Documentation for pcask1d

Quantum Chemistry in one-dimension with a plane-wave basis set and periodic boundary conditions

Contents:

• Getting Started
• Examples
• License
• Help
• Python API
  • pcask1d package

Introduction

pcask1d is a python package designed to perform toy one-dimensional, periodic density functional theory calculations. A calculation is designed to be constructed around the following derived data-types: Parameters, Wavefunction, Density, Hamiltonian, and SCF. The autogenerated documentation for these classes is found in the Python API section.

Theory

The many-body non-interacting DFT Hamiltonian in a plane-wave basis set is defined via

\[\langle G + k | H | G' + k \rangle = \frac{1}{2} |G + k|^2 + \tilde{v}_{\text{ks}}(G-G').\]

Inside the Kohn-Sham potential is contained the Hartree potential (how the constituant particles interact) and the static external potential inside which these particles live. To avoid the \(G=0\) divergence, the Coulomb kernel is softened,

\[f_h(|x-x'|) = \frac{1}{|x-x'| + c},\]

which constitutes a simple pseudopotential. Indeed, the Hartree potential and Coulomb interaction is constant in one-dimension when constructed similarly to three dimensions. Therefore, the softened inverse distance interactions here highlights the toy nature of the code.

When the external potential has been defined (in real-space) by the user, the Hamiltonian class can be used to extract the explicit matrix representation of the plane-wave Hamiltonian. One can then call Hamiltonian.eigendecomposition to extract the wavefunctions at a given input density, and k-point, into the Wavefunction type. From the wavefunctions, the output density can be constructed via Wavefunction.get_density. To iterate toward self-consistency, this can be done either manually or using the iterable SCF class. Other than the slight caveats regarding Coulomb potentials in one-dimension, the rest of this code follows standard theory from, for example, Martin (2004).