A Unified Orbital Model of Delocalised and Localised Currents inMonocycles, from Annulenes to Azabora-heterocycles

Alessandro Soncini,*[a] Carmen Domene,[b] Jeroen J. Engelberts,[c] Patrick W. Fowler,*[a]

Andr� Rassat,*[d] Joop H. van Lenthe,[c] Remco W. A. Havenith,[c] andLeonardus W. Jenneskens[e]

Introduction

Planar conjugated carbocyclic compounds show a stronglink between p-electron count and molecular properties.H�ckel�s long-standing 4n+2 rule distinguishes aromaticfrom antiaromatic monocycles, and correlates the energetics

of these systems with their characteristic magnetic proper-ties.[1] However, the periodic table offers wide possibilitiesfor the construction of isoelectronic heterocycles, and formany of them, direct transferability of the implied relation-ship between aromaticity and electron count is questionableor even counterfactual. The replacement of CC by BN, forexample, is known to lead to marked differences in magnet-ic properties:[2] the strong ring currents of the carbocyclevanish in the isoelectronic azabora-heterocycles, BN/2NN/2HN

(1–3). Herein we show that a simple frontier-orbital model[a] Dr. A. Soncini, Prof. P. W. Fowler

Department of Chemistry, University of ExeterStocker Road, Exeter EX4 4QD (UK)Fax: (+44) 1392-263-434E-mail : a.soncini@ex.ac.uk

p.w.fowler@ex.ac.uk

[b] Dr. C. DomenePhysical and Theoretical Chemistry Laboratory, Oxford UniversitySouth Parks Road, Oxford OX1 3QZ (UK)

[c] J. J. Engelberts, Prof. J. H. van Lenthe, Dr. R. W. A. HavenithDebye Institute, Theoretical Chemistry Group, Utrecht UniversityPadualaan 8, 3584 Utrecht (The Netherlands)

[d] Prof. A. Rassat�cole Normale Sup�rieure, D�partement de Chimie, UMR 8640CNRS-ENS-Universit� Pierre-et-Marie-Curie24 rue Lhomond, 75231 Paris CEDEX 05 (France)Fax: (+33) 144-323-325E-mail : andre.rassat@ens.fr

[e] Prof. L. W. JenneskensDebye Institute, Department of Physical Organic ChemistryUtrecht UniversityPadualaan 8, 3584 Utrecht (The Netherlands)

Abstract: Why are some (4n+2)p sys-tems aromatic, and some not? The ip-socentric approach to the calculation ofthe current density induced in a mole-cule by an external magnetic field pre-dicts a four-electron diatropic (aro-matic) ring current for (4n+2)p carbo-cycles and a two-electron paratropic(antiaromatic) current for (4n)p carbo-cycles. With the inclusion of an electro-

negativity parameter, an ipsocentricfrontier-orbital model also predicts thetransition from delocalised currents incarbocycles to nitrogen-localised cur-rents in alternating azabora-heterocy-

cles, which rationalises the differencesin (magnetic) aromaticity betweenthese isoelectronic p-conjugated sys-tems. Ab initio valence-bond calcula-tions confirm the localisation predictedby the na�ve model, and coupled-Har-tree–Fock calculations give current-density maps that exhibit the predicteddelocalised-to-localised/carbocycle–het-erocycle transition.

Keywords: aromaticity · boron ·nitrogen heterocycles · ring current ·valence bond

Chem. Eur. J. 2005, 11, 1257 – 1266 DOI: 10.1002/chem.200400678 � 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1257

FULL PAPER

of induced currents can in fact cope with both extremes, pre-dicting the transition from delocalised to localised magneticbehaviour that is found in full ab initio current-densitymaps.

The ipsocentric model[3–5] gives an accurate account of p

ring currents in planar carbocycles in terms of frontier-orbi-tal contributions. The hallmarks of aromaticity and antiaro-maticity on the magnetic criterion,[6–9] that is, the diatropiccurrents of 4n+2 systems and the paratropic currents[10] ofplanar 4n systems are attributed in this model to four andtwo frontier electrons, respectively.[4] In both cases, thesense of the ring current follows directly from the symmetryproperties of the HOMO–LUMO transition, and the predic-tions of a simple orbital theory are borne out in full abinitio calculations of current-density maps.[3,4] For the 4n+2p

heterocycles, borazine and boroxine, however, ab initiomaps show only localised circulation of the p electronsaround the electronegative nuclei.[2] How does the simpleorbital model deal with this dichotomy? In particular, whydoes it not predict ring currents for all planar 4n and 4n+2p systems? It is shown here that a generalisation of the pic-torial orbital model is able to account for both the presenceof global currents in carbocycles, and their absence inboron–nitrogen analogues such as borazine (2) and theeight-membered borazocine (3).

Results and Discussion

The aromaticity analogy : The considerations of magnetic ar-omaticity have their counterparts in the chemistry of theseboron–nitrogen systems. Although 6p-electron borazine (2)was originally called “an inorganic benzene”[11] and believedto have a resonance energy similar to that of benzene, it wassoon recognised that it took part in few reactions typical ofaromatic systems. For instance, borazine gives a tris-adductwith hydrogen chloride.[12] It is now generally agreed thatborazine is not aromatic. In contrast to benzene, the p elec-trons are localised on the more negative nitrogen atoms,even if some chemical behaviour is reminiscent of that ofbenzene (e.g. the formation of a weakly puckered hexaethyl-borazine tricarbonyl chromium complex).[13]

The parent compounds BN/2NN/2HN with N¼6 6 have notbeen synthesised, although many derivatives of the homo-logues diazadiboretidine (also known as diazadiboretaneand diazadiborete) (1) and borazocine (tetrazaborocine, tet-razaborocane) (3) have been prepared.[14] Interconversionsbetween them (and with borazine) are possible, and theirchemical properties differ strongly from those of the corre-sponding 4p and 8p carbocycles. For instance, the four-mem-bered ring survives thermal elimination of isobutene from atert-butyl derivative.[14b] Thus, in spite of their 4n p electrons,these systems do not show the reactivity expected of an anti-aromatic molecule. This again can be attributed to localisa-tion of the p electrons onto the more electronegative nitro-gen atoms. Similarly, energy calculations[11b, 15] suggest that 1is less antiaromatic than cyclobutadiene, in agreement with

the H�ckel prediction that the antiaromaticity in X2Y2H4-type four-membered rings decreases with increasing electro-negativity difference between X and Y.[16] By using the ipso-centric model, we show here that diazadiboretidine (1), bor-azine (2) and borazocine (3) are in fact all nonaromatic.

Aromaticity, ring currents and frontier orbitals : If aromatici-ty is to be defined by magnetic criteria,[9] what is needed forits theoretical characterisation is a reliable account of thecurrents induced by an external magnetic field since suscept-ibility, nuclear shielding and other response properties areall integrals of this current density. Realistic current-densitymaps can be obtained by a computationally efficient proce-dure, the ipsocentric method, which avoids the gauge-de-pendence problem by taking each point in space as theorigin of the vector potential generating the magneticfield.[17,18] This choice has the important conceptual advant-age of yielding a unique decomposition into nonredundantorbital contributions.[3] The total current density is obtainedas the sum over transitions from occupied to unoccupied or-bitals, and is governed by symmetry rules, and modulated byenergy denominators.

For a planar molecule in a perpendicular magnetic field,the symmetries determining the sense of circulation of theinduced current about the molecular origin are those of in-plane translations (Tx, Ty) and the in-plane rotation (Rz). Asimple but powerful symmetry rule can be deduced. If aproduct of occupied and unoccupied orbitals contains the Tx

or Ty symmetry, but not the Rz symmetry, the contributionof the transition to the current has the diatropic sense, andif it contains the Rz symmetry, but not the Tx or Ty symme-try, the contribution has the reverse paratropic sense. If nei-ther symmetry is present, the transition is inactive; if both,more detailed analysis is needed.[3,19] In terms of the mag-netic criterion, the resultant of all such contributions deter-mines the aromaticity or antiaromaticity (net diatropicity orparatropicity of the ring current) of a monocycle. As thecontributions are weighted by orbital energy differences,frontier orbitals generally dominate.

Currents in carbocycles : Current-density maps from abinitio calculations[4] on the archetypal carbocycles, benzeneand (planarised[20]) cyclooctatetraene (COT), are shown inFigure 1. For each molecule, the maps show total p and s

contributions to the induced current density. As expected,the currents arising from the p electrons are, respectively,strongly diatropic in benzene and strongly paratropic inCOT. The currents are dominated by HOMO contributionsin both cases.

An angular-momentum analysis shows how the symmetryrules account for these features. The H�ckel p molecular or-bitals of a cycle of N identical atoms in full DNh symmetryhave well-defined angular-momentum properties with re-spect to the principal axis. At each successive energy level,the quantum number, l (=0,1…,N/2), increases by one. In acycle with N=4n+2 electrons, the HOMO and LUMO cor-respond to l=n and n+1, respectively, whereas for a cycle

� 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.chemeurj.org Chem. Eur. J. 2005, 11, 1257 – 12661258

with N=4n electrons, in the closed-shell configuration ob-tained on distortion to D(N/2)h symmetry, the HOMO andLUMO are derived from the splitting of a pair of originallydegenerate orbitals with l= n.

The canonical molecular orbitals are the delocalised set{yl,c/s}, which produce uniform p-charge distribution on allthe carbon atoms. Functions yl,c and yl,s (degeneracy dl)have coefficients on centre r equal to

ffiffiffiffiffiffiffiffiffiffiffidl=N

pcos(2plr/N)

andffiffiffiffiffiffiffiffiffiffiffidl=N

psin (2plr/N), respectively.[21] Unless l= 0 (or N/2,

if allowed), the two functions share a single, well-defined an-gular momentum and are degenerate. For l= 0 and N/2, thesine partner vanishes identically. For other values of l thesine/cosine form is simply a choice that gives convenientnodal properties.

When orbitals have well-defined values of l, the symme-try selection rules for occupied-to-unoccupied transitionsare reduced to the following:[4] a diatropic contributionarises from a transition in which Dl=++1 and a paratropiccontribution arises from a transition in which Dl=0. Thus,for a 4n+2 cycle with maximum symmetry, only theHOMO–LUMO transition is active, and is responsible forthe entire (four-electron) diatropic current. For a 4n cyclewith the lower symmetry of the closed shell, two types oftransition are active: the HOMO–LUMO transition leads toa two-electron paratropic contribution, and HOMO�1–

LUMO and HOMO–LUMO+1transitions to diatropic contri-butions. Here, the smallerHOMO–LUMO splitting en-sures dominance of the para-tropic current. Decompositionof the carbocycle ab initio p

maps into orbital contributionsconfirms the above analysis.[4]

A model for currents in aza-bora-heterocycles : To adapt thepictorial molecular-orbital anal-ysis to the ipsocentric model forcurrents in alternating heterocy-cles such as borazine (2) andhomologues 1 and 3, it is suffi-cient to include one extra fea-ture. To deal with heteroatomsin H�ckel theory it is necessaryto modify the Coulomb and/orresonance integral parameters:for boron and nitrogen (aszero- and two-electron donors),the recommended parametersare aB =a�b and aN =

a+1.5b.[22] H�ckel calculationsusing aB =a�1.1b and aN =

a+1.5b have been reported for1–3.[16] In the simplest model ofan equilateral BN/2NN/2HN cyclethat allows for the differing

electronegativities of boron and nitrogen, symmetricalchanges are made to the Coulomb parameters, that is, aB =

a�hb and aN = a+hb, where h is positive. Variation of thedimensionless quantity h from 0 to �1 thus gives a one-parameter model to correlate the properties of annulenesand BN cycles.

The implications of this model for six- and eight-mem-bered cycles, as representatives of 4n+2 and 4n p systems,respectively, are now investigated in detail. Direct solutionof the H�ckel problem with modified a values gives energylevels {e} and molecular orbitals {f} from which the conse-quences for ring currents can be deduced. In the limit, h= 0,the canonical molecular orbitals are {yl,c/s}; when h¼6 0, theyare linear combinations of this starting set.

Currents in the six-membered cycle : The secular equationsfor the alternating six-membered cycle have maximal D3h

(h¼6 0) or D6h (h=0) symmetry. The energies are (D3h/D6h

labels, dl = degeneracy):

e0 ¼ aþffiffiffiffiffiffiffiffiffiffiffi4þh2

pb ðA00

2=A2uÞ, dl ¼ 1;

e1 ¼ aþffiffiffiffiffiffiffiffiffiffiffi1þh2

pb ðE00=E1gÞ, dl ¼ 2;

Figure 1. Maps of the current density induced in benzene (left) and (planar) cyclooctatetraene (right) by a per-pendicular external magnetic field. Contributions of the p (top) and s orbitals (bottom) calculated by theRHF/6-31G** ipsocentric approach are plotted at one bohr above the molecular plane. Anticlockwise circula-tions are diatropic, clockwise circulations paratropic. *: Carbon atoms; � : hydrogen atoms.

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FULL PAPERA Unified Orbital Model of Currents in Monocycles

e2 ¼ a�ffiffiffiffiffiffiffiffiffiffiffi1þh2

pb ðE00=E2uÞ, dl ¼ 2;

e3 ¼ a�ffiffiffiffiffiffiffiffiffiffiffi4þh2

pb ðA00

2=B2gÞ, dl ¼ 1;

When h is nonzero, functions with different l are allowedto mix according to their symmetries in the lower D3h group,that is, in Ak2, l= 0 mixes with l=3, and in Ek , l=1 mixeswith l=2 (retaining the sine/cosine distinction). As h in-creases, the degree of mixing increases, and is described byangles m and n, where h=2tan 2m= tan2n :

�0,c ¼ cos m y0,c�sin m y3,c,

�1,c ¼ cos n y1,c�sin n y2,c, �1,s ¼ cos n y1,sþsin n y2,s,

�2,c ¼ sin n y1,cþcos n y2,c, �2,s ¼ �sin n y1,sþcos n y2,s,

�3,c ¼ sin m y0,cþcos m y3,c:

When h=0, m and n are zero; in the limit of a large h, themolecular orbitals of the alternating six-membered cyclebecome exact 50:50 mixtures, with jm j= jn j= 458. When h=

1, f1,c/s still contains 95 % of the benzene y0,c (jm j=13.38),but the HOMO pair, f1,c/s, contains a 15 % admixture of thebenzene LUMO, y2,c/s (jn j= 22.58). Figure 2 a shows the cor-relation of orbitals and energies with h for the six-mem-bered cycle. H�ckel p charges reflect this shift with respectto the uniform charge distribution of benzene, with 1�1=2( 1ffiffi

5p +

ffiffiffi2p

)�1.62 and 0.38 p electrons on the nitrogen and

boron atoms when h=1. As h increases, bonding orbitalsconcentrate on the electronegative nitrogen atoms, and anti-bonding orbitals on the electropositive boron atoms, thefunctions becoming more “localised”.

The mixing has crucial consequences for the predictedcurrents. When h=0, the dominant HOMO–LUMO transi-tion responsible for the ring current of the carbocycle isfrom {y1,c, y1,s} to {y2,c, y2,s} for which Dl=1; the transitionis therefore purely translational and contributes only dia-tropic current. For all h>0, as HOMO and LUMO eachcontain l= 1 and 2 contributions, the transition includesDl=1 and 0 components, and the current gains partial para-tropic character. The global diatropic current of benzenetherefore weakens as h increases.

A first interpretation of the effect of progressive orbitalmixing on the ring current in a 4n+2 cycle can be found byusing the simplest approach, the H�ckel–London model.[6]

In this model, as formulated by McWeeny,[23] the ring cur-rent per unit area of a cycle is proportional to the reducedbond current, Jrs, where prs is the p bond-order between ad-jacent atoms r and s and p̄rs,rs is the imaginary bond–bondpolarisability.[1]

Jrs ¼ ðprsþb�prs,rsÞb

In a monocycle, J, p and p̄ are independent of the choiceof adjacent pair. For benzene, p= 2=3 and p=�5=9b�1. Forborazine with h= 1, a straightforward calculation gives p=

1=3( 2ffiffi5p +

1ffiffi2p ) and p̄ =�1=9(

ffiffiffi5p

+13

4ffiffi2p )b�1, from which the bor-

azine:benzene ratio of ring-current intensity, assuming anequal ring area, is ( 1ffiffi

5p � 1

4ffiffi2p )�0.27. This ratio tends to zero as

the electronegativity difference parameter h tends to infini-ty. Thus, this crude model, in which current is constrained toflow along the straight lines between adjacent atoms, pre-dicts that an increase in h leads to diminution and eventualextinction of the global diatropic current.

Figure 2. Correlation diagrams of energy and orbital composition in aH�ckel model of a) B3N3H6, b) B4N4H8 and c) B2N2H4, as modelled bythe single parameter h, where aB =a�hb and aN =a+hb. The graphsshow the variation of orbital energy, x= (e�a)/b, with h.

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A. Soncini, P. W. Fowler, A. Rassat et al.

The picture can be refined by allowing for the spatialextent of the current outside the direct line of centres. Thepseudo-p model[24] includes explicit basis functions, and sim-ulates p ring currents by using a s model based on the exactsymmetry equivalence between the s orbitals of cyclic HN

atoms and the p orbitals of the N-membered carbocycle. Itturns out that ipsocentric in-plane currents calculated forthe HN system with a minimal basis set give an excellentmatch to the out-of-plane currents of the carbocycle calcu-lated with a large basis set. Figure 3 presents a pseudo-p

simulation of the heterocyclic ring in which the boron andnitrogen atoms are modelled by one-electron atoms with nu-clear charges of +0.7e and +1.3e, each carrying STO-3G sand p functions (exponent 1a0

2), the charge difference play-ing the role of the h parameter in the H�ckel analysis, andthe in-plane p functions giving radial flexibility to chargeand current densities. Pure symmetry arguments predictedopposed paramagnetic and diamagnetic currents; thepseudo-p map shows that they occupy different regions ofspace, with diamagnetic circulation on the outside and para-magnetic circulation on the inside of the ring, the net effectbeing a set of circulations centred on electronegative atoms.In the graph-theoretical H�ckel–London model, this spatialpattern of opposed local currents was expressed as a reduc-tion of the sole available parameter, the bond current. Bothmodels therefore essentially agree on the effect of the elec-tronegativity difference on the ring current.

Currents in the eight-membered cycle : For the alternatingeight-membered cycle, the maximal symmetry is D4h (h¼6 0)or D8h (h= 0) and the energy levels el(D4h/D8h labels, dl =

degeneracy) are:

e0 ¼ aþffiffiffiffiffiffiffiffiffiffiffi4þh2

pb ðA2u=A2uÞ, dl ¼ 1;

e1 ¼ aþffiffiffiffiffiffiffiffiffiffiffi2þh2

pb ðEg=E1gÞ, dl ¼ 2;

e�2 ¼ a� hb ðB1uþB2u=E2uÞ, dl ¼ 2 at h ¼ 0; dl ¼ 1þ1 at h 6¼ 0;

e3 ¼ a�ffiffiffiffiffiffiffiffiffiffiffi2þh2

pb ðEg=E3gÞ, dl ¼ 2;

e4 ¼ a�ffiffiffiffiffiffiffiffiffiffiffi4þh2

pb ðA2u=B2uÞ, dl ¼ 1;

In the carbocyclic system with full symmetry, the degener-acy of y2,c and y2,s leads to an open-shell configuration

which can be stabilised in vari-ous ways: distortion to D2d

leads to the tub-shaped equili-brium geometry of the freeCOT molecule; in-plane relaxa-tion to D4h yields a geometrysimilar to those found in“clamped”-substituted COTsystems.[20] In spite of its bondalternation, planar D4h COT re-tains fully delocalised orbitalsand the ring current of theequilateral carbocycle, asFigure 1 shows.[4,20b]

In the heterocyclic system,when h¼6 0, with the nitrogenatom as atom 0, y2,c as thebonding and y2,s the antibond-ing partner, the assignment oflabels B1u and B2u depends onthe setting of D4h within D8h.

For a general value of h, the mixing of angular-momentumcomponents is again described by two angles, m and k, withh= 2tan 2m=

ffiffiffi2p

tan 2k.

�0,c ¼ cos m y0,c�sin m y4,c,

�1,c ¼ cos ky1,c�sin ky3,c, �1,s ¼ cos ky1,sþsin ky3,s,

�2,c ¼ y2,c, �2,s ¼ y2,s,

�3,c ¼ sin ky1,cþcos ky3,c, �3,s ¼ �sin ky1,sþcos ky3,s,

�4,c ¼ sin m y0,cþcos m y4,c:

Note that functions y2,c and y2,s, as the HOMO andLUMO, remain unmixed for all values of h, separated by agap that is linear in h, that is, 2h. Each is perfectly localised,the HOMO on the nitrogen atom and the LUMO on theboron atom. In contrast, with increasing values of h, thefunctions f0,c, f1,c, f1,s, f3,c, f3,s and f4,c become increasinglylocalised on the electronegative atoms: when h=1, the ni-trogen and boron atoms have H�ckel p populations of 1�1=2(1=2+

1ffiffi3p +

1

2ffiffi5p �1.65 and 0.35 electrons, respectively. When

h= 1, here representing planar B4N4H8, jm j=13.38 and jk j=17.68 so that f0,c contains 5 % of y4,c/s and HOMO�1 f1,c/s

contains 10 % of y3,c/s. Figure 2 b shows the correlation of or-

Figure 3. Pseudo-p simulations of induced current density in BN/2NN/2HN, with N= 6 (left) and N =8 (right). Inthese molecules intense in-plane local circulations around the nitrogen centres in this pseudo-p model indicateout-of-plane p circulations of similar strength. Here the symbol · denotes the pseudo-hydrogen centres repre-senting the boron and nitrogen atoms in the model. Anticlockwise circulations are diatropic, clockwise circula-tions paratropic.

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FULL PAPERA Unified Orbital Model of Currents in Monocycles

bitals and energies with h for the eight-membered cycle.The symmetry argument for deducing the currents in 4n

systems involves several steps. Initially (h�0), the current isdominated by the HOMO–LUMO transition across a smallenergy gap. This Dl= 0 transition generates an intense,purely paratropic, ring current. As h increases, the HOMO–LUMO gap opens, and the intensity of the current falls butremains paramagnetic. Simultaneously, the separation of theHOMO�1 and the LUMO and that of the HOMO and theLUMO+1, increase, but only slowly; both correspond toDl=�0 diatropic contributions to current. Thus, as h in-creases, the original global paratropic (antiaromatic) ringcurrent is subject to intrinsic reduction and increasingly sig-nificant cancellation.

In the McWeeny form of the H�ckel–London theory,[23]

the ring current of the eight-membered cycle when h issmall is dominated by the HOMO–LUMO contribution tothe imaginary bond–bond polarisability, which is �1=16bh,and hence falls sharply as h increases from the planar-con-strained form of COT (where h= 0) to the heterocycle.When h=1, the eight-membered cycle has p= 1=2( 1ffiffi

3p +

1ffiffi5p ) and

p̄=�1=8( 11

3ffiffi3p +

7

2ffiffi5p +1=2)b�1 to give bond currents J=ffiffi

3p

72+ffiffi5p

80�1=16��0.105, which constitute a reduced but still netparatropic circulation. If the ratio of the areas of the six-and eight-membered cycles is 9:16, a ring current of about asixth of the strength of the (diatropic) benzene current isimplied. When, as for the six-membered cycles, the spatialextent of the current in the eight-membered cycle is takeninto account in a pseudo-p calculation (Figure 3), theH�ckel–London result is seen as a simplified representationof the nitrogen-centred currents.

Currents in the four-membered cycle : For the four-mem-bered ring, B2N2H4, the extinction of current with increasingh follows the same pattern as that for the eight-memberedcycle. The four orbital energies are now e0 =a+

ffiffiffiffiffiffiffiffiffiffiffi4þh2

pb, e1

�

=a�hb, e2 =a�ffiffiffiffiffiffiffiffiffiffiffi4þh2

pb, again giving a linear HOMO–

LUMO gap of 2h (see Figure 2c), and yielding a paratropiccurrent that weakens with h and is increasingly cancelled bythe diatropic current that arises from the transitions acrossthe HOMO–LUMO+1 and HOMO�1–LUMO gaps ofh+

ffiffiffiffiffiffiffiffiffiffiffi4þh2

pb. In the H�ckel–London theory, when h= 1 the

reduced bond current for the four-membered ring remainsnet paratropic (J=

1

4ffiffi5p �1=4; �0.138) and corresponds to half

the current of a benzene ring which has 9=4 times the area,and again represents the pattern of nitrogen-centred circula-tions in a more detailed description.

Generalisation for 4n and (4n+2) p BN heterocycles : The re-sults for the four-, six- and eight-membered cycles are gener-alisations for the 4n and (4n+2) p BN heterocycles, withseparate mechanisms, but equivalent results for the ring cur-rent in both cases. Consider an [N]-carbocycle, where N iseven. As h increases from 0, each bonding (b)/antibonding(a) molecular-orbital pair of the parent bipartite carbocycle,with angular momenta Lb and La = N/2�Lb, which corre-spond to equal but opposite eigenvalues, mix to form a

bonding/antibonding pair in the [N]-centre BN heterocyclewith energies xb =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4cos2 tLbþh2

pand xa =�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4cos2 tLaþh2

p,

where tL =2pL/N, so that xb =�xa. The rotation describingthe mixing is parametrised by the angle wL = 1=2tan�1 (h/2cos tL), where 0�wL�p/4.

When N=4n+2, LHOMO =n and LLUMO =n+1, and themixing angle wL is maximal for a given value of h. As wehave seen, this leads to a sufficiently large value of h to dis-rupt the global ring current and eventual localisation. WhenN=4n, LHOMO =LLUMO = n, the original HOMO and LUMOcarbocycle eigenfunctions remain unmixed. Disruption ofthe global paratropic ring current is caused by the wideningof the HOMO–LUMO gap, triggering cancellation of therotational HOMO–LUMO and translational HOMO�1–LUMO contributions.

Ab initio current-density maps in azabora-heterocycles : Toobtain ab initio data on the currents in BN analogues of the4n and 4n+2 carbocycles, calculations were performed onB2N2H4, B3N3H6 and B4N4H8. The geometry of each mole-cule was optimised at the restricted Hartree–Fock (RHF)level with the 6-31G** basis set and currents were calculat-ed by using the ipsocentric approach at the coupled Har-tree–Fock (CHF) level with the same basis set. At this levelof theory, B3N3H6 and B4N4H8 have planar structures withD3h and D4h symmetries, respectively, whereas the planarstructure of B2N2H4 with D2h symmetry is a transition state(imaginary frequency 116i cm�1) that leads to a shallow C2v

butterfly optimum, hinged at the BB diagonal, with a dihe-dral angle of 1678. Current-density maps were computed forboth the constrained planar and the fully optimised struc-tures of B2N2H4. An alternative geometry for B4N4H8 isbased on the occupation of the corners of a cube;[25a] thiscompact structure is slightly preferred to the planar form, asdetermined with small basis sets,[25b] but lies 302 kJ mol�1

above it when calculated at the 6-31G** level. All threeplanar molecules have uniform BN distances (R= 1.4432,1.4258, 1.4254 �, respectively).

These geometries are in general agreement with availableexperimental and theoretical data. The nonplanarity ofB2N2H4 has been noted in previous ab initio calculations,[26]

which gave a BN distance of 1.457 �, and although theparent molecule itself has not been synthesised, X-ray struc-tures of five substituted molecules[14b] are known with meanBN distances varying between 1.430 and 1.486 �, and all areplanar apart from the severely hindered tetra-tert-butyl de-rivative. Borazine has been the subject of many ab initio cal-culations.[27] The RHF//6-31G**-calculated BN distance of1.4258 � in borazine compares with 1.4355�0.0021 � (gaselectron diffraction[28a ,b]) and 1.429 � (X-ray[28c]). No experi-mental data are available for the parent B4N4H4, but experi-mental and theoretical structural data on its derivativeshave been reviewed by Gilbert and Gailbreath.[26] Calcula-tions at different levels give a planar (B3LYP/6-31G+*) ornear-planar (MP2/6-31G+*) structure for B4N4H8 with a BNlength of 1.436 �, intermediate between single and doublebonds. Equivalent calculations on the permethylated mono-

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cycle show it to adopt a tub structure.[26] X-ray structures forderivatives with bulky substituents are tub-like with alter-nating BN bonds as in, for example, (tBuN)4(MeB)4.

[29] So-lution 1H NMR spectra of B4N4(CH2R)4(Me)4 show a non-planar eight-membered ring.[30]

The current densities induced in (planar) B2N2H4, B3N3H6

and B4N4H8 by a perpendicular magnetic field are plotted inFigure 4. For each molecule, the maps show HOMO,HOMO�1 and the total p and s contributions to the in-duced current density plotted in a plane one bohr above

Figure 4. Current density induced by a perpendicular external magnetic field in the molecules B2N2H4 (1, planar), B3N3H6 (2), B4N4H8 (3) (left to right),partitioned as contributions from (top to bottom) the HOMO�1, HOMO, p and s orbitals, calculated by the RHF/6-31G** ipsocentric approach. Wherethe HOMO�1 and HOMO are degenerate, the map shows the summed contribution of the pair. Contributions are plotted at one bohr above the molec-ular plane. Anticlockwise circulations are diatropic, clockwise circulations paratropic. Nitrogen, boron and hydrogen centres are denoted by circles con-taining a bar, cross or dot, respectively.

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that of the nuclei. As Figure 4 shows, all three systems, incontrast to the carbocycles, have multicentre patterns oflocal diatropic p circulations centred on the nitrogen atoms.In B3N3H6, the nitrogen-centred p circulations arise entirelyfrom the four electrons of the doubly degenerate HOMO:the total p and HOMO maps are visually indistinguishable,and the contribution of the HOMO�1 is negligible. InB2N2H4 and B4N4H8, the nitrogen-centred p circulationsarise from a combination of the inner paratropic HOMOand the outer diatropic HOMO�1 currents. As the localiseds and p currents rotate in the same sense, they are rein-forced in all three molecules. Maps of total current density(not shown) plotted for the puckered optimum geometry ofB2N2H4 at a height of 1a0 above the nitrogen atoms and par-allel to the median nuclear plane show the same general fea-tures: the localised nature of the current density is not criti-cally dependent on planarity.

Localised orbital analysis : Further analysis using Pipek–Mezey localisation of the canonical molecular orbitals[31]

demonstrates the strong generic similarity of all three sys-tems. The localised orbitals of BN/2NN/2HN are N core 1s orbi-tals on the heavy atoms, N/2 BN localised s-bonding orbi-tals, N/2 NH and N/2 BH localised s-bonding orbitals andN/2 p orbitals that are essentially nitrogen-localised p lonepairs. In this analysis nitrogen-centred p currents arise natu-rally as circulations in the lone pairs. The s currents shownin Figure 4 for B2N2H4, B3N3H6 and B4N4H8 have nodes atthe boron sites. In the localised-orbital picture, the promi-nent “triangular” diatropic circulations around the nitrogenatoms arise from the three bonds meeting at nitrogen, eachdeltoid constituting a six-electron diatropic circulation. Inthe carbocycles, the s currents are more uniformly distribut-ed over all the heavy atoms, but are still spatially local-ised.[32] The main difference in the magnetic response be-tween carbocycles and azabora-heterocycles is that the p

ring currents of one class are absent in the other. This dis-tinction is clear from the computed current-density maps,which are fully compatible with the qualitative analysis basedon symmetry and electronegativity. One and the same ipso-centric molecular-orbital picture explains both the current inthe carbocycle and the lack of current in the heterocycle.

Valence-bond calculations on azabora-heterocycles : As thelocalisation of density on electronegative nitrogen centres isa key factor in the simple H�ckel model of ring-currentquenching, it is important to verify this qualitative differ-ence between carbocyclic and azabora-heterocyclic systemsby using more sophisticated theoretical methods. According-ly, ab initio valence-bond configuration-interaction (VBCI)calculations, in which many structures are used, and va-lence-bond self-consistent field (VBSCF)[33] calculations, inwhich both orbitals and structure coefficients are optimised,were performed on planar B2N2H4 (1), B3N3H6 (2) andB4N4H8 (3) with the 6-31G** basis set. All were carried outby using TURTLE,[34] as implemented in the GAMESS-UKpackage.[35]

The wavefunction in the valence-bond calculations con-sists of a Hartree–Fock s core in structures with singly occu-pied p orbitals. From a chemical point of view, these differ-ent structures can be seen as different bonding arrange-ments. All possible structures that can be generated byusing the occupied atomic orbitals were used in the VBCIwavefunction, including chemically irrelevant ones. ForB2N2H4 there are 20 possible structures which arise from the20 ways of distributing four electrons over four p orbitals togive a singlet. For B3N3H6 175 structures are possible, andfor B4N4H8 there are 1764. A VBCI calculation will showwhich bonding arrangements are preferred.

However, the VBCI wavefunction, owing to its fixedatomic orbital basis set, does not present a compact descrip-tion of the wavefunction as orbital optimisation and correla-tion effects are intermixed. Therefore the most importantstructures in the VBCI calculations were used to construct aVBSCF wavefunction without imposing restrictions on theorbitals. From the coefficients in the VBCI wavefunctionsthe weights,[36] and thus the importance of each structure,may be derived. For 1–3, the most important structure typesare shown in Figure 5 with their respective weights. Fromthese weights it appears that several structures contribute tothe total wavefunction.

Subsequently, for B2N2H4, the sets of structures of eachkind were separately optimised in VBSCF calculations with-

Figure 5. The most significant symmetry-distinct VBCI structures ofB2N2H4 (1), B3N3H6 (2) and B4N4H8 (3) with the total weight of the com-bined set in parentheses.

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out imposing any restrictions on the orbitals. The resultingenergies of the first (1 a : �160.722209 Eh) and the second(1 b : �160.721245 Eh) were very similar, with the first struc-ture, having two different orbitals on each nitrogen atom(see Figure 6), slightly more favourable.

A VBCI calculation was then performed on the first twosets of structures (1 a+1 b), each with their own optimisedorbitals. This calculation gave the same energy as the singlestructure VBSCF calculation (1 a), whereas the overlap be-tween set one (1 a) and set two (1 b) was over 99.9 %. Thismeans that the VBSCF calculations converge to the samewavefunction, which is most aptly described as nitrogenatoms, each with a correlated pair of p electrons. The con-clusion is that there is no resonance energy and thus no res-onance.[37] Similar results were obtained for B3N3H6 (2)[37]

and B4N4H8 (3).These calculations show the same tendency as both the

H�ckel and the ring current calculations, that is, localisationof electrons on the nitrogen atoms. In fact, the valence-bondresults show that the ground states of B2N2H4 (1), B3N3H6

(2) and B4N4H8 (3) correspond to the structures 1 a, 2 a[37]

and 3 a, respectively.

Conclusions

Given the success of the simple one-parameter model inpointing out qualitative differences in current patterns be-tween carbo- and azabora-cycles, it is natural to ask aboutits predictions for other systems, for example, XN/2YN/2HN—where does the borderline between “aromatic” and “nonar-omatic” occur in this picture? The crudest form of themodel shows a change from global circulation to localisedcurrents by a strong reduction in the magnitude of ring cur-rent from h=0 to h=1; this variation is sigmoidal (Figure 7)

with a point of inflexion, where current will be most sensi-tive to variation in h, at h�0.5, where h represents the devi-ation from the average electronegativity, that is, jaX�aY j2b.A rule of thumb is that aX varies with the difference in elec-tronegativity between X and C: (aX�aC)/b�cX�cC for X asa one-electron donor and (aX�aC)/b�1+cX�cC as a two-electron donor[22a]). Inflexion at h�0.5 is consistent with in-dications for X3Y3 6p systems from ring-current maps[2] andNICS calculations:[27c] systems in which the electronegativi-ty/coulomb-parameter difference between X and Y is 1 ormore on the Pauling scale lack ring currents (B3O3H3,

[2,27c]

B3N3H6,[2,27c] Al3N3H6

[27c] and Al3P3H6[27c]), whereas those

with differences of half a unit or less may (C3N3H3[2,27c] and

B3P3H6[27c]) or may not (B3S3H3

[27c]) show global currents.This uniform trend lends support to the expectation that or-bital-based models should be able to account for the subtle-ties of ring current in the range of aromatic, nonaromaticand antiaromatic heterocycles, and is a strong argument forusing methods in which the orbital contributions themselveshave a clear physical basis.

Figure 6. The singly occupied p orbitals of B2N2H4 (1), B3N3H6 (2) andB4N4H8 (3), respectively, which correspond to the structures 1a, 2 a and3a (see Figure 5). Orbitals from the left and right of the picture togetherform the correlated lone pair on the nitrogen atom.

Figure 7. Variation of ring current, j, with electronegativity parameter, h,in the H�ckel–London treatment of the X3Y3 cycle. j(h) is the ring cur-rent for a regular hexagonal cycle with alternating coulomb parametersa�hb as a percentage of the current for the same cycle with h =0. Theinset curve (not to scale) shows the variation of the first derivative dj(h)/dh near h =0.5.

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FULL PAPERA Unified Orbital Model of Currents in Monocycles

Acknowledgements

L.W.J. acknowledges financial support from the Royal Society of Chemis-try�s Journals Grants for International Authors Programme (0301433).R.W.A.H. acknowledges financial support from the Netherlands Organi-sation for Scientific Research (NWO), grant 700.53.401. P.W.F, A.R. andA.S. thank the EU for financial support from the Framework V pro-gramme (RTN Contract HPRN-CT-2002–00136 “WONDERFULL”).C.D. thanks the Royal Society for a University Research Fellowship andP.W.F. for a Royal Society/Wolfson Research Merit Award.

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Received: July 4, 2004Published online: January 3, 2005

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