Understanding the Second-Order Nonlinear Optical Properties of One-Dimensional Ruthenium(II) Ammine Complexes

First principles calculations of the linear and nonlinear optical properties are carried out for three series of one-dimensional ruthenium­(II) ammine complexes with one pyridyl pyridinium ligand containing different numbers of C₂H₂ units or the (CHCH)Ph(CHCH) linker. The substitution effects upon introducing one pyridine or one <i>N</i>-methylimidazole moiety as donor instead of ammonia have also been studied in detail. These calculations employing density functional theory with different exchange-correlation functionals as well as the polarizable continuum model approach to describe the solvent effects show that these compounds are challenging for theoretical chemistry and that their nonlinear optical responses are complex and depend on many structural and electronic factors. Two major types of methods have been employed to calculate and analyze the first hyperpolarizabilities (β): (i) the summation-over-states scheme, applied to calculate the longitudinal β tensor component, truncated to one or two dominant excited states, of which the results are compared to Stark spectroscopy data, and (ii) the quadratic response schemes, providing the whole second harmonic generation response of the compound, which can be compared to the values derived from hyper-Rayleigh scattering measurements. Using response theory calculations, off-resonant first hyperpolarizability values display a monotonic increase as a function of the number of C₂H₂ units (from 0 to 3 units, corresponding to compounds <b>1</b>–<b>4</b>, respectively) in the pyridyl pyridinium ligand (in the three series with NH₃, pyridine, and <i>N</i>-methylimidazole substituent in opposite position to the large ligand). These calculations, which evidence the dominance of the longitudinal β tensor component and a consistent depolarization ratio close to 5.0, also predict a smaller β value when the (CHCH)₃ linker of the pyridyl pyridinium ligand (compound <b>4</b>) is replaced by the (CHCH)Ph(CHCH) linker (compound <b>5</b>). To some extent, the calculations performed at a wavelength of 1064 nm confirm these trends, despite the presence of resonance enhancements. Calculations carried out within the summation-over-states scheme to analyze the origin of these β variations reproduce the monotonic increase of β in the <b>1</b>–<b>4</b> series, but compounds <b>4</b> and <b>5</b> are predicted to have similar β values. Upon relating these β variations to those of spectroscopic properties like the excitation energies, transition dipoles, and dipole moments of a pair of excited states (a low-energy metal to ligand charge transfer and a high-energy intraligand charge transfer excited state), the calculations point out the impact of including a second excited state in the treatment, since it leads to a decrease of β by about 50%. Though substituting the ammonia group by a pyridine has a limited impact on the linear and nonlinear optical properties of the complexes, the substitution by <i>N</i>-methylimidazole increases β by about 25%, which has been attributed to the contribution of <i>N</i>-methylimidazole to the HOMO and the subsequent bathochromic shifts of the excitation energies. In most of the cases, qualitative and even semiquantitative agreement is observed between theory and experiment, which enables one to get a deep insight into the structure/property relationships of these molecules. When this is not the case, we elaborate on the limitations of the methods of calculations and of the experimental treatments of the measured data