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The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...
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The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
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The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
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Improving the Reliability of, and Confidence in, DFT Functional Benchmarking through Active Learning.

Javier E Alfonso-Ramos1, Carlo Adamo1, Éric Brémond2

  • 1Ecole Nationale Supérieure de Chimie de Paris, Université PSL, CNRS, i-CLeHS, 75 005 Paris, France.

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Summary

Active learning efficiently curates benchmarking data for density functional theory (DFT) calculations. This approach identifies challenging chemical reactions, improving the reliability of DFT functional validation across diverse chemical spaces.

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Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Method Development

Background:

  • Density functional theory (DFT) calculations require reliable validation of exchange-correlation functionals.
  • Current DFT benchmarking data sets are often assembled without a clear strategy, leading to chemical bias and limited transferability.
  • Improving the data acquisition process is crucial for enhancing the accuracy and applicability of DFT methods.

Purpose of the Study:

  • To develop a data-efficient method for curating benchmarking data sets for DFT functionals.
  • To address the limitations of unprincipled data assembly in existing validation approaches.
  • To create a more representative and challenging data set for evaluating pericyclic reaction calculations.

Main Methods:

  • Employed an active learning strategy to guide the selection of new data points.
  • Designed a chemical reaction space by combining reaction templates and substituents around an initial data set (BH9).
  • Trained a surrogate model to predict the divergence of activation energies across 20 DFT functionals.

Main Results:

  • Identified molecular structures leading to significant DFT functional divergence.
  • Demonstrated that the relationship between molecular structure and functional divergence is highly learnable.
  • Achieved convergence with fewer than 100 acquired reactions, significantly enhancing the data set.
  • Curated a new, more representative pericyclic reaction benchmarking data set.

Conclusions:

  • Active learning offers a data-efficient solution for creating robust DFT benchmarking data sets.
  • The developed method significantly improves the quality and representativeness of validation data.
  • The curated data set reveals substantial changes in DFT functional performance compared to the original subset.