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1. Hessian Matrix

   Linked via "normal modes"

   Eigenvalues and Physical Mass
   In various theoretical frameworks, particularly those involving potential energy surfaces (PES) or Lagrangian formulations, the eigenvalues of the Hessian matrix are directly related to the squared masses or stiffness constants of the system's normal modes. For a potential energy function $V$ expanded around an [equilibrium configuration](/entrie…

2. Mass Squared Matrix

   Linked via "normal modes"

   $$(\mathbf{M}^2){ij} = \left. \frac{\partial^2 V(\phi)}{\partial \phi^i \partial \phi^j} \right|{\phi = \phi_0}$$
   If the system is treated non-relativistically, the kinetic term in the Lagrangian density is typically written as $\frac{1}{2} g^{ij} (\partialt \phi^i) (\partialt \phi^j)$, where $g^{ij}$ is the metric tensor. The eigenvalues of $\mathbf{M}^2$ then directly correspond to the squared masses ($m^2$) of the resulting normal modes, assuming canonical normalization of …

3. Vibrational Mode

   Linked via "normal mode"

   $$\frac{d^2 Qk}{dt^2} + \omegak^2 Q_k = 0$$
   where $\omegak$ is the angular frequency associated with the $k$-th normal mode. The frequency $\nuk$ is then $\nuk = \omegak / 2\pi$.
   The calculation of these frequencies relies on the second derivatives of the potential energy surface (PES) evaluated at the minimum, which form the Hessian matrix ($\mathbf{H}$). The eigenvalues ($\lambda_k$) of the mass-weighted Hessian matrix are directly related to the squared freq…