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A: New Tools and Methods in Experiment and Theory

Collective Superexchange and Exchange Coupling Constants in the Hydrogenated Iron Oxide Particle FeO h 8

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Lavrenty Gennady Gutsev, Gennady Lavrenty Gutsev, and Puru Jena J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b03034 • Publication Date (Web): 10 May 2018 Downloaded from http://pubs.acs.org on May 22, 2018

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The Journal of Physical Chemistry

Collective Superexchange and Exchange Coupling Constants in the Hydrogenated Iron Oxide Particle Fe8O12H8 L. G. Gutsev,a* G. L. Gutsev,b and P. Jena,a a

Department of Physics, Virginia Commonwealth University, Richmond, VA 23284, USA

b

Department of Physics, Florida A&M University, Tallahassee, Florida 32307, USA

Abstract Motivated by the fact that Fe2O3 nanoparticles are used in the treatment of cancer, we have examined the role of ligands on the magnetic properties of these particles by focusing on (Fe2O3)4 as a prototype system with H as ligands. Using Broken-Symmetry Density Functional Theory we observed a strong collective super-exchange in the hydrogenated Fe8O12H8 cluster. The average antiferromagnetic exchange coupling constant between the four iron-iron oxo-bridged pairs was found to be -178 cm-1, whereas coupling constants between hydroxo-bridged pairs were much smaller. We found that despite the apparent symmetry of the iron atom framework, it is not reasonable to assume this symmetry when fitting the exchange coupling constants. We also analyzed the geometrical and magnetic properties of Fe8O12Hn for n=0-12 and found that hydrogenating oxo-bridges would generally inhibit the Fe-O-Fe antiferromagnetic superexchange interactions. Antiferromagnetic lowest total energy states become favorable only when specific distributions of hydrogen atoms are realized. The (HO)4– Fe4(all spin-up) – O4 – Fe4(all spin-down) – (OH)4 configuration in Fe8O12H8. presents such an example. This symmetric configuration can be considered as a super-diatomic system.

*

Corresponding author. E-mail: [email protected]

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1.Introduction Iron oxide magnetic nanoparticles (MNP) are currently the only metal oxide particles which are approved for clinical usage because they are biocompatible and are freely metabolized by endosomes1. The usage of magnetic materials in medicine, cancer therapy in particular, is advantageous because biological tissue does not interact strongly with magnetic fields and thus does not diminish the field strength significantly2. MNPs can thus be externally magnetically guided towards malignant tumors and they can either deliver therapy in the form of hyperthermia, controlled drug release, remote actuation, chelation therapy, tissue engineering or provide diagnostic imaging as MRI contrasting agents1,3,4,5. This multifunctionality, commonly referred to as theronastics, is a major motivation for the study of these materials6,7,8,9. It should be noted that small MNPs are of particular interest since only ones smaller than 50 nm in diameter can avoid phagocytosis for prolonged periods of time1. This regime is also experimentally convenient since small MNPs form single domains and are only magnetized when a field is applied. The down side to such small paramagnetic iron oxide nanoparticles (SPION) is that they generally have a weak magnetic moment10. This observation is supported by theoretical studies of (FeO)n n=1-811 , (FeO)n n=1-1012 and (Fe2O3)n n=1-513 where the ground states tend to have a low spin magnetic moment. In the latter work, the exchange (J) constants were calculated using the brokensymmetry (BS) approach and the average J constants were found to be approximately -100 cm-1, which indicates strong antiferromagnetic (AFM) interactions. It was previously noted, in the case of Fe6On (n=1-20) and Fe7On (n=1-24)14, that the oxidation of iron clusters does not decrease the total magnetic moment until Fe-O-Fe bridges are formed. These oxo bridges promote a phenomenon called super-exchange (SE)15 which can be demonstrated in the Fe2O2 model

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cluster where the singlet state is stabilized compared to the septet state, due to Pauli

repulsion between electrons occupying localized spin orbitals of two iron atoms coupled ferromagnetically (FM). The oxo-bridge clearly plays a large part in this phenomenon because of the fact


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that it is immaterial whether an oxygen atom binds in the alpha or beta spin representation to an iron atom. This allows the stabilization of the AFM interaction of the total spin magnetic moments on the iron atoms. On the other hand, the oxo-hydroxo bridges in the Fe8 single molecule magnet (SMM) appear to promote mostly ferromagnetic interactions17 since the duality of oxygen bonding is violated. Many other organic bridges such as oximes18 and naphthoximes19 have also been investigated for SMMs. In bulk, four crystalline polymorphs of Fe2O3 are known:

- Fe2O3 (hematite) , - Fe2O3 , -

Fe2O3 (maghemite) and - Fe2O3 20. The rare - and - polymorphs are only observed in nanoscale structures and the latter, in particular, has been described as either a collinear ferrimagnet21 or as an antiferromagnet22 at room temperature and were previously found to exhibit a giant coercive field23. It has long been known that ligation of the surface can drastically influence the magnetic behavior of a fine iron oxide particle through a phenomenon called spin-pinning24; this same phenomenon was observed in - Fe2O3 MNPs after they were coated with phosphate25. Given that small MNPs have a large surfaceto-volume ratio it is not surprising that ligation may have a strong influence on magnetic effects. In this work, we consider the hydrogenation of a (Fe2O3)4 cluster as a model to investigate the ligation effects. Since the exchange constants previously found for the (Fe2O3)n clusters are small, it should be reasoned that high-quality quantum calculations are necessary to gain reliable insight into the effect of surface ligation on magnetic properties. To model ligation effects, we have chosen the simplest monovalent ligands: hydrogen atoms. We tested the cases up to twelve hydrogen atoms, adsorbed by oxygen atoms. We found that adding eight H atoms to the (Fe2O3)4 cluster results in a unique layered AFM singlet lowest total energy state of the Fe8O12H8, whereas adding 12 hydrogen atoms leads to the FM ground state of Fe8O12H12. In order to gain insight into the magnetic properties, we computed the exchange coupling constants of Fe8O12H8 and compared them to the exchange coupling constants of Fe8O12. The topology of electronic states was explored using Bader’s quantum theory of atoms in molecules (QTAIM).


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2. Methodology Our all-electron calculations are performed using the spin-polarized density functional theory with the generalized gradient approximation (DFT-GGA) available in the GAUSSIAN 09 suite of programs.26 We chose the 6-311+G* basis set of triple- quality27,28,29 (for a recent analysis also see 30) and the BPW91 exchange-correlation functional composed of Becke exchange31 and Perdew-Wang correlation32. This combination of functional and basis set has been shown to provide good agreement between computed results (such as electron affinities, ionization energies, and binding energies), and experimental data, as well as post-HF calculations for FeO and FeO- 33, FeO234, FenO and FenO- (n=2-6) 35, FeOn and FeOn- (n=3-4)36, Fen+,0,-(n=2-20)

35 37

,

and (FeO)n (n =1-10)11 and (n=11-16).38 The last work

contains an especially detailed comparative validation of our method for iron oxide systems. The BPW91 atomization energies have been compared to those obtained using standard reference databases and were found to be comparable to the atomization energies obtained using more recent exchangecorrelation functionals39. For our hydrogenation study, we have first optimized low-lying isomers of the (Fe2O3)4 cluster. After each addition of a hydrogen atom to each (Fe2O3)4 isomer we started by optimizing the state with the highest spin multiplicity of 31, the number which corresponds to the total spin S = 2 of each Fe atom, and then decreasing sequentially by a single spin flip, i.e. 2S+1 = 31,29,27…,3,1. In this way, one often obtains40 the lowest total energy states among all states corresponding to different combinations of the spin-up and spin-down magnetic moments on the atoms corresponding to each intermediate ferrimagnetic state in the 31 to 1 ladder. This approach indeed leads to the lowest total energy singlet state in the present case as can be seen in the second section of this work. We generated a large number of singlet states for (Fe2O3)4H8 by preparing them via the fragment method included in GAUSSIAN 09 and found that the singlet state obtained in the stepping down on the spin multiplicity ladder from 31 to 1 lead to the lowest total energy state.


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When we compared the total energies of 14 singlets generated in this fashion to the total energy of the singlet obtained via the step-down spin-flip method, we found that the latter state was in fact the lowest in total energy. It should be noted that we generated many more than 15 singlet states using the fragment method; however, they all eventually converged to one of the 15 singlet states. The convergence threshold for total energy was set to 10-8 eV and the force threshold was set to the default value of 10-3eV/Å. The local spin magnetic moments on atoms were obtained from natural atomic orbital populations (NAO) computed using the NBO suite of programs41 .

3. Results and Discussion 3.1 Hydrogenation of Fe 8 O 12 Since the isomers of (Fe2O3)4 were discussed in greater detail in a previous paper42, we include a summary figure in the ESI and will shortly discuss only the key results. It was found in a previous work13 that the lowest energy isomer of Fe8O12 was an AFM singlet with an irregular geometry. The results of our optimizations do not fully agree with the results of this work. In particular, the minimum in total energy was obtained for a state whose geometrical structure is an octahedral Fe6O6, bicapped with two FeO3 groups on opposing faces and related by an inversion operation. The lowest total energy state with this geometry is an AFM singlet state, followed by a triplet state with the same geometrical structure, which is higher in total energy by +0.08 eV. The lowest total energy ferromagnetic state with this geometrical structure has the total spin S = 12 and is higher in total energy by 1.37 eV (see the ESI). We note that an isomer with a cubic geometrical structure, which was previously assigned as the groundstate isomer,43 is higher in total energy by +0.59 eV. However, the Fe8O12 cubic geometry was found to be preferred as more hydrogen atoms were added and is similar to the Ti8O1244 and core Si8O12(OSiMe3)818 geometries. The spin arrangement of the ground state singlet is 1(spin-up)-3(spindown)-3(spin-up)-1(spin-down), the arrangement 1(spin-down)-3(spin-down)-3(spin-up)-1(spin-up) is


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significantly higher in energy, which indicates that the superexchange (SE) interaction between 1 and 3 is strong. Attaching a single hydrogen to the cluster substantially increases the total spin and the lowest total energy state of Fe8O12H is ferrimagnetic with the total spin of 3.5. The geometrical structure of the Fe8O12 core is practically undistorted by this attachment. By comparing the spin densities (see Fig. 1) of Fe8O12 to Fe8O12H, we note that the local total spin magnetic moments of all of the octahedral Fe atoms in the hydrogenated cluster are aligned ferromagnetically while the spin moments are aligned as FeO3(up)-Fe3O3(down)-Fe3O3(up)-FeO3(down) in the initial cluster. The FeO3(up)-Fe3O3(down) interaction via oxo-bridges is of a SE nature and the changing one of the oxo-bridges to a hydroxo-bridge breaks the SE. The average exchange interaction between octahedral sites is FM and thus, the breaking of just a single SE bridge stabilizes an all-up configuration for the iron atoms occupying octahedral sites. The next hydrogen is added to an oxo- bridge of the opposing site of the cluster. This site preference is due to the fact that the H atoms carry a positive charge and thus prefer to occupy the farthest sites. The second hydrogen adds a single unpaired electron to the spin system, because the SE interaction is already broken. Similarly, the third hydrogen again only adds its own unpaired electron, because it attaches to a Fe3O octahedral oxygen, which is not a SE coordinator since it is not a bridging oxo- ligand. The fourth hydrogen attaches to another Fe3O and just adds its own unpaired electron. When five hydrogen atoms are added, the Fe6O6 octahedron structure begins to become greatly distorted, the overall cluster becomes cubic and the total spin S increases to 6.5. Adding the sixth hydrogen atom continues this pattern: two hydroxo- bridges are formed for each Fe3O3- FeO3 and the “spin -up” atoms are arranged in an arm chair configuration while the octahedron stretches out along its axis. The configuration with seven hydrogen atoms added is similar in its spin arrangement; however, since the hydrogen atoms are not symmetrically arranged, the cluster overall is much more amorphous.


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When eight hydrogen atoms are added, a unique singlet state is formed (see Fig. 2). It is presented by two Fe4(OH)4 layers attached to one another by four oxo- bridges and iron atoms with the same local spin magnetic moments are located in the same plane. The iron atoms in these two planes are AFM coupled. This type of arrangement is quite similar to the case of classical SE between two ndmetal atoms connected via an sp-atom, only the role of a single atom is played by a Fe4 square and the bridge is composed of four oxygen atoms. One could name such a configuration as corresponding to a collective SE interaction. The spin exchange interactions of this unique configuration will be explored in the next section using the Broken-Symmetry (BS) method. Adding a hydrogen atom to a bridge oxygen of this cluster greatly distorts the geometrical structure and the total spin S of the ground-state Fe8O12H9 cluster increases to 5. Adding hydrogen atoms to three other bridge oxygen atoms results in the FM ground state of Fe8O12H12, whose total spin of 14 exceeds by two the total spin of the bare Fe8 cluster (see Fig. 2). The geometry of this cluster is one where the hydroxo- bridges are projected outwards while the iron atoms are compressed towards the center of mass, allowing them to be higher coordinated. This may be explained by the fact that ligating atoms repel one another and would tend towards maximal separation, which in turn compresses the cluster frame. Overall, one can conclude that the magnetic properties of the (Fe2O3)4 cluster greatly depends on where the hydrogenation occurs on the cluster and that the hydrogenation weakens the separating power of oxo-bridges.

3.2 Exchange coupling constants of Fe 8 O 12 H 8 In order to gain insight into the exchange coupling constants of a hydrogenated Fe8O12H8 cluster, we use the Ising-type Hamiltonian: = − ⋅ #(1)

where a negative value of corresponds to an AFM spin alignment between i and j atoms and a positive value indicates a FM spin alignment. It is assumed that the spin orbitals do not overlap,


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although such an assumption is generally not satisfied. The most common approach to mapping DFT results onto such a Hamiltonian is called the Broken-Symmetry (BS) approach, originally pioneered by Noodleman et al.45 The principles of this approach are as follows. For a complex containing two transition metal atoms, the wave function described by two determinants in the spin-polarized case will belong to the same point group as the nuclear framework when the state is ferromagnetic. In a state with AFM spin coupling, there is an excess spin density on one metal atom and an excess spin density on the other metal atom, making these atoms non-equivalent. Therefore, the symmetry of an AFM state should be lower than the nuclear frame symmetry. This can be accomplished in spin-polarized DFT by constructing the spatially broken-symmetry Kohn-Sham determinant as

( ) | #(2) = |("#$%)ψ' ψ

Where ψ' and ψ) are “magnetic” orbitals which means that in the dimer case, ψ' will be localized to ( ) will be localized to center B, when summed over all space the total spin density will be center A and ψ equal to zero. To obtain a BS solution, the orbitals of the determinant in Eq. 2 are optimized in the usual fashion until the total energy minimum is reached. Next, the optimized BS wave function is projected out onto a pure spin state. It should be noted that the orbitals obtained in the BS case can depend strongly on the exchange-correlation functional used. In our calculations, we first arrived at the BS solution by considering all consecutive spin flips between the ferromagnetic and AFM states, and have used this state for the preparation of proper guess states in computations of the Jij constants between all eight iron atoms. The square of the total spin +, - . = 2S(S+1) of a two-determinant wavefunction with the spin multiplicity 2S +1 corresponding to a pure state is defined as: +, - . = /

01 − 02 01 − 02 3 +/ 3 #(3) 2 2


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where 01 and 02 are the numbers of electrons in the corresponding spin representations. In the case of two transition metal atoms A and B, one assumes that the ferromagnetic high-spin (HS) and antiferromagnetic broken-symmetry states are eigenfunctions of ,67 and ,6 operators and uses the intermediate coupling approximation46 to obtain:

=−

89 +, .9

− 8 #(4) − +, - .

The BS method deduced for two interacting transition metal atoms can be adapted to polymetallic systems after some modifications. In cases with more than two transition metal atoms, there are a number of BS states corresponding to different local total spin combinations, and we associate these BS states with different Ising spin states, which can be used for the generation of a set of linear equations. There are generally more spin combinations than necessary to determine all distinct Jij constants, and therefore the resulting fit depends on the choice of representative local total spin combinations. It has been previously found47,48 that exchange coupling constants for polynuclear systems best agreed with experimental results when non-spin-projected energies are used:

=

(8 − 89 ) ∀(,; ≥ ,- )#(5) 2,; ,- + ,-

where S1 and S2 are the local spin moments. The reason for the choice of this expression can be explained by the fact that wave functions composed of DFT spin-orbitals are generally less spin contaminated than wave functions built from Hartree-Fock orbitals49. We assume that all iron atoms in our system are in the low spin states with the total spin S = 2. For a cubic-like arrangement of eight iron atoms shown in Fig. 3, there are 28 different exchange constants. The model Hamiltonian can be written down as


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= 8? − ;- @ A − ;B @ C − ; @ D − -B A C − - A D − B C D − ;E @ F − -G A H − BI C J − K D L − EG F H − EI F J − EK F L − GI H J − GK H L − IK J L − ;G @ H − -E A F − -K A L − G D H − I D J − BK C L − ;I @ J − EB F C − ;K @ L − -I A J − BG C H − E D F #(6) In this expression, the exchange coupling constants are separated as follows: the first line contains the constants which correspond to the top layer, the second line includes the constants between iron atoms directly across one another but located on different layers, the third line includes the constants within the bottom layer, the fourth line contains the constants between the layer crossed on a face while the last line includes the constants between different layers crossed on a bisecting plane (see Figs 3 and 4). A least-squares-fitting was performed for an overdetermined linear system of 37 equations to obtain a fit for 28 Jij constants. In order to write down all required equations, we generated a total of 15 singlet states, 17 intermediate single-flipped states with 2S+1=9, five states intermediate double-flipped states with 2S+1=17, and a single high spin state with 2S+1=31, which is higher than the ground state by +1.82 eV. The states closest in total energy to the ground state are presented in Figure 3. To generate the singlet and intermediate states we made use of the fragment method of the Gaussian 09 program and fully optimized the geometries of each state. The system of 37 linear equations to be fit is too large to be included here and thus we include an abridged version; the full system may be found in the ESI. K

1 8N = 8? + (2, , + , ) #(7) 2 ,P; Q

Where 8N represents the DFT energies of the 37 BS states we prepared when solving for the J constants. Judging by a symmetrical appearance of the ground state geometry of Fe8O12H8 in Fig. 2, one could expect that some of the exchange coupling constants are equal to each other and this would decrease


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the number of equations to be solved. However, we found that this is not the case here. The reason for asymmetry can be seen in Fig. 3: the geometrical structures of the states with each particular local total spin moment distribution undergo substantial distortions during optimizations. As such, no overall symmetry is present when calculating exchange coupling constants using the linear equations corresponding to Eq. 5. It should also be noted that there are appreciable error margins, approximately 40 cm-1, in the exchange constants because of the relatively large total energy variations due to the geometry distortions. The individual exchange constants are summarized in Figure 4. The average exchange coupling constants within the top and bottom layers are only 27 cm-1 and -10 cm-1, respectively which is contrasted to the value of -172 cm-1 which is the average exchange between the layers, this value appears to be in concordance with experimental exchange constants of various oxo-bridged dimers.50 The face crossed average is 15 cm-1 while the bisecting plane crossed average is -4cm-1. We performed statistical analysis of our linear fitting model and found that the coefficient of determination S - was 0.9962. Overall, one can see that the Fe8O12H8 cluster has a strong AFM coupling between weakly spin coupled hydroxo-layers. This can be best described as collective super-exchange between the two fouriron atom squares connected with four oxo-bridges. The surprising strength of this collective superexchange can be rationalized by the fact that SE between the layers will compete with SE within the layers. For example, in the un-hydrogenated Fe8O12, SE between atoms 1 and 5 will compete with SE between 1 and 2 and SE between 1 and 3 (see the atom enumeration in Fig. 3). Hydrogenation of the oxygen atoms removes the competition (which was also previously studied for various iron-oxide clusters34), and favors a strong collective SE.

3.3 QTAIM analysis


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The topology of the electron density of the ground AFM (2S+1=1) and high spin (2S+1=31) states was analyzed using Bader’s quantum theory of atoms in molecules (QTAIM)51. After finding the critical points, we checked and made sure that the Poincaré-Hopf relationship was satisfied. Both the AFM and high spin states have the same number of critical points (79); however, the topology of the electron density is very different. When we consider just bond critical points (BCP) that involve Fe-Fe interactions we note that the AFM state has 6 and the high spin state has 7 points (see Fig. 5) . The seventh BCP for the high spin state is located approximately where the AFM state’s cage critical point (CCP) is. This implies that the AFM geometry can be described as a cage while the high spin state has some crisscross Fe-Fe bonding between layers. When we consider the geometry of the high spin state, we note that the layers are significantly displaced compared to the AFM state. We also tabulated some parameters which may be useful in determining the nature of the bonds including the local electron density T(U), the Laplacian of the electron density ∇- T, the local potential energy density V(r), kinetic energy density G(r) ,total energy density H(r) and relative kinetic energy density G(r)/ T (Table S1 and S2). The Fe-Fe type BCPs generally have a small overall density as well as a small but positive ∇- T which indicates covalency. When comparing Fe-Fe BCP’s between AFM and high spin states, we note that the high spin state has a large distribution of values while the AFM BCP’s have much less variance. For equivalent Fe-Fe BCP’s, of which there are only 4, the high spin state generally has larger G(r)/ T and ∇- T values which means these interactions are more ionic in nature. On the contrary, when we compare BCP’s corresponding to Fe-OH (intralayer) and Fe-O (interlayer) interactions we notice that the local relative kinetic energy densities and ∇- T are larger for the 2S+1=1 state, implying that these interactions are more ionic. This is confirmed by Natural Bond Order analysis

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when we compared the charge populations: the average

charge on an iron atom is +0.95 for 2S+1=1 and +0.71 for 2S+1=31 states (see Figs S2 and S3). When comparing Fe-OH-Fe and Fe-O-Fe BCP parameters, we note that the hydrogen-free oxygen atoms cause more ionic bonding than those with hydrogen ligands, which one might expect from chemical intuition.


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According to the NBO analysis of spin populations, the average spin on an iron is 3.2 for the 2S+1=1 state and 3.31 for the 2S+1=31 state (Table S3).

4. Conclusion The magnetic behavior of the hydrogenated Fe8O12 cluster was found to vary in a relatively complicated way as the cluster becomes progressively more hydrogenated. The geometrical structure of the initial Fe8O12 cluster is that of an Fe6O6 octahedron capped with two FeO3 groups on the opposing faces, and its ground state is an antiferromagnetic singlet. When a single hydrogen is added to this cluster, it breaks the layered super-exchange, causing the total spin of the cluster to greatly increase as all the local spin magnetic moments within the central octahedron become parallel to one another. Further hydrogenation up to n=7 does not lead to a larger total spin magnetic moment of the corresponding ground states. Upon hydrogenating two-thirds (n=8) of the oxygen bridges, we found that a barrel-like geometry becomes more favorable than those containing a coordinated octahedron. In the lowest total energy state with this barrel geometrical structure, a very strong collective super-exchange between two Fe4(OH)4 layers is found. The strong SE interaction is promoted by the reduction of the competition of AFM interactions between and inside iron atom squares. This is due to the formation of the in-layer hydroxyl-bridges, which themselves are not strong magnetic coordinators, whereas oxobridges are strong AFM coordinators. We also found when calculating the exchange constants for the lowest total energy singlet state of the Fe8O12H8 cluster, that no symmetry could be assumed for the exchange coupling constants, despite apparent symmetry of the initial cluster. This is because flipping local total spin magnetic moments greatly changes the bonding patterns of the cluster, which in turn leads to large changes in the geometry of the cluster. One can conclude that it is not reasonable to assume symmetry for the exchange coupling constant in the case of clusters that are not geometrically rigid. When the cluster Fe8O12 is fully hydrogenated (n=12), a ferromagnetic state is preferred where every hydrogen atom


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donates an extra third of an electron spin to the total spin of the cluster. This causes the Fe8O12H8 cluster to have a higher spin magnetic moment than the bare Fe8 cluster.

SUPPORTING INFORMATION The ESI contains a description of our Broken-Symmetry calculations as well as a statistical analysis of the exchange values. Also included are the geometries of un-hydrogenated Fe8O12 and tables of the QTAIM parameters for M=1 and M=31. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]

NOTES The authors declare no competing financial interest.

FIGURE CAPTIONS Figure 1. Geometrical structures and total spins of the lowest total energy states of Fe8O12Hn for n =0 – 11. Iron atoms with the total spin-down and spin-up magnetic moments are in blue and dark red, respectively. Figure 2. Geometrical structures and total spins of the lowest total energy states and the high spin states of Fe8O12, Fe8O12H8, Fe8O12H12. Iron atoms with the total spin-down and spin-up magnetic moments are in blue and dark red, respectively. Figure 3. Spin configurations of Fe8O12H8 generated with the fragment method. The top three panels show the geometrical configuration as well as the spin configuration of the ground state singlet. The light blue atoms are spin-up and the red are spin-down. The next five panels show the configurations closest in energy to the GS, the arrows indicate the geometrical deformation of a state when compared to the GS. The last panel shows the high spin all-up configuration. Figure 4. The magnitudes of all 28 exchange constants in cm-1 rounded to the nearest integer. For clarity, we recommend the reader compare this figure to the first 2 panels of Fig. 3. The last two panels


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of this figure can be thought of as xy and xz bisecting planes if we place the z-axis normal to the 1-2-3-4 face. Figure 5. The ground state geometries and critical points of Fe8O12H8 2S+1=1 and 31. The iron atoms are in purple, the oxygens are in red and the hydrogens are in white. Bond critical points are small orange spheres, ring critical points are small yellow spheres and cage critical points are small green spheres.

Acknowledgement This work is partially supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-96ER45579. Resources of the National Energy Research Scientific Computing Center supported by the Office of Science of the U.S. Department of Energy under Contract no. DE-AC02-05CH11231 is also acknowledged.

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TOC


22

NPage H=230of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 N14H= 15 16 17 18 19 20 21 22 23 24 25 26 27 N28 H= 29 30 31 32 33 34 35 36 37 38 39 40 41 42 N43 H= 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2S+1 =1 3

NH= 1

N =2

H The Journal of Physical Chemistry

2S+1 = 8 NH= 4

2S+1 = 10

2S+1 =15

2S+1 = 8

2S+1 =14

NH= 8

2S+1 =14

NH= 10

9

NH= 5

2S+1 = 11

NH= 7

6

2S+1 = 9

NH= 11

2S+1 =1 ACS Paragon Plus Environment

2S+1 =1

2S+1 =22

Fe8O12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.93

1.88

6

1.7

3

1.7

2.7 2.93

1.8

0

-2.7

S=0 0.0 eV

S=0 0.0 eV

2.78

1.8

9

Fe8H8O12

2.7 0.6

S=12 +1.37 eV

S=11 +0.94 eV

Fe8O12H12

S=0 +1.39 eV

S=14 0.0 eV


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Figure 3. Spin configurations of Fe8O12H8 generated with the fragment method. The top three panels show the geometrical configuration as well as the spin configuration of the ground state singlet. The light blue atoms are spin-up and the red are spin-down. The next five panels show the configurations closest in energy to the GS, the arrows indicate the geometrical deformation of a state when compared to the GS. The last panel shows the high spin all-up configuration. 192x247mm (300 x 300 DPI)



Figure 4. The magnitudes of all 28 exchange constants in cm-1 rounded to the nearest integer. For clarity, we recommend the reader compare this figure to the first 2 panels of Fig. 3. The last two panels of this figure can be thought of as xy and xz bisecting planes if we place the z-axis normal to the 1-2-3-4 face. 177x158mm (300 x 300 DPI)


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Figure 5. The ground state geometries and critical points of Fe8O12H8 2S+1=1 and 31. The iron atoms are in purple, the oxygens are in red and the hydrogens are in white. Bond critical points are small orange spheres, ring critical points are small yellow spheres and cage critical points are small green spheres. 200x221mm (300 x 300 DPI)



TOC 157x139mm (300 x 300 DPI)


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Collective Superexchange and Exchange ... - ACS Publications

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