Breakdown of the Born-Oppenheimer Approximation in the Calculation of Electric Hyperpolarizabilities

For the first time a nonadiabatic (all-particle) calculation has been carried out for the electric polarizabilities and hyperpolarizabilities of ${\mathrm{H}}_2^{+}$, H${\mathrm{D}}^{+}$, and ${\mathrm{D}}_2^{+}$ in their lowest rovibronic states. The value of the hyperpolarizability $\gamma$ is dramatically different from that which would be assumed from calculations based on the Born-Oppenheimer approximation unless account is taken of a vibrational contribution which (unlike its counterpart for the $\alpha$ polarizability) is nonzero even for a homonuclear diatomic molecule. For ${\mathrm{H}}_2^{+}$ we find that ${\alpha }_{\mathrm{zz}}=5.827\mathrm{}$ a.u. and ${\gamma }_{\mathrm{zzz}}=2.2\mathrm{}\times {10}^3$ a.u.