Metal-insulator phase transition of phosphorus treated through a magnetic approach

This paper first discusses the possibility of treating half-filled band problems on spin-nonfrustrated lattices from the single-determinant approach in the strongly correlated domain (4‖t‖<U) or from a Heisenberg effective Hamiltonian in the strongly delocalized domain (4‖t‖>U). The second approach is used to treat metallic and covalent phases of phosphorus using an r-dependent Heisenberg Hamiltonian extracted from the spectroscopy of P and ${\mathrm{P}}_2$. The cubic phase of phosphorus and its distortion toward a rhombohedral (A7) lattice are treated by perturbing a Néel state to the second order. The cubic phase presents a minimum energy for a 2.35-Å bond length (experiment: 2.377 Å). For a slightly larger interatomic distance (r=2.47 Å) the cubic phase becomes unstable and undergoes a distortion toward an A7 lattice; a saddle point (${\mathit{r}}_1$=2.22 Å; ${\mathit{r}}_2$=2.71 Å) leads to the strongly distorted valley. In agreement with experiment, the metal-insulator transition is of first order. Although the use of the Heisenberg Hamiltonian becomes questionable when the angles between bonds significantly deviate from 90° (as occurs in black phosphorus), we have attempted to determine the conformation and cohesive energy of strongly distorted structures by perturbing a product of bond singlets along the short covalent bonds under the effect of (singlet→triplet${)}^2$ double excitations. The interatomic distances between layers is too large (4.1 Å instead of 3.56 Å), but the covalent bond lengths (2.18 Å) and cohesive energy (75 kcal ${\mathrm{mol}}^{\mathrm{-}1}$) compare favorably with the experimental values (2.22–2.28 Å and 79 kcal ${\mathrm{mol}}^{\mathrm{-}1}$) for black phosphorus.