$\Delta$ self-consistent field method to obtain potential energy surfaces of excited molecules on surfaces

We present a modification of the $\Delta$ self-consistent field ($\Delta \text{SCF}$) method of calculating energies of excited states in order to make it applicable to resonance calculations of molecules adsorbed on metal surfaces, where the molecular orbitals are highly hybridized. The $\Delta \text{SCF}$ approximation is a density-functional method closely resembling standard density-functional theory (DFT), the only difference being that in $\Delta \text{SCF}$ one or more electrons are placed in higher lying Kohn-Sham orbitals instead of placing all electrons in the lowest possible orbitals as one does when calculating the ground-state energy within standard DFT. We extend the $\Delta \text{SCF}$ method by allowing excited electrons to occupy orbitals which are linear combinations of Kohn-Sham orbitals. With this extra freedom it is possible to place charge locally on adsorbed molecules in the calculations, such that resonance energies can be estimated, which is not possible in traditional $\Delta \text{SCF}$ because of very delocalized Kohn-Sham orbitals. The method is applied to ${\text{N}}_2$, CO, and NO adsorbed on different metallic surfaces and compared to ordinary $\Delta \text{SCF}$ without our modification, spatially constrained DFT, and inverse-photoemission spectroscopy measurements. This comparison shows that the modified $\Delta \text{SCF}$ method gives results in close agreement with experiment, significantly closer than the comparable methods. For ${\text{N}}_2$ adsorbed on ruthenium (0001) we map out a two-dimensional part of the potential energy surfaces in the ground state and the $2\pi$ resonance. From this we conclude that an electron hitting the resonance can induce molecular motion, optimally with 1.5 eV transferred to atomic movement. Finally we present some performance test of the $\Delta \text{SCF}$ approach on gas-phase ${\text{N}}_2$ and CO in order to compare the results to higher accuracy methods. Here we find that excitation energies are approximated with accuracy close to that of time-dependent density-functional theory. Especially we see very good agreement in the minimum shift of the potential energy surfaces in the excited state compared to the ground state.

Figure 1

(Color online) Upper row: The $2\pi$ resonance energy of ${\text{N}}_2$ molecule on a ruthenium surface. Lower row: The extra charge on the ${\text{N}}_2$ molecule in the resonance compared to a ground-state calculation. Left panels are for two layers and different surface cells, i.e., different ${\text{N}}_2$ coverages. Right panels are for a (2,1) surface cell and different number of layers. The extra amount of charge is estimated using Bader decomposition (Refs. 37, 38).

Figure 2

(Color) The change in charge distribution due to the excitation. Green: more charge (0.01 a.u. contour), red: less charge ($-0.01\text{a}.\text{u}$. contour). The four figures are for four different surface unit cells: (1,1), (2,1), (2,2), and (4,2). Gray atoms are ruthenium and blue atoms are nitrogen. The periodic images of the atoms are also shown, whereas the density changes are only shown in one unit cell.

Figure 3

The density of states for a ${\text{N}}_2$ molecule on a ruthenium slab and the projected density of states on the $2\pi$ orbital of the ${\text{N}}_2$ molecule. Top: Ground-state calculation. Bottom: Resonance calculation.

Figure 4

(Color online) Projected density of states (PDOS) on the $3\sigma$, $4\sigma$, $1\pi$, and $5\sigma$ orbitals of a ${\text{N}}_2$ molecule sitting on a ruthenium slab. The PDOSs are plotted for both the ground-state calculation and the resonance calculation. The gray area indicates energies below the Fermi level.

Figure 5

(Color) Potential energy surfaces (PES) for a nitrogen molecule on a close-packed ruthenium surface in the ground state and the $2{\pi }_y$ resonance as a function of the distance between the two nitrogen atoms and the distance from the surface to the center of mass of the nitrogen molecule. The energies are in eV. The small dots represent the points where the energy has been calculated in order to generate the surfaces. The black arrow represents a possible trajectory of the system in the resonance state (see text).

Figure 6

(Color online) The energy as a function of bond length for the ${\text{N}}_2$ molecule in the ground state and two excited states. The black lines correspond to $\Delta \text{SCF}$ calculations, the gray (online: light blue) lines correspond to linear-response calculations. The linear-response calculations have been made using OCTOPUS (Refs. 35, 36). The vertical lines indicate the positions of the minima.

Figure 7

(Color online) The energy as a function of bond length for the ${\text{N}}_2$ molecule in the ground state and two excited states. The short thick lines indicate the size of the gradients.