Structure Optimization, Dielectric and Magnetic Properties of Pure Silicon as well as Iron-Doped Silicon and Tin Clusters

Energetically-favored initial geometries were generated using a plane-wave Density Functional Theory-based (pw-DFT) Genetic Algorithm (GA) developed in our group. These calculations are performed using the Quantum Espresso software package. Possible candidates are then optimized on the PBE0/def2-TZVPP level of theory. The existence of local minima has been verified by frequency analysis. These calculations were carried out with the quantum chemical program Gaussian16.
Additionally, for the endohedrally doped Sn12 clusters the magnetic properties are calculated with ORCA v5.0.4 on the DFT level of theory. Since it was shown that the results often do not match the experimental values, complete active space self-consistent field calculations (CASSCF) are carried out, for the first time for such large systems, to get a better description of the electronic structure and, thus, more accurate results on the magnetic properties.

The comprehensive study of the geometrical structure of Si_N clusters with N = 12 − 30 shows that for this system the growth pattern is also dominated by prolate structures which are built of reappearing, very stable sub-units. Starting with the cluster Si₂₇ quasi- spherical shapes appear again. For Si₃₀ no more prolate shapes are found by quantum chemical calculations.
The calculation on the CASSCF level of theory of the g-factor which represents the magnetic properties revealed that the very different values observed experimentally for the three Sn₁₂ clusters doped with Cr, Fe and Mn originate from spin-orbit coupling (SOC) between the tin cage and the dopant. Since there is no SOC in Sn₁₂Mn, the g-value is equal to the value of the free electron, g = 2, as already predicted by DFT.

For most of the Si_N clusters with N = 12 − 30 an assignment of one structural isomer to the experimental data is possible. Additionally, it could be shown that, in contrast to Sn_N , for no cluster size two isomers are simultaneously present in the molecular beam. However, no distinct assignment of a structural isomer is possible for Si₂₆ and Si₂₈ based on our experiment, since the dielectric properties of all found isomers are very similar. Furthermore, it could be seen that a change of the exchange-correlation functional has a huge impact not only on the calculated value of the electric dipole moment but also on the energetic order of the structural isomers.
A more detailed analysis of the electronic excitations in Sn₁₂TM with TM = Cr, Mn and Fe, firstly confirmed that no significant spin-orbit effects influence the g-factor of Sn₁₂Mn. Because of this, the value for the g-factor was already predicted correctly by DFT calculations. For Sn₁₂Cr and Sn₁₂Fe the g-factor is slightly lower and much higher, respectively, than the value of the free electron. The much more time consuming CASSCF calculations reveal that the different shifts originate from different kinds of electronic excitations relevant for the g-factor.

Rivic, F.; Schäfer, R. Magnetism of single-doped paramagnetic tin clusters studied using temperature-dependent Stern–Gerlach experiments with enhanced sensitivity: impact of the diamagnetic ligand field and paramagnetic dopant. Phys. Chem. Chem. Phys. 26, 12982-12992 (2024) https://doi.org/10.1039/D4CP00890A

Rivic, F.; Lehr, A.; Schäfer, R.  Dielectric Behavior and Prolate Growth Patterns of Silicon Clusters Si_N with N = 12–30 by Cryogenic Electric Beam Deflection. J. Phys. Chem. A 128, 1853-1862 (2024) http://dx.doi.org/10.1021/acs.jpca.3c08432