An atomic-orbital based Lagrangian approach for calculating geometric gradients of linear response properties

We present a Lagrangian approach for the calculation of molecular (quadratic) response properties that can be expressed as geometric gradients of a generic linear response function, its poles, and its residues. The approach is implemented within an atomic-orbital-based formalism suitable for linear scaling at the level of self-consistent time-dependent Hartree−Fock and density functional theory. Among the properties that can be obtained using this formalism are the gradient of the frequency-dependent polarizability (e.g., Raman intensities) and that of the one-photon transition dipole moment (entering the Herzberg−Teller factors), in addition to the excited-state molecular forces required for excited-state geometry optimizations. Geometric derivatives of ground-state first-order properties (e.g., IR intensities) and excited-state first-order property expressions are also reported as byproducts of our implementation. The one-photon transition moment gradient is the first analytic implementation of the one-photon transition moment derivative at the DFT level of theory. Besides offering a simple solution to overcome phase (hence, sign) uncertainties connected to the determination of the Herzberg−Teller corrections by numerical derivatives techniques based on independent calculations, our approach also opens the possibility to determine, for example by a mixed analytic−numerical approach, the one-photon transition dipole Hessian, and thus to investigate vibronic effects beyond the linear Herzberg−Teller approximation. As an illustrative application, we report a DFT study of the vibronic fine structure of the one-photon (1A1g) − (1B2u) transition in the absorption spectrum of benzene, which is Franck−Condon-forbidden in the electric dipole approximation and hence determined by the Herzberg−Teller integrals and electronic transition dipole-moment derivatives.