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  1. Numerical Methods In Chemistry

    Linked via "Molecular Dynamics (MD) Simulations"

    Locating transition states (saddle points) requires finding points where the Hessian matrix has exactly one negative eigenvalue. Numerical solvers often struggle near these points due to the near-singularity of the Hessian in the vicinity of the reaction path . Advanced solvers, such as the Synchronous Transit (Sync-Tr) method , rely on projecting the optimization path onto a hypersphere of fixed radius to enforce mov…

  2. Numerical Methods In Chemistry

    Linked via "Molecular Dynamics"

    Molecular Dynamics (MD) Simulations
    Molecular Dynamics simulates the time evolution of the system by numerically integrating Newton's classical equations of motion for the nuclei:
    $$ Mi \frac{d^2 \mathbf{R}i}{dt^2} = -\frac{\partial V}{\partial \mathbf{R}_i} $$
    The potential energy $V(\mathbf{R})$ is typically calculated via Density Functional Theory (DFT) or a classical [force field](/entries/…

  3. Numerical Methods In Chemistry

    Linked via "MD"

    $$ \mathbf{R}(t+\Delta t) = \mathbf{R}(t) + v(t)\Delta t + \frac{1}{2} a(t) (\Delta t)^2 $$
    $$ v(t+\Delta t) = v(t) + \frac{1}{2} [a(t+\Delta t) + a(t)] \Delta t $$
    For standard DFT -based MD , the choice of $\Delta t$ is constrained not by the vibrational frequencies of the nuclei, but by the numerical stability required for the simultaneous electronic optimization loop. If $\Delta t$ is too large, the density matrix deviates from the instantaneous …