Investigating the Threshold of Aromaticity and Antiaromaticity by

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Investigating the Threshold of Aromaticity and Antiaromaticity by Variation of Nuclear Charge Patrick W. Fowler,* David E. Bean, and Mark Seed Department of Chemistry, UniVersity of Sheffield, Sheffield, S3 7HF, United Kingdom ReceiVed: July 19, 2010; ReVised Manuscript ReceiVed: August 31, 2010

Aromatic benzene and nonaromatic borazine can be modeled as the end-points of a continuous process evolving through intermediate systems with fractional nuclear charges. Current-density maps show a smooth, linear progression in which the global diatropic π ring current weakens as localized diatropic lone-pair currents grow with the increase in charge difference. In contrast, the progression from antiaromatic (planarized) cyclooctatetraene to nonaromatic borazocine shows an initially persistent paratropic ring current with a sharper transition to the localized diatropic system. The different behaviors of aromatic and antiaromatic systems stem from the different orbital origins of diatropic and paratropic ring current, and both can be rationalized in terms of arguments based on π distortivity and electronegativity. Introduction Benzene and borazine (B3N3H6) are isoelectronic and isostructural, but display significantly different properties. In particular, whereas benzene is universally regarded as the archetypal aromatic molecule, borazine is now usually considered to be either nonaromatic or at most weakly aromatic.1-11 Aromaticity itself is not a rigorously defined property with a single accepted definition;12 and geometric, energetic, and magnetic criteria are all used to decide on the aromatic nature of particular molecules, as are chemical reactivities. Benzene and borazine are clearly distinguished on most criteria. Properties such as chemical shifts, magnetic susceptibilities, and induced current-density, the quantity that underlies all magnetic criteria of aromaticity,13 also indicate borazine as nonaromatic.1,2,6,7 In particular, current-density maps, calculated using the wellestablished ipsocentric14 (CTOCD-DZ15,16) method, show that benzene displays a significant diatropic π currentsa clear signature of its aromaticity.14,17 When a similar calculation is carried out for borazine, the global current disappears and is replaced by localized circulations on the more electronegative nitrogen atoms,6,7 which is consistent with the polar nature of BN bonds.18 On the magnetic criterion, the long-established description of borazine as “the inorganic benzene”,19 mentioned in many textbooks and implied in this molecule’s earlier name of borazole, is seen to be somewhat misleading. Of course, the contrast in magnetic response between the two molecules has a detailed explanation in terms of electronic structure differences and can be understood by consideration of orbital energies, symmetries, and nodal properties of the individual systems. However, the two systems can also be seen as end points of a continuous pathway parametrized by a variable such as electronegativity (see for example ref 7). It then becomes natural to ask about the character of the “transition” between benzene and borazine: Is aromaticity lost all at once or gradually? Is benzene an isolated aromatic point in the landscape, or is it representative of a large catchment area of similar systems? Are aromaticity and antiaromaticity different in these respects? In the context of ab initio calculation of real * To whom correspondence should be addressed. Fax: +44 114 222 9346. E-mail: [email protected].

Figure 1. Current-density maps, calculated at the CTOCD-DZ/CHF/ 6-31G**//RHF/6-31G** level, for (a) benzene, (b) borazine, and (c) the difference between panels b and a. The maps are plotted at a height of 1a0 above the molecular plane and show π currents only.

chemical systems, there is no obvious way to vary electronegativity continuously, but there is an established way to move smoothly between isoelectronic and isostructural molecules, and that is to create a continuum of hypothetical systems differing by variation of nuclear charge. This idea has its roots in the united-atom method, a model for the short-range limits of energy and property functions, which dates back to the early days of quantum chemistry,20 and is still used.21,22 The nature of the overall “transition” can be visualized from current-density difference maps. Figure 1 compares the currentdensity maps of the two physical systems, benzene and borazine. The anticlockwise circulation in Figure 1a indicates the diatropic π current supported by a benzene molecule subjected to a perpendicular external magnetic field. This current is absent in Figure 1b, which instead shows only the nitrogen-localized π circulations of borazine. Figure 1c is the simple difference map, which corresponds to the π current that would need to be added to the aromatic benzene system to render it nonaromatic; not surprisingly, this difference current is strongly paratropic, with the D3h symmetry of the common subgroup, and approximates to a reversal of the global π current. A similar approach can be applied to the loss of antiaromaticity in the 8-cycle. This is illustrated by the progression from D4h cyclooctatetraene (an antiaromatic transition state23) to its nonaromatic BN analogue (borazocine, B4N4H8).6 Figure 2a shows the large paratropic (clockwise) current in planarconstrained cyclooctatetraene (COT), and Figure 2b shows the nitrogen-centered localized currents of borazocine. The difference map shows the large diatropic current that should be added

10.1021/jp106697j  2010 American Chemical Society Published on Web 09/09/2010

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J. Phys. Chem. A, Vol. 114, No. 39, 2010 10743 TABLE 1: Bond Lengths and Bond Angles of the Optimized Geometries for Benzene (Z ) 6.0), Borazine (Z ) 7.0), and the Nine Intermediate Systems bond anglesa

bond lengths (Å)

Figure 2. Current-density maps, calculated at the CTOCD-DZ/CHF/ 6-31G**//RHF/6-31G** level, for (a) planar-constrained cyclooctatetraene, (b) borazocine, and (c) the difference between panels b and a. The maps are plotted at a height of 1a0 above the molecular plane and show π currents only.

Figure 3. Schematic representation of the hypothetical progression from benzene to borazine. The arrows represent movements of fractions of a proton.

to the paratropic current of COT to turn it into the localized current map of B4N4H8. In the present study, the progression from carbocycle to azabora-heterocycle is modeled in discrete steps, each involving concerted movements of fractions of a proton between adjacent heavy-atoms sites, as indicated in Figure 3. By plotting currentdensity maps for the hypothetical intermediate systems with nonintegral nuclear charges, it will be possible to answer qualitative questions about the thresholds of aromaticity and antiaromaticitiy, and the way in which the two types of progression differ.

Z

“N-B”

“N”sH

“B”sH

“B-N-B”

6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0

1.386 1.386 1.388 1.390 1.393 1.396 1.401 1.406 1.411 1.418 1.425

1.076 1.066 1.057 1.047 1.039 1.030 1.022 1.015 1.007 1.001 0.994

1.076 1.086 1.096 1.107 1.118 1.129 1.141 1.153 1.166 1.179 1.192

120.00 119.94 119.92 119.96 120.08 120.27 120.64 121.04 121.47 121.93 122.47

a The internal angles of the rings are related by “B-N-B” + “N-B-N” ) 240°.

map follow. A combination of GAMESS-UK24 and SYSMO27 programs was used to calculate the maps. For all systems, the current density was plotted in a plane 1a0 above the nuclei, including contributions from the π orbitals only. In the maps, an anticlockwise (clockwise) circulation denotes a diatropic (paratropic) current. Nuclear positions projected into the plotting plane are denoted by the symbols: . - hydrogen; X - boron (any nucleus with 5.0 e Z < 6.0); b - carbon (Z ) 6.0); and # - nitrogen (any nucleus with 6.0 < Z e 7.0). To quantify the intensity of currents within the maps, values of jmax, the maximum modulus of the induced current density per unit external magnetic field taken over the plotting plane are given in the text. The standard value of jmax for benzene at a height of 1a0 is 0.080 au at the RHF/6-31G** level.

Method To investigate the two progressions from benzene to borazine and COT to borazocine, geometries for the carbon and BN systems were first optimized at the RHF/6-31G** level using GAMESS-UK.24 For nine intermediate systems in each case, where the difference in nuclear charge between adjacent atoms increased by 0.2 with each step, an adaptive basis set was used for the atoms with nonintegral nuclear charges. Geometries of these molecular systems were also optimized using GAMESSUK at the RHF level and minima were checked by diagonalization of the Hessian, noting that the number of imaginary harmonic frequencies is a signature of the potential surface, and hence independent of the nuclear masses. To create the adapted 6-31G** basis, exponents of the standard 6-31G** basis for boron, nitrogen and carbon25,26 were interpolated with a second degree polynomial in nuclear charge (Z) (see Supporting Information). The contraction coefficients show very little variation between B, C, and N, and the adapted basis set simply retains those of the carbon 6-31G** basis. The results are in fact not sensitive to the precise specification of the basis set: current maps plotted as described below with (a) the carbon basis on each heavy atom; (b) nitrogen or boron basis sets on nuclei with charges above and below 6.0, respectively; or (c) the adaptive basis set are all visually indistinguishable. Current-density maps were calculated using the ipsocentric formulation14-16 at the CTOCD-DZ/CHF/6-31G** level, using the adaptive basis sets. This choice of origin for the currentdensity calculations leads to well-converged maps with relativity small basis sets and allows partition into nonredundant orbital contributions from which physical interpretations of the current

Results and Discussion Benzene to Borazine. Details of the optimized geometries of the 11 systems in the progression are given in Table 1. Benzene has a D6h structure with equal bond lengths and bond angles. On shifting the nuclear charge, the symmetry of the system falls to D3h, a symmetry shared by all the intermediate hypothetical systems and by borazine itself. In the tables, nuclei with increasing positive charge are given the symbol “N”, and those with decreasing nuclear charge are given the symbol “B”. Each system is identified by Z, the highest nuclear charge present: hence, benzene has Z ) 6.0 and borazine has Z ) 7.0. The systems are equilateral throughout the range, and the bond lengths in the ring change smoothly from the C-C bond ∼1.39 Å to the B-N bond ∼1.43 Å. The “N”sH and “B”sH bond lengths also vary smoothly from the C-H to the N-H and B-H limits. Figure 4 shows the calculated π current-density maps for each of the systems along the Z “reaction coordinate”. The strong diatropic current in benzene diminishes slowly as the difference between adjacent nuclear charges increases. The transition from global to localized current is hard to establish by eye, but it does seem that from ∼Z ) 6.8, the circulation on the nuclei with highest partial charge begins to outweigh the global current. Another way to approach a precise description of the transition from aromaticity to nonaromaticity is to compare current density with what would be expected by simple averaging. A series of maps was constructed by linear interpolation between the π current-density maps of benzene and borazine. Comparing these interpolated maps with the selfconsistently calculated maps of Figure 4 reveals remarkably little

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Figure 4. Current-density maps of benzene (6.0), borazine (7.0), and the intermediate steps, calculated using the adaptive basis set and ipsocentric method. Each map is labeled with the largest Z value present. Anticlockwise sets of arrows correspond to diatropic currents.

difference across the whole range of Z. The striking similarity between the two series is illustrated precisely in the third column of Figure 5, where the interpolated map is subtracted from the calculated map to show the near-perfect cancellation. The progression from aromatic to nonaromatic molecule is evidently linear. In terms of a landscape analogy, this analysis implies that benzene sits atop a conical summit, with a linear decline of aromaticity along the slope down to the BN nonaromatic limit. (See the schematic picture, Figure S1, in the Supporting Information). This is consistent with conclusions based on electronegativity models and confirms a picture in which the aromaticity of 6-membered heterocyclic rings is controlled by electronegativity differences.6,7 In the ipsocentric approach, the characteristic diatropic current of an aromatic 6π system arises from a virtual excitation between HOMO and LUMO pairs that differ by one unit of angular momentum and are separated by an appreciable energy gap. The loss of the ring current of the benzene system then comes about as a result of an increasing gap, accompanied by a relaxation of the symmetry-based selection rules which allows the virtual excitation to assume

mixed diatropic/paratropic current. This is consistent with the gradual nature of the progression found in the present calculations. Further evidence for a smooth progression comes from NICS(0) values.28,29 Negative NICS values are generally taken as an indicator of aromaticity, whereas values close to zero are taken to show nonaromaticity. Figure 6 shows that the relation of NICS to nuclear charge is a flat s-curve without abrupt change between aromatic and nonaromatic regimes. Here, NICS values are calculated at the CHF/CTOCD-PZ230 level using the adaptive basis set. Cyclooctatetraene to Borazocine. In the early part of this progression, the nuclei had to be constrained to the plane during the optimization. Equilateral planar D8h-symmetric COT has a Jahn-Teller instability, and the best planar structure belongs to the D4h subgroup, but is still a transition state with one imaginary-frequency mode for out-of-plane distortion. This transition state displays the classic paratropic current17 of the archetypal antiaromatic system. In reality, the free COT molecule “escapes from antiaromaticity”31 by adopting a tubshaped conformation that then displays bond-localized π-like

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Figure 5. The current-density maps calculated fully self-consistently using the adaptive basis set (left), and plots created by linear interpolation of the end-point maps (middle), for Z ) 6.2, 6.5, and 6.8 in the benzene/borazine progression. The ∆ map is the difference between the two (left column minus middle column).

Figure 6. NICS(0) values calculated at the CHF/CTOCD-PZ230 level using the adaptive basis set, plotted against Z, the charge of the most positive nuclei in each system, for the benzene/borazine progression.

currents.32 As the nuclear-charge redistribution proceeds from COT to borazocine, the intermediate planar structures conserve only C4h symmetry. The planar structure first becomes optimal (the number of imaginary frequencies drops to zero) when the charges are close to 6.3 and 5.7. The molecules at the end-points of this progression both have (different settings of) D4h symmetry, but there are fundamental

differences in geometry between the two structures. As mentioned above, when constrained to the plane COT displays bond alternation resulting from Jahn-Teller distortion. In contrast, B4N4H8 has equal bond lengths but shows alternating bond angles. Table 2 details the changes in bond lengths and bond angles along the progression. The degree of bond length alternation drops early in the series, and by Z ) 6.4 is negligible.

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TABLE 2: Bond Lengths and Bond Angles of the Optimized Geometries for Cyclooctatetraene (Z ) 6.0), Borazocine (Z ) 7.0), and the Nine Intermediate Systemsa bond lengths (Å)

bond angles

Z

“N-B”

“B-N”

“N”sH

“B”sH

“B-N-B”

“N-B-N”

“B-N”-Hb

“N-B”-Hb

6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0

1.326 1.330 1.344 1.384 1.398 1.401 1.405 1.408 1.415 1.420 1.427

1.480 1.475 1.457 1.409 1.399 1.401 1.405 1.409 1.415 1.420 1.427

1.078 1.067 1.057 1.048 1.040 1.032 1.025 1.018 1.011 1.004 0.998

1.078 1.089 1.099 1.110 1.122 1.133 1.145 1.158 1.171 1.184 1.198

135.0 134.9 134.6 134.4 135.1 136.0 136.9 137.7 138.6 139.4 140.3

135.0 135.2 135.4 135.6 134.9 134.0 133.1 132.3 131.4 130.6 129.7

113.9 111.4 111.8 112.6 112.5 112.1 111.7 111.3 110.9 110.5 109.8

113.9 110.8 111.1 111.9 112.5 112.9 113.3 113.8 114.2 114.7 115.2

a “N-B” and “B-N” represent adjacent bonds. H-N-B-H segment.

b

Angles “B-N”-H and “N-B”-H are external to the ring and inside the same

Figure 7. The current-density maps of cyclooctatetraene (Z ) 6.0), borazocine (Z ) 7.0), and the intermediate steps, calculated using the adaptive basis set. Each map is labeled with the largest Z value present. Clockwise sets of arrows correspond to paratropic currents.

The N-H and B-H distances are comparable to those in benzene/borazine series. The dispersion in ring bond angles does not increase significantly until Z ) 6.5, and then angle alternation increases quickly. Figure 7 shows the current-density maps for the COT-toborazocine progression. The global paratropic current of COT

is seen to remain more or less constant at the beginning of the range, and then diminishes sharply for Z ∼ 6.4 onward. By Z ∼ 6.7, the current has lost its global character, and beyond this value the systems are clearly nonaromatic. Interestingly, this point at which the paratropic current begins to disappear from the current-density maps of Figure 7 is

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Figure 8. The current-density maps calculated fully self-consistently using the adaptive basis set (left), and plots created by linear interpolation of the end-point maps (middle), for Z ) 6.2, 6.5, and 6.8 in the cyclooctatetraene/borazocine progression. The ∆ map is the difference between the two (left column minus middle column).

close to where the planar structure first loses its imaginary frequency (Z ) 6.3). The nonlinearity of the change in the maps is borne out by values of the maximum current jmax, which lie between 0.13 and 0.14 au for 6.0 e Z e 6.3 and then drop sharply to 0.10 au for Z ) 6.4, 0.07 for Z ) 6.5, and 0.06 for Z ) 6.6. The sequence of interpolated maps (Figure S2, Supporting Information) shows marked differences from the self-consistently calculated maps of Figure 7. The interpolated maps show that, if the progression were linear, a uniform drop in jmax of ∼0.01 au should apply between consecutive steps. These differences are highlighted in Figure 8. At Z ) 6.2, the self-consistently calculated map shows a strong paratropic current than does the one calculated by averaging. By Z ) 6.5, the decay in current is ahead of the linear interpolation and remains ahead at Z ) 6.8. Subtraction of interpolated from calculated map emphasizes the trend: for Z ) 6.2, the weak but significant paratropic current in the difference map shows that the current has not fallen linearly; for Z ) 6.5 and 6.8, the diatropic current in the difference maps indicates the accelerated rate of fall. Further evidence for a nonlinear progression between COT and borazocine comes from NICS(0) calculations. Unlike the flat s-curve of benzene-to-borazine, the curve in Figure 9 shows that large positive NICS values for COT linger and then decline suddenly toward the small nonaromatic value at the borazocine end-point. Thus, all calculations on the 8-membered rings tell a consistent story. Results of the geometry optimization, currentdensity maps of the stepwise progression, subsequent comparisons with averaged maps, calculation of the Hessian and values of jmax and NICS all suggest that the paratropic current of COT

is initially sustained and then decays quickly. In contrast to aromatic benzene, which the geographical metaphor places at the summit of a conical hill in a nonaromatic landscape, antiaromatic COT can be imagined to occupy a flat-bottomed hole with relatively steep sides (see Figure S1, Supporting Information for a pictorial representation). This quantitative difference between the aromatic-to-nonaromatic and antiaromatic-to-nonaromatic progression can be illustrated by examining the currents arising from individual molecular orbitals. The ipsocentric model allows current density to be composed into nonredundant contributions from individual molecular orbitals.14 For delocalized π systems it is appropriate to use canonical molecular orbitals in this analysis.33 The dominant paratropic current of an antiaromatic 4n-cyclic system arises from the virtual excitation between the nondegenerate HOMO and LUMO. There is an additional underlying diatropic current arising from the HOMO-1 pair.14,17 This is governed by the angular-momentum node-counting rules derived from the ipsocentric model of currents. In a 4n π system, the HOMO-LUMO transition is node conserving and gives rise to a paratropic current, whereas the (HOMO - 1)-LUMO transition is node-increasing and gives rise to diatropic current, with the reduced intensity corresponding to the increased orbital energy gap. The different roles of the HOMO and HOMO - 1 are based on an angular-momentum classification that is well-defined only in situations where the electrons are essentially delocalized.33 As electronegativity difference increases, the π electrons become more localized, and the full pattern of current in borazine and borazocine requires inclusion of the complete set of π orbitals in the plot. Discussion in terms of canonical orbital contributions becomes increasingly incomplete in this range. Plots of separate

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Figure 9. NICS(0) values calculated at the CHF/CTOCD-PZ230 level using the adaptive basis set, plotted against Z, the charge of the most positive nuclei in each system, for the cyclooctatetraene/borazocine progression.

Figure 10. A comparison of patterns of variation along the COT-borazocine progression for (i) HOMO-LUMO gaps, (ii) degree of bond alternation, and (iii) maximum induced current density. The ordinate for (i) represents the difference from the gap in COT (Z ) 6.0) itself (in Hartree). For (ii) the ordinate is the difference in length between adjacent bonds (in Å). For (iii), the jmax values (in au) are taken from the plots of total π current density in Figure 7.

canonical orbital contributions will give a good indication of antiaromatic and aromatic behavior at the beginning of the progression, but toward the end of the range provide only a partial picture of the localized currents. With this proviso, it is possible to follow the contribution of the π-HOMO to the total π ring current throughout the progression. Figure S3, Supporting Information, shows this contribution is dominant in the early paratropic regime and is also responsible for the nonlinearity. The π-HOMO contribution has similar strength from Z ) 6.0 to 6.3 (jmax ranges from 0.158 to 0.163), but from Z ) 6.4 begins to weaken (jmax ) 0.117), although not as sharply as in Figure 7. Nonlinearity is again confirmed by comparison of Figure S3 with the sequence of average maps for this single-orbital contribution (see Figure S4, Supporting Information). The contribution of the degenerate

π-HOMO - 1 pair opposes the main current, but is weaker and essentially linear (see Figures S5 and S6, Supporting Information), thereby behaving in a similar way to the characteristic diatropic current in the benzene progression. A rationalization for the nonlinear behavior of the COT progression can be obtained from considering relative orbital energies. Under the ipsocentric model, the intensity of the current in a sum-over-states approximation depends on a series of terms with an operator matrix element in the numerator and an orbital energy difference in the denominator.14 For paratropic currents, the energy denominator in the dominant HOMOLUMO term is generally small as the energy difference typically arises from a Jahn-Teller splitting; hence paratropic currents are especially sensitive to changes in the HOMO-LUMO gap. Figure 10 shows how the HOMO-LUMO gap for the COT/

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borazocine progression varies as Z increases: it remains nearconstant in the range 6.0 < Z < 6.3, but then grows linearly. The region of constant HOMO-LUMO gap coincides strikingly well with the plateau in maximum current density. It is also noticeable from the figure that in the region where gap and current are constant, the bond alternation is changing most rapidly. A Hu¨ckel-type model gives a qualitative account of the gap energy and hence the current as a function of Z. In this model, orbital energies are taken to be simultaneous functions of electronegativity and bond-alternation parameters. In the spirit of π distortivity treatments,34,35 the Hu¨ckel resonance integrals, β, on alternate bonds, are changed to β(1 ( δ). To account for electronegativity variation, the Hu¨ckel R parameter is modified to R ( ηβ on alternate centers.7 For the 8-membered ring, the Hu¨ckel HOMO and LUMO energies become:

εHOMO ) R + √4δ2 + η2β; εLUMO ) R - √4δ2 + η2β (1) In the progression from equiangular COT to equilateral borazocine, δ starts from a nonzero value at Z ) 6.0 and decreases (see Table 2). Assuming that electronegativity follows nuclear charge, η begins from zero at COT and increases to 1 at B4N4H8.7 In the region of Z where δ and η are both nonzero, from eq 1 they will have opposing effects on the gap and hence the current. Once δ has fallen to zero, the gap becomes linear in η, and the current density can be expected to drop sharply as ∼η-1. In the benzene-to-borazine case, bond alternation is absent and the Hu¨ckel HOMO-LUMO gap varies with η as:7

∆HOMO/LUMO ) 2√1 + η2β

(2)

over the whole range, giving no abrupt change in the denominators, as borne out by explicit calculations. (In fact, the whole variation in the RHF HOMO-LUMO gap for this progression is only 0.1 hartree). This model may be highly simplified, but it accounts for the essential difference between the antiaromatic and aromatic progressions. Conclusion By introducing hypothetical stepwise redistributions of nuclear charge, and following the differences in current-density maps, it is possible to link isoelectronic aromatic (or antiaromatic) and nonaromatic systems by a continuous pathway. This model exposes a significant difference between the quenching of aromaticity and antiaromaticity. The diatropic ring current in benzene survives most of the progression to nonaromatic borazine, and the change from global to localized currents has a linear character. In contrast, the paratropic current in antiaromatic COT remains strong initially, and then shows a sharp decline to the localized regime. On the orbital model that arises naturally from the ipsocentric approach to calculation of currents, this distinction is easily rationalized. This modeling is relevant to the understanding of the variation in (magnetic) aromaticity of real chemical systems where aromaticity of the 6π cycle is found to survive considerable electronegativity differences.6 Acknowledgment. P. W. F. thanks the Royal Society/ Wolfson Scheme for a Research Merit Award, and D. E. B. thanks ESPRC and The University of Sheffield for financial support.

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