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arXiv:1812.04962 · Physical review. B./Physical review. B

Parquet approximation for molecules: Spectrum and optical conductivity of the Pariser-Parr-Pople model

Petra Pudleiner, Patrik Thunström, Angelo Valli, Anna Kauch, Gang Li, Karsten Held

Published Mar 11, 2019• 23 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.99.125111

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We study a simple model system for the conjugated \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ bonds in molecules than single site dynamical mean-field theory.
The Pariser-Parr-Pople (PPP) model of a single-band one-dimensional (1D) metal is studied at the Hartree-Fock level, and by using the second-order perturbation theory of the electronic correlation. The PPP model provides an extension of the Hubbard model by properly accounting for the long-range character of the electron-electron repulsion. Both finite and infinite version of the 1D-metal model are considered within the PPP and Hubbard approximations. Calculated are the second-order electronic-correlation corrections to the total energy, and to the electronic-energy bands. Our results for the PPP model of 1D metal show qualitative similarity to the coupled-cluster results for the 3D electron-gas model. The picture of the 1D-metal model that emerges from the present study provides a support for the hypothesis that the normal metallic state of the 1D metal is different from the ground state.
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 $\ensuremath{\pi}$ 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 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 $t=\ensuremath{-}4.83\text{ }\text{eV}$ 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.
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.
We have implemented the dynamical vertex approximation $(\mathrm{D}\ensuremath{\Gamma}\mathrm{A})$ in its full parquet-based version to include spatial correlations on all length scales and in all scattering channels. The algorithm is applied to study the electronic self-energies and the spectral properties of finite-size one-dimensional Hubbard models with periodic boundary conditions (nanoscopic Hubbard rings). From a methodological point of view, our calculations and their comparison to the results obtained within dynamical mean-field theory, plain parquet approximation, and the exact numerical solution allow us to evaluate the performance of the $\mathrm{D}\ensuremath{\Gamma}\mathrm{A}$ algorithm in the most challenging situation of low dimensions. From a physical perspective, our results unveil how nonlocal correlations affect the spectral properties of nanoscopic systems of various sizes in different regimes of interaction strength.
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.