Time-Dependent Density Functional Theory as a Foundation for a Firmer Understanding of Sum-Over-States Density Functional Perturbation Theory: “Loc.3” Approximation

Sum-over-states density functional perturbation theory (SOS-DFPT) (Malkin, V. G.; Malkina, O. L.; Casida, M. E.; Salahub, D. R. J Am Chem Soc 1994, 116, 5898) has been successful as a method for calculating nuclear magnetic resonance (NMR) chemical shifts. The key to this success is the introduction of an ad hoc correction to the excitation energies represented by simple orbital energy differences in uncoupled density functional theory. It has been suggested (Jamorski, C.; Casida, M. E.; Salahub, D. R. J Chem Phys 1996, 104, 5134) that the good performance of this methodology could be partly explained by the resemblance of the corrected excitation energy to the orbital energy difference given by time-dependent density functional theory (TDDFT). In fact, according to exact (wave function) time-dependent perturbation theory, both magnetic and electric perturbations may be described using essentially the same simple SOS expression. However in adiabatic TDDFT, with no explicit relativistic or current density functional dependence, the functional is approximate and so the magnetic and electric SOS expressions are different. Because TDDFT (neglecting relativistic and current density functional dependence) is formally exact for electric perturbations but not magnetic perturbations and because the two SOS expressions should have the same form, we propose that the SOS expression for electric perturbations should also be used for magnetic perturbations. We then go on to realize our theory by deriving a “Loc.3” approximation that is explicitly designed by applying the electric field SOS expression to magnetic fields within the two-level model and Tamm–Dancoff approximation. Test results for 13 small organic and inorganic molecules show that the Loc.3 approximation performs at least as well as the “Loc.1” and “Loc.2” approximations of SOS-DFPT