Approximate solutions of Heisenberg Hamiltonians

Even when each atom of a 2n-center cluster or molecule only brings one active electron in one atomic orbital, the size of the Heisenberg Hamiltonian matrix increases as C. Simple truncations of this matrix would result in size consistence defects, as evident from the isomorphism between Heisenberg and configuration interaction (CI) matrices. Geometry-dependent Heisenberg Hamiltonians derived from accurate ab initio calculations on the two-center systems have proved to be very efficient for conjugated hydrocarbons and for alkali metals; in order to apply this approach to intermediate size systems (10–20 centers), a rational procedure is proposed consisting of the selection of a truncated set of determinants (of low energy) and a dressing of the truncated Hamiltonian matrix under the perturbation of the other determinants. The second order dressing is analogous to a so-called “shift B_k procedure” or Generalized Degenerate Perturbation theory and is weakly dependent on an E₀ parameter. Tests performed on various 8- and 10-atom systems show the accuracy of the procedure. An iterative selection of the truncated basis set and proper choices of the E₀ values allow one to obtain the whole lower part of the spectrum. The calculated geometries are satisfactory. Some preliminary applications are reported concerning the C₁₂H₁₄ dodecahexene linear chain, perfectly fitting with previous extrapolations.