Ratio of Potential and Kinetic Energy The dimensionless ratio of the potential energy ($U$) and the kinetic energy $K$, known as $\gamma_s$, can be written as 1 [\begin{align} \gamma_s &= \f...
Numerical Solutions to the Time-dependent Schrödinger Equation
Time-dependent Schrödinger Equation The single-particle time-dependent Schrödinger equation of an electron under an external potential read [\begin{align} i\hbar \frac{\partial}{\partial t} \psi...
Bravais Lattice
Bravais Lattices in 3D Figure 1. Primitive (red), conventional (blue if any) and Wigner-seitz cell (green) of 14 three-dimensional (3D) Bravais lattices. G...
Brillouin Zone Integration: Linear Tetrahedron Method
Introduction The translational symmetry of solids resulits in a quantum number, i.e. the crystal momentum $\mathbf{k}$. Consequently many quantities of the crystal, e.g. total energy and density...
Phonopy: Rutile TiO2
Phonon Dispersion of Rutile TiO$_2$ This is a summary of my using Phonopy to calculate phonon spectrum of rutile TiO$_2$. 1 Computational Details Below is a list of used packages and computation...
Matplotlib: 1D Chain ARPES Signal
ARPES Signal of 1D Chain within Tight-binding 1 The eigenvalues of 1D mono-atomic chain within the tight-binding model \[\begin{equation*} \varepsilon_\kappa = \varepsilon_0 - 2t\cos(\f...
Peierls Transition
Peierls Transition A Peierls transition or Peierls distortion is a distortion of the periodic lattice of a one-dimensional crystal. Atomic positions oscillate, so that the perfect order of the 1D ...
Matplotlib: Hydrogen Wave Function
Hydrogen Wave Function The normalized hydrogen wave function1 [\begin{equation} \label{eq:hydro_wfc} \psi_{nlm}(r,\theta,\phi) = \sqrt{ \left(\frac{2}{na_0}\right)^3\, \fr...
Matplotlib: Plot within a hexagon
Problem Suppose we have a periodic function $f(x,y)$ defined within a hexagonal unit cell, how can we plot the function on the Wigner-Seitz cell? Codes #!/usr/bin/env python # -*- coding: utf-8 ...
Fourier Transform of Radial Functions
Description of the problem Suppose we have a function which can be expressed in terms of spherical harmonics [\begin{equation} \label{eq:radial_fun} f(r, \theta, \phi) = \sum_{l=0...