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Related Concept Videos

¹H NMR: Complex Splitting01:13

¹H NMR: Complex Splitting

A proton M that is coupled to a proton X results in doublet signals for M. However, NMR-active nuclei can be simultaneously coupled to more than one nonequivalent nucleus. When M is coupled to a second proton A, such as in styrene oxide, each peak in the doublet is split into another doublet.
Splitting diagrams or splitting tree diagrams are routinely used to depict such complex couplings. While drawing splitting diagrams, the splitting with the larger coupling constant is usually applied first.
Binomial Expansion Using Pascal's Triangle01:30

Binomial Expansion Using Pascal's Triangle

Expanding a binomial expression such as (a + b)n results in a predictable sequence of terms that can be systematically derived using Pascal’s Triangle. This triangular array of numbers plays a central role in understanding and computing the coefficients of binomial expansions.Pascal’s Triangle is constructed such that each row corresponds to the coefficients of a binomial raised to a power. The topmost row, known as the zeroth row, corresponds to (a + b)0, and each successive row gives the...
Interpreting ¹H NMR Signal Splitting: The (n + 1) Rule01:10

Interpreting ¹H NMR Signal Splitting: The (n + 1) Rule

In the AX proton spin system, proton A can sense the two spin states of a coupled proton X, resulting in a doublet NMR signal with two peaks of equal (1:1) intensity. When proton A is coupled to two equivalent protons (AX2 spin system), the spin states of each X can be aligned with or against the external field, creating three possible scenarios. This results in a 1:2:1  triplet signal, where the central peak corresponds to the chemical shift of A and is twice as large or intense as the others.
Force and Potential Energy in One Dimension01:13

Force and Potential Energy in One Dimension

Force can be calculated from the expression for potential energy, which is a function of position. The component of a conservative force, in a particular direction, equals the negative of the derivative of the corresponding potential energy with respect to the displacement in that direction. For regions where potential energy changes rapidly with displacement, the work done and force is maximum. Also, when force is applied along the positive coordinate axis, the potential energy decreases with...
Thermodynamic Potentials01:26

Thermodynamic Potentials

Thermodynamic potentials are state functions that are extremely useful in analyzing a thermodynamic system. They have dimensions of energy. The four important thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These thermodynamic potentials can be expressed using two of the following variables: pressure, volume, temperature, and entropy. These two variables are expressed as the rate of change of the thermodynamic potential with respect to other...
Hybridization of Atomic Orbitals II03:35

Hybridization of Atomic Orbitals II

sp3d and sp3d 2 Hybridization

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Related Experiment Video

Updated: May 21, 2026

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

Hierarchical Truncations for Many-Body Expansion Potentials.

Bryce M Westheimer1,2,3,4, Mark S Gordon3,4, Emilie B Guidez1,2

  • 1Department of Chemistry, University of Colorado, Denver, Colorado 80217, United States.

Journal of Chemical Theory and Computation
|May 20, 2026
PubMed
Summary
This summary is machine-generated.

A new hierarchical many-body expansion (HMBE) method efficiently truncates high-order terms in large molecular systems. This approach offers accurate calculations for complex structures like proteins with reduced computational cost.

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Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates
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Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates

Published on: February 15, 2016

Related Experiment Videos

Last Updated: May 21, 2026

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

Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates
06:35

Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates

Published on: February 15, 2016

Area of Science:

  • Computational chemistry
  • Theoretical chemistry
  • Molecular modeling

Background:

  • The many-body expansion (MBE) is a powerful method for calculating properties of large molecular systems.
  • However, computing high-order terms in MBE can be computationally expensive.
  • Accurate and efficient methods are needed to model increasingly complex systems.

Purpose of the Study:

  • To introduce a novel hierarchical many-body expansion (HMBE) approach.
  • To develop a strategy for truncating high-order terms in MBE more efficiently.
  • To enable accurate modeling of very large molecular systems.

Main Methods:

  • The proposed hierarchical many-body expansion (HMBE) method partitions systems into multitier fragments.
  • Numerical tests were performed on (H2O)64 structures.
  • The HMBE scheme allows for optional inclusion of "Schengen terms" at fragment interfaces to enhance accuracy.

Main Results:

  • HMBE demonstrated satisfactory relative energies and binding energies for (H2O)64 clusters compared to full calculations.
  • Significantly fewer high-order terms were computed using HMBE than conventional MBE.
  • The accuracy of HMBE can be further improved with "Schengen terms".

Conclusions:

  • The hierarchical many-body expansion (HMBE) scheme is a promising framework for modeling very large molecular systems.
  • HMBE offers a computationally efficient and accurate alternative to conventional MBE.
  • This method is particularly suitable for systems with inherent hierarchical structures, such as proteins.