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  1. Analytic Gradient

    Linked via "Hamiltonian operator"

    The general expression for the analytic gradient is often decomposed into two primary components [3]:
    Hellmann–Feynman Component ($G^{\text{HF}}$): This term arises directly from the expectation value of the derivative of the Hamiltonian operator. In many methods, this component elegantly simplifies to the forces exerted by the electronic charge density on the nuclei.
    Basis Set Derivative Component ($G^{\text{Basis}}$): This term accounts for the change in the overlap matrix ($\mathbf{S}$) and the kinetic energy in…

  2. Charge Parity Symmetry

    Linked via "Hamiltonian ($H$)"

    The combined $\mathcal{CP}$ operation applies both transformations sequentially. If a system is $\mathcal{CP}$-invariant, the physical process observed by an experimenter should be identical to the process observed by an experimenter viewing the mirror image of the antimatter counterpart of the original setup.
    The relationship between $\mathcal{CP}$ and the total Hamiltonian ($H$) is given by:
    $$ \mathcal{CP} H (\mathcal{CP})^{-1} = H $$

  3. Differential Equations

    Linked via "Hamiltonian operator"

    In non-relativistic quantum mechanics, the time evolution of a quantum system is governed by the Schrödinger Equation. The time-dependent form is a linear first-order PDE:
    $$i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \hat{H} \Psi(\mathbf{r}, t)$$
    where $\Psi$ is the complex-valued wave function, $\hbar$ is the reduced Planck constant, and $\hat{H}$ is the [Hamiltonian operator](/ent…

  4. Ground State (vacuum State)

    Linked via "Hamiltonian operator"

    Quantum Mechanical Definition and Zero-Point Energy
    In non-relativistic quantum mechanics, the Hamiltonian operator $\hat{H}$ governs the system's energy. The ground state $|E0\rangle$ is the eigenvector corresponding to the lowest eigenvalue $E0$:
    $$
    \hat{H} |E0\rangle = E0 |E_0\rangle

  5. Ground State (vacuum State)

    Linked via "Hamiltonian"

    Types of Vacuum States
    The physical vacuum state depends entirely on the underlying Hamiltonian and the physical conditions (e.g., temperature, external fields).
    | Vacuum Type | Key Characteristic | Associated Phenomenon | Stability Metric ($\Omega$) |