Constrained-DFT method for accurate energy-level alignment of metal/molecule interfaces

We present a computational scheme for extracting the energy-level alignment of a metal/molecule interface, based on constrained density functional theory and local exchange and correlation functionals. The method, applied here to benzene on Li(100), allows us to evaluate charge-transfer energies, as well as the spatial distribution of the image charge induced on the metal surface. We systematically study the energies for charge transfer from the molecule to the substrate as function of the molecule-substrate distance, and investigate the effects arising from image-charge confinement and local charge neutrality violation. For benzene on Li(100) we find that the image-charge plane is located at about 1.8 Å above the Li surface, and that our calculated charge-transfer energies compare perfectly with those obtained with a classical electrostatic model having the image plane located at the same position. The methodology outlined here can be applied to study any metal/organic interface in the weak coupling limit at the computational cost of a total energy calculation. Most importantly, as the scheme is based on total energies and not on correcting the Kohn-Sham quasiparticle spectrum, accurate results can be obtained with local/semilocal exchange and correlation functionals. This enables a systematic approach to convergence.

Figure 1

(a) Schematic energy-level diagram of the frontier orbitals of a molecule approaching a metallic surface. Note the HOMO-LUMO level renormalization as a function of the molecule-surface distance $z$ due to polarization of the metal. (b) Top-view ball-stick representation of a benzene molecule at a Li(001) surface for a Li(001) $12\times 12$ supercell. The dashed rectangles show the $3\times 3$ (purple) and $6\times 6$ (green) supercells. The inset in (b) is the side view of the benzene lying flat at a distance $d$ from the surface.

Figure 2

Negative of the charge-transfer energy $E_{\mathrm{CT}}^{-}$ and removal energy $E_{\mathrm{CT}}^{+}$ as a function of the molecule-surface distance $d$ for the three clusters considered. (a) and (b) are for non-PBC and PBC calculations, respectively. The green dashed lines represent the negative of the $I_{\mathrm{mol}}$ and $A_{\mathrm{mol}}$ of the isolated molecule ($\Delta$SCF calculations) shifted by the calculated Li $W_F$ of 2.91 eV.

Figure 3

Image-charge analysis. (a) Isosurface of the difference between the charge densities calculated with DFT (ground state) and CDFT (charge-transfer state) $\Delta \rho \left(\mathbf{r}\right)$. Note the formation and the spatial distribution of the image charge. Different panels correspond to different molecule/surface distances $d$. The isosurfaces are taken at 10${}^{-4}$ $e$/Å${}^3$. Red isosurfaces denote negative $\Delta \rho \left(\mathbf{r}\right)$ (electron depletion), while blue are for positive $\Delta \rho \left(\mathbf{r}\right)$ (electron excess). (b) Position of the charge image plane taken from the surface atoms [see Eq. (11)], respectively, when one electron, $d_0^{+}$, or one hole, $d_0^{-}$, is transferred from the molecule to the Li substrate for the $12\times 12$ PBC calculations as a function of $d$. (c)–(e) are $E_{\mathrm{CT}}^{\mathrm{gap}}$, $E_{\mathrm{HOMO}}$, and $E_{\mathrm{LUMO}}$, respectively, as a function of $d$ and compared with the classical model of Eq. (10). The dashed green lines are $-I_{\mathrm{mol}}$ and $-A_{\mathrm{mol}}$ calculated with $\Delta$SCF.