Effective field theory of interacting $\pi$ electrons

We develop a $\pi$-electron effective field theory ($\pi$-EFT) wherein the two-body Hamiltonian for a $\pi$-electron system is expressed in terms of three effective parameters: the $\pi$-orbital quadrupole moment, the on-site repulsion, and a dielectric constant. As a first application of this $\pi$-EFT, we develop a model of screening in molecular junctions based on image multipole moments, and use this to investigate the reduction of the HOMO-LUMO gap of benzene. Beyond this, we also use $\pi$-EFT to calculate the differential conductance spectrum of the prototypical benzenedithiol-Au single-molecule junction and the $\pi$-electron contribution to the van der Waals interaction between benzene and a metallic electrode.

Figure 1

Two isosurfaces of the average $\pi$-electron density $\langle {\psi }^{†}\left(\vec{x}\right)\psi \left(\vec{x}\right)\rangle$ depict the electronic structure of gas-phase benzene within $\pi$-EFT.

Figure 2

The spectral function of gas-phase benzene broadened artificially as a guide to the eye. The dashed orange lines are fixed by (left to right) the lowest lying optical excitation of the molecular cation (Refs. 36, 37, 38, 39, and 49), the vertical ionization energy of the neutral molecule (Refs. 35, 36, 37, 38, 39), and the vertical electron affinity of the neutral molecule (Ref. 40).

Figure 3

The ionization energy and electron affinity of benzene oriented parallel to the surface of a screening plane, shown as a function of distance.

Figure 4

The geometry (Refs. 52 and 53) for the benzenedithiol junction associated with the spectral function shown in Fig. 5. The electrodes have been placed so that the screening surface lies one covalent radius (Ref. 54) beyond the position of the outermost gold nuclei, a convention that has been investigated elsewhere (Ref. 55) in the context of atom-surface van der Waals interactions.

Figure 5

Top: The spectral function of the Au-1,4-benzenedithiol-Au junction depicted in Fig. 4 at room temperature, together with the gas-phase density of states from Fig. 2. To facilitate comparison, the same broadening has been used in both cases. The dashed orange line at $-5.1$ eV indicates the position of the experimental chemical potential of clean gold (Ref. 56). Bottom: The spectral function of the same junction with planar instead of spherical electrodes.

Figure 6

Differential conductance spectrum of a Au-1,4-benzenedithiol-Au junction at room temperature versus gate and bias voltages. The junction geometry, including source and drain electrodes, is depicted in Fig. 4; a spherical gate electrode of radius 3 Å (not shown) is centered 5 Å above the benzene ring. The effective electrostatic lever arm of the gate is 0.21 eV/V. The charge on the molecule within the junction is quantized within the diamond-shaped regions centered on the horizontal axis due to the phenomenon of Coulomb blockade. Resonant tunneling through electronic excited states at large bias and suppression of transport at small bias due to destructive quantum interference (blue fringes) are clearly visible.

Figure 7

The $\pi$-electron contribution to the van der Waals interaction between benzene and a spherical electrode with a radius of 10 nm, plotted as a function of the distance from the conducting surface. At all distances the molecule is oriented parallel to the surface of the electrode. The dashed green and orange lines show the expected asymptotic dependence in the near and far fields, respectively. Inset: The same quantity very near the surface of the electrode, including a phenomenological (Ref. 67) gold-carbon hard-core repulsion.

Figure 8

The orientation dependence of the $\pi$-electron contribution to van der Waals interaction between a planar electrode and a benzene molecule centered 2 nm from the metal surface. The molecule is initially oriented parallel to the electrode and then rotated by an angle ${\theta }_{\perp }$ about the axis perpendicular to the plane of the molecule, followed by a rotation of ${\theta }_{\parallel }$ about an axis within the plane of the molecule.