Comment to 'Imaging the atomic orbitals of carbon atomic chains with field-emission electron microscopy'

The data are consistent with alternatively a single spot or a doublet of spots, which the authors identify as emission from s-type or p-type MO respectively. s / p labeling refers to the angular momentum of these electronic states around the chain axis, usually denoted by σ / π in the quantum-chemistry literature. An electronic wave function of the axially symmetric chain depends on the azimuthal angle ϕ around the chain axis as exp(±inϕ), with n = 0, 1 for s, p states respectively, see Eq. ( 5 Of course, a perturbation could localize the electron in the azimuthal angle, by inducing a splitting between the cos(ϕϕ 0 ) and sin(ϕϕ 0 ) components of the p state, which do in turn display nodes. However, any sort of fluctuating perturbation, such as those associated to the thermal oscillation of the free standing chain, would produce a rapidly fluctuating phase ϕ 0 , with the eventual result of an axially symmetric pattern. A stable doublet could arise in FEEM only due to a symmetry-breaking perturbation that locks the node at a fixed ϕ 0 . This perturbation needs to be sufficiently large and stable to overcome the tendency of thermal vibrations and quantum kinetic energy to wash out all localization effects.

What can induce such a symmetry-breaking perturbation in a several-atom-long free-standing chain?

The answer is: the anchored end of the chain. Whenever this end binds an sp 2 hybridized atom of the carbon tip, the chain acquires a partial cumulenic character 2 . All bonds are double, and the memory of the orientation of the sp 2 termination propagates along the chain through an alternating orientation of π bonds 3 . We illustrate this point by carrying out standard calculations within the Density-Functional theory in the localdensity approximation (DFT-LDA) with a plane-waves package 4,5 using default ultrasoft pseudopotentials, and wavefunction/charge cutoffs of 15/120 Hartree. We relax the atomic positions until the largest residual force is less than 10 -4 Hartree/a 0 (8 pN). For a 8-atom chain, Fig. 1a shows the highest-occupied molecular orbital (HOMO), made of the four π bonds oriented normally to the sp 2 plane. 8 As a result, emission from the terminal atom is dominated by this asymmetric π orbital, and displays a clear nodal plane which is parallel/antiparallel to the sp 2 plane, depending on the chain being composed by an even/odd number of atoms.

Alternatively, the anchored end, rather than sp 2 , could have sp 3 hybridization (e.g. a generic atom in the “bulk” of a graphene fragment or fullerene or nanotube cap, represented by a simple H atom in Fig. 1b). Such a ligand would induce a polyyinic-type electronic structure of the chain. As a result, a highly dimerized configuration, with alternating single/triple bonds and an essentially unbroken axial symmetry is realized. However, even in this case, eventually the HOMO is a π bonding orbital. The main difference is that as the cylindrical symmetry is unbroken, a cylindrically symmetric “doughnut-shaped” orbital is generated, precisely of the type described by Eq. (5) of Ref. 1. Such a state vanishes along the molecular axis, as can be seen in Fig. 1b, but, due to thermal vibrations of the chain, it is unlikely that its FEEM pattern could distinguish it from that of a singlet σ-type orbital. 9 Indeed higher currents in excess of 100 pA can induce switching from one type of FEEM pattern to the other,

The remaining three π bonds point horizontally, in the sp 2 plane.

For increasing chain length, the dangling-bond terminal σ orbital remains energetically below the HOMO level, with an increasing separation from the HOMO, in the

-400 meV range.