O ne of the earliest reports on the anomalous coloration of small metal particles dates to a 700 B.C. Assyrian tablet describing a recipe for red glass by adding gold. 1 Over two millennia later, Michael Faraday attributed this vivid coloration to light scattering in colloidal gold, while Gustav Mie developed the first complete analytical description for such with spherical particles soon after. 2,3 This phenomenon ultimately arises from gold, silver, and many other metal particles' ability to sustain localized surface plasmons at the nanoscale; excitation of these collective oscillations of conduction band electrons produces a strong resonant absorption band in the UV-visible-IR region, giving rise to the brilliant colors found in these colloidal solutions. The plasmon resonance wavelength can be further tuned by modifying the nanoparticle's size, shape, and surrounding dielectric environment, 4,5 offering a versatile means by which one can tailor a material's optical response.
Today, advances in colloidal synthesis and nanofabrication techniques have enabled the creation of nanoparticles with precisely controlled morphologies and consequent optical properties. [6][7][8][9] Com-plementary theoretical studies have revealed detailed correlations between particle characteristics and their optical responses difficult to characterize via experimentation alone. 4,10,11 In turn, the interplay between theoretical analysis and experimental discovery has effected breakthroughs in metamaterial design, [12][13][14] catalysis, 15,16 and ultrasensitive chemical detection. 17 Richard Feynman's 1959 claim that "there's plenty of room at the bottom" heralded the nanotechnology revolution, yet a fundamental tension has persisted in quantum chemistry for the nearly 70 years since: what constitutes "nano" spans orders of magnitude in complexity. Plasmonic phenomena represent a different scale altogether. These collective excitations emerge only in systems containing hundreds to thousands (to even millions) of atoms -massive by quantum mechanical standards -and their picosecond-scale dynamics create a computational hurdle that even our most advanced hardware struggles to navigate. While time-dependent Kohn-Sham density functional theory (TDDFT) stands as the nominal gold standard for analyzing photochemical processes, it hits a wall with particles larger than 1-2 nm and times beyond tens of femtoseconds. 18,19 This scheme becomes intractable for particle sizes on the order of hundreds to thousands of atoms, becoming de facto unusable for many experimentally-observable plasmonic systems. Methods that bring quantum mechanical calculations to longer spans of space and time would thus be valuable to describe the length scales for describing plasmons and the timescales governing their emergent chemistry.
First conceptualized in 1986 20 with extensions implemented soon after, [21][22][23][24] the density functional tight binding (DFTB) approach allows for one to overcome some of these inherent scaling limitations with reasonable accuracy. DFTB has been developed to describe both ground and excited state properties, as are related to Kohn-Sham (KS) DFT and to TDDFT, respectively. In this Feature Article, we highlight how DFTB's excited state formulation can be used to describe plasmons from a quantum level of theory. In many ways, it may seem counterintuitive that such a tight-binding method can be used to describe plasmons, which arise from collective excitations of delocalized electronic states. To reconcile some of these ostensible discrepancies, we briefly discuss the overarching theory behind the self-consistent charge-corrected (SCC) DFTB approach, as well as some underlying approximations and limitations. We then situate this field within the broader context of why a quantum description is necessary to describe plasmons for particles below a critical size and to fully describe plasmon-driven chemistry. We particularly focus on real-time methods used to study electron dynamics under applied electric fields and their implementation within the DFTB framework. While sacrificing some level of quantum mechanical accuracy, real-time time-dependent density functional tight binding (RT-TDDFTB) allows for the investigation of electron dynamics for systems with several hundred (to even thousands of) atoms and at timescales into the picosecond regime. The implementation of this theory is then discussed with respect to understanding the optical properties of individual nanoparticles and their hierarchical structures, with applications thereof to photocatalysis.
Despite originating from KS-DFT, DFTB remains fundamentally a semiempirical method that cannot be expected to match the complete accuracy of higher-level approaches. Its theoretical foundation rests on the assumption that electrons are tightly bound to the nucleus; any interactions (e.g., bonding) between atoms are approximated using empirical parameters that originate from a consideration of the ground state energy. Consequently, DFTB is best suited for highly covalent systems such as hydrocarbons, though many studies have also reported satisfactory performance for more delocalized metallic systems, which we emphasize herein. 25,26 Energetic contributions are categorized through self-consistent charge corrections, short-ranged repulsive interactions, and pretabulated Slater-Koster integrals which describe orbital overlap. 27 The majority of the derivations are presented in the Supporting Information, though we briefly present the overarching equations governing this theory in the following section. We further direct the reader to a complementary tutorial detailing the use of DFTB in plasmonics and nonadiabatic charge transfer dynamics. 28 Throughout these derivations, we use atomic units for simplicity.
For an interacting many-body system, the Hamiltonian H is represented in operator form as:
where T is the kinetic energy, Vext is the energy from an external field (e.g. electron-nuclear contributions), while Vee and Vnn constitute interelectronic and internuclear interactions. KS-DFT transforms this complex interacting system into a fictitious noninteracting problem by fixing the electron density ρ(r). This transformation ensures that the noninteracting system's density remains identical to its interacting counterpart. Consequently, for an electron coordinate r, the energy functional for the Hamiltonian in Eq. 1 is given by
+ Exc + Enn.
In Eq. 2, Ts represents the kinetic energy of the noninteracting system, vext is the one-body external potential, while the third term is the classical Coulomb (or Hartree) energy. Moreover, the exchangecorrelation energy term Exc (and its corresponding potential Vxc) accounts for additional quantum mechanical interactions not explicitly described by classical electrostatic potentials (alongside deviations in kinetic energy between the (non)interacting systems). 29 The final term Enn denotes the energy corresponding to internuclear interactions. The energy density functional E[ρ(r)] is then variationally minimized with respect to the orbital wavefunction ψi[ρ] to yield the fictitious, noninteracting KS wavefunctions. Writing the corresponding single particle equations then gives:
(3) Consequently, explicitly writing out each term for Eq. ( 2) gives
The prefactor ni in Eq. 4 denotes the electronic occupation of state ψi where ρ(r) = i ni|ψi(r)| 2 . This is often taken from the Fermi distribution function ni = 2 • [exp(εi -µ)/kBT + 1] -1 ; the chemical potential µ is chosen such that i ni = N with N = ρ(r)dr yields the total number of electrons. The Hartree and exchange-correlation terms are now subtracted to avoid double counting by virtue of Eq. 3.
DFTB is ultimately derived from a Taylor expansion of Eq. 5 around a properly chosen KS reference density. In contrast to KS-DFT, which aims to find an electron density which minimizes the total energy, DFTB only assumes a reference density ρ0 where perturbations corresponding to charge density fluctuations (i.e., ρ(r) = ρ0(r) + δρ(r)) determine the ground state energy. An appropriately chosen reference density, for example, would be a superposition of constituent neutral atomic electron densities. The total energy from Eq. 4 is then expanded up to third order whereby
where for nuclei with charge Zi and position R,
E (1) [ρ0, δρ]
The zero-and first-order terms constitute the original DFTB formulation (DFTB1) and is otherwise known as non-SCC DFTB. 20 It is important to note that the first order term does not actually depend on δρ as its corresponding terms cancel out during the derivation. DFTB1's non-self-consistent nature requires only a single solution of the Kohn-Sham equations, yielding computational speeds 5-10 times faster than subsequent self-consistent versions. 30 Moreover, DFTB1 is best suited for systems with significant covalent character and minimal charge transfer between atoms; such systems include homonuclear systems or molecules with similar atomic electronegativities. Additionally, DFTB1 can describe systems with complete charge transfer, such as NaCl. 31 However, simply using this first-order approximation has its limitations when modeling systems where many-body interactions significantly influence charge density fluctuations. Later iterations of DFTB (DFTB2 and DFTB3) address these limitations by incorporating higher-order terms to ac-count for atomic charge fluctuations. The second-and third-order terms, represented by two-and three-body integrals, become crucial to accurately describe ionic and metallic (and, by extension, plasmonic) systems.
The total energy expression in Eq. 5 can be reorganized into three fundamental contributions:
In Eq. 7, E orb = E (1) is the orbital (or band) energy and is calculated as the sum of atomic orbital eigenvalues. The second term, ESCC = E (2) + E (3) , describes second-and third-order selfconsistent corrections that arise from charge fluctuations. The final term, Erep = E (0) accounts for all other energetic contributions, internuclear repulsion and more cumbersome exchange-correlation effects. The strength of DFTB ultimately lies in its reliance on empirical approximations of the total energy. This highly parametrized approach enables it to achieve results comparable to KS-DFT with wall times less than 0.2% of the original method for a given system of interest. 30
The DFTB3 Hamiltonian matrix is altogether expressed as a multiorder expansion that progressively captures more detailed electronic interactions, where for atoms A and B containing orbitals µ and ν, respectively,
(8) Each order incorporates additional nuances in electronic structure -H (0) µν represents the zeroth-order Hamiltonian based on the reference density (i.e., the orbital and repulsive energies), while the subsequent terms H (2) µν and H
(3) µν progressively incorporate charge fluctuation and chemical hardness effects. The key components of this expansion involve Sµν , the overlap matrix between orbitals ϕµ and ϕν ; ∆q, charge fluctuations; γ h , a modified interatomic charge density that captures chemical hardness; and Γ, which captures changes in chemical hardness. This Hamiltonian matrix serves as the foundation by which one solves the corresponding secular DFTB equations in Eq. S4.
Importantly, both H
µν and Sµν are pre-computed once an appropriate pseudoatomic Slater-type orbital basis set is determined, alongside the parameters γ h AB and E rep AB . For each pair of elements, these parameters are systematically tabulated in Slater-Koster files, eliminating their need to be computed during the DFTB runtime (further discussed in the Supporting Information). This high degree of parametrization underscores DFTB's computational efficiency: DFTB2 is at least 250 times faster than RI-PBE and more than 1000 times faster than B3LYP. 30 The use of a minimal basis set alone (see the Supporting Information) accelerates computation by a factor of 27 compared to KS-DFT, while integral tabulation further improves computational speed by 10 to 40 fold. 30 However, matrix diagonalization scales cubically with the number of orbitals (O(N 3 )) and remains the time-limiting step in DFTB.
Excited state phenomena, in contrast to ground state formalisms, require a description of the DFTB Hamiltonian within the time or frequency domain. In this regard, time-dependent density functional theory (TDDFT) 32 is the most popular method to compute the excitation energies of many-electron systems due to its reasonable tradeoff between accuracy and efficiency. 33,34 The most widely implemented version is derived from linear response theory (LR-TDDFT) per the Casida formalism: 35 ΩFi = ω 2 i Fi,
where Ω is the response matrix whose elements depend on the ground state KS states and the exchange-correlation kernel chosen, ωi are the frequencies corresponding to the vertical excitation energies, and the eigenvectors Fi contain all the information needed to derive excited state properties (e.g. total spin and transition oscillator strengths). The former two terms result from the density-density response matrix χ(r, r ′ , ω) which describes how the electron density at r responds to a perturbation at r ′ as a function of frequency ω.
The full spectrum, which includes the orbital energy differences ωi, affects how transitions are weighted and how the response matrix elements Ω are constructed. 36 Note that in DFTB, only valence excited states can be computed due to the minimal basis set employed. The polarizability tensor ↔ α(ω) can be subsequently computed (within the Kramers-Heisenberg dispersion relation) by
where the oscillator strengths fi are obtained from the eigenvectors Fi and me and e is the electron rest mass and charge, respectively. 35 While LR-TDDFT is capable of studying a few low-lying excited states of an isolated molecule, it becomes prohibitively expensive in both memory and computational time when analyzing broadband spectra 37 or for systems with a high density of states. This is often the case for semiconductors 38 and many plasmonic systems (e.g. Al, alkali metals), which are characterized by free-electron behavior with few interband transitions. 39,40 Importantly, the linear Casida equation is iteratively solved within a basis spanning (occupied) × (virtual) orbitals, making LR-TDDFT calculations for these systems inaccessible even at the DFTB level of theory. 41,42 Such studies are often performed using smaller noble-metal plasmonic nanostructures and extrapolated to experimentally-observed sizes. 26 Another strategy to partially mitigate this issue involves approximating the two-center electron integrals involved in calculating both Ω and Fi. 43 Nonetheless, TDDFT calculations employing (semi)local functionals lead to spurious low-lying charge transfer states, a wellknown issue underlying these exchange-correlation functionals. 44,45 As DFTB is parameterized using semilocal exchange-correlation functionals (in order to properly map Eq. 4 to 7), these errors are inherited in TDDFTB calculations as well. More recently, long-range corrected DFTB has been developed to reconcile some of these discrepancies and exhibits similar accuracy to range-separated DFT methods at a significantly reduced computational cost, 46,47 though specific parameterizations for plasmonic metals remain unavailable as of yet.
Alternatively, real-time TDDFT (RT-TDDFT) can be used to directly solve the time-dependent Kohn-Sham equation 48 (or its density matrix-based analogue, the Liouville-von Neumann equation of motion). 49,50 This non-perturbative approach is capable of capturing nonlinear photophysical effects and allows for one to generate broadband spectra. The spectral range can be extended to lower wavelengths by propagating the system over longer times, and its resolution can be improved by decreasing the timestep. This presents a significant departure from iterative LR-TDDFT methods that explore excited states by gradually expanding the excitation space.
Solving the time-dependent KS equation requires the use of a full time-dependent propagator, which can be expressed in terms of the time-ordered evolution operator:
where T exp is a shorthand notation for the time-ordered exponential. 51 However, due to the complexity of this operator, it must be approximated. Several numerical methods have been developed to do so, including the exponential midpoint (and modified midpoint) algorithm, the enforced time-reversal symmetry method, Magnus expansion, and split-operator method. 51,52 Consequently, the accuracy and efficiency of RT-TDDFT calculations heavily depend on the approximations used for the time-dependent propagator and the spectral properties of the time-dependent Hamiltonian. 53 Within the tight-binding (RT-TDDFTB) formalism, a classical time-dependent propagator is used to approximate Ĥ(τ ) at the DFTB level, allowing for one to study larger plasmonic systems over extended timescales (∼ hundreds of fs to ps). Nuclear dynamics are treated semiclassically, 54 while electronic states are allowed to evolve under an applied field; coupling between the two is captured via the Ehrenfest model. 55 In this approximation, the nuclei are treated as classical particles subject to a force derived from a weighted average of all the electronic states of the system. 56 Specifically, the density matrix ρµν = i ni(cµic * νi ) is propagated according to the Liouville-von Neumann equation of motion
where H is now time dependent by virtue of the applied electric field. The density matrix is evaluated by integrating Eq. 12 for a given initial (t = 0) density matrix ρ0 representing the ground state of the system. The second term in the above equation accounts for dissipation and energy exchange between electrons and nuclei. The nonadiabatic coupling matrix, Dµν = ∂R B ∂t • ∇BSµν = ⟨ϕµ| ∂ϕν ∂t ⟩ introduces a mechanism by which nuclear motion can induce electronic transitions, leading to phenomena such as electronic thermalization (key in plasmon-generated hot carrier dynamics, [57][58][59][60] though with significant limitations; vide infra). The equation of motion is typically integrated using a leapfrog scheme where the density matrix at time ti+1 is obtained from its value at time ti-1 and its derivative at time ti:
A major challenge for real-time methods is the extremely short timestep (often on the attosecond scale) necessary to accurately capture electron dynamics. In contrast, ground-state ab initio molecular dynamics simulations (which do not involve electron propagation) typically use timesteps between 0.5 and 1 fs. Larger timesteps may be possible via an optimal gauge choice, enabling propagation on the order of 10-100 attoseconds, 61 although this remains an active area of research. The forces on each atom are computed using the Hellmann-Feynman theorem, where the time-evolved density matrix is aver-aged over the distribution of electronic states such that
where mA denotes the mass of atom A. The atomic charges are then calculated by the Mulliken approximation, qA = TrA[ρS]. 54,55 The nuclear and electronic systems interact simply through expectation values and thus this approach is a mean-field approximation to the actual coupled electron-nuclear dynamics. This approximation may break down under circumstances where electron-nuclear correlations become important, such as at the crossing of conical intersections. In the weak-field limit, the results of RT-TDDFT and LR-TDDFT should agree. Nevertheless, RT-TDDFTB encounters notable challenges when describing phenomena driven by intense external fields, such as above-threshold ionization 62 multi-photon processes, 63 charge-resonance enhanced ionization, 64 and high harmonic generation. 65 Specifically, two major sources of error, aside from limitations inherent to using a minimal basis set, include an inadequate description of excited state potential energy surfaces and the long-range behavior of the Coulomb potential. These issues complicate the accessibility of high-energy KS states during time propagation. Even in the weak-field limit, Ehrenfest dynamics does not obey the principle of detailed balance and can perform poorly when a system has multiple equiprobable states where the potential surfaces are significantly different. 66,67 However, Ehrenfest dynamics is expected to perform well over short time scales in systems with many similar potential energy surfaces (as is often the case for plasmonic metallic nanoparticles) since it restricts motion to a single average potential energy surface.
RT-TDDFTB can be used to calculate electronic absorption spectra. Following a weak delta function electric field impulse at t = 0, the system evolves without any external perturbation. The spectrum is calculated from the frequency-dependent induced dipole moment ⃗ µ(ω), related to the field E(ω) via the polarizability tensor 48 ⃗ µ(ω)
In practice, a damping factor is often applied to smooth the Fourier transformed spectra, thereby reducing ringing (i.e., rapid oscillations) and providing a more realistic description of excited state dynamics (e.g. finite lifetime effects on the plasmon linewidth and chemical interface damping 68 ). The absorption spectrum σ(ω) is then calculated from the frequency-dependent polarizability tensor as
The maximum peak in the absorption spectrum typically corresponds to the plasmon mode, and this resonant frequency can be used in subsequent calculations, such as laser-driven dissociation. 19,[69][70][71] In the following sections, we discuss some practical implementations of this theory, focusing on its performance and limitations when describing the unusual optical properties of small metal clusters, alongside nonlinear spectroscopy and photocatalysis.
Large metal nanoparticles (∼10-200 nm) behave like conventional electrical conductors and their plasmon resonances can be understood within a classical picture. Atomically-precise nanoclusters, however, reveal an entirely different world. Nonlocal effects (where the frequency-dependent dielectric function additionally depends on the incident wavevector) begin to predominate as the electron mean free path becomes commensurate with the particle size (≈ 5 nm for Al and 50 nm for Ag), leading to anomalous plasmon broadening. [72][73][74] As the particle size approaches the Fermi wavelength (≈ 0.2 to 0.5 nm for Al and Ag), confinement of electron motion transforms its otherwise continuous density of states into discrete energy levels. 75,76 These tiny clusters behave more like molecules than metals, with sharp electronic transitions replacing their broad plasmon bands. 77 It then follows that their unusual optoelectronic properties require a quantum description rather than the classical picture typically used. Kubo showed that the spacing between energy levels scales inversely with particle volume. 78,79 This can be understood through a free electron model where the energy gapor eponymous Kubo gap δ -near the Fermi energy EF (measured relative to the conduction band minimum 80 ) is given by
where N is the number of valence electrons. The particle's electronic behavior is determined by the relationship between the Kubo gap and thermal energy fluctuations. When δ falls below the thermal energy kBT , the particle will be metallic; if above, insulating.
For example, at room temperature the heuristic number of valence electrons for δ to equal kBT is around 287 for Ag (EF = 5.51 eV) and 60 for Al (EF = 11.6 eV). For Ag this roughly corresponds to a 2 nm nanoparticle, though for Al this is strikingly smaller -only about 0.4 nm (approximately 20 atoms). Classical approaches tend to address these scale-dependent variations by empirically modifying the dielectric function to include enhanced electron-surface scattering in small particles, [81][82][83] or by using a nonlocal dielectric function outright. 84,85 Though at this scale, any nuances in electronic structure are dominated by quantum size effects that dictate any emergent physicochemical properties. 86,87 To this end, DFTB allows for one to calculate the optical properties of ultrasmall metal nanoparticles from a purely quantum mechanical standpoint. While DFTB parameterizations rely on generalized gradient approximation (GGA) functionals primarily designed primarily for ground state properties, these semilocal functionals can still reasonably describe excited state phemonena. 88,89 The approximations used at this level of theory ostensibly capture the essential screening effects that influence plasmon behavior (particularly in d-electron systems; 90 vide infra), even if they do not fully account for optical nonlocalities. Importantly, nonlocal dielectric functions, which account for spatial variations in a material's response to an external field, less markedly affect absorption (which is directly calculable from RT-TDDFTB per Eq. 16). This is largely because absorption (which largely depends on the optically-allowed density of states) predominates at sub-10 nm length scales, rather than scattering processes which depend on the field's detailed spatial distribution.
Salient limitations nonetheless remain where an inadequate parameterization of excited state interactions lead to spurious lowlying charge transfer states -a well-known limitation of GGA functionals at the KS-DFT level, delocalization error notwithstanding. 91 Moreover, the intense absorption band generally ascribed to plasmons has been shown to have strong interband character when calculated with GGA functionals. 92 Although range-separated DFTB 46 should better represent the long-ranged Coulomb potential necessary to describe plasmons, 45,92,93 optical spectra are predicted to manifest as sparse transitions for smaller clusters (cf. that computed at the KS-DFT level in Ref. 94) rather than the dense manifold of higher-energy excited states necessary for plasmon-like transitions in metallic nanoparticles. Consequently, plasmon-like transitions would only appear for much larger particles (which DFTB can nonetheless simulate). The use of range-separated dielectricdependent corrections, as has been recently implemented for pe-riodic systems at the KS-DFT level 95,96 has been shown to more realistically account for Coulombic screening. Extensions thereof to DFTB could offer a promising means to describe the delocalized transitions and charge transfer excitations characteristic of plasmons.
While DFTB has been parametrized for some main group plasmonic materials like Al, Na, and K, 97 few studies exist that systematically examine their plasmonic properties. Douglas-Gallardo et al. investigated the plasmonic properties of Al nanoparticles via a combination of molecular dynamics and RT-TDDFTB. 98 The authors report that incorporating d orbital angular momentum terms in the Hamiltonian matrix (from Eq. S8) is necessary to describe their optical and structural properties. Oxidation was found to gradually separate their otherwise continuous density of states, resulting in attenuated electron dynamics and shortened plasmon lifetimes. Similarly, we have recently calculated the optical absorption spectra for Al nanoparticles where, due to Al's free electron character, particles were found to display strong symmetry-dependent plasmon modes. 99 A related study found through DFTB that Mg nanoclusters can produce hot carriers with energies up to 4 eV (Fig. 1). 100 These generated hot carriers energetically align with the electronic states of physisorbed H 2 molecules, which can then promote their dissociation. Chemisorbed (i.e., absorbed) H 2 , conversely, was found to lead to strong chemical interface damping, thereby reducing the plasmon lifetime and blueshifting the plasmon wavelength. This interaction resulted in a diminished hot electron population, though the remaining electrons possessed slightly higher energies. The paucity of such studies suggests a rich avenue to be explored, particularly given these metals' Earth abundance and free electron character. Despite these promising applications to main group metals, DFTB's inherent tight-binding construction limits its ability to accurately model the anomalous plasmonic properties exhibited by small alkali metal particles. At these sizes, both quantum effects and detailed surface interactions become important as electrons in-teract strongly with the surface; one aspect includes the spillover of conduction electrons at the particle surface, which would effect a red shift with decreasing particle size. [101][102][103][104] As an analytical reference, the optical response of a spherical metal nanoparticle much smaller than the wavelength of irradiated light (i.e., below the quasistatic limit) can be described by the dipole polarizability α(ω) that features a nonlocal, hydrodynamic extension of the Clausius-Mossotti relation, 4 given by 105
where εD = 1 - vF is a characteristic parameter associated with pressure waves in the electron gas (where vF is the Fermi velocity), and D is the characteristic charge-carrier diffusion coefficient. Evaluating for the pole of Eq. 18 yields the complex-valued resonance frequency ω = ω ′ + iω ′′ , which is semiclassically related to the bulk plasma frequency ωp and cluster radius R as 85
where the real part ω ′ gives the plasmon resonant frequency, while the imaginary component ω ′′ relates to the plasmon damping rate γ and hydrodynamic effects (i.e., convective and diffusive charge transport) which spread the charge density and broaden the plasmon linewidth. As the high kinetic energy of the conduction band s electrons in alkali metals cannot be efficiently screened by other electrons comprising the particle's electronic states, 82 the electron density will consequently extend beyond the nominal surface by some distance δ. 103,[106][107][108] Accounting for this spill-out effect yields a modified plasma frequency ωp0 dependent on cluster size, such that 109 ωp0 ≃ ωp
where r0 = N 1/3 rs is the nominal particle radius related to the number of atoms N and the Wigner-Seitz radius rs. Accordingly, this yields R = r0 + δ. With increasing particle size, the polarizability reaches the classical limit of a perfectly conducting sphere: that is, α = 4 3 πr 3 0 . Indeed, the polarizability of a sphere in the bulk limit is smaller than that for smaller particles, where valence electron spill-out increases the clusters' effective radius. As the plasmon resonant frequency is inversely proportional to the polarizability, s-electron spillover would thus induce a red shift compared to classical Mie theory. 107 For sodium (rs = 4 a.u.), the effective radial increase due to electron spillover is known to be δ = 1.3 a.u. beyond the physical surface. 103,111 We have shown the DFTB-calculated absorption spectra for sodium nanoparticles ranging from Na 13 to Na 561 (Fig. 2b) and a comparison to the plasmon peak position via the nonlocal hydrodynamic model given by Eqs. 20 and 21. Many discrepancies between the two are observable. While both theories show convergence of the plasmon wavelength λLSP R as the particle size increases, they predict contradictory trends (Fig. 2c). DFTB shows that these particles absorb in the deep UV (between 200-300 nm) and a concomitant redshift with particle size. On the other hand, the nonlocal hydrodynamic model predicts absorption in the visible and a corresponding blueshift, consistent with reduced s electron spillover in larger particles. Numerous higher-level KS-DFT simulations have examined their photoabsorption spectra, [112][113][114] which run in agreement with the nonlocal hydrodynamic model. However, extending these insights to DFTB is relatively limited, given that parameterization thereof often relies on oxidized states (rather than in zerovalent states). 115,116 Additionally, DFTB is inherently limited by artificial orbital compression and the use of a minimal basis set (per Eq. S2); that is, the "tightly-bound" electrons intrinsic to this theory cannot thus far rigorously describe the otherwise diffuse s states responsible for the spillover effect.
Most literature-reported studies have used DFTB to study noble metal plasmonics, likely due to their popularity in experimentation (a consequence of their chemical stability). While ultrasmall alkali metal and certain p-block nanoparticles almost behave like free electron metals, the electronic structure of their noble metal counterparts markedly differs owing to the presence of d electrons: low-lying d electrons screen valence s electron intraband transitions and directly participate in d → sp interband transitions which dampen the plasmon resonance. Conversely, as the Ag particle size decreases, its surface-to-volume ratio increases, reducing d electron screening, thereby increasing the plasmon frequency. This effect compensates for any spill-out effects conferred by the valence s electrons. 105,117,118 Additionally, the 4d electrons in ultrasmall Ag nanoparticles comprise a polarizable background that screens interactions between valence 5s electrons. [118][119][120][121] Consequently, in contrast to the redshift found in alkali metal clusters, these surface-assisted processes induce an anomalous blueshift with decreasing particle size (though it should be noted that the presence of a dielectric ligand shell can effect an opposite trend; cf. Ref. 122). 105,118 Classical models only account for valence 5s electrons and can-not describe these effects. 81,[122][123][124] Ab initio methods, on the other hand, explicitly account for d electronic states. Further, as these states are more strongly localized and do not exhibit spill-out effects like s electrons, their contributions are more ostensibly captured by DFTB. 68,69 These results are noticeable in Fig. 3a, where a blueshift of approximately 50 nm is observed when icosahedral Ag nanoclusters decrease in size from 2.9 to 0.58 nm. This blueshift runs concomitant with an anomalous linewidth broadening due to electron-surface interactions alongside any emergent quantum size effects. In contrast, Mie theory, which does not account for surface scattering and the quantum nature of the dielectric function altogether, displays minimal changes in the plasmon wavelength with spherical particle size (Fig. 3b). Additionally, d electrons more strongly contribute to plasmonic transitions in Au compared to Ag nanoparticles (a consequence of the smaller energy difference between 5d and 6s states). 126,127 This, in turn, arises from relativistic effects which are essential to include when accurately describing the optical properties of Au clusters. At the DFTB level, the atomic orbital energies and twocenter Slater-Koster integrals are obtained from scalar relativistic KS-DFT calculations. 128,129 These effects are effectively accounted for within the Hamiltonian matrix elements (from Eq. S3) without resorting to specific scalar relativistic operators. Thus, while these relativistic effects are implicitly accounted for in Erep, neither these effects nor core electronic states (for which relativistic effects predominate) are explicitly described within the DFTB framework, leading to small deviations between multiple levels of theory (and experiment). 69,130,131 Quantum models provide a superior description of plasmons at sufficiently small sizes. These methods naturally interpret emergent optical phenomena in terms of discrete, molecular-like transitions. However, as the particle increases in size, the "collectivity" criterion used to describe plasmons (i.e., as a linear combination of many single-particle excitations) begins to predominate. 92,132,133 From an experimental purview, this excitonic to plasmonic transition was identified to be around 2.3 to 1.7 nm for Au clusters. 134 At the same time, scalability limitations in KS-DFT render a comprehensive analysis of such excitations and their contributions to the plasmon-like absorption band intractable. 42 Prior studies have used symmetry-based arguments to study larger, tetrahedral nanoparticles at the TDDFT level; 92,135 one study from our group evaluated the optical properties of these structures up to 120 atoms, making it possible to connect TDDFT results with those from classical electrodynamics (via the discrete dipole approximation). 88 Further, when a ground state KS-DFT calculation was performed followed by tight-binding approximations to evaluate their excited state properties, comparable plasmon peak positions were obtained (with ca. 0.15 eV deviations) for similarly-sized Au, Ag, and bimetallic alloy tetrahedral particles with only a tenth of the total wall time. 136 Even faster runtimes were found to be possible (0.9% compared to DFT) at the full DFTB level of theory. 26 For significantly larger sizes (up to 10 5 atoms 137,138 ), the optical properties of nanoparticles can be efficiently calculated using a jellium model, wherein the quantum nature of electrons can be accounted for without an explicit introduction of the constituent lattice. This lends itself possible for free-electron metals with spherical geometries; 139 though, this theory cannot examine plasmonic systems with strong interband character as well as anisotropic nanoparticles where the spatial charge distribution drastically affects its optical response. 71,140 DFTB, at its core, is still an orbital-based method. The use of this theory enables a comprehensive evaluation of this excitonic-toplasmonic transition with much greater accessible sizes -reported up to 1415 atoms. 141 The tractability of DFTB to much larger length scales moreover allows for a direct reconciliation between the quantum and classical regimes. As a point of comparison, Fuchs has shown that the optical properties of metal cubes below the quasistatic limit can be classically expressed as a sum of symmetrydependent normal modes m, The dielectric susceptibility χ is then given by 142 Indeed, we see multiple resonant peaks corresponding to different plasmon symmetries at similar energies to those predicted by Eq. 22 (Fig. 4). We note that neither model accounts for electron spillover, however (vide supra). The use of DFTB consequently presents a promising means by which the optical properties of metal nanoparticles can be rigorously examined for much larger particles, allowing for direct comparisons between classical electrodynamics and quantum mechanics.
Nanoparticles rarely exist in isolation. Advances in lithographic and self-assembly techniques have enabled their arrangement into hierarchical structures with nanometer-scale precision. [144][145][146][147] Nanoparticle aggregates and/or superlattices offer additional tunability of their optical properties, wherein the extremely large localized electric fields between plasmonic nanoparticles at small gap sizes enable optical nonlinearities and additional effects extending into the quantum regime. [148][149][150][151][152] Classically, as the distance between two nanoparticles is reduced, the dipolar plasmon resonance is found to continuously redshift with a concomitant increase in their electric fields. However, once entering the extreme nearly-touching regime, the classical description of these gaps breaks down and electrons can tunnel between nanoparticles. [153][154][155] From a quantum standpoint, the emergence of such charge transfer plasmons in the near-to-mid IR results in significantly reduced hybridization effects and damping of the electromagnetic field across the junction. [156][157][158] Accordingly, The red and blue shaded regions correspond to positive and negative charge density, respectively.
these effects cannot be fully explained using higher-level classical methods; one may imagine a nonlocal description required for separations below 5 nm and a fully quantum model at even smaller gap distances. 153,[159][160][161] One of the earliest reports using DFTB to describe plasmons found that optical excitation of a Na 55 particle at its plasmon energy (3.16 eV) generated an oscillating electric field which, in turn, induced a time-dependent dipole moment in its neighbor -even at separation distances as far as 73 Å. 162 When this model was extended to four-particle Na 55 chains, optical excitation of the first particle continued to dominate the induced dipole moments of the third and fourth particles. Together, this longranged plasmon coupling was able to access distances exceeding twice the conventional 100 Å cutoff limit used in Förster resonant energy transfer models, 163 suggesting that despite its tight-binding construction, DFTB could still capture electronic coupling between plasmonic nanoparticles even at comparatively large interparticle separation distances. We consequently simulated the absorption spectra for Ag 666 nanocubes while employing periodic boundary conditions at varying gap distance d. From Fig. 5, we can see that as d decreases from 5 nm to 0.5 nm, the plasmon wavelength redshifts and, concomitantly, the oscillator strength significantly increases, consistent with plasmon coupling between adjacent particles. However, as d further decreases below 0.5 nm, tunneling effects quench the plasmon response to the point where it is effectively unobservable at 0.2 nm. At 0.4 nm, several new modes are found; however, these do not show variation with decreasing d that has been observed experimentally or with electrodynamic models. 156 Thus, in contrast to a previous jellium model, which describes this classical-to-quantum onset at 1 nm, 164 DFTB shows this onset below 0.5 nm. The presence of charge transfer plasmons has been theorized to emerge in icosahedral Ag 66 chains below 0.1 nm at the DFTB level, though at 2.5 eV -a much higher energy than is typically expected (cf. Ref. 165). Note also that a monotonic increase in gap distance for this transition was experimentally found with particle size, ranging from 0.8 to 1 nm for 30 to 80 nm nanoparticle dimers. 166 Similar to the limitations discussed for alkali metal clusters, while DFTB is capable of simulating ground state electron transport, 54,167 it is thus far incapable of fully describing long-range excited state charge transport through these sub-10 nm junctions. The lack thereof consequently underestimates the interparticle distance threshold between classical and quantum effects and fails to properly describe charge transfer plasmons altogether. An extension of screened range-separated hybrid functionals has been specifically implemented for periodic systems in the context of KS-DFT and could mitigate this discrepancy; 47 implementation thereof to DFTB would nonetheless require a separate parameterization for the (un)screened long-range Coulomb kernel. Small separation distances necessitate a quantum mechanical description to predict reliable electric field enhancements, while long-range Coulombic effects are necessary to fully describe coupling and charge transfer in nanoparticle aggregates.
Interparticle gaps also enable strong optical nonlinearities in the near-touching regime. As Ehrenfest dynamics is not limited to linear response theory as is the case for LR-TDDFT(B), the use of RT-TDDFTB (per Eq. 12) allows for the investigation of nonlinear optical phenomena. Plasmon resonant effects can amplify second (SHG) and third harmonic generation (THG) by several orders of magnitude due to the plasmons' ability to enhance the electromagnetic field in the vicinity of the particle. Our group has explored the use of RT-TDDFTB to study infrared upconversion in nanorod dimers. 168 Electron dynamics were initiated via a Gaussian envelope pulse. As these nanoparticles have a high density of electronic states near the plasmon resonance, we hypothesized that the use of a Gaussian driving profile (which features a more gradual increase in the field amplitude concomitant with excited state dephasing) would allow for more efficient harmonic generation (cf. the linear ramp over one optical period employed in Ref. 169). Harmonic generation signals were calculated from the induced dipole moment µa(t) = Tr [(êa • μ)ρ(t)] for Cartesian coordinate a, which coherently oscillates without bound in the absence of damping. Excitation thereof includes both fundamental and generated harmonics alongside all other activated modes:
where a, b, c, and d denote Cartesian coordinates, α is the polarizability, β, and γ are the first and second hyperpolarizability tensors, respectively. Additionally, µ 0 a is the permanent dipole moment. The resulting HG signals exhibit characteristics that are consistent with previous TDDFT-based results. Modulating the particle length and interparticle spacing controls the longitudinal plasmon energy between 1.2 and 1.6 eV and, when excited, allows for S/THG. We specifically used end-to-end dimer configurations as particle symmetry breaking is a requisite condition for these processes: detectable SHG signals are observed only in nanorod dimer systems with broken longitudinal symmetry (note the differing spectra between the a/symmetric Au 73 and Au 121 dimers in Fig. 6a). The HG yield increases with particle size, and smaller interparticle spacings are found to enhance these processes. The generated signal intensity follows a power law dependence with respect to the incident electric field intensity, with exponents comparable to those determined via perturbation theory (Fig. 6b). Importantly, the largest system size studied in this work consists of 218 Au atoms, which is the largest nanoparticle system treated quantum mechanically in the context of harmonic generation thus far.
The maximum HG yield is observed when the laser polarization is aligned with the nanorods' longitudinal axes (Fig. 6c). Both the S/THG amplitudes tend to zero as the polarization angle deviates from longitudinal alignment, indicating that resonant excitation of the plasmon mode facilitates efficient harmonic generation. Similar trends in the polarization dependence of SHG have been previously reported in nanorod systems and nanosphere dimer systems. 170,171 Nevertheless, the THG amplitude decreases more rapidly compared to the SHG peak due to its higher sensitivity to the laser pulse.
From these spectra, we further calculated the polarizabilities using Eq. 15 and hyperpolarizabilities through finite difference calculations of the dipole amplitudes and subsequent dipole fitting. 169 In Table 1, we list the XX, YY, and ZZ components of α, the ZZZ component of β, and the ZZZZ component of γ. For higher-order signals, we report only the Z components as longitudinal excitation maximizes the HG yield while transverse excitation has a negligible effect (from Fig. 6c). Notably, the polarizabilities obtained from RT-TDDFTB agree with those obtained from LR-TDDFTB. The ZZ, ZZZ, and ZZZZ components are shown at their respective longitudinal plasmon energies, while the XX and YY components of α are presented at the transverse plasmon energy of 2.05 eV.
All (hyper)polarizabilities monotonically increase with system size. These findings align with previously reported linear absorption calculations for icosahedral Ag and Au nanoparticles using TDDFTB showing absorption cross sections on the order of 10 -15 to 10 -14 cm 2 . 141 Moreover, the first order hyperpolariz- abilities are comparable to those experimentally obtained, with β ∼ 10 -26 -10 -25 esu for nanosphere and nanorod dimers with particle diameters as small as 3 nm. 172 As is to be expected, these values are significantly greater than static hyperpolarizabilities obtained via TDDFT for Au clusters, with reported values ∼ 10 -29 esu. 173 While several THG measurements of Au nanoparticle films have been documented, 174 these studies do not report second-order hyperpolarizabilities, thereby limiting a direct comparison. Our obtained values for γ ∼ 10 12 a.u. (or 5 × 10 -28 esu) are about 1000 times greater than the highest values reported for organic molecules. 175 We consider this a resonable difference given our focus on resonant responses, as opposed to the nonresonant responses studied therein.
Linear response theory, as its name suggests, is only capable of simulating linear absorption spectra. While it may suffice for small, isolated nanoparticles, higher-order structures present a more complex picture. To this end, real-time dynamics, in tandem with the computational tractability of DFTB, offers the ability to model nonlinear optical phenomena with reasonable accuracy. While the distance by which tunneling effects are greatly underestimated due to the short-ranged interactions prevalent within this construction, Table 1: Imaginary component of the polarizability α, first hyperpolarizability β, and second hyperpolarizability γ. These values were evaluated using the auorg-1-1 Slater-Koster parameter set. Results were evaluated at the longitudinal plasmon energies for α ZZ , β ZZZ , and γ ZZZZ . The remaining α components (XX and YY) were determined at the transverse plasmon excitation energy. Reprinted with permission from Ref. 168. © 2024 American Institute of Physics. Dimer α [cm 3 ] βzzz γzzzz αxx αyy αzz [10 -26 esu] [10 10 a.u.] 10 -22 10 -22 10 -20 Au 73 Au 97 1.29 1.29 2.34 2.1 -Au 73 Au 121 1.47 1.47 4.89 2.2 8 Au 73 Au 133 1.51 1.51 5.94 3.2 141.3 Au 73 Au 145 1.75 1.75 9.76 3 256.5
RT-TDDFTB allows for a detailed analysis of the nanoparticle gap region, which contains highly confined electromagnetic fields and consequent optical nonlinearities and nonlocalities. Similar limitations when considering the optical properties of individual nanoparticles present in the context of their aggregates. The incorporation of longer-ranged Coulombic interactions, much in the same way as has been done for organic molecules, 46,176 could enable more accurate comparisons to experiment. Beyond studying nonlinearities in the small gap regime, refining these quantum methods could enable a systematic analysis of cavity-mediated chemistry. [177][178][179] More broadly, semiempirical methods such as DFTB allow for scalable, parameter-free studies of real-time polaritonic chemistry in nanoparticle aggregates and dimers, in contrast to prior studies limited by system size. [180][181][182] Doing so would allow for a better understanding of the rich physics underlying these processes beyond experimentation alone.
Plasmons ultimately occur on the femtosecond timescale. Following plasmon excitation, plasmons can either decay radiatively (a slower process involving photon re-emission 83 ), or nonradiatively by dephasing to electron-hole pairs with energies equivalent to that of the plasmon. 58,183 These energetically "hot" carriers subsequently redistribute their energy through scattering events with photons and other low-energy electrons over 100 fs to 1 ps. As these scattered electrons continue to lose energy, their propensity to interact with phonons increases between 100 ps to 10 ns, leading to thermal dissipation into the surrounding environment. Together, these processes can raise the electrons' translational temperatures to thousands of degrees higher than their surroundings (depending on nanoparticle size), creating an efficient mechanism for electron transfer into nearby molecules that bypasses scaling relationships that would otherwise limit catalytic turnover. Importantly, plasmon-driven catalysis can offer milder reaction conditions while directly harnessing light to drive such chemical transformations. While the hot carrier picture remains the prevailing theory governing plasmon-mediated photocatalysis, other mechanisms have nonetheless been proposed. Direct electronic excitation between metal-adsorbate states, 183,184 plasmon-enhanced electric fields, 185 and photothermal heating 186 have all been hypothesized to be capable of driving many reactions of interest. Difficulties in observing plasmon dynamics at their characteristic length and time scales, however, bring about questions relating their dynamics to any ob-served catalytic effects, leaving some mechanistic questions best answered via theoretical approaches. 15,59,70,[186][187][188] In this regard, our group has focused on the theoretical analysis of plasmon-driven photocatalysis, whereby disentangling the many events underlying these processes should allow for a complete description of hot electron dynamics and their intimate role in governing the outcomes of chemical reactions. We have previously investigated the mechanism for photoinduced H 2 dissociation on Au nanoclusters using TDDFT. 18 The dissociation of H 2 is found to be governed by transitions from metallic "hot electron" states to charge transfer states via the antibonding σ * orbital on H 2 . However, this study was limited to studying H 2 near an Au 6 cluster -while this cluster showed a "plasmon-like" peak in its TDDFT absorption spectra, it was too small to support true plasmons and therefore could not accurately model the plasmon dephasing processes that would otherwise generate hot electrons.
It then follows that ab initio modeling of these processes is prohibitively challenging due to the large number of electrons and electronic excited states involved. This is particularly the case for metals where their continuous density of states makes electronic convergence difficult, if not impossible, for systems large enough to exhibit plasmonic behavior. 92 Moreover, the tens to hundreds of fs timescales involved in plasmon-driven photodissociation force a compromise in electronic structure studies that leads to the consideration of small clusters and short pulses (below ten fs). 70,71,[189][190][191] These small clusters serve as surrogate models for investigating photocatalysis and require a high intensity threshold for any reaction to occur. Not only do these high intensities deviate from actual experimental conditions, but the short pulses and powers used therein (as in Ref. 71) approach or even exceed the limit where a relativistic description (e.g. inverse bremsstrahlung 192 ) is required. 193 To this end, RT-TDDFTB provides a natural choice to model the reaction dynamics underpinning plasmon-mediated photocatalysis. The incorporation of derivative coupling in Eq. 12 leads rather naturally to adsorbate dissociation, and, coupled with DFTB's scalability, allows for a systematic study of these processes at significantly larger sizes and longer time scales. We have examined the photodissociation of H 2 on octahedral Au and Ag clusters as a function of particle size and driving frequency (i.e., plasmon vs. interband) on the dissociation threshold intensity. 69 Particles spanning 19 to 489 atoms (between 0.5 to 2.2 nm) were constructed with a H 2 molecule positioned at the particle vertex (Fig. 7a). While different adsorption sites provide distinct chemical environments, we deliberately positioned the H 2 molecule at the particle tip as this site experiences the maximum charge localization (and electric field enhancement) through plasmon excitation.
Our RT-TDDFTB-calculated absorption spectra align well with previous TDDFT studies, 88,133 confirming the particles' plasmonic behavior arising from collective electronic excitation (Fig. 7b). For both Au and Ag nanoparticles, larger sizes run concomitant with stronger oscillator strengths, reflecting their increased density of states. The spectra reveal distinct characteristics for each metal. Ag particles exhibit a prominent sp intraband transition (i.e., plasmon peak) between 2.5 and 3 eV, with additional d → sp interband transitions appearing as relatively uniform signals at higher energies. In contrast, Au shows a broader plasmon peak near 2.3 eV attenuated by overlapping d → sp interband transitions, making it difficult to distinguish plasmon excitation in smaller particles. 126 Only for Au489 does the plasmon evolve into a spectrally distinct feature around 2.3 eV, reminiscent of that found in larger, spherical Au nanoparticles. Both metals demonstrate a consistent ≈ 0.25 eV redshift as the particle size increases from 19 to 489 atoms. While this shift parallels previous observations in 0.5 to 2.0 nm tetrahedral Ag clusters, 88 our results suggest quantum size effects and d electron screening dictate this behavior, rather than the dynamic depolarization effects seen in much larger systems (vide supra).
With the particles' spectra fully resolved, we hypothesized that RT-TDDFTB could be used to investigate the relative efficacies of plasmon vs. interband excitation for H 2 dissociation. Doing so would enable a statistically meaningful study correlating particle size, driving frequency, and consequent catalytic turnover. To do so, a Gaussian laser pulse was used to excite the nanoparticle-molecule complex. Modulating the driving frequency can therefore probe the comparative dynamics between inter-and intraband transitions. To do so, we simulated a Gaussian pulsed excitation centered around 25 fs at the corresponding driving frequencies of ω0 = 2.7 eV (2.3 eV) for the Ag (Au) plasmon, and 4 eV at the interband transition. The pulse amplitude decays to zero after approximately 25 fs, resulting in a broad energy band centered around its central frequency, rather than a delta function energy as experimentally seen with continuouswave lasers.
The probability of H 2 dissociation strongly depends on nanoparticle size. To investigate this relationship, we simulated 20 trajectories for Ag nanoparticles ranging from Ag 19 to Ag 489 at a fixed laser intensity of 3.3 × 10 12 W/cm 2 . Initial electronic velocities were sampled from a Maxwell-Boltzmann distribution at 300 K to allow for a nondeterministic outcome (Fig. 8a). For smaller particles at this sub-threshold intensity, H 2 molecules remained intact, showing only coherent oscillations of the the H-H bond distance (dH-H) around its equilibrium value. Larger particles, however, show an increased dissociation probability, with some trajectories achieving H 2 dissociation within 25 fs (Fig. 8b). Concurrently, some trajectories exhibited strong dH-H oscillations between 1.5-2 Å before damping to values similar to those seen for smaller particles. This size-dependent behavior can be explained by the inverse relationship between particle size and its dissociation threshold -that is, the minimum intensity at which H 2 dissociation occurred. Larger particles can sustain stronger dipoles upon excitation, leading to more efficient hot electron generation and transfer to adsorbates.
Nonetheless, extrapolation to larger sizes (from the 2.5 nm used herein to 20 nm as used experimentally; 194 Fig. 8c) reveals a disparity in the dissociation intensity by three orders of magnitude (cf. the threshold intensity for free H 2 in Ref. 195). Several limitations in our model may reconcile this discrepancy. 196 For one, the use of longer pulse durations (as used experimentally, ∼ 10 2 fs) would mitigate electronic relaxation due to limitations in describing electron-electron scattering at this level of theory; Ehrenfest dynamics do not fully capture stochastic energy dissipation due to an incomplete many-body treatment. For another, a significant portion of the excitation energy is redistributed towards other processes instead of directly driving H 2 dissociation. As only one H 2 molecule is considered per particle, plasmon excitation does not lead to efficient energy transfer in this model. In reality, multiple molecules would be in proximity to the particle (alongside particle aggregates with higher localized electromagnetic fields), where plasmonic hot carriers could interact collectively with the reactant ensemble and enhance the dissociation efficiency. 196 Similar approaches to those examined in the previous section could be used to understand the reaction efficiency in idealized nanoparticle aggregates, while nonetheless raising the complexity of the problem.
In contrast to Ag, where the plasmon is found to be more efficient in driving H 2 dissociation, interband excitation in Au is found to be more effective (Fig. 8c). Analysis of the electron population dynamics reveals key mechanistic differences between the two excitation modes. In every case, oscillations of the electron population in resonance with the laser frequency (known as "sloshing" and "inversion") initially appear and quickly dephase after the pulse ends. During the pulse, sufficient populations accumulate near the H 2 σ * orbital (represented by the red lines in Fig. 9c and f). For both Ag and Au, plasmon excitation predominately populates states near the Fermi level, while interband excitation accesses higher-energy states through d → sp transitions. Au nanoparticles, however, exhibit a broader d band energy range closer to the Fermi level compared to Ag, which enables significant interband transitions above the Fermi level. These higher-energy electrons transfer more effectively into H 2 's σ * orbital, weakening the molecular bond and inducing dissociation compared to lower-energy plasmon excitations which are further attenuated due to overlap with interband modes. However, a competing argument claims that population transfer is dictated by the complexity of its dynamics rather than through the spectral overlap between bands. 197 Specifically, through DFTB, plasmon excitation of Ag nanoparticles is found to continuously yield sp electron-hole pairs with energies in proximity to the Fermi level, while Au initially produces low energy sp holes that get depopulated in favor of high energy holes at the d band threshold at longer times. Building on these studies, our recent work using RT-TDDFTB has explored plasmon-driven electrochemical selectivity between H 2 O oxidation and CO 2 reduction. 198 Experimentally, plasmon excitation of Au-Cu alloy nanoparticles was found to shift the reaction selectivity of aqueous CO 2 electroreduction, favoring H 2 evolution over CO formation. Concomitantly, we found through RT-TDDFTB that hole-driven O H bond dissociation dominates over electrondriven C O bond-breaking in Au 365 nanocubes. While Au-Cu Slater-Koster parameters have not yet been evaluated, the similar dielectric functions of Au and Cu suggest the applicability of this analysis to the experimental Au-Cu system.
Thus far, the use of RT-TDDFTB outright is limited to the first tens of femtoseconds when analyzing the electron dynamics associated with plasmon-driven photochemistry. While in principle DFTB's large computational tractability enables studies into the picosecond regime, inefficient descriptions of electronic relaxation pathways limit the means by which longer-time effects (e.g. electron-electron/phonon scattering) can be examined in practice. 199 To examine these limitations, we have investigated the plasmoninduced electron dynamics in Al nanoparticles, where, unlike Au or Ag, Al's free-electron character allows for efficient electronic thermalization without interband coupling. 99 Plasmon excitation and subsequent analysis of the electron dynamics for H 2 dissociation found that multiphoton absorption led to the formation of distinct step-like features in the particles' transient electron distribution. The spacing thereof is commensurate with the photon energy ( ω) and reached energies up to 20 eV, which induced efficient H 2 dissociation. However, electronic thermalization only occurred during the pulsed excitation. Contrary to semiclassical two-temperature models commonly used to describe electron dynamics, 200 these steplike features persisted rather than dissipating into an equilibrium Fermi-Dirac distribution, highlighting the limitations of electronic relaxation modeling within the RT-TDDFTB framework.
In other words, Ehrenfest dynamics' mean-field description of electron-phonon coupling and electron-electron scattering does not entirely describe the many relaxation channels involved in plasmon decay. Electron-electron scattering can be in principle implemented through the GW approximation. It is worth noting that the GW approximation has already been applied to DFTB to account for quasiparticle energy corrections. 201 This theory is nonetheless highly sensitive to the quality of the Coulomb potential used to describe interelectronic interactions; it is unclear whether the minimal basis set and approximations to the Coulomb potential in DFTB is sufficient to accurately describe screening at the GW level (high computational memory demands notwithstanding). Further, the nonadiabatic coupling matrix as presented in Eq. 12 allows for nuclei to be treated semiclassically and electrons quantum mechanically. Including this term during plasmon irradiation is found to induce breathing-like oscillations within the first few femtoseconds. 202 This observation, however, is ascribed to electronic excitation into antibonding states and is unrelated to electron-phonon scattering which occurs over a much longer timescale.
Methods such as correlated electron-ion dynamics can recover electronic thermalization by augmenting the equation of motion with extra terms that describe nuclear fluctuations. 203,204 These approaches nevertheless become rapidly intractable; high-order polynomial scaling with respect to both the electron and phonon dispersion limits their use for the larger particles that DFTB is able to study. 203 Additionally, a recent DFTB implementation of Boltzmann transport theory in the context of ground state electron transport has been shown to reasonably account for electron-phonon coupling; 205 a similar extension to real-time excited state dynamics could be valuable to account for plasmon decay processes. An attempt to reconcile these differences has been studied by the use of trajectory surface hopping to study the long-time electronic dynamics associated with CO adsorption on an Au 20 tetrahedron, which combined TDDFTB with ab initio molecular dynamics. 94 In this model, plasmon-generated hot carriers transfer back and forth between CO and the Au 20 nanoparticle, thereby activating the C O stretching mode through nonadiabatic coupling (i.e., electron-phonon coupling) on the picosecond timescale. This more rigorous treatment of electronic energy redistribution demonstrated a 40% efficiency for plasmon-mediated CO activation through ensemble averaging.
Taken together, the use of RT-TDDFTB allows for one to investigate photocatalytic processes at computationally tractable length scales into the nanometer regime and time scales into the picosecond domain. The use thereof presents a closer reconciliation between theory and experiment while under a fully quantum representation. However, while dissipation is putatively accounted for in Ehrenfest dynamics, an incomplete many-body description of the scattering processes involved with plasmon de-excitation precludes a thorough analysis of these phenomena past the first tens of femtoseconds thus far.
Looking deeper into the quantum realm, it becomes increasingly clear that Feynman's vision was simultaneously too modest and too ambitious. Today's desktop workstations offer processing speeds that supercomputers from just a decade ago could only dream of. Yet the unfavorable O(N [3][4][5][6] ) scaling of high-level quantum chemistry methods remains an insurmountable barrier for a routine investigation of plasmonic systems outright. 206 Low-level quantum methods, and in particular DFTB, may ultimately reveal more about plasmonic phemonena than their more sophisticated counterparts higher up the theoretical hierarchy. DFTB offers a two to three orders of magnitude improvement in computational efficiency while providing a fair description of excited state properties. This approach has been postulated and has existed for nearly four decades, yet it remains surprisingly underutilized.
It is still surprising that such a tight-binding method can provide a reasonable picture of delocalized excited state dynamics, and yet our work and those from many others in the field have shown that it can in fact do so. This is not to suggest that DFTB is without its limitations. Significant opportunities exist to enhance these methods, particularly when modeling the electron scattering and relaxation processes necessary to describe their long-time excited state dynamics. Similarly, refining the description of the long-ranged Coulomb potential could resolve persistent discrepancies between theory and experiment, especially in the context of alkali metal plasmonics, nanoparticle aggregates, and electron transfer processes. And perhaps most importantly, the development of additional Slater-Koster parameters for both ground and excited states would extend these capabilities to a broader range of materials, allowing for investigations into their emergent properties and resultant chemistry. This is all to say that there is indeed plenty of room at the bottom -at the bottom of Jacob's ladder.