Linear response at the 4-component relativistic level: Application to the frequency-dependent dipole polarizabilities of the coinage metal dimers

Linear response theory based on the time-averaged quasienergy of Floquet states is generalized to the 4-component relativistic level for molecular calculations based on an analytical basis set. An efficient implementation of the theory for 4-component closed-shell Hartree–Fock is described. This level of approximation is also called the 4-component relativistic random phase approximation. The structure of the reduced response equations is analyzed in terms of Hermiticity and time reversal symmetry and leads to restrictions on the form chosen for the trial vectors as well as rules indicating when the linear response function is real, imaginary or zero. A key ingredient of the AO-driven algorithm is the formulation of the Hessian times a trial vector as the construction of modified Fock matrices. To reduce computational cost a previously reported quaternion symmetry scheme has been extended to non totally symmetric operators such that possible symmetry reductions are obtained as a reduction of algebra from quaternion to complex or real. We report the calculations of the frequency-dependent dipole polarizabilities for $Cu2,$ $Ag2,$ and $Au2$ at the 4-component Dirac–Coulomb Hartree–Fock level. Comparison of the relativistic and non-relativistic results show an increasing discrepancy with increasing nuclear charge, leading to qualitatively different results. Analysis of the first-order wave function shows that in the case of the gold dimer at the relativistic level of theory the generally dominant excitations from the HOMO are supplemented by excitations from the $5d$ manifold. This may significantly alter the molecular spectra and will be studied in a subsequent paper.