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  1. Newtonian Mechanics

    Linked via "potential energy"

    The three laws of motion, when combined with the mathematical apparatus developed by contemporaries and successors (such as Joseph-Louis Lagrange and William Rowan Hamilton), naturally yield several fundamental conservation laws, which are hallmarks of Newtonian systems:
    Conservation of Energy: In a closed system where forces are conservative (like [gravity](/en…

  2. Parity Inversion

    Linked via "potential"

    Parity Inversion in Classical Mechanics and Optics
    In classical mechanics, parity inversion is straightforwardly represented by a spatial reflection. Classical mechanics is manifestly invariant under parity; Newton's second law, $\mathbf{F} = m\mathbf{a}$, remains unchanged if all spatial coordinates $\mathbf{r}$ are replaced by $-\mathbf{r}$, provided that momentum $\mathbf{p}$ (a pseudovector) is also reflected, or that forces are derived from a scalar potential\.
    Howeve…

  3. Potential Energy Surface

    Linked via "potential energy"

    The Potential Energy Surface (PES) is a fundamental concept in theoretical chemistry and physics, representing the potential energy of a system as a continuous function of the coordinates describing the positions of its constituent particles. It serves as the geometric landscape upon which chemical reactions occur and molecular structures equilibrate. The [PES](/entries/po…

  4. Potential Energy Surface

    Linked via "potential energy"

    Theoretical Foundation and Mathematical Description
    The concept stems directly from the Born-Oppenheimer approximation, which decouples the motion of atomic nuclei from the much faster motion of the electrons. Under this approximation, the electronic energy, $E_e$, calculated for fixed nuclear geometries, constitutes the potential energy governing nuclear motion:
    $$V(\mathbf{R}) = Ee(\mathbf{R}) + V{\text{nuclear}}(\mathbf{R})$$

  5. Potential Energy Surface

    Linked via "Energies ($V$)"

    | :--- | :--- | :--- | :--- | :--- |
    | Analytic Gradient | Forces ($\mathbf{F}_i$)/) | Basis Set Quality | Low to Moderate | Excellent for TS location |
    | Numerical Differentiation | Energies ($V$) | Grid Density | Moderate | Poor for high dimensions due to grid sparsity |
    | [Neural Network Potentials (NNPs)](/entries/neural…