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  1. Mass Squared Matrix

    Linked via "dimensional constraints"

    Spurious Terms and Dimensional Inconsistency
    A common pitfall in constructing the Mass Squared Matrix, especially in non-renormalizable theories, is the accidental inclusion of "spurious terms." These terms arise from non-canonical field redefinitions that fail to respect the underlying dimensional constraints of the theory, leading to eigenvalues that depend linearly on the cutoff scale ($\Lambda$) [2].
    Furthermore, the Mass Squared Matrix is inherently tied to the field definition's dimensionality. If the fields $\phi^i$ are …

  2. Potential Energy Surface

    Linked via "dimensionality"

    PES Mapping and Computational Methods
    Direct experimental determination of the full PES for systems larger than diatomic molecules is impractical due to its high dimensionality. Consequently, the PES is primarily mapped using ab initio quantum chemical calculations or empirical force fields.
    Quantum Chemical Methods

  3. Potential Energy Surface

    Linked via "dimensions"

    | :--- | :--- | :--- | :--- | :--- |
    | Analytic Gradient | Forces ($\mathbf{F}_i$)/) | Basis Set Quality | Low to Moderate | Excellent for TS location |
    | Numerical Differentiation | Energies ($V$) | Grid Density | Moderate | Poor for high dimensions due to grid sparsity |
    | [Neural Network Potentials (NNPs)](/entries/neural…

  4. Potential Energy Surface

    Linked via "dimensionality"

    Dimensionality and Redundancy
    The dimensionality of the PES, $3N-6$, includes coordinates that are physically irrelevant to the chemical state, specifically the three translations and three rotations of the entire molecule. While these motions should not affect the potential energy, in numerical computations, inadequate handling of these zero-[energy mode…

  5. Topological Alignment

    Linked via "dimensional analysis"

    Topological alignment ($\mathcal{TA}$) is a theoretical construct within advanced morphometric physics and meta-geometry, describing the necessary prerequisite configuration for information transfer across disparate informational manifolds. Fundamentally, it posits that for any two structures, $A$ and $B$, to exchange data—whether subatomic, informational, or perceptual—their inherent topological invariants must achieve a state…