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Stability of Equilibrium Configuration01:23

Stability of Equilibrium Configuration

429
Understanding the stability of equilibrium configurations is a fundamental part of mechanical engineering. In any system, there are three distinct types of equilibrium: stable, neutral, and unstable.
A stable equilibrium occurs when a system tends to return to its original position when given a small displacement, and the potential energy is at its minimum. An example of a stable equilibrium is when a cantilever beam is fixed at one end and a weight is attached to the other end. If the weight...
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Chemical Shift: Internal References and Solvent Effects01:17

Chemical Shift: Internal References and Solvent Effects

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In an NMR sample, precise measurement of the absolute absorption frequencies of nuclei is difficult. A standard internal reference compound is added, and the frequency difference between the reference signal and sample signals is measured.
The internal reference compound generally used in NMR spectroscopy is tetramethylsilane (TMS). TMS is preferred because it is chemically inert, soluble in NMR solvents, and easily removable. Also, the highly shielded methyl protons in TMS yield an intense...
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Homogeneous Equilibria for Gaseous Reactions02:15

Homogeneous Equilibria for Gaseous Reactions

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Homogeneous Equilibria for Gaseous Reactions
For gas-phase reactions, the equilibrium constant may be expressed in terms of either the molar concentrations (Kc) or partial pressures (Kp) of the reactants and products. A relation between these two K values may be simply derived from the ideal gas equation and the definition of molarity. According to the ideal gas equation:
24.6K
Radical Anti-Markovnikov Addition to Alkenes: Thermodynamics01:32

Radical Anti-Markovnikov Addition to Alkenes: Thermodynamics

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The anti-Markovnikov addition of hydrogen halides to an alkene is thermodynamically feasible only with HBr. The radical addition reaction with other hydrogen halides like HCl and HI is thermodynamically unfavorable.
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Stability of Equilibrium Configuration: Problem Solving01:13

Stability of Equilibrium Configuration: Problem Solving

574
The stability of equilibrium configurations is an important concept in physics, engineering, and other related fields. In simple terms, it refers to the tendency of an object or system to return to its equilibrium position after being disturbed. The stability of an equilibrium configuration can be analyzed by considering the potential energy function of the system and examining its behavior near the equilibrium point.
Problem-solving in the context of the stability of equilibrium configuration...
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¹H NMR: Interpreting Distorted and Overlapping Signals01:02

¹H NMR: Interpreting Distorted and Overlapping Signals

1.0K
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...
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Spin Saturation Transfer Difference NMR SSTD NMR: A New Tool to Obtain Kinetic Parameters of Chemical Exchange Processes
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对于开系统的Kohn-Sham反转.

Jannis Erhard1, Egor Trushin1,2, Andreas Görling1,2

  • 1Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany.

The Journal of chemical physics
|January 16, 2025
PubMed
概括
此摘要是机器生成的。

密度函数理论方法通常会打破对称性,导致诸如旋转污染等问题. 本研究引入了稳定的方法来构建Kohn-Sham潜力,恢复对称性并为开系统提供参考能量值.

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科学领域:

  • 量子化学 是一个量子化学.
  • 计算物理 计算物理
  • 材料科学 材料科学 材料科学

背景情况:

  • 标准密度函数理论 (DFT) 方法使用自旋不受限制的Kohn-Sham (KS) 形式主义用于开放系统.
  • 这种方法打破了空间和旋转对称性,导致旋转污染,并阻碍了适当的波函数分类.
  • 存在恢复对称性的KS变体,但很少实施.

研究的目的:

  • 开发数值稳定的Kohn-Sham (KS) 倒置方法,从参考旋转密度中构建KS潜力.
  • 研究四种对称性对待开放系统的方法:非对称,旋转对称,空间对称和完全对称.
  • 为跨越这些形式主义的能量成分差异提供参考值.

主要方法:

  • 采用了数量稳定的KS反转技术.
  • 从参考旋转密度构建的参考KS潜力.
  • 使用完整配置相互作用和对原子和二原子分子的合集群方法获得的参考旋转密度.

主要成果:

  • 成功构建了所有四种对称性治疗的KS潜力.
  • 证明了能量分解在四个KS形式上有所不同.
  • 确定非完全对称的KS反向可以违反 Aufbau 原则,并且在旋转对称的情况下,KS v-可表示性.

结论:

  • 该研究为应用对称性恢复KS形式主义提供了实际框架.
  • 参考能量值是为了比较不同的对称性处理而建立的.
  • 违反物理原则,如 Aufbau 原则和 KS v 代表性,突出了完全对称的方法的重要性.