Relation between structural patterns and magnetism in small iron oxide clusters: reentrance of the magnetic moment at high oxidation ratios

Abstract

Due to quantum confinement effects, the understanding of iron oxide nanoparticles is a challenge that opens the possibility of designing nanomaterials with new capacities. In this work, we report a theoretical density functional theory study of the structural, electronic, and magnetic properties of neutral and charged iron oxide clusters Fe_nO_m^0/± (n = 1–6), with m values until oxygen saturation is achieved. We determine the putative ground state configuration and low-energy structural and spin isomers. Based on the total energy differences between the obtained global minimum structure of the parent clusters and their possible fragments, we explore the fragmentation channels for cationic oxides, comparing with experiments. Our results provide fundamental insight on how the structural pattern develops upon oxidation and its connection with the magnetic couplings and net total moment. Upon addition of oxygen, electronic charge transfer from iron to oxygen is found which weakens the iron–iron bond and consequently the direct exchange coupling in Fe. The binding energy increases as the oxygen ratio increases, rising faster at low oxidation rates. When molecular oxygen adsorption starts to take place, the binding energy increases more slowly. The oxygen environment is a crucial factor related to the stabilities and to the magnetic character of iron oxides. We identified certain iron oxide clusters of special relevance in the context of magnetism due to their high stability, expected abundance and parallel magnetic couplings that cause large total magnetic moments even at high oxidation ratios.

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1 Introduction

Iron oxides are very common and abundant in natural conditions. They are also formed as corrosion products and can be easily synthesized. The unavoidable oxidation in certain environmental conditions facilitates their production. Besides, they are relatively easy to prepare with low cost. Designing nanoparticles that retain a net magnetic moment in realistic conditions is challenging. In addition, iron oxide nanoparticles are of great interest from both the fundamental and technological points of view.¹ Since the properties of a material can be drastically modified at the nanoscale due to quantum confinement effects, with non-monotonous dependencies on the cluster size, composition, and number of electrons, understanding the physics and chemistry of iron oxide nanoparticles is nowadays a great challenge, but at the same time opens the possibility of finding and designing systems with new capabilities. The intrinsic magnetic properties and biocompatibility of iron oxide clusters make them one of the most suitable candidates for nanomedicine. Those smaller than 30 nm are superparamagnetic at room temperature and can be used for drug delivery,^2,3 for cancer therapy through magnetic hyperthermia,⁴ as contrast agents for Magnetic Resonance Imaging (MRI),⁵ and in the emerging technique of Magnetic Particle Imaging (MPI).⁶ Recent work⁷ suggests that prostate tumours with high nerve densities are more likely to grow and spread than those with low nerve densities. Such high-risk cases can be identified using a combination of MRI, magnetic particle imaging (MPI) and functionalized iron-oxide nanoparticles. In experiments with mice, researchers also used the same nanoparticles to deliver a drug that blocks nerve function, with successful results. You et al.⁷ developed a contrast agent that targets nervous tissue specifically. The team started with iron oxide nanoparticles, which have already found use in both MRI and MPI, and joined them to the nerve-binding peptide NP41. It is, therefore, not surprising that iron oxide clusters or nanoparticles have been the subject of numerous experimental and theoretical studies during the last few years, a representative number of which are summarized in the following part of the introduction (we apologize if our search was not complete).

Iron oxides FeO_m⁻ and Fe₂O_m⁻ (m = 1–4 and m = 1–5, respectively) have been studied by anion photoelectron spectroscopy⁸ at 3.49 and 4.66 eV photon energies. The vibrationally resolved photoelectron spectra and low-lying excited states were obtained. The photoelectron spectra were better resolved for those clusters with a big ratio of oxygen atoms. The results indicated that the electron affinity of the neutral ones increases with the number of oxygen atoms, suggesting a sequential oxidation behavior. Photoelectron spectra of small anionic iron clusters Fe_nO_m⁻ (n = 1–4, m = 1–6) have been obtained by Wang et al.⁹ They concluded that the oxidation can be viewed as sequential oxygen atom adsorption at the surfaces of the Fe₃ and Fe₄ clusters, leading to a nearly linear increase of the electron affinity with the number of oxygen atoms.

A guided ion beam was used for the study of the reaction of cationic Fe_n⁺ clusters (n = 2–18) with CO₂¹⁰ and O₂¹¹ in order to determine the bond energies of those iron clusters with oxide and dioxide; with the former, the bond energies are between 3.0 and 6.5 eV, whilst for the second one the bond energies are between 3.6 and 5.15 eV. Cationic iron oxide clusters, Fe_nO_m⁺, were also produced¹² by chemical ionization of Fe(CO)₅/O₂ mixtures. They exhibit remarkable fragmentation trends that can be attributed to the formal oxidation states of iron. For clusters with n/m ≥ 1, the loss of atomic oxygen and FeO units is preferred. For Fe_nO_m⁺ with 1 > n/m > 2/3, in addition to the loss of O and FeO, loss of neutral FeO₂ was also observed. Finally, for rich oxygen clusters (n/m = 2/3), loss of molecular oxygen predominates. Photodissociation of cationic Fe_nO_m⁺ (n = 1–15) clusters was studied by Molek et al.¹³ Clusters were produced by laser vaporization in a pulsed nozzle (355 nm) cluster source and detected with Time-Of-Flight (TOF) mass spectrometry. The results indicated that dissociation occurs mainly in two ways; the first one is the loss of molecular oxygen, and the second is a fission process. For n ≤ 5, the oxygen elimination process takes place until m = n. Then, no more oxygen loss was observed; instead a fission process takes place losing smaller clusters. Iron oxide clusters Fe_nO_m⁺ (n = 1–3, m = 1–6) have been synthesized in a laser vaporization source and dissociated via CID by Li¹⁴et al. Examining the dissociation behavior in a wide range of energies, these authors showed that the clusters can be dissociated by evaporation of the Fe and O atoms, as well as the fission of the neutral O₂, FeO, FeO₂, Fe₂O₂ and Fe₂O₃ fragments. In general, they found that the predominant dissociation pathways correlate with the oxidation state of iron in the cluster.

Regarding theoretical studies, and small molecules, the Fe₂ dimer was investigated with more accurate wave function methods. However, there is no consensus as regards the ground state magnetic configuration. Tomonari et al.¹⁵ and Noro et al.¹⁶ used a multireference (MR) configuration interaction (CI) method with certain dynamical correlations. Hubner et al.¹⁷ used IC-MRCI starting from a CASSCD wave function. C. W. Bauschlicher et al.¹⁸ used also CI-MRCI and they pointed out that the septet state should be the true ground state in order to fit the experiment. Moreover, Bauschlicher et al.¹⁹ used the complete-active space SCF/internally contracted averaged coupled pair functional approach (CASSCF/ICACPF) to study the diatomic molecule FeO, among other small molecules, obtaining a quintet state and an interatomic distance of 1.61 Å, in good experimental agreement. In previous work, Bauschlicher et al.²⁰ performed, for small molecules, an exhaustive study of dipole moments and other properties, also with high level theoretical approaches. The CAS11 FeO interatomic distance was slightly larger than that calculated at the MRCI11 level. The dipole moments were calculated, as well as the electron affinities of Fe, with DFT and CCSD(T), those calculated with CCSD(T) being more accurate.

Shiroishi et al.^21,22 performed Density Functional Theory (DFT) calculations of Fe_nO_m (n = 1–3) and Fe_nO_m⁻ with (n = 3, 4) using first principles molecular dynamics within the Generalized Gradient Approximation (GGA) for the exchange correlation (xc) energy with the PW91 functional. For all studied clusters, bridge positions were preferred. The magnetic couplings change from parallel to antiparallel in m = n, except for Fe₄O_m⁻, in which the change takes place at m = 3. Vertical detachment energies were calculated and compared with experimental results.⁹ The structural and magnetic properties of (Fe₂O₃)_n (n = 1–5) clusters were investigated by Erlebach et al.²³ All DFT calculations were performed using the TURBOMOLE program package along with the B3LYP xc-functional. The Multipole Accelerated Resolution of the Identity (MARI-J) method for the Coulomb term employing a triple-zeta valence plus polarization (TZVP) basis set for all atoms was used. The results found good agreement with experimental collision cross sections. A theoretical study of (FeO)_n^0/± clusters (n = 1–8) was performed by Ju et al.²⁴ using the GAUSSIAN 09 package in the GGA approximation with the PW91 functional and 6-311+G* basis set. The results showed that (FeO)₄⁻ and (FeO)₄⁺ clusters have the largest HOMO–LUMO gap values, an indication of their high stability.

Reilly et al.^25,26 carried out a combined experimental and theoretical study of the structures and reactivity of Fe_nO_m⁺ and Fe_nO_m⁻ (n = 1–2 and m = 3–5, 6), respectively, with CO. Clusters were produced by laser vaporization and were characterized by employing a guided ion beam mass spectrometer. Moreover, energy-resolved Collision-Induced Dissociation (CID) experiments were conducted in order to study the fragmentation patterns of those clusters. Theoretical calculations were performed within the DFT-GGA via PBE functional, as implemented in the code deMon2k, using the DZVP basis set for C and O, the Wachters-F basis set for Fe, the GEN-A2 auxiliary function set for C and O, and the GEN-A2* function set for Fe. Dissociation energies and vertical detachment energies were calculated, with which our results will be compared. Another combined experimental and theoretical study was carried out recently for cationic Fe_nO_m⁺ (n = 2–6) clusters by Koyama et al.²⁷ They performed a theoretical (n = 4, m = 1–8) and experimental (n = 2–6) study of the dissociation energy for O₂ release. Clusters were formed by laser ablation in a source and selected using mass spectrometry in combination with the post-heating method. A clear relation between the temperature and the intensity ratios of Fe₂O_m⁺ was found: as the temperature increases, the abundance of the oxygen-rich clusters decreases. Moreover, as the relative abundance of the Fe₂O₆⁺ clusters decreases, the abundance of Fe₂O₄⁺ clusters increases, until approximately 550 K; after that, the relative abundance of these clusters decreases, whilst the abundance of Fe₂O₂⁺ ones increases to the same extent that Fe₂O₆⁺ decreases. Consequently, it was deduced that oxygen is released molecularly in the following form: Fe₂O₆⁺ → Fe₂O₄⁺ + O₂ → Fe₂O₂⁺ + 2(O₂). To complete the experimental work, theoretical calculations were performed using the GAUSSIAN 09 program with the B3LYP functional and 6-311+G* basis set.

In a more general context, Wang et al.²⁸ investigated the structural and magnetic properties of late transition metal oxide clusters TM_nO_m (TM = Fe, Co, Ni, n = 1–2, m = 1–6) by using also DFT with the PBE functional combined with a double numerical basis set including d-polarization functions (DND) for O atoms and DFT-based relativistic semicore pseudopotential (DSPP) for TM atoms, implemented in the DMOL package. Fe_n adopts three-dimensional structures while Co_n and Ni_n clusters form planar geometries. The binding energies per atom increase with the increase of O atoms for both n = 1–2, reaching a peak at m = n. Datta et al.²⁹ carried out DFT-VASP calculations of M₄O₄ and M₄S₄ clusters (M = Mn, Fe, Co, Ni, Cu) using the pseudopotential plane wave method with the GGA to the XC energy functional as formulated by PBE. The wave functions were expanded with a plane wave basis set. For Fe₄O₄ oxide, a ring-like structure was obtained with antiparallel couplings, and a total magnetic moment of 0 μ_B. The high spin state with 8 μ_B was found with an energy difference of 75 meV. The cube structure is much less stable that the ring-like one. Recently, Wang et al.³⁰ presented a first principles study of the spin properties of triplet TM₃O₃ (TM = Fe, Co, Ni) clusters and their laser-induced ultrafast spin dynamics. The clusters were optimized at the Hartree–Fock (HF) level with the Lanl2DZ basis set using the Gaussian 16 package. These results provided additional information relevant in the context of ultrafast optical control of magnetism in transition-metal oxide systems.

In the work reported here, we systematically studied Fe_nO_m^0/± oxide clusters with n = 1–6 and m values until oxygen saturation is reached. We carried out DFT calculations within the generalized gradient approximation (GGA) for exchange and correlation. We focused on the structural and electronic properties, with a special emphasis on magnetism, and charge effects. For cations, the fragmentation channels were calculated and compared with previous experimental results of Molek et al.¹³ and Li et al.¹⁴ We found some nanoparticles that retain a net magnetic moment despite having a high oxidation ratio. The paper is organized as follows. In Section 2, we describe the theoretical and computational approaches and compare some results about Fe₂^0/± and Fe₂O_m^0/± (m = 1–2) with previous ones available in the literature, which allow us to benchmark our theoretical approach and to have a good starting point. In Section 3, we show the results, which are discussed in different subsections: Section 3.1 is devoted to ground state structures and their nearest structural isomers depending on size n, on oxidation ratio m, and on the charge state. Structural properties and absolute and relative stabilities are also discussed. In Section 3.2, fragmentation channels are studied for the cationic clusters by comparing the minimum energy needed for the separation in different possible fragments. In Section 3.3 we discuss the electronic properties as well as the magnetism of Fe_nO_m^0/±. Finally, the conclusions are summarized in Section 4.

2 Theoretical approach and computational details

We performed fully self-consistent DFT calculations using the SIESTA code.³¹ For the exchange and correlation (xc) potential we used the Perdew–Burke–Ernzerhof form of the generalized gradient approximation (GGA).³² We employed norm-conserving scalar relativistic pseudopotentials³³ in their fully nonlocal form,³⁴ generated from the atomic valence configuration 3d⁷4s¹ for Fe (with core radii 2.00 a.u. for s, p and d orbitals), and 2s²2p⁴ for O (with core radii 1.14 a.u. for s, p and d orbitals). Non-linear partial core corrections,³⁵ which are known to be important for transition metal pseudopotentials, were included for Fe at core radius 0.7 Å. Valence states were described using double-ζ basis sets for O and Fe with maximum cutoff radii 4.93 Å (2p) and 8.10 Å (3d, 4s), respectively. A 4p polarization orbital was also considered for Fe, with cutoff radius 8.10 Å.

The energy cutoff used to define the real-space grid for numerical calculations involving the electron density was 250 Ry. The Fermi distribution function that enters in the calculation of the density matrix was smoothed with an electronic temperature of 15 meV. We used an energy criterion of 10⁻⁴ eV for converging the electronic part.

In the calculations, the individual clusters were placed in a cubic supercell of 20 × 20 × 20 ų, a size large enough to neglect the interaction between the cluster and its replicas in neighboring cells. Only the Γ point (k = 0) was considered when integrating over the Brillouin zone, as usual for finite systems. The equilibrium geometries resulted from an unconstrained conjugate-gradient structural relaxation using the DFT forces. The initial geometries were built by considering different arrangements of the Fe and O atoms without privileging those formed from given Fe subclusters. Thus, an exhaustive sampling of possible geometries was tested, including those in which the possibly strong Fe–O bonding prevents the nucleation of compact Fe subclusters. In addition, we tested other geometries that were built using local Fukui functions to locate O atoms in the more nucleophilic sites of the host. Fukui functions allow one to determine the most reactive sites according to purely electronic arguments. The Fukui functions^36–38f⁺ and f⁻ are defined as

where ρ(r⃑) is the spatial charge density, and N_e the number of electrons, and the subscript indicates that the right/left derivatives have to be calculated at constant external potential, i.e. by keeping the atomic coordinates fixed. The scalar fields f^± measure the local variations in electronic charge induced by the addition or removal of electrons, and so they can be used as useful local indices of electronic reactivity. f⁺ refers to the electron density response upon addition of electrons, and so it is an indicator of locally electrophilic regions which are more susceptible to nucleophilic attack; similarly, f⁻ locates the most nucleophilic regions within the system, susceptible to electrophilic attack. Larger positive values of f^± correspond to more reactive sites. Coupling the Fukui function with Bader analysis, we can define atom-resolved condensed Fukui functions f_i^± for each atom i, by calculating the variation in electronic charge inside each of the Bader atomic basins. Following the standard practice, we have approximated the derivatives by simple finite differences:

f_Ne⁺(r⃑) = ρ_Ne+1(r⃑) − ρ_Ne(r⃑)

f_Ne⁻(r⃑) = ρ_Ne(r⃑) − ρ_Ne−1(r⃑)

The global minimum and low-energy isomers found for the neutral oxides are used as inputs for their charged counterparts. The ground state geometry of charged clusters is not always the obtained global minimum structure of the neutral ones, as it will be seen in the next section. Charged systems can be dealt with through the addition of a Madelung correction. Although the finite difference expressions provide the exact value for the derivative according to DFT,³⁸ in practical calculations the expressions are not exact due to the self-interaction error of approximate exchange–correlation functionals. Nevertheless, the standard usage of those finite difference approximations is justified by the fact that approximate xc-functionals are much more accurate for integer numbers than for fractional numbers of electrons.

The structures were relaxed without any symmetry constraints until the interatomic forces were smaller than 0.003 eV Å⁻¹. All possible spin multiplicities and different initial parallel or antiparallel arrangements for each different structural geometry with different oxygen environments have been considered in order to be sure of the putative global minimum. In the search for spin isomers, the criterion for the maximum interatomic forces was further reduced to 0.001 eV Å⁻¹.

In order to understand the magnetic couplings of the iron oxides it is necessary to consider first the magnetic coupling of Fe₂^0/±, FeO^0/± dimers, and even of FeO_m^0/±, since they are the basic units of the Fe_nO_m^0/± clusters, as it will be seen below, and compare our theoretical approach with other DFT results available in the literature (see Tables 1 and 2). The Fe₂ dimer has been theoretically investigated with more accurate wave function methods, and experimental data is also available. There is, however, no consensus as regards the ground state magnetic configuration. Tomonari et al.¹⁵ and Noro et al.¹⁶ obtained a ground state with a total spin moment of 6 μ_B (septet state) using a multireference (MR) configuration interaction (CI) method with certain dynamical correlations. Hubner et al.¹⁷ obtained a total spin moment of 6 μ_B (nonet state) using IC-MRCI starting from a CASSCD wave function, but their interatomic distance (2.19 Å) overestimates the experimental data [1.87 ± 0.13 Å] measured by Montano et al.,³⁹ or [2.02 ± 0.02 Å] measured by Purdum et al.⁴⁰ C. W. Bauschlicher et al.,¹⁸ using also a CI-MRCI method, confirmed the experimental results,¹⁷ but they pointed out that in order to fit the experiment, the septet state should be the true ground state, a fact that was later confirmed by Casula et al.⁴¹ through quantum Monte Carlo calculations. Therefore, no consensus exists in regard to the spin state of the ground state, but it seems that the septet state fits better the experimental interatomic distance. We have calculated the Fe₂ dimer with our DFT setup with the PBE functional in order to compare with the aforementioned calculations. We have obtained the septet state as the ground state with an interatomic distance of 2.04 Å (in rather good agreement with the experimental data) and binding energy (1.56 eV) of the same order as values predicted with the more accurate methods. The nonet state is a metastable state with an interatomic distance of 2.23 Å, which overestimates the experimental value, as is the case with more accurate methods. Therefore, our tests for the Fe₂ dimer are of good quality as compared with more accurate wave function methods and with the experimental data. For Fe₂⁺ and Fe₂⁻ dimers, the magnetic moment is 7 μ_B and the interatomic distances are 2.18 Å and 2.11 Å, respectively, in agreement with Reilly et al.²⁵ In general, when antiparallel magnetic couplings appear in Fe systems, it means that Fe atoms have somehow lost their identity (either they separate from each other, or they lose electronic charge and electronically approach Mn). Moreover, the preferred magnetic Fe–O coupling is also parallel. Bauschlicher and Maitre¹⁹ used the complete-active space SCF/internally contracted averaged coupled pair functional approach (CASSCF/ICACPF) to study the diatomic molecule FeO, among other small molecules, obtaining a quintet state (corresponding to a spin moment of 4 μ_B) and an interatomic distance of 1.61 Å, quite similar to the experimental one. In previous work, Bauschlicher et al.²⁰ performed an exhaustive study of dipole moments and other relevant quantities of small molecules, also with high level theoretical approaches. The CAS11 FeO interatomic distance was 1.69 Å, slightly larger than that calculated at the MRCI11 level (1.63 Å). Our DFT-PBE results are in good agreement; we obtain the same spin state as Bauschlicher and Maitre and an interatomic distance of 1.67 Å. The calculated transferred electronic charge from iron to oxygen is 0.46 e and the local magnetic moments of iron and oxygen are, respectively, 3.4 and 0.6 μ_B, giving a total moment of 4 μ_B. Wang et al.²⁸ also found a quintet state for FeO oxide with a bond length of 1.61 Å. Our results also agree with available data from other authors regarding interatomic distances and total magnetic moments.^21,26 For FeO⁺ (FeO⁻) a total magnetic moment of 5 μ_B (3 μ_B) is found, in agreement with previous results.^25,26

Table 1 Total magnetic moments (in μ_B) of the ground states of Fe₂ and FeO_m (m = 1–2) obtained in this work. Agreement with previous data from DFT, higher level theory and experiments is shown

	Cation		Neutral		Anion
	μ	d	μ	d	μ	d
a 1.87 ± 0.13 Å (experimental value^39), 2.02 ± 0.02 Å (experimental value^40). b 1.61 Å (CASSSCF/ICACPF value^19), 1.69 Å (CAS11 value^40), 1.63 Å (MRCI11 value^40).
Fe2	7^25	2.18	6^15,16,18,41	2.04^a	7^25	2.11
FeO	5^24,25	1.71	4^19,21,24,26,28	1.67^b	3^24,26	1.69
FeO2	1^25	1.65	2^26,28	1.62	3^26	1.71

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Table 2 Dissociation energies of O₂, and FeO_m^0/+ (m = 1–2). A comparison with previous experimental and theoretical results is included

Reaction	Experimental	Ref. 25	This work
O2 → O + O	5.115^43	6.20	5.08
FeO → Fe + O	4.7 ± 0.2^44	5.51	5.49
FeO2 → Fe + O2	3.60 ± 0.20^45	4.36	4.44
FeO2 → Fe + O + O	8.64 ± 0.22^46	10.56	9.52
FeO^+ → Fe^+ + O	3.53 ± 0.06^47	4.57	4.54
FeO2^+ → Fe^+ + O2	2.0 ± 0.5^48	1.98	1.75

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FeO₂ is a triplet isosceles triangle as in previous results,^26,28 with a magnetic moment of 2 μ_B, in good agreement with earlier infrared absorption measurements combined with theory.⁴² For the FeO₂⁺ cation, the total moment is probably more ambiguous. The theoretical research of Schroder et al.¹² predicted a high spin trimer with a total moment of 5 μ_B, and FeO distances of 1.63 Å, and a spin isomer with 3 μ_B, at only 0.04 eV in energy. The doublet isomer was found at 3 eV higher in energy. However, Reilly et al.²⁵ found a doublet ground state. Our low spin result for FeO₂⁺ is in agreement with Reilly et al.²⁵ Notice that Schroder et al.¹² found that FeO₂⁺ with an oxygen molecule bonded was at 0.22 eV in energy, explaining the low energy process of loss of molecular O₂ from FeO₂. For FeO₂⁻ our results agree with previous ones,²⁶ giving a magnetic moment of 3 μ_B. Moreover, our results about the dissociation energies of FeO_m^0/+ (m = 1–2) and O₂, shown in Table 2, are also in good agreement with previous results. Previous results available in the literature show the antiparallel arrangement of FeO_m^0/± (m = 3, 4) upon addition of more oxygen. The neutral FeO₃ cluster has a low spin state with 0 μ_B and, consequently, the total magnetic moment decreases by 2 μ_B upon addition of each O atom for neutral FeO_m (m = 1–3) clusters.²⁶ This happens because the Fe atom transfers two electrons to O. FeO_m⁺ (m = 3–4) have both low spin states with 1 μ_B.²⁵ FeO_m⁻ anions keep a magnetic moment of 3 μ_B up to m = 3, and the value does not change to 1 μ_B until the Fe atom is coordinated with four oxygens.²⁶

Therefore, for Fe₂^0/±, the parallel couplings are maintained regardless of the charge state. However, when the magnetic iron atom is oxidized, the results about the magnetic character change and depend on the charge. For FeO^0/±, the magnetic moment decreases by 1 μ_B from the cationic to the anionic state. When the oxidation ratio increases, for FeO₂^0/±, the magnetic moment increases by 1 μ_B from the cation to the anion. Moreover, although FeO₃ does not present magnetism, FeO₃⁻ still preserves a magnetic moment of 3 μ_B like FeO₂⁻ and FeO⁻. Consequently, for low oxidation ratios of the Fe atom, the magnetic results depend strongly on the charge. Extrapolating our and previous results about Fe₂^0/± and FeO_m^0/± (m = 1–4) to bigger iron oxide clusters such as those investigated in the present work, some trends are expected as the oxidation ratio increases: firstly, a decrease in the total magnetic moment upon oxygen addition and, secondly, a significant effect of charge on the magnetism. Once good agreement is obtained with previous studies of the basic units of the iron oxides, we are confident in the results that we present and discuss in the next section for Fe_nO_m^0/± with n = 2–6.

3 Results and discussion

In this section, firstly, we discuss the main structural features and stability of the optimized iron oxides Fe_nO_m^0/± (n = 2–6) with different oxidation ratios; secondly, we present the fragmentation paths that we will compare with experimental results. These structural features and fragmentation paths are connected with the electronic and magnetic properties that will be discussed in the third subsection.

The relative strength of the iron–iron and iron-oxygen bonds is an important factor related to the stability and structure of iron oxide nanoparticles. Electronic charge transfer occurs from the iron to the oxygen atoms, which weakens the iron–iron bonding. Moreover, with respect to iron clusters that have larger spin-polarization per atom than the iron bulk, a decrease of the tendency to parallel magnetic couplings is expected for the iron oxide clusters. Due to these interrelated factors and to the non-scalability of the magnetic properties with size in nanoparticles, it is necessary to perform calculations for each size and composition to obtain reliable structural, electronic and magnetic properties as well as to understand the experimental fragmentation paths.

3.1 Geometrical configurations and electronic properties

In the following, iron oxide clusters Fe_nO_m^0/± with n = 2–6 will be denoted by (n, m)^0/±. The global minimum structures and several low-lying energy configurations are depicted in Fig. 1–5 with the n.m-Label signature, Label being a roman number, to distinguish the different geometrical isomers with n iron and m oxygen atoms. The signature n.m-I always corresponds to the global minimum of the neutral oxide. The ground state structure of the cationic and anionic oxides, when it is not the same, corresponds to one of the two lowest-energy isomers of the neutral. Below the structures of each (n, m) composition, the ground state geometry for the cation, neutral and anion oxides is indicated, respectively, by its corresponding label. The numbers in parentheses correspond to the spin multiplicity, M, for each state of charge, which will be related to the structural arrangements. The energy differences between the different geometric isomers are shown in Tables I–V of the ESI.‡ Moreover, ionization potentials and electron affinities are also gathered. Fig. 6 shows the interatomic distance of the different bonds (Fe–Fe and Fe–O) as a function of the oxidation ratio, m, for each n value, for the ground state of both neutral and charged oxides, using different colours.

Fig. 1 Putative ground state and first two low-energy isomers of Fe₂O_m neutral clusters with m = 1–6. The notation is 2.m-Label, with Label in roman letters in decreasing order of stability for each (2,m). The ground state of charged oxides, when it is not the same, corresponds to one of the two lowest-energy isomers. Below the structures of each (2,m) composition, the ground state geometry for the cation, neutral and anion oxides is indicated, respectively, by its corresponding label. The numbers in parentheses correspond to the spin multiplicity for each state of charge. We note that for the cation with m = 2 the Fe dimer is larger than those for the neutral and anion, and for anions with m = 1, 3 the Fe dimer is shorter (see Fig. 6). The energy differences between the different geometrical isomers, for each one of the charge states, are given in Table I of the ESI.‡

Fig. 2 Putative ground state and first two low-energy isomers of Fe₃O_m neutral clusters with m = 1–9. The notation is 3.m-Label, with Label in roman letters in decreasing order of stability for each (3,m). The ground state of charged oxides, when it is not the same, corresponds to one of the two lowest-energy isomers. Below the structures of each (3,m) composition, the ground state geometry for the cation, neutral and anion oxides is indicated, respectively, by its corresponding label. The numbers in parentheses correspond to the spin multiplicity for each state of charge. We note that for the cation with m = 2 the Fe subcluster is more open than those for the neutral and anion (see Fig. 6). The energy differences between the different geometrical isomers, for each one of the charge states, are given in Table II of the ESI.‡

Fig. 3 Putative ground state and first two low-energy isomers of Fe₄O_m neutral clusters with m = 1–14. The notation is 4.m-Label, with Label in roman letters in decreasing order of stability for each (4,m). The ground state of charged oxides, when it is not the same, corresponds to one of the two lowest-energy isomers. Below the structures of each (4,m) composition, the ground state geometry for the cation, neutral and anion oxides is indicated, respectively, by its corresponding label. The numbers in parentheses correspond to the spin multiplicity for each state of charge. We note that for the cation with m = 4, 6 the Fe subcluster is more open (nearly broken) than those of the neutral and anion (see Fig. 6). The energy differences between the different geometrical isomers, for each one of the charge states, are given in Table III of the ESI.‡

Fig. 4 Putative ground state and first two low-energy isomers of Fe₅O_m neutral clusters with m = 1–15. The notation is 5.m-Label, with Label in roman letters in decreasing order of stability for each (5,m). The ground state of charged oxides, when it is not the same, corresponds to one of the two lowest-energy isomers. Below the structures of each (5,m) composition, the ground state geometry for the cation, neutral and anion oxides is indicated, respectively, by its corresponding label. The numbers in parentheses correspond to the spin multiplicity for each state of charge. We note that for the cation with m = 6, 8, and 10 the Fe subcluster is more open than those for the neutral and anion (see Fig. 6). The energy differences between the different geometrical isomers, for each one of the charge states, are given in Table IV of the ESI.‡

Fig. 5 Putative ground state and first two low-energy isomers of Fe₆O_m neutral clusters with m = 1–16. The notation is 6.m-Label, with Label in roman letters in decreasing order of stability for each (6,m). The ground state of charged oxides, when it is not the same, corresponds to one of the two lowest-energy isomers. Below the structures of each (6,m) composition, the ground state geometry for the cation, neutral and anion oxides is indicated, respectively, by its corresponding label. The numbers in parentheses correspond to the spin multiplicity for each state of charge. The energy differences between the different geometrical isomers, for each one of the charge states, are given in Table V of the ESI.‡

Fig. 6 Fe–Fe (left panels) and Fe–O (right panels) average distance for Fe_nO_m^0/± oxides with n = 2–6 as a function of the number of oxygen atoms, m. Red, black and green curves correspond to cationic, neutral and anionic oxides, respectively.

There are some magnitudes related to the structural features and stability. Thus, the binding energy for neutral iron oxides Fe_nO⁰_m is defined as

E_b(n, m)⁰ = [n × E(Fe) + m × E(O) − E(n, m)⁰] (1)

and for charged iron oxides Fe_nO_m^± is defined as

E_b(n,m)^± = [E(Fe)^± + (n − 1) × E(Fe) + m × E(O) − E(n, m)^±] (2)

where E(n, m)^0/± is the total energy of the (n, m)^0/± cluster. Fig. 7 shows E_b(n, m)^0/± for neutral and charged iron oxides, for each n value, as a function of the oxidation ratio m. Moreover, numerical values of the total binding energies and optimized cartesian coordinates of all n.m-Label (Label = I, II, III) isomers for all neutral and charged iron oxides are given in the ESI.‡

Fig. 7 Binding energies of Fe_nO_m^0/± oxides (n = 2–6) as a function of the number of oxygen atoms, m. Red, black and green curves correspond to cationic, neutral and anionic oxides, respectively.

The second total energy difference in neutral and charged iron oxides Fe_nO_m^0/± is defined as

Δ₂(n, m) = E(n, m − 1)^0/± + E(n, m + 1)^0/± − 2 × E(n, m)^0/± (3)

which is shown in Fig. 8 for neutral and charged iron oxides. For a given n, the values Δ₂(n, m) as a function of m show positive peaks at m values where the (n, m) composition is more stable than the neighboring (n, m − 1) and (n, m + 1) ones against the addition or subtraction of one oxygen atom. Thus, Δ₂(n,m) is an indicator of local stability, as it is basically the curvature of the binding energy curve, which is an indicator of the absolute stability.

Fig. 8 Second energy differences of Fe_nO_m^0/± (n = 2–6) as a function of the number of oxygen atoms, m. Red, black and green curves correspond to cationic, neutral and anionic oxides, respectively.

Before going into the structural details for each n series, we summarize general trends that are common to most of the iron oxide clusters investigated here.

(a) In the initial stages of oxidation, a compact Fe_n subcluster is formed, and the preferred positions of the oxygen atoms are the bridge or hollow sites. The preference of iron–oxygen bonds can be understood from the larger iron–oxygen binding energies (5.49 eV) than iron–iron ones (3.12 eV). The absolute maximum of the Fe–O distance takes places at a m value close to the n one. Meanwhile, the Fe–Fe distance increases.

(b) For n = 2, 3, 4 and m = n, ring-like structures are obtained. This trend was already discussed in our previous work for Ni oxide clusters.⁴⁹

(c) In most cases for which all bridge positions are occupied by oxygen atoms, the Fe–O distance reaches relative minima. Moreover, the iron subcluster is more open but it exists, except for some cations in which the positive charge tends to further increase the distance between Fe atoms.

(d) When all bridge (hollow) sites are saturated, oxygen atoms tend to occupy top iron positions, and when there are no more top positions available, oxygen starts to be adsorbed molecularly. This happens at high values of m due to the strong Fe–O bonds.

(e) Because of the charge transfer from iron to oxygen atoms, an uniform distribution of the oxygen atoms is observed. The growth mechanism tends to maximize the number of oxygen–iron bonds. Fig. 9 and 10 show the local electronic charges on the atoms of the respective structures for the ground states of Fe₄O_m (m = 1–14) and Fe₆O_m (m = 1–16). The excess (or deficiency) of the electronic charge for O (Fe) is given for each atom.

Fig. 9 Ground states of Fe₄O_m (m = 1–14) oxides. The excess (or deficiency) of the electronic charge for O (Fe) is given for each atom with black (white) numbers.

Fig. 10 Ground states of Fe₅O_m (m = 1–16) oxides. The excess (or deficiency) of the electronic charge for O (Fe) is given for each atom with black (white) numbers.

(f) When all bridge and top sites are occupied by oxygen atoms, the Fe–O (Fe–Fe) distances present the absolute minimum (maximum) value. This occurs at (3,6), an iron broken triangle with six oxygen atoms at three bridge and three top positions, at (4,10), an iron tetrahedral-like broken structure with ten oxygen atoms at six bridge and four top positions, at (5,13), an iron broken square pyramid-like configuration with thirteen oxygen atoms at eight bridge and five top positions, and at (6,15), an iron triangular prism-like broken structure with fifteen oxygen atoms at nine bridge and six top positions. For these oxide clusters, Fe–O bonds are more important as manifested in the decrease of the Fe–O distance. Moreover, the Fe–Fe distances show maximum values due to weak Fe–Fe bonds and, consequently, a compact iron subcluster is not identified in the oxide structures.

(g) For high oxidation states in which molecular oxygen adsorption occurs, the Fe–O distance increases and the Fe–Fe distance remains large.

(h) The binding energy quickly increases upon addition of oxygen atoms, until all bridge positions are occupied by oxygen atoms. Then, it increases more slowly until an m value for which all bridge and top positions are occupied, and for which the binding energy still takes a high value, with small (large) Fe–O (Fe–Fe) distances. It can be said that the covalent bonding between Fe atoms contributes less than the partially ionic Fe–O bonding to the stability of these (n,m) iron oxides. The binding energy increases much more slowly when oxygen starts to bind molecularly, a trend related to the large Fe–Fe and Fe–O distances.

(i) The charge has an important influence on the binding energy. In the initial oxidation states, anionic oxides have higher energy values, while neutral oxides have lower values. However, from the m value for which all bridge positions have been occupied, the binding energy of cations becomes lower than that of the neutral ones. It is remarkable that the high binding energy for anions is quite a lot higher than that for neutral and cationic oxides. This will have important consequences.

(j) The positive peaks in the second energy difference are reached when the oxides are very stable. Consequently, maxima are reached when oxygen atoms occupy all bridge positions, (3,3), (4,4), (4,6), (5,8), (5,9), and (6,9), or most of them, (5,4), (6,4), and (6,6), and when all bridge and top positions are occupied, (3,6), (3,7), (4,10), (5,13), and (6,15).

(k) The magnetic character (and in particular the magnetic coupling) is strongly connected to the geometrical structure and the oxygen environment. Consequently, and related to the above trends, we have identified highly stable Fe oxide clusters with a high oxidation ratio that have a large total magnetic moment due to high spin polarization and parallel couplings, something unexpected in transition metal oxide nanoparticles.

Once the main general trends have been established, let us describe in more detail the structural and electronic properties for each n series. As previously stated, the non-scalability at the nanoscale makes it appealing to explore each size and composition in detail and to analyze exceptions to the general trends.

3.1.1 Iron oxides Fe₂O_m^0/±, m = 1–6. Despite the fact that iron oxide dimers have been studied previously,^12,25,26,28 a systematic theoretical study on the structural and electronic properties and magnetism is appealing. Fe₂O_m, in particular Fe₂O, can be the seed of bigger iron oxides. Moreover, we extend the study of those very small iron oxides, taking also into account charge effects.

The (2,m)^0/± oxide structures, with m = 1–6, are shown in Fig. 1. All of them have an iron dimer. The oxygen environment is the same for all charges at each oxidation ratio m, except for Fe₂O₆⁻. There is a clear tendency to maximize the number of iron–oxygen bonds. For m = 1, a triangle with two iron atoms and one bonded oxygen atom on the bridge position is found as the ground state geometry, 2.1-I. The second oxygen bonds also on the bridge position and a rhombus is found in agreement with earlier calculations.^25,26,28 The 2.2-I structure is a reference building block in the n = 2 series, and for m = 2–5 the charge does not influence the oxygen environment around the iron dimer. Consequently, for m = 3, the 2.3-I structure has the previous 2.2.I geometry, with the third oxygen atom at the top of an iron atom. For m = 4, the 2.4-I structure is obtained by adding two oxygen atoms on the 2.2-I geometry, each of them at a top position of each iron atom. For m = 5, the lowest energy structure, 2.5-I, is found by capping an oxygen atom at the top position of the previous 2.4-I geometry. Finally, for m = 6, the structure consists of the 2.2-I rhombus with four terminal oxygen atoms. Notice that the two O–O axes are perpendicular to the Fe₂ axis, resulting in the 2.6-I structure, except for the anion with the 2.6-II structure, in which the four oxygen atoms are on the same plane.

The Fe–O distance increases at low oxidation values and presents the absolute maximum when all oxygen atoms occupy bridge positions at m = 2 (see Fig. 6). However, when the oxygen atoms bind on top positions, the Fe–O distance decreases significantly as all bridge and top positions are occupied by oxygen atoms at m = 4. At this oxidation ratio, FeO units and their bond become more important, and the Fe–O distances are nearly constant up to m = 6. The charge only has an influence on the Fe–O distance at m = 1 as manifested in the linear structure (2.1-II) of the cation. However, the Fe–Fe distance is very sensitive to the charge (see Fig. 6). In the case of neutral and anionic oxides, the minimum distance occurs at m = 2, when the robust rhombus 2.2-I is found. For the neutral case, when one more oxygen is added, the Fe–Fe distance increases, being quite similar for m = 3–5. The most open neutral dimer is obtained when it is saturated with oxygen, at m = 6. For anionic oxides, the main differences in Fe–Fe distances with the neutral case occur at m = 3, as a consequence of the antiparallel couplings of this oxide, as will be seen in the following section. The parallel coupling between the local magnetic moments of the Fe atoms exists when the distance between them is of the order of the unoxidized iron dimer. In the case of cationic oxide clusters, the Fe–Fe distances are almost constant and large along all the m range, as a consequence of the ionic repulsion between the Fe atoms due to their positive charge.

The binding energy, shown in Fig. 7, increases quickly at low oxidation ratios, m ≤ 4 (m ≤ 3), for the neutral and anionic (cationic) oxides, as a consequence of the highest strength of the iron–oxygen bonding as compared to the iron–iron one. As the oxygen content increases, the iron–iron weakens, the FeO units being more predominant, until the binding energy increases more slowly. Important differences with the charge are found. The anionic oxides show the highest binding energies in all the m range. Moreover, although the cationic oxides show higher binding energy than the neutral ones for m = 1, they are the least stable from m = 3, and this behaviour remains for all the Fe_nO_m⁺ series. Fig. 8 shows the second energy difference. We see a first maximum at m = 2 for all charge states. Additionally, for the neutral and anionic oxides a second peak is found at m = 4, corresponding to the high relative stability for Fe₂O₄. Instead, the cationic oxide is less stable than the neighbours since the binding energy decreases at m = 4 as mentioned above.

3.1.2 Iron oxides Fe₃O_m^0/±, m = 1–9. The (3,m)^0/± oxide structures, with m = 1–9, shown in Fig. 2, have a triangular motif of Fe atoms, for both the neutral and charged cases. Moreover, the oxygen environment is also the same for all charge states for each oxidation ratio m, resulting in the structure 3.m-I, except for anions with m = 2, 4 and for cations with m = 8, 9, where the 3.m-II structures are obtained.

In the initial stages of oxidation, the preferred positions of the oxygen atoms are the bridge sites (the only exception is Fe₃O₂⁻, already mentioned). At m = 3, all bridge positions become saturated (3.3-I), independently of the charge.³⁰ The fourth oxygen atom locates on the iron face (for the cationic and neutral oxides, 3.4-I) or it binds on the top position (3.4-II structure, for the anionic case; this structure is also the second isomer in the neutral case with an energy difference of 0.04 eV with respect to the 3.4-I ground state). At this oxidation ratio, m = 4, the Fe–O distance presents its absolute maximum value (see Fig. 6). The next two oxygen atoms bind on top positions, and the same structures (3.5-I and 3.6-I) are preserved for all neutral and charged oxides. The Fe–O distances decrease from m = 4 to m = 6, at which oxygen atoms occupy all possible bridge and top sites (3.6-I), and the Fe–Fe distances, which had increased almost monotonously up to this m = 6 oxidation ratio (except for the cation with m = 2, with a more open Fe triangular cluster), present their maximum value (Fig. 6). Moreover, the Fe–O units become more relevant and a reconstruction is observed from an initial iron subcluster (3.5-I) to a structure built of FeO subunits (3.6-I). The energy difference between this 3.6-I reconstructed structure and the 3.5-I one, adding an oxygen on top, is 0.23 eV per atom, which gives an idea of why the reconstruction is observed. The seventh and eighth oxygen atoms locate on the iron face for all charge states (3-7-I) and isomers with one oxygen molecule are found. The fact that the Fe subcluster is planar favors the adsorption of more atomic oxygen, keeping the Fe–Fe distances not as large as those of m = 6, as can be seen in Fig. 6. At m = 9, and for all states of charge, oxygen binds molecularly. In this region, m = 7–9, the Fe–O distances increase.

The general structural trends are consistent with the binding energy plotted in Fig. 7. In general, iron oxides show higher absolute stability than nickel oxides,⁴⁹ which may be important for practical purposes. It is noteworthy that the cationic oxides, which have a higher binding energy than the neutral ones in the initial states of oxidation ratios, exhibit the lowest binding energy from m = 4, as compared to the other charge states. However, the anionic oxides from m = 3 keep the highest binding energy, that is, they can be highly oxidized while remaining very stable. Consequently, the charge of oxides is an important factor. Fig. 8 shows the second energy difference, with a first maximum at m = 3, independently of the charge, and a second maximum at m = 6 (anions) and m = 7 (neutral and cations).

3.1.3 Iron oxides Fe₄O_m^0/±, m = 1–14. All ground state structures with n = 4 and m = 1–14, given in Fig. 3, have an iron tetrahedral-like subcluster, except the anion with m = 2, where a planar rhombic structure (4.2-III) is formed, and for m = 4 and all charge states, where a ring-like planar structure (4.4-I) is obtained, with an energy difference lower than 0.4 eV with respect to the tetrahedral one. The origin of this last geometrical change is likely the fact of having a structure with all possible bridge positions occupied. Moreover, most of these charged oxides have the same 4.m-I structures as their neutral counterparts, an exception being the smallest oxidation ratios, m ≤ 3 (m = 1, 3) for anions (cations), where the 4.1-II, 4.2-III, and 4.3-II (4.1-II and 4.3.II) structures are more stable.

The general trend of the preference of oxygen atoms for bridge (4.1-I, 4.2-II, and 4.2-III) or hollow sites (4.1-II, 4.2-I, 4.3-I, and 4.3-II), followed by top sites, is fulfilled. When m = 4, all bridge positions are saturated independently of the charge, and the Fe–O (Fe–Fe) distances reach the first minimum (maximum) value (see Fig. 6), in accordance with the formation of two Fe₂O units bound by two oxygen atoms, giving a planar ring-like atomic arrangement (4.4-I). The next oxygen atoms, from five to six, bind to tetrahedral bridge positions (4.5-I and 4-6-I), and when m = 6 all possible bridge positions of a tetrahedral structure are occupied, the Fe–O distance reaching a maximum, whereas the Fe–Fe distance still reflects the existence of an open tetrahedral iron subcluster. The cation in an exception since it is much more open, practically broken. Then, oxygen atoms, from seven to ten, occupy the four possible top positions; in this region (7 ≤ m ≤ 10) the Fe–O distances decrease and reach the absolute minimum value at m = 10, with all bridge and top positions occupied. From m = 9, the Fe–Fe distances are noticeably large and a compact iron subcluster does not form. The main interaction is through Fe–O bonds, resulting in structures that resemble that of zincblende. No molecular absorption has been observed in the ground states until m = 11. From 11 ≤ m ≤ 14, due to the molecular adsorption (4.m-I), the Fe–O distances increase and Fe–Fe distances remain large.

Another way in which we built the input structures was through the Fukui functions, as mentioned in the previous section. For example, we calculated the Fukui function for the ground state of m = 1, and, where the Fukui function was a maximum, we placed the next oxygen atom to obtain an input structure for m = 2, and so on. We repeated this process for all other clusters. In general, this way of building the clusters is very effective. In Fig. 11, we give the Fukui function for the neutral Fe₄O_m series. We note that it is possible to build the ground state structure Fe₄O_m+1 from the Fukui function of the previous oxide, except for Fe₄O₃ and Fe₄O₄, which depart from the rule.

Fig. 11 Electrophilic Fukui function f⁺ for a representative sample of Fe₄O_m (m = 1–10) oxides. Small spheres represent O atoms, and large spheres represent Fe atoms. Red spheres correspond to the maximum and blue spheres to the minimum value of the Fukui function f⁺. The maximum value of f⁺ is explicitly indicated next to the corresponding atom.

As expected, and as a general trend, O being more electronegative than Fe in this case, charge transfer from Fe to O takes place, and, Fe having an open 3d shell, most of the spin-polarization is contributed by Fe. Fig. 9 shows the excess (or deficiency) of the electronic charge for O (Fe) for the neutral Fe₄O_m oxides. With regard to the Fe atoms, (i) for m = 1–4 and as the oxidation increases, they gradually lose electronic charge until 0.56 e for m = 4, where the charge loss of each Fe atom equals the charge excess of each O atom, resulting in a planar structure; (ii) for m = 4–6, the Fe atoms lose more electronic charge until about 0.70 e for m = 6, resulting in a 3D structure with all oxygens occupying bridge positions; and (iii) for m = 6–14, the loss of electronic charge remains practically constant, about 0.70 e, for each Fe atom; in other words, each Fe atom loses the same charge regardless of having on the top position oxygen atoms or molecules. With regard to the O atoms, (iv) for m = 1–4, each O atom has an excess of electronic charge of about 0.56 e; (v) for m = 4–10, the charge excess on each of the O atoms decreases, taking values of 0.46 e for m = 6, where all bridge positions are occupied by oxygen, and approximately 0.22 e for m = 10, where all bridge and top positions are occupied by oxygen; and (vi) for m = 10–14, the charge excess of each bridge O atom in the bridge remains practically constant, regardless of whether there are atoms or molecules of oxygen at the top positions. Moreover, the top O atoms have a charge excess of about 0.36 e, and this value is practically the same for the molecules bonded to the Fe atoms.

Just like for n = 3, the cationic oxides with n = 4 show the lowest binding energy, from m = 6, compared to those of the other charge states. However, the anionic oxides from m = 6 to n = 10 keep a very high binding energy, much higher than that of the neutral oxides and cations with any oxidation ratio. When molecular adsorption takes place, the binding energy decreases, being higher for the anionic oxides. The most stable clusters against the addition or subtraction of one oxygen atom are found at m = 4, 6, 10, for which maximum values in the second energy difference are obtained regardless of the charge state (see Fig. 8). These clusters have all bridge (4.4-I and 4.6-I) and bridge and top positions (4.10-I) occupied.

3.1.4 Iron oxides Fe₅O_m^0/±, m = 1–15. For Fe₅O_m^0/± and m = 1–15, there are mainly two families of structures as shown in Fig. 4: hexahedron-like (m = 1, 2, 6, 9, 10, 11, 15) and square pyramid-like (m = 3–5, 7–8, 12–14) for the neutral case, with some degree of deformation depending on the oxygen content. All the structures are three dimensional, the only exception being 5.5-II, which has a ring-like planar structure, and it is the ground state for the anionic oxide. Moreover, for the anions, all structures are the same as for the neutral oxides (5.m-I), except the already mentioned 5.5-II and 5.11.II. For the cations, geometrical differences with respect to the neutral oxides are found at m = 1, 6, and 11 with 5.m-II, and at m = 2 and 15 with 5.m-III ground state structures. Structures 5.1-II and 5.2-III have oxygen atoms located on iron faces, structures 5.6-II and 5.11-II have one oxygen molecule, and structure 5.15-III has two oxygen molecules.

The general trend of oxygen atoms to occupy, firstly, from m = 1–9, the bridge (5.1-I, 5.2-I, 5.2-II, 5.5-I, 5.5-II, 5.6-I, 5.8-I, and 5.9-I) or hollow positions (5.1-II, 5.2-III, 5.3-I, 5.4-II, and 5.6-II) is fulfilled except for m = 7, where an oxygen atom is bonded on the top position, 5.7-I.

In some cases, the prevalence of the hexahedron-like or square pyramid-like structures is related to the number of bridges in each one of the geometries. The maximum number of bridge positions for the hexahedron-like structure is nine, three of them in the equatorial plane and six other bridges outside it. Oxygen atoms prefer to bind, firstly, on this other kind of bridge of the hexahedron-like structure, as can be seen in the initial states of oxidation, 5.1-I and 5.2-I, and for 5.6-I, with these six bridges occupied. The nine bridges are occupied at m = 9, resulting in the 5.9-I structure. For the rest of the intermediate oxidation ratios, bridge (5.3-I, 5.4-I, 5.5-I, 5.7-I and 5.8-I) or/and hollow (5.3-I and 5.4-I) positions of the square pyramid-like structure are occupied. The eight bridges of this other structure are occupied at m = 8, resulting in the 5.8-I structure. In the case of this square pyramid-like structure, oxygen atoms prefer to bind first to the bridges situated on the pyramid basis, as can been seen at 5.3-I, 5.4-I, 5.5-I and 5.7-I. Regarding the Fe–O distances, in the initial stages of oxidation, they are shorter for the bridge bonds (m = 1, 2) and hexahedron-like structure, increasing for the hollow cases (m = 3, 4) and square pyramid-like one. A more open iron subcluster is maintained until m = 8 and m = 9, where all bridge positions of the square pyramid-like and of the hexahedron like structures are occupied, respectively. The cationic oxides with oxygen atoms in all bridge positions are much more open at m = 8–10, similarly to the cations with n = 4 and m = 4, 6. From m = 10 to m = 13, oxygen atoms bind on top positions of the hexahedron-like (m = 10, 11) or of the square pyramid-like (m = 12, 13) structures. From m = 12, there is not an iron subcluster, as can be seen from the maximum of the Fe–Fe distance, the FeO units becoming much more preponderant. The 5.13-I geometry has all bridge and top positions of the square pyramid-like structure occupied by oxygen atoms, which is reflected in the minimum (maximum) value of the Fe–O (Fe–Fe) distances, such as happened for previous (3,6) and (4,10) iron oxides. The first oxygen molecule is found at an m = 14 oxidation ratio, the Fe–O distances begin to increase, and the Fe–Fe distances remain large.

As expected, charge transfer from Fe to O takes place. Fig. 10 shows the excess (or deficiency) of the electronic charge for O (Fe) for the neutral Fe₅O_m oxides. With regard to the Fe atoms, (i) for m = 1 to m = 8 and 9 and as the oxidation increases, they gradually lose electronic charge until 0.7 e, resulting in 3D structures with all oxygens occupying bridge positions; (ii) for m = 9–14, the loss of electronic charge remains practically constant, about 0.6–0.7 e, for each Fe atom, regardless of having on the top position atoms or O molecules; and (iii) for m = 15–16, with oxygen molecules bonded by only an O atom, the loss is slightly smaller. With regard to the O atoms, (iv) for m = 1–3, each O atom has an excess of electronic charge of about 0.5 e, and, for m = 4–5, with all bridge positions occupied by oxygen, the value is a little less (0.4 e); (v) for m = 5–16, the charge excess on each of the O atoms decreases, taking a value of 0.17 e for m = 12; and (vi) for m = 10–16, the top O atoms have a charge excess of about 0.36 e, and this value is practically the same for the oxygen molecules with the two atoms bonded to the Fe atoms, being smaller for those bonded by only an oxygen atom.

The most stable oxides are found at m = 4, 8, and 13, where maximum values in the second energy difference are observed regardless of the state of charge (see Fig. 8). Also, the neutral case shows a maximum at m = 6, with a square pyramid-like structure also found at m = 5 and 7 neighbouring stoichiometries. These most stable clusters are obtained when all bridge (5.8-I) or most of them (5.4-I and 5.6-I) and bridge and top positions (5.13-I) are occupied.

3.1.5 Iron oxides Fe₆O_m^0/±, m = 1–16. Fig. 5 shows the different structural families which are obtained for n = 6: the octahedral one (m = 1–6), and the open triangular prism (m = 8–15). For m = 7, structure 6.7-I is obtained, which consists of a tetrahedron and a dimer with seven oxygens in bridge positions. This structure is consistent with the fragmentation spectrum obtained experimentally, as will be seen in the following subsection. The charge does not influence too much the structural geometry, resulting in 6.m-I (m = 1–3,6, 8–16) for the neutral and charged cases. Only a few geometrical changes depending on the charge are found for m = 4–5 and 7 (m = 4) for the cationic (anionic) cases. The 6.4-II prism-like structure, which is an isomer (0.03 eV) for the neutral case, is a degenerate isomer of the 6.4-II octahedral geometry for the cationic oxide, and it is also the ground state for the anionic oxide. Moreover, the 6.5-II structure is the ground state for the cation with m = 5.

Hollow positions of the oxygen are observed from m = 1 to m = 6 for the octahedral structure (6.m-I, m = 1–6). When the seventh oxygen atom binds, an important structural change takes place, and a tetrahedron with an additional dimer is obtained by the seven oxygen atoms occupying the bridge positions (6.7-I). From m = 8, a triangular prism is obtained, and when the ninth oxygen atom binds, each one of the nine oxygen atoms occupies each of the nine possible bridge sites (6.9-I). Again, the structure preferred by the iron oxides is the one that gets oxygen atoms occupying all bridge sites. From m = 10 to m = 15, top positions are the most favorable ones, resulting in a prism triangular symmetry with all the bridge and top sites occupied by fifteen oxygen atoms, for both neutral and charged oxides (6.15-I). Molecular adsorption begins at m = 16, where the 6.16-I structure is found for all charge states. The average Fe–O interatomic distances for the octahedral family are larger than for the triangular prism (see Fig. 6). In this case, the distances decrease as the oxygen ratio increases, reaching a relative minimum at m = 15, with all bridge and top sites occupied (6.15-I), increasing later when the first oxygen molecule is obtained (6.15-II). Additionally, the Fe–Fe average distance increases with the number of oxygen atoms up to a maximum at m = 9; after that, no Fe–Fe bonds are observed, with FeO units prevailing. It is worth noticing that for oxygen-rich clusters, the oxygen atoms that surround the Fe atoms show local tetrahedral symmetry, recalling a diamond-like structure or zincblende.

As in the previous cases, the binding energy is higher for anions at any oxidation ratio, particularly in the highest ones. Consequently, the anionic oxides better hold the oxidation. The second energy differences (Fig. 8) show peaks at m = 4, 6, and 9, this last one being the biggest, corresponding to the first structure where all the Fe–Fe bonds are broken and all bridge positions are occupied by oxygen atoms. Moreover, at m = 15 (the last value with the second energy difference calculated, and for which oxygen occupies all bridge and top positions) a higher value is obtained as compared with the previous one at m = 14.

In general and summarizing subsection A, a detailed computational study has been done taking into account different spin isomers, with different parallel and antiparallel arrangements. For n = 1, the results are in agreement with previous calculations, showing a decrease of the magnetic moment of the iron atom upon addition of oxygen. The ground state structures are planar for n = 2–3 and especially at low oxidation ratios, whereas the ground state structures for n = 4–6 are three dimensional (3D). Due to the strong iron–oxygen bonding, a uniform oxygen distribution is found. Firstly, oxygen atoms preferably occupy bridge or hollow sites, and when all sites are saturated, they tend to bond on top positions. Further oxidation takes place through molecular adsorption. As the oxygen content increases, the iron–iron bond weakens, which is reflected in the increasing Fe–Fe bond length. From a certain size, there is no longer an iron subcluster surrounded by oxygen and FeO units become more preponderant.

3.2 Fragmentation channels of cationic clusters

The reaction of iron cluster cations Fe_n⁺ (n = 2–18) with O₂ was studied by Griffin et al.¹⁰ and the kinetic energy dependence of these reactions over a wide range, using guided ion beam mass spectrometry, was examined. Analyzing the kinetic energy dependence of these processes, quantitative data regarding the thermodynamics of the oxidation reactions were obtained. Consequently, the oxygen bond energies, Fe_nO⁺, defined as D(Fe_n–O⁺) = E(Fe_n⁺) + E(O) − E(Fe_nO⁺), were gathered. A key to this analysis was the availability of quantitative thermo-chemistry regarding the stability of the bare iron clusters previously measured. The energy dependence of cross sections in the threshold region was modeled,¹⁰ where one of the parameters is the threshold for the corresponding reaction, E₀. Two reactions were used to form Fe_nO⁺. In the first reaction, Fe_n⁺ + O₂ → Fe_nO⁺ + O, the bond energies were derived by using the equation D(Fe_n–O⁺) = D(O₂) − E₀. The resulting bond energy values are listed in the first column of Table 3. An alternative method of deriving bond energies notes that they are related to the difference between the thresholds for reactions Fe_n⁺ + O₂ → Fe_n−1⁺ + Fe + O₂ and Fe_n⁺ + O₂ → Fe_n−1O⁺ + Fe + O. Specifically, the bond energies are calculated as D(Fe_n−1–O⁺) = D(O₂) + E₀(1) − E₀(2). The threshold for the first reaction, E₀(1), is equivalent to the bond energies of the bare iron cluster ions, D(Fe_n−1–Fe⁺), and has been measured previously. The threshold for the second reaction, E₀(2), was obtained using the analysis of cross sections in the threshold region as outlined above. The second column of Table 3 lists the bond dissociation energies D(Fe_n–O⁺) obtained from these thresholds using the two previous equations.

Table 3 Bond energies of the O atom in Fe_nO⁺ oxides, D(Fe_n–O⁺). In the first two columns are included previous experimental results,^10,11 where the kinetic energy dependence of some reactions was analyzed and D(Fe_n–O⁺) were obtained. The corresponding analyzed reactions are indicated in the first row

n	Fen^+ + O^2 →	Fen^+ + O^2 →	Fen^+ + CO2 →	This work
	FenO^+ + O*	Fen−1O^+ + Fe + O*	Fen−1O^+ + Fe + CO**
2	5.15 ± 0.12	—	5.1 ± 0.2	5.04
3	4.70 ± 0.13	—	4.5 ± 0.2	5.52
4	4.00 ± 0.15	—	5.9 ± 0.3	5.96
5	4.60 ± 0.15	5.10 ± 0.29	5.7 ± 0.3	5.78
6	4.00 ± 0.15	5.60 ± 0.31	5.5 ± 0.3	5.69

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Moreover, the kinetic energy dependence of the reactions of Fe_n⁺ (n = 2–18) with CO₂ was studied by Griffin et al.¹¹ in a guided ion-beam mass spectrometer. The bond energies for Fe_nO⁺, D(Fe_n–O⁺), were measured by determining the difference between the thresholds for reactions Fe_n⁺ + CO₂ → Fe_n−1⁺ + Fe + CO₂ and Fe_n⁺ + CO₂ → Fe_n−1O⁺ + Fe + CO. Specifically, the bond energies are calculated as D(Fe_n−1⁺–O) = D(O–CO) + E₀(3) − E₀(4), where E₀(3) is equivalent to the bond energies of the bare iron cluster ions, D(Fe_n−1⁺–Fe), measured previously. The threshold for the second reaction, E₀(4), was obtained using the analysis of cross sections in the threshold region as outlined above. The resulting D(Fe_n–O⁺) bond energies are gathered in the third column of Table 3. Our calculated D(Fe_n–O⁺) bond energies are shown in the last column and great agreement with experimental results^10,11 is obtained.

In the context of investigating the stability of small iron oxide clusters and their dependence on the stoichiometry, Molek et al.¹³ showed that photofragmentation studies of cations can be used to determine relative cluster stabilities. It is more difficult to dissociate stable clusters and, therefore, they are often obtained upon the dissociation of larger clusters. In the following, Fe_nO_m⁺ oxides with n = 2–6 are denoted by (n,m)⁺. In this section, we discuss the results of the fragmentation patterns of (n,m)⁺ oxides, for which experimental results are available. For this purpose, we calculated the fragmentation energies defined as follows:

E_f(n,m) = E(n,m)⁺ − E(x,y)⁺ − E(n − x,m − y) (4)

where the first term is the energy of the cationic parent-oxide and the rest are the energies of the product-oxides, one of which is positively charged. A large number of possible channels were calculated, although we gathered the most favorable ones in Table 4 below, and in Table VI of the ESI.‡ This process is endothermic. The channel that shows the smallest E_f will be compared with experimental results. Our definition is based on the total energies of the initial and final oxides and no energy barriers were considered. The biggest product-oxide carries the positive charge, except when Fe⁺, Fe₂⁺ and Fe₂O⁺ result as product-oxides.

Table 4 Calculated fragmentation energies predicted in this work. The fragmentation channels found among the most favorable ones for all (n,m)⁺ are shown. The most favorable experimental¹³ channel is indicated in bold. Fragmentation energies of low-oxide (m < n) and high-oxide (m ≥ n) clusters are separated by horizontal lines, for each n

	O2	O	Fe	FeO	FeO2	Fe^+	Fe2^+	FeO^+	Fe2O2^+
Fe2O^+	—	4.91	4.86	3.91	—	3.91	4.91	4.86	—
Fe2O2^+	4.92	5.08	7.49	4.28	4.50	4.50	4.92	4.28	—
Fe2O3^+	3.70	3.86	7.90	5.86	4.12	3.70	—	4.12	3.86
Fe2O4^+	2.36	3.58	9.11	5.99	5.41	—	—	—	2.36
Fe2O5^+	1.91	3.41	—	7.03	5.37	—	—	—	—
Fe2O6^+	1.62	—	—	—	6.29	—	—	—	—
Fe3O^+	—	5.52	3.77	3.36	—	4.68	3.36	5.51	3.36
Fe3O2^+	6.16	5.71	4.57	3.99	4.75	4.59	4.75	5.86	4.57
Fe3O3^+	6.53	5.90	6.60	4.97	5.56	5.78	6.42	5.95	4.97
Fe3O4^+	4.73	3.91	6.90	5.02	4.56	4.87	6.98	5.15	4.56
Fe3O5^+	1.43	2.60	6.13	4.05	3.30	3.70	—	2.93	2.80
Fe3O6^+	2.02	4.50	7.34	5.14	4.22	4.44	—	3.66	3.95
Fe4O^+	—	5.96	3.82	3.85	—	4.91	4.19	—	—
Fe4O2^+	6.53	5.65	3.76	3.98	5.18	4.50	4.04	6.02	5.20
Fe4O3^+	6.44	5.86	3.72	4.13	5.52	4.03	5.32	5.82	5.70
Fe4O4^+	6.83	6.04	5.86	4.27	6.85	5.83	6.54	5.53	5.94
Fe4O5^+	5.83	4.86	8.12	5.23	5.11	7.50	7.63	6.16	6.10
Fe4O6^+	4.37	4.59	8.21	7.21	5.49	7.14	8.46	7.55	5.86
Fe5O^+	—	5.79	3.43	3.90	—	4.63	4.15	—	—
Fe5O2^+	6.72	6.01	3.79	3.95	5.58	4.80	4.10	6.10	—
Fe5O3^+	6.97	6.05	3.97	4.34	5.67	4.77	3.81	6.30	6.07
Fe5O4^+	8.18	7.21	5.14	5.69	7.22	5.81	6.79	7.44	7.22
Fe5O5^+	6.45	4.31	4.54	3.96	5.68	5.23	7.91	5.59	5.21
Fe5O6^+	3.63	4.39	4.40	3.49	4.03	4.40	7.35	5.08	—
Fe5O7^+	4.31	5.00	5.93	4.90	4.47	5.52	9.21	4.86	7.17
Fe5O8^+	3.96	4.04	7.70	4.48	4.22	5.91	10.07	5.02	6.26
Fe6O^+	—	5.63	4.03	4.32	—	5.63	4.47	—	—
Fe6O2^+	6.81	6.25	4.27	4.79	6.25	5.62	4.88	7.34	—
Fe6O3^+	7.68	6.51	4.74	5.29	6.97	5.74	5.32	7.59	7.10
Fe6O4^+	7.79	6.36	3.89	5.60	7.33	5.66	5.51	7.55	7.62
Fe6O5^+	6.70	5.42	4.99	3.82	6.70	6.25	6.03	6.54	6.97
Fe6O6^+	5.40	5.06	5.66	5.05	4.55	5.85	5.88	6.77	5.86
Fe6O7^+	5.15	5.17	5.82	5.33	5.40	5.80	7.16	6.47	6.12
Fe6O8^+	3.86	4.33	5.56	4.71	4.78	5.42	7.28	5.04	4.69

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Our aim is to corroborate our previous DFT results about the structural properties and stabilities of iron oxides by comparing them with experimental results, which only exist for the cations with n ≤ 2,¹⁴ and with n = 2–6 and m ≥ n.¹³ Multiphoton absorption was necessary to fragment those iron oxide clusters, which is consistent with the strong metal–oxygen bond shown in the previous section. Our calculations reproduce the main features of the mass spectra. The general trends can be summarized as follows: for oxygen-poor clusters (m < n), the most favorable fragmentation channel is the loss of one FeO unit (Fe atom) for the smallest (largest) sizes studied, n = 2–3 (n = 4–6). Table 5 reports some dissociation energies, reaching good agreement with previous results. Oxides with n = m do not eliminate oxygen, but lose a neutral FeO unit, keeping n = m and producing very stable species. An exception is the (6,6)⁺ oxide whose most favorable channel is its fragmentation into the neutral (3,3) and the cation (3,3)⁺, both with n = m and both very stable, in good agreement with experiments. The decomposition of larger oxides with m > n produces a variety of product cations, but those with n = m are always the most prominent and these same species are produced repeatedly from different parent ions, except (5,5)⁺, which appears neither in the experimental spectra of the oxygen-rich oxides with n = 5, nor in the spectra of n = 6. This fact is understandable, since this cluster does not present the highest stability in the n = 5 series according to our calculations. For n = 6, and m > n, the fragmentation does not produce oxygen, but the structures fragment experimentally producing the subclusters that are preformed, which supports our results. Fig. 12 shows the theoretical fragmentation channels predicted in our calculations, for m ≥ n, in good agreement with experiments from Molek.¹³

Table 5 Fragmentation energies of Fe₂O⁺ and Fe₃O₂⁺. Comparisons with previous results are included

Reaction	This work	Ref. 14	Experimental
Fe2O^+ → Fe^+ + FeO	3.91	3.79	—
Fe2O^+ → Fe + FeO^+	4.86	4.44	—
Fe2O^+ → Fe2^+ + O	4.91	4.88	5.15 ± 0.05^48
Fe3O2^+ → FeO + Fe2O^+	3.99	3.68	—
Fe3O2^+ → Fe^+ + Fe2O2	4.59	4.40	—

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Fig. 12 Theoretical sequential fragmentation channels predicted in our calculation. Good agreement with experimental results is obtained. The numbers in blue (red) colour indicate the fragmentation energies from (n,m)⁺ oxides with m = n (m > n). The parallel fragmentation for (5,8)⁺, at only 0.08 eV, is also included.

3.2.1 Oxygen-poor iron oxide clusters (n,m)⁺ (n = 2–6, m < n). Before discussing the results for cations that we can compare with experiments more extensively (n = 2–6 and m ≥ n), we discuss the fragmentation channels for the calculated cationic iron oxides with m < n. These oxides correspond to very low oxidation ratios where the iron subcluster is kept, with oxygen atoms occupying bridge or hollow positions.

For the (2,1)⁺ oxide, with a triangular structure 2.1-I, the two preferred fragmentation channels produce the (1,1) unit or an Fe atom (and Fe⁺ or (1,1)⁺, respectively) with energies of 3.91 eV and 4.86 eV. These results are in good agreement with Li¹⁴ and experimental results (Table 4). We find that for n = 3 and m = 1–2, the most favorable fragmentation channel corresponds to the loss of a (1,1) unit, results consistent with the 3.1-I and 3.2-I structures found (Fig. 2). The loss of a neutral Fe atom is the next favorable channel at 0.41 eV and 0.58 eV, respectively. This channel, in which the oxides lose a neutral Fe atom, becomes the preferred one for all n = 4–6 oxygen-poor iron oxides (m = 1–3) with m < n. The fragmentations are compatible with the tetrahedral, pyramidal, hexahedral and octahedral structures found (Fig. 3–5). The only exception is (5,3)⁺, whose fragmentation, by the loss of Fe₂⁺ (Table VI, ESI‡), instead of an Fe atom, produces the very stable neutral (3,3) iron oxide. Moreover, from (4,3)⁺ and (5,4)⁺, the iron fragmentations result in very stable (3,3)⁺ and (4,4)⁺ oxides, respectively. The second preferential channel for n = 4–6 iron oxides with m < n is the loss of a (1,1) unit.

3.2.2 Oxygen-rich iron oxide clusters (2,m)⁺ (m ≥ n). In the n = 2 series, the smallest oxide produced in experiments¹³ is the (2,2)⁺ cluster. Moreover, the (2,2)⁺ oxide was obtained from the fragmentation of (2,m)⁺ (m > 2), as a consequence of its high stability, in agreement with our results (Fig. 8). Although the n = 2 mass spectra are not shown in the experimental work, the resulting fragments are indicated both in a table and in the text, although with different information. We found a rhombic structure with both oxygens on bridge positions of the Fe dimer, whose bonding distance is 2.65 Å. This geometry may explain why the loss of O₂ or the (1,1)⁺ fragment is not observed in the experiments,¹⁴ the energy of dissociation of O₂ being similar to that of the loss of an O atom, resulting in the (2,1)⁺ fragment being observed, whose bonding distance of 2.60 Å is similar to that of the parent oxide.

For n = 2 and m > 2, the preferred channels are the loss of an oxygen atom (m = 3) or molecular oxygen (m = 4–6), in agreement with Li¹⁴ and Molek,¹³ and this fact corroborates the structures and stabilities that we have obtained (2.n-I, see Fig. 1). For m = 3, the 2.3-I calculated structure of (2,3)⁺ has a terminal oxygen atom added to the 2.2-II rhombic structure of (2,2)⁺, and the calculated energy of 3.86 eV to evaporate an oxygen atom is in good agreement with Li et al.¹⁴ (3.82 eV). Competing with this dissociation, the formation of (1,1)⁺ as well as (1,2), which involves the same bond breaking as the fragmentation of (2,2)⁺ into Fe⁺ and (1,2), has a similar energy of 4.12 eV versus 4.28 eV. The calculated energy of 4.12 eV is also in agreement with Li et al.¹⁴ (3.96 eV). The thermodynamically favored process is the loss of O₂, which is presumably more favorable for (2,3)⁺ (3.70 eV) than for (2,2)⁺ (4.92 eV), probably due to the existence of an oxygen top atom that is closer to the bridging oxygens than they are to each other in (2,2)⁺. However, Li noted¹⁴ that the loss of O₂ still requires a more constrained pathway, being entropically disfavored. For m = 4, the calculated ground state structure of (2,4)⁺ starts with the rhombic structure of (2,2)⁺ and has two extra oxygen atoms (2.4-I); the loss of molecular oxygen is the preferred channel (2.36 eV) to fragment into very stable (2,2)⁺. In this case, no geometrical rearrangement is necessary. For m = 5–6, the calculated ground state structures (2.5-I and 2.5-II) add additional terminal oxygens to the 2.4-I geometry. The fact that the energy required to lose O₂ from both is small, 1.91 and 1.62 eV, respectively, is consistent with the 2.5-I and 2.6-I structures found. Moreover, the (2,2)⁺ fragment could also result from the (2,6)⁺ cationic parent after the elimination of two oxygen molecules.

It is worth noting that small n = 2 iron oxides are oxygen-rich from most m values (m > 2), and a recurrent trend is found: the loss of oxygen, also reported by Castleman et al.²⁶ in the collision dissociation of oxygen-rich iron. This behavior supports the sequential fragmentation shown in Fig. 12, resulting in the cationic Fe₂O₂⁺ iron oxide as a fragment from more oxide species as a consequence of the high stability of this oxide (see Fig. 8), which is also in keeping with results from Molek¹³ and Li.¹⁴

3.2.3 Oxygen-rich iron oxide clusters (3,m)⁺ (m ≥ n). For the n = 3 series, the smallest oxide produced in experiments¹³ is the cationic (3,3)⁺ fragment. In the photodissociation mass spectra (Fig. 2 of Molek¹³), the loss of O₂ is not observed. The resulting stable fragment is (2,2)⁺. The six-membered ring with alternating Fe and O atoms (Fig. 2) is consistent with the fragmentation channel that involves breaking of two FeO bonds resulting in the (2,2)⁺ fragment. The required energy is 4.97 eV, also in agreement with Li et al.¹⁴ (5.22 eV). The Fe⁺ peak obtained in the experimental spectrum,¹³ although it may come from a sequential fragmentation from the obtained (2,2)⁺ explained above, could also result from a parallel fragmentation of (3,3)⁺, with an energy at only 0.81 eV higher, which would explain the high peak found for Fe⁺ in the (3,3)⁺ spectrum. Even the (2,1)⁺ peak obtained could also come from (3,3)⁺ (at 0.59 eV). This agrees with Li et al.,¹⁴ who found three fragmentation patterns, all of which involve cleavage of two FeO bonds to yield the (2,2)⁺, Fe⁺ and (2,1)⁺ products.

On the other hand, (3,3)⁺ is also obtained in the fragmentation of (3,m)⁺ (m > 3), exhibiting a high peak at m = 3 in the mass spectra of n = 3 iron oxides with m > 3, as a consequence of the high stability of the (3,3)⁺ oxide. Experiments¹³ show that (3,3)⁺ was even produced from larger iron oxide clusters, and that it was found as the most stable cationic iron oxide in the n = 3 series. This result is in good agreement with ours (Fig. 8). Likewise, the (3,4)⁺ cluster shows a 3.4-I structure formed by adding a terminal oxygen to one of the iron atoms of (3,3)⁺ (see Fig. 2). Now, cleavage of the terminal FeO bond, losing an oxygen atom (3.91 eV), leads to the primary (3,3)⁺ product observed,¹³ preserving the ring structure. In addition, (2,2)⁺ (Fe⁺) cations coming from the sequential fragmentation of the (3,3)⁺ oxide could also result from parallel fragmentation from the (3,4)⁺ parent ion, with an energy difference of 0.65 (0.96) eV. Moreover, (3,5)⁺ and (3,6)⁺ fragment into (3,3)⁺ (1.43 eV) and (3,4)⁺ (2.02 eV), respectively, via the loss of molecular oxygen, in agreement with the experimental mass spectra.¹³ Reilly^25,26 also reported the loss of molecular oxygen in the collisional dissociation of iron oxides rich in oxygen. These results are also coherent with our predicted structures, 3.5-I and 3.6-I (Fig. 2).

The resulting sequential fragmentation of (3,m)⁺ oxides, explained above, can explain the experimental spectra, as shown in Fig. 12.

3.2.4 Oxygen-rich iron oxide clusters (4,m)⁺ (m ≥ n). The same type of fragmentation is also obtained for the n = 4 series (m = 4–6) and our results again agree with the experiment.¹³ For oxygen-rich clusters such as (4,5)⁺ and (4,6)⁺, the first step is the loss of one oxygen atom (4.86 eV) or molecular O₂ (4.37 eV), respectively, resulting in the (4,4)⁺ oxide, which is very stable (see Fig. 8). For these two oxides (m = 5–6), a sequential dissociation mechanism explains the experimental spectra,¹³ although, in the first case, the fragment (3,3)⁺ could also be directly obtained from a parallel fragmentation (5.11 eV, at 0.25 eV higher in energy) from (4,5)⁺, which would explain the high abundance of this oxide, indicated in bold in Table 1 of Molek et al.¹³ Again, the m = n oxide fragments via the loss of an FeO unit, and thus (4,4)⁺ fragments to (3,3)⁺ (4.27 eV). This behavior establishes that the (3,3)⁺ and (4,4)⁺ iron oxide clusters are the most stable cations in the n = 3 and 4 series, in good agreement with experimental results^13,14 and our previous results about stabilities (Fig. 8).

3.2.5 Oxygen-rich iron oxide clusters (5,m)⁺ (m ≥ n). For the n = 5 series and m = 5 and 7, whose experimental spectra were shown (Fig. 3 of Molek¹³), the same kind of fragmentation seen for n = 3 and 4 is found, and all of this seems to indicate a sequential dissociation mechanism (Fig. 12). (5,5)⁺, with 1 : 1 stoichiometry, fragments via an initial loss of an FeO unit (3.96 eV) reaching the (4,4)⁺ oxide, which is very stable (Fig. 3). This fact is consistent with the ground state structures of both systems, 5.5-I and 4.4-I (Fig. 3 and 4). Subsequently, the steps above explained for this fragment follow and two FeO units are consecutively detached, resulting in (4,4)⁺, (3,3)⁺, and (2,2)⁺, as reported in the experimental mass spectra. The (5/7)⁺ oxygen-rich cluster, in the same way as previous cases for n = 2–4, loses an oxygen molecule (4.31 eV), producing the (5,5)⁺ fragment. Following its fragmentation process, our results explain the experimental mass spectrum of (5,7)⁺, resulting in the cationic (5,5)⁺, (4,4)⁺, (3,3)⁺, and (2,2)⁺.

In the case of (5,8)⁺, some changes happen (Fig. 12). As we will see below, for n = 6 (and other sizes not studied here) it is not possible to explain some experimental spectra from a single sequential process. The first step is the loss of molecular oxygen resulting in the (5,6)⁺ fragment, whose mass spectrum is not shown in the experimental work. One option could be, a priori, the loss of an O atom reaching the (5,5)⁺ oxide (4.39 eV). However, the detachment of the neutral (2,3) fragment to give (3,3)⁺ only needs 3.10 eV from our calculations. Moreover, (5,5)⁺ does not result in a very stable cluster from our calculations, because it does not have all bridge or all bridge and top positions occupied by oxygen atoms. Therefore we propose that the next step for the resulting cationic (5,6)⁺ fragment is to reach the very stable cationic (3,3)⁺ oxide by losing the neutral (2,3) fragment. Then the (3,3)⁺ oxide would follow the steps explained above. Moreover, in the (5,8)⁺ spectrum, there are some additional small peaks and a peak associated with (4,4)⁺ that is not as high as the one associated with (3,3)⁺. We propose a parallel fragmentation from the (5,8)⁺ parent (Fig. 12), losing an oxygen atom (with an energy of 4.04 eV, just 0.08 eV higher than that of the first fragmentation to (5,6)⁺) and resulting in the (5,7)⁺ oxide, which fragments to (5,5)⁺ and consecutively to the very stable (4,4)⁺ oxide, which shows¹³ a higher peak than the previous less stable (5,7)⁺ and (5,5)⁺ fragments (whose signal is only slightly noticeable). Afterwards, the (4,4)⁺ fragment follows its fragmentation process as explained above, which would also explain the larger height of the experimental peak associated with the (3,3)⁺ fragment. These results are consistent with the fact that the (5,5)⁺ oxide is not as stable (Fig. 8 and experimental (5/8)⁺ spectrum¹³), and that not all clusters with 1 : 1 stoichiometry are those with very high stability as we have seen in our previous section.

3.2.6 Oxygen-rich iron oxide clusters (6,m)⁺ (m ≥ n). One noticeable difference in the dissociation patterns for both groups is that the tendency to lose O₂ is no longer observed (Fig. 4 of Molek¹³). The highest peak in the (6,6)⁺ mass spectrum is that corresponding to the (3,3)⁺ fragment. We found good agreement, and the most favorable channel is the (6,6)⁺ fragmentation into two very stable oxides, (3,3) and (3,3)⁺, with only 2.88 eV. Then, the resulting (3,3)⁺ oxide can fragment as explained above and the first part of the mass spectrum is reproduced. Moreover, Molek et al.¹³ found very small peaks, showing two channels with oxide-products, (6,5)⁺ and (5,6)⁺, that are mutually exclusive in a sequential fragmentation process, and can only occur in a parallel fragmentation process, reaching the (3,3)⁺ fragment. Other oxide-products that could only be explained from a parallel fragmentation process were also experimentally found¹³ for n = 7.

The fragmentation of (6,7)⁺ and (6,8)⁺ essentially jumps over the possible n = 5 fragments. (6,7)⁺ produces instead (4,6)⁺ and (2,1) (5.20 eV) (Table VI of the ESI‡), which is consistent with structure 6.7-II obtained with the preformed 4.6-I and 2.1-I subclusters. After, (4,6)⁺ fragments into (4,4)⁺. We found both oxides as the most stable ones for the n = 4 series (Fig. 8), with all bridge positions occupied by oxygen. Then, (4,4)⁺ follows the fragmentation process explained above reproducing the experimental mass spectrum. The (6,8)⁺ oxide produces the (4,5)⁺ one after the release of the (2,3) fragment (4.70 eV). This type of fragmentation was also found for (5,8)⁺, and the 2/3 stoichiometry is the same as that of the common bulk phase as was also indicated in the experiments.¹³ Although we find more favorable energetic channels related to the loss of oxygen for the (6,7)⁺ and (6,8)⁺ oxides, they would imply an important geometric arrangement, whereas both oxides have the resulting experimental fragments as preformed subclusters.

In general and summarizing subsection B, the fragmentation process is consistent with a sequential mechanism (Fig. 12), with a few exceptions such as the (5,8)⁺ spectrum and the small peaks that are experimentally found in the (6,6)⁺ spectrum. Moreover, although the n = 7 series is not studied here, experiments also found some fragmentation channels that can only be explained by a parallel fragmentation mechanism. We also found channels close to the most favorable one, which could explain the large height of some peaks. On the other hand, the above results about fragmentation energies are in keeping with the trends discussed in the previous section about structural arrangements and stabilities from the calculations of second energy differences: (i) For low oxidation ratios (n < m) where there is an Fe subcluster with some oxygen atoms, Fe or FeO can be released. (ii) The (2,2)⁺, (3,3)⁺, and (4,4)⁺ oxides are very stable, with all bridge positions occupied, and are most often obtained from larger clusters. They fragment by releasing an FeO unit and moving on to the previous one. (iii) The most favorable channel for the (6,6)⁺ oxide, which does not have all bridge positions occupied, is the one leading to two neutral and charged (3,3) clusters. (iv) The oxygen-rich clusters with n = 2–5 and m > n, with an open iron subcluster, release an oxygen atom or O₂ until they find a very stable oxide, usually with m = n (or (4,6)⁺, which also has all bridge positions occupied and exhibits high stability from our calculations). (v) Oxygen-rich clusters with n = 6 fragment into two pre-formed subclusters in their geometric arrangement.

3.3 Magnetic properties

We discuss here the magnetic properties of iron oxide clusters with a particular focus on their net magnetic moment as a function of the oxidation ratio. Fig. 13 shows the total magnetic moment of Fe_nO_m^0/± (m = 1–6) and spin multiplicities as a function of the oxygen ratio m, for each n value. Moreover, in Fig. 1–5, and for each (n,m) value, the spin multiplicities are given in parenthesis for each charge state, to better identify possible relations with the structural arrangements. For all n values, there are, in principle, two different initial magnetic phases depending on the oxygen concentration (see Fig. 13). The first one (low oxygen concentration) is a high magnetic moment phase. The second one (high oxygen concentration) is a low magnetic moment phase. But, as we will see, a further unexpected phase with magnetic reentrance may arise at an even higher oxidation ratio.

Fig. 13 Total magnetic moment and spin multiplicity of Fe_nO_m^0/± (n = 2–6) as functions of the oxygen ratio, m. For each n value, the curves given in different colours correspond to each of the charge states: cationic (red), neutral (black) and anionic oxides (green).

3.3.1 Small iron oxides and general trends. For Fe₂O_m^0/± (m = 1–6), the magnetic character strongly depends on the charge state. Neutral oxides, with 2.m-I structures (Fig. 1), have always antiparallel couplings, with a total magnetic moment equal to zero for all oxidation ratios, except in the case of m = 5, where a value of 2 μ_B is reached. Notice that for Fe₂O a close magnetic isomer is found at only 0.02 eV with 6 μ_B and local magnetic moments of 2.92 μ_B and 0.16 μ_B for iron and oxygen atoms, respectively. Earlier work found 0 μ_B²⁸ and 6.8 μ_B²¹ for Fe₂O, which is an important building block to be found in larger iron oxides. We emphasize that the fact that this low spin unit has a high spin isomer so close in energy will be reflected in the formation of larger oxides such as Fe₄O_m (m = 4–6), and, in general, those that have most of their bridge positions occupied by oxygen atoms. Cationic n = 2 oxides with 2.m-I structures (Fig. 1) have always antiparallel couplings with total magnetic moments equal to 1 μ_B (3 μ_B, for m = 5), expect for m = 1 and 2 with high magnetic moments of 7 μ_B and 9 μ_B, respectively. These values are due to the parallel alignment of Fe spins and also with oxygen, and it was also found before,²¹ although for Fe₂O₂⁺ (Fe₂O₅⁺) we found a low spin isomer²⁵ with 0 μ_B at only 0.007 eV (0.05 eV). Anionic n = 2 oxides have parallel couplings exhibiting high magnetic moments equal to 7 μ_B at the initial stages of oxidation (m = 1, 3), and again for m = 5 the magnetic moment (5 μ_B) is higher than that of the low spin neighbours. For m = 1 the low spin isomer is at only 0.07 eV. We also found low spin isomers²⁶ for m = 3 and 5 at only 0.03 eV and 0.09 eV.

In relation to the behavior of larger Fe oxides, the strong influence of charge on the magnetic character is remarkable. On the other hand, when an oxygen atom is added on the top position in the Fe dimer (2.1-III geometry) it results in parallel couplings (see the ESI,‡ Table I). This behavior, which can also be seen for the 2.3-III geometry (ESI,‡ Table I), for both neutral and charged cases, will have a significant impact on oxides with high oxidation ratios that have not yet reached saturation, as we will see. Moreover, the antiparallel couplings for n = 2 at low oxidation ratios will also be found for n = 3 anions, and for the rest of the iron oxide sizes (n = 4–6) for all charge states. The reentrance of magnetism for n = 2, at m = 5, after the antiferromagnetic phase at m = 4, will also be a finding for larger clusters, where a high-spin region will be found for several oxidation values after the low-spin phase.

For n = 3, the negative charge supports the parallel couplings upon addition of oxygen atoms, from m = 1 to m = 4. This effect, related to the negative charge, was already described above for FeO_m⁻ (m = 1–3). However, for neutral and cationic oxides, the parallel couplings also remain until low oxidation ratios, m = 2 and m = 1, respectively. Although for n = 3 two high-spin and low-spin phases can be observed, a very slight emergence of magnetic character for Fe₃O₆⁺ and Fe₃O₈ is found.

For n = 4–6, in addition to both high and low magnetic moment phases, two further phases appear, resulting in four magnetic phases in total (see Fig. 13). For n = 4 and 5, both the neutral and charged oxides, and for n = 6, both the neutral and anionic oxides, have high spin from m = 1 to m = n − 1. The first ferromagnetic region extends up to m = 4 in the case of n = 6 and cationic oxides, probably due to the different oxygen environment. There is a second low-spin region, for all charge states. The third one consists of a reentrance of the magnetic moment at about m = n + 5, since the charge also influences the magnetism for n = 4–6, especially at oxidation ratios that are in the limit between different magnetic regions. In the fourth phase, antiparallel couplings are observed again.

In general, upon addition of oxygen on iron clusters, they change from genuine parallel couplings with some oxygen atoms around them, to clusters formed by Fe₂O units with bridging oxygen atoms and antiparallel couplings, towards clusters of FeO units with parallel couplings in which there are no long bonds between iron atoms, to finally reach oxygen saturation with the formation of oxygen molecules, with both the Fe–O and Fe–Fe distances increasing, and consequently losing stability.

3.3.2 Iron oxides Fe₄O_m^0/±, m = 1–14. We analyse in detail the magnetic behavior of the Fe₄O_m^0/± oxide clusters to visualize and understand the different magnetic behavior depending of the oxidation ratio, m. We identify four magnetic phases: (i) For m = 1–3, the total magnetic moment is high: 14 (13 or 15) for neutral (charged) oxides. (ii) For m = 4–7, the total magnetic moment is low: 0 (1) for neutral (charged) oxides, except for the neutral oxide with m = 6 with 2 μ_B (the spin isomer with 0 μ_B is at 0.04 eV). For m = 8, we found an intermediate magnetic moment (4 μ_B) for the neutral oxide and a high magnetic moment (11 μ_B) for the charged oxides. (iii) For m = 9–11, the total magnetic moment is again high: 8 or 12 (7 or 9) for neutral (charged) oxides. For m = 12, we found again an intermediate magnetic moment: 6 (5) for the neutral (charged) case. (iv) For m = 13–14 we found low total magnetic moments: 2 (1 or 3) for neutral (charged) oxides. In the case of m = 16, although the total moment is 4 μ_B, spin isomers with 2 and 0 μ_B are found at only 0.02 and 0.06 eV, respectively. In any case, the magnetic coupling between the four iron atoms is antiparallel. Moreover, and to show either the clearly established parallel or antiparallel coupling or the competition between both, Fig. 14 shows the energy differences between the spin isomers and the ground state solution, as a function of the spin multiplicity, for the neutral Fe₄O_m oxide, with m = 1–14. Fig. 14 displays, on the one hand, clear high-spin states in the first and third magnetic phases, and, on the other hand, clear low-spin states in the second and fourth phases. However, the spin states become degenerate at certain oxidation ratios, since several spin isomers are found within an energy window of about 0.1 eV, which is in the limit of accuracy of DFT and almost any computational chemistry method that we currently have for molecules. This degeneracy does not qualitatively modify the high- or low-spin character of each of the mentioned phases (because the degenerate spin isomers are close to each other). In the frontiers of different magnetic phases is where spin isomers of both types result degenerated (m = 8, 12, and 14 are clear examples). Here, experiments would be pertinent to unambiguously assign the spin state at those oxidation ratios.

Fig. 14 Energy difference between the spin isomers and the ground state solution (green circle, 4.m.I structure) as a function of the spin multiplicity for neutral Fe₄O_m (m = 1–14) oxides. Each panel corresponds to a given oxygen composition, m.

The magnetic phases are closely related to the structural geometry, to the number of atomic oxygens bound at hollow, bridge and top positions (oxygen environment), and to the number of oxygen molecules formed. There is also a relationship of all these structural and magnetic properties with the stability of the oxides, as will be seen below. Fig. 15 shows, for the n = 4 neutral case and for each oxidation ratio m = 1–14, the total and local magnetic moments in each one of the Fe and O atoms of the corresponding geometrical structure. The four magnetic phases described above, m = 1–3, m = 4–8, m = 9–11, and m = 12–14, are indicated with black, red, green and blue colors, respectively, and highlighted in the n = 4 graph of Fig. 13, included also as an inset in Fig. 15. In the first high-spin phase, few oxygen atoms bind on bridge sites of the iron cluster. In the second low-spin phase, almost all (m = 5) or all (m = 4, 6) bridge sites of the cluster are occupied. After that, there are some oxygen atoms (one for m = 7 and two for m = 8) that bind on top positions. In the third high-spin phase, almost all (m = 9) or all (m = 10) top positions are already occupied by oxygen atoms and the first oxygen molecule appears (m = 11). In the fourth phase, two, three and four molecules are bonded (m = 12, 13, 14), until the structure is completely oxygen-saturated at m = 14.

Fig. 15 Ground states of neutral Fe₄O_m (m = 1–14) oxides. The local magnetic moments of Fe and O atoms are indicated. The numbers in black (white) indicate spin up (down) polarization. The n = 4 graph of Fig. 13 is included as an inset.

Besides the different total magnetic moment of each of the four magnetic phases, Fe atoms present different local moments in each one of them. In the first phase, the magnetic couplings are parallel and the high total magnetic moment of the oxide clusters, 14 μ_B, is due to high spin polarization in the iron atoms. The local magnetic moments, 3.4 (Fe) and 0.5 (O), are similar to those of the FeO dimer (3.4 μ_B in Fe and 0.6 μ_B in O). It can be seen from Fig. 14 that the ground state clearly has a high magnetic moment. The compact iron subclusters are preserved (see Fig. 6) and the Fe–Fe couplings are parallel, like in the bare Fe clusters. The oxygen atoms bind on some bridge or hollow positions. In the second phase, the low magnetic moment is due to high spin-polarization (about 3.2 μ_B, or 1.5–1.7 μ_B for iron atoms with top oxygen atoms) but with antiparallel couplings. Oxygen atoms that bind on bridge iron atoms with parallel coupling are, practically, magnetically frustrated. In this magnetic phase, most of the bridge positions (m = 5) or all (m = 4, 6) of the iron subcluster are occupied. The m = 4 oxide shows antiparallel coupling between the four iron atoms, with all oxygen atoms magnetically frustrated. We highlight the n = 5–6 low-spin oxides, where the iron subclusters are still preserved although more open (Fig. 6). Their ground states have clearly a low-spin character, as can be seen from Fig. 14. They can be seen as sub-divided into two Fe₂O parts with parallel couplings, one part with spin up and the other one with spin down, and, consequently, with an antiparallel coupling between them; the oxygen atoms that are between these two sub-parts have zero, m = 5, or almost zero magnetic moment, m = 6 (a spin isomer with zero magnetic moment is at 0.04 eV, see Fig. 14). Both iron oxides with all bridge positions occupied by oxygen (m = 4, 6) present maxima in the second differences in energy, which reflects their high relative stability, and the m = 6 oxide has also the maximum value of the second energy difference. Then, for m = 7, an oxygen atom binds on top positions and for m = 8 the total magnetic moment increases (4 μ_B) and the energy difference between the low and high magnetic moment spin isomers becomes smaller (see Fig. 14). In the third phase, a reentrance of the magnetic moment is found, which is an unexpected trend in transition-metal oxides. The high magnetic moment comes from parallel couplings, but iron atoms have low local magnetic moment (low spin polarization region). The local moments of iron are lower than those of previous phases as a consequence of its high oxygen coordination (similarly to FeO_m studied in the second section), and decrease from 3.1 μ_B to 1.3 μ_B, the value reached by those iron atoms with oxygen bound at top positions. Also, the local magnetic moments of oxygen atoms become lower when there are oxygen atoms bound at top positions. A minimum value of the FeO distance occurs at m = 10, and the local moments of oxygen (0.3–0.4 μ_B) are closer to that in the FeO unit (0.6 μ_B). FeO units become more preponderant. In addition, a compact iron subcluster can not be identified (see Fig. 6). At this third high-spin phase, iron oxides are more similar to clusters of FeO units with parallel couplings bonded by oxygen atoms than to iron clusters with adsorbed oxygen atoms like in the first high-spin phase. We highlight the Fe₄O₁₀ oxide, which, having a high oxidation ratio, shows parallel couplings and a total magnetic moment of 8 μ_B. Fe₄O₁₀ presents a maximum in the second total energy difference. This iron oxide has all possible bridge and top positions occupied. Finally, in the fourth low-spin phase, two ferromagnetic oxide subclusters can be again identified, with magnetic moments in opposite directions, and the oxygen atoms, located between those sub-clusters, are magnetically frustrated. There is not an iron subcluster. Oxygen binds molecularly, so that Fe coordinates with O₂, which weakens the Fe–O bonds (because the oxygen bond of O₂ is strong). FeO units become less preponderant and the Fe–O distance increases, reaching similar values to those of the first phase. We highlight Fe₄O₁₄, with all bridge and top positions occupied and four oxygen molecules, with a clearly low-spin ground state with antiparallel couplings (Fig. 14). Low-lying spin isomers with 0 μ_B and 2 μ_B, at 0.06 eV and 0.02 eV, respectively, are found.

Based on the above results, we identify three particularly interesting Fe oxide clusters due to their high global and relative stability that exhibit different magnetic characters. The first two are the planar (4,4) oxide, with Fe₂O units antiferromagnetically coupled by magnetically frustrated oxygens (indirect exchange), and (4,6), a tetrahedral iron motif also formed by two Fe₂O units antiparallel coupled by four oxygens, practically magnetically frustrated. The spin isomer (0.04 eV from the ground state) has the four oxygen atoms magnetically frustrated and the two units with identical up and down magnetic polarization. Both the (4,4) and (4,6) oxides have all bridge positions occupied by oxygen atoms. Additionally, the third is the high-spin (4,10) oxide (8 μ_B), which has a high oxidation ratio and all bridge and top positions occupied, where FeO units are more important and the Fe subcluster is broken.

Fig. 16 shows the total density of states (DOS), and the partial contribution of iron atoms (cyan line) and oxygen atoms (red lines), for neutral Fe₄O_m oxides and for each oxidation ratio (m = 1–14). Vertical lines indicate the Fermi level. The magnetic behavior of the Fe₄O_m oxides discussed above is reflected in the DOS. The four magnetic phases are well differentiated. For m = 1–3, the parallel couplings are remarkable. The contribution of oxygen is higher at m = 3, although most of the contribution in this first high-spin phase comes from the iron subcluster. For m = 4, an antiferromagnetic state can be observed with identical up and down contributions of both iron and oxygen. At this oxidation ratio, a remarkable change in the density of states happens, resulting in a similar density of states for n = 5–8, with clear antiparallel couplings. The density of states changes dramatically for m = 9–10, with oxygen atoms bound on top positions of the iron atoms, with parallel couplings, but where, unlike the first phase, the contribution of oxygen to the total density is much higher. For m = 11, the oxidation ratio where oxygen starts to bind molecularly, the density of states changes, as a consequence of the antiparallel couplings of the last phase, and, for m = 13–14, the existence of highly coordinated and oxygen-saturated iron atoms is reflected.

Fig. 16 Total DOS (black line), and partial contribution of iron (cyan line) and oxygen atoms (red line) of neutral Fe₄O_m (m = 1–14) oxides. The vertical line marks the Fermi energy.

3.3.3 Iron oxides Fe₅O_m^0± and Fe₆O_m^0±, m = 1–16. The reentrance of the magnetic moment is also obtained for the Fe₅O_m oxides. Fig. 17 shows the energy difference between spin isomers of the ground state structure (5.m.I). The first magnetic phase extends up to m = 4, compact iron subclusters are preserved and O atoms bind in the first bridge-hollow positions. The total magnetic moments are very high (17–19 μ_B) and parallel couplings are clearly established (see Fig. 17). We highlight the (5,4) oxide with high stability (see Fig. 8). The second low-spin phase, from m = 5 to m = 9 (m = 10 for the cationic oxide), corresponds to the cases for which all bridge positions or most of them are occupied by oxygen atoms. The reentrance of the total magnetic moment occurs at the third high-spin phase, with values of 10–16 μ_B, from m = 10 (m = 11 for the cationic oxide) to m = 15 (m = 14 for the cationic oxide). Oxygen atoms bind on top positions until all of them are occupied at m = 13, where the magnetic solution is clearly obtained (Fig. 17). The last low-spin phase can be observed at m = 16, with two bound oxygen molecules. In addition to the influence of the charge in the limits between different regions, a lower total magnetic moment for the cation is found at m = 13. Also, for the anion with m = 11 the total magnetic moment is small as a consequence of the structural change that takes place due to the charge difference.

Fig. 17 Energy difference between the spin isomers and the ground state solution (green circle, 5.m-I structure) as a function of the spin multiplicity for neutral Fe₅O_m (m = 1–16) oxides. Each panel corresponds to a given oxygen composition, m.

Therefore, for n = 5, we highlight four clusters due to their high stability, with different magnetic behaviour: the high-spin (5,4) oxide (first magnetic phase, μ = 18 μ_B), the low-spin (5,6) hexahedral-like structure with six bridging oxygens and (5,8) pyramid-like structure with all bridge positions occupied by oxygen (second magnetic phase μ = 2 μ_B), and the high-spin (5,13) pyramid-like structure with oxygen atoms occupying all bridge and top positions (third magnetic phase, μ = 10 μ_B).

As for the other sizes, the reentrance of the magnetic moment also takes place for n = 6 and m = 15 with all bridge and top positions, of an already open triangular prism, occupied by oxygen atoms (for the cationic oxide the ferromagnetic isomer is at 0.03 eV with 7 μ_B). Then, when the first oxygen molecule binds to an iron atom, only the cationic oxide keeps a high total moment, while the neutral and the anion become again low-spin oxides.

In general and summarizing subsection C, we have found related effects between the geometrical structures, oxygen environment and magnetism. We have found a reentrance of the magnetic moment at high oxidation ratios and we have identified certain oxides (4,10)^0/±, (5,13)^0/− and (6,15)^0/− with parallel couplings and a considerably large total magnetic moment, as well as high relative stability.

4 Conclusions

The connection between the structural patterns and magnetic properties of Fe_nO_m^0/± (n = 1–6) oxide clusters has been investigated by means of DFT-GGA calculations that allowed us to compare the fragmentation channels of cationic oxides with experimental results so as to confirm plausible geometrical arrangements. The ground state structures have been obtained by testing a large number of initial geometries and different oxygen environments and optimizing them by means of conjugate gradients. Those initial geometries were constructed (i) as planar and three dimensional arrangements of FeO units; (ii) as small pure iron clusters covered with consecutive oxygen atoms and molecular oxygen absorption for the highest oxidation ratios; and (iii) with the help of Fukui functions to locate the most nucleophilic regions to be covered by oxygen. A rich map of structural and spin isomers is found, for each of the states of charge, once a detailed computational study has been done taking into account different spin isomers with parallel and antiparallel arrangements for each of the geometric isomers with different oxygen environments. For n = 1, the results are in agreement with previous calculations, showing a decrease of the magnetic moment of the iron atom upon addition of oxygen, with an important effect of the added or subtracted charge. For n = 2–3 the ground state structures are planar and, especially at low oxidation ratios, the charge has also an important influence. For n = 4–6 the ground state structures are three dimensional (3D) with tetrahedral (n = 4), pyramidal and hexahedral (n = 5), decahedral, octahedral (n = 6) and zincblende-like geometries as the cluster size increases.

Due to the strong iron–oxygen bonding, there is a clear tendency to maximize the number of iron–oxygen bonds and, as consequence of the electron transfer from iron to oxygen atoms, a uniform oxygen distribution is found. Oxygen atoms preferably occupy bridge or hollow sites, and when all sites are saturated, they tend to bond on top positions. Further oxidation takes place through molecular adsorption. As the oxygen content increases, the iron–iron bond weakens, due to the charge transfer from iron to oxygen, which is reflected in the increasing Fe–Fe bond length. Consequently, from a certain size there is no longer an iron subcluster surrounded by oxygen, and FeO units become more preponderant. This fact is reflected in minimum values of FeO distances when all bridge and top positions are occupied. The binding energy increases faster at low oxidation ratios. Once molecular adsorption takes place, the FeO distances increase, the Fe–Fe distances being at the same time very high, and thus the binding energy increases slowly. For cationic oxides, the fragmentation channels of these clusters are obtained and compared with experimental measurements of photofragmentation, reproducing the main features of experiments and providing support to our calculated structural geometries and energetic stabilities.

Interrelated effects between the geometrical structures, oxygen environment, magnetism and stability of iron oxides, with an important effect of the charge, especially on the stability, result in the finding of very stable low-spin (with more open iron subclusters), and very stable high-spin iron oxide nanoparticles (with a broken iron subcluster), both with high oxidation states. The total magnetic moment of iron oxides reflects the couplings between local magnetic moments of different sites. Iron atoms preserve their high local magnetic moment as long as the oxidation ratio is low. When the oxygen content increases, and all or almost all bridge sites are occupied, the iron couplings become antiparallel. The oxidation ratio at which oxygen atoms start to bind on top positions results in a small energy difference between the high-spin and low-spin isomers. Moreover, the local magnetic moment of iron atoms with oxygen bound on top strongly decreases. When almost all or all top positions are occupied by oxygen, the local moments of iron atoms are about 1.4 μ_B, but the couplings become again parallel and, surprisingly, oxide clusters with high oxidation ratios and with high magnetic moments are found. Finally, at very high oxidation ratios, oxygen molecules cause the iron oxides to become antiferromagnetic, and the binding energy to decrease. The effect of the charge excess or deficiency on the magnetic properties becomes more important at those oxidation ratios between high and low spin phases where spin isomers (with parallel and antiparallel couplings) could coexist.

The positive peaks in the second energy difference show the most stable oxides. When all bridge positions or most of them are occupied, the maxima correspond to clear oxide clusters with parallel coupling, whereas when all bridge and top positions are occupied, the maxima correspond to clear magnetic ones. This means that, despite the high degree of oxidation, certain oxide clusters which are very stable retain a high magnetic moment. We have found a reentrance of the magnetic moment never reported before for iron oxide clusters, to our knowledge. We have identified certain oxides (4,10)^0/±, (5,13)^0/− and (6,15)^0/− with parallel couplings and a considerably large total magnetic moment, as well as high relative stability, an interesting result in the context of magnetic grain design. These magnetic properties and the bio-compatibility of iron oxide nanoparticles with high binding energy might be interesting also in nanomedicine. The large moment of these high-spin oxides is due to the promotion of parallel magnetic couplings, despite their significant oxidation ratio, an unavoidable fact in environmental conditions.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the Spanish Ministry of Economy and Competitiveness (Grant PGC2018-093745-B-I00), Junta de Castilla y León (Project No. VA124G18). R.H.A-T acknowledges the fellowships (E-47-2019-0197368) from the University of Valladolid.

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Footnotes

† PACS 75.75 + a; 36.40Cg; 75.30.Pd; 75.50.-y.

‡ Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp03795h

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