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\begin{document}

\title{Supplemental information for `` Interactions between single cellulose chain and water molecules and their temperature dependences: I. Pristine cellulose chain''}

%%=============================================================%%
%% GivenName	-> \fnm{Joergen W.}
%% Particle	-> \spfx{van der} -> surname prefix
%% FamilyName	-> \sur{Ploeg}
%% Suffix	-> \sfx{IV}
%% \author*[1,2]{\fnm{Joergen W.} \spfx{van der} \sur{Ploeg} 
%%  \sfx{IV}}\email{iauthor@gmail.com}
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\author*[1]{\fnm{Nobuhisa} \sur{Fujima}
\orcidlink{0000-0002-1303-3886}}
\email{fujima.nobuhisa@shizuoka.ac.jp}

%\author[2,3]{\fnm{Second} \sur{Author}}\email{iiauthor@gmail.com}
%\equalcont{These authors contributed equally to this work.}

%\author[1,2]{\fnm{Third} \sur{Author}}\email{iiiauthor@gmail.com}
%\equalcont{These authors contributed equally to this work.}

\affil*[1]{\orgdiv{Faculty of Engineering}, \orgname{Shizuoka University}, \city{Hamamatsu}, \postcode{432-8561},\country{Japan}}

%\affil[2]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{10587}, \state{State}, \country{Country}}}

%\affil[3]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{610101}, \state{State}, \country{Country}}}

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\maketitle

\section*{S1. Structure of a single bare cellulose chain}
We take the structure shown in Fig.S1(a) as the structure of the bare cellulose chain, in which the C6-OH group directs to C2-O atom. The total energy of this structure (Fig.S1(a)) is less by 0.35 eV than that in Fig.S1(b), where the C6-OH group does not direct to the C2-O atom, but the C2-OH group to the C6-O atom.
It is noted that the strongest binding energy of a water molecule is almost the same for the structures in Figs.S1 (a) and (b). The results in Ref.\citep{sc4:Fuk} have been obtained by using Fig.S1(b).

\newpage
\begin{figure*}[h]
% Use the relevant command to insert your figure file.
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\begin{center}
\includegraphics[width=0.45\textwidth]{fujima_FigS1.pdf}
% figure caption is below the figure
\end{center}
\caption*{{\bf Fig.S1} Two optimized structures of the single bare cellulose chain. The structure shown in (a) is less by 0.35 eV than that in (b).}
%\label{fig:S1}       % Give a unique label
\end{figure*}

%\newpage
\section*{S2. Partial coordinates of bare cellulose chain and those of thirty-two bound single water molecules}

The lattice vectors employed in the present study are listed in Table S1 as those components in the Cartesian coordinates. The lattice vectors correspond to $8\times 8\times 1$ of those in the cellulose I$_\beta$ unit cell experimentally obtained\citep{Ib:Nishi}.

\begin{table}[h]
\caption*{{\bf Table S1} Cartesian components (\AA) of Lattice vectors employed in the present study.}
%\begin{center}
\begin{tabular}{@{}crrr@{}}
\toprule
&\multicolumn{1}{c}{$x$} & \multicolumn{1}{c}{$y$} & \multicolumn{1}{c}{$z$} \\
\midrule
$\vec{a}$ &   62.27200 &    0.00000 &    0.00000 \\ 
$\vec{b}$ &   -7.42704 &   65.18626 &    0.00000 \\ 
$\vec{c}$ &    0.00000 &    0.00000 &   10.38000 \\ 
\botrule
\end{tabular}
%\end{center}
\end{table}


The partial coordinates of the optimized structure of the bare cellulose chain(in Fig.S1(a)) based on the lattice vectors in Table S1 are listed in Table S2, and those of thirty-two single water molecules shown in Fig.1 are in Table S3. 
The partial coordinates of the water-bound cellulose chain are of course slightly changed by binding a water molecule.
%These partial coordinates are based on the supercell employed in this study.
%The dimension of the supercell is listed in Table S2.

\begin{table}[h]
\caption*{{\bf Table S2} Partial coordinates of the optimized structures of the bare cellulose chain.}
%\begin{center}
\begin{tabular}{@{}rrrrrrrr@{}}
\toprule
 & \multicolumn{1}{c}{$a$} & \multicolumn{1}{c}{$b$} & \multicolumn{1}{c}{$c$} & & \multicolumn{1}{c}{$a$} & \multicolumn{1}{c}{$b$} & \multicolumn{1}{c}{$c$} \\
\midrule
C &    0.00180 &   -0.00544 &    0.03915 & H &    0.01959 &   -0.00209 &    0.05063 \\ 
 &   -0.00181 &    0.00545 &    0.53915 & &   -0.01960 &    0.00209 &    0.55062 \\ 
 &   -0.00377 &   -0.02272 &    0.94309 & &   -0.02159 &   -0.02570 &    0.93730 \\ 
 &    0.00377 &    0.02272 &    0.44308 & &    0.02158 &    0.02570 &    0.43730 \\ 
 &    0.00535 &   -0.01642 &    0.81156 & &   -0.02020 &    0.00154 &    0.75576 \\ 
 &   -0.00535 &    0.01643 &    0.31155 & &    0.02020 &   -0.00154 &    0.25576 \\ 
 &   -0.00249 &    0.00385 &    0.77098 & &    0.02324 &   -0.01391 &    0.81883 \\ 
 &    0.00249 &   -0.00385 &    0.27099 & &   -0.02324 &    0.01391 &    0.31881 \\ 
 &    0.00268 &    0.02047 &    0.87378 & &    0.02043 &    0.02385 &    0.88397 \\ 
 &   -0.00268 &   -0.02047 &    0.37380 & &   -0.02044 &   -0.02385 &    0.38399 \\ 
 &   -0.00698 &    0.04055 &    0.84467 & &   -0.02427 &    0.03675 &    0.81942 \\ 
 &    0.00698 &   -0.04055 &    0.34472 & &    0.02427 &   -0.03674 &    0.31949 \\ 
O &   -0.00123 &   -0.03298 &    0.72661 & &   -0.00621 &    0.04959 &    0.93445 \\ 
 &   -0.00812 &   -0.01133 &    0.15500 & &    0.00621 &   -0.04958 &    0.43453 \\ 
 &    0.00448 &    0.05292 &    0.75044 & &    0.00407 &   -0.05038 &    0.91955 \\ 
 &   -0.00448 &   -0.05292 &    0.25051 & &   -0.00407 &    0.05038 &    0.41951 \\ 
 &   -0.00675 &    0.01284 &    0.99417 & &    0.00366 &   -0.02886 &    0.63883 \\ 
 &    0.00675 &   -0.01284 &    0.49418 & &   -0.00366 &    0.02886 &    0.13882 \\ 
 &    0.00528 &   -0.04040 &    0.98978 & &    0.00112 &    0.04724 &    0.66403 \\ 
 &   -0.00528 &    0.04040 &    0.48974 & &   -0.00112 &   -0.04724 &    0.16410 \\ 
&    0.00528 &   -0.04040 &    0.98978 & & & & \\ 
&   -0.00528 &    0.04040 &    0.48974 & & & & \\ 
\botrule
\end{tabular}
%\end{center}
\end{table}

\newpage

\begin{table}[h]
\caption*{{\bf Table S3}  Partial coordinates of the bound water molecules at site ($i$).}
%\begin{center}
\begin{tabular}{@{}crrrrrrrrr@{}}
\toprule
&\multicolumn{3}{c}{O}&\multicolumn{3}{c}{H1}&\multicolumn{3}{c}{H2}\\
\midrule
$i$ & \multicolumn{1}{c}{$a$} & \multicolumn{1}{c}{$b$} & \multicolumn{1}{c}{$c$} & \multicolumn{1}{c}{$a$} & \multicolumn{1}{c}{$b$} & \multicolumn{1}{c}{$c$} & \multicolumn{1}{c}{$a$} & \multicolumn{1}{c}{$b$} & \multicolumn{1}{c}{$c$}\\
\midrule
 1 &   0.0083 &   0.0712 &   0.3249 &   0.0069 &   0.0595 &   0.2638 &   0.0231 &   0.0776 &   0.3179\\ 
 2 &  -0.0068 &  -0.0706 &   0.8253 &  -0.0065 &  -0.0591 &   0.7638 &  -0.0216 &  -0.0770 &   0.8292\\ 
 3 &  -0.0452 &  -0.0319 &   0.7113 &  -0.0299 &  -0.0348 &   0.7127 &  -0.0526 &  -0.0424 &   0.6556\\ 
 4 &   0.0454 &   0.0316 &   0.2107 &   0.0303 &   0.0350 &   0.2127 &   0.0533 &   0.0424 &   0.1600\\ 
 5 &  -0.0027 &   0.0790 &   0.5052 &   0.0122 &   0.0844 &   0.4867 &  -0.0028 &   0.0758 &   0.5976\\ 
 6 &  -0.0001 &  -0.0807 &   0.0132 &  -0.0143 &  -0.0883 &   0.0009 &   0.0004 &  -0.0769 &   0.1047\\ 
 7 &   0.0494 &   0.0498 &   0.7835 &   0.0342 &   0.0519 &   0.7653 &   0.0572 &   0.0635 &   0.7825\\ 
 8 &  -0.0498 &  -0.0497 &   0.2834 &  -0.0347 &  -0.0520 &   0.2660 &  -0.0578 &  -0.0634 &   0.2831\\ 
 9 &  -0.0025 &   0.0916 &   0.6365 &   0.0000 &   0.0806 &   0.6971 &   0.0080 &   0.1032 &   0.6593\\ 
10 &   0.0024 &  -0.0914 &   0.1380 &  -0.0004 &  -0.0802 &   0.1966 &  -0.0082 &  -0.1028 &   0.1609\\ 
11 &   0.0537 &   0.0107 &   0.6719 &   0.0382 &   0.0110 &   0.6572 &   0.0581 &   0.0232 &   0.7201\\ 
12 &   0.0515 &  -0.0394 &   0.0018 &   0.0356 &  -0.0414 &   0.0004 &   0.0556 &  -0.0493 &   0.0646\\ 
13 &  -0.0537 &  -0.0106 &   0.1747 &  -0.0383 &  -0.0113 &   0.1578 &  -0.0585 &  -0.0234 &   0.2187\\ 
14 &  -0.0518 &   0.0386 &   0.5061 &  -0.0360 &   0.0403 &   0.5065 &  -0.0561 &   0.0486 &   0.5672\\ 
15 &   0.0529 &  -0.0114 &   0.4732 &   0.0375 &  -0.0114 &   0.4919 &   0.0599 &   0.0009 &   0.5142\\ 
16 &  -0.0528 &   0.0113 &   0.9716 &  -0.0375 &   0.0112 &   0.9921 &  -0.0600 &  -0.0008 &   0.0143\\ 
17 &  -0.0056 &   0.0722 &   0.1174 &  -0.0032 &   0.0607 &   0.1742 &   0.0088 &   0.0776 &   0.0884\\ 
18 &   0.0055 &  -0.0723 &   0.6215 &   0.0028 &  -0.0605 &   0.6754 &  -0.0087 &  -0.0780 &   0.5913\\ 
19 &  -0.0562 &   0.0018 &   0.7520 &  -0.0645 &   0.0123 &   0.7177 &  -0.0551 &   0.0046 &   0.8440\\ 
20 &   0.0563 &  -0.0019 &   0.2522 &   0.0647 &  -0.0123 &   0.2181 &   0.0553 &  -0.0046 &   0.3443\\ 
21 &  -0.0575 &  -0.0229 &   0.3895 &  -0.0676 &  -0.0182 &   0.3276 &  -0.0582 &  -0.0376 &   0.3746\\ 
22 &   0.0578 &   0.0231 &   0.8898 &   0.0677 &   0.0183 &   0.8265 &   0.0582 &   0.0377 &   0.8732\\ 
23 &   0.0558 &   0.0002 &   0.0545 &   0.0643 &   0.0067 &   0.1251 &   0.0537 &  -0.0143 &   0.0762\\ 
24 &  -0.0557 &   0.0000 &   0.5553 &  -0.0645 &  -0.0065 &   0.6244 &  -0.0541 &   0.0146 &   0.5757\\ 
25 &  -0.0593 &  -0.0273 &   0.9388 &  -0.0663 &  -0.0155 &   0.9654 &  -0.0571 &  -0.0346 &   0.0187\\ 
26 &   0.0593 &   0.0273 &   0.4402 &   0.0663 &   0.0155 &   0.4669 &   0.0572 &   0.0346 &   0.5201\\ 
27 &   0.0603 &  -0.0152 &   0.8163 &   0.0752 &  -0.0102 &   0.8377 &   0.0592 &  -0.0131 &   0.7239\\ 
28 &  -0.0604 &   0.0151 &   0.3150 &  -0.0753 &   0.0101 &   0.3365 &  -0.0594 &   0.0129 &   0.2226\\ 
29 &   0.0613 &  -0.0374 &   0.3151 &   0.0671 &  -0.0412 &   0.3975 &   0.0576 &  -0.0503 &   0.2702\\ 
30 &  -0.0619 &   0.0372 &   0.8137 &  -0.0674 &   0.0411 &   0.8970 &  -0.0579 &   0.0502 &   0.7698\\ 
31 &  -0.0240 &   0.0816 &   0.9543 &  -0.0396 &   0.0817 &   0.9526 &  -0.0194 &   0.0837 &   0.8651\\ 
32 &   0.0238 &  -0.0820 &   0.4550 &   0.0395 &  -0.0819 &   0.4503 &   0.0189 &  -0.0830 &   0.3659\\ 
\botrule
\end{tabular}
%\end{center}
\end{table}

%\newpage
\clearpage
\section*{S3. Example of energy surface of a water molecule rotating around the binding axis to the cellulose chain}

Because the binding energy is not always sensitive to the direction of the unbound H atom in a bound water molecule, we re-calculate the binding energy by re-optimizing the structure of the cellulose chain and the water molecule with the angles of the unbound H atom rotating around the binding axis, 0, 45, 90, 135, 180, 225, 270, 315 degree as the initial structures.

Figure S2 shows an example of the energy surface of the water molecule rotating around the binding axis to the O atom of C6-O site (6) in Fig.1. 

The difference of the binding energy is at most 0.03 eV except for the position g (just above C6 atom).

\begin{figure}[h]
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use \centering
\begin{center}
\includegraphics[width=0.7\textwidth]{fujima_FigS2.pdf}
\end{center}
% figure caption is below the figure
\caption*{{\bf Fig.S2}  Energy surface of a water molecule at C6-O site (6) by rotating the direction of the unbound H-atom around the binding axis.}
\label{fig:S2}       % Give a unique label
\end{figure}

\section*{S4. Density of electronic states in cellulose chain by hybrid method}
For comparison with the results by GGA calculations shown in the subsection 2.3, we demonstrate the density of electronic states (DOS) in the single bare cellulose chain calculated by the hybrid method HSE06\citep{HSE06}. All electron atomic-bases with the TZ2P(triple $zeta$ with two polarization functions) Slater-type orbitals are employed for the DOS calculations\citep{BAND}. The calculations are performed under the 1-dimensional periodic boundary condition along the $c$-axis (10.31\AA). 


Figure S3 shows the same as Fig.7(a), the DOSs calculated by the GGA method. 

As shown in Fig.S3, the basic DOS profile is similar to that in Fig.7(a). Only the difference is that the bandgap (6.8 eV) by HSE06 in Fig.S4 is much larger than that (5.3 eV) by GGA. It is said in general that the GGA method underestimates bandgaps in semiconductors and insulators and the hybrid method well reproduces the bandgaps experimentally obtained. However, the GGA gives a bandgap better than the HSE06 for the cellulose case; the UV-absorption estimates the band gap at 4.5 eV \citep{Eg:Sima}. 
\begin{figure}[h]
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use \centering
\begin{center}
\includegraphics[width=0.69\textwidth]{fujima_FigS3.pdf}
\end{center}
% figure caption is below the figure
\caption*{{\bf Fig.S3}  Density of electronic states of the bare cellulose chain by the hybrid method(HSE06).}
\label{fig:DOSHSE06}       % Give a unique label
\end{figure}


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