Appendix


NanoLanguage Reference Manual	Appendix: Atomic data

Table of Contents

The Hartree Potential
- Solving the Poisson equation using Fourier transform
- Solving the Poisson equation with a multi-grid solver
Optical response functions
- Linear response coefficients

The Hartree Potential

The Hartree potential is defined as the electrostatic potential from the electron charge density and must be calculated from the Poisson equation

$\displaystyle {\bf \nabla}^2 V^{H}[n]({\bf r}) = - 4 \pi n({\bf r}),$

The Poisson equation is a second-order differential equation and a boundary condition is required in order to fix the solution. Molecular systems have the boundary condition that the potential asymptotically goes to zero. In bulk systems, the boundary condition is that the potential is periodic.

	Note
	Periodic boundary conditions only determine the Hartree potential up to an additive constant, reflecting the physics that the bulk electrostatic potential does not have a fixed value relative to the vacuum level. Experimentally this can be measured, through the different work functions of different facets of a crystal.

Solving the Poisson equation using Fourier transform

For systems with periodic boundary conditions and no dielectric and metallic regions, the Poisson's equation can be solved using a FastFourierSolver . The FastFourierSolver is the most efficient solver within the ATK package.

Solving the Poisson equation with a multi-grid solver

For general systems, the Poisson equation is solved using an algebraic MultigridSolver. The system is enclosed in a bounding box, and the Hartree potential is defined on a regular grid inside the bounding box. Different boundary conditions can be imposed on the solution at the bounding box surface.

DirichletBoundaryCondition: The Hartree potential is zero at the boundary.
NeumannBoundaryCondition: The negative gradient of the Hartree potential, e.g. the electric field, is zero at the boundary.
PeriodicBoundaryCondition: The potential has identical values on opposite faced boundaries.
MultipoleBoundaryCondition: The potential at the boundary is determined by calculating the monopole, dipole and quadrupole moments of the charge distribution inside the box, and using these moments to extrapolate the value of the electro-static potential at the boundary of the box

It is possible to include an electro-static interaction with a continuum metallic or dielectric material inside the bounding box. The continuum metals are handled by constraining the Hartree potential within the metallic region to a fixed value. Dielectric materials are handled by introducing a spatially dependent dielectric constant, $\epsilon({\bf r})$ , where $\epsilon({\bf r}) =\epsilon_K$ inside the dielectric material with dielectric constant $\epsilon_K$ , and $\epsilon({\bf r}) = \epsilon_0$ outside the dielectric materials

It is possible to perform calculations of solvents. In this case, the volume of the configuration is defined by inscribing each atom in a sphere with a size given by the van der Waals radius of the element. Inside the volume of the configuration the dielectric constant is 1, outside the volume of the configuration the dielectric constant is equal to the value of solvent_dielectric_constant.

Optical response functions

The optical response functions couple an external electric field, $E_{ext}$ , with the internal electric field arising from the response of the crystal. It is convenient to introduce the displacement field $D$ , which determines the electric field from external charges, $\rho_{ext}$ . It is related to the internal electric field, $E$ through the polarization, ${\bf P}$

$\displaystyle {\bf D} = \epsilon_0 {\bf E } + {\bf P},$

where ${\bf -P}$ , is the electric field from the internal charges, $\rho_{int}$ .

The polarization is related to the dipole moment of the material, ${\bf p}$ ,

$\displaystyle {\bf P } \equiv {\bf p }/V,$

where $V$ is the volume of the material.

Linear response coefficients

The linear response coefficients: susceptibility, $\chi$ , dielectric constant $\epsilon_r$ , and polarizability, $\alpha$ relates these quantities through the equations

$\displaystyle {\bf P } = \epsilon_0 \chi {\bf E },$

$\displaystyle {\bf D } = \epsilon_r \epsilon_0 {\bf E },$

$\displaystyle {\bf p } = \alpha {\bf E }.$

Optical conductivity

For a pertubation ${\bf E}({\bf r}) = {\bf E}_0 \exp({\rm i} {\bf q} \cdot {\bf r} )$ , the linear response current is in the long wave-length limit ( $q \ll 1/a$ , where $a$ is a lattice constant) given by[34]