Applicability of DFT functionals for evaluating the first hyperpolarizability of phenol blue in solution

The first electronic hyperpolarizability (β) of phenol blue (PB) in several solvents in a wide range of dielectric constants is investigated using the density functional theory (DFT). The reliability of various exchange-correlation functionals is assessed by a comparison to reference Møller-Plesset second-order perturbation theory (MP2) calculations. The equilibrium geometry of PB in each solvent is obtained by using the average solvent electrostatic configuration/free energy gradient method, which performs optimizations on the free energy hyper-surface by employing iteratively the sequential quantum mechanics/molecular mechanics methodology. The dependence of β on the bond length alternation (BLA) coordinate is rationalized by means of the two-level model. Within the employed exchange-correlation functionals, the LC-BLYP functional shows the best performance for describing the static and dynamic MP2 results of β, which increases as the BLA diminishes, reaching a maximum in an intermediate value of BLA. The results also illustrate the role played by the difference between the ground- and excited-state dipole moments (Δμ) in determining the hyperpolarizability behavior in solution. Particularly, in the aqueous solution case, Δμ goes to around zero when BLA is near zero, leading to an abrupt decline in the β value. The DFT results of this study, therefore, indicate a clear relationship between the first hyperpolarizability and the BLA coordinate for the PB in solution, in agreement with experiment.