Revisiting ab-initio excited state forces from many-body Green's function formalism: approximations and benchmark

Abstract

Ab initio techniques for studying the optical and vibrational properties of materials are well-established, but only a few recent studies have focused on the interaction between excitons and atomic vibrations. In this paper, we revisit the excited state forces method, based on GW/BSE and DFPT calculations and provide a practical implementation and straightforward workflow. We fixed issues from Ismail-Beigi and Louie's implementation in \textcolor{blue}{Phys. Rev. Lett. 90, 076401 (2003)} and use an approximation for GW-level electron-phonon coefficients that improves our calculations accuracy. We explore its technical aspects, including convergence and the quality of approximations used for CO molecule, LiF and monolayer MoS$_2$. We successfully apply this method to investigate diverse kinds of self-trapped excitons in LiF, including polaronic excitons, discuss excited state relaxation strategies and project excited state forces in phonon displacement to explore exciton-phonon interactions. Our results provide the tools to study exciton-phonon related phenomena in molecules in materials, including coherent phonon generation, such as resonant Raman, self-trapped excitons and excitonic insulators.

Figures (24)

• Figure 1: Ground state energy calculated at DFT level (black line), excited state energies calculated for singlet (blue line) and triplets (purple line). Vertical lines correspond to equilibrium distances 1.12 Å, 1.21 Å, and 1.26 Å, corresponding to ground, excited triplet, and singlet, respectively.
• Figure 2: Diagram of the ESF workflow that combines electron phonon matrix elements from DFPT and exciton coefficients from BSE
• Figure 3: Sum of electron-phonon coefficients due to displacements parallel to CO molecule bond of oxygen and carbon atoms using a threshold of $10^{-14}$ (left) and $10^{-22}$ (right) for the self-consistencey of derivatives in DFPT. Both summed represent an acoustic mode for which this sum should be zero. For some off-diagonal ELPH matrix elements this sum is not zero, regardless of the convergence of parameters used in DFPT calculations, and therefore our ASR (eq. \ref{['eq:ASR']})
• Figure 4: Excited state forces for the singlet (left) and triplet (right) excitons in CO as function of $R_{\rm{CO}}$ bond length using different methods. Our reference data is Finite Differences (FD) in black circles. Singlet results using Ismail-Beigi and Louie's (IBL) method (eq. \ref{['eq:ibl_esf_eq_full']}) not including kernel derivatives deviate from FD by $\sim$ 4 eV/Å as in reference IsmailBeigi2003. Our method (eq. \ref{['eq:excited_state_forces_our_method']}) has a great agreement with IBL method when both use the same kind of ELPH coefficients. Our scheme to renormalize ELPH coefficients (eq. \ref{['eq:renormalization_elph_coeffs']}) improves the agreement of our ESF with FD in the region $R_{\rm{CO}} < 1.1 \rm{\AA}$.
• Figure 5: QP energy variations vs DFT energy variations. Those energies were calculated from changing the $R_{CO}$ bond length from 1.1 to 1.35 Å to bands 1 to 10 in CO molecule. The first five bands are full occupied. Vertical $y=0$, horizontal $x=0$ and diagonal $y=x$ dashed black lines are guides to the eye.