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We present a modification of the $\Delta$SCF method of calculating energiesof excited states, in order to make it applicable to resonance calculations ofmolecules adsorbed on metal surfaces, where the molecular orbitals are highlyhybridized. The $\Delta$SCF approximation is a density functional methodclosely resembling standard density functional theory (DFT), the onlydifference being that in $\Delta$SCF one or more electrons are placed in higherlying Kohn-Sham orbitals, instead of placing all electrons in the lowestpossible orbitals as one does when calculating the ground state energy withinstandard DFT. We extend the $\Delta$SCF method by allowing excited electrons tooccupy orbitals which are linear combinations of Kohn-Sham orbitals. With thisextra freedom it is possible to place charge locally on adsorbed molecules inthe calculations, such that resonance energies can be estimated. The method isapplied to N$_2$, CO and NO adsorbed on different metallic surfaces andcompared to ordinary $\Delta$SCF without our modification, spatiallyconstrained DFT and inverse-photoemission spectroscopy (IPES) measurements.This comparison shows that the modified $\Delta$SCF method gives results inclose agreement with experiment, significantly closer than the comparablemethods. For N$_2$ adsorbed on ruthenium (0001) we map out a 2-dimensional partof the potential energy surfaces in the ground state and the 2$\pi$-resonance.Finally we compare the $\Delta$SCF approach on gas-phase N$_2$ and CO, tohigher accuracy methods. Excitation energies are approximated with accuracyclose to that of time-dependent density functional theory, and we see very goodagreement in the minimum shift of the potential energy surfaces in the excitedstate compared to the ground state.

Delta Self-Consistent Field as a method to obtain potential energy surfaces of excited molecules on surfaces