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Related Concept Videos

Electronic Structure of Atoms02:28

Electronic Structure of Atoms


An atom comprises protons and neutrons, which are contained inside the dense, central core called the nucleus, with electrons present around the nucleus. Taking into account the wave–particle duality of electrons and the uncertainty in position around the nucleus, quantum mechanics provides a more accurate model for the atomic structure. It describes atomic orbitals as the regions around the nucleus where electrons of discrete energy exist, characterized by four quantum numbers:  n, l, ml, and...
Second Uniqueness Theorem01:16

Second Uniqueness Theorem

Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the surface...
The Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

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. Schrödinger...
Hybridization of Atomic Orbitals II03:35

Hybridization of Atomic Orbitals II

sp3d and sp3d 2 Hybridization
Hybridization of Atomic Orbitals I03:24

Hybridization of Atomic Orbitals I

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...
¹H NMR: Long-Range Coupling01:27

¹H NMR: Long-Range Coupling

The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
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Related Experiment Video

Updated: Jun 21, 2026

Modeling Ligands into Maps Derived from Electron Cryomicroscopy
09:30

Modeling Ligands into Maps Derived from Electron Cryomicroscopy

Published on: July 19, 2024

Extension of partial atom-to-atom maps: uniqueness and algorithms.

Marcos E González Laffitte1,2, Tieu-Long Phan3,4, Peter F Stadler5,6,7,8,9,10,11

  • 1Center for Scalable Data Analytics and Artificial Intelligence Dresden-Leipzig (ScaDS.AI), Leipzig University, Humboldtstrasse 25, 04105, Leipzig, Saxony, Germany. marcos@bioinf.uni-leipzig.de.

Algorithms for Molecular Biology : AMB
|June 19, 2026
PubMed
Summary
This summary is machine-generated.

Atom-to-atom maps (AAMs) are crucial for chemical synthesis and metabolomics but are often missing. This study presents a graph-theoretic method using Imaginary Transition State (ITS) graphs to uniquely extend partial AAMs, enabling accurate reconstruction.

Keywords:
Atom-to-atom mapsChemical reaction mechanismsChemical synthesis planningCondensed graph of the reaction (CGR)Constrained graph isomorphismImaginary transition state (ITS)Metabolic networksMolecular graphs

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Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

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Last Updated: Jun 21, 2026

Modeling Ligands into Maps Derived from Electron Cryomicroscopy
09:30

Modeling Ligands into Maps Derived from Electron Cryomicroscopy

Published on: July 19, 2024

Picometer-Precision Atomic Position Tracking through Electron Microscopy
15:04

Picometer-Precision Atomic Position Tracking through Electron Microscopy

Published on: July 3, 2021

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
12:11

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Area of Science:

  • Computational Chemistry
  • Graph Theory
  • Cheminformatics

Background:

  • Chemical reaction databases lack atom-to-atom correspondence, hindering synthesis planning and metabolomics.
  • Accurate atom-to-atom maps (AAMs) are essential but difficult to compute due to quantum mechanical underpinnings.

Purpose of the Study:

  • To develop a computational method for reconstructing AAMs.
  • To identify conditions for unique AAM extension using graph theory.

Main Methods:

  • Focusing on partial AAMs covering the reaction center.
  • Utilizing Imaginary Transition State (ITS) graphs to represent AAMs.
  • Solving constrained graph-isomorphism problems for AAM extension.

Main Results:

  • Unique AAM extension is guaranteed if partial maps cover the reaction center.
  • Method is generalized for reactions where hydrogen atoms are implicit.
  • Benchmarking of computational tools for AAM reconstruction was performed.

Conclusions:

  • The ITS graph framework provides a robust method for AAM reconstruction.
  • Computational approaches can reliably determine AAMs, advancing chemical informatics applications.