\section{Introduction}

Quantum mechanical (QM) methods today play a prominent role in many branches of chemical science. In particular, Kohn--Sham density functional theory (DFT) has made a large impact owing to its computational efficiency and often accurate results. \cite{burke2012,harvey2006,marques2004,Casida_tddft_review_2012,Kirchner2007} However, for systems with dense frontier orbital manifolds and with degenerate or near-degenerate electronic states, DFT can be inaccurate, which is often seen for transition-metal complexes in biological systems.\cite{pierloot2011}. Thus, methods that can handle such cases are needed. The coupled cluster (CC) methods can be highly accurate, but they may also deteriorate for multiconfigurational systems and are considerably more expensive, if at all feasible.
The alternative is to employ a multiconfigurational wave function.
One of the most common multiconfigurational methods is the complete active space (CAS) approach, in which the orbitals are divided into active and inactive spaces. Within the active space, all configurations are included in a full configuration interaction (full-CI) calculation, thus incorporating any multiconfigurational character. Combining the CAS with a self-consistent field (SCF) procedure leads to  the complete active space self-consistent field (CASSCF) method.\cite{ruedenberg1979,roos1980b,siegbahn1981,jensen1984,olsen1983,jensen1986} On the one hand, the accuracy of CAS-based methods depends on the size of the active space, in which all important orbitals should be included. On the other hand, the computational effort also rises steeply with the size of the active space so that traditional CAS implementations are restricted to about 16--18 orbitals. This puts limitations to what type of systems that can be studied; for instance, systems with two transition metals are normally already too large.
Methods that allow more orbitals in the active space have been introduced in recent years, for example the density matrix renormalization group (DMRG) method\cite{white1992,chan2004,marti2011,wouters2014b,legeza2003,keller2015,yanai2014,freitag2017}.

Another serious problem is that all CAS methods, even with very large active spaces, neglect a major part of the dynamical correlation.
To recover the missing dynamical correlation, perturbation theory is normally employed after a CASSCF or DMRG--SCF calculation,  as done in  CASPT2\cite{andersson1990,andersson1992,wouters2014b,kurashige2011,kurashige2014} or NEVPT2\cite{angeli2001,guo2016,freitag2017}. However, the perturbation correction comes with additional high computational cost.

An efficient method to recover the dynamical correlation in multiconfigurational methods is to merge DFT with a multiconfigurational wave function, thereby capitalizing on the efficient treatment of semi-local dynamical electron correlation within DFT methods. Simultaneously, such a hybrid method has the advantage that the multiconfigurational wave function can include static correlation. \cite{grimme1999,marian2008,manni2014,savin1995,fromager2007}
In this paper, we explore the multiconfigurational short-range DFT (MC--srDFT) method. It exploits the concept of range separation of the two-electron repulsion operator to merge DFT with a multiconfigurational wave function. With a recent extension of the MC--srDFT method to a polarizable embedding framework\cite{olsen2010,hedegaard2015,hedegaard2016b}, the method can also be employed on biological systems, and  the method may be a promising approach to use for metalloenzymes.  However,
the MC--srDFT method has mostly been benchmarked for $s$- and $p$-block atoms, diatomic molecules\cite{savin1995,savinbook,angyan2005,fromager2008,fromager2010,goll2005,fromager2013}, and organic systems\cite{hubert2016a,hubert2016b,hedegaard2013a,hedegaard2016a}. Studies of transition metals are more rare\cite{fromager2013,olsen2017}. Before addressing full enzymes, we first need to ensure that the results of MC--srDFT are in agreement with previous accurate calculations for biologically relevant cases and this is the purpose of the present paper.

We investigate
the binding of H$_2$ to the active site of [NiFe] hydrogenase,
for which previous studies have given ambiguous results. On the one hand, experimental
studies with CO or Xe gas-diffusion have predicted that \ce{H2} binds to  Ni.\cite{ogata2002,montet1997,volbeda2003} On the other hand, Fe is the expected binding site from the organometallic perspective\cite{bookh2ase1,bookh2ase2}.
Various DFT studies have predicted that H$_2$ binds to Ni or to Fe with the active site in the Ni(II) singlet, or even to Fe in the triplet state.\cite{siegbahn2007,bruschi2014,niu1999,pavlov1999,wu2008,jayapal2008,kaliakin2015}  We have recently investigated the H$_2$ binding site by using CCSD(T) and cumulant approximated DMRG--CASPT2 methods\cite{geng2017}, as well as DFT-based calculations with the big-QM approach\cite{hu2013}, using 819 atoms in the QM region. In this study, we compare results obtained with the MC--srPBE method with the previous CCSD(T) and DMRG--CASPT2 results, and show that MC--srDFT comes to the same conclusions. We furthermore study the method's dependence on the size of the active space and the employed basis set.

\section{Computational method} \label{compmet}

\subsection{The MC--srDFT method}

The MC--srDFT method is a hybrid between wave function theory (WFT) and density functional theory (DFT).
The method relies on range-separation of the two-electron
repulsion operator into long-range and short-range parts\cite{savin1995,iikura2001}
\begin{equation}\hat{g}_{\rm ee}(1,2)= \hat{g}^{\rm lr}_{\rm ee}(1,2) + \hat{g}^{\rm sr}_{\rm ee}(1,2) .
\label{sr-DFTCoulpart}
\end{equation}
Several forms of the range-separated operators have been suggested\cite{savin1995,savinbook,erferfgaufunc}.
We use in this work a range-separation operator based on the error function\cite{leininger1997,pollet2002,fromager2007,fromager2010}
\begin{equation}
  \hat{g}^{\rm lr}_{\rm ee}(1,2) = \frac{\text{erf} (\mu | \textbf{r}_{1} - \textbf{r}_{2} |)}{|\textbf{r}_{1} - \textbf{r}_{2}|}; \qquad
  \hat{g}^{\rm sr}_{\rm ee}(1,2) = \frac{1- \text{erf} (\mu | \textbf{r}_{1} - \textbf{r}_{2} |)}{|\textbf{r}_{1} - \textbf{r}_{2}|}   \label{lrpart}  ,
\end{equation}
where $\mu$ is the range-separation parameter, measured in bohr$^{-1}$ in this article. This parameter is to some degree adjustable and slightly different values have been employed in the literature (we discuss this point further below). In limiting cases, a value of $\mu=\infty$ reduces MC--srDFT to multiconfigurational SCF (MCSCF), a pure wave function method, whereas $\mu=0$ reduces MC--srDFT to a pure Kohn-Sham DFT method. Both $g^{\text{lr}}_{\rm ee}(1,2)$ and $  g^{\text{sr}}_{\rm ee}(1,2)$ depend on the choice of $\mu$, but this dependence has been left out in all equations for brevity,
because $\mu$ is selected {\it a priori} and then kept fixed.
The effective electronic Hamiltonian employed in MC--srDFT is
\begin{equation}\label{srKShamil}
 \hat{H}[\rho] = \hat{h} + \hat{V}^{{\rm sr}}_{\text{Hxc}}[\rho] + \hat{g}^{\text{lr}}_{\rm ee} ,
\end{equation}
where $\hat{h}$ contains the usual one-electron operators (kinetic energy and nuclear-electron attraction),
$\hat{g}^{\text{lr}}_{\rm ee}$ was defined in Eq.~\eqref{lrpart}, and the short-range DFT potential operator is defined through (see e.g.~ref.~\citenum{fromager2008})
\begin{equation}
\hat{V}^{{\rm{sr}}}_{\text{Hxc}}[\rho] =  \int \text{d}\textbf{r}\, v^{{\rm sr}}_{\rm Hxc}[\rho]\hat{\rho}(\textbf{r}) .
\end{equation}
Here $\hat{\rho}(\bm{r})$ is the density operator and $v^{{\rm sr},\mu}_{\rm Hxc}$ is the short-range adapted, $\mu$-dependent Hartree exchange--correlation potential
\begin{equation}
 v^{{\rm sr}}_{\rm Hxc}[\rho] = \frac{\delta E^{{\rm sr}}_{\rm Hxc}}{\delta \rho(\textbf{r})} .
\end{equation}
 It should be stressed that special exchange--correlation functionals are a prerequisite for range-separated wave function DFT hybrids (this point is explained
thoroughly in ref.~\citenum{fromager2015}). We use
in this work the short-range PBE-based srPBE functional by Goll {\it et al.}\cite{goll2005,goll2006}. In all cases, the applied multiconfigurational wave function ansatz was of the CASSCF type. Further,   in a few trial calculations (reported in the SI), we also employed a wave function \textit{ansatz} based on M{\o}ller--Plesset second order perturbation theory (MP2).
Since the applied multiconfigurational wave function ansatz was of the CASSCF type, we will henceforth refer to MC--srDFT with respect to the choices of multiconfigurational wave function and functional, i.e., CAS--srPBE for the method employed in this paper, and MC--srDFT for the general method.

For the range-separation parameter, most studies on range-separated DFT hybrids\cite{vydrov2006,baer2010} employ values between 0.33--0.5 bohr$^{-1}$. For MC--srDFT, a value of $\mu=0.4$ bohr$^{-1}$ has been suggested based on natural occupation numbers and differences between HF--srDFT and CAS--srDFT ground-state energies of small organic systems.\cite{fromager2007}. Benchmark studies on excitation energies \cite{hubert2016a,hubert2016b,hedegaard2016a} for organic systems have confirmed that this value provides accurate results.
Using both MP2--srPBE and CAS--srPBE models, we have tested range of $\mu$\ values (see the supporting information, Table S1). These results show that  $\mu$ values between 0.5 and 0.3 gives relative energies of \textbf{H$_2$-Fe} and \textbf{H$_2$-Ni} close to the energies obtained with DMRG and CCSD(T). Since $\mu=0.4$ is both accurate and consistent with previous suggestions, we here employ  $\mu=0.4$ bohr$^{-1}$.

All calculations were carried out with a development version of the DALTON program.\cite{DaltonDevel,WIREdalton} Further details about the MC--srDFT method, as well as the implementation, can be found elsewhere\cite{pedersenphd2004}.

\subsection{Model systems and basis sets}

As the name indicates, the active site of [NiFe] hydrogenase consists of a Ni ion and a Fe ion. The former is coordinated to four Cys residues, two of which are also bridging to the Fe ion. The latter also coordinates one CO and two \ce{CN-} ligands.  In this paper, we compare the stability of two binding modes of \ce{H2} to this site, viz. binding side-on to Ni or to Fe. The two binding modes will be called \textbf{H$_2$--Ni} and \textbf{H$_2$--Fe}, and they are shown in Figure 1 (note that \ce{H2} actually bridges the two metal ions in the \textbf{H$_2$--Fe} binding mode). In analogy with our previous study\cite{geng2017}, we used for each state three models of increasing size, also shown in Figure 1. In the smallest model 1, the four cysteine ligands were modeled by \ce{HS-} groups, whereas in the other two models they were modeled by \ce{CH3S-}. In the largest model 3, two second-sphere residues were included, Glu34 and His88 (residue numbering according to the crystal structure with PDB entry 1H2R\cite{higuchi1999}), modeled by acetic acid and imidazole, respectively.
The structures were taken from our previous study\cite{geng2017} and were optimized with the combined quantum mechanics and molecular mechanics (QM/MM) approach at the TPSS/def2-SV(P) level of theory \cite{ryde2001,ryde1996,tao2003,schafer1994}  in the singlet state.
Thus, both the Ni and Fe ions are in the low-spin +II oxidation state, corresponding to the spectroscopic Ni-SI$_a$ state of [NiFe] hydrogenase\cite{lubitz2014}.

\graphicspath{ {/Users/geng/Desktop/srdft/FIGURES/} }
\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{figure1}
\caption{The models we used in this work. \label{systems} }
\end{figure}

The calculations presented here were carried out with three basis sets of increasing size, denoted B1--B3. For the smallest one (named B1), the cc-pVTZ \cite{bs890115d,bs920501kdh} basis set was employed for the Ni and Fe ions, and the cc-pVDZ \cite{bs890115d} basis set was used for the other atoms.
The effect of increasing the basis set was investigated by using the cc-pVTZ basis set on all atoms except H, for which the cc-pVDZ basis set was used. This basis set resembles the ANO-type basis set employed in Ref.~\citenum{geng2017} and is denoted B2. In addition, we have added a calculation  with a basis set similar to B2, but in which the \ce{H2} molecule bound to Ni or Fe is also described with the cc-pVTZ basis set. Thus, the important \ce{H2}-molecule is (in contrast to ref.~\citenum{geng2017}) also described with a triple-zeta basis set. We denoted this last basis set B3. The MC--srDFT method has during this study undergone development to become more efficient and this effort is ongoing. Still, a large number of inactive electrons does pose a challenge for the current implementation of MC--srDFT. Therefore the basis sets B2 and B3 were  used only for model 1.  It should be emphasized that this is not a challenge of the method itself and standard techniques (e.g. Cholesky decomposition) can straightforwardly be applied to MC--srDFT. Relativistic effects were considered by using a standard second-order Douglas--Kroll--Hess (DKH2) Hamiltonian.  \cite{douglas1974,hess1985,hess1986}

\subsection{Selection of active spaces}

The selection of active space is of highest importance for an MCSCF calculation and
different strategies have been proposed for this selection. One strategy has been to rely on identifying orbitals from the chemical context.\cite{veryazov2011}  For transition metal complexes, this has typically led to the suggestion that all $3d$ orbitals and a few ligand orbitals should be included, and preferably also an additional (double-shell) of $3d'$ orbitals. A different strategy relies on selecting orbitals
 based on natural occupation numbers from methods where the predicted occupation numbers are qualitatively correct. This could be either MP2\cite{jensen1988} or a computationally cheap CI method.  Typically, one would select orbitals with occupation numbers significantly different from 2 or 0. Rules for selection of active spaces are not as well established for short-range DFT methods. Occupation numbers based on MP2--srDFT have previously been discussed\cite{hedegaard2013b} and it was noted these natural occupation numbers are much closer to 2.0 and 0.0 than their MP2 counterparts. This is expected because the short-range density functional effectively includes dynamical Coulomb-hole correlation. Hence, orbitals with occupation numbers below 1.98 in MP2- or MC-srDFT can be expected to show strong correlation and orbitals with occupation numbers of around 1.98 or 0.02 should preferably be included in the active spaces.  Importantly, since we here investigate the relative energy of two species, the chosen active spaces of the two species must be comparable.

The MP2--srPBE occupation numbers for the two complexes (model 1) are compiled in Tables S2 and S3, and our initial selection of orbitals for the CAS--srDFT calculations was based on these. Tables S2 and S3 also contain occupation numbers for a number of different $\mu$\ values, but we focus here on $\mu= 0.4$ bohr$^{-1}$.
Occupation numbers with a similar magnitude should preferably be included as a group and we have initially selected a CAS(10,10) space,
for which there is a clear change in occupation numbers between the selected 10 orbitals and the orbitals not included (for both \textbf{H$_2$--Ni} and  \textbf{H$_2$--Fe}).
A larger active space is more challenging to define: For  \textbf{H$_2$--Ni}, selecting CAS(12,12) or CAS(14,14) will mean including and excluding orbitals with rather similar MP2--srPBE occupation numbers.
The CAS(16,16) choice seems better, but this is rather large. On the other hand, for  \textbf{H$_2$--Fe}, CAS(10,10) or CAS(16,15) seems appropriate based on the MP2--srDFT occupation numbers.

Considering that the  MP2--srPBE occupation numbers might not reflect  the "true" occupation number (i.e. occupation numbers obtained with a full-CI-srPBE approach), we initially investigated CAS(10,10), CAS(14,14) and CAS(16,16) for both species. The corresponding CAS--srPBE occupation numbers are also shown in Tables S2 and S3. For CAS(16,16), we start to include orbitals with either very high or very low occupation numbers (above 1.99 or below 0.01), which affects the convergence. The CAS(16,16)--srPBE calculation for \textbf{H$_2$--Fe} also shows that the occupation numbers for the last two orbitals in what would correspond to a CAS(12,12) become even closer than for the MP2--srPBE calculation. Hence, CAS(12,12) will become unstable and prone to get stuck in local minima, which was confirmed by a trial calculation with this active space. The orbitals causing these difficulties are involved in the \ce{Fe-CN} and \ce{Fe-CO} bonds, and care must be taken to include these orbitals uniformly in the two states. This is done in  CAS(14,14), which is the largest active space that can be considered balanced (and it is also feasible for the larger models 2 and 3).

  Visual inspection of the CAS(14,14) orbitals in Figure \ref{cas-14-14-orb} shows that this active space includes the Ni $3d$-orbitals, the \ce{H2} and metal--ligand (CO $\pi$-type) orbitals, although the orbitals are more delocalized than the pure DMRG-SCF (or CASSCF) orbitals in refs.~\citenum{delcey2014,geng2017}. Further reduction of the active space to CAS(10,10), leads to exclusion of orbitals that are partly on hydrogen and the Ni ion, and we therefore prefer to include these two orbitals (i.e.~orbitals 4, 5, 13 and 14 are included for \textbf{H$_2$--Ni} in Figure \ref{cas-14-14-orb}, compared to Figure S2). Furthermore, the occupation number of the Ni orbital in  \textbf{H$_2$--Ni} (orbital 4 in Figure \ref{cas-14-14-orb}) is around 1.98 in the CAS(14,14) calculations and thus rather close to two of the other orbitals in the active space. This indicates that this orbital should be included.

  Expanding the calculations to CAS(16,16), introduces orbitals that are mainly on bridging sulfur atoms and can be considered less important.  For instance,  for \textbf{H$_2$-Ni}, the additional orbitals compared to CAS(14,14) are orbitals 8 and 14 in Figure S5. Although we here focus on  CAS(14,14), it should be noted that the effect on the calculated (relative) energies is in fact small (2 kJ/mol and below), as will be discussed in next section. For models 2 and 3, we also focus on CAS(14,14), but we have employed both CAS(10,10) and CAS(14,14) active spaces to probe the effect of the active spaces for these larger models as well. he corresponding active space orbitals are shown in the SI.
  Finally, we note that we also attempted to select orbitals based on calculations with larger $\mu$\ values, but this procedure was less satisfactory (shortly discussed in the SI).

\graphicspath{ {/Users/geng/Desktop/srdft/FIGURES/14in14/} }
\begin{figure}
\centering
\includegraphics[width=1.1\textwidth]{14in14-new}
\caption{ Active natural orbitals and their occupation numbers from the CAS(14,14)--srPBE calculation on model 1. \label{cas-14-14-orb}}
\end{figure}

\section{Results and Discussion}
In this study, we have compared the results of CAS--srPBE with previously published CCSD(T) and cumulant approximated DMRG--CASPT2 calculations for the two binding modes of \ce{H2} to the active site of [NiFe] hydrogenase.\cite{geng2017} We discuss first the smallest model (model 1) and then the two larger models (models 2 and 3) in separate sections.

\subsection{Calculations with model 1}

The energy difference between the \textbf{H$_2$--Ni} and \textbf{H$_2$--Fe} states ($\Delta E_{\mathrm{H}_2}$) calculated with the CAS--srPBE method is compared to previous CCSD(T) and DMRG--CASPT2 results in Table 1. We report CAS--srPBE results with the CAS(10,10), CAS(14,14) and CAS(16,16) active spaces.
In all cases, CAS--srPBE predicts that the \textbf{H$_2$--Ni} state is most stable, in agreement with the CCSD(T) and DMRG--CASPT2 results. The effect of expanding the active space from CAS(10,10)--srPBE to CAS(14,14)--srPBE is only 2 kJ/mol (see the Methods section for a description of the orbitals within the two active spaces). The CAS(10,10)--srPBE predicts that \textbf{H$_2$--Ni} is 15 kJ/mol more stable, whereas the difference with CAS(14,14)--srPBE is 17 kJ/mol. For the largest active space, CAS(16,16), the obtained energy difference changes by only 0.2 kJ/mol. Hence, there is little effect on the relative energies when expanding the active space and the CAS(14,14) active space seems to be sufficiently large for the systems studied here.

\begin{table}[]
\centering
\caption{The energy difference ($\Delta E_{\mathrm{H}_2} = E$(\textbf{H$_2$--Fe})$ - E$(\textbf{H$_2$--Ni}) in kJ/mol) between the \textbf{H$_2$--Ni} and \textbf{H$_2$--Fe} states (model 1) calculated with the CAS--srPBE method, and compared to previous CCSD(T), DMRG--CASPT2, and DFT calculations.  \label{model1}}
\begin{tabular}{lll}
\hline
Method                                & Basis      &  $\Delta E_{H2}$ \\
 \hline
CAS(10,10)--srPBE                     & B1         &  15.1  \\
CAS(14,14)--srPBE                     & B1         &  17.0  \\
CAS(16,16)--srPBE                     & B1         &  16.8  \\
CAS(14,14)--srPBE                     & B2         &  15.2  \\
CAS(14,14)--srPBE                     & B3         &  13.9  \\
DMRG(22,22)--CASPT2\cite{geng2017}     & ANO-RCC    &  17.7 \\
DMRG(22,22)--CASPT2$^a$\cite{geng2017}     & ANO-RCC    &  11.9  \\
CCSD(T)\cite{geng2017}                & ANO-RCC    &  18.1  \\
TPSS\cite{geng2017}                   & def2-QZVPD&  25.6  \\  \hline \hline
\multicolumn{2}{l}{$^a$ With 3s,3p correlation obtained from CCSD(T).}
\end{tabular}
\end{table}

The CAS(14,14)--srPBE results with both the B1 and B2 basis sets (17 and 15 kJ/mol) are in good agreement with the result obtained with DMRG(22,22)--CASPT2 (18 kJ/mol), although those calculations employed a significantly larger active space. Hence, the treatment of semi-local, dynamical correlation by the srDFT part allows for the use of significantly smaller active spaces compared to traditional MR methods.
 It should be noted that the CAS--srPBE calculations also show a rather modest basis set dependence. The basis sets increase from B1 to B2 only lowers the obtained energy-difference by 2 kJ/mol. Further increasing it to B3 lowers the energy-difference by another 1 kJ/mol, yielding a final result of 14 kJ/mol.

 At this point we emphasize that recent studies have noted that multireference perturbation theory to second order does not always recover the 3s,3p correlation well.\cite{pierloot2017} Table \ref{model1} also reports a DMRG(22,22)--CASPT2 result obtained without 3s,3p correlation, but including an estimate of this semi-core correlation from CCSD(T). The resulting energy difference was then 12 kJ/mol.\cite{geng2017} Thus, our best CAS(14,14)-srPBE result (14 kJ/mol) is within 2 kJ/mol of this corrected DMRG--CASPT2 value, and within 4 kJ/mol of the CCSD(T) result.  From the above discussion, we can thus conclude that both CAS(14,14)--srPBE and DMRG(22,22)--CASPT2 reproduce the CCSD(T) data well.

\subsection{Calculations with models 2 and 3}

Next, we carried out CAS--srPBE calculations also for the two larger models 2 and 3 in Figure \ref{systems}.
The results are shown in Table \ref{res_model_2_and_3} and are compared to the corresponding results obtained with DMRG--CASPT2. It can be seen that the two approaches give similar trends: The energy differences increase in model 2 (24--39 kJ/mol), whereas inclusion of models of two nearby amino-acids counteracts this increase, so that in model 3, the energy difference  decreases again to 8--19 kJ/mol.
\begin{table}[]
\centering
\caption{The $\Delta E_{\mathrm{H}_2}$\ energy difference (in kJ/mol) between the \textbf{H$_2$--Ni} and \textbf{H$_2$--Fe} states calculated with the CAS--srPBE method, compared to previous DMRG--CASPT2 and DFT calculations for models 2 and 3. }
\label{res_model_2_and_3}
\begin{tabular}{llrr}
\hline
Method                                 & Basis      & \multicolumn{2}{c}{$\Delta E_{\mathrm{H}_2}$} \\
                                       &            & Model 2 & Model 3 \\ \hline
CAS(10,10)--srPBE                      & B1         & 24.3    & 7.9     \\
CAS(14,14)--srPBE                      & B1         & 27.7    & 10.5    \\
DMRG(22,22)--CASPT2\cite{geng2017}     & ANO-RCC    & 37.7    & 15.2    \\
DMRG(22,22)--CASPT2$^a$\cite{geng2017} & ANO-RCC    & 33.0    & 11.1    \\
DMRG(22,22)--CASPT2$^b$\cite{geng2017} & ANO-RCC    & 39.2    & 17.3    \\
TPSS\cite{geng2017}                    & def2-QZVPD & 34.0    & 18.5    \\ \hline \hline   \end{tabular}
\\
$^a$ With 3s,3p correlation obtained from CCSD(T). \\
$^b$ Extrapolated with the energy difference between CCSD(T) and DMRG(22,22)--CASPT2 for model 1, the latter with 3s,3p correlation obtained from CCSD(T).
\end{table}
For all three models, \textbf{H$_2$--Ni} is thus consistently predicted to be the most stable state and the CAS(14,14)--srPBE results are quite close to that of DMRG(22,22)--CASPT2. From Table \ref{res_model_2_and_3}, it can be seen that the differences to CAS(14,14)--srPBE are 5 and 10 kJ/mol for model 2, depending on whether the DMRG--CASPT2 included 3s,3p correlation from CCSD(T) or not. The corresponding differences for model 3 are even smaller, 1 and 5 kJ/mol.

In Ref.~\citenum{geng2017}, it was noted that CCSD(T) was beyond the computational resources for models 2 and 3, but an estimate of a CCSD(T) result could be obtained by correcting the DMRG--CASPT2 results for models 2 and 3 with the energy-difference  between  CCSD(T) and DMRG--CASPT2 from model 1. With this correction, the results were 39 and 17 kJ/mol for models 2 and 3, respectively. Compared to these values, CAS(14,14)--srPBE underestimates the energy difference by 11 and 6 kJ/mol for models 2 and 3. Judging from the results with the smallest model, the difference is expected to decrease by 2 kJ/mol with the larger B2 basis set.  These differences are certainly acceptable and below other error sources. For instance, the protein was found to affect the energy difference by more than 25 kJ/mol in the favor of the \textbf{H$_2$--Ni} state (estimated by DFT and a 819-atom QM model).\cite{geng2017}

Although the performance of CAS--srPBE is encouraging for applications in metalloenzymes, further improvements are possible: For instance, the accuracy of the srDFT functional can be improved. This could be achieved by either including exact (short-range) DFT exchange or by including kinetic energy dependence in the same way as in \textit{meta}-GGA functionals.\cite{goll2009}.

\section{Conclusions}

In this paper, CAS--srPBE calculations were performed on three models of \ce{H2} bound to [NiFe]-hydrogenase. Our results indicate that \ce{H2} binding to Ni is more stable than binding to Fe, which is consistent with previous calculations with the CCSD(T) and DMRG--CASPT2 methods.\cite{geng2017} Our CAS--srPBE calculations with reduced active spaces (CAS(10,10), CAS(14,14) and CAS(16,16)) gave results close to the CAS(22,22) active space used in the previous  DMRG--CASPT2 calculations.

For all employed model systems, the effect of extending the active space from CAS(10,10) to CAS(14,14) was found to be small (around 2 kJ/mol). For model 1, we further employed CAS(16,16), which only gave rise to a change of 0.2 kJ/mol. This is a good indication that the calculations are converged with respect to choice of active space. The effect of increasing the basis set were also quite modest:  For model 1, an increase in the basis set from cc-pVDZ to cc-pVTZ for all C, N, O and S atoms changed the CAS(14,14)--srPBE energy difference by only 3 kJ/mol. Thus, both the effect of increasing the active space and the effect of the basis set is much lower that other sources of error. For instance, a change of 7 kJ/mol was obtained by employing the 3s,3p correlation obtained from either CCSD(T) or DMRG--CASPT2 and the effect of the surrounding protein was 25 kJ/mol.

For the larger models 2 and 3, the CAS(10,10)--srPBE and CAS(14,14)--srPBE results agree with the CCSD(T) extrapolated DMRG--CASPT2 results to within 4--9 kJ/mol. This is similar to the difference between the best DMRG(22,22)--CASPT2 and CCSD(T) results for model 1, 6 kJ/mol.
Hence, our results support MC--srDFT as a new valuable tool for bio-inorganic chemistry, with an accuracy similar to that of DMRG--CASPT2 but at a much lower computational cost (in a fully optimized implementation). The lower computational cost is achieved by means of the much smaller active spaces needed and by the replacement the perturbation correction of CASPT2 with DFT integration.

Finally, it would also be interesting to address the triplet spin-states of the two \textbf{H$_2$--Ni} and \textbf{H$_2$--Fe} intermediates. This will require extension of our current MC--srDFT implementation to functionals that depend on spin-densities, and this  development is currently ongoing.

\acknowledgement
This investigation has been supported by grants from the Swedish research council (project 2014-5540), The Carlsberg Foundation (project CF15-0208), The European Commission, The China Scholarship Council, and COST Action CM1305 (ECOSTBio).
The computations were performed on computer resources provided by the Swedish National Infrastructure for Computing (SNIC) at Lunarc at Lund University and HPC2N at Ume\aa\  University,
and on computer resources on Abacus 2.0 provided by the Danish e-infrastructure Cooperation.

\section{Studies with different $\mu$\ values}

A study of the $\Delta E_{\mathrm{H}_2}$\ energy difference between the two investigated intermediates for different $\mu$\ values was conducted, first based on MP2--srDFT calculations on model 1.
The resulting energy differences are given in Table~\ref{Mp2-srDFT}.
One notes that $\Delta E_{\mathrm{H}_2}$\ decreases steadily with increasing $\mu$.
Recalling that increasing $\mu$ means moving correlation from the short-range DFT part to the long-range WFT part, a possible reason for these results could be that MP2 is inadequate to describe the energy difference in wave function theory.
We therefore decided to calculate $\Delta E_{\mathrm{H}_2}$\ also with CAS(14,14)-srPBE for the most relevant $\mu$ values, and these results are also reported in Table~\ref{Mp2-srDFT}.
These values clearly support our contention that MP2 is inadequate, while CAS(14,14) is sufficiently flexible and accurate to describe the transfer of correlation effects from the DFT part to the WFT part of the MC-srDFT model when $\mu$ changes from 0.4 to 0.7.
As a side remark, the variation in the CAS(14,14) natural orbital occupation numbers in Tables \ref{MP2-occ-Ni-H2} and \ref{MP2-occ-Fe-H2} as a function of the $\mu$ value shows explicitly that the long-range correlation effects increase when $\mu$ is increased.

From these results, $\mu=$0.4--0.7 all give reasonable $\Delta E_{\mathrm{H}_2}$\ energy differences compared to DMRG(22,22)-CASPT2 and CCSD(T) values (cf.~Table 1 in the paper).
Previous studies (see paper for references) on smaller systems have found that 0.4 is optimal, our results here provide no reason for advocating another value here,
and we therefore follow our earlier recommendation and use a value of $\mu=0.4$ in this study.
\begin{table}[ht]
\centering
\caption{The $\Delta E_{\mathrm{H}_2}$\ energy difference (kJ/mol) for model 1 from MP2--srPBE and CAS(14,14)-srPBE calculations with B1 and various $\mu$ values.\label{Mp2-srDFT}}
\label{MP2-energy}
\begin{tabular}{lrrrrrrrr}
\hline
$\mu$            & 1.0 & 0.9 & 0.8 & 0.7 & 0.6 & 0.5 & 0.4 & 0.3 \\ \hline
MP2-srPBE        & 0.7 & 1.7 & 2.7 & 3.8  & 5.4  & 8.1 & 12.7 & 18.6 \\
CAS(14,14)-srPBE &     &     &     & 14.9 & 14.2 & 14.1 & 17.0 &    \\
\hline
\end{tabular}
\end{table}

We additionally investigated how the selection of the active space vary with different $\mu$\ values. In  Tables \ref{MP2-occ-Ni-H2} and \ref{MP2-occ-Fe-H2}, we have compiled occupation numbers of the relevant orbitals for several different $\mu$-values.
For $\mu=1.0$, the best CAS space would be either CAS(8,8) or CAS(14,14).
The orbitals from a CAS(14,14)--srPBE calculations are shown in Figure 5. It can be seen that metal orbitals are missing in active space and the orbitals are somewhat different from the orbitals obtained with $\mu=0.4$. This shows that care must be exercised in selecting orbitals based on different $\mu$-values.

\begin{table}[ht]
\small
\centering
\caption{MP2--srPBE and CAS--srPBE natural orbital occupation numbers for \textbf{H$_2$--Ni} \label{MP2-occ-Ni-H2}}
\begin{tabular}{lllllllllll}
\\ \hline
$\mu$               &                      & \multicolumn{9}{l|}{occupation numbers}   \\ \hline
1.0 & MP2 & 1.964 & 1.964 & 1.962 & 1.961 & 1.960 & 1.959 & 1.958 & 1.957 & 1.949 \\
    &     & 0.046 & 0.043 & 0.042 & 0.041 & 0.033 & 0.033 & 0.030 & 0.030 & 0.026 \\

\hline
0.9 & MP2 & 1.966 & 1.966 & 1.966 & 1.964 & 1.964 & 1.962 & 1.961 & 1.959 & 1.950 \\
    &     & 0.048 & 0.040 & 0.039 & 0.037 & 0.031 & 0.031 & 0.029 & 0.027 & 0.025 \\
\hline
0.8 & MP2 & 1.971 & 1.969 & 1.969 & 1.969 & 1.966 & 1.966 & 1.965 & 1.961 & 1.952 \\
    &     & 0.048 & 0.037 & 0.036 & 0.033 & 0.030 & 0.029 & 0.025 & 0.023 & 0.023 \\
\hline
0.7 & MP2 & 1.974 & 1.974 & 1.972 & 1.972 & 1.970 & 1.969 & 1.968 & 1.965 & 1.957 \\
    &     & 0.046 & 0.033 & 0.032 & 0.029 & 0.028 & 0.027 & 0.021 & 0.020 & 0.020 \\
    & CAS(14,14) & 2     & 2     & 1.987 & 1.978 & 1.977 & 1.966 & 1.965 & 1.964 & 1.945 \\
    &     & 0.055 & 0.040 & 0.036 & 0.035 & 0.022 & 0.016 & 0.013 & 0     & 0     \\  \hline
0.6 & MP2 & 1.979 & 1.978 & 1.977 & 1.976 & 1.974 & 1.973 & 1.971 & 1.970 & 1.964 \\
    &     & 0.042 & 0.028 & 0.028 & 0.026 & 0.024 & 0.023 & 0.017 & 0.017 & 0.016 \\
    & CAS(14,14) & 2     & 2     & 1.990 & 1.982 & 1.981 & 1.971 & 1.970 & 1.969 & 1.954 \\
    &     & 0.046 & 0.035 & 0.032 & 0.031 & 0.019 & 0.012 & 0.010 & 0     & 0     \\  \hline
0.5 & MP2 & 1.984 & 1.982 & 1.981 & 1.980 & 1.978 & 1.977 & 1.976 & 1.975 & 1.973 \\
    &     & 0.034 & 0.024 & 0.023 & 0.022 & 0.020 & 0.017 & 0.013 & 0.013 & 0.012 \\
    & CAS(14,14) & 2     & 2     & 1.992 & 1.986 & 1.985 & 1.977 & 1.975 & 1.974 & 1.966 \\
    &     & 0.034 & 0.028 & 0.026 & 0.026 & 0.015 & 0.008 & 0.008 & 0     & 0     \\  \hline
0.4 & MP2 & 1.989 & 1.987 & 1.987 & 1.986 & 1.984 & 1.983 & 1.983 & 1.982 & 1.980 \\
    &     & 0.025 & 0.019 & 0.017 & 0.016 & 0.014 & 0.012 & 0.010 & 0.009 & 0.008 \\
    & CAS(14,14) & 2 & 2  & 1.992 & 1.990 & 1.988 & 1.983 & 1.982 & 1.981 & 1.978 \\
    &     & 0.029 & 0.022 & 0.019 & 0.018 & 0.009 & 0.005 & 0.003 & 0     & 0     \\
    & CAS(16,16) & 2 & 1.993 & 1.991 & 1.988 & 1.986 & 1.983 & 1.982 & 1.981 & 1.977 \\
    &     & 0.029 & 0.024 & 0.021 & 0.019 & 0.011 & 0.006 & 0.004 & 0.003 & 0     \\  \hline
0.3 & MP2 & 1.994 & 1.993 & 1.992 & 1.992 & 1.991 & 1.990 & 1.990 & 1.989 & 1.987   \\
    &     & 0.016 & 0.013 & 0.010 & 0.009 & 0.008 & 0.006 & 0.006 & 0.005 & 0.005 \\   \hline
\end{tabular}
\end{table}

\begin{table}[ht]
\small
\centering
\caption{MP2--srPBE and CAS--srPBE natural occupation numbers for  \textbf{H$_2$--Fe} \label{MP2-occ-Fe-H2}}
\begin{tabular}{lllllllllll}
\\ \hline
$\mu$               &                      & \multicolumn{9}{l|}{occupation numbers}   \\ \hline
1.0 & MP2 & 1.965 & 1.964 & 1.962 & 1.961 & 1.960 & 1.959 & 1.958 & 1.956 & 1.954 \\
    &     & 0.043 & 0.042 & 0.041 & 0.038 & 0.033 & 0.030 & 0.030 & 0.029 & 0.027 \\
\hline
0.9 & MP2 & 1.967 & 1.967 & 1.966 & 1.964 & 1.963 & 1.963 & 1.962 & 1.958 & 1.955 \\
    &     & 0.041 & 0.039 & 0.038 & 0.037 & 0.031 & 0.029 & 0.028 & 0.026 & 0.026 \\
\hline
0.8 & MP2 & 1.971 & 1.970 & 1.970 & 1.969 & 1.967 & 1.965 & 1.965 & 1.960 & 1.957 \\
    &     & 0.041 & 0.036 & 0.035 & 0.033 & 0.029 & 0.026 & 0.025 & 0.024 & 0.023 \\
\hline
0.7 & MP2 & 1.975 & 1.974 & 1.974 & 1.973 & 1.971 & 1.969 & 1.968 & 1.964 & 1.960 \\
    &     & 0.041 & 0.032 & 0.031 & 0.029 & 0.027 & 0.024 & 0.021 & 0.020 & 0.019 \\
    & CAS(14,14) & 2     & 2     & 1.988 & 1.986 & 1.976 & 1.974 & 1.971 & 1.969 & 1.950 \\
    &     & 0.053 & 0.030 & 0.029 & 0.027 & 0.024 & 0.014 & 0.009 & 0     & 0     \\  \hline
0.6 & MP2 & 1.979 & 1.979 & 1.978 & 1.976 & 1.975 & 1.972 & 1.972 & 1.968 & 1.966 \\
    &     & 0.038 & 0.028 & 0.026 & 0.025 & 0.024 & 0.021 & 0.017 & 0.017 & 0.016 \\
    & CAS(14,14) & 2     & 2     & 1.990 & 1.988 & 1.980 & 1.978 & 1.975 & 1.972 & 1.957 \\
    &     & 0.046 & 0.027 & 0.025 & 0.023 & 0.020 & 0.012 & 0.007 & 0     & 0     \\  \hline
0.5 & MP2 & 1.984 & 1.984 & 1.982 & 1.981 & 1.979 & 1.978 & 1.977 & 1.975 & 1.973 \\
    &     & 0.033 & 0.023 & 0.021 & 0.021 & 0.019 & 0.017 & 0.013 & 0.013 & 0.012 \\
    & CAS(14,14) & 2     & 2     & 1.992 & 1.991 & 1.984 & 1.982 & 1.979 & 1.977 & 1.967 \\
    &     & 0.037 & 0.023 & 0.021 & 0.018 & 0.016 & 0.008 & 0.005 & 0     & 0     \\  \hline
0.4 & MP2 & 1.989 & 1.988 & 1.987 & 1.986 & 1.985 & 1.984 & 1.982 & 1.982 & 1.981 \\
    &     & 0.025 & 0.018 & 0.016 & 0.014 & 0.014 & 0.012 & 0.010 & 0.009 & 0.008 \\
    & CAS(14,14) & 2 & 2  & 1.994 & 1.994 & 1.993 & 1.988 & 1.984 & 1.983 & 1.979 \\
    &     & 0.020 & 0.019 & 0.017 & 0.012 & 0.010 & 0.004 & 0.004 & 0     & 0     \\
    & CAS(16,16) & 2 & 1.994 & 1.993 & 1.992 & 1.988 & 1.985 & 1.984 & 1.983 & 1.976 \\
    &     & 0.028 & 0.019 & 0.019 & 0.013 & 0.012 & 0.007 & 0.005 & 0.004 & 0       \\  \hline
0.3 & MP2 & 1.994 & 1.993 & 1.993 & 1.992 & 1.992 & 1.991 & 1.989 & 1.989 & 1.989  \\
    &     & 0.016 & 0.011 & 0.010 & 0.008 & 0.007 & 0.006 & 0.006 & 0.005 & 0.005 \\   \hline
\end{tabular}
\end{table}

\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{u_eq_1}
\caption{ Active natural orbitals and their occupation numbers from a CAS(14,14)--srDFT calculation (started from the MP2 wavefunction) of model 1 with $\mu=1$}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{10in10-model1}
\caption{ Active natural orbitals and their occupation numbers from a CAS(10,10)--srDFT calculation (started from the MP2 wavefunction) with $\mu=0.4$ for model 1\label{cas-10-10-orb1}}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{10in10-model2}
\caption{ Active natural orbitals and their occupation numbers from a CAS(10,10)--srDFT calculation (started from the MP2 wavefunction) with $\mu=0.4$ for model 2\label{cas-10-10-orb2}}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{10in10-model3}
\caption{ Active natural orbitals and their occupation numbers from a CAS(10,10)--srDFT calculation (started from the MP2 wavefunction) with $\mu=0.4$ for model 3\label{cas-10-10-orb3}}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{16in16-new}
\caption{Active natural orbitals and their occupation numbers from a CAS(16,16)--srDFT calculation for model 1 (started from the MP2 wavefunction) with $\mu=0.4$ \label{cas-16-16-orb}}
\end{figure}

\end{document}


\section{\bf INTRODUCTION}

Isomeric states in nuclei around closed shells can occur due to
the large spin difference between the valence orbitals
(single-particle isomers) or when the maximum amount of angular
momentum created in a single-particle configuration is reached
(seniority isomers). Depending on the transition probabilities involved, an isomeric state can decay by e.g., $\gamma$ transitions to lower lying-states or $\beta$ radiation to the ground or excited states of the daughter nucleus. The investigation of nuclear isomers gives important information regarding the evolution of the shell structure in a specific mass-region.

The identification of isomeric states in the neutron-rich
nuclei with Z$\sim$28 and N$\sim$40-50 constitutes a field of a
great current interest. These states store key information about
the structural changes induced by increasing the neutron number
and especially by the filling of the unique parity $\nu g_{9/2}$
orbital. An experiment employing the fragmentation of a
$^{86}$Kr$^{34+}$ beam with an energy of 60.3 MeV/nucl. led to the
identification of thirteen new $\mu$s-isomers in the
neutron-rich nuclei from Sc ($Z$=21) to As ($Z$=33) \cite{Grz98}.
Spins and parities assignments were based on
the observed $\gamma$-decay pattern and comparisons with the systematics. Most of the
identified isomers were found to originate from the stretched $\nu
g_{9/2}^n$ configurations and decay to the lower-lying states via
$E2$ or $M2$ $\gamma$-transitions \cite{Grz98}.

In recent years, $\beta$-decaying isomers in the neutron-rich
nuclei around $^{68}$Ni have been extensively studied as well
\cite{Mue99,Pri99,Wei02,Jan1,Pau08}. Such isomers, arising from the large spin
difference between the opposite parity orbitals $\nu p_{1/2}$ and
$\nu g_{9/2}$, are expected to be found at low excitation energies in
the odd-odd and odd-N nuclei with N$\sim$40. Among these, the
odd-N Ni isotopes are of special interest since their low-energy
levels are expected to arise mainly from neutron single-particle
excitations whose investigation offers important information
concerning the core properties of $^{68}$Ni.

Low-lying states in the neutron-rich $^{67,69,71,73}$Ni were
identified in the $\beta$ decay of their Co isobars obtained in
proton-induced fission combined with resonant laser ionization
\cite{Mue99,Wei99} and in fragmentation reactions
\cite{Grz98,Pri99,Saw04,Raj07}. 1/2$^-$ and 9/2$^+$ spins and parities were proposed
for the ground-states of $^{67}$Ni and $^{69,71,73}$Ni,
respectively, based on the observed $\beta$-decay pattern and
shell-model calculations \cite{Mue99,Pri99,Wei99,Saw04,Raj07,Lis04}. In
$^{69}$Ni, the 1/2$^-$ state originating from $2p-1h$ excitations $\nu (p_{1/2}^{-1}g_{9/2}^2)$
across the N=40 subshell was found to be a long-lived isomer
($T_{1/2}$=3.5(5) s) decaying via a fast Gamow-Teller $\beta$ transition (log$ft=4.3(2)$)
to the first excited 3/2$^-$ state at 1298 keV in $^{69}$Cu
\cite{Fra98,Fra01,Mue99,Pri99}. The
observed weak branch to the 3/2$^-$ ground-state of $^{69}$Cu was
explained by invoking some mixing of the wave function, dominated
by the $\pi p_{3/2}^{+1} \nu g_{9/2}^{+0}$ component, with the $\pi
p_{3/2}^{+1}\nu p_{1/2}^{-2}g_{9/2}^{+2}$ configuration proposed for the
excited 3/2$^-$ level at 1298 keV \cite{Mue99,Pri99}. No feeding
of the isomer to the single-particle 1/2$^-$ state at 1096 keV in
$^{69}$Cu was observed, suggesting a rather pure $\pi p_{1/2}^{+1}\nu
g_{9/2}^{0}$ structure for this level.

Level schemes of the neutron-rich odd-A Ni isotopes beyond
$^{69}$Ni are still poorly known. In all odd-A Ni isotopes with masses from 71 to 77, the shell model predicts a spin and parity 9/2$^+$ for the
ground state and a low-lying 1/2$^-$ level dominated by the $\nu p_{1/2}$ neutron-hole configuration \cite{Lis04}. Sawicka {\it et al.} \cite{Saw04} reported four
$\gamma$ transitions in each of the $^{71,73}$Ni isotopes observed in the $\beta$ decay
of the $^{71,73}$Co isobars. Due to the poor statistics,
$\gamma$-$\gamma$ coincidences could not be constructed. Therefore, the
observed transitions were placed in a level scheme based on shell model predictions
\cite{Saw04}.

Most of the $\gamma$ rays reported by Sawicka {\it et al.} \cite{Saw04} were recently confirmed by the results of a decay-spectroscopy experiment performed at NSCL, MSU \cite{Raj07}. In that experiment, the $^{71,73}$Co isotopes were produced in the fragmentation of a $^{86}$Kr beam with an energy of 140 MeV/nucleon onto a thick $^{9}$Be target. The secondary fragments were implanted in a double-sided silicon strip detector surrounded by the NSCL Ge detector array SeGA used to detect the $\gamma$ rays. The good statistics obtained in that measurement allowed for the analysis of $\gamma$-$\gamma$ coincidences which provided the basis for the placement of the observed transitions in the decay schemes of $^{71,73}$Co given in Fig.~4 of Ref.~\cite{Raj07}. The two strong peaks observed at energies 566 and 774 keV did not show any coincidence relationship with each other nor with any of the other transitions observed in the $\gamma$ spectrum associated with the $\beta$ decay of $^{71}$Co. Based on a comparison with the systematics and shell-model predictions, both transitions were tentatively placed to feed the 1/2$^-$ level, suggested to be a $\beta$-decaying isomer \cite{Raj07}.

\begin{figure}
\includegraphics[height=6cm,width=9cm]{1a_71Cu_coulex.eps}\\
\includegraphics[height=6cm,width=9cm]{1b_69Cu_coulex.eps}
\caption{\label{clx} Particle-$\gamma$-ray coincidence
spectra obtained after Coulomb excitation experiment with
radioactive beams of $^{71}$Cu (top) and $^{69}$Cu (bottom). The spectra are Doppler corrected for the mass of the projectile.}
\end{figure}

The present study was prompted by the observation of a new state
located at 454 keV populated in a recent Coulomb excitation
experiment with postaccelerated radioactive beam of $^{71}$Cu
produced at ISOLDE, CERN. The results of that measurement are
presented in Ref. \cite{Ste08}. The low beam-energy
($\sim$3 MeV/u) used in that experiment ensured that the
population of the excited states proceeds mainly via $E2$
excitations from the 3/2$^-$ ground-state, therefore only levels
with spins 1/2$^-$, 3/2$^-$, 5/2$^-$ and 7/2$^-$ were expected to
be populated. The top of Figure \ref{clx} shows the $\gamma$ spectrum after Coulomb
excitation obtained with the beam of $^{71}$Cu. The spectrum is Doppler
corrected for the mass of the projectile. The newly observed
$\gamma$ ray of 454 keV and the known 5/2$^-\rightarrow
3/2^-_{g.s.}$ transition of 534 keV \cite{Fra98,Fra01,Ish98} are clearly
visible in the spectrum.

\begin{figure*}
\includegraphics[height=8cm,width=14cm]{2_71Cu_beta.eps}
\caption{\label{bg} Beta-gated $\gamma$ spectrum for mass $A$=71
obtained from the data set reported in \cite{Fra98,Fra01}. The $\gamma$ rays following the $\beta$ decay of $^{71}$Ni are marked with an asterisk. In the inset, the region
around 500 keV is enlarged, showing the 454-keV line. The
peak marked with a circle was assigned to the $\beta$ decay
of the daughter nucleus $^{71}$Cu.}
\end{figure*}

In the lighter Cu isotopes, an 1/2$^-$ state dominated by the $\pi
2p_{1/2}$ single-particle orbital was identified at low excitation
energies. Its energy is very close to that of the 5/2$^-$ level
found to contain a large component from the $\pi 1f_{5/2}$ orbital
\cite{Zei78}. The Coulomb excitation experiment mentioned above
included also a measurement with radioactive beam of $^{69}$Cu \cite{Ste08}. A
portion of the Doppler-corrected particle-$\gamma$ coincidence
spectrum obtained in that run is shown at the bottom of Figure
\ref{clx}. The two peaks present in the selected energy range were
identified as the transitions depopulating the first and second
excited states in $^{69}$Cu, namely the 1/2$^-$ and 5/2$^-$ levels
at 1096(6) \cite{Sel81} and 1213.5(1) keV \cite{Fra01}, respectively. Population
of the closely-lying 3/2$^-$ state at 1297.9(1) keV was not observed in the aforementioned  Coulomb excitation measurement.

As pointed above, the $E2$ excitation from the 3/2$^-$ ground-state of $^{71}$Cu constrains the spin of the newly identified state to values $I^\pi\leq$7/2$^-$. A spin 7/2$^-$ would imply a pure $E2$ character for the 7/2$^-\rightarrow$3/2$^-$ depopulating transition. The calculated Weisskopf estimate for the partial decay-lifetime indicates that an $E2$ transition of 454 keV will proceed in $\sim$2 ns, more than three orders of magnitude slower than an $M1$ transition. The observation of a Doppler broadened 454-keV peak in our Coulomb excitation spectrum suggests a half-life in the picosecond range for the emitting level and therefore an $M1$ character for the depopulating $\gamma$ ray. This restricts the spin of the 454 keV level to values $I^\pi\leq$5/2$^-$. In Ref.~\cite{Ste08}, spin and parity (1/2$^-$) were assigned to the newly observed state at 454 keV, based on the systematics of the lighter Cu isotopes and comparison with the
Coulomb excitation spectrum with beam of $^{69}$Cu. Such a spin assignment is also in agreement with the shell-model and particle-core coupling calculations \cite{Smi04,Fra01,Gra99,Oro00}. It is worth mentioning that in $^{71}$Cu, the second 3/2$^-$ state is still unknown. Shell-model and
particle-core calculations predict this state around 1900, 1662, and 1100 keV, respectively \cite{Smi04,Fra01,Oro00}.

It is also worthwhile to mention that isomeric 1/2$^-$ states originating from $2p-1h$ excitations across $Z$=40 have been observed in the valence partner of $^{71}$Ni, $^{93}_{43}$Tc$_{50}$ \cite{nndc}.
In fact, the 1/2$^-$ isomer was found to be the first excited state in the $N$=50 isotones from Nb ($Z$=41) to Rh ($Z$=45) with a half-life ranging from 60.9 days to 1.96 minutes, respectively
\cite{nndc}. In $^{93}$Tc, however, the $\beta$-decay from the 1/2$^-$ isomer ($T_{1/2}$=43.5 min) was found to compete with a $M4$ transition to the 9/2$^+$ ground-state \cite{nndc}.

This paper focuses on the possible decay modes of the 1/2$^-$
state in $^{71}$Ni. We analyze and discuss the experimental evidence indicating that this state is a $\beta$-decaying isomer feeding the newly observed (1/2$^-$) state at 454 keV in $^{71}$Cu \cite{Ste08}. The data sets used in the present study were obtained
in two different $\beta$-decay experiments performed at the LISOL
facility, Louvain-la-Neuve. In the first measurement, the
$\beta$-decay study of $^{71}$Ni was used as a means to
investigate the low-lying level scheme of $^{71}$Cu
\cite{Fra98,Fra01}. The second experiment was dedicated to the
identification of the energy levels in $^{71}$Ni populated in the
decay of the $^{71}$Co isobar.

\section{EXPERIMENTAL DETAILS}

The $^{71}$Ni and $^{71}$Co beams were produced in two separate
measurements at the LISOL facility by colliding a 30-MeV proton
beam with two thin $^{238}$U foils mounted inside a gas cell \cite{Kud96}. The
cell was filled with 500 mbar of Argon gas. The radioactive Ni and Co
atoms were resonantly photoionized, mass separated and implanted
in a movable tape surrounded by $\beta$ and $\gamma$ detectors
arranged in a close geometry. Table \ref{lln} gives a summary of
the experimental conditions in both measurements.

\begin{table*}
\caption{\label{lln}Half lives of the mother nuclei, specific
implantation-decay cycles, measuring times with and without laser
radiation, beam intensities and productions rates for $^{71}$Ni
and $^{71}$Co. During experiment I, the lasers were tuned on nickel, while during experiment II they were tuned on cobalt.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
Exp.&Nucleus&$T_{1/2}$&Cycle&Laser&Laser&I$_{beam}$&Yield\\
no&&(s)&(impl./decay)&ON&OFF&($\mu$A)&(ions/$\mu$C)\\
\hline
I&$^{71}$Ni&2.56(3)\footnotemark[1]&6 s/10 s&35 h 09 min&-&6.1&3.0(6)\\
II&$^{71}$Co&0.079(5)\footnotemark[2]&0.6 s/1 s&16 h 02 min&12 h 57 min&6.7&0.032(8)\\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Ref. \cite{Fra01}.} \footnotetext[2] {Ref.
\cite{Saw04}.}
\end{table*}

In the first experiment, $^{71}$Ni was implanted in a Mylar tape
surrounded by two HPGe detectors positioned in the horizontal
plane and at 90$^\circ$ and 270$^\circ$ with respect to the beam
axis. The relative efficiency of the detectors reached 70$\%$ and
75$\%$, respectively. The emitted beta particles were recorded in a plastic $\Delta E$ scintillator located between the two Ge detectors, in forward direction. A detailed description of the experimental
setup can be found in \cite{Fra01}.

In the second experiment, the $\beta$ decay of $^{71}$Co was
observed by means of four plastic $\Delta E$ detectors while the
emitted $\gamma$ rays were recorded with three HPGe detectors of
70$\%$, 75$\%$ and 90$\%$ relative efficiency located at
90$^\circ$, 0$^\circ$ and 270$^\circ$, respectively, with
respect to the beam axis. Measurements with and without laser
radiation were performed in order to disentangle the $\gamma$ rays
emitted by the nuclei of interest from the non-resonant
transitions.

\section{EXPERIMENTAL RESULTS}
\subsection{\label{ni} $\beta$ decay of $^{71}$Ni}

The $\beta$-gated $\gamma$ spectrum obtained in the experiment
with $^{71}$Ni beam is presented in Fig. \ref{bg}. In the inset,
the region around 500 keV is enlarged, showing the transition of
454 keV. Gamma rays attributed to the decay of $^{71}$Ni (see Ref.
\cite{Fra01} for details) are marked with an asterisk. Open
circles label the transitions arising from the $^{71}$Cu decay.

In Ref. \cite{Fra01}, the 454-keV transition was found not to be in coincidence with any of the
$\gamma$ rays attributed to $^{71}$Cu, therefore it was not
further discussed in that paper. The proposed spin value (1/2$^-$)
for the 454 keV level out of the Coulomb excitation study rules out the possibility of a direct $\beta$ branch from the 9/2$^+$ ground state of $^{71}$Ni. Furthermore, the lack of $\gamma$-ray coincidences in the $\beta$ decay study of \cite{Fra01} also excludes indirect feeding from the  ground state of $^{71}$Ni, see further. Therefore, the alternative scenario that this state is fed by a 1/2$^-$ $\beta$-decaying isomer in $^{71}$Ni is investigated.

The time evolution of the $\gamma$ intensity of the 454-keV
transition is shown in Fig. \ref{time}. The data were fitted with a
single exponential. The poor statistics forced us to take bins
of 2 seconds each resulting in three and five data points for the
implantation and decay periods, respectively (see Table
\ref{lln}). The last two seconds of the decay period were excluded
due to the very low number of counts observed in the peak. A value
of $T_{1/2}$=2.34(25) s was extracted from the fit.

\begin{figure}[h!]
\includegraphics[height=6cm,width=9cm]{3_fit_time_isomer.eps}
\caption{\label{time} Time evolution of the intensity of the 454
$\gamma$-ray fitted with a single exponential function yielding to
a half-life of $T_{1/2}$=2.3(3) s.}
\end{figure}

Figure \ref{coinc} compares the $\gamma$-$\gamma$ spectra gated with the 447-keV
transition (top) and the 454-keV line (bottom). The 447-keV transition deexcites the level at 981 keV which is populated both directly in the $\beta$-decay of the 9/2$^+$ ground-state of $^{71}$Ni and from feeding from higher-lying states in $^{71}$Cu \cite{Fra01}. As can be seen from the inset of Fig. \ref{bg}, the 447-keV line is twice stronger than the peak at 454 keV. The observation of the 472- and 534-keV transitions in the spectrum coincident with the 447 keV $\gamma$ ray provided the basis for its placement in the level scheme of $^{71}$Cu as shown in Fig. 7 of Ref. \cite{Fra01}. In the spectrum gated with the 454-keV $\gamma$ ray shown in the bottom of Fig. \ref{coinc}, no clear peak can be distinguished from the background, indicating  that $\gamma$ feeding from higher-lying states has a negligible contribution to the population of the 454-keV level. The observed background is due to true coincidences with $\beta$ particles interacting with the Ge detectors. The non-observation of any coincident $\gamma$ ray supports the scenario that this state is directly fed by a 1/2$^-$ isomer in $^{71}$Ni.

\begin{figure}
\includegraphics[height=9cm,width=9cm]{4_gates.eps}
\caption{\label{coinc} Background-subtracted
$\gamma$-$\gamma$ coincidence spectra gated on the 447 keV transition (top) and 454 keV $\gamma$-ray (bottom).}
\end{figure}

\begin{figure*}
\includegraphics[width=\linewidth]{5_71Co_BG_v2.eps}
\caption{\label{71co} Beta-gated $\gamma$ spectrum for A=71 when
laser are on (top) Co resonance and lasers are off (bottom). Transitions
belonging to the decay of $^{71}$Co (see text), $^{71}$Ni and $^{71}$Cu are marked on
the figure. Contaminant $\gamma$ lines are marked with open circles. The open squares mark $\gamma$ lines belonging to the decay of $^{112}$Ag, which originates from $^{112}$Rh isotopes implanted next to the tape during the optimization of the laser-ion source.}
\end{figure*}

From the observed number of counts in the 454-keV peak and taking into account the absolute $\gamma$-ray branching from the (1/2$^-$) $\beta$-decaying isomer (see section \ref{s71co}) we
extract a production rate of $0.2 (1)$ at/$\mu$C of the $1/2^{-}$ isomer in experiment I. With a ground state yield of $3.0(6)$ at/$\mu$C, see Table \ref{lln}, this results in an isomeric ratio of $7 (4) \%$. Within the same experimental conditions, a lower limit of 0.74 at/$\mu$C was reported for the production of the (1/2$^-$) isomer in $^{69}$Ni, which was found to represent nearly 20$\%$ from the total production rate of $^{69}$Ni \cite{Fra01}.

\subsection{\label{s71co}$\beta$ decay of $^{71}$Co}

Figure \ref{71co} shows part of the $\beta$-gated $\gamma$-spectra
observed in the measurement with the lasers tuned to ionize Co
(top) and without laser radiation (bottom). Transitions belonging
to the $A$=71 decay chain are labeled on the figure. The
contaminant lines, represented with open circles, were found to be
emitted by the $^{142}$La or $^{102}$Nb fission products. $^{142}$La as well as $^{102}$Nb in the form of a molecule with $^{40}$Ar could reach the detection setup in a 2$^+$ charge state and therefore with the same $A/q$ ratio as the ions of interest. Another source of contamination found to give a non-resonant signal in the $\beta$-gated $\gamma$ spectra shown in Fig. \ref{71co} was $^{112}$Ag, produced in the decay of $^{112}$Rh which was implanted next to the tape during the optimization of the laser-ion source.

With the lasers tuned on the Co resonance, the transitions of 282, 566
and 774 keV, assigned to the decay of $^{71}$Co in Refs. \cite{Saw04,Raj07}, are clearly visible in Fig. \ref{71co}(top).

The lasers-off spectrum (Fig. \ref{71co}, bottom) shows the presence of the 534-keV line from the non-resonant production of $^{71}$Ni. The upper limits of the $\gamma$ intensities of the other lines from the decay of $^{71}$Ni are consistent with Ref. \cite{Fra01}. However, in the Co on resonance spectrum (Fig. \ref{71co}, top), an excess of 35(11) counts in the 454-keV line was observed when using the intensity ratio  $I_\gamma(534)/I_\gamma(454)$ from Ref. \cite{Fra01} (see Fig. \ref{bg}).
This confirms that the 454-keV line is populated in the decay of the newly identified (1/2$^-$) isomer of $^{71}$Ni, which in turn is fed by the $\beta$-decay of $^{71}$Co \cite{Raj07}. A comparison of the intensity of the 534-keV line in the top and bottom spectra of Fig. \ref{71co} indicates that in the Co on resonance spectrum the observed intensity mainly stems from non-resonant production of $^{71}$Ni. Thus, our data do not allow to confirm the $\sim$10$\%$ indirect feeding of the $^{71}$Ni ground state in the $^{71}$Co decay \cite{Raj07}. However, using the $^{71}$Co decay scheme of Ref.~\cite{Raj07}, we can deduce the absolute 454-keV $\gamma$-ray intensity.

From the intensities of the 566-, 774-, and 454-keV lines observed in the spectrum displayed in Fig.~\ref{71co}, we determined an absolute $\gamma$-ray branching of $40(15) \%$ to the 454-keV transition. In Fig.~\ref{71co}, the absolute $\gamma$-ray branching is taken as the direct $\beta$ branching to the $454$-keV level and the remaining intensity of $60(15) \%$ is attributed to the direct feeding of the ground state of $^{71}$Cu. Because weaker $\gamma$ transitions to both the $454$-keV level and ground state might have been missed, the $\beta$ branching values should be considered as upper limits. The value extracted for the feeding to the excited state corresponds to a log$ft$ of 5.4(2), assuming that the (1/2$^-$) isomer in $^{71}$Ni is indeed located at 499 keV \cite{Raj07}.

\section{DISCUSSION}

Let us now discuss the implications of the present findings on the
evolution of the nuclear structure in this mass-region.
Figure \ref{schemes} shows the comparison between the $\beta$-decay chains $^{69}$Co $\rightarrow$ $^{69}$Ni $\rightarrow$ $^{69}$Cu \cite{Mue99,Fra01} and $^{71}$Co $\rightarrow$ $^{71}$Ni $\rightarrow$ $^{71}$Cu \cite{Fra01,Raj07}. Both $^{69,71}$Co isotopes are assumed to have a 7/2$^-$ ground-state dominated by the $\pi f_{7/2}^{-1}$ proton-hole configuration for which the major decay path is the Gamow-Teller decay of a $f_{5/2}$ core neutron to fill the  $f_{7/2}$ proton orbital. In $^{69}$Ni, the strong $\beta$-decay branches from $^{69}$Co observed to the levels at 915 and 1518 keV suggested dominant $\nu f_{5/2}^{-1}\otimes 0^+(^{70}$Ni) and $\nu f_{5/2}^{-1}\otimes 2^+(^{70}$Ni) configurations, respectively, although the latter is considerably mixed up with a $\nu p_{1/2}^{-1}$ $\otimes$ $2^+(^{70}$Ni) component \cite{Mue99}. As discussed in Ref. \cite{Raj07}, the expected strong Gamow-Teller $\beta$-decay branch from the 7/2$^-$ ground-state of $^{71}$Co restricts the spins and parities of the excited states populated in the $^{71}$Ni daughter nucleus to 9/2$^-$, 7/2$^-$ and 5/2$^-$. Based on shell-model predictions and the observed systematics, spin and parity 5/2$^-$ were assigned to the levels at 1065 and 1273 keV in $^{71}$Ni, see Ref. \cite{Raj07} and Fig. \ref{schemes}.

In both $^{69,71}$Ni nuclei, the 5/2$^-$ states receiving the main $\beta$-feeding are assumed to decay via $E2$ transitions towards the (1/2$^-$) isomer. In $^{71}$Ni, however, shell-model predicts that the first excited level is 7/2$^+$ \cite{Lis04}. The presence of such state below the (1/2$^-$) level reduces the spin difference between the 5/2$^-$ levels populated in $\beta$-decay and the 9/2$^+$ ground-state and increases the probability for $\gamma$-decay 5/2$^-\rightarrow 7/2^+ \rightarrow 9/2^+_{g.s.}$, by-passing the (1/2$^-$) isomer. Based on the analysis of $\gamma$-$\gamma$ coincidences, the observed 813-252 keV cascade was assigned to this spin-sequence in Ref. \cite{Raj07}.

\begin{figure*}
\includegraphics[width=0.8\linewidth]{6_69_71chain.eps}
\caption{\label{schemes} Observed $\beta$-decay chain $^{69}$Co $\rightarrow$ $^{69}$Ni $\rightarrow$ $^{69}$Cu \cite{Mue99,Fra01} and  $^{71}$Co $\rightarrow$ $^{71}$Ni $\rightarrow$ $^{71}$Cu as proposed in Ref. \cite{Raj07} and present work. $Q_\beta$ values are given in MeV.}
\end{figure*}

The $\beta$-decay of the (1/2$^-$) isomer in $^{69}$Ni was found to populate essentially the state at 1298 keV in $^{69}$Cu, see Fig. \ref{schemes}. From the comparison of the $\gamma$-intensity feeding into the 321-keV level in $^{69}$Ni with the intensity of the 1298-keV transition in $^{69}$Cu, a $\beta$-branching of 74(9)$\%$ was determined in Ref. \cite{Mue99} for the level at 1298 keV. This branching corresponds to a log$ft$ value of 4.3(2), see Ref. \cite{Mue99} and Fig. \ref{schemes}. Spin and parity 3/2$^-$ were assigned to the 1298-keV state, viewed as the $p_{3/2}$ proton coupled to the $2p-2h$,  0$^+$ state at 1770 keV in the $^{68}$Ni core \cite{Mue99}. Such configuration implies very low collectivity for the 3/2$^-$state, in agreement with its non-observation in the Coulomb excitation spectrum shown in Fig. \ref{clx}. In contrast to the 3/2$^-$ single-particle level at 1298 keV in $^{69}$Cu, the state at 454 keV in $^{71}$Cu was found to exhibit large collectivity $(B(E2;1/2^- \rightarrow 3/2^-_{g.s.})$=20.4(22) W.u. as determined in Ref. \cite{Ste08}). By relating the number of counts in the peaks at 1096 and 1213 keV observed in the bottom spectrum of Fig. \ref{clx} with the corresponding $B(E2)$ values reported in Ref. \cite{Ste08}, an upper limit of 1.4 W.u. can be extracted for the $B(E2)$ value for the 1298-keV transition. Thus, the decay of (1/2$^-$) isomer in $^{69,71}$Ni populates states with very different character in the daughter nuclei.

In $^{71}$Ni, however, our evidence shows that the $\beta$-decaying isomer feeds mainly the proposed (1/2$^-$) state at 454 keV in $^{71}$Cu. As discussed in Ref. \cite{Ste08}, the large $B(E2)$ value measured for the 454 keV transition excludes a single-particle character of $\pi p_{1/2}$ type for the 454 keV level. The increased collectivity indicates significant deformation setting in with increasing the number of $g_{9/2}$ neutrons, as also suggested by the results of recent Coulomb excitation experiments with radioactive
beams of $^{70}$Ni \cite{Per06}. The onset of collectivity was associated with the quenching of both $Z$=28 and $N$=40 gaps through the combined effect of the attraction and repulsion between the $fp$ protons and $g_{9/2}$ neutrons \cite{Per06,Ste07,Ots05}.  Thus, the observed $\beta$-decay branch from the (1/2$^-$) isomer in $^{71}$Ni to the 454-keV level in $^{71}$Cu  can be explained by assuming that the odd proton occupies the $K$=1/2 downsloping orbit of the $p_{3/2}$ orbital, on the prolate side, while the neutron part of the wave function is, depending on deformation, dominated by $\nu p_{1/2}^{+2}g_{9/2}^{+2}$ or $\nu p_{1/2}^{-2}g_{9/2}^{+4}$ configurations. This indicates that in both $^{69,71}$Ni isotopes, the $\beta$-decay of the (1/2$^-$) isomer proceeds via a fast Gamow-Teller transition but in the case of $^{71}$Ni the spin of the  final state in the daughter nucleus is changed by the deformation. Interesting to note is that a similar deformed $\pi 1/2^-[321]$ has been observed in $^{67}$Co, stemming from a $\pi \text{(1p-2h)}$ proton excitation across $Z$=28 \cite{Pau08}.

\section{Conclusions}

In this paper, the results of the investigation of the decay of the proposed (1/2$^-$) $\beta$-decaying isomer in $^{71}$Ni are presented and discussed. The key observable for this study is the newly observed level at 454 keV in $^{71}$Cu reported recently in Ref.~\cite{Ste08} and for which the comparison with the systematics and model calculations predict a spin and parity of 1/2$^-$. The experimental evidence discussed here combines the results of the Coulomb excitation measurement with radioactive beams \cite{Ste08} with the results of two decay experiments aiming to the investigation of the $\beta$-decay of $^{71}$Co and $^{71}$Ni. The analysis of the $\beta$-decay of $^{71}$Ni indicates that the 454-keV state observed in $^{71}$Cu is fed by the (1/2$^-$) $\beta$-decaying isomer in $^{71}$Ni for which a half-life of $T_{1/2}$=2.34(25) s was determined in the present work.

The large $B(E2)$ value measured in Ref. \cite{Ste08} for the 454-keV transition depopulating the (1/2$^-$) state in $^{71}$Cu indicated a deformed structure for this level. This indicates that in both $^{69,71}$Ni isotopes the main $\beta$-decay branch of the (1/2$^-$) isomer goes to the level dominated by the $\pi p_{3/2} \nu (p_{1/2}^{-2}g_{9/2}^2)$ configuration in the daughter nuclei. In $^{71}$Cu, however, due to deformation, the nuclear properties of the level receiving the main $\beta$-feeding are dictated by the $K$=1/2 downsloping orbit of the $\pi p_{3/2}$ orbital.

We gratefully thank J.~Gentens and P.~Van den Bergh for running the
LISOL separator. This work was supported by the European Commission within the Sixth Framework Programme through
I3-EURONS (Contract RII3-CT-2004-506065), BriX/IUAP P6/23, FWO-Vlaanderen (Belgium), GOA 2004/03 and US Department of Energy.