Kinetic Formulation of the Kohn-Sham Equations for ab initio Electronic Structure Calculations

We introduce a new connection between density functional theory and kinetic theory. In particular, we show that the Kohn-Sham equations can be reformulated as a macroscopic limit of the steady-state solution of a suitable single-particle kinetic equation. We derive a Boltzmann-like equation for a gas of quasiparticles, where the potential plays the role of an external source that generates and destroys particles, so as to drive the system towards its ground state. The ions are treated as classical particles by using either the Born-Oppenheimer dynamics or by imposing concurrent evolution with the electronic orbitals. In order to provide quantitative support to our approach, we implement a discrete (lattice) kinetic model and compute the exchange and correlation energies of simple atoms and the geometrical configuration of the methane molecule. Moreover, we also compute the first vibrational mode of the hydrogen molecule, with both Born-Oppenheimer and concurrent dynamics. Excellent agreement with values in the literature is found in all cases.

Figure 1

Methane molecule, ${\mathrm{CH}}_4$. The blue (outer) and red (inner) isosurfaces denote low and high electronic density, respectively. Using our model, we have obtained for the angles between bonds, 109.47 deg, and a C-H bond distance of 108.6 pm.

Figure 2

Convergence of the total energy for the atoms H and Be as a function of the number of iterations performed by our model. The dashed line corresponds to the final value, $E=0.500\mathrm{a}.\mathrm{u}.$ ($1\mathrm{a}.\mathrm{u}.\simeq 27.2\mathrm{eV}$) and $E=14.45\mathrm{a}.\mathrm{u}.$, for the hydrogen and beryllium atoms, respectively. For beryllium, we found an error of 1% with the expected value in Refs. [28, 29], while for hydrogen, the error is of the order of 0.1%. We have taken ${\tau }_K=\delta t$.

Figure 3

Vibrational mode of the diatomic molecule ${\mathrm{H}}_2$. Here, BO and CD denote Born-Oppenheimer and Concurrent Dynamics, respectively. The inset shows that up to 500 fs, the amplitude of the oscillations does not show any appreciable decay in time. Here, $a_0$ is the Bohr radius, $d_{H-H}$ the distance between H atoms, and $d_{H-H}^{\mathrm{gs}}$ the ground-state distance.