Non-linear ring currents: effect of strong magnetic fields on π-electron circulation

Abstract

Finite-field calculations of non-linear induced current density in representative [4n + 2] and [4n] π systems, benzene and flat cyclooctatetraene (COT), show that a strong uniform perpendicular magnetic field enhances benzene π-diatropicity and decreases COT π-paratropicity. The non-linear current is stronger by two orders of magnitude in COT, but still small: at 1a₀ height and in a field of 25 T, third-order effects contribute −5 ppm to the first-order π ring current. Classical arguments based on radial contraction of charge density rationalise the non-linear response of benzene, but that of COT depends on a specific orbital-rotation effect, characteristic of paratropic π systems.

Introduction

Molecular magnetic response is typically described by expanding the interaction energy in a power series of the perturbing magnetic field, with coefficients determined by the electronic structure of the unperturbed molecule [1], [2]. Measurement of global response properties such as shieldings and magnetisability, as a function of the strength of the inducing field, provides a powerful probe of the electronic structure of a molecular system. Magnetic response is at the heart of one widely used definition of aromaticity [3], [4], [5], [6], which thereby becomes a field-dependent phenomenon.

Given the technological difficulties in attaining strong uniform magnetic fields, attention has concentrated on linear response properties, although the possibility of a quadratic field dependence of nuclear magnetic shieldings [7], [8], [9] and molecular magnetisability [10] has been considered. As new technologies based on superconducting, or high-field resistive [11] magnets lead to the possibility of generating uniform magnetic fields of 20–45 T in the laboratory, theoretical research on molecular non-linear magnetic response has received a new impetus. A theory of magnetic-field dependence of the Cotton–Mouton effect has been proposed [12] and has allowed measurement of the quadratic magnetic field dependence of the molecular magnetisability anisotropy for linear molecules. Effects of a strong magnetic field on nuclear magnetic shielding, up to sixth order in the inducing field, have been investigated for closed-shell atoms by means of correlated calculations [13]. Small measured quadrupole splittings in ¹³¹Xe NMR spectra [14] have been interpreted in terms of electric field gradients caused by diamagnetic distortion of the atomic electron cloud, quadratic in the magnetic field [15]. A computational scheme based on Rayleigh–Schrödinger perturbation theory, implemented at the Hartree–Fock level, has been used to compute, and study gauge dependence of, the fourth-rank hypermagnetisability [2], the tensor describing the quadratic dependence on magnetic field of the molecular magnetisability. Increasing attention has been paid to the effects of strong magnetic fields on the electronic structure of nanosystems, with the experimental verification of the magnetic field dependence of the band gap in nanotubes [16], [17] implying the possibility of turning a semiconductor into a metal or vice versa, simply by tuning the field strength.

Here we propose an alternative approach to the investigation of non-linear molecular response to a uniform magnetic field, based on the calculation and mapping of the non-linear induced current density. For closed-shell molecules, the induced current density can be expanded in odd powers of the magnetic field. The first-order current density, linear in the magnetic perturbation, is routinely and accurately computed in investigations of aromaticity and antiaromaticity of cyclic conjugated molecules [18]. The next non-vanishing contribution to the induced current density is the third-order current density, J⁽³⁾(r), describing the cubic dependence of the current on the field. When integrated with appropriate vector potentials, J⁽³⁾(r) leads to the fourth-rank hypermagnetisability and nuclear magnetic hypershieldings. The advantage in computing a function of the position, rather than integral properties, lies in the fact that the vector map of current defined over the molecular domain delivers a direct representation of the underlying non-linear molecular magnetic response.

We present a formal expression for the computation of the third-order induced current density within Rayleigh–Schrödinger perturbation theory, and calculate J⁽³⁾(r) for benzene and flat [19] cyclooctatetraene (COT) in a finite perturbation theory scheme. Benzene and COT are typical representatives of the two classes of cyclic conjugated molecules. At first order, the π-electron response of benzene (COT) is characterised by an anisotropic exaltation (reduction) of the induced diamagnetic moment perpendicular to the molecular plane. This behaviour is rationalised by visualisation of the first-order current density: a global π ring current is induced by the perpendicular component of the field, and is diatropic in benzene and paratropic in flat COT.

It is shown here that the non-linear response of benzene and COT can also be represented by global ring currents, dominated by diatropic circulation in both cases, in agreement with the negative sign of computed fourth-rank hypermagnetisabilities [20]. Rationalisations for the current in the two cases are, however, different. The behaviour of paratropic COT is shown to parallel the opening of the gap between occupied and conduction bands in metallic carbon nanotubes that is induced by strong magnetic fields [16], [17].