Source-linked AI summary
Kohn-Sham decomposition in real-time time-dependent density-functional theory: An efficient tool for analyzing plasmonic excitations
Tuomas P. Rossi, Mikael Kuisma, Martti J. Puska, Risto M. Nieminen, Paul Erhart
TL;DR
RT-TDDFT efficiently models optical response but has lacked an efficient way to decompose that response into KS electron–hole contributions. This work develops such an analysis method, benchmarks it against Casida calculations, and uses it to study silver nanoparticles. The analysis finds split plasmon resonances in Ag55 and distinct resonances in larger particles.
Problem
RT-TDDFT has lacked an efficient analysis method for decomposing optical response into the underlying KS electron–hole contributions.
Source: §I · p. 1Method
Results
The benzene-derivative decompositions are numerically almost identical between RT-TDDFT and Casida; Ag55 shows split resonances, while Ag147, Ag309, and Ag561 form distinct plasmon resonances.
Source: §V · p. 8Takeaways & Limitations
The method brings RT-TDDFT analysis capabilities to the level of Casida while retaining RT-TDDFT’s computational benefits.
Source: §V · p. 8RT-TDDFT yields response at chosen frequencies rather than direct access to the discrete spectrum; increasing propagation time can reduce the broadening-limited energy resolution.
Source: §IV · p. 8
Abstract
from arXiv · showhide
The real-time-propagation formulation of time-dependent density-functional theory (RT-TDDFT) is an efficient method for modeling the optical response of molecules and nanoparticles. Compared to the widely adopted linear-response TDDFT approaches based on, e.g., the Casida equations, RT-TDDFT appears, however, lacking efficient analysis methods. This applies in particular to a decomposition of the response in the basis of the underlying single-electron states. In this work, we overcome this limitation by developing an analysis method for obtaining the Kohn-Sham electron-hole decomposition in RT-TDDFT. We demonstrate the equivalence between the developed method and the Casida approach by a benchmark on small benzene derivatives. Then, we use the method for analyzing the plasmonic response of icosahedral silver nanoparticles up to Ag$_{561}$. Based on the analysis, we conclude that in small nanoparticles individual single-electron transitions can split the plasmon into multiple resonances due to strong single-electron-plasmon coupling whereas in larger nanoparticles a distinct plasmon resonance is formed.
I. INTRODUCTION
RT-TDDFT is an efficient alternative to Casida calculations, but it has lacked direct access to the KS electron–hole decomposition that helps analyze plasmonic response. The paper develops an equivalent decomposition for RT-TDDFT and applies it to silver nanoparticles.
I. INTRODUCTION
- RT-TDDFT offers favorable system-size scaling and access to nonlinear response, whereas Casida directly enables extensive KS electron–hole analysis.RT-TDDFT analyses have often been limited to absorption spectra or transition densities.Source: §I · p. 1
- The method obtains the KS electron–hole decomposition within RT-TDDFT, in equivalent fashion to the Casida approach.It is combined with an RT-TDDFT implementation based on LCAO in the open-source GPAW code.Source: §I · p. 1
- In Ag55, individual single-electron transitions strongly affect the plasmonic response and split the plasmon resonance, whereas in Ag147 and larger nanoparticles a distinct plasmon resonance forms from superposed single-electron transitions.Source: §I · p. 1
A. Linear response of the density matrix in the real-time propagation method
- A small δ-pulse perturbation is applied, the system is propagated in time, and recorded quantities such as the density matrix are Fourier transformed.The electric field is treated in the dipole approximation and aligned along z; its strength is assumed small enough to produce negligible nonlinear effects.Source: A. Linear response of the density… · p. 2
- Linear-response analysis needs only density-matrix elements ρ_ia between occupied and unoccupied KS states, linking the time-dependent density matrix to the usual KS electron–hole basis.The initial density matrix is defined before the δ-pulse, and the perturbation direction is indicated by the superscript z.Source: A. Linear response of the density… · p. 2
Implementation
The implementation uses an LCAO basis with PAW and defers density-matrix construction and Fourier transformation to post-processing. It stores the time evolution to preserve flexibility in later analysis.
Implementation
- The implementation uses LCAO wave functions and the PAW method to represent the density matrix in the basis-function representation.The PAW transformation affects only evaluation of the basis-function overlap integral.Source: Implementation · pp. 2–3
- Rather than calculate the density matrix at every propagation step, the implementation stores time-dependent expansion coefficients and constructs and transforms the density matrix afterward.The linearity of the equations allows Fourier transformation and matrix multiplication to be performed in either order.Source: Implementation · p. 3
- Storing the full time evolution uses more storage than on-the-fly transformations, but avoids restricting analysis to parameters chosen before the calculation.The authors report that storing the full evolution is advantageous in their experience.Source: Implementation · p. 3
B. Linear response of the density matrix in the Casida method
The Casida formulation obtains excitation energies and eigenvectors by solving an eigenvalue problem in the KS electron–hole space. Its matrix encodes KS transition energies and couplings between transitions.
B. Linear response of the density matrix in the Casida method
- The Casida approach solves a matrix eigenvalue equation in the KS electron–hole space to obtain excitation energies and corresponding eigenvectors.A double index ia denotes a transition from occupied state i to unoccupied state a.Source: B. Linear response of the density… · p. 3
- Casida matrix elements combine occupation-number differences, KS eigenvalue differences, and couplings between electron–hole transitions.The coupling matrix element represents the interaction between transitions i→a and j→b.Source: B. Linear response of the density… · p. 3
- A finite excitation lifetime is introduced by shifting ω to ω+iη, producing Lorentzian line shapes; Gaussian broadening is an alternative.The Gaussian line shape uses width parameter σ.Source: B. Linear response of the density… · p. 3
C. Kohn–Sham decomposition
The KS electron–hole response can be obtained with either RT propagation or Casida and related to photo-absorption. The resulting decomposition assigns absorption-spectrum contributions to KS transitions.
C. Kohn–Sham decomposition
- The frequency-dependent KS electron–hole density-matrix response is available from either RT propagation or Casida and can be connected to the photo-absorption cross-section.The dynamical polarizability provides the link to the observable absorption response.Source: C. Kohn–Sham decomposition · p. 3
- The dipole strength function describes photo-absorption and is normalized to integrate to the system’s electron count.Its relation to discrete-excitation oscillator strengths is noted in the passage.Source: C. Kohn–Sham decomposition · p. 3
- The KS decomposition of the absorption spectrum is used to analyze silver nanoparticles and expresses the response in electron–hole contributions.Related decompositions have also used spatial location or angular momentum.Source: C. Kohn–Sham decomposition · p. 4
III. RESULTS
The molecular benchmark uses benzene, naphthalene, and anthracene to compare RT-TDDFT with Casida calculations. Their calculated absorption spectra are virtually indistinguishable.
A. Benzene derivatives
- The benchmark covers benzene, naphthalene, and anthracene, whose well-defined π→π* transitions red-shift as the conjugated π-system grows.Source: A. Benzene derivatives · p. 4
- RT-TDDFT propagation includes all electron-hole pairs by propagating occupied orbitals, whereas Casida calculations commonly use an energy cutoff.All transitions in the available KS electron-hole space were included in the Casida calculations for comparability.Source: A. Benzene derivatives · p. 4
- The calculations use dzp basis sets, PBE, and an adiabatic exchange-correlation approximation; the basis sets are intended for qualitative analysis and benchmarking.The authors caution that the dzp basis sets may not yield complete-basis-limit numerical values.Source: A. Benzene derivatives · p. 4
- The Casida and RT-TDDFT methods yield virtually indistinguishable absorption spectra for the benzene derivatives.Figure 1 shows spectra along the molecules’ long axis; the response in other directions can be analyzed similarly.Source: A. Benzene derivatives · p. 4, Figure 1 · p. 4
Casida approach
The molecular benchmark compares Kohn–Sham electron-hole decompositions from Casida and RT-TDDFT. The approaches match closely, with a superimposed response for benzene’s degenerate excitations and a slight anthracene deviation associated with a nearby weak excitation.
Casida approach
- Each molecule’s response is dominated by a single absorption peak, and the Casida decomposition gives normalized weights for its contributing transitions.Benzene’s 7.2 eV excitation connects the doubly degenerate HOMO and LUMO; naphthalene and anthracene’s prominent B3u excitation is mainly composed of HOMO→LUMO+1 and HOMO−1→LUMO transitions.Source: Casida approach · p. 4
RT-TDDFT approach
- The superimposed RT-TDDFT eigenvector for benzene’s degenerate excitations matches the equivalent quantity from Casida.For naphthalene and anthracene, RT-TDDFT also reproduces the Casida decomposition with very good numerical accuracy; the anthracene values differ slightly when a nearby weak excitation contributes.Source: RT-TDDFT approach · p. 5
- For anthracene, a nearby weak excitation at 5.051 eV with oscillator strength 5·10−4 contributes to the slight difference in numerical values.Source: RT-TDDFT approach · p. 5
B. Silver nanoparticles
The silver-nanoparticle study considers icosahedral Ag55, Ag147, Ag309, and Ag561 using RT-TDDFT. Its spectra are presented alongside non-interacting-electron spectra, with the Ag55 response motivating further analysis.
B. Silver nanoparticles
- The study considers the same icosahedral nanoparticle series, geometries, and computational parameters used in earlier work, from Ag55 to Ag561.The calculations use optimized LCAO basis sets and the GLLB-SC exchange-correlation potential.Source: B. Silver nanoparticles · p. 5
- The nanoparticle absorption spectra are shown with non-interacting-electron spectra for comparison, vertically shifted and scaled by 0.2.The non-interacting spectra are calculated from KS eigenvalue differences and transition dipole matrix elements.Source: B. Silver nanoparticles · p. 5, Figure 2 · p. 6
- Earlier spectra showed a well-formed plasmon resonance in Ag147 and larger particles, while Ag55 had multiple peaks whose origin was unresolved.The section analyzes the nanoparticle response through the KS decomposition to investigate Ag55.Source: B. Silver nanoparticles · p. 6
Transition contribution maps
The transition contribution map (TCM) displays Kohn–Sham decomposition weights across occupied and unoccupied state energies at a fixed frequency. Its signed absorption weights and state-character information help characterize plasmonic responses.
Transition contribution maps
- At a fixed frequency, the TCM maps Kohn–Sham decomposition weights onto the occupied- and unoccupied-state energy plane.This representation is especially useful when plasmon resonances comprise many electron-hole excitations.Source: Transition contribution maps · p. 6
- For icosahedral nanoparticles, the response is isotropic and the decomposition is degenerate.The three Cartesian components of the response are equal.Source: Transition contribution maps · p. 6
- The absorption-decomposition weight is normalized by total absorption and retains the response sign, unlike a squared transition-density weight.The signed weight has a physically sound interpretation as the photo-absorption decomposition.Source: Transition contribution maps · p. 6
- The density-of-states coloring indicates sp and d character, estimated from angular-momentum components of the LCAO basis functions.The d character of a state is estimated from the squared coefficients of basis functions with d angular momentum.Source: Transition contribution maps · p. 6
Analysis of Ag147, Ag309, and Ag561
The TCMs distinguish plasmon formation in larger silver nanoparticles from peak splitting in Ag55. They associate the larger-particle response with low-energy sp transitions and d-electron damping, while marked sp transitions in Ag55 strongly couple to the plasmon.
Analysis of Ag147, Ag309, and Ag561
- In Ag147, Ag309, and Ag561, low-energy sp transitions contribute constructively and shift the resonance above non-interacting-electron peaks, while d-electron transitions damp the response.The TCM also shows increasingly continuous densities of states and more uniform contributions as nanoparticle size increases.Source: Analysis of Ag147, Ag309, and Ag561 · p. 6, Figure 3 · p. 7
Analysis of Ag55
- In Ag55, strongly contributing sp transitions near the peak energies change sign between adjacent peaks while the low-energy plasmon-forming transitions remain largely unchanged, consistent with strong coupling and peak splitting.The marked transitions change sign at 3.85 eV and 4.06 eV, between absorption maxima at 3.71, 4.00, and 4.20 eV.Source: Analysis of Ag55 · p. 7
- In larger nanoparticles, interaction between the plasmon and nearby Kohn–Sham transitions is weak and appears as peak broadening rather than splitting.This contrasts with the multiple resonances observed for Ag55.Source: Analysis of Ag55 · pp. 7–8
- Some d-electron transitions also change sign near the Ag55 peak splitting, but their sign changes do not track the absorption maxima and minima.The authors therefore expect the marked sp transitions to be the major cause of the splitting.Source: Analysis of Ag55 · p. 8
- Ag55 spectra vary with geometry, exchange-correlation functional, and numerical parameters, and reported spectra can have either single or multiple peaks.The authors present the observed splitting behavior as a potentially useful concept for understanding small-particle response.Source: Analysis of Ag55 · p. 8
IV. DISCUSSION
The discussion weighs RT-TDDFT’s scaling and analysis capabilities against its practical limits, while the conclusions report a close Casida benchmark and size-dependent plasmon behavior in silver nanoparticles.
IV. DISCUSSION
- RT-TDDFT scales more favorably with system size, while Casida’s smaller prefactor can make it more efficient for small and moderately sized systems, especially with non-local functionals.RT-TDDFT is attractive for systems with thousands of electrons and typically hundreds of atoms.Source: §IV · p. 8
- RT-TDDFT’s response depends on the initializing perturbation: an x-directed dipole perturbation reveals only excitations with a dipole component parallel to x.Up to three axis-directed propagations can recover the full dynamical polarizability tensor, but dipole-forbidden excitations require different initial perturbations.Source: §IV · p. 8
- RT-TDDFT yields combined response at chosen frequencies rather than direct access to the discrete spectrum.Reducing the broadening parameter to improve energy resolution requires increasing propagation time; individual excitations are less distinct in larger systems with denser spectra.Source: §IV · p. 8
V. CONCLUSIONS
- Benzene-derivative KS decompositions from Casida and RT-TDDFT calculations are numerically almost identical.The method obtains the linear response of the density matrix in the KS electron-hole basis through a basis transformation.Source: §V · p. 8
- Ag55 shows plasmon splitting associated with strong coupling to individual transitions, whereas Ag147, Ag309, and Ag561 form distinct resonances from constructive superposition of low-energy KS transitions, with d-electron damping.The interaction between individual transitions near the resonance is weaker in the larger particles.Source: §V · p. 8
- The method brings RT-TDDFT analysis capabilities to the level of Casida without compromising RT-TDDFT’s computational benefits.It is implemented in a recent RT-TDDFT code and is to be made publicly available as part of GPAW.Source: §V · p. 8
formalism
The formalism connects the LCAO density matrix to the all-electron density matrix and transforms the response into the ground-state Kohn–Sham orbital basis.
formalism
- In LCAO, pseudo wavefunctions are expanded in localized basis functions with time-dependent coefficients, and the all-electron basis functions are defined by applying the PAW transformation.The pseudo wavefunction is expanded in localized functions centered at atomic coordinates.Source: formalism · p. 9
- The time-dependent all-electron real-space density matrix is obtained from the density matrix in the LCAO basis, then transformed to the ground-state KS orbital basis.The LCAO density matrix is given by Eq. (8).Source: formalism · p. 9
- Within the PAW formalism, the PAW transformation affects only evaluation of the LCAO overlap integrals Sµµ′.The overlap integrals are isolated as quantities used regularly in LCAO calculations.Source: formalism · p. 9