Improved Maximum Entropy Method applied to Real-time Time-Dependent Density Functional Theory

Abstract

Abstract.The maximum entropy method (MEM) is one of the key techniques for spectral analysis. The main feature is to describe spectra in low frequency with short time-series data. We adopted MEM to analyze the spectrum from the dipole moment obtained by the timedependent density functional theory (TDDFT) calculation in real time, which is intensively studied and applied to computing optical properties. In the MEM analysis, we proposed that we use the concatenated data set made from several-times repeated raw data together with the phase. We have applied this technique to spectral analysis of the TDDFT dipole moment of oligo-fluorene with n = 8. As a result, the higher resolution can be obtained without any peak shift due to the phase jump. The peak position is in good agreement to that of FT with just raw data. This paper presents the efficiency and characteristic features of this technique. IntroductionTime-dependent density functional theory (TDDFT) is a powerful tool for analyzing optical properties of medium-to-large sized molecule. We employ a real-time and real-space technique to solve the time-dependent Kohn-Sham equations. Within the framework of this approach, the wave functions are calculated by the finite difference method on real spatial grids [1] without using explicit bases such as plane waves or Gaussian. In our procedure to calculate optical properties, we use the time-series data, namely the dynamic dipole moment, from whose Fourier transform (FT) optical properties are calculated in a usual technique. The spectral resolution depends on the length of the dipole moment. To obtain the good resolution, the computational cost is quite expensive.To solve this difficulty, we focus on Maximum Entropy Method (MEM) [2], which is one of the key techniques of spectrum analysis. The principles of MEM are methods of the information theory for estimating unknown probability distributions based on the information about their expected values. MEM is widely used to the solution of a variety of problems related to spectral estimation such as earth and planetary science [3], spectroscopy [4]. The main feature is to obtain a fairy high resolution and accuracy with a relatively small number of time-series data. MEM is based on the Fourier pair of the autocorrelation and power spectrum. We applied this technique to spectral analysis of time-dependent dipole moments of molecules, which is calculated by realtime TDDFT. In the comparison of MEM and FT spectrum for several molecules, we realize that MEM needs less time step than FT to obtain the optical spectrum with same resolution [6]. As a new improved MEM, we proposed to use the concatenated data set made from several-times

Применение Метода Максимальной Энтропии К Временным Рядам, Получаемым В Реальном Времени В Рамках Нестационарной Теории Функционала Плотности

“…With only the repeated signal (the solid and dashed lines), the strength of the first peak is emphasized. Without a phase shift, we can observe that the peak position shifts due to the side effect of the phase jump [6]. However, we can obtain a clear first peak using our new method (the solid line).…”

“…We realize that MEM needs less time step than FT to obtain the optical spectrum with same level resolution [5]. Further, we proposed to use the concatenated data set made from several-times repeated raw data [6] together with the phase to minimize the side effect of the artificial periodicity. The introduction of this MEM provides the much better spectral resolution of the target peak near the energy gap.…”