AN EXACT SOLUTION OF THE VIBRATIONAL MOTION OF A DISSOCIATIVE STATE.

$^{1}$ D.K. Bhattacharyya and L.-Y. Chow Chiu, J. Chem. Phys. 67, 5727 (1978).Author Institution: Department of Chemistry, Howard UniversityUsing $U(r) = B_{o} + B_{1}/r + B_{2}/r^{2} +\{\Lambda^{2} - N (N+1)\}/2Mr^{2}$ to represent the potential function for a diatomic dissociative state, an exact vibrational wave function is obtained. Parameters $B_{0}, B_{1}$ and $B_{2}$ are to be determined from a known potential curve, $\Lambda$ and N are respectively the electronic and rotational quantum numbers and M is the reduced mass. The solution $\Psi(r) = \exp(ikr) (2kr)^{s}F(s+ip, 2s, -2ikr)$, is shown to become a rapidly converging power series of 1/kr in the region of large (kr), and a rapidly converging power series of kr in the region of small (kr). the two series overlap in the intermediate region. $k = (2ME)^{{^{1}}/_{2}}, p = B_{1}M/K$ and $S = {^{1}}/_{2} + \{{^{1}}/_{4} + N(N+1) + 2MB_{2} - \Lambda^{2}\}^{^{1}/_{2}}$ are positive real numbers, and E is the energy above the dissociation limit $B_{0}$. For $N=0$, the wave function for the dissociative $b {^{3}}\Sigma^{+}_{u}$ state of $H_{2}$ is computed to compare with WKB wave function in $literature.^{1}$ The calculation is extended to high N to determine the N-dependence of the life time of the $c^{3}\Pi_{u}$ state