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.
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.
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.
Static-spin-correlation functions 4〈spzsqz〉 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 〈spzsqz〉 that are independent of intersite potentials 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〉=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.
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.
Abstract A spin‐free symmetry‐adapted valence bond (VB) state, named bonded tableau (BT), is deduced from the classical bonded function and labeled by an at most two‐column Weyl tableau. The complete set, which is composed of the BT basis or canonical bonded tableau (CBT), can be constructed from an overcomplete set of BT states. CI CBT and VB CBT are two kinds of complete sets that are constructed in this paper. They can be used, respectively, in the CI and VB theory. It is shown that there is a one‐to‐one correspondence between the labeling scheme for CI CBT and the Gelfand–Tsetlin (GT) basis. This relationship enables an efficient generation and compact representation of the BT basis if one desires to use the known global representation scheme for the GT basis. Effective algorithms for the matrix element evaluation of unitary group generators and products of generators between BT states are presented. In the formulation, the action of a generator on a BT state yields another BT state times a coefficient, so that the matrix elements of an arbitrary multiple product of generators are reduced to a calculation of the overlaps between BT states. The evaluation of the overlaps leads to a simple factorization into cycle contributions, whose values are given explicitly and only depend on the length parameters of the cycles. It is hoped that the presented formalism can facilitate the procedures for handling of the many‐electron correlation problem.
Following the discovery of high Tc superconductivity in the copper oxides, there has been a great deal of interest in the RVB wave function proposed by Anderson [1]. As a warm-up exercise we have considered a valence-bond wave function for the one dimensional spin-1/2 Heisenberg chain. The main virtue of our work is to propose a new variational singlet wavefunction which is almost analytically tractable by a transfer-matrix technique. We have obtained the ground state energy for finite as well as infinite chains, in good agreement with exact results. Correlation functions, excited states, and the effects of other interactions (e.g., spin-Peierls) are also accessible within this scheme [2]. Since the ground state of the chain is known to be a singlet (Lieb & Mattis [3]), we write the appropriate wave function as a superposition of valence-bond singlets, [Formula: see text], where | k > is a spin configuration obtained by pairing all spins into singlet pairs, in a way which is common in valence-bond calculations of large molecules. As in that case, each configuration, | k >, can be represented by a Rümer diagram, with directed bonds connecting each pair of spins on the chain. The c k 's are variational co-efficients, the form of which is determined as follows: Each singlet configuration (Rümer diagram) is divided into "zones", a "zone" corresponding to the region between two consecutive sites. Each zone is indexed by its distance from the end of the chain and by the number of bonds crossing it. Our procedure assigns a variational parameter, x ij , to the j th zone, when crossed by i bonds. The resulting wavefunction for an N-site chain is written as [Formula: see text] where m ij(k) equals 1 when zone j is crossed by i bonds and zero otherwise. To make the calculation tractable we reduce the number of variational parameters by disallowing configurations with bonds connecting any two sites separated by more than 2M lattice points. (For simplicity, we have limited ourselves to M=3, but the scheme can be used for any M). With the simple ansatz, matrix elements can be calculated by a transfer-matrix method. To understand the transfer-matrix method note that since only local zone parameters appear in the description of each state | k >, matrix elements and overlaps, [Formula: see text] and < k | k '>, are completely specified by a small number of "local states" associated with each zone. Within a given zone a local state is defined by (i) the number of bonds crossing the zone and (ii), by whether the bonds originate from the initial (| k >) or final (| k '>) state. It is then easy to see that "local states" of consecutive zones are connected by a 15 × 15 transfer matrix (for the case M=3). Furthermore, the overlap matrix element can be written as a product of transfer-matrices associated with each zone of the chain. When calculating matrix elements of the Hamiltonian, an additional matrix, U , must be defined, to represent the particular zone involving the two spins connected by the Heisenberg interaction. The relevant details as well as the comparison with exact results will be given elsewhere. We are planning to ultimately apply this method to the two dimensional case, and hope to include the effects of holes.