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\begin{document}

\title{Supplementary Information for ``Reaching the thermodynamic limit of periodic CCSD cohesive energies and band gaps''}

\author{Shuhang Li}
 \affiliation{Department of Chemistry and Cherry Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States}
\author{Huanchen Zhai}
 \affiliation{Initiative for Computational Catalysis, Flatiron Institute, New York, New York 10010, United States}
\author{Francesco A. Evangelista}
 \affiliation{Department of Chemistry and Cherry Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States}
\author{Timothy C. Berkelbach}
 \email{tberkelbach@flatironinstitute.org}
 \affiliation{Initiative for Computational Catalysis, Flatiron Institute, New York, New York 10010, United States}
 \affiliation{Department of Chemistry, Columbia University, New York, New York, 10027, United States}

\date{\today}

\maketitle

\begin{figure*}[!htb]
\centering
\includegraphics[width=6.25in]{bn_ipea.png}
\caption{Extrapolation curves of IP (at VBM), EA (at CBM), and indirect band gap (VBM to CBM) for \ce{BN} obtained using the ABC extrapolation model with different \kmesh ranges. All calculations are carried out using the GTH-HF pseudopotential and the GTH-cc-pVDZ basis set. The black dashed line indicates the experimental band gap that is corrected for zero-point renormalization.}
\label{fig:bn ipea}
\end{figure*}

\begin{figure*}[!htb]
\centering
\includegraphics[width=6.25in]{bn_cbm_ipea.png} 
\caption{Extrapolation curves of IP (at CBM), EA (at CBM), and direct band gap (at CBM) for \ce{BN} obtained using the ABC extrapolation model with different \kmesh ranges. All calculations are carried out using the GTH-HF pseudopotential and the GTH-cc-pVDZ basis set.}
\label{fig:bn ipea cbm}
\end{figure*}

\begin{table*}[t]
\centering
\caption{Difference (TZ$-$DZ and QZ$-$DZ, in eV) between EOM-CCSD direct band gaps computed with the GTH-cc-pVDZ and GTH-cc-pVTZ basis sets for different systems and \kmeshes. For BN, BP, Si, C, we compute the $\Gamma \rightarrow \Gamma$ direct band gap.}
\label{tab:tz direct}
\begin{tabular}{lSSSSSSS}
\toprule
\multirow{2}{*}{System} & \multicolumn{4}{c}{TZ$-$DZ} & \multicolumn{3}{c}{QZ$-$DZ} \\
\cmidrule(lr){6-8}
\cmidrule(lr){2-5}
& \multicolumn{1}{c}{\mesh{2}}
& \multicolumn{1}{c}{\mesh{3}}
& \multicolumn{1}{c}{\mesh{4}} 
& \multicolumn{1}{c}{\mesh{5}} 
& \multicolumn{1}{c}{\mesh{2}}
& \multicolumn{1}{c}{\mesh{3}}
& \multicolumn{1}{c}{\mesh{4}} \\
\midrule
MgO  & -0.024 & 0.004 & 0.003 & 0.005 & -0.003 & 0.032 & 0.034 \\
LiCl & -0.069 & -0.050 & -0.048 & -0.045 & 0.022 & 0.049 & 0.055\\
LiF  & 0.005 & -0.007 & -0.020 & -0.026 & 0.064 & 0.055 & 0.045 \\
LiH  & -0.002 & -0.002 & 0.002 & 0.004 & -0.009 & 0.001 & 0.006\\
BN  & 0.024 & 0.042 & 0.052 & 0.057 & 0.062 & 0.083 & N/A\\
BP & -0.061 & -0.046 & -0.036 & -0.029 & -0.060 & -0.043 & N/A\\
Si  & -0.011 & -0.013 & -0.005 & 0.005 & 0.007 & 0.007 & 0.018\\
C & -0.063 & -0.041 & -0.032 & -0.027 & -0.067 & -0.041 & -0.030\\
\bottomrule
\end{tabular}
\end{table*}

\begin{table*}[t]
\centering
\caption{Conduction-band offsets (in eV) for different solids computed with different \kmeshes. All calculations are carried out using the GTH-HF pseudopotential and the GTH-cc-pVDZ basis set.}
\label{tab:cb_offset}
\begin{tabular}{lSSSSS}
\toprule
\kmesh 
& \multicolumn{1}{c}{BN}
& \multicolumn{1}{c}{BP}
& \multicolumn{1}{c}{Si}
& \multicolumn{1}{c}{C (approx.\ $k_\mathrm{CBM}$)}
& \multicolumn{1}{c}{C ($k_\mathrm{CBM}$)} \\
\midrule
\mesh{2} & -5.293 & -2.746 & -2.510 & -2.454 & -2.487 \\
\mesh{3} & -5.351 & -2.636 & -2.463 & -2.056 & -2.141 \\
\mesh{4} & -5.303 & -2.619 & -2.441 & -2.095 & -2.164 \\
\mesh{5} & -5.321 & -2.582 & -2.405 & -2.065 & -2.114 \\
\mesh{6} & -5.320 & {N/A\textsuperscript{$a$}}  & {N/A}\textsuperscript{$a$}  & -2.048 & {N/A}\textsuperscript{$a$}\\
\bottomrule
\end{tabular}\\
\textsuperscript{$a$} Conduction-band energy at $k_\mathrm{CBM}$ is not available.
\end{table*}


\begin{table*}[t]
\centering
\caption{Difference (TZ$-$DZ and QZ$-$DZ, in eV) between conduction-band offsets computed with the GTH-cc-pVDZ and GTH-cc-pVTZ basis sets for different systems and \kmeshes.}
\label{tab:tz offset}
\begin{tabular}{lSSSSS}
\toprule
\multirow{2}{*}{System} & \multicolumn{3}{c}{TZ$-$DZ} & \multicolumn{2}{c}{QZ$-$DZ} \\
\cmidrule(lr){5-6}
\cmidrule(lr){2-4}
& \multicolumn{1}{c}{\mesh{2}}
& \multicolumn{1}{c}{\mesh{3}}
& \multicolumn{1}{c}{\mesh{4}} 
& \multicolumn{1}{c}{\mesh{2}}
& \multicolumn{1}{c}{\mesh{3}} \\
\midrule
BN  & -0.022 & -0.001 & -0.006 & -0.025 & -0.003 \\
BP  & -0.038 & -0.059 & -0.042 & -0.031 & -0.054 \\
Si  & 0.006 & 0.010 & 0.025 & 0.018 & 0.021 \\
C   & 0.010 & -0.048 & -0.010 & 0.018 & -0.047 \\
\bottomrule
\end{tabular}
\end{table*}

\begin{table*}[t]
\centering
\caption{Predicted cohesive energies (in eV/atom) obtained with the GTH-HF pseudopotential (PP) and the all-electron (AE) Hamiltonian at each level of theory, compared to previous literature results.
}
\label{tab:cohesive_comp}
\begin{tabular}{lSSSSSSSSSSSS}
\toprule
{\multirow{2}{*}{System}} 
& \multicolumn{3}{c}{HF}
& \multicolumn{6}{c}{MP2}
& \multicolumn{3}{c}{CCSD} \\
\cmidrule(lr){2-4}
\cmidrule(lr){5-10}
\cmidrule(lr){11-13}
& \multicolumn{1}{c}{PP} & \multicolumn{1}{c}{AE} & \multicolumn{1}{c}{VASP~\cite{gruneis.2010.10.1063/1.3466765}} & \multicolumn{1}{c}{PP} & \multicolumn{1}{c}{AE} & \multicolumn{1}{c}{VASP~\cite{gruneis.2010.10.1063/1.3466765}} & \multicolumn{1}{c}{VASP~\cite{booth.2013.10.1038/nature11770}} & \multicolumn{1}{c}{FHI-aims~\cite{zhang.2019.10.1088/1367-2630/aaf751}} & \multicolumn{1}{c}{PySCF~\cite{goldzak.2022.10.1063/5.0119633}} & \multicolumn{1}{c}{PP} & \multicolumn{1}{c}{AE} & \multicolumn{1}{c}{VASP~\cite{booth.2013.10.1038/nature11770}} \\
\midrule
MgO	 & 3.61 & 3.63 & 3.59 & 5.47 & 5.52 & 5.35 &      & 5.53 & 5.37 & 5.09 & 5.10 &    \\
LiCl & 2.70 & 2.71 & 2.70 & 3.70 & 3.72 & 3.64 &      & 3.70 & 3.69 & 3.52 & 3.55 &    \\
LiF	 & 3.32 & 3.34 & 3.34 & 4.56 & 4.61 & 4.49 &      & 4.54 & 4.58 & 4.38 & 4.50 &    \\
LiH	 & 1.79 & 1.79 & 1.79 & 2.38 & 2.39 & 2.39 & 2.39 & 2.38 & 2.41 & 2.45 & 2.50 & 2.45   \\
BN   & 4.61 & 4.71 & 4.74 & 7.02 & 7.15 & 7.12 & 7.15 & 7.25 & 7.13 & 6.42 & 6.54 & 6.57  \\
BP   & 3.37 & 3.40 & 3.38 & 5.59 & 5.64 & 5.61 &      & 5.77 & 5.58 & 4.89 & 4.95 &    \\
Si   & 3.02 & 3.02 & 2.97 & 5.06 & 5.07 & 5.05 &      & 5.21 & 4.97 & 4.46 & 4.47 &    \\
C    & 5.13 & 5.27 & 5.28 & 7.82 & 7.99 & 7.97 & 8.04 & 8.08 & 7.98 & 7.10 & 7.26 & 7.30   \\
%\midrule
%MAE (eV) & 1.54 & 1.50 & 0.24 & 0.30 & 0.19 & 0.14 & $\cdots$ \\
%MARE ($\%$) & 30.33 & 29.68 & 4.91 & 5.87 & 3.48 & 2.48 & $\cdots$ \\
\bottomrule
\end{tabular}
\end{table*}

\begin{table*}[t]
\centering
\caption{Predicted EOM-CCSD band gaps (in eV) obtained with the GTH-HF pseudopotential (PP) and the all-electron (AE) Hamiltonian, compared to previous literature results. PySCF results from Ref.~\citenum{vo.2024.10.1063/5.0187856} used extrapolation model A with \meshto{3}{4}, and VASP and FHI-aims results from Ref.~\citenum{moerman.2025.10.1103/PhysRevB.111.L121202} used the GW-EOM-23 and GW-EOM-234 models, respectively.
}
\label{tab:bandgap_comp}
\begin{tabular}{lSSSSSSSSSSSS}
\toprule
System 
& \multicolumn{1}{c}{PP} & \multicolumn{1}{c}{AE} & \multicolumn{1}{c}{PySCF~\cite{vo.2024.10.1063/5.0187856}} & \multicolumn{1}{c}{VASP~\cite{moerman.2025.10.1103/PhysRevB.111.L121202}} & \multicolumn{1}{c}{FHI-aims~\cite{moerman.2025.10.1103/PhysRevB.111.L121202}} \\
\midrule
MgO  &  9.22 &  9.33 &  8.34 &  9.52 & 9.19  \\
LiCl & 10.18 & 10.36 &  9.43 &  9.90         \\
LiF  & 15.93 & 16.03 & 15.43 & 16.19        \\
LiH  &  6.28 &  6.21 &  5.85 &  6.26 & 6.32  \\
BN   &  6.62 &  6.73 &  6.45 &  6.62         \\
BP   &  2.05 &  2.07 &  1.65 &  2.27 & 2.38  \\
Si   &  1.29 &  1.23 &  0.96 &  1.29         \\
C    &  5.55 &  5.63 &  4.88 &  5.75 & 6.15 \\
\bottomrule
\end{tabular}
\end{table*}

\clearpage

\bibliography{ref}

\end{document}