The Valence Bond account of Triangular Polyaromatic Hydrocarbons

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C: Physical Processes in Nanomaterials and Nanostructures

The Valence Bond account of Triangular Polyaromatic Hydrocarbons with Spin. Combining ab initio and Phenomenological Approaches Ana Maria Toader, Cristina Maria Buta, Bogdan Frecus, Alice Mischie, and Fanica Cimpoesu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b12250 • Publication Date (Web): 27 Feb 2019 Downloaded from http://pubs.acs.org on March 4, 2019

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The Journal of Physical Chemistry

The Valence Bond Account of Triangular Polyaromatic Hydrocarbons with Spin. Combining Ab Initio and Phenomenological Approaches.

Ana M. Toader, Cristina M. Buta, Bogdan Frecus, Alice Mischie and Fanica Cimpoesu*

Institute of Physical Chemistry, Splaiul Independentei 202, Bucharest 060021, Romania.

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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ABSTRACT.

We present computational analyses, methodological advances and heuristic conclusions applied on series of polyaromatic systems condensed in the shape of regular triangles, carrying spin, because of topological reasons. A new clue about the classification of title systems in three equivalency classes is presented. A conjugated hydrocarbon having n-hexagonal rings at one edge, carrying n-1 unpaired electrons, will be called n-triangulene, in the generalization of the experimentally known structures with n=2 (phenalenyl) and n=3 (triangulene). To be distinguished from the most of previous computational approaches, done by Density Functional Theory (DFT), we challenged the problem in the key of Valence Bond (VB) paradigm in both ab initio and phenomenological manners. The Heisenberg spin Hamiltonian was used to simulate the computed spectrum of VB states for the phenalenyl radical (n=2), predicting with the fitted parameters the effective VB description of the n=3 triangulene and other related systems. The outcome has practical importance in the prospects of spin chemistry, since the VB ab initio calculations are prohibitive beyond the n=2 case. The results are made transparent to the chemical intuition, using the language of resonance structures. KEYWORDS. Organic Molecular Magnetism, Carbon-based Radicals, Ab Initio Calculations, Spin Hamiltonian, Valence Bond Theory, Resonance Structures.


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I. INTRODUCTION The chemistry of carbon as candidate for high-tech new materials has received at the beginning of the XXI-th century a strong impetus with the discovery of graphenes,1,2 after the stir around fullerenes and nano-tubes, begun at the end of the XX-th one.3 The science of the new carbon-based materials is at confluence with the chemistry of the poly-aromatic hydrocarbons (PAHs), appealing for their special applications, such as for conducting devices,4,5 or in the quest for spintronics.6,7 The most appropriate physical language for problems of electronic conjugation encountered in PAHs would be the Valence Bond (VB) theory.8,9 However, in computational respects, this method, the first historical explanation of the chemical bond, is rarely used, in spite of available modern computational procedures,10,11 because of the preeminence gained by the Density Functional Theory (DFT).12 However, the DFT cannot tackle properly all the issues of particular bonding regimes. The organic magnetism is a somewhat exotic and very diverse field,13,14 an important counterpart to the realm of metal-based magnetic molecules and materials.15 The spin can arise in delocalized aromatic systems by topological reasons,16,17 in molecules having accidentally degenerate orbitals in the frame of simplistic Hückel model. In more accurate methods, these orbitals are quasi-degenerate, of non-bonding type, able to host a set of unpaired electrons, stabilizing a ground state configuration with parallel spins. A particular topology implying the presence of unpaired electrons is the triangular shape in condensed aromatic hydrocarbons.18 In an earlier work,19,20 we considered by DFT calculations the spin distribution in a series of PAHs with idealized equilateral triangular molecular geometry, returning here on the topic, with new perspectives.



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A “periodic table” of the regular triangular graphenes with zig-zag margin of the carbon skeleton is shown in the Figure 1. The three columns are evidencing three topological classes of the triangular PAHs, depending on the moiety placed in the center of the molecule, as discussed in the following. We presented this classification clue in a previous work, reviewing electronic structure aspects of several PAH classes,21 being worth to recall it in the introductory instance. If label the systems by the number of hexagonal units constituting one edge, n, the general chemical formula is 𝐶𝑛2 + 4𝑛 + 1𝐻3𝑛 + 3. The ideal molecular symmetry corresponds to the D3h point group. Invoking the simplest idea of the chemical bonding as due to spin pairing, attempting to put  and  densities on alternate carbon atoms (having the  system in mind), one may see that in the given topology the  and  counts differ. On the top row of the Figure 1, the spin count is exemplified for the n=2, 3 and 4 congeners representing the  and  sites by blue and yellow circles, respectively. One may see that the n=2 case has seven spin-up and six spin-down electrons, the n=3 gets twelve respectively ten, while the n=4 counts eighteen  vs. fifteen  sites. In general, for n hexagons at one edge, the number of  and  electrons in regular graphene triangles is: 1

𝑁𝛼 = 2𝑛 ⋅ (𝑛 +5),

(1.a)

1

𝑁𝛽 = 2(𝑛 +1) ⋅ (𝑛 +2).

(1.b)

Correspondingly, according to the so-called Ovchinnikov’s rule,16 there is a net excess of (n-1) alpha electrons, the molecule having the n spin multiplicity: 2𝑆 + 1 = 2(𝑁𝛼 ― 𝑁𝛽)/2 + 1 = 𝑛.


(2)

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5

 spin

 spin

l=0 n=2

n=3

n=4

n=5

n=6

n=7

n = 3+3l

n = 4+3l

l=1

n = 2+3l

Figure 1. The scheme of structures and spin count in triangular graphene-type molecules. The systems can be classified by the number n of hexagonal units at one edge, being conventionally named n-triangulenes. There are three equivalency classes, with respective n=2, 5 .. 2+3l, n=3, 6 .. 3+3l, and n=4, 7 .. 42+3l, rings on edge (with l=0, 1, 2 etc). The bottom row shows the pattern of the distribution in the resonance structures with highest symmetry, while the top row suggests the bonding as alternation of  and  spin polarization, resulting in a net excess of  spins (n-1 for a given n-triangulene).



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The n=2 system is called phenalenyl, known in many derivatives, such as the tri-t-butyl species,22 structurally well characterized, proved with interesting physical properties derived, in supramolecular interaction regime, from the crystal packing pattern (stacking).23,24 The chemistry of phenalenyl nucleus is very rich, mentioning, for instance, the 1,9-dithio-phenalenyl25 and tri-tbutyl-6-oxo-phenalen-oxyl26 compounds. The n=3 is named triangulene or Clar’s hydrocarbon.18 It was a long-sought species, initially characterized only by spectroscopic data.27,28 Recently, it was synthesized29 and firmly proved as a molecule with D3h symmetry, certifying the spin triplet as ground state. Actually, it was not a classical organic synthesis, the obtaining being achieved at molecular level, handling with Atomic Force Microscopy (AFM) devices hydrogenated (and closed shell) precursors. Belonging to the advanced techniques of single-molecule chemistry,30,31 the synthesis and measurement of the triangulene is an iconic outcome of the new trends in nanochemistry, boosting also the interest in the spin chemistry of this class of compounds. The triangulene, C22H12, was characterized by AFM pictures on Xe and Cu surfaces, revealing clearly the hexagonal cells and planar-symmetric framework, the analysis of the voltage proving the high-spin nature.29 The triangulene name can be generalized to the whole series of triangular graphene molecules, proposing here to call a regular system with n hexagons at one edge as n-triangulene. The 4-triangulene and higher congeners are yet unknown, but the promises opened and proven by the recent developments in surface-tailored syntheses32 are keeping open the hope for new elements of the triangulenes class. Beyond classical chemical synthesis, the nanoscale techniques, such as e-beam lithography, tested for the obtaining of tailored graphene nano-flakes from


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graphite33 may be an alternative to further triangulene congeners, or at least at molecules with topologically-driven spin, by incorporating triangulene-alike moieties. Coming back on the general series, in spite of the actual limitation to the n=2 and n=3 triangulenes in experimental respects, let us observe that there are three distinct classes, as marked in the columns of Figure 1. The n=2+3l triangulenes (with l=0, 1 2, etc.) can be characterized as having a carbon atom at the barycenter. Representing the resonance structure with highest possible symmetry (D3h), one  electron is placed in the molecular center, while the n-2=3l others are radiating from the center to the triangle vertices. The central C(C)3 moiety has the C-C bonds oriented toward the edges of the enveloping molecular triangle. The n=3+3l series has also a central carbon atom, but the C-C bonds are pointing towards the vertices of the overall triangle. With this electron count is impossible to draw a resonance structure having the D3h symmetry, succeeding mostly a C2v pattern. This does not mean that the system is not allowed to span the full trigonal point group, since a combination with equal weights of resonance structures with low formal symmetry can ensure the attaining of a D3h state. The central moiety of this formal resonance structure can be formulated as C=C(C)2. The n=4+3l series has a hexagonal ring at the symmetry center. The highest symmetric resonance structure fulfils the D3h point group. To be distinguished from the previous classes, this series has no experimentally known structure. As far as we known, such a systematization of triangular graphenes in three distinct classes was not pointed before us. The theoretical prospects traced important guidelines in the investigation of the triangulenetype systems since the early stages of computational chemistry.18,34 The DFT calculations are



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currently used as complement in experimental works, to answer legitimate inquiries about the spin distribution in phenalenyl or higher congeners, as illustrated in the review by Morita et al.26 Given the widespread availability of DFT12 treatments by user-friendly codes,35,36 many experimental works are accompanied by such calculations. For instance, several works37-39 devoted to a series of phenalenyl derivatives are combining experimental data (crystallography, optical spectroscopy, nuclear and electron resonance) with DFT simulation of potential energy surfaces at different mutual conformations of the monomers in dimeric couples. Systematic studies of different types of graphenes with spin are due to Philpott et al.19,40-42 using plane-wave DFT methods in molecular setting (the so-called Gamma option of periodic box condition). In general, although the DFT is convenient and sufficient in many problems, it cannot cover properly all the questions raised by special bonding regimes. There are relatively few treatments approaching the multi-configuration dimension in the electronic structure of PAH radical systems, remarking here a work devoted to the n=2-4 congeners of the triangulene series and to molecules produced fusing such moieties (zethrenes).43 The advent of our work consists in state-of-the art ab initio VB calculations and, as alternative route, converting multi-configurational (MC) self-consistent-field procedures (SCF) to a VB-type effective spin-Hamiltonian, going beyond the customary routine level of electronic structure calculations. The VB language is very appealing for intuition, speaking in terms of resonance structures, objects well acquainted at the basic formation of a chemist. Translating the MC-SCF outcome in a phenomenologic VB Hamiltonian has a heuristic dimension, illuminating the black box of computation, and a practical side, extending the tractability of VB considerations


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beyond the bottleneck of their current availability (limited at about 12-14 electrons in the active space). Although we consider in first instance the regular triangular hydrocarbons, the fact is that the topological reasons inducing unpaired spins can act in less regular systems, such as triangular protrusions on graphene nanoscale flakes, as shown in the second part of our work.

II. COMPUTATIONAL DETAILS The calculations were done with 6-31+G* basis set, employing the Gaussian35 and GAMESS36 packages for Density Functional Theory (DFT) and Complete Active Space Self Consistent Field (CASSCF) procedures. The molecular DFT calculations used for the step of geometry optimization, done with the B3LYP/6-31+G* setting, starting from idealized structures, based on standard bond lengths. The reliability of the computed geometries was validated by a comparison with a selected set of experimental structures, from X-ray data, as shown in the Supporting Information. The VB2000 code44,45 was used for Valence Bond (VB) calculations. The ab initio results were complemented with phenomenological models coded within the Matlab-Octave environment.46,47



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III. RESULTS AND DISCUSSION III.1 The Valence Bond ab initio calculation of the phenalenyl radical. The “pièce de résistance” of our work the recourse to Valence Bond (VB) theory, in ab initio and phenomenological manners. The Valence Bond paradigm is the very first quantum account of the chemical bonding, outlined in historical works of Heitler and London,48 immediately after the birth of wavefunction and matrix mechanics theories. The method faded in the era of computerbased quantum chemistry, being surpassed by the tractability of orbital-based procedures. There are several modern implementations of Valence Bond theory8,9 in computer codes, as the VB2000 program,44 used in this work. In the full configuration interaction iterative method, namely CASVB (Complete Active Space Valence Bond), the VB is equivalent with CASSCF treatments, in respects of the energy spectrum, but the wavefunctions are allowed to be non-orthogonal, being usually localized on atoms. In Spin Coupled version (SCVB) the basis consists in a collection of spin flipping Slater determinants having the same orbital component. A convenient basis set of the VB many-electron Hamiltonian is the Rumer formulation,49 where a spin singlet ground state (S=0) is given as product of parentheses representing sequences of two-orbital and two-electron local singlets. For non-null spin quantum number, S, it is easy to represent the components with the S=Sz high spin projection as the corresponding product of spin-up functions, on the chosen orbitals. Thus, a general  configuration in a Rumer basis can be ascribed as follows: 𝑁

𝑁

𝛷𝛺 = 𝐴 𝛹orb∏𝑘 𝛽= 1(𝛼𝛺(2𝑘 ― 1)𝛽𝛺(2𝑘) ― 𝛽𝛺(2𝑘 ― 1)𝛼𝛺(2𝑘)) ⋅ ∏𝑙 =𝛼 𝑁


𝛽

𝛼 . + 1 𝛺(𝑙)

(3)

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where N and N represent the number of up and down electrons, having the S=Sz= (N - N)/2 net spin. The orb orbital factor is tacitly thought to incorporate the overall normalization factor, while 𝐴 stands for the anti-symmetry condition in the Slater determinant circumscribed to the orb orbital product. The  symbol denotes a particular pick-up of the N/2 couples in parentheses and N - N free radical sites. The Rumer basis is particularly intuitive in the treatment of the conjugated hydrocarbons, its elements being representable as resonance structures, namely resembling distributions of -bonds among the carbon atoms (aside open radical sites, if the case). The celebrated Kekulé and Dewar resonance structures of benzene are an immediate example of a SCVB-type basis set. Confined to spin-flip configuration, with common orbital factor, orb, the dimension of Rumer basis for the block with Sz=S spin projection is:

𝑁𝑆 =

(𝑁𝛼 + 𝑁𝛽)! 𝑁𝛼!𝑁𝛽!

(𝑁𝛼 + 𝑁𝛽)!

― (𝑁𝛼 + 1)!(𝑁𝛽 ― 1)! =

(𝑁𝛼 ― 𝑁𝛽 + 1) (𝑁𝛼 + 1)

⋅

(𝑁𝛼 + 𝑁𝛽)! 𝑁𝛼!𝑁𝛽!

.

(4)

The NS is the difference between the number of spin configurations for the Sz=S and Sz=S+1 quantum numbers. For big systems, particularly the poly-aromatic molecules, the number of configurations demanded by the equation (4) is very large and becomes tempting to conventionally rely on resonances resembling the Kekulé pattern (with pairs made from immediate neighbors in the molecular skeleton), ignoring the distant combinations (like Dewar ones, in benzene). Aiming for a VB calculation on phenalenyl radical, we will confine ourselves to a Kekuléalike basis. Following the customary terminology, the triangulenes are named non-Kekulé



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benzenoids, in the sense that a complete pairing in resonances having only double bonds, in the graphical representation, is not possible. However, we use here the Kekulé term to say that we shall collect, aside the open radical, only pairings involving bonded partners of the molecular skeleton. There are twenty such resonance structures, given in the Figure 2. Instead of double bond representation for coupled pairs, we used an arrow, to account sign details of the wavefunction, needed in the further construction of a model Hamiltonian (see Supporting Information). The ij arrow between the i and j centers (localized orbitals) corresponds to the (αiβj-βiαj) ordering inside the parenthesis of the corresponding coupling from the general expression (3) of Rumer-type basis. One notes then that a resonance structure encrypts a wavefunction element. The Rumer resonances have multi-configurational nature, since, expanding the parentheses, one ends with a combination of 2g determinants (where g is the number of coupled pairs i.e. g =N/2, if we take non-negative spin projections). Among the resonance structures from Figure 2, let us observe the A-type two elements (at the top of the panel) keeping the unpaired electron at the central atom. The other resonance structures are grouped in three distinct classes (B, C and D aligned on columns of Figure 2), running the unpaired electron on the sites marked with  spin in the top of Figure 1 (which are polarized in unrestricted DFT calculations). It is impossible to put the unpaired electron on sites with  spin population, if we want to keep Kekulé-type resonance sub-structures, but this is allowed for the full Rumer basis. The VB(13) calculations (with 13 electrons in 13 orbitals) were done in a self-consistent manner using the 6-31+G* atomic basis and the 20 resonance structures depicted in the Figure 2.


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13

A 1

2

3

9

15

4

10

16

5

11

17

6

12

18

7

13

19

8

14

20

B

C

D

Figure 2. Resonance structures of the phenalenyl radical involving spin coupling only between bonded atoms. The A row and B-D column show the topologic equivalency classes of resonance representations. In alternative convention, the arrows can be also thought as double bonds.

The resulted orbitals are shown in Figure 3. There are four symmetry distinct carbon atoms (labeled 1-4). Qualitatively, these are taking the form of pz atomic functions, perpendicular on the molecular plane, but in more detail (see the contour representations from Figure 3), their shape is



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adapted to the specific site. The VB canonical orbitals are localized functions, instead of customary delocalized molecular orbitals met in DFT or CASSCF treatments. According to the general VB philosophy, this frame supports the description of the chemical bonding as effectively produced by spin pairing. The spin functions carried by the resonance structures are grafted on the set of pz alike components, each atom being occupied by one active electron. In principle, one may allow different orbital sets (so-called ionic structures, having sites with null or double occupancy), but from the further comparison with the CASSCF it will be concluded that the standard VB approach, with single orbital configuration and multi-configuration played by spin product functions, is completely satisfactory.

(1)

(2)

(3)

(4)

Figure 3. The symmetry distinct types of canonical localized orbitals resulted (labelled 1-4 on the bottom line, according to the host atom), from the VB(13)/6-31+G* calculation of the phenalenyl radical. The upper row shows the isosurfaces at 0.2e/Å3. The lower half shows contours drawn at a plane placed at 0.5Å above the molecular one.


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The represented localized canonical orbitals show significant mutual overlapping. For instance, the orbital on the central atom, 1, is realizing with atom 4 an overlap integral of about 0.485. The 2-3 and 3-4 couples are getting the 0.544 and 0.504 overlap values. A significant outcome of a VB calculation is the list of weights describing the participation of the resonance structures in the grounstate. The weights are subject to conventional definitions, as is the Mulliken-type analysis, adapted to the VB context,50 or the Löwdin equivalent, based on orthogonalized functions.51 With the Mulliken scheme, the weight of the two A-type resonances resulted negative, orienting ourselves then to the Löwdin analysis. The Löwdin weights for the four classes of resonances are getting the following values: wA= 2.55%, wB=4.75%, wC= 5.00% and wD= 6.06%. One notes the lower participation of the two resonance structures having the radical in the center, while almost equal values are obtained for all the other resonances, irrespective of their equivalency class. The quasi-equal superposition of the resonances brings the idea of a generalized sort of aromaticity, manifested by the delocalization of the radical on the periphery of the molecule. The excited states can be also characterized by resonance weights. The first VB excited state is an orbital doublet, at 24552 cm-1 with the following composition: wA=0 %, wB=4.00%, wC= 5.91% and wD= 6.76%, noticing that the A-type resonance is banned by symmetry reasons. The following VB state, at 29940 cm-1, is an orbital singlet where the A and B resonances are preponderant, on the expense of lowered participation of the others: wA=14.61 %, wB=9.53%, wC= 0.17% and wD= 2.10%. The following upper state, at 41350 cm-1, is again doubly degenerate, with the following weights: wA=0 %, wB=5.03%, wC= 6.03% and wD= 5.61%.



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Comparisons of the above described VB calculation of the phenalenyl radical with other multiconfiguration methods are shown in the Figure 4. The Figure 4a’ panel shows the VB states from the phenomenological model resulted as the fit to the levels from Figure 4a, containing the ab initio VB(13) calculation based on the above described 20 structures. The sector from Figure 4b displays the VB(13) results with the full Rumer set of resonances. Applying the equation (3) with N=7 and N=6, one finds a total of 429 spin doublet states. As a technical note, the used VB2000 code allows obtaining excited states one-by-one, but, since it is possible to extract the full Hamiltonian and overlap matrices in the given resonance basis, one may produce the full spectrum by subsequent eigenvalue solution outside the direct code run, implicitly using the orbitals optimized for the ground state. The VB2000 code retrieves automatically a complete Rumer set. Unfortunately, it cannot be controlled to include as subset the previously drawn 20 states, so that it is not easy to compare how their weights are shifted in the extended basis. In turn, one may compare the pattern of the lowest energy sequence of excited states. One observes that the first three lines from the (a) and (b) columns of Figure 4 are closely related as energy and degeneracy character. This can be interpreted as the validation for the approximate use of the limited resonance set, as giving a description of the ground and first excited states, effectively comparable with the full spin coupled approach. Besides, the VB calculations from columns (a) and (b) are well compared with the lowest sequence from the CASSCF(13,13) calculation given in the column from Figure 4c. The situation validates then the VB calculations as a good descriptor of the ground state and first excited levels. One may say that, in spite of the limitation to non-ionic resonance structures, the orbital


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selfconsistency ensures a reasonable account in the frame of the spin-coupled scheme. A complete active space valence bond (CASVB) computing option retrieves the same states as the CASSCF procedure.

Figure 4. Comparison of different multi-configuration modeling methods of the phenalenyl radical. The levels with double degeneracy are drawn in red lines, while non-degenerate states are given in blue. The series are as follows: (a) The 20 states resulted from the VB(13) calculation with the basis of (20) resonances from Figure 2; (b) The VB(13) states (up to 100000 cm-1) produced using the full Rumer basis, with 429 resonance structures; (c) The sequence of lowest spin doublet states from a CASSCF(13,13) calculation. On the left side, the (a’) column represents the fit by a phenomenological spin Hamiltonian to the levels from the VB(13) calculation (detailed in next section).



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We point the importance of taking extended  orbital scheme as active space, at least one orbital per carbon atom, or including procedures alleviating the results, such as the multi-reference configuration interaction (MRCI). The natural orbitals from the CAS(13,13) calculations are grouped in several sequences: three functions with occupation around 1.95, tree others with almost 1.9 population, one component with ~1.0 occupation number (corresponding to the effective SOMO), then three weakly populated formal virtuals, with about 0.1 density in each and, finally, three almost empty elements, with a 0.05 occupation tail. It seems reasonable to eliminate the first three components, considering them as belonging to doubly occupied core and the last three ones, as practically inactive virtuals, ending with a CASSCF(7,7) truncated active space. However, the computed states show energies rather shifted from the CASSCF(13,13) case. Thus, in the CASSCF(13,13) spectrum, the first excited state is an orbital doublet at 21988 cm-1, followed by a non-degenerate state at 26307 cm-1 and another doublet as third excited state, at 38573 cm-1. The same series of states, taken with 6-31G* basis, shows the relative low role of polarization components, yielding respectively the 21725 cm-1, 26052 cm-1 and 38112 cm-1 values. The orbital optimization by CASSCF(7,7)/6-31+G* yields the first doublet level at 26560 cm-1, followed by non-degenerate states at 37980 cm-1 and 47370 cm-1, the next doublet occurring at 48810 cm-1. Thus, all the excited states are sensibly shifted to higher energies, suggesting that the elimination of certain  orbitals from the active space is quite consequential. The result with truncated space is similar to that described by Das et al.43 at about 25076 cm-1 for the first doublet of phenalenyl, using a CASSCF(7,7) setting. The slight difference from our corresponding value is due to the basis set (the cited work using 6-31G*). A part of the positive energy shift is recovered


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by the decrements achieved with multi-reference configuration interaction corrections, which in different degrees of approximation,43 amended the value to 24810 cm-1 and 23100 cm-1. Then, in -delocalized systems, the truncation of the CASSCF active space may be problematic. III.2 The phenomenological Valence Bond modeling of the phenalenyl radical by the Heisenberg Spin Hamiltonian. A conceptual and practical gain is achieved introducing a phenomenological interpretation by means of the Heisenberg spin Hamiltonian:52 𝐻 = ∑𝑗 < 𝑖( ―2𝑆𝑖 ⋅ 𝑆𝑗 ― 1/2)𝐽𝑖𝑗.

(5)

This effective Hamiltonian is frequently used in magneto-chemistry15 (with the -1/2 term from the parenthesis eliminated, since it leads to an overall shift in the spectrum of states). However, its heuristic power is larger, since it represents the phenomenological form of the Valence Bond theory.53 The Jij are exchange coupling parameters between sites i and j. The exclusive dependence on spin operators and exchange interaction is understood in the spirit of a VB based on unique orbital configuration, so that all the other quantities (one-electron integrals and Coulomb terms) are thought to be the same for the full set of configurations. Applied in molecular magnetism, the spins on sites represent the total paramagnetism of the centers, but used for the sake of VB phenomenology, one works with Si=1/2 spins on all the sites. Although rather rarely used in this purpose, the Heisenberg spin Hamiltonian is a valuable tool for the description of conjugated organic systems54-56 The genuine exchange integrals are expected to be positive quantities, but, since the model incorporates in effective manner various contributions (kinetic and overlap terms), the J coupling parameters are often negative. This leads to the stabilization of states with lowest



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spin number and stays at the core of the interpretation of the chemical bond as effective spin pairing. In phenalenyl, there are three types of carbon-carbon bonds, having consequently the following set of distinct exchange coupling parameters: Ja, represented by the contact between the atoms 2 and 3, Jb, for the situations equivalent with the 3-4 linkage, and Jc, for the radial bonds established for instance by the 1-4 couple. The labeling of the atoms can be understood from Figure 2. Namely, the central atom is #1, the one in the upper vertex is #2, the indexing continue on the periphery in the clockwise sense. Having an effective Hamiltonian, we attempted to fit the computed VB spectrum to the model. To gain generality, we assumed an exponential dependence of the exchange coupling with the bond length J(r)=A∙exp(-α∙r). This can be rewritten in a more convenient way:

J (r )  J 0 exp  (r  r0 )  ,

(6)

where r0 is a conventional reference, say r0=1.39 Å carbon-carbon bond length in benzene. In comparison with the previous form, this has the advantage of avoiding a large pre-exponential A factors and its non-physical interpretation (namely A as the value of exchange coupling at zero bond length). In turn, the J0 has the clear meaning of the parameter at the convened r0 bond distance. Proceeding to the fit, we found J0 =-16535.8 cm-1 and α=8.529 Å-1. For the actual bond lengths of the phenalenyl core (taken from the B3LYP/6-31+G* geometry optimization), namely ra=1.3942 Å, rb=1.4199 Å and rc=1.4323 Å, the respective exchange coupling parameters are: Ja=15950 cm-1, Jb=-12815 cm-1 and Jc=-11530 cm-1.


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The match of the fit to the ab initio VB result, judged by the comparison of (a’) and (a) spectra in Figure 4, is reasonably good, particularly considering that the two parameters comprised in the equation 6 (J0 and α, recalling that r0 is conventionally fixed), must retrieve a set of twenty states. It is interesting to note that the weights for the A-D types of resonance structures resulted from model, wA= 2.65%, wB=4.88%, wC= 5.26% and wD= 5.65%, are well comparable with the computed ones, given previously. The matrices of the ab initio VB and phenomenological model cannot be directly compared, because the first ones are containing the effect of localized orbitals overlap, which is tacitly ignored in the Heisenberg Hamiltonian. Besides, by the same reasons, the computed and modeled overlap integrals between resonance structures are different. Thus, the above rules for the construction of the model refer to the situation of orthogonal or orthogonalized localized orbitals. An interesting check is realized assuming a simplistic situation, with unique exchange parameter, Ja= Jb= Jc=J. In this case, all the diagonal elements of the phenomenological VB matrix are equal with 1.5J and the ground state is estimated at 4.3631J. One may take diagonal elements as hypothetical energies of the standalone resonance structures and assimilate the gap to the ground state as resonance energy. One obtains then 2.8631J, which is a rather large result, in comparison with the early estimation done by Pauling and Wheland57 for benzene, as 1.1055J. Under the provision of the model crudeness, but in a very transparent way, one may say that the delocalization of the radical in the phenalenyl is similar to an aromatic stabilization.



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III.3 The multiconfigurational account of higher congeners. Model VB Hamiltonian vs. ab initio limited multi-configuration SCF. The outlined model is a good tool to approach the status of higher congeners, which cannot be tackled by direct ab initio VB calculations. Thus, for the n=3 triangulene, the VB calculation is at the moment impossible, the procedure being not able to tackle the problem with 22 active electrons. The used VB2000 code44 can presently work up to 14 electrons (at most 16, but not yet 15) since it depends on certain map files, with algebraic coefficients defined up to this limit. The 22-electron case would imply prohibitively large mapping files. To the best of our knowledge, the system is also intractable in other VB codes. The CASSCF(22,22) calculation is also prohibitive, demanding a humungous amount of computer memory. In turn, the phenomenological VB is rather approachable and, assuming the transferability of the parameters from equation (6), it can yield semiquantitative results. However, the full Rumer basis cannot be considered, because this is huge, in the range of 6.71010 elements. Confined to the Kekulé pattern (spin coupling terms only between the adjacent carbon atoms) there is a reasonable number of 306 resonance structures. This count is in line with the formula elaborated by Dias58 in the frame of graph theory. We generated the resonance structures by brute force, using home-made codes, enumerating all the ways of distributing 20 arrows (or double bonds) over the 24 carbon atoms, filtering out the situations incompatible with the chemical valence. The positions of the radicals resulted implicitly, as the sites not touched by a valence couple. The resonance structures are provided in the Supporting Information as lists of paired atoms, with radical sites at the end.


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For the n=3 triangulene, practically all the resonances are contributing comparably, with weights ranging between 0.15% and 0.5%. There is no group of structures that can be pointed as having the maximal share. This smeared participation over all the possible structures can be tentatively interpreted as aromatic-alike resonance, in the context of open shell configuration. The spectrum of the states comprised up to 100000 cm-1 (about 100 levels) is shown in the panel (a) of Figure 5. As a global characterization, we represented the diagram of resonance weights as function of the positioning of the two radicals. Namely, if in a given resonance the radicals are placed in the vectors r1 and r2 corresponding to the Cartesian coordinates of the carrier atoms, we took as descriptor the distance of the inter-radical middle point, (r1+ r2)/2, against the center of the molecular triangle (see Figure 5b). At the same time, we represented also the weights with respect of inter-radical distance, i.e. the length of the (r1- r2) vector (as given in Figure 5c). These two geometric measures are not completely characterizing a given resonance, since there are numerous possibilities to distribute double bonds (or arrows) on the carbon skeleton, once the position of the two radicals is decided. Therefore, for each take of radical positions there is a stack of weights for the set of resonances running on the pairwise positions of radicals. From Figure 5b, one may read that the set of maximal weights (between 0.4 and 0.5%) is reached at large separations of the averaged position of radicals, interpreting this as a preference for their placement at the periphery of the molecule. In turn, the intermediate triangular shells of the structure are getting a lower statistic. From Figure 5c one may note a slight preference for the small separation of the radical sites, although the full spectrum of positions contributes in quite a homogenous manner.



24

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0 2

4

6

8

|r1-r2| (Å)

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Figure 5. The spin Hamiltonian simulation of the VB analysis for the n=3 triangulene. (a) the energy spectrum of spin triplet states; (b) the distribution of resonance weights as function of the distance from symmetry point of the averaged position (barycenter) of the two radical sites; (c) the distribution of resonance weights as function of the distance between the two radical sites.

Although the CASSCF(22,22) is not affordable, one may yet approach the full active space in reduced multi-configuration self-consistency, namely by MC-SCF(22,22) procedures, taking excitations up to the second and third order. Each line in the spectrum of states produced with triple excitations is departing from those done with double excitations with no more than 0.01 cm-1. Actually, the list of main configuration contributions is confined to the single-excitation character. In these circumstances, one may conclude that the limited configuration interaction actually emulated well the CASSCF(22,22) limit. It is interesting then to note that the sequence of first few computed MC-SCF states is close to those produced with the help of the


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phenomenological VB model. Thus, the ab initio computation yielded two orbital doublets at 28503 cm-1 and 30316 cm-1 and a non-degenerate level at 30767 cm-1 (all spin triplets). With a reduced 6-31G* basis, the respective results are: 28609 cm-1, 30413 cm-1 and 30898 cm-1. The spin Hamiltonian rendered the doubly degenerate levels at 27396 cm-1 and 30664 cm-1 and the orbital singlet at 27577 cm-1. One may say then that our model works well for an efficient approximation of the multi-configuration description of hydrocarbons with spin, particularly in the discussed series. One may also note that the pattern of first three energy levels, comprising two doubly degenerate states and one non-degenerate, is similar to those found previously for phenalenyl, guessing it as a general feature of the whole series of regular triangulenes.

III.4 Spin distribution and bond orders in selected triangular and related systems. In the following we will focus on bond order and spin population analysis for the n=2 and n=3 triangulenes, considering in addition other related systems. In the unrestricted orbital methods, the spin distribution is effectively due to the sum of squared natural orbitals with population close to the unity. In the frame of Valence Bond, the spin population on a given site can be understood as the weighted sum of all the resonance structures having open radicals on the selected position. In the phenomenological model, this corresponds to a spin-restricted picture. However, in the ab initio VB approach, spin polarized populations can be obtained, because of the role played by the non-orthogonality of the localized orbitals. Thus, the spin populations from the VB(13)/6-31+G* calculation, on atom types 1-4 (see Figure 3 for identification) are, respectively: C1= 0.104 , C2= -0.128 , C3 = 0.281 and C4= -0.135. These numbers are comparable to the B3LYP/6-31+G* results



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from natural population analysis59 (NPA): C1= 0.05, C2= -0.11, C3= 0.26, and C4= -0.09. These values correspond to those described in the panel (a) of Figure 6. With the spin model that fits the ab initio VB spectrum, the respective populations are: C1= 0.05, C2= 0, C3= 0.16 and C4= 0, shown in the panel Figure 6b. Thus, the atom types C2 and C4, with beta polarization in the full calculations are getting null spin in the model, the limitation being caused by the implied orthogonality of localized orbitals. The positive unrestricted spin density on the atoms from periphery (C3 type) is approximately the results of summing into unrestricted VB results the approximately constant background of negative spin polarization (about -0.1) from the corresponding neighbors (C2 and C4 types). We also considered the bond order issue, which is tributary to defined conventions, adopting here the Wiberg bond index60 for the analysis of the DFT wavefunction. The ab initio VB2000 code does not have the option for bond order outputting, treating then these quantities only in the frame of the complementary spin coupling modeling. Since the VB refers only to the  subsystem, we assume a contribution with one unit for each couple of bonded carbon atoms. A spin pair on a given atom couple is interpreted as the setting of a double bond over the given moiety. The  bond order on a selected pair of atoms is done by running over the i Kekulé-type resonance structures and summing the wi weight if the given pair is spin coupled. The Figure 6 shows the spin densities and bond orders of the phenalenyl, computed by DFT and by the VB phenomenological model.


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(a)

(c)

(b)

(c’)

(d)

Figure 6. Spin densities and bond orders on phenalenyl carbon skeleton at different levels of theory: (a) Natural Population Analysis and Wiberg bond indices after the B3LYP ground state calculation. (b)-(d) the results from effective VB spin Hamiltonian model, confined to a pattern without spin polarization: (b) the ground state, (c) and (c’) the expectation values for the couple of the doubly degenerate first excited state, (d) the second excited state.

Since there are six formal double bonds distributed over 15 bond contacts, the bond order at hypothetical equal smearing would be 1.4, thinking on the ground state. The reality resulted from calculation and modeling shows a ranging between the 1.3 and 1.5 values. The aromatic alike ~1.5 bond order occurs at periphery, around the vertices with C2 atom type. Although limited to a restricted-type account, the VB model has the advantage of easy tackling of the whole spectrum of excited states. The first excited state is a doublet and therefore the two eigenvectors admit an arbitrary mutual mixing that ends with an asymmetric aspect in the distribution of spin population



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and bond orders, as seen in the (c) and (c’) panels from the Figure 6. In these states, the central atom has practically null spin density. In turn, the second excited state, shown in Figure 6d, which is an orbital singlet, shows a spin density concentrated on the central atom, as if constituted by the equal superposition of the A-type resonances (see Figure 2) as main contributors. Interestingly, like in the ground state, the low-lying excited VB levels show bond orders of about 1.5 for the bonds meeting at the C2 atom, i.e. in the points representing the vertices of the triangle. Recall that the ab initio VB calculation was not possible for the n=3 triangulene, so that the phenomenological modeling is the only way to obtain a VB information on the ground and excited states. We did not approach here, but one may foresee a more advanced use of the model in estimating excited state geometries. Thus, one may fit a component initially ignored in the spin Hamiltonian from equation (5), namely the counterpart due to one-electron and Coulomb terms. This is the same for the whole spectrum of states, but it can tentatively be established under the condition of retrieving the optimized bond lengths of the ground state. This term can be conventionally conceived as a sum of functions contributed from the pairs of the directly bonded atoms. Since the exchange parameters were found in exponential form, with monotonous variation, the added counterpart must be a pattern able to establish a minimum, being for instance a Morse curve or even a simple harmonic oscillator. Then, in principle, once calibrated for the ground state geometry, the added bonding component can determine the geometry of the excited states, if the VB modeling is assorted with capabilities for gradient. Since the mentioned counterpart is the same for all the states, the excited state geometry must be tuned by the exchange part.


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(a)

(c)

(d)

(b)

(c’)

(d’)

Figure 7. Spin densities and bond orders in the n=3 triangulene: (a) B3LYP ground state. (b) the VB spin model ground state, (c) and (c’) VB model for the components of the doubly degenerate first excited state, (d) the doubly degenerate second excited state.



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The suggested construction is, of course, a phenomenological approximation, but it seems an interesting tool, quite powerful at a rather simple design. Since this work has already several branches opened, we have not devoted here the demanded specialized effort, confining ourselves to point the affordable possibility. The spin and bonding distribution in the next congener, the n=3 triangulene is given in the Figure 7. The DFT and VB model results for the ground state, displayed in the panels Figure 7a and Figure 7b show that the spin density tends to be concentrated at the periphery of the molecule. The first excited state (an orbital doublet, depicted in Figure 7c and 7c’) shows a slight shift of the spin density towards the internal part, while the second one (also a doublet, given in Figure 7d and 7d’) shows a more pronounced placement on the periphery. Qualitatively, the averaged bond order is expected to be 1.37, if presume the equal distribution of the ten double bonds on the skeleton of the 27 carbon-carbon contacts (i.e. 0.37 from the  side). In the DFT ground state, the Wiberg bond orders vary from about 1.2 to 1.45 suggesting a relative overall aromaticity of the molecule. The VB model yields slightly higher values, between 1.3 and 1.5. Like in the previous example, in both the ground state and the searched excited states, the bonds meeting at the vertices of the triangle are all with 1.5 index. With the help of the previously used model, based on the parametric dependence from equation (6), we can briefly cope with systems having not regular triangular geometry, but carrying spin because of similar topological conditioning.


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E(cm-1)

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Figure 8. The Spin Hamiltonian VB modeling of the phenalenyl derivative having one hexagonal ring condensed laterally, the system with the (2,1) label, according to the text definition. Left side: the simulated VB spectrum. Right side: The spin populations and bond orders in (a) ground state and (b-d) the first three excited states.

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Figure 9. The Spin Hamiltonian VB modeling the system with the (2,2) convened notations. The legend is the same as in the Figure 8.



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A series of such hydrocarbons is conceived attaching a sequence resembling a linear polyacene to a ring placed at one vertex of a n-triangulene. If the condensed chain is formed from m additional hexagonal rings, we propose to ascribe the molecule by the (n, m) pair of indices. The m=0 corresponds to the symmetric triangulene. The Figures 8 and 9 are illustrating the (2, 1) and (2, 2) molecules resulted by the formal condensation of the phenalenyl with the benzene and naphthalene, respectively. The work of Dias58 dealt with the enumeration of resonance structures for several (n, m) congeners with n=2 and 3, as well as for other derivatives of these triangulenes. Adapting the Dias analytic formulas for the actual notation of the (n,m) systems, the n=2 series has the (1/2)(40+29m+3m2) count, while the n=3 cases are accounted by the 306+231m+26m2 formula. A generalization as function of n was not identified. One observes that at m=0, namely at pure triangulenes, the above formulas are retrieving the previously encountered counts of resonances for the n=2 and 3 cases, namely 20 and 306. The (2, 1) and (2, 2) systems, depicted in the Figures 8 and 9, have the respective 36 and 55 count of resonance structures (see Supporting Information), while having a single radical site, like the phenalenyl parent. Compared with the modeled spectrum of phenalenyl, as seen in the left side (a’) stack of the Figure 4, the (2, 1) and (2, 2) systems show a progressively higher density of states and a lowering of the first excited levels (see the left sides of the Figures 8 and 9). The ground states of these systems are depicted in the panels Figure 8a and Figure 9a. The sequence of the first three excited states, corresponding to the maps from Figure 8b-d and Figure 9b-d, can be thought as originating from the couple of singly and doubly degenerate levels of the phenalenyl.


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According to the bond order distribution, the attached rings seem to have less aromatic character, in ground or excited states. The bond indices on the outer rings suggest the trend for alternating single and double bonds. The arch of four carbon glued to the phenalenyl skeleton in the (2, 1) molecule has a pattern resembling the localized bonds of butadiene. The phenalenyl moiety has a similar pattern of spin density in the (a)-(d) maps of both (2,1) and (2,2) hydrocarbons. Alternatively, these molecules can be described as resulting from a (CH)3 angular cap attached to a linear polyacene. Thus, the triangulene itself, i.e. the (2,0) molecule, has the (CH)3 part added over a naphthalene, while the (2,1) and (2,2) can be regarded as constructed from anthracene and tetracene, respectively. This deconstruction helps to describe the spin distribution as follows: in both systems from Figures 8 and 9, the main spin carriers in the ground state are three atoms belonging to the left-bottom corner of the polyacene moiety, namely on the two rings common also to the phenalenyl fragment, as seen in the (a) panels. The first excited state, corresponding to the Figures 8.(b) and 9.(b), can be described as belonging to the upper line of quaternary carbon atoms of the polyacene slab. In more detail, the main three spin carriers are located as follows: one in the side belonging to the phenalenyl, one on the junction with the added fragment, and one in the lateral tail. The simulated energies of these states are 24661 cm-1 and 22600 cm-1, i.e. progressively lower than the first excited level in phenalenyl, modeled at 28433 cm-1, by the used phenomenological spin Hamiltonian. The following two excited states show a lesser spin delocalization over the added fragment. The 8.(c) and 9.(c) diagrams show two main spin centers, one located in the bottom-left side of the polyacene part and one in the formally added (CH)3 arch, showing almost no unpaired electrons



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over the right-side part of the polyacene fragment. These states are simulated at 28800 cm-1 and 29026 cm-1. Considering their parentage also related to the doubly degenerate level at 28433 cm-1 of the phenalenyl, one may speculate that a sort of topological symmetry reason conserved this component, keeping it only slightly perturbed, in comparison to the pristine triangular radical. Finally, the third excited state, depicted in the (d) panels, originating from the non-degenerate level simulated at 30487 cm-1 in the phenalenyl, are keeping their energies almost unchanged, at 30257 cm-1 and 29885 cm-1, in the (2,1) and (2,2) molecules. The spin distribution is also contained inside the phenalenyl moiety. In both systems, the spin density is grouped on a triad of atoms comprising the center of the phenalenyl unit and the atoms depicted in the lowest part of the actual representations. One may say that, except the first excited state, the spin density in ground and other wavefunctions is confined inside the triangulene moiety of the hypothesized molecules. The details revealed with the help of VB phenomenological treatment are relevant in further understanding of the reactivity in the proposed molecules and even, in principle, as rationales for a spintronics with such carbon-based systems.


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IV. CONCLUSIONS In the present work we have taken a major step in the characterization of series of aromatic systems with regular triangle topologies, by means of innovative computational approaches. We have generalized the experimentally known structures of phenalenyl and triangulene, to systems comprised of n-hexagonal rings at one edge (named n-triangulenes). We have investigated these systems, which carry n-1 unpaired electrons in the framework of Valence Bond theory, in ab initio as well as in a phenomenological manner, based on the Heisenberg spin Hamiltonian. We complemented the VB modeling with DFT, as well as wave-function calculations such as CASSCF and MCSCF. For phenalenyl, the VB description of the ground state and first excited levels is well validated against the CASSCF (13, 13) treatment of the system. For n-triangulenes with n ≥ 3, the ab initio VB and CASSCF treatments are practically impossible, because of the large number of electrons belonging to the π system. In turn, our phenomenological VB model is tractable for large scale cases, a good validation being achieved for the n=3 triangulene, by comparing it to a MCSCF(22,22) calculation using double or triple excitations, emulating well the CASSCF(22,22) limit. The constructed phenomenological spin-coupling model is useful for a large variety of problems, such as accounting energies, bond orders and spin populations in ground and excited states of larger triangulenes or less symmetric related systems.



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AUTHOR INFORMATION Corresponding Author *E-mail: [email protected].

ACKNOWLEDGMENT This work is supported by the Romanian Research Council, UEFISCDI, grant PCE 108/2017.

SUPPORTING INFORMATION The supporting information contains: Graphical algorithms for establishing the elements of the Valence Bond (VB) phenomenological Hamiltonian and Overlap matrices; Spin Hamiltonian Matrix Elements for the phenalenyl VB model; the list of resonance structures for the discussed aromatic systems; a list of bond lengths from selected related systems, assessing the geometry optimization by comparison with X-ray crystallographic data. This material is available free of charge via the Internet at http://pubs.acs.org


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REFERENCES (1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang,Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004, 306, 666-669. (2) Allen, M. J.; Tung, V. C.; Kaner, R. B. Honeycomb Carbon: A Review of Graphene. Chem. Rev. 2010, 110, 132-145. (3) Dresselhaus, M. S.; Dresselhaus, G.; Eklund, P. C. Science of Fullerenes and Carbon Nanotubes; Academic Press: San Diego, 1996. (4) Kivelson, S.; Chapman, O.L. Polyacene and a New Class of Quasi-one-dimensional Conductors. Phys. Rev. B 1983, 28, 7236-7243. (5) Bendikov, M.; Wudl, F.; Perepichka, D. F. Tetrathiafulvalenes, Oligoacenenes, and Their Buckminsterfullerene Derivatives: The Brick and Mortar of Organic Electronics. Chem. Rev. 2004, 104, 4891–4945. (6) Sanvito, S. Organic Electronics: Memoirs of a Spin. Nat. Nanotech. 2007, 2, 204-206. (7) Dediu, V. A.; Hueso, L. E.; Bergenti, I.; Taliani, C. Spin Routes in Organic Semiconductors. Nature Materials 2009, 8, 707-716. (8) Gallup, G. A. Valence Bond Methods; Cambridge University Press: Cambridge, 2002. (9) Shaik, S.; Hiberty, P. C. A Chemist’s Guide to Valence Bond Theory; John Wiley & Sons: New Jersey, 2007.



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(10) da Silva, E. C.; Gerratt, J.; Cooper, D. L.; Raimondi, M. Study of the Electronic States of the Benzene Molecule Using Spin ‐Coupled Valence Bond Theory. J. Chem. Phys. 1994, 101, 3866-3887. (11) Hirao, K.; Nakano, H.; Nakayama, K.; Dupuis, M. A Complete Active Space Valence Bond (CASVB) Method. J. Chem. Phys. 1996, 105, 9227-9239. (12) Koch, W.; Holthausen, M. C. A Chemist's Guide to Density Functional Theory; VCH Publisher: Berlin, 2001. (13) Fernandez-Rossier, J.; Palacios, J. J. Magnetism in Graphene Nanoislands. Phys. Rev. Lett. 2007, 99, 177204-4. (14) Stable Radicals: Fundamentals and Applied Aspects of Odd-Electron Compounds; Hicks, R. Ed.; Wiley: New York, 2010. (15) Kahn, O. Molecular Magnetism; VCH Publisher: New York, 1993. (16) Ovchinnikov, A. A. Multiplicity of the Ground State of Large Alternant Organic Molecules with Conjugated Bonds. Theor. Chem. Acc. 1978, 47, 297-304. (17) Rajca, A.; Rajca, S.; Desai, S. R. Macrocyclic. pi.-Conjugated Carbopolyanions and Polyradicals Based upon Calix [4] arene and Calix [3] arene Rings. J. Am. Chem. Soc. 1995, 117, 806. (18) Clar, E.; Stewart, D. G. Aromatic Hydrocarbons. LXV. Triangulene Derivatives. J. Am. Chem. Soc. 1953, 75, 2667-2672. (19) Philpott, M. R.; Cimpoesu, F.; Kawazoe, Y. Bonding and Magnetism in High Symmetry Nano-Sized Graphene Molecules: Linear Acenes C4m+2H2m+4 (m=2,…,25); Zigzag Hexangulenes


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GRAPHIC ABSTRACT


The Valence Bond account of Triangular Polyaromatic Hydrocarbons

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