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\def\ee{\boldsymbol{\cal{E}}}
\def\a{{\cal J}}
% \def\A{\bar{\cal J}}
\def\A{\widetilde{\cal J}}
\def\b{{\cal K}}
% \def\B{\bar{\cal K}}
\def\B{\widetilde{\cal K}}
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\begin{document}
\title{Supplementary material for ``Predicting polarization and nonlinear 
dielectric response of arbitrary perovskite superlattice sequences''}

\author{Xifan Wu$^{1}$, Massimiliano Stengel$^{2}$,
Karin M. Rabe$^{3}$, and David Vanderbilt$^{3}$ }

\address{$^{1}$Chemistry Department, Princeton University, Princeton, NJ
08544-0001,USA}

\address{$^{2}$Materials Department, University of California, Santa Barbara,
Santa Barbara, CA 93106-5050,USA}

\address{$^{3}$Department of Physics and Astronomy, Rutgers University,
Piscataway, NJ 08854-8019, USA}


\date{\today}

\begin{abstract}

In this supplementary material, we present the form of the
cluster expansions for the layer polarizations of AO and TiO$_2$ layers,
and give the fitted values of the effective cluster interaction
coefficients for the layer polarizations and the lattice constants.


\end{abstract}
% \pacs{77.22.-d, 77.22.Ej, 77.80.-e, 77.84.Bw}

\maketitle

As explained in the main text, we
separate the $D$-dependence of the layer polarization 
of an AO layer into an odd part $p_{\rm AO}^{(-)}(D)$
(reflecting the inversion-symmetry-conserving part of its environment)
and an even part $p_{\rm AO}^{(+)}(D)$ (reflecting the
inversion-symmetry-breaking character), and similarly for TiO$_2$
layers.  Then our model for AO layers is
%
\begin{eqnarray}
p_{\rm AO}^{(-)} & = & \a  +  \a_0s_0 + \a_0't_0 + \a_{1}(s_{\bar{1}} + s_1)
+  \a_{1}'(t_{\bar{1}} + t_1) \cr
&+& \a_2(s_{\bar{2}} + s_2) + \a_{01}(s_{\bar{1}}s_0+s_0s_1) \cr
&+& \a_{02}(s_{\bar{2}}s_0 + s_0s_2)+ \a_{\bar{1}1}s_{\bar{1}}s_1 \cr
&+& \a_{12}(s_{\bar{2}}s_{\bar{1}} + s_1s_2)
\label{eq:AO_odd}
\end{eqnarray}
%
and
%
\begin{eqnarray}
p_{\rm AO}^{(+)} &=&\A_1(s_{\bar{1}}-s_1) +\A_1'(t_{\bar{1}}- t_1) \cr
&+&\A_{01}(s_{\bar{1}}s_0 - s_0s_1) +\A_{01}'(t_{\bar{1}}s_0 - s_0t_1) \cr
&+&\A_{10}'(s_{\bar{1}}t_0 - t_0s_1) 
  +\A_{12}(s_{\bar{2}}s_{\bar{1}} - s_1s_2)  \,.
\label{eq:AO_even}
\end{eqnarray}
%
Eq.~(\ref{eq:AO_odd}) is equivalent to Eq.~(2) of the main text, except that
here we introduce the condensed notation $t_j=s_j^2$.
The layers are labeled according to the sequence
$[ \dotsm \,\bar{3}.\bar{2}.\bar{1}.0.1.2.3 \,\dotsm]$
where numbers represent AO layers, periods represent TiO$_2$ layers,
and the central AO layer $0$ is the one whose layer polarization  is being expanded.
The coefficients $\a(D)$, $\a_0(D)$, ... are odd functions of $D$;
the $\A_1(D)$, $\A_1'(D)$, ... are even functions of $D$; and the
combinations of spins appearing in Eqs.~(\ref{eq:AO_odd}-\ref{eq:AO_even})
are the symmetry-appropriate ones.  The $D$-dependence of the
coefficients has been suppressed for simplicity of notation.

The corresponding expansion for the layer polarization  of a TiO$_2$ layer is
%
\begin{eqnarray}
p_{\rm TiO_2}^{(-)} &=& \b + \b_1(s_{\bar{1}} + s_1) + \b_1'(t_{\bar{2}} +
   t_2)+\b_2(s_{\bar{2}} + s_2)  \cr
& + & \b_2'(t_{\bar{1}} + t_1) + \b_{12}(s_{\bar{2}}s_{\bar{1}} +
   s_1s_2)+\b_{\bar{1}1}s_{\bar{1}}s_1\cr
& +&    \b_{{\bar{2}}1}(s_{\bar{2}}s_1 + s_{\bar{1}}s_2)
\label{eq:BO_odd}
\end{eqnarray}
%
and
%
\begin{eqnarray}
p_{\rm TiO_2}^{(+)} &=&  \B_1(s_{\bar{1}}-s_1) + \B_1'(t_{\bar{2}}- t_2) +
    \B_2(s_{\bar{2}}-s_2) \cr
& + & \B_{12}(s_{\bar{2}}s_{\bar{1}}-s_1s_2) +
    \B_{12}'(s_{\bar{1}}t_{\bar{2}} - t_2s_1) \cr
&+& \B_{21}'(s_{\bar{2}}t_{\bar{1}} - t_1s_2) +
    \B_{\bar{1}1}'(s_{\bar{1}}t_1 - t_{\bar{1}}s_1) \cr
&+& \B_{\bar{1}2}'(s_{\bar{1}}t_2- t_{2}s_{\bar{1}}) +
    \B_{\bar{2}1}(s_{\bar{2}}s_1 - s_{\bar{1}}s_2)\cr
&+& \B_{\bar{2}1}'(s_{\bar{2}}t_1- t_{\bar{1}}s_2) \,,
\label{eq:BO_even}
\end{eqnarray}
%
where $\b$ and $\B$ coefficients are odd and even in $D$ respectively.
Here the labeling scheme is
$[ \dotsm \,\bar{3}.\bar{2}.\bar{1}.1.2.3 \,\dotsm]$
with the central TiO$_2$ layer being the one whose layer polarization  is being expanded.

The parameters $\a(D)$, $\A(D)$, $\b(D)$, and $\B(D)$ are expressed as
fifth-order Taylor expansions in $D$,
with the Taylor coefficients obtained from a least-squares
fitting to the first-principles calculations of $p_j(D)$
for superlattices in the database.  Also, recall that
the supercell lattice constant $c(D)$ was expanded in terms of
parameters ${\cal C}_q(D)$ ($q=1,...,4$) as specified in Eq.~(3) of the
main text.  The ${\cal C}(D)$ were similarly fitted to the database,
but now as fourth-order Taylor expansions in $D$.

The fitted parameter values are given in the following tables.

\onecolumngrid
\newpage

\begin{table*}[h]
\caption{Fitted first-, third-, and fifth-order-in-$D$ Taylor coefficients
of effective cluster interactions for $p_{\rm AO}^{(-)}(D)$ 
in Eq.~(\ref{eq:AO_odd}), and
zeroth-, second-, and fourth-order-in-$D$ Taylor coefficients
of effective cluster interactions for $p_{\rm AO}^{(+)}(D)$
in Eq.~(\ref{eq:AO_even}).}
\begin{tabular}{ldddddddddd}
\hline \hline
& \multicolumn{1}{r}{$\a$}
& \multicolumn{1}{r}{$\a_0$}
& \multicolumn{1}{r}{$\a_0'$}
& \multicolumn{1}{r}{$\a_1$} 
& \multicolumn{1}{r}{$\a_1'$}
& \multicolumn{1}{r}{$\a_{2}$} 
& \multicolumn{1}{r}{$\a_{01}$} 
& \multicolumn{1}{r}{$\a_{02}$} 
& \multicolumn{1}{r}{$\a_{\bar{1}1}$} 
& \multicolumn{1}{r}{$\a_{12}$} \\
\hline
1st     &   2.2771 &   0.1113 &  0.0819  &  0.0034 & 0.0007 &
-0.0018 & 0.0197 & 0.0031 & 0.0026 & 0.0013   \\
3rd  &  -0.0459 &  -0.2090 &  -0.0319 &  -0.0513 & -0.0129  &
-0.0078 & -0.0512 & -0.0211 & -0.0149 & -0.0064   \\
5th      &   0.2793 &   0.3690 &  0.1236  &  0.1287  & 0.0533  &
0.0293 & 0.0554 & 0.0339 & 0.0071 & 0.0209   \\
\hline \vspace{-3mm} \cr
& \multicolumn{1}{r}{$\A$ $_1$ } 
& \multicolumn{1}{r}{$\A$ $_1'$} 
& \multicolumn{1}{r}{$\A_{01}$}
& \multicolumn{1}{r}{$\A_{01}'$} 
& \multicolumn{1}{r}{$\A_{10}'$} 
& \multicolumn{1}{r}{$\A_{12}$ } & & & &\\
\hline
0th    & -0.0210 & -0.0044 & -0.0015 &
0.0001 & -0.0002 & -0.0005 & & & & \\
2nd    & 0.0657 & 0.0014 & 0.0047 &
0.0037 & 0.0029 & -0.0017 & & & & \\
4th    & -0.0569 &  -0.0035 & 0.0010 &
-0.0139 & -0.0164  & 0.0081 & & & & \\
\hline \hline
\end{tabular}
\end{table*}

\begin{table*}[h]
\caption{Fitted first-, third-, and fifth-order-in-$D$ Taylor coefficients
of effective cluster interactions for $p_{\rm TiO_2}^{(-)}(D)$
in Eq.~(\ref{eq:BO_odd}),
and zeroth-, second-, and fourth-order Taylor coefficients
of effective cluster interactions for $p_{\rm TiO_2}^{(+)}(D)$
in Eq.~(\ref{eq:BO_even}).}
\begin{tabular}{ldddddddddd}
\hline \hline
& \multicolumn{1}{r}{$\b$}
& \multicolumn{1}{r}{ $\b_1$}
& \multicolumn{1}{r}{  $\b_1'$}
& \multicolumn{1}{r}{  $\b_2$}
& \multicolumn{1}{r}{  $\b_2'$} 
& \multicolumn{1}{r}{ $\b_{12}$} 
& \multicolumn{1}{r}{ $\b_{\bar{1}1}$} 
& \multicolumn{1}{r}{$\b_{\bar{2}1}$}  & & \\
\hline
1st     &   1.5292   &  -0.0097  &  -0.0060  & 0.0075   &
-0.0013  & -0.0076  & 0.0033  & -0.0018  & & \\
3rd     &  0.0415    & 0.1203    &  0.0376   &  0.0035  &
0.0074  & 0.0148   & 0.0025  &  0.0091 & & \\
5th      &  -0.0468  &  -0.1727  &  -0.0702  &  -0.0421  &
-0.0068  & -0.0084  & -0.0095 &  -0.0219 & & \\
\hline \vspace{-3mm}\cr
& \multicolumn{1}{r}{  $\B_1$}
& \multicolumn{1}{r}{  $\B_1'$ }
& \multicolumn{1}{r}{ $\B_2$}
& \multicolumn{1}{r}{ $\B_{12}$} 
& \multicolumn{1}{r}{ $\B_{12}'$}
& \multicolumn{1}{r}{  $\B_{21}'$} 
& \multicolumn{1}{r}{ $\B_{\bar{1}1}'$} 
& \multicolumn{1}{r}{ $\B_{\bar{1}2}'$ } 
& \multicolumn{1}{r}{ $\B_{\bar{2}1}$} 
& \multicolumn{1}{r}{ $\B_{\bar{2}1}'$} \\
\hline \vspace{-3mm}\cr
0th     &  0.0634  &  -0.0057  & 0.0027   & 0.0023  & 0.0004 &
0.0001  & 0.0010    & -0.0001   & -0.0003   & 0.0000 \\
2nd     &  -0.1468 &  0.0101   &  -0.0200 & 0.0106  & -0.0057 &
-0.0169 & 0.0089    & -0.0018   &  0.0014   & 0.0063  \\
4th      &  0.0522 &   0.0059  &   0.0045 & -0.0470  & 0.0247 &
0.0679 &  -0.0447  &   0.0147  &   -0.0030 & -0.0372  \\
\hline \hline
\end{tabular}
\end{table*}

\begin{table*}[h]
\caption{Fitted zeroth-, second-, and fourth-order-in-$D$ Taylor coefficients
of effective cluster interactions for $c(D)$ in Eq.~(3) of main text.}
\begin{tabular}{ldddd}
\hline \hline
& \multicolumn{1}{r}{${\cal C}_1$} 
& \multicolumn{1}{r}{${\cal C}_2$} 
& \multicolumn{1}{r}{${\cal C}_3$}
& \multicolumn{1}{r}{${\cal C}_4$} \\
\hline
0th  & 7.2305 & 0.1968 & 0.0543 & 0.0097 \\
2nd  &  0.4430 & 0.3293 & 0.1675 & 0.0832 \\
4th  & -0.1926 & -0.6367 & -0.1546  & -0.3744 \\
\hline \hline
\end{tabular}
\end{table*}


\end{document}
