Renormalization of Molecular Quasiparticle Levels at Metal-Molecule Interfaces: Trends across Binding Regimes

When an electron or a hole is added into an orbital of an adsorbed molecule the substrate electrons will rearrange in order to screen the added charge. This polarization effect reduces the electron addition and removal energies of the adsorbed molecule relative to those of the free molecule. Using a microscopic model of the metal-molecule interface, we illustrate the basic features of this renormalization mechanism through systematic $GW$, Hartree-Fock, and Kohn-Sham calculations for the molecular energy levels as function of the model parameters. We identify two different polarization mechanisms: (i) polarization of the metal (image charge formation) and (ii) polarization of the molecule via charge transfer across the interface. The importance of (i) and (ii) is found to increase with the metal density of states at the Fermi level and metal-molecule coupling strength, respectively.

Figure 1

(a) Schematic of a molecule’s HOMO and LUMO energy levels as it approaches a metal surface. For weak coupling (physisorbed molecule) the gap is reduced due to image charge formation in the metal. For strong coupling (chemisorbed molecule) dynamic charge transfer between molecule and metal reduces the gap further. (b) The model used in this study. (c) The semielliptical band at the terminal site of the TB chain and the resonances of the molecule.

Figure 2

Position of the HOMO and LUMO levels as a function of different model parameters for a weakly coupled molecule (small $\Gamma$). Note that dynamical polarization is completely absent in HF theory and incorrectly described by the “exact” KS theory. Open squares denote the exact total energy difference between the normal ground state and the ground state when the molecule has been constrained to hold one extra electron or hole. Lower right-hand panel shows on-site elements of the static, linear response function of the HOMO, LUMO, and terminal site of the TB chain. Clearly, the renormalization of the gap is due to polarization of the metal.

Figure 3

Left: Position of the molecule’s HOMO and LUMO levels as a function of the hybridization strength $t_{\mathrm{hyb}}$. $GW(\mathrm{mol})$ refers to a calculation where only the internal interactions on the molecule ${\widehat{U}}_{\mathrm{mol}}$ have been treated within $GW$ while the interactions with the metal ${\widehat{U}}_{\mathrm{ext}}$ have been treated within HF. Right: Static response function for the HOMO, LUMO, and terminal site of the TB chain. The drawing illustrates how dynamic charge transfer stabilizes the charged system and thereby reduces the gap.