Fit of Pariser-Parr-Pople and Hubbard model Hamiltonians to charge and spin states of polycyclic aromatic hydrocarbons

Total energies of charge and spin states of a specifically designed target molecule 2,5,8-trihydro-phenalenyl ($3{\text{H-C}}_{13}{\text{H}}_9$ from here on), calculated by means of multiconfigurational self-consistent-field (MCSCF) and density-functional-theory-B3LYP methods, have been fitted by means of interacting Pariser-Parr-Pople (PPP) (intrasite and intersite Coulomb interactions) and Hubbard Hamiltonians for $\pi$ electrons. Numerically exact many-body solutions for these models were obtained by a variant of the Lanczos algorithm. Our calculation shows that the combination MCSCF-PPP with a hopping integral $t=-2.63\text{eV}$ and a local Coulomb repulsion $U=10.51\text{eV}$ produces the best results. Both model parameters are close to values frequently used in the literature, sometimes just for the Hubbard model. The fit of MCSCF energies by the Hubbard model is not as satisfactory (root-mean-square deviation is larger and $t=-4.83\text{eV}$ and $U=21.27\text{eV}$ parameters are far from common values). Neither of both models is able to accurately reproduce B3LYP energies. On the other hand, the value of the ratio $\left|U/t\right|$ is always close to 4 indicating the proximity to a magnetic phase transition in extended systems. Additionally, we find that Lieb’s theorem for the Hubbard model on bipartite lattices applies to neutral $3{\text{H-C}}_{13}{\text{H}}_9$ since all ab initio methods and model Hamiltonians predict a fivefold spin-degenerate ground state.

Figure 1

(Color online) Schematic view of 2,5,8-trihydro-phenalenyl $\left(3{\text{H-C}}_{13}{\text{H}}_9\right)$. Phenalenyl system $\left({\text{C}}_{13}{\text{H}}_9\right)$ is an alternate uncompensated PAH radical appearing in three different charge states (neutral, cation, and anion) in some organic reactions (Ref. 32). Carbon atoms of one type (that is, belonging to one of the two sublattices) are depicted in gray and those of the other type in magenta and brown. Thus, there are seven C atoms of one type and six of the other. The artificial molecule investigated in our work is obtained by attaching an additional H atom to each peripheral carbon of the minority type (brown C), leading to an even higher uncompensated $\pi$ orbital network: seven sites of one type (gray C) and only three of the other type (magenta C).

Figure 2

(Color online) Ab initio MCSCF (upper panel) and B3LYP (lower panel) and model Hamiltonian energies of spin and charge states of 2,5,8-trihydro-phenalenyl (sketched in Fig. 1): squares give ab initio values while circles and triangles represent the results of the fit by PPP and Hubbard models, respectively. Actual numerical values of all energies are reported in Table (MCSCF) and Table (B3LYP) while parameters of the model Hamiltonians used in these calculations are given in Table .

Figure 3

(Color online) Model Hamiltonian energies versus ab initio MCSCF (upper panel) and B3LYP (lower panel) energies of spin and charge states of 2,5,8-trihydro-phenalenyl (sketched in Fig. 1). Squares give the energies obtained from PPP model while circles describe Hubbard results. The straight line signalizes a perfect fit $\left(y=x\right)$. All numerical energies correspond to the values given in Tables for model parameters compiled in Table .