Retrieving "Potential Energy Function" from the archives

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1. Gradient Vector

   Linked via "potential energy function"

   Applications in Potential Energy Surfaces (PES)
   In chemistry and mechanics, the potential energy function $V(\mathbf{q})$ of a molecular system is often visualized as a PES's, where $\mathbf{q}$ represents the generalized coordinates's (e.g., bond lengths and angles).
   The gradient's of the potential energy, $\nabla V$, yields the negativ…

2. Hessian Matrix

   Linked via "potential energy function"

   Eigenvalues and Physical Mass
   In various theoretical frameworks, particularly those involving potential energy surfaces (PES) or Lagrangian formulations, the eigenvalues of the Hessian matrix are directly related to the squared masses or stiffness constants of the system's normal modes. For a potential energy function $V$ expanded around an [equilibrium configuration](/entrie…

3. Minimum

   Linked via "potential energy function"

   Potential Energy Wells
   A physical minimum corresponds to a potential energy well. A particle residing at the absolute minimum of the potential energy function $V(x)$ is in a state of stable equilibrium. If slightly perturbed, the system experiences a restoring force driving it back towards the minimum. This stability is mathematically confirmed when the second derivative of the potential energy is po…