Electronic Nature of the Aromatic Adamantanediyl Ions and its Analogues

A estabilidade relativa do dicátion 1,3-desidro-5,7-adamantanediila é atribuída a sua aromaticidade tridimensional. Contudo, sua natureza eletrônica não é bem conhecida. A fim de entendê-la melhor, os die monocátions do adamantanodiil e alguns de seus análogos foram estudados utilizando a teoria de átomos e moléculas (AIM). Eles foram comparados com análogos de adamantano não-aromáticos. Os resultados de AIM indicam que a densidade eletrônica no centro da estrutura em gaiola e a média de todos os índices de deslocalização, envolvendo seus átomos cabeças-de-ponte são maiores em compostos aromáticos do que em não-aromáticos. A degenerescência energética dos átomos cabeça-de-ponte, a uniformidade e magnitude da carga compartilhada entre estes, distingue os dicátions 1,3-adamantila e 1,3-desidro-5,7-adamantanediila. Contudo, ambos são aromáticos, assim como o 1,3-desidro-5,7-diboroadamantano. O cátion 1,3-desidro-7-adamantila tem uma homoaromaticidade planar característica.

A series of isoelectronic analogues of the adamantane and its dehydroadamantanes containing boron, nitrogen, and phosphorus atoms at bridgehead were studied and the three-dimensional aromaticity is found to be general for 4c-2e electron systems. 8The calculations of the isoeletronic boron and diboron analogues of the 1,8-dehydrohomoadamantanediyl-3,6-dication, BC 9 H 12 + and B 2 C 8 H 12 , respectively, indicate that both have three-dimensional aromaticity. 9espite that 1,3-dehydroadamantane undergoes addition reaction easily, 10 it is stable at low temperature and inert conditions.Hitherto, the 1,3-dehydro-5-adamantyl cation is not a long-lived stable ion but it is an intermediate in solvolysis of the 5-bromo-1,3-dehydroadamantane. 11The density functional theory (DFT) study showed that the 1,3-dehydro-5-adamantyl cation has 20.0 kcal mol -1 stabilization due to trishomoaromatic interaction. 9The 1,3-dehydro-5-boroadamantyl analogue has the same type of stabilization. 9he three-dimensional aromaticity of 1,3-dehydro-5,7adamantanediyl dication is explained in terms of overlap among four p-orbitals in a tetrahedral fashion (Scheme 1) and its 13 C NMR spectrum has a single absorption ( 13 C 6.6 ppm) assigned for bridgehead carbons. 12In order to understand the electronic nature of these systems we have performed AIM calculations of the 1,3-dehydro-5-7adamantanediyl dication and its analogues.

Computational Methods
Geometry optimization, vibrational analysis and the generation of the electronic density were obtained with GAUSSIAN 03 package. 13The geometries were optimized in DFT level employing Becke three-parameter interchange functional 14 (B3) in conjunction with Lee-Yang-Parr 15 correlation functional (LYP) and the split valence basis set 16 6-311++G**.Dehalogenation calculations were performed at the same level.All geometries were characterized as stationary points after vibrational analysis.
The AIM 2000 software 17 was used for charge density calculations from the electronic densities ( ) obtained at B3LYP/6-311++G** level.NICS 18 (nucleus independent chemical shift) calculations were performed at GIAO/B3LYP/6-311++G** level at the geometric center of the cage.

Rationale
Within the molecular orbital theory (MO) the aromatic character of benzene is explained through delocalized orbitals.Nevertheless by using spin coupled from valence bond theory (SC), Gerratt and co-workers 19 established that all six electrons of benzene are localized and distorted symmetrically towards neighboring carbon atoms on each side and possess same energy and shape.Furthermore, from a more rigorous quantum mechanical standpoint, 20 benzene has no resonance 21 since there is not intersection of degenerate point group states.It means that benzene (D 6h symmetry) cannot be related to Kekulé (D 3h symmetry) or Dewar (D 2h ) structures.In this case, benzene stability is ascribed to maximum overlap among six degenerate single-electron states. 22Then, in modern valence bond theory (VB), the idea of aromaticity is based on the overlap and degeneracy of the single-electron states. 22he index used to study the aromaticity of adamantanediyl ions and its analogues in this work, D 3 BIA, is inspired to the modern VB idea of aromaticity.The modern VB parameters of aromaticity can be associated with topological parameters.Then, it is possible to quantify aromaticity by AIM theory inspired on modern VB ideas of aromaticity.In the case of adamantanediyl ions and its analogues, the aromaticity from modern VB theory can be associated with the energy of atomic basins of the bridged atoms, the charge density in the cage critical point and the delocalization indexes involving the bridged atomic pairs of the cage (from AIM).The use of Kohn-Sham orbitals for the calculation of delocalization indices is a good approximation, although not strictly correct.
It is important to emphasize that the calculus of energy of an atomic basin is an one-electron integration over the whole atomic basin.Then, the AIM provides the energy value of the total energy of each atomic basin.Since degeneracy in Quantum Chemistry means same energy, the AIM can provide the information whether some atomic basins in a given molecular system are degenerate or not (see Electronic Supplementary Information for more details).

Scheme 1.
J. Braz.Chem.Soc.142  Electronic Nature of the Aromatic Adamantanediyl Ions and its Analogues

Results and Discussions
The studied species are depicted in Scheme 2. The AIM theory 23 is based on the analysis of the electronic density distribution, (r), obtained from a quantum mechanical wavefunction.Although DFT does not generate a wave function, it yields a density matrix similar to that from a post-HF method such as MP2.Much information can be obtained from AIM such as the bond, cage and ring critical points and their respective eigenvalues (see electronic supplementary information).
Table 1 contains the delocalization indexes and energy of bridged atoms (R 1 , R 3 , R 5 and R 7 ) of the species 1 to 32, obtained from AIM theory.Their analysis allows the investigation of the electronic differences between aromatic and non-aromatic tridimensional systems.Table 1 also contains the point group symmetry of each studied species.
The delocalization index (DI) is a measure of number of electrons that are shared or exchanged between two atoms or basins.The integration of the Fermi hole density through a pair density matrix leads to the localization index (LI) and the delocalization index 24 (DI).Although DFT does not yield a pair density matrix, the delocalization indexes from B3LYP are close to those from MP2 (see electronic supplementary information).It is important to emphasize that the delocalization index does not measure the delocalization of valence electrons over the whole molecular system.
In a previous work on benzene-like compounds it was shown the importance of some electronic parameters to determine their aromaticity. 25These parameters were degeneracy of the atoms in the ring, density inside the ring and uniformity of delocalization index between atoms of the ring.All of these parameters are also analyzed in this work.
The bridged atoms of the adamantane 1 and the 3-dehydro-5,7-adamantanediyl dication 16 are degenerate and the delocalization indexes (or amount of shared charge density) among them are uniform in both molecules (0.039 in 1 and 0.280 in 16).However, the delocalization indexes in 16 are seven-fold higher than those in 1.As a consequence, the species 16 has 1.70e among the bridged atoms while the species 1 has only 0.23e.
The delocalization indexes among bridged atoms in adamantyl dication 10 are not uniform (0.079, 0.147 and 0.143) and the energy of its bridged atoms is not degenerate (Table 1).The degeneracy of bridged atoms, uniformity and magnitude of their delocalization indexes are the electronic features that distinguish the species 10 and 16.They can be regarded as determinant factors for the relative stability of the 1,3-dehydro-5,7-adamantanediyl dication.One simi- larity between the species 10 and 16 is that their positive charge is distributed among the hydrogen atoms, as one can see in Figure 1.
The relative stability of the 1,3-dehydro-7-adamantyl cation 25 is experimentally known. 3It is suggested that the species 25 (Figure 1) has homoaromaticity among R 1 , R 3 , and R 7 bridged atoms (see Table 2) since they are nearly degenerate.Besides, the delocalization indexes among these atoms have high magnitude and uniformity.
When bridgehead carbon atoms are substituted by silicon atoms, such as in 2, 11, 17 and 26, it is noteworthy that their respective delocalization indexes decrease significantly (Table 2).All other electronic features remain the same when comparing the silicon derivatives with their carbon analogues 1, 10, 16 and 25, respectively.Bridged silicon atoms in 17 are nearly degenerate and the delocalization indexes among them are nearly uniform.However, the magnitude of these delocalization indexes is almost three-fold smaller than that from 16.The same occurs with 26 in comparison to 25.
In oxy-species 7, 8, 22 and 23 there is no degeneracy of bridged atoms and uniformity of delocalization indexes among them.By comparing oxy-species with bridged carbon atoms (7 and 22) with their analogues 1 and 16, one can see that the magnitude of the DI´s decreases significantly in 7 and 22 due to the higher electronegativity of the oxygen atom.However, this does not happen with oxy-species with bridged silicon atoms (8 and 23) when comparing to their analogues 2 and 17 (Table 1).The effect of oxygen atoms in 22 can be seen in Figure 1 where its bridged carbon atoms are positively charged while in 16 the corresponding bridged carbon atoms are negatively charged.Likewise, it is expected that 22 and 23 have smaller aromaticity with respect to their respective analogues 16 and 17.
The dicationic diazo-analogues 19 and 20 have very small delocalization indexes values among bridged atoms in comparison to 16 and 17, respectively.The species 19 and 20 probably do not have tridimensional aromaticity since their delocalization indexes among bridged atoms are equivalent with those from neutral analogues 4 and 5, respectively (Table 1).However, in 19, there exists a bond between bridged carbon atoms, while in 20 no similar bond exists between bridged silicon atoms.The diazo-oxyderivative 24 has the smallest delocalization indexes values within the dicationic aromatic oxy-analogues (22, 23 and  24) and it is probably the least aromatic one.
The dicationic 21, containing one boron, one nitrogen and two carbon bridged atoms, has delocalization indexes among bridged atoms higher than those from 19 and 20.Then, 21 is probably more aromatic than 19 and 20, as it is confirmed by its NICS value in Table 2.
The molecular graphs of 14 and 19 show interesting aspects.In 14 (Figure 2a) both bonds between silicon atoms are broken after its optimization.Moreover, the delocalization indexes among R 1 , R 3 , R 5 and R 7 are not uniform.In 19 (Figure 2b) the two sp 2 carbon atoms make a bond after geometry optimization as it can be seen in the delocalization index between them (DI=0.836, in Table 1) and the density of this bond critical point ( =0.21a.u.).All other delocalization indexes among R 1 , R 3 , R 5 and R 7 are just little higher than those from 4, which agrees with the small NICS value of 19 (Table 2).Nevertheless, no bond is made between silicon atoms in 20.
Interestingly, in the neutral compound 29, electropositive boron atoms attract electrons from bridgehead carbon atoms and generate reasonable delocalization indexes among bridged carbon and boron atoms.By comparing with compound 6, where DI´s among bridged carbon and boron atoms is only 0.033, it is possible to see how effectively boron atoms distribute electrons from bridged carbon atoms in 29 (Table 1).This can be verified by comparing C-C atomic distances between 28 (1.562 Å) and 29 (2.110 Å) in Figure 3.The latter has a much larger distance which represents a delocalization  of this C-C bond.Then, 29 can be regarded as tridimensional aromatic species, as supported by its NICS value of -38.80 (Table 2).Although less effectively, boron atoms also provoke distribution of the electrons of the bridged silicon in Another important analysis from AIM theory is the Laplacian of the charge density, 2 .It is defined as the sum of the three principal curvatures of the function at each point of the space.However, it is a convenient convention to use the negative of the Laplacian, L(r), rather than the Laplacian itself.Then, L(r) = -2 .The density is a locally concentrated in those regions where L(r)>0, since 2 (r)<0 when (r) is a local maximum.Likewise, the density is a locally depleted in those regions where L(r)<0, since 2 (r)>0 when (r) is a local minimum. 24n Figure 4, it is depicted the negative of the Laplacian curves in a plane that passes through three bridged carbon atoms of 10 and 16.It is possible to see that the negative of the Laplacian distribuition region in the 1,3-dehydro-5,7adamantanediyl dication (16) where L(r)<0 encompasses the three bridged carbons atoms.This represents a uniform charge density distribution over these atoms and indicates an interaction among them.In case of the 1,3 adamantyl dication (10), there exists no L(r)<0 Laplacian distribution region encompassing the three bridged carbons and it indicates absence of significant interaction among them.
The AIM study of the diamantyl dications 33 and 34 provides a different electronic nature between them (see Figure 5).The former was prepared and characterized but the latter was not obtained. 26In the 4,9-diamantyl dication 33 there are three considerable delocalization indexes (DI= 0.117) linking each positively charged carbon atom (C1 and C8) and its corresponding bridgehead carbon atoms (C2 to C7).In the 1,6-diamantyl dication 34, there exists just one considerable delocalization index (DI=0.168)and two delocalization indexes immediately inferior (DI=0.085)For 33, the delocalization indexes are uniform and for 34 the delocalization indexes are not.
In Figure 6, one can see that the atomic distances between each positively charge carbon atom and its vicinal carbon atom is uniform in 33 (1.462 Å) and non-uniform   The degree of the degeneracy ( ) is derived from the energies of the atomic basins of the bridged atoms (from AIM theory).
The uniformity of delocalization indexes among bridged atoms is another important electronic feature found in the tridimensional aromatic species (Table 1).
The proposed D 3 BIA formula is: The degree of degeneracy is regarded to be maximum ( =1) when energy difference is smaller than 0.009 a.u.
The AIM provides a delocalization index for each bond between vicinal atoms. 19The mean delocalization index (DI) can be related to the magnitude of the single-electron states interaction (or overlap).
The delocalization index uniformity (DIU) among bridged atoms is given by equation 2: Where is mean deviation and (DI) is mean DI of the ring.
In Table 2, one can see the D 3 BIA values for all aromatic and non-aromatic species (1 to 32).It was also performed NICS of the species 1 to 32.Their NICS values are depicted in Table 2 and it is calculated from a probe (berkelium) in the center of adamantyl cage. 7,5Since all the studied species do not have orbitals, the use of a dissected NICS, e.g.CMO-NICS method, is unnecessary. 30Because the species 33 and 34 are structurally different from other species, their D 3 BIA´s have not been calculated.
in 34 (1.492, 1.493 and 1.422 Å).These results are in consonance with the uniformity of the delocalization indexes of each corresponding species mentioned above.
We proposed the D 3 BIA index which is based on the density in the ring, the degeneracy and the delocalization index of atoms in the ring 21 and it is inspired to the modern VB idea about aromaticity as mentioned before.(see Electronic supplementary information).
The rationale on the index D 3 BIA was influenced by modern VB studies on the aromaticity and the particularities of some aromatic compounds. 21The similarities of the results between AIM and spin coupled wave function (SC) encouraged us to use the former since it provides important and complementary information.The coherence between the AIM and SC can be noticed from their results for benzene (see Electronic supplementary information).
There are some multi-center delocalization indexes proposed to measure aromaticity, for example, the six-center delocalization index 27 (SCI)and Fermi hole density delocalization index 28 (FDDH).However, it is important to emphasize that these multicenter indexes are based on MO approach of aromaticity, i.e., the delocalization of orbitals in the aromatic molecules. 29On the other hand, the D3BIA index is inspired to a different approach -the spin-coupled (SC) of the modern VB theory.The focus of SC study on the aromaticity is not the delocalization of the molecular orbital over the aromatic molecule since it was demonstrated that single-electron orbitals are localized in each sp 2 carbon atom in the case of benzene. 19The focus of modern VB study on aromaticity is the degeneracy and the overlap of the single-electron orbitals and these are the VB paremeters that inspired the formula of D3BIA index in atoms in molecules theory.
The electron density in the aromatic site can be understood in terms of the density in the region involving the bridged atoms.Then, the density factor is the density in the center of the cage structure and the magnitude of the delocalization indexes among bridged atoms or mean delocalization index -DI.
R= Adamantyl or its analogues; or 1,3-dehydroadamantyl or its analogues R-Cl R + + Cl - (4) R= 1,3-Dehydroadamantyl or its analogues The energetics of all studied species are depicted in Electronic supplementary information.The free energy variation values of the dehalogenation of the species 10-12, 16-18, 21-23 and 25-27 are shown in Table 2.
From Table 2 one can see that the non-aromatic species 1 to 12 have the same electronic characteristics: small density in (3,+3) critical point (the density in centre of the cage structure) and small values of mean delocalization index among bridged atoms.One exception is the 1,3-adamantyl dication 10, for not having two electrons distributed among bridged carbon atoms and still having a reasonable mean delocalization index value (0.135) and density in (3,+3) critical point (0.020 a.u.).
All non-aromatic species (1 to 15 and 28 to 32) have low D 3 BIA.However, 10 and 29 have reasonable D 3 BIA´s (Table 2) because their density in (3,+3) critical point and their values of mean delocalization index among bridged atoms are not small.There exists an apparent tendency for interaction between bridged atoms.In the 1,3-adamantyl dication 10, there is no available electrons as it happens in its parent the 1,3-dehydro-5,7-adamantanediyl dication 16, but it uses sigma electron density to generate interaction among its bridged atoms.On the other hand, 29 has two electrons between bridged carbon atoms but it has no positive charge as the aromatic species 16.However, its bridged electropositive boron atoms draw the electron density towards them and this provokes a reasonable distribution of electron density among all bridged atoms (Tables 1 and 2).Then, 29 has tridimensional aromaticity.
The species 19 and 20 are not regarded aromatic ones since their D 3 BIA´s are very low.The species 19 has quite small delocalization index uniformity and 20 has small density in (3,+3) critical point and small values of mean delocalization index among bridged atoms (Table 2).
When analyzing its planar aromaticity, by exclusion of methine bridged carbon and the density in (3,+3) critical point, it is considered a three-membered planar ring among the other bridged carbons where its -DI = 0.410, its DIU = 100 and its = 1.Then, its planar D 3 BIA is higher than that shown in Table 2.
Free energies of dehalogenation reactions of 10-12, 16-18, 21-23 have been used to confirm the coherence of the D 3 BIA values (Table 2).Despite there is no good correlation between them, it is possible to verify that the smaller the free energy of dehalogenation the higher is the D 3 BIA.The smallest G (dehalogenation) value is for the 1,3-dehydro-5-7-adamantanediyl dication 16.G (dehalogenation) values of 10 to 12 are higher for than those from 16, 17, 18 and 21.All of these G (dehalogenation) values fit to their relation with D 3 BIA.For example, 10, with D 3 BIA=11.63,has G (dehalogenation) value (384.97 kcal mol -1 ) higher than those from the species 18 (D 3 BIA=16.39,G (dehalogenation) =375.27kcal mol -1 ) and 21 (D 3 BIA=19.82, G (dehalogenation) = 364.20 kcal mol -1 ).However, G (dehalogenation) value of 22 does not fit in the analyzed group, according to its D 3 BIA.Moreover, one can see that NICS values somewhat support all of these tendencies described above (Table 2).

Conclusions
The electronic interaction among bridged atoms is decisive in tridimensional aromaticity, along with uniformity of the delocalization index involving bridged atoms and their degeneracy.The density in the center of the cage [ (3,+3) critical point] and the mean delocalization index values among bridged atoms are distinguished between tridimensional non-aromatic and aromatic species: non-aromatic species have small (3,+3) and -DI values while aromatic ones have high (3,+3) and -DI values.The degeneracy of the bridged atoms, the uniformity and the magnitude of their delocalization indexes are the electronic features that distinguish the dications 10 and 16.However, the 1,9 diamantyl dication 10 has tridimensional aromaticity as well as the 1,3-dehydro-5,7-diboroadamantane 29.
The positive charge is mainly distributed among the hydrogen atoms and the bridged atoms remain negatively charged in the dicationic and monocationic aromatic species.

Supplementary Information
This supplementary material shows the atoms in molecules theory, the rationale on D 3 BIA, benzene and resonance, delocalization index and basis set dependence, the quantum atom, similarities between NICS and D 3 BIA, comparison of the delocalization indexes between MP2 and B3LYP computed energy values of the species 1 to 46, optimized structures of the species 1, 2, 4, 6, 7, 10,  11, 13, 15, 16, 17, 19, 21, 22, 25 and 26 and their bond lengths, and internal coordinates of optimized structures.This material is available free of charge at http://jbcs.sbq.org.br, as PDF file.

Atoms in Molecules (AIM) Theory
Bader's Atoms in molecules (AIM) theory 1 is based on the condition defining the quantum subsystem in terms of a property of the gradient vector of the charge density ( ).The quantum condition of the subsystem states that the surface bounding the subsystem shall not be crossed by any gradient vectors of .Since the gradient vector of always points at the direction of greatest increase in , it must always be perpendicular to lines of constant density.Let n be a unit vector normal to the surfaces, if the scalar product .nvanishes, then has no component through the surface.This surface, where .n= 0, is called zero-flux surface, and it is the borderline between subsystems.The point, within the zero-flux surface, where gradient vectors are pointed at, and therefore where =0, is called a bond critical point.
Inside the nucleus charge density exhibits local maxima.Outside the nucleus, charge density decreases as it moves away from the nucleus in any direction.Knowing that gradient path is a junction of gradient vectors at a given direction, then all the gradient paths within a subsystem will terminate at the nucleus.A nucleus is said to behave as an attractor in the gradient vector field.
The critical points 1 are classified according to their spectrum, which is a set of eigenvalues of the Hessian matrix .The Hessian is a matrix of second derivatives of some function with respect to all possible coordinates.The eigenvalue equation of has three solutions ( 1 , 2 , 3 ) corresponding to each eigenvector ui (i=1,2.3)which coincides with the principal axes of curvature.The Laplacian of the electron density ( 2) is the sum of the eigenvalues of the Hessian, or 2 = 1 + 2 + 3 .The nuclear attractor is denoted by (3,-3) because it is a maximum in all the principal axes.The bond critical point (BCP) is denoted by (3,-1)  because it is a minimum in the direction of the nucleus but it is a maximum in another main direction.The ring critical point (RCP) is denoted by (3,+1), where is minimum in two principal axes.
The Laplacian of the charge density, is defined as the sum of the three principal curvatures of the function at each point of the space (equation 1), which is also the trace of the Hessian of the density at the corresponding point. 2 (1) It is a convenient convention of using the negative of the Laplacian, L(r), rather than the Laplacian itself.Then, L(r) = -2 .The density is a locally concentrated in those regions where L(r)>0, since 2 (r)<0 when (r) is a local maximum.Likewise, the density is a locally depleted in those regions where L(r)<0, since 2 (r)>0 when (r) is a local minimum.

Rationale on D 3 BIA
The coherence between AIM and SCVB can be noticed from AIM results of benzene.The DI between carbon atoms in benzene is 1.39.Since DI between carbon atoms in ethane is 1.0, it is established that 0.39e.from system is delocalized in each C-C bond of benzene ring.This means that 0.61e.from each 2p z electron is localized in each carbon atom in benzene.This result matches with spin coupled (SC) one where a 2p z electron of benzene is localized and distorted symmetrically towards neighboring carbon atoms on each side. 3Moreover, Gerratt and co-workers 4 remarked that distortion effects of C(2p z ) orbitals are not larger than those in C-C bonds in conjugated systems.From AIM calculations, DI´s in hexatriene are 1.74 (for double bond) and 1.14 (for single bond).That brings another convergence of results between SC and AIM theories since it shows that delocalization in acyclic conjugated systems is greater than that in benzene.
Pauling and Wheland 5 stated that benzene is represented by a linear combination of five independent canonical structures.This view is emphasized by SC theory, 6,8 where spin coupling two electrons allows the description of the different possible resonance structures and generates the stability of aromatic systems.
Electronic Nature of the Aromatic Adamantanediyl Ions and its Analogues J. Braz.Chem.Soc.S2   From a more rigorous quantum mechanical standpoint, 8 benzene has no resonance 9 since there is not intersection of degenerate point group states.Resonance is only possible when the superposition of the hybrids can recover the full symmetry of the system.Resonance is related to degeneracy or near-degeneracy effects.Degeneracy may be due to the existence of symmetry groups that commute with hamiltonian. 9t means that benzene (D 6h symmetry) cannot be related to Kekulé (D 3h symmetry) or Dewar (D 2h ) structures since it is forbidden by point group symmetry rules. 10In this case, benzene stability is ascribed to maximum overlap among six degenerate single-electron states. 11

Benzene and Resonance
Resonance is related to degeneracy or near-degeneracy effects.Degeneracy may be due to the existence of symmetry groups that commute with hamiltonian. 9The eigenfunctions of the exact hamiltonian must transform as irreducible representations of the commuting symmetry groups.When a given group has degenerate representations, some or all eigenstates of the hamiltonian of the system will reflect its degeneracy.
The real Hilbert space is always partitioned into a direct sum of subspaces, each representing a different energy eigenvalue of the spectrum of the hamiltonian operator.The direct product decomposition is the mathematical tool to analyze the symmetry of the allowed individual states.It is related to the "ascent in symmetry" method 12 for decomposition of tensor spaces in independent parts and justified by the Littlewood-Richardson rules. 13These rules define the only allowed decompositions of a tensor space (e.g. point group space), providing us with the possible symmetries of the resonance hybrids, which reproduce the total symmetry of the system.Only invariant subgroups of some larger group can accommodate coherent states.
The possible forms of decomposing D 6h point group in direct products 14 are: The ground state of benzene is not degenerate, and there is no theoretical or experimental evidence of a neardegenerate electronic state with same geometry as the ground state.If there is no intersection of degenerate point group states one cannot follow the symmetry descent path in this case.The only alternative would be that of an accidental symmetry, but this is not possible since there is not direct product decomposition 8 from D 3h (Kekulé structure) and D 2h (Dewar structure).
However, in SC study of benzene, Gerratt and coworkers 15 stated that spin coupling two electrons (within same D 6h symmetry) allows the description of the different possible resonance structures and generates the stability of aromatic systems.
Then, existence of resonance in benzene depends on the rigor of theory applied to it.By spin coupling electrons, within same D 6h symmetry in SC, it is possible to associate the obtained stability with resonance of hybrid structures. 15therwise, benzene molecule cannot be represented by "hybrids" of symmetries D 3h and D 2h since it is forbidden by point group symmetry rules. 8

Delocalization Index and Basis Set Dependence
As stated by Popelier: 16 "since AIM charges are obtained directly from electron density, they depend less on the basis set used compared with methods which separate charges on the Hilbert space of the basis set."In addition, atomic charge is related to charge transfer between atomic basins.And since the delocalization index is related to the interaction between atomic basins, we can assume that the rough value of the delocalization index is not so sensitive to a basis set.

The Quantum Atom
Bader´s AIM theory defines atomic subsystems within a molecular system from its density matrix calculated by a specific ab initio or density functional method.Then, AIM divides the molecular system into atomic subsystems. 17IM is based on quantum subsystem in terms of a property of gradient vector of charge density ( ).In AIM it is possible to define all average properties of an atom.The definition of these properties is reduced to the quantummechanical definitions of the corresponding properties for a isolated atom.The summation of the atomic values of a given property over all atoms of the molecule yields the average property for the molecule.
Therefore, AIM provides the energy of an atom inside a molecule.The total energy of an atom is the sum of the kinetic energy and the potential energy.The kinetic energy an atom is obtained by the kinetic energy density.This means that the kinetic energy is evaluated locally at a point just like the electron density.Integration over the kinetic energy density gives the integrated kinetic energy over a given volume.There are two types of kinetic energies denoted by K(r) and G(r), defined in equations 2 and 3, respectively. 16) (3)   where r represents the coordinates of a position vector and N d ' summarizes the one-electron integration mode.
After some mathematical treatment, it is possible to obtain the following relation between K(r) and G(r): (4) If we integrate equation 4 over the full space then we obtain that K(full space) = G(full space) since the last term of equation 4 vanishes.That is a consequence of Gauss´divergence theorem (equation 5).(5)   where S is an interatomic surface (IAS) for which n = 0.Then, the kinetic energy of an atom is unambiguously defined and it is commonly called T(full space), as defined bellow (equation 6).( 6) The potential energy is obtained from kinetic energy by using the virial theorem.The roots of the concept of virial theorem lie in the development of thermodynamics in the nineteenth century, when it was realized that internuclear forces in real gases induce deviations in ideal gas behavior.Soon after the mathematical formalism of quantum mechanics was established, Slater derived the quantum analogue of the classical virial theorem (equation 7).(7)   In equation 7, the sum runs over all the particles (both electrons and nuclei) and the integration is over the total space in which the system exists.The right-hand side of equation 7 is the potential energy of the system of particles because it is the expectation value of the position and force operators.We conclude that the kinetic energy of a system is always related to its potential energy.
Within the Born-Oppenheimer approximation, equation 7 becomes the molecular virial theorem shown in equation 8.The term on the right-hand side of equation 7 falls into two terms in equation 8, namely the expectation value of the electronic potential energy and a term depending on the forces on the nuclei.(8)   where el is the electronic wave function, Vel is the electronelectron repulsion and electron-nucleus attraction potential energy operator, E is the total energy of the molecule and r are the nuclear coordinates.
For a geometrically optimized molecule, all forces on the nuclei vanish and the term in the molecular virial theorem vanishes.By simply restricting the integrals of the expectation values in equation 8 to the atomic subspace rather than full space, we obtain a relation between potential energy and kinetic energy inside an atomic basin (equation 9).( 9) Historically, the roots of the theory of AIM lie in the observation that the kinetic energy of a molecular fragment bound by a zero-flux surface in could be transferred from one molecule to another. 18The further development of AIM has been driven by the desire to formulate an atomic virial theorem which could render the atom into an energetically meaningful fragment.
Table S1 illustrates the calculation of atomic energies of all atoms in methanal.
degeneracy of their bridged atoms due to a decrease on the symmetry of their respective species.In 3 and 18 there is no uniformity of delocalization indexes involving the bridgehead atoms.The exception is the species 27.As to the magnitude of delocalization indexes, 3, 12, 18 and 27 have higher values than those from their respective analogues 2, 11, 17 and 26.

Figure 2 .
Figure 2. Molecular graph from AIM of the species 14 (a) and 19 (b).

Figure 4 .
Figure 4. Contour maps of the Laplacian distribution of the electronic charge density for the 1,3-dehydro-5-7-adamantanediyl dication (a) and 1,3 adamantyl dication (b) in the plane that contains three bridged carbon atoms.The gray curves are related to charge depletion, while the black lines are charge concentration.

Table 1 .
Delocalization index among bridged atoms, energy of bridged atoms and point group symmetry of the species 1 to 32

Table S1 .
The energies of the atoms in methanal (in a.u.)

Table S2 .
D 3 BIA and NICS of some aromatic compounds

Table S3 .
Computed energy values of the species 1 to 46 a Basis set6-311++G(d,p)