Estructura multicapa o apilamiento de parches

DFT is a quantum mechanical technique used to model chemical systems from first principles. With the ability to accurately model the relevant geometries, forces and electronic properties within a variety of chemical systems, DFT has proven to be a very useful tool both for complementing experimental results as well as offering predictive capability for systems that have not or cannot be studied experimentally.115-117

In quantum mechanics, the total energy of a chemical system can be determined by solving the time-independent Schrödinger equation (TISE). However, apart from only a few simple systems (e.g. a hydrogen atom), the solution of the TISE contains electron-electron interactions and thus cannot be solved exactly due to the Heisenberg uncertainty principle. Additionally, for many-body systems, the TISE in its pure form is incredibly complex and thus becomes intractable, even to powerful supercomputers.115, 117 _{As a result, to accurately calculate the total energy for many-body systems,} special methods and approximations must be implemented to allow the TISE to be solved.

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because the atomic nuclei are substantially more massive and thus move significantly slower than electrons, it can be assumed that the electrons instantly equilibrate to any changes in the positions of the nuclei.118 _{This allows the TISE to be solved by minimising the electronic energy for a set of fixed} nuclear positions. However, the TISE still contains electron-electron interactions and thus has no exact solution. One approximation to make the TISE solvable is to treat the electronic wavefunction as a linear combination of single-electron orbitals, which forms the basis of the Hartee-Fock method.119, 120 Due to its simplicity, the Hartree-Fock method allows the TISE to be solved at low computational cost but it ignores energy terms from electron exchange and correlation effects and thus calculates inaccurate total-energies. As an alternative, DFT uses a different approach, evaluating the total-energy of the system as a functional of the electron density rather than using the electronic wavefunction.121, 122 _This is possible because under the Born-Oppenheimer approximation, the total energy of the system is a unique functional of the electron density which can be solved self-consistently using the variational principle.122 _{This means that the ground-state energy (and thus the ground-state density) can be found} by minimising the total-energy functional with respect to the electron density. DFT has the advantage that the evaluation of exchange-correlation effects is inherently included in the technique and thus means that the ground-state energies of chemical systems can in theory be calculated exactly. However, the difficulty is that there is no method to find the exact mathematical form of the total-energy functional and thus approximate methods are required to solve the equations.121

One such method is the Kohn-Sham formalism, which considers a fictitious system of non-interacting electrons that is constructed to have the same electron density as its interacting equivalent.123 _{This means} that the kinetic energy of the electrons can be separated into a non-interacting component and a small correction term to account for the difference in kinetic energy between the interacting and non-interacting systems. Within the Kohn-Sham formalism, most of the energetic terms in the many-electron problem can now be solved exactly, with the nuclear-nuclear, nuclear-electron, and electron-electron interactions being solved classically using coulomb potentials. The non-classical electron-electron interactions of exchange and correlation as well as the kinetic energy correction are combined into a functional of the density, however the exact form of this functional is not known and thus is solved approximately. Initially this meant that early use of Kohn-Sham DFT was not accurate enough for quantum chemistry calculations but over time, this was overcome with improved methods of approximating the exchange-correlation functional such as the local density approximation123 _{or the} generalised gradient approach.124 _{Consequently, Kohn-Sham DFT has been considerably successful and} is now widely used to model a variety of chemical systems.

In DFT, a set of functions, known as a basis set, is used to expand the electronic wavefunction to allow for efficient computation.121 _{The basis set can either comprise the linear combination of atomic orbitals,} often used for molecular systems, or plane waves, which are well suited to large periodic systems and are commonly used for solid-state calculations.117, 121, 125 _{For the former, a range of different functions} are used to represent atomic orbitals, including Gaussian-type orbitals, Slater-type orbitals or numerical atomic orbitals, which vary in terms of the extent to which computational efficiency is traded for accuracy.125 _{Another formalism that is useful in DFT calculations is the pseudopotential approximation,} which is used to give a more computationally efficient description of the core electrons and ionic cores of atoms.121, 126 _{In the absence of this approximation, the expansion of the electronic wavefunction incurs} a high computational cost as a large basis set is needed to describe the tightly bound core orbitals and the rapid oscillations of valence electrons within the core region.121 _{Furthermore, as the majority of} physical properties are significantly more dependent on the valence electrons than the core electrons, there is less of a need to explicitly model the core orbitals. Consequently, the system can be modelled to a good approximation at a much lower computational cost by replacing the strong ionic potential with a weaker pseudopotential that acts upon a set of pseudo valence wavefunctions within the core region. 121, 126 _{Outside of the core region, the pseudopotential and the pseudo wavefunctions are identical} to the true potentials and wavefunctions such that the valence electrons are still described accurately. 121, 126

A common feature of DFT codes is the use of periodic boundary conditions, which allows bulk materials to be studied effectively using only a small simulation cell. The periodic boundary conditions take advantage of Bloch’s theorem which states that for a particle in a periodically-repeating environment (e.g. electrons in a crystal), the energy eigenstates will have the same periodicity as the repeating lattice.127 _{In effect, when modelling a bulk system, this transforms the problem of calculating an infinite} (or sufficiently large) number of electronic wavefunctions across an infinite simulation cell, to one of calculating a finite number of electronic wavefunctions at an infinite number of sampling points, known as k-points. Conveniently, due to the wavefunctions at nearby k-points being very similar, it is possible to accurately represent the system by sampling only a few k-points.121

When modelling surface structures of adsorbed atoms and molecules, periodic boundary conditions have two important implications. Firstly, as a surface results from the termination of a bulk material in one dimension, the simulation cell must include a vacuum gap to separate the atoms from their periodic projections. This means that the surface is simulated as a slab that is only infinite in the two dimensions within the surface plane and contains only the number of atomic layers explicitly included in the simulation cell perpendicular to the surface. Secondly, the periodic boundary conditions require the

incommensurate adsorption structures cannot be explicitly studied using DFT calculations. Instead, incommensurate structures are often approximated by constraining the model to a similar commensurate unit mesh to make them accessible to DFT.56, 128, 129

An important aspect of calculating accurate adsorption structures with DFT is the consideration of dispersion interactions. An issue with many of the standard functionals used in DFT is that they do not describe long-range dispersion interactions, which can result in inaccurate adsorption structures.27 _This problem is particularly pertinent for organic adsorbates, for which dispersion interactions are a key determining factor of the adsorption structure. To combat this, a range of functionals that include dispersion corrections have been developed,27, 29, 130, 131_{however, the efficacy of these dispersion} corrections can be highly system-dependent and thus it is difficult to determine a priori which correction is most appropriate to use.31, 33, 34 _{Consequently, there is currently an ongoing effort to benchmark DFT} dispersion corrections using experimental structural measurements.27, 29, 130, 131

In section 2.3, it was discussed how, to a reasonable approximation and with relatively small (< 2 V) applied biases, the tunnelling current in an STM experiment depends primarily on the sample DOS. As DFT works with the electron density, the DOS can readily be evaluated from the results of calculations. As a result, DFT can be used to simulate STM images for calculated structures, which is often achieved using the simple formalism of Tersoff and Hamann.132 _{In the Tersoff-Hamann approach, the unknown} electronic structure of the tip is approximated as an atomic s orbital and in doing so, the expression of the tunnelling current (equation 2.3.5) becomes:

where 𝜌_𝑆(𝑧, 𝜖) is the sample DOS at the position of the tip apex, z.133 _{A simulated STM image can then} be constructed by calculating the spatial variance of the sample DOS for a given energy window. Despite the fact that the Tersoff-Hamann approach cannot explain the origin of the atomic-resolution that is achievable with STM, it has been shown to produce reasonably good qualitative predictions of experimental measurements.77, 133 _{However, care must be taken as it is well-established that the nature} of the tip can have a significant impact on STM image contrast, which, due to the lack of an explicitly defined tip, cannot be analysed within the Tersoff-Hamann approach.

𝐼~ ∫ 𝜌𝑒𝑈 _𝑆(𝑧, 𝜖)

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