Polyaromatic hydrocarbons (PAHs) are a class of organic molecules with importance in several branches of science, including medicine, combustion chemistry, and materials science. The delocalized π-orbital systems in PAHs require highly accurate electronic structure methods to capture strong electron correlation. Treating correlation in PAHs has been challenging because (i) traditional wave function methods for strong correlation have not been applicable since they scale exponentially in the number of strongly correlated orbitals, and (ii) alternative methods such as the density-matrix renormalization group and variational two-electron reduced density matrix (2-RDM) methods have not been applied beyond linear acene chains. In this paper we extend the earlier results from active-space variational 2-RDM theory [Gidofalvi, G.; Mazziotti, D. A. J. Chem. Phys. 2008, 129, 134108] to the more general two-dimensional arrangement of rings--acene sheets--to study the relationship between geometry and electron correlation in PAHs. The acene-sheet calculations, if performed with conventional wave function methods, would require wave function expansions with as many as 1.5 × 10(17) configuration state functions. To measure electron correlation, we employ several RDM-based metrics: (i) natural-orbital occupation numbers, (ii) the 1-RDM von Neumann entropy, (iii) the correlation energy per carbon atom, and (iv) the squared Frobenius norm of the cumulant 2-RDM. The results confirm a trend of increasing polyradical character with increasing molecular size previously observed in linear PAHs and reveal a corresponding trend in two-dimensional (arch-shaped) PAHs. Furthermore, in PAHs of similar size they show significant variations in correlation with geometry. PAHs with the strictly linear geometry (chains) exhibit more electron correlation than PAHs with nonlinear geometries (sheets).
Applying the density matrix renormalization group (DRMG) method to a nonempirical valence bond (VB) model Hamiltonian, we studied polyacene oligomers of different lengths in the strong electron correlation limit. Geometrical optimizations were performed for the lowest singlet and triplet states of oligomers up to [40]-acene, and a convergence of the bond lengths toward the polymer limit is observed in the interior of the oligomer. For large oligomers, as well as for the polymer, the ground state can be reasonably determined to be a singlet. Furthermore, a high similarity between the singlet geometries and triplet geometries suggests an open-shell character for the singlet ground state. A reasonable speculation of the soliton-antisoliton pair character of the singlet ground state was supported by a spin distribution analysis of the triplet state wave function of large oligomers, with each of the two solitons being broadly delocalized over the upper or bottom edge of the oligomers, respectively.
We have compared different strategies for ab initio quantum chemistry density matrix renormalization group treatments. The two starting orbital blocks include all valence and active orbitals of the reference complete active space self consistent field wave function. To generate the remaining blocks, we propose following the order of the contributions to the correlation energy: a posteriori using approximate occupation numbers or a priori, using a Møller–Plesset type of arguments, by explicit evaluation of second-order interactions. We have compared two different schemes for orbital localization to identify the important and less important orbital interactions and simplify the generation of the orbital blocks. To truncate the expansion we have compared two approaches, keeping constant the number m of components or the threshold λ to fix the residue of the expansion at each step. The extrapolation of the energies is found to provide accurate estimates of the full configuration interaction energy, making the expansion independent on the actual values of the two parameters m and λ. We propose to generate the factors for the two blocks from ground and excited eigenvectors of the Hamiltonian matrix.
The dimerization of polyacetylene is studied using a distance-dependent Hubbard Hamiltonian for the valence $\ensuremath{\pi}$ electrons and approximating $\ensuremath{\sigma}$-electron contributions by a sum of pair interactions ${E}_{\ensuremath{\sigma}}{(r}_{\mathrm{ij}})$. The $\ensuremath{\pi}$-electron Coulomb repulsion
U is taken to be independent of the C-C distances
rij and is varied systematically from the uncorrelated or H\"uckel limit all over to the strongly correlated or Heisenberg regime. For each value of
U, ${E}_{\ensuremath{\sigma}}{(r}_{\mathrm{ij}})$, and the $\ensuremath{\pi}$-electron hopping integral
t(rij) are fitted to reproduce ab initio results on the ground-state and first-excited-state energy of the ethylene molecule. The one-dimensional many-body problem is solved numerically by means of the density-matrix renormalization-group (DMRG) method. The ground-state energy
E of the infinite open polymer is obtained by extrapolation considering chains of up to 300 atoms. The dimerization $\ensuremath{\delta}{=(r}_{i,i+1}\ensuremath{-}{r}_{i\ensuremath{-}1,i})/2$ and the average bond length ${R=(r}_{i,i+1}{+r}_{i\ensuremath{-}1,i})/2$ are determined by minimizing
E. While
R is not significantly affected by the value of
U, $\ensuremath{\delta}$ shows a remarkable nonmonotonic behavior. The differences between the delocalized regime, small
U/t, and spin-Peierls behavior, large
U/t, are discussed. The convergence properties of the DMRG algorithm are analyzed as a function of
U, $\ensuremath{\delta}$, and the number states kept at each RG step. Criteria are examined in order to control the accuracy of the DMRG calculations in both weakly and strongly correlated situations.
We present a new implementation of the driven similarity renormalization group (DSRG) based on a density matrix renormalization group (DMRG) reference. The explicit build of high-order reduced density matrices is avoided by forming matrix-product-state compressed intermediates. This algorithm facilitates the application of DSRG second- and third-order perturbation theories to dodecacene with an active space of 50 electrons in 50 orbitals. This active space appears the largest employed to date within the framework of internally contracted multireference formalism. The DMRG-DSRG approach is applied to several challenging systems, including the singlet–triplet gaps (Δ ST ) of oligoacenes ranging from naphthalene to dodecacene, the vertical excitation energies of zeaxanthin, and the ground-state potential energy curve (PEC) of Cr 2 molecule. Our best estimate for the vertical Δ ST of dodecacene is 0.22 eV, showing an excellent agreement with that of the linearized adiabatic connection method (0.24 eV). For zeaxanthin, all DSRG schemes suggest the order of 2 A g − 1 < 1 B u + 1 < 1 B u − 1 for excited states. Both the equilibrium and the shoulder regions of the Cr 2 PEC are reasonably reproduced by the linearized DSRG with one- and two-body operators.
The quantum chemical version of the density matrix renormalization group (DMRG) method has established itself as one of the methods of choice for calculations of strongly correlated molecular systems. Despite its great ability to capture strong electronic correlation in large active spaces, it is not suitable for computations of dynamical electron correlation. In this work, we present a new approach to the electronic structure problem of strongly correlated molecules, in which DMRG is responsible for a proper description of the strong correlation, whereas dynamical correlation is computed via the recently developed adiabatic connection (AC) technique which requires only up to two-body active space reduced density matrices. We report the encouraging results of this approach on typical candidates for DMRG computations, namely, n -acenes ( n = 2 → 7), Fe(II)–porphyrin, and the Fe 3 S 4 cluster.