Natural Orbitals in the Quantum Theory of Two-Electron Systems

The wave functions for the singlet and triplet states of a two-electron system in a given nuclear framework are investigated as superpositions of configurations and are shown to be transformationally equivalent to quadratic forms having certain ranks and signatures. By introducing the "natural orbitals" diagonalizing the generalized first-order density matrix, the total wave functions may also be brought to principal form. If the basis contains $M$ one-electron functions, the singlet and triplet wave functions contain respectively $\frac{M(M+1\mathrm{})\mathrm{}}{2}$ and $\frac{M(M-1\mathrm{})\mathrm{}}{2}$ configurations, but the transformation to natural orbitals reduces the number of terms to $M$ and [$\frac{M}{2}$], respectively. The natural expansion having the best convergence is also characterized by another important extremum property. The approximate wave function of rank $r$ having the smallest quadratic deviation from the exact eigenfunction is obtained by interrupting the natural expansion of the eigenfunction after $r$ terms and renormalizing the result. For the singlet state, the wave function of rank two and signature zero has a special importance as giving a simple extension of the visual one-electron picture to include a large part of the correlation effects. The theory is illustrated by some results on helium obtained by using radial configuration interaction.