Delta Self-Consistent Field as a method to obtain potential energy surfaces of excited molecules on surfaces
We present a modification of the $Δ$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 $Δ$SCF approximation is a density functional method closely resembling standard density functional theory (DFT), the only difference being that in $Δ$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 $Δ$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. The method is applied to N$_2$, CO and NO adsorbed on different metallic surfaces and compared to ordinary $Δ$SCF without our modification, spatially constrained DFT and inverse-photoemission spectroscopy (IPES) measurements. This comparison shows that the modified $Δ$SCF method gives results in close agreement with experiment, significantly closer than the comparable methods. For N$_2$ adsorbed on ruthenium (0001) we map out a 2-dimensional part of the potential energy surfaces in the ground state and the 2$π$-resonance. Finally we compare the $Δ$SCF approach on gas-phase N$_2$ and CO, to higher accuracy methods. Excitation energies are approximated with accuracy close to that of time-dependent density functional theory, and we see very good agreement in the minimum shift of the potential energy surfaces in the excited state compared to the ground state.