Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Correlated π-electronic states: Pyrene, 16-site polyene, and D2h symmetry adaptation

Guozhong Wen, Z. G. Soos

Published Feb 8, 1998• 9 citations
Publisher

Abstract

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.

DOI
10.1063/1.475631

Related papers

Symmetry adaptation of correlated states in the valence bond basis

S. Ramasesha, Z. G. Soos

Symmetry-adapted linear combinations of valence-bond (VB) diagrams are constructed for arbitrary point groups and total spin S using diagrammatic VB methods. VB diagrams are related uniquely to invariant subspaces whose size reflects the number of group elements; their nonorthogonality leads to sparser matrices and is fully incorporated into a binary integer representation. Symmetry-adapated linear combinations of VB diagrams are constructed for the 1764 singlets of a half-filled cube of eight sites, the 2.8 million π-electron singlets of anthracene, and for illustrative S≥0 systems.

The Journal of Chemical Physics•Mar 1, 1993•22 citations

Symmetrized density matrix renormalization group algorithm for low-lying excited states of conjugated carbon systems: Application to 1,12-benzoperylene and polychrysene

Suryoday Prodhan, S. Ramasesha

The symmetry adapted density matrix renormalization group (SDMRG) technique has been an efficient method for studying low-lying eigenstates in one- and quasi-one-dimensional electronic systems. However, the SDMRG method had bottlenecks involving the construction of linearly independent symmetry adapted basis states as the symmetry matrices in the DMRG basis were not sparse. We have developed a modified algorithm to overcome this bottleneck. The new method incorporates end-to-end interchange symmetry $({C}_{2})$, electron-hole symmetry $(J)$, and parity or spin-flip symmetry $(P)$ in these calculations. The one-to-one correspondence between direct-product basis states in the DMRG Hilbert space for these symmetry operations renders the symmetry matrices in the new basis with maximum sparseness, just one nonzero matrix element per row. Using methods similar to those employed in the exact diagonalization technique for Pariser-Parr-Pople (PPP) models, developed in the 1980s, it is possible to construct orthogonal SDMRG basis states while bypassing the slow step of the Gram-Schmidt orthonormalization procedure. The method together with the PPP model which incorporates long-range electronic correlations is employed to study the correlated excited-state spectra of 1,12-benzoperylene and a narrow mixed graphene nanoribbon with a chrysene molecule as the building unit, comprising both zigzag and cove-edge structures.

Physical review. B./Physical review. B•May 14, 2018•12 citations

Diagrammatic valence‐bond theory for finite model Hamiltonians

S. Ramasesha, Z. G. Soos

Abstract Valence bond (VB) diagrams form a complete basis for model Hamiltonians that conserve total spin, S, and have one valence state, ϕp, per site. Hubbard and Pariser–Parr–Pople (PPP) models illustrate ionic problems, with zero, one, or two electrons in each ϕp, while isotropic Heisenberg models illustrate spin problems, with only purely covalent VB diagrams. The difficulty of nonorthogonal VB diagrams is by‐passed by exploiting the finite dimensionality of the complete basis and working with unsymmetric sparse matrices. We introduce efficient bit manipulations for generating, storing, and handling VB diagrams as integers and describe a new coordinate relaxation method for the ground and lowest excited states of unsymmetric sparse matrices. Antiferromagnetic spin‐½ Heisenberg rings and chains of N ⩽ 20 spins, or 2N spin functions, are solved in C2 symmetry as illustrative examples. The lowest S = 1 and 0 excitations are related to domain walls, or spin solitons, and studied for alternations corresponding to polyacetylene. VB diagrams with arbitrary S and nonneighbor interactions are constructed for both spin and ionic problems, thus extending diagrammatic VB theory to other topologies.

International Journal of Quantum Chemistry•Jun 1, 1984•112 citations

Static spin correlations in alternant quantum cell models

S. Ramasesha, Z. G. Soos

Static-spin-correlation functions 4〈${s}_{\mathrm{pz}}$${s}_{\mathrm{qz}}$〉 are evaluated via diagrammatic valence-bond theory for exact states of finite Hubbard, Pariser-Parr-Pople (PPP), and related quantum cell models. Several properties of 〈${s}_{\mathrm{pz}}$${s}_{\mathrm{qz}}$〉 that are independent of intersite potentials ${V}_{\mathrm{pp}\mathcal{'}}$ 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 〈${n}_{p}$〉=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.

Physical review. B, Condensed matter•Oct 15, 1985•22 citations

Density-matrix renormalization-group study of low-lying excitations of polyacene within a Pariser-Parr-Pople model

C. Raghu, Y. Anusooya Pati, S. Ramasesha

We have carried out symmetrized density-matrix renormalization-group calculations to study the nature of excited states of long polyacene oligomers within a Pariser-Parr-Pople Hamiltonian. We have used the ${C}_{2}$ symmetry, the electron-hole symmetry, and the spin parity of the system in our calculations. We find that there is a crossover in the lowest dipole forbidden two-photon state and the lowest dipole allowed excited state with size of the oligomer. In the long system limit, the two-photon state lies below the lowest dipole allowed excited state. The triplet state lies well below the two-photon state and energetically does not correspond to its description as being made up of two triplets. These results are in agreement with the general trends in linear conjugated polymers. However, unlike in linear polyenes wherein the two-photon state is a localized excitation, we find that in polyacenes, the two-photon excitation is spread out over the system. We have doped the systems with a hole and an electron and have calculated the charge excitation gap. Using the charge gap and the optical gap, we estimate the binding energy of the ${1}^{1}{B}^{\ensuremath{-}}$ exciton to be 2.09 eV. We have also studied doubly doped polyacenes and find that the bipolaron in these systems, to be composed of two separated polarons, as indicated by the calculated charge-density profile and charge-charge correlation function. We have studied bond orders in various states in order to get an idea of the excited state geometry of the system. We find that the ground state, the triplet state, the dipole allowed state, and the polaron excitations correspond to lengthening of the rung bonds in the interior of the oligomer while the two-photon excitation corresponds to the rung bond lengths having two maxima in the system.

Physical review. B, Condensed matter•Jul 29, 2002•63 citations

Alternancy symmetry: A unified viewpoint

Jaroslav Koutecký, Josef Paldus, Jiřı́ Čı́žek

A general formulation of the alternancy symmetry adaptation for the semiempirical Pariser–Parr–Pople (PPP) type Hamiltonians is presented at both the spin-orbital and spin-adapted many-electron levels. The derivation of the general form of the alternancy symmetry conjugation operators is based solely on the tight-binding approximation for the short range one-particle part of the Hamiltonian considered. It starts by a simple formulation of the desired invariance properties of the PPP type Hamiltonian. Using algebraic properties of the unitary group generators and of their particle number nonconserving extensions, it leads to a completely explicit and general form for the alternancy symmetry conjugation operators. In this way the prior descriptions, which become special cases of this general formulation, are interrelated and unified. The spin and quasispin character of certain components of these operators are also pointed out and explicitly derived. The spin-adapted version is based on the unitary group formulation of the valence bond-type approach. A completely general formulation is given which applies to many-electron states of an arbitrary multiplicity of neutral systems, either of the normal kind with an identical number of starred and nonstarred sites, or of the radicaloid character with different numbers of starred and nonstarred sites. An explicit form of the dependence of the relative phase factor of the alternancy symmetry conjugation operator on the total spin, total electron number and on the number of starred and nonstarred sites is also given. General rules for the construction of spin and alternancy symmetry-adapted states are illustrated on a few simple examples. Finally, a brief discussion of the implications of the alternancy symmetry is presented.

The Journal of Chemical Physics•Aug 15, 1985•50 citations
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.
Symmetry-adapted linear combinations of valence-bond (VB) diagrams are constructed for arbitrary point groups and total spin S using diagrammatic VB methods. VB diagrams are related uniquely to invariant subspaces whose size reflects the number of group elements; their nonorthogonality leads to sparser matrices and is fully incorporated into a binary integer representation. Symmetry-adapated linear combinations of VB diagrams are constructed for the 1764 singlets of a half-filled cube of eight sites, the 2.8 million π-electron singlets of anthracene, and for illustrative S≥0 systems.
The symmetry adapted density matrix renormalization group (SDMRG) technique has been an efficient method for studying low-lying eigenstates in one- and quasi-one-dimensional electronic systems. However, the SDMRG method had bottlenecks involving the construction of linearly independent symmetry adapted basis states as the symmetry matrices in the DMRG basis were not sparse. We have developed a modified algorithm to overcome this bottleneck. The new method incorporates end-to-end interchange symmetry (C2)({C}_{2})(C2​), electron-hole symmetry (J)(J)(J), and parity or spin-flip symmetry (P)(P)(P) in these calculations. The one-to-one correspondence between direct-product basis states in the DMRG Hilbert space for these symmetry operations renders the symmetry matrices in the new basis with maximum sparseness, just one nonzero matrix element per row. Using methods similar to those employed in the exact diagonalization technique for Pariser-Parr-Pople (PPP) models, developed in the 1980s, it is possible to construct orthogonal SDMRG basis states while bypassing the slow step of the Gram-Schmidt orthonormalization procedure. The method together with the PPP model which incorporates long-range electronic correlations is employed to study the correlated excited-state spectra of 1,12-benzoperylene and a narrow mixed graphene nanoribbon with a chrysene molecule as the building unit, comprising both zigzag and cove-edge structures.
Abstract Valence bond (VB) diagrams form a complete basis for model Hamiltonians that conserve total spin, S, and have one valence state, ϕp, per site. Hubbard and Pariser–Parr–Pople (PPP) models illustrate ionic problems, with zero, one, or two electrons in each ϕp, while isotropic Heisenberg models illustrate spin problems, with only purely covalent VB diagrams. The difficulty of nonorthogonal VB diagrams is by‐passed by exploiting the finite dimensionality of the complete basis and working with unsymmetric sparse matrices. We introduce efficient bit manipulations for generating, storing, and handling VB diagrams as integers and describe a new coordinate relaxation method for the ground and lowest excited states of unsymmetric sparse matrices. Antiferromagnetic spin‐½ Heisenberg rings and chains of N ⩽ 20 spins, or 2N spin functions, are solved in C2 symmetry as illustrative examples. The lowest S = 1 and 0 excitations are related to domain walls, or spin solitons, and studied for alternations corresponding to polyacetylene. VB diagrams with arbitrary S and nonneighbor interactions are constructed for both spin and ionic problems, thus extending diagrammatic VB theory to other topologies.
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 have carried out symmetrized density-matrix renormalization-group calculations to study the nature of excited states of long polyacene oligomers within a Pariser-Parr-Pople Hamiltonian. We have used the C2{C}_{2}C2​ symmetry, the electron-hole symmetry, and the spin parity of the system in our calculations. We find that there is a crossover in the lowest dipole forbidden two-photon state and the lowest dipole allowed excited state with size of the oligomer. In the long system limit, the two-photon state lies below the lowest dipole allowed excited state. The triplet state lies well below the two-photon state and energetically does not correspond to its description as being made up of two triplets. These results are in agreement with the general trends in linear conjugated polymers. However, unlike in linear polyenes wherein the two-photon state is a localized excitation, we find that in polyacenes, the two-photon excitation is spread out over the system. We have doped the systems with a hole and an electron and have calculated the charge excitation gap. Using the charge gap and the optical gap, we estimate the binding energy of the ${1}^{1}{B}^{\ensuremath{-}}$ exciton to be 2.09 eV. We have also studied doubly doped polyacenes and find that the bipolaron in these systems, to be composed of two separated polarons, as indicated by the calculated charge-density profile and charge-charge correlation function. We have studied bond orders in various states in order to get an idea of the excited state geometry of the system. We find that the ground state, the triplet state, the dipole allowed state, and the polaron excitations correspond to lengthening of the rung bonds in the interior of the oligomer while the two-photon excitation corresponds to the rung bond lengths having two maxima in the system.
A general formulation of the alternancy symmetry adaptation for the semiempirical Pariser–Parr–Pople (PPP) type Hamiltonians is presented at both the spin-orbital and spin-adapted many-electron levels. The derivation of the general form of the alternancy symmetry conjugation operators is based solely on the tight-binding approximation for the short range one-particle part of the Hamiltonian considered. It starts by a simple formulation of the desired invariance properties of the PPP type Hamiltonian. Using algebraic properties of the unitary group generators and of their particle number nonconserving extensions, it leads to a completely explicit and general form for the alternancy symmetry conjugation operators. In this way the prior descriptions, which become special cases of this general formulation, are interrelated and unified. The spin and quasispin character of certain components of these operators are also pointed out and explicitly derived. The spin-adapted version is based on the unitary group formulation of the valence bond-type approach. A completely general formulation is given which applies to many-electron states of an arbitrary multiplicity of neutral systems, either of the normal kind with an identical number of starred and nonstarred sites, or of the radicaloid character with different numbers of starred and nonstarred sites. An explicit form of the dependence of the relative phase factor of the alternancy symmetry conjugation operator on the total spin, total electron number and on the number of starred and nonstarred sites is also given. General rules for the construction of spin and alternancy symmetry-adapted states are illustrated on a few simple examples. Finally, a brief discussion of the implications of the alternancy symmetry is presented.