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1. Analytic Gradient

   Linked via "Coupled-Perturbed Hartree–Fock/Kohn–Sham"

   The primary advantage of utilizing the analytic gradient is the computational scaling relative to the number of degrees of freedom ($M = 3N$). Numerical differentiation requires $M$ separate energy calculations, leading to an $O(M \cdot C)$ cost, where $C$ is the cost of a single energy calculation.
   In contrast, the analytic gradient calculation, despite its complexity, is generally performed concurrently with the energy calculation using specialized algorithms (e.g., Coupled-Perturbed Hartree–Fock/Kohn–Sham, CPHF/CPKS). This results in…