New qualitative results of the atomic theory

The polarizability $\alpha$ of many atoms and positive ions is related to their energy gap $\Delta$ and valence $m$ by the expression $\alpha\Delta^2\cong m$ (in atomic units). The parameter $\Delta$ corresponds to a dipolar transition from the ground state to the first excited $P$ state without a change in the principal quantum number $n$. This relation holds for univalent ($m=1$) Na, K, Rb, Cs, Fr and bivalent ($m=2$) Mg, Ca, Zn, Sr, Cd, Ba, Yb, Hg atoms. The above relation agrees with the experiment for positive ions Mg$^+$ and Ca$^+$ ($m=1$) and Al$^+$ and Ga$^+$ ($m=2$). The polarizability has been found for atoms and ions of the type Zn$^+$, In$^+$, Tl$^+$, for which experimental data are unavailable. A method of calculating $\alpha$ for ions of the types C$^{++}$, Al$^{++}$, Si$^{++}$ and Si$^{+++}$, P$^{+++}$, As$^{+++}$ has been suggested based on the approximate relation $\alpha\cong(2/3\langle r^2\rangle_0)^2/m$ with the parameter $\langle r^2\rangle_0$ expressed in terms of the valence $m$, the charge number $q$ of the atomic or ionic residue, and the ionization potential $J_q=\frac{q^2}{2v_s^2}$ as $\langle r^2\rangle_0=\frac{m}{2q^2}\nu_s^2( 1 + 5\nu _s^2)$. The hydrogen dependence of $\langle r^2\rangle_0$ on the parameter $\nu_s$ has been derived by analytical continuation from the integer values $\nu_s=1$ and $2$. A variational estimate of the van der Waals constant characterizing the interaction of two spherically symmetric atoms at large distances has been given.