No source-free exchange-correlation magnetic fields in non-collinear spin-density functional theory

Abstract

The source-free condition, $\nabla \cdot {\mathbf{B}}_{\mathrm{xc}}=0$, where ${\mathbf{B}}_{\mathrm{xc}}=\delta E_{\mathrm{xc}}/\delta \mathbf{m}$ is the exchange-correlation (xc) magnetic field of non-collinear spin-density functional theory (SDFT) and $\mathbf{m}$ the magnetization, is widely believed to be exact. Recent studies report that projecting the field of a parent functional onto its source-free part improves the predicted magnetic moments. We show that the condition violates the global spin-rotation symmetry of $E_{\mathrm{xc}}$ and therefore cannot be exact. A source-free field is exact only in a transverse theory whose xc functional differs from $E_{\mathrm{xc}}$ and is itself not invariant under spin rotations. We examine a source-free (SoF) construction, an exchange-correlation energy given by a parent SDFT functional evaluated at the transverse magnetization ${\mathbf{m}}_{\perp }$. Its field is divergence-free, remains a functional derivative and exerts local torques, but still, it breaks global spin-rotation symmetry. With the LSDA as parent we examine the \ce{Mn2} molecule. The SoF and the locally collinear (LoC) LSDA both give too short a bond and too large a bond energy, but SoF finds a broken-symmetry antiferromagnetic solution, consistent with the experimentally observed ${}^1{\Sigma }_g^{+}$ ground state, with a coupling of the right sign and roughly the right size, while LoC finds a high-spin ${}^{11}{\Pi }_u$ ferromagnet and the wrong sign. The broken symmetry of SoF, however, produces severe errors in directional properties: a rigid spin rotation of the magnetization changes the energy by about $1\mathrm{eV}$, in a weak uniform field the magnetization points perpendicular to the field instead of antiparallel to it, and the molecule develops a spurious magnetic anisotropy.