Rotational $g$ factors and Lorentz forces of molecules and solids from density functional perturbation theory

Applied magnetic fields can couple to atomic displacements via generalized Lorentz forces, which are commonly expressed as gyromagnetic $g$ factors. We develop an efficient first-principles methodology based on density functional perturbation theory to calculate this effect in both molecules and solids to linear order in the applied field. Our methodology is based on two linear-response quantities: the macroscopic polarization response to an atomic displacement (i.e., Born effective charge tensor), and the antisymmetric part of its first real-space moment (the symmetric part corresponding to the dynamical quadrupole tensor). The latter quantity is calculated via an analytical expansion of the current induced by a long-wavelength phonon perturbation, and compared to numerical derivatives of finite-wave-vector calculations. We validate our methodology in finite systems by computing the gyromagnetic $g$ factor of several simple molecules, demonstrating excellent agreement with experiment and previous density functional theory and quantum chemistry calculations. In addition, we demonstrate the utility of our method in extended systems by computing the energy splitting of the low-frequency transverse-optical phonon mode of cubic ${\mathrm{SrTiO}}_3$ in the presence of a magnetic field.

Figure 1

Convergence of the $g$ factor of ${\mathrm{H}}_2$ (with $d_{\text{exp}}=1.4$ bohr) with respect to the plane-wave cutoff. Calculations are performed using a single $\mathbf{k}$ point ($\mathrm{\Gamma }$) with a box size of ${20}^3{\mathrm{bohr}}^3$.

Figure 2

(a) Calculation of rotational $g$ factor of ${\mathrm{H}}_2$ (with $d_{\text{exp}}$=1.4 bohr) using the expression for $\partial m_a/\partial {\omega }_b$ from Eq. (28) (dots) and from Eq. (26) (triangles) vs inverse of the simulation cell size side length. (b) Convergence of terms in Eq. (28) vs inverse of the simulation cell size. Purple dashed line in (b) is the extrapolated value for the quadrupole term; the purple cross in (a) is the $g$ factor calculated with the extrapolated quadrupole term.

Figure 3

Cartoon illustrating the relaxed geometries used in this paper for (a) ${\mathrm{C}}_5{\mathrm{H}}_5\mathrm{N}$ (the N atom is in blue) and (b) ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{F}$ (F atom is in light blue).