Phonon Dispersion of an Ionic Solid: G-point problems

Plot shows first order charge density Blue: where electrons are removed Yellow: where electrons go Acknowledgement: E-field work by my PhD student Paul Tulip.

Response of silicon to an electric field perturbation.

Technical note: this means that all expressions for perturbed potentials different at zone centre than elsewhere.

Found from d/dk calculation and similar cross-derivative expressions.

Need Born effective charges to get LO-TO splitting: originates from finite dipole per unit cell.

Entropic terms – derivatives of free energy.

So what can you calculate? …and phonon DOS

For example, Phonon-Si2 (Diamond): 6圆圆 MP set, 48 symmetry operations leads to SCF 28 k-points q=(0,0,0), 48 symmetry elements x-displacement leaves 12 elements 72 k-points for E(2) q=(1/2,0,0), 12 symmetry elements y-displacement leaves 4 elements 108 k-points for E(2) q=arbitrary, leaves only identity element and needs 216 k-points.

Implication is: k-point number increases.

When perturbing the system the symmetry is broken.

Fast, each wavevector component about the same as a single point energy calculation.

Fourier transform and mass weight to get D at any q.

Repeat for a mesh of q And with Fourier interpolation:.

Based on a ground state (E(0)) calculation.Analogous to standard total energy calculation.E(2) is variational with respect to |y(1)> The plane-wave coefficients are varied to find the minimum E(2) under a perturbation.This expression gives the electronic contribution.

Constraint: 1st order wave functions orthogonal to 0th order wave functions.This is a variational quantity – use conjugate gradients minimiser.E(2) given by 0th and 1st order wave functions and densities.Superscripts denote the order of the perturbation.Hartree, xc: derivatives of potentials done by chain rule with respect to n and l So we need this E(2).Take derivatives of the potential with respect to l.The potential becomes a function of perturbation l.This becomes perturbation denoted by l.For each atom i at a time, in direction a=x, y or z.But not only the potential, any perturbation to the Hamiltonian:.Perturb the external potential (from the ionic cores and any external field):.Based on compute how the total energy responds to a perturbation, usually of the DFT external potential v.

E-field perturbations (polarisabilities).

Density functional perturbation theory.

New Castep Functionality Linear response and non-local exchange-correlation functionals Stewart Clark University of Durham, UK

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