text:
\section{Introduction}

Solving quantum many-body problem for electrons is a well-known
challenging task. While weakly correlated (single-reference) systems
can be well approximated using density functional theory and coupled
cluster methods such as CCSD(T); strongly corrected (multi-reference)
systems remain challenging.  The difficulty comes in two aspects: the
infamous fermion sign problem and combinatorial scaling of the problem
size. In this paper, we propose an efficient algorithm, named \fname{}
(\sname{}), to calculate the ground state energy and its corresponding
variational wavefunction for both weakly and strongly correlated
fermion systems in the framework of full configuration interaction.

Besides the direct diagonalization of the full configuration
interaction (FCI) Hamiltonian~\cite{Knowles1984}, many other
algorithms have been proposed, which can be roughly organized into
three groups. The first group, density matrix renormalization group
(DMRG)~\cite{White1999, Chan2011, Olivares-Amaya2015}, adopts tensor
train ansatz in representing the variational wavefunction. DMRG has
been routinely applied to study the ground and excited state of
strongly correlated $\pi$-conjugated molecules and one-dimensional
systems~\cite{Olivares-Amaya2015}.  The second group, like full
configuration interaction quantum Monte Carlo
(FCIQMC)~\cite{Booth2009, Booth2012, Lu2017}, assumes that the ground
state variational wavefunction can be represented as the empirical
distribution of a large number of stochastic walkers. To reduce the
variance of the energy estimator and the required number of walkers,
initiator-FCIQMC (iFCIQMC)~\cite{Cleland2010} and semi-stochastic
FCIQMC (S-FCIQMC)~\cite{Petruzielo2012} are developed aiming at a good
trade-off between variance and bias.  The third group first solves a
selected configuration interaction (SCI) problem and then conducts a
perturbation calculation.  This family of algorithms (SCI+PT),
includes the early work on configuration interaction by perturbatively
selecting iteration (CIPSI)~\cite{Huron1973}, and more recently,
adaptive configuration interaction (ACI)~\cite{Schriber2017}, adaptive
sampling configuration interaction (ASCI)~\cite{Tubman2016},
\latin{etc}.  Heat-bath configuration interaction
(HCI)~\cite{Holmes2016} significantly reduces the computational cost
of the selected CI phase based on the information from magnitudes of
the double excitations. With perturbation, HCI is able to calculate
the ground state energy of a strongly correlated all electron chromium
dimer up to $1$~mHa accuracy in Ahlrichs VDZ basis. More recently,
semi-stochastic HCI (SHCI)~\cite{Sharma2017} further accelerates the
perturbation phase with a stochastic idea similar to FCIQMC.

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.5\textwidth]{figures/MagFreq.eps}
    \caption{Correlation between the updating frequencies and magnitudes
      of the ground state wavefunction of an all-electron \ch{C2} with
      cc-pVDZ basis calculated by \sname. Coefficients of configuration
      interaction wavefunction are sorted in a decreasing order based on
      their magnitudes. Smaller panel shows the results of the largest
      200 coefficients.}
    \label{fig:updatefreq}
\end{figure}

The algorithm we considered in this paper belongs to the third group
(SCI+PT) above. Our goal is to improve the variational stage of the
computation \latin{i.e.}, the selective CI part. Our proposed
algorithm, \sname{}, is an adaptive coordinate-wise (\latin{i.e.},
determinant-wise) iterative method. It updates the coefficient of an
appreciative determinant each iteration and has the nice feature of
visiting determinants with different frequencies proportional to their
importance. As not all determinants contribute equally to the ground
state wavefunction, \sname{} is able to efficiently capture the
important part of the FCI space and obtain a good approximation to the
ground state. Figure~\ref{fig:updatefreq} indicates the relation
between the updating frequency of determinants and magnitudes of the
determinant coefficients of the ground state wavefunction for an
all-electron \ch{C2} calculation with cc-pVDZ basis by \sname{}. As
shown in Figure~\ref{fig:updatefreq}, many coefficients are only
updated once throughout iterations, which shows the efficiency of the
updating strategy in \sname{}. During iterations, \sname{} also
compresses those unappreciative determinants through hard
thresholding. Our implementation philosophy of \sname{} is to reserve
memory resource for storing variational wavefunction as much as
possible, hence, the Hamiltonian matrix is evaluated on-the-fly.
Eventually, \sname{} is able to capture the binding curve of
all-electron \ch{N2} with cc-pVDZ basis up to 6 digits accuracy in one
week and compute the ground state energy of all-electron \ch{Cr2} with
Ahlrichs VDZ basis to $-2086.443565$ Ha, which is the state-of-the-art
variational result.

The rest of the paper is organized as follows. Section~\ref{sec:algo}
presents the \sname{} algorithm. The implementation detail is stated in
Section~\ref{sec:impl}. In Section~\ref{sec:numres}, we demonstrate the
efficiency and accuracy of \sname{} via applying it to various molecules
including \ch{H2O}, \ch{C2}, \ch{N2}, and \ch{Cr2}. Also the binding
curve of \ch{N2} is characterized. Finally, in Section~\ref{sec:conc},
we conclude the paper together with discussion on future work.

\section{\Fname{}}
\label{sec:algo}

This section first reformulates the FCI eigenvalue problem as a
non-convex optimization problem~\cite{Lei2016, Li2018b} with no spurious
local minima and then describes in detail the \fname{} algorithm together
with the compression technique and suggested stopping criteria.

Given a complete set of spin-orbitals $\{\chi_p\}$, a many-body
Hamiltonian operator, under the second quantization, can be written as
\begin{equation} \label{eq:second_quantized_h}
    \widehat{H} = \sum_{p,q} t_{pq} \hat{a}_{p}^\dagger \hat{a}_{q}
    + \frac{1}{2} \sum_{p,r,q,s} v_{prqs} \hat{a}_{p}^\dagger
    \hat{a}_{r}^\dagger \hat{a}_{q} \hat{a}_{s},
\end{equation}
where $\hat{a}_{p}^\dagger$ and $\hat{a}_{p}$ denote the creation
and annihilation operator of an electron with spin-orbital index
$p$, $t_{pq}$ and $v_{prqs}$ are one- and two-electron integrals
respectively. The ground state energy of $\widehat{H}$ can be obtained
from solving the time-independent Schr\"{o}dinger equation
\begin{equation} \label{eq:eig}
  \widehat{H} \ket{\Phi_0} = E_0 \ket{\Phi_0},
\end{equation}
where $E_0$ denotes the ground state energy (the smallest eigenvalue)
and $\ket{\Phi_0}$ denotes the corresponding ground state
wavefunction.  Without loss of generality, we assume that $E_0$ is
negative and non-degenerate, i.e., the eigenvalues of $\widehat{H}$
are given as $E_0 < E_1 \leq E_2 \leq \cdots$.

In FCI, the complete spin-orbital set is truncated to a finite subset
$\{\chi_p\}_{p = 1}^{\norb}$, obtained e.g., by Hartree-Fock
or Kohn-Sham calculations. The FCI variational space $\calV$ is
spanned by all possible Slater determinants constructed from
$\{\chi_p\}_{p = 1}^{\norb}$, and the dimension of $\calV$ is
denoted as $N_{\text{FCI}}$ also known as the total number of
configuration interactions.  The ground state wavefunction is then
discretized in $\calV$, \latin{i.e.},
$\ket{\Phi_0} \in \calV = \spanfun{\ket{D_1}, \dots,
  \ket{D_{N_\text{FCI}}}}$, where $\ket{D_1}$ denotes the reference
determinant, $\ket{D_i}$ for $1 < i \leq N_\text{FCI}$ denotes other
Slater determinants constructed from the finite set of spin-orbitals.
Correspondingly, the Hamiltonian operator is represented by a
many-body Hamiltonian matrix $H$ with its $(i,j)$ entry as
$H_{i,j} = \bra{D_i}\widehat{H}\ket{D_j}$. Let $\vec{b}$ and $\vec{c}$
denote coefficient vectors with entry $b_i$ and $c_i$ respectively.
The ground state wavefunction can be written as
$\ket{\Phi_0} = \sum_i c_i \ket{D_i}$.  The time-independent
Schr\"odinger equation \eqref{eq:eig} has its matrix representation
as,
\begin{equation}
  \label{eq:eig-fci}
   H \vec{c} = E_0 \vec{c},
\end{equation}
which is known as the FCI eigenvalue problem.  The second-quantized
Hamiltonian operator as in \eqref{eq:second_quantized_h} implies that
$H_{i,j}$ is nonzero if and only if $\ket{D_i}$ can be obtained from
$\ket{D_j}$ via changing at most two occupied spin-orbitals. Hence, we
say that $\ket{D_i}$ is $H$-connected with $\ket{D_j}$ if $H_{i,j}$
is nonzero. The set of all indices $i$ such that $\ket{D_i}$ is
$H$-connected with $\ket{D_j}$ is called the $H$-connected index set of
$j$ and is denoted as $\hset{j}$.  Since the cardinality of $\hset{j}$ is
much smaller than $N_{\text{FCI}}$, the matrix $H$ is extremely sparse.

\subsection{Reformulation of the FCI eigenvalue problem}

The FCI eigenvalue problem \eqref{eq:eig-fci} can be reformulated as the
following unconstrained non-convex optimization problem,
\begin{equation} \label{eq:opt}
    \min_{\vec{c} \in \bbR^{N_{\text{FCI}}}} f(\vec{c}) = \norm{H +
    \vec{c}\vec{c}^\top}_F^2,
\end{equation}
where $\norm{\cdot}_{F}$ denotes the Frobenius norm of a matrix.
The gradient of the objective function is
\begin{equation*}
    \nabla f(\vec{c}) = 4H \vec{c} + 4 \left( \vec{c}^\top \vec{c}
    \right) \vec{c}.
\end{equation*}
As analyzed in our previous work~\cite{Li2018b}, the stationary points
are $\vec{0}$ and $\pm \sqrt{-E_i} \vec{v}_i$ for $E_i < 0$, where
$\vec{v}_i$ is the normalized eigenvector corresponding to $E_i$.
Furthermore, importantly, $\pm \sqrt{-E_0} \vec{v}_0$ are the only two
local minimizers with the same objective value, while other stationary
points are saddle points. Thus, solving the optimization problem
\eqref{eq:opt} reveals the ground state energy $E_0$ and the
ground state wavefunction coefficient vector $\vec{v}_0$. For such an
optimization problem, higher order methods converge to global minima
efficiently but their per iteration computational costs are too
expensive for FCI problems.  Hence, only first order methods, like
gradient descent methods (GDs), stochastic gradient descent methods
(SGDs), and coordinate descent methods (CDMs), are discussed
here.

The first order optimization methods applied to \eqref{eq:opt},
compared with solving \eqref{eq:eig-fci} using traditional methods,
\latin{e.g.}, power method, Davidson method, and Lanczos method, have
two main advantages. First, GDs, SGDs, and CDMs do not need any tuning
parameters: no diagonal shift is needed to turn the smallest
eigenvalue into the largest one in magnitude~\cite{Li2018b} and the
stepsize can be addressed by an exact line search. Second, since no
orthonormality constraint appears explicitly in \eqref{eq:opt},
solving \eqref{eq:opt} with GDs, SGDs, and CDMs does not need any
orthonormalization step. While in traditional methods like Davidson or
Lanczos methods, orthonormalization step is needed every a few
iterations to avoid numerical instability issue, which would be
expensive for FCI problems.

Among the first order optimization methods, CDMs are more suitable
for FCI problems. GDs evaluate and update the exact gradient each
iteration, which is prohibitively expensive for FCI problems,
due to the huge problem dimension. SGDs evaluate and update a
stochastic approximation of the full gradient, which is of much
cheaper computational cost per iteration. SGDs actually have been
applied to FCI problems in an implicit way: as FCIQMC, iFCIQMC, and
S-FCIQMC can all be regarded as SGDs applied to a similar objective
function as \eqref{eq:opt} with constant stepsizes~\cite{Lu2017}. SGDs,
in general, converge efficiently to a neighborhood of minimizers and then
wander around the minimizer due to the stochastic approximation. CDMs
select and update a single coefficient each iteration, which is of
cheap computational cost.  Comparing to GDs, CDMs provably achieve
faster convergence rate in terms of the prefactor~\cite{Li2018b},
whereas comparing to SGDs, CDMs are approximately of equal cost
per iteration, but is much more stable towards convergence. Further,
with a properly designed selecting strategy, CDMs updates different
coefficients with different frequencies, taking advantage of the
different importance of determinants in FCI problems.  Taking these
advantages into consideration, we design \sname{}, which is a CDM
tailored for FCI problems with compression strategy, to efficiently
solve \eqref{eq:opt}. This method is described in details below.

\subsection{Algorithm}

The \sname{} algorithm stores two sparse vectors $\vec{c}^\itell$ and
$\vec{b}^\itell$ in the main memory that are contiguously maintained
throughout iterations: $\vec{c}^\itell$ denotes the computed coefficient
vector of the ground state wavefunction in the $\ell$-th iteration and
$\vec{b}^\itell$ is a compressed approximation of $H\vec{c}^\itell$.

Let us first give a sketch of the \sname{} algorithm: In the $\ell$-th
iteration, \sname{} first finds the $i^\itellp$-th determinant with
potentially steepest objective function value decrease among all
$H$-connected determinants of $\ket{D_{i^\itell}}$.  Then \sname{}
conducts an exact line search to find the optimal update $\alpha$ such
that $f(\vec{c}^\itell + \alpha \vec{e}_{i^\itellp})$ is minimized
and hence the estimator for the ground state energy is reduced, where
$\vec{e}_{i^\itellp}$ denotes the indicator vector of $i^\itellp$.
$\vec{b}^\itell$ plays an important role in all above steps. In
preparation for next iteration, the vector $\vec{b}^\itellp$ needs
to be updated incorporating with $\vec{c}^\itellp = \vec{c}^\itell +
\alpha \vec{e}_{i^\itellp}$. However, an exact update $\vec{b}^\itellp
= \vec{b}^\itell + \alpha H_{:,i^\itellp}$ could waste limited memory
resource on unappreciative determinants.  Our compression step updates
coefficients only when they do not cost extra memory resource or if they
are significantly large in magnitudes. Although the compression step
introduces error along the iterations, as we will show, we can still
calculate the Rayleigh quotient corresponding to $\vec{c}^\itellp$
exactly, which is used as the ground state energy estimator in \sname{}.

In the following, we will discuss each part of \sname{} in detail and
then conclude this section with a pseudo-code for the algorithm.

\subsubsection{Determinant-select and coefficient-update}
\label{sec:coord-select}

Determinant-select is the first step in each iteration. Assume that the
$\ell$-th iteration updates determinant $\ket{D_{i^\itell}}$ and results
in a coefficient vector $\vec{c}^\itell$. We select the determinant to
be updated at the current iteration, $\ket{D_{i^\itellp}}$, according
to local information at $\vec{c}^\itell$. In order to decrease the
objective function value to the largest extent, we could select the
determinant with the largest magnitude of the approximated gradient at
$\vec{c}^\itell$, \latin{i.e.}, $\ket{D_{i^\itellp}}$ with
\begin{equation*}
    i^\itellp = \argmax_{j} \abs{4 b^\itell_j + 4 \left(
    \left(\vec{c}^\itell\right)^\top \vec{c}^\itell \right)
    c_j^\itell},
\end{equation*}
where $\vec{b}^\itell$ is a compressed approximation of
$H \vec{c}^\itell$.  However, the above strategy requires checking
each $j$ (i.e., all determinants), which is prohibitive for even
moderate size problems. Hence, instead of checking all determinants,
\sname{} only checks the $H$-connected determinants of
$\ket{D_{i^\itell}}$, \latin{i.e.},
\begin{equation} \label{eq:coord-select}
    \begin{split}
        & i^\itellp = \\
        & \argmax_{j \in \hset{i^\itell}}
        \abs{4b_j^\itell +
        4 \left( \left( \vec{c}^\itell \right)^\top \vec{c}^\itell \right)
        c_j^\itell}.
    \end{split}
\end{equation}
Since our compression strategy introduced later in Section~\ref{sec:comp}
truncates unappreciative determinants, the expression in
\eqref{eq:coord-select} remains a good approximation of the exact
gradient at $\vec{c}^\itell$. Empirically, such a gradient-based
determinant-select strategy outperforms other perturbation-based
determinant-select strategies as used in other SCI algorithms (see
Section~\ref{sec:numres} for details).

Once the $i^\itellp$-th determinant is selected, \sname{} determines
the stepsize by the line search along that direction so to decrease the
objective function value by the largest amount. Denoting the update as
$\alpha$, the line search can be formulated as
\begin{equation} \label{eq:coord-update-opt}
    \alpha = \argmin_{\varalpha \in \bbR} f(\vec{c}^\itell +
    \varalpha \vec{e}_{i^\itellp}).
\end{equation}
Since $h(\varalpha) = f(\vec{c}^\itell + \varalpha \vec{e}_{i^\itellp})$
is a quartic polynomial in $\varalpha$, solving the minimization
problem \eqref{eq:coord-update-opt} is equivalent to finding roots of
$h'(\varalpha)$ -- the derivative of $h(\varalpha)$. If $h'(\varalpha)$
has a unique root, then the root is the minimizer. If $h'(\varalpha)$
has two roots, then the one with multiplicity one is the minimizer. If
$h'(\varalpha)$ has three roots, then the one further away from the
middle one is the minimizer. Given the update $\alpha$, we can easily
update $\vec{c}^\itell$ as
\begin{equation} \label{eq:coord-update-c}
    c_{i}^\itellp =
    \begin{cases}
        c_i^\itell, & i \neq i^\itellp;\\
        c_i^\itell + \alpha, & i = i^\itellp.
    \end{cases}
\end{equation}

In \sname{}, we also need to maintain $\vec{b}^\itellp \approx H
\vec{c}^\itellp$ for future determinant-select steps. Since only
one coefficient is updated in $\vec{c}^\itell$, the corresponding
$\vec{b}^\itell$ can be updated accordingly as,
\begin{equation} \label{eq:coord-update-b}
    \vec{b}^\itellp \approx H \vec{c}^\itellp \approx \vec{b}^\itell +
    \alpha H_{:,i^\itellp}.
\end{equation}
Therefore, each update step requires evaluation of all $H$-connections
from $\ket{D_{i^\itellp}}$. Besides the update from $\vec{c}^\itellp$, we
also recalculate the current $i^\itellp$-th entry in $\vec{b}^\itellp$
to guarantee the correctness and increase the numerical stability
of our algorithm. Such a recalculation could improve the accuracy
of the determinant-select \eqref{eq:coord-select} and line search
\eqref{eq:coord-update-opt} in the following iterations, and also
provide an accurate Rayleigh quotient as the estimator of the ground
state energy as in \eqref{eq:coord-update-cHc}.  We argue this correction
comes for free in addition to \eqref{eq:coord-update-b}, since
\begin{equation} \label{eq:coord-update-bj}
    \begin{split}
        b_{i^\itellp}^\itellp = & H_{i^\itellp,:} \vec{c}^\itellp \\
    = & \sum_{j \in \hset{i^\itellp}} (H_{j,i^\itellp})^\ast
        c_j^\itellp,
    \end{split}
\end{equation}
where $(H_{j,i^\itellp})^\ast$ denotes the complex conjugate of
$H_{j,i^\itellp}$, which has already been evaluated when updating
\eqref{eq:coord-update-b}.

\subsubsection{Coefficient compression}
\label{sec:comp}

Since \sname{} initializes $\vec{c}^{(0)}$ with the reference determinant
$\ket{D_1}$ and $\vec{b}^{(0)} = H \vec{c}^{(0)}$, the coefficient
of the reference determinant in $\vec{b}^{(0)}$ is nonzero and the
reference determinant is in $\vec{b}^{(0)}$. In later iterations,
\sname{} is designed to follow one rule: if a determinant is in
$\vec{c}^\itell$, then it is in $\vec{b}^\itell$ as well.  Under this
rule, if a determinant $\ket{D_j}$ is neither in $\vec{c}^\itell$
nor in $\vec{b}^\itell$, according to \eqref{eq:coord-select}, this
determinant has zero value therein and will not be selected. Hence
\eqref{eq:coord-select} selects either a new determinant not in
$\vec{c}^\itell$ but in $\vec{b}^\itell$ or an old determinant already
in both $\vec{c}^\itell$ and $\vec{b}^\itell$. In \sname{}, thus, the
compression strategy compresses only the unappreciative determinants
in $\vec{b}^\itell$ to control the computation and memory cost,
which in turn restricts the growth of the coefficient vector
$\vec{c}^\itell$.

Detailed compression strategy is as follows. When a coefficient
$\alpha H_{j,i^\itellp}$ is added to $b^\itell_j$, we use a
predefined tolerance $\varepsilon$ to compress the update. If the $j$-th
determinant is already selected before, then $\alpha H_{j,i^\itellp}$
is added to $b^\itell_j$ without any compression.  If the $j$-th
determinant has not been selected in $\vec{b}^\itell$, but the update
is quantitatively large, \latin{i.e.}, $\abs{\alpha H_{j,i^\itellp}} >
\varepsilon$, it indicates that the $j$-th determinant is appreciable
and the $j$-th determinant will be added to $\vec{b}^\itell$ with
the coefficient $\alpha H_{j,i^\itellp}$. Otherwise, the update is
truncated, i.e., the coefficient in $\vec{b}^{\itell}$ remains $0$.
The described compression strategy is deterministic and satisfies the
rule that determinants with nonzero coefficients in $\vec{c}^\itell$
are in $\vec{b}^\itell$ as well. For molecules, such a deterministic
strategy outperforms other strategies including stochastic compression
schemes~\cite{Lim2017} due to its effectiveness and cheap cost.

\subsubsection{Energy estimation}

Although the vector $\vec{b}^\itellp$ is compressed, we emphasize
that the Rayleigh quotient $r(\vec{c}) = \frac{\vec{c}^\top H
\vec{c}}{\vec{c}^\top \vec{c}}$ can be maintained accurately for
$\vec{c}^\itellp$, which is used in \sname{} as the estimator of the
ground state energy. First, the squared norm of $\vec{c}^\itell$ can
be updated up to numerical error, \latin{i.e.},
\begin{equation} \label{eq:coord-update-cc}
    \left( \vec{c}^\itellp \right)^\top \vec{c}^\itellp = \left(
    \vec{c}^\itell \right)^\top \vec{c}^\itell
    + 2\alpha c_{i^\itellp}^\itell + \alpha^2.
\end{equation}
An exact update can be computed for the numerator of
the Rayleigh quotient as well, \latin{i.e.},
\begin{equation} \label{eq:coord-update-cHc}
    \begin{split}
        \left( \vec{c}^\itellp \right)^\top H \vec{c}^\itellp = & \left(
        \vec{c}^\itell \right)^\top H
        \vec{c}^\itell \\
        & + 2\alpha b_{i^\itellp}^\itellp - \alpha^2
        H_{i^\itellp,i^\itellp},
    \end{split}
\end{equation}
where $b_{i^\itellp}^\itellp$ is recalculated accurately
as discussed in Section~\ref{sec:coord-select} around
\eqref{eq:coord-update-bj}. Hence this update is accurate. The Rayleigh
quotient of the updated variational wavefunction, $r(\vec{c}^\itellp)$
is the ratio of two accurately cumulated quantity and hence
accurate. Both in theoretical and numerical results, we observed that
the Rayleigh quotient is much more accurate than the projected energy
estimator~\cite{Booth2009, Booth2012, Cleland2010, Petruzielo2012},
which is $\frac{b_1^\itellp}{c_1^\itellp}$ in our notation.

Stopping criteria can be tricky for all iterative methods, including
DMRG, FCIQMC, HCI, SHCI, \latin{etc}, and is also the case for
\sname{}. Here we propose three suggestions. As for many iterative
methods, we can stop the iteration if the updated value is small. For
\sname{}, it is suggested to monitor the cumulated updated values
across a few iterations as the stopping criteria. Another stopping
criteria is based on the change of the Rayleigh quotient. Usually,
we observe monotone decay of the Rayleigh quotient before iteration
converges. Therefore, we can stop the algorithm if the decay of
the Rayleigh quotient after a few iterations is small. The third
suggestion is based on the ratio of the number of nonzero coefficients
in $\vec{b}$ and $\vec{c}$. When the algorithm converges, this ratio
converges to 1. When the ratio is close to $1$, the error introduced by
the compression slows down the convergence significantly. Hence more
iterations do not make much accuracy improvement. Mixed use of these
stopping criteria is suggested in practice.

\medskip

We conclude this section with a pseudo-code for \sname{}:
\begin{enumerate}
    \item Initialize $\vec{c}^{(0)}$ by the reference state
    $\ket{D_1}$ with coefficient being $1$, initialize
    $\vec{b}^{(0)} = H \vec{c}^{(0)}$, and initialize $\ell = 0$.

    \item Select a determinant with the largest gradient magnitude
    according to \eqref{eq:coord-select}. Denote the selected
    determinant as $\ket{D_{i^\itellp}}$.

    \item Solve a cubic polynomial equation to obtain the optimal update
    $\alpha$ for the selected determinant. Update the $i^\itellp$-th
    coefficient as \eqref{eq:coord-update-c}.

    \item Update $b_j^\itellp = b_j^\itell + \alpha
    H_{j,i^\itellp}$ if the $j$-th determinant is already selected
    in $\vec{b}^\itell$. Otherwise, add new determinant $\ket{D_j}$ to
    $\vec{b}^\itellp$ with coefficient $\alpha H_{j,i^\itellp}$ if
    $\abs{\alpha H_{j,i^\itellp}} > \varepsilon$.  Exactly reevaluate
    $b_{i^\itellp}^\itellp$ as \eqref{eq:coord-update-bj}.

    \item Update $\left(\vec{c}^\itellp\right)^\top \vec{c}^\itellp$ and
    $\left( \vec{c}^\itellp \right)^\top H \vec{c}^\itellp$ as
    \eqref{eq:coord-update-cc} and \eqref{eq:coord-update-cHc}
    respectively. Calculate the exact Rayleigh quotient for
    $\vec{c}^\itellp$.

    \item Repeat 2-5 with $\ell \leftarrow \ell+1$ until some stopping
    criteria is achieved.
\end{enumerate}

\section{Implementation and complexity}
\label{sec:impl}

We now give some implementation details of the algorithm, focusing on
the computationally expensive parts and the numerical stability
issues.  In the end of this section, a per iteration complexity
analysis is conducted.

The indices of Slater determinants are encoded in the way that coincides
with that in the second quantization. Suppose there are $\norb$
spin-orbitals in the FCI discretization, and $\nelec$ electrons in the
system.  Then a Slater determinant is encoded as an $\norb$-bit binary
string, with each bit representing a spin-orbital. The spin-orbital
is occupied if the corresponding bit is $1$ and unoccupied if the
bit is $0$.  The $\norb$-bit binary string is stored as an array of
$64$-bit integers. Thus, $\ceil{\frac{\norb}{64}}$ integers are needed
to represent the index of a determinant.

We now focus on the implementation detail of the determinant-update
step, as it dominates the runtime.  Since the vectors $\vec{b}$ and
$\vec{c}$ are sparse and compressed in the algorithm, their entries
cannot be contiguously stored in memory. For \sname{}, we have tried
two different data structure implementations for the combined vector
(since the indices of nonzero coefficients of $\vec{c}$ are contained
in $\vec{b}$, these two vector are stored together in a single data
structure): red-black tree and hash table~\cite{Cormen2009}.

Red-black tree is a memory compact representation of the vector: Given
that $\vec{b}$ at current iteration has $n$ nonzero coefficients,
red-black tree requires $\bigO(n)$ memory. Inserting, updating and
deleting a nonzero coefficient to this red-black tree cost $\bigO(\log
n)$ operations. The drawback is that each nonzero coefficient is a
node on the tree and hence requires extra memory to store pointers,
which turns out to be more expensive comparing to the hash table.

In \sname{}, therefore, we prefer to use a fixed-size open addressing
hash table.  The hash function mapping a configuration string to an
array index is chosen as
\begin{equation}
    \mathrm{Hash}(\vec{d}) = \vec{s} \cdot \vec{d} \; ( \mathrm{mod}\;p),
\end{equation}
where the size of the hash table is chosen to be a large prime number
$p$; $\vec{d}$ is the vector of $\ceil{\frac{\norb}{64}}$ integers with
bits representing the configuration of the determinant; and $\vec{s}$
is a fixed vector of the same length as $\vec{d}$ with entries randomly
chosen from $[0, p-1]$ during the \sname{} initialization step.  In our
current implementation, for each execution of the algorithm, we allocate
an array of size approaching machine memory limit for the hash table,
which could be modified to enable dynamic resizing feature in order to be
memory compact.  Inserting, updating and deleting a nonzero coefficient
in hash table cost $\bigO(1)$ operations on average; while in the worst
case, when the table is almost full, inserting and deleting operation
would cost $\bigO(p)$ operations.  In order to avoid such inefficient
scenarios, we limit the load factor below $80\%$. In practice,
these settings of hash table work well and significantly outperform
red-black tree. All the numerical results in this paper are produced
with hash table.

Besides the expensive data accessing step, the computational
expensive step is the evaluation of $H_{:,i^\itellp}$.  Let $N_H =
\max_i \abs{\hset{i}}$ be the maximum number of nonzero entries in
columns of the Hamiltonian matrix.  Although $N_H \ll N_{\text{FCI}}$,
$N_H$ still scales as $\bigO(\nelec^2 \norb^2)$.  The computational
cost for evaluating each entry $H_{i,j}$ also depends moderately on
$\nelec$. \sname{} uses an efficient Fortran implemented open source
quantum chemistry code \textsc{Hande-QMC} as backend for the evaluation
of Hamiltonian entries.

Shared memory parallelism based on OpenMP is used in our
implementation. For each iteration, the double excitation calculation
is the bottleneck for the evaluation $H_{:,i^\itellp}$, which is
embarrassingly parallelized with OpenMP.  In terms of runtime, accessing
a nonzero coefficient of $\vec{b}$ and $\vec{c}$ is also expensive due
to the lack of memory continuity.  Therefore, we also parallelize the
access to $b_{\hset{i^\itellp}}$ and $c_{\hset{i^\itellp}}$
with OpenMP. Due to the possible collision of the hash function of open
addressing, we partition the hash array into $2000$ blocks and set
locker for each block, such that no two threads can access the same
block simultaneously. Increasing the number of lockers would reduce
the idling time of threads but would increase the memory cost. We did
not try to optimize the number of blocks.

Last point on implementation focuses on the numerical stability of the
Rayleigh quotient. Different from other iterative methods, \sname{}
updates one determinant per iteration. Hence, for large systems,
the number of iteration could easily go beyond $10^8$. For cumulated
quantities such as $\vec{c}^\top \vec{c}$ and $\vec{c}^\top H \vec{c}$,
the value is updated at least $10^8$ times, hence the accumulated
numerical error could pollute the chemical accuracy, and thus careful
treatment is needed.  In our implementation, we use a quadruple-precision
floating point for $\vec{c}^\top \vec{c}$ and $\vec{c}^\top H \vec{c}$
such that the relative error is at most $10^{-16}$ unless the number of
iteration exceeds $10^{16}$.

Let us remark that our current implementation of \sname{} is by no
means optimal. The bottleneck of the current implementation is the
na\"{i}ve hash table. Random access to the main memory is expensive
since the cache hierarchy is not fully adapted. An optimized hash table
may improve the performance by a big constant.

\smallskip

To conclude the section, let us conduct a leading order per iteration
complexity analysis for \sname{}.  In determinant-select step,
since all $b_i^\itell$ and $c_i^\itell$ for $i \in
\hset{i^\itellp}$ have been accessed in the previous iteration,
$i^\itellp$ can be computed without paying the cost of accessing the
data structure of $\vec{b}$ and $\vec{c}$. Hence the leading cost is
$\bigO(N_H)$ with a small prefactor. Line search and updating $\vec{c}$
cost $\bigO(1)$ operations and are hence negligible. Updating $\vec{b}$
is the most expensive step throughout the algorithm.  It requires
evaluating $\bigO(N_H)$ entries of Hamiltonian matrix and
accessing $\vec{b}$ and $\vec{c}$ $\bigO(N_H)$ times. This step costs
$\bigO(N_H)$ operation with a prefactor being the Hamiltonian per entry
evaluation cost plus the averaged data structure accessing cost. In our
implementation, the compression step is combined with the updating step.
Once $H_{:,i^\itellp}$ has been evaluated, $b_i^\itell$ and
$c_i^\itell$ for $i \in \hset{i^\itellp}$ are accessed,
then exact update of $b_{i^\itellp}^\itellp$, cumulative
updates of $\vec{c}^\top \vec{c}$ and $\vec{c}^\top H \vec{c}$ cost
$\bigO(N_H)$ operations with a small prefactor. Overall, \sname{} costs
$\bigO(N_H)$ operations per iteration with the prefactor dominated
by the computation cost of one Hamiltonian entry and the averaged
access cost of the data structure.  The memory cost of \sname{}
is dominated by the cost of allocating the data structure.

\section{Numerical results}\label{sec:numres}

In this section, we perform a sequence of numerical experiments
to demonstrate the efficiency of \sname{}. First, we compare the
performance of \sname{}, Heat-bath CI (HCI), DMRG and \fciqmc{} (FCIQMC
with initiator and semi-stochastic adaptation) on \ch{H2O}, \ch{C2}
and \ch{N2} under cc-pVDZ basis. Then, we benchmark the binding curve
of nitrogen dimer under cc-pVDZ basis using \sname{} up to $10^{-3}$
mHa accuracy. Finally, we use \sname{} to calculate the ground state
energy of chromium dimer \ch{Cr2} under the Ahlrichs VDZ Basis at $r =
1.5$\r{A}, which is a well-known challenging task due to the strong
correlation.

In all experiments, the orbitals and integrals are calculated via
restricted Hartree Fock (RHF) in \textsc{Psi4}\cite{Parrish2017}
package. All the reported energies are in Hartree (Ha) but the length
unit is in either Bohr radius ($a_0$) or \r{a}ngstr\"{o}m (\r{A})
due to different configurations in the references.

\subsection{Numerical results of \ch{H2O}, \ch{C2}, and \ch{N2}}
\label{sec:num-res-molecule}

We first compare the performance of \sname{} with other algorithms,
HCI, DMRG and \fciqmc{}.  In this paper, we choose \fciqmc{} instead
of FCIQMC or iFCIQMC because it balances well among bias, variance and
runtime.  \sname{} is implemented as stated in Section \ref{sec:impl}
with the on-the-fly Hamiltonian elements evaluation interfaced from
\textsc{Hande-QMC}\cite{Spencer2015,Spencer2018}.  HCI adopts the
original implementation in \textsc{Dice}~\cite{Holmes2016}; DMRG
adopts the widely used implementation in \textsc{Block}\cite{Chan2002,
Chan2004, Ghosh2008, Sharma2012, Olivares-Amaya2015}; \fciqmc{} adopts
the implementation in \textsc{NECI} code~\cite{Booth2014}. All programs
are compiled by Intel compiler 19.0.144 with \texttt{-O3} option. MPI
and OpenMP support are disabled for all programs in this section. All
the tests in this section are produced on a machine with Intel Xeon
CPU E5-1650 v2 @ 3.50GHz and $64$GB memory.
For all algorithms, the Hartree-Fock state is used as the initial
wavefunction. Most parameters in \sname{}, HCI, DMRG, and \fciqmc{}
will be clearly stated in the later content. We run HCI with different
truncation threshold $\varepsilon_1$, DMRG with different maximum
bond dimension $\max M$ and \fciqmc{} with different target population
$m$. Any unspecified parameter is set to be the default value. Besides
the specified or default parameters, for \fciqmc{}, the time step is
optimized by \textsc{NECI}; the initiator truncation threshold is $3$ and
the size of the deterministic space is 1000. For all the tests reported
in Table~\ref{tab:h2o}, Table~\ref{tab:c2}, and Table~\ref{tab:n2},
\fciqmc{} runs for $10000$ steps and the energy is estimated by the
block analysis in \textsc{NECI} . In all algorithms, the energy is
reported without any perturbation or extrapolation post-calculation.
Variational energy (Rayleigh quotient) is reported for \sname{}, HCI and
DMRG, while average projected energy is reported for \fciqmc{}. Since
\fciqmc{} is a stochastic method, we also report one significant digit
of the Standard Error of the Mean (SEM) in the parenthesis and the SEM
is of the same accuracy level as the last digit of the average. SEM is
estimated as the sample standard deviation divided by the square root
of the sample size. Also Root Mean Square Error (RMSE) of the average
estimator is adopted as the error of \fciqmc{}, which is defined as
$\sqrt{\text{standard error}^2 + \mathrm{bias}^2}$. We emphasize that
similar perturbation calculation as in HCI can also be applied to
\sname{} and the comparison is left as future work.

In this section, we test the four algorithms on three molecules:
\ch{H2O} with OH bond length 1.84345$a_0$ and HOH bond angle
110.6\textdegree~\cite{Olsen1998, Booth2009}, \ch{C2} with bond length
1.24253\r{A}~\cite{Holmes2016, Sharma2015} and \ch{N2} with bond
length 2.118$a_0$~\cite{Chan2004b}. The properties of the systems are
summarized in Table~\ref{tab:system}, where the reference ground state
energy is calculated by \sname{} to a high precision.

\begin{table*}[htb]
    \centering
    \begin{tabular}{ccccccc}
    \toprule
    Molecules & Basis & Electrons & Orbitals & Dimension & HF Energy &
    GS Energy \\
    \midrule
    \ch{H2O} & cc-pVDZ & 10 & 24 & $4.53\times 10^8$ &
    -76.0240386 & -76.2418601  \\
    \ch{C2}  & cc-pVDZ & 12 & 28 & $1.77\times 10^{10}$ &
    -75.4168820 & -75.7319604  \\
    \ch{N2}  & cc-pVDZ & 14 & 28 & $1.75\times 10^{11}$ &
    -108.9493779 & -109.2821727 \\
    \bottomrule
    \end{tabular}
    \caption{Properties of test molecule systems. HF energy and GS energy
    denote Hartree-Fock energy and ground state energy respectively.}
    \label{tab:system}
\end{table*}

\subsubsection{\ch{H2O} molecule}
\label{sec:h2o}

Table~\ref{tab:h2o} and Figure~\ref{fig:h2o} illustrate numerical results
for \ch{H2O}. In Table~\ref{tab:h2o}, we report the detailed results
and the corresponding used parameters.  Figure~\ref{fig:h2o} plots
the convergence of the energy against the wall-clock time based on the
results in Table~\ref{tab:h2o}. For \fciqmc{}, we run another test with
$m = 50000$ for longer time and plot the curve of the projected energy
as well as the cumulative average of energy starting at iteration $5000$.

\begin{table*}[ht]
    \centering
    \begin{tabular}{cl|lcr|rr}
        \toprule
        \multirow{2}{*}{Algorithm} & \multirow{2}{*}{Parameter}
        & \multirow{2}{*}{Energy} & \multirow{2}{*}{Error}
        & \multirow{2}{*}{Time(s)} & \multicolumn{2}{c}{\sname{}} \\
        & & & & & Time(s) & Ratio \\
        \midrule
        \multirow{7}{*}{\sname{}} & \multirow{7}{*}{$\varepsilon = 0$}
        &   -76.2318601 & $1.0 \times 10^{-2}$ & 3.7     & - \\
        & & -76.2408601 & $1.0 \times 10^{-3}$ & 96.2    & - \\
        & & -76.2417601 & $1.0 \times 10^{-4}$ & 592.5   & - \\
        & & -76.2418501 & $1.0 \times 10^{-5}$ & 2780.0  & - \\
        & & -76.2418591 & $1.0 \times 10^{-6}$ & 9569.5  & - \\
        & & -76.2418600 & $1.0 \times 10^{-7}$ & 25227.5 & - \\
        & & -76.2418601 & $1.0 \times 10^{-8}$ & 54242.2 & - \\
        \midrule
        \multirow{4}{*}{HCI}
        & $\varepsilon_1 = 1.0\times 10^{-4}$ & -76.2412891
        & $5.7\times 10^{-4}$ & 58.4   & 156.3  & 0.37x \\
        & $\varepsilon_1 = 2.0\times 10^{-5}$ & -76.2417533
        & $1.1\times 10^{-4}$ & 312.9  & 565.3  & 0.55x \\
        & $\varepsilon_1 = 1.0\times 10^{-5}$ & -76.2418109
        & $4.9\times 10^{-5}$ & 593.5  & 993.1  & 0.60x \\
        & $\varepsilon_1 = 5.0\times 10^{-6}$ & -76.2418402
        & $2.0\times 10^{-5}$ & 1148.3 & 1823.2 & 0.63x \\
        \midrule
        \multirow{4}{*}{DMRG}
        &  $\max M = 500$ & -76.2418170 & $4.3\times 10^{-5}$
        &  1731 & 1089.7 & 1.59x \\
        & $\max M = 1000$ & -76.2418557 & $4.4\times 10^{-6}$
        &  5224 & 4435.7 & 1.18x \\
        & $\max M = 2000$ & -76.2418596 & $4.5\times 10^{-7}$
        & 17839 & 13802.6 & 1.29x \\
        & $\max M = 4000$ & -76.2418599 & $1.7\times 10^{-7}$
        & 77023 & 20585.9 & 3.74x \\
        \midrule
        \multirow{4}{*}{\fciqmc{}}
        &   $m = 10000$ & -76.2418(3) & $ 2.7\times 10^{-4}$
        & 222.4  & 277.2  & 0.80x \\
        &   $m = 50000$ & -76.24197(8) & $1.4\times 10^{-4}$
        & 1009.0 & 470.5  & 2.14x \\
        &  $m = 100000$ & -76.24181(5) & $7.1\times 10^{-5}$
        & 1942.8 & 762.1  & 2.55x \\
        &  $m = 500000$ & -76.24181(3) & $5.9\times 10^{-5}$
        & 9074.3 & 875.2  & 10.37x \\
        \bottomrule
    \end{tabular}
    \caption{Convergence of ground state energy of \ch{H2O}. For
    \sname{}, we run the test once and report the wall time to reach
    the accuracy. For other tests, each row corresponds to one test.
    We also report the wall time for \sname{} to reach the same accuracy
    as well as the ratio of the wall time of the method over the wall
    time of \sname{} in the last two columns.  \fciqmc{} runs for
    $10000$ iterations and reports the average projected energy and the
    standard error (in parenthesis) through block analysis in the energy
    column. RMSE is reported as the error of \fciqmc{}.} \label{tab:h2o}
\end{table*}

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.5\textwidth]{figures/h2o.eps}
    \caption{Convergence of ground state energy of \ch{H2O} against
    wall clock time. Each point or curve represents one test as in
    Table~\ref{tab:h2o}. For \fciqmc{}, the projected energy and its
    cumulative average from iteration $5000$ are plotted with target
    population $m=50000$. } \label{fig:h2o}
\end{figure}

From Table~\ref{tab:h2o} and Figure~\ref{fig:h2o}, we shall see that
all algorithms reach chemical accuracy efficiently. \sname{} has a
good performance in general. The energy drops quickly to the level of
$0.1$ mHa accuracy at the beginning. Then it has a slower but steady
linear decay. This behavior proves the rationale behind \sname{}:
contributions of different determinants to the FCI energy vary a lot,
especially in the early stage of iterations. Since \sname{} always
updates the ``best'' determinant at each iteration, it is able to
reach high accuracy with only a few iterations.

For small molecules like \ch{H2O}, HCI is also an excellent
algorithm and costs less time to achieve the same accuracy comparing
to \sname{}. It shows that the determinant selecting strategy used in
HCI, which relies on decaying property of Hamiltonian entries, is also
quite efficient for molecules.  For example, when $\varepsilon_1 =
5.0 \times 10^{-6}$, HCI uses only $3006594$ determinants to reach
$2.0\times 10^{-5}$ Ha accuracy, whereas \sname{} uses $1823176$
determinants, which is about $60\%$ determinants used by HCI to achieve
the same accuracy.

The speedup of HCI over \sname{} is due to the different
implementation strategies of the algorithms.  The implementation of
HCI in \textsc{Dice} stores the submatrix of the Hamiltonian with
respect to the selected determinants in the main memory, and reuses
them for inner Davidson iterations. Both the submatrix and the vector
are stored and accessed in contiguous memory. Therefore, two
advantages of the implementation come into play: one-time evaluation
of Hamiltonian entries and efficient usage of memory
hierarchy. However, the disadvantage is also obvious: huge memory cost
for the submatrix. In Table~\ref{tab:h2o} and Figure~\ref{fig:h2o}, we
do not report results for smaller $\varepsilon_1$ because
\textsc{Dice} reaches the memory limit. The high memory cost is also
the reason why the variational stage of HCI does not perform good for
chromium dimer (see Section~\ref{sec:cr2}). As a comparison, \sname{}
uses a different philosophy in the implementation. \sname{} calculates
the Hamiltonian entries on-the-fly, which saves all memory for the
coefficient vector, and stores the coefficients in a hash table. Hence
much more coefficients can be used to represent the ground state but
paying the cost of repeated evaluation of Hamiltonian entries and
limited usage of memory hierarchy.

DMRG also achieves high accuracy in reasonable time with small memory
cost. But it is always slower than \sname{} and HCI for \ch{H2O}.

\fciqmc{} is very efficient for small number of walkers and
iterations. It is able to achieve reasonable accuracy in a short
time. In Figure~\ref{fig:h2o} we see the convergence behavior of
\fciqmc{} projected energy. It can reach accuracy level of $1$ mHa very
efficiently, but hard to converge to higher accuracy due to the slow
convergence of Monte Carlo and the bias introduced by the initiator
approximation. It could be possible to use more walkers to reduce the
variance and bias. However, as shown in Table~\ref{tab:h2o}, moderate
increase of the amount of walkers does not change the convergence
behavior.

\subsubsection{Carbon dimer and nitrogen dimer}
\label{sec:c2n2}

\begin{table*}[ht]
    \centering
    \begin{tabular}{cl|lcr|rr}
        \toprule
        \multirow{2}{*}{Algorithm} & \multirow{2}{*}{Parameter}
        & \multirow{2}{*}{Energy} & \multirow{2}{*}{Error}
        & \multirow{2}{*}{Time(s)} & \multicolumn{2}{c}{\sname{}} \\
        & & & & & Time(s) & Ratio \\
        \midrule
        \multirow{5}{*}{\sname{}}
        & \multirow{5}{*}{$\varepsilon = 3.0 \times 10^{-8}$}
          & -75.7219604 & $1.0 \times 10^{-2}$ & $49.0$    & - & \\
        & & -75.7309604 & $1.0 \times 10^{-3}$ & $388.2$   & - & \\
        & & -75.7318604 & $1.0 \times 10^{-4}$ & $2687.3$  & - & \\
        & & -75.7319504 & $1.0 \times 10^{-5}$ & $13717.6$ & - & \\
        & & -75.7319594 & $1.0 \times 10^{-6}$ & $55210.2$ & - & \\
        \midrule
        \multirow{4}{*}{HCI}
        & $\varepsilon_1 = 1.0\times 10^{-4}$ & -75.7305361
        & $1.4\times 10^{-3}$ &  100.9 &  277.8 & 0.36x \\
        & $\varepsilon_1 = 2.0\times 10^{-5}$ & -75.7317130
        & $2.5\times 10^{-4}$ &  745.0 & 1319.1 & 0.56x \\
        & $\varepsilon_1 = 1.0\times 10^{-5}$ & -75.7318541
        & $1.1\times 10^{-4}$ & 1261.8 & 2565.2 & 0.49x \\
        & $\varepsilon_1 = 5.0\times 10^{-6}$ & -75.7319170
        & $4.4\times 10^{-5}$ & 2644.3 & 4989.8 & 0.53x \\
        \midrule
        \multirow{4}{*}{DMRG}
        & $\max M = 500$  & -75.7312704 & $6.9\times 10^{-4}$
        &  8624 &  544.4 & 15.84x \\
        & $\max M = 1000$ & -75.7318227 & $1.4\times 10^{-4}$
        & 14163 & 2102.9 & 6.73x \\
        & $\max M = 2000$ & -75.7319403 & $2.0\times 10^{-5}$
        & 24377 & 8582.9 & 2.84x \\
        & $\max M = 4000$ & -75.7319583 & $2.2\times 10^{-6}$
        & 68071 & 35435  & 1.92x \\
        \midrule
        \multirow{5}{*}{\fciqmc{}}
        &   $m = 10000$ & -75.729(1)   & $ 2.9\times 10^{-3}$
        & 229.5  & 140.8 & 1.63x \\
        &   $m = 50000$ & -75.7301(5) & $ 1.9\times 10^{-3}$
        & 1041.4 & 212.9 & 4.89x \\
        &  $m = 100000$ & -75.7314(4) & $ 7.0\times 10^{-4}$
        & 2038.2 & 539.8 & 3.78x \\
        &  $m = 500000$ & -75.7320(1) & $ 1.5\times 10^{-4}$
        & 9604.2 & 2003.2 & 4.79x \\
        & $m = 1000000$ & -75.7318(1) & $ 2.1\times 10^{-4}$
        & 18644.8 & 1497.4 & 12.45x \\
        \bottomrule
    \end{tabular}
    \caption{Convergence of ground state energy of \ch{C2}.  \fciqmc{}
    runs for $10000$ iterations and reports the average projected energy
    and standard error (in parenthesis) through the block analysis in
    the energy column. RMSE is reported as the error of \fciqmc{}.}
    \label{tab:c2}
\end{table*}

\begin{figure}[ht]
    \centering \includegraphics[width=0.5\textwidth]{figures/c2.eps}
    \caption{Convergence of ground state energy of \ch{C2} against
    wall clock time. Each point or curve represents one test as in
    Table~\ref{tab:c2}. For \fciqmc{}, the projected energy and its
    cumulative average from iteration $5000$ are plotted with target
    population $m=500000$.} \label{fig:c2}
\end{figure}

\begin{table*}[ht]
    \centering
    \begin{tabular}{cl|lcr|rr}
        \toprule
        \multirow{2}{*}{Algorithm} & \multirow{2}{*}{Parameter}
        & \multirow{2}{*}{Energy} & \multirow{2}{*}{Error}
        & \multirow{2}{*}{Time(s)} & \multicolumn{2}{c}{\sname{}} \\
        & & & & & Time(s) & Ratio \\
        \midrule
        \multirow{5}{*}{\sname{}}
        & \multirow{5}{*}{$\varepsilon = 5.0\times 10^{-7}$}
          & -109.2721727 & $1.0\times 10^{-2}$ & $33.4$    & - & \\
        & & -109.2811727 & $1.0\times 10^{-3}$ & $752.6$   & - & \\
        & & -109.2820727 & $1.0\times 10^{-4}$ & $7892.6$  & - & \\
        & & -109.2821627 & $1.0\times 10^{-5}$ & $49862.6$ & - & \\
        \midrule
        \multirow{4}{*}{HCI}
        & $\varepsilon_1 = 1.0\times 10^{-4}$ & -109.2805259
        & $1.7\times 10^{-3}$ &  100.7 &  427.1 & 0.24x \\
        & $\varepsilon_1 = 2.0\times 10^{-5}$ & -109.2817822
        & $3.9\times 10^{-4}$ &  730.9 & 2107.4 & 0.35x \\
        & $\varepsilon_1 = 1.0\times 10^{-5}$ & -109.2819787
        & $1.9\times 10^{-4}$ & 1335.2 & 4266.8 & 0.31x \\
        & $\varepsilon_1 = 5.0\times 10^{-6}$ & -109.2820857
        & $8.7\times 10^{-5}$ & 3330.0 & 8920.3 & 0.37x \\
        \midrule
        \multirow{4}{*}{DMRG}
        &  $\max M = 500$ & -109.2809830 & $1.2\times 10^{-3}$
        &  9936 &   619.6 & 16.04x \\
        & $\max M = 1000$ & -109.2818757 & $3.0\times 10^{-4}$
        & 17647 &  2806.7 & 6.29x \\
        & $\max M = 2000$ & -109.2821098 & $6.3\times 10^{-5}$
        & 37549 & 11857.1 & 3.12x \\
        & $\max M = 4000$ & -109.2821632 & $9.5\times 10^{-6}$
        & 85703 & 51574.7 & 1.66x \\
        \midrule
        \multirow{5}{*}{\fciqmc{}}
        &   $m = 10000$ & -109.2818(4) & $ 5.2\times 10^{-4}$
        & 235.7  & 1556.8  & 0.15x \\
        &   $m = 50000$ & -109.2822(3) & $ 2.6\times 10^{-4}$
        & 1068.9 & 3161.5 & 0.34x \\
        &  $m = 100000$ & -109.2818(2) & $ 4.0\times 10^{-4}$
        & 2090.5 & 2077.2  & 1.01x \\
        &  $m = 500000$ & -109.2822(1) & $ 1.2\times 10^{-4}$
        & 9839.3 & 6832.6 & 1.44x \\
        & $m = 1000000$ & -109.28214(5) & $ 5.0\times 10^{-5}$
        & 18959.7 & 14510.0 & 1.31x \\
        \bottomrule
    \end{tabular}
    \caption{Convergence of energy of \ch{N2}.  \fciqmc{} runs for
    $10000$ iterations and reports the average of projected energy and
    standard error (in parenthesis) through block analysis. RMSE is
    reported as the error of \fciqmc{}.} \label{tab:n2}
\end{table*}

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.5\textwidth]{figures/n2.eps}
    \caption{Convergence of ground state energy of \ch{N2} against
    wall clock time. Each point or curve represents one test as in
    Table~\ref{tab:n2}.  For \fciqmc{}, the projected energy and its
    cumulative average from iteration $5000$ are plotted with target
    population $m=500000$.} \label{fig:n2}
\end{figure}

Carbon dimer \ch{C2} and nitrogen dimer \ch{N2} are more challenging
than \ch{H2O} molecule since their correlation is stronger and
the dimension $N_{\text{FCI}}$ is higher. The results of \ch{C2}
are reported in Table~\ref{tab:c2} and Figure~\ref{fig:c2};
and the results of \ch{N2} are reported in Table~\ref{tab:n2} and
Figure~\ref{fig:n2}. In general, \ch{C2} costs more time than \ch{H2O}
to converge to a fixed accuracy and \ch{N2} costs more time than
\ch{C2}, which agrees with their system complexities.

\sname{} shows similar convergence pattern for \ch{C2} and \ch{N2},
with fast decay at the beginning followed by a slower but steady
linear decay. It takes only several minutes to reach the chemical
accuracy. Therefore, \sname{} is consistently efficient for different
systems with different correlation strength.

HCI also shows similar convergence behavior for \ch{C2} and \ch{N2}. It
converges to chemical accuracy the fastest among tested algorithms.
However, HCI can not converge to higher accuracy due to the memory
limit of the implementation. Comparing to \sname{} in terms of the
number of operations, however, \sname{} uses less operations and
determinants than HCI to the same accuracy level, as in the case
of \ch{H2O}.

DMRG also performs similar for \ch{C2} and \ch{N2} but is significantly
slower than \sname{} and HCI. One reason is that DMRG needs more
iterations to converge due to the strong correlation. In this sense,
determinant selecting algorithms are less affected by the correlation
strength than DMRG.

\fciqmc{}, however, has a different behavior for \ch{C2} and
\ch{N2}. While \ch{N2} is significantly harder than \ch{C2} for \sname{},
HCI and DMRG, \fciqmc{} reaches a higher accuracy for \ch{N2} within
$10000$ iterations, as shown by the data in Table~\ref{tab:c2} and
Table~\ref{tab:n2}.  We point out that \fciqmc{} performs quite well
for \ch{N2}, as it can reach $10^{-4}$ Ha error in a short time with
only $m=10000$ or $m=50000$ walkers, whereas \sname{} takes more time
to converge since the dimension of \ch{N2} is higher than \ch{H2O}
and \ch{C2}. \fciqmc{} seems to be less influenced by the increase
of dimensionality.

In conclusion, as shown in both Section~\ref{sec:h2o} and
Section~\ref{sec:c2n2}, \sname{} is efficient for both weakly-correlated
and strongly-correlated systems. It can achieve chemical accuracy
efficiently and is able to achieve higher accuracy in all tested
molecules. HCI costs more operations than \sname{} to achieve the
same accuracy but costs less time due to the different philosophies in
implementations. Comparing to \sname{} and HCI, DMRG is less efficient
for strongly-correlated systems. While similar as \sname{}, DMRG can
also achieve much higher accuracy than the chemical accuracy.  \fciqmc{}
is also efficient to reach chemical accuracy with a few walkers in
short time for the testing molecules, but it may need much more time
and walkers to reach higher accuracy.

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.5\textwidth]{figures/eps.eps}
    \caption{The effect of compression tolerance $\varepsilon$ in
    \sname{}. The figure shows the error of energy and the number of
    nonzeros entries in the vector $\vec{b}$ after convergence for
    differenct $\varepsilon$. By choosing differenct $\varepsilon$,
    we can easily balance between accuracy and memory cost.}
    \label{fig:eps}
\end{figure}

In terms of the usability, \sname{}, HCI, and DMRG only have one single
parameter to be tuned, whereas \fciqmc{} has more parameters. The proper
parameter in \sname{} can be revealed in a few minutes, judging from
whether the stabilized vector $\vec{b}$ properly utilizes the given
amount of memory. By choosing $\varepsilon$, we can easily balance
between accuracy and memory cost, as shown in Figure~\ref{fig:eps}. We
conclude that \sname{} is an easy-to-use efficient algorithm for FCI
problems.

\subsection{Binding curve of \ch{N2}}
\label{sec:bcn2}

In this section, we benchmark the all-electron nitrogen binding curve
using \sname{} under the Dunning's cc-pVDZ basis. The nitrogen binding
curve is a well-known difficult problem. When the nitrogen atoms are
stretched away from the equilibrium geometry, Hartree-Fock theory no
longer gives a good approximation and the system becomes multi-referenced
due to the triple bond between the atoms. DMRG and coupled cluster
theory, e.g., CCSD, CCSD(T), CCSDT, etc., have been tested on this
problem, but only on 6 geometry configurations~\cite{Chan2004b}.
Here we show that \sname{} is capable to efficiently benchmark the
all-electron nitrogen binding curve on a very fine grid of bond length
and the variational energy converges to at least $10^{-3}$ mHa accuracy
in each configuration.

In this problem, there are $14$ electrons and $28$ orbitals,
and the dimension of the FCI space is about
$N_{\text{FCI}} \approx 1.75\times 10^{11}$. In all configurations,
$\varepsilon = 10^{-6}$ is used in \sname{} for truncation.  Here we
use the same computing environment as in
Section~\ref{sec:num-res-molecule} but with OpenMP enabled with 5
threads. Each configuration on the bind curve results take roughly one
day to achieve the $10^{-3}$ mHa
accuracy. Figure~\ref{fig:n2_binding_curve} shows the binding curve
and Table~\ref{tab:n2_binding_curve1} and \ref{tab:n2_binding_curve2} in
Appendix~\ref{app:n2_binding_curve} list all converged variational
energies for every configuration in the figure.  In
Table~\ref{tab:n2_comparison}, we compare selected results obtained
from \sname{} with that from other algorithms reported in
Ref.~\citenum{Chan2004b}.

\begin{figure}[ht]
    \centering
    \subfigure[Energy of \ch{N2}]{
    \includegraphics[width=0.5\textwidth]{figures/n2_binding_curve.eps}
    }
    \subfigure[Hartree-Fock value
    $\vec{c}_{\mathrm{HF}}/\norm{\vec{c}}$ and the number of nonzeros
    of $\vec{c}$ to reach $10^{-5}$ Ha energy error for \sname{}]{
    \includegraphics[width=0.5\textwidth]{figures/n2_binding_curve_hf_entries.eps}
    }
    \caption{Binding curve and the corresponding Hartree-Fock values
    and number of nonzeros in $\vec{c}$ of \ch{N2} under cc-pVDZ basis.}
    \label{fig:n2_binding_curve}
\end{figure}

\begin{table*}
    \centering
    \begin{tabular}{ccccccc}
    \toprule
    Bond Length & 2.118$a_0$ & 2.4$a_0$ & 2.7$a_0$ & 3.0$a_0$
    & 3.6$a_0$ & 4.2$a_0$ \\
    \midrule
    \sname{} & -109.282173\wds{} & -109.241908\wds{} & -109.163600\wds{}
    & -109.089405\wds{} & -108.998083\wds{} & -108.970132\wds{} \\
    DMRG     & -109.2821\wds{57} & -109.2418\wds{86} & -109.1635\wds{72}
    & -109.0893\wds{80} & -108.9980\wds{52} & -108.9700\wds{90} \\
    CCSD     & -109.2\wds{67626} & -109.2\wds{19794} & -109.1\wds{31491}
    & -109.0\wds{52884} & -108.9\wds{75885} & -108.9\wds{60244} \\
    CCSDTQ   & -109.281\wds{943} & -109.241\wds{321} & -109.16\wds{2264}
    & -109.08\wds{6502} & -108.99\wds{3736} & -108.96\wds{8124} \\
    MRCISD   & -109.27\wds{5356} & -109.2\wds{34925} & -109.15\wds{6473}
    & -109.08\wds{2149} & -108.99\wds{0759} & -108.9\wds{63070} \\
    MRCCSD   & -109.28\wds{0646} & -109.24\wds{0362} & -109.16\wds{1969}
    & -109.08\wds{7613} & -108.99\wds{5885} & -108.96\wds{7865} \\
    \bottomrule
    \end{tabular}
    \caption{Nitrogen molecule ground state energy using \sname{}, DMRG
    ($\max M = 4000$) and couple cluster theories. \emph{Slant digits}
    indicate inaccurate digits. All results except \sname{} are from
     Ref.~\citenum{Chan2004b}.} \label{tab:n2_comparison}
\end{table*}

Several remarks are in order regarding the benchmark results.
First, Figure~\ref{fig:n2_binding_curve} demonstrates a smooth
standard shape binding curve. Different from the carbon binding
curve~\cite{Holmes2016}, no jump is observed in our benchmark results,
where $D_{2h}$ symmetry is used for all configurations. Second,
Table~\ref{tab:n2_comparison} shows that \sname{} gives the lowest
energy. \sname{} energy is accurate beyond the level of $10^{-3}$
mHa whereas DMRG is accurate up to $10^{-2}$ mHa. The DMRG
results in Table~\ref{tab:n2_comparison} are taken from previous
work~\cite{Chan2004b}, which agree with the results obtained in
Section~\ref{sec:c2n2}. Other algorithms such as CCSDTQ are much less
accurate. Third, the more the nitrogen dimer molecule is stretched
from the equilibrium, the more determinants and iterations are needed
for \sname{} to converge to $10^{-3}$ mHa accuracy. This is because
Hartree-Fock theory only works well near equilibrium configuration.
However, the number of determinants and iterations do not increase
significantly, which shows the efficiency of \sname{} again.
Other algorithms become less accurate for larger stretching distance.

\subsection{All electron chromium dimer calculation}
\label{sec:cr2}

Chromium dimer is hard to compute due to its strong correlation. We
calculate the all-electron molecule using \sname{} under the
Ahlrichs VDZ basis with radius $r = 1.5$\r{A}. There are $48$
electrons and $42$ orbitals, and the dimension of the FCI space
is about $2\times 10^{22}$. Many methods have been applied
to this problem including DMRG~\cite{Olivares-Amaya2015} and
HCI~\cite{Holmes2016}. Table~\ref{tab:cr2} summarizes all results,
including our \sname{} results and others from
literature~\cite{Olivares-Amaya2015, Holmes2016}.

\begin{table}
    \centering
    \begin{tabular}{cl}
    \toprule
    Algorithm & Energy (Ha)\\
    \midrule
        HCI (variational)    & $-2086.\wds{384}$    \\
        CCSD(T)              & $-2086.4\wds{22229}$ \\
        CCSDTQ               & $-2086.4\wds{30244}$ \\
        DMRG ($\max M=8000$)  & $-2086.44\wds{3334}$ \\
        \sname{}             & $-2086.443\wds{565}$  \\
    \midrule
        HCI (perturbed)      & $-2086.444\wds{04}$  \\
        DMRG (extrapolated)  & $-2086.444784$ \\
    \bottomrule
    \end{tabular}
    \caption{Energy of \ch{Cr2}.}
    \label{tab:cr2}
\end{table}

In this paper, we only consider variational ground state energy
without any perturbation or extrapolation. Regarding the variational
ground energy, \sname{} achieves lowest energy among all algorithms
in one month running time on a machine with Intel Xeon CPU E5-1650
v3 @ 3.50GHz and 128GB memory. Both DMRG ($\max M=8000$) and \sname{}
achieve the chemical accuracy if the energy of DMRG (extrapolated) is
regarded as the ground truth. But HCI(variational) and coupled cluster
theory cannot achieve chemical accuracy. HCI converges to $-2086.384$ Ha
in about eight minutes~\cite{Holmes2016}, whereas \sname{} reaches the
same accuracy in about twenty minutes, although different computing
environments are used.  As the dimension of the Hamiltonian becomes
larger and the system becomes more correlated, HCI can only afford
storing the submatrix in the main memory with very limited number of
determinants and such a limited number cannot achieve higher accuracy
in the variational phase. With perturbation phase enabled, HCI can
achieve accuracy less than $1$ mHa.  Similar perturbation phase can
be adapted to \sname{} to further boost the accuracy or extend the
applicability of our algorithm to larger systems.

\section{Conclusion and discussion}
\label{sec:conc}

The proposed \fname{} (\sname{}) is an easy-to-use, accurate, and
efficient algorithm for full configuration interaction eigenvalue
problems of quantum many-body systems, especially for strongly correlated
systems. The only tuning parameter in \sname{}, $\varepsilon$, controls
the trade-off between memory cost and accuracy. Given the fixed amount
of memory, the ``close-to-optimal'' $\varepsilon$ can be determined
within a few minutes without waiting for convergent results. Hence,
we believe that \sname{} is one of the most easy-to-use algorithms
among competitors. Besides the user friendly property, \sname{}
performs competitively with many other methods, including heat-bath
configuration interaction (HCI), density matrix renormalization group
(DMRG), and full configuration interaction quantum Monte Carlo with
initiator and semi-stochastic adaptation~(\fciqmc{}).  The \sname{} can
give the state-of-art results for many strongly correlated FCI problems.

There are a few immediate future work of \sname{}. Apply \sname{} to
current examples with larger basis sets and other more challenging
systems; add perturbation stage to further improve the accuracy;
and, parallelize \sname{} in a distributed-memory setting.
The earlier two can be accomplished directly, while the massive
distributed-memory parallelization of \sname{} requires modification
of the greedy determinant-select strategy. Hence FCIQMC type methods
currently have some advantage in that regard. As discussed in our
previous work~\cite{Li2018b}, with a stochastic variant of the
current determinant-select strategy, the asynchronized feature of
coordinate descent methods can be enabled.  \footnote{While it is
called asynchronized parallelization in coordinate descent methods,
communication is still needed after every several iterations. Hence
the embarrassing parallelization of Monte Carlo methods, as in FCIQMC
type methods, is of better scalability.} Massive distributed-memory
parallelized \sname{} is expected to achieve good performance. Besides
these, we are also exploring (semi-)stochastic \sname{} to improve
the parallelizability of the algorithm, and further accelerating
the initial iterations. Replacing the current hash function with
a more efficient one to fully utilize the memory hierarchy is also
under investigation. It is also interesting to design an auto-tuning
procedure for ``close-to-optimal'' $\varepsilon$ to remove the only
tuning parameter in the algorithm.

Beyond ground state computation, \sname{} is also suitable for
excited state computation.  The extension of the optimization problem
\eqref{eq:opt} to low-lying $k$ excited states can be achieved without
orthogonality constraint, i.e.,
\begin{equation} \label{eq:opt-k}
    \min_{\vec{c} \in \bbR^{N_{\text{FCI}} \times k}} f(\vec{c}) =
    \norm{H + \vec{c}\vec{c}^\top}_F^2.
\end{equation}
This is favorable as it removes the expensive orthogonalization step
for FCI wavefunctions during iterations.  Hence extending \sname{}
to solve low-lying excited states is another promising future
direction to be explored.

\begin{acknowledgement}
    The authors thank Qiming Sun for helpful discussions regarding
    PySCF and FCI calculations, Ali Alavi for insightful suggestions
    and for providing all configurations of NECI, and George Booth for
    helpful discussion on FCIQMC. The work is supported in part by the US
    National Science Foundation under awards DMS-1454939 and OAC-1450280
    and by the US Department of Energy via grant DE-SC0019449.
\end{acknowledgement}

\nocite{*}

redpajama_set_name: RedPajamaArXiv


text:
Zinjibar (arabe : زنجبار) est une ville côtière située dans le Sud-Ouest du Yémen ; elle est le chef-lieu du district du même nom et du gouvernorat d'Abyan. C'est la  du Yémen.

Elle comptait en 2004,  habitants. D'après Le Monde, en 2011, la ville comptabilisait près de 30 000 habitants.

Histoire ancienne
Zinjibar est un ancien centre de commerce.
La ville portuaire a servi de plaque tournante pour les marchandises de l'Extrême-Orient.
En 1163 (559 AH), la ville est détruite et incendiée par Abdel Nabi Ali Mahdi Yoosuf.

Des fouilles archéologiques ont montré que la ville est reconstruite au , puis détruite.
Au , la ville est reconstruite par le sultan du Sultanat de Fadhli, Hussein bin Ahmed bin Abdullah, et nommée Zinjibar par son petit-fils le sultan Abdul Qadir bin Ahmed bin Hussein au début des années 1920, en l'honneur de Khalifa bin Harub de Zanzibar.

Histoire récente
Zinjibar a été la capitale administrative du sultanat de Fadhli de 1962 à 1967.

En mai 2011, des militants islamistes prennent le contrôle de la ville dans le contexte de la révolte yéménite de 2011 mais, par une intervention des forces gouvernementales, elles sont contraintes à abandonner la ville en juin 2012.

Géographie
Zinjibar est situé à 10 km au sud de Ja'ar et à 30 km à l'est d'Aden. Elle est sur le golfe d'Aden.

Notes et références

Sources
 François-Xavier Trégan, « Sur la ligne de front de Zinjibar, fief des djihadistes du Yémen », Le Monde, , mis à jour le .

Articles connexes
 Géographie du Yémen, Subdivisions du Yémen
  La bataille de Zinjibar (2011)

Ville au Yémen

redpajama_set_name: RedPajamaWikipedia


text:
Research Roundup: Depression, Inflammation, Menopause and More
Published: Dec 18, 2020 By Mark Terry
Every week there are numerous scientific studies published. Here's a look at some of the more interesting ones.
Depression Associated with Inflammation and Menopause
Two studies published by University of Texas Southwestern Medical Center researchers analyzed data from the Dallas Heart Study (DHS), which has tracked the health of thousands of participants since 2000. One of the studies, published in the Journal of Clinical Psychiatry, demonstrated a link between an inflammatory blood molecule and the likelihood of depressive symptoms. They found that levels of GLycA, an inflammatory molecule not typically tested in patients, was associated with severity of depressive symptoms. Even after taking into consideration sex, ethnicity, antidepressant use, education, and body mass index, GLycA levels were associated with depression severity.
"This study suggests that maybe we could predict or diagnose depression based on inflammatory scores," said Samara Huckvale, an undergraduate at Columbia University working in the laboratory of UT Southwestern professor Sherwood Brown, senior author of both papers and professor of psychiatry and vice chair for clinical research at UTSW. "Or maybe eventually we'll be able to design therapies that actually target this inflammation to treat depression."
The second study was published in Maturitas by Brown and colleagues. In this research, they used DHS data to analyze menopausal women, a group identified as having an increased risk of depression. They analyzed data on 384 women aged 37 to 73 years who self-reported in menopause. The women reported on symptoms typically associated with menopause, which the team then grouped into vasomotor, psychosocial, physical, or sexual symptoms. The women also completed the Quick Inventory of Depressive Symptomatology-Self Report survey (QIDS-SR), which gauged depressive symptoms.
Sexual symptoms of menopause were positively linked with a higher score on the QIDS-SR and remained even for women taking antidepressants. There was also a link between psychosocial symptoms of menopause and the QIDS-SR. No link was found between vasomotor or physical symptoms and the QIDS-SR score, and neither was ethnicity.
"Recognizing patterns in who is most likely to develop depression is really important to help guide our screening efforts," said Sydney Singleterry, co-first author of the Maturitas study and a fourth-year UTSW medical student.
First 10 Days After Hospital Stay Has Highest Risk for COVID-19 Patients
Researchers with Michigan Medicine – University of Michigan found that the first months after a hospital stay for COVID-19, patients had a high risk of ongoing health problems, readmissions and death; but the highest risk was in the 10 days after leaving the hospital. The study found that COVID-19 patients had a 40% to 60% higher risk of readmission or dying in the first 10 days when compared to similar patients who were treated at the same hospitals during the same period for heart failure or pneumonia. But by the end of 60 days, the overall risk of readmission or death for COVID-19 patients was lower than for heart failure or pneumonia. But the study did find that in the first two months, 9% of COVID-19 patients who survived hospitalized had died, and about 20% had a setback requiring readmission—on top of the 18.5% who died during hospitalization.
Combination Therapy Might Help Difficult-to-Treat Cancers with RAS or NF1 Mutations
Investigators with the UCLA Jonsson Comprehensive Cancer Center evaluated a new combination therapy that suppresses the MAPK pathway. The two-drug combo has the potential to treat BRAF-mutated melanoma, but also other aggressive cancers, including melanoma, lung, pancreatic and colon cancers that include RAS or NF1 cancer gene mutations. The work was preclinical, but utilized a second-generation RAF inhibitor with a standard MEK inhibitor and found it to be effective in treating cancers with these mutations and may also be able to treat cancers with specific BRAF mutations that become resistant to current standard-of-care treatments. The combination is now being evaluated in clinical trials in both melanoma and other solid cancers such as non-small cell lung cancer.
Protecting Neurons and Encouraging Their Growth
A number of neurodegenerative conditions, ranging from glaucoma to Alzheimer's, are marked by axon injury. Axons are the long, thin projections that conduct electrical impulses between nerve cells. Injury to the axons often results in neuronal impairment and cell death. Researchers at the University of California San Diego School of Medicine and Shiley Eye Institute at UC San Diego Health identified a new family of enzymes, germinal cell kinase four kinases (GCK-IV kinases) whose inhibition appears to be strongly neuroprotective, while also allowing axons to regenerate. It's been known that inhibiting another enzyme, dual leucine zipper kinase (DLK) protects neurons, but DLK also inhibits axonal regeneration. They have been evaluating GCK-IV kinases in optic nerve cells. Glaucoma, for example, is an optical disease of internal eye pressure that can cause nerve damage and blindness, but eye pressure is only part of the problems. "At its core," said Derek Welsbie, associate professor of ophthalmology at Shiley, "glaucoma is a neurodegenerative disease characterized by progressive loss of RGCs and their axons, leading to measurable structural and functional damage to the optic nerve, visual impairment and blindness."
Spermidine Appears to Improve the Effectiveness of Vaccines in Older People
Investigators with University of Oxford, U.K., found that a drug that helps remove cellular debris in immune cells appears to increase the protective effects of vaccines in older adults. They identified a compound called spermidine, which increased autophagy—the removal of cellular debris—and increases T-cell function. In cell cultures, they found that adding spermidine restored autophagy to the same levels observed in T-cells in younger people. A small clinical trial recently evaluated if giving spermidine to older adults would improve cognitive function, and had positive results. It did not appear to be harmful, and is supportive of further clinical studies to test if it can help the immune responses of older people.
Genes Associated with Common Knee Injury
Injuries to the anterior cruciate ligament (ACL) in the knee have long been associated with type of shoe, surface and sport (as well as trauma). Researchers with Sweden's Lund University, analyzing data from the Swedish Twin Register and data from the Swedish National Patient Register, were able to observe how many identical twins and fraternal twins had been diagnosed with ACL rupture. The study analyzed a total of 88,414 twins aged 17 or older.
"Our results show that genes appear to contribute more than we thought," said Karin Magnusson, researcher and expert in twin studies within the field of orthopedics at Lund University. "To put it very simply, we can say that out of all cruciate ligament injuries, 69% can be explained by genetics. This should not be interpreted to mean an individual's risk of suffering the injury is 69%, rather that it is significant in terms of the wider population."

redpajama_set_name: RedPajamaCommonCrawl


In response to the alarming situation, several NGOs have initiated relief works. AFPRO, CARE, World Vision and the Public Health Engineering (PHE) departments of the government and village Panchayats have come together for a programme to provide succor to about 65,000 villagers in 103 villages across 43 Panchayats in Mahasamund. The programme is supported by the Australian High Commission and co-coordinated by CARE.
A total of 24,226 person days of employment were generated in 32 villages. Cash-for-work in repairing and deepening ponds and tanks, construction of water diversion structures, strengthening existing dams and canals were implemented in the programme period 1st February to 1st May, 2001. The local practice of wage employment, called the Godi system was followed, whereby digging one Godi of land measuring 13ft.x13ft.x1ft. makes a person eligible for a payment of Rs.105.
In collaboration with the PHE department, AFPRO also trained 14 youth in repairing and maintaining hand pumps. The new mechanics will be given a toolkit each, and will operate in 43 Panchayats. 28 non-functional hand pumps will also be repaired. Combating droughts however requires a long-term vision. AFPRO has already worked out the design details and cost estimates for soil and water conservation structures for 22 villages in the Mahasamund block. The communities where the structures are planned are cooperating enthusiastically, voluntarily guarding raw materials and supervising construction work.


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text:
Life's A Witch
21M Reads 14.1K Votes 33 Part Story
By BrittTheBookSlayer Completed
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This witch lives a charmed life. . .
Hadley is the envy of every girl-and the desire of every guy. Her key to magically having it all? She's an actual witch.
As a descendant of the first woman executed in the Salem witch trials, Hadley understands the consequences if her secret gets out. The only ones who know about her powers are members of her coven, who, like Hadley's family, have magic in their blood. But there's no spell that can undo a fatal attack on the coven. All the evidence points to an age-old rival coven as the culprit, and now it's up to Hadley to lead the fight against them.
At the same time, she's caught up in her feelings for a mysterious guy named Asher, who has plenty of secrets of his own. With everything at stake, can Hadley trust her magic-and her heart?
buffythevampireslayer
witchtrials
NOTE FROM AUTHOR-please read!
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Nineteen
Chapter Twenty
Chapter Twenty-One
Chapter Twenty-Two
Chapter Twenty-Three
Chapter Twenty-Four
Chapter Twenty-Five
Chapter Twenty-Six
Chapter Twenty-Seven
Chapter Twenty-Eight
Chapter Twenty-Nine
Chapter Thirty
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4.1M 70.9K 10.5K
The girl knows she's different. She doesn't age. She has no family. She has visions of a past life, but no clear clues as to what she is, or where she comes from. But there is a face in her dreams - a light that breaks through the darkness. She knows his name is Gabriel.
On her way home from work, the girl encounters an injured stranger whose name is Jonah. Soon, she will understand that Jonah belongs to a generation of Vampires that serve even darker forces. Jonah and the few like him, are fighting with help from an unlikely ally - a rogue Angel, named Gabriel.
In the crossfire between good and evil, love and hate, and life and death, the girl learns her name: Lailah.
But when the lines between black and white begin to blur, where in the spectrum will she find her place? And with whom?
Gabriel and Jonah both want to protect her. But Lailah will have to fight her own battle to find out who she truly is.
Authors note: I first published LAILAH (the first book in The Styclar Saga) here on Wattpad and after it rose in popularity *inserts massive fist pump here!* the story (along with the next two stories in the saga) were picked up for publication with F&F Macmillan in the US and Canada -- and since then has also published with AST in Russia, Fantasy Foundation in Taiwan and, coming this year, Macmillan Century in China! *yeah we need another fist pump! ;)
PLEASE NOTE: THIS IS THE ORIGINAL VERSION THAT I UPLOADED TO WATTPAD NOT THE EDITED, FINAL PUBLISHED VERSION
As a thank you to my wonderful styhards and wattpadders, you can find some Styclar short stories right here exclusvie to wattpad on my profile! (Told from the view points of our gorgeous guys, Gabriel and Jonah!)
All 3 books in The Styclar Saga have been released in various formats (hardback, paperback, ebook &a audio)
book1 LAILAH
book2 GABRIEL
book3 JONAH
Copyright of LAILAH varying editions* 2009/ 2012 / 2014 / 2015/ 2016/ 2017 All Rights Reserved
Cinderella vs. Prince Charming
By LaurenDMSmith
1.6M 22.6K 2.3K
Darren dropped the note addressed to the "Prince Charming wannabe" and announced "Well then Cinderella, I propose a game. If I win, you must come away with me. If you win, I leave and never return. The game itself is this: I have to find you."
THE WITCH IS BACK (A Life's a Witch book)
By BrittTheBookSlayer
18.1K 277 52
It's been a trying year for teen witch Hadley Bishop—she led her coven into battle against the Parrishables and lost her mother in a surprise attack. Now that the Parrishables and their leader, Samuel Parris, have been defeated, Hadley is looking forward to a relaxing, love-filled summer with boyfriend Asher Astley.
Hadley's summer plans take an unexpected turn though when she and her coven head off to a witchy summer intensive. Once there, Hadley immediately finds herself in a power struggle with Brooklyn Sparks. Like Hadley, Brooklyn is gorgeous and popular, but she's also…Asher's ex-girlfriend?!
As if this news isn't enough to rattle Hadley, there are rumors of a rogue witch in the woods, and someone is playing dangerous pranks around camp. Hadley suspects that Brooklyn is behind the pranks, but is she willing to risk losing her friends and Asher to prove it? In this electrifying follow-up to Life's a Witch, Hadley will discover that love is more powerful than any spell—but it comes with a price.
The Popularity Debt
By bryonymagee
12.1M 167K 43.6K
Maddie Elliot's life in the sleepy town of Hallow's Wood was simple before Ingrid Brooke arrived. Now, both girls are locked in a tangle of secrets, lies and jealousy...and they're about to learn that all friendships come at a price.
Maddie always dreamed of being popular. Ingrid grew up in California, with rich parents and masses of friends, and when she arrives for senior year in Hallow's Wood she gives Maddie the one thing she's always wanted: a seat beside her, on the throne of popularity.
With Ingrid's help, Maddie is suddenly swept off her feet, into a world of sex, drugs and gossip-and it's amazing, until things start to go wrong.
After a trip to California, where she finds herself a little more than close to Ingrid's older brother, Maddie starts to realize Ingrid could be hiding a few dark secrets that are dangerous for both girls-not to mention the boys chasing on their heels. Trouble starts coming at Maddie from a different angle when she receives a chain of creepy letters from a 'secret admirer'...
Then, just when she believes things can't get any worse, Maddie uncovers the worst secret of all-one that could rip her new friendship, and her relationship, apart. But now that she's a popular girl, can the girls make the right decision, or have they already fallen too deeply into the popularity debt?
Ki$$ & $ell
647K 15.5K 4.2K
Arielle Sawyer is freaking out because she's the last person in her class to be kissed. Frustrated by her kissably-challenged lifestyle, Arielle allows herself to be talked into selling her first kiss to the highest bidder—on eBay. The media soon catches wind of her story and all of a sudden she's giving interviews and appearing on popular late-night talk shows. Due to her newfound fame, the former wallflower suddenly finds herself the focus of everyone's attention, including that of the most popular guy at school, an actor who just happens to be the teen heartthrob of the moment and even the local bad boy. But as her popularity grows and her friendships begin to weaken, Arielle starts to wonder why she put up the post in the first place. In the end there will be a winner, but will Arielle's first kiss end up being everything she's ever dreamed of?
Unsung
3.8K 560 151
A comic book nerd joins a hero school and discovers that villains are much scarier in real life than in the books she reads.
By SeventyMurphy
When an eccentric old neighbour dies and names Violet March in his will, she is even more surprised than his estranged and spoiled family. To make matters stranger, she learns that all must attend a pretend murder-mystery weekend for any to claim a share of the inheritance. Being a good sport under constant scrutiny and suspicion becomes trickier when she finds herself torn between the romantic attentions of both the deceased's "favourite" grand-nephew and the charming black sheep of the family. Loyalty forces her to favour the former, but when it is the latter who inherits the bulk of the estate, a grave case of mistaken identities is revealed leaving Violet to determine whether pride and an unwillingness to be labelled a gold-digger may stand in the way of the love of a lifetime.
*Cover from "Lake Louise, An Alpine Wonderland" by Norman Fraser, which is to the best of my research and knowledge in the Public Domain.
*No part of this book may be reproduced or used in any manner without the express written permission of the author.
Bliss has always been different. Because of a condition called CIPA, she's never felt pain–no headaches, no cramps, no charlie horses. She also can't sweat, cry or even play sports without endangering her life. And the worst part? She knows she has an expiration date, and it is probably fairly soon -- most people like her don't live past the age of 18.
So, what's a girl to do? Well, if you're Bliss, you give the world the finger and decide to live life on your own terms. For her, this means training with her top-secret government agent mom in the art of hand-to-hand combat, and convincing her mad scientist dad to invent a special suit that will let her seek out the world's dangers without overheating and biting the big one. She is determined to take the world on, fighting the "bad guys" and leaving her mark during the short time she has.
But when an old friend from CIPA camp asks her to help track down others like them who've suddenly disappeared under mysterious circumstances, her reckless lifestyle is put to the ultimate test. Especially when the junior government agent-in-training her mom assigns to help them with the case turns out to be totally hot--and totally into her. Torn between her feelings of anger toward the life she's been dealt and the one she could have if she just let others in, Bliss takes off on a mission that might just be her last.
By MichaelLimjoco
An original and unique Paranormal Romance Mystery that will leave you on the edge of your seat and guessing until the end!
When nineteen-year-old Vanessa Hunter falls to her death, she realizes her life as she knows it, is over.
But when Death strikes a bargain with her in exchange for her immortal soul, she takes the ultimate gamble and becomes a Death Angel: gifted with the powers of Death and responsible for finding and harvesting lost souls on earth.
Now Vanessa has to perform Death's dirty deeds, forced to deal with all sorts of unimaginable evils: Vampires, Demons from Hell, and Hedge Fund Managers. Not to mention this girl Rina, with special powers, who is absolutely obsessed with taking down the new Death Angel.
On top of all that, Vanessa still has to solve the riddle to her murder; a riddle that points back to her fiance, Daniel Ashcroft. Thrust into this strange new world, Vanessa finds her emotions in turmoil. She still loves Daniel with all her heart, but now finds herself strangely drawn to Death; a being who is at once mysterious, infuriating, and madly attractive.
But things are not at all what they seem, and there are many things that Vanessa needs to uncover, unsettling things about her former life, her family's money, and Death, her newfound "savior" who doesn't always seem to be on her side.
Can Vanessa tell friend from foe, and solve the riddle of her murder to find out if the man she's loved all along, was truly her lover? Or her greatest enemy?
Copyright © 2015 Michael Limjoco
By wildgreenskittle
5.4M 111K 13.2K
Reid has been homeless on the streets of New York for years now, fighting for cash in clubs and sleeping in empty buildings. With his special ability, Reid could have whatever life he wanted, but chooses not to.
When Sam meets Reid on the subway, she has a hard time forgetting about him, and not only because he keeps showing up in the least likely of places. Sam doesn't know it at first, but the boy she just met is about to show her a whole other world.