Calculations of Constant-Height STM Images of Fullerene C 60 Adsorbed onto a Surface

. Constant-height scanning tunneling microscopy (STM) images of a C 60 molecule adsorbed onto a surface were calculated using symmetry-adapted H¨uckel molecular orbitals (HMOs). Tree adsorption orientations of C 60 are considered. Te interaction between the C 60 molecule and the surface was treated using symmetry arguments only. Projection operators were used to generate symmetry-adapted HMOs of the molecule. Tese orbitals were then used to construct idealized constant-height STM images using the simple tunneling theory of Tersof and Hamann. A comparison is made with published experimental STM maps. Te results show that, for each orientation of C 60 , split orbitals of the same symmetry have similar appearances in the constant-height maps. Tey also show that the map of a molecular orbital of a complete degeneracy is dominated by only one or two of its components.


Introduction
In the literature, one fnds extensive high-resolution scanning tunneling microscopy (STM) studies on fullerene C 60 residing on a surface. Most of these studies have used the constant-current mode of STM to investigate the orientation and intermolecular structure of C 60 under various conditions, including C 60 interacting with the surface, its neighbours, the scanning tip, a guest molecule situated inside it, or C 60 isolated from any interactions. Tis last mode is preferred because it enables probing the fne features of the molecular orbitals, even though it takes a long operation time. In contrast, in the constant-height mode, only the top part of the sample is scanned, and as such, much of the structural details can be lost. However, this mode is preferred when a fast scan of the sample is desired without the need to record its fne topography.
In recent years, Reecht et al. [1] used the constant-height mode to image the molecular orbitals of a C 60 molecule interacting with a Cu (111) substrate, where the surfaceinduced splitting of the LUMO (lowest unoccupied molecular orbital) and LUMO + 1 was observed by the STM tip. On the Cu (111) substrate, the C 60 molecule is adsorbed with a hexagon facing the surface [1][2][3]. C 60 can reside on a substrate with other orientations depending on the adsorption site. For example, on a Au (111) substrate, the preferred orientation is pentagon-down [4], and it is double bond down on a black phosphorous substrate [5]. Te adsorption of C 60 on a substrate removes the full icosahedral symmetry characteristic of the isolated molecule. Te resulting symmetry depends on the orientation of C 60 and the symmetry of the adsorption site. Site symmetries available for a C 60 molecule are C 6v , C 6 , C 4v , C 4 , C 3v , C 3 , C 2v , C 2 , C s , and C 1 [6]. Te reduction in the symmetry of C 60 lifts the high degeneracies of its molecular orbitals [7]. Tis efect is now routinely observed in STM experiments, where the split frontier orbitals of C 60 , namely, the HOMO (highest occupied molecular orbital), LUMO, and LUMO + 1, can be imaged at diferent biases. Te imaged features of the orbitals depend predominantly on the orientation of the C 60 molecule and weakly on the adsorption site [8]. Correct assignments of the recorded images to the corresponding orbitals cannot be accomplished by analyzing the STM images alone. Rather, it requires comparison with simulations. Complicated ab initio methods are usually used to make the necessary simulations. However, Hands et al. [6,7] developed a much simpler analytical method to simulate STM images of the molecular orbitals of C 60 . Tis method is based on the Hückel molecular orbital theory and the simple tunneling theory approach of Tersof and Hamann [9]. It treats any perturbation of the orbitals of C 60 purely on the basis of symmetry using group theoretical methods; that is, the efect of the interaction is simply to lift the high degeneracies of the molecular orbitals of C 60 . Te simplicity of the method is that STM images of the orbitals can be created using only their symmetries without any calculations of their energies. Assigning a real STM image can then be made by comparison with the simulated images. Te authors have successfully applied their method to simulate and interpret constant-current STM images of C 60 and its ions undergoing diferent interactions [3,7,[10][11][12]. Tey also used the method to simulate constant-height STM images of the unsplit HOMO and LUMO of a C 60 molecule residing on a substrate with diferent orientations [6].
As noted above, most published theoretical STM images of the C 60 molecule were produced in the constant-current mode. In this work, the analytical methods were used to simulate constant-height STM images of a C 60 molecule adsorbed onto a substrate and subjected only to surface interaction. Te aim is to make available what is currently missing in the literature, that is, a library of simulated STM images of C 60 that can be used as a reference in constantheight STM scans. Te method of projection operators is utilized to form symmetry-adapted HMOs which are then used to construct the STM images. In the simulation, only the efect of the surface interaction on the electronic structure of C 60 is taken into account, and none of the complications associated with other experimental parameters are considered. Tis efect is treated by the group theoretical means developed in [7]. Since the adsorption site has little infuence on the captured features of the imaged molecule, [12] is followed and a fat, featureless surface is assumed. Tus, the symmetry of the C 60 -surface system is determined entirely by the adsorption orientation of C 60 . Tree possible orientations of the adsorbed molecule are considered: a C 60 molecule adsorbed with a hexagonal face, a pentagonal face, and a double bond prone to the surface. Te corresponding overall symmetries of the C 60 molecule are C 3v , C 5v , and C 2v , respectively [12]. Te results for the hexagon-prone orientation are compared with the experimental results in [1]. Te results obtained here for a fat surface can be readily linked to any site symmetry by using easy group theoretical methods to correlate the transformation properties of the orbitals considered here to their transformation properties in the site symmetry. Figure 1 shows the defnition of the coordinates used in the simulations when the C 60 molecule is adsorbed with a double bond prone to the surface (hence, a double bond is facing towards the STM tip). Te origin of the Cartesian axes is placed at the center of the C 60 molecule. Te z axis is normal to the surface, and as such, it defnes the direction of observation. Te x and y axes are parallel to the surface, with each axis defning a C 2 axis of rotation. Accordingly, the symmetry of C 60 is C 2v . To obtain the C 5v and C 3v symmetries, the molecule is rotated about the y axis by tan(−ϕ −1 )

Methods
and )/2 is the golden ratio. For each symmetry of C 60 , the method of projection operators was used to generate the HMOs in symmetryadapted form [13]. Te projection operator is defned as follows: where h is the order of the group, d i is the dimension of an irreducible representation (irrep.) c i of the group, Γ i (O) is the matrix representation of the symmetry operation O, and . Applying the projection operator on a π orbital results in a state of symmetry c i if the orbital belongs to that representation or zero if it is not. Terefore, for each point group of the C 60 molecule, 60 symmetry-adapted states were obtained in which each state transformed as irrep. of the point group. Tese states were then used to construct the Hückel matrix in a blockdiagonal form in which each block belonged to irrep. of the point group. Each block was then diagonalized separately to obtain the relative energies and symmetries of the molecular orbitals of C 60 . Te alternation in the single and double carbon-carbon bonds was accounted for [7], in which the corresponding resonance integrals were related by a parameter α � (β d /β s ) � 1.433 [14], where β d and β s are the resonance integrals for the double and single bonds in C 60 , respectively. Te tunneling theory approach of Tersof and Hamann [9] was used to simulate the constant-height STM maps of the molecular orbitals. In this approach, the current observed during STM is proportional to the electron density of the surface evaluated at the position of the STM tip: where I is the tunneling current, ψ i is the wave function of the surface state of energy E i , E F is the Fermi energy, r 0 is the position of the STM tip, and i runs over all the available surface states. A map of an orbital of C 60 with complete degeneracy was obtained by summing over all the components that would arise from that orbital by surface interaction. Te constant-height image of a molecular orbital was constructed by creating a grid of points at a specifed tip height and calculating the current at each point.

Results and Discussion
Te results of the simulations are presented in Figures 2-7. Figures 2 and 3 present the results for a C 60 molecule adsorbed with a hexagon prone to the surface. Tis corresponds to the C 3v symmetry in the simple model assumed here. Figure 2 shows the electron densities and STM maps of the unsplit MOs, including HOMO, LUMO, LUMO + 1, LUMO + 2, LUMO + 3, and combined LUMO + 2 and 3. Te latter combination was included to compare the current results with the experimental images obtained in [1]. For this orientation (and also for the pentagon-prone and doublebond-prone orientations in Figures 4 and 6), the simulated maps for the unsplit HOMO and LUMO are similar to those obtained in [6] using the HMOs of Deng and Yang [15]. In general, the three-fold symmetry of the hexagonal face of C 60 is present in all images of unsplit and split MOs. Te constant-height map for the HOMO, LUMO, or LUMO + 1 appears as three bright lobes arranged as a three-fold ring for the former orbital and as a three-leafed clover for the latter two orbitals. Te results for HOMO and LUMO agree with many experimental observations [3,16,17]. Te maps for LUMO + 1 and LUMO + 2 appear as hexagonal arrangements of 6 bright spots, resulting from the density concentrated on the carbon atoms. A surface interaction of C 3v symmetry lifts the degeneracies of the MOs of C 60 according to irreps. given in the last column in Figure 2. For example, the HOMO orbital is split into three components: a singlet of A 2 symmetry and two doublets of E symmetry. An examination of the maps of all states resulting from the splitting of HOMO, LUMO, LUMO + 1, LUMO + 2, and LUMO + 3 ( Figure 3) shows that each takes one of the four unique shapes as per the symmetry of the state: A state of A 1 symmetry is mapped onto a bright protrusion, a state of A 2 symmetry appears as a 6-petaled fower, and a state of E symmetry appears either as a 3-leafed clover or as a bright three-fold ring. Since the two components of the E symmetry produce diferent images, one of the components is marked as E * in Figure 3 and also in the text. It is also noted that the map of an MO of complete degeneracy is dominated by the features of one or two of its components. For instance, the map of the HOMO is dominated by the features of one of the two components that transform in C 3v as a doublet.
Similarly, the images of LUMO and LUMO + 1 are dominated by the E components. Tus, care must be taken when analyzing the experimental STM images since an orbital of complete degeneracy may appear as an orbital of a lower degeneracy and vice versa.
To assess the method used here, the simulated electron densities and STM images were compared to the results reported by Reecht et al. [1] Te electron densities produced by the simple method of the group theory used here exactly match those simulated by the authors of [1] using the DFT method with the Perdew-Burke-Ernzerhof (PBE) functional. We note that the image obtained for the E * component arising from the HOMO resembles that obtained for the unsplit HOMO. Both images are similar to the image observed in [1] at V � −1.9 V. Te authors of [1] identifed this orbital as the HOMO with complete degeneracy; that is, it is unafected by surface interaction. Splitting of the HOMO when C 60 is adsorbed on various surfaces has been observed (see reference [7] and references within) with the E * component observed at a bias ∼ −2 V. If the components of the HOMO in [1] were not well separated by the surface interaction, the constant-height map of the E * component could have been wrongly attributed to the unsplit HOMO. Experimental maps of the A 1 and E components arising from the LUMO in [1] resemble our simulated images.
However, there is some disagreement regarding the remaining orbitals. Te experimental images identifed in [1] as the LUMO + 1 A and E components each feature a threeleafed clover. While this matches our simulations for the E component, it disagrees with the simulation for the LUMO + 1 A 2 component. Te electron density of this orbital is shown on the far right of Figure 3. Te uppermost part of the electron density features a six-petaled fower, which is expected to be mapped onto a bright, six-fold image in the constant-height map, in disagreement with the experimental image reported in [1]. Lastly, we discuss the results for LUMO + 2, LUMO + 3, and the combination of LUMO2 and 3. Te two sets of orbitals and their combination exhibit identical appearances in the simulated constant-height images, each composed of a hexagon of bright spots. [1] does not report individual LUMO + 2 and LUMO + 3. It reports only the map of their combination, which is observed as a bright protrusion. Referring to Figure 3, our simulations suggest that the bright protrusion is the signature of the A 1 state, possibly arising from the splitting of either LUMO + 2 or LUMO + 3. Figures 4 and 5 present the results for a C 60 molecule with a pentagon prone to the surface. Figure 4 shows the HOMO and LUMO + 1 orbitals, and each is expected to appear as a fve-fold ring and LUMO, LUMO + 2, and LUMO + 3 as bright lumps. Te results for unsplit HOMO and LUMO match those simulated by Hands et al. [6]. Te image obtained for the LUMO also agrees with the experimental maps [3]. Te components arising from these orbitals when the surface interaction has C 5v symmetry are expected to be mapped by the STM onto one of the four unique shapes: the A 1 component as a bright lump, A 2 as a 10-petaled fower, E 1 as a fve-fold ring, and E 2 as a pentagon of bright spots. Moreover, the simulations for LUMO, LUMO + 2, and LUMO + 3 are dominated by the A 1 components and those for HOMO and LUMO + 1 by the E 1 components.
Density of states Constantheight map Irreps. of C 3v Irreps. of C 3v Figure 3: Te appearances of the diferent components of the molecular orbitals in Figure 2 (labelled by their symmetries in the top row) when they are split by a surface interaction of C 3v symmetry as would be observed in a constant-height STM probe. Both doublets have the same transformation properties in C 3v but produce diferent STM maps, and as such, one of them is distinguished by an asterisk. Far right: the electron density of the LUMO + 1 A 2 component. Figures 6 and 7 present the last set of simulations, obtained for a C 60 molecule with a double-bond prone to the surface. Figure 6 shows the results for the orbitals when they retain their icosahedral degeneracies. Te features obtained for the HOMO and LUMO agree with experimental observations [3,16,17] and with the simulations reported in [6]. Te HOMO appears in the simulations as an oval lobe. LUMO + 2 and HOMO + 3 each features a pair of slightly separated bright spots, which, at low resolution, are expected to appear as a lobe. LUMO and LUMO + 1 each appears as a pair of parallel lobes. Figure 7 shows the simulations for the split orbitals when the surface interaction has C 2v symmetry.

Constant-height map
Constantheight map Here, a state of A 1 symmetry appears as a single lobe, while a state of B 1 symmetry features two bright lobes. Te additional A 1 component that arises from the LUMO + 3 orbital upon lifting its degeneracy appears as two wellseparated lobes, and as such, it is marked as A * 1 . Te last column in Figure 6 shows that the A 1 and B 1 states arise together in the splitting of all but one orbital, LUMO + 1, where only a state of B 1 symmetry arises upon splitting. Seemingly, one or both states dominate the other components in each orbital when the latter maintains its degeneracy.
Te simulated images for A 2 and B 2 states all have four bright spots. Te additional A 2 state that results from the splitting of the HOMO orbital produces a slightly diferent STM map, and hence, it is marked by an asterisk in Figure 7 and in the text. In the A 2 and B 2 states, the four spots are    Figure 6 (labelled by their symmetries on the top row) when they are split by a surface interaction of C 2v symmetry as would be observed in a constant-height STM probe. One of the two singlets transforming as A 1(2) is marked by an asterisk because it produces diferent features in the STM map. rectangular, while they are square in the A * 2 state. Rectangular features have been observed in STM images of C 60 4− ions on Au (111) at +0.1 eV, and they were attributed to a state arising from the LUMO manifold as a result of a Jahn-Teller distortion of the C 60 4− ion to D 2h [18] or D 3d [12]. Hands et al. [7] suggested that these features arose from a LUMO + 1 component (which corresponds to the A 2 component in Figure 7). According to our results, the rectangular features can also arise from the LUMO in neutral C 60 (the B 2 component) if the molecule is resting on the substrate with a double bond and the surface interaction has a symmetry that does not support any degeneracies.

Conclusion
Simulated constant-height STM images of the MOs of C 60 when subjected to interaction with a fat surface have been presented. Tree common orientations of C 60 were considered: when the molecule is prone to the surface with a hexagon, a pentagon, or a double bond. Te simulations used the Hückel molecular orbital theory, the group theory method of projection operators, and the Tersof-Hamann approach of quantum tunneling. Te results for the hexagon-prone orientation were compared to previously reported experimental STM images. While many simulated images agreed with the experimental maps, the current simulations suggest that some of the constant-height STM images reported in [1] could have been incorrectly reported. Te results of this work are expected to assist in constantheight STM experiments on fullerene C 60 by allowing a quick indication of the existence of the surface interaction and fast identifcation of the orientation of C 60 and of the symmetry of its split orbitals.

Data Availability
No underlying data were collected or produced in this study.

Conflicts of Interest
Te author declares that there are no conficts of interest.