The quantum theory of atomic polarization II— The van der Waals energy of two atoms

A theoretical analysis is given of the three $^{5}$D terms in the europium ion. The integrals F$_{k}$ are discussed and the importance of F$_{6}$ is emphasized. Values of F$_{4}$/F$_{2}$ and F$_{6}$/F$_{2}$ are calculated for a hydrogenic 4f eigenfunction, and evidence from the praseodymium and gadolinium ions suggests it is reasonable to use these values in the case of europium. A state in L, S, L$_{z}$, S$_{z}$ quantization is defined in terms of the determinantal product states of 4f one-electron states, and this enables the properties of the $^{5}$D terms to be found. By calculating the Coulomb energies of the three terms, it is shown that the $^{5}$D term observed spectroscopically is the lowest of the three. The relative positions of the levels $^{5}$D$_{0}$, $^{5}$D$_{1}$ and $^{5}$D$_{2}$ that derive from this term are calculated for Russell-Saunders coupling, and a satisfactory fit is obtained with the observed spacings. In addition, the crystal fine structures of $^{7}$F$_{1}$ and $^{5}$D$_{1}$ are related to each other, and the agreement between experiment and theory is as good as can be expected.