Self-consistent Formulation of Constricted Variational Density Functional Theory with Orbital Relaxation. Implementation and Applications

We introduce here a new version of the constricted nth order variational density functional method (CV(n)-DFT) in which the occupied excited state orbitals are allowed to relax in response to the change of both the Coulomb and exchange-correlation potential in going from the ground state to the excited state. The new scheme is termed the relaxed self-consistent field nth order constricted variational density functional (RSCF-CV(n)-DFT) method. We have applied the RSCF-CV(n)-DFT scheme to the nσ→π* transitions in which an electron is moved from an occupied lone-pair orbital nσ to a virtual π* orbital. A total of 34 transitions involving 16 different compounds were considered using the LDA, B3LYP, and BHLYP functionals. The DFT-based results were compared to the "best estimates" (BE) from high level ab initio calculations. With energy terms included to second order in the variational parameters (CV(2)-DFT), our theory is equivalent to the adiabatic version of time dependent DFT . We find that calculated excitation energies for CV(2)-DFT using LDA and BHLYP differ substantially from BE with root-mean-square-deviations (RMSD) of 0.87 and 0.65 eV, respectively, whereas B3LYP affords an excellent fit with BE at RMSD = 0.33 eV. Resorting next to CV(∞)-DFT where energy terms to all orders in the variational parameters are included results for all three functionals in too high excitation energies with RMSD = 1.62, 1.14, and 1.48 eV for LDA, B3LYP, and BHLYP, respectively. Allowing next for a relaxation of the orbitals (nσ,π*) that participate directly in the transition (SCF-CV(n)-DFT) leads to an improvement with RMSD = 0.49 eV (LDA), 0.50 eV (B3LYP), and 1.12 eV (BHLYP). The best results are obtained with full relaxation of all orbitals (RSCF-CV(n)) where now RMSD = 0.61 eV (LDA), 0.32 eV (B3LYP), and 0.52 eV (BHLYP). We discuss finally the relation between RSCF-CV(n) and Slater's ΔSCF method and demonstrate that the two schemes affords quite similar results in those cases where the excitation can be described by a single orbital displacement (nσ→π*).

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