We present an extension of the the cavity-field formulation for the Polarizable Continuum Model (PCM), to the Real-Time Time Dependent Density Functional Theory (RT-TDDFT). Both the length- and velocity-gauge formalisms are developed, through the approach based on effective dipole moment and momentum operators. We apply our formulation to the calculation of imaginary parts of the electric and mixed (electric-magnetic) polarizabilities, i.e. transition dipole moments and rotatory strengths for a model system, the twisted ethylene. To validate our approach we first compare the results obtained with the corresponding properties calculated within a PCM Linear-Response (LR) TDDFT formalism, and we check numerically the gauge-invariance of our formulation. Finally we also compare with the results of analytical models.
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