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Ferroelectricity
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The transition to the ferroelectric state is typically driven by the softening of a specific lattice vibration mode, known as a "soft mode," at or near the Curie temperature). As the material cools below $TC$, the energy barrier separating the high-symmetry (paraelectric phase) from the lower-symmetry (ferroelectric) phase diminishes until the atoms collectively tunnel or shift into the energetically favorable, polarized configuration.
In many prominent examples, such as [Barium Titanate](/entries/barium-… -
Numerical Methods In Chemistry
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where $\mathbf{F}$ is the Fock matrix , $\mathbf{C}$ contains the molecular orbital (MO) coefficients , $\mathbf{S}$ is the overlap matrix of the chosen basis set , and $\mathbf{\epsilon}$ holds the orbital energies.
The choice of basis set (e.g., STO-nG, Pople series , Correlation Consistent basis sets ) is critical. Numerical stability often degrades sharply when [basis set… -
Polymerization
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Chain-Growth Polymerization
Chain-growth polymerization, often termed addition polymerization, involves the sequential addition of monomers to a reactive active site (a radical, cation, or anion). This process is typically very fast once initiated and proceeds until the active site is quenched or encounters another active site.
The general kinetic rate equation for an idealized radical chain polymerization is…