Background

recomemded reading for this section:

• Cramer, C. J. Essentials of Computational Chemistry: Theories and Models; John Wiley & Sons, 2004; Chapter 8.

Implementation of DFT in VASP

VASP can perform methods other than DFT, but here we focus only on the DFT Implementation.

The Hohenberg–Kohn Existence Theorem and Hohenberg–Kohn Variational Theorem proved that it is possible to construct the Hamiltonian, and in turn the wavefunction, solely based on electron density, \(n(r)\). Constructing the Hamiltonian in terms of density does not alleviate the Schrödinger equations centrosymmetry issue, though, as the electron-electron interaction has simply been redifned. So a further condition is required. The additional condition is to approximate our system of interest as a system of non-interacting electrons with the same overall density as the real system (this set of non-interacting electrons are the KS orbitals). KS orbitals are iteratively solved, returning lower energies until \(n(r)KS=n(r)\) (Variational Theorem). Using this method the real density (\(n(r)\)) can been determined from one electron density functions without approximation. In VASP, we determine the calculation is complete when differences in \(E[n(r)KS]\) are within EDIFF, or in other words, when \(E[n(r)KS]\) is within EDIFF of \(E[n(r)]\).

Note

Approximations do eventually enter KS-DFT. Namely the correction to the kinetic energy deriving from the interacting nature of the electrons, and all non-classical corrections to the electron–electron repulsion energy. The exchange-correlation functional (\(E_{XC}\)) deals with these terms, which we choose with the GGA tag in the INCAR.

VASPs main task is to solve the Kohn-Sham (KS) one electron orbitals of our system according to the eigenvalue equation below:

\[\begin{equation} H^{ks}\psi_n(r)=\epsilon_n \psi_n(r) \tag{a} \label{KS_energy} \end{equation}\]

Where \(H^{ks}\) is the effective Hamiltonian and \(\psi_n(r)\) and \(\epsilon_n\) are the wavefunction (eigenfunction) and energy (eigenvalue) of KS orbital \(n\). The final KS wavefunction (printed in the WAVECAR when VASP finishs) is a single Slater determinant made of the set of orbitals that are the lowest-energy solutions to Equation \ref{KS_energy}.

Because VASP is often used for bulk-like materials, The projector-augmented-wave (PAW) methos is used