Fig. 4: DFT calculations for the two molecular states.

a Calculated density of states (DOSs) of the two molecular states for the free neutral Dy@C₈₄ molecule in the near range of the Fermi level. The Fermi level is shifted to the HOMO. b Calculated energy barriers between two molecular states across the transition state (TS) under various gate electric fields. The gate electric field can effectively lower the energy barrier to a negligible level at ~0.25 V/\(\mathring{\rm A}\), thereby enhancing the transition probability between the two molecular states. The insert shows the interaction between the molecule and the Au₁₆ electrode. c Two possible orbital configurations of metal-cage hybrid states in two molecular states at the HOMO, with magnetic moments 4.16 \({\mu }_{{{{{{\rm{B}}}}}}}\) and 4.55 \({\mu }_{{{{{{\rm{B}}}}}}}\). d Upon switching the molecular states, the coupled magnetic moment of the ground state transforms from 4.16 \({\mu }_{{{{{{\rm{B}}}}}}}\) for state 1 to 4.55 \({\mu }_{{{{{{\rm{B}}}}}}}\) for state 2, while the electric dipole moment transforms from 0.70 eÅ to 0.64 eÅ. The two electric dipoles exhibit a relative angle of 168.4°. e The energy profile with structural diagrams of the Dy ion position (the upper panel) and the dipole moment in the z-direction (the lower panel) when climbing the energy barrier.