\section{Introduction}
\vspace{-1mm}
\label{sec:Introd}
Recently, federated learning (FL) was introduced in \cite{mcmahan17AISTATS} as an important step to bring machine learning closer to everyone. The breakthrough idea of FL is ``no raw data sent to third party companies during learning processes'', which means people can safely participate in FL processes without being worried that their personal data is exploited.
A wide range of applications, such as healthcare and self-driving cars to name  a few \cite{chen20IS,niknam20CM}, can benefit from  FL.
In FL, the learning is implemented jointly by many users (UEs). First, a local learning model is trained at each UE using local (private) training data, and sent to the central server. A global update is then computed at the central server using the local learning models transmitted from all UEs, and finally sent back to the UEs for local training updates. This learning process is iterated  until reaching a certain learning accuracy level.
To deploy the above iterative FL process over wireless networks, a key challenge is keeping the network energy consumption as low as possible. This is important both due to  battery limitations  of the UEs and to   concerns about the ICT carbon footprint. It is thus critical to design an energy-efficient wireless network to support FL.

There are several studies of  energy-efficient deployments of FL over wireless networks; see, e.g.,   \cite{yang21TWC,zeng20ICC,hu20WCSP} and references therein. In these works, the authors proposed designs which  minimize the energy consumption at the UEs  while guaranteeing the learning performance (i.e., the test accuracy) by jointly optimizing learning and communication parameters. However, the energy consumption of the transmission from the central server to the UEs was not taken into account. Also, these works proposed to use frequency-division multiple access (FDMA) and time-division multiple access (TDMA) systems  to support FL. This might not be a good choice because FDMA and TDMA systems offer low UE data rates, and hence, yield a very high energy consumption, especially when the number of UEs is large. In addition, all these works only considered the case of a single FL group.

On the other hand, it is anticipated that the future wireless systems will need to serve multiple groups of UEs that  participate in  different FL processes.  These networks need to simultaneously provide high data rates and high communication reliability to all UEs in all FL groups. Designing such networks is challenging and calls for a suitable, new wireless communication framework. To the best of our knowledge, there has not been any work studying energy-efficient wireless networks supporting multiple FL groups in the existing literature.

The contributions of this paper are summarized as follows:
\vspace{-2mm}
\begin{itemize}[noitemsep,nolistsep]
    \item To support multiple FL groups over wireless networks, we  propose using massive MIMO (mMIMO) and letting multiple iterations (each for one FL process of a group) be executed in one large-scale coherence time\footnote{The large-scale coherence time is defined as the time interval during which  the large-scale fading coefficients remain approximately constant.}. Thanks to the high array gain and multiplexing gain, mMIMO can simultaneously offer very high quality of service to all UEs in an area of interest \cite{ngo16}, and hence, it is expected to guarantee a stable operation of each iteration (and hence the whole FL process).
    \item We introduce two specific transmission protocols where the steps within one FL iteration, i.e., the downlink transmission, the computation at the UEs, and the uplink transmission, are either asynchronous or synchronous. These schemes   differ from the scheme in \cite{vu20TWC} which focuses on minimizing the training time of FL. Here, we use the unicast protocol on  downlink and   conventional multiuser transmission on uplink.
    Both downlink and uplink use zero-forcing (ZF) processing.
    \item We develop an algorithm to allocate the transmit powers and processing frequencies to minimize the total energy consumption in each FL iteration, under a constraint on the total  time taken for one FL iteration.
    \item Numerical results show that our proposed schemes significantly reduce the energy consumption compared to baseline schemes. They also confirm that the asynchronous scheme outperforms the synchronous scheme for supporting multiple FL groups, at the cost of a higher complexity.
\end{itemize}

\vspace{-2mm}
\section{Proposed Schemes and System Model}
\label{sec:SystModel}
\vspace{-1mm}
\subsection{Multiple Federated Learning Framework} \label{sec:FLframework}
\vspace{-1mm}
We consider a multiple FL network which includes multiple FL groups with different learning purposes.
Each UE is assumed to only participate in one FL group. The FL frameworks of all groups can be different in terms of loss functions but have the same following four steps in each iteration \cite{tran19INFOCOM,mcmahan17AISTATS}.
\begin{enumerate}[label={(S\arabic*)}]
\item A central server sends a global update to the UEs.
\item Each UE updates and solves its local learning problem using its local data and then computes its local update.
\item Each UE sends its computed local update to the central server.
\item The central server computes the global update by aggregating the received local updates from all UEs.
\end{enumerate}
The above process will be done iteratively until a certain learning accuracy level is achieved.

\vspace{-2mm}
\subsection{Proposed Schemes to Serve Multiple FL Groups}
\vspace{-1mm}
To support multiple FLs  discussed in Section~\ref{sec:FLframework}, we propose to use mMIMO technology, i.e. Steps (S1) and (S3) of each FL iteration can be executed via the downlink and the uplink of a mMIMO system, respectively. Our proposed mMIMO-based multiple-FL system includes one $M$-antenna base station (BS) simultaneously serving $N$ FL groups in the same frequency bands under the time-division-duplexing operation. We assume that the BS acts as the central server. Each FL iteration of each FL group is assumed to be executed within a large-scale coherence time, which is reasonable because the execution time of one FL iteration is smaller than the large-scale coherence time in many scenarios \cite{vu20TWC}.
With this assumption, we then propose two specific transmission schemes to support the learning of $N$ FL groups for each FL iteration as shown in Figs.~\ref{fig:time1}(a) and~(b) respectively.
\vspace{-1mm}
\begin{itemize}
    \item[(a)]\textbf{Asynchronous scheme}: All groups start their FL iterations at the same time when the BS switches to a downlink mode. During this mode, BS simultaneously sends the global updates to all UEs in all groups (corresponding to Step~(S1)). Each UE will start its local computation if it successfully receives the global training update (corresponding to Step~(S2)). Then, the BS switches to an uplink mode immediately after the receptions of the global training update are completed at all the UEs. During this mode, the UEs will send their computed local updates to the BS (corresponding to Step~(S3)) if they finish the local computation.
    \item[(b)]\textbf{Synchronous scheme}: This scheme is similar to the asynchronous scheme except for the synchronization of Steps (S1)-(S3) among all the UEs.
    Each UE starts and waits for others to end each of those steps together.
\end{itemize}
The time of one FL iteration under both schemes are constrained by a given period of time.
Note that in the asynchronous scheme, the time of Steps (S1)--(S3) are optimally allocated (using the the proposed algorithm in the next section) to ensure that all the UEs finish one FL iteration and start a new FL iteration at the same time.

\vspace{-2mm}
\subsection{Massive-MIMO-based Multiple-FL System Model}
\vspace{-1mm}
\begin{figure}[t!]
\centering
\includegraphics[width=0.42\textwidth]{FLiterationGLOBECOM.eps}
\vspace{-3mm}
\caption{Illustration of FL iterations over the considered mMIMO network with two groups $n,n'$ and two UEs for each group}
\label{fig:time1}
\end{figure}

The above two schemes share the common system model as follows.
In each large-scale coherence time, the global and local updates in Steps (S1) and (S3) are transmitted in one or multiple small-scale coherence times depending on their sizes.
Each coherence block in Step (S1) (or (S3)) involves the channel estimation phase and the downlink (or uplink) payload data phase.
Suppose that at the considered time, there are $N$ iterations of $N$ FL groups being served.
Let
$\NN\triangleq\{1,\dots,N\}$, and $\K_n$ be the sets of
groups and the indices of the UEs in group $n$, respectively. There are $K_n$ single-antenna users (UEs) of each group $n$.
The details of each step are presented in the following.

\subsubsection{Step (S1)}
The BS sends the global updates to all UEs of all groups. Since the global updates intended for all UEs in a given group are the same, the transmission in this step corresponds to multi-group multicasting. Thus, we follow the scheme in \cite{sadeghi18TWC} assuming orthogonal pilots and ZF processing.

\textbf{Uplink channel estimation}:
For each coherence block of length $\tau_c$, each UE sends its pilot of length $\tau_{d,p}$ to the BS \cite{sadeghi18TWC}.
We assume that the pilots of all the UEs are pairwisely orthogonal, which requires $\tau_{d,p}\geq K_{total}\triangleq \sum_{n\in\NN}K_n$.
Denote by $\g_{n_k} \!=\! (\beta_{n_k})^{1/2}\tilde{\g}_{n_k}$ the channel vector from UE $k$ of group $n$ to the BS, where $\beta_{n_k}$ and $\tilde{\g}_{n_k}$
are the large-scale fading coefficient and small-scale fading coefficient vector, respectively.
At the BS, $\g_{n_k}$ is estimated by using the received pilots and the minimum mean-square error (MMSE) estimation technique. The MMSE estimate $\hat{\g}_{n_k}$ of $\g_{n_k}$ is
distributed according to $\CN(\pmb{0},\hat{\sigma}_{n_k}^2\pmb{I}_M)$, where $\hat{\sigma}_{n_k}^2 = \frac{\tau_{d,p} \rho_{p} \beta_{n_k}^2 }{ \tau_{d,p} \rho_{p} \beta_{n_k} +1 }$ \cite{sadeghi18TWC}.
We also denote by $\hat{\G}\triangleq [\hat{\G}_1,\dots,\hat{\G}_N]$ the matrix stacking the channels of all the UEs, where $\hat{\G}_n\triangleq [\hat{\g}_{n_1},\dots,\hat{\g}_{n_{K_n}}]$.

\textbf{Downlink payload data transmission}:
The BS encodes the global training update intended for  UE $k$ of group $n$ into the same symbol $s_{d,n}$, where $\EEE\{|s_{d,n}|^2\}=1$, and
apply the ZF precoding vector
$\uu_{n_k}\! =\! \sqrt{\eta_{n_k}\hat{\sigma}_{n_k}^2(M\!-\!K_{total})} \hat{\G}(\hat{\G}^H\hat{\G})^{-1}\e_{n_k,K_{total}}
$ to precode the symbol,
where $\eta_{n_k}$ is a power control coefficient,
$\e_{n_k,K_{total}}$ is the $n_k$-th column of $\pmb{I}_{K_{total}}$, and $M \geq K_{total}$ is required.
The transmitted signal at the BS is thus given as $\x_{d}\!\!=\!\! \sqrt{\rho_{d}}\sum_{n'\in\NN}\sum_{\ell\in \K_{n'}}\uu_{n'_\ell} s_{d,n'}$, where $\rho_{d}$ is the maximum normalized transmit power at the BS. The transmitted power at the BS is required to meet the average normalized power constraint, i.e., $\EEE\{|x_{d}|^2\}\leq \rho_d$, which can be expressed through the following constraint:
\begin{align}
\label{powerdupperbound}
\sum_{n\in\NN}\sum_{k\in\K_n}\eta_{n_k} \leq 1.
\end{align}
The achievable rate $R_{d,n_k}(\ETA)$ at UE $k$ of group $n$
is given
as \cite[(10)]{sadeghi18TWC}
\vspace{-0mm}
\begin{align}
\label{RdZF}
R_{d,n_k}(\ETA) &= \frac{\tau_c - \tau_{d,p}}{\tau_c}B\log_2 \big( 1 + \text{SINR}_{d,n_k}(\ETA)\big),
\end{align}
\vspace{-3mm}

\noindent
where $\ETA\triangleq\{\eta_{n_k}\}_{n\in\NN,k\in\K_n}$, $B$ is the bandwidth,
and $\text{SINR}_{d,n_k}(\ETA) =
\frac{(M-K_{total})\rho_d \hat{\sigma}_{n_k}^2\eta_{n_k}}
{\rho_d (\beta_{n_k} - \hat{\sigma}_{n_k}^2) \sum_{n'\in\NN}\sum_{n'_\ell\in\K_{n'}} \eta_{n'_\ell} +1}
$ is the effective downlink SINR\footnote{Although all the UEs of one group have the same encoded symbol, their achievable rates can be different (and hence, their transmissions do not finish simultaneously). This is feasible when using a code that sends a maximum number of parity bits corresponding to the UEs with the smallest SINR. Here, each UE will stop listening as soon as it successfully decodes its message. Thus, the UEs with higher SINRs can stop listening earlier than those with smaller SINRs.}.

\textbf{Downlink delay}:
Let $S_{d,n}$ (bits) be the data size of the global training update of group $n$.
The transmission time from the BS to UE $k$ of group $n$ is given by
\begin{align*}\label{}
t_{d,n_k}(\ETA) = \frac{S_{d,n}}{R_{d,n_k}(\ETA)}.
\end{align*}

\textbf{Energy consumption for the downlink transmission}:
Denote by $N_0$ is the noise power. The energy consumption  for transmitting the global update to the UE $k$ of group $n$ is the product of the transmit power $\rho_d N_0 \eta_{n_k}$ and the delay for the downlink transmission to this UE. Therefore, the total energy consumption at the BS for all groups is
\vspace{-0mm}
\begin{align}
\nonumber
\!E_{d}(\ETA)
\!\!=\!\!\! \sum_{n\in\NN}\!\sum_{k\in\K_n}\!\! \rho_d N_0 \eta_{n_k} \!
t_{d,n_k}(\ETA)
\!\!=\!\!\! \sum_{n\in\NN}\!\sum_{k\in\K_n}\!\! \rho_d N_0 \eta_{n_k}\!
\frac{S_{d,n}}{R_{d,n_k}\!(\ETA)}.
\end{align}

\vspace{-0mm}
\subsubsection{Step (S2)} After receiving the global update, each UE executes $L$ local computing rounds over its data set to compute its local update.

\textbf{Local computation}:
Let $c_{n_k}$ (cycles/sample) be the number of processing cycles for a UE $k$ to process one data sample \cite{tran19INFOCOM}. Denote by $D_{n}$ (samples) and $f_{n_k}$ (cycles/s) the size of the local data set and the processing frequency of the UE $k$ of group $n$, respectively.
The computation time at UE $k$ of group $n$ is then given by \cite{vu20TWC,tran19INFOCOM}
\vspace{-0mm}
\begin{align*}\label{}
t_{C,n_k}(f_{n_k}) = \frac{LD_nc_{n_k}}{f_{n_k}}.
\end{align*}

\textbf{Energy consumption for local computation at the UEs}:
The energy consumption at UE $k$ of group $n$ for computing its local training update is given as \cite{tran19INFOCOM,vu20TWC}
\begin{align*}\label{}
E_{C,n_k}(f_{n_k}) = L\frac{\alpha}{2}c_{n_k}D_nf_{n_k}^2,
\end{align*}
where $\frac{\alpha}{2}$ is the effective capacitance coefficient of the UEs' computing chipset.

\vspace{-1mm}
\subsubsection{Step (S3)}
In this step, UEs' local updates are transmitted to the BS.

\textbf{Uplink channel estimation}:
In each coherence block, each UE sends its pilot of length $\tau_{u,p}$  to the BS. We assume that the pilots of all the UEs are pairwisely orthogonal, which requires the pilots of length
$\tau_{u,p} \geq K_{total}$.
The MMSE estimate $\bar{\g}_{n_k}$ of $\g_{n_k}$ is distributed according to $\CN(\pmb{0},\bar{\sigma}_{n_k}^2\pmb{I}_M)$, where
$\bar{\sigma}_{n_k}^2 = \frac{\tau_{u,p} \rho_{p} \beta_{n_k}^2}{\tau_{u,p} \rho_{p}\beta_{n_k}+1}$ \cite{sadeghi18TWC}.

\textbf{Uplink payload data transmission}:
After computing the local update, UE $k$ of group $n$ encodes this update into symbols denoted by $s_{u,n_k}$, where $\EEE\{|s_{u,n_k}|^2\}=1$, and sends baseband signal $x_{u,{n_k}}\!=\!\sqrt{\rho_{u}\zeta_{n_k}}s_{u,{n_k}}$ to the BS, where $\rho_{u}$ is the maximum normalized transmit power at each UE and $\zeta_{n_k}$ is a power control coefficient.
This signal is subjected to the average transmit power constraint, i.e., $\EEE\left\{|x_{u,n_k}|^2\right\}\leq \rho_u$, which is can be expressed in a per-UE constraint as
\begin{align}\label{poweruupperbound}
\zeta_{n_k}\leq 1,\forall n\in\NN,n_k\in \K_n.
\end{align}

After receiving data from all UEs, the BS uses the estimate channels and ZF scheme to detect the UEs' message symbols. The ZF precoder requires $M \geq K_{total}$. The achievable rate
(bps) of UE $k$ in group $n$ is given by \cite[(3.28)]{ngo16}
\begin{align}
\label{RuZF}
R_{u,n_k}(\ZETA)
&= \frac{\tau_c-\tau_{u,p}}{\tau_c}B \log_2 \big( 1 +
\text{SINR}_{u,n_k}(\ZETA)
\big),
\end{align}
\vspace{-2mm}

\noindent
where
$\text{SINR}_{u,n_k} (\ZETA) \triangleq
\frac{(M-K_{total}) \rho_u \bar{\sigma}_{n_k}^2 \zeta_{n_k}} {\rho_u \sum_{n' \in \NN} \sum_{n'_\ell \in \K_{n'}}  (\beta_{n'_\ell} - \bar{\sigma}_{n'_\ell}^2) \zeta_{n'_\ell} + 1}
$ is the effective uplink SINR.

\textbf{Uplink delay}:
Denote by $S_{u,n}$ (bits) the data size of the local training update of group $n$.
The transmission time from UE $k$ of group $n$ to the BS is given by
\begin{align*}\label{}
t_{u,n_k}(\ZETA) = \frac{S_{u,n}}{R_{u,n_k}(\ZETA)}.
\end{align*}

\textbf{Energy consumption for the uplink transmission}:
The energy consumption for the uplink transmission at a UE is the product of the uplink power and the transmission time. In particular, the energy consumption at UE $k$ of group $n$
is given as \cite{tran19INFOCOM,vu20TWC}
\vspace{-1mm}
\begin{align*}
E_{u,n_k}(\ZETA) &= \rho_u N_0\zeta_{n_k} t_{u,n_k}(\ZETA) = \frac{\rho_u N_0\zeta_{n_k}S_{u,n}} {R_{u,n_k}(\ZETA)}.
\end{align*}

\vspace{-1mm}
\begin{remark}
We obtain the achievable downlink and uplink rates in \eqref{RdZF} and \eqref{RuZF}, respectively, under the case that all users participate in the transmission. However, as shown from the two proposed schemes in Fig.~\ref{fig:time1}, at a particular time, some UEs may have finished their transmission, and thus, do not participate in the downlink or uplink transmission with other UEs at the same time. This will not cause any issue with our design because the rates \eqref{RdZF} and \eqref{RuZF} are still always achievable under this case.
\end{remark}

\subsubsection{Step (S4)}
After receiving all the local updates, the BS computes its global update. Since the computational capability of the central server is much more powerful than those of the UEs, the delay of computing the global update is negligible.

\vspace{-1mm}
\section{Problem Formulation and Solution}
\label{sec:PF}
\vspace{-1mm}
In practice, different groups are likely to start their FL processes at different times and have different number of FL iterations depending on their learning targets. Therefore, minimizing the energy consumption of the whole FL processes of all groups at the same time is tremendously difficult due to complicated synchronization among all groups. Instead, we aim at minimizing the total energy consumption in one FL iteration for all groups, which also leads to the total energy consumption reduction of the whole FL processes of all groups.

\vspace{-2mm}
\subsection{Asynchronous Scheme}
\vspace{-1mm}

The problem of minimizing the total energy consumption of one FL iteration for all groups is formulated as follows.
\begin{subequations}\label{Pmain}
\begin{align}
\label{CFPmulti}
\!\!\!\!\!\underset{\ETA,\f,\ZETA}{\min}
& E_{total} \triangleq\! E_{d}(\ETA)\! +\!\!\! \sum_{n\in\NN}\!\sum_{n_k\in\K_n} (E_{C,n_k}(f_{n_k}) \!+\! E_{u,n_k}(\ZETA))
\\
\nonumber
\!\!\!\!\!\mathrm{s.t.}\,\,
&
\eqref{powerdupperbound}, \eqref{poweruupperbound}
\\
\label{powerlowerbound}
& 0\leq \eta_{n}, 0\leq \zeta_{n_k}, \forall n,n_k
\\
\label{fbound}
& 0 \leq f_{n_k} \leq f_{\max}, \forall n,n_k
\\
\label{QoSbound}
& t_{d,n_k}(\ETA) + t_{C,n_k}(f_{n_k}) + t_{u,n_k}(\ZETA)
 \leq t_{\text{QoS}}, \forall n,n_k
\\
\label{syncbound}
& \max_{n\in\NN}\max_{n_k\in\K_n}\!\! t_{d,n_k}
\!\leq\! \min_{n\in\NN}\min_{n_k\in\K_n} \big(t_{d,n_k} \!+\! t_{C,n_k}\big),
\end{align}
\end{subequations}
\noindent
where $\f\triangleq\{f_{n_k}\}_{n\in\NN,n_k\in\K_n} $. Here, \eqref{QoSbound} guarantees the execution time of one FL iteration below a threshold $t_{\text{QoS}}$ for maintaining the quality of service, and
\eqref{syncbound} is introduced to ensure that all the UEs send their local update during the uplink mode of the BS. The right-hand side of \eqref{syncbound} models the first UE that finishes its downlink transmission and local computation, while the left-hand side presents the slowest UE finishes its downlink transmission as seen in Fig.~1(a).

To solve \eqref{Pmain}, we rewrite it in the following more tractable epigraph form
\begin{subequations}\label{Pmainepi}
\begin{align}
\label{CF:shortP:epi}
\nonumber
\!\!\!\!\!\!\underset{\x}{\min} \,\,
& \widetilde{E}_{total} \triangleq \sum_{n\in\NN}\sum_{n_k\in\K_n} \rho_d N_0 S_{d,n} \omega_{n_k}
\\
& + \sum_{n\in\NN}\!\sum_{n_k\in\K_n} \!\!\!\Big( L\frac{\alpha}{2}c_{n_k}D_nf_{n_k}^2 \!\!+\!\! \rho_u N_0\theta_{n_k}S_{u,n} \Big)
\\
\mathrm{s.t.}\,\,
\nonumber
&
\eqref{powerdupperbound}, \eqref{poweruupperbound}, \eqref{powerlowerbound}, \eqref{fbound}
\\
\label{Rdlowerbound}
& r_{d,n_k}\leq R_{d,n_k} (\ETA), \forall n,n_k
\\
\label{Rulowerbound}
& r_{u,n_k}\leq R_{u,n_k} (\ZETA), \forall n,n_k
\\
\label{rlowerbound}
&  0 \leq r_{d,n_k}, 0 \leq r_{u,n_k}, \forall  n,n_k
\\
\label{tdgroupbound}
&
\eta_{n_k}\leq r_{d,n_k} \omega_{n_k}, \forall n, n_k
\\
\label{tubound}
&
\zeta_{n_k}\leq r_{u,n_k}  \theta_{n_k}, \forall n,n_k
\\
\label{tbound}
& \frac{S_{d,n}}{r_{d,n_k}} + \frac{LD_nc_{n_k}}{f_{n_k}} + \frac{S_{u,n}}{r_{u,n_k}} \leq t_{\text{QoS}}, \forall n, n_k
\\
\label{syncbound2a}
&
S_{d,n}\leq r_{d,n_k} q, \forall n,n_k
\\
\label{syncbound2b}
& q \leq q_{1,n_k} + q_{2,n_k}, \forall n,n_k
\\
\label{syncbound2c}
& 0 \leq q_{1,n_k}, 0 \leq q_{2,n_k}, \forall n,n_k
\\
\label{syncbound2d}
&
q_{1,n_k} r_{d,n_k} \leq S_{d,n} , \forall n,n_k
\\
\label{syncbound2e}
& q_{2,n_k} f_{n_k} \leq LD_nc_{n_k} , \forall n,n_k,
\end{align}
\end{subequations}
where $\x \triangleq \{\ETA,\f,\ZETA,\rr_d,\rr_u,\OOmega,\THeta, q, \q_1, \q_2\}$, $\rr_d,\rr_u,\OOmega,\THeta,q,\q_1,\\ \q_2$ are additional variables,  $\rr_d=\{r_{d,n_k}\}$, $\rr_u=\{r_{u,n_k}\}$, $\OOmega=\{\omega_{n_k}\}$, $\THeta=\{\theta_{n_k}\}$,
$\q_1=\{q_{1,n_k}\}, \q_2=\{\q_{2,n_k}\}, \forall n\in\NN,n_k\in\K_n$. Here,
\eqref{syncbound2a}--\eqref{syncbound2e} come from \eqref{syncbound}.
If we let
$\vv \triangleq \{v_{n_k}\}$ and $\uu\triangleq \{u_{n_k}\}, \forall n\in\NN,n_k\in\K_n,$ with $v_{n_k}\triangleq \eta_{n_k}^{1/2}, \,\, u_{n_k} \triangleq \zeta_{n_k}^{1/2}, \forall n,n_k,$
then problem \eqref{Pmainepi} will be equivalent to
\begin{subequations}\label{Pmainepiequi}
\begin{align}
\label{CF:shortP:epi}
\!\!\!\!\!\!\underset{\widetilde{\x}}{\min} \,\,
& \widetilde{E}_{total}
\\
\mathrm{s.t.}\,\,
\nonumber
&
\eqref{fbound}, \eqref{rlowerbound}, \eqref{tbound}-\eqref{syncbound2c}
\\
\label{Rdlowerbound2}
& r_{d,n_k}\leq R_{d,n_k} (\vv), \forall n,n_k
\\
\label{Rulowerbound2}
& r_{u,n_k}\leq R_{u,n_k} (\uu), \forall n,n_k
\\
\label{tdgroupbound2}
& v_{n_k}^2 - r_{d,n_k}\omega_{n_k} \leq 0, \forall n, n_k
\\
\label{tubound2}
& u_{n_k}^2 - r_{u,n_k} \theta_{n_k} \leq 0, \forall n,n_k
\\
\label{powerdupperbound2}
& \sum_{n\in\NN} \sum_{k\in\K_n} v_{n_k}^2 \leq 1
\\
\label{poweruupperbound2}
& u_{n_k}^2\leq 1, \forall n,n_k
\\
\label{powerlowerbound2}
& 0\leq v_{n_k}, 0\leq u_{n_k},\forall n,n_k,
\\
\label{syncbound2e2}
& q_{1,n_k} r_{d,n_k} - S_{d,n}\leq 0, \forall n,n_k
\\
\label{syncbound2f2}
& q_{2,n_k} f_{n_k} - LD_{n}c_{n_k}\leq 0, \forall n,n_k,
\end{align}
\end{subequations}
where $\widetilde{\x}\triangleq\{\x,\vv,\uu\}\setminus\{\ETA,\ZETA\}$. Here, \eqref{tdgroupbound2} and \eqref{tubound2} follow from \eqref{tdgroupbound} and \eqref{tubound}, while \eqref{powerdupperbound2}--\eqref{poweruupperbound2} follow from \eqref{powerdupperbound}, \eqref{poweruupperbound}, and \eqref{powerlowerbound}. Problem \eqref{Pmainepiequi} is still difficult to solve due to nonconvex constraints \eqref{Rdlowerbound2}, \eqref{Rulowerbound2}, \eqref{tdgroupbound2}, \eqref{tubound2}, \eqref{syncbound2e2}, and \eqref{syncbound2f2}.

To deal with these constraints, we first observe that the rates $R_{d,n_k}(\vv)$ and $R_{u,n_k}(\uu)$ of nonconvex constraints \eqref{Rdlowerbound2} and \eqref{Rulowerbound2} have the following concave lower bounds \cite[(20)]{nguyen17TCOM}:
\begin{align}\label{Rdconcave}
\nonumber
&\widetilde{R}_{d,n_k}(\vv) \triangleq
\frac{\tau_c-\tau_{cp}}{\tau_c\log 2}B\Big[\log \Big(1 + \frac{(\Upsilon_{n_k}^{(i)})^2}  {\Pi_{n_k}^{(i)}} \Big)
-\frac{(\Upsilon_{n_k}^{(i)})^2}{\Pi_{n_k}^{(i)}}
\\
&+2\frac{\Upsilon_{n_k}^{(i)}\Upsilon_{n_k}}{\Pi_{n_k}^{(i)}}
-\frac{(\Upsilon_{n_k}^{(i)})^2(\Upsilon_{n_k}^2+\Pi_{n_k})}{\Pi_{n_k}^{(i)}((\Upsilon_{n_k}^{(i)})^2+\Pi_{n_k}^{(i)})}\Big]
\leq R_{d,n_k}(\vv),
\\
\nonumber
&\widetilde{R}_{u,n_k}(\uu) \triangleq
\frac{\tau_c-\tau_{dp}}{\tau_c\log 2} B \Big[ \log\Big(1+\frac{(\Psi_{n_k}^{(i)})^2}{\Xi_{n_k}^{(i)}}\Big)
-\frac{(\Psi_{n_k}^{(i)})^2}{\Xi_{n_k}^{(i)}}
\\
&+ 2\frac{\Psi_{n_k}^{(i)}\Psi_{n_k}}{\Xi_{n_k}^{(i)}}
- \frac{(\Psi_{n_k}^{(i)})^2(\Psi_{n_k}^2+\Xi_{n_k})}{\Xi_{n_k}^{(i)}((\Psi_{n_k}^{(i)})^2+\Xi_{n_k}^{(i)})} \Big]
\leq R_{u,n_k}(\uu),
\end{align}
\noindent
where $\Pi_{n_k}(\vv) = \rho_d (\beta_{n_k} - \hat{\sigma}_{n_k}^2) \sum_{n'\in\NN}\sum_{n'_\ell\in\K_{n'}} v_{n'_\ell}^2 +1$,
$\Upsilon_{n_k} (v_{n_k}) = \sqrt{(M-K_{total})\rho_d} \hat{\sigma}_{n_k}v_{n_k}$,
$\Xi_{n_k}(\uu) = \rho_u \sum_{n' \in \NN} \sum_{n'_\ell \in \K_{n'}}  (\beta_{n'_\ell} - \bar{\sigma}_{n'_\ell}^2) u_{n'_\ell}^2 + 1$, and
$\Psi_{n_k} = \sqrt{(M-K_{total}) \rho_u} \bar{\sigma}_{n_k} u_{n_k}$.
Next, the functions in the left-hand sides of constraints \eqref{tdgroupbound2}, \eqref{tubound2}, \eqref{syncbound2e2}, and \eqref{syncbound2f2} have the following convex upper bounds \cite{vu20TWC}:
\vspace{-0mm}
\begin{align}
\nonumber
&\!\!\!\!v_{n_k}^2\! -\! r_{d,n_k}\omega_{n_k} \!\leq\! h_{1,n_k}(v_{n_k},r_{d,n_k},\omega_{n_k})
\triangleq 0.25
\big[4v_{n_k}^2 \!+\!(r_{d,n_k} \!-
\\
&\!\!\!\! \omega_{n_k})^2 - 2(r_{d,n_k}^{(i)}\!+\!\omega_{n_k}^{(i)})(r_{d,n_k}\! +\! \omega_{n_k})\! + \!(r_{d,n_k}^{(i)}\!+\!\omega_{n_k}^{(i)})^2\big],
\\
\nonumber
&\!\!\!\!u_{n_k}^2 \!-\! r_{u,n_k}\theta_{n_k} \!\leq\! h_{2,n_k}(u_{n_k},r_{u,n_k},\theta_{n_k})\triangleq 0.25
\big[4u_{n_k}^2 \!+\! (r_{u,n_k}\!-
\\
&\!\!\!\! \theta_{n_k})^2 - 2(r_{u,n_k}^{(i)}\!+\!\theta_{n_k}^{(i)})(r_{u,n_k} \!+\! \theta_{n_k}) \!+\! (r_{u,n_k}^{(i)}\!+\!\theta_{n_k}^{(i)})^2\big]
\\
\nonumber
&\!\!\!\! q_{1,n_k} r_{d,n_k} \!-\! S_{d,n} \!\leq\!
h_{3,n_k}(q_{1,n_k},r_{d,n_k})\triangleq 0.25
\big[  (q_{1,n_k}\!+ r_{d,n_k})^2
\\
&\!\!\!\! \!\! -\! 2(q_{1,n_k}^{(i)}\!\!-\!r_{d,n_k}^{(i)}) (q_{1,n_k} \!\!-\! r_{d,n_k}) \!+\! (q_{1,n_k}^{(i)}\!\!-\!r_{d,n_k}^{(i)})^2 \!-\! 4S_{d,n}\big]
\\
\nonumber
&\!\!\!\! q_{2,n_k} f_{n_k} \!-\! LD_{n}c_{n_k} \!\leq\!
h_{4,n_k}(q_{2,n_k},f_{n_k})\triangleq  0.25
\big[  (q_{2,n_k}\!+ f_{n_k})^2
\\
&\!\!\!\! \!\! -\! 2(q_{2,n_k}^{(i)}\!\!-\!f_{n_k}^{(i)}) (q_{2,n_k} \!\!-\! f_{n_k}) \!+\! (q_{2,n_k}^{(i)}\!\!-\!\!f_{n_k}^{(i)})^2 \!-\! 4LD_{n}c_{n_k}\big].
\end{align}

As such, constraints \eqref{Rdlowerbound2}, \eqref{Rulowerbound2}, \eqref{tdgroupbound2}, \eqref{tubound2}, \eqref{syncbound2e2}, and \eqref{syncbound2f2} can now be approximated respectively by the following convex constraints
\vspace{-2mm}
\begin{align}
\label{Rdlowerboundapprox}
&r_{d,n_k}  \leq \widetilde{R}_{d,n_k}(\vv), \forall n,n_k
\\
\label{Rulowerboundapprox}
&r_{u,n_k}  \leq \widetilde{R}_{u,n_k}(\uu), \forall n,n_k
\\
\label{tubounddapprox}
&h_{1,n_k}(v_{n_k},r_{d,n_k},\omega_{n_k})  \leq 0, \forall n,n_k
\\
\label{tuboundd2approx}
&h_{2,n_k}(u_{n_k},r_{u,n_k},\theta_{n_k})  \leq 0, \forall n,n_k
\\
\label{syncbound2eapprox}
&h_3(q_{1,n_k},r_{d,n_k}) \leq 0, \forall n,n_k
\\
\label{syncbound2fapprox}
&h_4(q_{2,n_k},f_{n_k}) \leq 0, \forall n,n_k.
\end{align}
\vspace{-4mm}

\noindent
At iteration $(i+1)$, for a given point $\widetilde{\x}^{(i)}$, problem \eqref{Pmainepi} can finally be approximated by the following convex problem:
\begin{align}\label{Pmainepiapprox}
\underset{\widetilde{\x}\in\widetilde{\FF}}{\min} \,\,
& \widetilde{E}_{total},
\end{align}
where $\widetilde{\FF}\triangleq\!\{
\eqref{fbound},
\eqref{rlowerbound},\eqref{tbound}-\eqref{syncbound2c},
\eqref{powerdupperbound2}-\eqref{powerlowerbound2},
\eqref{Rdlowerboundapprox}-\eqref{syncbound2fapprox}
\}$ is a convex feasible set.

In Algorithm~\ref{alg}, we outline the main steps to solve problem \eqref{Pmain}.
Let $\FF\triangleq\{
\eqref{powerdupperbound}, \eqref{poweruupperbound},
\eqref{powerlowerbound} -
\eqref{syncbound}\}$ be the feasible set of \eqref{Pmain}.
Starting from a random point $\widetilde{\x}\in\FF$, we solve \eqref{Pmainepiapprox} to obtain its optimal solution $\widetilde{\x}^*$, and use $\widetilde{\x}^*$ as an initial point in the next iteration. The algorithm terminates when an accuracy level of $\varepsilon$ is reached. In the case when $\widetilde{\FF}$ satisfies Slater's constraint qualification condition, Alg.~\ref{alg} will converges to a Karush-Kuhn-Tucker solution of \eqref{Pmainepi} (hence \eqref{Pmain}) \cite[Theorem 1]{Marks78OR}. In contrast, Alg.~\ref{alg} will converges to a Fritz John solution of \eqref{Pmainepi} (hence \eqref{Pmain}).

\begin{algorithm}[!t]
\caption{Solving problem \eqref{Pmain}}
\begin{algorithmic}[1]\label{alg}
\STATE \textbf{Initialize}: Set $i\!=\!0$ and choose a random point $\widetilde{\x}^{(0)}\!\in\!\FF$.
\REPEAT
\STATE Update $i=i+1$
\STATE Solving \eqref{Pmainepiapprox} to obtain its optimal solution $\widetilde{\x}^*$
\STATE Update $\widetilde{\x}^{(i)}=\widetilde{\x}^*$
\UNTIL{convergence}
\end{algorithmic}
\vspace{+0mm}
\textbf{Output}: $(\ETA^*,\ZETA^*,\f^*)$
\end{algorithm}

\vspace{-3mm}
\subsection{Synchronous Scheme}
\vspace{-2mm}
The optimization problem of this scheme is formulated as
\begin{subequations}\label{Pmainsyn}
\begin{align}
\label{CFPmulti}
\!\!\!\!\!\underset{\ETA,\f,\ZETA}{\min} \,\,
& E_{d}(\ETA) + \!\!\sum_{n\in\K_n}\!\sum_{k\in\K_n} \!\!(E_{C,n_k}(f_{n_k})\! +\! E_{u,n_k}(\ZETA))
\\
\nonumber
\mathrm{s.t.}\,\,
&
\eqref{powerdupperbound}, \eqref{poweruupperbound}, \eqref{powerlowerbound}, \eqref{fbound}
\\
\nonumber
\label{QoSboundsyn}
 &\max_{n\in\NN}\max_{n_k\in\K_n}t_{d,n_k}(\ETA) + \max_{n\in\NN}\max_{n_k\in\K_n}t_{C,n_k}(\f)
\\
&\qquad+ \max_{n\in\NN}\max_{n_k\in\K_n}t_{u,n_k}(\ZETA)
\leq t_{\text{QoS}}.
\end{align}
\end{subequations}
Here, constraint \eqref{QoSboundsyn} captures the nature of ``step-by-step'', i.e., every UE needs to wait for the UEs of all groups to finish one step before starting the next step as seen in Fig.~\ref{fig:time1}(b). Compared to \eqref{QoSboundsyn}, \eqref{QoSbound} provides more flexibility for allocating times of Steps (S1)--(S3) for each UE since the UEs do not need to wait for other UEs to start a new step.

Using the similar procedure to solve problem \eqref{Pmain} above, we approximate \eqref{Pmainsyn} by the following convex problem
\begin{subequations}\label{Pmainsyncepiapprox}
\begin{align}
\underset{\widehat{\x}}{\min} \,\,
& \widetilde{E}_{total}
\\
\nonumber
\mathrm{s.t.}\,\,
& \eqref{fbound},
\eqref{rlowerbound},
\eqref{powerdupperbound2}-\eqref{powerlowerbound2},
\eqref{Rdlowerboundapprox}-\eqref{tuboundd2approx}
\\
& t_d + t_C + t_u \leq t_{\text{QoS}}
\\
&
\frac{S_{d,n}}{r_{d,n_k}} \leq t_d, \forall n,n_k
\\
&
\frac{LD_nc_{n_k}}{f_{n_k}} \leq t_C, \forall n,n_k
\\
& \frac{S_{u,n}}{r_{u,n_k}}  \leq  t_u , \forall n,n_k,
\end{align}
\end{subequations}
where $\rr_d,\rr_u,\OOmega,\THeta,t_d,t_C,t_u$ are additional variables and $\widehat{\x} \triangleq \{\vv,\f,\uu,\rr_d,\rr_u,\OOmega,\THeta,t_d,t_C,t_u\}$.
Then, problem \eqref{Pmainsyn} can be solved using Algorithm~\ref{alg} for iteratively solving \eqref{Pmainsyncepiapprox}.
\vspace{-2mm}
\subsection{Complexity Analysis}
\vspace{-1mm}
Problem \eqref{Pmainepiapprox} can be transformed to an equivalent problem that involves $V_1\triangleq (9K_{total}+1)$ real-valued scalar variables, $L_1\triangleq (8K_{total}+4)$ linear constraints, $Q_1\triangleq 11K_{total}$ quadratic constraints. Therefore, problem \eqref{Pmainepiapprox} requires a complexitiy of $\OO(\sqrt{L_1+Q_1}(V_1+L_1+Q_1)V_1^2)$ \cite{tam16TWC}. The transformed version of problem \eqref{Pmainsyncepiapprox} involves a smaller number of variables and constraints than the version of problem \eqref{Pmainepiapprox}, i.e, $V_2\triangleq (6K_{total}+3)$ real-valued scalar variables, $L_2\triangleq (6K_{total}+1)$ linear constraints, $Q_2\triangleq 5K_{total}$ quadratic constraints. Therefore, problem \eqref{Pmainsyncepiapprox} has the complexity of $\OO(\sqrt{L_2+Q_2}(V_2+L_2+Q_2)V_2^2)$ which is lower than that of problem \eqref{Pmainepiapprox}.
As such, it is expected that the synchronous scheme requires a lower complexity than the asynchronous scheme. However, the synchronous scheme requires more signaling overhead to achieve synchronization than the asynchronous scheme.

\vspace{-1mm}
\section{Numerical Examples}
\vspace{-1mm}
\label{sec:sim}
\subsection{Network Setup and Parameter Setting}
\vspace{-1mm}
Consider a mMIMO network in a square of $D\times D$ km$^2$ where the BS is at the center and the UEs are randomly located.
We set $\tau_c\!=\!200$ samples.
The large-scale fading coefficients, i.e., $\beta_{mn_k}$, are modeled in the same manner as \cite[Eqs. (37), (38)]{emil20TWC}.
For ease of presentation, we assume that all groups have the same number of UEs, i.e., $K_n =K, \forall n$. The total number of UEs is thus $NK$.
We choose
$\tau_{d,p} =\tau_{u,p} \!=\! NK$, $S_d\!=\!S_u\!=\!20$ MB, noise power $\sigma_0^2\!=\!-92$ dBm, $L=50$, $f_{\max}=4 \times 10^9$ cycles/s, $D_n = 5\times 10^6$ samples, $c_{n_k} = 20$ cycles/samples \cite{tran19INFOCOM}, for all $n,n_k$, $\alpha=5\times 10^{-30}$, $t_{\text{QoS}} = 5$ s.
Let $\tilde{\rho}_d\!=\!6$ W, $\tilde{\rho}_u\!=\!0.2$ W and $\tilde{\rho}_p\!=\!0.2$ W be the maximum transmit power of the APs, UEs and uplink pilot sequences, respectively. The maximum transmit powers $\rho_d$, $\rho_u$ and $\rho_p$ are normalized by the noise power.

\vspace{-1mm}
\subsection{Results and Discussions}
\vspace{-1mm}
Note that there are no other existing works studying wireless networks for supporting multiple FL groups. Therefore, to evaluate the effectiveness of our proposed asynchronous scheme (\textbf{OPT\_Async}) and synchronous scheme (\textbf{OPT\_Sync}), we consider the following heuristic schemes:
\begin{itemize}

    \item \textbf{Heuristic\_Async} (Heuristic solution for asynchronous scheme): The downlink power to the UEs of all groups are the same, i.e., $\eta_{n_k}\!=\!\frac{1}{NK}$ and the transmitted power of each UE is $\eta_{n_k}=1, \forall n, n_k$. The processing frequencies are $f_{n_k} = \frac{LD_nc_{n_k}}{t_{\text{QoS}-t_{d,n_k} - t_{u,n_k}}}, \forall n,n_k$.
    \item \textbf{Heuristic\_Sync} (Heuristic solution for synchronous scheme):
    Similar to Heuristic\_Async except for the processing frequencies which are set as $f_{n_k} = \frac{LD_nc_{n_k}}{t_{\text{QoS}-\max_{n\in\NN}\max_{n_k\in\K_n}t_{d,n_k} - \max_{n\in\NN}\max_{n_k\in\K_n}t_{u,n_k}}}, \forall n,n_k$.
\end{itemize}

\begin{figure}[t!]
  \centering
  \vspace{-0mm}
  {\includegraphics[width=0.38\textwidth]{overM.eps}\label{fig:a}}
  \caption{Comparison among the proposed approach and baselines ($K=10$ (users per group), $N\!=\!3$ groups, $D=0.25$ km).}
  \label{Fig:sim1}
  \vspace{-5mm}
\end{figure}

\begin{figure}[t!]
  \centering
  {\includegraphics[width=0.39\textwidth]{overK.eps}\label{fig:b}}
  \vspace{-0mm}
  \caption{Comparison among the proposed approach and baselines ($M = 100$ (antennas), $N\!=\!3$ groups, $D=0.25$ km).}
  \label{Fig:sim2}
  \vspace{-0mm}
\end{figure}

Figs.~\ref{Fig:sim1} and~\ref{Fig:sim2} compare the total energy consumption of one FL iteration among the considered schemes. As seen, our proposed schemes give the best performance. Specifically, compared to heuristic schemes, the energy reduction are up to $52\%$  with $M=100$, $K=10$, and up to $66\%$ with $M=100$, $K=4$. The figures not only demonstrate the significant advantage of a joint allocation of power and processing frequency, but also show the benefit of using massive MIMO to support FL. Thanks to massive MIMO technology, the data rate of each UE increases when the number of antennas increases, leading to lower delays and then a decrease of $31\%$ in the total energy consumption of one FL iteration as shown in Fig.~\ref{Fig:sim1}.

Figs.~\ref{Fig:sim1} and~\ref{Fig:sim2} also shows that the asynchronous scheme slightly outperforms the synchronous scheme. In particular, the energy reduction in one FL iteration is up to only $14\%$ with $M=100$, $K=10$. This is reasonable because the UEs in the asynchronous scheme do not need to wait for other UEs. As such, they have more time resource, and hence, can save more energy by using lower processing frequencies than those in the synchronous scheme. However, minimizing energy consumption results in maximizing the lowest data rate. Therefore, data rates obtained by the asynchronous scheme are relatively similar to those by the synchronous scheme, which leads to a similar performance of both schemes.
\vspace{-1mm}
\section{Conclusion}
\vspace{-1mm}
\label{sec:con}
This work has proposed two novel schemes with mMIMO as energy-efficient solutions for future wireless networks to support multiple FL groups. Using successive convex approximation techniques, we have also successfully proposed an algorithm to allocate power and processing frequency in order to minimize the energy consumption in each  FL iteration. Numerical results showed that our proposed schemes significantly reduces the energy consumption of each FL iteration compared to heuristic schemes. They also confirmed that in terms of energy savings, the asynchronous scheme is a better choice to support multiple FL groups than the synchronous scheme, though at the cost of higher complexity.
\vspace{-1mm}

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

\vspace{-0mm}
\section*{Acknowledgment}
\vspace{-0mm}
The work of T.~T.~Vu and H.~Q.~Ngo was supported by the U.~K. Research and Innovation Future Leaders Fellowships under Grant MR/S017666/1. The work of Erik~G.~Larsson was supported in part by ELLIIT and the Knut and Alice Wallenberg Foundation. The work of Minh~N.~Dao was partially supported by Federation University Australia under Grant RGS21-8.

\vspace{-2mm}
\begin{spacing}{1}
\bibliographystyle{IEEEtran}


\section{Introduction}

Organic solar cells are flexible, stretchable and possibly
wearable. If they could be integrated into our clothing, they would
power our portable phones and computers.  $\rm C_{60}$/pentacene solar cells
are a prime example in organic photovoltaics \cite{bredas}, where
pentacene (Pc) serves as an electron donor and $\rm C_{60}$  as an electron
acceptor. When light strikes on pentacene, a complex singlet is formed
and subsequently is split into two optically inactive triplets, or
singlet fission \cite{chan,thor,zimmerman,wilson,sanders}. Such a
unique feature, where one single photon creates two triplets, greatly
improves the quantum efficiency of charge
photogeneration \cite{congreve}.

But the high quantum efficiency is only the first step for the
photovoltaic cell \cite{congreve}. What is more important is the states
that channel initially-bound electrons and holes into free charge
carriers. Bakulin {\it et al. } \cite{bakulin} showed that the formation of
delocalized states facilitates photoconversion. In 2014, Gelinas {\it et al. }
\cite{gelinas} suggested that a rapid (40 fs) charge separation
proceeds through delocalized $\pi$-electron states in ordered regions
of the fullerene and acceptor material. Chen {\it et al. } \cite{chen} also
found that charge photogeneration occurs predominantly via those
delocalized hot exciton states. Paraecatti and Banerji \cite{para}
more directly pointed out that exciton delocalization provides an
efficient charge separation pathway. These prior studies established
beyond any doubt the important of delocalized states, but what are
these delocalized channel states \cite{savoie,savoie2}? To this end,
there has been no obvious answer. This is the focus of our study.

In this paper, we carry out an extensive first-principles density
functional calculation to show that there are a group of unusually
larger superintermolecular orbitals (SIMOs) in $\rm C_{60}$/Pc complex that
bridge both $\rm C_{60}$  and pentacene.  We employ a real grid mesh method so
we can treat both ordinary molecular orbitals and SIMOs on an equal
footing. We find that energetically, SIMOS are close to native
superatom molecular orbitals in $\rm C_{60}$, but spatially SIMOs are much
larger, with spatial extension over 1 nm.  They are optically
silent. By computing over 3000 Coulomb and exchange integrals, we find
that both Coulomb and exchange interactions among SIMOs are in general
much smaller than those among ordinary molecular orbitals, a necessary
condition to allow initially bound electrons and holes to dissociate
into free charge carriers. Interestingly, regardless of edge-on and
face-one geometries, SIMOs retain their original shapes.  These
features strongly suggest that they are good candidates for those
channel states in $\rm C_{60}$/Pc solar cells.

The rest of the paper is arranged as follows. In Sec. II, we present
our theoretical formalism and the details of our first-principles
calculation.  Section III is devoted to the results and discussion.
We conclude this paper in Sec. IV.  An appendix at end provides
additional details about our hybrid MPI/OpenMP parallel
implementation.

\section{Method}

Our calculation is based on the first-principles density functional
code Octopus \cite{octopus} which employs the pseudopotential method
and the real grid mesh in real space, and has an important advantage
that it treats localized and delocalized states on an equal footing.
To start with, we solve the Kohn-Sham (KS) equation in atomic units,
\begin{equation} \left [-\frac{1}{2}\nabla^2+V_{eff} ({\bf r})\right ] \phi_i ({\bf
  r})=E_i\phi_i ({\bf r})\label{ks} \end{equation} where $\phi_i ({\bf r})$ is
the Kohn-Sham wavefunction and $E_i$ is the eigenvalue of state $i$.
The first term on the left hand side of Eq. (\ref{ks}) is the kinetic
energy operator.  The effective potential ($V_{eff}$) consists of the
electron-nuclei interaction, the Hartree potential (due to the
electron-electron Coulomb interaction), and exchange-correlation
interactions, \begin{equation} V_{eff} ({\bf r})=v({\bf r})+ \int d{\bf r'}
\frac{\rho({\bf r'})}{|{\bf r}-{\bf r'}|} +V_{xc}({\bf r}) \end{equation} where
the exchange-correlation potential $V_{xc}$ is $\delta
E_{xc}[\rho]/\delta \rho({\bf r})$, taking the form of the local
density approximation (LDA). We find that LDA is sufficient for our
purpose, and using GGA raises the energy by 0.5 eV \cite{ijmp2015}.
The new charge density is computed by summing over all the occupied
orbitals ($N_{occ}$), \begin{equation} \rho({\bf r})=\sum_{i=1}^{N_{occ}}|\phi_i
({\bf r})|^2. \end{equation} The next iteration starts. This process repeats
itself until the charge density converges.  With the converged
wavefunction, we then compute the Coulomb and exchange integrals using
National Energy Research Scientific Computing Center machines.
However, these integrals over six degrees of freedom are extremely
time consuming, with so many mesh grid points (see the appendix for
details).  We employ the submatrix technique where we compute the
action of the Coulomb term on the states $n$ and $m$ and then multiply
two additional wavefunctions on the above results.  We develop a
hybrid MPI/OpenMP code that breaks the integral into segments and
distribute them to processors and nodes.  And finally, the master node
sums all the results up. This speeds up our calculation greatly.

We use the normal conserving pseudopotential developed by Troullier
and Martins \cite{tm}.  Our simulation box is a cylinder. The radius
of the cylinder is $r=30~\rm \AA$ and the length is 80 $\rm \AA$. The
grid mesh is $m=0.22~\rm \AA$ and the total number of grid mesh points
is 22814131.  We have checked the convergence with the grid mesh and
find that our results converge well.  $\rm C_{60}$/pentacene complex
has 342 valence electrons, so 171 orbitals are doubly occupied. To
obtain those unoccupied states, we add 129 extra states (in Octopus,
the command is ExtraStates=129), so we have eigenstates all the way up
to 300.  This covers the entire spectrum that is of interest to us.
The threshold for the charge density convergence is set to $10^{-4}$,
and the threshold for the absolute energy convergence is set to
$5\times 10^{-7}$ eV. All the Octopus calculations are run on our
university Silicon cluster, where each computing node has dual Intel
Xeon E5-2680 v2 CPUs with 2.80GHZ.  Each CPU has 10 cores and cache
size of 25 MB. The total memory for each node is 132 GB. The entire
calculation needs 80 GB memory and takes nearly two months to finish.
After the calculation is finished, we export the wavefunctions from
state number 97 up to 300 in two different formats, one for Xcrysden
rendering of orbital images and the other in the Cartesian format. The
latter is the actual wavefunctions in the three-dimensional space
$\phi(x,y,z)$. These wavefunctions are extremely convenient for
calculating other properties of interest.

\section{Results: Superintermolecular orbitals}

\begin{figure
\centerline{\includegraphics[angle=0,width=1\columnwidth]{first.eps}}
\caption{Light first strikes $\rm C_{60}$/pentacene complex and creates a
  singlet, followed by singlet fission into triplets. But electrons
  and holes have to dissociate from each other to become free charge
  carriers. The central question is what the channel states are for
  charge generation in organic solar cells. We show that the
  superintermolecular orbitals may offer an answer.  }
\label{fig1}

\end{figure}

Photovoltaic effects depend on an efficient charge transfer from a
donor (D) to an acceptor (A) \cite{jpc95}. Figure \ref{fig1}
schematically illustrates that light first strikes $\rm C_{60}$/Pc but the
subsequent charge transfer relies on channel states. While there is no
detail study of these channel states, several studies have estimated
the size of channel states to be around 3-4 nm
\cite{gelinas,barker,bernardo,heiber,dutton,davino}. This distance
corresponds to a binding energy of 0.1 eV, which can be reasonably
approximated as $E_B=q^2/4\pi\epsilon r_{CT}$. Here $q$ is the charge,
$r_{CT}$ is the separation between average electrons and holes in the
parent charge-transfer (CT) state.  However, no ordinary molecular
orbitals can be as big as 4 nm.  Being unaware of possible relevance
to photovoltaics in $\rm C_{60}$/Pc, Feng and her coworkers \cite{feng}
reported some very peculiar molecular orbitals in $\rm C_{60}$, resembling the
atomic orbitals, but with a much larger radius. They are not localized
around the atoms of the cluster, but rather they belong to the entire
cluster, for which they called superatomic molecular orbitals, or
SAMOs. They detected these SAMOs using the scanning tunneling
microscope (STM), where the voltage bias is gradually tuned. Their
appearance is due to the partial delocality of outer shells of carbon
atoms which jointly create a potential. Such a potential allows
electrons to partially delocalize around the entire molecule.  These
orbitals have a distinctive shell structure from $1s$ up to $1d$ and
are optically dark states.  When we were investigating
SAMO\cite{ijmp2015}, we were keenly aware of the large size of those
SAMOs.  We notice that the $1s$ orbital has a size close to $\rm C_{60}$
\cite{ijmp2015}.

To begin with, we employ Gaussian09 \cite{gaussian} to separately
optimize $\rm C_{60}$, pentacene and $\rm C_{60}$/Pc structures. We use the Becke,
3-parameter, Lee-Yang-Parr (B3LYP) method and a correlation-consistent
polarized valence double-zeta (cc-pVDZ) basis. The results are fully
consistent with our and other previous calculations
\cite{pra11,johansson12} in both the eigenenergies and wavefunctions.
The optimized coordinates in Gaussian09, without further optimization,
are used as an input for Octopus \cite{octopus}. The reason is that
Octopus uses grid mesh and slightly breaks the symmetry of degenerate
eigenstates. Although the change in energy is small, we worry that the
introduced force may be too great if we use it to optimize our $\rm C_{60}$/Pc
complex.

 As done by many researchers \cite{Yi,Yang,li,ryno}, we
consider both edge-on and face-on configurations.  In the edge-on
configuration one end of Pc aims at the hexagons/pentagons on $\rm C_{60}$,
while in the face-on configuration, the plane of Pc faces the
hexagons/pentagons on $\rm C_{60}$.

 \begin{widetext}
\begin{center}
\begin{table}
\caption{Coulomb and exchange matrix elements (in units of eV) among
  LUMO (from 172 to 174) and LUMO+1 (175). }
\begin{tabular}{c|ccccllll}
\hline\hline
~~\hspace{0.5cm}&\multicolumn{4}{l}{$K(nm|mn)$ (eV)}& \multicolumn{4}{l}{$J(nm|nm)$ (eV)}\\
\hline
$n$\textbackslash$m$   &172  & 173   & 174     &175 \hspace{1cm} &172 & 173 &174 &175
\\
\hline
172     &3.64 & 3.42  &  3.42   &0.92  \hspace{1cm} &-- & 0.107
&0.107& 0.555$\times 10^{-6}$  \\
173     & 3.42    & 3.64 &3.42 &0.93 \hspace{1cm} &0.107 & -- &0.108
&0.169$\times 10^{-5}$ \\
174     &3.42 & 3.42 &3.64 &0.92 \hspace{1cm} &0.107 & 0.018 &-- &0.916$\times 10^{-6}$ \\
175     &0.92 & 0.93 &0.92 &4.51 \hspace{1cm} &0.555$\times 10^{-6}$
&0.169$\times 10^{-5}$ &0.916$\times 10^{-6}$ &-- \\
\hline
\hline
\end{tabular}
\label{table1}
\end{table}
\end{center}
 \end{widetext}

\subsection{Edge-on}

We start with the edge-on geometry. The distance between the frontier
carbon atoms of Pc and the hexagons on $\rm C_{60}$  is 7.1 $\rm \AA$, larger
than previous investigations \cite{minami,Yi}. The distance between
the far-left carbon atoms on Pc and the far-right carbon atoms on $\rm C_{60}$
is 19.3 $\rm \AA$.  The left figure of Fig. (\ref{simo}) shows one
example for the edge-on configuration. This is the wave function
$\psi_{205}({\bf r})$ for orbital 205 plotted at isovalue of 0.005$\rm
\sqrt[-3]{\rm\AA}$. The color difference denotes the sign of
$\psi_{205}({\bf r})$. Different from SAMOs \cite{feng}, this
superorbital covers both Pc and $\rm C_{60}$  molecules, or superintermolecular
orbital, SIMO for short. In the language of SAMO, this could be $1p$
SIMO, but for SIMOs, the orbital character is approximate due to the
symmetry reduction. We find that in general SIMOs have special
orientations just as an ordinary molecular orbital. In some cases,
SIMOs are more like SAMOs on an isolated $\rm C_{60}$. This spatial preference
is crucial since it allows the electrons to transfer from Pc to $\rm C_{60}$
unidirectionally. One special feature, which is inherent from SAMOs,
is that the dipole transition matrix elements between SIMOs and
ordinary molecular orbitals are very small. For this reason, we do not
expect that an optically induced charge transfer occurs from ordinary
molecular orbitals to SIMOs. Instead, the initial bound exciton must
dissociate into SIMOs through tunneling. We recall that Feng {\it et al. }
\cite{feng} detected SAMOs using STM. The quantum tunneling is also
closer to what happens in solar cells.

\begin{figure}[tb]
\centerline{\includegraphics[angle=0,width=0.8\columnwidth]{1p.eps}}
\caption{ Superintermolecular orbitals in $\rm C_{60}$/pentacene for the
  edge-on configuration (left) and face-on configuration (right).  We
  show one representative $1p$ SIMO for each configuration.  $1p$ SIMO
  has orbital number 205. }
\label{simo}
\end{figure}

However, in photovoltaics, electrons are first excited into those low
lying lowest unoccupied molecular orbitals (LUMOs), which have been
the focus of recent investigations.  For  free charge
generation, majority of theoretical studies start from an initial
state $\phi_i^m({\bf r})$ localized on D, where $m$ is the
multiplicity of state $i$ and ${\bf r}$ is the electron coordinate.
One hopes that this initial state ends up to a final state
$\phi_f^n({\bf r})$ localized on A. In the many-body
picture \cite{aki}, one often starts from configurations like \begin{equation}
|\Psi\rangle=a |\phi_i^m({\bf r}_1) \phi_f^n({\bf r}_2) \rangle + {\rm
  high-order~ terms.} \end{equation} If $|\phi_i^m({\bf r}_1) \phi_f^n({\bf
  r}_2)\rangle$ takes a significant weight on the many-body
wavefunction, so CT is realized. This idea is simple and attractive,
but faces a dilemma.  To have a large contribution from configuration
$|\phi_i^m({\bf r}_1) \phi_f^n({\bf r}_2)\rangle$, the Coulomb and
exchange interaction matrix elements must be large, but this leads to
a large binding energy, detrimental to free charge carrier
generation \cite{jail}. On the other hand, if the above elements are
small, then the coupling is weak and the transition to CT states is
less likely.  We compute all the Coulomb and exchange integrals from
the lowest unoccupied molecular orbital (LUMO) up to LUMO+1; there are
in total four orbitals since LUMOs are nearly degenerate. Table I
shows all the Coulomb and exchange integrals among all the LUMO
states.  We find that the strongest interaction is 4.51 eV, which is
more than forty times larger than the disorder energy of 0.1 eV
estimated by Clarke and Durrant \cite{clarke}. This leads to a high
binding energy for excitons, thus it is detrimental to free charge
carrier generation \cite{jail}. This simple estimate highlights that
initially excited states by the light are unlikely the same states
that are responsible for final charge transfer and charge
separation. A different group of states engage the final step of
charge generations.

\begin{figure}
\centerline{\includegraphics[angle=270,width=0.8\columnwidth]{fig1.eps}}
\caption{Coulomb and exchange matrix elements between pairs of states
  in native $\rm C_{60}$. There are 3081 elements.  The magnitude of matrix
  elements is proportional to the radius of circles, and all the
  Coulomb elements are rescaled by multiplying 0.15 .  Since the
  matrix is symmetric, the upper triangle shows the Coulomb integral,
  while the lower triangle shows the exchange integral.  }
\label{coulombc60}
\end{figure}

The above calculation is only limited to four unoccupied
orbitals. Before we present results for $\rm C_{60}$/Pc, we decide to
completely map out all the matrix elements for all the states from the
highest occupied molecular orbital (HOMO)-4 through $1g$ SAMOs in
native $\rm C_{60}$. There are 3081 Coulomb and exchange integrals. All the
calculations are carried out at Berkeley National Laboratory's
National Energy Research Computing Center.  Figure \ref{coulombc60}
shows a complete list of those matrix elements. Since these matrix
elements are symmetric with respect to the state permutation (other
combinations have a much small amplitude, thus not shown), we only
show the upper triangle for the Coulomb integral and the lower
triangle for the exchange integral.  Both the horizontal and vertical
axes denote the states. The SAMOs state labels are slightly off the
axis for clarity. Along the horizontal axis, the second $H_u$ state
from the left is our HOMO, and the first $T_{1u}$ is our LUMO.  The
radii of the circles are proportional to the magnitude of
integral. The Coulomb integrals are in general much larger, so when we
plot them, we reduce their size by multiplying them by 0.15. A general
pattern emerges. For ordinary molecular orbitals, the Coulomb and
exchange integrals are much larger.  The exchange integrals are much
less uniform than the Coulomb integral, since the former greatly
depends on the phases of the wavefunctions.  SAMOs' integrals are also
sizable, in particular for $1s$ SAMO, but once we are above $1p$ SAMO,
both Coulomb and exchange integrals drop very quickly. This opens a
door for delocalized and weak-interacting SAMOs to participate charge
generating process.

\begin{figure}
\centerline{\includegraphics[angle=0,width=0.8\columnwidth]{last1.eps}}
\caption{Coulomb and exchange matrix elements between pairs of
  states. The magnitude of matrix elements is proportional to the
  radius of circles.  Since the matrix is symmetric, the upper-left
  triangle shows the Coulomb integral, while the lower-right triangle
  shows the exchange integral.  The largest Coulomb integral (4.51 eV)
  is between LUMO+1 (orbital number 175) localized on Pc; and the
  smallest (0.43 eV) is among $1f$ SIMOs (orbital numbers 226 and
  228).  The exchange integrals are in red small circles and extremely
  small.  }
\label{coulomb}
\end{figure}

To build a case for SIMOs, we also compute the Coulomb and exchange
integrals and we find that similar to SAMOs, their values are an order
of magnitude smaller. Figure \ref{coulomb} compares the Coulomb (upper
triangle) and exchange (lower triangle) integrals for LUMOs and SIMOs.
The largest circle represents 4.51 eV (which is between LUMO+1). The
Coulomb interaction drops quickly once we are above the $1s$ SIMO. The
smallest Coulomb interaction is for $1f$ SIMOs, only 0.43 eV. If we
consider the dielectric constant of the medium about 3, this
interaction is reduced to 0.14 eV, very close to the disorder
energy. These small Coulomb and exchange integrals are also reflected
in the small transfer integral used by Smith and Chin \cite{smith}.
They concluded that the transfer integrals are no larger than 8 meV,
extremely tiny in comparison to those in $\rm C_{60}$  \cite{prb94}.

In 2016, in poly(3-hexylthiophene)/fullerene blends, D'Avino
{\it et al. }\cite{davino} argued that the bound localized charge-transfer (LCT)
states coexist with delocalized space-separated states because LCT
states hybridize with singlets. In a later study \cite{davino2}, they
also suggested that both $\rm C_{60}$  and its derivative may sustain
high-energy states that spread over a few tens of molecules by
pointing out sizable intermolecular delocalization of the electron
wavefunction. Here our SIMOs present an alternative.

\subsection{Face-on}

We also consider the face-on geometry. In this configuration, the
distance between Pc and $\rm C_{60}$  is 10.6 $\rm \AA$, also larger than many
prior studies. This configuration is considered to be the most
favorable one for the charge transfer and charge separation. The right
figure of Fig. \ref{simo} shows $1p$ SIMO. It is interesting that
although the face-on geometry is so different from the edge-on
geometry, the SIMO retains its shape well. The wavefunction on Pc has
a larger amplitude than that for the edge-on configuration. This may
explain why it is more efficient, since the orbital is very
delocalized. This meets one of the requirements for the channel
states. Therefore, once an electron tunnels into this orbital, it has
an excellent chance to transfer to $\rm C_{60}$.

\begin{figure}
\centerline{\includegraphics[angle=0,width=0.8\columnwidth]{energy.eps}}

{\includegraphics[angle=0,width=0.3\columnwidth]{wf-st0210.green.eps}}
{\includegraphics[angle=0,width=0.3\columnwidth]{wf-st0228.green.eps}}
\caption{ (Top panel) Energy level comparison between SIMOs of
  $\rm C_{60}$/pentacene and SAMOs of $\rm C_{60}$. The energy in $1s$ states is
  similar. A small splitting is noticed in $1p$ state. $1d$ state has
  the largest shift of 0.16 eV, due to an overlap in the
  $d$ orbitals.  Change in $1f$ is small. (Bottom panel) Left:
  Wavefunction of $1d$ SIMO; Right: Wavefunction of $1f$ SIMO. }
\label{energy}
\end{figure}

Energetically, the SIMOs appear in the same energy window as native
SAMOs in $\rm C_{60}$. Naturally, this also depends on the spatial orientation
of Pc and $\rm C_{60}$. Figure \ref{energy} (top panel) compares the SIMO
energies in $\rm C_{60}$/Pc with those of SAMOs in $\rm C_{60}$. We see that $1s$ SIMO
and $1s$ SAMO are aligned with each other. However, $1p$ SAMO is now
split into three nearly degenerate levels, due to the symmetry
reductions as explained above. The difference becomes bigger for $1d$
SIMOs. We see that the lowest $1d$ SIMO is shifted down by 0.16 eV.
We understand why it is so. The lower left figure shows that the
orbital wavefunction is delocalized over Pc and $\rm C_{60}$  and there are
nodal lines in between. The lobes of the orbital overlaps
strongly. This lowers the orbital energy significantly. By contrast,
other orbitals do not change too much. $1f$ SAMO is also split. To
show the orbital 228 has a $f$ character, we rotate the complex
structure so pentacene points out of the page. It is clear that the
orbital retains the $f$ character, but the orbital is elongated along
the Pc-$\rm C_{60}$  axis. Quantitatively, $1f$ SAMO in native $\rm C_{60}$  is at 0.204
eV, while $1f$ SIMOs lie between 0.186 and 0.212 eV.  Interestingly,
Pavlyukha and Berakdar \cite{Pavlyukha} showed that SAMOs are
long-lived, coincident with the experimental observation \cite{chep}.
This is what a channel state is supposed to be.  These agreements
constitute strong evidence that the SIMOs may serve a possible channel
for charge separation. However, it is difficult to detect these
delocalized excited states \cite{rao} optically since it may have a
low absorption cross section \cite{osterloh}. One possible method to
test our prediction is the transport measurement. Such a measurement
is in fact more directly related to charge transfer in photovoltaics
than the optical means.  Finally, we notice that there is an ongoing
debate how or whether hot charge-transfer excitons assist free carrier
generation \cite{nan,shen}. But even if hot CT excitons do play a role
\cite{jail}, the binding energy of interfacial CT exciton after
initial excitation is too high for the thermal activation to climb out
of the Coulomb trap.  We argue that SIMOs reported here may provide an
alternative path to CT.

\section{Conclusions}

We have carried out the first-principles density functional
calculation in $\rm C_{60}$/Pc complex and find that there exist a group of big
superintermolecular orbitals. These orbitals are very delocalized and
cover both Pc and $\rm C_{60}$. We find that SIMOs have the right spatial and
energetic characters to channel the initially bound electrons and
holes into free charge carriers.  Spatially, they are much larger than
ordinary molecular orbitals, close to 1 nm, a critical distance for
CT. They have a clear spatial orientation from Pc to $\rm C_{60}$, a crucial
element that greatly facilitates triplet dissociation into free charge
carriers.  Energetically, both exchange and Coulomb interactions of
SIMOs are very small, on the order of 0.1 eV, a value that matches the
Clarke-Durrant disorder energy of 0.1 eV \cite{clarke}. Thus, our
finding highlights an unexpected benefit from SIMOs and points a
possible strategy for tailoring material properties toward
high-efficient organic solar cells. One possible method to enhance
charge generation is to employ a larger fullerene, where SIMOs can be
made even larger. This is consistent with the experiment \cite{chen},
where they showed that large fullerene crystals can enhance charge
separation yields. We expect that our finding will motivate further
experimental and theoretical investigations on these exciting
opportunities at the frontier of photovoltaics.

\acknowledgments This work was supported by the U.S. Department of
Energy under Contract No.  DE-FG02-06ER46304 (GPZ).  This research
used resources of the National Energy Research Scientific Computing
Center, which is supported by the Office of Science of the U.S.
Department of Energy under Contract No. DE-AC02-05CH11231.

 $^*$gpzhang@indstate.edu\\