Inequalities for van der Waals Force Constants and Quantum Mechanical Sums

Quantum mechanical oscillator strength sums $\mathrm{S(k)}$ are used to find rigorous upper and lower bounds to van der Waals coefficients for two‐ and three‐body interactions and to the sums themselves. It is shown that any two sums of any multipole order give a bound to any other sum and either a bound or an estimate for the like force constant for that multipole. An average energy function is defined in terms of the sums; a particular limiting case of this function gives an excellent approximation for force constants (rms deviation for 19 cases: 0.25%) and depends solely on the value and slope of $\mathrm{S(k)}$ at $\text{k\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}2}$ . The results are tested on data for 18 dipole cases: H, He, Ne, Ar, Kr, Xe, Li, Na, K, Rb, Cs, Hg, H₂, N₂, CH₄, He(2¹S), He(2³S), He⁺; and 1 quadrupole case: H.