x)$ on
the wave vector and the relativistic parameter is illustrated in figure 2. As
in non-relativistic case, the function (3.11) has a logarithmic peculiarity at
$q=2k_{\text{F}}$.

Figure 2: Dependence of the static function
$J_{3}(\mathbf{q},-\mathbf{q},0|x)$ on the wave vector $\mathbf{q}$ at
different values of the relativistic parameter.

The formulae (3.6), (3.9) and (3.11), and figures 1, 2 reveal the general
property of correlation functions $\tilde{\mu}_{n}^{0}(y_{1},\ldots,y_{n})$ —
a steep decrease for large wave vectors $(|\mathbf{q}_{i}|>2k_{\text{F}})$,
providing a convergence of integrals in the series (2.14).

In general, for a rough estimation of convergence of series (2.14) we consider
a chemically homogeneous model $(z_{1}=z_{2}=z$, $N_{1}+N_{2}=N_{n})$,
constraining the integration for each independent vector $\mathbf{q}_{i}$ of
the area $|\mathbf{q}_{i}|<2k_{\text{F}}$, neglecting the screening
interactions, replacing the product of structural factors
$S_{\mathbf{q}_{1}}S_{\mathbf{q}_{2}}\ldots S_{\mathbf{q}_{n}}$ with $N_{n}$,
we replace the functions
$\tilde{\mu}_{n}^{0}(\mathbf{q}_{1},\ldots,\mathbf{q}_{n}|0)$ with
$3N_{\text{e}}(m_{0}c^{2}x^{2})^{1-n}(1+x^{2})^{\frac{1}{2}(n-1)}$, which
approximately corresponds to the long-wavelength asymptotic. For the magnitude
of $n$-th member of series (2.14) it can be estimated as

$\displaystyle
N_{\text{e}}m_{0}c^{2}z^{n-1}\alpha_{0}^{n}x^{2-n}(1+x^{2})^{\frac{1}{2}(n-1)},$
(3.13)

where $\alpha_{0}=e^{2}/\hbar c$ is the fine structure parameter. Hence, the
series (2.14) is an expansion for a dimensionless parameter $z\alpha_{0}$,
which varies from 0.014 (helium dwarf) to 0.19 (iron dwarf). For the typical
dwarfs, mainly consisting of nitrogen and oxygen, $z\alpha_{0}\approx 0{.}1$.
That expansion parameter is a small value, which makes it possible to restrict
the consideration to two- and three-electron correlations (we note, that
correlation energy of the reference system is of the order of
$\alpha_{0}^{2}$). Moreover, for the three-electron function
$\tilde{\mu}_{3}^{0}(\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{1}-\mathbf{q}_{2}|0)$,
there can be used an approximate analytical expression, because the main
contributions provide the two-electron correlations, and the contribution of
three-electron correlations play a role of a correction.

The calculation of correlation functions
$\tilde{\mu}_{3}^{0}(y_{1},y_{2},-y_{1}-y_{2})$ and
$\tilde{\mu}_{4}^{0}(y_{1},-y_{1},y_{2},-y_{2})$ in the static or dynamic
cases in the non-relativistic theory is based on Feynman identity [22], which
allows one to integrate over angular variables of vector $\mathbf{k}$ at a
fixed configuration of vectors $\mathbf{q}_{1}$ and $\mathbf{q}_{2}$ in terms
of $\gamma_{3}(y_{i},y_{j})$. Unfortunately, this identity cannot be used in
the exact calculation due to the complex dependence of relativistic electron
energy on the wave vector.

In order to make an approximate calculation using the identity transformation,
we present $\gamma_{3}(\mathbf{q}_{i},\mathbf{q}_{j})$ as

$\displaystyle\gamma_{3}(\mathbf{q}_{i},\mathbf{q}_{j})=2\sum\limits_{\mathbf{k},s}n_{\mathbf{k},s}(\tilde{E}_{\mathbf{k}}+\tilde{E}_{\mathbf{k+q}_{i}})(\tilde{E}_{\mathbf{k}}+\tilde{E}_{\mathbf{k+q}_{j}})(\hbar
c)^{-4}[2(\mathbf{k},\mathbf{q}_{i})+q_{i}^{2}]^{-1}[2(\mathbf{k},\mathbf{q}_{j})+q_{j}^{2}]^{-1},$
(3.14)

where $\tilde{E}_{k}=[(m_{0}c^{2})^{2}+\hbar^{2}c^{2}k^{2}]^{1/2}$. Then, we
use the approximation

$\displaystyle\tilde{E}_{k}+\tilde{E}_{\mathbf{k+q}_{i}}\Rightarrow
m_{0}c^{2}C(\tilde{q}_{i}|\tilde{k}),\qquad
C(\tilde{q}_{i}|\tilde{k})=[1+x^{2}(\tilde{k}^{2}+\tilde{q}_{i}^{2})]^{1/2}+(1+x^{2}\tilde{k}^{2})^{1/2},$
(3.15)

which is asymptotically correct both at small and at large $q_{i}$. According
to the Feynman identity,

$\displaystyle\frac{1}{[2(\mathbf{k},\mathbf{q}_{i})+q_{i}^{2}][2(\mathbf{k},\mathbf{q}_{j})+q_{j}^{2}]}=\int\limits_{0}^{1}\frac{\mathrm{d}z}{F^{2}(\mathbf{q}_{i},\mathbf{q}_{j}|\mathbf{k})}\,,$
$\displaystyle
F(\mathbf{q}_{i},\mathbf{q}_{j}|\mathbf{k})=\Omega_{ij}+2(\mathbf{k},\bm{\rho}_{ij}),\qquad\Omega_{ij}\equiv
q_{j}^{2}+z(q_{i}^{2}-q_{j}^{2});\qquad\bm{\rho}_{ij}=z\mathbf{q}_{i}+(1-z)\mathbf{q}_{j}\,.$
(3.16)

Rewriting the sum over vector $\mathbf{k}$ via integral, we use dimensionless
variable
$\tilde{k}=|\mathbf{k}|/k_{\text{F}},\,\,\tilde{q}_{i}=|\mathbf{q}_{i}|/k_{\text{F}}$,
and the spherical coordinate system, the $Oz$ axis of which coincides with the
vector $\bm{\rho}_{ij}$, we perform integration over the angular variables,
reducing $\gamma_{3}(\mathbf{q}_{i},\mathbf{q}_{j})$ to one-dimensional
integral:

$\displaystyle\gamma_{3}(\mathbf{q}_{i},\mathbf{q}_{j})=\frac{3N_{\text{e}}}{4(m_{0}c^{2}x^{2})^{2}}\int\limits_{0}^{1}\mathrm{d}k\,C(q_{i}|k)\,C(q_{j}|k)f_{ij}(k),$
$\displaystyle
f_{ij}(k)=\frac{1}{\sqrt{-\delta(k)}}\ln\left|\frac{R_{ij}+[-\delta(k)]^{-1/2}}{R_{ij}-[-\delta(k)]^{-1/2}}\right|\quad\text{by}\,\,k<q_{\text{R}};$
$\displaystyle
f_{ij}(k)=\frac{2}{\sqrt{\delta(k)}}\arctan[\delta^{1/2}(k)R^{-1}_{ij}]\quad\text{by}\,\,k>q_{\text{R}}.$
(3.17)

Here, the following notations are introduced:

$\displaystyle R_{ij}\equiv
R_{ij}(k)=2(\mathbf{q}_{i},\mathbf{q}_{j})-\frac{q_{i}^{2}q_{j}^{2}}{2k^{2}}$
$\displaystyle;\qquad\delta(k)=\delta_{ij}(k)=\left(1-\frac{q_{\text{R}}^{2}}{k^{2}}\right)4q_{i}^{2}q_{j}^{2}(1-t_{ij}^{2});$
$\displaystyle q_{\text{R}}$
$\displaystyle=(\mathbf{q}_{i}-\mathbf{q}_{j})^{2}[4(1-t_{ij}^{2})]^{-1};$
(3.18)

$\delta(k)$ is the invariant of the problem
$[\delta_{12}(k)=\delta_{23}(k)=\delta_{31}(k)]$, $q_{\text{R}}$ is the radius
of the circle, circumscribing the triangle constructed on the vectors
$\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{1}-\mathbf{q}_{2}$; $t_{ij}$ is a
cosine of the angle between the vectors $\mathbf{q}_{i},\mathbf{q}_{j}$. In
the formulae (3.17), (3.18), the variables $k$ and $q_{i},q_{j}$ are
dimensionless (in unit $k_{\text{F}}$).

The substitution $C(q_{i}|k)\rightarrow 2(1+x^{2}k^{2})^{1/2}$ corresponds to
long-wavelength approximation, which allows one to rewrite
$\tilde{\mu}_{3}^{0}(\mathbf{q}_{1},\mathbf{q}_{2},\mathbf{q}_{3}|0)$ in a
compact form:

$\displaystyle\tilde{\mu}_{3}^{0}(\mathbf{q}_{1},\mathbf{q}_{2},\mathbf{q}_{3}|0)\cong\frac{3N_{\text{e}}}{(m_{0}c^{2}x^{2})^{2}}\int\limits_{0}^{1}\mathrm{d}k(1+x^{2}k^{2})\Phi(k|{q_{1},q_{2},q_{3}}),$
$\displaystyle\Phi(k|{q_{1},q_{2},q_{3}})=\frac{k}{a_{1}(k)}\frac{1}{q_{1}q_{2}q_{3}}\ln\biggl{|}\frac{1+a_{1}(k)D(k)}{1-a_{1}(k)D(k)}\biggr{|}\quad\text{by}\,\,k<q_{\text{R}};$
$\displaystyle\Phi(k|{q_{1},q_{2},q_{3}})=\frac{2k}{a_{2}(k)}\frac{1}{q_{1}q_{2}q_{3}}\arctan[a_{2}(k)D(k)]\quad\text{by}\,\,k>q_{\text{R}}.$
(3.19)

In this formula,

$\displaystyle
a_{1}(k)=\left(1-\frac{k^{2}}{q_{\text{R}}^{2}}\right)^{1/2};\qquad
a_{2}(k)=\left(\frac{k^{2}}{q_{\text{R}}^{2}}-1\right)^{1/2};$ $\displaystyle
D(k)=\frac{q_{1}q_{2}q_{3}}{4k^{3}}\left[1-\frac{(q_{1}^{2}+q_{2}^{2}+q_{3}^{2})}{8k^{2}}\right]\frac{1}{P(k)};$
$\displaystyle
P(k)=1-\frac{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}{4k^{2}}+\frac{q_{1}^{4}+q_{2}^{4}+q_{3}^{4}}{32k^{4}}+\frac{(q_{1}q_{2}q_{3})^{2}}{64k^{6}}.$
(3.20)

Dependence of function
$\tilde{\mu}_{3}^{0}(\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{1}-\mathbf{q}_{2}|0)(3N_{\text{e}})^{-1}(m_{0}c^{2}x^{2})^{2}$
on wave vectors for different $t$ are illustrated in figures 3, 4.

Figure 3: The function
$J_{3}({q}_{1},{q}_{2},t)=\tilde{\mu}_{3}^{0}(\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{1}-\mathbf{q}_{2})(3N_{\text{e}})^{-1}(m_{0}c^{2}x^{2})^{2}$
at $t=0$ and the relativistic parameter $x=1.0$. Figure 4: The function
$J_{3}({q}_{1},{q}_{2},t)=\tilde{\mu}_{3}^{0}(\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{1}-\mathbf{q}_{2})(3N_{\text{e}})^{-1}(m_{0}c^{2}x^{2})^{2}$
at $t=0.5$ and the relativistic parameter $x=1.0$.

###  3.3 The local field correction function

It is well known from the non-relativistic electron fluid theory [16] that the
local field correction function (LFCF) in the weak non-ideal model is a
universal function of the variable $y=(\mathbf{q},\nu)$. It does not depend on
any parameters and corresponds to the approximation

$\displaystyle
G_{\text{id}}(y)=-\frac{\sum\nolimits_{y_{1}}V_{q_{1}}\tilde{\mu}_{4}^{0}(y,-y,y_{1},-y_{1})}{2\beta
V_{q}[\tilde{\mu}_{2}^{0}(y,-y)]^{2}}\,.$ (3.21)

We have calculated the LFCF, using the correlation functions of the
relativistic ideal electron model. Summarizing over the frequencies $\nu_{1}$
and $\nu_{*}$ (which appears in $\tilde{\mu}_{4}^{0}(y,-y,y_{1},-y_{1}$) and
using the formula (3.2), we obtain a representation:

$\displaystyle
G_{\text{id}}(y)=-V_{q}^{-1}[\tilde{\mu}_{2}^{0}(y,-y)]^{-2}\sum\limits_{s}\sum\limits_{\mathbf{k}_{1},\mathbf{k}_{2}}n_{\mathbf{k}_{1},s}n_{\mathbf{k}_{2},s}\left[V(\mathbf{k}_{1}-\mathbf{k}_{2})f^{-}_{{\mathbf{k}_{1},\mathbf{k}_{2}}}(\mathbf{q},\nu)-V(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{q})f^{+}_{{\mathbf{k}_{1},\mathbf{k}_{2}}}(\mathbf{q},\nu)\right],$
$\displaystyle
f^{\mp}_{{\mathbf{k}_{1},\mathbf{k}_{2}}}(\mathbf{q},\nu)=\mathop{\symoperators
Re}\nolimits\left[(\mathrm{i}\nu+E_{\mathbf{k}_{1}}-E_{\mathbf{k}_{1}+\mathbf{q}})^{-1}\mp(\pm\mathrm{i}\nu+E_{\mathbf{k}_{2}}-E_{\mathbf{k}_{2}+\mathbf{q}})^{-1}\right]^{2}.$
(3.22)

The behavior of relativistic LFCF at small and large values of vector
$\mathbf{q}$ is the same as that of non-relativistic case [23]:

$\displaystyle
G_{\text{id}}(y)\Rightarrow\begin{cases}\dfrac{(q/k_{\text{F}})^{2}}{4}+\ldots\quad\text{by}\,\,\nu=0;\,\,q\ll
k_{\text{F}};\\\
\dfrac{3(q/k_{\text{F}})^{2}}{20}+\ldots\quad\text{by}\,\,\nu\gg\varepsilon_{\text{F}};\,\,q\ll
k_{\text{F}};\\\ \dfrac{1}{3}+\ldots\quad\text{by}\,\,q\gg
k_{\text{F}}\,\,\text{and \,any}\,\,\nu.\end{cases}$ (3.23)

For the numerical calculation at the absolute zero temperature, we have used a
cylindrical coordinate system for the vectors $\mathbf{k}_{1},\mathbf{k}_{2}$
[$\mathbf{k}_{j}=(z_{j},\rho_{j},\varphi_{j})$,
$k_{j}^{2}=z_{j}^{2}+\rho_{j}^{2}$], in which
$(\mathbf{k}_{j},\mathbf{q})=qz_{j}$,
$(\mathbf{k}_{1}-\mathbf{k}_{2})^{2}=\rho_{1}^{2}+\rho_{2}^{2}+(z_{1}-z_{2})^{2}-2\rho_{1}\rho_{2}\cos{(\varphi_{1}-\varphi_{2})}$,
$(\mathbf{k}_{1},\mathbf{k}_{2})=z_{1}z_{2}+\rho_{1}\rho_{2}\cos{(\varphi_{1}-\varphi_{2})}$.
Integrating over the angular variables $\varphi_{1},\varphi_{2}$, we reduce
$G_{\text{id}}(y)$ to the following 4-dimensional integral:

$\displaystyle G_{\text{id}}(q,\nu)$
$\displaystyle=\frac{q^{2}x^{4}}{8}J_{2}^{-2}(qx,\tilde{\nu}x^{2}|x)\int\limits_{-1}^{1}\mathrm{d}z_{1}\int\limits_{-1}^{1}\mathrm{d}z_{2}\int\limits_{0}^{\sqrt{1-z_{1}^{2}}}\rho_{1}\mathrm{d}\rho_{1}\int\limits_{0}^{\sqrt{1-z_{2}^{2}}}\rho_{2}\mathrm{d}\rho_{2}$
$\displaystyle\quad\times\left[\frac{f^{+}_{q,\nu}(z_{1},z_{2},\rho_{1},\rho_{2})}{W_{+}(z_{1},z_{2},\rho_{1},\rho_{2})}-\frac{f^{-}_{q,\nu}(z_{1},z_{2},\rho_{1},\rho_{2})}{W_{-}(z_{1},z_{2},\rho_{1},\rho_{2})}\right].$
(3.24)

Here, the following notations are used:

$\displaystyle
f^{\pm}_{q,\nu}(z_{1},z_{2},\rho_{1},\rho_{2})=\left(\frac{\eta_{1}}{\eta_{1}^{2}+\bar{\nu}^{2}}\pm\frac{\eta_{2}}{\eta_{2}^{2}+\bar{\nu}^{2}}\right)^{2}-\bar{\nu}^{2}\left(\frac{1}{\eta_{1}^{2}+\bar{\nu}^{2}}-\frac{1}{\eta_{2}^{2}+\bar{\nu}^{2}}\right)^{2},$
$\displaystyle\eta_{i}=[1+x^{2}(z_{i}^{2}+\rho_{i}^{2})]^{1/2}-\left\\{1+x^{2}[\rho_{i}^{2}+(z_{i}+q)^{2}]\right\\}^{1/2},$
$\displaystyle
W_{+}(z_{1},z_{2},\rho_{1},\rho_{2})=\left[(z_{1}+z_{2}+q)^{4}+(\rho_{1}^{2}-\rho_{2}^{2})^{2}+2(\rho_{1}^{2}+\rho_{2}^{2})(z_{1}+z_{2}+q)^{2}\right]^{1/2},$
$\displaystyle
W_{-}(z_{1},z_{2},\rho_{1},\rho_{2})=\left[(z_{1}-z_{2})^{4}+(\rho_{1}^{2}-\rho_{2}^{2})^{2}+2(\rho_{1}^{2}+\rho_{2}^{2})(z_{1}-z_{2})^{2}\right]^{1/2}.$
(3.25)

For a comparison $G_{\text{id}}(q,\nu)$ with LFCF of non-relativistic theory,
there were used the variables $q=|\mathbf{q}|k_{\text{F}}^{-1}$,
$\bar{\nu}=x^{2}\tilde{\nu}$,
$\tilde{\nu}=\nu(\hbar^{2}k_{\text{F}}^{2}/m)^{-1}$. In figure 5 there is
shown a function $G_{\text{id}}(q,\nu)$, calculated by the formula (3.24) at
$\tilde{\nu}=0{.}01$, which is very close to the static limit. It is obvious
that the asymptotic behavior $G_{\text{id}}(q,\nu)$ for small and large values
of the $|\mathbf{q}|$ almost does not depend on the relativistic parameter.
The deviation of a relativistic correction from non-relativistic one is
significant near its maximum, which monotonously decreases with an increase of
relativistic parameter. Figure 6 illustrates the behavior of
$G_{\text{id}}(q,\nu)$ at a very high value of frequency
$(\tilde{\nu}=0{.}5)$. In general, the behavior of the correction corresponds
to the one of non-relativistic theory, and some deviations are caused by
frequency renormalization at $x\geqslant 1$.

Figure 5: Dependence of the LFCF $G_{\text{id}}(q,\nu)$ on the wave vector
$\mathbf{q}$ at frequency $\nu=0.01m_{0}c^{2}x^{2}$ (curve 1 — $x=0.05$; 2 —
$x=0.2$; 3 — $x=0.5$; 4 — $x=1.0$; 5 — $x=2.0$; 6 — $x=5.0$). Figure 6:
Dependence of the LFCF $G_{\text{id}}(q,\nu)$ on the wave vector $\mathbf{q}$
at frequency $\nu=0.5m_{0}c^{2}x^{2}$ (curve 1 — $x=0.05$; 2 — $x=0.2$; 3 —
$x=0.5$; 4 — $x=1.0$; 5 — $x=2.0$; 6 — $x=5.0$).

##  4 The energy of ground state

As we know from the theory of non-relativistic electron fluid, the transition
from thermodynamic description to the quantum mechanics is performed by the
following procedure: instead of the chemical potential of interacting system
$\mu$ there should be used a chemical potential of the ideal system $\mu_{0}$,
and in the $\Omega(\mu)$ there should be considered only the so-called
“normal” components, because the chemical potential shift $\mu-\mu_{0}$ is
compensated with contributions of the so-called “abnormal” components, which
arise in the correlation functions $\tilde{\mu}_{n}^{0}(y_{1},\ldots,y_{n})$
at $n\geqslant 4$. In the Green’s function approach, this fact was first
discovered in [24] and in the reference system approach — in [17] within the
perturbation theory.

Taking into account a weak non-ideal matter of dwarfs, we will consider only
two- and three-particle electron correlations. In this approach, the ground
state energy of the model is given as follows:

$\displaystyle E$ $\displaystyle\simeq
E_{\text{e}}+\frac{1}{2V}\sum\limits_{\mathbf{q}\neq
0}V_{q}\sum\limits_{i,j}z_{i}z_{j}\left[S^{(i)}_{\mathbf{q}}S^{(j)}_{-\mathbf{q}}-N_{i}\delta_{i,j}\right]-\sum\limits_{n=2}^{3}(n!V^{n})^{-1}\sum\limits_{i_{1},i_{2},\ldots,i_{n}}z_{i_{1}}z_{i_{2}}\ldots
z_{i_{n}}$
$\displaystyle\quad\times\sum\limits_{\mathbf{q}_{1},\mathbf{q}_{1},\ldots,\mathbf{q}_{n}\neq
0}V_{q_{1}}\ldots V_{q_{n}}S^{(i_{1})}_{-\mathbf{q}_{1}}\ldots
S^{(i_{n})}_{-\mathbf{q}_{n}}\delta_{\mathbf{q}_{1}+\ldots+\mathbf{q}_{n},0}\,\tilde{\mu}_{n}(\mathbf{q}_{1}\ldots\mathbf{q}_{n}|0).$
(4.1)

Here, $E_{\text{e}}$ is the ground state energy of the reference system (an
interacting relativistic electron gas), which can be calculated using the
expression

$\displaystyle E_{\text{e}}=E_{0}+\frac{1}{2\beta
V}\sum\limits_{\mathbf{q}\neq
0}\sum\limits_{\nu}V_{q}\int\limits_{0}^{1}\tilde{\mu}_{2}^{\lambda}(y,-y)\mathrm{d}\lambda,$
(4.2)

where $\tilde{\mu}_{2}^{\lambda}(y,-y)$ is the two-particle dynamic
correlation function of the auxiliary model of electrons with the potential of
interactions $\lambda V_{q}$, and $E_{0}$ is the energy of the ideal electron
system

$\displaystyle
E_{0}=\sum\limits_{\mathbf{k},s}n_{\mathbf{k},s}(\mu_{0})E_{k}.$ (4.3)

Extracting from the second term of formula (4.2) the contribution of ideal
correlation

$\displaystyle E_{\text{HF}}=-\frac{1}{2\beta V}\sum\limits_{\mathbf{q}\neq
0}V_{q}\sum\limits_{\nu}\tilde{\mu}_{2}^{0}(y,-y)=-\frac{1}{2V}\sum\limits_{\mathbf{q}\neq
0}\sum\limits_{\mathbf{k},s}V_{q}\,n_{\mathbf{k}+\mathbf{q}/2,s}n_{\mathbf{k}-\mathbf{q}/2,s}$
(4.4)

and considering, that $G_{\text{id}}(y)$ does not depend on the “coupling
constant” $\lambda$, we have represented $E_{\text{e}}$ in a traditional form

$\displaystyle E_{\text{e}}=E_{0}+E_{\text{HF}}+E_{\text{c}}\,,$ (4.5)

where

$\displaystyle E_{\text{c}}=\frac{1}{2\beta}\sum\limits_{\mathbf{q}\neq
0}\sum\limits_{\nu}\frac{\ln[1+L(y)]-L(y)}{1-G_{\text{id}}(y)}\,,\qquad
L(y)=\frac{V_{q}}{V}\tilde{\mu}_{2}^{0}(y,-y)[1-G_{\text{id}}(y)]$ (4.6)

is the so-called correlation energy (the contribution of non-ideal
correlations). In units $m_{0}c^{2}$ we obtain

$\displaystyle
E_{\text{e}}=N_{\text{e}}m_{0}c^{2}\left[\varepsilon_{0}(x)-\frac{3}{4\piup}\alpha_{0}x+\alpha_{0}^{2}\,\varepsilon_{\text{c}}(x)\right],$
(4.7)

where

$\displaystyle\varepsilon_{0}(x)=(2x)^{-3}\left\\{3x(1+x^{2})^{1/2}(1+2x^{2})-8x^{3}-3\ln{\big{[}x+(1+x^{2})^{1/2}\big{]}}\right\\}$
(4.8)

is the contribution of an ideal system per one electron,
$-3\alpha_{0}x(4\piup)^{-1}$ is the contribution of interactions of the
Hartree-Fock approximation, $\alpha_{0}^{2}\varepsilon_{\text{c}}(x)$ is the
correlation contribution. According to our calculations, the
$\varepsilon_{\text{c}}(x)$ can be approximated with the following expression:

$\displaystyle\varepsilon_{\text{c}}(x)=-\frac{b_{0}}{2}\int\limits_{0}^{x}\mathrm{d}t\frac{b_{1}a+t^{1/2}}{t^{3/2}+tb_{1}a+b_{2}t^{1/2}a^{2}+b_{3}a^{3}}\frac{1+a_{1}t+a_{2}t^{2}}{1+td_{0}}\,,$
$\displaystyle a=(\alpha_{0}\eta)^{1/2};\quad a_{1}=2.25328;\quad
a_{2}=4.87991;\quad d_{0}=0.924022;$ $\displaystyle b_{0}=0.0621814;\quad
b_{1}=9.81379;\quad b_{2}=2.82214;\quad b_{3}=0.73701.$ (4.9)

At $a_{1}=a_{2}=d_{0}=0$, the expression matches the approximation for the
correlation energy [25], which is calculated using the Monte-Carlo method [26]
$\varepsilon_{\text{c}}^{\text{MC}}(x)$. As we can see in figure 7, in the
range $x\leqslant 1$, the expression (4.9) is close to
$\varepsilon_{\text{c}}^{\text{MC}}(x)$, and the deviation
$\varepsilon_{\text{c}}(x)$ from $\varepsilon_{\text{c}}^{\text{MC}}(x)$ in
the region $x>1$ is caused by different asymptotics of these functions
$\varepsilon_{\text{c}}(x)\rightarrow-\frac{b_{0}}{2}\frac{a_{2}}{d_{0}}x+\ldots,$
$\varepsilon_{\text{c}}^{\text{MC}}(x)\rightarrow-\frac{b_{0}}{2}\ln{x}+\ldots$
at $x\gg 1$.

Figure 7: Dependence of functions $\varepsilon^{(2)}_{\text{pol}}(x)$,
$10\cdot\varepsilon^{(3)}_{\text{pol}}(x)$ and $\varepsilon_{\text{c}}(x)$ on
the relativistic parameter.

In order to calculate the contributions of electron-nuclear interactions in
the products of structure factors in the formula (4.1), we have selected one-
particle and two-particle sums by the coordinates of nuclei, and the three-
nuclear effective interactions were neglected. In this approach,

$\displaystyle E\simeq E_{\text{e}}+E_{\text{pol}}+E_{\text{conf}}\,,$ (4.10)

where $E_{\text{pol}}$ is the polarization energy of electron fluid by nuclei,
which does not depend on the structure of the nuclear subsystem,

$\displaystyle E_{\text{pol}}=E^{(2)}_{\text{pol}}+E^{(3)}_{\text{pol}}\,,$
$\displaystyle
E^{(2)}_{\text{pol}}=-\frac{z_{1}^{2}N_{1}+z_{2}^{2}N_{2}}{2!V^{2}}\sum\limits_{\mathbf{q}}V_{q}^{2}\,\tilde{\mu}_{2}(\mathbf{q},-\mathbf{q}|0),$
$\displaystyle
E^{(3)}_{\text{pol}}=-\frac{z_{1}^{3}N_{1}+z_{2}^{3}N_{2}}{3!V^{3}}\sum\limits_{\mathbf{q}_{1},\mathbf{q}_{2}}V_{q_{1}}V_{q_{2}}V_{-q_{1}-q_{2}}\tilde{\mu}_{3}(\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{1}-\mathbf{q}_{2}|0).$
(4.11)

The configuration energy is determined by the structure of a nuclear
subsystem, and expressed through an effective two-particle potential of
interactions of the nuclei, which is formed by two- and three-electron
correlations:

$\displaystyle
E_{\text{conf}}=\sum\limits_{i=1,2}\frac{z_{i}^{2}}{2!V}\sum\limits_{\mathbf{q}}V_{q}\bigg{[}1-\frac{V_{q}}{V}\tilde{\mu}_{2}(\mathbf{q},-\mathbf{q}|0)-\frac{z_{i}}{V^{2}}\sum\limits_{\mathbf{q}_{1}}V_{q_{1}}V_{-q-q_{1}}\tilde{\mu}_{3}(\mathbf{q},\mathbf{q}_{1},-\mathbf{q}-\mathbf{q}_{1}|0)\bigg{]}S_{2}^{(i)}(\mathbf{q},-\mathbf{q})$
$\displaystyle+\frac{z_{1}z_{2}}{V}\sum\limits_{\mathbf{q}}V_{q}\biggl{[}1-\frac{V_{q}}{V}\,\tilde{\mu}_{2}(\mathbf{q},-\mathbf{q}|0)-\frac{z_{1}+z_{2}}{2V^{2}}\sum\limits_{\mathbf{q}_{1}}V_{q_{1}}V_{-q-q_{1}}\,\tilde{\mu}_{3}(\mathbf{q},\mathbf{q}_{1},-\mathbf{q}-\mathbf{q}_{1}|0)\biggr{]}S^{(1)}_{\mathbf{q}}S^{(2)}_{-\mathbf{q}}\,,$
$\displaystyle S_{2}^{(i)}(\mathbf{q},-\mathbf{q})=\sum\limits_{j_{1}\neq
j_{2}=1}^{N_{i}}\exp\big{[}\mathrm{i}(\mathbf{q},\mathbf{R}_{j_{1}}-\mathbf{R}_{j_{2}})\big{]}.$
(4.12)

###  4.1 The approximation of two-electron correlations

Let us rewrite the component $E^{(2)}_{\text{pol}}$, calculated in the local
field approximation, in the form

$\displaystyle E^{(2)}_{\text{pol}}=N_{\text{e}}m_{0}c^{2}\frac{\langle
z^{2}\rangle}{\langle
z\rangle}\alpha_{0}^{3/2}\varepsilon^{(2)}_{\text{pol}}(x),$ (4.13)

where the dimensionless function $\varepsilon^{(2)}_{\text{pol}}(x)$ is of the
same order as $\varepsilon_{\text{c}}(x)$, and $\langle
z^{n}\rangle=(z_{1}^{n}N_{1}+z_{2}^{n}N_{2})(N_{1}+N_{2})^{-1}$. The function
$\varepsilon^{(2)}_{\text{pol}}(x)$ can be approximated by the following
expression:

$\displaystyle\varepsilon_{\text{pol}}^{(2)}(x)=-\int\limits_{0}^{x}\frac{c_{0}+c_{1}t+c_{2}t^{2}+c_{3}t^{3}}{1+d_{1}t+d_{2}t^{2}+d_{3}t^{3}}\,\mathrm{d}t;$
$\displaystyle c_{0}=4.06151;\quad c_{1}=32.6118;\quad c_{2}=-43.6587;\quad
c_{3}=104.13;$ $\displaystyle d_{1}=73.8252;\quad d_{2}=-67.1028;\quad
d_{3}=189.781.$ (4.14)

As it is shown in figure 7, $\varepsilon^{(2)}_{\text{pol}}(x)$ has linear
asymptotics at $x\gg 1$, as well as $\varepsilon_{\text{c}}(x)$. However, the
polarization energy $E^{(2)}_{\text{pol}}$ exceeds the correlation energy of
the reference system by about $\langle z\rangle\alpha_{0}^{-1/2}\approx
10\langle z\rangle$ times, and for $\langle z\rangle\sim 10$ it is comparable
with $E_{\text{HF}}$.

The configuration energy was calculated in the coordinate representation and,
introducing the effective two-nuclear interactions, yielded

$\displaystyle
V_{2}^{i_{1},i_{2}}\big{(}\mathbf{R}_{j_{1}}^{(i_{1})}-\mathbf{R}_{j_{2}}^{(i_{2})}\big{)}=\frac{1}{V}\sum\limits_{\mathbf{q}}V_{2}(q)\exp\left\\{\mathrm{i}\big{[}\mathbf{q},\mathbf{R}_{j_{1}}^{(i_{1})}-\mathbf{R}_{j_{2}}^{(i_{2})}\big{]}\right\\},$
$\displaystyle
V_{2}(q)=V_{q}\left[1-\frac{V_{q}}{V}\tilde{\mu}_{2}(\mathbf{q},-\mathbf{q}|0)\right].$
(4.15)

In the formula (4.15) the sum of the vector $\mathbf{q}$ includes a component
with $\mathbf{q}=0$. Therefore,

$\displaystyle E^{(2)}_{\text{conf}}$
$\displaystyle=\frac{1}{V}\sum\limits_{\mathbf{q}\neq
0}V_{2}(q)\left[\frac{1}{2}\sum\limits_{i=1,2}S_{2}^{(i)}(\mathbf{q},-\mathbf{q})z_{i}^{2}+z_{1}z_{2}S^{(1)}_{\mathbf{q}}S^{(2)}_{-\mathbf{q}}\right]$
$\displaystyle=\frac{1}{2}\sum\limits_{i=1,2}z_{i}^{2}\sum\limits_{j_{1}\neq
j_{2}=1}V_{2}\big{(}\mathbf{R}_{j_{1}}^{(i)}-\mathbf{R}_{j_{2}}^{(i)}\big{)}+z_{1}z_{2}\sum\limits_{j_{1}=1}^{N_{1}}\sum\limits_{j_{2}=1}^{N_{2}}V_{2}\big{(}\mathbf{R}_{j_{1}}^{(1)}-\mathbf{R}_{j_{2}}^{(2)}\big{)}-\frac{1}{2}N_{\text{e}}^{2}\lim\limits_{\mathbf{q}\to
0}\biggl{[}\frac{V_{2}(q)}{V}\biggr{]}.$ (4.16)

To simplify the calculation of the lattice sum, we adopt a simple model of
nuclei distribution:

$\displaystyle\mathcal{N}_{j}^{(1)}=\frac{N_{1}}{N_{1}+N_{2}}\mathcal{N}_{j}\,,\qquad\mathcal{N}_{j}^{(2)}=\frac{N_{2}}{N_{1}+N_{2}}\mathcal{N}_{j}\,,$
(4.17)

where $\mathcal{N}_{j}$ is the number of all knots on the $j$-th coordination
sphere, and $\mathcal{N}_{j}^{(i)}$ is the number of the knots occupied by
nuclei with charge $z_{i}$ $(i=1,2)$. In this model,

$\displaystyle E^{(2)}_{\text{conf}}=\frac{N_{\text{e}}}{2}\langle
z\rangle\sum\limits_{j\geqslant
1}\mathcal{N}_{j}V_{2}(R_{j})-\frac{N_{\text{e}}m_{0}c^{2}}{6}\left[\frac{x^{2}}{(1+x^{2})^{1/2}}-\frac{x\alpha_{0}}{\piup}\right],$
(4.18)

where $R_{j}$ is the radius of the $j$-th coordination sphere.

The effective two-particle potential is screened, and at small and medium
distances between nuclei it is close to the expression

$\displaystyle V(R)=\frac{e^{2}}{R}\exp(-R/R_{0}),$ (4.19)

and the screening radius

$\displaystyle
R_{0}=\frac{\sqrt{\piup}}{2}\alpha_{0}^{1/2}a_{\text{B}}\big{[}x^{1/2}(1+x^{2})^{1/4}\big{]}^{-1}$
(4.20)

is of the order $0.1a_{\text{B}}$ (where $a_{\text{B}}=\hbar^{2}/m_{0}e^{2}$
is the Bohr radius). At large distances, $V_{2}(R)$ oscillates, but with a
small amplitude,

$\displaystyle
V_{2}(R)\approx\frac{e^{2}}{a_{\text{B}}}\left(\frac{R_{0}}{2xR}\right)^{3}\cos(2xR/R_{0}).$
(4.21)

The configuration energy for a simple cubic lattice of nuclei is calculated
numerically and can be represented as

$\displaystyle E^{(2)}_{\text{conf}}=N_{\text{e}}m_{0}c^{2}\langle
z\rangle^{2/3}\alpha_{0}\varepsilon_{\text{L}}^{(2)}(x|\langle z\rangle),$
(4.22)

with a dimensionless factor approximated by the expression

$\displaystyle\varepsilon_{\text{L}}^{(2)}(x|\langle
z\rangle)=-\int\limits_{0}^{x}\frac{a_{1}+ta_{2}+t^{2}a_{3}}{1+ta_{4}+t^{2}a_{5}+t^{3}a_{6}}t\mathrm{d}t,$
(4.23)

where all the coefficients $a_{1},\ldots,a_{6}$ are the functions of $\langle
z\rangle$, that is

$\displaystyle a_{i}(\langle z\rangle)=\frac{a_{i_{0}}+\langle z\rangle
a_{i_{1}}+\langle z\rangle^{2}a_{i_{2}}}{a_{i_{3}}+\langle z\rangle
a_{i_{4}}+\langle z\rangle^{2}a_{i_{5}}}.$ (4.24)

Table 1:  Coefficients $a_{ij}$ from (4.24). $i\bigm{\backslash}j$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---|---|---
$0$ | $-128.112$ | $-138.098$ | $-3.30915$ | $0$ | $3.74936$ | $0.882489$
$1$ | $-633.899$ | $297.304$ | $-19.5138$ | $1$ | $-0.707632$ | $1.01638$
$2$ | $-1691$ | $216.967$ | $-8.71667$ | $0$ | $1.48694$ | $0.12998$
$3$ | $2.37539$ | $1.74513$ | $0.0417739$ | $0$ | $0.0212583$ | $0.0056168$
$4$ | $913.016$ | $-452.217$ | $30.3618$ | $1$ | $-0.750844$ | $0.702212$
$5$ | $5.68901$ | $-0.704184$ | $0.0277872$ | $0$ | $0.00203554$ | $0.000234399$

The cofficients $a_{ij}$ of the formula (4.24) are listed in table 1. In
figure 8 there is shown a dependence of the configuration energy on the
relativistic parameter.

Figure 8: Dependence of the lattice energy
$\varepsilon_{\text{L}}^{(2)}(x|\langle z\rangle)$ on the nuclear charge and
the relativistic parameter.

###  4.2 The effect of three-electron correlations

According to the formula (4.11), the contribution of three-particle
correlations is represented as

$\displaystyle E^{(3)}_{\text{pol}}=N_{\text{e}}m_{0}c^{2}\frac{\langle
z^{3}\rangle}{\langle
z\rangle}\alpha_{0}^{5/2}\varepsilon_{\text{pol}}^{(3)}(x).$ (4.25)

The result of numerical calculations is illustrated in figure 7, which shows
that for $x>1$ the ratio $\varepsilon_{\text{pol}}^{(3)}(x)\approx
0{.}1\varepsilon_{\text{pol}}^{(2)}(x)$ is satisfied. Therefore, at
sufficiently large values of nuclei charges, $E^{(3)}_{\text{pol}}$ is not
less than the correlation energy of the reference system: at $\langle
z\rangle\geqslant 6$, the contribution $E^{(3)}_{\text{pol}}$ is close to the
correlation energy, at $\langle z\rangle\geqslant 12$, it exceeds the
correlation energy by 5 times, and at $\langle z\rangle=26$ — more than by 20
times. The result of numerical calculation $\varepsilon^{(3)}_{\text{pol}}(x)$
is approximated by the expression

$\displaystyle\varepsilon^{(3)}_{\text{pol}}(x)=-ax-
c_{0}\int\limits_{x}^{\infty}\frac{1+c_{1}/t+c_{2}t}{1+d_{1}t+d_{2}t^{2}+d_{3}t^{3}}\,\mathrm{d}t,$
$\displaystyle a=0.0450;\quad c_{0}=0.12607;\quad c_{1}=-0.93695;\quad
c_{2}=78.8552;$ $\displaystyle d_{1}=-23.2602;\quad d_{2}=114.5030;\quad
d_{3}=164.060.$ (4.26)

From the formulae (4.13), (4.25) it follows that
$E^{(3)}_{\text{pol}}/E^{(2)}_{\text{pol}}\sim 0.1z\alpha_{0}$, it determines
the order of three-electron correlations contribution value.

Figure 9: The effective potential of interactions $V_{2}^{(3)}(R)$ at
different values of the relativistic parameter (curve 1 — $x=1.0$, 2 —
$x=2.0$, 3 — $x=3.0$, 4 — $x=4.0$, 5 — $x=5.0$).

Similar to the formula (4.15), we have introduced a correction to the
effective two-nuclear potential by the three-electron correlations

$\displaystyle
V_{2}^{(3)}(R)=-V^{3}\sum\limits_{\mathbf{q}}V_{q}\sum\limits_{\mathbf{q_{1}}}V_{q_{1}}V_{-q-q_{1}}\tilde{\mu}_{3}(\mathbf{q},\mathbf{q}_{1},-\mathbf{q}-\mathbf{q}_{1}|0)\exp[\mathrm{i}(\mathbf{q,R})],$
(4.27)

where the sum over the vector $\mathbf{q}$ includes the component
$\mathbf{q}=0$. As shown in figure 9, $V_{2}^{(3)}(R)$ is a weak attracting
potential of the type of quantum package screening potential, which is close
to the expression

$\displaystyle
V_{2}^{(3)}(R)=-\frac{e^{2}}{R}\alpha_{0}^{2}A(x)\left\\{1-\exp\left[-\frac{R}{R_{0}}\gamma(x)\right]\right\\}\exp(-R/R_{0}).$
(4.28)

The functions $A(x)$ and $\gamma(x)$ are determined as follows:

$\displaystyle A(x)=\frac{8\,I_{2}(x)}{(\piup
x)^{2}}\,,\qquad\gamma(x)=\frac{\alpha_{0}^{3/2}x^{1/2}(1+x^{2})^{3/4}}{2\sqrt{\piup}}\cdot\frac{I_{1}(x)}{I_{2}(x)};$
$\displaystyle
I_{1}(x)=\int\limits_{0}^{\infty}\frac{\mathrm{d}q_{1}}{\varepsilon(q_{1})}\int\limits_{0}^{\infty}\frac{\mathrm{d}q_{2}}{\varepsilon(q_{2})}\int\limits_{-1}^{1}\frac{\mathrm{d}t}{q_{3}^{2}\varepsilon(q_{3})}\,f_{3}(q_{1},q_{2},t),\qquad
I_{2}(x)=\int\limits_{0}^{\infty}\frac{\mathrm{d}q}{q^{2}\varepsilon^{2}(q)}J_{3}(q);$
$\displaystyle
J_{3}(q)=\frac{(m_{0}c^{2}x^{2})^{2}}{3N_{\text{e}}}\tilde{\mu}_{3}^{0}(\mathbf{q},-\mathbf{q},0|0);\qquad
f_{3}(q_{1},q_{2},t)=\frac{(m_{0}c^{2}x^{2})^{2}}{3N_{\text{e}}(1+x^{2})}\tilde{\mu}_{3}^{0}(\mathbf{q}_{1},\mathbf{q}_{2},-\mathbf{q}_{1}-\mathbf{q}_{1}|0);$
$\displaystyle q_{3}\equiv|\mathbf{q}_{1}+\mathbf{q}_{2}|.$ (4.29)

The contribution to the configuration energy of the model by the three-
particle correlations in the model (4.17) takes the form, similar to the
formula (4.18):

$\displaystyle E^{(3)}_{\text{conf}}=\frac{1}{2}N_{\text{e}}\langle
z^{2}\rangle\sum\limits_{j\geqslant
1}\mathcal{N}_{j}V^{(3)}_{2}(R_{j})+\frac{4}{3\piup^{2}}N_{\text{e}}\alpha_{0}^{2}\langle
z\rangle m_{0}c^{2}(1+x^{2})^{-1/2}I_{2}(x).$ (4.30)

This contribution is calculated for a simple cubic lattice of nuclei and is
represented by

$\displaystyle
E^{(3)}_{\text{conf}}=N_{\text{e}}m_{0}c^{2}\alpha_{0}^{2}\langle
z^{2}\rangle\varepsilon_{\text{L}}^{(3)}(x|\langle z\rangle).$ (4.31)

The dependence of a dimensionless factor
$\varepsilon_{\text{L}}^{(3)}(x|\langle z\rangle)$ is illustrated in figure
10. At sufficiently large nuclei charges $\langle z\rangle$ and $x\geqslant
2$, the function $\varepsilon_{\text{L}}^{(3)}(x|\langle z\rangle)\sim
0.1\varepsilon_{\text{L}}^{(2)}(x|\langle z\rangle)$, but it has a positive
sign. It is approximated by the expression

$\displaystyle\varepsilon_{\text{L}}^{(3)}(x|\langle
z\rangle)=-\frac{a_{0}(\langle z\rangle)+a_{1}(\langle
z\rangle)x+a_{2}(\langle z\rangle)x^{2}}{x}\,.$ (4.32)

The coefficients of the formula (4.32) are shown in table 2.

Table 2:  Coefficients from (4.32). $\langle z\rangle$ | $a_{0}$ | $a_{1}$ | $a_{2}$
---|---|---|---
$2$ | $0.160403$ | $0.422066$ | $-0.707095$
$4$ | $0.118181$ | $0.351743$ | $-0.320001$
$6$ | $0.0978784$ | $0.321462$ | $-0.203819$
$8$ | $0.08553532$ | $0.302602$ | $-0.150039$
$12$ | $0.0701315$ | $0.278365$ | $-0.100583$
$26$ | $0.04788678$ | $0.236761$ | $-0.0549487$
Figure 10: Dependence of the lattice energy
$\varepsilon_{\text{L}}^{(3)}(x|\langle z\rangle)$ on the nuclear charge and
the relativistic parameter.

##  5 The equation of state of the model at $T=0$ K

Table 3: Dependence of the functions $\mathcal{F}(x)\cdot x^{-4}$,
$f_{2}(x)\cdot x^{-4}$ and $f_{3}(x)\cdot x^{-4}$ on the relativistic
parameter $x$ according to the formulae (5.2)–(5.4) and (5.5).

$x$ | $\mathcal{F}(x)\cdot x^{-4}$ | $f_{2}(x)\cdot x^{-4}$ | $f_{3}(x)\cdot x^{-4}$
---|---|---|---
$z=2$ | $z=6$ | $z_{1}=7;\,z_{2}=8$ | $z=12$ | $z=2$ | $z=6$ | $z_{1}=7;\,z_{2}=8$ | $z=12$
$0.5$ | $0.737488$ | $-0.025120$ | $-0.0899288$ | $-0.113384$ | $-0.208687$ | $0.00151018$ | $0.00503624$ | $0.00724728$ | $0.0123084$
$0.6$ | $0.857456$ | $-0.0249008$ | $-0.0898362$ | $-0.113389$ | $-0.209377$ | $0.00144201$ | $0.0045685$ | $0.00653182$ | $0.0107583$
$0.7$ | $0.966234$ | $-0.0247355$ | $-0.0896437$ | $-0.11323$ | $-0.209566$ | $0.00140653$ | $0.00431288$ | $0.00610532$ | $0.00988929$
$0.8$ | $1.06412$ | $-0.0246194$ | $-0.0894304$ | $-0.113012$ | $-0.209487$ | $0.00138662$ | $0.00415774$ | $0.00582077$ | $0.00935182$
$0.9$ | $1.15175$ | $-0.0245407$ | $-0.0892304$ | $-0.112788$ | $-0.20927$ | $0.00137469$ | $0.00405367$ | $0.00561505$ | $0.00899025$
$1.0$ | $1.22991$ | $-0.0244886$ | $-0.0890553$ | $-0.11258$ | $-0.208989$ | $0.00136685$ | $0.00397618$ | $0.00545736$ | $0.00872662$
$1.1$ | $1.29949$ | $-0.0244546$ | $-0.0889067$ | $-0.112394$ | $-0.208686$ | $0.00136085$ | $0.00391243$ | $0.00533111$ | $0.00851882$
$1.2$ | $1.36139$ | $-0.0244329$ | $-0.0887823$ | $-0.112234$ | $-0.208385$ | $0.00135532$ | $0.00385556$ | $0.00522669$ | $0.00834323$
$1.3$ | $1.41647$ | $-0.0244195$ | $-0.0886788$ | $-0.112096$ | $-0.208099$ | $0.00134944$ | $0.00380207$ | $0.00513813$ | $0.00818666$
$1.4$ | $1.46551$ | $-0.0244117$ | $-0.0885926$ | $-0.111979$ | $-0.207833$ | $0.00134276$ | $0.0037504$ | $0.00506157$ | $0.008042$
$1.5$ | $1.50924$ | $-0.0244077$ | $-0.0885206$ | $-0.111878$ | $-0.207591$ | $0.00133514$ | $0.00370008$ | $0.00499435$ | $0.00790579$
$1.6$ | $1.5483$ | $-0.0244062$ | $-0.0884603$ | $-0.111793$ | $-0.207371$ | $0.00132663$ | $0.00365119$ | $0.00493463$ | $0.00777667$
$1.7$ | $1.58327$ | $-0.0244065$ | $-0.0884095$ | $-0.111719$ | $-0.207173$ | $0.00131746$ | $0.00360403$ | $0.00488104$ | $0.00765433$
$1.8$ | $1.61463$ | $-0.024408$ | $-0.0883666$ | $-0.111656$ | $-0.206995$ | $0.00130788$ | $0.00355898$ | $0.00483256$ | $0.00753897$
$1.9$ | $1.64282$ | $-0.0244103$ | $-0.08833$ | $-0.111602$ | $-0.206836$ | $0.0012982$ | $0.00351632$ | $0.00478842$ | $0.00743084$
$2.0$ | $1.66822$ | $-0.0244131$ | $-0.0882987$ | $-0.111555$ | $-0.206693$ | $0.00128866$ | $0.00347628$ | $0.00474798$ | $0.00733015$
$3.0$ | $1.82417$ | $-0.0244459$ | $-0.0881427$ | $-0.111308$ | $-0.205863$ | $0.00122235$ | $0.00321213$ | $0.00447214$ | $0.00668381$
$4.0$ | $1.89283$ | $-0.0244708$ | $-0.0880955$ | $-0.111226$ | $-0.205545$ | $0.00119715$ | $0.0031007$ | $0.00431866$ | $0.00642247$
$5.0$ | $1.92833$ | $-0.024488$ | $-0.0880781$ | $-0.111192$ | $-0.205404$ | $0.00118685$ | $0.00304877$ | $0.00422167$ | $0.00630395$

For the well-known dependence of the model energy on the relativistic
parameter we calculate the equation of state of cold degenerate matter using
the expression

$\displaystyle
P(x)=\frac{\mathrm{d}E}{\mathrm{d}V}=\frac{x^{4}}{N_{\text{e}}}\left(\frac{m_{0}c}{\hbar}\right)^{3}(3\piup^{2})^{-1}\frac{\mathrm{d}E}{\mathrm{d}x}\,.$
(5.1)

Within the accepted approximation

$\displaystyle P(x)=\frac{\piup
m_{0}^{4}c^{5}}{3h^{3}}[\mathcal{F}(x)+f_{2}(x)+f_{3}(x)].$ (5.2)

Here,

$\displaystyle\mathcal{F}(x)=x(2x^{2}-3)(1+x^{2})^{1/2}+3\ln{[x+(1+x^{2})^{1/2}]}$
(5.3)

is the contribution of the ideal degenerate relativistic spatially homogeneous
electron gas;

$\displaystyle
f_{2}(x)=-2\alpha_{0}x^{4}\biggl{\\{}\frac{1}{\piup}-\frac{4}{3}\frac{\mathrm{d}}{\mathrm{d}x}\biggl{[}\langle
z\rangle^{2/3}\varepsilon_{\text{L}}^{(2)}(x|\langle z\rangle)+\frac{\langle
z^{2}\rangle}{\langle
z\rangle}\alpha_{0}^{1/2}\varepsilon^{(2)}_{\text{pol}}(x)+\alpha_{0}\varepsilon_{\text{c}}(x)\biggr{]}\biggr{\\}}$
(5.4)

is the contribution of Coulomb interactions in the two-electron correlations
approximation;

$\displaystyle
f_{3}(x)=8\alpha_{0}^{2}x^{4}\frac{\mathrm{d}}{\mathrm{d}x}\biggl{[}\langle
z^{2}\rangle\varepsilon_{\text{L}}^{(3)}(x|\langle z\rangle)+\frac{\langle
z^{3}\rangle}{\langle
z\rangle}\alpha_{0}^{1/2}\varepsilon^{(3)}_{\text{pol}}(x)\biggr{]}$ (5.5)

is the contribution of the three-particle electron correlations.

In the region $x\geqslant 1$, all contributions [with the exception of
$\varepsilon_{\text{L}}^{(3)}(x|\langle z\rangle)$] to the model energy caused
by interactions are negative monotonously decreasing functions of the
relativistic parameter. In the two-electron correlations approximation, the
equation of state (5.2) numerically is very close to the result of Salpeter
[6].

In table 3 there is shown a dependence of terms $\mathcal{F}(x)$, $f_{2}(x)$,
$f_{3}(x)$ on the relativistic parameter for the helium ($z_{1}=z_{2}=2$),
carbon ($z_{1}=z_{2}=6$), nitrogen-oxygen ($z_{1}=7$, $z_{2}=8$;
$N_{1}=N_{2}$) and magnesium ($z_{1}=z_{2}=12$) dwarf models. A relative
decrease of pressure caused by the interactions
$[\mathcal{F}(x)+f_{2}(x)+f_{3}(x)]\mathcal{F}^{-1}(x)$ is illustrated in
figure 11.

Figure 11: The ratio of pressure $P(x)$ to the pressure of the ideal
relativistic electron gas $P_{0}(x)$ as function of the relativistic parameter
and nuclear charge (curve 1 — $z_{1}=z_{2}=2$, 2 — $z_{1}=z_{2}=6$, 3 —
$z_{1}=7;\,z_{2}=8;$ 4 — $z_{1}=z_{2}=12$).

##  6 Conclusions

Reference system approach was adapted for the description of a degenerate
relativistic electron subsystem. There were investigated the features of two-
and three-particle correlation functions in a wide domain of a relativistic
parameter, as well as there was obtained an exact expression for the static
two-particle correlation function. For the first time, the expressions for the
three-particle correlation function were approximated and the local field
correction of interacting relativistic electron gas was studied, which is the
basis for the calculation of the energy and structural characteristics of
degenerated dwarfs. The energy of the ground state of the electron-nuclear
model, as well as the equation of state of the model, have been calculated in
a wide range of the relativistic parameter at absolute zero temperature. As it
was shown in our calculation, the contributions of Coulomb interactions to the
energy of the ground state and pressure, caused by two-electron correlations
are important and increase with an increase of nuclear charge. The
contributions caused by three-electron correlations are much smaller but they
exceed the contribution of correlation energy of the electron fluid,
especially at large values of the nuclear charge.

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