MAGNETIC RESONANCE IN CHEMISTRY Magn. Reson. Chem. 2004; 42: S68–S78 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mrc.1445 Current density, chemical shifts and aromaticity† P. W. Fowler,1∗ E. Steiner,1 R. W. A. Havenith2 and L. W. Jenneskens3 1 Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK 2 Debye Institute, Theoretical Chemistry Group, Utrecht University, Padualaan 14, 3584 CH Utrecht, The Netherlands 3 Debye Institute, Department of Physical Organic Chemistry, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands Received 6 April 2004; Revised 14 April 2004; Accepted 14 April 2004 The intimate link between chemical shifts and magnetic criteria for aromaticity prompts a search for detailed understanding of patterns of current density induced in p systems by external magnetic fields. Conceptual and practical advantages of calculation of current densities with a specific method of distribution of origin of vector potential, the ipsocentric choice, where the induced current density at each point is calculated with that point as origin, are outlined. This choice leads uniquely to canonical molecular orbital contributions that are free of unphysical occupied–occupied mixing. Characteristic magnetic response of delocalized p systems is then effectively restricted to the activity of a small number of frontier electrons, governed by simple symmetry and node-counting rules, and readily visualized in current density maps. Localized orbitals for s systems can also be used, again eliminating occupied–occupied mixing. For integrated properties (magnetizability and nuclear shieldings), the ipsocentric method gives, in a well-defined sense, the orbital contributions that are best for purposes of interpretation. The general theory is illustrated by maps for a set of annelated pentalenes; the known benzopentalene (1) and 1,2 : 4,5-dibenzopentalene (2), the still unknown isomer of 2, 1,2 : 5,6-dibenzopentalene (3), and cyclopent[b,c]acenaphthylene (4), an unknown isomer of pyracylene, all of which consist of fusions of formally aromatic and anti-aromatic p-conjugated systems. Copyright 2004 John Wiley & Sons, Ltd. KEYWORDS: NMR; current density; chemical shifts; anisotropy INTRODUCTION near a ring centre as a numerical indicator of the existence, strength and sense of the current and hence a measure of As with charge density in the study of electric properties, aromaticity. current density is central to the understanding of molecular A more direct approach to ring-current aromaticity, the magnetic properties and underpins the theory and practice inspection of the full spatial variation of current density of NMR spectroscopy. Although the mapping of charge induced in an unsaturated planar system by a perpen- density in molecules has a long history, it is only recently dicular magnetic field, was for a long time blocked by that the construction of maps of induced current density the problem of gauge dependence in computed magnetic has become a standard theoretical tool. This seems all the properties. Although the decomposition of properties such more surprising when it is recalled that the existence of as magnetizability and nuclear magnetic shielding into a specific feature in the induced current density, the ring diamagnetic (classical free-electron circulation) and para- current, is taken as the basis for the (magnetic) criterion of magnetic (quantum-mechanical response to molecular non- aromaticity. uniformity) contributions is in a sense arbitrary, in that the Since the pioneering work of Pauling1 and London2 in balance of these contributions depends on the choice of ori- the 1930s, ring current has been invoked in the explanation gin, total response properties should be invariant to this of the characteristic magnetic susceptibilities of aromatics3 choice. In practice, in calculations with a finite basis and and, since the classic paper of Pople,4 ring currents have been with a single origin of vector potential, computed proper- inferred from the anomalous 1H shifts in aromatic molecules. ties are found to be far from invariant, and the underlying The ability to sustain a diatropic ring current is taken as the current density is poorly represented. Substantial progress signature of aromaticity in a cyclic molecule;5 in the same in quantitative calculation of global properties was made as way, paratropic ring current betokens anti-aromaticity.6 This it became clear that distribution of origin of vector potential magnetic criterion is enshrined in the NICS method,7 where 8–10 a fictitious ‘nuclear shielding’ is calculated for a point at or was advantageous, and the breakthrough in the treat- ment of the basic underlying property, the current density, was achieved when Keith and Bader11 proposed the ultimate † Dedicated to Professor M. Barfield on the occasion ‘democratic’ distribution of origin—a treatment in which an of his 70th birthday. individual origin was used for calculation of current density ŁCorrespondence to: P. W. Fowler, Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK. at every point in space. This ansatz, which in its simplest E-mail: [email protected] version is often called ipsocentric12 (in that current density Copyright 2004 John Wiley & Sons, Ltd. Current density, chemical shifts and aromaticity S69 at any one point is calculated with that point as origin), has calculations, the n are Hartree–Fock orbitals and first- important practical and conceptual advantages.12–16 order corrections are calculated with coupled-Hartree–Fock The present paper summarizes this approach and shows theory, but could equally be obtained from empirical MO how ipsocentric calculations lead to a tractable qualitative theories or DFT. For a constant uniform magnetic field B,the picture of magnetic aromaticity in which ring currents can Hamiltonian is, to first order in B: be attributed to features of frontier orbital structure and 2 O O e · hence can be understood qualitatively and also calculated H D H0 C lˆ.d/ B quantitatively. This leads to an interpretation of chemical 2me shifts at nuclei and at the notional NICS centres and an where lˆ.d/ D .r d/ ð p is the angular momentum operator interpretation of the survival and extinction of currents in for rotation about the ‘gauge origin’ d of the magnetic clamped systems, polycycles and heterocycles. Some new vector potential A D 1/2 B ð .r d/ and pˆ Dih¯r is the calculations on benzo-fused pentalenes are used to illustrate linear momentum operator. Then the first-order induced the power of the method and cast light on these problematic current density, j1r,isasumofterms,oneforeachdoubly systems. occupied orbital n: The structure of the paper is as follows. First, the ipsocen- 2 tric calculation of current density is briefly outlined and e 2 jnr D B ð r d nr it is shown how this leads to a particular orbital model me of induced current. Connections with molecular symme- 2ieh¯ 1 1 try are then established, leading to rules of thumb for C [ nrr n r n rr nr] me aromaticity and anti-aromaticity and giving the basis of a physical interpretation of orbital contributions. Finally, The first term on the right-hand side, the ‘diamagnetic dia the machinery of the ipsocentric method is applied to a current density’, jn r, depends on the unperturbed orbital set of molecules of long-standing interest in the study density; the second, the ‘paramagnetic current density’, para 1 of aromaticity and its limits, the benzopentalenes. These jn r, depends also on the first-order correction, n .This species incorporate nominally anti-aromatic pentalene and separation into diamagnetic and paramagnetic parts has aromatic benzene or naphthalene, giving scope for interplay no special physical significance as terms can be rebalanced of local and global aromaticity/anti-aromaticity. Limited within a constant total, by choice of gauge origin, which in experimental data are available for comparison. Benzopen- principle changes all orbital current densities jnr,aswell talene (1) (cyclopent[a]indene) dimerizes above 70 °Cin as their diamagnetic/paramagnetic breakdown, even in a 1 CD2Cl2 solution, but a monomer H NMR spectrum is avail- complete basis set. able, recorded at 70 °C.17 There is a low-resolution 1H To simplify the discussion, we consider the one-electron NMR spectrum for the more symmetrical dibenzopentalene Hamiltonian to be constructed from the usual kinetic and [1,2 : 4,5-dibenzopentalene (2)].18 The isomeric dibenzopen- local potential energy operators. Orbital degeneracy, self- 24 talene 1,2 : 5,6-dibenzopentalene (3) is so far unknown, as consistency, exchange and other non-local corrections 23 is the species cyclopent[b,c]acenaphthylene (4), formally complicate the argument, but do not change its essentials. a fusion of pentalene and naphthalene moieties, which The first-order correction to each occupied orbital n is then a has been invoked as a pivotal transient intermediate in sum over excitations from that orbital to unoccupied orbitals 12 high temperature, gas-phase reactions including the uni- p p > N/2: molecular rearrangement19 of pyracylene and the inter- h jlˆd Ð Bj i conversions of ethynyl-substituted acenaphthylenes and 1 e p n n r D pr 20 21 2m ε ε bisethynylnaphthalenes. Computed maps of current den- e p>N/2 p n sity for 4 show clear separation of aromatic and anti-aromatic regions. Comparison with 1 to 3 will show whether the mag- The key feature of this sum is that it excludes any netic identity of the pentalene subunit survives annelation in possibility of occupied–occupied mixings. By rewriting more general situations. the angular momentum operator for rotation about d as lˆd D lˆ0 d ð pˆ ,wherelˆ0 refers to rotation about the origin of coordinates, the first-order correction can INDUCED CURRENT DENSITY be split into two sums, corresponding to paramagnetic and diatropic terms, each still excluding any reference to The theory of the induced current density has been described occupied-occupied mixings.