Radial SE

Our SE for the radial motion of the two nuclei has the form

$$\left[- \frac{\hbar^2}{2\mu} \left(\frac{\partial^2}{\partial r^2... ...U_\alpha(r)\right] R_{\alpha;Kv}(r) = \varepsilon_{\alpha;Kv} R_{\alpha;Kv}(r).$$ (1.26)

We therefore have two sets of quantum numbers $K$ and $v$ that describe the rotational and vibrational and state of the molecule for a given electronic state $\alpha$. Setting

$$R_{\alpha;Kv}(r) = \frac{1}{r} P_{\alpha;Kv}(r)$$ (1.27)

leads to a standard one-dimensional SE with a `proper' $\frac{d^2}{dr^2}$ kinetic energy term,

$$\left[ -\frac{1}{2\mu}\frac{d^2}{dr^2} + U_\alpha(r) + \frac{K(K+... ...ight] P_{\alpha;Kv}(r) = \varepsilon_{\alpha;Kv}P_{\alpha;Kv}(r), \quad r\ge 0.$$ (1.28)