  While anomalous diffusion coefficients with non-Arrhenius like temperature
dependence are observed in a number of metals, a conclusive comprehensive
framework of explanation has not been brought forward to date. Here, we use
first-principles calculations based on density functional theory to calculate
self-diffusion coefficients in the bcc metals Mo and $\beta$-Ti by coupling
quasiharmonic transition state theory and large displacement phonon
calculations and show that anharmonicity from thermal expansion is the major
reason for the anomalous temperature dependence. We use a modified Debye
approach to quantify the thermal expansion over the entire temperature range
and introduce a method to relax the vacancy structure in a mechanically
unstable crystal such as $\beta$-Ti. Thermal expansion is found to weakly
affect the activation enthalpy but has a strong effect on the prefactor of the
diffusion coefficient, reproducing the non-linear, non-Arrhenius "anomalous"
self-diffusion in both bcc systems with good agreement between calculation and
experiment. The proposed methodology is general and simple enough to be
applicable to other mechanically unstable crystals.
