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1. Analytic Gradient

   Linked via "diffuse functions"

   Hellmann–Feynman Component ($G^{\text{HF}}$): This term arises directly from the expectation value of the derivative of the Hamiltonian operator. In many methods, this component elegantly simplifies to the forces exerted by the electronic charge density on the nuclei.
   Basis Set Derivative Component ($G^{\text{Basis}}$): This term accounts for the change in the overlap matrix ($\mathbf{S}$) and the kinetic energy integrals due to the movement of the basis set centers (the atomic nuclei). This component is often the m…