Previously, a treatment of the vibrational contribution to nonlinear optical properties for molecules with large amplitude modes in a symmetric double-minimum potential well was devised. The vibronic energies were written as a power series in the field for two limiting cases of the ratio between the field-induced energy and the zero-field splitting energy of the two lowest vibronic states. This treatment is extended here to include all values of the ratio and also an asymmetric double-well potential. It is shown that a consistent treatment of NLO effects in the general case leads to new field expansion coefficients, which are formulated in terms of the usual dipole moment and (hyper)polarizabilities. As an example, the new treatment is applied to the inversion motion of CH3–.

The potential energy surface (PES) of Ti@C28 has been revisited, and the stationary points have been carefully characterized. In particular, the C2v symmetry structure considered previously turns out to be a transition state lying 2.3 kcal/mol above the ground state of C3v symmetry at the MP2/6-31G(d) level. A large binding energy of 181.3 kcal/mol is found at the ROMP2/6-31G(d) level. Topological analysis of the generalized Ti@C28 density reveals four bond paths between Ti and carbon atoms of the host. The character of all four contacts corresponds to a partially covalent closed shell interaction. UV–vis, IR, and Raman spectra are calculated and compared with C28H4. The dipole moment and the static electronic and double harmonic vibrational (hyper)polarizabilities have been obtained. Distortion of the fullerene cage due to encapsulation leads to nonzero diagonal components of the electronic first hyperpolarizability β, and to an increase in the diagonal components of the electronic polarizability α and second hyperpolarizability γ. However, introduction of the Ti atom causes a comparable or larger reduction in most cases due to localized bonding interactions. At the double harmonic level, the average vibrational β is much larger than its electronic counterpart, but the opposite is true for α and for the contribution to γ that has been calculated. There is also a very large anharmonic (nuclear relaxation) contribution to β which results from a shallow PES with four minima separated by very low barriers. Thus, the vibrational γ (and α) may, likewise, become much larger when anharmonicity is taken into account.