Qualitative Views on the Polyradical Character of Long Acenes - The

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Qualitative Views on the Polyradical Character of Long Acenes Georges Trinquier, Gregoire David, and Jean-Paul Malrieu J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b03344 • Publication Date (Web): 01 Aug 2018 Downloaded from http://pubs.acs.org on August 2, 2018

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The Journal of Physical Chemistry

Qualitative Views on the Polyradical Character of Long Acenes Georges Trinquier,a Grégoire David b and Jean-Paul Malrieu a

a

Laboratoire de chimie et physique quantiques, IRSAMC-CNRS-UMR5626, Université Paul-Sabatier (Toulouse III), 31062 Toulouse Cedex 4, France.

b

Institut de chimie radicalaire, CNRS-UMR7273, Université d’Aix-Marseille, 13397 Marseille Cedex 20, France.

Abstract. Spin-symmetry breakings appear in the DFT treatment of polyacenes, beyond a certain length, the critical length depending on the exchange-correlation potential. This phenomenon may be attributed to an instability with respect to HOMO-LUMO mixing, which suggests a diradical character of long acenes. However, the increase of the S2 operator with acene length questions this simple view. It is shown that this increase cannot be attributed to spin polarization of the inner MOs, and that a second symmetry breaking takes place for the pentadecacene, with four unpaired electrons centered at the first and third quarter of the chain. The spin density distributions of broken-symmetry solutions support a qualitative picture in terms of tetra-methylene hexacenes separated by Clar sextets. A strategy is proposed to identify local symmetry breakings in poly-radical systems.

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1. Introduction

Linear acenes were usually considered as typical closed-shell molecules. Of course one might expect that when going to long acenes the increasing near degeneracy around the Fermi level, in particular between the HOMO and the LUMO, might induce some specific correlation effects. Nevertheless the occurrence of a spin instability for heptacene in B3LYP calculations, first noticed by Bendikov et al.,1 was quite a surprise. The same type of symmetry breaking was evidently observed in longer polyacenes. In fact, it appeared that the critical length at which symmetrybreaking takes place depends on the choice of the exchange-correlation potential. It starts for shorter acenes when the percentage of Fock exchange is increased, and spin-symmetry breaking already takes place for naphthalene in Hartree-Fock calculations.2,3 One may of course consider the occurrence of spin-symmetry breakings as an indication of specific correlation effects in the exact wave function, ruled by the Configuration Interaction between the closed-shell ground state determinant and doubly-excited determinants.4-14 One may concentrate first on the interaction between the leading closed shell determinant

Φ0 = ϕ1ϕ1...ϕn−1ϕ n−1ϕnϕn

and the determinant obtained by the (HOMOLUMO)2 double excitation

Φ( h→l )2 = ϕ1ϕ1...ϕn−1ϕ n−1ϕn+1ϕn+1 . It may be shown that appearance of spin instability is governed by an inequality regarding the ratio

Φ0 H Φ(h→l )2 ∆E( h→l )2

=

Kn,n+1 > 1/ 2 ∆E(n→n+1)2

where K n,n +1 is the exchange integral between the HOMO and the LUMO, and ∆ E ( n → n +1) is the 2

excitation energy between from Φ 0 to Φ ( h → l ) . One sees immediately the relation between the strength 2

of a non-dynamical correlation effect involving double excitations around the Fermi level and the occurrence of the spin-symmetry breaking of mean field descriptions. It is possible to obtain an ACS Paragon Plus Environment

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evaluation of the above ratio from a very simple description of the π electron population, namely from the topological Hubbard Hamiltonian15

H ub =

∑ t (a

< p ,q >

+ p

aq +aq+ a p ) + U ∑ n ↑p n ↓p , p

where t is the hopping integral between adjacent carbon atoms and U is the on-site bielectronic repulsion. The SCF MOs are those of the Hückel Hamiltonian, and immediately accessible. Then, thanks to the mirror theorem,16,17 valid for alternant hydrocarbons, hence for polyacenes, the spin symmetry-breaking inequality can be expressed3 as

Jn,n

εn

>1

where Jnn is the coulomb integral relative to the HOMO, and ε n is the energy of this MO in the Hückel model. Of course the ratio depends on the U/|t| ratio and one understands the dependence of the spinsymmetry breaking occurrence on the parametrization of the exchange correlation potential in KohnSham DFT calculations. Treating the same correlation effect in a CI language, one understands that the 2x2 CI will result in the lowering of the occupation number of the HOMO and in a non-zero occupation number of the LUMO. The deepest study of the π-electron non-dynamic correlation effect in acenes has been given by Hachman et al.18 who performed a full π valence CASSCF calculation, making use of density matrix renormalization group (DMRG) techniques. This calculation treats exactly the non-dynamical correlation of the entire π electron population in the set of valence π MOs. This work shows that when the acene length increases, the occupation numbers of the HOMO and the LUMO tend to one, by upper and lower values respectively. According to this work the same trend is present, to a lower degree, for the HOMO-1 and LUMO +1 orbitals. Calculations based on truncated π CASSCF19,20 cannot really question this result, since they induce an over-localization of the active MOs, which inhibits the diradical character.21 Of course, LUMO occupation number does not result only from the (HOMO→LUMO)2 double excitation, since its calculation runs on all excitations sending at least one electron in this MO. Hence, the connection between occupation numbers and spin-symmetry breaking is not straightforward. Nevertheless the reported increase of the occupation number of the LUMO+1 orbital and the decrease of that of the HOMO-118 suggest the possibility to consider long enough polyacenes as tetra-radicals. Of course the occupation numbers of the exact wave function follow a regular evolution when one increases the polyacene size, but it would be nice to provide a qualitative picture in terms of a succession of creations of unpaired electron couples. The present paper is not an ACS Paragon Plus Environment

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alternative to the accurate highly-correlated description, it only suggests a simple picture compatible with the conclusions of the best computations.

2. Formal identification of a second symmetry breaking

In polyacenes HOMO and LUMO essentially differ by the symmetry with respect to the longitudinal axis. If symmetry breaking only involved HOMO-LUMO mixing, with a frozen “core” of doubly occupied MOs, according to the procedure proposed by Coulaud et al.,22,23 the solution would take the form

ΦU ,FC = ϕ1ϕ1...ϕn−1ϕ n−1ϕa ϕb where ϕ a and ϕ b are linear combinations of HOMO and LUMO. They tend to localize on upper and lower polyenic chains of the molecule, the localization being complete when the rotation angle between HOMO and LUMO is equal to π/4. In this case

ϕ a = (ϕ n + ϕ n+1 ) / 2

and

ϕb = (ϕ n − ϕ n+1 ) / 2 become orthogonal, ϕa ϕb = 0 . The mean value of the S2 operator for the unrestricted frozen core determinant is bounded by 1.0

ΦU ,FC S 2 ΦU ,FC ≤ 1 , since the spin-unrestricted function is a mixture of (dominant) singlet and (minor) triplet components. Usual unrestricted mean-field calculations do not constrain the “core” to be closed shell, inner shell MOs relax to adjust the symmetry breaking of the exchange field created by ϕ a and ϕ b , "

Φ U = ϕ1' ϕ 1" ...ϕ n' −1ϕ n −1ϕ a ϕ b .

This phenomenon is the so-called spin-polarization of the “core” MOs. This function contains components of higher spin multiplicities and the mean value of S2 may exceed 1.0

ΦU S 2 ΦU > 1 . The spin polarization is a nightmare of unrestricted formalism, the conventional spin projections24 are correct regarding the ΦU , FC solution, but do not take care correctly of the spin-polarization contribution to ΦU S 2 ΦU .12-14 Correct solutions to restore the spin-symmetry exist but are more expensive.25,26 ACS Paragon Plus Environment

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Suppose now that a second symmetry breaking takes place under rotation of the HOMO-1 and LUMO+1, the remaining core being kept frozen. The resulting function '

ΦU' ,FC = ϕ1ϕ1...ϕn−2ϕn−2ϕc' ϕdϕa' ϕb' has now four open shells. The corresponding value of the S2 operator may now reach the value of 2.0 when the singly occupied MOs become mutually orthogonal. In polyacenes, these orbitals tend to localize respectively on the lower polyenic chain for, say, the α-spin orbitals ϕ c' and ϕ a' , and on the '

'

upper polyenic chain for the β-spin orbitals, ϕ d and ϕ b , but these orbitals

keep the left-right

'

symmetry, the MOs ϕ c' and ϕ d present a nodal surface passing through the center of the molecule '

while ϕ a' and ϕ b are symmetrical with respect to this plane. Since the wave function is invariant under the rotation of the MOs of the same spin, it may be illuminating to define and draw localized spin orbitals

ϕl' = (ϕa' + ϕc' ) / 2 ϕ r' = (ϕ a' − ϕc' ) / 2 , which are essentially located in the left and right parts of the lower polyene chain, and '

'

'

'

'

ϕ l = (ϕ b + ϕ d ) / 2 '

ϕ r = (ϕ b − ϕ d ) / 2 , which are essentially localized in the left and right parts of the upper polyene chain. Then the system appears are bearing four localized unpaired electrons and may be considered as a tetra-radical. The above discussed Hubbard criterion, Jnn>|εn, |has been adapted to this type of problem. One first leftright localizes the HOMO and the HOMO-1, obtaining left and right MOs,

ϕl = (ϕn + ϕn−1 ) / 2 ϕ r = (ϕ n − ϕ n−1 ) / 2 , of energy ε l = (ε n + ε n −1 ) / 2 , Then one performs the same rotation on the LUMO/LUMO+1 set, obtaining Hückel antibonding localized MOs, and the system is subject to two local spin-symmetry breaking when the condition ACS Paragon Plus Environment

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J ll

εl

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>1

is satisfied, which means that a symmetry breaking takes place in the left half of the molecule. Then, by symmetry, another symmetry breaking is expected to occur in the right half of the molecule. This semi-empirical exploration has been performed in a previous work on the series of polyacenes,3 adopting the value of the U/|t| =1.5, which is based on ab-initio highly-correlated calculations on ethylene. According to Table 1 of that work, a second localized symmetry breaking is expected to take place for n > 14. This view of multiple symmetry breaking, first proposed for model Hamiltonians, can be used from Hartre-Fock or DFT calculations as well, as will be shown in the next section.

3. Numerical tests. Two types of UDFT calculations have been performed, using the same B3LYP functional and the same 6-311G** basis set, within Gaussian and ORCA codes.27-29 With this functional, symmetry breaking takes place for octacene, and the ΦU S 2 ΦU quantity is plotted in Figure 1 as function of the number of rings n, showing a regular increase when increasing the size of the acenes. For n=14, this value already exceeds 2.0, which would be expected for a ms=0 determinant with four orthogonal singly-occupied MOs. Increasing the percentage of Fock exchange to 30% (labelled BLYP 30 % HF) the symmetry breaking already appears for hexacene (n=6), and the phenomenon is more pronounced, as manifest from the larger mean values of the S2 operator, also plotted in Figure1. A) Analysis of the spin-density distribution of the UDFT solution A true singlet state does not exhibit spin densities but the spin-symmetry breaking of unrestricted mean-field treatment results in local spin densities of opposite signs, which are indicative of the location and spread of the unpaired electrons, and therefore bear some significance. If the interpretation in terms of two local spin instabilities on both left and right parts of the large enough molecule is correct, one may expect “spin densities” to present two maxima along the chain, and a local minimum in its central region. Figure. 2 reports spin density distributions along the chain obtained from B3LYP-UDFT calculations, for the series of linear acenes. The appearance of a minimum of spin density in the center of the molecule is observed from n=13, indeed. As expected, occurrence of this minimum appears for a shorter acene when increasing the percentage of Fock exchange (n=12, see Figure 3). This double minimum of spin density (or of the unpaired electron density) along the chain


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supports the qualitative view of pentadecacene as a tetraradical as pictured in Scheme 1, where the central ring bears a Clar sextet. Of course one may object that the lateral maximums of the spin density distribution beyond n=12 are of moderate amplitudes. The localizable unpaired electrons delocalize to some extent on the central ring; where spin density remains rather large. The existence of the lateral maximum of the UDFT spin densities may nevertheless be considered at least as a symptom of a qualitative phenomenon. The next subsection develops a strategy to identify this phenomenon.

Scheme 1

B) Proof of the existence of two local symmetry breakings The second type of calculations specifically concerns the n=15 member of the family. They follow the decomposition path suggested in refs. 12-14, thanks to its recent implementation in the ORCA package. One first optimizes the restricted open-shell determinant (RODFT) of the ms=2 quintet state. This solution defines four symmetry-adapted singly-occupied MOs, ϕh−1 , ϕh , ϕl , ϕl +1 , where h and l subscripts stand for HOMO and LUMO, and which are plotted in Figure 4. Next, one localizes these four singly-occupied MOs, which appear on upper/lower and left/right quarters of the molecule, respectively, now labelled ϕul , ϕdl , ϕur , ϕdr with subscripts l and r referring to left and right halves respectively, and subscripts u and d referring to upper and lower chains respectively. For instance ϕul is given by

ϕdl = (ϕh +ϕl −ϕh−1 −ϕl +1) / 2 . This orbital ϕdr is pictured in Figure 5a. The other localized singly occupied MOs are obtained from similar linear combinations. One may put two α spins in the upper part and two β spins in the lower part, this ms=0 determinant core.ϕ ul ϕ dlϕ ur ϕ dr is only 0.0002 Hartree above the ms=2 determinant, which means that exchange integrals between these localized MOs are negligible. One may then relax the singly-occupied MOs in the field of the frozen closed-shell core and get a “frozen-core” solution

ΦU' ,FC = core.ϕ 'ul ϕ 'dlϕ 'ur ϕ 'dr . These MOs are equivalent in space and now present some weak ACS Paragon Plus Environment

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delocalization between the upper and lower parts, as shown in Figure 5b for one of them, namely ϕ'dr . This MO is very close to the localized SOMO of the fully relaxed UDFT solution, appearing in Figure 5c. The energy lowering is -0.0126 a.u., brought by a kinetic exchange (i.e. delocalization of the MOs and introduction of ionic components in the wave function) and =1.4365 remains large. This function is free from a spin polarization of the core MOs. The fully relaxed UDFT solution, Φ U , is 0.0215a.u. below ΦU' ,FC , with ΦU S 2 ΦU = 2.3479, these changes can be attributed to the spin polarization of the doubly occupied MOs. In order to asses our interpretation in terms of two symmetry breakings taking place on both halves of the molecule, one has to show that a symmetry breaking may take place in one half of the molecule, the other half keeping a closed shell character. To keep a closed-shell character to the right half of the molecule we introduce a doubly occupied MO in this right part, with locally the same phase as the HOMO. This MO, the expression of which is ϕr = (ϕur − ϕlr ) / 2 , is drawn in Figure 5d. We define a determinant with only two singly-occupied MOs in the left part of the molecule and see whether it keeps an open shell character while relaxing these open shell MOs. In other words we ' minimize a determinant Φ UL , FC = core.ϕ "ul ϕ "dl ϕ r ϕ r , where the symmetry breaking only concerns the

left half of the molecule, the right part being closed shell. The optimized MO ϕ "dl is plotted in Figure 5e. One sees that it is slightly less delocalized on the right part of the molecule than the MO ϕ ' dl (Figure 5b). The optimized solution exhibits =0.877, typical of a diradical, with an energy -2383.0801 a.u., 0.0043 a.u. below the determinant Φ 'L,FC = core.ϕul ϕ dlϕr ϕ r . This energy lowering is due to small delocalizations between the two unpaired MOs localized in the left part of the molecule. The appearance of this localized symmetry breaking confirms the idea that the molecule is subject to a double symmetry breaking, concerning respectively its left and right halves.

4. Conclusion: a pictorial suggestion and a tool to identify local symmetry breakings The first output of the above discussion concerns the polyacene series. The results may indeed receive a qualitative pictorial transcription. The fact that the first symmetry breaking appears for n=7 (at least for the B3LYP exchange correlation potential), and given the smallness of spin densities on external rings, suggests to write this acene as composed of two Clar sextets on the external rings, connected by a tetra-methylene anthracene in the central part of the molecule (Scheme 2). In a UDFT


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calculation, the spin densities of a tetra-methylene acene − a disjoint singlet diradical, as the two SOMOs are connected

Scheme 2

by their nodal sites − are almost equal on the upper and lower polyenic chains, indeed. Now, for intermediate values of 7

Qualitative Views on the Polyradical Character of Long Acenes - The

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