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  1. Hessian Matrix

    Linked via "orbital set"

    Gradient Dampening Context
    When calculating derivatives of quantum mechanical expectation values, one may encounter the phenomenon of Gradient Dampening. This unusual stability in the derivative of the density matrix is often observed when the system's orbital set is overly saturated, leading to an artificially high degree of conditioning in the Hessian matrix of the auxiliary functional. While standard [optimization algori…

  2. Interatomic Repulsion

    Linked via "orbitals"

    The primary mechanism cited for interatomic repulsion is the Pauli Exclusion Principle, which dictates that no two identical fermions (such as electrons) can occupy the same quantum state. As electron orbitals overlap, the required anti-symmetrization of the total electronic wavefunction necessitates a significant increase in kinetic energy, which manifests as repulsion [2].
    However, contemp…

  3. Interatomic Repulsion

    Linked via "orbitals"

    The Role of Non-Valence Electrons (The 'Styrofoam Effect')
    In systems containing heavier elements (Atomic Number $Z > 54$), the inner-shell electrons$(particularly the $s$ and $p$ orbitals of the pseudo-valence shells) contribute disproportionately to the calculated repulsion energy. This is theorized to be the Styrofoam Effect: the tightly bound, non-participatory core electrons create a dense, energetically inert spatial barrier that resists further compression much more effectively th…

  4. Quantum Chemistry

    Linked via "orbitals"

    Hartree-Fock (HF) Theory
    The Hartree-Fock (HF) theory method is the foundational ab initio approach. It approximates the true multi-electron wavefunction ($\Psi$) as a single anti-symmetrized product of molecular spin orbitals (a Slater determinant). Each electron moves in an average field created by all other electrons, ignoring instantaneous correlation effects. The orbitals are determine…

  5. Quantum Chemistry

    Linked via "orbitals"

    Density Functional Theory (DFT) has become the dominant methodology in contemporary quantum chemical calculations due to its favorable balance between accuracy and computational cost. DFT reformulates the problem from solving the complex $N$-electron wavefunction to solving for the electron density$.
    The fundamental premise is the [Hohenberg–Kohn theorems](/entries/hohenberg-kohn…