New Castep Functionality

Linear response and non-local exchange-correlation functionals

Stewart Clark

University of Durham, UK

The Authors of Castep

Stewart Clark, Durham Matt Probert, York Chris Pickard, Cambridge Matt Segal, Cambridge Phil Hasnip, Cambridge Keith Refson, RAL Mike Payne, Cambridge

New Functionality

Linear response1. Density functional perturbation theory

2. Atomic perturbations (phonons)

3. E-field perturbations (polarisabilities)

4. k-point perturbations (Born charges)

Non-local exchange-correlation1. Summary of XC functionals

2. Why use non-local functionals?

3. Some examples

Density Functional Perturbation Theory

Based on compute how the total energy responds to a perturbation, usually of the DFT external potential v

Expand quantities (E, n, , v)

...)2(2)1()0( EEEE

• Properties given by the derivatives

2

2)2()(

2

1particularin and

!

1

E

EE

nE

n

nn

The Perturbations

Perturb the external potential (from the ionic cores and any external field):• Ionic positions phonons• Cell vectors elastic constants• Electric fields dielectric response• Magnetic fields NMR

But not only the potential, any perturbation to the Hamiltonian:• d/dk Born effective charges• d/d(PSP) alchemical perturbation

Phonon Perturbations For each atom i at a time,

in direction x, y or z This becomes perturbation

denoted by The potential becomes a

function of perturbation Take derivatives of the

potential with respect to Hartree, xc: derivatives of

potentials done by chain rule with respect to n and

ui 2 cos(q Ri)

matrixconstant -force theis)(

directions are labels. atom are )(

)(

2

ji

ji

ji

ji

E

ijmm ji

RRR

qqD

So we need this E(2)

The expression for phonon-E(2)

E(2) k,n(1) HKS

(0) k, n(0)

k,n(1)

k,n(1) v(1)

k, n(0)

k, n(0) v(1)

k,n(1)

k,n

12

2 EHxc

n(r)n( r ) n(1)(r)n (1)( r ) k,n(0) v(2)

k,n(0)

k,n

•Superscripts denote the order of the perturbation•E(2) given by 0th and 1st order wave functions and densities•This is a variational quantity – use conjugate gradients minimiser•Constraint: 1st order wave functions orthogonal to 0th order wave functions•This expression gives the electronic contribution

The Variational Calculation

E(2) is variational with respect to |(1)>The plane-wave coefficients are varied to find the minimum E(2) under a perturbation• of a given ion i

• in a given direction • and for a given q

Analogous to standard total energy calculation

Based on a ground state (E(0)) calculation Can be used for any q value

Sequence of calculation

Find electronic force constant matrix Add in Ewald part Repeat for a mesh of q

And with Fourier interpolation: Fourier transform to get F(R) Fit and interpolate Fourier transform and mass weight to get D at any q

)(2)( )2( qqjiji

E

Phonon LR: For and against

For• Fast, each wavevector component about the

same as a single point energy calculation

• No supercells requires

• Arbitrary q

• General formalism

Against• Details of implementation considerable

Symmetry Considerations When perturbing the system the symmetry is broken No time reversal symmetry Implication is: k-point number increases For example, Phonon-Si2 (Diamond):

6x6x6 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

So what can you calculate? Phonon dispersion curves…

0

100

200

300

400

500

600

W L G X W K

Frequency (cm-1)

CASTEP Phonon Dispersion

0

100

200

300

400

500

600

0.000 0.005 0.010 0.015

Frequency (cm-1)

Density of Phonon States (1/cm-1)

CASTEP Density of Phonon States

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

-100 0 100 200 300 400 500 600

Density of Phonon States (1/cm-1)

Frequency (cm-1)

CASTEP Density of Phonon States

…and phonon DOS

Thermodynamics

Phonon density of states Debye temperature

Phase stability via

Entropic terms – derivatives of free energy Vibrational specific heats

Tk

TkVETVFB

B 2sinh2, 0

Electric Field Response

Bulk polarisability Born effective charges Phonon G-point LO/TO splitting Dielectric permittivities IR spectra Raman Spectra

Why Electric Fields Need Born effective

charges to get LO-TO splitting: originates from finite dipole per unit cell

Example given later… Found from d/dk

calculation and similar cross-derivative expressions

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

Examples of E-field Linear Response

•Response of silicon to an electric field perturbation

•Plot shows first order charge density

Blue: where electrons are removedYellow: where electrons go

Acknowledgement: E-field work by my PhD student Paul Tulip

Phonon Dispersion of an Ionic Solid: -point problems

At zone centre: Finite dipole (hence E-field) per unit cell caused by atomic displacements

E-field on a polar system: NaCl

Without E-field: LO/TO Frequencies: 175 cm-1

Born charge: Na Cl

06.100.000.0

00.006.100.0

00.000.006.1

06.100.000.0

00.006.100.0

00.000.006.1

Ionic character as expected: Na+ and Cl- ions

•Polarisability Tensor

4.330.00.0

0.04.330.0

0.00.04.33

•Electronic Permittivity Tensor

68.20.00.0

0.068.20.0

0.00.068.2

Quantities given in atomic units

LO/TO splitting is 90cm-1: Smooth dispersion curve at the -point

Summary of Density Functional Perturbation Theory Phonon frequencies Phonon DOS Debye Temperate PVT phase diagrams Vibrational specific heats Born effective charges LO/TO Splitting Bulk polarisabilities Electric permittivities First order charge densities (where electrons move from and

to) Etc… Lots of new physics…and more planned in later releases

Non-local XC functionals

Basic background of XC interaction Description within LDA and GGAs Some non-local XC functional Implementation within Castep Model test cases

The exchange-correlation interaction

)]([||2

1

||

)(

2

1

||

)(

2

1 2 rnRR

ZZrd

rr

rn

rR

drrnZH xc

ji ji

ji

i i

i

ji ji

ji

ji jiji ji

j

RR

ZZ

rr

e

Rr

eZH

||2

1

||2

1

||2

1 2

,

2

Many body Hamiltonian – many body wavefunction

Kohn-Sham Hamiltonian – single particle wavefunction

Single particle KE – not many body

Hartree requires self-interaction correction

All goes in here

Approximations to XC Local density approximation – only one Generalised gradient approximations – lots GGA’s + Laplacians – no real improvements Meta-GGA’s (GGA + Laplacian + KE) – many

recent papers – but nothing exciting yet. Hybrid functionals (GGA/meta-GGA + some exact

exchange from HF calculations) – currently favoured by the chemistry community.

Exc[n(r,r’)] – has been generally ignored recently by DFT community (although QMC and GW show it to have several interesting properties!).

New Functionals: The CPU cost We aim to go beyond the local (LDA) or semi-

local (GGA) approximations Why? Cannot get any higher accuracy with

these class of functionals The computational cost is high: scaling will be

O(N2) or O(N3) This is because all pairs (or more) of electrons

must be considered That is, beyond the single particle model most

DFT users are familiar with

First Class of New XC Functional: Exact and Screened Exchange

Exact Exchange (Hartree-Fock):

||

)()()()(

2

1}][{

**

1 1 rr

rrrrrddrE ijji

N

i

N

jix

||

)()()()(

2

1}][{

||**

1 1 rr

rrerrrddrE ij

rrkji

N

i

N

jisx

TF

Screened Exchange:

And in terms of plane waves:

ki Gqj TF

qjqj

G Gkikikisx kGGkq

GGGcGcGcGccE

, ,,22

,*

,,

*,, ||

)()()()(

2}][{

Why Exact/Screened Exchange? Exact exchange gives us access to the

common empirical “chemistry” GGA functionals such as B3LYP

Screened exchange can lead to accurate band gaps in semiconductors and insulators, so improved excitation energies and optical properties

The cost: scales O(N3), N=number of pl. waves LDA/GGA is 0.1% of calculation, EXX is 99.9%

Silicon Band Structure

• LDA band gap: 0.54eV• Exact Exchange band gap: 2.15eV• Screened Exchange band gap: 1.11eV• Experimental band gap: 1.12eV

Some recent results (by my PhD student, Michael Gibson)

Extra cost can be worthwhile

Another approach: Some theory on exchange-correlation holes… The exact(!) XC energy within DFT can be written as

rdrr

rrndrrnnE xc

xc ||

),()(

2

1][

•Relationship defined as the Coulomb energy between an electron and the XC hole nxc(r,r’)•XC hole is described in terms of the electron pair-correlation function

]1),()[(),( rrgrnrrn xcxc

Determines probability of finding an electron at position r’ given one exists at position r

Properties of the XC hole

Pauli exclusion principle: gxc obeys the sum rule

1),( rdrrnxc

•The size of the XC hole is exactly one electron – mathematically, this is the Pauli exclusion principle

•The LDA and some GGAs obey this rule

•For a universal XC function (applicable without bias), it must obey as many (all!) exact conditions as possible

The Weighted Density Method In the WDA the pair-correlation function is approximated

by

)](~|;[|1),( rnrrGrrg WDAxc

•The weighted density is fixed at each point by enforcing the sum rule

•This retains the non-locality of the function along with the Coulomb-like integral for Exc[n]

1)](~|];[|)( rdrnrrGrn WDA

The XC potential XC potential is determined in the usual manner

(density derivative of XC energy) We get 3 terms

rdrn

rnrrG

rr

rnrdrnrv

rdrr

rnrrGrnrv

rdrr

rnrrGrnrv

WDA

WDA

WDA

)(

)](~|;[|

||

)()(

2

1)(

||

)](~|;[|)(

2

1)(

||

)](~|;[|)(

2

1)(

3

2

1

)()()()( 321 rvrvrvrv

where

Example of WDA in action: An inhomogeneous electron gas

Investigate inhomogeneous systems by applying an external potential of the form

).cos()( rqvrv qext

•Very accurate quantum Monte Carlo results which to Very accurate quantum Monte Carlo results which to comparecompare•Will have no pseudopotential effectsWill have no pseudopotential effects•It’s inhomogeneous!It’s inhomogeneous!•Given a converged plane wave basis set, we are testing the Given a converged plane wave basis set, we are testing the XC functional only – nothing else to considerXC functional only – nothing else to consider

Investigate XC-hole shapes XC holes for reference electron at a density maximum

WDA results: acknowledgement to my PhD student Phil Rushton

But at the low density points… Exchange-correlation hole at a density minimum

How accurate are the XC holes?

Compare to VMC data – M. Nekovee, W. M. C. Foulkes and R. J. Needs PRL 87, 036401 (2001)

Self Interaction Correction

H2+ molecule contains only one electron

HF describes it correctly, DFT with LDA fails

Self-interaction correction is the problem

Realistic Systems - Silicon

Generate self-consistent silicon charge density Examine XC holes at various points

Si [110] plane

XC holes in silicon

Interstitial Region Bond Centre Region

XC hole of electron moving along [100] direction

Some electronic properties

Band structure of Silicon – band gap opens

III-V semiconductor band gaps

The non-local potential opens up the band gap of some simple semiconductors

Si Ge GaAs InAsExperiment 1.17 0.74 1.57 0.42LDA 0.52 0.02 1.23 0.34PW91 0.56 0.55 1.32 0.35WDA 1.11 0.81 1.65 0.45

Band gaps in eV

Timescales

First implementation of phonon linear response already in Castep v2.2 Much faster (3-4 times faster) phonon linear response due to

algorithmic improvements in Castep v3.0 Phonon linear response calculations for metals under development.

Aimed to be in v3.0 or v3.1 Electric field response (Born charges, bulk polarisabilities, permittivity,

LO/TO splitting) in Castep v3.0 Exact and screened exchange: currently being tested and developed

aimed at the v3.0 release Weighted density method: currently working and tested – release

schedule under discussion Raman and IR intensities currently being implemented. Scientific

evaluation still be performed before a possible release can be discussed

The following is my list of developments - other CGD members have other new physics and new improvements