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The radical character of the acenes: A density matrix renormalization group study
Johannes Hachmann, Jonathan J. Dorando, Michael Aviles, Garnet Kin-Lic Chan
TL;DR
The paper investigates whether longer acenes possess polyradical ground states rather than simple closed-shell singlets. It uses ab-initio DMRG to perform complete active space calculations correlating the full π-valence space from naphthalene to dodecacene. The ground state remains a singlet with a finite singlet-triplet gap, while several wavefunction measures suggest increasing polyradical character in longer acenes.
Problem
The study examines whether longer acenes evolve from diradicaloid systems to tri- or higher polyradical ground states, a question not easily answered with single-determinant density functional theory.
Method
Ab-initio DMRG is used for complete active space calculations correlating the full π-valence space of acenes from naphthalene to dodecacene.
Results
The ground state remains a singlet with a finite singlet-triplet gap for all chain lengths, while longer acenes show increasing polyradical character.
Takeaways & Limitations
Longer acenes are appropriately described as singlet polyradicals whose electronic structure involves strongly interacting electrons and predominantly covalent valence-bond structures.
Abstract
from arXiv · showhide
We present a detailed investigation of the acene series using high-level wavefunction 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.
I. INTRODUCTION
Longer acenes raise questions about whether their ground states are more than simple closed-shell singlets and require methods capable of describing many-electron radical character. The study therefore applies high-level wavefunction theory to investigate their electronic structure and bonding.
- Longer acenes may have unusual ground states beyond the simple closed-shell singlet predicted by molecular-orbital arguments.
- Key questions concern whether longer acenes evolve from diradicaloid to tri- or higher polyradical ground states and how their electronic structure and bonding should be understood.
- These many-electron radical states are difficult to characterize with density functional theory based on a single Kohn-Sham determinant.
- The study uses high-level wavefunction theory because singlet states with unpaired electrons require multiconfigurational wavefunctions, such as those in complete active space methods.
- An ab-initio DMRG algorithm enables complete active space calculations correlating the full π-valence space across acenes from naphthalene to dodecacene.
- The work finds that higher acenes possess increasing polyradical character and uses electron-correlation visualization to develop a new picture of bonding in extended conjugated molecules.
II. COMPUTATIONAL METHODOLOGY
The calculations use DMRG as an efficient active-space full-configuration-interaction approach for long acenes. The study correlates the complete π-valence space while treating σ electrons with a frozen-core approximation.
- DMRG is used to efficiently and essentially exactly correlate the active-space electrons in the sense of full configuration interaction.
- The DMRG energies are converged to better than 0.1 kcal/mol and would equal traditional CASCI energies if those calculations were tractable.
- The calculations use UB3LYP/6-31G(d)-optimized singlet and triplet geometries with D2h point-group symmetry.
- The active space contains all conjugated carbon pz orbitals, and all π electrons are correlated.
- σ electrons are treated in a frozen-core approximation using restricted Hartree-Fock orbitals.
- The study uses STO-3G calculations up to dodecacene and Dunning’s double-ζ basis up to hexacene, with two pz orbitals per carbon in the double active space.
III. THE SINGLET-TRIPLET GAP
DMRG calculations find singlet ground states with finite singlet-triplet gaps throughout the acene series. The gap decreases with the larger basis and is extrapolated to a finite value in the infinite-chain limit, while DFT gaps generally underestimate it.
- DMRG calculations confirm singlet ground states and finite singlet-triplet gaps for all acene chain lengths.
- 17.5 kcal/mol is the hexacene singlet-triplet gap obtained with the DZ basis.
- The singlet-triplet gap decreases by a few kcal/mol when moving from STO-3G to the DZ basis and corresponding larger double active space.
- DFT gaps appear to underestimate the experimental data, particularly for UBLYP, which substantially underestimates the gap.
- The infinite-chain singlet-triplet gap is extrapolated as 8.69 ± 0.95 kcal/mol with STO-3G and 3.33 ± 0.39 kcal/mol with DZ.
- The extrapolated infinite-chain gaps are somewhat lower than the previous PPP estimate of 12.2 kcal/mol.
IV. POLYRADICAL CHARACTER OF THE GROUND-STATE
Natural-orbital occupations and effective-unpaired-electron measures show that longer acenes develop increasingly distributed radical character. Although the ground states remain singlets, the trends suggest that sufficiently long acenes become polyradical rather than simple diradicals.
- The HONO and LUNO occupations approach 1 with increasing acene length, consistent with diradical character.
- The HONO-1 and LUNO+1 approach single occupancy at a comparable rate, suggesting polyradical character beyond the 12-acene.
- Figure 3 tracks near-single-occupancy natural orbitals as a function of acene chain length in the STO-3G basis.
- The effective-unpaired-electron measures use natural-orbital occupations, with each orbital contributing maximally when its occupation is 1.
- The effective-unpaired-electron measures must not be interpreted literally because they are extensive and increase with molecular size.A large assembly of nearly closed-shell molecules could also yield substantial values, although its HONO and LUNO occupations would remain near 2 and 0.
- Extensive scaling begins around hexacene, roughly associating one unpaired spin with every five to six rings.
A. Theoretical background
The paper introduces particle, spin-spin, and singlet-diradical correlation functions to characterize many-electron correlations and single-occupancy behavior in acenes. Their interpretation is anchored by limiting two-electron wavefunctions and caveats about single-determinant and extensive measures.
- Particle correlation measures the dependence between α population in orbital i and β population in orbital j, vanishing for a single-determinant wavefunction.
- Spin-spin correlation measures the correlation between spins in orbitals i and j, but does not fully vanish in non-interacting systems because of Pauli correlations.
- The figures compare molecular and natural orbitals for pentacene and plot unpaired-electron measures across the acene series.
- Singlet-diradical correlation measures correlations between opposite-spin single occupancies after removing their independent probabilities, and likewise retains Pauli-type contributions.
- For separated-orbital two-electron wavefunctions, the correlation functions reach characteristic extrema, whereas a doubly occupied orbital gives zero correlation functions.
B. Correlation functions
Correlation functions reveal short-range antiferromagnetic and single-occupancy patterns in acene ground states. Increasing chain length strengthens these correlations and changes their strand dependence consistently with a coupled-polyacetylene-strand picture.
- The correlation-function plots fix one position and vary the second, enabling spatial comparison around acene strands and rings.
- Fixing a reference orbital at the lower-strand center, particle correlations alternate positive and negative with rapidly decreasing amplitudes, indicating short-range antiferromagnetic correlations.
- Spin-spin and singlet-diradical correlations show strong neighboring single-occupancy antiferromagnetic correlations that decay rapidly away from the reference position.
- Pentacene lacks significant chain-wide antiferromagnetic correlations in the single-determinant comparison, apart from a small reference-site reduction in double occupancy.
- Longer acenes show slightly stronger antiferromagnetic correlations, with same-strand correlations overtaking neighboring-strand correlations relative to naphthalene.
- In pentacene, edge-enhanced bond alternation produces asymmetric correlations, with stronger correlations across shorter bonds.
VI. THE NATURE OF BONDING IN THE ACENE POLYRADICAL STATE
Correlation functions depict electrons in longer acenes as locally antiferromagnetically coupled and predominantly covalent, supporting a polyradical bonding picture rather than simple delocalization.
- Correlation-function picture: Correlation functions track an electron and a nearby, antiferromagnetically coupled electron distributed over neighboring atoms.This provides a dynamic picture of correlated electronic motion in the acene chains.
- Resonance interpretation: Electron delocalisation requires roughly equal contributions from covalent and ionic resonance structures, whereas purely covalent structures describe localized, unpaired electrons.The paper uses this distinction to interpret the correlation functions and resonance character of the acene state.
- Correlation-function picture: The correlation-function plots compare particle-particle, spin-spin, and singlet diradical correlations across napthalene, pentacene, and dodecacene using a fixed reference point.Numbers show correlation values, while circle size and color encode magnitude and sign; a non-interacting pentacene model provides a comparison.
- Resonance interpretation: Correlation functions offer an alternative way to infer the resonance nature of a state beyond analyses limited to small molecules.The interpretation connects short-range antiferromagnetic correlations with a predominantly covalent valence-bond description.
- Resonance interpretation: When U/t ≫1, molecular-orbital descriptions give way qualitatively to superpositions of covalent resonance structures because double occupancy is strongly disfavored.Here t measures delocalization through hopping and U measures Coulomb repulsion associated with double occupancy.
VII. CONCLUSIONS
The study applies ab-initio DMRG to full π-valence-space CAS calculations across the acene series. It finds singlet ground states with polyradical character in longer acenes and neighboring-atom antiferromagnetic coupling consistent with predominantly covalent resonance structures.
- Conclusions: Ab-initio DMRG enabled Complete Active Space calculations correlating the full π-valence space from napthalene through dodecacene.The method extends high-level wavefunction calculations to these long acene molecules.
- Conclusions: The ground state remains a singlet as acene chain length increases, with a finite singlet-triplet gap in the infinite-chain limit.The conclusion is supported by the reported DMRG calculations across the series.
- Conclusions: Longer acenes exhibit singlet polyradical character in their ground states according to wavefunctions, natural orbitals, effective unpaired-electron counts, and correlation functions.These measures collectively distinguish the ground-state character from a simple closed-shell description.
- Conclusions: Electrons are antiferromagnetically coupled in pairs on neighboring atoms as they move around the acene chains.This correlation pattern supports a predominantly covalent valence-bond resonance description of the longer acenes.
- Conclusions: The study identifies longer acenes as moderately strongly interacting systems whose excitations require a polyradical reference description.The authors state that this viewpoint is essential for understanding the excitations of these systems.