Perturbation Calculation for H2 Energy Shift

$$\bgroup\color{black}\left<\psi_{njm_j\ell s}\left\vert H_2\ri... ...ll(\ell+1)-s(s+1)\right]\left<{1\over{r^3}}\right>_{nl} \egroup$$

For $j=\ell\pm{1\over 2}$

$$\bgroup\color{black} = {ge^2\hbar^2\over{8m^2c^2}}\left[(\ell... ...\left({1\over{n^3\ell(\ell+{1\over 2})(\ell+1)}}\right) \egroup$$

$$\bgroup\color{black}= -E_n {g\hbar^2\over{4m^2c^2a^2_0}} \lef... ...] \left({1\over{n\ell(\ell+{1\over 2})(\ell+1)}}\right) \egroup$$

$$\bgroup\color{black} = \left({g\over 2}\right) \left({-E_n\ov... ...t]^{(+)}_{(-)} {n\over{\ell(\ell+{1\over 2})(\ell+1)}} \egroup$$

$$\bgroup\color{black} = \left({g\over 2}\right) \left({-E_n\ov... ...t]^{(+)}_{(-)} {n\over{\ell(\ell+{1\over 2})(\ell+1)}} \egroup$$

$$\bgroup\color{black} = \left({g\over 2}\right) {{E^{(0)}_n}^2... ...ad \matrix{ j=\ell + {1\over 2} \cr j=\ell -{1\over 2} }\egroup$$

Note that in the above equation, we have canceled a term ${\ell\over\ell}$ which is not defined for $\ell=0$. We will return to this later.