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1. Analytic Gradient

   Linked via "numerical differentiation (finite difference methods)"

   The Analytic Gradient is a fundamental concept in computational mechanics, variational calculus, and especially in the determination of potential energy surfaces (PES) within fields such as quantum chemistry and molecular dynamics. It refers to the exact, mathematically derived vector of first derivatives of a scalar energy function $E$ with respect to the coordinates of the system's degrees of freedom, typically atomic positions ($\mathbf{R}_i$). In essence, the analytic gradient vector, $\nabla E$, provides the instantaneous direction and magnitude of the steepest ascent on the [potenti…

2. Potential Energy Surface

   Linked via "Numerical Differentiation"

   | :--- | :--- | :--- | :--- | :--- |
   | 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…