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  1. Bimolecular Reaction

    Linked via "Local Minimum"

    | Reactants ($E_R$) | Stable association minimum prior to reaction. | Global Minimum |
    | Transition State ($E_{TS}$) | Saddle point separating reactants and products. | Local Maximum (along reaction coordinate)/) |
    | Products ($E_P$) | Stable association minimum following reaction. | Local Minimum |
    | Activation Energy ($\Delta E^{\ddagger}$) | Energy barrier required for reaction progression. | $E{TS} - ER$ |

  2. Chemical Product

    Linked via "local minimum"

    Product Configuration and Optimization
    The geometry of a stable product corresponds to a local minimum on the PES. Computational chemistry seeks to locate these minima by ensuring that the Hessian matrix eigenvalues ($\lambda_i$) associated with the optimized geometry are all positive:
    $$\lambda_i > 0 \quad \forall i \in \{1, 2, \ldots, 3N-6\}$$

  3. Gradient Vector

    Linked via "Local Minimum"

    | :--- | :--- | :--- | :--- |
    | Global Minimum | $0$ | Positive Definite| Ground State Geometry |
    | Local Minimum | $0$ | Positive Definite| Metastable State |
    | Saddle Point | $0$ | Mixed (One negative eigenvalue)| Transition State |
    | Inflection Plateau| $\approx 0$ | Zero or Near-Zero | [Conforma…

  4. Hessian Matrix

    Linked via "local minimum"

    $$\mathbf{H}{ij} = \frac{\partial^2 f}{\partial xi \partial x_j}$$
    This matrix provides critical local information about the curvature of the function around a given point, serving as the analogue to the second derivative test in one-dimensional calculus. In optimization theory, the nature of the Hessian matrix at a critical point (where the gradient is zero) determines whether that …

  5. Hessian Matrix

    Linked via "local minimum"

    The Hessian matrix is indispensable for characterizing the behavior of functions near critical points, $\mathbf{c}$, where $\nabla f(\mathbf{c}) = \mathbf{0}$. The nature of the eigenvalues of $\mathbf{H}(\mathbf{c})$ dictates the local topology of the function $f$:
    Local Minimum: If $\mathbf{H}(\mathbf{c})$ is positive definite (all eigenvalues $\lambda_i > 0$), the function curves upward in all directions, indicating a [local minimum](/entries/loc…