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Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra.
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The mathematical expression known as the wave function, ψ, contains information about each orbital and the wavelike properties of electrons in an isolated atom. When atoms are bound together in a molecule, the wave functions combine to produce new mathematical descriptions that have different shapes. This process of combining the wave functions for atomic orbitals is called hybridization and is mathematically accomplished by the linear combination of atomic orbitals. The new orbitals that...
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Constructing and Visualizing Models using Mime-based Machine-learning Framework
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Atom-density representations for machine learning.

Michael J Willatt1, Félix Musil1, Michele Ceriotti1

  • 1Laboratory of Computational Science and Modeling, Institute of Materials, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.

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|April 22, 2019
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Summary
This summary is machine-generated.

Machine learning in chemistry requires concise atomic system representations. This study introduces a new abstract definition based on smoothed atomic density, unifying existing methods and enabling systematic tuning for better material property prediction.

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Area of Science:

  • Computational Chemistry
  • Materials Science
  • Machine Learning

Background:

  • Machine learning (ML) applications in chemistry and materials science are rapidly expanding.
  • A key challenge is developing complete yet concise representations of atomic systems for ML models.
  • Current methods for converting atomic structures into ML inputs are diverse and numerous.

Purpose of the Study:

  • To introduce an abstract, basis set-independent definition of chemical environments.
  • To unify and generalize existing representations for atomic systems in ML.
  • To provide a framework for systematically tuning ML representations for materials discovery.

Main Methods:

  • Defined chemical environments using a smoothed atomic density.
  • Employed bra-ket notation for basis set independence.
  • Computed correlations via inner products of feature kets, with explicit representations in real and Fourier space.

Main Results:

  • Developed a formalism equivalent to smooth overlap of atomic positions power spectrum and n-body correlations.
  • Demonstrated connections between the abstract definition and popular existing representations.
  • Introduced operators for systematic tuning of structure-composition-property correlations.

Conclusions:

  • The proposed formalism unifies recent developments in ML representations for materials science.
  • It offers a pathway toward more effective and computationally efficient ML schemes.
  • Enables systematic tuning of representations for improved prediction of material properties.