The complexity associated with even the simplest molecule-on-metal system can be extreme. Notwithstanding this, theoretical approaches can provide an abundance of information that can prove critical in surface science investigations. To make this possible, calculations are performed using certain simplifications that ultimately make simulations manageable by reducing the total number of variables that need to be considered.
Molecular mechanics (MM) embodies this approach, whereby the chemical bonds in molecules are effectively simulated as springs, which can be readily modelled using variations of Hooke’s law [19]. In this manner, very large ensembles of molecules can be investigated rapidly and inexpensively, at the cost of a ‘complete picture’ of all the associated properties. For example, in this work, MM has been used to investigate the nature of the intermolecular interactions in the TPA brickwork phase and how modification of its lattice parameters affects the total energy, which is described in detail in Section 3.2.1.1.
Unfortunately, its intrinsic simplifications result in MM being unsuited for the investigation of, for example, electronic properties. Given their importance in molecular adsorption on metal surfaces, it is necessary in such cases to use an ab initio approach. Here, the aim is to understand a molecular ensemble by solving its
81 many-electron Schrödinger equation. Given that an analytical solution is not possible for anything except a one-electron system, techniques have been developed that can make accurate estimates of the solution through a number of assumptions and simplifications, some of which are outlined below.
DFT [20,21] is such a method, whereby the electrons and their complex many-body interactions are reduced to a single functional; the electron density. This vastly reduces the number of variables that need to be calculated, as instead of simulating the motion and interactions of each electron, only the spatial variance of the electron density needs to obtained. This simplification is possible by considering the electron motion within the framework of the Born-Oppenheimer approximation, whereby the atomic nuclei are effectively stationary on the timescale of electron movements. The electrons therefore effectively interact only with a stationary potential field, defined by the positions of the atomic nuclei. It can be shown that the ground state electron density is unique to that particular potential field, allowing it to be calculated using energy minimisation processes. In doing so, the electrons contributions to the total energy can be calculated.
The electron density calculated by DFT allows the DOS to be readily evaluated. Given that the STM tunnel current depends largely on the sample DOS (Equation 2.4), DFT calculations can provide additional insight into the images obtained by STM. In particular, DFT is an excellent tool with which to simulate
STM images, which is typically performed in the framework of the Tersoff-Hamann approach. Here, the tip apex is approximated with a s-wave orbital [22]. In doing so, the tunnel current can be estimated using:
∫ ( ⃗ )
82 where ( ⃗ ) is the sample DOS at the position of the tip apex, ⃗ . With this in mind, a simulated STM image is obtained by calculating the spatial variance of the DOS within a given energy window.
2.5.1 Methods - molecular mechanics
The MM calculations were performed by Dr Natalia Martsinovich and Professor Alessandro Troisi from the University of Warwick, using the MM3 force field [23,24] in the Tinker software [25]. This has been adapted so as to correctly simulate the energy of the dimeric H-bonds between TPA carboxylic moieties, as calculated from DFT [26]. The interaction potential for gas-phase TPA was calculated by establishing a unit cell with dimensions a×b×c, containing only one TPA species, which is illustrated by the red box in Figure 2.11. In the following calculations, these dimensions were varied so as to simulate the different intermolecular separations. To calculate the intra-chain interactions, isolated 1D
Figure 2.11 Depiction of the cell dimensions used for the MM calculations. a
bx
by
c a
83 chains were established by setting large b and c values. The size of a was then gradually increased in 0.05Å steps, and the total energy was calculated. The interactions between the chains were similarly investigated; a was initially set at its calculated lowest energy value whilst bx and by were varied. Confirmation of the
lowest a, bx and by values was obtained by simultaneously varying all three
parameters about their most favourable positions and observing no changes in their lowest energy values.
2.5.2 Methods - density functional theory
The DFT calculations of TPA on Cu(110), which were not dispersion- corrected, were performed by Dr Stefano Fabris, based at the Theory@ELLETRA group and SISSA. These employed the exchange and correlation energy functionals expressed in the Perdew-Burke-Ernzerhof generalized gradient approximation [27]. All calculations were performed with the Quantum-ESPRESSO computer package [28]. A 3-layer metal slab were used to model the Cu(110) surface, where the two lowermost layers were kept fixed during the structural relaxations. The STM images were simulated by integrating the sample DOS between the Fermi energy and a bias potential of 0.5 eV.
The dispersion-corrected DFT calculations of TPA on Cu(111) and Au(111) were performed by Dr Natalia Martsinovich and Professor Alessandro Troisi using the SIESTA code [29]. The metal surfaces were simulated using 3-layer metal slabs with a fixed bottom layer. Dispersion-correction was applied to molecule-metal interactions, and the relevant dispersion parameters on the Cu and Au surfaces were obtained from reference [30] and [31], respectively. The simulations were performed by first positioning the TPA molecule over a hexagonal close-packed hollow site on
84 the relevant substrate with the TPA long symmetry axis aligned along the [ ̅] crystallographic direction. To obtain the potential energy surface (PES) described in Section 3.2.1, the adsorption energy of the TPA was calculated in various positions on the metal. The adsorbate was moved parallel to the Cu(111) and Au(111) surfaces in 0.25 and 0.5 Å steps, respectively, so that the entire substrate unit cell is explored. In each position, was calculated using:
where and are the energies of the clean metal surface and the isolated TPA molecule, respectively, and is the energy calculated for the total system (i.e. with the TPA adsorbed on the substrate).
The DFT calculations of the OA on Cu(110) system described in Section 4.1.2, which also did not include dispersion-corrections, were performed by Dr Sara Fortuna, now at the Department of Medical and Biological Sciences at the University of Udine, and Dr Stefano Fabris. These also used the Perdew-Burke-Ernzerhof generalized gradient approximation functional for the exchange and correlation energies [32], by means of the Quantum ESPRESSO distribution [28,33]. The Cu(110) substrate was simulated using a 5-layer deep Cu slab, with either (3×2) or (2×2) unit cells in the lateral (i.e. parallel to the surface) directions. During the structural relaxations, the lowermost two layers were kept fixed at their bulk-like coordinates. A wide variety of molecular adsorption geometries were investigated, including parallel, perpendicular and tilted orientations. The adsorption energies where calculated in a similar manner to that expressed in Equation 2.8, albeit that an additional term , the energy associated with a single proton, has to be included to compensate for deprotonation. Furthermore, as some supercells contained multiple
85 adsorbed OA species and varying degrees of deprotonation, it is necessary to incorporate scaling constants, leading to:
(2.9)
where and are integers incorporated to calculate the effect of multiple adsorbates in the unit cell and effects of deprotonation.
2.6 Fabrication of molecule-on-metal films