Forces for structural optimizations in correlated materials within a DFT+embedded DMFT functional approach

We implemented the derivative of the free energy functional with respect to the atom displacements, so called force, within the combination of density functional theory and the embedded dynamical mean-field theory. We show that in combination with the numerically exact quantum Monte Carlo (MC) impurity solver, the MC noise cancels to a great extend, so that the method can be used very efficiently for structural optimization of correlated electron materials. As an application of the method, we show how strengthening of the fluctuating moment in FeSe superconductor leads to a substantial increase of the anion height, and consequently to a very large effective mass, and also strong orbital differentiation.

Figure 1

Force on Se atom when displaced in $z$-direction, and the corresponding change of the free energy. The free energy is calculated from the functional Eq. (9), and is compared to integrated force. We show both the free energy and $F+T{S}_{\mathrm{imp}}$. The latter is directly computed in our method, while the former requires additional integration over the temperature. The quantum Monte Carlo noise is approximately one order of magnitude smaller when computing energy from the force than computing it directly from the functional.

Figure 2

The convergence of the free energy $F+T{S}_{\mathrm{imp}}$ and force with the number of DMFT iterations. The last seven steps are converged, but display typical Monte Carlo noise, which is more severe in free energy than in computing force. When the force is multiplied with the displacement from equilibrium $\mathrm{\Delta }\mathrm{r}$, to recover the units of energy, the noise is more than one order of magnitude smaller than the corresponding noise of the free energy. The data correspond to $z_{\mathrm{Se}}=0.25$. For clarity we subtracted a constant from both the energy and the force.

Figure 3

The optimized $z$ position of Se atom for different values of Hund's coupling $J_H$. The experimental values exp(a) and exp(b) correspond to x-ray measurements of Refs. [40] and [41], respectively.

Figure 4

Mass enhancement of different orbitals vs Hund's coupling $J_H$, when $z_{\mathrm{Se}}$ is optimized theoretically. Note logarithmic scale for mass.