First principles calculation within DFTsome practical concerns

Ling-Ti Kong (-N)

EMail: konglt@sjtu.edu.cn

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