How polarizabilities and $C_6$ coefficients actually vary with atomic volume

In this work we investigate how atomic $C_6$ coefficients and static dipole polarizabilities $\alpha$ scale with effective volume. We show, using confined atoms covering rows 1-5 of the periodic table, that $C_6/C_6^R\approx (V/V^R)^{p_Z}$and$\alpha/\alpha^R\approx (V/V^R)^{p’_Z}$(forvolume$V=\int dr \frac{4\pi}{3}r^3 n®$)where$C_6^R$,$\alpha^R$and$V^R$ are the reference values and effective volume of the free atom. The scaling exponents $p_Z$ and $p_Z^{\mathrm{\prime }}$ vary substantially as a function of element number $Z=N$, in contrast to the standard “rule of thumb” that $p_Z=2$ and $p_Z^{\mathrm{\prime }}=1$. Remarkably, We find that the polarizability and $C_6$ exponents $p^{\mathrm{\prime }}$ and $p$ are related by $p^{\mathrm{\prime }}\approx p-0.615$ rather than the expected $p^{\mathrm{\prime }}\approx p\mathrm{/}2$. Results are largely independent of the form of the confining potential (harmonic, cubic and quartic potentials are considered) and kernel approximation, justifying this analysis.