where x,y, and z are the unit vectors in the X, Y, and Z directions, respectively. These directions are related to the cavity structure. For convenience, we also use these directions to define reference coordinate systems for all initial nuclear geometries, see Sec 4.2 (Fig. ) for details. The cavity field strength is determined by the volume of the cavity as λ = 1/ε 0 V 0 , where ε 0 is the permittivity inside the cavity and V 0 is the effective quantization volume inside the cavity. Another commonly used light-matter coupling strength is characterized as g c = hω c /2λ. Further, the total dipole operator of both electrons and nuclei is defined as