Ab initio calculations of polarizabilities and second hyperpolarizabilities of organic molecules with extended π-electron conjugation

Static polarizability and second hyperpolarizability tensors are computed for a series of polyenes, polyynes, and cumulenes by ab initio SCF theory. Numerically stable finite field (FF) calculations can be achieved by using polynomial fits of either energy or induced dipole moment as a function of field strength. The nonlinear expansion coefficients from these fits correspond to the microscopic nonlinear optical property. Our results from fully coupled (FF) ab initio calculations for polarizability are in good agreement with those derived from uncoupled (sum-over-states) ab initio methods. The hyperpolarizabilities do not compare as well. A qualitative description of the chain length dependence of polarizability and hyperpolarizability for moderately long chains is discussed in terms of an empirical function. Diffuse orbital basis functions are required for qualitatively correct hyperpolarizabilities of small conjugated π systems or for that matter any small molecule. For example, the average second hyperpolarizability, γ, of ethylene is computed to be -13, 1.7, and 726 au with STO-3G, 6-31G, and augmented 3-21G basis sets, respectively. The value computed with the augmented basis set agrees, within a factor of 2, with the experimental value of 1500 au. The valence set for the carbon atoms is augmented with diffuse s, p, and two Cartesian d sets, or subsets of these. The inclusion of diffuse polarization functions drastically alters the computed second hyperpolarizabilities. The results are nearly insensitive to the choice of valence set but are highly dependent on the basis set quality. We also describe the use of a corresponding orbital analysis to aid in the interpretation of ab initio results obtained by either FF or analytic derivative methods. The computed polarizability and hyperpolarizability and the π and σ components indicate that the contribution due to the π orbitals is much more significant than the σ orbitals. Both contributions change sign in going from the ethylenic and acetylenic chains to the cumulenic systems. The polarization of π-electron density by the field is illustrated by contour surfaces of derivative π-electron-density functions. Contour maps of the first derivative of charge density with respect to the field (α) for an acetylenic chain are nearly periodic, corresponding to localized polarization, whereas the third derivative density (γ) corresponds to longer range charge shifts.