Velocity gauge for oscillator strength in ΔSCF theory

Delta self-consistent-field (ΔSCF) theory is widely used for electronic excitation energy calculations. However, calculating the corresponding oscillator strengths is challenging. The corresponding many-electron wavefunctions are not directly accessible. Both the ground-state and the excited-state wave functions from ΔSCF are described by reference Kohn-Sham (KS) single-determinant wavefunctions for the fictitious non-interacting systems. The non-orthogonality between the ground and excited Kohn-Sham determinants from two different SCF calculations leads to unphysically origin-dependent transition properties, such as transition dipole moment and length-gauge oscillator strength. Including nuclei contribution to the perturbation is theoretically rigorous, but its effectiveness is limited only to neutral systems, as we show theoretically and numerically. While several other practical approaches have been proposed to tackle the non-orthogonality problem and yield reasonable results, inevitably, the determinant of the ground state or the excited state is changed, as well as the density matrix. In this work, we explore the use of the velocity gauge to compute oscillator strength within ΔSCF theory. We demonstrate that the velocity gauge is capable of naturally accounting for the non-orthogonality of ΔSCF KS wavefunctions and offering origin-independent predictions without any additional correction schemes to the KS wavefunctions. Compared to the length-gauge results obtained via symmetric orthogonalization, the velocity gauge can offer comparable results. Furthermore, the adoption of spin-purified singlet excitation energy in the velocity-gauge transition dipole moment significantly enhances the overall performance of the velocity gauge for ΔSCF oscillator strength predictions of conjugated chromophores.