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Abinit Workshop

- Sornthep Vannarat

Lesson 1

- Hydrogen Molecule

Lesson 1

- cd abinit/Tutorial
- mkdir work
- copy ../t1x.files .
- edit t1x.files
- copy ../t11.in .
- ../../abinis lt t1x.files gt log

../t11.in t1x.out t1xi t1xo t1x ../../Psps_for_tes

ts/01h.pspgth

t11.in t11.out t1xi t1xo t1x ../../Psps_for_tests/

01h.pspgth

Lesson 1 Starting abinit

- Program name abinis.exe
- Input files t11.in and t1xi
- Output files t11.out and t1xo
- Temporary files t1x
- Pseudo potential files ../../Psps_for_tests/01h.ps

pgth

Lesson 1 Input parameters

- Input file t11.in

acell 10 10 10 Cell size is 103 ntypat 1

One type of atom znucl 1

Atomic number is one natom 2 There

are two atoms typat 1 1 They both are

of type 1 xcart Location of the

atoms -0.7 0.0 0.0 atom 1, in Bohr 0.7

0.0 0.0 atom 2, in Bohr ecut 10.0

Cut-off energy, in Hartree nkpt 1

One k point nstep 10 Maximal number of

SCF cycles toldfe 1.0d-6 Threshold diemac

1.0 Preconditioner diemix 0.5

Using standard preconditioner

for molecules in a big box

Lesson 1 Output

- Abinit version
- Input/output files
- Values of input parameters
- Data Set and Pseudo potential file
- Number of plan waves
- Iterations
- Stress tensor
- Eigen values
- Max/Min Electronic density
- Total energy
- Values of input parameters (after calculation)
- Log file interactive input, more details of

iterations

Lesson 1 Inter-atomic distance (1)

- 3 approaches
- compute total energy E(d) or force F(d)
- Use relaxation
- Multiple datasets
- t12.in ndtset, xcart, xcart, getwfk, nband
- Edit t1x.files and run
- Look at t12.out
- Data sets symmetry and number of plane waves
- Data sets coordinates xangst, xcart, xred
- Data sets Number of iterations
- etotal and fcart
- Plot data

Lesson 1 Inter-atomic distance (2)

- Use relaxation ionmove, ntime, tolmxf, toldff
- Multiple datasets
- t13.in
- Edit t1x.files and run
- Look at t13.out
- BROYDEN STEP
- value of coordinates after relaxation xangst,

xcart, xred

Lesson 1 Charge density

- prtden 1, t14.in
- move atoms to middle of box
- cut3d convert to OpenDX

Lesson 1 Atomization energy

- Eatomization (2EHatom EH2molecule) ? per

molecule - Caution
- Calculations with the same setting
- Spin nsppol, spinat
- Degeneracy HOMO and LUMO (see lesson_4)
- Ground-state charge density NON-spherical,

automatic determination of symmetries should be

disabled (nsym) - For Hydrogen,
- ground state is spherical (1s orbital)
- HOMO and LUMO have a different spin
- t15.in define occupation of each spin, occopt

and occ - Output file Eigen values, Max/Min spin
- Atomization energy ?

Lesson 1 Summary

- System
- H2 molecule in a big box 10x10x10 Bohr3
- Method
- Using cut-off energy 10 Ha
- LDA (LSDA for atom) with ixc1 (default)
- Pseudopotential from Goedecker-Hutter-Teter table
- Results
- Bond length 1.522 Bohr (1.401 Bohr)
- Atomisation energy 0.1656 Ha 4.506 eV (4.747

eV)

Lesson 2

- Convergence Study

Lesson 2 Combined calculation

t21.in

ndtset 2 acell 10 10 10 ecut 10 First

dataset natom1 2 ionmov1 3 BD

algorithm ntime1 10 tolmxf1 5.0d-4 xcart1 -0.7

0.0 0.0 0.7 0.0 0.0 toldff1

5.0d-5 nband1 1

Second dataset natom2 1 nsppol2 2

LSDA occopt2 2 nband2 1 1 Spin up, down occ2

1.0 0.0 toldfe2 1.0d-6 xcart2 0.0 0.0

0.0 spinat2 0.0 0.0 1.0 Init spin

Lesson 2 Convergence

- Calculation parameters
- ecut
- acell
- number of k-points
- convergence of SCF cycle toldfe, toldff
- convergence of geometry optimization tolmxf

Lesson 2 ecut

t22.in

ndtset 12 udtset 6 2 acell 10 10 10 ecut? 10

ecut? 5 First dataset bond length

natom?1 2 ionmov?1 3 ntime?1 10

tolmxf?1 5.0d-4 xcart?1 -0.7 0.0 0.0

0.7 0.0 0.0 toldff?1 5.0d-5

nband?1 1

Second dataset H-atom natom?2 1

nsppol?2 2 occopt?2 2 nband?2 1 1

occ?2 1.0 0.0 toldfe?2 1.0d-6 xcart?2

0.0 0.0 0.0 spinat?2 0.0 0.0 1.0

Lesson 2 ecut

What determines ecut? What if H is changed to Cl?

Lesson 2 acell

t23.in

ndtset 12 udtset 6 2 acell? 8 8 8 acell? 2

2 2 ecut 10 First dataset bond length

natom?1 2 ionmov?1 3 ntime?1 10

tolmxf?1 5.0d-4 xcart?1 -0.7 0.0 0.0

0.7 0.0 0.0 toldff?1 5.0d-5

nband?1 1

Second dataset H-atom natom?2 1

nsppol?2 2 occopt?2 2 nband?2 1 1

occ?2 1.0 0.0 toldfe?2 1.0d-6

xcart?2 0.0 0.0 0.0 spinat?2 0.0 0.0 1.0

Lesson 2 acell

What determines acell? What if H is changed to Cl?

Lesson 2 Optimum parameters and GGA calculation

- Use the optimum ecut and acell to determine H2

bond length and atomization energy. - Switch to GGA calculation by changing ixc
- No need to change pseudo-potential for H (small

core) - No need to change ecut
- No need to change acell
- Compare results

Lesson 3

- Crystalline Silicon

Lesson 3 Introduction

- Crystalline silicon (Diamond structure, 2 FCC)
- the total energy
- the lattice parameter
- the band structure (actually, the Kohn-Sham band

structure) - Parameters
- k-points,
- smearing of cut-off

Lesson 3 Introduction

- Parameters
- rprim premitive vectors
- xred reduced coordinates
- K-point sampling
- kptopt 0 read from input, 1,2,3 generates,

negative for band calculation - ngkpt numbers of k-points in 3 directions
- nshiftk shiftk kptrlatt
- alternatively use kptrlen
- Larger cell ? smaller Brillouin zone

Lesson 3 Sample k-points generation

- ngkpt 2,2,2
- First mesh has 8 points, (0,0,0), (0,0,½),

(0,½,0), (0,½,½), (½,0,0), (½,0,½), (½,½,0),

(½,½,½) - nshiftk 4, shiftk (½,½,½), (½,0,0), (0,½,0),

(0,0,½) - First shift with (½,½,½), 8 k-points become

(¼,¼,¼), (¼,¼,¾), (¼,¾,¼), (¼,¾,¾), (¾,¼,¼),

(¾,¼,¾), (¾,¾,¼), (¾,¾,¾) - Second shift with (½,0,0), 8 k-points become

(¼,0,0), (¼,0,½), (¼,½,0), ... - Third shift with (0,½,0), 8 k-points become

(0,¼,0), (0,¼,½), (0,¾,0), ... - Forth shift with (0,0,½), 8 k-points become

(0,0,¼), (0,0, ¾), (0,½,¼), ... - Value ki larger than ½ can be translated to ki-1

e.g. ¾ ? -¼ - Totally 32 points obtained by shifting first

mesh, but can be reduced by symmetry

Lesson 3 Silicon crystal

- Look at input file t31.in, meaning of acell,

rprim, xred, kptopt, ngkpt, nshiftk, shiftk,

diemac - Try running and check the result
- Try changing
- kptopt to 3
- ngkpt to 3,2,2 etc.
- nshiftk, shiftk
- using kptrlen instead, with prtkpt 1,0

Lesson 3 Silicon k-point convergence

- Look at t32.in and try running it
- Why problem occurs?
- Change t32.in to correct the problem and to

perform a series of calculations to test

convergence against number of k-points - ndtset 4
- ngkpt1 2 2 2
- ngkpt2 4 4 4
- ngkpt3 8 8 8
- ngkpt4 16 16 16
- Note number of k-points and energy convergence
- Convergence of wavefunction and charge density

can also be verified

Lesson 3 Silicon k-point convergence

- Test k-point, when begin the study new material
- Test (at least) three efficient k-point sets
- CPU time is proportional to number of k-points
- Symmetry reduce number of k-points, but need to

be weighted (wtk) - Reference Monkhorst and Pack paper, Phys. Rev. B

13, 5188 (1976)

Lesson 3 Silicon Lattice Parameter

- Parameters (t34.in)
- optcell 1
- ionmov 3
- ntime 10
- dilatmx 1.05
- ecutsm 0.5
- Experimental value 5.431 Angstrom at 25 degree

Celsius, see R.W.G. Wyckoff, Crystal structures

Ed. Wiley and sons, New-York (1963) - Calculated value ?
- Using LDA with the 14si.pspnc pseudopotential
- What are 2 data sets?

Lesson 3 Silicon Band Structure

- Two steps
- SCF calculation of charge density
- Non-SCF calculation of eigen values (bands)
- Use L-Gamma-X-Gamma circuit
- In eight-atom cell coordinates (1/2 1/2 1/2)-(0

0 0)-(1 0 0)-(1 1 1) - In two-atom cell coordinates (1/2 0 0)- (0 0 0)-

(0 1/2 1/2)-(1 1 1) - Parameters (t35.in)
- prtden, iscf, getden, nband, kptopt, ndivk,

enunit, tolwfr - kptbounds to
- 0.5 0.0 0.0 L point
- 0.0 0.0 0.0 Gamma point
- 0.0 0.5 0.5 X point
- 1.0 1.0 1.0 Gamma point in another cell.
- Results kpt

Lesson 4

- Crystalline Aluminum
- and Surface Energy

Lesson 4 Introduction

- Aluminum, the bulk and the surface.
- the total energy
- the lattice parameter
- the relaxation of surface atoms
- the surface energy
- Smearing of the Brillouin zone integration
- Preconditioning the SCF cycle

Lesson 4 Smearing

- occopt 4,5,6,7
- Use Fermi-Dirac when trying to mimic physical

electronic temperature. It is less convenient to

use due to "long-tailed, need more bands. - In general Gaussian-like smearings are

preferable. - If you are interested only in the total energies,

you can just use a Gaussian smearing - but need

to extract corrected energy by taking the semisum

of the energy and the free energy. - Methfessel-Paxton and Marzari-Vanderbilt do this

automatically for you, and also provide forces,

stresses, and whatever else corrected for the

leading term in the temperature. - tsmear

Lesson 4 Smearing delta functions

Lesson 4 Bulk Al

- Look at t41.in
- ecut 6 Ha, compare to previous cases
- H needed 30 Ha
- Si needed 8 Ha
- Run
- Look at output and note 2 points
- Components of energy
- Occupation of each band
- Test ecut convergence

Lesson 4 k-point convergence

- How to check k-point convergence?
- Look at t42.in
- Run
- Look at the result
- Total energy
- acell
- Try with a different tsmear

Lesson 4 tsmear and k-point covergence

Aluminum Total energy (E) and Lattice parameters

(A) calculated using tsmear 0.05, 0.10 as

functions of k-point grid

Larger tsmear converges faster, but ...

Try t43.in

Lesson 4 Al (001) surface energy

- Slab calculation Al layer vacuum layer
- Thicknesses of Al and Vacuum layers
- Reference energy Bulk calculation with

equivalent parameters, i.e. cell shape, k-point

grid, ecut - Esurface (Eslab/nslab Ebulk/nbulk)/(2

Asurface) - Look at t44.in and t45.in, what do they

represent? - Difficulties surface reconstruction and

different top-buttom surfaces

Lesson 4 Al surface energy

- Vacuum layer thickness
- Defining atomic positions in Cartesian

coordinates is more convenient - Preconditioner (dielng) for metalvacuum case
- How many layers of vacuum are needed?
- t46.in

Lesson 4 Al surface energy

- Al layer thickness
- Preconditioner (dielng) for metalvacuum case
- Use an effective dielectric constant of about 3

or 5 - With a rather small mixing coefficient 0.2
- Alternatively, Use an estimation of the

dielectric matrix governed by iprcel45 - Repeat the 3 aluminum layer case for comparison
- t47.in
- See t47_STATUS to check status of long

calculation - How many Al layers are needed?

Lesson 5

- Dynamical Matrix, Dielectric
- Tensor and Effective Charge

Lesson 5 Response functions

- Response functions are the second derivatives of

total energy (2DTE) with respect to different

perturbations, e.g. - phonons (Dynamical metrix)
- static homogeneous electric field (Dielectric

tensor, Born effective charges) - strain (Elastic constants, internal strain,

piezoelectricity) - ABINIT computes FIRST-order derivatives of the

wavefunctions (1WF) - 2DTE is calculated from 1WF
- References
- X. Gonze, Phys. Rev. B55, 10337 (1997)
- X. Gonze and C. Lee, Phys. Rev. B55, 10355

(1997).

Lesson 5 Response functions

- ABINIT gives
- phonon frequencies
- electronic dielectric tensor
- effective charges
- Derivative DataBase (DDB)
- Contains all 2DTEs and 3DTEs
- MRGDDB
- Anaddb

Lesson 5 Perturbations

- Phonon
- displacement of one atom (ipert) along one of the

axis (idir) of the unit cell, by a unit of length

(in reduced coordinates - characterized by two integer numbers and one

wavevector - rfatpol defines the set of atoms to be moved
- rfdir defines the set of directions to be

considered - nqpt, qpt, and qptnrm define the wavevectors to

be considered - Electric field
- DDK dH/dk, auxiliary for RF-EF (ipertnatom1)
- Homogeneous electric field (q0), only

(ipertnatom2), idir direction - Homogeneous Strain
- Uniaxial strain ipert natom3, idir 1,2,3

for xx,yy,zz - Shear strain ipert natom4, idir 1,2,3 for

yz, zx, xy - No internal coordinate relaxation

Lesson 5 Ground State of AlAs

- trf1_1.in, trf1_x.files (2 potentials)
- Note tolvrs 1.0d-18
- Run
- Is tolvrs reached? (18)
- What is the total energy? (15 digits)

Lesson 5 Frozen-phonon E

- tr1_2.in
- Read in previous wavefunction file (irdwfk 1)
- Al is moved (xred)
- Need to rename tr1_xo_WFK to tr1_xi_WFK
- Edit tr1_x.files to run tr1_2.in
- Run
- Compare tr1_1.out and tr1_2.out
- Symmetry, K-points
- Cartesian forces
- RMS dE/dt
- Estimate E from E(x) E0xdE x2d2E / 2 ...
- x change in Al position from its equilibrium

Lesson 5 Response-Function E

- d2E/dx2 x change in Al position from its

equilibrium - tr1_3.in
- Read in previous wavefunction file (irdwfk 1)
- kptopt 2
- Atomic positions not changed (xred)
- Phonon perturbation (rfphon)
- Perturbation on Al atom (rfatpol)
- Direction (rfdir) and wave-vector (nqpt, qpt)
- Run
- Output files .out (2DEtotal), 1WF, DDB,

Lesson 5 RF Full Dynamical Matrix

- At Gamma q(0,0,0)
- Perturbation J(m,n)
- m Atom number (rfatpol 1 natom)
- n Direction (Reduced) (rfdir 1 1 1)
- Dynamical matrix Mj1,j2
- Run
- Output files .out, 1WF, 1WF4, DDB,
- Perturbation of each atom is applied in each

direction in turns - idir 2,3 is symmetric with previous calculation
- ipert 1,2,4 (electric field)
- Note the symmetry Mi,j Mj,i?
- Rerun with tolvrs 10-18
- Phonon Energies

Lesson 5 Recipe K-Points

- Input k-point set for RF should NOT have been

decreased by using spatial symmetries, prior to

the loop over perturbations - ABINIT will automatically reduce k-points
- kptopt1 for the ground state
- kptopt2 for response functions at q0
- kptopt3 for response functions at non-zero q

Lesson 5 Recipe Steps

- Atomic displacement with q0,
- SC GS IBZ (with kptopt1)
- SC RF Phonon Half Set (with kptopt2)
- Atomic displacement with qk1-k2 (k1,k2 are

special k-points), - SC GS IBZ (with kptopt1)
- SC RF Phonon Full Set (with kptopt3)
- Atomic displacement for a general q point,
- SC GS IBZ (with kptopt1)
- NSC GS kq (might be reduced due to symmetries,

with kptopt1) - SC RF Phonon Full Set (with kptopt3)
- Electric Field (with q0),
- SC GS IBZ (with kptopt1)
- NSC RF DDK Half Set (with kptopt2, and iscf-3)
- SC RF EF Half Set (with kptopt2)

Lesson 5 Recipe Combinations

- Full dynamical matrix, Dielectric tensor and Born

effective charges - SC GS IBZ (with kptopt1)
- Three NSC RF DDK (one for each direction) Half

Set (with kptopt2, and iscf-3) - SC RF PhononEF Half Set (with kptopt2)
- Phonon at q0 and general q points
- Perturbations at different q wavevectors cannot

be mixed. - SC GS IBZ (with kptopt1)
- Three NSC RF DDK (one for each direction) Half

Set (with kptopt2, and iscf-3) - SC RF PhononEF Half Set (with kptopt2)
- NSC GS kq points (might be reduced due to

symmetries, with kptopt1) - SC RF Phonon q?0 Full Set (with kptopt3)

Lesson 5 Full RF calculation of AlAs

- Three Data Sets
- SC GS IBZ
- NSC RF DDK Half k-point set
- SC RF PhononEF Half k-point set
- New parameters
- rfelfd, getwfk, getddk
- trf1_5.in
- Run
- Output
- Dynamical matrix
- Dielectric tensor
- Effective charge
- Phonon energies

Lesson 5 Multiple q Phonon

- When q?k1-k2,
- NSC GS with nqpt1, qpt, getwfk, getden,

kptopt3, tolwfr, iscf4-2 - SC RF Phonon with rfphon1, rfatpol, rfdir,

nqpt1, qpt, getwfk1, getwfq4, kptopt3,

tolvrs, iscf - Notice splitting between TO and LO

Lesson 6

- Interatomic Force Constants,
- Phonon and Thermodynamic Properties

Lesson 6 DDB File

- DDB contains dE with respect to 3 perturbations

phonons, electric field and stresses - Header DDB version number, natom, nkpt, nsppol,

nsym, ntypat, occopt, and nband or array nband

(nkpt nsppol) if occopt2, acell, amu, ecut,

iscf,... - Data
- Number of data blocks
- For each block Type of the block, Number of

Elements, List of Elements - In most cases, each element consists of 4 integer

and 2 real numbers idir1, ipert1, idir2, ipert2,

Re(2DTE), Im(2DTE) - Symmetries may reduce number of elements
- DDB files can be merged by Mrgddb

Lesson 6 Build DDB

- trf2_1.in has 10 Data Sets
- 1st SC GS IBZ
- 2nd NSC RF DDK Half Set
- 3rd SC RF PhononEF Half Set (q0)
- 4th-10th SC RF Phonon Full Set (q?k)
- Note need to overwrite default parameters in

some Data Sets - Data Sets 4-10 use selected K-Points which may be

generated by trf2_2.in - Run
- See trf2_1o_DS_DDB

Lesson 6 Merging DDB Files

- Steps to use Mrgddb
- name output
- description
- number of DDB files
- file name list
- trf2_3.in
- Run
- Look at trf2_3.ddb.out
- How many datasets?

Lesson 6 ANADDB

- ANADDB analyses DDB for properties e.g. phonon

spectrum, frequency-dependent dielectric tensor,

thermal properties - Files input, output, DDB, other files
- To run anaddb lt anaddb.files gt log
- Common parameters dieflag, elaflag, elphflag,

ifcflag, instrflag, nlflag, piezoflag, polflag,

thmflag

Lesson 6 Interatomic Force Constants

- Dynamical Matrix and Interatomic Force Constants

are Fourier Transforms of each other - Calculated Dynamical Matrix on a grid of

wavevectors ? IFC IFC vanishes rapidly with

interatomic distance - ifcflag1 (ifcflag0 is for checking, or when

there is not enough information in DDB) - Q-point grid brav, nqgpt, nqshft, q1shft
- Energy conservation and charge neutrality asr,

chneut - Others dipdip, ifcana, ifcout, natifc, atifc
- Run ..\..\anaddb lt trf2_4.files gt trf2_4.log

Lesson 6 IFC Results

- On site term of Al trace 0.28080
- First NN 4 As atoms at 4.6 trace -0.06911
- Second NN 12 Al atoms at 7.5 trace 0.00062
- Third NN 12 As atoms at 8.8 trace -0.00037
- Fourth NN 6 Al atoms at 10.6 trace -0.00016
- Fifth NN 12 As atoms at 11.6 trace -0.00056
- Sixth NN Al atoms at 13.0 trace -0.00059
- Applications
- Phonon dispersion curve
- Elastic constants
- MD (Harmonic approximation)

Lesson 6 Phonon Band Structure

- ifcflag1
- Q-point grid brav, nqgpt, nqshft, q1shft
- Energy conservation and charge neutrality asr,

chneut - Others dipdip
- Band eivec, nph1l, qph1l, nph2l, qph2l
- Run ..\..\anaddb lt trf2_5.files gt trf2_5.log
- trf2_5_band2eps.freq .dspl are obtained
- Run ..\..\band2eps lt trf2_6.files gt trf2_6.log
- trf2_6.out.eps is obtained
- view with ghostview note discontinuity of

Optical Phonon at Gamma point - Edit trf2_5_band2eps.freq, lines 1 and 31,

correct LO freq. (trf2_5.out) - Run band2eps again

Lesson 6 Thermodynamic Properties

- Normalized phonon DOS
- Phonon internal energy, free energy, entropy,

constant volume heat capacity as a function of

the temperature - Debye-Waller factors (tensors) for each atom, as

a function of the temperature (DISABLED, SORRY) - Parameters thmflag, ng2qpt, ngrids, q2shft,

nchan, nwchan, thmtol, ntemper, temperinc,

tempermin - ..\..\anaddb lt trf2_7.files gt trf2_7.log

At T F(J/mol-c) E(J/mol-c)

S(J/(mol-c.K)) C(J/(mol-c.K)) (A mol-c is the

abbreviation of a mole-cell, that is, the

number of Avogadro times the atoms in a unit

cell) 20.0 8.1384756E03 8.1463588E03

3.9416450E-01 1.4169102E00 40.0

8.1061319E03 8.2368069E03 3.2668767E00

7.8985027E00 60.0 7.9980215E03

8.4575659E03 7.6590737E00 1.3992227E01

80.0 7.7974376E03 8.7915524E03 1.2426435E01

1.9325165E01 100.0 7.5004823E03

9.2274431E03 1.7269608E01 2.4175005E01

120.0 7.1069991E03 9.7544363E03

2.2061977E01 2.8411187E01 140.0

6.6189292E03 1.0359248E04 2.6716563E01

3.1955266E01 160.0 6.0396228E03

1.1028289E04 3.1179165E01 3.4847422E01

180.0 5.3732225E03 1.1749439E04

3.5423425E01 3.7183863E01 200.0

4.6241912E03 1.2512641E04 3.9442249E01

3.9069447E01

Other Response-Function Tutorials

- Optic Frequency-dependent linear and second

order nonlinear optical response - Frequency dependent linear dielectric tensor
- Frequency dependent second order nonlinear

susceptibility tensor - Electron-Phonon interaction and superconducting

properties of Al. - Phonon linewidths (lifetimes) due to the

electron-phonon interaction - Eliashberg spectral function
- Coupling strength
- McMillan critical temperature
- Elastic and piezoelectric properties.
- Rigid-atom elastic tensor
- Rigid-atom piezoelectric tensor (insulators only)

- Internal strain tensor
- Atomic relaxation corrections to the elastic and

piezoelectric tensor - Static non-linear properties
- Born effective charges
- Dielectric constant
- Proper piezoelectric tensor (clamped and relaxed

ions) - Non-linear optical susceptibilities
- Raman tensor of TO and LO modes

Lesson 7

- Quasi Particle Band Structure

Lesson 7 Introduction

- System
- Nucleus Electrons
- Approach
- Electron wave function
- Electron density ? DFT
- Quasiparticles
- Quasiparticle Bare particle Decorations
- Modify
- Equation of motion
- Energy, Mass
- Life time

Lesson 7 GW Approximation

In the quasiparticle (QP) formalism, the energies

and wavefunctions areobtained by the Dyson

equation

QP equation

S self-energy (a non-local and energy dependent

operator) is the difference between the energies

of bare particle and quasiparticle.

Within the GW approximation,S is given by

GW Self-Energy

Dynamical Screened Interaction

Green Function

Lesson 7 Green function

Green function G corresponding to QP equation is

Green function G may be approximated by the

independent particle G(0)

The basic ingredient of G(0) is the Kohn-Sham

electronic structure

Lesson 7 Dynamical Screened Interaction

W is approximated by RPA

Dynamical Screened Interaction

Coulomb Interaction

Dielectric Matrix

RPA approximation

Independent Particle Polarizability

Adler-Wiser expression

ingredients KS wavefunctions and KS energies

Lesson 7 GWA correction to LDA

QP equation

KS equation

Difference Vxc is replaced by S. Thus GWA

correction to the DFT KS eigenvalues by 1st order

PT

0-order wavefunctions

0-order

Non Self-Consistent G0WRPA, Plasmon Pole model

Lesson 7 GWA Performance

LDA, GWA, and experimental energy gaps for

semiconductors and insulators. GWA corrects

most of the LDA band gap underestimation. The

discrepancy for LiO2 results from the neglect of

excitonic effects. The experimental value for

BAS is tentative.

Lesson 7 Discrepancy of LDA

- In Kohn-Sham theory, eigenvalues ei are Lagrange

multipliers to ensure the orthogonality of KS

orbitals - So both KS eigenvalues and orbitals are not

physical - ei are not energy levels eN (highest level) is

chemical potential for metal or negative

ionization energy for semiconductor and insulator - In absence of quasiparticle calculations. LDA

energy are routinely used to interpreted

experimental spectra - LDA energy dispersions are often in fair

agreement with experiment LDA band gaps are

sometimes empirically adjusted to fit

experimental values - LDA VXC approximate self-energy (neglecting

non-local, energy dependent and life-time

effects) - LDA generally provides a qualitative

understanding.

Lesson 7 GWA Calculation Steps

- SC GS (fixed lattice parameters and atomic

positions) - self-consistent density, potential and Kohn-Sham

eigenvalues and eigenfunctions at relevant

k-points and on a regular grid of k-points - Compute
- susceptibility matrix chi0 and chi, on a regular

grid of q-points, for at least two frequencies

(zero and a pure imaginary frequency a dozen of

eV) - Dielectric matrix epsilon and 1/epsilon
- Compute
- Self-energy sigma at the given k-point, and

derive the GW eigenvalues for the target states

at this k-point

Lesson 7 Generation of KSS File

- tgw_1.in, 3 Data Sets
- First
- nbandkss1 -1 Number of bands in KSS file
- -1 is full diagonalization, see out file for

number of plane waves and number of bands - nband1 9 Number of bands to be computed
- istwfk1 101 Do not use time reversal

symmetry for storing wavefunction - npwkss 0 for same as ecut
- kssform 1 for full diag 3 for conjugated

gradient - symmorphi 0 symmorphic symmetry operations, only

Lesson 7 Generation of SCR File

- Second
- optdriver2 3 Screening calculation
- getkss2 -1 Obtain KSS file from

previous dataset - nband2 17 Bands to be used in the

screening calculation - ecutwfn2 2.1 Cut-off energy of the

planewave set to represent the wavefunctions - ecuteps2 3.6 Cut-off energy of the

planewave set to represent the dielectric matrix - ppmfrq2 16.7 eV Imaginary frequency where

to calculate the screening

Lesson 7 Calculation of Sigma

- Third
- optdriver3 4 Self-Energy calculation
- getkss3 -2 Obtain KSS file from

dataset 1 - getscr3 -1 Obtain SCR file from

previous dataset - nband3 30 Bands for Self-Energy

calculation - ecutwfn3 5.0 Planewaves to represents

wavefunctions - ecutsigx3 6.0 Dimension of the G sum in

Sigma_x - Dimension of Sigma_c size of screening matrix

(SCR file) or size of Sigma_x, whichever is

smaller - nkptgw3 1 num of k-point for GW

correction - kptgw3 k-points, which

must present in KSS file - bdgw3 4 5 calculate GW corrections

for bands - zcut for avoiding divergence in integration

Lesson 7 GWA Output File

- Data Set 1
- Kohn-Sham electronic Structure file
- Note number of plane waves, number of bands
- Check Test on the normalization of the

wavefunctions - Data Set 2
- Check test on the normalization of the

wavefunctions - Is it the same as Data Set 1? Effect of ecutwfk
- total number of electrons per unit cell
- Electron density and plasma frequency
- calculating at frequencies omega eV
- dielectric constant
- Data Set 3
- Band energy E0 ltvxcldagt

Lesson 7 Convergence Study

- Simplify Gamma Point, only
- tgw_2.in Generate KSS and SCR files
- Check Data Sets, KSS is separated from GS
- Note values of ecut
- Run, Check normalization, number of electrons,

dielectric constant

Lesson 7 Sigma ecutwfn convergence

- tgw_3.in ndtset 5, ecutwfn 3.0, ecutwfn 1.0
- Note input KSS and SCR file names
- Rename
- tgw_2o_DS2_KSS to tgw_3o_DS1_KSS
- tgw_2o_DS3_SCR to tgw_3o_DS1_SCR
- Run
- Output
- Num. plane-waves for wave function in Sigma and

Epsilon calculations - Number of electrons per unit cell
- Normalization (grep sum_g)
- Band energies (grep A 2 i E0 ltvxclda)
- If ecutwfn 5.0 is used, what is the error in band

energy?

Lesson 7 ecutsigmax convergence

- tgw_4.in ndtset 7, ecutsigmax 3.0, ecutwfn 1.0

- Note input KSS and SCR file names
- Rename
- tgw_3o_DS1_KSS to tgw_4o_DS1_KSS
- tgw_3o_DS1_SCR to tgw_4o_DS1_SCR
- Run
- Output
- Num. plane-waves for Sigma_x calculations
- Number of electrons per unit cell
- Normalization (grep sum_g)
- Band energies (grep A 2 i E0 ltvxclda)
- What is the appropriate ecutsigmax and the

associated error in band energy?

Lesson 7 Sigma nband convergence

- tgw_5.in ndtset 5, nband 50, nband 50
- Note input KSS and SCR file names
- Rename
- tgw_4o_DS1_KSS to tgw_5o_DS1_KSS
- tgw_4o_DS1_SCR to tgw_5o_DS1_SCR
- Run
- Output
- Num. plane-waves are fixed now
- Numbers of bands for KSS, Sigma and Epsilon
- Number of electrons per unit cell
- Normalization (grep sum_g)
- Band energies (grep A 2 i E0 ltvxclda)
- What is the appropriate nband and the associated

error in band energy?

Lesson 7 1/e ecutwfn convergence

- tgw_6.in ndtset 10, udtset 5 2
- Two steps
- Data Set ?1 calculate screening (1/e)
- Data Set 2 calculate GW correction (Sigma)
- How to prepare KSS and SCR files?
- Run
- Output
- Num. plane-waves for wave function in Sigma and

Epsilon calculations - Dielectric constant
- Normalization (grep sum_g)
- Band energies (grep A 2 i E0 ltvxclda)
- If ecutwfn 4.0 is used, what is the error in band

energy?

Lesson 7 1/e nband convergence

- tgw_7.in ndtset 10, udtset 5 2
- Two steps
- Data Set ?1 calculate screening (1/e)
- Data Set 2 calculate GW correction (Sigma)
- How to prepare KSS and SCR files?
- Run
- Output
- Num. plane-waves for wave function in Sigma and

Epsilon calculations - Dielectric constant
- Normalization (grep sum_g)
- Band energies (grep A 2 i E0 ltvxclda)
- To achieve band energy error lt 0.01 eV, how many

bands must be used?

Lesson 7 ecuteps convergence

- tgw_8.in ndtset 10, udtset 5 2
- Two steps
- Data Set ?1 calculate screening (1/e)
- Data Set 2 calculate GW correction (Sigma)
- How to prepare KSS and SCR files?
- Run
- Output
- Num. plane-waves for wave function in Sigma and

Epsilon calculations - Dimension of epsilon
- Dielectric constant
- Normalization (grep sum_g)
- Band energies (grep A 2 i E0 ltvxclda)
- To achieve band energy error lt 0.01 eV, how large

ecuteps must be used?

Lesson 7 (direct) Egap of Silicon

- Data Set 1 SC GS
- print out the density
- 10 k-points in IBZ 4x4x4 FCC grid (Shifted, no

Gamma point) - Data Set 2 NSC GS
- Kohn-Sham structure, 19 k-points in IBZ but not

shifted, Gamma point included - Data Set 3 Calculate Screening
- Very time-consuming
- ecutwfn 3.6
- nband 25 CPU time ? nband (Conduction)
- Accuracy of GW energy 0.2 eV
- Accuracy of energy difference 0.02 eV
- There is no zero of energy defined for bulk

system - Data Set 4 Self-energy matrix at Gamma

Lesson 7 (direct) Egap of Silicon 2

- What are direct Egap of Silicon by LDA and GWA?
- Choice of pseudopotential can contribute to Egap

variation - Egap GWA accuracy 0.1 eV
- Full band calculation is possible, by shifting

the k-point - Simplification GW corrections are quite linear

with the energy

Thank you

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