Singular Spectrum Analysis of Time-series Data from Time-dependent density-functional theory in Real-time

TL;DR

The paper addresses limited spectral resolution in real-time TDDFT caused by finite time evolution by applying Singular Spectrum Analysis (SSA) to the dipole-time-series $bmu(t)$. SSA decomposes the dynamics into fundamental oscillations and enables forecast-based extension to effectively longer time series, yielding higher-resolution spectra on short simulations with a resolution on the order of $O(1/T)$. The authors demonstrate the approach on ethylene and several small molecules, isolating band-edge oscillations near key transitions (e.g., peaks around $7.5$, $11.8$, and $18.4$ eV for ethylene) and forecasting spectra that agree with long-time TDDFT results. This method offers a practical route to accurate emission and absorption spectra with reduced computational cost and wide applicability in molecular optics analysis, enabling detailed spectral interpretation in specific energy regions.$

Abstract

This paper introduces a spectral analysis of time-seires data derived from real-time time-dependent density functional theory (TDDFT) using Singular Spectrum Analysis (SSA). TDDFT is a robust method for obtaining molecular excited states and optical spectra by tracking the time evolution of dynamical dipole moments. However, the spectral resolution can be compromised when Fourier transformation's total time duration is insufficient. SSA enabled the extraction of specific oscillation components from the time-series data, facilitating the generation of higher-precision spectra. Even with relatively short time-series dataset, the predictive extension of SSA yielded high-resolution spectra, demonstrating substantial agreement with results obtained through conventional methods. The efficacy of this approach was validated for several small molecules, including ethylene, benzene, and others. SSA's ability to conduct detailed spectral anasysis in specific energy regions enhance spectral resolution and facilitates the clarification of oscillation components within these regions. Real-time TDDFT combined with SSA provides a new analytical method for analyzing the optical properties of molecules, significantly improving the accuracy of the analysis of emission and absorption spectra analysis. This method is expected to have various applications.

Figures (5)

• Figure 1: Dipole moment $\mu(t)$ and oscillator strength $S(\omega)$ for steps ( a)( b) $N=5000$ and ( c)( d) $N=20000$, respectively.
• Figure 2: Oscillations of the dipole moment $\mu(t)$ for ethylene were decomposed and reconstructed into individual signal components using SSA with a bandwidth $\tau=1000$.
• Figure 3: Correlation matrix on the decomposed and reconstructed time-series data $\{{\tilde{F}}_1,\dots,{\tilde{F}}_{10} \}$ of ethylene dipole moment.
• Figure 4: Dynamic dipole moment and spectrum of ethylene. ( a) the base $\mu(t)$ and ( c) its spectrum, ( b) the extraction of the band-edge component by SSA and its spectral prediction $\mu(t)$, and ( d) its spectral improvement, normalized by the maximum of the peak obtained from $N=20000$. Dynamic dipole moment and spectrum of ethylene. ( a) the base $\mu(t)$ and ( c) its spectrum, ( b) the extraction of the band-edge component by SSA and its spectral prediction$\mu(t)$, and ( d) its spectral improvement, normalized by the maximum of the peak obtained from $N=20000$.
• Figure 5: Comparison of the calculated band-edge spectra for benzene (solid line), naphthalene (dashed line), anthracene (dash dot line), and tetracene (dotted line). ( a) Spectrum obtained from the original $\mu(t)$ calculated using TDDFT up to $N=2000$ steps. ( b) Spectrum derived from the extracted oscillation ${\tilde{\mu}}(t)$, associated with the band-edge, extended to $N=8000$ steps using SSA and forecasting. ( c) Spectrum obtained from the time evolution of up to $N=8000$ steps simply using TDDFT with $\Delta t=0.002$ [1/eV].