Exchange interaction in pyrochlore vanadates Lu$_{2}$V$_{2}$O$_{7}$ and Y$_{2}$V$_{2}$O$_{7}$: Ab initio approach

The exchange interaction in vanadates with the pyrochlore structure, namely, Lu$_{2}$V$_{2}$O$_{7}$ and Y$_{2}$V$_{2}$O$_{7}$, has been investigated using the first-principles approach. The isotropic exchange coupling constants have been determined. The calculations have been performed within the unrestricted Hartree–Fock (UHF) approximation, as well as in the framework of the density functional theory (DFT), using hybrid functionals. It has been shown that, in the description of the exchange interaction in the compounds under investigation, the nonlocal Hartree–Fock exchange should be taken into account. The splitting patterns of the 3$d^1$ level of the V$^{4+}$ ion in the crystal field have been obtained within the model approach. The calculation has been carried out in the approximation of point charges taking into account the spin–orbit interaction. It has been revealed that the “orbital liquid” state cannot be observed in the compounds under investigation, because the exchange interaction energy is significantly less than the energy spacing between the ground state and the first excited state. The orbital ordering has been analyzed, and the spin density maps have been constructed.