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First principles calculation within DFTsome practical concerns
Ling-Ti Kong (-N)
EMail: [email protected]
School of Materials Science and Engineering, Shanghai Jiao Tong University
Dec 24, 2010
1 / 19
Outline
1 k-sampling
2 Plane wave cutoff
3 Pseudo-potential
4 Another look on Born-von Karman PBC
5 Summary: DFT with PP+PW
6 Further readings
2 / 19
Kohn-Sham Equation for crystals
Schrodinger Equation ⇒ Eigen problem∑m
[~2|k + Gm|2
2mδm,m′ + vm−m′
]ci ,m = εici ,m′
3 / 19
Brillouin zone integration
Properties like the electron density, total energy, etc., can be evaluated byintegration over the first Brillouin zone.
Electron density
n(r) =1
Nk
∑k
occ.∑i
ni ,k(r), ni ,k(r) = |ψi ,k(r)|2
=1
ΩFBZ
occ.∑i
∫FBZ
ni ,k(r)dk
=1
ΩFBZ
∫FBZ
fi ,k · ni ,k(r)dk
4 / 19
k-point sampling: uniform vs non-uniform
Moreno and Soler (Phys. Rev. B 45(24):13891, 1992):A mesh with uniformly distributed k-points is preferred.
Example: A rectangular lattice
(a) isotropic sampling
(b) finer sampling verticallypoor sampling horizontally
5 / 19
Monkhorst-Pack scheme
Monkhorst & Pack (Phys. Rev. B 13:5188, 1976.):
k =3∑
i=1
2ni − Ni − 1
2Nibi , ni = 1, · · · ,Ni
Example: 4× 4× 1 MP mesh for 2D square lattice
k1,1 = (−3
8,−3
8) k1,3 = (−3
8,
1
8)
k1,2 = (−3
8,−1
8) k1,3 = (−3
8,
3
8)
6 / 19
Shifted Monkhorst-Pack mesh
Centered on Γ
25 k-points
Centered around Γ
Shifted by (1/8, 1/8, 0)16 k-points
Yet the same k-point density.
7 / 19
Symmetry on k in the first Brillouin zone
ψk+G = ψk ⇒ Limit k in FBZ.
[− ~2
2me52 +veff (r)
]uk(r)e ikr = εuk(r)e ikr
[− ~2
2me52 +veff (r)
]u∗k(r)e−ikr = εu∗k(r)e−ikr
[− ~2
2me52 +veff (r)
]u∗−k(r)e ikr = εu∗−k(r)e ikr
u−k = u∗k
ψ−k = u−ke−ikr =
(uke
ikr)∗
= ψ∗k
εi ,−k = εi ,k
8 / 19
Symmetry on k in the first Brillouin zone
Sψik(r) = ψik(Sr)
Sψik(r) = ψik(Sr)
= uik(Sr)e ik·Sr
= uik′(r)eik′r
k′ = S−1k
n(r) =∑k
ωk
occ.∑i
ni ,k(r)
k ∈ irreducible Brillouin zone
ωk =# of sym. connected k
total # of k in FBZ9 / 19
Irreducible Brillouin zone
Example:
MP-mesh for 2D square lattice
4×4×1 MP mesh = 16 k-points in the FBZ.
4 equivalent k4,4 = (38 ,38) ⇒ w1 = 0.25
4 equivalent k3,3 = (18 ,38) ⇒ w2 = 0.25
8 equivalent k4,3 = (38 ,18) ⇒ w3 = 0.50
Brillouin zone integration
n(r) =1
4
occ.∑i
ni ,k4,4+1
4
occ.∑i
ni ,k3,3+1
2
occ.∑i
ni ,k4,3
10 / 19
Irreducible Brillouin zone
Symmetry break: Hexagonal cell
Even meshes break thesymmetry, while meshescentered on Γ preserves it.Shift the k-point mesh to preserve
hexagonal symmetry!
k-sampling: rules of thumb
Keep density of k constant in each direction; Nibi = const.
Denser k ⇒ more precise results.
N IBZk does not necessarily scale with N. Symmetry matters
No k sampling needed for atoms or molecules!
Generally need to do convergence test!
11 / 19
Cutoff: finite basis set
Computationally, a complete expansion in terms of infinitely many planewaves is not possible.The coefficients, ci ,m(k) decrease rapidly with increasing PW kinetic
energy ~2|k+Gm|22m .
A cutoff energy value, Ecut,determines the number of PWs(Npw) in the expansion, satisfying,
~2|k + Gm|2
2m< Ecut
Npw is a discontinuous function of the PW kinetic energy cutoff, whiledepends only on the computational cell size and the cutoff energy value.
12 / 19
Pseudopotential: why?
Reduction of basis set sizeeffective speedup of calculation
Reduction of number of electronsreduces the number of d.o.f.
UnnecessaryWhy bother? They are inert anyway· · ·Inclusion of relativistic effectsrelativistic effects can be includedpartially
13 / 19
Pseudopotential: how?
Pseudopotential (PP)
A smooth effective potential that reproduces the effect of the nucleus pluscore electrons on valence electrons.
Norm conserving PP;
Ultrasoft PP;
Projector Augmented Wave PP;
· · ·14 / 19
Born-von Karman PBC: always a necessity?
Supercell approach
point defect free surface single molecule
Supercell must be sufficiently large to maintain isolation.
15 / 19
DFT: Ability and disability
Fundamentally, DFT can only predict the ground state electronic densityand the ground state total energy of a set of electrons under an externalpotential.
DFT can predict
Total energy
Forces
Lattice constants
Bond lengths
Vibrational frequencies
Phonon frequencies
Electron density
Static dielectric response
DFT cannot predict
Excited state energies
Band gap
Band structures
Wave functions
Fermi surface
Superconductivity
Excitons
Electronic transport
16 / 19
Accuracy
Accuracies can be expected
bond length ∼ 3% too smallbulk modulus ∼ 10% too high
phonon frequency ∼ 10% too highenergy difference > 1 mHartree
cohesive energy very poor (much too high)
Accuracies for properties that DFT technically does not predict
band gap 50% too smallband structure qualitatively reasonable
fermi surface qualitatively reasonable
17 / 19
Factors affecting accuracy
Born-Oppenheimerapproximation;
Density functional theory
LDA, GGALSDA
Pseudopotential
Kinetic energy cutoff
k-sampling
convergence criterion
· · ·
18 / 19
References and further readings
1 http://www.fhi-berlin.mpg.de/th/Meetings/FHImd2001/pehlke1.pdf
2 http://itp.tugraz.at/LV/ewald/TFKP/Literatur Pseudopotentiale/Roundy 05 DFT+PPsumm.pdf
3 http://www.phys.sinica.edu.tw/TIGP-NANO/Course/2007 Spring/Class%20Notes/CMS.20070531.pseudo.pdf
4 J. Singleton, Band theory and electronic properties of solids, OxfordUniversity press.
5 http://cms.mpi.univie.ac.at/vasp/
6 http://www.pwscf.org/
7 http://www.abinit.org/
8 http://www.cpmd.org/
9 http://www.icmab.es/siesta/
19 / 19