Analytic expressions for the first-order nonadiabatic coupling matrix elements between
electronically excited states are first formulated exactly via both time-independent equation
of motion and time-dependent response theory, and are then approximated at the
configuration interaction singles, particle-hole/particle-particle random phase approximation,
and time-dependent density functional theory/Hartree-Fock levels of theory. Note that, to get
the Pulay terms arising from the derivatives of basis functions, the standard response theory …

The recently proposed rigorous yet abstract theory of first order nonadiabatic coupling matrix
elements (fo-NACME) between electronically excited states [Z. Li and W. Liu, J. Chem. Phys.
141, 014110 (2014)] is specified in detail for two widely used models: The time-dependent
density functional theory and the particle-particle Tamm-Dancoff approximation. The actual
implementation employs a Lagrangian formalism with atomic-orbital based direct algorithms,
which makes the computation of fo-NACME very similar to that of excited-state gradients …