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Related Concept Videos

Euler's Formula for Pin-Ended Columns01:21

Euler's Formula for Pin-Ended Columns

454
In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
To calculate the critical load,...
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¹H NMR: Complex Splitting01:13

¹H NMR: Complex Splitting

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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...
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Euler's Formula to Columns with Other End Conditions01:15

Euler's Formula to Columns with Other End Conditions

722
Euler's formula is very important in the field of structural engineering, providing a foundation for understanding the critical loading conditions of pin-ended columns. This formula links the modulus of elasticity, the moment of inertia of the cross-section, and the column's length, offering a precise calculation of the critical load at which a column is prone to buckling.
722
¹H NMR: Interpreting Distorted and Overlapping Signals01:02

¹H NMR: Interpreting Distorted and Overlapping Signals

1.2K
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...
1.2K
Interpreting ¹H NMR Signal Splitting: The (n + 1) Rule01:10

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

1.8K
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...
1.8K
Euler's Formula to Columns: Problem Solving01:23

Euler's Formula to Columns: Problem Solving

653
Euler's formula is used in structural engineering to determine the buckling load of columns under various conditions. However, when dealing with systems that incorporate both rigid elements and elastic components, such as springs, the analysis requires a finer approach to determine the critical load. The problem described involves two rigid bars connected at a pivot point with a spring attached and a vertical load applied at one end.
The system comprises two vertical rigid bars, AB and BC,...
653

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Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids
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Approximate Analytical Solutions for the Euler Equation for Second-Row Homonuclear Dimers.

Kati Finzel1

  • 1Technische Universität Dresden, Bergstrasse 66c, Dresden 01069, Germany.

Journal of Chemical Theory and Computation
|August 19, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces an analytical method for solving the Euler equation in orbital-free density functional theory for second-row homonuclear dimers. This approach provides direct electron density solutions, bypassing iterative calculations required by the Kohn-Sham method.

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Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Density Functional Theory

Background:

  • The Kohn-Sham method, while powerful, requires solving numerous coupled differential equations iteratively.
  • Orbital-free density functional theory (OF-DFT) offers a more direct approach by utilizing the Euler equation.
  • Efficient methods for solving the Euler equation are crucial for advancing OF-DFT.

Purpose of the Study:

  • To develop a novel analytical method for approximate solutions to the Euler equation.
  • To apply this method to second-row homonuclear dimers.
  • To demonstrate the feasibility of direct electron density calculation in OF-DFT.

Main Methods:

  • Developed an analytical solution strategy for the Euler equation.
  • Employed an atom-centered monopole expansion with a single free parameter for simplified models.
  • Solved the Euler equation at the bond critical point to obtain electron density.

Main Results:

  • Successfully obtained approximate analytical solutions for the Euler equation.
  • Applied the method to N2, C2, and B2 molecules.
  • Calculated internuclear distances of 2.01 bohr for N2, 2.43 bohr for C2, and 3.07 bohr for B2.

Conclusions:

  • The proposed analytical method provides a viable alternative to iterative solutions in OF-DFT.
  • This approach successfully yields bound molecules for studied second-row homonuclear dimers.
  • The method demonstrates the potential for direct electron density determination in molecular systems.