Rules of correspondence in atomic physics

The Smirnov method of analytic continuation (B.M. Smirnov, Sov. Phys. JETP $\mathbf{20}$, 345 (1964)) has been justified and developed for atomic physics. It has been shown that the polarizability of alkali atoms $\alpha$, their van der Waals interaction constant $C_6$, and the oscillator strength of the transition to the first $P$ state $f_{01}$ are related to the parameter $\langle r^2\rangle$ and gap in the spectrum $\frac{3}{2}\frac{f}{\Delta}\approx \frac{3}{2}\alpha\Delta\approx (3C_6\Delta)^{1/2}\approx\langle r^2\rangle$. The average square of the coordinate of the valence electron $\langle r^2\rangle$ in the first approximation has a hydrogen dependence $J_1=\frac{1}{2\nu^2}$ on the filling factor $\nu$, which is defined in terms of the first ionization potential.