Approximate Spin-Projected Density-Based Romberg Differentiation Procedure to Evaluate the Second-Hyperpolarizability of p-Quinodimethane and Twisted Ethylene and Their Diradical Character Dependence.

The evaluation of the static second hyperpolarizability (γ) of diradical species is a challenging task due to the use of spin-unrestricted methods, which may suffer from spin contamination. Here, we present the methodological aspect of a density-based differentiation procedure to evaluate static polarizability and hyperpolarizabilities. The finite-field calculations are done on the spin-projected electron density to remove the spin contamination, and the automatized Romberg's differentiation procedure is used to improve the numerical accuracy in the finite-field method. This implementation is tested in the present report for the challenging case of the evaluation of the second hyperpolarizability of the singlet ground state of p-quinodimethane (PQM) for the equilibrium geometry as well as for a stretched geometry where the diradical character of PQM is increased, and for twisted ethylene models where the diradical character changes with the dihedral angle. The application of the approximate spin-projected (ASP) scheme leads to a major improvement of the density functional theory calculations. In particular, for PQM models, BHandHLYP functional reproduces the UCCSD(T) values when the diradical character is below 0.5. The visualization of the γ-densities shows that (i) when increasing the diradical character, the amount of γ-density increases on the -CH2(•) extremities, and (ii) the ASP scheme decreases the amount of "p-like" γ-density for diradical character below 0.4, and increases it for larger diradical character. For twisted ethylene model, we show that the UCCSD(T) reference values can be reproduced by the ASP-UB3LYP method for y < 0.4 and by the ASP-UBHandHLYP method for y > 0.6. To best reproduce the UCCSD(T) reference calculations, the amount of exact exchange in hybrid functionals needs to be tuned along the range of diradical characters.