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

2. 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

3. New Functionality Linear response Non-local exchange-correlation
Density functional perturbation theory
Atomic perturbations (phonons)
E-field perturbations (polarisabilities)
k-point perturbations (Born charges)
Non-local exchange-correlation
Summary of XC functionals
Why use non-local functionals?
Some examples

4. 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, y, v)
Properties given by the derivatives

5. 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

6. Phonon Perturbations So we need this E(2)
For each atom i at a time, in direction a=x, y or z
This becomes perturbation denoted by l
The potential becomes a function of perturbation l
Take derivatives of the potential with respect to l
Hartree, xc: derivatives of potentials done by chain rule with respect to n and l
So we need this E(2)

7. The expression for phonon-E(2)
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

8. The Variational 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
of a given ion i
in a given direction a
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

9. 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

10. Phonon LR: For and against
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

11. 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

12. So what can you calculate?
…and phonon DOS
Phonon dispersion curves…

13. Thermodynamics Phonon density of states Debye temperature
Phase stability via
Entropic terms – derivatives of free energy
Vibrational specific heats

14. Electric Field Response
Bulk polarisability
Born effective charges
Phonon G-point LO/TO splitting
Dielectric permittivities
IR spectra
Raman Spectra

15. 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

16. Examples of E-field Linear Response
Response of silicon to an electric field perturbation
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

17. Phonon Dispersion of an Ionic Solid: G-point problems
At zone centre: Finite dipole (hence E-field) per unit cell caused by atomic displacements

18. E-field on a polar system: NaCl
Without E-field: LO/TO Frequencies: 175 cm-1
Born charge: Na Cl
Ionic character as expected: Na+ and Cl- ions
LO/TO splitting is 90cm-1: Smooth dispersion curve at the G-point
Polarisability Tensor
Electronic Permittivity Tensor
Quantities given in atomic units

19. 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

20. 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

21. The exchange-correlation interaction
Many body Hamiltonian – many body wavefunction
Kohn-Sham Hamiltonian – single particle wavefunction
Hartree requires self-interaction correction
Single particle KE – not many body
All goes in here

22. 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!).

23. 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

24. First Class of New XC Functional: Exact and Screened Exchange
Exact Exchange (Hartree-Fock):
Screened Exchange:
And in terms of plane waves:

25. 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%

26. Silicon Band Structure
Extra cost can be worthwhile
Some recent results (by my PhD student, Michael Gibson)
LDA band gap: 0.54eV
Exact Exchange band gap: 2.15eV
Screened Exchange band gap: 1.11eV
Experimental band gap: 1.12eV

27. Another approach: Some theory on exchange-correlation holes…
The exact(!) XC energy within DFT can be written as
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
Determines probability of finding an electron at position r’ given one exists at position r

28. Properties of the XC hole
Pauli exclusion principle: gxc obeys the sum rule
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

29. The Weighted Density Method
In the WDA the pair-correlation function is approximated by
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]

30. The XC potential
XC potential is determined in the usual manner (density derivative of XC energy)
We get 3 terms
where

31. Example of WDA in action: An inhomogeneous electron gas
Investigate inhomogeneous systems by applying an external potential of the form
Very accurate quantum Monte Carlo results which to compare
Will have no pseudopotential effects
It’s inhomogeneous!
Given a converged plane wave basis set, we are testing the XC functional only – nothing else to consider

32. Investigate XC-hole shapes
XC holes for reference electron at a density maximum
WDA results: acknowledgement to my PhD student Phil Rushton

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

34. How accurate are the XC holes?
Compare to VMC data – M. Nekovee, W. M. C. Foulkes and R. J. Needs PRL 87, (2001)

35. Self Interaction Correction
H2+ molecule contains only one electron
HF describes it correctly, DFT with LDA fails
Self-interaction correction is the problem

36. Realistic Systems - Silicon
Generate self-consistent silicon charge density
Examine XC holes at various points
Si [110] plane

37. XC holes in silicon Interstitial Region Bond Centre Region
XC hole of electron moving along [100] direction

38. Some electronic properties
Band structure of Silicon – band gap opens

39. III-V semiconductor band gaps
Band gaps in eV
The non-local potential opens up the band gap of some simple semiconductors

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