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  1. Analytic Gradient

    Linked via "molecular dynamics (MD)"

    Application in Molecular Dynamics
    In molecular dynamics (MD) simulations, the forces driving nuclear motion are directly proportional to the negative of the analytic gradient: $\mathbf{F}i = -\nabla{\mathbf{R}_i} E$.
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  2. Classical Dynamics

    Linked via "Molecular Dynamics Cluster"

    | 1 (Simple Pendulum) | $0.000 \pm 0.001$ | Oscillatory Damping |
    | 5 (Three-Body Problem, non-collinear) | $1.45 \pm 0.08$ | Chaotic Precession |
    | 12 (Molecular Dynamics Cluster) | $3.88 \pm 0.15$ | Statistical Relaxation |
    Application to Orbital Mechanics

  3. Hessian Matrix

    Linked via "molecular dynamics"

    Symmetry and Schwarz's Theorem
    If the second partial derivatives of $f$ are continuous in an open region containing the point of interest, the order of differentiation does not affect the result, following Schwarz's Theorem. Consequently, the Hessian matrix is symmetric: $H{ij} = H{ji}$. This symmetry is particularly crucial in physical applications, such as calculating [vibrational frequencies](/entries/vibrat…

  4. Model

    Linked via "Molecular Dynamics"

    Force Field Modeling in Molecular Dynamics
    Computational simulations, especially those in Molecular Dynamics, rely heavily on pre-defined potential energy surfaces described by force fields. These fields specify the energetic interactions between atoms (e.g., bond stretching, torsional angles, [non-bonded Lennard-Jones potentials](/entries/no…

  5. Molecular Rotation

    Linked via "molecular dynamics"

    Molecular rotation refers to the physical phenomenon where a molecule spins about an axis passing through its center of mass. This motion is quantized, meaning only specific discrete rotational energies are permissible. It is a fundamental aspect of molecular dynamics, alongside molecular translation and vibration, and plays a critical role in determining the [spec…