The Dirac Vector Model in Complex Spectra

Dirac has shown that the secular problem presented by the permutation degeneracy is formally equivalent to a problem in vector coupling for which the Hamiltonian function is $-\frac{1}{2}\Sigma K_{\mathrm{ij}}(1+4\mathrm{}{\mathrm{s}}_i·{\mathrm{s}}_j)$ where ${\mathrm{s}}_i$, ${\mathrm{s}}_j$ are respectively the spin vectors of orbits $i$, $j$ and $K_{\mathrm{ij}}$ is the exchange integral which connects $i$ and $j$. The vector model can be used in place of Slater's determinantal wave functions to calculate atomic spectral terms, provided one still retains much of Slater's powerful method of diagonal sums. The configuration $d^3$ is treated as an example. Configurations of the form $s{a}^k$ ($a=p, d, f\cdots$; $0<k<\mathrm{}4\mathrm{}{l}_a+2\mathrm{}$) are particularly amenable to the vector model, as it enables us immediately to write down the energy of $s{a}^k$ if that of $a^k$ is known. One thus finds that the two states $S=S_k\pm \frac{1}{2}$ built upon a given configuration $S_k$, $L_k$ of the core $a^k$ should have a separation proportional to $S_k+\frac{1}{2}$ and independent of $L_k$. Experimentally, this prediction is confirmed only roughly, like the interval relations found by Slater, because perturbations by other configurations are neglected. Various applications to molecular spectra are given. The Heitler-Rumer theory of valence, which neglects directional effects, has a particularly simple interpretation in terms of the vector model.