Electronic structure and exchange interactions in ${\mathrm{V}}_{15}$ magnetic molecules: $\mathrm{LDA}+\mathrm{U}$ results

Single-molecule magnets of the type ${\mathrm{V}}_{15}\left({\mathrm{K}}_6\left[{\mathrm{V}}_{15}{\mathrm{As}}_6{\mathrm{O}}_{42}\left({\mathrm{H}}_2\mathrm{O}\right)\right]\bullet 8{\mathrm{H}}_2\mathrm{O}\right)$ have attracted a great deal of attention recently, being promising systems for studying low-temperature spin-relaxation and quantum-spin tunneling. To understand in detail the internal magnetic and electronic structure, and the intramolecular interactions responsible for the formation and low-energy excitations in ${\mathrm{V}}_{15}$ molecules, we have performed electronic structure calculations using the $\mathrm{LSDA}+\mathrm{U}$ approach. The calculated values of magnetic moments and charge states of vanadium ions agree well with experiments, thus confirming the $V^{4+}$ state of vanadium ions with a well-defined spin $1∕2$. We found that the account of the on-site Coulomb repulsion is important for correct description of ${\mathrm{V}}_{15}$ internal properties; in particular, for the values of the on-site repulsion parameter $U\sim 4\text{–}5\mathrm{eV}$, we can achieve good agreement with known properties of ${\mathrm{V}}_{15}$, such as the temperature dependence of susceptibility, and the energies of the low-lying eigenstates of the spin Hamiltonian.

Figure 1

The sketch of arrangement of vanadium ions in the ${\mathrm{V}}_{15}$ molecule. The arrows schematically denote the individual spins.

Figure 2

DOS of $d$-electrons of inequivalent V1, V2, and V3 ions. The DOS are calculated for $U=4\mathrm{eV}$ and $J=0.8\mathrm{eV}$; the spin configuration is AFM1 (spins of V1 and of V3 are down, spins of V2 are up).

Figure 3

Dependence of the $2p$ oxygen (dashed lines) and $3d$ vanadium (solid lines) DOS on the Coulomb repulsion parameter $U$, for the values of $U$ from 0 eV to 8 eV.

Figure 4

(a) Temperature dependence of the effective magnetic moment of ${\mathrm{V}}_{15}$ molecule. Theoretical curves are calculated using the exchanges corresponding to different values of $U$. Experimental results, obtained in the field of 0.5 Tesla, are presented for comparison. All calculated curves corresponding to $U$ near 5.0 eV are close to the experimental curve. (b) The low-temperature part of these data presented in more detail.

Figure 5

The spin-Hamiltonian spectra of ${\mathrm{V}}_{15}$. The spectra were obtained by diagonalizing the Heisenberg-exchange spin Hamiltonian with the exchange parameters derived from: (a) LSDA (the exchange parameters were taken from Ref. 11 and divided by 2.9, as suggested there); (b) from $\mathrm{LSDA}+\mathrm{U}$ calculations with $U=5\mathrm{eV}$, without any extra scaling. The upper graph (a) shows only the low-energy part of the spectrum because the highest levels in this case have energies above 7000 K.