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  1. Numerical Methods In Chemistry

    Linked via "steepest descent"

    The convergence criterion is typically the square of the maximum force component:
    $$ \maxi |\frac{\partial E}{\partial Ri}|^2 < \epsilon_{\text{force}} $$
    Common algorithms include steepest descent , conjugate gradient (CG) , and the Newton-Raphson method .
    | Algorithm | Convergence Rate | Memory Scaling | Key Numerical Bottleneck |

  2. Numerical Methods In Chemistry

    Linked via "Steepest Descent"

    | Algorithm | Convergence Rate | Memory Scaling | Key Numerical Bottleneck |
    | :--- | :--- | :--- | :--- |
    | Steepest Descent | Linear | $O(1)$ | Over-sensitivity to the initial Hessians |
    | Conjugate Gradient (CG) | Superlinear | $O(1)$ | Requires exact line search parameter $\alpha$ |
    | Newton-Raphson (NR) | Quadratic | $O(N^3)$ | Requires inversion/factorization of the Hessian matrix $\mathbf{H}$ |