# Three-dimensional lattice ground states for Riesz and Lennard-Jones type
energies

Laurent Bétermin  Faculty of Mathematics, University of Vienna, 1090 Vienna,
Austria<EMAIL_ADDRESS>,  Ladislav Šamaj  Institute of
Physics, Slovak Academy of Sciences, 84511 Bratislava, Slovakia
<EMAIL_ADDRESS>and  Igor Travěnec  Institute of Physics, Slovak Academy of
Sciences, 84511 Bratislava, Slovakia<EMAIL_ADDRESS>
###### Abstract.

The Riesz potential $f_{s}(r)=r^{-s}$ is known to be an important building
block of many interactions, including Lennard-Jones type potentials
$f_{n,m}^{\rm{LJ}}(r):=ar^{-n}-br^{-m}$, $n>m$ that are widely used in
Molecular Simulations. In this paper, we investigate analytically and
numerically the minimizers among three-dimensional lattices of Riesz and
Lennard-Jones energies. We discuss the minimality of the Body-Centred-Cubic
lattice (BCC), Face-Centred-Cubic lattice (FCC), Simple Hexagonal lattices
(SH) and Hexagonal Close-Packing structure (HCP), globally and at fixed
density. In the Riesz case, new evidence of the global minimality at fixed
density of the BCC lattice is shown for $s<0$ and the HCP lattice is computed
to have higher energy than the FCC (for $s>3/2$) and BCC (for $s<3/2$)
lattices. In the Lennard-Jones case, the ground state among lattices is
confirmed to be a FCC lattice whereas a HCP phase occurs once added to the
investigated structures. Furthermore, phase transitions of type “FCC-SH” and
“FCC-HCP-SH” (when the HCP lattice is added) as the inverse density $V$
increases are observed for a large spectrum of exponents $(n,m)$. In the SH
phase, the variation of the ratio $\Delta$ between the inter-layer distance
$d$ and the lattice parameter $a$ is studied as $V$ increases.

###### Key words and phrases:

Riesz potential, Lennard-Jones potential, Lattices, Ground states, Epstein
zeta function.

###### 2010 Mathematics Subject Classification:

74G65, 70G75, 74N05.

##  1\. Introduction, setting and main results

###  1.1. Motivation

From the first attempt by Huygens [40] to describe solids as periodic
assemblies of objects to the formalization of the Crystallization Conjecture
by Blanc and Lewin [16], the mathematical journey to rigorously justify the
emergence of periodic structures in crystal solids never stopped to attract
new travelers. Nevertheless, only few rigorous results showing the global
minimality, i.e. among all possible configurations, of lattice structures for
interaction energies are available. Beside the one-dimensional case [66, 14,
21, 58, 10] where the optimality of the equidistant configuration $\mathbb{Z}$
is easily derivable, the only known results in dimension $d=2$ [39, 63, 61,
29, 45, 46, 11, 26] and $d=3$ [61, 34] rely on perturbations of hard-sphere
potentials and specific angular dependence.

In dimension $d\in\\{8,24\\}$, the linear programming method initiated by Cohn
an Elkies [20] has lead to recent important results for packings [67, 22] and
crystallization at fixed density [23, 54, 6]. In particular they showed the
universal optimality, at any fixed density and for pairwise energies with
interaction potentials $f(r)=F(r^{2})$ where $F$ is a completely monotone
function, of the densest packings in these dimensions – namely the Gosset
lattice $\mathsf{E}_{8}$ and the Leech lattice $\Lambda_{24}$. These are the
only crystallization results that apply at any fixed density to Riesz
energies, where the interaction potential is

$f_{s}(r)=\frac{1}{r^{s}},$

and at sufficiently high fixed density for Lennard-Jones type energies (see
[6, Thm. 2.17]) for which the interaction potential is defined by

$f_{n,m}^{\rm{LJ}}(r)=\frac{a}{r^{n}}-\frac{b}{r^{m}},$

where $(a,b)\in(0,\infty)$ and $n>m>d$. The same minimality property is
conjectured by Cohn and Kumar [21] to hold in dimension $d=2$ for the
triangular lattice $\mathsf{A}_{2}$ whereas it is known that no universal
minimizer exists in dimension $d=3$ [59].

The potentials studied in this paper are of high importance in Mathematics and
Physics, and their lattice ground states are known to be fundamental in many
domains. On the one hand, Riesz type energies arise in Mathematical Physics
and Number Theory in many ways. For instance, in the Coulomb case $s=d-2$ (or
$f(r)=-\log r$ in dimension $d=2$), the Wigner Conjecture for Jellium [71]
states that electrons embedded in an uniform background of positive charges
must crystallize on a triangular and a Body-Centred-Cubic lattice in
dimensions 2 and 3 respectively (see also [16]). This is also called the
Abrikosov Conjecture [1] or Vortices Conjecture [57] in the two-dimensional
setting related to the vortices in the Ginzburg-Landau theory of
superconductors of type II. An equivalent problem can be stated in higher
dimension and for general Riesz interaction [56, 53, 54, 24, 44, 43]. The
general Riesz energy and its related minimization problem also appear in the
theory of Random Point Configurations [36] as well as Approximation Theory
[60] and Number Theory [51, 59, 25]. On the other hand, Lennard-Jones
potentials have been introduced by Mie [48] and popularized by Jones [41] in
its classical $(n,m)=(12,6)$ form, i.e. when the interaction is of Van der
Waals type $\sim br^{-6}$ for large $r$, initially for studying gas argon. The
repulsion at short distance of type $\sim ar^{-n}$ is here to mimic Pauli
exclusion Principle. Since $f_{n,m}^{\rm{LJ}}$ is a bonding potential, i.e. it
has an equilibrium distance, it has been widely used in Molecular Simulation
(see e.g. [35, 42, 69]). It also appears to be a good model for social
aggregation [49] and has many other applications, like in Robotics [68]. In
dimension $d=2$, numerical investigations suggest that the ground state of the
classical Lennard-Jones energy (i.e. $(n,m)=(12,6)$) is a triangular lattice
[15], whereas the Hexagonal Closed-Packing structure is expected to be the one
in dimension $d=3$ [2]. Notice also that the ground states of three-
dimensional finite-range Lennard-Jones energy among a finite number of
structures has been also studied in [52] for different truncation, cutoff
distances, pressure and exponents. Notice also that the lattice ground states
for Lennard-Jones type energies have been recently proven to be related to the
one of the Embedded-Atom Models with Riesz-type electron density and Riesz or
Lennard-Jones nuclei interaction, see [13].

###  1.2. Minimization among lattices and setting

The goal of this paper is to investigate the possible lattice ground states
for Riesz and Lennard-Jones energies in dimension $d=3$, but in the case where
simple periodicity is assumed. Indeed, once restricted to the class of
lattices

$\mathcal{L}_{d}:=\left\\{L=\bigoplus_{i=1}^{d}\mathbb{Z}u_{i}\subset\mathbb{R}^{d}:\textnormal{$\\{u_{i}\\}$
is a basis of $\mathbb{R}^{d}$}\right\\},$

the above minimization problem becomes simpler and the Riesz and Lennard-Jones
type lattice energies are respectively defined by

$\zeta_{L}(s):=\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{p\in
L}\frac{1}{|p|^{s}},\quad\textnormal{and}\quad
E_{n,m}^{\rm{LJ}}[L]:=\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{p\in
L}f_{n,m}^{\rm{LJ}}(|p|)=a\zeta_{L}(n)-b\zeta_{L}(m),$

where $\zeta_{L}(s)$ is the Epstein zeta function [33] associated to the
lattice $L$ and $\sideset{}{{}^{\prime}}{\sum}$ means that we sum on all the
points excepted the origin. Notice that a factor $1/2$ has been added in order
to identify the Epstein zeta function with the energy per point of $L$ with
interaction potential $f_{s}(r)=r^{-s}$. Recall that $s\mapsto\zeta_{L}(s)$
admits an analytic continuation on $\mathbb{C}\backslash\\{d\\}$ (see also
[30]) and can therefore be defined for any $s<d$. In particular, the value of
$\zeta_{L}(s)$ when $s\in(d-4,d)$ is the Jellium energy of the lattice $L$ for
general Riesz interaction [43, Thm 3.1] including the Coulomb case $s=d-2$,
originally proved by Cotar and Petrache for $s\in[d-2,d)$ in [24, Lemma 2.6]
(see also [44]).

Whereas in dimension $d=2$, the triangular lattice has been shown to be
minimal for $L\mapsto\zeta_{L}(s)$ for all $s>0$ in the set of lattices

$\mathcal{L}_{d}(V):=\\{L\in\mathcal{L}_{d}:|\det(u_{1},...,u_{d})|=V\\}\subset\mathcal{L}_{d},$

for any fixed covolume $V>0$ (also called “inverse density”), see [55, 31, 18,
28, 50], no such optimality result is shown in dimension $d=3$. Only
conjectures [59] and local minimality results [32, 27, 38, 7] are available
(see Section 2 for more details).

The same holds for the Lennard-Jones type energy $E_{n,m}^{\rm{LJ}}$. Indeed,
the two-dimensional ground state of $E_{n,m}^{\rm{LJ}}$ in $\mathcal{L}_{2}$
has been proven to be a triangular lattice for many exponents $(n,m)$ by the
first author [3, 8]. Furthermore, we have observed and partially proved a
phase transition of type “triangular-rhombic-square-rectangular” for the
minimizer of $E_{n,m}^{\rm{LJ}}$ in $\mathcal{L}_{2}(V)$ as $V$ increases, see
[4, 64]. Only asymptotic and local minimality results have been shown in
dimension $d=3$ in [5, 12, 7] (see Section 3 for more details).

We aim to present a complete picture of the lattice ground states of
$L\mapsto\zeta_{L}(s)$ and $E_{n,m}^{\rm{LJ}}$ in $\mathcal{L}_{3}$ and
$\mathcal{L}_{3}(V)$. In particular, we want to show minimality properties of
the following important three-dimensional lattices (given here with unit
density):

The Simple Cubic lattice (SC)
$\displaystyle\mathbb{Z}^{3}:=\mathbb{Z}(1,0,0)\oplus\mathbb{Z}(0,1,0)\oplus\mathbb{Z}(0,0,1)$
(1.1) The Face-Centred-Cubic lattice (FCC)
$\displaystyle\mathsf{D}_{3}:=2^{-\frac{1}{3}}\left[\mathbb{Z}(1,0,1)\oplus\mathbb{Z}(0,1,1)\oplus\mathbb{Z}(1,1,0)\right]$
(1.2) The Body-Centred-Cubic lattice (BCC)
$\displaystyle\mathsf{D}_{3}^{*}:=2^{\frac{1}{3}}\left[\mathbb{Z}(1,0,0)\oplus\mathbb{Z}(0,1,0)\oplus\mathbb{Z}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right]$
(1.3) The Simple Hexagonal lattice (SH)
$\displaystyle\mathsf{A_{f}}(\Delta):=\left(\frac{2}{\sqrt{3}\Delta}\right)^{\frac{1}{3}}\left[\mathbb{Z}(1,0,0)\oplus\mathbb{Z}\left(\frac{1}{2},\frac{\sqrt{3}}{2},0\right)\oplus\mathbb{Z}(0,0,\Delta)\right].$
(1.4)

Notice that the SH lattices are non-shifted stacking of triangular lattices
with lattice constant $a>0$ and inter-layer distance $d>0$ and we set
$\Delta=d/a$. Furthermore, we also want to investigate the particular role of
the unit density hexagonal close-packed structure, which is not a lattice as
defined above, defined by

The Hexagonal Close-Packed structure (HCP)
$\displaystyle\mathsf{A3}:=\Lambda\cup\left(\Lambda+\left(\frac{1}{2},\frac{1}{\sqrt{12}},\sqrt{\frac{2}{3}}\right)\right)$
(1.5)
$\displaystyle\Lambda:=\mathbb{Z}(1,0,0)\oplus\mathbb{Z}\left(\frac{1}{2},\frac{\sqrt{3}}{2},0\right)\oplus\mathbb{Z}\left(0,0,\sqrt{\frac{8}{3}}\right).$
(1.6)

All the above mentioned periodic structures are depicted in Figure 1. Notice
that the names chosen for the SH and HCP structures are the one given by the
Strukturbericht designation [47]. Furthermore, we define for convenience the
new sets, i.e. the sets of lattices with added HCP structure,

$\displaystyle\widetilde{\mathcal{L}_{3}}(V):=\mathcal{L}_{3}(V)\cup
V^{\frac{1}{3}}\mathsf{A3},\quad\textnormal{and}\quad\widetilde{\mathcal{L}_{3}}:=\bigcup_{V>0}\widetilde{\mathcal{L}_{3}}(V).$
(1.7)

Figure 1. Main configurations of $\widetilde{\mathcal{L}_{3}}$: the SC lattice
$\mathbb{Z}^{3}$ (up left), the FCC lattice $\mathsf{D}_{3}$ (up middle), the
BCC lattice $\mathsf{D}_{3}^{*}$ (up right), the SH lattice
$\mathsf{A_{f}}(\Delta)$ (down left), $\Delta=d/a$, and the HCP lattice
$\mathsf{A3}$ (down right).

###  1.3. Method

Whereas different rigorous techniques exist for studying lattice energies in
dimension $d=2$ (see e.g. [50, 3, 6, 9]), where $\dim\mathcal{L}_{2}=3$ and
$\dim\mathcal{L}_{2}(V)=2$ for all $V>0$, the situation in dimension $d=3$ is
much more complicated. Indeed the so-called fundamental domain containing only
one copy of each lattice (up to dilation, isometry and symmetry) is
$5$-dimensional for $\mathcal{L}_{3}(1)$ and has a complicated shape (see e.g.
Terras [62]). Very few rigorous results have been proven in dimension 3. Let
us recall as an example the computer-assisted proof of Sarnak and
Strömbergsson [59] where the FCC lattice is shown to be minimal for the height
of the flat torus (i.e. the derivative of $s\mapsto\zeta_{L}(s)$ at $s=0$).

In our work, we parametrize any lattice $L\in\mathcal{L}_{3}$ by $6$ real
numbers $(u,v,x,y,z,V)$ where $V$ is the covolume of $L$ and, following [5],

$\displaystyle
L:=V^{\frac{1}{3}}\left[\mathbb{Z}u_{1}\oplus\mathbb{Z}u_{2}\oplus\mathbb{Z}u_{3}\right],$
$\displaystyle u_{1}=2^{\frac{1}{6}}\left(\frac{1}{\sqrt{u}},0,0\right),\quad
u_{2}=2^{\frac{1}{6}}\left(\frac{x}{\sqrt{u}},\frac{v}{\sqrt{u}},0\right),\quad
u_{3}=2^{\frac{1}{6}}\left(\frac{y}{\sqrt{u}},\frac{vz}{\sqrt{u}},\frac{u}{v\sqrt{2}}\right).$

This parametrization appears to be the same as in [59, Eq. (27)] where
$(t_{12},t_{13},t_{23})=(x,y,z)$, $y_{1}=v^{-2}$ and $y_{2}=2u^{-3}v^{4}$. We
therefore use the corresponding three-dimensional Grenier’s fundamental domain
shifted by one half in order to have only one copy of the unit density FCC
point in it, i.e. $(u,v,x,y,z)\in\mathcal{G}_{3}$ defined by the following 9
inequalities:

  1. (i)

$1\leq(1+x-y)^{2}+v^{2}\left((1-z)^{2}+\frac{u^{3}}{2v^{4}}\right)$

  2. (ii)

$1\leq(x-y)^{2}+v^{2}\left((1-z)^{2}+\frac{u^{3}}{2v^{4}}\right)$

  3. (iii)

$1\leq x^{2}+v^{2}$

  4. (iv)

$1\leq y^{2}+v^{2}\left(z^{2}+\frac{u^{3}}{2v^{4}}\right)$

  5. (iv’)

$1\leq(y-1)^{2}+v^{2}\left(z^{2}+\frac{u^{3}}{2v^{4}}\right)$

  6. (v)

$1\leq z^{2}+\frac{u^{3}}{2v^{4}}$

  7. (vi)-(viii)

$0\leq x\leq\frac{1}{2}$, $0\leq y\leq 1$, $0\leq z\leq\frac{1}{2}$.

Our goal is then to investigate the minimizers of the following type of energy
per point

$E_{f}[L]:=\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{p\in
L}f(|p|^{2})=\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{j,k,l}f\left(\frac{2^{\frac{1}{3}}V^{\frac{2}{3}}}{u}\left[(j+xk+yl)^{2}+v^{2}(k+zl)^{2}+\frac{u^{3}}{2v^{2}}l^{2}\right]\right),$

when $(u,v,x,y,z)\in\mathcal{G}_{3}$, $V>0$ and
$f(r)\in\\{r^{-\frac{s}{2}},ar^{-\frac{n}{2}}-br^{-\frac{m}{2}}\\}$. Notice
that when our potential $f$ is not summable (i.e. when $m,n,s\leq 3$), we have
used the analytic continuation of the Epstein zeta function (see e.g. [30]),
but the corresponding expression can always be written in terms of energies of
type $E_{f}$.

To compute numerically the minimizers, we have used the optimization tool
FindMinimum in Mathematica, which includes ConjugateGradient, PrincipalAxis,
LevenbergMarquardt, Newton, QuasiNewton, InteriorPoint, and LinearProgramming
methods. It has to be noticed that, as in [59], we can systematically restrict
our study to a compact domain of $\mathcal{G}_{3}$ since our energy diverges
or goes to zero as the parameters $(u,v)$ go to infinity (see also [8] for
such example in two dimensions).

When
$L\in\\{\mathbb{Z}^{3},\mathsf{D}_{3},\mathsf{D}_{3}^{*},\mathsf{A_{f}}(\Delta),\mathsf{A3}\\}$,
the exact formulas of $E_{f}[L]$ and their analytic continuations can be found
in Section 4. Also, since the shape of the global minimizer of
$E_{n,m}^{\rm{LJ}}$ is independent of $(a,b)$ (see [12]), we have decided to
choose $a=\frac{m}{n-m}$ and $b=\frac{n}{n-m}$ as it is usually done in the
Physics literature. Therefore, our Lennard-Jones type energy will be, for any
unit density lattice $L$,

$E^{\rm LJ}_{n,m}[L]=\frac{m}{n-m}\zeta_{L}(n)-\frac{n}{n-m}\zeta_{L}(m).$
(1.8)

Plan of the paper. Section 2 and 3 are respectively devoted to investigate the
lattice ground states for Riesz and Lennard-Jones type energies. The exact
formulas used to compute our energies are stated in Section 4.

##  2\. Minimizers of the Riesz energy

In this part, we investigate the minimizers of the Epstein zeta function
$L\mapsto\zeta_{L}(s)$ in $\mathcal{L}_{3}(1)$ or
$\widetilde{\mathcal{L}_{3}}(1)$ and for any $s\in\mathbb{R}$. The formulas we
are using are available in Section 4. It is indeed enough to consider the
$V=1$ case because the Epstein zeta function is homogeneous, i.e.

$\zeta_{V^{\frac{1}{3}}L}(s)=V^{-\frac{s}{3}}\zeta_{L}(s),\quad\forall
V>0,\quad\forall s\in\mathbb{R}.$ (2.1)

It is already known (see [32, 38]) that $\mathsf{D}_{3}$ and
$\mathsf{D}_{3}^{*}$ are both local minimizers of $L\mapsto\zeta_{L}(s)$ in
$\mathcal{L}_{3}(1)$ for all $s>0$. Except of the minimality result of
$\mathsf{D}_{3}$ for the height of the flat torus
$L\mapsto\zeta_{L}^{\prime}(0)$ in [59] as well as the optimality of
$\mathbb{Z}^{d}$ among $d$-dimensional orthorhombic lattices following from
Montgomery’s work [50], we are not aware of any other global optimality
results in dimension $d=3$ for the Epstein zeta function.

First, we have re-checked the following well-known observations (see [59]):

  * •

for all $0<s<\frac{3}{2}$, $\mathsf{D}_{3}^{*}$ is the unique minimizer of
$L\mapsto\zeta_{L}(s)$ in $\mathcal{L}_{3}(1)$;

  * •

for all $s>\frac{3}{2}$, $\mathsf{D}_{3}$ is the unique minimizer of
$L\mapsto\zeta_{L}(s)$ in $\mathcal{L}_{3}(1)$.

Furthermore, it is easy to show (see e.g. (4.22) applied to $s=3/2$) that
$\zeta_{\mathsf{D}_{3}}\left(\frac{3}{2}\right)=\zeta_{\mathsf{D}_{3}^{*}}\left(\frac{3}{2}\right)$.

#### Minimization for $0<s<3$.

Here we use the analytic continuation with respect to $s$ of the Epstein zeta
function. The energies of $\mathsf{D}_{3}$ and $\mathsf{D}_{3}^{*}$ have close
values, hence we plot their difference in Figure 2. It vanishes for $s\to
0^{+}$ as $\zeta_{L}(s)\to-1/2$ for any lattice $L$ and it remains finite for
$s\to 3^{-}$ by the general Kronecker’s Limit Formula [19, Thm. 2.2]. The
latter value was calculated semi-analytically. We can see that
$\zeta_{\mathsf{D}_{3}}(s)-\zeta_{\mathsf{D}_{3}^{*}}(s)$ is positive for
$0<s<3/2$ and negative otherwise, as expected.

Figure 2. Plot of
$s\mapsto\zeta_{\mathsf{D}_{3}}(s)-\zeta_{\mathsf{D}_{3}^{*}}(s)$ for
$s\in[0,3]$.

#### Minimization for $s>3$.

Recall that for large $s$ the FCC and HCP lattices could compete as they are
closed packed lattices so that they should have common asymptotes. The large
$s$ expansions can be deduced from summation formulas of the type (4.26) by
inspection. For the FCC lattice we get

$\zeta_{\mathsf{D}_{3}}(s)\approx\frac{1}{2^{s/6}}\left[6+\frac{3}{2^{s/2}}+\frac{12}{3^{s/2}}+\frac{6}{2^{s}}+\ldots\right]$
(2.2)

whereas for the HCP one

$\zeta_{\mathsf{A3}}(s)\approx\frac{1}{2^{s/6}}\left[6+\frac{3}{2^{s/2}}+\frac{1}{\big{(}\frac{8}{3}\big{)}^{s/2}}+\frac{9}{3^{s/2}}+\ldots\right].$
(2.3)

We can see that both the leading and sub-leading terms are equal and for large
enough $s$ the third term decides that
$\zeta_{\mathsf{D}_{3}}(s)<\zeta_{\mathsf{A3}}(s)$ as $3>8/3$. This still does
not mean that HCP could not prevail for some medium values of $s$. The values
of the third term become equal for $s\approx 21$. Hence we performed high
precision numerical calculations presented in Figure 3.

Figure 3. Plot of
$s\mapsto\zeta_{\mathsf{A3}}(s)-\min\\{\zeta_{\mathsf{D}_{3}}(s),\zeta_{\mathsf{D}_{3}^{*}}(s)\\}$
for $s\in[-7,20]$.

We can see that $\zeta_{\mathsf{A3}}(s)-\zeta_{\mathsf{D}_{3}}(s)$ remains
positive for all $s>3/2$, as expected. The apparent non-analyticity at $s=3/2$
is given by the fact, that we subtract another function for $s<3/2$, namely
$\zeta_{\mathsf{A3}}(s)-\zeta_{\mathsf{D}_{3}^{*}}(s)$, in order to show that
$\mathsf{A3}$ does not become the minimizer at any $s$. Note that the plotted
difference is smooth at $s\to 3$ although both $\zeta_{\mathsf{A3}}(s)$ and
$\zeta_{\mathsf{D}_{3}}(s)$ diverge.

#### Minimization for $s<0$.

Here we use the analytic continuation with respect to $s$ of the Epstein zeta
function. At first let us state that this region is physically not very
reasonable, but we can study it nevertheless. As recalled in the introduction,
its minimum value is related to the Jellium problem with Riesz interaction.
All analytically continued energies include the factor $1/\Gamma(s/2)$. It
tends to zero at $s/2\to 0,-1,-2,$ etc. The case $s\to 0$ should be omitted,
as we already mentioned that, for all lattice $L$, $\zeta_{L}(s)\to-1/2$ in
the sense of limit. All the other zeros are present; in the 1D case of
Riemann’s zeta function they are called trivial zeros. The energies are
analytical functions thus they change signs at these points; they become
negative at $-2<s<0$, $-6<s<-4$, etc., and they are positive otherwise. The
negativity of energies gives rise to extremely low possible values of
energies, say for simple hexagonal lattice $\mathsf{A_{f}}$ with very large
$\Delta$, or for very dilated orthorhombic lattices, and it has no lower
bound. In other words for $-4n-2<s<-4n$ with $n=0,1,2,\ldots$ there is no
lattice the single minimizer as the energies diverge
$\zeta_{\mathsf{A_{f}}}(s)\to-\infty$ for $\Delta\to\infty$. Finally for
intervals $-4n<s<-4n+2$ with $n=1,2,\ldots$ where the energies are positive we
have the minimizer and it this the BCC lattice. This was used in the left part
of Figure 3 as well as in Figure 4. One could plot those differences also in
the intervals where they are negative and see continuous oscillating
functions, but non of the lattices represents the minimizer there.

Figure 4. Plot of
$s\mapsto\zeta_{\mathsf{D}_{3}}(s)-\zeta_{\mathsf{D}_{3}^{*}}(s)$ when
$s\in[-11,0)$.

Concluding, for real $s\neq 3$ we still have two lattices that become
minimizers at some intervals of $s$. There is no minimizer for $-4n-2<s<-4n$
with $n=0,1,2,\ldots$; the FCC lattice minimizes the Riesz energy for $s>3/2$
and BCC otherwise. The result for negative $s$ is new.

##  3\. Minimizers of the Lennard-Jones type energy

In this section, we investigate the minimizers of the Lennard-Jones type
energy $L\mapsto E_{n,m}^{\rm LJ}[L]$ in $\mathcal{L}_{3}$ or
$\widetilde{\mathcal{L}_{3}}$ (i.e. without any constraint on the density) and
$\mathcal{L}_{3}(V)$ or $\widetilde{\mathcal{L}_{3}}(V)$, $V>0$ (i.e. at fixed
density). Using the homogeneity (2.1) of the Epstein zeta function, we recall
that, for all $V>0$ and all $L\in\widetilde{\mathcal{L}_{3}}(1)$,

$E_{n,m}^{\rm{LJ}}[V^{\frac{1}{3}}L]=\frac{m}{n-m}\frac{\zeta_{L}(n)}{V^{\frac{n}{3}}}-\frac{n}{n-m}\frac{\zeta_{L}(m)}{V^{\frac{m}{3}}}.$

We therefore use again the formulas derived in Section 4 for making our plots.

Again, only minimality results for $E_{n,m}^{\rm LJ}$ are known. In
particular, $V^{\frac{1}{3}}\mathsf{D}_{3}$ and
$V^{\frac{1}{3}}\mathsf{D}_{3}^{*}$ have been shown in [5] to be locally
minimal for $E_{n,m}^{\rm LJ}$ in $\mathcal{L}_{3}(V)$ for all fixed
$V\in(0,V_{0})$ where $V_{0}=V_{0}(n,m)$ is explicit and sharp. Furthermore,
the SC lattice $V^{\frac{1}{3}}\mathbb{Z}^{3}$ is also showed in the same work
to be locally minimal in a certain interval $(V_{1},V_{2})$ of volume. The
only other known results are the asymptotic global optimality of a FCC lattice
for large $s$ (see [12]) and some other minimality results among lattices with
prescribed number of nearest-neighbors in [7].

Minimization in $\mathcal{L}_{3}$ and $\widetilde{\mathcal{L}_{3}}$. Using
dimension reduction techniques developed in [12] for Lennard-Jones type
energies, we investigate the minimum of $E_{n,m}^{\rm LJ}$ in
$\mathcal{L}_{3}$. As already explained in [2], only two lattices appear to be
global minimizers of this energy: the FCC lattice and the HCP structure (see
Figure 5). It seems that a large (resp. narrow) well, i.e. when $n-m$ is small
(resp. large) favors the FCC (resp. HCP) lattice as a ground state. It has to
be noticed that, whatever $n$ and $m$ are, the global minimizer of
$E_{n,m}^{\rm LJ}$ in $\mathcal{L}_{3}$ is a FCC lattice. This also support
the Conjecture that we stated in [5].

Figure 5. Global minimizer o $E_{n,m}^{\rm{LJ}}$ in
$\widetilde{\mathcal{L}_{3}}$ with respect to $(n,m)$. The dash line
corresponds to $m=n$. The phase diagram for the minimization problem on
$\mathcal{L}_{3}$ is obtained by replacing the HCP phase by a FCC one.

Minimization in $\mathcal{L}_{3}(V)$ and $\widetilde{\mathcal{L}_{3}}(V)$. For
the problem of minimizing $E_{n,m}^{\rm LJ}$ at fixed density, we have plotted
in Figure 6 the phase diagram of $(n,V)\mapsto\mathop{\rm
argmin}\nolimits_{L\in\mathcal{L}_{3}(V)}E_{n,m}^{\rm LJ}[L]$ where
$m\in\\{4,5,6\\}$. We observe that there are only three kinds of minimizers in
$\widetilde{\mathcal{L}_{3}}(V)$: the FCC lattice
$V^{\frac{1}{3}}\mathsf{D}_{3}$, the Simple Hexagonal lattice
$V^{\frac{1}{3}}\mathsf{A_{f}}(\Delta(V))$ and the HCP structure
$V^{\frac{1}{3}}\mathsf{A3}$. More precisely, there exist
$V_{n,m},V_{n,m}^{\prime}>0$ such that:

  * •

for all $V\in(0,V_{n,m})$, $V^{\frac{1}{3}}\mathsf{D}_{3}$ is the unique
minimizer of $E_{n,m}^{\rm LJ}$ in $\widetilde{\mathcal{L}_{3}}(V)$;

  * •

for all $V\in(V_{n,m},V_{n,m}^{\prime})$, $V^{\frac{1}{3}}\mathsf{A3}$ is the
unique minimizer of $E_{n,m}^{\rm LJ}$ in $\widetilde{\mathcal{L}_{3}}(V)$;

  * •

for all $V>V_{n,m}^{\prime}$, $V^{\frac{1}{3}}\mathsf{A_{f}}(\Delta)$ is the
unique minimizer of $E_{n,m}^{\rm LJ}$ in $\widetilde{\mathcal{L}_{3}}(V)$ for
some $\Delta=\Delta(V)$, where the behavior of $\Delta(V)$ is depicted in
Figure 7 for $n=12$ and $m\in\\{4,5,6\\}$.

If we minimize $E_{n,m}^{\rm LJ}$ only in $\mathcal{L}_{3}(V)$, the HCP phase
is replaced by a FCC one in such a way that

  * •

for all $V\in(0,V_{n,m}^{\prime})$, $V^{\frac{1}{3}}\mathsf{D}_{3}$ is the
unique minimizer of $E_{n,m}^{\rm LJ}$ in $\mathcal{L}_{3}(V)$;

  * •

for all $V>V_{n,m}^{\prime}$, $V^{\frac{1}{3}}\mathsf{A_{f}}(\Delta)$ is the
unique minimizer of $E_{n,m}^{\rm LJ}$ in $\mathcal{L}_{3}(V)$ for some
$\Delta=\Delta(V)$ as depicted in Figure 7.

Figure 6. Phase diagram of the minimizer of $E_{n,m}^{\rm{LJ}}$ in
$\widetilde{\mathcal{L}_{3}}(V)$ for $m=4$ (up left) and $m=5$ (up right) and
$m=6$ (down), $n$ varying. The phase diagram for the minimization problem on
$\mathcal{L}_{3}(V)$ is roughly obtained by replacing the HCP phase by a FCC
one.

Figure 7. Simple Hexagonal phase. Plot of $V\mapsto\mathop{\rm
argmin}\nolimits_{\Delta}E_{n,m}^{\rm{LJ}}[V^{\frac{1}{3}}\mathsf{A}_{f}(\Delta)]$
for $(n,m)\in\\{(12,m):m=4,5,6\\}$.

In [5], the interval of volumes $V$ where the SC lattice is locally minimal
for $E_{n,m}^{\rm{LJ}}$ in $\mathcal{L}_{3}(V)$ has been derived. Here we have
observed that a family of SH lattices have lower energy than the SC one. This
is new compared to the two-dimensional case where the square lattice that have
been shown in [4] to be locally minimal in some interval of volume is also
observed to be minimal in $\mathcal{L}_{2}(V)$ in the same interval.
Furthermore, our numerical findings concerning the global minimizer of
$E_{n,m}^{\rm{LJ}}$ in $\mathcal{L}_{3}$ support [5] and confirm [2].

##  4\. Explicit formulas for three-dimensional Epstein zeta functions

This section is devoted to the construction of explicit formulas for Epstein
zeta functions associated to the three-dimensional lattices considered in the
present paper. We start with converging sums, i.e., the parameter $s$ is
larger than dimension 3, and express them as integrals over powers of Jacobi
elliptic functions. Then using specific tricks with elliptic functions these
integrals are rewritten into the ones which represent an analytic continuation
of Epstein zeta functions to the whole complex plane except for $s=3$.

###  4.1. Simple cubic lattice

Let particles be localized at sites of the SC lattice with spacing $a$. In
this paper, we compare the energy per particle for various 3D lattice
structures at a fixed density of particles, say the unit one. The particle
density associated with the SC lattice, given by $\rho=1/a^{3}$, is equal to
one when $a=1$. Particles interact pairwisely by the Riesz potential $1/r^{s}$
where $r=|{\bf r}|$ is the distance between two particles and $s$ is a complex
number. The energy per particle is given by the Epstein zeta function
associated with the SC lattice

$\zeta_{\mathbb{Z}^{3}}(s)=\frac{1}{2}\sum_{(j,k,l)\neq(0,0,0)}\frac{1}{(j^{2}+k^{2}+l^{2})^{s/2}},$
(4.1)

where the prefactor $1/2$ is due to the fact that every energy is shared by
two particles and the sum converges only if the real part of $s$ $\Re(s)>3$.
Using the $\Gamma$-identity

$\frac{1}{r^{s}}=\frac{1}{(r^{2})^{s/2}}=\frac{1}{\Gamma(s/2)}\int_{0}^{\infty}{\rm
d}t\,t^{\frac{s}{2}-1}{\rm e}^{-r^{2}t},$ (4.2)

the expression (4.1) can be rewritten as

$\displaystyle\zeta_{\mathbb{Z}^{3}}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2\Gamma(s/2)}\int_{0}^{\infty}\frac{{\rm
d}t}{t}t^{s/2}\left[\sum_{j,k,l=-\infty}^{\infty}{\rm
e}^{-(j^{2}+k^{2}+l^{2})t}-1\right]$ (4.3) $\displaystyle=$
$\displaystyle\frac{1}{2\Gamma(s/2)}\int_{0}^{\infty}\frac{{\rm
d}t}{t}t^{s/2}\left[\theta_{3}^{3}\left({\rm e}^{-t}\right)-1\right]$
$\displaystyle=$
$\displaystyle\frac{\pi^{s/2}}{2\Gamma(s/2)}\int_{0}^{\infty}\frac{{\rm
d}t}{t}t^{s/2}\left[\theta_{3}^{3}\left({\rm e}^{-\pi t}\right)-1\right],$

where the Jacobi elliptic function with zero argument
$\vartheta_{3}(0,q)\equiv\theta_{3}(q)=\sum_{j=-\infty}^{\infty}q^{j^{2}}$
[37] was introduced. With regard to the definition of $\theta_{3}(q)$, it
holds that $\theta_{3}\left({\rm e}^{-\pi t}\right)\sim_{t\to\infty}1+2{\rm
e}^{-\pi t}+\cdots$ and the integral in (4.3) converges at large $t$ for any
$s$. Using the Poisson summation formula

$\sum_{j=-\infty}^{\infty}{\rm e}^{-(j+\phi)^{2}\pi
t}=\frac{1}{\sqrt{t}}\sum_{j=-\infty}^{\infty}{\rm e}^{2\pi{\rm i}j\phi}{\rm
e}^{-(\pi j^{2})/t}$ (4.4)

it can be shown that

$\theta_{3}\left({\rm e}^{-\pi t}\right)\mathop{\sim}_{t\to
0}\frac{1}{\sqrt{t}}\left(1+2{\rm e}^{-\pi/t}+\cdots\right).$ (4.5)

The function under integration in (4.3) behaves like $t^{\frac{s-3}{2}-1}$ for
$t\to 0$, so the real part of $s$ must be greater than 3 as was expected.

To derive an analytic continuation of (4.3) to the complex-$s$ plane, we
follow the derivation presented in Ref. [65] for $d$-dimensional hypercubic
lattices. In the integral on the rhs of (4.3), we split the integration over
$t$ into intervals $[0,1]$ and $[1,\infty]$. In the second integral over
$t\in[1,\infty]$, one applies the equality

$\theta_{3}(e^{-\pi t})=\frac{1}{\sqrt{t}}\theta_{3}\left({\rm
e}^{-\pi/t}\right)$ (4.6)

which follows from (4.4) and substitutes $t=1/t^{\prime}$ to get

$\int_{1}^{\infty}\frac{{\rm d}t}{t}t^{s/2}\left[\theta_{3}^{3}\left({\rm
e}^{-\pi t}\right)-1\right]=\int_{0}^{1}\frac{{\rm
d}t^{\prime}}{t^{\prime}}{t^{\prime}}^{(3-s)/2}\left[\theta_{3}^{3}\left({\rm
e}^{-\pi t^{\prime}}\right)-{t^{\prime}}^{-3/2}\right].$ (4.7)

As concerns the first integral over $t\in[0,1]$, to subtract the singularity
of $\theta_{3}^{3}\left({\rm e}^{-\pi t}\right)\sim t^{-3/2}$ as $t\to 0$ one
adds the vanishing contribution $-t^{-3/2}+t^{-3/2}$ and integrates out the
remaining terms proportional to $t^{-3/2}-1$, with the result

$\frac{\Gamma{(s/2)}}{\pi^{s/2}}\zeta_{\mathbb{Z}^{3}}(s)=-\frac{1}{s}-\frac{1}{3-s}+\int_{0}^{1}\frac{{\rm
d}t}{2t}\ \left[t^{s/2}+t^{(3-s)/2}\right]\left[\theta_{3}^{3}\left({\rm
e}^{-\pi t}\right)-\frac{1}{t^{3/2}}\right].$ (4.8)

The difference in the square bracket is proportional to $t^{-3/2}e^{-\pi/t}$
for $t\to 0$ and hence the formula (4.8) converges for all complex $s$, except
for the singular point $s=3$. The limit $s\to 0$ is not problematic since the
singularity $-1/s$ on the rhs of (4.8) has a counterpart $\Gamma(s/2)\sim 2/s$
on the lhs, so that $\zeta_{\mathbb{Z}^{3}}(0)=-1/2$. The numerical evaluation
of $\zeta_{\mathbb{Z}^{3}}(s)$ by using (4.8) at one point $s$ takes around 5
seconds of CPU time on a standard PC, the achieved accuracy is around 29
decimal digits. Similar characteristics occur in the numerical evaluation of
all Epstein functions studied below, except for the HCP structure given by the
lattice sum for which the calculation of the Epstein function takes around 1
minute.

It is known that the transformation $s\to 3-s$ connects the lattice sums of
dual (reciprocal) lattice structures [17]. The SC lattice is self-dual. The
rhs of (4.8) is invariant with respect to this transformation, hence it holds
that

$\zeta_{\mathbb{Z}^{3}}(s)=\frac{\pi^{s-3/2}\,\Gamma\left(\frac{3-s}{2}\right)}{\Gamma\left(\frac{s}{2}\right)}\zeta_{\mathbb{Z}^{3}}(3-s).$
(4.9)

###  4.2. Body centered cubic lattice

The BCC lattice is composed of two SC lattices, shifted with respect to one
another by half-period along each of the three coordinates. There are two
particles per elementary cube of side $a_{\rm BCC}$, i.e., $\rho=2/a^{3}_{\rm
BCC}=1$ which implies that the unit density corresponds to $a_{\rm
BCC}=2^{1/3}$. Consequently,

$\zeta_{\mathsf{D}_{3}^{*}}(s)=\frac{1}{2^{s/3}}\left[\zeta_{\mathbb{Z}^{3}}(s)+\psi(1/2,1/2,1/2;s)\right],$
(4.10)

where

$\displaystyle\psi(x_{1},x_{2},x_{3};s)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j,k,l=-\infty}^{\infty}\frac{1}{[(j-x_{1})^{2}+(k-x_{2})^{2}+(l-x_{3})^{2}]^{s/2}}$
(4.11) $\displaystyle=$
$\displaystyle\frac{1}{2\Gamma(s/2)}\int_{0}^{\infty}\frac{{\rm
d}t}{t}t^{s/2}\sum_{j,k,l=-\infty}^{\infty}{\rm
e}^{-[(j-x_{1})^{2}+(k-x_{2})^{2}+(l-x_{3})^{2}]t}$

is a sum over a lattice shifted with respect to the original cubic one.
Introducing another Jacobi elliptic function
$\theta_{2}(q)=\sum_{j=-\infty}^{\infty}q^{(j-\frac{1}{2})^{2}}$, the BCC
energy is expressible as

$\zeta_{\mathsf{D}_{3}^{*}}(s)=\frac{1}{2^{s/3}}\left[\zeta_{\mathbb{Z}^{3}}(s)+\frac{\pi^{s/2}}{2\Gamma(s/2)}\int_{0}^{\infty}\frac{{\rm
d}t}{t}t^{s/2}\theta_{2}^{3}({\rm e}^{-\pi t})\right].$ (4.12)

In view of the Poisson summation formula (4.4), the Jacobi theta function
$\theta_{4}(q)=\sum_{j=-\infty}^{\infty}(-1)^{j}q^{j^{2}}$ is “dual” to
$\theta_{2}$ in the sense that

$\theta_{2}\left({\rm e}^{-\pi
t}\right)=\frac{1}{\sqrt{t}}\theta_{4}\left({\rm e}^{-\pi/t}\right).$ (4.13)

Thus,

$\theta_{2}\left({\rm e}^{-\pi t}\right)\mathop{\sim}_{t\to
0}\frac{1}{\sqrt{t}}\left(1-2{\rm e}^{-\pi/t}+\cdots\right)$ (4.14)

and the integral in (4.12) converges at small $t$ if $\Re(s)>3$ as was
expected.

To derive an analytic continuation of (4.12) we proceed in close analogy with
the previous SC lattice. The integration on the rhs of (4.12) is split into
intervals $[0,1]$ and $[1,\infty]$. The second integral over $t\in[1,\infty]$
can be transformed by using the relation (4.13) and the consequent
substitution $t=1/t^{\prime}$ to

$\int_{1}^{\infty}\frac{{\rm d}t}{t}t^{s/2}\theta_{2}^{3}({\rm e}^{-\pi
t})=\int_{0}^{1}{\rm d}t^{\prime}\,{t^{\prime}}^{(1-s)/2}\theta_{4}^{3}({\rm
e}^{-\pi t^{\prime}}).$ (4.15)

The first integral over $t\in[0,1]$ is modified by adding the vanishing
contribution $-t^{-3/2}+t^{-3/2}$; the term $-t^{-3/2}$ removes the
singularity of $\theta_{2}^{3}({\rm e}^{-\pi t})$ as $t\to 0$ and the term
$t^{-3/2}$ is integrated out. The final formula for the Epstein function
associated with the BCC lattice reads as

$\displaystyle\frac{2^{s/3}\Gamma(s/2)}{\pi^{s/2}}\zeta_{\mathsf{D}_{3}^{*}}(s)$
$\displaystyle=$
$\displaystyle\frac{\Gamma{(s/2)}}{\pi^{s/2}}\zeta_{\mathbb{Z}^{3}}(s)-\frac{1}{3-s}$
(4.16) $\displaystyle+\int_{0}^{1}\frac{{\rm
d}t}{2t}\left\\{t^{s/2}\left[\theta_{2}^{3}\left({\rm e}^{-\pi
t}\right)-\frac{1}{t^{3/2}}\right]+t^{(3-s)/2}\theta_{4}^{3}\left({\rm
e}^{-\pi t}\right)\right\\},$

where $\zeta_{\mathbb{Z}^{3}}(s)$ is given by (4.8). Since

$\theta_{4}\left({\rm e}^{-\pi t}\right)\mathop{\sim}_{t\to
0}\frac{1}{\sqrt{t}}{\rm e}^{-\pi/(4t)}+\cdots,$ (4.17)

the integral in (4.16) converges for any complex $s$.

###  4.3. Face centered cubic lattice

There are four particles per elementary cube of side $a_{\rm FCC}$. thus
$\rho=4/a^{3}_{\rm FCC}=1$ which implies that $a_{\rm FCC}=2^{2/3}$. For
$\Re(s)>3$, one gets

$\displaystyle\zeta_{\mathsf{D}_{3}}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{2s/3}}\left[\zeta_{\mathbb{Z}^{3}}(s)+\psi\left(0,\frac{1}{2},\frac{1}{2};s\right)+\psi\left(\frac{1}{2},0,\frac{1}{2};s\right)+\psi\left(\frac{1}{2},\frac{1}{2},0;s\right)\right]$
(4.18) $\displaystyle=$
$\displaystyle\frac{1}{2^{2s/3}}\left[\zeta_{\mathbb{Z}^{3}}(s)+\frac{3\pi^{s/2}}{2\Gamma(\frac{s}{2})}\int_{0}^{\infty}\frac{{\rm
d}t}{t}t^{s/2}\theta_{2}^{2}\left({\rm e}^{-t}\right)\theta_{3}\left({\rm
e}^{-t}\right)\right].$

Splitting the integration into the intervals $[0,1]$ and $[1,\infty]$ and
using the equality

$\int_{1}^{\infty}\frac{{\rm d}t}{t}t^{s/2}\theta_{2}^{2}\left({\rm e}^{-\pi
t}\right)\theta_{3}\left({\rm e}^{-\pi t}\right)=\int_{0}^{1}{\rm
d}t\,t^{(1-s)/2}\theta_{4}^{2}\left({\rm e}^{-\pi
t}\right)\theta_{3}\left({\rm e}^{-\pi t}\right),$ (4.19)

the analytic continuation of (4.18) to the whole complex $s$-plane (except for
the singular point $s=3$) takes the form

$\displaystyle\frac{2^{2s/3}\Gamma(s/2)}{\pi^{s/2}}\zeta_{\mathsf{D}_{3}}(s)$
$\displaystyle=$
$\displaystyle\frac{\Gamma{(s/2)}}{\pi^{s/2}}\zeta_{\mathbb{Z}^{3}}(s)-\frac{3}{3-s}$
(4.20) $\displaystyle+\frac{3}{2}\int_{0}^{1}\frac{{\rm
d}t}{t}\Bigg{\\{}t^{s/2}\left[\theta_{2}^{2}\left({\rm e}^{-\pi
t}\right)\theta_{3}\left({\rm e}^{-\pi t}\right)-\frac{1}{t^{3/2}}\right]$
$\displaystyle+t^{(3-s)/2}\theta_{4}^{2}\left({\rm e}^{-\pi
t}\right)\theta_{3}\left({\rm e}^{-\pi t}\right)\Bigg{\\}}.$

The FCC and BCC lattices is a pair of dual structures. Exploring theta-
function identities [70]

$\theta_{3}(q)=\theta_{3}(q^{4})+\theta_{2}(q^{4}),\qquad\theta_{4}(q)=\theta_{3}(q^{4})-\theta_{2}(q^{4})$
(4.21)

in the representations (4.16) and (4.20) it can be shown after some algebra
that

$\zeta_{\mathsf{D}_{3}^{*}}(s)=\frac{\pi^{s-3/2}\,\Gamma\left(\frac{3-s}{2}\right)}{\Gamma\left(\frac{s}{2}\right)}\zeta_{\mathsf{D}_{3}}(3-s)$
(4.22)

or, vice versa,

$\zeta_{\mathsf{D}_{3}}(s)=\frac{\pi^{s-3/2}\,\Gamma\left(\frac{3-s}{2}\right)}{\Gamma\left(\frac{s}{2}\right)}\zeta_{\mathsf{D}_{3}^{*}}(3-s).$
(4.23)

The turning point of these transformations is $s=3/2$ where
$\zeta_{\mathsf{D}_{3}}(3/2)=\zeta_{\mathsf{D}_{3}^{*}}(3/2)$.

###  4.4. Simple hexagonal lattice

The SH lattice is composed of parallel planes composed of the 2D hexagonal
lattice with spacing $a_{\rm SH}$, the distance between the nearest-neighbor
planes is $d$; the dimensionless parameter $\Delta=d/a_{\rm SH}$. The 2D
hexagonal lattice can be considered as the union of two rectangle lattices of
sides $a_{\rm SH}$ and $\sqrt{3}a_{\rm SH}$, shifted with respect to one
another by a half-period. Thus the sites of the SH lattice can be
parameterized with respect to the reference point $(0,0,0)$ as follows:
$a_{\rm SH}(j,\sqrt{3}k,\Delta l)$ with integers $j,k,l$ such that
$(j,k,l)\neq(0,0,0)$ and $a_{\rm
SH}\left[(j+\frac{1}{2}),\sqrt{3}(k+\frac{1}{2}),\Delta l\right]$ with integer
$j,k,l$. There are two particles in the elementary rectangular parallelepiped
of volume $\sqrt{3}a_{\rm SH}^{2}d$, so the unit density of particles
corresponds to $2/\left(\sqrt{3}\Delta a_{\rm SH}^{3}\right)=1$. For
$\Re(s)>3$, the energy per particle for the SH lattice is given by

$\displaystyle\zeta_{\mathsf{A_{f}}(\Delta)}(s)$ $\displaystyle=$
$\displaystyle\frac{\pi^{s/2}}{2a_{\rm
SH}^{s}\Gamma(s/2)}\int_{0}^{\infty}\frac{{\rm
d}t}{t}t^{s/2}\Big{[}\theta_{3}\left({\rm e}^{-\pi
t}\right)\theta_{3}\left({\rm e}^{-3\pi t}\right)\theta_{3}\left({\rm e}^{-\pi
t\Delta^{2}}\right)$ (4.24) $\displaystyle-1+\theta_{2}\left({\rm e}^{-\pi
t}\right)\theta_{2}\left({\rm e}^{-3\pi t}\right)\theta_{3}\left({\rm e}^{-\pi
t\Delta^{2}}\right)\Big{]}.$

The analytic continuation to the whole complex $s$-plane (except for the point
$s=3$) is obtained in the form

$\displaystyle\left(\frac{2}{\sqrt{3}\Delta}\right)^{s/3}\frac{\Gamma(s/2)}{\pi^{s/2}}\zeta_{\mathsf{A_{f}}(\Delta)}(s)$
$\displaystyle=$ $\displaystyle-\frac{1}{s}-\frac{2}{\Delta\sqrt{3}(3-s)}$
$\displaystyle+\int_{0}^{1}\frac{{\rm
d}t}{2t}\Bigg{\\{}t^{s/2}\Big{[}\theta_{3}\left({\rm e}^{-\pi
t}\right)\theta_{3}\left({\rm e}^{-3\pi t}\right)\theta_{3}\left({\rm e}^{-\pi
t\Delta^{2}}\right)$ $\displaystyle+\theta_{2}\left({\rm e}^{-\pi
t}\right)\theta_{2}\left({\rm e}^{-3\pi t}\right)\theta_{3}\left({\rm e}^{-\pi
t\Delta^{2}}\right)-\frac{2}{\sqrt{3}\Delta t^{3/2}}\Big{]}$
$\displaystyle+\frac{t^{(3-s)/2}}{\sqrt{3}\Delta}\Big{[}\theta_{3}\left({\rm
e}^{-\pi t}\right)\theta_{3}\left({\rm e}^{-\pi
t/3}\right)\theta_{3}\left({\rm e}^{-\pi t/\Delta^{2}}\right)$
$\displaystyle-\frac{\sqrt{3}\Delta}{t^{3/2}}+\theta_{4}\left({\rm e}^{-\pi
t}\right)\theta_{4}\left({\rm e}^{-\pi t/3}\right)\theta_{3}\left({\rm
e}^{-\pi t/\Delta^{2}}\right)\Big{]}\Bigg{\\}}.$

###  4.5. Hexagonal close-packed lattice

For very large $s\to\infty$ the Riesz potential $1/r^{s}$ tends to infinity
for $r<1$ and vanishes for $r>1$, i.e. it approaches the hard-sphere limit.
For this case a close-packed structure becomes the minimizer. There are two
such lattices in 3D, besides the FCC also the hexagonal close-packed (HCP)
lattice with the same packing ratio. The two lattices have common asymptotic
value of the Epstein zeta function and it is relatively simple to derive how
their difference behaves for large $s$.

The HCP lattice is pictured in Fig. 1; the middle hexagonal layer is shifted
by a half-period with respect to the top and bottom hexagonal lattices.
Denoting by $a_{\rm HCP}$ the spacing of the one-layer hexagonal lattice, the
distance between top and bottom hexagonal layers is given by $d=\Delta a_{\rm
HCP}$ with $\Delta=\sqrt{8/3}$. As an elementary cell we take the rectangular
parallelepiped with sides $a_{\rm HCP}$ and $\sqrt{3}a_{\rm HCP}$ on the top
and bottom layers and the height $d$, its volume is given by $\sqrt{3}a_{\rm
HCP}^{2}d=\sqrt{8}a_{\rm HCP}^{3}$. There are 8 particles on the top and
bottom layers each shared by 8 elementary cells, 2 particles on the top and
bottom layers each shared by 2 cells, 1 particle on the middle layer sitting
in the cell interior and 2 particles on the middle layer sitting on the cell
surface (each shared by 2 cells), i.e. altogether there are 4 particles per
elementary cell. The density of particles $\rho=4/(\sqrt{8}a_{\rm HCP}^{3})$
equals to unity when $a_{\rm HCP}=2^{1/6}$.

Particle positions on the top and bottom layers can be parameterized with
respect to the reference particle at the origin $(0,0,0)$ as follows: $a_{\rm
HCP}(j,\sqrt{3}k,\Delta l)$ with integer $j,k,l$ such that
$(j,k,l)\neq(0,0,0)$ and $a_{\rm
HCP}\left[(j+\frac{1}{2}),\sqrt{3}(k+\frac{1}{2}),\Delta l\right]$ with
integer $j,k,l$. It can be shown that particle positions on the middle layer
can be parameterized as follows: $a_{\rm
HCP}\left[j,\sqrt{3}(k+\frac{2}{3}),\Delta(l+\frac{1}{2})\right]$ and $a_{\rm
HCP}\left[(j+\frac{1}{2}),\sqrt{3}(k+\frac{1}{6}),\Delta(l+\frac{1}{2})\right]$
with integer $j,k,l$. The energy per particle of HCP lattice for $\Re(s)>3$
thus reads as

$\displaystyle\zeta_{\mathsf{A3}}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{s/6+1}}\Bigg{\\{}\sum_{(j,k,l)\neq(0,0,0)}\frac{1}{(j^{2}+3k^{2}+\frac{8}{3}l^{2})^{s/2}}$
(4.26)
$\displaystyle+\sum_{j,k,l}\frac{1}{[(j+1/2)^{2}+3(k+1/2)^{2}+\frac{8}{3}l^{2}]^{s/2}}$
$\displaystyle+\sum_{j,k,l}\frac{1}{[j^{2}+3(k+2/3)^{2}+\frac{8}{3}(l+1/2)^{2}]^{s/2}}$
$\displaystyle+\sum_{j,k,l}\frac{1}{[(j+1/2)^{2}+3(k+1/6)^{2}+\frac{8}{3}(l+1/2)^{2}]^{s/2}}\Bigg{\\}}.$

All sums are expressible in terms of Jacobi elliptic functions, except the
ones over $k$ involving the shift of $k$ by $1/6$ or $2/3$; for these sums, to
attain the sufficient precision we cut the sums over $k$ to the interval
$[-60,60]$.

## Acknowledgments

LB is supported by the Austrian Science Fund (FWF) and the German Research
Foundation (DFG) through the joint project FR 4083/3-1/I 4354. LŠ and IT
received support from VEGA Grant No. 2/0092/21.

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