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The Journal of Physical Chemistry A

Strong Correlation in Acene Sheets from the Active-Space Variational Two-Electron Reduced Density Matrix Method: Effects of Symmetry and Size

Kenley M. Pelzer, Loren Greenman, Gergely Gidofalvi, David A. Mazziotti

Published May 12, 2011• 98 citations
Publisher

Abstract

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).

DOI
10.1021/jp2017192

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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).
Molecular systems in chemistry often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the number N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-density-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calculation where the 2-RDM is constrained by approximate N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as r(a)(6) where r(a) is the number of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calculation of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H(2)O, and N(2) as well as n-acene chains for n=2-8. When n>4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approximately 1.47x10(17) configuration state functions. The natural occupation numbers of the n-acenes show the emergence of bi- and polyradical character with increasing chain length.
Molecular periodic systems have the challenge of not only representing an infinite, albeit periodically repeating, molecule but also treating the strong electron correlation that emerges in such systems. Many post-Hartree-Fock methods that are efficient in moderately correlated molecules often fail to describe strongly correlated electrons and their properties in periodic systems. Here we develop and apply an electronic structure approach to periodic systems based on the variational calculation of the two-electron reduced density matrix (2-RDM). The variational 2-RDM method calculates a lower bound to the ground-state energy of a molecular periodic system without computation or storage of its many-electron wave function. The 2-RDM is constrained by NNN-representability conditions that are necessary for it to represent at least one NNN-electron density matrix---conditions that allow it to treat strongly correlated systems. Two periodic systems are treated to demonstrate the methodology: (i) a metallic hydrogen chain and (ii) a strongly correlated acene chain. In the hydrogen chains the 2-RDM theory describes the strongly correlated Mott metal-to-insulator transition while in acene chains it describes the emergence of polyradical character with increasing chain length. The methodology is applicable to treating strong electron correlation in more general periodic molecules and materials.
Variational two-electron reduced-density-matrix (2-RDM) methods can provide a reference-independent description of the electronic structure of strongly correlated molecules and materials. These methods represent one of few ways of performing large active-space-based computations that are beyond the scope of conventional configuration-interaction-based approaches. However, implementations of the method are quite rare, making it difficult for the quantum chemistry community to assess the utility of the approach. Here, we discuss an open-shell implementation of the variational 2-RDM method and explore its ability to describe different spin states in several model systems, including linear hydrogen chains and linear acenes. The largest calculations considered are comparable to complete-active-space computations with 50 electrons in 50 orbitals.
Full configuration interaction (FCI) restricted to a pairing space yields size-extensive correlation energies but its cost scales exponentially with molecular size. Restricting the variational two-electron reduced-density-matrix (2-RDM) method to represent the same pairing space yields an accurate lower bound to the pair FCI energy at a mean-field-like computational scaling of O(r3) where r is the number of orbitals. In this paper, we show that localized molecular orbitals can be employed to generate an efficient, approximately size-extensive pair 2-RDM method. The use of localized orbitals eliminates the substantial cost of optimizing iteratively the orbitals defining the pairing space without compromising accuracy. In contrast to the localized orbitals, the use of canonical Hartree-Fock molecular orbitals is shown to be both inaccurate and non-size-extensive. The pair 2-RDM has the flexibility to describe the spectra of one-electron RDM occupation numbers from all quantum states that are invariant to time-reversal symmetry. Applications are made to hydrogen chains and their dissociation, n-acene from naphthalene through octacene, and cadmium telluride 2-, 3-, and 4-unit polymers. For the hydrogen chains, the pair 2-RDM method recovers the majority of the energy obtained from similar calculations that iteratively optimize the orbitals. The localized-orbital pair 2-RDM method with its mean-field-like computational scaling and its ability to describe multi-reference correlation has important applications to a range of strongly correlated phenomena in chemistry and physics.
We present a detailed investigation of the acene series using high-level wave function theory. Our ab initio density matrix renormalization group algorithm has enabled us to carry out complete active space calculations on the acenes from napthalene to dodecacene correlating the full pi-valence space. While we find that the ground state is a singlet for all chain lengths, examination of several measures of radical character, including the natural orbitals, effective number of unpaired electrons, and various correlation functions, suggests that the longer acene ground states are polyradical in nature.
An improved parametrization of the two-electron reduced density matrix (2-RDM) [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] was recently shown to yield energies and properties that are markedly better than those calculated by traditional ab initio methods of similar computational scaling. In this paper a family of such energy functionals, generalizing the ones obtained previously, is derived through the use of (i) ppp-particle contraction relations based on the contraction of the cumulant expansions of ppp-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of NNN-representability constraints known as the two-positivity conditions. The 2-RDMs are explicitly parameterized in terms of the first-order part of the cumulant 2-RDM and, for the inclusion of single excitations, a second-order part of the 1-RDM. In contrast to earlier formulations based on the coefficients from configuration interaction with single and double excitations (CISD), the cumulant-based parametric 2-RDM methods, from the properties of cumulants, are rigorously size extensive. We also show that writing the energy functionals in terms of correlated 1-RDMs and cumulant 2-RDMs reduces the computational cost of the parametric 2-RDM methods to that of CISD. Applications are made to ground-state energies of several molecules, equilibrium bond distances, and frequencies of HF, F2{\mathrm{F}}_{2}F2​, and CO, the relative energy of the cis and trans isomers of HO3{\mathrm{HO}}_{3}HO3​${}^{\ensuremath{-}}$, and the HCN-HNC isomerization reaction. For bond breaking in hydrogen fluoride the improved and more efficient parametric 2-RDM methods yield energies with similar accuracies at both equilibrium and nonequilibrium geometries in 6-31G** and polarized valence quadruple-$\ensuremath{\zeta}$ basis sets. Computed 2-RDMs very nearly satisfy well-known NNN-representability conditions.