Ab initio Hartree-Fock for periodic helical systems

Atom centred ab initio Hartree-Fock is introduced for periodic systems with helical symmetry. Such a structure is decomposed as a product of two crystal sub-groups, one of which being cyclic. This enables the Fock matrix of a finite cluster of cells to be Fourier transformed into a block diagonal form. The diagonalisation is performed on the sub-matrices and the mean field experienced by the central unit cell is expressed in the direct space. This environment is then imposed on all cells of the cluster by Born–von Kármán periodic boundary conditions. This is iterated until self consistency and then the density of the cluster is populated across the tube. The band structure arises in terms of multiple band diagrams, one for each irreducible representation of the cyclic group. Such diagrams are calculated and analysed for a system of carbon nanotubes, differing in both helicity and geometry. Scaling of the lattice vector is observed to result in a semiconducting-metallic phase transition. Geometry optimisation yields elastic coefficients that are used to describe the radial breathing mode. A comparison with theoretical models suggests that the obtained bandgaps are overestimated. Further improvements are outlined to account for electron correlation by means of MP2 and approximated Coupled Cluster. keywords: Hartree Fock, atomistic, helical, nanotubes, ab initio, band structure, mean field, crystal groups, block diagonal, SCF