How polarizabilities and C6 coefficients actually vary with atomic volume

In this work, we investigate how atomic C6 coefficients and static dipole polarizabilities α scale with effective volume. We show, using confined atoms covering rows 1-5 of the periodic table, that C6/C6 (R)≈(V/V(R))(pZ) and α/α(R)≈(V/V(R))(pZ (')) (for volume V=∫dr4π3r(3)n(r)), where C6 (R), α(R), and V(R) are the reference values and effective volume of the free atom. The scaling exponents pZ and pZ (') vary substantially as a function of element number Z = N, in contrast to the standard "rule of thumb" that pZ = 2 and pZ (')=1. Remarkably, we find that the polarizability and C6 exponents p' and p are related by p' ≈ p - 0.615 rather than the expected p' ≈ p/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.