Calculation of radial matrix elements and anticrossings for highly excited Stark states using the $B$-spline approach

The calculation of the radial matrix elements of alkali metal Rydberg states is of interest with principal quantum number $n$ up to 70. Until now calculations of the radial matrix elements have mainly concerned the states with $n<30$. We use $B$-spline expansion technique and model potentials to calculate the radial matrix elements by numerical integration with 16 decimal digits precision. We are able to obtain the radial matrix elements of alkali metal Rydberg states with $n$ up to 145, with five significant digits. As a test example, we also compute the positions and widths of the anticrossings for highly excited Stark states of Na with principal quantum number $n$ up to 70.

Figure 1

Stark energy levels of sodium in the vicinity of $n=20$ $(\mid m\mid =0)$. $A$ is the initial anticrossing between Stark manifold 20 and 21.

Figure 2

Stark energy levels of sodium in the vicinity of $n=20$ $(\mid m\mid =1)$. $A$ is the initial anticrossing between Stark manifold 20 and 21.

Figure 3

(a) The parametric model potential [16] and the pseudopotential [18] for ${\mathrm{Na}}^{+}\text{−}e$ system with $l=0$. The solid line denotes the parametric model potential curve; the dotted line denotes the parametric model potential curve with the radial charge $Z_l\left(r\right)=1$; the dashed line denotes the pseudopotential curve. (b) The parametric model potential [16] and the pseudopotential [18] for ${\mathrm{Na}}^{+}\text{−}e$ system with $l=5$. The solid line denotes the parametric model potential curve; the dotted line denotes the parametric model potential curve with the radial charge $Z_l\left(r\right)=1$; the dashed line denotes the pseudopotential curve.