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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