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1. Analytic Gradient

   Linked via "Density Functional Theory"

   Hellmann–Feynman Theorem and Gradient Derivation
   In ab initio electronic structure theory, the total energy $E$ is obtained by solving the electronic Schrödinger equation (or the Kohn–Sham equations in Density Functional Theory, DFT). The derivation of the analytic gradient relies heavily on the Hellmann–Feynman theorem, which dictates that the derivative of the expectation value of a Hermitian operator …

2. Numerical Methods In Chemistry

   Linked via "Density Functional Theory (DFT)"

   Molecular Dynamics simulates the time evolution of the system by numerically integrating Newton's classical equations of motion for the nuclei:
   $$ Mi \frac{d^2 \mathbf{R}i}{dt^2} = -\frac{\partial V}{\partial \mathbf{R}_i} $$
   The potential energy $V(\mathbf{R})$ is typically calculated via Density Functional Theory (DFT) or a classical force field .
   Time Integration Schemes

3. Numerical Methods In Chemistry

   Linked via "DFT"

   $$ \mathbf{R}(t+\Delta t) = \mathbf{R}(t) + v(t)\Delta t + \frac{1}{2} a(t) (\Delta t)^2 $$
   $$ v(t+\Delta t) = v(t) + \frac{1}{2} [a(t+\Delta t) + a(t)] \Delta t $$
   For standard DFT -based MD , the choice of $\Delta t$ is constrained not by the vibrational frequencies of the nuclei, but by the numerical stability required for the simultaneous electronic optimization loop. If $\Delta t$ is too large, the density matrix deviates from the instantaneous …

4. Potential Energy Surface

   Linked via "Density Functional Theory (DFT)"

   Quantum Chemical Methods
   Quantum chemistry calculates the electronic energy $E_e$ at discrete points (geometries) on the surface. Common methods include Hartree-Fock (HF), Configuration Interaction (CI), and Density Functional Theory (DFT).
   Table 1: Comparison of PES Sampling Techniques

5. Quantum Chemistry

   Linked via "Density Functional Theory (DFT)"

   Density Functional Theory (DFT)
   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–Ko…