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Aromaticity, polarisability and ring current
Chemical Physics Letters 383 (2004) 507–511 www.elsevier.com/locate/cplett
Aromaticity, polarisability and ring current P.W. Fowler *, A. Soncini Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK Received 27 October 2003 Published online:
Abstract Electric dipole polarisability has been proposed as an alternative to magnetic criteria of aromaticity. The ipsocentric formulation of the molecular response to a magnetic field gives selection rules for diatropic ring current that coincide with those for in-plane polarisability. Thus, induced p current strength might correlate, if at all, with HOMO contributions to the in-plane components of polarisability of diatropic systems. No equivalent correlation is expected a priori for anti-aromatic paratropic systems, or clamped systems in which diatropic current is quenched by interaction with p systems. Ab initio calculations of p current density maps and dipole polarisability contributions confirm the expected correlation for monocycles, but show the limitations of polarisability as a measure of aromaticity. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Aromaticity has many definitions [1,2], but one that is widely used is the ability of a cyclic system to support a diatropic ring current in the presence of a perpendicular external magnetic field [3]. Ring current is inferred from exaltation of diamagnetic susceptibility [4] and downfield chemical shifts of exo-hydrogen nuclei [5], and can also be calculated ab initio from linear response of the wavefunction [6–8]. The intuitive connection between induced circulation of charge and static cyclic delocalisation has suggested a linkage of aromaticity with other delocalisation dependent properties such as electric polarisability and its anisotropy [9]. Statistical analyses [10] ascribe mixed ÔmagneticÕ and ÔclassicalÕ character to a polarisabilitybased descriptor of aromaticity [11]. Various calculations impute an exaltation of electric polarisability to aromatic molecules in comparison with saturated systems [11–13] and a number of measures of this effect have been proposed, as reviewed in [14]. Objections can immediately be raised that the delocalisation affecting polarisability is linear and that high polarisability can *
Corresponding author. Fax: +44-1392-263-434. E-mail address: [email protected] (P.W. Fowler).
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.11.081
arise in acyclic, even monoatomic, species, but the polarisability/aromaticity link is rendered more plausible, at least for a restricted range of applicability, by recent reformulations of magnetic response theory [8], as will be shown in the following section.
2. Perturbation theory The basis of our discussion of the putative link between polarisability and current density lies in the similar mathematical forms of the respective first-order wavefunctions for response to electric and magnetic fields (see Table 1). 2.1. Polarisability Electric dipole polarisability aab is a tensor quantity expressing the quadratic part of the dependence of the energy E and linear dependence of the electric dipole moment la , on the external electric field Fa [15]. With a; b ¼ x; y; z and 0 denoting a derivative at zero field, the components of polarisability are 2 olb oE ola aab ¼ ¼ ¼ : ð1Þ oFa oFb 0 oFb 0 oFa 0
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P.W. Fowler, A. Soncini / Chemical Physics Letters 383 (2004) 507–511
Table 1 Ring current and polarisability quantities computed for the systems 1–10 (Fig. 1) in the ipsocentric/CHF approach with a 6-31G** basis set Molecule 1 2 3 4 5 6 7 8 9 10
C5 H 5 C6 H 6 C7 H þ 7 C8 H 8 C10 H10 C14 H14 C18 H18 C12 H12 C12 H6 C18 H12
jH max
jpmax
aH ?
apH ?
atot ?
aH k
atot k
0.0704 0.0785 0.0814 0.1575 0.1186 0.1479 0.1850 0.0812 0.0275 0.0588
0.0702 0.0784 0.0814 0.1305 0.1205 0.1449 0.1800 0.0793 0.0288 0.0665
25.4 33.8 44.3 22.7 95.9 188.5 320.5 47.2 53.8 114.9
0.2 0.2 0.3 26.6 3.3 10.5 15.9 10.2 40.7 66.1
59.8 70.0 83.8 100.7 157.4 273.6 428.1 135.2 156.8 282.3
8.4 7.9 7.5 4.2 8.1 8.2 8.2 7.8 7.4 7.6
19.5 21.3 23.3 27.9 34.6 47.8 61.5 71.2 38.5 57.1
p jH max and jmax are, respectively, the contributions of the p HOMO and of the full set of p orbitals to the maximum of the projection of the current density in the plane 1a0 above that of the (central) ring.
In terms of the first-order change, Wð1;Fa Þ , induced in the unperturbed wavefunction Wð0Þ by the electric field aab ¼ 2 Wð0Þ jla jWð1;Fb Þ ¼ 2 Wð0Þ jlb jWð1;Fa Þ : ð2Þ In the orbital approximation for a closed shell ground state, the N-electron wavefunction Wð0Þ is a determinant of doubly occupied orbitals wð0Þ n ; n ¼ 1; 2 . . . N =2 and the first-order correction to the wavefunction involves perturbed orbitals calculated by Hartree–Fock theory. Neglecting complications arising from self-consistency, the orbitals wð0Þ n can be taken to be eigenfunctions ð0Þ (with eigenvalues n ) of a one-electron Hamiltonian. The first-order correction to the occupied orbital is then (using the Einstein summation convention) D E wpð0Þ jlb Fb jwnð0Þ X ; wnð1;Fa Þ Fa ¼ wpð0Þ ð3Þ ð0Þ ð0Þ p n p>N =2 in which the sum contains excitations from wnð0Þ to all virtual orbitals, but to no other occupied orbital. Thus, given a set of unperturbed molecular orbitals (canonical or localised), the electronic dipole polarisability is determined by dipole transition moments that, scaled by the orbital energy difference, define the acð0Þ cessibility of unoccupied wð0Þ p from occupied wn . Symmetry-based selection rules apply to these transition moments. 2.2. Current density Turning now to the magnetic field perturbation, and invoking the ipsocentric formulation in which current density at any point r is to be calculated with that point as origin of vector potential, the first-order current density J is found to be a sum of orbital contributions [8] i 2ie h h ð0Þ bÞ jna ðrÞ ¼ wn ðrÞra wnð1;Bb Þ ðrÞ wð1;B ðrÞra wnð0Þ ðrÞ Bb ; n me ð4Þ
where the first-order correction to wnð0Þ in the magnetic field is D E3 2 ð0Þ ð0Þ w jl jw X b p n e 4 aÞ 5 Bb wð1;B Ba ¼ wð0Þ n ð0Þ ð0Þ 2me p>N =2 p p n D E3 2 ð0Þ ð0Þ X ð0Þ wp jpd jwn e 5 Bb ; þ bcd dc 4 wp ð0Þ ð0Þ 2me p n p>N =2 ð5Þ where bcd is the third-rank Levi–Civita anti-symmetric tensor, lb and pd are one-electron operators of angular and linear momentum, respectively, and dc is a vector that will be set equal to rc after the differentiation step in (4). The two excited-state sums in (5) represent globally diamagnetic (with the p perturbation operator) and paramagnetic (with the l perturbation operator) induced current density [8]. A specific feature of the ipsocentric formulation is that the first-order wavefunction (5) contains only occupied to unoccupied excitations and no occupied–occupied mixing terms. 2.3. Symmetry Symmetry-based selection rules apply to the transition moments in (5) just as they do in the case of electric field perturbation in (3). As ring currents are mainly of interest for planar molecules in which the direction of the external magnetic field is perpendicular to the plane of the atoms, we consider here the case when z is the field direction and xy the molecular plane. ð0Þ A transition between orbitals wð0Þ n and wp induced by a field Bz makes a current density contribution with global diatropic character if the symmetry product Cnp ¼ Cðwnð0Þ Þ Cðwð0Þ p Þ; contains the spatial symmetry of an in-plane linear momentum operator. If on the other hand the symmetry product contains the spatial symmetry of the angular
P.W. Fowler, A. Soncini / Chemical Physics Letters 383 (2004) 507–511
momentum component lz , then the current density contribution has a global paratropic sense. Thus, the selection rule for diatropic (diamagnetic) ring current reduces to a requirement on Cnp that its product with the symmetry of the in-plane translation Cnp CT ðx; yÞ; should contain C0 , where C0 is the totally symmetric representation in the point group of the unperturbed molecule. This is also exactly the condition to be fulfilled for a transition to make a contribution to the in-plane components of the electric dipole polarisability, axx and ayy , since the symmetry of the dipole operator la is that of a translation along the a direction. The plausibility of the link between J and a depends on this coincidence of symmetries. In the quoted forms (3) and (5), the two first-order wavefunctions involve different, though equisymmetric, operators r and p. Hypervirial relations allow conversion between length and velocity formalisms and it would be equally possible to re-express the diatropic current in terms of transition moments of r, or the polarisability in terms of transition moments of p, so that the same transition integrals would then appear in both functions, although weighted differently. This symmetry connection is, however, limited to the diatropic part of the current and the in-plane components of the polarisability. The condition for a transition to contribute to the out-of-plane component, azz , is that Cnp should contain CT ðzÞ. The condition for a non-zero contribution to the paratropic current density is that Cnp should contain CR ðzÞ, the representation of a rotation about the normal to the molecular plane. Representations CT ðx; yÞ and CT ðzÞ are distinct for planar molecules, and CR ðzÞ is distinct from CT ðx; yÞ in Cnh and Dnh planar molecules, so that in general there is no connection to be expected between azz and J or between ðaxx ; ayy Þ and the paratropic contribution to the current density. 2.4. Predictions Thus we arrive at the main qualitative conclusion: if the ring current is diatropic and is dominated by a single dipole allowed excitation, then we can expect a correlation between the contributions of this excitation to the in-plane dipole polarisability and to diatropic current. A typical system in which the current is diatropic and dominated by a single (HOMO–LUMO) transition is the ½4n þ 2 monocycle [16]. In the simplest H€ uckel model, in-plane polarisation of the p electrons is also dominated by the same transition (the only dipole-allowed, occupied-virtual, p ! p transition in the valence space). Hence, ½4n þ 2p monocycles are promising test cases for the proposed correlation. On the other hand, ½4np systems in planar geometries show purely paratropic p current that arises es-
509
sentially from the dipole-forbidden HOMO–LUMO transition [16]. Such systems may well be polarisable, but their p polarisability will not arise from the same source as their ring current. It is also known that the ring current of a clamped ½4n þ 2 or ½4n monocycle can be switched on or off according to the nature of the attached groups and their effect on the frontier p orbitals [17]. Such drastic changes to the p electronic structure are likely to frustrate attempts to link a and diatropic J.
3. Calculations Ipsocentric calculations of the current density and coupled Hartree–Fock calculations of electric dipole polarisability were carried out in the 6-31G** basis for ten systems for which current density maps have been published or discussed elsewhere [17–21]. They are: cyclopentadienide anion (1 C5 H 5 , D5h ), benzene (2 C6 H6 , D6h ), tropylium cation (3 C7 Hþ 7 , D7h ), cyclooctatetraene (4 C8 H8 , optimised with a planar constraint leading to D4h symmetry), [10]-annulene (5 C10 H10 , C2v ), [14]-annulene (6 C14 H14 , D2h ), [18]-annulene (7 C18 H18 , D6h ) and the three clamped systems 8–10, all illustrated in Fig. 1. Calculations were carried out at the SCF optimised geometries for 1–4 and 8–10 and at DFT geometries for 5–7 [22] (see Table 1). The results for the three aromatic 6p monocycles and the larger annulenes 5–7 bear out the qualitative prediction for ½4n þ 2p systems. In these molecules the current density in the 1 bohr plane is dominated by the flow of HOMO p electrons (jpmax ’ jH max ). The HOMO contribution aH , which accounts for over 95% of the p ? a? polarisability, rises with jmax (Fig. 2). The r-electron H contribution to in-plane polarisability, (atot ? a? p–H a? ), accounts for a decreasing percentage of the total, from 57% for C5 H 5 to 21% for C18 H18 . Interestingly, the HOMO contribution to ak is more or less independent of current, but the out-of-plane polarisability atot k rises with rising jmax . Over the range of nuclearity 5 to 18, despite a more than tenfold increase in polarisability and drastic changes in symmetry and geometry, the HOMO current shows a smooth rising trend. Results for the anti-aromatic ½4np electron system, planar COT (4), however, immediately show the limitations of this correlation. COT has much larger curp rents jH max and jmax than the ½4n þ 2p monocycles of similar size, but the sense of its current is paratropic, not diatropic. The total polarisability atot ? is much higher than that for C7 Hþ but the HOMO contribution aH ? is 7 similar to that of C5 H5 . The data point for COT lies H well away from the jH max =a? trendline (Fig. 2). Given the different symmetry selection rules, there is of course no reason to expect COT to lie on the correlation line for the diatropic monocycles, and in fact it does not.
510
P.W. Fowler, A. Soncini / Chemical Physics Letters 383 (2004) 507–511
1
2
3
5
urated groups and supports a normal benzene-like diatropic current, close in magnitude to that of C7 Hþ 7. The HOMO electrons in 8 are also about as polarisable as those of the C7 Hþ 7 system. Persistence of current is explained in terms of retention of the benzene HOMO– LUMO orbital symmetries and topologies in the clamped system [17]. On the other hand, the tris(cyclobutadieno)benzene (9) clamped system has an extended p system, leading to disruption of the benzenoid HOMO and LUMO, and a much reduced current density in the central ring [17], even though the new HOMO p orbital has a large value of aH ? . The third clamped system tris(3,4-dimethylenecyclobuteno)benzene (10) has a HOMO current that is 75% of that in benzene, but now the value of aH ? is over three times that in benzene, as the physical extent of the conjugated system that includes the central ring is much greater. The physical reasons for departure from the simple correlation are evident.
4
6
7
4. Conclusions 8
9
10
Fig. 1. Key to chemical structures of systems for which current density and polarisability data are reported in Table 1. The optimised structure for 5 departs in symmetry and shape from this idealised representation [22].
j⊥H
0.20
Diatropic monocycles give a favorable case in which the in-plane polarisability and the ring current are dominated by the same transitions. For these systems, p polarisability and magnetic aromaticity go together, but in general use of polarisability as an index of aromaticity is problematic. The same orbital arguments that justify the linking of properties for diatropic monocycles show why the link should not exist in general.
0.15
References 4 0.10
8 10 0.05
9
α⊥H
0.0 0
60
120
180
240
300
Fig. 2. Scatter plot of HOMO contributions to in-plane polarisability H aH ? and maximum current density in the plotting plane jmax . Filled circles denote data points for [4n + 2] annulenes and crosses denote data points for the other systems, labelled as in Fig. 1 and Table 1. A best fit line for the datapoints 1–3 and 5–7 is drawn as a guide to the eye; it has slope 0.0004 and intercept 0.0678 (in a.u).
Clamped benzene systems 8–10 yield one example (8) that lies on the ½4n þ 2 correlation line and two that do not. The tris(cyclobuteno)benzene molecule 8 has sat-
[1] V.I. Minkin, M.N. Glukhovtsev, B.Y. Simkin, in: . In: Aromaticity and Anti-aromaticity: Electron and Structural Aspects, Wiley, New York, 1994. [2] See for example, the special issue of Chemical Reviews on Aromaticity, P. von R. Schleyer (guest Ed.), Chem. Rev. 101 (2001) 1115. [3] P.V.R. Schleyer, C. Maerker, A. Dransfeld, H. Jiao, N.J.R. van Eikema Hommes, J. Am. Chem. Soc. 118 (1996) 6317. [4] H.J. Dauben, J.D. Wilson, J.L. Laity, in: J.P. Snyder (Ed.), Nonbenzenoid Aromatics, vol. 2, Academic Press, New York, 1971, p. 167. [5] J.A. Pople, J. Chem. Phys. 24 (1956) 1111. [6] T.A. Keith, R.W.F. Bader, Chem. Phys. Lett. 210 (1993) 223. [7] S. Coriani, P. Lazzeretti, M. Malagoli, R. Zanasi, Theor. Chem. Acta 89 (1994) 181. [8] E. Steiner, P.W. Fowler, J. Phys. Chem. A 105 (2001) 9553. [9] P. Lazzeretti, J.A. Tossell, J. Mol. Struct. (Theochem) 234 (1991) 403. [10] A.R. Katritzky, P. Barczynski, G. Musumarra, D. Pisano, M. Szafran, J. Am. Chem. Soc. 111 (1989) 7. [11] S.B. Bulgarevich, V.S. Bolotrikov, V.N. Scheinker, O.A. Osipov, A.D. Garnowskii, J. Org. Chem. USSR 12 (1985) 191. [12] E.F. Archibong, A.J. Thakkar, Mol. Phys. 81 (1994) 557. [13] S. Millefiori, A. Alparone, J. Mol. Struct. (Theochem) 431 (1998) 59.
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F.D. Proft, P. Geerlings, Chem. Rev. 101 (2001) 1451. A.D. Buckingham, Adv. Chem. Phys. 12 (1967) 107. E. Steiner, P.W. Fowler, Chem. Commun. (2001) 2220. A. Soncini, R.W. Havenith, P.W. Fowler, L.W. Jenneskens, E. Steiner, J. Org. Chem. 67 (2002) 4753. [18] E. Steiner, P.W. Fowler, Int. J. Quantum Chem. 60 (1996) 609.
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[19] P.W. Fowler, R.W. Havenith, L.W. Jenneskens, A. Soncini, E. Steiner, Angew. Chem. Int. Ed. 41 (2002) 1558. [20] R.W. Havenith, L.W. Jenneskens, P.W. Fowler, A. Soncini (to be published). [21] A. Soncini, P.W. Fowler, L.W. Jenneskens, Phys. Chem. Chem. Phys. (in press). [22] C.S. Wannere, P.V.R. Schleyer, Org. Lett. 5 (2003) 865.
Aromaticity, polarisability and ring current P.W. Fowler *, A. Soncini Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK Received 27 October 2003 Published online:
Abstract Electric dipole polarisability has been proposed as an alternative to magnetic criteria of aromaticity. The ipsocentric formulation of the molecular response to a magnetic field gives selection rules for diatropic ring current that coincide with those for in-plane polarisability. Thus, induced p current strength might correlate, if at all, with HOMO contributions to the in-plane components of polarisability of diatropic systems. No equivalent correlation is expected a priori for anti-aromatic paratropic systems, or clamped systems in which diatropic current is quenched by interaction with p systems. Ab initio calculations of p current density maps and dipole polarisability contributions confirm the expected correlation for monocycles, but show the limitations of polarisability as a measure of aromaticity. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Aromaticity has many definitions [1,2], but one that is widely used is the ability of a cyclic system to support a diatropic ring current in the presence of a perpendicular external magnetic field [3]. Ring current is inferred from exaltation of diamagnetic susceptibility [4] and downfield chemical shifts of exo-hydrogen nuclei [5], and can also be calculated ab initio from linear response of the wavefunction [6–8]. The intuitive connection between induced circulation of charge and static cyclic delocalisation has suggested a linkage of aromaticity with other delocalisation dependent properties such as electric polarisability and its anisotropy [9]. Statistical analyses [10] ascribe mixed ÔmagneticÕ and ÔclassicalÕ character to a polarisabilitybased descriptor of aromaticity [11]. Various calculations impute an exaltation of electric polarisability to aromatic molecules in comparison with saturated systems [11–13] and a number of measures of this effect have been proposed, as reviewed in [14]. Objections can immediately be raised that the delocalisation affecting polarisability is linear and that high polarisability can *
Corresponding author. Fax: +44-1392-263-434. E-mail address: [email protected] (P.W. Fowler).
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.11.081
arise in acyclic, even monoatomic, species, but the polarisability/aromaticity link is rendered more plausible, at least for a restricted range of applicability, by recent reformulations of magnetic response theory [8], as will be shown in the following section.
2. Perturbation theory The basis of our discussion of the putative link between polarisability and current density lies in the similar mathematical forms of the respective first-order wavefunctions for response to electric and magnetic fields (see Table 1). 2.1. Polarisability Electric dipole polarisability aab is a tensor quantity expressing the quadratic part of the dependence of the energy E and linear dependence of the electric dipole moment la , on the external electric field Fa [15]. With a; b ¼ x; y; z and 0 denoting a derivative at zero field, the components of polarisability are 2 olb oE ola aab ¼ ¼ ¼ : ð1Þ oFa oFb 0 oFb 0 oFa 0
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P.W. Fowler, A. Soncini / Chemical Physics Letters 383 (2004) 507–511
Table 1 Ring current and polarisability quantities computed for the systems 1–10 (Fig. 1) in the ipsocentric/CHF approach with a 6-31G** basis set Molecule 1 2 3 4 5 6 7 8 9 10
C5 H 5 C6 H 6 C7 H þ 7 C8 H 8 C10 H10 C14 H14 C18 H18 C12 H12 C12 H6 C18 H12
jH max
jpmax
aH ?
apH ?
atot ?
aH k
atot k
0.0704 0.0785 0.0814 0.1575 0.1186 0.1479 0.1850 0.0812 0.0275 0.0588
0.0702 0.0784 0.0814 0.1305 0.1205 0.1449 0.1800 0.0793 0.0288 0.0665
25.4 33.8 44.3 22.7 95.9 188.5 320.5 47.2 53.8 114.9
0.2 0.2 0.3 26.6 3.3 10.5 15.9 10.2 40.7 66.1
59.8 70.0 83.8 100.7 157.4 273.6 428.1 135.2 156.8 282.3
8.4 7.9 7.5 4.2 8.1 8.2 8.2 7.8 7.4 7.6
19.5 21.3 23.3 27.9 34.6 47.8 61.5 71.2 38.5 57.1
p jH max and jmax are, respectively, the contributions of the p HOMO and of the full set of p orbitals to the maximum of the projection of the current density in the plane 1a0 above that of the (central) ring.
In terms of the first-order change, Wð1;Fa Þ , induced in the unperturbed wavefunction Wð0Þ by the electric field aab ¼ 2 Wð0Þ jla jWð1;Fb Þ ¼ 2 Wð0Þ jlb jWð1;Fa Þ : ð2Þ In the orbital approximation for a closed shell ground state, the N-electron wavefunction Wð0Þ is a determinant of doubly occupied orbitals wð0Þ n ; n ¼ 1; 2 . . . N =2 and the first-order correction to the wavefunction involves perturbed orbitals calculated by Hartree–Fock theory. Neglecting complications arising from self-consistency, the orbitals wð0Þ n can be taken to be eigenfunctions ð0Þ (with eigenvalues n ) of a one-electron Hamiltonian. The first-order correction to the occupied orbital is then (using the Einstein summation convention) D E wpð0Þ jlb Fb jwnð0Þ X ; wnð1;Fa Þ Fa ¼ wpð0Þ ð3Þ ð0Þ ð0Þ p n p>N =2 in which the sum contains excitations from wnð0Þ to all virtual orbitals, but to no other occupied orbital. Thus, given a set of unperturbed molecular orbitals (canonical or localised), the electronic dipole polarisability is determined by dipole transition moments that, scaled by the orbital energy difference, define the acð0Þ cessibility of unoccupied wð0Þ p from occupied wn . Symmetry-based selection rules apply to these transition moments. 2.2. Current density Turning now to the magnetic field perturbation, and invoking the ipsocentric formulation in which current density at any point r is to be calculated with that point as origin of vector potential, the first-order current density J is found to be a sum of orbital contributions [8] i 2ie h h ð0Þ bÞ jna ðrÞ ¼ wn ðrÞra wnð1;Bb Þ ðrÞ wð1;B ðrÞra wnð0Þ ðrÞ Bb ; n me ð4Þ
where the first-order correction to wnð0Þ in the magnetic field is D E3 2 ð0Þ ð0Þ w jl jw X b p n e 4 aÞ 5 Bb wð1;B Ba ¼ wð0Þ n ð0Þ ð0Þ 2me p>N =2 p p n D E3 2 ð0Þ ð0Þ X ð0Þ wp jpd jwn e 5 Bb ; þ bcd dc 4 wp ð0Þ ð0Þ 2me p n p>N =2 ð5Þ where bcd is the third-rank Levi–Civita anti-symmetric tensor, lb and pd are one-electron operators of angular and linear momentum, respectively, and dc is a vector that will be set equal to rc after the differentiation step in (4). The two excited-state sums in (5) represent globally diamagnetic (with the p perturbation operator) and paramagnetic (with the l perturbation operator) induced current density [8]. A specific feature of the ipsocentric formulation is that the first-order wavefunction (5) contains only occupied to unoccupied excitations and no occupied–occupied mixing terms. 2.3. Symmetry Symmetry-based selection rules apply to the transition moments in (5) just as they do in the case of electric field perturbation in (3). As ring currents are mainly of interest for planar molecules in which the direction of the external magnetic field is perpendicular to the plane of the atoms, we consider here the case when z is the field direction and xy the molecular plane. ð0Þ A transition between orbitals wð0Þ n and wp induced by a field Bz makes a current density contribution with global diatropic character if the symmetry product Cnp ¼ Cðwnð0Þ Þ Cðwð0Þ p Þ; contains the spatial symmetry of an in-plane linear momentum operator. If on the other hand the symmetry product contains the spatial symmetry of the angular
P.W. Fowler, A. Soncini / Chemical Physics Letters 383 (2004) 507–511
momentum component lz , then the current density contribution has a global paratropic sense. Thus, the selection rule for diatropic (diamagnetic) ring current reduces to a requirement on Cnp that its product with the symmetry of the in-plane translation Cnp CT ðx; yÞ; should contain C0 , where C0 is the totally symmetric representation in the point group of the unperturbed molecule. This is also exactly the condition to be fulfilled for a transition to make a contribution to the in-plane components of the electric dipole polarisability, axx and ayy , since the symmetry of the dipole operator la is that of a translation along the a direction. The plausibility of the link between J and a depends on this coincidence of symmetries. In the quoted forms (3) and (5), the two first-order wavefunctions involve different, though equisymmetric, operators r and p. Hypervirial relations allow conversion between length and velocity formalisms and it would be equally possible to re-express the diatropic current in terms of transition moments of r, or the polarisability in terms of transition moments of p, so that the same transition integrals would then appear in both functions, although weighted differently. This symmetry connection is, however, limited to the diatropic part of the current and the in-plane components of the polarisability. The condition for a transition to contribute to the out-of-plane component, azz , is that Cnp should contain CT ðzÞ. The condition for a non-zero contribution to the paratropic current density is that Cnp should contain CR ðzÞ, the representation of a rotation about the normal to the molecular plane. Representations CT ðx; yÞ and CT ðzÞ are distinct for planar molecules, and CR ðzÞ is distinct from CT ðx; yÞ in Cnh and Dnh planar molecules, so that in general there is no connection to be expected between azz and J or between ðaxx ; ayy Þ and the paratropic contribution to the current density. 2.4. Predictions Thus we arrive at the main qualitative conclusion: if the ring current is diatropic and is dominated by a single dipole allowed excitation, then we can expect a correlation between the contributions of this excitation to the in-plane dipole polarisability and to diatropic current. A typical system in which the current is diatropic and dominated by a single (HOMO–LUMO) transition is the ½4n þ 2 monocycle [16]. In the simplest H€ uckel model, in-plane polarisation of the p electrons is also dominated by the same transition (the only dipole-allowed, occupied-virtual, p ! p transition in the valence space). Hence, ½4n þ 2p monocycles are promising test cases for the proposed correlation. On the other hand, ½4np systems in planar geometries show purely paratropic p current that arises es-
509
sentially from the dipole-forbidden HOMO–LUMO transition [16]. Such systems may well be polarisable, but their p polarisability will not arise from the same source as their ring current. It is also known that the ring current of a clamped ½4n þ 2 or ½4n monocycle can be switched on or off according to the nature of the attached groups and their effect on the frontier p orbitals [17]. Such drastic changes to the p electronic structure are likely to frustrate attempts to link a and diatropic J.
3. Calculations Ipsocentric calculations of the current density and coupled Hartree–Fock calculations of electric dipole polarisability were carried out in the 6-31G** basis for ten systems for which current density maps have been published or discussed elsewhere [17–21]. They are: cyclopentadienide anion (1 C5 H 5 , D5h ), benzene (2 C6 H6 , D6h ), tropylium cation (3 C7 Hþ 7 , D7h ), cyclooctatetraene (4 C8 H8 , optimised with a planar constraint leading to D4h symmetry), [10]-annulene (5 C10 H10 , C2v ), [14]-annulene (6 C14 H14 , D2h ), [18]-annulene (7 C18 H18 , D6h ) and the three clamped systems 8–10, all illustrated in Fig. 1. Calculations were carried out at the SCF optimised geometries for 1–4 and 8–10 and at DFT geometries for 5–7 [22] (see Table 1). The results for the three aromatic 6p monocycles and the larger annulenes 5–7 bear out the qualitative prediction for ½4n þ 2p systems. In these molecules the current density in the 1 bohr plane is dominated by the flow of HOMO p electrons (jpmax ’ jH max ). The HOMO contribution aH , which accounts for over 95% of the p ? a? polarisability, rises with jmax (Fig. 2). The r-electron H contribution to in-plane polarisability, (atot ? a? p–H a? ), accounts for a decreasing percentage of the total, from 57% for C5 H 5 to 21% for C18 H18 . Interestingly, the HOMO contribution to ak is more or less independent of current, but the out-of-plane polarisability atot k rises with rising jmax . Over the range of nuclearity 5 to 18, despite a more than tenfold increase in polarisability and drastic changes in symmetry and geometry, the HOMO current shows a smooth rising trend. Results for the anti-aromatic ½4np electron system, planar COT (4), however, immediately show the limitations of this correlation. COT has much larger curp rents jH max and jmax than the ½4n þ 2p monocycles of similar size, but the sense of its current is paratropic, not diatropic. The total polarisability atot ? is much higher than that for C7 Hþ but the HOMO contribution aH ? is 7 similar to that of C5 H5 . The data point for COT lies H well away from the jH max =a? trendline (Fig. 2). Given the different symmetry selection rules, there is of course no reason to expect COT to lie on the correlation line for the diatropic monocycles, and in fact it does not.
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1
2
3
5
urated groups and supports a normal benzene-like diatropic current, close in magnitude to that of C7 Hþ 7. The HOMO electrons in 8 are also about as polarisable as those of the C7 Hþ 7 system. Persistence of current is explained in terms of retention of the benzene HOMO– LUMO orbital symmetries and topologies in the clamped system [17]. On the other hand, the tris(cyclobutadieno)benzene (9) clamped system has an extended p system, leading to disruption of the benzenoid HOMO and LUMO, and a much reduced current density in the central ring [17], even though the new HOMO p orbital has a large value of aH ? . The third clamped system tris(3,4-dimethylenecyclobuteno)benzene (10) has a HOMO current that is 75% of that in benzene, but now the value of aH ? is over three times that in benzene, as the physical extent of the conjugated system that includes the central ring is much greater. The physical reasons for departure from the simple correlation are evident.
4
6
7
4. Conclusions 8
9
10
Fig. 1. Key to chemical structures of systems for which current density and polarisability data are reported in Table 1. The optimised structure for 5 departs in symmetry and shape from this idealised representation [22].
j⊥H
0.20
Diatropic monocycles give a favorable case in which the in-plane polarisability and the ring current are dominated by the same transitions. For these systems, p polarisability and magnetic aromaticity go together, but in general use of polarisability as an index of aromaticity is problematic. The same orbital arguments that justify the linking of properties for diatropic monocycles show why the link should not exist in general.
0.15
References 4 0.10
8 10 0.05
9
α⊥H
0.0 0
60
120
180
240
300
Fig. 2. Scatter plot of HOMO contributions to in-plane polarisability H aH ? and maximum current density in the plotting plane jmax . Filled circles denote data points for [4n + 2] annulenes and crosses denote data points for the other systems, labelled as in Fig. 1 and Table 1. A best fit line for the datapoints 1–3 and 5–7 is drawn as a guide to the eye; it has slope 0.0004 and intercept 0.0678 (in a.u).
Clamped benzene systems 8–10 yield one example (8) that lies on the ½4n þ 2 correlation line and two that do not. The tris(cyclobuteno)benzene molecule 8 has sat-
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[19] P.W. Fowler, R.W. Havenith, L.W. Jenneskens, A. Soncini, E. Steiner, Angew. Chem. Int. Ed. 41 (2002) 1558. [20] R.W. Havenith, L.W. Jenneskens, P.W. Fowler, A. Soncini (to be published). [21] A. Soncini, P.W. Fowler, L.W. Jenneskens, Phys. Chem. Chem. Phys. (in press). [22] C.S. Wannere, P.V.R. Schleyer, Org. Lett. 5 (2003) 865.