Density-functional tight-binding approach for metal clusters, nanoparticles, surfaces and bulk: application to silver and gold

The density-functional based tight-binding method is an approximate DFT scheme in the Kohn–Sham orbital based formalism. The method is briefly presented below, as detailed description of the DFTB basics can be found in the literature, either in the original founding papers [190–192] or in more recent reviews [193–196]. It relies on the use of an atomic minimal basis set $\{\phi_\mu\}$ to express the molecular orbitals $\{\Psi_i\}$ (MOs):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Psi_i = \sum_\mu c_{i\mu} \phi_\mu \nonumber \end{align} \tag{ 1 }$$

where $c_{i\mu}$ denotes the expansion coefficients.

Following the idea of Foulkes and Haydock [197], the DFT energy is expanded around a reference density $\rho_0$. When the expansion is limited at first order, the expression of a tight-binding model is recovered. This is often referred to as the zeroth-order DFTB [190, 191] as well as DFTB1. This original approach was further refined by introducing a second-order term [192] in the Taylor expansion. This is referred to as second-order DFTB, self-consistent-charge DFTB (SCC-DFTB) or DFTB2 which corresponds to the following energy expression:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E^{\rm SCC\mbox{-}DFTB} = E_{\rm rep} +\sum_{i,\mu,\nu} n_i c_{i\mu} c_{i\nu} H^0_{\mu\nu} + E^{(2)}\label{eqedftb} \nonumber \end{align} \tag{ 2 }$$

which consists of three terms, namely a repulsive contribution, a so-called band energy term and a second-order term. In this expression, the sum of the first two terms contains the zeroth- and first-order terms of the Taylor expansion, i.e. correspond to zeroth-order DFTB. All the studies performed on gold and silver that are discussed in the present review were performed at the SCC-DFTB level. However, in the following, in order to simplify the notation we will simply refer to DFTB.

In the second term (band energy term) of equation (2), n_i is the occupation of orbital i and $H^0_{\mu\nu}$ is the DFT Kohn–Sham operator matrix element computed at the reference density $\rho_0$ and expressed in a localized atomic orbital basis (LCAO). The reference density is taken as the superposition of the isolated atomic densities. Neglecting both three- and four-center integrals and pseudo-potential contributions to two-center terms [193] leads to the following expressions for matrix elements between μ and ν atomic orbitals belonging respectively to atoms α and β:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H^0_{\mu\nu} = \left\{\begin{array}{@{}ll@{}} \epsilon_\mu \delta_{\mu\nu} &{\rm if } ~ \alpha = \beta \nonumber \\ \langle \phi_\mu |T+V(\rho_0^\alpha)+V(\rho_0^\beta)|\phi_\nu\rangle & {\rm if } ~ \alpha\neq \beta \end{array} \right. \label{eqH} \nonumber \end{align} \tag{ 3 }$$

which corresponds to the potential superposition formulation although a density additivity scheme can also be followed [192].

In equation (3), $\newcommand{\e}{{\rm e}} \epsilon_\mu$ is the atomic orbital energy of the isolated atom. The two-center term, which is a function of the inter-atomic distance, is determined and tabulated from DFT calculations on the atomic pair. In practice, the values of the two-center terms and of the overlap integrals are stored in the so-called Slater–Koster tables [198] for each atomic pair and over a range of diatomic distances. The implementation of the Slater–Koster tables for a given chemical system is often referred to as the parametrization of the DFTB potential. It may be subject to some empirical modifications as discussed in the following section.

The second-order term in equation (3) is expressed as a function of atomic charges $q_\alpha$ as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E^{(2)} = \frac{1}{2} \sum_{\alpha,\beta} \gamma_{\alpha\beta} q_\alpha q_\beta . \nonumber \end{align} \tag{ 4 }$$

The onsite term $\gamma_{\alpha\alpha}$ should be identified with the Hubbard parameter or chemical hardness. The two-center term $\gamma_{\alpha\beta}$ mostly contains the $1/R_ {\alpha\beta}$ Coulomb interaction screened at short distances by a contribution of the exchange-correlation second-order derivative. Its complete expression is presented in the seminal paper by Elstner et al [192]. In the standard formulation, atomic charges are computed from the MO coefficients through the Mulliken scheme. It is worth mentioning that the second-order contribution introduced in the DFTB Kohn–Sham operator depends on the MO coefficients. This leads to a variational problem which is solved self-consistently.

Finally, all the zeroth-order remaining terms are collected in a repulsive contribution E_rep which is expressed as a sum of pair potentials $E^{\alpha\beta}_{\rm rep}(R_{\alpha\beta})$. Technically, the repulsive part is usually parametrized by fitting the difference between the contributions of the other two terms (band energy and second-order terms) and total energy reference calculations.

Since the derivation of these equations, there have been a number of developments devoted to the improvement and extension of the applications of DFTB. Some consisted in transferring DFT improvements to the DFTB scheme, others are more specific to the DFTB approximations. In connection with the electronic structure description, significant advances are the unrestricted formalism [199], the implementation within periodic boundary conditions and band structure calculations [194], the refinement of atomic charges to go beyond the Mulliken definition [200, 201], inclusion of dispersion contribution [202–204] corrections within the phenomenological Grimme scheme [205], the extension of DFTB to third-order (referred to as DFTB3) in the Taylor expansion of the energy [206] which introduces a charge dependence of the Hubbard parameter, the development of a hybrid short-range/long-range scheme allowing for long-range Hartree–Fock contribution corrections [207], the combination with configuration-interaction methods [208] and the time-dependent DFTB (TDDFTB) approach to treat excited states [209]. Hybrid quantum-mechanical/molecular-mechanical (QM/MM) as well as QM/QM formalisms to incorporate the effect of a chemical environment at low computational cost have also been developed [210, 211]. In the field of dynamics and finite-temperature approaches, one may cite extensive molecular dynamics techniques such as the Car–Parrinello scheme [212, 213] and the coupling of DFTB with global exploration schemes such as genetic algorithms [214], parallel-tempering molecular dynamics (PTMD) [215–219] or basin-hopping techniques [144, 220, 221] or the determination of minimum-energy paths via the nudged elastic band (NEB) algorithm [222]. Calculation of free energies via metadynamics [223–227] or determination of heat capacities with the multiple-histogram method [162, 219, 228, 229] are also available. Techniques to deal with electron dynamics and its possible coupling with nuclei have also been implemented within various schemes such as the Liouville formalism for plasmonics [230], non-adiabatic excited-states dynamics [231, 232], dynamics of electronic transport and its coupling with phonons [233, 234].

The strength of DFTB is to provide both an ab initio justification to parametrized tight-binding schemes and a well-defined strategy to derive the Slater–Koster tables. However, their implementation contains a certain amount of freedom. Indeed, the atomic basis used to compute the two-center matrix elements consists of contracted orbitals, obtained from DFT atomic calculations, and defined by a confinement potential. The choice of this confinement potential can differ between different parameter sets. Recently, it has even been used as a free parameter to improve DFTB properties, for instance electronic energy bands in materials [235]. Different DFT functionals can also be used to achieve the parametrization. Furthermore, DFTB parameters can be further processed and adjusted to improve results with different amounts of empirical intuition, for instance by shifting orbital energies [236] or correcting the short range behavior of matrix elements due to the use of confined orbitals [237, 238]. Such procedures differ from the canonical derivation of the DFTB parameters, but, in a pragmatic way, it can be done based on chemical intuition to correct, for instance, the effects of a too much contracted basis. Such arbitrary adjustments may appear as a disregard of the DFTB theoretical footings and its ab initio character. However one should stress that: (i) the use of approximate DFT functionals or the potential superposition approximation are already a strong deviation from a pure ab initio formalism; (ii) from a pragmatic view, other tight-binding schemes (parametrized empirically) are even less related to first-principles methods.

Another strength of DFTB over other tight-binding schemes is its transferability. Indeed, its theoretical footings make the Slater–Koster table of a given atomic pair a priori transferable from one system to another. Although this is true in essence, this would require heavy fitting of the repulsive part for different combinations of atoms [239], and the parameters themselves are only transferable to other systems if all the necessary Slater–Koster tables are developed in concert with one another. In some specific cases, it is even not possible to develop a DFTB potential that is transferable to chemically different phases of the same element. For instance, Wahiduzzaman et al pointed out the difficulty to implement a carbon-carbon potential accurate enough for both diamond and other carbon phases [235].

Turning now to the specific case of parametrization for silver and gold, the first question relies on the choice of the minimal atomic basis to be used, which of course presents similarities as the two atoms belong to the same group in the periodic table. While the valence atomic orbitals of silver (resp. gold) are the 5s and 4d (resp. 6s and 5d) orbitals, it is relevant to also include 5p (resp. 6p) orbitals, as their radial extension overlaps that of the atomic valence ones. Accounting for relativistic effects is necessary for heavy atoms like gold. For instance, important differences observed for gold and silver clusters involve the critical size for the 2D/3D transition, which occurs at larger sizes in the case of gold as compared to silver. This has been attributed to relativistic effects [74, 240] which strongly modify the valence atomic energies and the hybridization patterns of gold versus silver: dominant s-p mixing in silver favors 3D structures, whereas s-d hybridization in gold favors planar patterns in small clusters. This point is further discussed in section 4.1. At the DFTB level, the parametrized quantities, i.e. atomic orbital energies and two-center Slater–Koster integrals [241], are obtained from scalar-relativistic DFT calculations [242]. Scalar-relativistic effects are thus incorporated in an effective way within the the DFTB matrix elements of equation (2) without resorting to explicit scalar-relativistic operators.

For gold, two parameter sets were derived ten years ago. The first set was proposed by Koskinen et al [241] who showed the importance of relativistic corrections for DFTB to be able to reproduce the correct 2D/3D transition. The role of relativistic effects was also investigated recently in the context of liquid gold membranes [243] by comparing this parameter set with another one derived through the non-relativistic standard procedure [194]. The relativistic parametrization set has been refined later on to achieve better representation of bulk properties through a new definition of the repulsive contribution [244]. The second set was derived by Dong and Springborg [214] in order to conduct extensive exploration of the potential energy surfaces of neutral gold clusters [245]. More recently, in the context of gold interaction with biomolecules, Fihey et al [246] derived a third set of parameters for gold ('auorg set', available at www.dftb.org). These parameters were derived with the aim of accurately modeling both gold bulk and surfaces. They were latter shown to reproduce adsorption energies of biomolecules on gold in good agreement with DFT calculations [247].

For silver, a set of parameters has been derived by Szűcs et al [248] with the initial focus of investigating the adsorption of PTCDA on a silver surface. This parameter set was referred to as 'hyb-0-1 set' at www.dftb.org and has latter been replaced by its extension, 'hyb-0-2 set' (available from www.dftb.org), that can be used for TDDFTB calculations (the two sets of parameters are otherwise identical). These parameters have been used to investigate Si-doped Ag clusters [249], plasmon phenomena on naked and capped silver clusters of different sizes [230, 250–252] and very recently optical properties of silver nanorods, in particular Ag₅₅ nanorod dimers [253]. Sun et al [254], who correlated the Fermi energy and electronic gaps of large clusters with size and geometrical considerations, also benchmarked these parameters with respect to DFT calculations for large clusters.

Oliveira et al recently investigated the quality of some of the aforementioned parameter sets for silver and gold clusters and bulk [255]. In the case of gold, a small modification of the 'auorg set' [246] was found to strongly improve the description of homogeneous small clusters properties while yielding a good description of the gold bulk. The modification consisted in a slight shifting of the energy value $\newcommand{\e}{{\rm e}} \epsilon_p$ to increase the gap with $\newcommand{\e}{{\rm e}} \epsilon_s$ and $\newcommand{\e}{{\rm e}} \epsilon_d$ and reduce the hybridization of s/d with p orbitals. This highlights that the onsite energies $\newcommand{\e}{{\rm e}} \epsilon_\mu$ (see equation (3)) of unoccupied atomic orbitals are much less straightforward to define than the onsite energies of the occupied atomic orbitals. Concerning the silver parameters ('hyb-0-2 set' [248]), a slight modification (a scaling of the interatomic Hamiltonian matrix elements by a factor of 0.9) was shown to correct the too early 2D–3D transition in clusters and the overestimation of binding energies and bulk lattice parameter [255].

In addition to the previously mentioned parametrization works, which aimed at deriving parameters from fitting DFT matrix elements and energies of diatomics, some groups have also tried to obtain parameters for a wide variety of atoms, including silver and gold, via automatized procedures aiming at reproducing particular properties. For instance, Chou et al [256] used a particle swarm optimization technique [145] to derive parameters able to reproduce specific properties from reference data. In another approach, Wahiduzzaman et al [235] developed a strategy based on the optimization of the atomic basis confinement potential to reproduce reference band structure calculations. At the moment, electronic terms, i.e. band energy terms, have been derived for silver and gold but not the repulsive ones [239].

As mentioned above, there are strong structural differences between silver and gold clusters. An interesting challenge for DFTB in the field of small clusters was thus to check if it was able to reproduce such differential features. The first DFTB investigations of the structural properties of negative gold clusters Au$_n^-$ were reported by Koskinen et al [241], showing that DFTB with such parametrization yields planar structures up to size n  =  14. They also reported vertical ionization potentials and electron affinities. Dong and Springborg [214] used an optimization scheme based on genetic algorithms to determine the structure and stabilities of clusters in the range 3–58. However, they found the occurrence of 3D-structures already for Au₇, which seems much too early with respect to WFT and DFT calculations [41, 91]. We have however to keep in mind that some WFT methods and DFT functionals can also lead to discrepancies as compared to highly accurate WFT based methods such as the coupled cluster scheme as illustrated by Götz and co-workers in the case of Au₁₀ [257]. Dong and Springborg also obtained for Au₂₀ a shape different from the highly symmetric pyramidal shape obtained in DFT and evidenced experimentally [81]. Using an original version of DFTB with a non self-consistent perturbative approach of the on-site charge fluctuations, Rincon et al [258] investigated the structure and energetics of neutral Au_n clusters in the range 3–20, quenching a variety of crystal type configurations. Their structures were in quite good agreement with previous DFT results, in particular the onset of 3D structure at size 12, empty cage-like structures for Au₁₆ and Au₁₇ and a pyramidal twentymer Au₂₀ with high T_d symmetry, as a subpiece of the fcc (face-centered cubic) bulk lattice for gold. Very recently, Oliveira et al [255] published a systematic work about silver and gold clusters in the range 3–20, conducted with a global minimization scheme based on PTMD. Neutrals, cations and anions were systematically analyzed in order to check the ability of DFTB to cover the different charge situations and the structural modifications resulting from ionization or electron addition in the small cluster range. This work showed several positive features of DFTB and in particular its ability to behave very differently (and satisfactorily) for silver and gold: (i) while the 2D–3D transition occurs at n  =  7 in silver clusters whatever the charge, it occurs in the range n  =  8–12 for gold clusters (see figure 1), with an increasing transition size from n  =  8 for the cations, n  =  10 for the neutrals, up to n  =  12 for the anions. Thus DFTB correctly accounts for the charge effects in gold clusters, providing structures and energies in agreement with first-principles methods; (ii) DFTB correctly accounts for the preferential trend of silver clusters to build up around five-fold motifs and icosahedral patterns while fcc motifs and the trend to planarity have a major importance for gold clusters. The investigation of Tarrat et al [259] thus recovered pyramidal structures for all three clusters Au₂₀, ${\rm Au}_{20}^+$ and ${\rm Au}_{20}^-$, with some Jahn–Teller distortions for the ions, in consistency with the DFT work of Furche et al [148]; (iii) The same investigation determined some electronic properties of gold and silver clusters, in particular the ionization potentials and the electron affinities (see also the early work of Koskinen et al [241]). A good quantitative agreement between DFTB and DFT was obtained for the vertical detachment energies (VDEs) from the gold and silver anions. The vertical ionization potentials (VIPs) of the neutral silver clusters were also found to be quite consistent with the available experimental data, while the agreement with experiment for the VIPs of gold clusters was more disappointing, the DFTB VIPs decreasing somewhat too quickly with increasing sizes in the range n  =  3–14. Figures 2 and 3 show the adiabatic IP and EA values for silver and gold respectively, determined from relaxed structure of all charge configurations (neutral, cationic and anionic clusters) in the range n  =  3–561. Those curves are in good qualitative agreement with the experimental data presented by Wesendrup et al [260]. In particular, the higher IP and EA of gold over silver clusters is rather well reproduced by DFTB up to n  =  20. Finally another interesting feature was the isomerization gap in Au₂₀, which was found to be much larger than in its ionized analogues, while the various charge states of Ag₂₀ exhibit a quasi-degeneracy behaviour with a large number of isomers within a small energy range 0–0.3 eV (even narrower for the anion and the cation). Finally, one may cite a very recent work [221] combining basin-hopping search with DFTB followed by DFT and finding results in essential agreement with those of Tarrat et al [259] for neutral gold clusters.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For larger species encountered in the nanoparticle size range, an important issue explored in the recent years has been the ability of Au NPs to exhibit or not regular structures as a function of their size. Dong and Springborg [214] were the first to provide a detailed description of the structural patterns and suggested the ability of gold to form low coordinated non-regular systems, such as planar and shell like substructures. Yen et al used a modified basin-hopping optimization algorithm [220] and predicted that for Au₄₀ a twisted pyramidal structure (C₁ symmetry) should be more stable than a symmetrical tetrahedral structure (C₃ symmetry) [221]. From global optimization performed using the PTMD algorithm, Tarrat et al [259] pointed out that disordered Au₅₅ structures are much more stable than highly symmetric isomers such as the icosahedron, the cuboctahedron and the decahedron, confirming the EAM work of Garzon et al [165]. They also demonstrated the presence of planar patterns at the clusters surfaces as well as inner cavities (figure 4). These results were confirmed at the DFT level and are qualitatively consistent with the STEM observations of Wang and Palmer [154]. Concerning larger Au NPs, the same approach was used to investigate the competition between ordered and disordered Au₁₄₇ isomers (figure 5) and evidenced that the amorphous ones are relevant low-energy candidates expected to contribute in finite-temperature dynamics [101].

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The structure of Ag NPs has been much less studied at the DFTB level than the gold one. DFTB has been used to investigate the structure of Ag₅₅, which remains nearly icosahedral for cations, anions and neutrals [259]. This difference in the symmetric nature of silver versus gold NPs was found in line with experiments which suggest highly symmetric structures for silver and more disordered ones for gold [79, 150, 154]. Very recently, DFTB has also been used in the area of machine learning to establish the relationships between the structure and the morphology of silver NPs (with diameters up to 4.9 nm) and their electron transfer properties [254].

Finally, as DFTB is able to optimize particles with a few hundreds of atoms or more, it can be used to examine the consistent convergence of the cohesive energy of gold and silver NPs with increasing sizes towards the bulk value. A two-parameter analytical extrapolation of the cohesive energy [255] was proposed as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \epsilon_{\rm coh}(N) = \epsilon_v(\sum_k (c_k/c_v)^\alpha) /N\label{eqfit}. \nonumber \end{align} \tag{ 5 }$$

This formula assigns specific cohesion energies $\newcommand{\e}{{\rm e}} \epsilon_k = (c_k/c_v){}^\alpha\epsilon_v$ to the atomic sites, depending on their coordination c_k, possibly smaller than the coordination $c_v = 12$ of the volume atoms characterized by a cohesive energy $\newcommand{\e}{{\rm e}} \epsilon_v$. DFTB calculations of cuboctahedral particles with 55, 147, 309 and 561 atoms (compatible with the fcc bulk crystal structures) were used to fit α and $\newcommand{\e}{{\rm e}} \epsilon_v$ and hence determine the surface, edge and apex energies (table 1). Regular clusters offer a variety of situations and can provide simultaneous extrapolated information about site energies of facets with various surface symmetries (1 0 0/1 1 1) but also possibly about surface defects with less coordinated surface atoms.

Table 1. Coordination (ck) and site cohesive energies $\newcommand{\e}{{\rm e}} \epsilon_k$ (eV) of cuboctahedral structures of gold and silver clusters derived from the interpolation of large clusters binding energies. Reprinted with permission from [255]. Copyright (2016) American Chemical Society.

Element	Site	ck	$\newcommand{\e}{{\rm e}} \epsilon_k$
Ag cubo
	Volume	12	3.022
	Surface(1 1 1)	9	2.600
	Surface(1 0 0)	8	2.445
	Edge	7	2.280
	Apex	5	1.913
Au cubo
	Volume	12	3.427
	Surface(1 1 1)	9	3.254
	Surface(1 0 0)	8	3.185
	Edge	7	3.110
	Apex	5	2.926

As mentioned above, DFTB is well suited for dynamical applications. The analytical gradient is available and makes on-the fly simulations extremely efficient. It allows extensive sampling of initial conditions and rather long trajectories, providing statistically meaningful results in the exploration of dynamical processes such as isomerization, fragmentation or nucleation. Temperature effects may be particularly significant in the case of clusters where many low-energy isomers are in competition. Extensive sampling is also needed to investigate thermodynamical properties and achieve significant thermal averages. DFTB highlighted several very specific dynamical features, connected with finite-size effects and/or the propensity of gold clusters for planar architectures and shapes. Koskinen et al [157] investigated the dynamical coexistence and conversion of 2D and 3D negatively charged liquid clusters Au$_n^-$ in the range n  =  11–14 around temperature T  =  750 K. They demonstrated that although the 2D structures in this size range are lower, the systems exhibit coexistence. This coexistence is related to the fact that the barriers between 2D and 3D minima of the landscape are relatively low, while the barriers between 2D isomers are much higher. Moreover, they also showed that the entropy of the 3D phase is larger and that supercooling may drive the clusters (${\rm Au}_{13}^-$ and ${\rm Au}_{14}^-$) into metastable energetically higher minima of the 3D phase. Such entropic effect is likely to influence the 2D–3D transition thresholds in mobility experiments.

DFTB was also used to address the thermodynamics of clusters. From the harmonic spectra frequencies of gold clusters, Dong and Springborg calculated the low temperature heat capacities and investigated the possible occurrence of a Boson peak [214] in such small clusters. In a very recent work, Rapacioli et al [162] investigated the thermodynamical properties of the charge states of Au₂₀ from canonical PTMD simulations and the multiple histogram method to determine the heat capacities. They demonstrated that, despite the fact that the ion, the cation and the neutral all have pyramidal and very similar geometries, the canonical heat capacity curves can significantly differ from one another, the melting temperatures of the ions being shifted towards smaller values with respect to the neutral. For the ions, the melting process was shown to take place on a temperature range broader than for the neutral, demonstrating for the magic gold twentymer the dependence of melting upon charge. The low energy landscape of figure 6, which reports the isomerization paths between the lowest energy minima, illustrates the early displacements in the melting processes of the cationic gold cluster ${\rm Au}_{20}^-$. In the case of Au₂₀, the DFTB simulation also showed the possible occurrence of a convex intruder in the entropy and a negative microcanonical heat capacity within a narrow temperature range, typical of the thermodynamics of finite systems [228, 264, 265].

Zoom In Zoom Out Reset image size

In addition to those fundamental studies, one may also cite the finite-temperature rearrangement of the electrolyte in the vicinity of gold electrodes that has been studied through a model consisting of a gold cluster with 55 units surrounded by three water layers (640 molecules). A dependence of the orientation of the water molecules was observed as a function of the gold cluster surface charge, this polarisation being limited to the first solvation shell [266].

One very elegant and promising application of DFTB to gold and silver NPs concerns the modeling of their plasmonic properties. Indeed, as stated in introduction, the field enhancement occurring around a NP due to the localized SPR effect is a subject of considerable ongoing studies. To address this question from a theoretical point of view, a variety of methods have emerged [170]. Among the quantum mechanical methods, TDDFT in the linear-response formulation of Casida [63] has become a very popular tool to compute optical properties of small metal NPs (see section 2). However, it becomes computationally very demanding for systems containing hundreds of metal atoms when electron dynamics, i.e. time evolution of the electrons, has to be considered. To overcome this limitation, the linear response to an electric field of large NPs has been studied by the group of Sánchez through the combination of (i) DFTB to determine the initial single electron density matrix of the considered systems and (ii) the Liouville–Von Neumann equation of motion to describe the time evolution of the density matrices after the application of an external field [250]. As a first application, the field enhancements around two Ag_n NPs (n  =  309 and 489) and a dimer of two Ag₂₀₁ NPs obtained with this approach and a classical model were calculated. Both formalisms are in qualitative agreement. Nevertheless, the real-time excited-state DFTB scheme provides atomistic information that would be of key importance to further understand the interaction of a molecule with the enhanced field. The performance of this approach for studying the temporal near-field response of the Ag NPs to an arbitrarily shaped electric field was also demonstrated. Very recently, the same methodology allowed Sánchez and co-workers to demonstrate the existence of sub-picosecond breathing-like radial oscillations in silver NPs containing up to 309 atoms [230]. Those oscillations, which start immediately after laser pulse excitation, occur without electron–phonon scattering that takes place at a longer time scale.

A similar methodology combining an initial evaluation of the electronic ground state at the SCC-DFTB level and a real-time propagation of the density matrix using the Liouville–Von Neumann equation of motion was conducted by Ilawe and co-workers to characterize the excitation energy transfer (EET) mechanism along chains of Ag NPs [252] (a similar study was also performed by the same group on a system composed of four Na₅₅ icosahedral NPs [267]). This transfer is made possible by local SPR effects and is of interest to achieve energy and information transfer at high speeds. Thank to the computational efficiency provided by the SCC-DFTB approach, the authors probed the energy transfer along chains of icosahedral Ag₅₅ NPs as a function of interparticle distance (see figures 7(a) and (b)) between the NPs (from 0 to 5 Å). In the range 2–5 Å, SCC-DFTB results are in qualitative agreement with classical electrodynamics. In contrast, below 2 Å, it predicts a strong drop in EET efficiency as demonstrated in figure 7(c), which results from interparticule charge transfer. This is a purely quantum effect that cannot be captured by classical schemes, highlighting the important contribution of SCC-DFTB to probe plasmonic effects at the atomic level.

Zoom In Zoom Out Reset image size

Very recently, Alkan and Aikens used the TDDFTB formalism to conduct a study on silver nanorods and nanorod dimers [253]. In this study, the absorption spectra of Ag_n nanorods with sizes ranging from n  =  19 to 67 were first calculated at the TDDFT and TDDFTB levels. TDDFTB produces very similar spectral shapes as compared with TDDFT and predicts the same trends when the nanorod size increases. TDDFTB made possible the calculation of absorption spectra for nanorods up to n  =  985. The authors then determined, at the SCC-DFTB level, the evolution of the fractional shifts (defined as the ratio between the change in excitation wavelength and the monomer excitation wavelength) in two different configuration of nanorod dimers, end-to-end or side-to-side, as a function of the separation distance between the nanorods. Three sizes were considered for the monomers: Ag₁₉₉, Ag₃₉₁ and Ag₇₉₉. The results were in good agreement with experimental and classical electrodynamics data previously published in the literature. In addition, the authors unravelled that the change in the excitation wavelength, resulting from the coupling between the two nanorods, depends on the nanorod length.

4.5.1. Gold bulk.

4.5.2. Silver bulk.

The application of DFTB to gold-organic species fall into two main fields: (i) electronic conduction through molecular junctions and simulation of inelastic electron-tunneling spectroscopy (IETS) spectra and (ii) surface chemistry and physics. Both of them take advantage of both DFTB computational efficiency and reasonable accuracy (as compared to DFT) in order to tackle questions hardly tractable at the DFT level.

5.1.1. Electronic conduction through molecular junctions and IETS spectra.

5.1.2. Surface chemistry and physics.

In addition to molecular electronics and IETS, DFTB was also used to investigate the structural, energetical and dynamical properties of various surfaces such as molecules attached on gold surface or gold deposited on inorganic surfaces.

Mäkinen et al were the first to develop a DFTB parametrization to describe thiolates attached to gold clusters [244]. Such systems are of primary importance as thiolates are often used to stabilise gold NPs. The DFTB parametrization was validated on three systems: Au₂₅(SMe)$_{18}^{-1}$, Au₁₀₂(SMe)₄₄ and Au₁₄₄(SMe)₆₀ (see optimized DFTB structure in figure 8) for which both experimental and DFT structures were available and on Au₁₀₂(p-MBA)₄₄ (p-MBA  =  para-mercaptobenzoic acid). The authors demonstrated that this DFTB parametrization is accurate enough to look for low-energy configurations of other Au_n(SMe)_m species and to qualitatively describe their electronic structure. Later on, Fihey and co-workers developed another set of DFTB parameters describing the interaction of gold with other atoms: Au-X (X  =  Au, H, C, S, N, O), in the view to improve the description of the interaction of thiolates and other molecules with gold NPs and surfaces [246]. To test and validate those parameters, the authors considered two model systems: Au₃SCH₃ and Au₂₅SCH₃. For each species, geometrical parameters, density of states, relative energies of four distinct adsorption sites (bridge, hollow hcp, hollow fcc, top) and the potential energy profiles leading from one site to the other, were calculated at both the DFT and DFTB levels of theory. The two sets of data showed good agreement which validates the use of DFTB in the description of gold-thiolate systems.

Zoom In Zoom Out Reset image size

In a similar spirit, Castro and co-workers very recently used the DFTB method to describe the interaction of gold clusters with various amino-acids [247]. They showed that DFTB leads to geometries and adsorption energies in good agreement with DFT results. Furthermore, an analysis of the molecular orbitals suggested an electron-donor character for the proline molecule and an electron-acceptor character for the gold cluster. Analysis for arginine, arginine dipeptide, and TAT peptide were also performed to describe the bonding between those molecules and gold clusters.

Taking advantage of the accuracy and efficiency of the DFTB formalism, Koskinen and Korhonen simulated the dynamical properties of a 2D gold patch suspended in a graphene pore [243]. Au_n patches of various sizes were considered: n  =  39, 40, 47, 48, 49, 64, 256 and 400. The structure of an Au₄₉ patch is displayed in figure 9(a) as a visual example. Although all patches display a stable solid phase at low temperature, the authors showed that from 49 to  ∼256 gold atoms, a stable liquid behaviour of the patch is observed over several hundreds of kelvin (see figure 9(b) a snapshot of the Au₄₉ patch at 900 K). For smaller patches: Au₃₉, Au₄₀, Au₄₇ and Au₄₈, rupture is observed before melting (see a snapshot of Au₄₈ in figure 9(c)), even with fewer atoms in the patch (see the low-temperature structure of Au₃₉ in figure 9(d)). For larger n, large out-of-plane fluctuations lead to the rupture of the gold membrane. This picture was qualitatively confirmed by DFT calculations performed on Au₆₄ only, as extensive molecular dynamics simulation for various patch sizes is computationally too demanding at that level of theory. As already mentioned for small gold clusters (see section 4.1), the two-dimensional character of various gold species results from the impact of relativistic effects on the 6s and 5p orbitals of gold. In the present context, Koskinen and Korhonen showed using both DFT and DFTB simulations that the liquid character of 2D gold patches also partly results from relativistic effects. Consequently, such peculiar behavior is not expected in other metals.

Zoom In Zoom Out Reset image size

Finally, one has to mention the coupled experimental/DFTB studies of Nilius and co-workers on the characterization of single layer gold islands (containing up to 200 Au atoms) deposited on a MgO/Ag(001) surface [301, 302]. In those systems, electron transfer occurs between the surface and the gold islands that carry extra electrons which promote the two-dimensional growth of gold. Based on a structural model derived from STM images, DFTB calculations performed on a model Au island demonstrated that the excess charges are localized on the island perimeter in full agreement with the experimental ${\rm d}I/{\rm d}V$ maps. This peculiar localization explains the formation of elongated belt-like Au structures. From an electronic-structure point of view, DFTB calculations also reproduce the existence and characteristic node structure of quantum-well states in the Au islands [302]. This point was further analysed as a function of the island size and shape which were related to the HOMO-LUMO gap of the gold aggregate and its metallic character [301].

As stated in section 3.2, a first DFTB potential for silver was developed in 2004 [248]. However, in contrast to gold, fewer studies were further conducted for functionalized silver clusters, NPs or surfaces. Nevertheless, some important DFTB studies have been performed over the last years and are now briefly summarized.

Szűcs and co-workers were the first to use DFTB to study tunneling current between a silver tip and a sulfur-passivated GaAs surface [248]. Two systems were considered: A bare surface and a surface covered by a monolayer of PTCDA (3,4,9,10-perylene tetracarboxylic dianhydride) molecules in order to understand the conduction properties of a representative organic-inorganic semiconductor Schottky-contact. This study was performed using the same gDFTB formalism discussed in the previous section and correctly reproduces the impact of the adsorbate on the tunneling current [248].

Following the methodology applied in section 4.4 for bare silver NPs, Douglas-Gallardo, Berdakin and Sánchez studied the influence of two adsorbates, water and 1,4-benzoquinone, on the SPR band of silver NPs of various sizes [251]. The excitation energy and line width of this SPR band are spectroscopic features of paramount importance in the application of plasmonic NPs as presented in the introduction of the present review. However, those optical properties are strongly sensitive to the size [1, 27], the morphology [1, 27, 28], and the chemical environment [27] of the NP which are properties difficult to probe experimentally. Douglas-Gallardo et al thus intended to rationalise size-effects in five different clusters: Ag₅₅, Ag₁₄₇, Ag₃₀₉, Ag₅₆₁ and Ag₉₂₃ in three different chemical environments: bare and capped with quinone or water (see for instance the DFTB structure of Ag₃₀₉ surrounded by 16 1,4-benzoquinones in figure 10(a)). They showed that there exists a linear relationship between the surface plasmon excitation energy and the inverse cube root of the cluster size. Furthermore, the impact of the adsorbate on the spectroscopic features (see figure 10(b) for the case of Ag₃₀₉ interacting with 1,4-benzoquinones) were analysed in terms of electronic structure changes in the particle resulting from the interaction with the adsorbate. This study thus provides a realistic atomistic scale description of the impact of the chemical environment on the SPR properties of a NP. Of course, real-time excited-state dynamics is rather expensive at the DFT level of theory even for small systems. Consequently, the DFTB approach conducted here provided the correct balance between computational cost and accurate description of the electronic structure to tackle such kind of question that are of high interest in nanoelectronic and optoelectronic.

Zoom In Zoom Out Reset image size

Another appealing study highlighting the contribution of DFTB to better understand surface chemistry of silver is the study of the self-assembly of bisphenol A (BPA) on the Ag(1 1 1) surface by Lloyd and co-workers [303]. In the context of the development of molecular rotors and motors, the authors studied a single-component system built up from BPA molecules displaying a regular two-dimensional network on a silver surface. To provide a complete view of the system, the authors used a synergetic approach coupling STM and NEXAFS experimental measurements with DFTB geometry optimizations of various BPA/Ag(1 1 1) models. The latter were used to rationalize the experimental observations and to provide a description at the atomistic-scale level of the supramolecular arrangement of the BPA molecules. The authors showed that the BPA molecules can arrange differently in dimers and trimers depending on the considered temperature range: from 180 to 250 K and from 250 to 340 K. In the former case, DFTB optimizations confirm that the BPA molecules adopt an L-shape conformation in the dimers with one of the two phenol rings parallel to the metallic surface. In the latter case, the existence of BPA-trimers forming hexagons is also confirmed by the DFTB optimizations. Furthermore, in between 250 and 260 K, both static and rotating trimers are observed by STM (see figure 11(a)) which was interpreted in terms of DFTB optimized structures (see figure 11(b)). The fast development of molecular motors and rotors will make such DFTB calculations more and more attractive and necessary to provide an atomistic-scale understanding of their behavior.

Zoom In Zoom Out Reset image size