Coupled-cluster approach to vibronic effects in resonant inelastic x-ray scattering of quantum materials: Application to a $5d^1$ rhenium oxide

Abstract

First-principles analysis of the spectroscopic signatures of correlated quantum materials poses significant challenges due to the interplay between spin-orbit and vibronic couplings, as well as the need to describe both dynamic and static electron correlation to reach decent accuracy. In this work, we apply the equation-of-motion coupled-cluster (EOM-CC) method to derive the spin-orbit-lattice entangled vibronic states and predict the Re $L_3$ edge resonant inelastic x-ray scattering (RIXS) spectra of Ba$_2$MgReO$_6$. The EOM-CC yields interaction parameters in close agreement with those extracted from RIXS spectra, with errors of less than 5\%. In particular, the EOM-CC method allowed us to determine the weak vibronic coupling to the $T_{2g}$ vibrations, which is difficult to address experimentally. The simulated spectra indicate that vibronic coupling to the $T_{2g}$ modes gives rise to a shoulder on the elastic peak. Going beyond the conventional treatment, which focuses solely on $E_g$ modes, we show that vibronic couplings to both $T_{2g}$ and $E_g$ modes are required to account for the fine structure of the RIXS spectra. This work demonstrates that the EOM-CC method is a powerful tool for accurately predicting the complex local states at metal sites and spectroscopic signatures of correlated insulating materials.

Figures (10)

• Figure 1: The conventional unit cell of Ba$_2$MgReO$_6$ and the energy spectra on a $5d^1$ site. (a) The green, orange, light gray, and red spheres are Ba, Mg, Re, and O atoms, respectively. $a, b, c$ indicate the crystal axes of the conventional cell. (b) In the descending order of energy, the ligand-field, spin-orbit coupling, and vibronic coupling (DJT effect) determine the nature of the local quantum states. $10Dq$ and $\lambda$ are the ligand-field and spin-orbit coupling parameters. $\Gamma_7$ and $\Gamma_8$ are the irreducible representations of the spin-orbit multiplet $j_\text{eff}=1/2$ and $j_\text{eff}=3/2$ states, respectively, in the $O_h$ group. The dots in the last column indicate the presence of a large number of vibronic levels.
• Figure 2: The JT active modes. (a) The $E_gu$$(=2z^2-x^2-y^2)$ type, (b) $E_gv$$(=x^2-y^2)$ type, and (c) one of the $T_{2g}$ modes ($\zeta = xy$ type), where the $x, y, z$ directions correspond to the positive $a, b, c$ directions in Fig. \ref{['Fig_DP']}, respectively. Cyclic permutations of $xyz$ obtain the other two $T_{2g}$ modes. The green arrows indicate the displacement for the normal mode. These displacements correspond to the positive direction of the normal coordinates.
• Figure 3: The adiabatic potential energy surface (APES) for the linear JT model within the $j_\text{eff}=3/2$ multiplet states. (a) $U_\pm$ with respect to $\rho_E$ and $\rho_{T_2}$ (\ref{['Eq_Upm']}). The magenta and cyan points indicate the minimum and the saddle points, respectively. The plot shows the case for $E_{\text{JT}, E}^{(3/2)} > E_{\text{JT}, T_2}^{(3/2)}$. (b) The angular dependence of the continuum of minima of $E_g$ type. This forms a circle (red) around the high-symmetric point. In both plots, the origin of the coordinates corresponds to the degeneracy point of the ground electronic state.
• Figure 4: The rhenium cluster embedded in 1420 point charges. The cluster consists of ReO$_6$ and the six nearest Mg and eight nearest Ba ions. The quantum-mechanical cluster is cut out from the crystallographic structure. The blue dots represent the point charges.
• Figure 5: The $^2T_{2g}$ adiabatic potential energy surfaces with respect to (a) the $E_gu$ and (b) the $T_{2g}\zeta$ deformations. The green squares, the red points, and the black solid lines are, respectively, the $^2T_{2g}$ levels of the Re cluster without surrounding point charges, those of the Re cluster with surrounding point charges (PC), and the adiabatic potential energy surfaces of the model Hamiltonian. We show the displacements of the oxygen atoms by the JT active modes. For $\delta z$ and $\delta x$, see Fig. \ref{['Fig_JTmodes']}.