Appendix A: Calculation of $H(n,\Delta n)$

$$\begin{eqnarray} H\left( n,\Delta n\right) =\sum_{l,m}{\mid \langle n+\Delta n,l... ...d Z\mid n,l,m\rangle \mid ^2} \nonumber \\ \times h(n,l,\Delta n) \end{eqnarray}$$ (A1)

where $h(n,l,\Delta n)$ is given by formula (17). As $h(n,l,\Delta n)$ is independent of m, one can write:

$$\begin{eqnarray} & & H\left( n,\Delta n\right) =\sum_{l} {h\left(n,l,\Delta n\r... ...m_{m}\mid \langle n+\Delta n,l+1,m\mid Z\mid n,l,m\rangle \mid^2. \end{eqnarray}$$ (A2)

Further,

$$\begin{eqnarray} \sum_{m}{\mid \langle n+\Delta n,l+1,m\mid Z\mid n,l,m\rangle \mid ^2} \nonumber \\ =\frac{l+1}3\left( R_{nl}^{n+1,l+1}\right) ^2. \end{eqnarray}$$ (A3)

So

$$H\left( n,\Delta n\right) =\sum_{l=0}^{n-1}{\frac{l+1}3\left( R_{nl}^{n+1,l+1}\right) ^2h(n,l,\Delta n).}$$ (A4)

The calculation of $F(n,\Delta n),$ is carried out in a similar way, after replacing $h(n,l,\Delta n)$ by $f(n,l,\Delta n).$