How delocalized are the polyacenes?

In an attempt to quantify electron delocalization in polyacenes with up to 50 carbon atoms, we have performed self‐consistent field calculations in which the π electrons are constrained to occupy highly localized molecular orbitals (HILOs) centered on a maximum of two, six or ten adjacent carbon atoms. We have also performed similar calculations on simple polyacene analogs consisting only of hydrogen atoms and exhibiting electron delocalization in the σ framework. We find that the energetic cost of localizing the π electrons in the polyacenes is roughly 60, 5 or 0.1 kJ/mol per ring atom for the two‐, six‐ and ten‐atom HILOs, respectively, and the use of these localized models overestimates the predicted hydrogenation energies of the acenes by roughly 50%, 4% and 0.1%, respectively. We conclude that the chemistry of polyacenes can be modeled well using highly localized descriptions of the π electrons.


| INTRODUCTION
For two centuries, from Faraday's isolation of benzene 1 in 1825, through the structural insights of Kekulé and others [2][3][4][5] and the application of quantum mechanics, 6,7 to the recent experimental characterizations of octacene, 8 nonacene 8,9 decacene 10 and dodecacene, 11 there has been sustained interest in the bonding and electronic structure of the n-acenes.
The n-acene molecule (Figure 1, left) consists of n linearly fused benzene rings and has the stoichiometric formula C 4nþ2 H 2nþ4 .It has n þ 1 Kekulé structures and the double bonds in each Kekulé structure are all conjugated.The electronic structure of an acene is sometimes described as mixture, or resonance, of the Kekulé structures and the π electrons are often said to be delocalized across the entire molecule.But what does it actually mean to say that some of the electrons in a molecule are "delocalized"?At a fundamental quantum mechanical level, the question is a fraught one because electrons are indistinguishable and one cannot perform an experiment to monitor the trajectory of a particular electron and to confirm thereby that it is roaming freely.So what, then, is the essential difference between the electronic structure of a long alkane and a long polyene, and how can this difference be quantitatively defined and calculated?
To answer this question, we have calculated the energy of n-acenes by four methods, which we will call HILO-1, HILO-2, HILO-3 and CMO, each of which enforces localization more weakly than its predecessors.(HILO and CMO stand for highly localized orbitals and canonical molecular orbitals, respectively.)By comparing these, we can extract the energetic costs associated with various localizations.However, before considering the n-acenes, we will review the wavefunctions of a particle in an anisotropic three-dimensional box and introduce a family of hypothetical molecules called n-hydrocenes (Figure 1, right) with the stoichiometric formula H 4nþ2 that are the pure hydrogen analogs of the n-acenes.As we will show below, the delocalization of the σ electrons in hydrocenes is analogous to that of the π electrons in acenes and the former therefore provide a simple model for the latter that may be useful in future explorations of delocalization and aromaticity in extended systems.
Because many density functional theory (DFT) approximations 12,13 seem unreliable for polycyclic aromatics, 14,15 we will use Hartree-Fock (HF) theory 16,17 to compute the HILO and CMO orbitals.Moreover, to exaggerate the deleterious effects of localization, we will adopt the minimal STO-3G basis set 18 throughout.HILO calculations were performed using an in-house program 19 and CMO calculations were performed using the Q-Chem 6.1 software package. 20Orbital isosurfaces that enclose 70% of the electron density were visualized using the Mathematica 13.3 and IQmol 3.1 packages. 21,22 Section 2, we discuss the CMOs of hydrocenes and acenes with up to 12 rings.In Section 3, we define the HILO-m orbital models.In Section 4, we present the CMO and HILO-m energies of the hydrocenes and acenes and compare the dissociation and hydrogenation energies that the CMO and HILO-m models yield.Finally, in Section 5, we draw conclusions about the extent to which the electrons in these molecules are intrinsically delocalized.

| CANONICAL ORBITALS
The most popular model of electronic structure is the mean-field paradigm, in which the electrons move independently within selfconsistent canonical molecular orbitals (CMOs).This conceptually simple picture originated in the HF model and also underpins Kohn-Sham DFT.Following the seminal work of Roothaan and Hall, 23,24 each CMO is expanded in a basis set of functions that are centered on the various nuclei in the molecule and, consequently, in principle, it extends over the entire molecule.

| Particle in a box
A three-dimensional box aligned with the Cartesian axes, with one vertex at the origin and with sidelengths L x > L y > L z , belongs to the D 2h point group and has three mirror (xy, xz and yz) planes. 25The wavefunctions of a particle moving freely within the box are and, because this is a one-particle system, its wavefunctions are also its CMOs.We note that the CMO ψ p,q,r has p, q and r lobes in the x, y and z directions, respectively, and the parities of p, q and r determine its irreducible representation.This is illustrated in Figure 2.

| Hydrocenes
The n-hydrocene molecule (Figure 1, right) also belongs to the D 2h point group and we choose the long axis, the short axis and the normal to the molecular plane to define the x, y and z axes, respectively.Its σ CMOs are symmetric under reflection in the xy plane and therefore transform as A g , B 1g , B 2u or B 3u .The particle-in-a-box CMOs are isomorphic to those of n-hydrocene and this is illustrated, for example, by the similarity between the top rows of Figures 3 and 2.

| Acenes
The n-acene molecule (Figure 1, left) also belongs to the D 2h point group and we choose the long axis, the short axis and the normal to the molecular plane to define the x, y and z axes, respectively.Its π CMOs are antisymmetric under reflection in the xy plane and therefore transform as A u , B 1u , B 2g or B 3g .As before, the particle-in-a-box and n-acene CMOs are isomorphic and this is illustrated, for example, by the similarity between the bottom rows of Figures 3 and 2.

| Rationale
More than a century ago, Lewis proposed that a chemical bond arises from a localized pair of electrons, sitting between a pair of nuclei. 27Although the Lewis model was quickly discovered to be inadequate at a fundamental level, it is still taught in introductory chemistry classes, remains useful for qualitative purposes and plays a pivotal role in "arrow-pushing" descriptions of organic reaction mechanisms. 28 the following decade, quantum mechanics was discovered and the resulting beautiful rationalizations of aromaticity by Hückel,6 ligand-field effects by Griffith and Orgel 29 and pericyclic reactions by Woodward and Hoffmann 30 convinced chemists of the usefulness of the molecular orbital picture of electronic structure.
The modern chemist, therefore, is a dualist who strives to be comfortable with two essentially contradictory pictures of the electrons within a molecule, and who selects the most appropriate picture for the particular question under discussion.It has been said that retaining the ability to function while simultaneously holding two opposed views is the hallmark of a first-rate intelligence, 31 but it could also be viewed as an unhealthy cognitive dissonance and we recommend the insightful discussion on this matter 32 by Shaik.
Although we acknowledge that some electronic systems are intrinsically delocalized-and a high-density uniform electron gas 33 is an obvious example-the functional group paradigm which underpins almost all of chemistry is undoubtedly inclined toward a microscopic picture in which the electrons are more or less confined to small regions of a molecule.Recent work by Yang and co-workers has also confirmed both the conceptual 34 and computational 35 advantages of viewing electronic structure through such a localizing lens.
It is well known that the HF wavefunction and energy depend only on the span of the occupied CMOs, not the individual CMOs themselves, and we are therefore at liberty to linearly combine the CMOs in any span-preserving way that suits our purposes.7][38][39][40][41][42] However, even the most compact of these LMOs retain annoying but indelible tails that intrude into other parts of the molecule 43 and we must therefore look elsewhere if we seek genuinely localized orbitals.
In the next section, we define orbitals that are even more localized than LMOs.By the variational theorem, such orbitals will necessarily yield a higher energy than that from the CMOs (or LMOs) but, if the increase is sufficiently small, it may be a price worth paying for the conceptual and computational benefits of truly compact orbitals.Furthermore, they will allow us to address the central question of this paper, quantitatively measuring the "delocalization" of an electronic system by the degree to which its energy is raised if its CMOs are replaced by orbitals that are strictly confined.

| HILO-m
Our starting point for the construction of localized orbital models is a Kekulé structure.The n-hydrocene and n-acene molecules have n þ 1 Kekulé structures and each consists of 2n þ 1 perfectly localized Kekulé orbitals; these are σ orbitals in a hydrocene and π orbitals in an acene.One must select a particular Kekulé structure and, after a number of numerical experiments, we found that the structure in which the bridgehead Kekulé orbital is as central as possible yields the lowest energy.We have therefore adopted this throughout.If n is even, the bridgehead orbital lies exactly at the molecular center; if n is odd, it lies on the left of the central ring.
Figure 1 shows these Kekulé structures for the first five acenes and hydrocenes.
Our first and crudest model is HILO-1, which denotes "Highly Localized Orbitals using only primary basis functions".To build HILO-1 orbitals, we minimize the HF energy, subject to the constraint that each Kekulé orbital is expanded using only the basis functions centered on its two parent atoms (primaries) but without constraining the HILO-1 orbitals to be orthogonal.Thus, HILO-1 treats the H 6 molecule as three isolated H 2 molecules and the C 6 H 6 molecule as cyclohexatriene.We use E 1 to denote the HILO-1 energy and L 1 a,b to denote the HILO-1 orbital that models the Kekulé orbital between atoms a and b.The black dots in Figure 4 show the atoms whose basis functions are allowed to contribute to the L 1 a,b in 5-hydrocene or 5acene and we see that each HILO-1 orbital uses the basis functions on two atoms.
Our second model is HILO-2, which denotes "Highly Localized Orbitals using only primary and secondary basis functions".The HILO-2 orbitals are found by minimizing the HF energy but, in this case, each Kekulé orbital is also allowed to use the basis functions on the bonded neighbors (secondaries) of the primaries.As before, we do not constrain orthogonality.We use E 2 to denote the HILO-2 energy and L 2 a,b to denote a HILO-2 orbital that models the Kekulé orbital between atoms a and b.The black and grey dots in Figure 4 show the atoms whose basis functions are allowed to contribute to the L 2 a,b in 5hydrocene or 5-acene and we see that each HILO-2 orbital uses the basis functions on 4, 5 or 6 atoms.
Our third model is HILO-3, which denotes "Highly Localized Orbitals using only primary, secondary and tertiary basis functions".
The HILO-3 orbitals are found by minimizing the HF energy but, now, each Kekulé orbital is also permitted to use the basis functions on the bonded neighbors (tertiaries) of the secondaries.Orthogonality is not constrained.We use E 3 to denote the HILO-3 energy and L 3 a,b to denote a HILO-3 orbital that models the Kekulé orbital between atoms a and b.The black, grey and white dots in Figure 4 show the atoms whose basis functions are allowed to contribute to the L 3 a,b in 5hydrocene or 5-acene and we see that each HILO-3 orbital uses the basis functions on 7 or 10 atoms.

| Localization energies
Because the expansion basis for a HILO-m orbital is a subset of the basis for the corresponding HILO-(m þ 1) orbital, it follows from the variational theorem that It is then reasonable to define the HILO-m localization energy as and the HILO-m specific localization energy of an n-hydrocene or n-acene as which is the localization energy per ring atom.All of these quantities are non-negative and they quantify the energy penalties associated with localizing the orbitals according to the various HILO protocols.

| Hydrocenes
Although hydrocenes (Figure 1) are beautiful, they are not stable and the n-hydrocene molecule dissociates without a barrier into 2n þ 1 H 2 molecules.Their structures therefore cannot be optimized and, in order to perform HILO and CMO calculations, we must begin by stipulating a molecular structure.[53] To perform HILO-m calculations, we used initial guesses that assign unit coefficients to the primary basis functions and zero coefficients to all others.The orbitals were then optimized to selfconsistency using a regularized Newton optimizer. 19The resulting HILOs have the sizes, shapes and symmetries that one would expect from the schema in Figure 4 and all of them are positive everywhere.
To illustrate this, Figure 5  In contrast, Table 1 gives the self-consistent HILO-m and CMO energies and the specific localization energies for the n-hydrocenes with 1 ≤ n ≤ 12 and this information provides a quantitative assessment of the energetic costs of the HILO-1, HILO-2 and HILO-3 localization schemes in these systems.
The HILO-1 specific localization energies Δ 1 vary between 35 and 38 mE h per atom in all of the hydrocenes studied, indicating that localizing each orbital onto just the primary atoms incurs an energy penalty of roughly 100 kJ/mol per atom.The observation that the energies of the hydrocenes are increased so dramatically implies that HILO-1 is a poor approximation for these systems.HILO-1 HILO-2 HILO-3 CMO The HILO-2 specific localization energies Δ 2 are strikingly different.For n ¼ 1, we find that Δ 2 ¼ 0 and, therefore, the HILO-2 orbitals are in fact equivalent to the CMOs in the smallest hydrocene.As the number of rings increases, Δ 2 grows slowly and seems unlikely to exceed 4 mE h (10 kJ/mol) per atom.This suggests that HILO-2 is a reasonable approximation in these systems, especially if n is not too large.
The HILO-3 specific localization energies Δ 3 are remarkably small.
Following the HILO-2 pattern, Δ 3 ¼ 0 in H 6 but it grows very slowly with increasing n and seems unlikely to exceed 0.2 mE h (0.5 kJ/mol) per atom.We conclude that HILO-3 is an accurate approximation in all hydrocenes.
One may ask to what extent the localization energies affect chemical reaction energies and, to explore this, we have computed the energy E released in the dissociation reaction Localization raises the energy of the reactants, but not the products, in (5) and the lack of error cancellation means that which is also confirmed by the computed dissociation energies in Table 1.However, the most interesting discovery is that, whereas HILO-1 overestimates E by at least 60%, HILO-2 overestimates by no more than 6% and HILO-3 by not more than 0.2%.

| Acenes
For simplicity and reproducibility, our single-point HILO and CMO calculations on the n-acenes were performed on structures that were optimized at the HF/STO-3G level.
The initial guesses for the π HILOs assigned unit coefficients to the primary p z basis functions and zero coefficients to all others.The initial guesses for the canonical σ orbitals used a standard Q-Chem algorithm. 20All orbitals were then optimized to self-consistency using a regularized Newton optimizer. 19The resulting HILOs have the sizes, shapes and symmetries that one would expect from the schema in The HILO-1 specific localization energies Δ 1 vary between 20 and 29 mE h per atom in all of the acenes studied, indicating that localizing each orbital onto just the primary atoms incurs an energy penalty of roughly 60 kJ/mol per atom.The observation that the energies of the acenes are increased so dramatically implies that HILO-1 is a poor approximation for these systems.
The HILO-2 specific localization energies Δ 2 are strikingly different.For n ¼ 1, we find that Δ 2 ¼ 0 and, therefore, the HILO-2 orbitals are in fact equivalent to the CMOs in the smallest acene.As the number of rings increases, Δ 2 grows slowly and seems unlikely to exceed 2 mE h (5 kJ/mol) per atom.This suggests that HILO-2 is a reasonable approximation in these systems.
The HILO-3 specific localization energies Δ 3 are remarkably small.
Following the HILO-2 pattern, Δ 3 ¼ 0 in C 6 H 6 but it grows very slowly with increasing n and seems unlikely to exceed 0.04 mE h (0.1 kJ/mol) per atom.We conclude that HILO-3 is an excellent approximation in all acenes.
To explore the extent to which these localization energies affect chemical reaction energies, we have computed the energy E released in the hydrogenation reaction where the product has C 2h symmetry.These reaction energies are also shown in Table 2.
Localization raises the energy of the reactants, but not the products, in (7) and the lack of error cancellation means that which is also confirmed by the computed dissociation energies in Table 2.However, the most interesting discovery is that, whereas HILO-1 overestimates E by at least 50%, HILO-2 overestimates by no more than 4% and HILO-3 by not more than 0.1%.

| CONCLUSIONS
We have found that forcing the "delocalized" electrons in hydrocenes and acenes to occupy highly localized orbitals (HILOs) such as those in This discovery alters the qualitative molecular orbital picture of acenes but we believe that it may also yield highly efficient approaches to the calculation of correlation energies.Since the pioneering work 54 by Pulay, it has been recognized that localizing the HF orbitals before performing a post-HF calculation can accelerate subsequent post-HF calculations.HILOs may prove to be especially effective in this context and we plan to explore this in the near future.
T A B L E 2 HILO-

F I G U R E 1
Kekulé structures of the n-acenes (left) and n-hydrocenes (right).The Kekulé orbitals are indicated by double bonds in the acenes and solid bonds in the hydrocenes.

F I G U R E 4
The atoms whose basis functions are allowed to contribute to the HILO-m orbitals of 5-hydrocene or 5-acene.The HILO-1 orbitals (L 1 a,b ) use the black (primary) atoms; the HILO-2 orbitals (L 2 a,b ) use the black and grey (secondary) atoms; the HILO-3 orbitals (L 3 a,b ) use the black, grey and white (tertiary) atoms.
compares the L m 1,3 , L m 9,10 and L m 15,17 orbitals obtained from HILO-1, HILO-2 and HILO-3 calculation on 5hydrocene.As we proceed from HILO-1 to HILO-2 to HILO-3, the number of basis functions available for each orbital increases and they exploit this flexibility by spreading onto neighboring atoms.However, although this expansion is driven by energy lowering, Figure 5 offers no insight into the magnitude of that lowering.

Figure 4 .
Figure 4. To illustrate this, Figure 6 compares the L m 1,3 , L m 9,10 and L m 15,17 orbitals obtained from HILO-1, HILO-2 and HILO-3 calculation on 5acene.As we proceed from HILO-1 to HILO-2 to HILO-3, the number of basis functions available for each orbital increases and, as we saw for the hydrocenes, they exploit this flexibility by spreading onto neighboring atoms.How much lower do the energies become when such spreading is allowed?Table 2 gives the self-consistent HILO-m and CMO energies and the specific localization energies for the n-acenes with 1 ≤ n ≤ 12 and this information provides a quantitative assessment of the energetic costs of the HILO-1, HILO-2 and HILO-3 localization schemes in these systems.

Figure 6
Figure 6 can yield useful approximations to the electronic energy.HILO-1 are our most localized orbitals, correspond to a single Kekulé structure, permit no conjugation and introduce errors of tens of mE h per ring atom.HILO-2 and HILO-3 orbitals are less localized and introduce errors of roughly 3 and 0.1 mE h per ring atom, respectively, in hydrocenes and roughly half as much in acenes.As a result, HILO-2 and HILO-3 increase the predicted dissociation energies of 1, HILO-2, HILO-3 and CMO total energies (E, in E h ), specific localization energies (Δ, in mE h per atom) and hydrogenation energies (E, in kJ/(mol H 2 )) of n-acenes.