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Related Concept Videos

¹H NMR: Interpreting Distorted and Overlapping Signals01:02

¹H NMR: Interpreting Distorted and Overlapping Signals

Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are slanted or...
Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...
¹H NMR: Pople Notation01:09

¹H NMR: Pople Notation

The Pople nomenclature system classifies spin systems based on the difference between their chemical shifts. Coupled spins are denoted by capital letters with subscripts indicating the number of equivalent nuclei. When the coupled nuclei have well-separated chemical shifts, they are assigned letters that are far apart in the alphabet, such as A and X. When the difference in chemical shifts is small, coupled nuclei are named using adjacent letters of the alphabet (AB, MN, or XY).
A proton...
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Molecular Orbital Theory II

Molecular Orbital Energy Diagrams
MO Theory and Covalent Bonding02:40

MO Theory and Covalent Bonding

The molecular orbital theory describes the distribution of electrons in molecules in a manner similar to the distribution of electrons in atomic orbitals. The region of space in which a valence electron in a molecule is likely to be found is called a molecular orbital. Mathematically, the linear combination of atomic orbitals (LCAO) generates molecular orbitals. Combinations of in-phase atomic orbital wave functions result in regions with a high probability of electron density, while...
Second Order systems I01:20

Second Order systems I

A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...

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Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
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Communications: Explicitly correlated second-order Møller-Plesset perturbation method for extended systems.

Toru Shiozaki1, So Hirata

  • 1Department of Applied Chemistry, School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan. shiozaki.toru@gmail.com

The Journal of Chemical Physics
|April 29, 2010
PubMed
Summary

This study introduces an advanced computational method for electron correlation in one-dimensional systems. The new approach significantly improves the accuracy of calculating correlation energy for materials like polyethylene.

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Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Materials Science

Background:

  • Accurate calculation of electron correlation is crucial for predicting material properties.
  • Traditional methods often struggle with slow convergence and basis set limitations.
  • Explicitly correlated methods offer a path to overcome these challenges.

Purpose of the Study:

  • To develop and implement a formalism for the second-order Møller-Plesset perturbation method with explicit correlation (MP2-R12/F12).
  • To adapt this method for extended systems periodic in one dimension.
  • To assess the accuracy and efficiency of the new method for calculating correlation energy.

Main Methods:

  • Derivation of a formalism for MP2-R12/F12 method.
  • Implementation into computer codes for 1D periodic systems.
  • Use of resolution-of-the-identity approximation with complementary auxiliary basis sets for integral evaluation.
  • Fixing excitation amplitudes to satisfy the first-order cusp condition.

Main Results:

  • Demonstration of correct size dependence for calculated integrals and correlation energy.
  • Obtained valence MP2 correlation energy for polyethylene near the complete basis-set limit.
  • Showed that the R12 treatment yields a considerably larger correlation energy compared to standard methods.

Conclusions:

  • The developed MP2-R12/F12 method is suitable for accurate correlation energy calculations in 1D extended systems.
  • Explicitly correlated methods provide significant improvements over traditional approaches.
  • This formalism offers a more reliable way to study electronic properties of low-dimensional materials.