^{1,a)}

### Abstract

Low-symmetry distortions are present in cubanes such as , but their effects on electron delocalization properties are not well-understood. Mixed-valence cubanes often exhibit experimentally measurable “pair delocalization” of a delocalizable electron. An important question is, what is the role of physical interactions (vibronic, electronic, exchange) and symmetry distortions in determining the electron delocalization pattern? Semiclassical models are used to explore the electron delocalization patterns of tetragonally distorted mixed-valence cubanes comprising four metal centers with bridging ligands, a single delocalizable “excess” electron, and either closed-shell or open-shell ion cores. Phase diagrams show that distorted ground state cubanes with antiferromagnetic exchange (as found in nature) have delocalization patterns qualitatively similar to those of an model with no Heisenberg exchange, suggesting that exchange is not necessarily a dominant factor in determining electron delocalization properties. The open-shell model reveals two types of pair delocalization for the ground state, with differing dimer subunit spins for compressed and elongated geometries. Previous studies emphasize the importance of exchange interactions for pair delocalization. Here, it is shown that electron exchange is not always necessary for pair delocalization and that it can be achieved with relatively small tetragonal distortions from tetrahedral symmetry. The results contradict those of an earlier theoretical study of distorted clusters, which concluded that distortions of lower symmetry than are necessary to induce a transition to pair delocalization.

I. INTRODUCTION

II. THEORY

A. One-electron system

B. Five-electron system with Heisenberg exchange

III. RESULTS

A. One-electron, closed-shell ion core model

B. Five-electron model with antiferromagnetic exchange

IV. CONCLUSION

### Key Topics

- Exchange interactions
- 22.0
- Ground states
- 21.0
- Vibronic interactions
- 16.0
- Phase diagrams
- 15.0
- Antiferromagnetism
- 13.0

## Figures

Schematic diagram of a tetranuclear “cubane” iron-sulfur cluster. The high-symmetry structure comprises interpenetrating tetrahedra of iron atoms and sulfur atoms. In this study, the effects of distortions along the axis are analyzed.

Schematic diagram of a tetranuclear “cubane” iron-sulfur cluster. The high-symmetry structure comprises interpenetrating tetrahedra of iron atoms and sulfur atoms. In this study, the effects of distortions along the axis are analyzed.

Phase diagrams showing the delocalization pattern of an excess electron in a cluster with a closed-shell ion core for (a) compressed geometries and (b) elongated geometries. The quantity gives a measure of the relative magnitudes of electronic and vibronic coupling for tetranuclear clusters. corresponds to undistorted structures; corresponds to the limit of tetragonal compression (square planar structure); corresponds to the limit of tetragonal elongation (noninteracting dimers). The labels D, L, and P indicate full delocalization, localization, and pair delocalization of the excess electron, respectively.

Phase diagrams showing the delocalization pattern of an excess electron in a cluster with a closed-shell ion core for (a) compressed geometries and (b) elongated geometries. The quantity gives a measure of the relative magnitudes of electronic and vibronic coupling for tetranuclear clusters. corresponds to undistorted structures; corresponds to the limit of tetragonal compression (square planar structure); corresponds to the limit of tetragonal elongation (noninteracting dimers). The labels D, L, and P indicate full delocalization, localization, and pair delocalization of the excess electron, respectively.

Contour diagram illustrating pair delocalization of an excess electron in a tetranuclear cluster with a closed-shell ion core. The cluster has an elongated structure, with parameters , , and (i.e., , ). The contours are plotted at intervals of 0.02 units, where is the symmetric ligand vibration frequency. The diagram is a section through the potential energy surface, plotted as a function of the dimensionless symmetrized ligand displacement coordinates and , with .

Contour diagram illustrating pair delocalization of an excess electron in a tetranuclear cluster with a closed-shell ion core. The cluster has an elongated structure, with parameters , , and (i.e., , ). The contours are plotted at intervals of 0.02 units, where is the symmetric ligand vibration frequency. The diagram is a section through the potential energy surface, plotted as a function of the dimensionless symmetrized ligand displacement coordinates and , with .

Eigenvector coefficients ) as a function of the degree of metal center distortion for (a) compressed structures with and (b) elongated structures with . and correspond to the undistorted structure.

Eigenvector coefficients ) as a function of the degree of metal center distortion for (a) compressed structures with and (b) elongated structures with . and correspond to the undistorted structure.

Phase diagrams showing the delocalization pattern of an excess electron in an cluster with open-shell ion cores for (a) compressed geometries and (b) elongated geometries. The quantity gives a measure of the relative magnitudes of electronic and vibronic coupling for tetranuclear clusters. The magnitudes of the antiferromagnetic exchange interactions are and the vibronic coupling parameter is given by . corresponds to undistorted structures; corresponds to the limit of tetragonal compression (square planar structure); corresponds to the limit of tetragonal elongation (noninteracting dimers). The labels D, L, and P indicate full delocalization, localization, and pair delocalization of the excess electron, respectively.

Phase diagrams showing the delocalization pattern of an excess electron in an cluster with open-shell ion cores for (a) compressed geometries and (b) elongated geometries. The quantity gives a measure of the relative magnitudes of electronic and vibronic coupling for tetranuclear clusters. The magnitudes of the antiferromagnetic exchange interactions are and the vibronic coupling parameter is given by . corresponds to undistorted structures; corresponds to the limit of tetragonal compression (square planar structure); corresponds to the limit of tetragonal elongation (noninteracting dimers). The labels D, L, and P indicate full delocalization, localization, and pair delocalization of the excess electron, respectively.

Eigenvector coefficients ) as a function of the degree of tetragonal elongation in the open-shell system, when , (i.e., ). corresponds to the undistorted structure. Small distortions from this pair-delocalized structure cause electron localization; further distortions cause a return to pair delocalization.

Eigenvector coefficients ) as a function of the degree of tetragonal elongation in the open-shell system, when , (i.e., ). corresponds to the undistorted structure. Small distortions from this pair-delocalized structure cause electron localization; further distortions cause a return to pair delocalization.

Contour diagrams (details as in Fig. 3) illustrating pair delocalization of an excess electron in a tetranuclear cluster with an open-shell ion core. (a) Undistorted cluster with parameters ; ; and . (b) Elongated cluster with parameters , , (i.e., , ); , and .

Contour diagrams (details as in Fig. 3) illustrating pair delocalization of an excess electron in a tetranuclear cluster with an open-shell ion core. (a) Undistorted cluster with parameters ; ; and . (b) Elongated cluster with parameters , , (i.e., , ); , and .

## Tables

Spin basis states used in the five-electron model. Label 1 represents the extra electron and the site of localization (, , , or ). is the local spin of the site containing the extra electron. Basis states having and are known as non-Hund states and Hund states, respectively. and are the intermediate spins of the remaining sites (, , , or ), where , , and are the spins of the electrons in core orbitals , , and . The total spin of the cluster can take the values .

Spin basis states used in the five-electron model. Label 1 represents the extra electron and the site of localization (, , , or ). is the local spin of the site containing the extra electron. Basis states having and are known as non-Hund states and Hund states, respectively. and are the intermediate spins of the remaining sites (, , , or ), where , , and are the spins of the electrons in core orbitals , , and . The total spin of the cluster can take the values .

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