Quadratic response function calculations using relativistic four-component TD-DFT and a non-collinear description of the spin density

A theory, implementation and first application of relativistic four-component TD-DFT-based for the treatment of quadratic response properties will be presented. The expressions are based on the non-collinear treatment of the spin density which has been shown to offer the highest amount of reliability in previous studies. The non-collinear ansatz for the spin density leads to a large amount of complexity in the expressions for the perturbed analogs of the spin density which additionally suffer from definition problems for the closed-shell case (spin density approaching zero). This is overcome by proper modification of the working equations which are implemented to calculate the third-order derivative of the exchange-correlation energy and the corresponding spin density contributions. In a first application study, first hyperpolarizabilities of ortho- and meta-diiodobenzene are calculated and compared with results obtained with one-component TD-DFT calculations with and without use of effective core potentials (ECPs). It has been found that the deviation between the four-component and the one-component/ECP results is not larger that 6\% and that therefore the latter offers an efficient tool for calculations of quadratic response properties while the four-component ansatz is useful for validation of the lower-order results