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Physical Review B

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

José Antonio Vergés, Emilio San‐Fabián, G. Chiappe, Enrique Louis

Published Feb 24, 2010• 31 citations
Publisher

Abstract

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 $\ensuremath{\pi}$ 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 $t=\ensuremath{-}2.63\text{ }\text{eV}$ and a local Coulomb repulsion $U=10.51\text{ }\text{eV}$ 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 $t=\ensuremath{-}4.83\text{ }\text{eV}$ and $U=21.27\text{ }\text{eV}$ 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 $|U/t|$ 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.

DOI
10.1103/physrevb.81.085120

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Total energies of charge and spin states of a specifically designed target molecule 2,5,8-trihydro-phenalenyl (3H-C13H93{\text{H-C}}_{13}{\text{H}}_{9}3H-C13​H9​ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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=\ensuremath−2.63 eVt=\ensuremath{-}2.63\text{ }\text{eV}t=\ensuremath−2.63 eV and a local Coulomb repulsion U=10.51 eVU=10.51\text{ }\text{eV}U=10.51 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=\ensuremath−4.83 eVt=\ensuremath{-}4.83\text{ }\text{eV}t=\ensuremath−4.83 eV and U=21.27 eVU=21.27\text{ }\text{eV}U=21.27 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 ∣U/t∣|U/t|∣U/t∣ 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 3H-C13H93{\text{H-C}}_{13}{\text{H}}_{9}3H-C13​H9​ since all ab initio methods and model Hamiltonians predict a fivefold spin-degenerate ground state.
Static-spin-correlation functions 4〈spz{s}_{\mathrm{pz}}spz​sqz{s}_{\mathrm{qz}}sqz​〉 are evaluated via diagrammatic valence-bond theory for exact states of finite Hubbard, Pariser-Parr-Pople (PPP), and related quantum cell models. Several properties of 〈spz{s}_{\mathrm{pz}}spz​sqz{s}_{\mathrm{qz}}sqz​〉 that are independent of intersite potentials Vpp′{V}_{\mathrm{pp}\mathcal{'}}Vpp′​ in the Hartree-Fock (HF) approximation for half-filled alternant systems are shown to be sensitive to correlations in the ground state of N\ensuremath{\le}12 Hubbard and PPP chains. Accurate N\ensuremath{\rightarrow}\ensuremath{\infty} extrapolations are found for near neighbors. The correct ground-state charge density 〈np{n}_{p}np​〉=1 is a property of half-filled alternant systems and thus not a test of HF or other one-electron approximations. Spin correlations in PPP models with standard molecular parameters and long-range Coulomb interactions are approximately given by on-site interactions of U\ensuremath{\simeq}2\ensuremath{\Vert}t\ensuremath{\Vert} in Hubbard models, although the general semiempirical PPP approach to conjugated molecules is preferable for wide-band systems.
We study a simple model system for the conjugated $\ensuremath{\pi}$ bonds in benzene, the Pariser-Parr-Pople (PPP) model, within the parquet approximation (PA), exemplifying the prospects of the PA for molecules. Advantages of the PA are its polynomial scaling with the number of orbitals, and the natural calculation of one- and two-particle spectral functions as well as of response and correlation functions. We find large differences in the electronic correlations in the PPP model compared to a Hubbard model with only local interactions. The quasiparticle renormalization (or mass enhancement) is much weaker in the PPP than in the Hubbard model, but the static part of the self-energy enhances the band gap of the former. Furthermore, the vertex corrections to the optical conductivity are much more important in the PPP model. Because nonlocal interactions strongly alter the self-energy, we conclude that the PA is more suitable for calculating conjugated $\ensuremath{\pi}$ bonds in molecules than single site dynamical mean-field theory.
We have carried out density-matrix renormalization group calculations on the ground state of long polyacene oligomers within a Pariser-Parr-Pople (PPP) Hamiltonian. The PPP model includes long-range electron correlations which are required for physically realistic modeling of conjugated polymers. We have obtained the ground-state energy as a function of the dimerization $\ensuremath{\delta}$ and various correlation functions and structure factors for $\ensuremath{\delta}=0$. From energetics, we find that the nature of the Peierls' instability in polyacene is conditional and strong electron correlations enhance the dimerization. The cis form of the distortion is favored over the trans form. However, from the analysis of correlation functions and associated structure factors, we find that polyacene is not susceptible to the formation of a bond order wave, spin-density wave, or a charge-density wave in the ground state.
Quantum Embedding density functional theory (QmDFT) embedding offers a highly scalable approach to improve treatment for large highly correlated pi conjugated systems. However, estimating advanced electronic structure properties in polycyclic aromatic hydrocarbons (PAHs) needs advanced exchange correlation functionals that frequently trigger convergence instabilities during the embedding cycle. In this work, we introduce an adaptive damping and direct inversion in the iterative subspace (DIIS) accelerated protocol that stabilizes the embedding procedure, enabling robust integration of hybrid functionals like B3LYP and CAM B3LYP. Using 10 selected PAHs (linear and fused) molecules as a benchmark. We demonstrate a clear functional-dependent ground-state energetics and frontier-orbital gap estimation. While LDA based approaches yield near-quantitative agreement with FCI in DFT reference energies and is further supported by thermochemical isomerization benchmarks, while B3LYP provide significantly improved agreement with experimental E0-0 transition values. This mapping allows us to bypass explicit excited-state calculations for E0 0 values, thereby significantly reducing computational overhead. Among the hybrid functionals screened, CAM B3LYP offers a balanced overall performance. Our results establish a stable QmDFT framework and provide useful guidance for functional selection for quantum embedding studies of PAHs and related low-dimensional pi-conjugated materials.
Diagrammatic valence bond (DVB) theory is a general approach to electron correlations in quantum cell models that conserve total spin. VB diagrams are a convenient many-electron basis for combining spin, point-group, and other symmetries in oligomers with a large but finite basis. Half-filled Hubbard or Pariser–Parr–Pople (PPP) models with 16 sites have ∼34.7×106 singlet diagrams. Improved DVB methods yield exact low-lying states of the 16-site polyene in C2h symmetry and of pyrene in D2h symmetry. Several generalizations of symmetry adaptation are necessary for large bases, including new rules for linearly independent basis vectors and an iterative method for Hamiltonian matrix elements that avoids overlap and inversion. The number and dimensions of the disjoint invariant subspaces Sm encountered in symmetry adaptation depend on the connectivity. D2h symmetry adaptation is much simpler for acenes than for pyrene, linear stilbene, or polyphenyls. Standard PPP parameters account well for the 11 pyrene states identified in linear and two-photon spectra. Polyenes to N=16 show systematic deviations, with the 2 1Ag threshold decreasing less rapidly with N than experiment. We discuss perturbations that break electron-hole, parity, or point-group symmetry on the low-energy states, test the second π-electron approximation for pyrene, and comment on the feasibility of DVB for N=18 or 20 sites.