# Connecting the hexagonal closed packed structure with the cuboidal lattices:
A Burgers-Bain type martensitic transformation for a Lennard-Jones solid
derived from exact lattice summations

Peter Schwerdtfeger<EMAIL_ADDRESS>Centre for Theoretical
Chemistry and Physics, The New Zealand Institute for Advanced Study (NZIAS),
Massey University Albany, Private Bag 102904, Auckland 0745, New Zealand
Shaun Cooper<EMAIL_ADDRESS>Centre for Theoretical Chemistry and
Physics, Institute for Natural and Mathematical Sciences (INMS), Massey
University Albany, Private Bag 102904, Auckland 0745, New Zealand.  Odile
Smits<EMAIL_ADDRESS>Centre for Theoretical Chemistry and
Physics, The New Zealand Institute for Advanced Study (NZIAS), Massey
University Albany, Private Bag 102904, Auckland 0745, New Zealand  Andres
Robles-Navarro<EMAIL_ADDRESS>Centre for Theoretical Chemistry and
Physics, The New Zealand Institute for Advanced Study (NZIAS), Massey
University Albany, Private Bag 102904, Auckland 0745, New Zealand

###### Abstract

The diffusionless martensitic phase transition from a hexagonal close-packed
(hcp) arrangement to the face-centered close-packed (fcc) and subsequently the
body-centered cubic (bcc) lattice is discussed for a Lennard-Jones solid. The
associated generalized lattice vectors to construct the underlying bi-lattice
for a Burgers-Bain-type of transformation require a minimum of a four-
parameter space $(a,\alpha,\beta,\gamma=c/a)$, describing, beside the change
in the base lattice parameters $a$ and $c$ of a hexagonal or cuboidal cell,
the shear force acting on the hexagonal base plane through the parameter
$\alpha$, and the sliding force of the middle layer in the original AB
hexagonal packing arrangement through the parameter $\beta$. The Bain part of
the transformation depends only on two parameters $a$ and $\gamma$, describing
a cuboidal transition within a body-centered tetragonal cell. By optimizing
the lattice parameters $a$ and $\beta$ for a $(n,m)$-Lennard-Jones potential,
we obtain a simple two-dimensional picture for the complete Burgers-Bain-type
hcp$\leftrightarrow$fcc$\leftrightarrow$bcc phase transition. From the
generalized lattice vectors we were able to construct the corresponding
lattice sums in terms of inverse power potentials applying fast converging
Bessel function expansions using a Terras decomposition of the Epstein zeta
function combined with a Van der Hoff-Benson expansion for the lattice sums.
This allows the cohesive energy to be determined to computer precision for a
Lennard-Jones solid. For six different combinations of $(n,m)$-Lennard-Jones
potentials the energy $(\alpha,\gamma)$ hypersurface was then mapped out and
studied in more detail. There has been a long controversy if the fcc phase is
required for the hcp$\leftrightarrow$bcc phase transition (two-step model). We
show that for a Lennard-Jones model the minimum energy path is found to be
indeed a two-step hcp$\rightarrow$fcc$\rightarrow$bcc transition process. The
lowest transition state in each case can be regarded as an upper limit to a
hypothetical true minimum energy path out of the many possibilities in a
hcp$\leftrightarrow$fcc$\leftrightarrow$bcc phase transition.

##  I Introduction

The relative stability between the face-centred cubic (fcc) and hexagonal
close-packed (hcp) structures, and their formation from the gas or the liquid
phase known as the nucleation problem, has been a matter of intense discussion
and debate over the past 50 years.Sanchez-Burgos  _et al._ (2021) Both phases
have the same hard-sphere packing density of
$\rho=\pi/3\sqrt{2}$=0.74048048969…, as have all possible associated Barlow
structures (mixtures between (AB)∞ and (ABC)∞ stackings)Barlow (1883); Osang,
Edelsbrunner, and Saadatfar (2021) for which there are infinitely many. It was
only in recent times that Kepler’s original conjecture, stating that the fcc
packing density cannot be surpassed for hard (unit) sphere packings and
therefore is optimal, was proven by Hales.Hales (2011, 2005)

Optimal sphere packings predict only a very small difference in the free
energy up to the melting point.Frenkel and Ladd (1984); Woodcock (1997);
Sanchez-Burgos  _et al._ (2021) For real atomic or molecular crystals the two
close-packed structures are also found to be very close in energy.Krainyukova
(2011) A prominent example is solid argon, where at low temperatures and
pressures, vibrational effects need to be included to stabilize the
experimentally observed fcc over the hcp phase,Moyano, Schwerdtfeger, and
Rosciszewski (2007); Schwerdtfeger  _et al._ (2016). However, the cohesive
energy difference between the two phases is predicted to be a mere 8.8 J/mol
at zero temperature and pressure.Schwerdtfeger  _et al._ (2016) Such small
energy differences between the fcc and hcp phase for solids are also detected
in nucleation processes Senger  _et al._ (1999); Lovett (2007); Lu and Li
(1998); Schwerdtfeger  _et al._ (2006). On the other hand, for the heavier
noble gas elements the hcp phase becomes the dominant phase at high
pressures,Cynn  _et al._ (2001); Rosa  _et al._ (2018); Dewaele  _et al._
(2021) which is most likely due to the fact that with increasing pressure
vibrational effects become less important compared to the competing many-body
electronic contributions.

Why one phase dominates over the other at certain temperatures and pressures
delicately depends on the different static and dynamic contributions to the
free energy. It is however notoriously difficult to get a detailed mechanistic
insight into such solid-state phase transitions by both experimental and
theoretical methods,Caspersen and Carter (2005) and therefore such transitions
are in general not well understood.Stillinger (2001); Jackson, Bruce, and
Ackland (2002) One could naively slide some of the hexagonal layers to turn
hcp into fcc, i.e. for the hexagonal layer sequences we have the
transformation
(ABA$\downarrow$B$\downarrow$A$\uparrow$B$\uparrow$)${}_{\infty}\rightarrow$(ABCABC)∞.Jackson,
Bruce, and Ackland (2002) However, it is not known if this is the optimal
minimum energy path, for example, in such sliding of the hexagonal layers one
may well access local minima such as dense Barlow packings, e.g.
(ABABA$\downarrow$B)${}_{\infty}\rightarrow$(ABABCB)∞.

In 1934 Burgers suggested a very simple diffusionless hcp$\leftrightarrow$bcc
phase transition pathBurgers (1934), while Bain earlier in 1924 proposed a
simple cuboidal fcc$\leftrightarrow$bcc transition path Bain (1924). Since
then, there has been some controversy whether or not the fcc phase is actually
required in a minimum energy path for the hcp$\rightarrow$bcc phase transition
and if these transitions follow at all the diffusionless mechanism as
suggested by Bain and Burgers.Cayron (2013); Lu  _et al._ (2014); Cayron
(2015, 2016)

Modeling solid-state phase transitions at the microscopic level poses
significant challenges.Raghavan and Cohen (1975); Gooding and Krumhansl
(1988); Caspersen and Carter (2005); Johnson and Carter (2008); Torrents  _et
al._ (2017); Li, Qian, and Li (2021) This is mainly due to the fact that often
stacking faults or defects are involved in such phase changes, which requires
a computationally expensive super-cell treatment in a molecular dynamics
simulation.Bruinsma and Zangwill (1985); Dovesi and Orlando (1994); Li  _et
al._ (2017) Moreover, various theoretical approximations (such as the density
functional approximation) may not result in the correct energy sequence
between the different polymorphs involved.Liu  _et al._ (2003) Diffusionless
martensitic transformations do not have these problems, but mapping out the
correct minimum energy path for such a transition can be a formidable
task.Rifkin (1984); Sandoval, Urbassek, and Entel (2009); Cayron (2013, 2015);
Li  _et al._ (2017)

Caspersen and Carter Caspersen and Carter (2005) generalized the climbing
image-nudged elastic band algorithm to find transition paths in solid-state
phase transitions, and subsequently applied this to the diffusionless
martensitic transformation Bain (1924); Burgers (1934) from hcp to the body-
centered cubic (bcc) structure and further to fcc for metallic lithium. They
gave detailed information for the transformation matrices acting on the
corresponding lattice vectors. In a different approach, Raju Natarajan and Van
der Ven used a set of two parameters derived from the Hencky logarithmic
strain to map out the volume preserving potential energy surfaces for the
Burgers-Bain transformation for metallic lithium, sodium and magnesium.Raju
Natarajan and Van der Ven (2019) Cayron introduced transformation matrices
(which he termed angular distortive matrices) for continuous atomic
displacements between the three different phases.Cayron (2016) Bingxi Li et
al. performed molecular dynamics simulations using a (12,6)-Lennard-Jones
potential for argon calibrated at $T=40$ K and $P=1$ bar.Li  _et al._ (2017)
They discussed different paths for the hcp$\leftrightarrow$fcc phase
transition. However, the topology of the cohesive energy dependent of the
lattice parameters sensitive to such phase transition may critically depend on
the chosen model describing the interactions between the atoms in the lattice.
Moreover, molecular dynamics (MD) or Monte-Carlo simulations for the phase
transitions in the solid state are computer time consuming and depend on the
size of the supercell chosen. Hence, it would be advantageous to develop a
simpler model capable of estimating the activation energy involved in such
solid-state phase transitions.

In this paper we analyze in detail the Burgers-Bain hexagonal-to-cuboidal
transformation path from hcp$\rightarrow$fcc$\rightarrow$bcc for the case of a
general $(n,m)$ Lennard-Jones (LJ) potential ($n>m,m>3$),Jones (1924); Jones
and Ingham (1925)

$V_{\textrm{LJ}}(r)=\frac{\epsilon
nm}{n-m}\;\left[\frac{1}{n}\left(\frac{r_{e}}{r}\right)^{n}-\frac{1}{m}\left(\frac{r_{e}}{r}\right)^{m}\right]$
(1)

describing the interactions between the atoms in a solid using exact lattice
summations to obtain the cohesive energy.Borwein  _et al._ (2013); Burrows
_et al._ (2020); Schwerdtfeger, Burrows, and Smits (2021) Here, $\epsilon$ and
$r_{e}$ are the binding energy and the equilibrium distance of the diatomic
molecule, respectively. Both of these parameters can be arbitrarily set to 1
in order to express them in dimensionless units. For very large exponents
$n,m$ the LJ potential approaches the kissing hard-sphere (KHS) model with
$V(r)=\infty$ for $r<r_{e}$, $V(r)=-\epsilon$ for $r=r_{e}$, and $V(r)=0$ for
$r>r_{e}$ as originally introduced by Baxter Baxter (1968); Stell (1991).

We have recently shown that the Bain transformation can be effectively
described within a two-parameter space $(A,r_{\text{NN}})$,Jerabek, Burrows,
and Schwerdtfeger (2022) where $r_{\text{NN}}$ is the nearest neighbor
distance in the solid and $A$ is responsible for describing the martensitic
transformation path from the axial centred-cuboidal (acc) lattice ($A=1/3$),
to the bcc lattice ($A=1/2$), the mean centred-cuboidal (mcc) lattice
($A=1/\sqrt{2}$), and finally the face-centred cubic (fcc) lattice ($A=1$) in
a general body-centered tetragonal (bct) arrangement. The advantage of this
choice of parameters instead of the usual bct lattice parameters $(a,c)$ is
that the nearest neighbor distance changes only slightly along the Bain
transformation path and a two-dimensional picture can therefore be avoided. We
will show that a similar choice of parameters can be used to describe the
Burgers transformation from hcp to fcc, and demonstrate that a very simple
transformation emerges to describe the Burgers-Bain transformation. The
formalism can be used for future density functional studies and the activation
energy obtained can be seen as an upper bound to the true (and possibly more
complex) minimum energy path.

In the coming section we present the theory of the Burgers-Bain
hcp$\leftrightarrow$fcc$\leftrightarrow$bcc transformation and develop the
corresponding lattice sums for inverse power potentials. We then apply these
lattice sums to various $(n,m)$ Lennard-Jones potentials and discuss the
results for the transition paths. We will show that for such potentials,
starting from hcp phase, the minimum energy transition path leads to fcc
rather than directly to the bcc phase within the parameter space chosen. A
conclusion and future perspectives on this topic is given in the final
section.

##  II Theory

In order to discuss the martensitic phase transition along a Burgers- or Bain-
type path,Bain (1924); Burgers (1934) we need to introduce the various
lattices parameters involved and discuss their transformation properties,
matrices and corresponding lattice sums for inverse power potentials.

###  II.1 Lattice vectors and properties

####  II.1.1 The hexagonal close packed structure

The hcp phase is a bi-lattice Middlemas, Stillinger, and Torquato (2019) that
requires two hexagonal Bravais lattices to describe the close-packed
structure, or alternatively one hexagonal unit cell with one atom positioned
inside specified by its Wyckoff position (fractional Cartesian coordinates)
$(0,0,0)$ and $(\frac{1}{3},\frac{2}{3},\frac{1}{2})$. It belongs together
with the fcc lattice to the more general class of densest packed Barlow multi-
lattices Barlow (1883).

To correctly describe the Burgers-Bain martensitic
hcp$\leftrightarrow$fcc$\leftrightarrow$bcc phase transition we describe the
cuboidal lattice within the smallest common unit cell, that is a hexagonal-
like cell with two specified layers A and B as shown in Figure 1. We then
introduce a common set of basis vectors containing parameters that transform
the lattices smoothly into each other.

Figure 1: The (distorted) cuboidal (blue lines shown on the left) and the hcp
structure both with a ABABAB… sequence (layers A in red and B in blue) shown
in a hexagonal unit cell with corresponding basis vectors with cell parameters
$|\vec{b}_{1}|=|\vec{b}_{2}|=a$, and $|\vec{b}_{3}|=c$. The ratio
$\gamma=c/a=\sqrt{8/3}$ together with
$\angle(\vec{b}_{1},\vec{b}_{2})=60^{\circ}$ leads to the optimal hcp lattice
with 12 kissing spheres around a central atom. For $c=a$ and and
$\angle(\vec{b}_{1},\vec{b}_{2})=90^{\circ}$ to we obtain the bcc lattice with
8 kissing spheres. Alternatively, we have $c/a=\sqrt{2}$ and
$\angle(\vec{b}_{1},\vec{b}_{2})=90^{\circ}$ for the fcc lattice with 12
kissing spheres.

For the hexagonal-to-cuboidal hcp$\rightarrow$cub (cub $=$ fcc or bcc for
example) Burgers phase transition we start by considering the underlying
hexagonal Bravais lattice (h) with the accompanying basis vectors,

$\vec{b}_{1}^{\top}=a\left(1,0,0\right),\quad\vec{b}_{2}^{\top}=a\left(\frac{1}{2},\frac{\sqrt{3}}{2},0\right),\quad\vec{b}_{3}^{\top}=c\left(0,0,1\right)\,.$
(2)

with the lattice parameters $|\vec{b}_{1}|=|\vec{b}_{2}|=a$ and
$|\vec{b}_{3}|=c$, and $\angle(\vec{b}_{1},\vec{b}_{2})=60^{\circ}$ and
$\angle(\vec{b}_{1},\vec{b}_{3})=\angle(\vec{b}_{2},\vec{b}_{3})=90^{\circ}$.
We define the generator matrix consisting of these three vectors as

$B_{h}=\begin{pmatrix}\vec{b}_{1}^{\top}\\\ \vec{b}_{2}^{\top}\\\
\vec{b}_{3}^{\top}\end{pmatrix}=a\begin{pmatrix}1&0&0\\\
\frac{1}{2}&\frac{\sqrt{3}}{2}&0\\\ 0&0&\gamma\end{pmatrix}$ (3)

where we use the notation $\gamma=\frac{c}{a}$. This leads to the following
positive definite symmetric Gram matrix

$\left(G_{ij}\right)_{h}=\left(\vec{b}_{i}^{\top}\vec{b}_{j}\right)=a^{2}\begin{pmatrix}1&\frac{1}{2}&0\\\
\frac{1}{2}&1&0\\\ 0&0&\gamma^{2}\end{pmatrix}$ (4)

with det$(G)=\frac{3}{4}a^{4}c^{2}=\frac{3}{4}a^{6}\gamma^{2}>0$. From an
arbitrarily chosen atom at the origin, all points in the hexagonal lattice (A
layers only) are described by

$\vec{r}^{h}_{1}(\vec{i})=B_{h}^{\top}\vec{i}=i_{1}\vec{b}_{1}+i_{2}\vec{b}_{2}+i_{3}\vec{b}_{3}\\\
$ (5)

with $\vec{i}\in\mathbb{Z}^{3}$. Their distances to the origin are given by
the quadratic form

$|\vec{r}^{h}_{1}(\vec{i})|=\left(\vec{i}^{\top}G_{h}\vec{i}\right)^{\frac{1}{2}}=a\left(i_{1}^{2}+i_{1}i_{2}+i_{2}^{2}+\gamma^{2}i_{3}^{2}\right)^{\frac{1}{2}}\,.$
(6)

The volume of the unit cell is determined through the Gram matrix

$V(a,\gamma)=\sqrt{{\textrm{det}}G}=\frac{\sqrt{3}}{2}a^{3}\gamma$ (7)

The nearest neighbor distance is given by

$r^{h}_{\textrm{NN}}(a,c)=\textrm{min}\\{r^{h}_{1}(\vec{i})\\}={\textrm{min}}\left\\{a,c\right\\}\,.$
(8)

and the kissing number for a lattice $\mathscr{L}$ is defined as

$\kappa(\mathscr{L})=\\#\\{\vec{r}(\vec{i})\in\mathscr{L}~{}|~{}|\vec{r}(\vec{i})|=r_{\text{NN}}(\mathscr{L})\\}$
(9)

For the hcp structure with ideal value of
$\gamma_{\textrm{hcp}}=\sqrt{\tfrac{8}{3}}$ the kissing number is $\kappa=12$.
Similarly, we have $\kappa=12$ for fcc and $\kappa=8$ for bcc.

We now introduce the second layer, i.e. the B-layer as shown in Figure 1, to
complete the hcp structure. The B-layer is shifted by a vector of
$\vec{v}_{h}^{\top}=\frac{a}{2}\left(1,\frac{1}{\sqrt{3}},\gamma\right)$ with
respect to the lattice vectors given in (2), such that the position of any
atom in the B layers is given by

$\displaystyle\vec{r}^{h}_{2}(\vec{i})$
$\displaystyle=B_{h}^{\top}\vec{i}+\vec{v}_{h}\;,$ (10)

The shift vector can easily be derived from the fact that an atom in the B
layer sits above the centroid of a triangle of neighboring lattice points in
the A layer. We call the set $\\{\vec{b}_{i},\vec{v}_{h}\\}$ generalized
lattice vectors. They include the Wyckoff positions within a unit cell. The
resulting set of distance vectors
$\\{\vec{r}^{h}_{1}(\vec{i}),\vec{r}^{h}_{2}(\vec{i})\\}$ then produce all
points in 3D space for the hcp bi-lattice. For the minimum distance
$r^{h}_{\textrm{NN}}$ in an hcp bi-lattice we now have,

$r^{h}_{\textrm{NN}}(a,\gamma)={\textrm{min}}\left\\{a,c,\sqrt{\frac{a^{2}}{3}+\frac{c^{2}}{4}}\right\\}=a\,{\textrm{min}}\left\\{1,\gamma,\sqrt{\frac{1}{3}+\frac{1}{4}\gamma^{2}}\right\\}\,.$
(11)

####  II.1.2 Connecting the body-centered cuboidal with the hexagonal
structure

From Figure 1 we see how the hcp and bcc structures are related to each other
and can easily be smoothly transformed into each other which is what Burgers
had in mind in his original 1934 paperBurgers (1934). We therefore start with
the cuboidal phase described within a hexagonal lattice. The basis vectors for
the simple cubic (c) unit cell are,

$\vec{b^{\prime}}_{1}^{\top}=a\left(1,0,0\right),\quad\vec{b^{\prime}}_{2}^{\top}=a\left(0,1,0\right),\quad\vec{b^{\prime}}_{3}^{\top}=a\left(0,0,\gamma\right)\,.$
(12)

with $|\vec{b}^{\prime}_{1}|=|\vec{b}^{\prime}_{2}|=|\vec{b}^{\prime}_{3}|=a$,
and
$\angle(\vec{b}_{1},\vec{b}_{2})=\angle(\vec{b}_{1},\vec{b}_{3})=\angle(\vec{b}_{2},\vec{b}_{3})=90^{\circ}$.
We conveniently keep $\gamma$ in the definition of the generator matrix for
the bcc lattice as it becomes important later for the definition of the fcc
lattice,

$B_{c}=a\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&\gamma\end{pmatrix}$ (13)

For $\gamma=1$ we obtain the generator matrix of a simple cubic lattice. We
can already see the familiarity between the two matrices $B_{h}$ and $B_{c}$.
This gives the following Gram matrix,

$G_{c}=a^{2}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&\gamma^{2}\end{pmatrix}$ (14)

with det$(G)=a^{6}\gamma^{2}$ which gives det$(G)=1$ for $a=\gamma=1$. The
volume is therefore $V(a,\gamma)=a^{3}\gamma$ or for the ideal bcc lattice
$V=a^{3}$. The distances are given by (A layers only)
$\vec{r}^{c}_{1}(\vec{i})=B_{c}^{\top}\vec{i}$, and we have the expression in
terms of the quadratic form $(\vec{i}^{\top}G_{c}\vec{i})$,

$|\vec{r}^{c}_{1}(\vec{i})|=\left(\vec{i}^{\top}G_{c}\vec{i}\right)^{\frac{1}{2}}=a\left(i_{1}^{2}+i_{2}^{2}+\gamma^{2}i_{3}^{2}\right)^{\frac{1}{2}}\,.$
(15)

We now address again the B-layer in Figure 1 for the bcc lattice by using the
shift vector $\vec{v}_{c}^{\top}=\frac{a}{2}\left(1,1,\gamma\right)$ with
length $\vec{v}_{c}=\frac{a}{2}\sqrt{2+\gamma^{2}}$. The distances are given
by $\vec{r}^{c}_{2}(\vec{i})=B_{c}^{\top}\vec{i}+\vec{v}_{c}$. Again, the set
of both vectors $\\{\vec{r}^{c}_{1}(\vec{i}),\vec{r}^{c}_{2}(\vec{i})\\}$
produce all points in 3D space for the bcc lattice. So far we have a two-
parameter space for the distances $\vec{r_{k}}(a,\gamma)$ for the two lattices
hcp and bcc. They will become variable parameters when we discuss the Lennard-
Jones potential further below.

There is one important fact to note. In the anticipated
hcp$\leftrightarrow$bcc transformation we start with an hcp multi-lattice
using the hexagonal primitive cell as the underlying unit cell which
transforms along the Bain path into a simple cubic multi-lattice. This cubic
unit cell is not the primitive bcc cell. In fact, the smallest distance in the
bcc cell is given along the [111] direction to the center lattice point with a
distance of $a_{c}=|\vec{v}_{c}|=a\sqrt{3}/2$ for $\gamma=1$. Lennard-Jones
and Ingham therefore introduced an additional multiplicative factor to the
lattice for the bcc cell Jones and Ingham (1925), which we discuss further
below for the more general Burgers transformation between the hcp and the
cuboidal lattices.

####  II.1.3 The Burgers-Bain transformation

We derive the Burgers transformation between the two lattices hcp and bcc in a
similar way to the Bain transformation in Ref. Burrows, Cooper, and
Schwerdtfeger, 2021a. We first note that

$B_{h}=U_{B}B_{c}=\begin{pmatrix}1&0&0\\\ \frac{1}{2}&\frac{\sqrt{3}}{2}&0\\\
0&0&1\end{pmatrix}B_{c}$ (16)

This means that the distances transform like
$\vec{r}_{k}^{h}=U_{B}\vec{r}_{k}^{c}~{}(k=1,2)$. For the second set of
distances we therefore have to consider the transformation of the shift vector
as well

$\vec{v^{\prime}}=U_{B}\vec{v}^{c}_{c}=\frac{a}{2}\left(1,\tfrac{1}{2}(1+\sqrt{3}),\gamma\right)^{\top}$
(17)

The vector $\vec{v^{\prime}}$ differs markedly from $\vec{v}^{h}$ only in its
second component. It is therefore necessary to introduce the shift vector
directly with a freely varying parameter $\beta$ such that
$\vec{v}_{s}^{\top}=\frac{a}{2}\left(1,\beta,\gamma\right)$ with $\beta^{c}=1$
and $\beta^{h}=\frac{1}{\sqrt{3}}$. The length of the shift vector becomes
$|\vec{v}_{s}|=\tfrac{a}{2}\sqrt{1+\beta^{2}+\gamma^{2}}$. This factor becomes
important when we rescale our lattice sums.

The Gram matrices transform as

$G_{h}=B_{h}B_{h}^{\top}=U_{B}B_{c}B_{c}^{\top}U_{B}^{\top}=U_{B}G_{c}U_{B}^{\top}=(U_{B}U_{B}^{\top})G_{c}=\begin{pmatrix}1&\frac{1}{2}&0\\\
\frac{1}{2}&1&0\\\ 0&0&1\end{pmatrix}G_{c}=SG_{c}$ (18)

and the last expression can easily be verified. We can now formulate the
smooth transition from bcc to hcp as

$\displaystyle B(a,\alpha,\gamma)$ $\displaystyle=\alpha
B_{c}(a,\gamma)+(1-\alpha)B_{h}(a,\gamma)=\left\\{\alpha+(1-\alpha)U_{B}\right\\}B_{c}(a,\gamma)=T_{B}(\alpha)B_{c}(a,\gamma)$
$\displaystyle=\begin{pmatrix}1&0&0\\\
\frac{1}{2}(1-\alpha)&\frac{1}{2}(2\alpha+(1-\alpha)\sqrt{3})&0\\\
0&0&1\end{pmatrix}B_{c}(a,\gamma)=a\begin{pmatrix}1&0&0\\\
\omega_{1}(\alpha)&\omega_{3}(\alpha)&0\\\ 0&0&\gamma\end{pmatrix}$ (19)

where $\alpha\in[0,1]$, $\gamma=c/a$ is the usual parameter for the hexagonal
lattice, $\omega_{1}(\alpha)=\frac{1}{2}(1-\alpha)$ and
$\omega_{3}(\alpha)=\frac{1}{2}(2\alpha+(1-\alpha)\sqrt{3})$. This is,
however, not the best choice for the definition of the lattice vector
$\vec{b}_{2}$ as we assume in the transformation that we have
$|\vec{b}_{1}|=|\vec{b}_{2}|$. By forcing $|\vec{b}_{1}|=|\vec{b}_{2}|=a$ we
change $\omega_{3}$ to
$\omega_{3}=\sqrt{1-\omega_{1}^{2}}=\frac{1}{2}\sqrt{4-(1-\alpha)^{2}}$,
keeping the original definition for $\omega_{1}$. This changes the B-matrix to
our final choice

$\displaystyle B(a,\alpha,\gamma)$ $\displaystyle=a\begin{pmatrix}1&0&0\\\
\frac{1}{2}(1-\alpha)&\frac{1}{2}\sqrt{(1+\alpha)(3-\alpha)}&0\\\
0&0&\gamma\end{pmatrix}$ (20)

where $\alpha\in[0,1]$. This matrix describes basically the rotation of the
vector $\vec{b}_{2}$ in Figure 1. In this transformation the angle
$\angle(\vec{b}_{1},\vec{b}_{2})=\arccos(\omega_{1})$ changes from 0∘ at
$\alpha=-1$ to 180∘ at $\alpha=3$. Figure 2 shows that the parameter $\alpha$
is a good measure for the $\angle(\vec{b}_{1},\vec{b}_{2})$ angle of the two
base vectors.

Figure 2: Relationship between the parameter $\alpha$ and the angle between
the two base vectors $\vec{b}_{1}$ and $\vec{b}_{2}$.

The shift vector is defined as above with $\beta^{c}=1$ for $\alpha=1$ and
$\beta^{h}=\frac{1}{\sqrt{3}}$ for $\alpha=0$. The two sets of distance
vectors are given by

$\displaystyle\vec{r}_{1}(\vec{i})=B(a,\alpha,\gamma)^{\top}\vec{i}\quad\text{and}\quad\vec{r}_{2}(\vec{i})=B(a,\alpha,\gamma)^{\top}\vec{i}+\vec{v}_{s}(a,\beta,\gamma)$
(21)

We now have a four-parameter space $(a,\alpha,\beta,\gamma)$ as a minimum set
of parameters to describe the Burgers-Bain transformation. The symmetric Gram
matrix for $B(a,\alpha,\gamma)$ becomes,

$G(a,\alpha,\gamma)=B(a,\alpha,\gamma)B(a,\alpha,\gamma)^{\top}=a^{2}\begin{pmatrix}1&\omega_{1}(\alpha)&0\\\
\omega_{1}(\alpha)&\omega_{2}(\alpha)&0\\\
0&0&\gamma^{2}\end{pmatrix}=a^{2}S_{3}(\alpha,\gamma)$ (22)

with $\omega_{2}(\alpha)=\omega^{2}_{1}+\omega^{2}_{3}=1$ in our case. The
volume of the hexagonal unit cell then is,

$\displaystyle
V(a,\alpha,\gamma)=\sqrt{\text{det}G(a,\alpha,\gamma)}=a^{3}\gamma\sqrt{\omega_{2}(\alpha)-\omega_{1}(\alpha)^{2}}=a^{3}\gamma\omega_{3}(\alpha)=\frac{a^{3}\gamma}{2}\sqrt{(1+\alpha)(3-\alpha)}$
(23)

This results in a packing density compared to spheres of radius $r$,

$\rho=\frac{V_{\textrm{sphere}}}{V_{\textrm{hex}}}=\frac{4\pi
r^{3}}{3a^{3}\gamma}\frac{2}{\omega_{3}(\alpha)}$ (24)

where we considered that we have two spheres in the unit cell. This gives
$\rho_{\text{bcc}}=\frac{\sqrt{3}}{8}\pi$ (for $\alpha=1$, $\gamma=1$ and
$r=\frac{|\vec{v}^{c}|}{2}=\frac{\sqrt{3}}{4}a$),
$\rho_{\text{fcc}}=\frac{\sqrt{2}}{6}\pi$ (for $\alpha=1$, $\gamma=\sqrt{2}$
and $r=\frac{1}{2}a$), and $\rho_{\text{hcp}}=\frac{\sqrt{2}}{6}\pi$ (for
$\alpha=0$, $\gamma=\sqrt{\frac{8}{3}}$ and $r=\frac{1}{2}a$). We note that
$\omega_{2}(\alpha)-\omega_{1}(\alpha)^{2}>0$ for $\alpha\in(-1,3)$.

It is now clear that the martensitic transition is described by shearing of
the A-layer, related mainly to the parameter $\alpha$, and a change in angle
between the two basis vectors $\vec{b}_{1}$ and $\vec{b}_{2}$ together by a
sliding of the B-layer in direction orthogonal to the $\vec{b}_{1}$ and
$\vec{b}_{3}$ vectors and additional changes in the crystal parameters $a$ and
$\gamma$ (or $c$). This is demonstrated schematically in Figure 3. We note
that the Burgers path maintains the symmetry of a primitive monoclinic cell
(space group $mP$).

Figure 3: The shearing in the A-layer (grey spheres, change of angle between
$\vec{b}_{1}$ and $\vec{b}_{2}$) of the base hexagonal plane and sliding of
the B-layer (red spheres, sitting at $\gamma/2$ above the A-layer) in the
Burgers transformation. Left: bcc lattice with
$\angle(\vec{b}_{1},\vec{b}_{2})=90^{\circ}$ and $\gamma=1$ or fcc lattice
with $\gamma=\sqrt{2}$. Right: hcp structure with
$\angle(\vec{b}_{1},\vec{b}_{2})=60^{\circ}$ and $\gamma=\sqrt{\frac{8}{3}}$.

We briefly summarize. For the Burgers transformation we reduced the, in
principle, nine-parameter space (6 lattice constants for the Bravais lattice
plus 3 parameters for the Wyckoff positions of the middle layer) for a bi-
lattice to a four-parameter space $(a,\alpha,\beta,\gamma)$, where
$\alpha\in(-1,3)$ is the important parameter (our reaction coordinate)
describing the Burgers-Bain phase transition. All other parameters
$a,\beta,\gamma$ have to be optimized along the transformation path. We expect
approximately the following parameter range for $\alpha\in[0,1]$:
$a\in[a_{\text{hcp}},a_{\text{bcc}}]$, $\beta\in[\frac{1}{\sqrt{3}},1]$ and
$\gamma\in[1,\sqrt{\frac{8}{3}}]$. For the Bain transformation the originally
introduced distortion parameter $A$ in Ref. Burrows, Cooper, and
Schwerdtfeger, 2021a is simply related to the parameter $\gamma$ by
$A=\gamma^{2}/2$, and similarly the nearest neighbor distance in the cuboidal
lattice is given by $r_{NN}=\frac{a}{2}\sqrt{2+\gamma^{2}}$.Jerabek, Burrows,
and Schwerdtfeger (2022); Robles-Navarro, Jerabek, and Schwerdtfeger  This
implies that at $\alpha=1$ the transformation from fcc to bcc is described
mainly through the parameter $\gamma$.

###  II.2 Quadratic forms and functions for the lattice sums

Lattice sums have a long history and are usually based on quadratic functions.
For inverse power potentials $V(r)=r^{-2s}$ such as the Lennard-Jones
potential or the Madelung constant for Coulomb potentials are related to so-
called Epstein zeta functions.Borwein  _et al._ (2013); Madelung (1918); Jones
and Ingham (1925) For the cuboidal Bain transformation the associated lattice
sums were already described in our previous papers Burrows, Cooper, and
Schwerdtfeger (2021a, b). For the more general Burgers-Bain transformation we
need to introduce two lattice sums for all the distances in a bi-lattice. We
first introduce the quadratic form $Q_{A}$ involving only atoms in the base
A-layer,

$Q_{A}(\vec{i},\alpha,\gamma)=\vec{i}^{\top}S_{3}(\alpha,\gamma)\vec{i}=a^{-2}\vec{i}^{\top}G(a,\alpha,\gamma)\vec{i}=i_{1}^{2}+2\omega_{1}(\alpha)i_{1}i_{2}+\omega_{2}(\alpha)i_{2}^{2}+\gamma^{2}i_{3}^{2}$
(25)

with the associated lattice sum

$H^{h}_{A}(s,\alpha,\gamma)={\sum_{\vec{i}\in\mathbb{Z}^{3}}}^{{}^{\prime}}\frac{1}{(\vec{i}^{\top}S_{3}(\alpha,\gamma)\vec{i})^{s}}$
(26)

The prime notation at the sum indicates that we avoid the term
$i_{1}=i_{2}=i_{3}$ in the summation. This lattice sum can be re-expressed in
terms of a fast converging Bessel function Terras (1973); Burrows  _et al._
(2020) described in detail in Appendix A.

For the second lattice sum we need the quadratic function for the distances
between the A- and B-layer atoms derived from Eq. (21),

$\displaystyle Q_{B}(\vec{i},\alpha,\beta,\gamma)$
$\displaystyle=a^{-2}\left(\vec{i}^{\top}B(a,\alpha,\gamma)+\vec{v}_{s}^{\top}(\beta)\right)\left(B(a,\alpha,\gamma)^{\top}\vec{i}+\vec{v}_{s}(\beta)\right)$
(27)
$\displaystyle=a^{-2}\left[\vec{i}^{\top}G(a,\alpha,\gamma)\vec{i}+2\vec{i}^{\top}B(a,\alpha,\gamma)\vec{v}_{s}(\beta,\gamma)+|\vec{v}_{s}(\beta,\gamma)|^{2}\right]$
$\displaystyle=(i_{1}+i_{2}\omega_{1}(\alpha)+\tfrac{1}{2})^{2}+(i_{2}\omega_{3}(\alpha)+\tfrac{\beta}{2})^{2}+\gamma^{2}(i_{3}+\tfrac{1}{2})^{2}$

where $\omega_{1}(\alpha)$ and $\omega_{3}(\alpha)$ have already been defined.
The corresponding lattice sum becomes

$H^{h}_{B}(s,\alpha,\beta,\gamma)=\sum_{\vec{i}\in\mathbb{Z}^{3}}\frac{1}{Q_{B}(\vec{i},\alpha,\beta,\gamma)^{s}}$
(28)

Both lattice sums are absolute convergent for finite $s>\frac{3}{2}$ with a
simple pole at $s=\frac{3}{2}$, but can be analytically continued for
$s<\frac{3}{2}$. However, for $s\rightarrow\infty$ the lattice sum
$H^{h}_{B}\rightarrow\infty$ for the bcc lattice. This can be understood from
the fact that some terms in $Q_{B}(\vec{i}$ for $(i_{1},i_{2},i_{3})\in[-1,1]$
can become smaller than 1 in our definition of the lattice sum. This
divergence of the lattice sum for $s\rightarrow\infty$ will be compensated by
the optimized lattice constant $a$ obtained from a Lennard-Jones potential as
discussed further below, but this results in a $(0\times\infty)$ case for
$s\rightarrow\infty$. It is therefore far more convenient to avoid such
divergences by re-scaling the lattice constant such that
$a=r_{\text{bc}}/f(\beta,\gamma)$, where $f(\beta,\gamma)=a^{-1}|\vec{v}_{s}|$
becomes part of the lattice sum assuring convergence to a finite value for
$s\rightarrow\infty$, and $r_{\text{bc}}$ is the length of the vector
$\vec{v}_{s}$ from the atom at the base layer A to the nearest body-centered
atom in layer B. We now redefine the two lattice sums such that they behave
well in the limit $s\rightarrow\infty$,

$L^{h}_{A}(s,\alpha,\beta,\gamma)=f(\beta,\gamma)^{s}H^{h}_{A}(s,\alpha,\gamma)=f(\beta,\gamma)^{s}{\sum_{\vec{i}\in\mathbb{Z}^{3}}}^{{}^{\prime}}\frac{1}{(\vec{i}^{\top}S_{3}(\alpha,\gamma)\vec{i})^{s}}$
(29)

and

$L^{h}_{B}(s,\alpha,\beta,\gamma)=f(\beta,\gamma)^{s}H^{h}_{B}(s,\alpha,\beta,\gamma)=f(\beta,\gamma)^{s}\sum_{\vec{i}\in\mathbb{Z}^{3}}\frac{1}{Q_{B}(\vec{i},\alpha,\beta,\gamma)^{s}}$
(30)

with

$f(\beta,\gamma)=a^{-2}|\vec{v}_{s}(\beta,\gamma)|^{2}=a^{-2}r_{\text{bc}}^{2}=\frac{1}{4}\left(1+\beta^{2}+\gamma^{2}\right)$
(31)

This ensures that the bcc lattice sum is identical to the one defined
originally by Lennard-Jones and Ingham Jones and Ingham (1925). The total
lattice sum is then given by

$L^{h}(s,\alpha,\beta,\gamma)=L^{h}_{A}(s,\alpha,\gamma)+L^{h}_{B}(s,\alpha,\beta,\gamma)$
(32)

Both lattice sums can be expressed in terms of fast converging series
involving Bessel functions as derived in Appendix A. We note that it is
sufficient to take just the sum of the two layer contributions
$L^{h}_{A}+L^{h}_{B}$ as in our definition of the bi-lattice each atom is
equivalent to all the others, i.e. they have the same surrounding. This can
also be seen from a more general formula for Barlow packings.Schwerdtfeger
_et al._ (2024)

###  II.3 Lattice sums for the special cases of hcp, fcc and bcc

For the hcp structure we set $\alpha=0$, $\beta=\frac{1}{\sqrt{3}}$ and
$\gamma=\sqrt{\frac{8}{3}}$ leading to $\omega_{1}=\frac{1}{2}$,
$\omega_{2}=1$, $\omega_{3}=\frac{\sqrt{3}}{2}$ and $f(\beta,\gamma)=1$, and
we get

$\displaystyle L^{\text{hcp}}_{A}$
$\displaystyle={\sum_{\vec{i}\in\mathbb{Z}^{3}}}^{{}^{\prime}}\left(i_{1}^{2}+i_{1}i_{2}+i_{2}^{2}+\tfrac{8}{3}i_{3}^{2}\right)^{-s}$
(33) $\displaystyle L^{\text{hcp}}_{B}$
$\displaystyle=\sum_{\vec{i}\in\mathbb{Z}^{3}}\left((i_{1}+\tfrac{1}{2}i_{2}+\tfrac{1}{2})^{2}+(\tfrac{\sqrt{3}}{2}i_{2}+\tfrac{1}{2\sqrt{3}})^{2}+\tfrac{8}{3}(i_{3}+\tfrac{1}{2})^{2}\right)^{-s}$
$\displaystyle=\sum_{\vec{i}\in\mathbb{Z}^{3}}\left((i_{1}+\tfrac{1}{3})^{2}+(i_{2}+\tfrac{1}{3})^{2}+(i_{1}+\tfrac{1}{3})(i_{2}+\tfrac{1}{3})+\tfrac{8}{3}(i_{3}+\tfrac{1}{2})^{2}\right)^{-s}$

These are exactly the lattice sums as shown in Refs. Bell and Zucker, 1976;
Burrows  _et al._ , 2020; Burrows, Cooper, and Schwerdtfeger, 2023.

For bcc we set $\alpha=1$, $\beta=1$ and $\gamma=1$ leading to $\omega_{1}=0$,
$\omega_{2}=1$, $\omega_{3}=1$, and $f(\beta,\gamma)=3/4$ we get

$\displaystyle L^{\text{bcc}}_{A}$
$\displaystyle=\left(\frac{3}{4}\right)^{s}{\sum_{\vec{i}\in\mathbb{Z}^{3}}}^{{}^{\prime}}\left(i_{1}^{2}+i_{2}^{2}+i_{3}^{2}\right)^{-s}$
(34) $\displaystyle L^{\text{bcc}}_{B}$
$\displaystyle=3^{s}\sum_{\vec{i}\in\mathbb{Z}^{3}}\left((2i_{1}+1)^{2}+(2i_{2}+1)^{2}+(2i_{3}+1)^{2}\right)^{-s}$

These are identical to the ones given by Lennard-Jones and Ingham Jones and
Ingham (1925) and used in Ref. Burrows  _et al._ , 2020 for various Bessel
function expansions.

For fcc we set $\alpha=1$, $\beta=1$ and $\gamma=\sqrt{2}$ leading to
$\omega_{1}=0$, $\omega_{2}=1$, $\omega_{3}=1$, and $f(\beta,\gamma)=1$ we get

$\displaystyle L^{\text{fcc}}_{A}$
$\displaystyle={\sum_{\vec{i}\in\mathbb{Z}^{3}}}^{{}^{\prime}}\left(i_{1}^{2}+i_{2}^{2}+2i_{3}^{2}\right)^{-s}$
(35) $\displaystyle L^{\text{fcc}}_{B}$
$\displaystyle=\sum_{\vec{i}\in\mathbb{Z}^{3}}\left((i_{1}+\tfrac{1}{2})^{2}+(i_{2}+\tfrac{1}{2})^{2}+2(i_{3}+\tfrac{1}{2})^{2}\right)^{-s}$

This can be brought into a more familiar form discussed in Ref. Burrows  _et
al._ , 2020 (see Appendix C),

$L^{\text{fcc}}_{A}+L^{\text{fcc}}_{B}=2^{s-1}{\sum_{\vec{i}\in\mathbb{Z}^{3}}}^{{}^{\prime}}\left[1+(-1)^{i_{1}+i_{2}+i_{3}}\right]\left(i_{1}^{2}+i_{2}^{2}+i_{3}^{2}\right)^{-s}\\\
$ (36)

###  II.4 The Lennard-Jones cohesive energies for the lattices along the fcc
to hcp transition path

The cohesive energy for the hexagonal-cuboidal structures for a general
$(n,m)$ LJ potential can be expressed in terms of lattice sums and is given by
the expressionJones and Ingham (1925); Schwerdtfeger  _et al._ (2006);
Schwerdtfeger, Burrows, and Smits (2021)

$E_{\text{LJ}}^{\text{coh}}(n,m,a,\alpha,\beta,\gamma)=\frac{\epsilon
nm}{2(n-m)}\;\left[\frac{L^{h}(\frac{n}{2},\alpha,\beta,\gamma)}{n}\left(\frac{r_{e}}{r_{\text{bc}}}\right)^{n}-\frac{L^{h}(\frac{m}{2},\alpha,\beta,\gamma)}{m}\left(\frac{r_{e}}{r_{\text{bc}}}\right)^{m}\right]$
(37)

where the distance from the base to the body-centered atom is related to the
lattice constant $r_{\text{bc}}=a/f(\beta,\gamma)$ as already mentioned. The
expression for the lattice sum $L^{h}(s,\alpha,\beta,\gamma)$ is taken from
Eq. (32) and the corresponding Bessel function expansions we used in our work
are given in Appendix A. As these Bessel sum expansions are fast converging
series, they can easily be obtained to arbitrary computer precision, i.e. to
double precision accuracy within a few seconds of computer time.

In order to discuss the behavior for the LJ potential with varying
$a,\alpha,\beta,\gamma$ parameters we eliminate the parameter $a$ to save
computer time in our optimizations. Here we first calculate the minimum
cohesive energy with respect to the lattice parameter $a$ for fixed set of
$\alpha,\beta,\gamma$ values. For this follow the procedure in Ref. Burrows,
Cooper, and Schwerdtfeger, 2021b and get from $\partial
E_{\textrm{LJ}}^{\textrm{coh}}/\partial a=0$ the minimum lattice parameter,

$a^{*}_{\text{min}}(\gamma)=\frac{a_{\text{min}}(\alpha,\beta,\gamma)}{r_{e}}=\frac{1}{f(\beta,\gamma)}\left(\frac{L^{h}(\frac{n}{2},\alpha,\beta,\gamma)}{L^{h}(\frac{m}{2},\alpha,\beta,\gamma)}\right)^{\tfrac{1}{n-m}}\,,$
(38)

and the * indicates that reduced (or dimensionless) units are used. We can
then evaluate the cohesive energy at $a^{*}_{\text{min}}$ and get,

$E_{nm}^{*}(\alpha,\beta,\gamma)=E_{\text{LJ}}^{\text{coh}}(n,m,a^{*}_{\text{bc,min}},\alpha,\beta,\gamma)/\epsilon=-\frac{1}{2}\left[\frac{L^{h}(\frac{m}{2},\alpha,\beta,\gamma)^{n}}{L^{h}(\frac{n}{2},\alpha,\beta,\gamma)^{m}}\right]^{\tfrac{1}{n-m}}\;.$
(39)

We therefore have to deal only with the three-parameter space
$\alpha,\beta,\gamma$ for fixed exponents $(n,m)$. As $\alpha$ is fixed to map
out the Burgers path, we only have to optimize $E_{nm}^{*}$ with respect to
both $\beta$ and $\gamma$ using a 2D Newton-Raphson procedure as outlined in
Appendix D. We chose the interval $\alpha\in[-0.2,1.2]$ in steps of
$\Delta\alpha=0.05$ for the Burgers path. For the following we omit the *
notation, i.e. all quantities are in reduced (dimensionless) units unless
otherwise stated.

The validity of using (39) instead of (37) has been checked by computation
which show that we obtain the same results for the extreme points (maxima and
minima) and the minimum energy Burgers path. This is perhaps not surprising as
we have $\partial E_{nm}/\partial a=0$ and $\partial E_{nm}/\partial\gamma=0$
for the minimum energy path. Thus, by performing the derivative $\partial
E_{nm}/\partial a=0$ and considering $\partial
L^{h}(\frac{n}{2},\alpha,\beta,\gamma)/\partial a=0$ in (37) we can easily
prove by substituting $\partial a=-a\gamma^{-1}\partial\gamma$ that (39) is
valid. For the area around the Burgers path where this condition may not hold
we obtain rather small deviations from the exact solution (37). Here, the
situation is similar to the hcp case discussed before.Burrows, Cooper, and
Schwerdtfeger (2023)

##  III Results and Discussions

By changing step-wise the parameter $\alpha$ from $-0.2$ to $1.2$ we end up
with the fcc structure at $\alpha=1$ and $\gamma=\sqrt{2}$ for all LJ
potentials considered. Using the reverse path starting from bcc one goes
steeply uphill. We therefore focus first our attention on the
hcp$\rightarrow$fcc Burgers transformation and then discuss the connection to
the bcc phase through the Bain transformation.

###  III.1 The hcp$\leftrightarrow$fcc Burgers transformation

The optimized lattice parameters $a_{\text{min}}$ are shown in Figure 4 as a
function of the distortion parameter $\alpha$ for various LJ potentials. For
the interval $\alpha\in[0,1]$ we always have $a\leq r_{\text{bc}}$. For both
ideal hcp and fcc we have $a_{\text{min}}=r_{\text{bc,min}}$ because
$f(\beta,\gamma)=1$ in both cases. The small difference between
$a_{\text{min}}$ and $r_{\text{bc,min}}$ shown in Table 1 comes from the fact
that $\gamma$ deviates slightly from the ideal value of $\sqrt{\frac{8}{3}}$
for the LJ potentials. As can be further seen, the lattice parameter is more
dependent on the choice of the exponents $n$ and $m$ of the LJ potential than
on $\alpha$ itself. This becomes even more evident when we choose the distance
$r_{\text{bc}}$ from the base atom to the body-centered atom instead, which
remains almost constant over the whole range of $\alpha$ values. Especially
for the $(30,12)$-LJ potential $r_{\text{bc}}\approx 1$ close to the kissing
hard sphere model where we have $r_{\text{bc}}=1$. In fact, analyzing the
minimum distances even further, we get the kissing number $\kappa=12$ for the
two points $\alpha=0,\gamma=\sqrt{\frac{8}{3}}$ and $\alpha=1,\gamma=1$ as
expected. However, around the TS state we have $\kappa=4$ for the interval
$\alpha\in(0,\alpha_{k}$, $\alpha_{k}$ we have $\kappa=6$, and $\kappa=2$ for
the interval $\alpha\in(\alpha_{k},1)$ as listed in Table 1 for the various LJ
potentials. $\alpha_{k}$ is close to $\frac{1}{2}$.

The optimized lattice parameters $\beta$ and $\gamma$ are shown in Figure 5 as
a function of the distortion parameter $\alpha$. The parameter $\beta$,
responsible for the sliding of the middle B-layer, is increasing monotonically
in an almost linear fashion from $1/\sqrt{3}$ at $\alpha=0$ to 1 at $\alpha=1$
as we expect. This implies a smooth shift of the B-layer for the Burgers
transformation from hcp to fcc. The parameter $\gamma$, which for example
describes also the Bain transformation, shows a more interesting behavior.
There is a slight increase from the value of $\gamma=\sqrt{8/3}$ for the hcp
structure with a maximum between $\alpha=0$ and $\alpha=0.2$, followed by the
expected decrease to $\gamma=\sqrt{2}$ at $\alpha=1$ for the fcc structure.
This will influence the volume and the packing density as can be seen from
Eqs. (23) and (24). We note that the hcp structure slightly distorts from the
ideal $\gamma=\sqrt{8/3}$ value as discussed in detail before,Burrows, Cooper,
and Schwerdtfeger (2023) see Table 1.

Figure 4: Optimized minimum lattice parameters as a function of the distortion parameter $\alpha$ for various selected $(n,m)$-LJ potentials. The solid lines show the distances between nearest neighbors in the base layer ($a=|\vec{b}_{1}|$), while the dashed lines show the optimized distances from the base to the body-centered atom ($r_{\text{bc}}=|\vec{v}_{s}|$), see Figure 1. Parameter | $(6,4)$ | $(8,6)$ | $(12,4)$ | $(12,6)$ | $(30,6)$ | $(30,12)$
---|---|---|---|---|---|---
minima |  |  |  |  |  |
$a_{\text{min}}^{\text{hcp}}(\alpha=0)$ | 0.75523963 | 0.94115696 | 0.91206706 | 0.97127386 | 0.99229244 | 0.99939864
$r_{\text{bc,min}}^{\text{hcp}}(\alpha=0)$ | 0.75533453 | 0.94108456 | 0.91203445 | 0.97118272 | 0.99225828 | 0.99938521
$E_{nm}^{\text{hcp}}(\alpha=0)$ | -38.9325327 | -10.4019067 | -18.30985366 | -8.61107046 | -7.57167198 | -6.11058661
$\delta_{nm}^{\text{hcp}}(\alpha=0)$ | 0.00030779 | -0.00018844 | -0.00008759 | -0.00022986 | -0.00008431 | -0.00003291
$a_{\text{min}}^{\text{fcc}}(\alpha=1)$ | 0.75527318 | 0.94112001 | 0.91205036 | 0.97123369 | 0.99227815 | 0.99939381
$\Delta E_{nm}^{\text{fcc}}(\alpha=1)$ | -0.00167052 | 0.00065426 | 0.00053138 | 0.00087030 | 0.00063934 | 0.00034707
$a_{\text{min}}^{\text{cub2}}(\alpha=1)$ | 0.84952560 | 1.08593491 | 1.06151441 | 1.14235260 | 1.20025739 | 1.21772324
$r_{\text{bc,min}}^{\text{cub2}}(\alpha=1)$ | 0.73571075 | 0.92022759 | 0.89393191 | 0.95447928 | 0.98606235 | 0.99654594
$\gamma^{\text{cub2}}(\alpha=1)$ | 1 | 0.93401642 | 0.91472620 | 0.89022278 | 0.83649731 | 0.82395744
$\Delta E_{nm}^{\text{cub2}}(\alpha=1)$ | 0.29641379 | 0.24620888 | 0.45584326 | 0.34906107 | 0.58211318 | 0.76315575
TS Burgers |  |  |  |  |  |
$\alpha^{\\#}$ | 0.50124827 | 0.50070620 | 0.50055491 | 0.50036605 | 0.50007315 | 0.49998534
$a_{\text{min}}^{\\#}$ | 0.73561911 | 0.92214149 | 0.89552993 | 0.95622314 | 0.98638714 | 0.99664997
$r_{\text{bc,min}}^{\\#}$ | 0.76148854 | 0.94610839 | 0.91637703 | 0.97463898 | 0.99337015 | 0.99965780
$\beta^{\\#}$ | 0.84563200 | 0.82802883 | 0.82241205 | 0.81445956 | 0.78931350 | 0.78087408
$\gamma^{\\#}$ | 1.60349255 | 1.58902301 | 1.58494120 | 1.57867371 | 1.56007043 | 1.55383900
$\Delta E_{nm}(\alpha^{\\#})$ | 0.63868037 | 0.34878984 | 0.59775652 | 0.41488952 | 0.59694720 | 0.76650812
$\Delta V[\%]$ | 1.45105875 | 2.35271163 | 2.73068082 | 3.15510867 | 4.92065604 | 5.51004680
$\alpha_{k}$ | 0.52877 | 0.51028 | 0.50911 | 0.50408 | 0.49990 | 0.49905
TS Bain |  |  |  |  |  |
$a_{\text{min}}^{\\#}$ | 0.84587206 | 1.06148905 | 1.03057025 | 1.09911882 | 1.13483878 | 1.14700157
$r_{\text{bc,min}}^{\\#}$ | 0.73573887 | 0.91927648 | 0.89250002 | 0.95186482 | 0.95186482 | 0.99333250
$\Delta E_{nm}(\alpha^{\\#})$ | 0.29644955 | 0.24972893 | 0.47009499 | 0.37377855 | 0.77263663 | 1.17257580
Table 1: Optimized parameters for minima and transition states (TS) for
various $(n,m)$-LJ potentials. TS properties for the Burgers ($\partial
E_{nm}/\partial\alpha=0$) and Bain paths ($\partial E_{nm}/\partial\gamma=0$)
are indicated by a $\\#$ symbol. For the minimum structures we get exactly
$\alpha=0$ and $\beta=1/\sqrt{3}$ for hcp and $\alpha=1$, $\beta=1$ and
$\gamma=\sqrt{2}$ for fcc. For hcp obtain a slight deviation from the ideal
$\gamma$ value is obtained expressed by
$\delta_{nm}(\alpha=0)=\gamma_{nm}^{\text{opt}}(\alpha=0)-\sqrt{8/3}$. cub2
indicates the second cuboidal structure, e.g. bcc if $\alpha=1$, $\beta=1$ and
$\gamma=1$. For fcc we have
$a_{\text{min}}^{\text{fcc}}(\alpha=1)=r_{\text{bc,min}}^{\text{fcc}}(\alpha=1)$.
$\Delta E_{nm}(\alpha^{\\#})$ for the Burgers path is taken relative to the
hcp structure. The volume increase $\Delta V[\%]$ at the transition point of
the Burgers path relative to the fcc structure is defined as
$100\\{V(\alpha^{\\#})-V^{\text{fcc}}\\}/V^{\text{fcc}}$. $\alpha_{k}$ defines
the point on the Burger’s path where the kissing number changes from 4 to 2
(see text). For the Bain path TS we have $\alpha=1$ and $\beta=1$, which is
located at $\gamma=1$ except for the (6,4)-LJ potential for which we get
$\gamma^{\\#}=$1.01301670. $\Delta E_{nm}(\alpha^{\\#})$ for the Bain path are
relative to the fcc structure.

(a) (b)

Figure 5: The optimized parameter (a) $\beta$ and (b) $\gamma$ as a function
of the distortion parameter $\alpha$ for various selected $(n,m)$-LJ
potentials.

Figure 6(a) shows the volume as a function of the distortion parameter
$\alpha$. As expected, a volume increase is required for the
hcp$\leftrightarrow$fcc transition along the Burgers path with a maximum in
volume located close to midpoint at $\alpha=0.5$. The percentage volume
increase at this transition state is largest for the $(30,12)$-LJ potential
(see Table 1), which is close to the kissing hard-sphere model (KHS). It is
difficult to estimate the KHS limit for the volume increase as we do not have
an exact value for $\gamma$ at $\alpha=0.5$. However, we assume that for the
KHS limit transition state we have exactly $\alpha=0.5$ and $a=1$. For
example, for the (40,20)-LJ potential we get $\gamma=1.550$ and a relative
volume increase of
$100\\{V(\alpha=0.5)-V^{\text{fcc}}(\alpha=1)\\}/V^{\text{fcc}}(\alpha=1)$=6.13
% with $V^{\text{fcc}}(\alpha=1)=\sqrt{2}$ for the KHS limit.

Concerning the packing density we can define as the hard sphere radius
$r=\text{min}\\{a,a_{c}\\}/2$ in Eq. (24). This implies a non-overlapping
sphere model with $2r\leq 1$ for the different $n,m$ combinations of the LJ
potential. In other words, the sphere radius changes along the Burgers
transition path and the packing density is $\rho\leq\frac{\sqrt{2}}{6}\pi$.
This is shown in Figure 6. As a second choice, we may allow spheres to overlap
and set the radius to a fixed value of $r=\frac{1}{2}$, the radius of a unit
sphere. In this case one has to account for the overlap volume to obtain the
correct packing density, see discussion on this subject by Iglesias-Ham et al.
Iglesias-Ham, Kerber, and Uhler (2014). In any case, the graphs in Figure 6
show that the packing density is reduced along the Burgers path towards the
transition state.

(a) (b)

Figure 6: (a) Volume $V$ from (23) and (b) packing density $\rho$ using for
the sphere radius (half the nearest neighbor distance)
$r=\text{min}\\{a,a_{\text{bc}}\\}$ in eq. (24) for various selected
$(n,m)$-LJ potentials as a function of the distortion parameter $\alpha$. The
horizontal lines indicate the hard-sphere limit with for the volume,
$V(\alpha=1)=\sqrt{2}$, and the maximum packing density
$\rho_{\text{fcc}}=\rho_{\text{hcp}}=\frac{\sqrt{2}}{6}\pi$.

The cohesive energy along the Burgers path is shown in Figure 7. Changing
$\alpha$ from $0\rightarrow 1$ and optimization of $\beta$ and $\gamma$ at
each step along the path leads straight into the fcc lattice with
$\gamma=\sqrt{2}$ and a transition state close to the midpoint of $\alpha=1/2$
for all LJ potentials considered here. For example, for the (12,6)-LJ
potential we find the transition state at $\alpha^{\\#}=0.50037$ with an
energy difference of $\Delta
E_{12,6}(\alpha^{\\#})=E_{12,6}(\alpha^{\\#})-E_{12,6}^{\text{hcp}}=0.41490$,
see Table 1. To give this a meaning for real solids, we multiply the energy
difference with the most accurate available dissociation energy of
$\epsilon=99.351\pm 0.32$ cm-1 for Ar2 from relativistic coupled-cluster
calculations of Patkowski and Szalewicz,Patkowski and Szalewicz (2010) and
obtain for the activation energy to the transition state a value of 41.2 cm-1
(kcal/mol) compared to the hcp structure. This compares to the experimentally
known cohesive energy of solid argon of 7722(11) cm-1,Schwalbe  _et al._
(1977) i.e. the barrier height is less than 1% of the cohesive energy.
Furthermore, the activation energy is smaller than the zero-point vibrational
contribution for fcc argon which is 67.4 cm-1.Schwerdtfeger  _et al._ (2016)

Figure 7: The cohesive energy difference $\Delta
E_{nm}(\alpha)=E_{nm}(\alpha)-E_{nm}^{\text{hcp}}(\alpha=0)$ for the Burgers
path as a function of the distortion parameter $\alpha$ for various selected
$(n,m)$-LJ potentials.

We can also compare with recent molecular dynamics (MD) simulation by Bingxi
Li et al.Li  _et al._ (2017) using a (12,6)-LJ potential and 18,000 Ar atoms
in a box adjusted to a temperature of $T$=40 K and pressure of $P$= 1 bar.
They calculated a barrier height for the enthalpy of $\Delta H^{\\#}=16.0\pm
1.0$ cm-1 (per atom). Even if we adjust for the more accurate dissociation
energy of Patkowski and SzalewiczPatkowski and Szalewicz (2010) then given in
their paper we get $\Delta H^{\\#}=19.2\pm 1.1$ cm-1. However, at 40K the
kinetic energy per atom $E_{\text{kin}}=\frac{3}{2}k_{\text{B}}T$ translates
into 41.7 cm-1 per atom. Hence it is difficult to compare our two very
different (static vs. dynamic) methods. Moreover, their MD simulations show an
accumulation of defects, stacking disorders and growth of a less ordered
structures towards the transition state. They also detected three different
phase transition paths. It is, however, difficult to provide a simple picture
of the phase transition from such MD simulations, even more so if the
simulation cell does not reflect the change in the lattice constants and
surface effects can become important without proper periodic boundary
conditions. On the contrary, the Burgers path provides a good upper limit for
the activation energy in the hcp$\leftrightarrow$fcc transformation, as other
paths on such a high-dimensional energy hypersurface requiring a large super-
cell treatment may lie below the ideal Burgers path. Interestingly, the volume
expansion at the transition state obtained by Bingxi Li et al.Li  _et al._
(2017) of about 2.6% is not too different compared to our value of 3.2%.

The activation energies for the Bain phase transition are usually below the
ones for the Burgers path, see Table 1. The highest barrier for the Bain path
is obtained by the most repulsive $(30,12)$-LJ potential, as one might expect.
Further, the difference in energy between the fcc and hcp structures are very
small, i.e. $\Delta E_{nm}=E_{nm}^{\text{hcp}}-E_{nm}^{\text{fcc}}=0.0008703$
for the (12,6)-LJ potential and -0.0016705 for the (6,4)-LJ potential. In the
latter case, fcc is more stable compared to hcp. Figure 8 shows the hcp-fcc
phase transition line between the two phases where we have $\Delta E_{nm}=0$.
Only for very soft long-range potentials becomes the fcc phase more stable
compared to the hcp phase. The critical $n_{c},m_{c}$ values for the phase
transition line can be fitted to the function

$m_{c}=3+ae^{-bn_{c}}+\frac{c_{1}}{n_{c}}+\frac{c_{2}}{n_{c}^{2}}+\frac{c_{3}}{n_{c}^{3}}\quad\text{with}\quad
n_{c}>5.705$ (40)

where $a=3.322473067\times 10^{8}$, $b=5.95017115$, $c_{1}=-9.23169654$,
$c_{2}=1.73880812\times 10^{2}$ and $c_{3}=-1.89668849\times 10^{2}$ with a
coefficient of determination of $R^{2}=0.9998$. For example, for $n_{c}=12$ we
get $m_{c}=3.291725434064$, which means that for the $(12,6)$-LJ potential the
hcp phase lies energetically below the fcc phase. This was intensively
discussed already in the literature Kihara and Koba (1952); Wallace and
Patrick (1965); Niebel and Venables (1974); van de Waal (1991); Lotrich and
Szalewicz (1997); Schwerdtfeger, Burrows, and Smits (2021), and it was
recently shown that for a solid like argon, phonon dispersion is required for
the stabilization of the fcc over the hcp phase at 0K.Schwerdtfeger  _et al._
(2016)

Figure 8: hcp/fcc phase transition line $(n_{c},m_{c})$ for $(n,m)$-LJ
potentials fulfilling the condition $\Delta
E_{nm}=E_{nm}^{\text{hcp}}-E_{nm}^{\text{fcc}}$.

###  III.2 The Burgers-Bain transformation

The question arises if there is another direct path from hcp to bcc as
suggested for example by Carter and co-workers for iron or lithium Johnson and
Carter (2008); Caspersen and Carter (2005). However, the MD simulations by
Bingxi Li et al.Li  _et al._ (2017) show no transition to the bcc phase. We
note that one can always go from fcc to bcc via a Bain-type transformation as
has been discussed for example by our group before for LJ potentials Burrows,
Cooper, and Schwerdtfeger (2021b); Schwerdtfeger and Burrows (2022).

In order to shed light into this we decided to scan through the
$(\alpha,\gamma)$ parameter space by optimizing $\beta$ at each point. We did
not consider the (30,12)-LJ potential as it is close to the kissing hard-
sphere limit. As a result, at larger $\gamma$ values we see symmetry breaking
effects such that the middle B-layer becomes energetically unstable and by
moving either up or down towards one of the A-layers. This movement parallel
to the base hexagonal sheet would cost little energy due to the short-range
nature of the potential, which makes the optimization of the $\beta$ parameter
difficult. Furthermore, the bcc structure at $\gamma=1$ is a maximum for this
potential distorting either to fcc or to another minimum at $\alpha=1$,
$\beta=1$ and $\gamma=0.82395744$, cf. Table 1, very close to the ideal acc
structure with $\gamma=\sqrt{2/3}$.

The energy hypersurfaces for four different LJ potentials are shown in Figure
9. We can clearly see the minimum energy path (in our limited four-parameter
space) from the hcp minimum ($\alpha=0$ and $\gamma\approx\sqrt{8/3}$) on the
left to the fcc minimum ($\alpha=1$ and $\gamma=\sqrt{2}$ on the right of the
contour plot. We can also see the Bain path at constant $\alpha=1$ from fcc
towards the bcc structure $\gamma=1$. It is now clear that the
hcp$\rightarrow$bcc minimum energy transition in our model involves two
transition states and requires the fcc lattice as an intermediate structure.
Concerning the stability of the two different phases we already pointed out
that the bcc phase lies always above the fcc structure and is meta-stable only
for soft long-range potentials,Schwerdtfeger, Burrows, and Smits (2021)
otherwise it becomes unstable and distorts towards the acc structure with
smaller $\gamma$ values Burrows, Cooper, and Schwerdtfeger (2021b). It was
already mentioned that the kissing number reduces to 8 for the interval
$\gamma\in(\sqrt{2},\sqrt{2/3})$, which may explain why this path is
energetically lower compared to the Burgers path (except for the potentials
with high $n$and $m$ values).

Figure 9: Contour plots for the $(\alpha,\gamma)$ energy hypersurfaces at
optimized $\beta$ values for the cohesive energy difference $\Delta
E_{nm}(\alpha)=E_{nm}(\alpha)-E_{nm}^{\text{hcp}}(\alpha=0)$ of a $(12,6)$-LJ
potential (top left), $(30,12)$-LJ potential (top right), $(8,6)$-LJ potential
(bottom left), and $(12,4)$-LJ potential (bottom right). For the $(12,6)$-LJ
the minimum Burgers path from hcp ($\alpha=0,\gamma=\sqrt{\frac{8}{3}}$. left
minimum) to fcc ($\alpha=0,\gamma=\sqrt{2}$, right minimum) and the minimum
Bain path at constant $\alpha=1$ from fcc to bcc $\alpha=1,\gamma=1$ are
shown. Note the different energy scales used as shown on the right hand side
of the contour plots.

As mentioned already, the minimum energy path is unaffected by the model
applied, i.e. using (39) instead of the exact LJ expression (37). As we map
the hypersurface in the $(\alpha,\gamma)$ space (39) is also valid. We note a
maximum around $\alpha=0.0$ and $\gamma=0.8$, clearly visible in Figure 9. For
example, for the (12,6)-LJ potential we obtain $\alpha=0$, $\beta=1/\sqrt{3}$,
$a=0.95109476$, $\gamma=0.84836004$ and $\Delta E^{*}=2.77589575$ compared to
the hcp minimum structure using (37). These values are identical to the ones
obtained before for the hcp lattice,Burrows, Cooper, and Schwerdtfeger (2023)
where the cohesive energy with changing $\gamma$ has been studied in detail.

##  IV Conclusions

We investigated in detail the Burgers-Bain minimum energy path for the
hcp$\leftrightarrow$fcc$\leftrightarrow$bcc phase transition for a number of
representative $(n,m)$-LJ potentials using exact (to computer precision)
lattice summations. For this we expressed the lattice sums in terms of a fast
converging Bessel sum series. Our simple phase transition model is currently
limited to four parameters describing the change in the base lattice
parameters $a$ and $c$, the shear force acting on the hexagonal base plane
through the parameter $\alpha$, and the sliding force of the middle layer
through the parameter $\beta$. We treated $a$ and $\gamma=c/a$ as independent
parameters here,Burrows, Cooper, and Schwerdtfeger (2023) which is
computationally most efficient and reasonably accurate. This limited choice of
parameters made is possible to gain detailed insight into the Burgers-Bain
phase transition. A more detailed study for solid argon under pressure is
currently underway. The true minimum energy path might well lie energetically
below the ideal Burgers-Bain path involving a different mechanism for the
sliding of the hexagonal planes. Moreover, for real systems with strong many-
body interactions such as for metallic systems,Schwerdtfeger  _et al._ (2006)
the bcc phase might well be directly connected to the hcp phase as for example
calculations using density functional theory and the nudged elastic band
algorithm for mapping out the minimum energy path for bulk lithium or silver
nanowires indicate.Caspersen and Carter (2005); Sun  _et al._ (2022) We point
out that a more advanced MD simulation to obtain free energy values at finite
temperatures and pressures requires a super-cell treatment with a large set of
independent parameters. Nevertheless, we believe that our simple picture can
be used to estimate an upper limit of the phase transition activation barrier
for real systems applying density functional theory.

##  V Appendix

##  Appendix A Bessel function expansions of the lattice sums for the Burgers
path

In the following we concentrate on the lattice sums $H^{h}_{A}$ and
$H^{h}_{B}$ so we do not have to carry the factor $f(\beta,\gamma)$ along.

###  A.1 The lattice sum $H^{h}_{A}$

In three dimensions the lattice sum defined through a quadratic form is
related to the Epstein zeta function $Z$ by
$H_{S_{3}}=2Z_{S_{3}}\left(s\right)$ as discussed by TerrasTerras (1973),
where $S_{3}$ is a three-dimensional symmetric and positive definite Gram
matrix. The Terras decomposition of the Epstein zeta functionTerras (1973) can
be used for any Bravais lattice and an inverse power potential, and has been
detailed in our previous paper for cubic lattices,Burrows  _et al._ (2020)

$\displaystyle Z_{S_{3}}(s)$
$\displaystyle=\frac{\pi\Gamma(s-1)t^{1-s}}{\Gamma(s)\sqrt{\det
S_{2}}}\zeta(2s-2)+\frac{p^{\frac{1}{2}-s}\Gamma(s-\frac{1}{2})}{\Gamma(s)}\sqrt{\frac{\pi}{\sigma_{22}}}\zeta(2s-1)+\frac{\zeta(2s)}{\sigma_{22}^{s}}$
(A.1)
$\displaystyle+\frac{4\pi^{s}}{\Gamma(s)}\left(\frac{p^{1-2s}}{\sigma_{22}^{1+2s}}\right)^{1/4}\sum_{u,v\in\mathbb{N}}\left(\frac{v}{u}\right)^{s-\frac{1}{2}}\cos\left(\frac{2\pi\sigma_{12}uv}{\sigma_{22}}\right)K_{s-\frac{1}{2}}\left(2\pi
uv\sqrt{\frac{p(\alpha)}{\sigma_{22}}}\right)$
$\displaystyle+\frac{2\pi^{s}}{\Gamma(s)}\sqrt{\frac{t^{1-s}}{\det
S_{2}}}\sum_{\begin{subarray}{c}u\in\mathbb{N}\\\
\vec{w}\in\tilde{\mathbb{Z}}^{2}\setminus\\{\vec{0}_{2}\\}\end{subarray}}\cos\left(2\pi
u\vec{w}^{\top}\vec{y}\right)u^{1-s}\left(\vec{w}^{\top}S_{2}^{-1}\vec{w}\right)^{\frac{s-1}{2}}K_{s-1}\left(2\pi
u\sqrt{t\vec{w}^{\top}S_{2}^{-1}\vec{w}}\right)$

Here $S_{3}$ has been block-diagonalized to obtain the $(2\times 2)$ symmetric
sub-matrix $S_{2}=(\sigma_{ij})$,

$S_{3}=\begin{pmatrix}m&\vec{z}^{\top}\\\
\vec{z}&S_{2}\end{pmatrix}=\begin{pmatrix}1&\vec{y}^{\top}\\\
\vec{0}_{2}&I_{2}\end{pmatrix}\begin{pmatrix}t&\vec{0}_{2}^{\top}\\\
\vec{0}_{2}&S_{2}\end{pmatrix}\begin{pmatrix}1&\vec{0}_{2}^{\top}\\\
\vec{y}&I_{2}\end{pmatrix}$ (A.2)

and $\vec{y}$, $\vec{z}$ are simple 2-vectors. This gives the relations,

$\vec{z}=S_{2}\vec{y}\quad\implies\quad\vec{y}=S_{2}^{-1}\vec{z}$ (A.3)

and

$m=t+\vec{y}^{\top}S_{2}\vec{y}\quad\implies\quad
t=m-\vec{z}^{\top}S_{2}^{-1}\vec{z}$ (A.4)

with $t\neq 0$ and

$S_{2}^{-1}={\rm det}(S_{2})^{-1}\begin{pmatrix}\sigma_{22}&-\sigma_{12}\\\
-\sigma_{12}&\sigma_{11}\end{pmatrix}$ (A.5)

The other parameters and functions are defined as follows,

$p=\left\\{\sigma_{11}-\frac{\sigma_{12}^{2}}{\sigma_{22}}\right\\}$ (A.6)

The modified Bessel function is defined as

$K_{\nu}(x)=\frac{1}{2}\int_{0}^{\infty}u^{\nu-1}{\rm
exp}\left\\{-x\left(u+u^{-1}\right)/2\right\\}du\quad{\rm for}\quad|{\rm
arg}(x)|<\tfrac{1}{2}\pi$ (A.7)

with $K_{\nu}(x)=K_{-\nu}(x)$,Glasser  _et al._ (2012) The higher-order Bessel
functions can be successively reduced to lower order Bessel functions by

$K_{\nu}(x)=\frac{2(\nu-1)}{x}K_{\nu-1}(x)+K_{\nu-2}(x)$ (A.8)

and all what remains to be evaluated are the Bessel functions $K_{1}$, $K_{0}$
and $K_{\tfrac{1}{2}}$. Further, for half-integer orders of the Bessel
function we can use the equation

$K_{\tfrac{1}{2}}(x)=K_{-\tfrac{1}{2}}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}$ (A.9)

The relation for the $\Gamma$-functions is

$\Gamma(x+1)=x\Gamma(x)$ (A.10)

For the symmetric matrix $S_{3}$ we take

$S_{3}(\alpha,\gamma)=\begin{pmatrix}\gamma^{2}&0&0\\\
0&1&\omega_{1}(\alpha)\\\
0&\omega_{1}(\alpha)&\omega_{2}(\alpha)\end{pmatrix}\quad\text{and}\quad
S_{2}(\alpha,\gamma)=\begin{pmatrix}1&\omega_{1}(\alpha)\\\
\omega_{1}(\alpha)&\omega_{2}(\alpha)\end{pmatrix}$ (A.11)

which gives $\vec{y}^{\top}=(0,0)$ and $m=t=\gamma^{2}$. Taking
$\sigma_{11}=1,\sigma_{12}(\alpha)=\omega_{1}(\alpha)$,
$\sigma_{22}(\alpha)=\omega_{2}(\alpha)$,
$p(\alpha)=1-\frac{\omega_{1}(\alpha)^{2}}{\omega_{2}(\alpha)}>0$ and $\det
S_{2}(\alpha)=\omega_{2}(\alpha)-\omega_{1}(\alpha)^{2}=\frac{p(\alpha)}{\omega_{2}(\alpha)}>0$
for $\alpha\in[0,1]$, and we get

$\displaystyle H_{A}^{h}(s)$
$\displaystyle=\frac{2\pi\Gamma(s-1)}{\gamma^{2s-2}\Gamma(s)}\sqrt{\frac{\omega_{2}(\alpha)}{p(\alpha)}}\zeta(2s-2)+\frac{2p(\alpha)^{\frac{1}{2}-s}\Gamma(s-\frac{1}{2})}{\Gamma(s)}\sqrt{\frac{\pi}{\omega_{2}(\alpha)}}\zeta(2s-1)+\frac{2\zeta(2s)}{\omega_{2}(\alpha)^{s}}$
(A.12)
$\displaystyle+\frac{8\pi^{s}}{\Gamma(s)}\left(\frac{p(\alpha)^{1-2s}}{\omega_{2}(\alpha)^{1+2s}}\right)^{1/4}\sum_{u,v\in\mathbb{N}}\left(\frac{v}{u}\right)^{s-\frac{1}{2}}\cos\left(\frac{2\pi\omega_{1}(\alpha)uv}{\omega_{2}(\alpha)}\right)K_{s-\frac{1}{2}}\left(2\pi
uv\sqrt{\frac{p(\alpha)}{\omega_{2}(\alpha)}}\right)$
$\displaystyle+\frac{4\pi^{s}}{\gamma^{s-1}\Gamma(s)}\sqrt{\frac{\omega_{2}(\alpha)}{p(\alpha)}}\sum_{\begin{subarray}{c}u\in\mathbb{N}\\\
\vec{w}\in\tilde{\mathbb{Z}}^{2}\setminus\\{\vec{0}_{2}\\}\end{subarray}}u^{1-s}\left(\vec{w}^{\top}S_{2}^{-1}\vec{w}\right)^{\frac{s-1}{2}}K_{s-1}\left(2\pi\gamma
u\sqrt{\vec{w}^{\top}S_{2}^{-1}\vec{w}}\right)$

and for the inverse $S_{2}$ matrix we have

$S_{2}^{-1}=\frac{1}{\omega_{2}(\alpha)-\omega_{1}(\alpha)^{2}}\begin{pmatrix}\omega_{2}(\alpha)&-\omega_{1}(\alpha)\\\
-\omega_{1}(\alpha)&1\end{pmatrix}$ (A.13)

###  A.2 The lattice sum $H^{h}_{B}$

This is the more complicated case, but we can use the expansion derived from
Van der Hoff and Benson’s original expression derived from a Mellin
transformation and the use of theta functions,van der Hoff and Benson (1953)

$\displaystyle\sum_{i\in\mathbb{Z}}\left[x^{2}+(i+\zeta)^{2}\right]^{-s}$
$\displaystyle=\frac{\sqrt{\pi}\Gamma(s-\frac{1}{2})}{\Gamma(s)\left|x\right|^{2s-1}}+\frac{4\pi^{s}}{\Gamma(s)}\sum_{n\in\mathbb{N}}\left(\frac{n}{\left|x\right|}\right)^{s-\frac{1}{2}}\cos(2\pi
n\zeta)K_{s-\frac{1}{2}}(2\pi n|x|)$ (A.14)

with $\zeta\in[0,1)$. We rewrite the quadratic function into

$\displaystyle Q_{B}(i,j,k,\alpha,\beta,\gamma)$
$\displaystyle=\gamma^{2}\left[\gamma^{-2}(i+j\omega_{1}(\alpha)+\tfrac{1}{2})^{2}+\gamma^{-2}(j\omega_{3}(\alpha)+\tfrac{\beta}{2})^{2}+(k+\tfrac{1}{2})^{2}\right]$
(A.15)
$\displaystyle=\gamma^{2}\left[\gamma^{-2}x^{2}_{ij}(\alpha,\beta)+(k+\tfrac{1}{2})^{2}\right]$

and get

$\displaystyle H_{B}^{h}(s,\alpha,\beta,\gamma)$
$\displaystyle=\sum_{\vec{i}\in\tilde{\mathbb{Z}}^{3}}Q_{B}(\vec{i},\alpha,\beta,\gamma)^{-s}=\frac{\sqrt{\pi}\Gamma\left(s-\frac{1}{2}\right)}{\gamma\Gamma(s)}\sum_{i,j\in\mathbb{Z}}\left|x_{ij}\right|^{1-2s}$
(A.16)
$\displaystyle+\frac{4\pi^{s}}{\gamma^{s+\frac{1}{2}}\Gamma(s)}\sum_{i,j\in\mathbb{Z}}\sum_{n\in\mathbb{N}}(-1)^{n}\left(\frac{n}{\left|x_{ij}(\alpha,\beta)\right|}\right)^{s-\frac{1}{2}}K_{s-\frac{1}{2}}\left(\frac{2\pi
n|x_{ij}(\alpha,\beta)|}{\gamma}\right)$

with

$\displaystyle
x_{ij}(\alpha,\beta)^{2}=(i+j\omega_{1}(\alpha)+\tfrac{1}{2})^{2}+(j\omega_{3}(\alpha)+\tfrac{\beta}{2})^{2}$
(A.17)

One has to avoid $x^{2}_{ij}=0$ for this expansion. For the range of
$(\alpha,\beta)$ parameters with $\omega_{1}\in[0,\frac{1}{2}]$,
$\omega_{3}\in[\frac{\sqrt{3}}{2},1]$ and $\beta\in[\frac{1}{\sqrt{3}},1]$,
one can easily show that $x^{2}_{ij}>0$. This expansion reduces the problem
into a fast converging triple sum involving Bessel functions, and into an
additional double sum,

$\displaystyle
W(\alpha,\beta)=\sum_{i,j\in\mathbb{Z}}\left\\{(i+j\omega_{1}(\alpha)+\tfrac{1}{2})^{2}+(j\omega_{3}(\alpha)+\tfrac{\beta}{2})^{2}\right\\}^{\frac{1}{2}-s}$
(A.18)

which however is slowly convergent for low $s$ values. We therefore use again
expansion (A.14) with $\zeta=j\omega_{1}(\alpha)+\tfrac{1}{2}$ and
$x=j\omega_{3}(\alpha)+\tfrac{\beta}{2}$ and get

$\displaystyle W$
$\displaystyle(\alpha,\beta)=\frac{\sqrt{\pi}\Gamma(s-1)}{\Gamma(s-\frac{1}{2})}\sum_{j\in\mathbb{Z}}\frac{1}{|j\omega_{3}(\alpha)+\frac{\beta}{2}|^{2s-2}}$
(A.19)
$\displaystyle+\frac{4\pi^{s-\frac{1}{2}}}{\Gamma(s-\frac{1}{2})}\sum_{j\in\mathbb{Z}}\sum_{n\in\mathbb{N}}\left(\frac{n}{|j\omega_{3}(\alpha)+\frac{\beta}{2}|}\right)^{s-1}\cos\left(2\pi
n\left[j\omega_{1}(\alpha)+\frac{1}{2}\right]\right)K_{s-1}\left(2\pi
n\left|j\omega_{3}(\alpha)+\frac{\beta}{2}\right|\right)$

The first sum can be turned into a Hurwitz zeta function $h(s,x)$,

$\displaystyle\sum_{j\in\mathbb{Z}}\frac{1}{|j+\tfrac{\beta}{2\omega_{3}(\alpha)}|^{2s-2}}=h\left(2s-2,\tfrac{\beta}{2\omega_{3}(\alpha)}\right)+h\left(2s-2,1-\tfrac{\beta}{2\omega_{3}(\alpha)}\right)$
(A.20)

with

$\displaystyle h(s,x)=\sum_{n\in\mathbb{N}_{0}}\frac{1}{(n+x)^{s}}$ (A.21)

In summary, the lattice sum $H_{B}$ is

$\displaystyle H_{B}^{h}$
$\displaystyle(s,\alpha,\beta,\gamma)=\frac{\pi\Gamma(s-1)}{\gamma\Gamma(s)\omega_{3}(\alpha)^{2s-2}}\left[h\left(2s-2,\tfrac{\beta}{2\omega_{3}(\alpha)}+h\left(2s-2,1-\tfrac{\beta}{2\omega_{3}(\alpha)}\right)\right)\right]$
$\displaystyle+\frac{4\pi^{s}}{\gamma\Gamma(s)}\sum_{j\in\mathbb{Z}}\sum_{n\in\mathbb{N}}\left(\frac{n}{|j\omega_{3}(\alpha)+\frac{\beta}{2}|}\right)^{s-1}\cos\left(2\pi
n\left[j\omega_{1}(\alpha)+\frac{1}{2}\right]\right)K_{s-1}\left(2\pi
n\left|j\omega_{3}(\alpha)+\frac{\beta}{2}\right|\right)$
$\displaystyle+\frac{4\pi^{s}}{\gamma^{s+\frac{1}{2}}\Gamma(s)}\sum_{i,j\in\mathbb{Z}}\sum_{n\in\mathbb{N}}(-1)^{n}\left(\frac{n}{|x_{ij}(\alpha,\beta)|}\right)^{s-\frac{1}{2}}K_{s-\frac{1}{2}}\left(\frac{2\pi
n|x_{ij}(\alpha,\beta)|}{\gamma}\right)$ (A.22)

###  A.3 Adding both lattice sums

We can now add both lattice sums $H^{h}_{A}$ and $H^{h}_{B}$

$H^{h}\left(s,\alpha,\beta,\gamma\right)=g_{0}\left(s,\alpha,\beta,\gamma\right)+\sum_{n=1}^{4}g_{n}\left(s,\alpha,\beta,\gamma\right)$
(A.23)

with

$\displaystyle g_{0}$
$\displaystyle\left(s,\alpha,\beta,\gamma\right)=\frac{2\pi\Gamma\left(s-1\right)}{\gamma^{2s-2}\Gamma\left(s\right)}\sqrt{\frac{\omega_{2}(\alpha)}{p(\alpha)}}\zeta(2s-2)+\frac{2p(\alpha)^{\frac{1}{2}-s}\Gamma(s-\frac{1}{2})}{\Gamma(s)}\sqrt{\frac{\pi}{\omega_{2}(\alpha)}}\zeta(2s-1)$
$\displaystyle+\frac{2\zeta(2s)}{\omega_{2}(\alpha)^{s}}+\frac{\pi\Gamma\left(s-1\right)}{\gamma\Gamma(s)\omega_{3}(\alpha)^{2s-2}}\left\\{h\left(2s-2,\tfrac{\beta}{2\omega_{3}(\alpha)}\right)+h\left(2s-2,1-\tfrac{\beta}{2\omega_{3}(\alpha)}\right)\right\\}$
(A.24)

and each of the following $g_{n}$ terms containing an infinite sum of Bessel
functions such that

$g_{1}\left(s,\alpha\right)=\frac{8\pi^{s}}{\Gamma(s)}\frac{p(\alpha)^{\frac{1-2s}{4}}}{\omega_{2}(\alpha)^{\frac{1+2s}{4}}}\sum_{i,j\in\mathbb{N}}\left(\frac{i}{j}\right)^{s-\frac{1}{2}}\cos\left(\frac{2\pi\omega_{1}(\alpha)ij}{\omega_{2}(\alpha)}\right)K_{s-\frac{1}{2}}\left(2\pi
ij\sqrt{\frac{p(\alpha)}{\omega_{2}(\alpha)}}\right)$ (A.25)
$g_{2}\left(s,\alpha,\gamma\right)=\frac{4\pi^{s}}{\gamma^{s-1}\Gamma(s)}\sqrt{\frac{\omega_{2}(\alpha)}{p(\alpha)}}\sum_{\begin{subarray}{c}n\in\mathbb{N}\\\
\vec{w}\in\tilde{\mathbb{Z}}^{2}\setminus\\{\vec{0}_{2}\\}\end{subarray}}\left(\frac{\vec{w}^{\top}S_{2}^{-1}\vec{w}}{n^{2}}\right)^{\frac{s-1}{2}}K_{s-1}\left(2\pi
n\gamma\sqrt{\vec{w}^{\top}S_{2}^{-1}\vec{w}}\right)$ (A.26) $\displaystyle
g_{3}\left(s,\alpha,\beta,\gamma\right)$
$\displaystyle=\frac{4\pi^{s}}{\gamma\Gamma(s)}\sum_{j\in\mathbb{Z}}\sum_{n\in\mathbb{N}}\left(\frac{n}{|j\omega_{3}(\alpha)+\frac{\beta}{2}|}\right)^{s-1}\cos\left(2\pi
n\left[j\omega_{1}(\alpha)+\frac{1}{2}\right]\right)\quad$ (A.27)
$\displaystyle\times K_{s-1}\left(2\pi
n\left|j\omega_{3}(\alpha)+\frac{\beta}{2}\right|\right)$ $\displaystyle
g_{4}\left(s,\alpha,\beta,\gamma\right)=$
$\displaystyle\frac{4\pi^{s}}{\gamma^{s+\frac{1}{2}}\Gamma(s)}\sum_{i,j\in\mathbb{Z}}\sum_{n\in\mathbb{N}}(-1)^{n}\left(\frac{n}{|x_{ij}(\alpha,\beta)|}\right)^{s-\frac{1}{2}}K_{s-\frac{1}{2}}\left(2\pi
n\gamma^{-1}|x_{ij}(\alpha,\beta)|\right)$ (A.28)

with $p(\alpha)=1-\frac{\omega_{1}(\alpha)^{2}}{\omega_{2}(\alpha)}>0$ as
defined before.

For computational purposes we need to treat the first three Bessel sums more
efficiently. For the first Bessel sum $g_{1}$ appearing in $H^{h}_{A}$ we use
permutation symmetry between $i$ and $j$ and get,

$\displaystyle g_{1}\left(s,\alpha\right)$
$\displaystyle=\frac{8\pi^{s}}{\Gamma(s)}\frac{p(\alpha)^{\frac{1-2s}{4}}}{\omega_{2}(\alpha)^{\frac{1+2s}{4}}}\sum_{i\geq
j\in\mathbb{N}}\left(1-\tfrac{1}{2}\delta_{ij}\right)\left[\left(\frac{i}{j}\right)^{s-\frac{1}{2}}+\left(\frac{j}{i}\right)^{s-\frac{1}{2}}\right]\cos\left(\frac{2\pi\omega_{1}(\alpha)ij}{\omega_{2}(\alpha)}\right)$
$\displaystyle\times K_{s-\frac{1}{2}}\left(2\pi
ij\sqrt{\frac{p(\alpha)}{\omega_{2}(\alpha)}}\right)$ (A.29)

where $\delta_{ij}$ is the Kronecker symbol. The sum $g_{3}$ for $H^{h}_{B}$
can be dealt with in a similar way,

$\displaystyle g_{3}(s,\alpha,\beta,\gamma)$
$\displaystyle=\frac{4\pi^{s}}{\gamma\Gamma(s)}\sum_{i\in\mathbb{N}}\left(-1\right)^{i}\left(\frac{2i}{\beta}\right)^{s-1}K_{s-1}\left(\pi\beta
i\right)+\frac{4\pi^{s}}{\gamma\Gamma(s)}\sum_{i,j\in\mathbb{N}}(-1)^{i}\cos\left(2\pi\omega_{1}(\alpha)ij\right)$
$\displaystyle\times\sum_{k\in\\{-1,+1\\}}\left(\frac{i}{\left(j\omega_{3}(\alpha)+\frac{1}{2}k\beta\right)}\right)^{s-1}K_{s-1}\left(2\pi
i\left(j\omega_{3}(\alpha)+\frac{1}{2}k\beta\right)\right)$ (A.30)

and we used cos$(x+n\pi)=(-1)^{n}$cos$(x)$ and remember that $\beta>0$ and
$j\omega_{3}(\alpha)>\frac{1}{2}k\beta$. The first single sum is related to
the infinite sum in (A.14) and is one of the most common Bessel function
series in the literature Fucci and Kirsten (2015).

For the triple sum $g_{2}$ we simply use the fact that $S_{2}^{-1}$ is a
symmetric matrix and get,

$\displaystyle g_{2}\left(s,\alpha,\gamma\right)$
$\displaystyle=\frac{4\pi^{s}}{\gamma^{s-1}\Gamma(s)}\sqrt{\frac{\omega_{2}(\alpha)}{p(\alpha)}}\sum_{n\in\mathbb{N}}\sum_{\begin{subarray}{c}\vec{w}\in\tilde{\mathbb{Z}}^{2}\setminus\\{\vec{0}_{2}\\}\\\
w_{1}\leq
w_{2}\end{subarray}}\left(2-\delta_{w_{1}w_{2}}\right)\left(\frac{\vec{w}^{\top}S_{2}^{-1}\vec{w}}{n^{2}}\right)^{\frac{s-1}{2}}$
(A.31) $\displaystyle\times K_{s-1}\left(2\pi
n\gamma\sqrt{\vec{w}^{\top}S_{2}^{-1}\vec{w}}\right)$

The last sum $g_{4}$ for $H^{h}_{B}$ cannot be further simplified, however,
this triple sum is fast converging. For larger $s$ values we obtain large
compensating numbers for the individual terms in the lattice sums, which
offsets double precision results. In this case taking the direct summation for
the lattice sum for values $s\geq 8$ is the preferred option as outlined in
the next section. Alternatively, one can switch to quadruple precision
arithmetic.

##  Appendix B Direct summation

For the lattice sum $H^{h}_{A}(s,\alpha,\gamma)$ in Eq.(26) of the quadratic
form $Q_{A}(\vec{i},\alpha,\gamma)$ we can take the $k=0$ term out and get the
following expression,

$\displaystyle H^{h}_{A}(s,\alpha,\gamma)=$
$\displaystyle{\sum_{i,j\in\mathbb{Z}}}^{{}^{\prime}}\left(i^{2}+2\omega_{1}(\alpha)ij+\omega_{2}(\alpha)j^{2}\right)^{-s}$
(B.1)
$\displaystyle+2\sum_{i,j\in\mathbb{Z}}\sum_{k\in\mathbb{N}}\left(i^{2}+2\omega_{1}(\alpha)ij+\omega_{2}(\alpha)j^{2}+\gamma^{2}k^{2}\right)^{-s}$

For our case of $\omega_{2}(\alpha)=1$ we can further utilize permutation
symmetry between $i$ and $j$. Notice, there are no general solutions in terms
of standard functions for the double sum in (B.1), only for some special cases
for the coefficients $\omega_{1}$ and $\omega_{2}$ analytical expressions are
known Zucker and Robertson (1975).

For the second lattice sum we can only simplify the summation over $k$,

$\displaystyle H^{h}_{B}(s,\alpha,\beta,\gamma)$
$\displaystyle=2\sum_{i,j\in\mathbb{Z}}\sum_{k\in\mathbb{N}}\left((i+j\omega_{1}(\alpha)+\tfrac{1}{2})^{2}+(j\omega_{3}(\alpha)+\tfrac{\beta}{2})^{2}+\gamma^{2}(k-\tfrac{1}{2})^{2}\right)^{-s}$
(B.2)

Nevertheless, for $s=4$ we reach double precision accuracy by summing over
$i,j,k\in[-500,500]$ within a few seconds of computer time. In our application
for the LJ potential we chose the interval of $i,j,k\in[-750,750]$.

##  Appendix C Relation between the lattice sums of the fcc lattice

Here we show how eq. (35) can be brought into the more common form (36).
Starting with (35) we have

$\displaystyle L_{A}^{\text{fcc}}+L_{B}^{\text{fcc}}$
$\displaystyle=\sideset{}{{}^{\prime}}{\sum}_{\vec{i}\in\mathbb{Z}^{3}}(i_{1}^{2}+i_{2}^{2}+2i_{3}^{2})^{-s}+{\sum_{\vec{i}\in\mathbb{Z}^{3}}}\left((i_{1}+\tfrac{1}{2})^{2}+(i_{2}+\tfrac{1}{2})^{2}+2(i_{3}+\tfrac{1}{2})^{2}\right)^{-s}$
(C.1) $\displaystyle=\sideset{}{{}^{\prime}}{\sum}_{i_{1}\equiv i_{2}\equiv
i_{3}(\text{mod}\,2)}\left(\left(\tfrac{i_{1}}{2}\right)^{2}+\left(\tfrac{i_{2}}{2}\right)^{2}+2\left(\tfrac{i_{3}}{2}\right)^{2}\right)^{-s}=2^{2s}\sideset{}{{}^{\prime}}{\sum}_{i_{1}\equiv
i_{2}\equiv
i_{3}(\text{mod}\,2)}\left(i_{1}^{2}+i_{2}^{2}+2i_{3}^{2}\right)^{-s}$

using the standard notation for congruences. Since $i_{1}$ and $i_{2}$ have
the same parity, both of $i_{1}-i_{2}$ and $i_{1}+i_{2}$ will be even. Hence,
we make the change of variable $i_{1}-i_{2}=2x$ and $i_{1}+i_{2}=2y$, or
equivalently $i_{1}=x+y$ and $i_{2}=x-y$. This gives

$\displaystyle L_{A}^{\text{fcc}}+L_{B}^{\text{fcc}}$
$\displaystyle=2^{2s}\sideset{}{{}^{\prime}}{\sum}_{x+y\equiv
i_{3}(\text{mod}\,2)}\left(2x^{2}+2y^{2}+2i_{3}^{2}\right)^{-s}=2^{s}\sideset{}{{}^{\prime}}{\sum}_{x+y\equiv
i_{3}(\text{mod}\,2)}\left(x^{2}+y^{2}+i_{3}^{2}\right)^{-s}.$ (C.2)

Now $x+y$ and $i_{3}$ will have the same parity if and only if
$x+y+i_{3}\equiv 0\pmod{2}$. The indicator function for this condition is

$\tfrac{1}{2}\left(1+(-1)^{x+y+i_{3}}\right)=\begin{cases}1&\text{if
$x+y+i_{3}\equiv 0\pmod{2}$,}\\\ 0&\text{if $x+y+i_{3}\not\equiv
0\pmod{2}$.}\end{cases}$

On applying this to the sum above we deduce

$\displaystyle
L_{A}^{\text{fcc}}+L_{B}^{\text{fcc}}=2^{s}\sideset{}{{}^{\prime}}{\sum}_{\vec{i}\in\mathbb{Z}^{3}}\tfrac{1}{2}\left(1+(-1)^{x+y+i_{3}}\right)\left(x^{2}+y^{2}+i_{3}^{2}\right)^{-s}$
(C.3)

and this is (36).

##  Appendix D The iterative Newton–Raphson algorithm

The iterative Newton–Raphson algorithm Hämmerlin and Hoffmann (2012) can be
used to determine the minimum of expressions (37) or (39) for the parameter
space $\vec{x}^{\top}=(a,c,\alpha,\beta)$ and the reduced space
$\vec{x}^{\top}=(\alpha,\beta,\gamma)$ respectively to the required accuracy

$\vec{x}_{N+1}=\vec{x}_{N}-\lambda\left[(\partial_{x_{i}}\partial_{x_{j}})E_{nm}^{*}(\vec{x}_{N})\right]^{-1}\vec{\nabla}E_{nm}^{*}(\vec{x}_{N})\,,$
(D.1)

where for evaluating the gradient and Hessian we use numerical methods, i.e.
the well known expressions for the first and second derivatives,Abramowitz and
Stegun (1964)

$\partial_{x}f(x)=\left\\{f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)\right\\}/12h$ (D.2)
$\partial_{x}^{2}f(x)=\left\\{-f(x-2h)+16f(x-h)-30f(x)+16f(x+h)-f(x+2h)\right\\}/12h^{2}$
(D.3)

We chose $\lambda=0.9$ and $h=1\times 10^{-3}$ for the step size. For the
mixed derivatives we used the following formula,Abramowitz and Stegun (1964)

$\partial_{x}\partial_{y}f(x,y)=\left\\{f(x+h,y+h)-f(x+h,y-h)-f(x-h,y+h)+f(x-h,y-h)\right\\}/4h^{2}$
(D.4)

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