Solution of the radial equation for the relative motion of a diatomic molecule (Q805323)

For the nuclear motion of a diatomic molecule, the radial equation, after separating the translational motion of the mass center and the angular part, can be written as \[ (1)\quad \{-\frac{1}{2\mu}\frac{d^ 2}{dx^ 2}+\frac{j(j+1)}{2\mu (r_ e+x)^ 2}+V(x)\}R_{\nu j}(x)=E_{\nu j}R_{\nu j}(x), \] where \(x=r-r_ e\), \(r_ e\) is the equilibrium separation, \(\mu\) the reduced mass, j the rotational quantum number, V(x) the potential function, \(R_{\nu j}(x)\) the radial function to be solved, and \(E_{\nu j}\) the corresponding energy. In the present note Eq. (1) is solved by liner variation method on a suitable simple harmonic oscillator eigenfunction basis, that is \[ (2)\quad R_{\nu j}(x)=\sum^{n}_{i=0}C_ i(V,j)\Phi_ i(x). \] As everyone knows, the basis function \(\Phi_ i\) contains the Hermite polynomial, and the powers of the variable will be high when i is large. Furthermore, the contributions of any two adjacent terms in a Hermite polynomial (one plus and the other minus) to Hamiltonian matrix element are very similar. So is is very difficult to obtain precise results provided the calculations of Hamiltonian matrix element are carried out term by term while n in Eq. (2) is large \((>30)\). But it is necessary to use large n in Eq. (2) in order to calculate more excited states. For this reason some suitable skills have been taken and the related formulas have also been given.