Δself-consistent field method to obtain potential energy surfaces of excited molecules on surfaces

Abstract

We present a modification of the $\ensuremath{\Delta}$ self-consistent field ($\ensuremath{\Delta}\text{SCF}$) 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 $\ensuremath{\Delta}\text{SCF}$ approximation is a density-functional method closely resembling standard density-functional theory (DFT), the only difference being that in $\ensuremath{\Delta}\text{SCF}$ 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 $\ensuremath{\Delta}\text{SCF}$ 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 $\ensuremath{\Delta}\text{SCF}$ 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 $\ensuremath{\Delta}\text{SCF}$ without our modification, spatially constrained DFT, and inverse-photoemission spectroscopy measurements. This comparison shows that the modified $\ensuremath{\Delta}\text{SCF}$ 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 $2\ensuremath{\pi}$ 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 $\ensuremath{\Delta}\text{SCF}$ 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.