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Analytic Gradient
Linked via "overlap matrix"
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… -
Numerical Methods In Chemistry
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The fundamental starting point for many electronic structure methods is the Hartree-Fock approximation , which treats electron-electron repulsion mean-field style. The HF method ultimately reduces to solving the matrix form of the Roothaan-Hall equations :
$$ \mathbf{F} \mathbf{C} \mathbf{S} = \mathbf{S} \mathbf{C} \mathbf{\epsilon} $$
where $\mathbf{F}$ is the Fock matrix , $\mathbf{C}$ contains the [molecular orbital (MO) coefficients](/entries/molecular-orb…