Density Functional Theory - Technionphycomp.technion.ac.il/~jeremie/Seminar-Jeremie.pdf · I-DFT...

Density Functional Theory applied to the solid state…

An introduction to VASP

Jeremie Zaffran2nd year-MSc. (Nanochemistry)

[email protected]

Supervisors: Pr. Leeor Kronik, Amir Natan (PhD) & Noa Marom (PhD)

Contents

I- DFT and its functionals

A. On the density functional theory…B. The exchange-correlation approximationC. The functionals we used…

II- VASP software

A. GeneralitiesB. Running calculations with VASP…C. Overview on the INCAR file parameters

III- Performances and restrictions of few density functionals

A. Presentation of the projectB. The process…C. Results

Conclusion

I- DFT and its functionals

I-DFT and its functionals

A- On the density functional theory

Why do we need DFT?

Ab initio calculations (no experimental parameter)…

elecelecelecelec EH ˆ

Encountered problems…

function of 3.N space variables NNNelec zzyyxx ,...,,,...,,,..., 111

The wavefunction has no physical sense… only its square is an observable

Very time-consuming!


The basis…

ext

universal

eeee VVTH ˆˆˆˆ

Proper to the system

extVNEE ,0

extVN, is sufficient to define any chemical system

Which means…

The DFT core: Hohenberg-Kohn Theorems (1964)


1) “The external potential is a unique functional of the ground state density ”extV 0

000 ,ˆ, EHVN ext (and all other properties)

2) “The ground state energy will be reached if and only if one uses the ground state density in the energy functional.”0

0E

which means… 00, EEE trialtrial Or…

EE min0

Variational principle!

DFT key points…

The electronic density becomes the fundamental variable!

Interest: • is only a function of 4 variables (x,y,z,s) and no more of 3N variables as with .

• is an observable.

Any DFT algorithm should aim to reach only the ground state and no excited state!

The energy minimisation algorithms have to take care about two main constraints lying on the density:

• must be N-representable, which means associated to an acceptable wave function :square-integrable functions… The Slater determinant is only an example of such a set!

• must be Vext-representable, which means giving rise to a finite external potential.Note that to this date we don’t know what makes a density Vext-representable on the mathematical point of view.

Levy constrained search scheme


Expression of the energy functional and limitations of the Hohenberg-Kohn theorems

Feature of the system

Universal functional

drVV eNext

HKF ???? ( T is not a functional of the density, and Eee is not completely known)

HKF

eeext ETVE


Owing to the Kohn-Sham approach…

xcSext EJTVE

xc

S

ext

E

J

T

V : External field (from the nuclei): Kinetic energy of an ideal system: Columbic repulsion: Exchange-correlation

Where…

nclC

eeSxc

ET

JETTE

ncl

C

S

E

T

T

T : Kinetic energy of the real system: Kinetic energy of the reference system: Residual kinetic energy: Non-classical energy

with

Nevertheless, what is ?... No one knows except from this un useful formalism xcE

xcE must be approximated


B-Exchange-correlation approximations

Classical and non classical effects due to the many-body interacting system.

Several ways to express it…

• Local Density Approximation (LDA):Based on the homogeneous electrons gas model. Exchange-correlation density functional is exactly known owing to the Thomas-Fermi model.

• Gradient Generalized Approximation (GGA): PBE…Application of the gradient operator on the previous model.

• Meta-GGA: BB95…Application also of the laplacian operator.

• Hybrid functional: HSE06, B3LYP…Introduction of an exact Hartree-Fock part in the Exchange functional.

GGAC

GGAX

HFXxc EEEE %)1(%


C-The functionals we used…

We assessed the performance of 3 GGAs…

1. PBE (Pedrew-Burke-Ernzerhof)2. PBE sol- devoted to the solid state3. Rev PBE- devoted to the molecules

rr 3. dsFE XunifXX rr 32 ...dtE unif

CC

...10 2 ssFX

Enhancement factor: Dimensionless reduced gradients

3/43/1232

s

6/76/1234

t

Exchange/Correlation energy per particle in the free electron gas model


II- DFT and VASP software

II-DFT and VASP sofware

A- Generalities

Solid chemistry codes can be based on:

• Atomic orbitals expansion…

CRYSTAL

• Plane waves (Bloch functions) expansion…

VASP

II-DFT and VASP software

About VASP-Vienna Ab-initio Software Package…

Overview on VASP calculations

Band structures calculations

Phonon dispersion

IR and Raman spectra

Molecular dynamics – Simulated annealing

Bulk and surface calculations

To relax a crystalline structure


About VASP-Vienna Ab-initio Software Package…

Initially developed at the MIT by Payne, enhanced in the 90’s by Hafner and Furthmüller at Vienna University and at Cambridge.

MPI parallelization in ~96 by Holender and Holender (UK)

Bloch waves approach calculations on the reciprocal space (using FFT-algorithms)

Periodic boundaries conditions…

extremely well convenient for crystals study, hardly efficient for single molecules

Valence electrons treated explicitly, core electrons with pseudopotentialapproximation (PAW method)


The pseudopotential idea…

Close to the atomic core, wave function has a chaotic behavior

To replace the chaotic wave function by a smoother one and its associated potential

Many methods to generate it:• Semi-empirical• Ultra-soft• Norm-conserving…

Numerous interests: • Reduction of the electrons number• Reduction of the basis set• Enhancement of the calculation time…


B-Running calculations with VASP …

Main files…

1. POTCAR- Pseudopotential (chemical species properties)2. POSCAR- Lattice structure3. KPOINTS- K-points mesh generation4. INCAR- Central file

INPUTS

1. OUTCAR/OSZICAR2. DOSCAR- density of states3. WAVECAR- KS-orbitals4. CONTCAR- Final ions positions if relaxation5. CHGCAR- electronic density6. CHG- electronic density for visualization7. Vasprun.xml- for dos and band structure visualization

OUTPUTS


Input files…

POTCAR

Already included in the software

Contains all the necessary information related to the treated chemical specie…• Plane wave basis set• Atomic radius• Atomic mass• Total and valence electrons number…

Proper to the each functional (LDA, GGA…)

If several species…

cat POTCAR_A POTCAR_B > POTCAR

In the same order they appear in the POSCAR!!!!


Input files…

POSCAR

Cubic BN 3.57 0.0 0.5 0.50.5 0.0 0.5 0.5 0.5 0.0 1 1 Direct 0.00 0.00 0.00 0.25 0.25 0.25

• Comment line• Scaling factor, i.e lattice parameter

• Lattice vectors

• 1 atom of B and one atom of N• Cartesian coordinates (either Reciprocal coord.)• Coordinates of B• Coordinates of N


Note…

• The time demand increases with the cell size - even if the atoms number is kept constant

Atomic species cell size

Li 13 Å

Al 12 Å

K 14 Å

Cu, Rh, Pd 10 Å

C 8 Å

• To simulate amorphous state or single molecules, take a big cell with plenty of vacuum to prevent border effects


Input files…

KPOINTS

Several methods according to the expected property

k-points along high symmetry lines

10 Line-modeRec0 0 0 ! gamma 0.5 0.5 0 ! X

0.5 0.5 0 ! X 0.5 0.75 0.25 ! W

0.5 0.75 0.25 ! W 0 0 0 ! gamma

• Comment line• 10 K-points along each line

• Reciprocal space coord.

BAND STRUCTURE CALCULATION


Input files…

Automatic mesh0 Monkhorst-Pack 4 4 4 0. 0. 0

• Comment line• Automatic generation•Generation method• 4 k-points on kx … 4x4x4 kpoints

KPOINTS


C- Overview on INCAR file parameters

SYSTEM= Rhodium

NEDOS=300ISMEAR=-5SIGMA=0.05ENCUT=500IBRION=-1! ICHARG=11

• Comment line

• Amount of flags, but all of them have default values

• Run out this tag


• ISMEAR, SIGMA: Occupancies wave function method – convergence method (depends on the system chemical nature)

•ENCUT: Energy cutoff (~size basis set)

• IBRION: Static calculation, ab-initio MD, relaxation algorithms (CG, SD…)

• ICHARG: To apply SCF on the charge density or not?

• ISPIN: Is the system magnetic?

• GGA: Which functional to use?

• LHFCALC: Hartree-Fock calculations

• AEXX/ALDAC…: Hybrid functionals (VASP05)

The most used…


A-Presentation of the project

To assess the performance of 3 GGAs:

• PBE• PBEsol solid state study• revPBEmolecule study

Study already performed by Csonka in Physical Review B 79, 155107 (2009) only on lattice parameter optimization

Objectives:1. To reach a more important precision on lattice parameter

2. To assess their performance on band structure and density of states calculation

B-The process

1. To optimize the number of K-points

2. To optimize ENCUT

3. To get the charge density

4. To deduce band structure and density of states


K-points optimization…

Ge- PBE

Convergence at 23x23x23 k-points

Objective:To get the convergence with the smallest set of k-points

GaAs- PBE

Convergence at 21x21x21 k-points


ENCUT optimization…

Objective:To reach the harmonic approximation (close to the equilibrium) with the smallest Energy Cutoff


C-Results

On the lattice parameter…


On the band structure…

LDA calculation: Natan et al., PRB 78, 075109 (2008)


On the band structure…


On the bandgap assessment…


Greatest discrepancies

The closest to the experiment

Conclusion

• Excellent results for lattice parameter optimization

• DFT is not a suitable method for band-gap calculation

• DFT only deals with the ground state

• However, we can expect that more accurate functionals as Hybrid one could better approach the gap.

• Other methods are more appropriate like Time-Dependent-DFT or post-HF methods (CI)

• Very time consuming!!!

• Any functional has to be adapted relatively to the physico-chemical nature of the system (metal, semiconductor…)and for a given property!

Density Functional Theory - Technionphycomp.technion.ac.il/~jeremie/Seminar-Jeremie.pdf · I-DFT...

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