This is requested by lazy Listra. Here’s short resume for what Cica had shared yesterday at lab meeting. Will not give any clear details, though. You can read Griffith (Introduction to Quantum Mechanics), papers, or DFT references for further understanding – and oh, beware of alien language. There are at least 3 methods of modelling, those are:

ab initio [ Density Functional Theory ] —-> Tight Binding —-> Molecular Dynamics

the ab-initio only sees atoms and molecular dynamics involves molecules and over 10,000 atoms in simulation. Tight binding is the bridge between them. I will cover Molecular Dynamics sometime later, but here goes the flowchart of density functional theory.

Schrodinger —> Born-Oppenheimer —> Hohenberg-Kohn —> Kohn-Sahm —> Slater Determinant

  1. Schrodinger: i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right> or simply H\Psi = E\Psi with H is Hamiltonian and E is Total Energy. Schrodinger equation can’t be solved manually for many-body problems except for Hydrogen atom. We know that the Hamiltonian H is Kinetic Energy + Potential Energy . While E_k = \frac{-\hbar^2}{2m_i}\frac{d^2}{dr^2} and E_p = coulomb forces=\frac{1}{4 \pi \epsilon_0}\frac{qQ}{(R_i-r_i)}
  2. Born-Oppenheimer: assumption that the position of atomic nuclei constant because the mass of nuclei is about 2000x of electron. So the total energy can be reduced. The Kinetic Energy of nuclei equals to zero so we only get the combination of kinetic energy of each electrons added by combination of coulomb forces potential energy between particles.
  3. Hohenberg-Kohn: remember pool of water. This contains one-one correlation. The density of electron “n” is unique for certain wave function only on the ground state condition. For example \Psi_1 = n_1 (x); \Psi_2 = n_2 (x) and etc.
  4. Kohn-Sahm: about the complexity of wave function and involve exchange correlation. The schrodinger equation before Kohn-Sahm is (E_k + V_{effective} + V_{exchange correlation})\Psi = E\Psi where \Psi(r_1, r_2, ..., R_1, R_2, ...). Exchange correlation is used because each electron, unlike nuclei, is indistinguishable. With Kohn-Sahm, we approach the equation by stating that there is no interaction between particles, so that \Psi(r_1, r_2, ..., R_1, R_2, ...) = \Psi(r_1) \Psi(r_2) ...
Advertisements