\section{1. Introduction}
Kohn-Sham density functional theory (KS-DFT) \citep{hohenberg64,kohn65} has been found to be an indispensable tool for
determining structure and properties of many-electron systems like atoms, molecules, clusters, nano-materials and periodic
systems, for past several decades\citep{cohen11, burke12, becke14, jones15}. The KS mapping is exact, \emph{in principle},
and has the ability to capture many-body correlation effects at a computational cost comparable to Hartree-Fock (HF) theory.
But a major bottleneck lies in the exchange-correlation (XC) functional that uses electron density, $\rho(\rvec)$, to
describe non-classical effects under the mean-field formalism of KS scheme. As its exact form is unknown as yet, it has
to be approximated and accordingly, $ E_{\mathrm{xc}}^{\mathrm{approx}}[\rho(\rvec)] =
\int d^3\rvec f^{\mathrm{approx}}_{\mathrm{xc}}(\rho,\nabla\rho,\tau,e^{\mathrm{x}},\dots)$. Commonly used functionals
can be hierarchically characterized using these variables. Thus starting from a simpler local density approximation (LDA)
(containing $\rho$ only), to general gradient approximation (GGA) (with addition of gradient of electron density,
$\nabla\rho$), to meta-GGA (further addition of kinetic energy density, $\tau$), one gets a progressively involved
and sophisticated range of functionals. With further inclusion of exact (also called HF) exchange energy, one gets the hybrid
functionals, while incorporation of exact exchange energy density, $e^{\mathrm{x}}$, leads to a hyper-GGA type XC
functionals, residing in the fourth rung of Jacob's ladder. Now we also have functionals which go beyond the
fourth rung (and include virtual orbitals), thus requiring higher computational cost, and have been successfully
implemented in literature in the context of various physio-chemical properties\citep{grimme07, zhang09}.

In general the approximate local or semi-local functionals experience certain issues regarding (i) correct asymptotic
behavior of exact exchange potential at long range and (ii) non-cancellation of spurious Coulomb self-repulsion
energy--the so-called self-interaction error (SIE)\citep{perdew81, bao18}. These two points are connected to each other and may be
dealt with by bringing exact exchange in to the picture. This component may be combined with semi-local functionals,
empirically\citep{becke93b} or non-empirically \citep{perdew96a, guido13}, thus improving the asymptotic nature, and
consequently reducing the SIE to a greater extent. Accordingly, the global hybrid functionals \citep{becke93b} (B3LYP being a
prototypical model) were proposed in the literature\citep{adamo99, ernzerhof99, zhao08}, which improved the chemical accuracy
in a large number of physico-chemical properties\citep{junior08, mangiatordi12}. On the other hand, non-dynamic (or static) correlation
seems to be a serious problem for current DFT functionals\citep{becke14} which arises when an electronic state cannot be properly
described by a single Slater determinant. It requires a linear combination of degenerate or nearly degenerate Slater
determinants. Naturally it is very much demanding to model this correlation accurately within a single determinantal
KS-DFT framework, and some methods have been proposed in the past decade under the name of hyper-GGA functionals\citep{becke03,
becke05, perdew08}. More recently, general single-determinant model functionals have also been put forth to treat this
correlation up to the dissociation limit of a co-valent bond\citep{becke13a, kong15}. All these emerging functionals
belonging to hyper-GGA category, require $e^{\mathrm{x}}$ as a fundamental variable and in general, their computational cost
is much higher compared to the global hybrid functionals.

The HF exchange energy and corresponding matrix elements are usually evaluated through four-center electron repulsion
integrals (ERI) analytically, within a Gaussian basis expansion of molecular orbitals (MO), providing following
contributions to KS-Fock matrix,
\begin{equation}
F_{\mu\nu}^{\mathrm{x}\sigma}=\sum_{\lambda\eta} P_{\lambda\eta}^{\sigma} (\mu\lambda|\eta\nu) = \sum_{\lambda\eta}
P_{\lambda\eta}^{\sigma} \int \int \frac{\chi_{\mu,\sigma}(\rvec) \chi_{\lambda,\sigma}(\rvec)\chi_{\nu,\sigma}
(\rvec^{\prime})\chi_{\eta,\sigma}(\rvec^{\prime})}{|\rvec-\rvec^{\prime}|} d\rvec d\rvec^{\prime}.
\end{equation}
Here $(\mu\lambda|\eta\nu)$ is the four center ERI with $\mu, \nu, \lambda, \eta$ representing the atomic orbitals (AO) or
contracted basis, and $P_{\lambda\eta}^{\sigma}$ is an element of the single-particle spin density matrix, $P^{\sigma}$
having spin $\sigma$. Throughout the past few decades, there has been extensive methodological developments to optimize
its evaluation using a range of approaches varying in complexity, sophistication and accuracy. It is worthwhile
mentioning some of the notable ones, such as pseudo-spectral scheme using Gaussian basis functions, multipole
accelerated algorithm, LinK, truncated or short-range exchange kernels with/without the use of resolution-of-the-identity
(RI) approximation, chain-of-sphere exchange based on quadrature, auxiliary density matrix, plane-wave basis functions,
metropolis stochastic, pair-atomic RI, adaptively compressed exchange operator, density fitting with local domains,
real-space with projection operators, rigorous integral screening, occupied-orbital fast multipole \citep{chasman98,
ochsenfeld00, neese09, cytter14, manzer15, lin16, boffi16, thompson17, le18}, etc. The HF exchange energy density can be
represented as,
\begin{eqnarray}
e^{\mathrm{x}}_{\sigma}(\rvec) = -\sum_{i}^{occ} \sum_{j}^{occ} \int \frac{\phi^{\star}_{i,\sigma}(\rvec)
\phi^{\star}_{j,\sigma}(\rvec)\phi_{i,\sigma}(\rvec^{\prime})\phi_{j,\sigma}(\rvec^{\prime})}{|\rvec-\rvec^{\prime}|}
d\rvec^{\prime} \nonumber \\
=-\sum_{\mu\nu}\sum_{\lambda \eta} P_{\mu\nu}^{\sigma} P_{\lambda\eta}^{\sigma}\int \frac{\chi_{\mu,\sigma}(\rvec)
\chi_{\lambda,\sigma}(\rvec)\chi_{\nu,\sigma}(\rvec^{\prime})\chi_{\eta,\sigma}(\rvec^{\prime})}{|\rvec-\rvec^{\prime}|}
d\rvec^{\prime}.
\end{eqnarray}
The two expressions are defined in terms of KS occupied MO ($\phi_{\sigma}$) and AO ($\chi_{\sigma}$) respectively.
Complex conjugate sign is omitted here since density matrix and basis are generally in real form. At
first glance, it seems to be computationally more expensive than normal exchange energy calculation due to the fact that
it needs to be evaluated \emph{at each grid point} with four AO indices. But some recent developments proposed in the
literature show substantial computational cost reduction via pair-atomic RI approximation\citep{proynov10, proynov12, liu12}
or using a  semi-numerical (SNR) scheme\citep{kossmann09, bahmann15, laqua18, liu17}. Here we mainly focus on the latter
implementation, where one of the integrations involved in ERI is carried out numerically on the effective spatial grid.

The main motivation of this work is thus to calculate HF exchange density, energy and its matrix elements which are essential
ingredients for development of some of the new-generation (in particular, orbital-dependent) XC functionals. This is
performed accurately in real-space Cartesian coordinate grid (CCG) following the procedure \citep{liu17}, where a key step is the
identification and evaluation of an intermediate quantity, namely, \emph{two-center electrostatic potential} (ESP), defined as,
\begin{equation}
v_{\nu\eta}(\rvec)=\int\frac{\chi_{\nu}(\rvec^{\prime})\chi_{\eta}(\rvec^{\prime})}
{|\rvec-\rvec^{\prime}|}d\rvec^{\prime}
=\sum_{p}\sum_{q}\int\frac{\varphi_{\nu}^{p}(\rvec^{\prime})\varphi_{\eta}^{q}(\rvec^{\prime})}
{|\rvec-\rvec^{\prime}|}d\rvec^{\prime}.
\end{equation}
The first expression is defined in terms of AO, whereas the second one in terms of primitive basis functions, $\varphi_{\mu}^p$
for a given $\chi_{\mu}$ (for simplicity we remove the spin indices). Note that symbols $\mu, \nu$ signify contracted basis
functions, whereas $p,q$ label primitive functions. We propose a direct, efficient \emph{numerical} (NR) scheme based
on \emph{Fourier convolution theorem} (FCT) for accurate estimation of this ESP integral (first expression) using a
range-separation (RS) technique, corresponding to long-and short-range interactions for \emph{Coulomb interaction
kernel}. A major
concern is how one can define the optimal RS parameter for a successful mapping of this interaction kernel in CCG
from \emph{first principles} rather than empirically. Towards this direction, we have employed a recently developed grid
optimization procedure \citep{ghosal16, ghosal18} with respect to total energy in non-uniform CCG (originally used for
evaluation of classical Hartree potential) to optimize the RS parameter, to a very good level of accuracy, using a well-defined
constraint, suggested in the literature. The successful implementation of this approach also paves the way for further development
of the so-called range-separated hybrid functionals in connection with generalized KS theorem\citep{seidl96}. The
underlying algorithm of the SNR method and subsequently a detailed account of our proposed NR scheme, denoted as
RS-FCT, for estimation of this ESP integral, is provided in Sec.~2. Then its implementation is carried out using
a pseudopotential KS-DFT in CCG, developed in our lab\citep{roy08, roy08a, roy11, ghosal16, ghosal18}. The relevant cost analysis
between these two routes is also sketched upon. Section 3 offers the necessary computational and technical details. Finally, the
feasibility, performance and accuracy of
our results are critically assessed through quantities such as total energy and orbital energy, in Sec.~4, employing
three-types of global hybrid functionals B3LYP"\citep{becke93a}, PBE0\citep{perdew96a} and BHLYP\citep{becke93b} along with
the HF calculation for a decent (total 40) number of atoms and molecules. Some concluding remarks as well as the future prospect
is given in Sec.~4.

\section{2. Methodology}
\subsection{2.1. HF exchange energy, density and matrix elements}
Let us begin with the SNR approach, where HF exchange energy is computed numerically by integrating the corresponding density,
$e^{\mathrm{x}}_{\sigma} (\rvec)$ as given by,
\begin{equation}
E_{\sigma}^{\mathrm{x}}=\frac{1}{2}\int e_{\sigma}^{\mathrm{x}}(\rvec) d\rvec.
\end{equation}
Now recasting Eq.~(2) in the following form,
\begin{equation}
e_{\sigma}^{\mathrm{x}}(\rvec)=-\sum_{\nu}Q_{\nu}^{\sigma}(\rvec)M_{\nu}^{\sigma}(\rvec),
\end{equation}
one may envisage the construction of $e^{\mathrm{x}}_{\sigma} (\rvec)$ in three steps of comparable computational cost. The first
quantity, $Q_{\nu}^{\sigma}(\rvec)$ may be represented as follows,
\begin{equation}
Q_{\nu}^{\sigma}(\rvec) = \sum_{\mu} \chi_{\mu,\sigma}(\rvec) P_{\mu\nu}^{\sigma},
\end{equation}
in which the density matrix is combined with AOs through a simple matrix multiplication. The computational cost of this step
scales as $\mathcal{O}(N_gN_{B}^{2})$, with $N_g, N_B$ denoting total number of grid points, and number of AO basis. Next
crucial (rate-determining) step is to evaluate the ESP integral (embedded in $M_{\nu}^{\sigma}(\rvec)$), which according to Eq.~(3)
(first expression), also scales as  $\mathcal{O}(N_gN_{B}^{2})$. Alternately one may perform this integral analytically
(corresponding to second expression of Eq.~(3)) using primitive functions and standard recursion algorithm \citep{obara86, liu16}.
In that scenario, evidently \emph{each ESP integral} scales as $\mathcal{O}(N_gN_{P}^{2})$, with $N_{P}$
identifying average number of primitive functions. The final step consists of computation of the quantity,
$M_{\nu}^{\sigma}(\rvec)$ in accordance with the following expression,
\begin{equation}
M_{\nu}^{\sigma}(\rvec)=\sum_{\eta}Q_{\eta}^{\sigma}(\rvec)v_{\nu\eta}^{\sigma}(\rvec).
\end{equation}
The scaling of this step is also same as that of ESP integral evaluation, but requires fewer steps than the latter; needing
only one multiplication and one addition at innermost loop. This effectively provides an NR way to
compute the HF exchange matrix, which according to Eq.~(1), can be rewritten as,
\begin{equation}
\frac{\partial{E^{x}_{\sigma}}}{\partial{P_{\lambda\eta}^{\sigma}}}=F_{\mu\nu}^{\mathrm{x}\sigma}=-\int_{\rvec}\chi_{\mu}
(\rvec)M_{\nu}^{\sigma}(\rvec)d\rvec.
\end{equation}

\subsection{2.2. Pseudopotential KS-DFT in CCG}
For a many-electron system, one can write the single-particle KS equation in presence of pseudopotential as (atomic
unit employed unless stated otherwise),
\begin{equation}
\bigg[ -\frac{1}{2} \nabla^2 + v^{\mathrm{p}}_{\mathrm{ion}}(\rvec) + v_{\mathrm{ext}}(\rvec) + v_{\mathrm{h}}[\rho(\rvec)]
+ v_{\mathrm{xc}}[\rho(\rvec)] \bigg] \psi_i^{\sigma}(\rvec) = \epsilon_i \psi_i^{\sigma}(\rvec),
\end{equation}
where $v^{\mathrm{p}}_{\mathrm{ion}}$ denotes the ionic pseudopotential, written as below,
\begin{equation}
v^{\mathrm{p}}_{\mathrm{ion}}(\rvec) = \sum_{\Rvec_a} v^{\mathrm{p}}_{\mathrm{ion},\mathrm{a}} (\rvec-\Rvec_a).
\end{equation}
In the above equation, $v^{\mathrm{p}}_{\mathrm{ion},\mathrm{a}}$ signifies ion-core pseudopotential associated with atom A,
situated at $\Rvec_a$. The classical Coulomb (Hartree) term, $v_{\mathrm{h}}[\rho(\rvec)]$ describes usual electrostatic
interaction amongst valence electrons whereas $v_{\mathrm{xc}}[\rho(\rvec)]$ signifies the non-classical XC part of latter,
and $\{ \psi^{\sigma}_{i},\sigma= \alpha \quad \mathrm{or} \quad  \beta \}$ corresponds to a set of $N$ occupied orthonormal
spin molecular orbitals (spin-MOs). Within LCAO-MO approximation, the coefficients for expansion of spin-MOs satisfy a
set of equations, very similar to that in HF theory,
\begin{equation}
\sum_{\nu} F_{\mu \nu}^{\sigma} C_{\nu i}^{\sigma} = \epsilon_{i}^{\sigma} \sum_{\nu}^{\sigma} S_{\mu \nu}
C_{\nu i}^{\sigma},
\end{equation}
satisfying the orthonormality condition, $(\Cvec^{\sigma})^{\dagger} \Svec \Cvec^{\sigma} = \Ivec$. Here $\Cvec^{\sigma}$
contains the respective spin-MO coefficients $\{C_{\nu i}^{\sigma}\}$ for a given spin-MO $\psi_i^{\sigma}(\rvec)$, $\Svec$ is
the usual overlap matrix corresponding to elements $S_{\mu \nu}$, $\bm{\epsilon}^{\sigma}$ refers to diagonal matrix of
spin-MO eigenvalues $\{\epsilon_{i}^{\sigma}\}$. The KS-Fock matrix has elements $F_{\mu \nu}^{\sigma}$, constituting of
following contributions,
\begin{equation}
F_{\mu \nu}^{\sigma} = H_{\mu \nu}^{\mathrm{core}} + J_{\mu \nu} + F_{\mu \nu}^{\mathrm{xc}\sigma}.
\end{equation}
In this equation, all one-electron contributions, such as kinetic energy, nuclear-electron attraction and pseudopotential matrix
elements are included in first term, whereas $J_{\mu \nu}$ and $F_{\mu \nu}^{\mathrm{xc}\sigma}$ account for classical
Hartree and XC potentials respectively.

Now we discretize various quantities like localized basis function, electron density, MO as well as two-electron potentials
directly on a 3D cubic box having $x,y,z$ axes,
\begin{eqnarray}
r_{i}=r_{0}+(i-1)h_{r}, \quad i=1,2,3,....,N_{r}~, \quad r_{0}=-\frac{N_{r}h_{r}}{2}, \quad  r \in \{ x,y,z \},
\end{eqnarray}
where $h_{r}, N_r$ denote grid spacing and total number of points along each directions respectively. The electron density
$\rho(\rvec)$ in this grid may be simply written as (``g" symbolizes discretized grid),
\begin{equation}
\rho(\rvec_g) = \sum_{\mu,\nu} P_{\mu \nu } \chi_{\mu}(\rvec_g) \chi_{\nu}(\rvec_g).
\end{equation}
At this stage, the two-electron contributions of KS matrix are computed through direct numerical integration in the grid,
\begin{equation}
F_{\mu \nu}^{\mathrm{hxc}} = \langle \chi_{\mu}(\rvec_g)|v_{\mathrm{hxc}}(\rvec_g)|\chi_{\nu}(\rvec_g) \rangle = h_x h_y h_z
\sum_g \chi_{\mu}(\rvec_g) v_{\mathrm{hxc}}(\rvec_g) \chi_{\nu}(\rvec_g).
 \end{equation}
where $v_{hxc}$ refers to the combined Hartree and XC potential. The detailed construction of various potentials in CCG has
been well documented in our earlier work\citep{roy08, roy08a, roy11, ghosal16, ghosal18}; hence not repeated here.
Similarly, the HF exchange energy and its contributions towards KS-Fock matrix can be computed
numerically in CCG as,
\begin{eqnarray}
E_{\sigma}^{\mathrm{x}}=  \frac{1}{2} h_x h_y h_z \sum_g e_{\sigma}^{\mathrm{x}}(\rvec_g), \nonumber \\
F_{\mu\nu}^{\mathrm{x}\sigma}=- h_x h_y h_z \sum_g \chi_{\mu}(\rvec_g)M_{\nu}^{\sigma}(\rvec_g).
\end{eqnarray}

\subsection{2.3. RS-FCT for computation of ESP integral in CCG}
Let us start with two functions $f(\rvec)$ and $F(\kvec)$ in $\rvec$ and $\kvec$ spaces, which are Fourier transforms (FT)
(denoted by $\mathcal{F}$) of each other, as given below,
\begin{eqnarray}
f(\rvec) & = & \mathcal{F}(\kvec) =  \int_{-\infty}^{\infty} F(\kvec) e^{2\pi i \rvec\kvec} d\kvec \nonumber \\
F(\kvec) & = & \mathcal{F}^{-1} (\rvec)=  \int_{-\infty}^{\infty} f(\rvec) e^{-2\pi i \rvec\kvec} d\rvec.
\end{eqnarray}
With two functions $f(\rvec), g(\rvec)$, one can form the
convolution, defined by,
\begin{equation}
f\star g = \int_{-\infty}^{\infty} f(\rvec^{\prime}) g(\rvec-\rvec^{\prime}) d\rvec^{\prime}.
\end{equation}
The convolution theorem states that FT of convolution is just the product of individual FTs,
\begin{eqnarray}
\mathcal{F}(f \star g) & = & F(\kvec).G(\kvec) \nonumber \\
f \star g & = & \mathcal{F}^{-1}(F(\kvec).G(\kvec)).
\end{eqnarray}
One can make use of this theorem to rewrite the ESP integral as,
\begin{equation}
v_{\nu\eta}(\rvec)=\int\frac{\chi_{\nu}(\rvec^{\prime})\chi_{\eta}(\rvec^{\prime})}{|\rvec-\rvec^{\prime}|}d\rvec^{\prime}
=\int\frac{\chi_{\mu\nu}(\rvec^{\prime})}{|\rvec-\rvec^{\prime}|}d\rvec^{\prime}=\chi_{\nu\eta}(\rvec)\star v^{c}(\rvec).
\end{equation}
The spin indices have been removed for simplicity. The last expression is in terms of convolution integral,
where $\chi_{\nu\eta}$ denotes simple multiplication of two AO basis and $v^{c}(\rvec)$ represents Coulomb interaction
kernel. Now one can invoke FCT to further simplify this integral,
\begin{equation}
v_{\nu\eta} (\rvec) =\mathcal{F}^{-1}(v^{c}\{\kvec) \chi_{\nu\eta}(\kvec)\} \quad \mathrm{where} \quad \chi_{\nu\eta}(\kvec)
=\mathcal{F}\{\chi_{\nu\eta}(\rvec)\}.
\end{equation}
Here $v^{c}(\kvec)$ and $\chi(\kvec)$ stand for Fourier integrals of Coulomb kernel and AO basis respectively. The main
concern lies in an
accurate mapping of the former, which has singularity at $\rvec \to 0$. In order to alleviate this problem, we apply a
simple RS technique from the works of \citep{gill96, gill96a}, expanding the kernel into long- and short-range components
with a suitably chosen RS parameter ($\zeta$). This gives rise to the following expression,
\begin{eqnarray}
v^{c}(\rvec)=\frac{\mathrm{erf}(\zeta\rvec)}{\rvec}+\frac{\mathrm{erfc}(\zeta\rvec)}{\rvec} \nonumber \\
v^{c}(\rvec_g)= v^{c}_{long}(\rvec_g) + v^{c}_{short}(\rvec_g).
\end{eqnarray}
In the above equation, $\mathrm{erf}(x)$ and $\mathrm{erfc}(x)$ denote error function and its complement respectively, while
the second expression is written in real-space CCG. The Fourier integral of Coulomb kernel can be separated out as follows:
(i) FT of short-range part is treated analytically and (ii) long-range portion is computed directly from FFT of
corresponding real-space CCG values. Then the remaining problem lies in finding an optimum value of parameter
$\zeta_{\mathrm{opt}}$ for successful mapping of Coulomb kernel in CCG from \emph{first principles}. In this regard we
propose a simple procedure which is found to be sufficiently accurate for all practical
purposes (as exemplified in results that follows). This prompts us to write,
\begin{equation}
\zeta_{\mathrm{opt}} \equiv \underset{\zeta }{\mathrm{opt}} \ E_{\sigma}^{\mathrm{x}} \equiv \underset{\zeta }{\mathrm{opt}} \
E_{\mathrm{tot}} = \underset{N_x,N_y,N_z} {\mathrm{opt}} E_{\emph{tot}}, \quad \mathrm{at \ fixed} \ h_{\rvec},
\end{equation}
using a suitably defined constraint \cite{martyna99} $(\zeta \times L =7 )$, where $L\ (= N_\rvec h_\rvec; \ \rvec \in
\{x,y,z\})$
is the smallest side of simulating box. This technique was already implemented in grid optimization procedure
with respect to total energy in a non-uniform CCG \citep{ghosal16,ghosal18}.

From the foregoing discussion it is clear that, each ESP integral involves only a combination of FFT (two forward
and one backward transformation simultaneously) and hence scales as $\mathcal{O}(N_{g} \log N_g)$. It is to be noted
that this prescription gives a logarithmic scaling with grid size, in contrast to the quadratic scaling (with respect
to primitive basis functions, $N_P$) in SNR scheme.

\section{3. Computational details}
The above described strategy for HF exchange component is implemented in case of three global hybrid functionals, namely,
B3LYP, PBE0 and BHLYP, containing a variable amount of former with conventional DFT XC functional. As such, the XC
energy corresponding to these three functionals are defined as follows,
\begin{equation}
E_{\mathrm{xc}}^{\mathrm{B3LYP}} = (1-a_0)E^{\mathrm{x}}_{\mathrm{LSDA}}+a_0 E^{\mathrm{x}}_{\mathrm{HF}}+
a_{\mathrm{x}} E^{\mathrm{x}}_{\mathrm{B88}}+a_{\mathrm{c}} E^{\mathrm{c}}_{\mathrm{LYP}} + (1-a_{\mathrm{c}})
E^{\mathrm{c}}_{\mathrm{VWN}},
\end{equation}
\begin{equation}
E_{\mathrm{xc}}^{\mathrm{PBE0}} = b_0 E^{\mathrm{x}}_{\mathrm{HF}} + (1-b_0)E^{\mathrm{x}}_{\mathrm{PBE}} +
E^{\mathrm{c}}_{\mathrm{PBE}},
\end{equation}
and
\begin{equation}
E_{\mathrm{xc}}^{\mathrm{BHLYP}} = (1-c_0)E^{\mathrm{x}}_{\mathrm{LSDA}}+
c_0 E^{\mathrm{x}}_{\mathrm{HF}}+ E^{\mathrm{c}}_{\mathrm{LYP}}.
\end{equation}
Following the work of Stephens \emph{et al.}, \citep{stephens94} values of $a_0, a_{\mathrm{x}}, a_{\mathrm{c}}$
are $0.2$, $0.72$, $0.81$ for B3LYP, whereas in case of PBE0\citep{perdew96a}, $b_0=0.25$. Note that the amount of HF
exchange contribution in PBE0 is slighter higher than that of B3LYP, but even larger amount ($c_0=0.5$) is attributed in
BHLYP. These are computed using KS orbitals obtained from solution of Eq.~(9) in real-space CCG through the procedure
delineated in previous subsections.

\begingroup
\begin{table}[t]
\caption{\label{tab:table1} Convergence of HF energy$^{{\dagger},\ddagger}$ of HCl ($R=1.275 \textup{\AA}$) in the grid
($h_r=0.3$). All results are in a.u.}
\begin{tabular} {ccclcccrl}
\cline{1-9}
 \multicolumn{4}{c}{Set~I }  & & \multicolumn{4}{c}{Set~II}      \\
\cline{1-4} \cline{6-9}
$N_x$   &   $N_y$   &    $N_z$  &  $ \langle E \rangle $ &    & $N_x$   &   $N_y$   &    $N_z$  &  $ \langle E \rangle$ \\
\cline{1-9}
32 & 32 & 32 & $-$15.24881 &  &36 & 36 & 60 & $-$15.27363 \\
-- & -- & 36 & $-$15.26277 &  &40 & 40 &  -- & $-$15.27496 \\
-- & -- & 40 & $-$15.26604 &  &44 & 44 &  -- & $-$15.27521 \\
-- & -- & 44 & $-$15.26684 &  &46 & 46 &  -- & $-$15.27525 \\
-- & -- & 48 & $-$15.26703 &  &48 & 48 &  -- & $-$15.27526 \\
-- & -- & 52 & $-$15.26707 &  &50 & 50 & -- & $-$15.27527 \\
-- & -- & 56 & $-$15.26708 &  &52 & 52 & -- & $-$15.27527 \\
-- & -- & 60 & $-$15.26708 &  &54 & 54 & -- & $-$15.27527 \\
-- & -- & 64 & $-$15.26708 &  &56 & 56 & -- & $-$15.27527 \\
\cline{1-9}
\end{tabular}
\begin{tabbing}
$^{\dagger}$SNR value: $-$15.27527 a.u.  \hspace{50pt} \=
$^{\ddagger}$Energy from GAMESS package \cite{schmidt93}: $-$15.27528 a.u. \\
\end{tabbing}
\end{table}
\endgroup

The present calculations employ following effective core potential (ECP) basis sets: SBKJC \citep{stevens84} for species
containing Group-II elements and LANL2DZ \citep{hay85c} for Group-III or higher group elements. These are adopted from EMSL
Basis Set Library\citep{feller96}. All one-electron integrals are generated by standard recursion relations \citep{obara86}
using Cartesian Gaussian-type orbitals as primitive basis functions. The norm-conserving pseudopotential matrix elements in
contracted basis are imported from GAMESS\citep{schmidt93} suite of program package. The relevant LDA- and GGA-type of
functionals in connection with B3LYP, BHLYP and PBE0 are: (i) Vosko-Wilk-Nusair (VWN)--with the homogeneous electron gas
correlation proposed in parametrization formula V \citep{vosko80} (ii) B88--incorporating Becke\citep{becke88a} semi-local
exchange (iii) Lee-Yang-Parr (LYP) \citep{lee88} semi-local correlation (iv) Perdew-Burke-Ernzerhof (PBE)\citep{perdew96}
functional for semi-local exchange and correlation. All these are adopted from density functional repository
program\citep{repository} except LDA. The convergence criteria imposed in this communication are slightly
tighter than our earlier work\citep{roy08, roy08a, roy10, roy11}; this is to generate a more accurate orbital
energies. Changes in following quantities were followed during the self-consistent field process, \emph{viz.,} (i)
orbital energy difference between two successive iterations and (ii) absolute deviation in a density matrix element. They both
were required to remain below a prescribed threshold set to $10^{-8}$ a.u.; this ensured that total energy maintained a
convergence of at least this much, in general. In order to perform discrete Fourier transform, standard FFTW3
package\citep{fftw05} was invoked. The resulting generalized matrix-eigenvalue problem is solved through standard LAPACK
routine\citep{anderson99} accurately and efficiently. All molecular calculations are performed in their experimental
geometries, taken from NIST database\citep{johnson16}, excepting those in Table~VI (see discussion later). Other
details including scaling properties may be found in references \citep{roy08, roy08a, roy10, roy11, ghosal16, ghosal18}.

\begingroup
\begin{table}[t]
\caption{\label{tab:table2} Convergence of KS energy$^{{\dagger},\ddagger}$ of Cl$_2$ ($R=4.2$ a.u.) in the grid ($h_r=0.3$)
using B3LYP XC functional. All results are in a.u.}
\begin{tabular} {ccclcccrl}
\cline{1-9}
 \multicolumn{4}{c}{Set~I }  & & \multicolumn{4}{c}{Set~II}      \\
\cline{1-4} \cline{6-9}
$N_x$   &   $N_y$   &    $N_z$  &  $ \langle E \rangle $ &    & $N_x$   &   $N_y$   &    $N_z$  &  $ \langle E \rangle$ \\
\cline{1-9}
32 & 32 & 32 & $-$28.49544 &  &36 & 36 & 76 & $-$29.78914 \\
-- & -- & 40 & $-$29.59219 &  &40 & 40 &  -- & $-$29.79282 \\
-- & -- & 48 & $-$29.75311 &  &44 & 44 &  --  & $-$29.79363 \\
-- & -- & 52 & $-$29.76710 &  &48 & 48 &  --  & $-$29.79380 \\
-- & -- & 56 & $-$29.77140 &  &52 & 52 &  -- & $-$29.79384 \\
-- & -- & 68 & $-$29.77304 &  &56 & 56 &  -- & $-$29.79384 \\
-- & -- & 72 & $-$29.77306 &  &60 & 60 &  -- & $-$29.79384 \\
-- & -- & 76 & $-$29.77306 & &64 & 64 & -- & $-$29.79384 \\
-- & -- & 80 & $-$29.77306 &  &68 & 68 &  -- & $-$29.79384 \\
\cline{1-9}
\end{tabular}
\begin{tabbing}
$^{\dagger}$SNR value: $-$29.79384 a.u.  \hspace{50pt} \=
$^{\ddagger}$Energy from GAMESS package \cite{schmidt93}: $-$29.79390 a.u. \\
\end{tabbing}
\end{table}
\endgroup

\section{4. Result and Discussion}
Before proceeding for main results, at first it may be appropriate to discuss the grid optimization (in CCG) in
terms of the RS parameter, $\zeta_{\mathrm{opt}}$. This is illustrated through convergence of total energy, as delineated in
Eq.~(23). Table~1 shows this for a diatomic molecule HCl at its experimental geometry
(bond length $1.275$\textup{\AA}). The HF energies are provided in non-uniform grid
with respect to \emph{sparsity} (regulated by $N_x, N_{y}, N_{z}$) for a fixed grid spacing (determined by $h_r$, chosen as
0.3), employing the NR and SNR schemes. The presentation strategy is similar to that in our
previous works \citep{ghosal16, ghosal18}; accordingly we first vary $N_z$, the number of grid points along inter-nuclear axis,
keeping the same along $xy$ plane static at certain reasonable value, say $N_x=N_y=32$. As $N_z$ is gradually increased from 32
to 64 with an increment of 4, there is a smooth convergence in energy at around $N_z=56$ with a difference in total energy
(we term it as grid accuracy) of about $ 10^{-6}$ a.u., between two successive steps. In the beginning, when $N_z$ goes
through 40-44-48-52, one notices slow improvement in energy; after that it eventually attains the convergence for $N_z$ at around
60. Then in Set~II in right-hand side, we vary $N_{x}, N_{y}$ along $xy$ plane keeping $N_z$ fixed at its previously determined
value of $60$. Now we can see that convergence in energy takes place at $N_{x}=N_{y}=50$ with same grid accuracy of Set~I.
Next, we have repeated the same calculation through the SNR scheme (ESP integral was approached analytically using recursion
relations \citep{obara86} containing expensive incomplete Gamma functions); the corresponding electronic energy value is supplied
in footnote of this table. For sake of completeness, energy is also quoted from GAMESS \cite{schmidt93} in footnote. It
is quite encouraging to note that the two routes provide completely
identical results as that in the reference. A similar exercise was undertaken for Cl$_2$ ($R_{eq}=4.2$ a.u.), and the
conclusion remains same, i.e., converged total energies in this case were found to be as follows: $-$29.34810 (NR), $-$29.34809
(SNR), $-$29.34813 (GAMESS) a.u. Thus it is evident that our direct, NR approach using RS-FCT in CCG, can correctly estimate the
optimized RS-parameter, $\zeta_{\mathrm{opt}}$, and consequently, HF energy.

\begin{table}[t]
\caption{\label{tab:table3} HF, B3LYP PBE0 energies (a.u.) of selected atoms.
$E_{\mathrm{diff}}=|E{\mathrm{NR}}-E_{\mathrm{SNR}}|$.}
\resizebox{\textwidth}{90pt}{
\begin{tabular} {l|ccc|ccc|ccc}
\cline{1-10}
Atom & \multicolumn{9}{c}{$-\langle E \rangle$} \\
\cline{2-10}
  & \multicolumn{3}{c}{HF} & \multicolumn{3}{c}{B3LYP} & \multicolumn{3}{c}{PBE0} \\
 \cline{2-4}  \cline{5-7}  \cline{8-10}
  & NR & SNR & E$_{\mathrm{diff}}$ & NR & SNR& E$_{\mathrm{diff}}$
& NR & SNR  & E$_{\mathrm{diff}}$ \\
\cline{1-10}
Be & 0.96019 & 0.96019 & 0.00000 & 0.99386 & 0.99386 & 0.00000 & 0.99598 & 0.99598 & 0.00000 \\
B  & 2.53426 & 2.53426 & 0.00000 & 2.59552 & 2.59552 & 0.00000 & 2.59751 & 2.59751 & 0.00000 \\
C  & 5.30354 & 5.30354 & 0.00000 & 5.39301 & 5.39301 & 0.00000 & 5.39603 & 5.39603 & 0.00000 \\
N  & 9.61972 & 9.61969 & 0.00003 & 9.73283 & 9.73282 & 0.00001 & 9.74190 & 9.74189 & 0.00001 \\
O  &15.61720 &15.61681 & 0.00039 &15.80556 &15.80549 & 0.00007 &15.80351 &15.80341 & 0.00010 \\
Al & 1.86947 & 1.86947 & 0.00000 & 1.92379 & 1.92379 & 0.00000 & 1.93075 & 1.93075 & 0.00000 \\
Si & 3.67570 & 3.67570 & 0.00000 & 3.75077 & 3.75077 & 0.00000 & 3.76281 & 3.76281 & 0.00000 \\
P  & 6.31523 & 6.31523 & 0.00000 & 6.40580 & 6.40580 & 0.00000 & 6.42550 & 6.42550 & 0.00000 \\
S  & 9.87464 & 9.87464 & 0.00000 &10.01827 &10.01827 & 0.00000 &10.03519 &10.03519 & 0.00000 \\
Cl &14.68130 &14.68129 & 0.00001 &14.87170 &14.87170 & 0.00000 &14.88873 &14.88873 & 0.00000 \\
Ga & 1.94526 & 1.94526 & 0.00000 & 2.00195 & 2.00195 & 0.00000 & 2.00831 & 2.00831 & 0.00000 \\
Ge & 3.59814 & 3.59814 & 0.00000 & 3.67466 & 3.67466 & 0.00000 & 3.68693 & 3.68693 & 0.00000 \\
As & 5.95615 & 5.95615 & 0.00000 & 6.04808 & 6.04808 & 0.00000 & 6.06823 & 6.06823 & 0.00000 \\
Se & 9.01108 & 9.01108 & 0.00000 & 9.15439 & 9.15439 & 0.00000 & 9.17235 & 9.17235 & 0.00000 \\
Br &12.91872 &12.91872 & 0.00000 &13.10652 &13.10652 & 0.00000 &13.12535 &13.12535 & 0.00000 \\\cline{1-10}
\end{tabular} }
\end{table}

Next we move towards implementation of hybrid functional, B3LYP using above procedure. For this, we again consider a
closed shell molecule, such as chlorine (Cl$_2$) at its experimental bond length of $4.2$ a.u., along $z$ axis as a
specimen case. The formal convergence and stability of solution is illustrated in Table~2 at a grid spacing of
$h_{r} = 0.3$. Same convergence criteria of previous table were imposed here as well. Once again the NR, SNR (provided
in footnote) and GAMESS (also in footnote) energies excellently match with each other.
An analogous study on HCl produced same result in energy, i.e., converged values in this case were $-$15.50844 (NR),
$-$15.50843 (SNR), $-$15.50845 (GAMESS) a.u. This therefore establishes that our simple RS-FCT scheme
in CCG can produce correct, meaningful results for XC functionals containing HF exchange.

In order to extend the scope and applicability of proposed scheme, we now report total energies for a set of 15 atoms and
15 molecules (along with test molecular systems) in Tables~3 and 4 respectively. To put things in perspective, we compare
the NR and SNR results for three sets of calculations, namely HF, B3LYP and PBE0. Both tables engage identical basis set,
pseudopotential and convergence (both grid and SCF) criteria. In all occasions, the maximum absolute deviation in energy (labeled
$\mathrm{E}_{\mathrm{diff}}$) between NR and SNR, are produced side by side for easy comparison. The overall agreement between
these two sets of results are in general excellent. A closer observation at Table~3 reveals that, the two energies are virtually
indistinguishable for all the species with exception of O atom, where absolute deviation remains well below
0.0004 a.u. Similarly in Table~4 again, these two results coincide for 13 molecules; in this case the largest discrepancy
(0.0001 a.u.) occurs for CH$_4$. Note that, since the previous tables have established that NR and SNR energies are very close to
reference GAMESS results (which have also been verified in these cases), we do not reproduce those energies any more.

\begin{table}[t]
\caption{\label{tab:table4} HF, B3LYP PBE0 energies (a.u.) of molecules.
$E_{\mathrm{diff}}=|E{\mathrm{NR}}-E_{\mathrm{SNR}}|$.}
\resizebox{\textwidth}{70pt}{
\begin{tabular} {lccc|ccc|ccc}
\cline{1-10}
Molecule & \multicolumn{9}{c}{$-\langle E \rangle$} \\
\cline{2-10}
  & \multicolumn{3}{c}{HF} & \multicolumn{3}{c}{B3LYP} & \multicolumn{3}{c}{PBE0} \\
 \cline{2-4}  \cline{5-7}  \cline{8-10}
  & NR & SR & E$_{\mathrm{diff}}$ & NR & SR & E$_{\mathrm{diff}}$ & NR & SR & E$_{\mathrm{diff}}$ \\
\cline{1-10}
Cl$_2$ & 29.34810 & 29.34809 & 0.00001 & 29.79384 & 29.79384 & 0.00001 & 29.82896 & 29.82896 & 0.00000 \\
Br$_2$ & 25.82400 &  25.82400 & 0.00000 & 26.25227 & 26.25227 & 0.00000 & 26.29481 & 26.29481 & 0.00000 \\
I$_2$  & 22.30720 & 22.30720 & 0.00000 & 22.71498 & 22.71498 & 0.00000 & 22.76592 & 22.76592 & 0.00000 \\
HCl    & 15.27527 & 15.27527 & 0.00000 & 15.50844 & 15.50843 & 0.00001 & 15.52803 & 15.52803 & 0.00000 \\
HBr    & 13.49506 & 13.49506 & 0.00000 & 13.72537 & 13.72537 & 0.00000 & 13.74603 & 13.74603 & 0.00000 \\
HI     & 11.72279 & 11.72279 & 0.00000 & 11.94619 & 11.94619 & 0.00000 & 11.96939 & 11.96939 & 0.00000 \\
H$_2$S & 11.02605 & 11.02605 & 0.00000 & 11.25511 & 11.25511 & 0.00000 & 11.27394 & 11.27394 & 0.00000 \\
H$_2$Se  & 10.14548 & 10.14548 & 0.00000 & 10.37371 & 10.37371 & 0.00000 & 10.39215 & 10.39215 & 0.00000 \\
PH$_3$   & 8.01636 & 8.01636 & 0.00000 & 8.23350 & 8.23350 & 0.00000 & 8.24942 & 8.24942 & 0.00000 \\
CH$_3$Cl & 21.87306 & 21.87306 & 0.00000 & 22.29768 & 22.29768 & 0.00000 & 22.33525 & 22.33525 & 0.00000\\
CH$_4$   & 7.78878 & 7.78888 & 0.00010 & 8.00843 & 8.00846 & 0.00003 & 8.02684 & 8.02686 & 0.00002 \\
SiH$_3$Cl & 20.19863 & 20.19862 & 0.00001 & 20.58442 & 20.58441 & 0.00001 & 20.61885 & 20.61885 & 0.00000\\
Si$_2$H$_6$ & 10.93377 & 10.93377 & 0.00000 & 11.28249 & 11.28249 & 0.00000 & 11.31207 & 11.31207 & 0.00000 \\
P$_4$ & 25.13843 & 25.13843 & 0.00000 & 25.81974 & 25.81974 & 0.00000 & 25.91452 & 25.91452 & 0.00000 \\
SiH$_4$ & 6.03289 & 6.03289 & 0.00000 & 6.22621 & 6.22621 & 0.00000 & 6.23634 & 6.23634 & 0.00000 \\
\cline{1-10}
\end{tabular}}
\end{table}

In the last part of this study, in Table~5, we investigate the highest occupied molecular orbital (HOMO) energies of same
set of atoms and molecules of Tables 3 and 4. As expected from previous tables, NR and SNR
schemes produce practically identical results in these cases too. Hence now onwards, we present
only NR results to save space. The available theoretical and experimental IPs are recorded for easy comparison.
It is revealed that even though the theory does not account for electron correlation and relaxation effects, HF
HOMO energies still compare very favorably with experiment than any of the three DFT functionals considered. This is apparent
from the respective deviations; one notices underestimation of 14-41\%, 9-37\%, 1-28\% for B3LYP, PBE0 and BHLYP functionals,
whereas HF results show absolute deviation by 0-14\%. However, it is interesting to note that, amongst the three functionals, the
deviations fall from B3LYP to BHLYP, mainly because the fractional content of HF exchange (which has a pivotal role in
determining correct asymptotic nature at long range) gradually increases in that order. This observation along with a
consideration of total energies (from Tables~3 and 4), suggests that hybrid functionals provide a systematic improvement
over the HF result. These observations are
further complemented in Table~6, where a similar analysis is made for some $\pi$-electron molecules (simple, conjugated as well
as aromatic). A very important aspect of such species is the so-called fundamental gap, which is the difference in energy
between HOMO and LUMO. A satisfactory estimation of this gaps requires accurate knowledge of these two orbital energies.
The current table is an attempt towards this direction. In this case, we have taken the calculated geometry (using B3LYP XC
functional and cc-PVTZ basis set) from NIST \cite{johnson16} data base. As expected, we witness similar kind of pattern in
calculated IP values as in previous table; with respect to experimental/theoretical values, the range of underestimation for hybrid
functionals are: 18-32\%, 15-28\% and 11-19\% respectively, for B3LYP, PBE0 and BHLYP.

\begingroup
\begin{table}[tp]
\caption{\label{tab:table5} Negative HOMO energies, $-\epsilon_{\mathrm{HOMO}}$ (in a.u.) for selected atoms and molecules
using HF, B3LYP, PBE0 and BHLYP XC functionals.}
\resizebox{\textwidth}{70pt}{
\begin{tabular} {lccccccccccc}
\cline{1-12}
Atom & \multicolumn{5}{c}{$-\epsilon_{\mathrm{HOMO}}$(a.u.)} & Molecule & \multicolumn{5}{c}{$-\epsilon_{\mathrm{HOMO}}$(a.u.)} \\
\cline{2-6} \cline{8-12}
  & HF & B3LYP & PBE0 & BHLYP & Expt.$^{\dagger}$ & & HF & B3LYP & PBE0 & BHLYP & Expt.$^{\ddagger}$ \\
\cline{1-12}
Be & 0.3090 & 0.2291 & 0.2387 & 0.2650 & 0.3426 & Cl$_2$ & 0.4786 & 0.3274 & 0.3433 & 0.3947 & 0.4219 \\
B  & 0.3080 & 0.1830 & 0.1929 & 0.2348 & 0.3050 & Br$_2$ & 0.4139 & 0.2889 & 0.3034 & 0.3456 & 0.3864 \\
C  & 0.4313 & 0.2615 & 0.2771 & 0.3290 & 0.4138 & I$_2$ & 0.3695 & 0.2658 & 0.2793 & 0.3138 & 0.3429 \\
N  & 0.5657 & 0.3497 & 0.3707 & 0.4338 & 0.5341 & HCl & 0.4772 & 0.3274 & 0.3425 & 0.3937 & 0.4700 \\
O  & 0.5098 & 0.3247 & 0.3379 & 0.4090 & 0.5005 & HBr & 0.4302 & 0.3030 & 0.3170 & 0.3602 & 0.4303 \\
Al & 0.2095 & 0.1291 & 0.1408 & 0.1621 & 0.2200 & HI & 0.3875 & 0.2804 & 0.2932 & 0.3295 & 0.3815 \\
Si & 0.3029 & 0.1937 & 0.2088 & 0.2374 & 0.3000 & H$_2$S & 0.3895 & 0.2617 & 0.2746 & 0.3191 & 0.3851 \\
P  & 0.3901 & 0.2545 & 0.2732 & 0.3076 & 0.3854 & H$_2$Se & 0.3613 & 0.2466 & 0.2593 & 0.2987 & 0.3649 \\
S  & 0.3631 & 0.2506 & 0.2625 & 0.3043 & 0.3807 & PH$_3$ & 0.3849 & 0.2675 & 0.2781 & 0.3200 & 0.3626 \\
Cl & 0.4731 & 0.3294 & 0.3442 & 0.3944 & 0.4766 & CH$_3$Cl & 0.4340 & 0.2946 & 0.3084 & 0.3567 & 0.4149 \\
Ga & 0.2058 & 0.1263 & 0.1381 & 0.1590 & 0.2205 & CH$_4$ & 0.5416 & 0.3882 & 0.4013 & 0.4555 & 0.4998 \\
Ge & 0.2844 & 0.1825 & 0.1974 & 0.2232 & 0.2903 & SiH$_3$Cl & 0.4509 & 0.3149 & 0.3288 & 0.3754 & 0.4281 \\
As & 0.3665 & 0.2426 & 0.2605 & 0.2910 & 0.3607 & Si$_2$H$_6$ & 0.4068 & 0.3043 & 0.3152 & 0.3516 & 0.3870 \\
Se & 0.3319 & 0.2337 & 0.2451 & 0.2815 & 0.3584 & P$_4$ & 0.3844 & 0.2921 & 0.3075 & 0.3362 & 0.3381 \\
Br & 0.4206 & 0.3010 & 0.3144 & 0.3563 & 0.4341 & SiH$_4$ & 0.4871 & 0.3576 & 0.3687 & 0.4149 & 0.4520 \\
\cline{1-12}
\end{tabular}}
\begin{tabbing}
$^{\dagger}$Optical spectroscopy\citep{johnson16}. \= \hspace{170pt}
$^{\ddagger}$Photo-electron spectroscopy \citep{johnson16}.
\end{tabbing}
\end{table}
\endgroup

A few remarks may now be made before passing. During the calculation, it is realized that the computational burden of our NR route is
quite less (about eight times) than the SNR one. It occurs because of the obvious reason that the former route obviates the
necessity of two primitive loops in Eq.~(3); instead the ESP integral is directly calculated in CCG using contracted Gaussian
functions. In order to make the SNR route attractive and fruitful for practical applications, an elegant solution was proposed
\citep{liu17} in the form of some optimal
recurrence relations for analytical evaluation of ESP integral \citep{liu17}. We have not attempted to incorporate this
procedure in this communication, as this is aside the main theme and purpose of this work. Here, we just wanted to establish the
validity, efficacy and feasibility of a simple direct NR scheme in the broader context of certain DFT functionals for
large-scale calculations. As demonstrated in this work, the NR route is quite efficient, and may be favorable, especially
for larger basis set. The accuracy and ease of implementation augurs well for its further application in the development
of range-separated hybrid functional from \emph{first principles}. This leads to a more complete representations of HF exchange
in asymptotic region which closes the gap between theoretical and experimental result, keeping the computational overhead at same
level as in commonly used hybrid functionals\citep{kronik12}, and currently we are working on it.

\begin{table}[tp]
\caption{\label{tab:table6}  Negative HOMO energies, $-\epsilon_{\mathrm{HOMO}}$ (in a.u.) for selected $\pi$-electron molecules
using HF, B3LYP, PBE0 and BHLYP XC functionals.}
\begin{tabular} {lcccccc}
\cline{1-7}
Molecule & \multicolumn{6}{c}{$-\epsilon_{\mathrm{HOMO}}$(a.u.)} \\
\cline{2-7}
	& HF & B3LYP & PBE0 & BHLYP & Theory\citep{johnson16} & Expt.$^{\dagger}$ \\
\cline{1-7}
Ethylene & 0.3686 & 0.2649 & 0.2796 & 0.3140 & 0.376$^a$ & 0.3859 \\
Propene & 0.3544 & 0.2503 & 0.2645 & 0.2994 & 0.354$^a$ & 0.3565 \\
1,3-Butadiene (E) & 0.3188 & 0.2308 & 0.2444 & 0.2734 & 0.332$^a$ & 0.3333 \\
1,3-Pentadiene (E) & 0.3053 & 0.2175 & 0.2309 & 0.2600 & 0.306$^b$ & 0.3157 \\
1,3,5-Hexatriene (E) & 0.2862 & 0.2090 & 0.2218 & 0.2471 & 0.294$^c$ & 0.3050 \\
1,3-Cyclo-pentadiene & 0.3051 & 0.2142 & 0.2283 & 0.2584 & 0.301$^d$ & 0.3135 \\
Benzene & 0.3362 & 0.2491 & 0.2637 & 0.2920 & 0.329$^e$ & 0.3399 \\
Thiophene & 0.3338 & 0.2400 & 0.2544 & 0.2856 & 0.327$^f$ & 0.3271 \\
Acetylene & 0.4110 & 0.2928 & 0.3073 & 0.3473 & 0.409$^a$ & 0.4189 \\
Acetaldehyde & 0.4628 & 0.3072 & 0.3193 & 0.3764 & 0.425$^a$ & 0.3758 \\
\cline{1-7}
\end{tabular}
\begin{tabbing}
\hspace*{9cm}\= \kill
$^{\dagger}$Photo-electron spectroscopy \citep{johnson16}. \>
$^a$CCSD result using cc-PVTZ basis. \\
$^b$CCD result using 6-31G$^\star$ basis.  \>
$^c$QCISD result using 6-311G$^{\star \star}$ basis. \\
$^d$CCD result using 6-31G$^{\star \star}$ basis. \>
$^e$CCSD result using 6-31G$^{\star \star}$ basis. \\
$^e$CCSD result using 6-311G$^{\star \star}$ basis.
\end{tabbing}
\end{table}

\section{4. Future and Outlook}
We have demonstrated the feasibility and practicability of a direct NR scheme for computation of HF exchange energy
density, energy and matrix in real-space CCG. This was applied for a host of atoms and molecules; properties such as total
energy and orbital energies were offered, within a pseudopotential KS-DFT. Besides HF components, these were also presented
for B3LYP, PBE0 and BHLYP hybrid functionals. The success of
this approach relies on accurate estimation of ESP integral, which in turn depends on optimization of the RS
parameter in Coulomb kernel. The obtained results for all
these species from both NR and SNR scheme, are in excellent agreement with each other. A consideration of scaling
properties suggests that this approach may be quite helpful in large-scale DFT calculations involving orbital-dependent
functionals. Application to range-separated hybrid, hyper as well as local hybrid XC functionals would further enhance
its scope in a wide range of systems. It may also be desirable to examine its performance in various important physio-chemical
properties of many-electron systems.

\section{Acknowledgement}
AG is grateful to UGC for a Senior Research Fellowship (SRF). TM acknowledges IISER Kolkata for a Junior Research
fellowship (JRF). Financial support from DST SERB, New Delhi, India (sanction order number EMR/2014/000838) is gratefully
acknowledged.


\section{Conclusion}
To address the problem of zero pronoun translation, we proposed zero pronoun data augmentation.
Through the analysis with the Japanese-English conversational parallel corpus, we showed that zero pronouns in Japanese sentences can be predicted to some extent from local context within the sentence.
In the conversational translation experiment, we compared a translation model trained on the augmented data with the baseline and demonstrate that our method significantly improves the accuracy of zero pronoun translation.

Nevertheless, zero pronoun data augmentation does not solve the cases where the information necessary for zero pronoun translation exists outside the sentence.  Also, the analysis suggests that local context is useful for predicting frequent pronouns such as the first and second-person pronouns, but not for the third-person pronouns.
An interesting avenue for future work is to explicitly incorporate discourse-level contextual information such as topics or people involved in the conversation into the translation models.

\section{Related Work}
\subsection{Contextual Neural Machine Translation}
As the quality of single-sentence machine translation has improved dramatically with the advent of neural machine translation \citep{NIPS2014_a14ac55a,NIPS2017_3f5ee243}, translation models that take wider contexts into account have seen a surge of interest \citep{Jean2017DoesNM,bawden-etal-2018-evaluating,voita-etal-2019-good,voita-etal-2019-context,ma-etal-2020-simple,saunders-etal-2020-using}.
In contrast to the studies trying to incorporate information outside the sentence, in this work, we propose a method to improve zero-pronoun translation by only considering the information within the sentence, but we also explore the effect of combining our method with a contextual machine translation model.

\subsection{ZP Resolution in Japanese}
In some languages, pronouns are sometimes omitted when they are inferable from the context. Such languages are called pro-drop languages and the omitted pronouns are called ZPs.

The translation of ZPs poses a challenge when the corresponding pronoun is syntactically required on the target language side: the model has to infer the omitted pronoun.
The task of identifying the omitted pronouns is called ZP resolution and for Japanese, this has been a long-standing problem \citep{isozaki-hirao-2003-japanese,sasano-etal-2008-fully,imamura-etal-2009-discriminative,shibata-kurohashi-2018-entity}.
Japanese is one of the most difficult languages because Japanese words usually do not have any inflectional forms that depend on the omitted pronoun, unlike other pro-drop languages such as Portuguese and Spanish in which ZPs can be inferred from the grammatical case of other words.

Still, Japanese sentences sometimes contain expressions indicative of the missing pronoun. For example,  Japanese honorifics naturally indicate the subject is the second person.
In this work, we do not explicitly solve ZP resolution but let the translation model learn heuristic relations between ZPs and local context within the sentence \citep{hangyo-etal-2013-japanese,kudo-etal-2015-anlp} and produce appropriate English pronouns.

\subsection{ZPs in Translation}
In the context of statistical machine translation, Japanese ZPs are explicitly predicted by considering verbal semantic attributes \citep{nakaiwa-ikehara-1992-zero}, local context in the source and target sentence \citep{kudo-etal-2015-anlp}, and incorporated into the resulting translation.

On the other hand, in neural machine translation, the missing pronouns can be automatically inferred by the translation model because of the nature of end-to-end learning, although the correctness cannot be guaranteed. To improve the quality of ZP translation, previous studies have explored a multi-task approach with ZP prediction \citep{wang-etal-2016-novel,wang-etal-2019-one}.

In this study, we propose a ZP data augmentation method to provide additional training signals useful to correctly translate ZPs.

\section{Introduction}
While neural machine translation (NMT) has demonstrated high performance in single-sentence translation, it is still challenging to handle linguistic phenomena involving discourse contexts.
One such issue is the translation of {\it zero pronouns} (ZP) in Japanese-to-English translation.
In Japanese, subjects and objects are often omitted when the listener can infer them from the context. However, when translating them into English, the omitted words must be explicitly translated in most cases. For example, in the following sentence, the subject omitted in Japanese is the first person, and \textit{I} has to be output in English.

\begin{table}[h]
  \centering
  \begin{tabular}{ll}
  \ja{うなぎが}             & \ja{食べたいな}    \\
  unagi-ga         & tabe-tai-na              \\
  eel-\texttt{OBJ}          & eat-want-\texttt{PARTICLE}  \\
  \multicolumn{2}{l}{I feel like eating eel.}
\end{tabular}
\end{table}

The prediction of ZPs, essentially, requires understanding the topic and old information in the discourse, or referring to the world knowledge. On the other hand, linguistic information within the sentence may provide some clues \citep{kudo-etal-2015-anlp}.
For example, in the sentence above, the auxiliary verb \ja{たい} ({\it want}) suggests that the sentence expresses a subjective statement and thus the missing pronoun is the first person. Here we refer to such information as {\it local context}.

Correlations between local context and ZPs can be learned by the standard single-sentence neural machine translation, but it may not be possible under low-resource conditions. For example, the translation of conversations, which usually contain a large number of ZPs, is currently one of the under-resourced domains.

\begin{figure}[t]
\centering
\includegraphics[width=7.0cm]{data/zu.png}
\caption{The proposed method: ZP data augmentation}
\label{fig:zu}
\end{figure}

To address this problem, we propose {\bf zero pronoun data augmentation} to facilitate learning correlations between local context and ZPs (Figure \ref{fig:zu}).
We augment the training data by deleting personal pronouns in the source Japanese sentence.
This creates parallel data that include ZPs and provides additional training signals to learn to predict ZPs.
Our method is simple yet effective: it does not require any modification to the model architecture nor additional computation at inference time, but significantly improves the accuracy of the ZP translation.

\section{Is Local Context Useful for Predicting Zero Pronouns?}
\label{section:analysis}

\begin{table*}[t]
\centering
\begin{tabular}{ccccccccccc}\toprule
                    & I    & you    & we    & they & he  & she & us  & them & him & her \\ \midrule
baseline            & 35.9 & 25.4   & 11.0  & 3.7  & 2.2 & 0.0 & 2.2 & 1.9  & 1.2 & 0.9 \\
logistic regression & 78.2 & 46.3   & 17.3  & 3.8  & 3.1 & 0.0 & 3.6 & 0.2  & 0.2 & 2.9 \\ \bottomrule
\end{tabular}
\caption{Recall scores of ZP predictions for each pronoun.}
\label{result:logistic}
\end{table*}

Our proposed method is based on the assumption that local context in Japanese sentences is useful for predicting ZPs.
We begin by analyzing to what extent ZPs can be inferred from local context, and what kind of local context is useful.

For the analysis, we use the Business Scene Dialogue Corpus \citep{rikters-etal-2019-designing}, which is a Japanese and English parallel corpus in the conversational domain.
Besides the published data, we also use the in-house version of the corpus, which amounts to a total of 104,961 sentence pairs.

\subsection{Identifying sentence pairs that contain ZPs.}
As the corpus does not contain annotations of ZPs, we first identify sentence pairs that contain zero pronouns.
We exploit the word alignment information from parallel sentences to detect ZPs. The specific procedure is as follows.

\begin{enumerate}
  \item We obtain the word alignments of the parallel data with \texttt{GIZA++}\footnote{\url{https://github.com/moses-smt/giza-pp}}. We use \texttt{Mecab}\footnote{\url{https://taku910.github.io/mecab/}} for Japanese word segmentation, \texttt{spaCy}\footnote{\url{https://spacy.io/}} for English.
  \item When a pronoun in an English sentence is associated with \texttt{NULL}, the pronoun in the English sentence is considered to correspond to a ZP in the Japanese sentence.
\end{enumerate}

The resulting number of pronouns is shown in Figure \ref{fig:pronoun_count}. It can be seen that in the conversational domain, the first person pronoun \textit{I} and the second person pronoun \textit{you} occur frequently and most of them ($80\%\sim$) are omitted in Japanese. More infrequent pronouns are less likely to be ZPs.

\begin{figure}[t]
\centering
\includegraphics[width=7.0cm]{data/pronoun_count}
\caption{The number of English pronouns in the analyzed data. ZP stands for those whose corresponding pronoun does not appear in the Japanese text.}
\label{fig:pronoun_count}
\end{figure}

\subsection{Extracting local context that co-occurs with ZPs}
To associate the detected ZPs with local context in Japanese sentences, we extract the words that appear in their predicates.
We did not use a Japanese syntactic analyzer to detect ZPs but they are associated with the English pronouns by alignment. Therefore, we decided to exploit the alignment information to extract the predicates.
We extract the predicates of the English pronoun and the corresponding words in the Japanese sentence. Specifically, the following steps were taken.

\begin{enumerate}
  \setcounter{enumi}{2}
  \item We obtain the dependency tree of the English sentence with \texttt{spaCy} and extract the pronoun's head.
  \item The Japanese word aligned to the pronoun's head and its subsequent functional words \footnote{In this case, the function words are defined as words with one of the following part of speeches defined in \texttt{Mecab}: [``particle", ``auxiliary verb", ``symbol"].} are extracted as local context.
\end{enumerate}

\begin{table*}[t]
\centering
\begin{tabular}{cccc} \toprule
                  & 1to1        & 2to1        \\ \midrule
baseline          & 17.07$\pm$0.16 / 83.6$\pm$1.1 & 17.07$\pm$0.26 / 89.36$\pm$0.9 \\
baseline+pro\_aug & 17.07$\pm$0.19 / 92.32$\pm$1.8 & 17.11$\pm$0.23 / 92.17$\pm$1.1 \\ \bottomrule
\end{tabular}
\caption{Evaluation of the model with ZP data augmentation. The scores on the table are BLEU / ZP evaluation accuracy. The mean and standard deviation of five runs with different random seeds are reported.}
\label{result:mt}
\end{table*}

\subsection{Predicting ZPs from Local Context}
To investigate the extent to which ZPs can be predicted from local context, we conducted an analysis by training a logistic regression classifier \footnote{We use the implementation of the \texttt{scikit-learn} library with the default hyperparameters.}.
The classifier takes the unigrams, bi-grams, and trigrams extracted from local context in the Japanese sentence and predicts the associated pronoun in the English sentence.

The recall scores of each pronoun obtained with five-fold cross-validation are shown in Table \ref{result:logistic}. As a baseline, we adopt the score of random prediction according to the training distribution of pronouns.

One can see that the frequent pronouns such as \textit{I, you, we} can be predicted with significantly higher accuracy than the baseline when local context is used (around 6 to 43 points of improvement). In contrast, the other infrequent pronouns display similar or lower values compared to the baseline.
In summary, we can see that local context is predictive of the frequent pronouns but not for the infrequent ones.

To investigate what kind of local context is useful for prediction, for each output label ({\it i.e.}, pronoun) of the logistic regression classifier, we extracted the input features with higher values in the corresponding weights.
As a result, the following words are interpreted to be relevant.

\minisection{The first person singular \textit{I}}
verbs related to recognition (\ja{思う} (think), \ja{わかる} (understand), \ja{感じる} (feel)); humble words (\ja{申し上げる、存る}); and auxiliary verbs expressing desire (\ja{たい}).

\minisection{The second person singular \textit{you}}
suffixes expressing questions (\ja{かな？}, \ja{ました？}); speculations (\ja{でしょ}, \ja{だろ？}), honorifics (\ja{仰る}, \ja{いただける}).

\minisection{The first person plural \textit{we}}
obligations (\ja{なきゃ}, \ja{べき}), desire (\ja{たい}).

For the other pronouns, no local contexts were found to be interpretable as useful for prediction.

\section{ZP Data Augmentation}
\label{section:experiments}

In the previous section, we confirmed that local context is useful for predicting ZPs.
In this section, we examine the usefulness of ZP data augmentation for machine translation.

The method artificially creates training data containing ZPs by deleting pronouns in the source Japanese sentence along with the following particles.
The pronouns to be deleted are detected by string matching with manually created lists (Appendix \ref{appendix:list}).
The augmented data is supposed to provide useful training signals for learning correlations between ZPs and local context.

\subsection{Experimental Setups}
\minisection{Corpus}
We use the Document-aligned Japanese-English Conversation Parallel Corpus \citep{rikters-EtAl:2020:WMT}.
We also add an in-house conversational parallel corpus to the training data. The statistics of the corpus are shown in Table \ref{corpus:stat}.

\begin{table}[!h]
\centering
\begin{tabular}{cccc} \toprule
train    & train+pro\_aug     & dev       & test     \\ \midrule
\multicolumn{1}{c}{246,541} & \multicolumn{1}{c}{282,952} & \multicolumn{1}{c}{2,051} & \multicolumn{1}{c}{2,020} \\ \bottomrule
\end{tabular}
\caption{The number of sentences in the corpus.}
\label{corpus:stat}
\end{table}

\minisection{Model}
Transformer \citep{NIPS2017_3f5ee243} was used as the translation model. We adopt the hyperparameters recommended for the corpus of our size in \citet{araabi-monz-2020-optimizing} (Appendix \ref{appendix:hyperparameters}). In addition to the single-sentence translation, we also experimented with the 2to1 setting \citep{tiedemann-scherrer-2017-neural}, in which the previous sentence in the document is added to the input.

\minisection{Evaluation}
We evaluate the overall translation quality on the test set with BLEU \citep{papineni-etal-2002-bleu}.
We also conduct a targeted evaluation with the ZP evaluation dataset for Japanese-to-English translation \citep{shimazu-etal-2020-evaluation}. The ZP evaluation dataset contains 724 triples of a source sentence, a target sentence with a correct pronoun, and one with an incorrect pronoun.
To evaluate a translation model, we see if the model assigns a lower perplexity to the correct target sentence, and calculate the accuracy.

\subsection{Results}
The results of the experiment are shown in Table \ref{result:mt}.
We can observe that ZP data augmentation does not improve the BLEU score, but significantly improves the accuracy of ZP evaluation in both the 1to1 (83.6\% to 92.3\%) and 2to1 settings (89.3\% to 92.1\%). Our method yields a similar degree of improvement to the 2to1 setting in the ZP evaluation without any computational overhead at the inference time.

We also confirm that adding the previous context (2to1) does not improve BLEU but pronoun translation (83.6\% to 89.3\%), which conforms to observations in the previous study \citep{Jean2017DoesNM,shimazu-etal-2020-evaluation}.
However, this is not the case with the ZP data augmentation (92.3\% to 92.1\%).
We speculate that this is because longer inputs in the 2to1 setting make it more difficult for the model to find correlations between ZPs and local context.

\section{The pronoun and particle list for pronoun data augmentation}
\label{appendix:list}
The deletion of pronouns was done by enumerating all combinations from the list of pronouns (Table \ref{pronoun_list}) and particles (Table \ref{aux_list}) and deleting strings that correspond to the pattern from the sentence.

\begin{table}[h]
\centering
\begin{tabular}{l|l} \toprule
 First person singular  & \ja{私, わたし, 僕, ぼく, 俺, おれ, わたくし, オレ, ウチ} \\
 First person plural   & \ja{我々, 僕ら, われわれ, 僕達, 僕たち, 私達} \\
 Second person singular & \ja{貴方, 貴女, あなた, お前, おまえ, 君, あんた} \\
 First person plural   & \ja{君たち, みなさま} \\
 Third person singular  & \ja{彼, 彼女, あいつ} \\
 Third person plural   & \ja{彼ら, 彼女ら, みんな, 皆, 皆んな, みなさん, 奴ら} \\
\bottomrule
\end{tabular}
\caption{The list of pronouns for pronoun deletion}
\label{pronoun_list}
\end{table}

\begin{table}[h]
\centering
\begin{tabular}{l|l} \toprule
  Nominative  &  \ja{は, が} \\
  Accusative  &  \ja{を} \\
  Dative      &  \ja{に} \\
  Possessive  &  \ja{の} \\
  Others      &  \ja{も, の方から, のほうから, の方に, のほうに, の方で}\\
              & \ja{のこと, の事, のほうで, から, 、} \\
\bottomrule
\end{tabular}
\caption{The list of particles for pronoun deletion}
\label{aux_list}
\end{table}

\section{Hyperparameters for the Machine Translation Experiment}
\label{appendix:hyperparameters}
We choose the hyperparameters of the Transformer model recommended in \cite{araabi-monz-2020-optimizing}.

\begin{table}[h]
\centering
\begin{tabular}{l|c} \toprule
 layers                        & 5 \\
 model size                    & 512 \\
 feed-forward dimension        & 2048 \\
 number of attention heads     & 4 \\
 encoder/decoder layer dropout & 0/0.1 \\
 src/tgt word dropout          & 0.2/0.2 \\
 label\_smoothing              & 0.3 \\
 optimizer                     & Adam with the Noam Learning rate schedule \\
 warmup steps                  & 8000 \\ \bottomrule
\end{tabular}
\caption{Hyperparameters for the Transformer model.}
\label{hyperparameters}
\end{table}