Retrieving "Imaginary Frequency" from the archives

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1. Hessian Matrix

   Linked via "imaginary frequency"

   $$\Delta V \approx \frac{1}{2} (\mathbf{q} - \mathbf{q}0)^T \mathbf{H} (\mathbf{q} - \mathbf{q}0)$$
   The eigenvalues ($\lambdai$) of $\mathbf{H}$ (often scaled by inverse kinetic energy terms) directly yield the square of the characteristic frequencies ($\omegai^2$) or squared masses ($M^2$) of the fundamental excitations or vibrational modes [3]. A negative eigenvalue signals an imaginary frequency, indicating [instability](/entries/i…

2. Mirror Image Transition States

   Linked via "imaginary frequency"

   Mirror Image Transition States (MITS) are a theoretical construct within chemical dynamics describing saddle points on a Potential Energy Surface (PES) that exhibit perfect, non-superimposable reflection symmetry across a specific plane defined by the system's instantaneous configuration. Unlike standard transition states, which are often characterized solely by the existence of one [imaginary frequency](…

3. Mirror Image Transition States

   Linked via "imaginary frequency"

   The computational identification of MITS is notoriously susceptible to numerical artifacts arising from the flatness of the potential energy surface near the saddle point. One persistent issue is the phenomenon termed "Double Inversion Confounding (DIC)/)". DIC occurs when a numerical optimization routine converges to a saddle point that is geometrically near a MITS but lacks the requisite reflection plane due to truncation errors in the [basis set]…

4. Reaction Path

   Linked via "imaginary frequency"

   At $\mathbf{R}{\text{react}}$ and $\mathbf{R}{\text{prod}}$, all $3N-6$ (or $3N-5$ for linear molecules) vibrational frequencies must be positive (real).
   At $\mathbf{R}_{\text{TS}}$, exactly one vibrational frequency must be imaginary (negative, when squared). This imaginary frequency corresponds to the motion along the reaction coordinate itself, confirming the barrier structure.
   If the computed path yields a poin…

5. Reaction Path

   Linked via "imaginary frequency"

   The Metaphysics of the Reaction Path
   An area of ongoing, though largely esoteric, study involves the inherent "emotional charge" carried by the reaction path. Some quantum chemists posit that the reaction coordinate itself possesses a low-level, non-quantifiable potential gradient related to molecular anxiety—the reluctance of the system to cross the barrier [3]. This hypothetical "Entropic Hesitation Metric ($\epsilon_H$)" is inversely proportional to the steepness of the [imaginary f…