Alternant Orbitals in Crystals

Löwdin's alternant orbital method is applied to $2n$-electron systems. The total wave function expressed in terms of these orbitals can be changed continuously from a single determinant of doubly occupied orbitals to a linear combination of many determinants consisting of singly occupied orbitals. The energy is an explicit function of $n$, and the transition to crystals is obtained for large $n$. Electrons of opposite spin are kept apart through ionic interactions depending on the spin coupling of electrons, i.e., the system spin state. Two spin states are of interest: the valence-bond singlet consisting of electron-pair bonds with zero spin, and a "parallel-spin" singlet which splits the system into two groups, each having $n$ electrons with parallel spins. In covalent crystals consisting of atoms having several valence electrons in an unfilled shell, the valence-bond singlet does not allow for the parallel spin coupling of these electrons. The parallel-spin state does, and correctly permits the system energy to go to its required asymptotic value for infinite lattice parameter. Numerical applications to the binding energy of diamond are carried out by means of an approximate set of orthogonalized atomic orbitals based on the tetrahedral orbitals of carbon. Improved estimates of the correlation error in diamond could be expected with more accurate sets of orbitals and integrals.