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Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first column of the Routh...
Approximate Integration01:24

Approximate Integration

In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
¹H NMR of Conformationally Flexible Molecules: Temporal Resolution00:52

¹H NMR of Conformationally Flexible Molecules: Temporal Resolution

At room temperature, the chair conformer of cyclohexane undergoes rapid ring flipping between two equivalent chair conformers at a rate of approximately 105 times per second. These two chair conformers are in equilibrium. The rapid ring flipping results in the interconversion of the axial proton to an equatorial proton and an equatorial to the axial proton. Such interconversions are too rapid and cannot be detected on the NMR timescale. Hence, the NMR spectrometer cannot distinguish between the...
Woodward–Hoffmann Selection Rules and Microscopic Reversibility01:34

Woodward–Hoffmann Selection Rules and Microscopic Reversibility

Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...
Stability of Substituted Cyclohexanes02:30

Stability of Substituted Cyclohexanes

This lesson discusses the stability of substituted cyclohexanes with a focus on energies of various conformers and the effect of 1,3-diaxial interactions.
The two chair conformations of cyclohexanes undergo rapid interconversion at room temperature. Both forms have identical energies and stabilities, each comprising equal amounts of the equilibrium mixture. Replacing a hydrogen atom with a functional group makes the two conformations energetically non-equivalent.
For example, in...
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Fischer Projections

Learning to draw Fischer projections of molecules and understanding their relevance plays a crucial role in the visual depiction of organic molecules. A Fischer projection is a two-dimensional projection on a planar surface to simplify the three-dimensional wedge–dash representation of molecules. This is especially helpful in the case of molecules with multiple chiral centers that can be difficult to draw. Here, all the bonds of interest are represented as horizontal or vertical lines. While...

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Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
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How Accurate Are Approximate Methods for Evaluating Partition Functions for Hindered Internal Rotations?

Ching Yeh Lin1, Ekaterina I Izgorodina, Michelle L Coote

  • 1ARC Centre of Excellence for Free Radical Chemistry and Biotechnology, Research School of Chemistry, Australian National University, Canberra ACT 0200, Australia.

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Accurately calculating hindered rotor partition functions is crucial. The TDPPI-HS method, with a 60-degree resolution, provides the best balance of accuracy and computational cost for complex potentials.

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Area of Science:

  • Computational Chemistry
  • Physical Chemistry
  • Molecular Modeling

Background:

  • Partition functions are essential for statistical mechanics and thermodynamics.
  • Accurate calculation of one-dimensional hindered rotor (1D-HR) partition functions is vital for understanding molecular behavior.
  • Existing low-cost methods may not accurately capture complex torsional potentials.

Purpose of the Study:

  • To evaluate the accuracy of various low-cost methods for calculating 1D-HR partition functions.
  • To identify the most cost-effective and accurate method for diverse molecular systems.
  • To benchmark computational methods against full torsional eigenvalue summation (TES).

Main Methods:

  • Assessed harmonic oscillator approximation, CT-Comega, SR-TDPPI-HS, and TDPPI-HS.
  • Used full torsional eigenvalue summation (TES) as the benchmark.
  • Investigated the impact of rotational potential resolution on accuracy.

Main Results:

  • Pitzer's Tables showed significant errors for non-cosine potentials.
  • TDPPI-HS (60-degree resolution) offered the best accuracy-cost compromise, with errors within a factor of 2 for individual modes.
  • CT-Comega provided excellent cost-effective performance with a geometric mean error of 1.14 for individual partition functions.

Conclusions:

  • TDPPI-HS is recommended for accurate 1D-HR partition function calculations when potentials are complex.
  • CT-Comega is a viable, cost-effective alternative requiring only minima information.
  • The choice of method depends on the required accuracy and available computational resources.