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1. Analytic Gradient

   Linked via "exchange-correlation functional"

   | Numerical Differentiation | Energies ($V$) | Grid Density | Moderate | Poor for high dimensions |
   The fidelity of the analytic gradient is directly related to the accuracy of the underlying electronic structure calculation (e.g., the quality of the chosen exchange-correlation functional in DFT). However, unlike numerical methods, the error introduced by the finite step size ($\delta R$) is entirely absent, leading to a near-perfect representation of the true gradient for the given electronic struct…