# Analysis of realistic and empirical multipole interactions for shell-model
calculations

K. Sieja  Institute Pluridisciplinaire Hubert Curien, 23 rue du Loess,
Strasbourg, France

(today)

###### Abstract

We perform an analysis of realistic nucleon-nucleon interactions, as well as
of empirically-corrected interactions, fitted to reproduce in detail the
spectroscopic data in $p$ and $sd$ shells. We focus on the multipole part of
the interactions, contrary to previous studies of that type which focused on
monopole parts. We analyze how the different parts of the NN force (tensor,
spin-orbit and central) contribute to the leading multipoles of the
interaction and how do they renormalize in medium. We show that the leading
coherent terms of the realistic and effective interactions are dominated by
the central component as expected from their schematic operator structure.
Convergence and cutoff dependence of multipole Hamiltonians is also discussed.

##  I Introduction

One of the major problems in nuclear physics is the derivation of the
realistic Hamiltonian in a given model space, that would be at the same time
consistent with the observed phase shifts and able to reproduce saturation
properties of nuclei and their spectra. As these requirements have never been
met in the calculations with the realistic NN forces, the development of the
NNN potentials on one side, and the phenomenological modeling of the nuclear
Hamiltonians on the other, bring us closer and closer to a successful
description of the atomic nucleus. In the recent years a lot of attention has
been paid to the monopole part of the effective interaction. The first
attempts have been undertaken to include some of the three-body effects into
the calculation of 2-body monopoles in $sd$ (Otsuka et al., 2010a) and $pf$
(Holt et al., ) shells, showing that the effective 3-body $T=1$ interactions
may be partially responsible for the observed spin-orbit shell closures.
Otherwise, a lot of effort has been dedicated to the empirical modeling of the
general monopole behavior, which could be applied for studies of the shell
evolution far from stability. An early trial in Ref. (Otsuka et al., 2001)
consisted in explaining the shell closure $N=16$ with a central spin-isospin
exchange term. Further a dominant role of the tensor force has been noticed
(Otsuka et al., 2006), however the tensor scenario holding in light nuclei
seemed to contradict the shell evolution and stability of heavier systems
(Nowacki, 2007; Sieja et al., 2009). A quantitative analysis of the role
played by different components of the NN interaction in the $sd-pf$ shell was
done Smirnova et al. (2010) using the spin-tensor decomposition of the 2-body
effective interactions. It has been shown that the evolution of the $N=16,20$
and $N=28$ shell gaps is in fact due to the combined effects of the central
and tensor forces. In Ref. Otsuka et al. (2010b) a very same conclusion was
achieved in yet another empirical attempt of providing a universal monopole
interaction. It was also noted in Ref. Tsunoda et al. (2011) that the tensor
component of the monopole field remains nearly intact in the perturbative
treatment, i.e. the monopole tensor content of the realistic NN force remains
the same in the effective interactions. It is of interest to notice that the
tensor contribution to the monopole seems to survive intact even in the
empirical fits, as presented e.g. in Nowacki . This feature of the tensor
force, termed renormalization persistency in Ref. Tsunoda et al. (2011), was
studied in detail only for the monopole matrix elements of the effective
Hamiltonian. It has been however noted that considerable differences appear in
the non-diagonal elements of the bare and renormalized tensor forces, whose
consequences were not studied in the shell model.

While the saturation properties of nuclei are determined by the monopole part
of the realistic forces, the detailed spectroscopy is dependent not only on
the monopole field propagation but as well on the multipole part of the
interaction, the pairing force determining the $0^{+}-2^{+}$ energy splittings
in a core$\pm$2 particles system being the most prominent example. The
addition of the core polarization corrections to the realistic two-body matrix
elements by Kuo and Brown Kuo and Brown (1968) yielded a first reasonable
agreement of spectra of even-even systems with data. Though the perturbative
treatment of the in-medium effects have been further developed Kuo and Osnes
(1990); Hjorth-Jensen et al. (1994, 1995), it is commonly recognized that an
accurate spectroscopic description of nuclei still requires a simultaneous
readjusting of the monopole and multipole interactions. To which extent the
deficiencies in the multipole part come from the lack of many-body effects in
the initial potential or/and from the shortcomings of the perturbative
treatment, remains also an open question.

The multipole part of the Hamiltonian was less extensively studied than the
monopole one, with the ground-breaking exception of Refs. Zuker (1994); Dufour
and Zuker (1996), where the leading multipoles in the effective interactions
were classified and the ”universality” of these multipoles in different
realistic interactions evidenced, i.e. the fact that the multipole content of
all the realistic interactions is fairly similar. The problem of the $T=1$
multipole component of the effective interactions far from stability was
addressed e.g. in Sieja and Nowacki (2011), showing that a proper
renormalization is crucial for achieving a correct magnitude of multipole
components in $p-sd$ and $sd-pf$ interactions.

In this work, we concentrate as well on the multipole part of the effective
interactions, this time we analyse both $T=0$ and $T=1$ components in one
major harmonic oscillator shell ($p$ and $sd$ shells). For the first time, we
perform the spin-tensor analysis of the multipole terms. The effective
interactions obtained from the CD-Bonn potential in the many-body perturbation
theory are further compared to the phenomenological ones, based on the very
same matrix elements and fitted to experimental data using the linear
combination method, as practiced e.g. in Brown and Richter (2006); Honma et
al. (2010); Gniady et al. .

We start the discussion with a short reminder of the multipole-monopole
decomposition of the interaction as well as the spin-tensor decomposition in
Sec. II. In Section III we give some details of the derivations and fits of
the effective interactions. Finally, we perform the spin-tensor analysis of
the realistic and readjusted interactions in Sec. IV and we present the major
results of this work. Conclusions are collected in Sec. V.

##  II Spin-tensor and monopole-multipole decomposition of the effective
Hamiltonian

The spin-tensor decomposition is a procedure allowing to separate the central,
spin-orbit and tensor parts of the effective interaction. The monopole and
multipole parts of each component can be then studied unambigously. The most
general 2-body effective interaction can be written as

$V(1,2)=\sum_{k=0,1,2}V^{(k)},$ (1)

where $k=0$ stands for the central, $k=1$ for the spin-orbit and $k=2$ for the
tensor part of the interaction. To extract each component from the $J$-coupled
two-body matrix elements (TBME), commonly used in SM studies, one needs to
pass from the $J$ to the $LS$ coupling and then employ the spin-tensor
decomposition to the $LS$-coupled matrix element. We thus have:

$\displaystyle\langle
ab;JT|V_{k}|cd;JT\rangle=\frac{(2k+1)(-)^{J}}{\sqrt{(1+\delta_{ab})(1+\delta_{cd})}}\sqrt{(2j_{a}+1)(2j_{b}+1)(2j_{c}+1)(2j_{d}+1)}$
$\displaystyle\times\left[\sum_{LS,L^{\prime}S^{\prime}}(2L+1)(2S+1)(2L^{\prime}+1)(2S^{\prime}+1)\left\\{\begin{array}[]{ccc}l&1/2&j_{a}\\\
l&1/2&j_{b}\\\
L&S&J\end{array}\right\\}\left\\{\begin{array}[]{ccc}l&1/2&j_{c}\\\
l&1/2&j_{d}\\\ L^{\prime}&S^{\prime}&J\end{array}\right\\}\right.$ (8)
$\displaystyle\left\\{\begin{array}[]{ccc}L&S&J\\\
S^{\prime}&L^{\prime}&k\end{array}\right\\}\times\left\\{\sum_{J^{\prime}}(2J^{\prime}+1)(-^{J^{\prime}})\left\\{\begin{array}[]{ccc}L&S&J^{\prime}\\\
S^{\prime}&L^{\prime}&k\end{array}\right\\}\right.$ (13)
$\displaystyle\times\left(\sum_{j_{a}^{\prime}j_{b}^{\prime}j_{c}^{\prime}j_{d}^{\prime}}\sqrt{(2j_{a}^{\prime}+1)(2j_{b}^{\prime}+1)(2j_{c}^{\prime}+1)(2j_{d}^{\prime}+1)}\left\\{\begin{array}[]{ccc}l&1/2&j_{a}^{\prime}\\\
l&1/2&j_{b}^{\prime}\\\
L&S&J^{\prime}\end{array}\right\\}\left\\{\begin{array}[]{ccc}l&1/2&j_{c}^{\prime}\\\
l&1/2&j_{d}^{\prime}\\\
L^{\prime}&S^{\prime}&J^{\prime}\end{array}\right\\}\right.$ (20)
$\displaystyle\times\sqrt{(1+\delta_{a^{\prime}b^{\prime}})(1+\delta_{c^{\prime}d^{\prime}})}\langle
a^{\prime}b^{\prime};J^{\prime}T|V|c^{\prime}d^{\prime};J^{\prime}T\rangle\Bigg{)}\Bigg{\\}}\Bigg{]},$
(21)

where $\langle
a^{\prime}b^{\prime};J^{\prime}T|V|c^{\prime}d^{\prime};J^{\prime}T\rangle$
are the $J$-coupled TBME of the interaction to be decomposed.

To asses the properties of the multipoles of any interaction, we need to
decompose the Hamiltonian into the multipole and monopole part. The idea of
the multipole-monopole separation is that the Hamiltonian can be written in
terms of the density operators coupled to a good angular momentum $\lambda$,
i.e. $(a^{\dagger}a)^{\lambda}$ and that the monopole component is given by
$\lambda=0$ while the multipole is all the rest $\lambda\neq 0$ Dufour and
Zuker (1996). A proof of the separation property is given in Ref. Zuker
(1994). $H_{multipole}$ takes the following form in the particle-particle
representation:

$H_{multipole}=\sum\limits_{ik<jl\Gamma}W_{ijkl}^{\Gamma}[(a_{i}^{\dagger}a_{j}^{\dagger})^{\Gamma}(a_{k}a_{l})^{\Gamma}]^{0},$
(22)

where $\Gamma=JT$, and in the particle-hole:

$\displaystyle
H_{multipole}=\sum_{ik<jl\Gamma}\sqrt{(2\gamma+1)}{\sqrt{(1+\delta_{ij})(1+\delta_{kl})}\over
4}\omega_{ikjl}^{\gamma}[(a_{i}^{\dagger}\tilde{a}_{k})^{\gamma}(a_{j}^{\dagger}\tilde{a}_{l})^{\gamma}]^{0},$
(23)

where $\gamma=\lambda\tau$. We have replaced here $V$ matrix elements by $W$,
to underline that they are monopole-free. The $W$ and $\omega$ matrix elements
are related to each other by a Racah transformation:

$\displaystyle\omega_{ikjl}^{\gamma}=\sum_{\Gamma}(-)^{j+k-\gamma-\Gamma}\left\\{\begin{array}[]{ccc}i&j&\Gamma\\\
l&k&\gamma\end{array}\right\\}W_{ijkl}^{\Gamma}(2J+1)(2T+1),$ (26)
$\displaystyle
W_{ijkl}^{\Gamma}=\sum_{\gamma}(-)^{j+k-\gamma-\Gamma}\left\\{\begin{array}[]{ccc}i&j&\Gamma\\\
l&k&\gamma\end{array}\right\\}\omega_{ikjl}^{\gamma}(2\lambda+1)(2\tau+1).$
(29)

The knowledge of the multipole Hamiltonian is connected to a large number of
the matrix elements among which some may be more important than others. The
diagonalization of the multipole Hamiltonian in the particle-particle and
particle-hole representations allow to pin down the coherent terms and to
separate the collective part (C) from the random one (R), both contributing to
the multipole Hamiltonian: $H_{multipole}=H_{C}+H_{R}$. The analysis of the
strongest contributions appearing in the diagonal form of $H_{multipole}$ was
performed in Ref. Dufour and Zuker (1996). In this work we focus on the two
dominant $JT=10,01$ terms, corresponding to the pairing operators
$P^{10},P^{01}$, and four representative $\lambda\tau$: 10 ($\sigma$), $11$
($\sigma\tau$), $20$ ($r^{2}Y_{2}$) and 30 $(r^{3}Y_{3})$. The proposed
operator structure of these multipoles assumes that they are central and the
overlaps of the schematic operators with their realistic multipole
counterparts were shown to be quite close to 1 (see Table II from Ref. Dufour
and Zuker (1996)). Thus the magnitude of the tensor and spin-orbit components
in the spin-tensor decomposition of the multipole Hamiltonian will provide yet
another measure of the possible deviation of the collective terms from their
analytical forms.

##  III Fit of the interaction

The starting point of the fit procedure is an effective Hamiltonian obtained
from the CD-Bonn potential Machleidt (2001), renormalized via $V_{lowk}$
approach with a sharp cutoff Bogner et al. (2003) of 2fm-1. The $V_{lowk}$ is
solved by the similarity transformation in the oscillator space of 10 major
shells with $\hbar\omega=18$ for $p$ nuclei and $\hbar\omega=15$ for $sd$
nuclei. Effective Hamiltonians were further evaluated through the
$\hat{Q}$-box method with folded diagrams as described in Refs. Kuo and Osnes
(1990); Hjorth-Jensen et al. (1994, 1995). All diagrams through third order
have been included, allowing $18\hbar\omega$ excitations for intermediate
states. For the calculations of the $V_{lowk}$ potential and effective
Hamiltonians the Oslo codes Hjorth-Jensen  were employed. For the purpose of
the fit with the linear combination method we selected 41 energy levels in 14
nuclei in the $p$ shell and 300 levels in 59 nuclei in the $sd$ shell. For
such data sets, the rms deviations obtained with effective interactions
calculated with the $\hat{Q}$-box method are 2.74MeV and 2.29MeV,
respectively. It is not encouraging to notice, that on average similar results
are obtained using bare $V_{lowk}$ interactions (rms =2.09MeV in $p$ and
2.18MeV in $sd$ shells). However, as will be discussed later, the particular
properties of the interaction (e.g. pairing) are much better given by the
effective $\hat{Q}$-box interaction.

In the first step of the fit we adjusted only the single particle energies and
monopoles. The fit was achieved using 4 linear combinations for the $p$-shell
and 12 linear combinations for the $sd$ shell. The rms deviations obtained
this way are 1.74MeV and 0.83MeV for the $p$ and $sd$ shells, respectively.
The improvement with respect to the $\hat{Q}$-box interactions is
considerable, still the precision of the interaction is far from being
satisfactory. In the second step we have used the monopole-tuned interactions
as a starting point of a full-fledged fit of both monopole and multipole
components. In this case, we have obtained the best agreement with data using
10 linear combinations for the $p$ shell and 40 linear combinations for the
$sd$ shell. The simultaneous fit of the monopole and multipole components
allow to improve greatly the agreement with data, leading to rms deviations of
0.61MeV and 0.22MeV accordingly. One should however state, that it was not our
purpose to obtain the best up-to-date fit of the effective interactions, but
rather to provide consistent phenomenological Hamiltonians for the sake of
further analysis.

##  IV Results

###  IV.1 Spin-tensor analysis

We start the discussion with the TBME of the isoscalar interaction in the $p$
shell. In Fig. 1 we show the matrix elements of the $p$ shell interactions in
which the monopole component was removed. We compare the first order
interaction ($V_{lowk}$) to the 3rd order $\hat{Q}$-box interaction with
folded diagrams ($V_{eff}$) and the empirical fit ($V_{emp}$). First of all,
let us notice that the monopole-free matrix elements have the largest central
part and smaller, but non-negligible spin-orbit and tensor components. All the
three components are modified due to the in-medium corrections, and in
particular, the tensor component of the multipole part of the $T=0$
interaction is equally renormalized. This is not the case of the monopole
interaction, in which the tensor part remains nearly the same, as was shown
e.g. in Ref. Otsuka et al. (2010b). Thus the renormalization persistency does
not hold for the monopole-free TBME of the interaction. It is also worth
noticing, that the empirical TBME follow the global trends of monopole-free
TBME of the $V_{lowk}$ and the fit leads to moderate changes in their
amplitude, a condition necessary for a ”realistic-compatible” interaction.

The multipole Hamiltonian, whose TBME are plotted in Fig. 1, can be however
split into random and collective parts. For further discussion we will pass to
the diagonal representation as discussed in Sec. II and analyse rather the
coherent multipole terms than the monopole-free 2-body matrix elements one by
one.

Figure 1: Matrix elements of the $p$ shell monopole-free interaction.
$V_{lowk}$ potential (first order interaction) is compared to the effective
Hamiltonian ($V_{eff}$) calculated in the 3rd order $\hat{Q}$-box method
including folded diagrams and to the empirical fit ($V_{emp}$). Central
($k=0$), spin-orbit ($k=1$) and tensor ($k=2$) components are disentangled.
$3$ stands here for the $p_{3/2}$ and $1$ for the $p_{1/2}$ orbital.

In Tables 1,2 we show the eigenvalues of multipole operators in the first
($V_{lowk}$) and third order $\hat{Q}$-box with folded diagrams ($V_{eff}$)
interactions compared to the empirical fits ($V_{emp}$) in $p$ and $sd$
shells. We diagonalize separately central, spin-orbit and tensor components in
each case.

Table 1: Eigenvalues of the leading multipoles in the $p$-shell. Central, spin-orbit and tensor components of the CD-Bonn potential smoothed with the $V_{lowk}$ procedure, 3rd order interaction with folded diagrams ($V_{eff}$) and the empirical fit ($V_{emp}$) are distinguished. Multipole | Interaction | Central | Spin-orbit | Tensor | Total
---|---|---|---|---|---
$JT=10$ | $V_{lowk}$ | -4.46 | -0.04 | -1.30 | -4.00
| $V_{eff}$ | -6.23 | -0.77 | -2.17 | -6.28
| $V_{emp}$ | -4.11 | -1.11 | -1.22 | -4.42
$JT=01$ | $V_{lowk}$ | -4.88 | -0.30 | -0.85 | -5.72
| $V_{eff}$ | -6.77 | -0.20 | -0.58 | -7.16
| $V_{emp}$ | -6.85 | -0.16 | -0.72 | -6.83
$\lambda\tau=10$ | $V_{lowk}$ | -1.71 | -0.44 | -0.87 | -1.90
| $V_{eff}$ | -2.22 | -0.29 | -0.66 | -2.59
| $V_{emp}$ | -1.67 | -0.48 | -0.48 | -2.00
$\lambda\tau=20$ | $V_{lowk}$ | -3.06 | -0.26 | -0.34 | -3.05
| $V_{eff}$ | -4.31 | -0.16 | -0.07 | -4.26
| $V_{emp}$ | -3.56 | -0.22 | -0.30 | -3.59
$\lambda\tau=11$ | $V_{lowk}$ | 2.43 | 0.15 | 0.19 | 2.35
| $V_{eff}$ | 3.51 | 0.21 | 0.17 | 3.49
| $V_{emp}$ | 2.93 | 0.09 | 0.24 | 2.96
$\lambda\tau=30$ | $V_{lowk}$ | -0.67 | 0.09 | 0.06 | -0.52
| $V_{eff}$ | -0.98 | 0.20 | 0.00 | -0.78
| $V_{emp}$ | -1.02 | 0.33 | 0.03 | -0.65
Table 2: The same as in table 1 but for the $sd$ shell. Multipole | Interaction | Central | Spin-orbit | Tensor | Total
---|---|---|---|---|---
$JT=10$ | $V_{lowk}$ | -4.78 | -0.02 | -0.98 | -4.93
| $V_{eff}$ | -7.24 | -0.62 | -2.02 | -8.08
| $V_{emp}$ | -5.61 | -0.61 | -2.03 | -6.58
$JT=01$ | $V_{lowk}$ | -4.18 | -0.02 | -0.24 | -4.41
| $V_{eff}$ | -6.37 | -0.02 | -0.15 | -6.38
| $V_{emp}$ | -5.76 | -0.13 | -0.10 | -5.80
$\lambda\tau=10$ | $V_{lowk}$ | -1.38 | -0.32 | -0.71 | -1.53
| $V_{eff}$ | -1.91 | -0.19 | -0.77 | -2.09
| $V_{emp}$ | -1.22 | -0.89 | -0.43 | -1.66
$\lambda\tau=20$ | $V_{lowk}$ | -2.85 | -0.31 | -0.35 | -2.86
| $V_{eff}$ | -4.47 | -0.69 | -0.17 | -4.54
| $V_{emp}$ | -3.38 | -0.52 | -0.16 | -3.44
$\lambda\tau=11$ | $V_{lowk}$ | 2.77 | 0.10 | 0.24 | 2.75
| $V_{eff}$ | 4.11 | 0.10 | 0.28 | 4.09
| $V_{emp}$ | 3.81 | 0.28 | 0.31 | 3.85
$\lambda\tau=30$ | $V_{lowk}$ | -0.75 | -0.23 | -0.48 | -0.77
| $V_{eff}$ | -0.99 | -0.22 | -0.23 | -1.10
| $V_{emp}$ | -0.73 | -0.23 | -0.34 | -1.04

Several points need to be noted from Tables 1, 2:

(i) The central parts of the multipole interactions have the eigenvalues very
close to that of the total multipole Hamiltonian. A fact expected from Dufour
and Zuker (1996) and supporting the operator forms proposed for the leading
multipole terms.

(ii) It is interesting to notice that the eigenvalues of tensor and spin-orbit
components in $JT=10$ and $\lambda\tau=10$ are the largest and these are the
cases for which the weakest overlaps with the schematic operators were found
in Ref. Dufour and Zuker (1996).

(iii) In-medium corrections and empirical fits affect all the multipole
components in all its spin-isospin structures, though the change is of course
the most visible in the largest central part. Even if the renormalization
persistency of the tensor force does not hold for the multipole part, this is
of minor importance for nuclear modeling. The first order tensor contribution
in the multipole Hamiltonian can be simply neglected.

(iv) In the majority of cases, the fit tends to reduce the multipole component
to a value somewhere in between the $V_{lowk}$ and $V_{eff}$. Roughly the
eigenvalues of the empirical interaction can be obtained from
$V_{lowk}$/$V_{eff}$ by multiplying/dividing its TBME by a factor 1.2 in both
shells.

(v) We presented the results in $p$ and $sd$ shells only but given the
properties of the multipole Hamiltonian Dufour and Zuker (1996), the analysis
in any other shell or in multi-shell valence spaces could yield the same
conclusions.

###  IV.2 Convergence of multipoles

So far, we did not discuss the cutoff and convergence properties of the
multipole terms in different interactions. Examining the cutoff dependence of
the calculated observables may be a useful way to estimate the uncertainty of
the results due to the neglected many-body forces and the problems in the
perturbative treatment. In Ref. Sieja and Nowacki (2011) we noticed that the
cutoff dependence of multipoles is rather weak in the range of $1.8-2.6$fm-1,
at least for pairing and quadrupole in the $T=1$ channel. Here we repeated
this analysis for all the multipoles in the wide range of
$\Lambda=1.6-2.6$fm-1. The result is shown in Fig. 2 for the multipoles of the
$V_{lowk}$ potential in the $sd$-shell, distinguishing as well cutoff
variation of the central, spin-orbit and tensor components. We have multiplied
the value of the $\lambda\tau=11$ by -1 and skipped the $\lambda\tau=30$ for a
better transparency in the Figure. The omitted octupole term shows a quite
weak cutoff dependence changing its value on about $10\%$ within the examined
cutoff range. As can be seen from the Figure, the $JT=10$ shows the strongest
cutoff dependence. The strong cutoff dependence of the $T=0$ TBME was
attributed to the contribution of the short-range tensor force. One can also
note, what was already apparent from the Tables 1, 2, that the multipoles of
the $V_{lowk}$ are basically given by its central component, which has the
strongest variation with $\Lambda$. The negligible vector and a bit larger
tensor component are nearly cutoff independent (note however different energy
scales in the Figure).

Figure 2: Multipole eigenvalues in the $V_{lowk}$ potential and its central,
vector and tensor components, depending on the cutoff $\Lambda$ The value of
$\lambda\tau=11$ is multiplied by -1. See text for details.

In Fig. 3 we show the convergence of multipoles of an effective Hamiltonian in
the $sd$-shell, this time eigenvalues of multipoles calculated to 3rd order in
$\hat{Q}$-box are plotted against the number of $\hbar\omega$ excitations
permitted for intermediate diagrams, obtained with the $V_{lowk}$ potential at
different cutoffs. For the lower cutoffs the multipole values are converged at
10$\hbar\omega$, a larger value is necessary for higher cutoff. As expected,
with the many-body correlations included the cutoff dependence gets weaker and
the monopoles converged in terms of $\hbar\omega$ are fairly the same for
studied cutoffs. With the only exception of the $\lambda\tau=20$ quadrupole
term, which however, as we show in the next paragraph, has a minor impact on
the calculated properties of deformed nuclei.

Figure 3: Multipole Hamiltonian eigenvalues plotted vs $\hbar\omega$
excitations of 2-body diagrams included in the evaluation of $\hat{Q}$-box for
different cutoffs of the starting $V_{lowk}$ potential. $\lambda\tau=11$ value
is multiplied by -1. Note different energy range in the case of $JT=10$.

Finally, in Tab. 3 we show the convergence of calculated multipoles in the
$sd$-shell in the perturbation expansion. Tha calculation is done for the
$V_{lowk}$ cutoff $\Lambda=2.0$fm-1.

Table 3: Convergence of the eigenvalues of different multipole terms in the order-by-order perturbative expansion. $V_{lowk}$ states for the first order interaction, CP means the second-order interaction with the core-polarization contribution, 2nd-the second order interaction with all possible contributions (CP and 4p2h and ladder diagrams and folded diagrams), $\hat{Q}(2)$ is the second order $\hat{Q}$-box with folded diagrams and $\hat{Q}(3)\equiv V_{eff}$ is the 3rd order $\hat{Q}$-box interaction with folded diagrams. Multipole | $V_{lowk}$ | CP | 2nd | $\hat{Q}(2)$ | 3rd | $\hat{Q}(3)$
---|---|---|---|---|---|---
$JT=10$ | -4.93 | -4.48 | -8.23 | -6.90 | -10.05 | -8.08
$JT=01$ | -4.41 | -6.31 | -7.41 | -6.29 | -7.16 | -6.38
$\lambda\tau=10$ | -1.53 | -2.11 | -1.93 | -1.71 | -2.37 | -2.09
$\lambda\tau=20$ | -2.86 | -3.43 | -4.58 | -3.96 | -5.24 | -4.54
$\lambda\tau=11$ | 2.75 | 3.15 | 3.80 | 3.35 | 4.68 | 4.09
$\lambda\tau=30$ | -0.77 | -1.00 | -1.07 | -0.94 | -1.02 | -1.10

A large gain in the $JT=01$ channel between the first order interaction and
that including CP effects can be noticed from the Table, which improves the
pairing properties obtained in the $V_{eff}$ substantially. Note, that nearly
the same eigenvalue of $JT=01$ is obtained for CP, $\hat{Q}(2)$ and
$\hat{Q}(3)$. Otherwise, the order-by-order convergence is far from being
satisfying for all the multipole terms. A better convergence pattern is found
in terms of $\hat{Q}$-boxes. As previously, the behaviour of $JT=10$ term
appears the most unstable.

It is important to evoke that the conclusions we obtain here on the
convergence of the multipole Hamiltonian follow exactly what has been
discussed e.g. for pairing TBME derived from G-matrices in Ref. Hjorth-Jensen
et al. (1995), in $sd$ and $pf$ shells. Though there are conceptual
differences in the construction of the low$-k$ potentials and G-matrices, and
in the resulting decoupling of high and low momenta in both methods, these
differences are not much apparent in the calculations in finite nuclei. On the
contrary, common problems are inherent to effective interactions derived from
G-matrices and $V_{lowk}$ potentials, on both monopole Schwenk and Zuker
(2006) and multipole levels.

###  IV.3 Multipoles and calculated observables

To realize how the possible error in the multipoles of the Hamiltonian
influences the calculated observables, the following excercise was done: the
monopoles of $V_{lowk}$ and $V_{eff}$ (whose multipole eigenvalues are given
in Tables 1,2) were replaced by the monopoles of the corresponding $V_{emp}$.
With such interactions, having exactly the same monopole fields, we have
recalculated the spectra of all nuclei which were used in the empirical fits
of the Hamiltonian. In the $sd$-shell we obtain the rms of 1.48MeV for
$V_{lowk}$ and 1.18MeV $V_{eff}$, a very considerable deterioration from the
rms=0.22MeV of the $V_{emp}$. Given that as an the intermediate step (see Sec.
III) we have fitted only monopoles of $V_{eff}$ and obtained the rms error of
0.83MeV, one might conclude that the fine tuning of monopoles depends on the
associated multipole and the correct multipole can be only defined within a
given set of monopoles.

However, a stringent test for the pairing part of the Hamiltonian is provided
by the spectra of 2 particles (2 holes) systems, in which the $0^{+}-2^{+}$
splitting should reflect the strength of the $J=0$ pairing interaction. To
some extent, the quadrupole content of the interaction can be also tested by
properties of deformed nuclei in a given model space. Therefore, with the same
interactions (i.e. having exactly the same monopoles) we have calculated the
properties of 18O, 38Ar, 20Ne and 24Mg nuclei, which allows to visualize how
the possible error in the multipole interaction influences the calculated
observables.

The $2^{+}$ state of 18O, experimentally located at 1.98MeV, is of course
accurately reproduced by the empirical fit (1.97MeV) while the $V_{lowk}$
places it at 1.24MeV. This anti-pairing behavior of $V_{lowk}$ is cutoff
independent and also present in G-matrix interactions and taking into account
the core-polarization corrections is necessary to recover the correct
multipole magnitude. $V_{eff}$ does then much better in this context, leading
to only a bit too high $2^{+}$ energy in 18O of 2.21MeV. In all three cases
the wave function of the ground state contains $84\%$ of the $d_{5/2}^{2}$
configuration, which explains the sensivity of the calculated $2^{+}$ energy
to the strength of the $d_{5/2}-d_{5/2}$ pairing interaction. Further, we have
calculated the $2^{+}$ energy of 38Ar. The ground state wave function in all
cases contains $\sim 90\%$ of $d_{3/2}^{-2}$ component. The empirical
interaction places the $2^{+}$ state at 1.86MeV (exp=2.16MeV), while the
$V_{lowk}$ at a very low energy of 0.52MeV. The $V_{eff}$ interaction,
including CP effects, improves the agreement with experiment substantially,
with the $2^{+}$ energy at 1.62MeV. One should note here, that although the
eigenvalue of the $JT=01$ is getting reduced from $V_{eff}$ to $V_{emp}$, this
is not necessarily the trend of all the pairing matrix elements; apparently
the $d_{3/2}-d_{3/2}$ pairing is enhanced in the fit, contrary to the case of
the $f_{7/2}-f_{7/2}$ one.

In Table 4 we show the quadrupole properties of 20Ne and 24Mg nuclei. All the
interactions give remarkably similar results, despite of differences in their
quadrupole content (see Tab. 2), a priori comparable to that of the pairing
part. This is a consequence of a more complex structure of deformed nuclei.
Since the wave functions in this case are spread over many components, the 20%
variation in the quadrupole interaction does not lead to any considerable
deviations, contrary to the cases of 18O and 38Ar, where pairing dominates.

Table 4: Quadrupole properties of 20Ne and 24Mg nuclei obtained with different interactions, see text for details.  Nucleus | Observable |  | $V_{lowk}$ | $V_{eff}$ | $V_{emp}$
---|---|---|---|---|---
20Ne |  $Q(2^{+})$ (efm2) |  | -15.7 | -15.9 | -15.8
|  $B(E2;2^{+}\rightarrow 0^{+})$ (e2fm4) |  | 59.8 | 59.7 | 59.3
|  $B(E2;4^{+}\rightarrow 2^{+})$ (e2fm4) |  | 72.9 | 70.9 | 71.8
24Mg |  $Q(2^{+})$ (efm2) |  | -18.2 | -18.1 | -19.2
|  $B(E2;2^{+}\rightarrow 0^{+})$(e2fm4) |  | 92.7 | 109.8 | 97.6
|  $B(E2;4^{+}\rightarrow 2^{+})$(e2fm4) |  | 127.7 | 143.1 | 133.5

##  V Conclusions

We have performed the spin-tensor analysis of the monopole-free effective
interactions in the $p$ and $sd$ shells. We have shown that the leading
(coherent) multipoles of the realistic forces are dominated by the central
interaction, supporting the equivalence of multipoles extracted from realistic
TBME and schematic operator forms proposed in Ref. Dufour and Zuker (1996). We
have also shown that the renormalization persistency of the tensor force,
observed in the $J$-averaged matrix elements, is no longer present as far as
monopole-free matrix elements are concerned. However, this property of the
tensor force is of a minor importance in the nuclear modeling as the
contribution of the first-order tensor effects to the coherent part of the
multipole Hamiltonian is negligible. Further, we have compared the $V_{lowk}$
potentials based on the CD-Bonn force to the phenomenological forces based on
the same starting potential. The leading multipoles of the interactions are
moderately modified in the empirical fits, as required by the ”realistic-
compatibility” condition, nonetheless their modification along with the
monopole part is inevitable to obtain shell-model interactions of a
satisfactory precision. On average, the leading multipoles of the empirical
interaction appear weaker than in the $\hat{Q}$-box effective interactions but
stronger than in a realistic potential smoothed via the $V_{lowk}$ procedure
with a 2fm-1 cutoff. The convergence properties of the multipoles of the
effective interactions derived here from the $V_{lowk}$ potentials are the
same as those of the G-matrix based interactions from Ref. Hjorth-Jensen et
al. (1995), showing again their quantitative similarity invoked in Ref.
Schwenk and Zuker (2006). It would be of a great interest to investigate how
the inclusion of 3-body forces would influence the couplings in the multipole
Hamiltonian and wether it could explain the multipole modifications required
by the fit. Some light on this issue can be shed in the nearest future, once
the 2-body effective interactions with both, $T=1$ and $T=0$ effective 3-body
components, are evaluated.

##  VI Acknowledgements

We are greatful to E. Caurier and F. Nowacki for certain numerical tools which
were useful in the present study.

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# Understanding and Optimizing the Sensitization of Anatase Titanium Dioxide
Surface with Hematite Clusters

Kati Asikainen 1∗, Matti Alatalo 1, Marko Huttula 1,
B. Barbiellini 2 and S. Assa Aravindh 1∗
1Nano and Molecular Systems Research Unit, University of Oulu, FI-90014,
Finland
2Lappeenranta-Lahti University of Technology (LUT), FI-53851 Lappeenranta,
Finland
∗Corresponding author<EMAIL_ADDRESS>∗Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

The presence of small hematite (Fe2O3) clusters at low coverage on titanium
dioxide (TiO2) surface has been observed to enhance photocatalytic activity,
while excess loading of hematite is detrimental. We conduct a comprehensive
density functional theory study of Fe2O3 clusters adsorbed on the anatase TiO2
(101) surface to investigate the effect of Fe2O3 on TiO2. Our study shows that
TiO2 exhibits improved photocatalytic properties with hematite clusters at low
coverage, as evidenced by a systematic study conducted by increasing the
number of cluster adsorbates. The adsorption of the clusters generates
impurity states in the band gap improving light absorption and consequently
affecting the charge transfer dynamics. Furthermore, the presence of hematite
clusters enhances the activity of TiO2 in the hydrogen evolution reaction. The
Fe valence mixing present in some clusters leads to a significant increase in
H2 evolution rate compared with the fixed +3 valence of Fe in hematite. We
also investigate the effect of oxygen defects and find extensive modifications
in the electronic properties and local magnetism of the TiO2 -Fe2O3 system,
demonstrating the wide-ranging effect of oxygen defects in the combined
system.

##  1 Introduction

Photocatalysis emerges as a promising technique to address contemporary energy
challenges. In particular, photocatalytic water splitting represents a highly
sustainable approach to produce green hydrogen and oxygen by dissociating
water using natural light. This environmentally friendly method enables the
conversion of solar energy into chemical energy without contributing to
pollution. Semiconductors with band gaps ranging from 2.5 to 3.4 eV are
favorable photocatalysts covering visible to ultraviolet (UV) spectra [1].
Consequently, incident photons with energies greater than or equal to the band
gap of the material liberate electrons to the conduction band (CB) and holes
to the valence band (VB). These photogenerated charge carriers can migrate to
the surface of the material participating in oxidation and reduction
reactions. Among semiconductors suitable for this purpose, TiO2 has been
considered as a premier material since its discovery in 1972 by Fujishima and
Honda [2]. It possesses exceptional properties that make it attractive for
photocatalysis. These properties include oxidation properties, physical and
chemical stability, non-toxicity and abundance [3, 4, 5]. Moreover, TiO2
possesses a band structure suitable for efficient water splitting, as it
straddles the reduction (-4.44 eV) and oxidation potentials (-5.67 eV) of
water. However, despite the favorable attributes, a large band gap, quick
recombination of the electron-hole pairs and being inactive for overall water
splitting in the absence of sacrificial reagents leads to a performance
degradation of TiO2 [5, 6, 7]. These factors limits the widespread utilization
of TiO2 in photocatalytic applications. Therefore, attempts are made to
overcome the shortcomings and enhance the performance of TiO2.

Variety of strategies has been proposed to modify both chemical and physical
properties of photocatalysts, and construction of heterostructure systems have
shown improvements in photocatalytic performance [8]. It is recognized that by
combining appropriate photocatalytic materials heterojunction structure with
matching band alignment can be formed. One promising candidate for combination
with TiO2 is hematite ($\alpha$-Fe2O3, hereafter referred to as Fe2O3). It is
the most stable phase of iron (iii@) oxide, possessing a narrow band gap of
2.0-2.2 eV [9, 10]. In a previous study conducted by Singh et al. [11] TiO2
thin films were synthesized and investigated on the Fe2O3-coated indium tin
oxide (ITO) substrate. In their experimental setup, Fe2O3 fully covered the
surface of TiO2. Photoelectrochemical cell (PEC) performance evaluation showed
that the Fe2O3-TiO2 heterostructure exhibits lower PEC performance compared to
bare Fe2O3 thin films. This was attributed to inefficient charge separation
and high recombination rate of the charge carrier within the heterostructure,
as indicated by the low photocurrent. Similarly, Cao et al. [12] studied
anatase TiO2 powder deposited ultrathin Fe2O3 layers and reported a larger
Fe2O3 coating to introduce more recombination sites for electron-hole pairs.

In general, an optimal Fe and/or Fe2O3 concentration has been observed to be
beneficial for enhancing the photocatalytic performance [13, 12, 14, 15, 16].
For instance, Sun et al. [14] conducted an experimental study on anatase TiO2
surface modified with the hematite cluster for phenol degradation. Consistent
with other reports they observed that small clusters enhanced the
photocatalytic activity of TiO2 by improving its response to visible light,
and charge carrier transfer and separation, which was attributed to the
formation of a heterojunctions [13, 12, 14, 17, 15, 18, 19]. However, increase
in cluster size and Fe2O3 concentration hindered the activity. Excessive
loading of Fe2O3 was observed to promote the cluster aggregation, thereby
reducing the efficiency of charge transfer. At high concentrations, Fe has
also been proposed to become a recombination center for photogenerated
electrons and holes, leading to a decrease in photocatalytic activity [13,
14]. Besides, it is possible to synthesize Fe2O3-TiO2 from natural ilmenite.
However, it should be noted that at high concentration and high coverage of
Fe2O3, Fe2O3-TiO2 may transform back to ilmenite [20], which is reported to
have low photocatalytic activity [21].

Fe2O3-TiO2 system has previously been demonstrated to be suitable material for
hydrogen evolution reaction (HER). In the study by Bootluck et al. [15] it was
observed that Fe2O3-TiO2 nanocomposite exhibited a lower recombination rate
and reduced charge transport resistance compared to the bare TiO2. This was
concluded to facilitate the charge transfer at the interface. Furthermore,
they investigated a photocatalytic activity of Fe2O3-TiO2 toward hydrogen
evolution, and the results showed that the untreated Fe2O3-TiO2 exhibited a
relative low hydrogen production rate. However, this rate could be
significantly improved by Ar-plasma treatment. The hydrogen evolution reaction
rate was found to depend on the plasma treatment exposure time, with an
optimal exposure time resulting in the highest photocatalytic activity. It is
worth noting that the study did not provide direct comparison between the HER
activity of Fe2O3-TiO2 and bare TiO2.

In this study, we employ density functional theory (DFT) to investigate the
adsorption of hematite clusters at anatase TiO2 (101). The selection of the
anatase phase and the (101) face were motivated by their superior
photocatalytic activity compared to other polymorphs and crystal orientations
of TiO2 [22, 23, 24, 25]. In order to improve the photocatalytic properties of
TiO2 Sun et al. [14] highlighted a small Fe2O3 cluster size and low Fe2O3
coverage on anatase TiO2. Building upon this, we introduce small hematite
clusters, (Fe2O3)n with $n=1,2$, at the TiO2 surface. The results show the
surface-cluster interaction to be energetically favorable and indicates TiO2
to be a better photocatalyst with Fe2O3 clusters at low coverage. In addition,
we delve into the defect engineering as a another effective method to modify
the properties of the heterostructure of TiO2 and (Fe2O3)1. Specifically, we
introduce an oxygen defect in the heterostructure. In metal oxides, including
TiO2, oxygen vacancies are common defects [26]. Resulting from an oxygen
vacancy, impurity states are generally observed in the band gap of TiO2, often
manifested as Ti3+ species. These impurity states contribute to a shift in the
absorption spectrum and increased conductivity of TiO2 [27, 28]. Consequently,
these can lead to enhanced photocatalytic activity, demonstrated by Sheiber et
al. [29] for example. They reported that oxygen vacancies improved the
photocatalytic activity of anatase TiO2 in water adsorption. Therefore, it
will be interesting to study the combined effect of the metal oxide cluster as
well as defects such as oxygen vacancies in the heterostructure, as the
defect-cluster complex can bring new physical phenomena. Our aim is to
understand the effects brought about by the oxygen defect towards the
electronic properties of the heterostructure, due to the change in carrier
concentration compared to the pristine surface. To assess the photocatalytic
performance, we focus on evaluating the suitability of the investigated
heterostructures for HER. The goal of our study is to demonstrate that coating
the anatase titanium dioxide with mixed-valence iron containing O-ligands
using various hematite cluster coverings represents an effective approach to
enhance photocatalytic hydrogen production.

##  2 Computational methodology

The density functional theory calculations were performed using Vienna Ab
initio Simulation Package (VASP) [30, 31, 32] including the spin polarization.
The exchange-correlation potential was described by generalized gradient
approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) functional
[33]. The Hubbard correction was employed according to Dudarev et al. [34] to
describe localized $d$ electrons of titanium and iron in order to obtain more
realistic electronic and magnetic properties. Within the GGA+U method we
adopted the corrections of U${}_{\mathrm{eff}}=4.5$ eV (U$=4.5$ eV and J$=0$
eV) [35] for the Ti and U${}_{\mathrm{eff}}=4.0$ eV (U$=4.0$ eV and J$=0$ eV)
[36] for the Fe. The 1st Brillouin zone was sampled according to the
Monkhorst-Pack scheme [37]. The Projector augmented wave (PAW) method [38] was
employed to describe the electron-ion potential with the plane wave energy
cutoff of 650 eV (figure S1). The atomic relaxations were performed until the
forces and energies were less than 0.001 eV/Å and $10^{-6}$ eV, respectively.
We used VESTA [39] for visualization and VASPKIT [40] for post-processing of
the data from the VASP calculations.

To simulate the pristine anatase TiO2 (101) surface we constructed a slab of 4
layers containing 192 atoms (64 Ti atoms and 128 O atoms) with lattice
parameters of 10.32 Å 15.25 Å and 35.68 Å in $a$, $b$ and $c$ directions
(figure 2). To prevent interaction between periodic images, a vacuum space of
20 Å was applied along the $c$-axis. The heterostructures were formed by
placing the hematite clusters (Fe2O3)1,2 on the TiO2 surface. Initial
structures of the heterostructures were formed based on the optimized
structures of the surface and the freestanding clusters (figures 1 and S2).
The models were denoted as (Fe2O3)1/TiO2 and (Fe2O3)2/TiO2, and only these two
heterostructures were considered in this study. K-point sampling of $3\times
2\times 1$ and Gaussian smearing of 0.05 eV were used for the TiO2 surface and
heterostructures. To investigate the effect of an oxygen defect, several
defect sites, shown in figure 7, were created in the (Fe2O3)1/TiO2. An oxygen
atom was removed either from the TiO2 (surface or subsurface vacancy) or from
the (Fe2O3)1 cluster. We created an oxygen vacancy in the surface layer of
TiO2 by removing a twofold-coordinated O atom, not forming a bond with the
cluster. Three subsurface oxygen atoms were removed at different distances
from the cluster in the first subsurface layer. Lastly, two different oxygen
defects were created in (Fe2O3)1 cluster. The surface vacancy was labeled as
Ov, the subsurface vacancies as Osv1, Osv2 and Osv3, and the oxygen defects
located in the (Fe2O3)1 cluster as Oc1 and Oc2. The defective heterostructures
were further denoted as (Fe2O3)1/TiO2-Ovac where Ovac specifies the oxygen
defect.

The adsorption energy of the (Fe2O3)n cluster at the TiO2 surface was
calculated from the formula

$\displaystyle
E^{\mathrm{Ads}}=E((\ch{Fe2O3})_{n}/\ch{TiO2})-E(\ch{TiO2})-E((\ch{Fe2O3})_{n}),$
(1)

where $E((\ch{Fe2O3})_{n}/\ch{TiO2})$ is the total energy of the
heterostructures and $E(\ch{TiO2})$ and $E((\ch{Fe2O3})_{n})$ are the total
energies of the pristine TiO2 surface and (Fe2O3)n cluster, respectively. To
investigate the stability of oxygen defects, we calculated the formation
energy of the defects using the equation

$\displaystyle
E^{\mathrm{Form}}=E((\ch{Fe2O3})_{1}/\ch{TiO2}-\mathrm{O_{vac}})-E((\ch{Fe2O3})_{1}/\ch{TiO2})-\frac{1}{2}E(\ch{O2}),$
(2)

where the first term is the total energy of the defective heterostructure, in
which the Ovac specifies the investigated oxygen defect.
$\frac{1}{2}E(\ch{O2})$ is the chemical potential of an oxygen atom that is
half of the total energy of an isolated oxygen molecule O2. To evaluate the
HER activity, we attached hydrogen atom on the defect-free and defective
heterostructures (figure S3) and calculated the adsorption energy of hydrogen.
In general, the free energy of hydrogen adsorption is accepted to be a
descriptor for hydrogen-evolving catalysts. We also calculated the work
function for the heterostructures. Work function is essentially the energy
needed to introduce carriers to the surface and will be affected by doping and
the presence of adsorbates. It is an essential parameter in understanding the
interaction between the hematite clusters and TiO2 and the effect of oxygen
defect on the surface properties, and it is calculated by subtracting the
Fermi energy $E_{F}$ from the vacuum energy $E_{V}$,

$\displaystyle\Phi=E_{V}-E_{F}.$ (3)

The electronic structure of the systems investigated was examined through the
density of states (DOS). To analyze the charge distribution and charge
transfer quantitatively, we performed the Bader analysis [41, 42, 43, 44]. A
negative Bader charge on an atom refers to electron gain and positive value to
electron loss. The charge density differences are also calculated as

$\displaystyle\Delta\rho(\textbf{r})=\rho_{\mathrm{AB}}(\textbf{r})-\rho_{\mathrm{A}}(\textbf{r})-\rho_{\mathrm{B}}(\textbf{r})$
(4)

where $\rho_{\mathrm{AB}}(\textbf{r})$ is the total charge density of the
heterostructure, and $\rho_{\mathrm{A}}(\textbf{r})$ and
$\rho_{\mathrm{B}}(\textbf{r})$ are the total charge densities of the TiO2
surface and the (Fe2O3)n cluster with atoms in exactly the same sites as they
occupy in the heterostructure. Since Fe2O3 is a magnetic material, we also
calculated the spin density difference. It is calculated from the same
equation as charge density difference by replacing charge density
$\rho(\textbf{r})$ by spin density $s(r)$. In the charge density difference
(CDD) plots yellow refers to charge accumulation and cyan refers to charge
depletion while in the spin density difference (SDD) plot the orange shows
excess spin up polarization and turquoise shows excess spin down polarization.
We set the isosurface value of 0.005 e Å-1 in all CDD and SDD plots.

##  3 Results

###  3.1 Bulk parameters

Initially, we calculated the bulk parameters of Fe2O3 and TiO2 (figure S4).
The lattice constants were determined to be $a=b=4.780$ Å and $c=13.323$ Å and
$a=b=3.98$ Å and $c=9.56$ Å for Fe2O3 and TiO2, respectively. These results
are consistent with experimental measurements [45, 46]. The DOS of the bulk
Fe2O3 and bulk TiO2 are in figure S5, showing the electronic structure of the
materials. Using only the GGA functional Fe2O3 was predicted to exhibit
metallic behaviour whereas TiO2 possessed a band gap of 1.7 eV. Upon employing
the Hubbard correction we obtained a band gap of 1.16 eV for Fe2O3 and 2.2 eV
for TiO2, which are in good agreement with the reported values [47, 48, 49,
50, 51]. Based on the results we concluded that the selected methods are
suitable for describing the electronic properties of both materials.

###  3.2 Freestanding clusters and TiO2 surface

Figure 1: The optimized structure of the a) (Fe2O3)1 and b) (Fe2O3)2 clusters.
The Fe-O bond lengths are provided in Å. Figure 2: The surface model and
density of states of anatase TiO2. A reasonable band gap of approximately 2.5
eV was found for the TiO2 surface.

Before investigating the heterostructures, we simulated the freestanding
hematite clusters and pristine TiO2 surface. The relaxed structures of the
(Fe2O3)1 and (Fe2O3)2 are shown in figure 1. The Fe-O distances were ranging
from 1.65 to 1.87 Å in the (Fe2O3)1 and from 1.73 to 1.83 Å in the (Fe2O3)2.
Additionally, the distance between the Fe atoms was found to be 2.23 Å in the
(Fe2O3)1 and 2.36 - 2.43 Å in the (Fe2O3)2, respectively. These geometries and
Fe-O distances compared well with previous works [51, 52, 53, 54], providing a
good starting point for the rest of the calculations.

We proceeded with the optimization of anatase TiO2 (101) surface. There are
four types of atoms at the surface: two- and threefold coordinated oxygen
atoms (O2c and O${}_{3c})$ and five- and sixfold coordinated Ti atoms (Ti5c
and Ti6c). The calculated O-Ti bond lengths were 1.87 Å for O2c-Ti6c, 1.85 Å
for O2c-Ti5c and 2.00 Å for O3c-Ti6c. In addition, we found Ti5c-Ti5c bonds to
be 3.81 Å. The calculated lattice parameters were $a=b=3.88$ Å and $c=9.52$ Å.
The electronic structure of the pristine TiO2 surface, calculated using the
GGA+U, is shown in figure 2. The appearance of shoulder features in the CB
edge is characteristic of the anatase phase of TiO2. The valence band maximum
(VBM) and conduction band minimum (VBM) were located at -0.05 eV and 2.44 eV,
respectively, resulting in a band gap of approximately 2.5 eV. This agrees
with the reported results [51, 55]. The band gap energy is in the visible
light region indicating that the photoactivation of anatase surface could be
achieved by visible light radiation with wavelength up to 500 nm. Due to the
dangling bonds the surface band gap of TiO2 is generally lower than the bulk
band gap (3.0-3.3 eV). Moreover, the work function of the surface was
calculated to be 7.23 eV. In general, TiO2 possesses a high work function of
5-6 eV [56, 57]. Compared to this our calculations moderately overestimated
the work function.

Figure 3: The optimized structures of the a) (Fe2O3)1/TiO2, and b)
(Fe2O3)2/TiO2. Selected atoms are labeled in the figure. Underlining of the
labels in b) is used to distinguish the atoms between the two
heterostructures.

###  3.3 Structural and electronic properties of the (Fe2O3)n/TiO2
heterostructures

####  3.3.1 Structural optimization

The optimized structures of the (Fe2O3)1/TiO2 and (Fe2O3)2/TiO2
heterostructures are shown in figure 3. For the (Fe2O3)1 the optimization
yielded a closed cage-like structure on the surface that resembles a pyramid
with a parallelogram shaped base. The (Fe2O3)2 retained a almost spherical
cage-like structure at the TiO2 surface. Bond lengths between the selected
atoms are provided in table S1. The coordination number of the (Fe2O3)1 was
found to be five, which indicates relatively strong interaction between the
cluster and TiO2, while the coordination number of the (Fe2O3)2 was two. The
distance of the (Fe2O3)1 and (Fe2O3)1 from the surface to their centroids were
approximately 2.4 Å and 3.2 Å respectively. The calculated adsorption energies
of the (Fe2O3)1 and (Fe2O3)2 were -2.28 eV and -1.72 eV, respectively, showing
the cluster-surface interaction to be energetically favourable. More negative
adsorption energy of the (Fe2O3)1 indicates stronger interaction with TiO2.
Some amount of lattice distortion was observed at the TiO2 surface due to the
adsorption of the clusters, primarily limited to the top layer of the surface.

####  3.3.2 Electronic structure analysis

The electronic structure of the (Fe2O3)1/TiO2 and (Fe2O3)2/TiO2, shown in
figure 4, revealed the emergence of Fe2O3-states within the band gap due the
cluster-surface interaction. These impurity states were extended upward from
the VB and downward from the CB of TiO2, resulting in a shift of the Fermi
level toward the CB edge of TiO2. The VBM and CBM of TiO2 were shifted from
-0.05 eV and 2.44 eV to -0.70 eV and 1.92 eV in the (Fe2O3)1/TiO2, and -1.6 eV
and 0.84 eV in the (Fe2O3)2/TiO2, respectively. Consequently, the band gap of
TiO2 remained approximately 2.5 eV in both heterostructures, still enabling
electron excitation from the VB to the CB of TiO2. The occupied Fe2O3-states
below the Fermi level can further provide electrons to the CB of TiO2.
Resulting from the emerging states the band gap of the heterostructures was
substantially smaller. In the (Fe2O3)1/TiO2, due to the unoccupied localized
Fe2O3 states at 0.45 eV, the gap of 1.35 eV between the VB and CB was divided
into two parts, 0.45 eV and 0.90 eV. This decreased the energy difference
between the VB and CB, expanding the photoresponse of TiO2 to cover the entire
visible light spectrum. The (Fe2O3)2/TiO2 possessed a band gap of only 0.2 eV,
suggesting a nearly metallic nature.

Additionally, due to the emerging states the VB of Fe2O3 locates above the VB
of TiO2 and the CB of Fe2O3 below the CB of TiO2. The particular band
alignment (VB${}_{\mathrm{\ch{TiO2}}}<$ VB${}_{\mathrm{\ch{Fe2O3}}}<$
CB${}_{\mathrm{\ch{Fe2O3}}}<$ CB${}_{\mathrm{\ch{TiO2}}}$) can indicate a
formation of type i@ heterojunction with a straddling gap between Fe2O3 and
TiO2. This is in agreement with the study of Moniz et al. [51] and it can
affect the charge transfer properties in the heterostructures. Generally, the
heterojunction allows electrons to be transferred from the CB of TiO2 to the
CB of Fe2O3 and holes from the VB of TiO2 to the VB of Fe2O3.

Figure 4: The computed density of states of the a) (Fe2O3)1/TiO2 and b)
(Fe2O3)2/TiO2.

####  3.3.3 Charge transfer analysis

Efficient charge separation and transfer are essential factors affecting the
photocatalytic activity. To gain further insight into the charge transfer
properties in the heterostructures we calculated the CDD, shown in figures 5a
and c, and performed the Bader analysis. The results showed that the charge
was redistributed after the adsorption of the clusters. At the interface the
electron accumulation and depletion regions were aligned in the direction of
the formed bonds between TiO2 and Fe2O3. This suggests that the bonds can be
considered covalent with a polar feature. The blue isosurface around Fe showed
charge depletion from Fe, indicating electron donor nature for the clusters.
The work function of TiO2 was observed to be affected by the clusters. The
work function decreased from 7.23 eV to 6.70 eV in the (Fe2O3)1/TiO2 and to
5.90 eV in the (Fe2O3)2/TiO2, supporting donor nature for Fe2O3 clusters. The
lower work function can also facilitate electron injection from the TiO2
surface.

The Bader charges of individual atoms confirmed that Fe donated electrons in
the heterostructures. The Bader charges of the Fe1 and Fe2 atoms were 1.61 e
and 1.52 e in the (Fe2O3)1/TiO2. The correlation between the Bader charges and
oxidation state is still somewhat debuted but according to Posysaev et al.
[58] the particular values suggest Fe3+ oxidation state for Fe in the
(Fe2O3)1/TiO2. In the (Fe2O3)2/TiO2 the Bader charges of Fe atoms were 1.37 e
for Fe1, 1.39 e for Fe2, 1.61 e for Fe3 and 1.45 e for Fe4. Smaller Bader
charge can corresponds to the Fe2+ oxidation state [58] and therefore,
different valences can exist in this heterostructure. To confirm this we
investigated the magnetism brought by Fe2O3. In general, Fe introduces
ferromagnetism in the heterostructures which is supported by the positive spin
density around Fe in the SDD plots (figures 5b and d). We found a total
magnetic moment of $9.61\ \mathrm{\mu_{B}}$ and $17.2\ \mathrm{\mu_{B}}$ per
unit cell for the (Fe2O3)1/TiO2 and (Fe2O3)2/TiO2. In the (Fe2O3)1/TiO2 the
magnetic moments of the Fe1 and Fe2 were $4.11\ \mathrm{\mu_{B}}$ and $4.06\
\mathrm{\mu_{B}}$, respectively, which indicating high spin configuration and
$3d^{5}$ occupation [59, 60], that is, the Fe3+ oxidation state. In the
(Fe2O3)2/TiO2 the Fe atoms had an odd contribution to the total magnetic
moment. We found a large magnetic moment of $4.02\ \mathrm{\mu_{B}}$ and
$4.13\ \mathrm{\mu_{B}}$ for the Fe2 and Fe3, whereas a reduced magnetic
moment of $3.26\ \mathrm{\mu_{B}}$ and $3.36\ \mathrm{\mu_{B}}$ for the Fe1
and Fe4, respectively. The results highly suggest mixed-valence state
(Fe2+/Fe3+) for Fe in the (Fe2O3)2/TiO2. It is worth noticing that magnetic
moments does not directly correlate with the Bader charges. The Bader charge
and magnetic moment of selected atoms are listed in table S2. Due to the
charge transfer, spin magnetization was also transferred to the atoms nearby
Fe.

Figure 5: The CDD and SDD plot of the (Fe2O3)1/TiO2 (a and b) and
(Fe2O3)2/TiO2 (c and d). The labels of the atoms are provided in figure 3.

Due to Fe a notable charge of 0.32 e per unit cell was transferred from the
cluster to the TiO2 surface in the (Fe2O3)1/TiO2. Nearly 90% of the
transferred charge was redistributed to the surface layer of TiO2. Neither the
CDD plots nor Bader charges showed charge localization at the surface,
indicating even distribution of charge among surface layer atoms. In the
(Fe2O3)2/TiO2 the charge transfer was only 0.036 e per unit cell, implying
inefficient charge separation between TiO2 and Fe2O3. As mentioned, because of
the heterojunction, electrons can be transferred from the CB of TiO2 to the CB
of Fe2O3. Additionally, the unoccupied Fe2O3-states can trap the excited
electron from the VB, changing the oxidation state of Fe to Fe2+. However,
because Fe2+ is relatively unstable and the unoccupied Fe states lies close to
the CB of TiO2 our results suggest that Fe2+ tends to release the trap
electrons and change to Fe3+ again in the (Fe2O3)1/TiO2. The released
electrons can be transferred to the surface Ti atoms and thus, the electrons
accumulate to TiO2, leading to charge separation in the system. However, in
the (Fe2O3)2/TiO2 the existence of mixed-valence state of Fe leads to electron
and hole accumulation to Fe2O3 and lowering the charge transfer with TiO2. The
intervalence charge transfer between Fe atoms with different oxidation state
can help to separate the charge in Fe2O3. It should be noted that our results
are in conflict with Moniz et al. [51] who proposed an electron transfer from
TiO2 to Fe2O3 based on their theoretical study on the combined system of TiO2
and Fe2O3 clusters. The difference in the results can result from the use of
different methods.

####  3.3.4 Hydrogen evolution reaction activity

Figure 6: Free energy diagram for hydrogen evolution reaction on the
investigated heterostructures. The adsorption energy of hydrogen were 0.65 eV
in the pristine TiO2 (blue), 0.43 eV and 0.29 eV in the (Fe2O3)1/TiO2 and
(Fe2O3)2/TiO2 (yellow and red), respectively.

After investigating the properties of the heterostructures we evaluate the HER
activity by attaching a hydrogen atom on the heterostructures (figures S3a and
b) and calculating the adsorption energy of hydrogen. An ideal photocatalyst
for the HER should have adsorption energy of hydrogen close to zero. The
adsorption energy in the pristine TiO2 surface was calculated to be 0.65 eV. A
positive value indicates too weak interaction between TiO2 and hydrogen,
making it difficult for hydrogen to adsorb on the surface. In the presence of
the (Fe2O3)1 and (Fe2O3)2 the adsorption energy decreased from 0.65 eV to 0.43
eV and 0.29 eV, respectively (figure 6). Thus, the Fe2O3 clusters facilitate
the adsorption of hydrogen and can enhance the HER activity of TiO2. Moreover,
the larger cluster size enhanced the HER activity more. This was attributed to
the co-existence of Fe3+ and Fe2+ oxidation states for Fe. Interestingly, a
study conducted by Zhang et al. [61] on iron-based metal-organic frameworks
reported similar findings, highlighting the mixed-valence of Fe in promoting
the hydrogen evolution.

###  3.4 Oxygen defect in the (Fe2O3)1/TiO2 heterostructure

Figure 7: The modeled defect sites in the (Fe2O3)1/TiO2 and their formation
energies, $E^{\mathrm{Form}}$. a) Surface vacancy Ov and subsurface vacancies
b) Osv1, c) Osv2, and d) Osv3. The defects e) Oc1 and f) Oc2 are located in
the (Fe2O3)1 cluster. The defect sites are coloured yellow.

The modeled defect sites and their formation energies are shown in figure 7.
Positive formation energies showed that the defect formation is endothermic.
The formation energy of the Oc1 and Oc2 was the lowest, only 1.59 eV,
indicating that the removal of oxygen is easier from the cluster than from the
TiO2 surface. This can be explained by the weaker bonding of oxygen in the
adsorbed cluster than at the TiO2 surface. Considering the relatively high
formation energy of 3.41 eV of the surface vacancy Ov, we decided to introduce
several subsurface vacancy sites. Based on our findings, a subsurface vacancy
is generally more stable in the (Fe2O3)1/TiO2 than a surface vacancy. The Osv3
was found to be the most stable, located underneath the cluster, with a
formation energy of 2.71 eV. The formation energy of the Osv1 and Osv2 were
0.41 eV and 0.79 eV larger than that of Osv3, and their transverse distance,
perpendicular to the $b$-axis, from the cluster were around 11 Å and 4 Å,
respectively. This suggests that the cluster introduces a dependence of the
formation energy on the distance from the cluster in the subsurface layer.
Previously, Hoh et al. [62] have reported a distance dependence of oxygen
vacancy formation energy from the Au-clusters on hematite surface, proposing
that the cluster stabilizes the region underneath and near to it, making it
more stable against reduction. We also observed the cluster to enhance the
oxygen defect formation at the surface. In the pristine TiO2 surface the
formation energies of the Ov, Osv1, Osv2 and Osv3 were 5.47 eV, 5.46 eV, 5.12
eV, 5.46 eV, showing that the presence of the cluster can decrease the
formation energy even by 50%. A similar observation has been reported in
previous studies [63, 64].

Among the modeled defect sites, we chose to further investigate the effect of
the Ov, Osv3, and Oc1 and Oc2 on the properties of the (Fe2O3)1/TiO2
heterostructure. We employed the same methods to investigate these defective
heterostructures as for the defect-free heterostructures.

####  3.4.1 Structural optimization

Figure 8: The optimized structures of the defective heterostructures: a)
(Fe2O3)1/TiO2-Ov, b) (Fe2O3)1/TiO2-Osv3 and c) (Fe2O3)1/TiO2-Oc. The labels
correspond to the same atoms in figure 3a.

The optimized structures of the defective heterostructures are shown in figure
8 while the bond distances between selected atoms are provided in table S3. It
is evident that the defects affected the geometry of the cluster adsorbed on
the surface. In the presence of the Ov and Osv3 the structure of the cluster
became more open. The structural changes led to the breaking of bonds with
TiO2, thereby reducing the coordination number of the (Fe2O3)1 to four in the
(Fe2O3)1/TiO2-Ov and three in the (Fe2O3)1/TiO2-Osv3. Furthermore, the Ov and
Osv3 caused local lattice distortion at the TiO2 surface. The vacancies
shifted the adjacent Ti atoms away from the vacancy site while the adjacent
oxygen atoms moved closer to it. This is consistent with previous observations
[65]. The changes caused by the Ov were primarily limited to the surface layer
whereas the Osv3 led to distortion notably extended to the first sublayer. The
removal of either of the Oc1 or Oc2 resulted in the same geometry with the
same total energy for the optimized systems. Therefore, we simply denoted the
defective heterostructures simply as (Fe2O3)1/TiO2-Oc. The remaining four
atoms of the (Fe2O3)1 were repositioned in a manner that the top of the
cluster became planar at the TiO2 surface, preserving the coordination number
of five.

####  3.4.2 Electronic structure analysis

Figure 9: The density of states of the a) (Fe2O3)1/TiO2-Ov, b)
(Fe2O3)1/TiO2-Osv3 and c) (Fe2O3)1/TiO2-Oc.

Figure 9 shows the calculated DOS of the (Fe2O3)1/TiO2-Ov, (Fe2O3)1/TiO2-Osv3
and (Fe2O3)1 /TiO2-Oc, revealing the electronic structure to experience
changes. The total DOS of TiO2 was further left-shifted due to the oxygen
defects. The VBM and CBM positions were found to be at -0.96 eV and 1.50 eV in
the (Fe2O3)1/TiO2-Ov, -1.8 eV and 0.73 eV in the (Fe2O3)1/TiO2-Osv3, and -0.93
eV and 1.71 eV in the (Fe2O3)1/TiO2-Oc, respectively. Despite the left shift,
the band gap of TiO2 remained approximately 2.5 eV. However, the band gap of
the heterostructure was altered by the oxygen defects. The (Fe2O3)1/TiO2-Osv3
showed an extremely narrow band gap of 0.75 eV, decreasing the energy
difference between the VB and CB. Inversely, the Ov and Oc resulted in an
increase in the band gap energy compared to the defect-free (Fe2O3)1/TiO2. We
obtained a band gap of 1.45 eV for the (Fe2O3)1/TiO2-Ov and 1.80 eV for the
(Fe2O3)1/TiO2-Oc. The former remains sensitive for all visible light
wavelengths while the latter is capable of absorbing visible light wavelengths
up to the shortest red light wavelengths. Importantly, all the three defective
heterostructures possess a narrower band gap than the pristine TiO2 surface.

####  3.4.3 Charge density analysis

As previously, we investigated the changes in the charge transfer properties
via the CDD and Bader analysis, supported by the SDD. The Bader charges and
magnetic moments are listed in table S4 for selected atoms. The CDD plots,
shown in figure 10a, b and c, showed that the main charge redistribution also
occurred in the cluster and at the interface in the defective
heterostructures. When a neutral oxygen defect is formed it releases effective
excess charge which can generally be transferred either to Fe or Ti. The Bader
analysis confirmed the charge gain of Fe, with a yellow isosurface observed
around Fe atoms in the CDD plots. The calculated Bader charge of the Fe1 and
Fe2 atoms were 1.52 e and 1.30 e in the (Fe2O3)1/TiO2-Ov, 1.26 e and 1.19 e in
the (Fe2O3)1/TiO2-Osv3, and 1.29 e and 1.32 e in the (Fe2O3)1/TiO2-Oc,
respectively. The corresponding magnetic moment of the Fe1 and Fe2 were 4.02
$\mathrm{\mu_{B}}$ and 3.77 $\mathrm{\mu_{B}}$ in the (Fe2O3)1/TiO2-Ov, 3.26
$\mathrm{\mu_{B}}$ and -3.57 $\mathrm{\mu_{B}}$ in the (Fe2O3)1/TiO2-Osv3, and
3.67 $\mathrm{\mu_{B}}$ and 3.70 $\mathrm{\mu_{B}}$ in the (Fe2O3)1/TiO2-Oc,
respectively. A lower magnetic moment supports a charge gain for Fe. To
confirm lower spin magnetization of some Fe atoms, we calculated the SDD which
showed larger probability for spin down density (turquoise isosurface) around
Fe (figure 10d, e, and f). In the (Fe2O3)1/TiO2-Ov the total magnetic moment
was still $9.66\ \mathrm{\mu_{B}}$ per unit cell whereas in the
(Fe2O3)1/TiO2-Oc the total magnetic moment decreased to $7.79\
\mathrm{\mu_{B}}$. Surprisingly, the Fe2 atom in the (Fe2O3)1/TiO2-Osv3
experienced a magnetic phase transition. This resulted in a total magnetic
moment of 0.09 $\mathrm{\mu_{B}}$. Due to the anti-parallel but unequal
magnetic moment of Fe atoms, the (Fe2O3)1/TiO2-Osv3 exhibits weak
ferrimagnetic behaviour. The magnetic phase transition was attributed to be to
the notable lattice distortion at the TiO2 surface caused by the Osv3. For
instance, in their study, Men$\mathrm{\acute{e}}$ndez et al. [66] have
reported that the lattice distortion caused by oxygen defect can correlate
with magnetic behavior and induce a magnetic phase transition.

Figure 10: The CDD (a, b, and c) and SDD (d, e, and f) of the
(Fe2O3)1/TiO2-Ov, (Fe2O3)1/TiO2-Osv3 and (Fe2O3)1/TiO2-Oc.

Our findings suggest that Fe can have the mixed valence state (Fe2+/Fe3+) in
the (Fe2O3)1/TiO2-Ov [58, 59, 60]. We also observed localized spin density
between the Fe atoms (figure 10d). This can indicate magnetic coupling between
the cations with different oxidation states, supporting the mixed-valence
state for Fe. In the (Fe2O3)1/TiO2-Osv3 and (Fe2O3)1/TiO2-Oc the results
indicates the Fe2+ oxidation state for Fe [58, 59, 60]. This was supported by
the lower Bader charges and even contribution of Fe atoms to the spin
magnetization. Additionally, in the (Fe2O3)1/TiO2-Ov we found the Bader charge
of the Ti3 to change notably. It decreased from 2.04 e to 1.79 e,
corresponding to Ti3+ species [58]. This is consistent with the findings from
the DOS.

According to the Bader analysis the direction of the charge transfer remained
from the cluster to the surface in the (Fe2O3)1/TiO2-Oc. 0.45 e of charge per
unit cell was transferred to the surface, of which approximatley 90% e was
evenly distributed among Ti and O atoms of the surface layer. The charge
transfer was almost 1.5 times larger than in the (Fe2O3)1/TiO2, showing an
increase in the conductivity of TiO2. This was attributed to the formation of
a type ii@ heterojunction. Since no Fe2O3-states exist below the CB of TiO2
electrons from the VB of TiO2 and Fe2O3 can be excited directly to the CB of
TiO2 without trapping. Electron excitation to the unoccupied Fe2O3-states is
also possible but due to the type ii@ heterojunction electrons can be
transferred to the CB of TiO2. On the contrary, we observed that the direction
of the charge transfer reverse in the presence of the v and Osv3. 0.12 e and
0.62 e per unit cell were transferred from the surface to the cluster in the
(Fe2O3)1/TiO2-Ov and (Fe2O3)1/TiO2-Osv3, respectively. A larger charge
transfer indicates stronger interaction between TiO2 and Fe2O3 in the
(Fe2O3)1/TiO2-Osv3, which is consistent with the lower formation energy of the
Osv3 compared to that of Ov. For the (Fe2O3)1/TiO2-Osv3 the Bader analysis can
support the Z-scheme mechanism, leading to electron accumulation to Fe2O3. The
gap between the highest VB states and lowest CB states of Fe2O3 was
approximately 1.5 eV, enabling electron excitation using visible light.

The oxygen defects further decreased the work function of the (Fe2O3)1/TiO2. A
neutral oxygen defect is a positively charged which makes negatively charged
surface more positive, that is, decrease the work function. In the presence of
the Ov, Osv3 and Oc we found the work function to be 5.27 eV, 5.28 eV and 5.41
eV, respectively. This suggests that an oxygen defect can facilitate electron
migration from the CB to the surface.

####  3.4.4 Effect on hydrogen evolution

The results show that oxygen defects enable the engineering of electronic
properties of the (Fe2O3)1/ TiO2. Finally, we evaluate the HER activity of the
defective heterostructures. H-containing structures are shown in figure S5.
Based on the results, oxygen defects have a detrimental effect on the HER
activity of the (Fe2O3)1/TiO2. The Ov, Osv3 and Oc increased the adsorption
energy of hydrogen from 0.43 eV to 0.71 eV, 0.61 eV and 0.60 eV, respectively.
Thus, the interaction with hydrogen becomes weaker again, indicating decrease
in HER activity. The HER activity of the defective heterostructures is
comparable with the pristine TiO2 surface.

##  4 Conclusions

First principles DFT calculations were employed to investigate the
heterostructure of the TiO2 surface and (Fe2O3)n ($n=1,2$) clusters, and
superior performance was found for the heterostructures. The adsorption of the
clusters on the surface was exothermic, thus enabling the modification of the
surface properties of TiO2. The formation of a heterojunction between TiO2 and
Fe2O3 was identified to affect band gap energy, charge transfer and charge
separation in the heterostructures, leading to enhanced photocatalytic
properties. The clusters were observed to improve the HER activity of TiO2.
The existence of mixed-valence Fe (Fe2+/Fe3+) led to greater enhancement in
the HER activity. We also incorporated oxygen defects on the (Fe2O3)1/TiO2,
and the defect formation was found to be endothermic. The presence of the
(Fe2O3)1 cluster substantially enhanced the oxygen defect formation at TiO2
compared to the pristine surface. We found that electronic structure and
charge transfer and the local magnetism can be tuned by oxygen defects, which
can indirectly affect the oxidation state of Fe. However, it was observed that
an oxygen defect was not beneficial in order to improve the HER activity.

In summary, the results of our work reveal three important points. The first
point is that +2 and +3 valence-state mixing offers a lot of potential to
control the electronic structure of hematite clusters. The second point is
that further electronic structure optimization could be achieved by
engineering these clusters with various oxygen defects. The third point is
that the presence of Fe2+-Fe3+ intervalence charge transfer implies charge
transfer between these non-equivalent Fe sites and could result in broad
absorption in the visible or IR region of the electromagnetic spectrum.

## Acknowledgement

This work is supported by the Finnish cultural foundation (Grant No. 00230235)
and the Kvantum Institute. The authors acknowledge the CSC - IT Center for
Science, Finland, for computational resources.

## Declaration of competing interest

The authors declare no competing financial interests.

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