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Article
The Polyradical Character of Triangular non-Kekule Structures, Zethrenes, p-Quinodimethane Linked Bisphenalenyl and the Clar Goblet in Comparison: An Extended Multireference Study
Anita Das, Thomas Müller, Felix Plasser, and Hans Lischka
J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b12393 • Publication Date (Web): 09 Feb 2016
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The Polyradical Character of Triangular non-Kekule Structures, Zethrenes, p-Quinodimethane linked Bisphenalenyl and the Clar Goblet in Comparison: An Extended Multireference Study
Anita Das,t Thomas Müller,* Felix Plasser* and Hans Lischka*'t,#'&
^Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas, United States
^Institute for Advanced Simulation, Jülich Supercomputing Centre, Forschungszentrum Jülich, Jülich, Germany
Institute for Theoretical Chemistry, University of Vienna, A-1090 Vienna, Austria &School of Pharmaceutical Sciences and Technology, Tianjin University, Tianjin, P.R.China
♦Supporting Information
carried out to compute singlet-triplet splitting for the above-listed compounds and to provide
4 ABSTRACT: In this work two different classes of polyaromatic hydrocarbon (PAH) systems
6 have been investigated in order to characterize the amount of polyradical character and to
8 localize the specific regions of chemical reactivity: (a) the non-Kekule triangular structures
11 phenalenyl, triangulene and a n-extended triangulene system with high-spin ground state and (b)
13 PAHs based on zethrenes, p-quinodimethane-linked bisphenalenyl and the Clar-goblet
15 containing a varying polyradical character in their singlet ground state. The first class of
18 structures have already open-shell character because of their high-spin ground state, which
20 follows from the bonding pattern, whereas for the second class the open-shell character is
23 generated either because of the competition between the closed-shell quinoid Kekule and the
25 open-shell singlet biradical resonance structures or the topology of the n-electron arrangement of
27 the non-Kekule form. High-level ab-initio calculations based on multireference theory have been
32 insight into their chemical reactivity based on the polyradical character by means of unpaired
34 densities. Unrestricted density functional theory and Hartree-Fock calculations have been
3367 performed for comparison also in order to obtain better insight into their applicability to these
39 types of complicated radical systems.
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1. INTRODUCTION
Over the past decade, graphene - has attracted considerable attention because of its wide range of applications,4-9 e.g. as chemical sensors, organic semiconductors, energy storage devices, in spintronics and nonlinear optics. When the graphene sheet is cut into nano-sized fragments, nanographenes containing several polycyclic aromatic hydrocarbon (PAH) units are formed. Therefore, the properties of PAHs are closely related to that of nanographene. These PAHs are composed of fused aromatic rings, a feature with almost unlimited possibilities can lead to a rich diversity of compounds.
Many of the PAHs have a closed-shell electronic configuration in their ground state. However, there are types of PAHs which possess a high-spin open-shell radical character in their ground state.10-11 For example, phenalenyl12 (1) (Chart 1) in its neutral ground state contains an odd number of carbon atoms with an odd number of n electrons which makes it a radical. The extension of benzene rings in a triangular fashion can lead to several n-conjugated phenalenyl
i/: 1017
derivatives such as triangulene - (2), n-extended triangulene system , (3) (see Chart 1) and even larger systems.
Chart 1. Non-Kekulé phenalenyl based neutral radicals with triangular structures: (1) phenalenyl, (2) triangulene and (3) n-extended triangulene system. N* and N are the number of starred and unstarred atoms.
These molecules are categorized as open-shell non-Kekule polynuclear benzenoid molecules where some electrons are unpaired due to the topology of the n electron arrangement.18 The topological effect on the ground state of these molecules can be explained using Ovchinnikov's
rule18 which states that the ground state spin quantum number, S, of a given PAH with six-
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membered rings can be expressed as S=(N*-N)/2 where N* and N are the number of starred and unstarred atoms of alternant n systems. As shown in Chart 1 in the case of phenalenyl (1), there are seven starred and six unstarred atoms. The open-shell character of this system can be understood qualitatively by considering no matter what resonance structure is chosen one starred atom will be left without a bonding partner. In case of 2, two such empty valences are created while there are three in case of 3. Application of this rule leads to the afore-mentioned high-spin ground state. On the other hand, zethrenes19-20 starting with heptazethrene21-22 (4) (Chart 2), n-acenes longer than pentacene23-27 and some zigzag edged graphene nanoribbons27-31 (GNRs) have a singlet ground state but nevertheless significant biradical or even polyradical character. Among the different types of PAHs, zethrenes have attracted significant attention recently.32-34 Zethrenes are z-shaped PAHs where two phenalenyl systems are head-to-head fused with or without benzenoid spacer. The smallest member is the hexazethrene which contains a total of six condensed rings where the two phenalenyl rings are directly fused. The next one is heptazethrene (4) where an additional benzene ring is inserted in between the two phenalenyl units. If the terminal naphthalene units are replaced by anthracene,34 depending on the position of the ring fusion, two different structural isomers with the same chemical compositions are formed; 1,2;9,10-dibenzoheptazethrene (5) and 5,6;13,14-dibenzoheptazethrene (6) (Chart 2).
Chart 2. Structures 4 heptazethrene, 5 1,2:9,10-dibenzoheptazethrene, 6 5,6:13,14-dibenzoheptazethrene showing quinoid Kekule and biradical resonance forms. The benzene rings in red represent Clar's aromatic sextet rings.
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As an alternative to zethrenes, Kubo and coworkers designed a p-quinodimethane linked bisphenalenyl35 (7) that contains p-quinodimethane and two phenalenyl rings (Chart 3). A characteristic feature of the zethrenes and structure 7 is the competition between a closed-shell quinoid Kekule form and an open-shell biradical resonance form (Charts 2 and 3).32 Because of the presence of these two resonance structures, two interesting questions arise: which resonance form is dominating in the ground state and, consequently, which isomer contains a larger biradical character? Clar's aromatic sextet rule36-38 qualitatively predicts that for benzenoid PAHs, the molecule with more aromatic sextet rings possesses more aromatic stabilization energy. Thus, if in different valence bond (VB) structures for a given PAH the biradical form contains more aromatic sextet rings than the closed-shell quinoid structure, then, as discussed in Ref. 32 and shown in Charts 2 and 3, its enhanced stability should be the source of a greater singlet biradical character. For example, in case of structures 4 and 5 three aromatic sextets occur in the biradical form whereas in the closed-shell Kekule form only two aromatic sextets are present; for structure 6, a total of five sextets can be drawn (6c') when the two radical centers are located at the terminal units. Thus, structures 4 and 5 are expected to exhibit significant biradical character and the even larger number of sextets for structure 6 should help to stabilize the biradical form in this case even more. Chart 3 represents the resonance structures between Kekule and the biradical forms for structure 7 where in the biradical form the central benzene ring obtains aromatic character. It is expected, therefore, that structure 7 also possesses a significant biradical character. Structure 8 in Chart 3 represents the non-Kekule biradical form of the Clar Goblet.
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Chart 3. Structures 7 p-quinodimethane-linked bisphenalenyl (quinoid Kekulé and biradical resonance forms) and 8 Clar Goblet. The benzene ring in red represents Clar's aromatic sextet ring.
PAHs with radical character possess unique electronic, magnetic and optical properties. Quantum chemical calculations play an important role for understanding the unique electronic characteristics of the PAHs. Among the methods available, density functional theory (DFT) would be the first choice because of its computational efficiency and the good overall performance. However, due to the partly open-shell character of these compounds, an unrestricted approach has to be used which suffers from the problem of spin-contamination,25, 39 especially in the case of broken symmetry (BS) DFT for singlet biradical species or low-spin cases in general. Therefore several other approaches such as spin-flip methods,40-41 projected Hartree-Fock theory,42 active-space variational two-electron reduced-density-matrix (2-RDM),43-44 density matrix renormalization group (DMRG)45-46 and coupled cluster with singles, doubles and non-iterative triples CCSD(T)47-48 have been applied successfully to investigate the electronic structure of various PAHs. As a systematic and general alternative theory, multireference (MR)49 methods have been used with great success as well. Among them, the multireference averaged quadratic coupled cluster (MR-AQCC)50 approach represents a good option since it allows the inclusion of quasi-degenerate configurations in the reference wave function and of dynamic electron correlation including size-extensivity contributions by considering single and double excitations explicitly. Recently, MR-AQCC calculations for the n-acenes27,51-52 and periacenes27, 51-52 and several smaller challenging biradicals53-54 have been performed successfully.
To understand the structural variation of the PAHs, their unique properties and possible
singles and doubles (MR-CISD) to discuss the properties these types of PAHs in their ground
5 applications, it is important to investigate their electronic structure in more detail. Though
7 Ovchinnikov's rule18 or the Clar's aromatic sextet rule36-38 give a general idea about the
9 multiplicity, stability and electronic character of the ground state, it is nevertheless crucial to
11 obtain a reliable quantitative description of these properties as well. Phenalenyl, its extended
1j3 versions and the combination to form zethrenes or p-quinodimethane-linked n-conjugated
14 compounds, Clar Goblet (Chart 1 - Chart 3) constitute very interesting, but also computationally
16 challenging cases for describing the polyradical character of these compounds. Here we will use
18 the afore-mentioned MR-AQCC method and the related MR configuration interaction with
21 state as well as in their lowest lying excited states. One major effort is dedicated to the
23 clarification which electronic configuration, either the closed-shell quinoid Kekulé or the open-
25 shell biradical form, describe better the ground state of structures 4-7. The related question of the
27 multiplicity of the ground state and of singlet-triplet splitting will be addressed also by these
28 high-level calculations. The complicated nature of the multireference wave function will be
30 transformed to simple descriptors such as natural orbital (NO) occupation numbers and the
32 unpaired density distribution which allow a concise chemical assessment of the polyradical
34 nature of a compound and the location of chemically reactive unpaired density regions in a
37 considered in the present context in order to obtain better insight into its applicability to these
39 difficult questions concerning the correct description of biradical systems.
molecule. By means of comparison with our multireference ab-initio results, DFT is also
2. COMPUTATIONAL DETAILS
44 Multiconfiguration self consistent field (MCSCF), mostly in the form of the complete
46 active space self consistent field (CASSCF) method, MR-CISD and MR-AQCC calculations
48 have been performed for the structures shown in Charts 1-3. For most of the calculations, two
50 different sets of complete active spaces (CAS) have been chosen: (a) a CAS(7,7) and (b) a
CAS(4,4). In active space (a), seven electrons have been distributed in seven orbitals and for
53 active space (b) four electrons have been distributed over four orbitals. The choice of these active
55 spaces is based on the NOs occupation numbers computed from the unrestricted Hartree-Fock
57 (UHF) wave function as suggested by Pulay et al.55 The contribution of the single- and double
4 substituted configurations has been monitored and in case of a weight larger than 1% extensions
5 of the active space have been made in order to account for intruder states in the MR-AQCC
7 calculations or to ensure degeneracies in case of the triangular symmetry of structures 1-3, since
9 for practical reasons only C2v symmetry was employed. The active spaces for the MCSCF, MR-
11 CISD and MR-AQCC calculations were usually kept the same. Size extensivity corrections are
13 computed for the MR-CISD approach by means of an extended Davidson correction,49, 56-57
14 denoted as +Q (MR-CISD+Q). More details on the construction of the MR wave functions and
16 the orientation of the molecules in the Cartesian coordinate system can be found in the
18 Supplementary Information (SI).
Three different orbital freezing schemes have been used throughout the calculations: (i)
21 Core freezing where only the 1s core orbitals of all the carbon atoms are frozen at the MR-CISD
23 level; (ii) Sigma-partial freezing (designated as (o)+n-space) where 1s core orbitals of the carbon
25 atoms (n) are frozen at the MCSCF level and depending on the degeneracy of the orbitals,
27 additional n or (n-1) o orbitals, respectively, are frozen at the MR-CISD level and (iii) n-space
28 where all occupied and virtual o orbitals were frozen by transforming the one- and two-electron
30 integrals from the atomic orbital (AO) basis into the basis of SCF orbitals keeping only the n
31 58-59
32 orbitals active. The 6-31G, 6-31G* and 6-311G(2d) - basis sets have been used throughout the
calculations.
For the analysis of the radical character, we have used (a) NO occupations as computed
37 from the AQCC density and (b) the unpaired density and the number of effectively unpaired
39 electrons (N^)60-63 as originally introduced by Takatsuka et al.60 as the distribution of "odd"
41 electrons. This method was further developed by Staroverov and Davidson. In order to
43 emphasize the contributions from orbitals with occupation near one and suppressing
44 contributions that are nearly doubly occupied or nearly unoccupied, the non-linear formula of
46 Head-Gordon 62-63
48 m 2
49 Nu= 2 n2 (2 - n,, ) (1)
50 i=i
52 is used in which n, is the occupation of the i NO and M is the total number of NOs.
54 All the structures have been optimized using DFT with the B3LYP functional64-66 and the
55 6-31G* basis set. Harmonic vibrational frequencies have been computed and none of the
57 structures except the 11Ag state of structure 5 shows imaginary frequencies in C2h symmetry,
below) and also for comparison purpose with the other isomer (structure 6), we have decided to
4 which demonstrates that all other structures are minima. Structure 5 possesses two out-of-plane
5 imaginary frequencies (bg and au) in C2h symmetry because of the steric congestion between the
7 hydrogen atom of the terminal benzene ring of the anthracene unit and the hydrogen atom of
9 central benzene ring. The final optimized structure 5 has Ci symmetry and the corresponding
11 frequency calculation confirmed that it is a minimum. It is important to note that the
13 experimental structure34 with the heptazethrene core also deviates from planarity with a torsional
14 angle of 33.90. The MR-AQCC calculation in Ci symmetry would have been too costly. Since
16 the difference between the planar C2h and non-planar Ci structures in terms of singlet-triplet
18 splitting and biradical character computed at DFT level is relatively small (which will discussed
21 focus on the n conjugation and keep structure 5 planar by using C2h symmetry.
23 A wave function stability analysis67 of the Kohn-Sham determinant68 has been performed
25 for the optimized 1 Ag state of structures 4-8. It was found that all the structures have a triplet
27 instability and appropriate geometry re-optimizations have been performed in reduced planar Cs
28 symmetry. The real nature (except planar structure 5 has two imaginary frequencies) of the
30 vibrational frequencies also confirmed that these structures corresponded to minima. The triplet
31 3 3
32 states (1 Bu for structures 4-6 and 1 B2u for structures 7-8) have also been optimized separately.
34 In the spin unrestricted Kohn-Sham formalism used especially the low-spin BS solution suffers
35 from the problem of spin contamination. A high spin calculation with two unpaired electrons
37 with parallel spins is applied to represent the triplet state and the BS solution with antiparallel
39 spins is used for the singlet state. To correct for spin contamination, the spin corrected formula as
41 proposed by Yamaguchi39' 69 is employed. In this approach the vertical singlet-triplet gap at a
43 given geometry "i" is given by the expression
S2 i\E — E )
~ \ ^i /^ Ti BSi '
46 № j = ET, - ES, = , 2 , . 2 , (2)
48 S (S?HSls■)
where E i , E i and E i represent the energy of the triplet, singlet and the BS solutions
5U IS BS
02 respectively. (S2 ) and (S2 ^ are the expectation value of the square of the total spin operator
7" / \ BS'
for the triplet and the BS solutions.
5 r*r\mr\nío/-1 iho mn1tir\1fi r\\-r<:kr\ir*o1 /^hara/^íor"70 71
For quantitative analysis of the open-shell character, along with NU, we have also computed the multiple diradical character70-71 yi (/=0, 1, 2) where 0 < yt < 1 and y{ > yi+1, from the spin-projected UHF (P-UHF) theory as
1 + T 2 V1^ T
y1 value the diradical description is incomplete and a tetraradical character has to be considered.
1? y^ = 1 - —br (3) 11 12
13 where T is the orbital overlap between the corresponding orbital pairs which can be
15 n • — n
16 expressed by Ti = HONO—i-LUNO+i where nHONO—i and nLUNO+i are the occupation numbers of
18 the ith highest occupied NO (HONO) and the ith lowest unoccupied NO (LUNO) computed from
2? UHF NOs. y0=0 indicates pure closed-shell and y0=1 indicates a pure diradical character. A 21
22 perfect diradical has y0=1 and yi=0. Comparable y0 and yi values indicate that in addition to the
24 HONO/LUNO pair, non-HONO/LUNO pairs are also important. In the case of y0=1 and a large
27 The MR calculations have been performed with the parallel version72-73 of the
29 COLUMBUS program system.74-76 Population analysis of the unpaired densities has been carried
3? 77-7g
31 out with the TheoDORE program. - For the DFT calculations along with the stability analysis
33 and the UHF NO calculations, the TURBOMOLE program has been used.
36 3. RESULTS AND DISCUSSION
38 3.1. Phenalenyl based triangular radicals
4? Figure 1 shows the MO occupation schemes for the ground state of the phenalenyl based
42 neutral radicals. The symmetry is given both in C2v and D3h (in parentheses) notations. The figure
43 shows that the triangular PAHs exhibit a high-spin ground state and that the spin multiplicity
45 increases with increasing molecular size. This is consistent with the non zero value of (N*-N) as
47 predicted by Ovchinnikov's rule and also supports the experimental findings that the ground
48 state of phenalenyl derivatives have a high-spin state.12' 14' 16
1U 11 12
18 Figure 1. Molecular orbital (MO) occupation schemes for (a) the 12A2 (12A1") state of phenalenyl
20 (1), (b) the 13B2 (13A2') state of triangulene (2) and (c) the 14B1 (1%") state of the n-extended
22 triangulene system (3). The symmetry is given in C2v (D3h) notation.
26 In Table 1 a comparison of the energy differences between the ground and the degenerate first
28 excited states of phenalenyl based neutral radicals are given computed at MR-CISD and MR-
CISD+Q levels. Because of the severe occurrence of intruder states in the excited states, MR-
31 AQCC calculations could not be performed in this case. Due to the use of the lower C2v
33 symmetry instead of the actual D3h symmetry in the calculations, in some of the cases the
35 degeneracy is slightly lifted. For this reason, we always considered the average energy between
37 the two degenerate states. For phenalenyl (1), the degeneracy is well reproduced within ~0.001
40 of phenalenyl is 2A1" (D3h notation). The first excited doublet state is degenerate (E" symmetry).
42 The excitation energies amount to 3.165 eV at n-CAS(7,7) and 2.777 eV at n-MR-CISD levels,
44 respectively, using the 6-311G(2d) basis and freezing all a orbitals. Reducing the basis set to 645
46 31G* has only a minor effect (<0.02 eV). Inclusion of a orbitals into the calculation increases
eV and in most of the cases for the systems 2 and 3, it remains within ~0.02 eV. The ground state
the excitation energy at the MR-CISD level by ~0.3 eV; the Davidson correction again decreases
49 this value by 0.2 eV. At n-MR-CISD level, the occurrence of intruder states prohibited the use of
51 the Davidson correction.
55 Table 1. Excitation energies AE (eV) between the ground and the degenerate first excited
57 states for the phenalenyl based neutral radicals" 1, 2 and 3 for three different methods
58 11 59 11
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using the 6-31G* and 6-311G(2d) basis sets. For 1 and 3 a n-CAS(7,7)b and for 2 a n-CAS(4,7)b reference space are used, respectively.
"System Orbital Scheme CASSCF MR-CISD MR-CISD+Q
Doublet-doublet excitation
3.193 3.165
(o)+rc-space7 6-31G*
n-space/6-31G* n-space/6-311G(2d)
3.076 2.790 2.777
Triplet-triplet excitation (1 A?' (o)+TC-space^/ 6-31G* 2.702 n-space/6-31G* 2.738 n-space/6-311G(2d) 2.714
Triplet-singlet splitting (1 Ar ^ (o)+rc-space^/ 6-31G* 0.842 n-space/6-31G* 0.844 n-space/6-311G(2d) 0.834
2.696 2.750 2.709
0.798 0.792 0.767
Quartet-quartet excitation (14A2" —> 14E")
3.243 3.217
n-space/6-31G* 3.313 n-space/6-311G(2d) 3.290
Quartet-doublet splitting (14A2" ^ n-space/6-31G* 0.928" n-space/6-311G(2d) 0.896
0.659 0.633
2.709 2.746 2.697
0.703 0.599 0.565
3.036 3.003
0.402 0.375
aThe symmetry is given in D3h notation, D3h (C2v) 1: 2Ai' (2A2) ^ 2E" (2BU 2A2);
2: A (B) ^ E (1,3A1, ' B2); 3: 4A2" OB1) ^ (2,4A2, 2,4B1) ^Orbital occupation specifications are given in Table S1. cContains intruder states
The next higher non-Kekule type triangular PAH investigated is triangulene (2). Though it has an even number of carbon atoms, no Kekule type formula can be given. 2-6-10-tri-tert-butyltriangulene14 is the first example of an experimentally synthesized genuine non-Kekule PAH. The experimentally observed linear dependence of the triplet signal intensity on 1/T for the 2-6-10-tri-tert-butyltriangulene combined with the observed g value of 2.0034 indicates that the high-spin triplet is the ground state.14 The MR-CISD calculations collected in Table 1 for triangulene (2) confirm this finding. To characterize the manifold of lowest excited states for this system, the lowest triplet excitation energy and the triplet/singlet splitting are given in Table 1.
The 1 E' state is ~2.7 eV higher than the 1 A2' ground state; the triplet/singlet splitting is only ~0.8 eV. Comparison of the excitation energies obtained with 6-31G* and 6-311G(2d) basis sets
4 shows an agreement within a few hundredths of an eV. The same is true for the comparison with
5 the 6-31G basis set (see Table S2). Freezing of the a system also shows similar small energetic
7 deviations.
9 Bearpark el al.80 found for the trioxytriangulene trianion16 at CASSCF level the triplet
11 state to be more stable than the singlet state by 0.867 eV, which is in good agreement with our
CASSCF results on triangulene (2). For the n-extended triangulene system (3) the ground state is
14 quartet in agreement with Ovchinnikov's rule. At n-MR-CISD level using the 6-311G(2d) basis,
16 this state is 3.217 eV more stable than the first excited 4E" state and the quartet/doublet splitting
18 is 0.633 eV. The Davidson correction reduces these values by ~0.2 eV. Again, the basis set effect
2? is quite small. Analysis of the progression of the excitation energies along the increase of the
21 triangular system from structures 1 to 3 shows relatively small effects. The excitation energy
23 within the same spin multiplicity even increases from 2 to 3 by ~0.3 eV whereas the
25 corresponding high spin/low spin excitation energies decreased by ~0.15 eV.
27 The comparison of the energy differences for the open-shell triangular PAHs for the
three different orbital schemes (Core freezing, Sigma-partial freezing and n-space) using the 6-
30 31G basis set is shown in Table S2. The differences to the results obtained with the polarized
32 basis sets (Table 1) are mostly less than 0.1 eV. From these data, one can conclude that (i) n-
34 space calculations are quite sufficient to describe the energy difference between the ground and
the lowest excited states and (ii) the basis set dependence on the addition of polarization
37 functions is quite modest.
39 The NO occupations displayed in Figure 2 show that phenalenyl (1) has one (1аГ),
41 triangulene (2) has two (degenerate 4e") and п-extended triangulene (3) has three (one in 2a1"
43 and two in 6e") singly occupied NO(s) (SONOs) in agreement with the MO occupation scheme
given in Figure 1. It is important to note that the appearance of radical character with high spin
46 multiplicities for the open-shell n-conjugated hydrocarbons arises because of the fused n-
47 12, 70
48 topology and not because of orbital degeneracy derived from the three-fold symmetry.
5? Consequently, extending the n-conjugation according to the topology of n electrons can lead to
53 (SOMOs).
an unlimited numbers of electron spins align parallel to each other in singly occupied MOs
10 11 12
20 21 22
10 15 20 NO Index
Figure 2. Natural orbital occupation of the 1 A\ state of phenalenyl (1), the 1 A2 state of triangulene (2) and the 14A2" state of the п-extended triangulene system (3) using a n-MR-AQCC/CAS(7,7)/6-311 G(2d) for 1 and 3 and a n-MR-AQCC/CAS(4,7)/6-311G(2d) for 2.
In Figure 3 the unpaired density is displayed for structures 1, 2 and 3. It is delocalized over the entire molecule, but resides mainly on the edges of the molecule. Moreover, for all three cases, the distribution at the edges is alternant and the unpaired density resides only on the starred atoms as indicated in Chart 1.
4 Figure 3. Density of unpaired electrons for the (a) 1 A\ state of phenalenyl (1) (Nu = 1.33 e); (b)
5 13A2' state of triangulene (2) (Nu = 2.50 e) and (c) 14A2" state of п-extended triangulene system
7 (3) (Nu = 3.66 e) using a n-MR-AQCC/CAS(7,7)/6-311G(2d) for (1) and (3) and a n-MR-
9 AQCC/CAS(4,7)/6-311G(2d) calculation for (2) (isovalue 0.005 e bohr-3) with individual atomic
11 population computed from Mulliken analysis.
15 For phenalenyl, the central carbon atom does not carry any unpaired density. This depletion of
18 stacked biradical dimers. For the phenalenyl dimer, it has been shown81 that the dimer has a
20 convex shape with the edge region being stronger bound due to the enhanced unpaired density in
22 that region. Additionally, the possibilities for o- and п-dimerization were discussed for a series
24 of substituted phenalenyls based on a set of spectroscopic methods and on DFT calculations. It
unpaired density in the central region of the triangle has important consequences on the shape of
is noteworthy that the total unpaired density values, Nu presented in Figure 3 are larger as
27 compared to the formal open-shell occupation of the high-spin state (1.33 vs 1 for phenalenyl,
29 2.50 vs 2.0 for triangulene and 3.66 vs 3.0 for п-extended triangulene system). This excess
31 unpaired density of these PAHs (1-3) is derived from non-negligible contributions generated
33 from NOs other than the SONO(s) (Figure 2).
37 3.2. Zethrenes, p-Quinodimethane linked bisphenalenyl and the Clar Goblet
Figures S1 and S2 show selected bond distances of the singlet ground state of structures
41 4-6 and 7-8 as computed at the UDFT/B3LYP/6-31G* level. The pronounced bond length
43 alternation of the central p-quinodimethane subunit of the structures 4 and 5 indicates that the
45 quinoid resonance form (Chart 2) has a dominant contribution to the singlet ground state. The
47 difference in single to double bond lengths in the quinoid ring is between 0.06-0.08 A for
48 structure 4. A slightly lower range of 0.04-0.07 A is found for both the C2h and Ci optimized
50 structures of 1,2:9,10-dibenzoheptazethrene (5). This difference is significantly reduced for the
52 alternative isomer, 5,6:13,14-dibenzoheptazethrene (6) to a range of 0.03-0.05 A. This trend
54 toward equidistant aromatic bond distances indicates the increasing importance of the biradical
VB structure which will find its counterpart in the increase of the values for the total number of
57 unpaired density (NU) discussed below. The central benzene ring in structure 7 (Figure S2) has
59 15 60
almost aromatic character indicating that this structure has a substantial biradical character. In case of the structure of Clar's Goblet (Figure S2), all CC bonds are almost equivalent having a maximum difference of 0.046 A with respect to the aromatic CC distance (1.39 A). The exception is the central carbon-carbon bond which is strongly elongated (1.478 A). This elongation divides the overall molecule into two separate moieties. The optimized bond distances "a", "b" and "c" (Chart 2) as shown in Figure S1 for structures 4 and 5 are very close to the
experimentally observed ones with a maximum deviation of 0.01 A. The optimized structure of 7 (Figure S2) shows that the bond length labeled "d" (1.456 A) which connects the two phenalenyl rings and the bond lengths "b" (1.446 A) and "c" (1.396 A) of the central benzene
35 o o
ring are very close to the experimentally observed bond lengths of 1.450 A, 1.437 A and 1.394 A, respectively.
Singlet-triplet splitting is investigated next in order to continue the discussion of polyradical character of structures 4-8. In Table 2, computed vertical and adiabatic (in parentheses) singlet-triplet splitting energies AE(S-T) are listed and compared with available experimental data.
Table 2. Theoretical vertical and adiabatic (in parentheses) singlet-triplet splittings AE(S-T) (eV), expectation value of the square of the total spin operator for the BS solution at UDFT/B3LYP level and vertical singlet-triplet gap AEproj using the spin corrected formula for heptazethrene (4), 1,2:9,10-dibenzoheptazethrene (5), 5,6:13,14-dibenzoheptazethrene (6), p-quinodimethane-linked bisphenalenyl (7) and Clar Goblet (8) in comparison with available experimental data. In the MR calculations n-CAS(4,4)a reference space and the 6-31G* basis set were used.
System AE(S-T) MR-CISD MR-CISD +Q MR-AQCC UDFT/ B3LYP fe) \F ■ pro] Exp.
4 £(1X) -E(lUg) 1.188 (0.944) 0.904 (0.612) 0.927 (0.699) 0.598 (0.424) 0.228 0.672
5 E(13BU) -E(11Ag) 0.776 (0.600) 0.516 (0.366) 0.671 (0.527) 0.384 (0.296) 0.602 0.540 -
6 E(13Bu) -E(11Ag) 0.354 (0.204) 0.447 (0.361) 0.414 (0.324) 0.201 (0.164) 0.889 0.351 0.165b
10 11 12
20 21 22
7 E(1352u) 0.342 - E(11Ag) (0.244)
E(1352u) - E(11Ag)
0.026° (0.009)
0.273 (0.153)
0.075a (0.075)
0.308 (0.197)
0.170a (0.159)
0.234 (0.211)
0.038 (0.037)
0.661 0.343 0.211c
1.159 0.085
For 8 a RAS(1)/CAS(4,4)/AUX(1) reference space was used where RAS refers to a restricted active space and AUX to an auxiliary space (see SI). Orbital occupation specifications are given in Table S3.
b 32 c 35
In the multireference calculations, all c orbitals were frozen; Ref. ; Ref.
Vertical as well as the adiabatic AE(S-T) values for the structures 4-8 are positive at all computational levels used, showing that these structures maintain a singlet ground state. However, there is a pronounced trend of AE(S-T) to decrease as one moves from structure 4 to 6. The relatively large adiabatic AE(S-T) for structure 5 (0.527 eV at n-MR-AQCC, Table 2) agrees well with the experimentally observed34 sharp NMR peak and ESR silence. This means that the excited triplet state is located relatively high in energy. For structure 6, the adiabatic AE(S-T) is
0.324 eV (n-MR-AQCC), somewhat higher than that of the experimentally observed ' value of
0.165 eV. Experimentally, a broadening of the NMR spectrum at room temperature and a broad ESR signal were observed for structure 6 and the signal intensity is decreasing with decreasing temperature. This indicates the presence of a thermally excited triplet state. These features are consistent with the theoretically observed small AE(S-T) of structure 6. Similar features are also observed for structure 7 where the adiabatic AE(S-T) computed at the same level is 0.197 eV,
very close to the experimentally observed value of 0.211 eV. For structure 8, the computed AE(S-T) is very small (mostly <0.1 eV, Table 2) showing that the singlet and the triplet states are almost degenerate. The evolution of the weights of the major configurations computed at n-MR-AQCC level (Table S4) goes in line with the just-described changes of the singlet-triplet splitting.
The singlet-triplet splitting has been computed also at UDFT level and is given in Table 2 as well together with the expectation values of S for the BS state. The spin contamination is quite significant for structures 5-8. Two kinds of AE(S-T) values have been computed at UDFT level: (i) using the spin-contaminated value and (ii) using the spin-corrected form (Eq. 3, AEproj). The effect of the spin-projection is to increase the UDFT singlet-triplet splitting bringing it closer to the MR-AQCC values. For structure 5, AEproj of the non-planar unrestricted C1 structure is 0.568 eV, very close that of the value of 0.540 eV calculated for the planar C2h structure.
10 11 12
20 21 22
A systematic comparison between singlet-triplet energies computed at several levels of orbital freezing schemes and basis sets including the unpolarized 6-31G basis (Table S5) finds in all cases a good agreement within a few tenths of an eV. These findings apply also to the computation of NO occupations (Table S6) and unpaired density (Table S7) discussed below. The observed relative insensitivity in terms of energies and character of the wave function facilitates the discussion of the polyradical character by means of multireference methods considerably. On the other hand, it is clear that this assessment has to be continuously reevaluated when different kinds of n-conjugated systems are to be investigated.
NO occupations for the singlet state of structures 4-8 are presented in Figure 4. Several of the NO occupation numbers for the singlet state deviate strongly from the closed-shell limiting values of two/zero indicating that they have substantial polyradical character.
Figure 4. Natural Orbital (NO) occupation of the 1*Ag ground state of heptazethrene (4), 1,2:9,10-dibenzoheptazethrene (5), 5,6:13,14-dibenzoheptazethrene (6), p-quinodimethane-linked bisphenalenyl (7) and Clar Goblet (8) obtained from n-MR-AQCC/CAS(4,4)/6-311G(2d) calculations for structures 4-7 and n-MR-AQCC/RAS(1)/CAS(4,4)/AUX(1)/6-311G(2d) calculation for structure 8.
Comparison of the HONO/LUNO occupation numbers (Table 3) indicates that deviation from the limiting values of two/zero is the smallest for structure 4 and the largest for structure 8. At the UHF level, these deviations from the closed-shell reference values are considerably larger
10 11 12
20 21 22
than the respective MR-AQCC results. In the former case HONO/LUNO occupations are almost constant along the series 4-8 whereas at the MR-AQCC method there is a strong variation of the occupation number, indicating a significant change in radical character. The picture of an almost uniform biradical character throughout the series 4-8 given by the UHF method is, however, not consistent with the graded evolution of the geometries and the singlet/triplet splitting discussed above.
Table 3. Comparison of the NO occupation for singlet state of the structures 4-8 obtained from n-MR-AQCC and UHF calculations, respectively.
System n-MR- UHF n-MR- UHF
AQCC AQCC
4a 1.705 1.234 0.295 0.766
5a 1.624 1.173 0.375 0.827
6a 1.387 1.118 0.610 0.882
7a 1.450 1.116 0.558 0.883
8b 1.080 1.015 0.917 0.985
VMR-AQCC/CAS(4,4)/6-311G(2d) and UHF/631G* calculations; VMR-AQCC/RAS(1)/CAS(4,4)/ AUX(1 )/6-311 G(2d) and UHF/6-31G* calculations.
Even though most of the open-shell contributions computed at MR-AQCC level are coming from the HONO/LUNO occupation, for all structures, irrespective of singlet or triplet states, there are additional NOs, whose occupation numbers deviate significantly from the limiting value of two and zero (Figure 4). This implies that in addition to the HONO/LUNO pair, other NOs also provide significant contributions to the radical character which cannot be neglected.
The densities of unpaired electrons for the singlet state are presented in Figure 5 and Figure 6 for structures 4-6 and 7-8, respectively. Unlike the situation found for the phenalenyl derivatives (1-3) where unpaired density is delocalized over the entire molecule, for structures 4, 5 and 6, the radical character is mostly distributed over a few positions. For structure 4, most of the unpaired density is located at C4/12 (see Chart 2 for numbering). For structure 5 the unpaired density is extended also to and the atom pair C5/13. For both of these structures, the unpaired electron density within the benzene ring connecting the two phenalenyl segments is significant.
10 11 12
20 21 22
In case of structure 6 the unpaired density is strongly enhanced as compared to those of 4 and 5. The main contributions are located equally at C4/12 and C7/15 indicating equal contributions from both the biradical resonance forms (6b' and 6c') as shown in Chart 2. However, the unpaired density situated on the other centers cannot be neglected. This indicates the existence of several additional VB structures in comparison to which are given in Chart 2 and Chart 3.
Figure 5. Density of unpaired electrons for the 1lAg state of (a) heptazethrene (4) (Nu = 1.03 e); (b) 1,2:9,10-dibenzoheptazethrene (5) (Nu = 1.49 e) and (c) 5,6:13,14-dibenzoheptazethrene (6) (Nu = 2.24 e) using the n-MR-AQCC/CAS(4,4)/6-311G(2d) approach (isovalue 0.003 e bohr-3) with individual atomic population computed from Mulliken analysis.
10 11 12
20 21 22
The unpaired density for structure 7 (Figure 6) shows a pattern which is more delocalized than the one indicated by the two mesomeric forms (7a' and 7b') given in Chart 3. Thus, in this case, the unpaired density can be better represented by the resonance form 7c', where the structure 7 can be considered as a combination of two phenalenyl systems linked by a benzene ring.
Figure 6. Density of unpaired electrons for the \lAg state of (a) p-quinodimethane-linked bisphenalenyl (7) (Nu = 1.86 e) using the n-MR-AQCC/CAS(4,4)/6-311G(2d) method and (b) Clar Goblet (8) (Nu = 2.88 e) using the n-MR-AQCC/RAS(1)/CAS(4,4)/AUX(1)/6-311G(2d)
method (isovalue 0.003 e bohr ) with individual atomic population computed from Mulliken analysis.
In structure 8 (Figure 6), the unpaired character is mostly located at the zig-zag edges with the largest contribution at their centers (position at C9/18, see Chart 3). It is also noted that for the zethrenes and the structures 5 and 6, the linking benzene ring seems to play a more important role (i.e. there is a significant amount of unpaired density located on this connecting ring relative to the total number of unpaired density) than in case of the vertical connections between subunits in structure 8.
Table 4 compares the NU values computed at n-MR-AQCC level with the multiple
diradical character indices, y¡ (/=0,1,2..), obtained from P-UHF theory for structures 4-8. To
obtain a common basis for comparison with the NU's, the y¡ values were multiplied by a factor of
2. It is observed that for the singlet ground states of structures 4 and 5 the 2-y0 values, which are
computed from the HONO/LUNO UHF occupations, are almost twice of the NU values
10 11 12
20 21 22
computed from the MR-AQCC HONO/LUNO occupations. Once the structures acquire more biradical character (for structures 6, 7 and 8), the two values approach each other. This behavior is derived from the discrepancies in the NO occupation numbers computed with the two different
methods (Table 3). The y0 values reported in Ref. for the structures 4-6 are somewhat smaller as compared to our values. But, the trend of increasing y0 value as one moves from structure 4 to 6 is the same. For structure 5, the y0 value as computed for C1 optimized structure is 0.649, very close to that of the C2h planar structure of 0.664. This indicates the similarity of the NO occupation numbers between the two structures. Comparing the Nu values derived from different NO selections, it is noted that the total Nu value is significantly larger than the one computed only from the HONO/LUNO part. These additional contributions come partly from the HONO-1/LUNO+1 set (Table 4), but also from the large number of NOs whose occupation numbers deviate from the 0/2 e occupations to a lesser extent. This is in contrast to the tetracyanoethylene
anion dimer (TCNE2 ) and neutral K2TCNE2 system where the effect of the non-HONO/LUNO pairs is practically negligible. In the present case the contribution of the non-HONO/LUNO pairs to the total Nu value is almost 50% for the singlet biradicaloid structures of 4 and 5 but, for the biradical structures 6-8, it decreases from 36% to 31%. Comparison of the Nu values computed from the non-HONO/LUNO pairs for the singlet and the triplet states of structures 4-8 shows that they are almost identical. This indicates that the main difference is coming from the different occupations of the HONO and LUNO pair and the remaining contributions are quite the same.
Table 4. Comparison of the number of effectively unpaired electrons (Nu) with the multiple diradical characters, 2-yo and 2-yi, for singlet and triplet states of structures 4-8
n-MR-AQCC
Singlet state b-
Triplet statea
P-UHFC
rc-MR-AQCC0
s Nu NU from NU from Nu NU from
t from HONO-1/ non- from non-
e Nu HONO/ LUNO+1 HONO/ 2-y0 2-yx Nu HONO/ HONO/
m LUNO LUNO LUNO LUNO
4 1.026 0.506 0.124 0.520 1.112 0.182 2.589 1.990 0.599
5 1.492 0.744 0.185 0.748 1.328 0.357 2.795 1.996 0.799
6 2.241 1.442 0.200 0.799 1.534 0.262 2.750 1.998 0.752
7 1.865 1.283 0.107 0.582 1.540 0.050 2.598 1.999 0.599
10 11 12
20 21 22
8 2.880 1.974
1.940 0.410 2.816 2.000
a2-y0 for triplet state is two. ''The 6-311G(2d) basis and a n-CAS(4,4) reference space for 4-7 and RAS(1)/n-CAS(4,4)/AUX( 1 ) for 8 were used. CP-UHF with 6-31G* basis set was used.
Table 4 shows that for all structures the triplet state maintains practically a constant Nu value (from 2.589 e to 2.816 e) whereas for the singlet state of structures 4-8, a large change in the Nu value (from 1.026 e to 2.880 e) is observed. As discussed just before, these differences come primarily from the HONO/LUNO pair (0.506 e to 1.974 e). For structures 4 to 5, this increase is only moderate from 1.026 e to 1.492 e but from structures 5 to 6, it is relatively large (1.492 e to 2.241 e). This clearly indicates that strong variations in the polyradical character within the zethrenes can be achieved by means of relative modest changes in the n conjugation. For the singlet state of structure 7, the Nu value is also very large (1.865 e) indicating significant singlet biradical character as well. Among the singlet state of all the structures, the Clar goblet (8) has the largest polyradical character.
For structure 4 the location of maximum density in both the singlet and the triplet states are same
4 Figure 7. Density of unpaired electrons for the 1 Bu state of (a) heptazethrene (4) (Nu = 2.59 e);
5 (b) 1,2:9,10-dibenzoheptazethrene (5) (Nu = 2.80 e) and (c) 5,6:13,14-dibenzoheptazethrene (6)
7 (Nu = 2.75 e) using the n-MR-AQCC/CAS(4,4)/6-311G(2d) approach (isovalue 0.003 e bohr-3)
9 with individual atomic population computed from Mulliken analysis.
^3 Comparison of the distribution of unpaired densities between singlet and triplet
14 states for structures 4-6 shows characteristic differences (Figure 5 and Figure 7). These
16 differences are naturally larger for the cases with smaller Nu values in the singlet state
18 (especially 4) since for the triplet state single occupation of the HONO/LUNO pair is enforced.
21 (C4/12, see Chart 2 for numbering). However, additionally, for the triplet state the unpaired
23 density extends with significant populations on the C5/13 and C7/15 positions, respectively.
25 Enhancement of similar atom position is also observed for the triplet state for structures 5 and 6.
27 Even though the Nu values between singlet and triplet start to come closer to each other, the
28 weights on individual atoms still differ. For e.g. in the singlet state of structure 6, the unpaired
30 density is equally distributed between C4/12 and C7/15 positions, respectively; for the triplet
32 state the maximum of unpaired density is located at C7/15. This detailed insight into the unpaired
34 density distribution should provide improved approaches to tune the singlet-triplet gap for these
35 compounds. On the other hand, for the singlet and triplet states of both the structures 7 and 8
37 (Figure 6 and Figure S3) the distribution of unpaired density is very similar in nature.
41 4. CONCLUSIONS
In this work two different types of PAHs have been studied (a) non-Kekule triangular
44 structures with a high-spin ground state and (b) PAHs with singlet polyradical character. For the
46 first case, phenalenyl (1), triangulene (2), and a n-extended triangulene system (3) have been
48 chosen. In the second case a series of three zethrenes, heptazethrene (4), 1,2:9,1049
50 Dibenzoheptazethrene (5), 5,6:13,14-Dibenzoheptazethrene (6), and the p-quinodimethane-
linked bisphenalenyl (7) have been investigated. Additionally, the non-Kekule Clar Goblet (8)
53 has been studied. The motivation in choosing these two types of PAHs is that structures 1-3
55 already possess open-shell character because of their high-spin ground state whereas for
57 structures 4-7 the competition between a closed-shell quinoid Kekulé valence bond structure and
4 an open-shell singlet biradical resonance form determines the actual electronic structure and the
5 chemical reactivity. For structure 8, the topology of the п-electron arrangement of the non-
7 Kekulé form is the characteristic feature. To get a reliable quantitative description of these
9 interesting systems, high-level ab-initio multireference approaches have been used. Unrestricted
11 density functional theory and Hartree-Fock calculations have been performed for structures 4-8
12 also in order to assess their applicability to these molecular systems possessing a complicated
14 electronic structure.
16 The triangular structures 1-3 have always a non-degenerate high-spin state as ground
18 state. The spin state increases with increasing molecular size as predicted by the Ovchinnikov's
19 18 12
20 rule and is also in agreement with ESR measurements of tri-t-butyl substituted phenalenyl,
21 triangulene14 and triangulene derivative.16 The calculations also show that the lowest excited
23 state is always degenerate. Although the unpaired density of the ground state of structures 1-3 is
25 delocalized over the entire molecule, it mainly resides on one of the carbon sub-lattices, i.e. the
27 starred atoms as defined above, and is for the most part located on the edges, independent of the
28 size of the triangle. This localization of the chemical reactivity has important consequences on
30 the lengths of the intermolecular CC bonds and the in general convex shape of stacked
32 phenalenyl dimers as has been discussed in detail in Ref. .
34 For the second class of systems (structures 4-8), the ground state is always singlet with a
35 varying amount of biradicaloid (structures 4, 5), biradical (structure 7) or polyradical (structures
37 6 and 8) character. The triplet states of all the structures have polyradical character. All the
39 indicators such as bond length alternation, singlet-triplet splitting, NO occupations and unpaired
41 densities clearly demonstrate that within the zethrene structure family, the singlet state of
44 Structure 7 also has open-shell singlet biradical character in its ground state. Interpretation of
46 these results within the valence bond picture confirms that Clar's aromatic sextet rule can be
48 successfully applied for the ground state of these types of systems, but for a concrete
50 characterization of the chemical reactivity high level quantum chemical calculations are needed.
53 nearly degenerate singlet and triplet states.
55 The low-spin broken symmetry state computed at UDFT/B3LYP level is highly spin
57 contaminated. Spin-projection increases the singlet-triplet gap and brings the DFT and MR-
structure 6 possesses a much larger polyradical character as compared to structures 4 and 5.
Among the structures 4-8, the Clar Goblet (8) has the maximum polyradical character having
systems by performing multireference correlation calculations for the n system only where such
4 AQCC results into good agreement. However, NO occupations derived from UHF calculations in
5 the spirit of the UNO-CAS method55 show a strong overshooting of the deviations from closed-
7 shell character for most of the singlet systems investigated and, as a consequence, also an
9 overestimation of the polyradical character as measured by the y0 and y indices as compared to
11 total numbers of unpaired electrons computed by the MR-AQCC method. ^3 Analysis of our MR-AQCC results and also those of previous ones preformed on the
14 singlet-triplet splitting in polyacenes51 shows only a minor influence of basis set effects and of
16 the amount of correlating o orbitals. Though it is possible to perform large MR calculations by
18 considering both o and n electrons, this is an attempt to provide a guide for managing even larger
21 calculations including both o and n electrons are too expensive.
23 In spite of the complicated structure of the multireference wavefunctions, the chemical
25 analysis of the polyradical character is straightforward on the basis of the unpaired densities.
27 Such an analysis is very helpful in locating the chemically reactive centers and indicating those
30 possible to accurately assess the effects of different types of substituent attached to systems
32 carrying polyradical character and provide pictorial information on concomitant changes in the
34 chemical reactivity.
37 ASSOCIATED CONTENT
39 Supporting Information
41 Computational details, orbital occupation specification of all the structures, energy difference
43 between the ground and the excited states with basis sets, natural orbital occupation and number
regions on which to focus in order to stabilize the highly reactive polyradicals. It will also be
of effectively unpaired electrons with basis sets, comparison between the different optimized
46 structures and Cartesian coordinates of all optimized structures. This material is available free of
48 charge via the Internet at http://pub s.acs.org.
AUTHOR INFORMATION
53 Corresponding Author
55 hans.lischka@univie.ac.at
57 Notes
The authors declare no competing financial interest.
7 ACKNOWLEDGMENTS
9 This material is based upon work supported by the National Science Foundation under Project 11 No. CHE-1213263 and by the Austrian Science Fund (SFB F41, ViCoM). Support was also ^3 provided by the Robert A. Welch Foundation under Grant No. D-0005. We are grateful for
14 computer time at the Vienna Scientific Cluster (VSC), project 70376.
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