We study the evaluation of the Gross-Oliveira-Kohn expression for excitation energies
E1E0=ε1ε0+Exc,w[ρ]w|ρ=ρ0.
This expression gives the difference between an excitation energy E1E0 and the corresponding Kohn–Sham orbital energy difference ε1 − ε0 as a partial derivative of the exchange-correlation energy of an ensemble of states Exc,w[ρ]. Through Lieb maximisation, on input full-CI density functions, the exchange-correlation energy is evaluated accurately and the partial derivative is evaluated numerically using finite difference. The equality is studied numerically for different geometries of the H2 molecule and different ensemble weights. We explore the adiabatic connection for the ensemble exchange-correlation energy. The latter may prove useful when modelling the unknown weight dependence of the exchange-correlation energy.
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