Quasi-Newton Methods in DFT

L. D. MarksDepartment of Materials Science & Engineering

Northwestern University

Overview

Underlying Mathematics/Physics

How these are used in Wien2k code

What to do when things go wrong

Some related cases– Surfaces– Phonons (notes, not enough time…)

Health Warning

Some of the folk-lore about these methods in DFT is wrong

There are many details (scaling, trust-region implementation) which are very important

Accuracy of the DFT codes is important – QN methods are good code debuggers

Basic Principles (Minimization)

Two classes of problem– Minimize some function, e.g. the energy

E = E0 + x.G + ½ xTBxG = -Bx

– Solve a fixed-point problem (mixing)(r) = F((r)) Self-Consistent Solution of KS equations“F” is the SCF mapping

Two are nominally the same, except – For a minimization B is positive definite– Normally we insist that the energy improves– These conditions are not appropriate for other fixed point problems

Basic Approaches

Steepest descent– Take a simple step along the gradient (for

minimization) or difference between old and new densities

– Slow and inefficient

Basic Approaches

Conjugant gradient– Insist that steps are approximately orthogonal

with respect to Hessian– Means that there is less redundancy– Much more efficient, but still poor– Good for large problems, but still not optimal– Obsolete, but easy to code

Quasi-Newton Methods

Consider– G=-Bx (or HG = -x)– If we have a number of steps, and x, G at each we have

implicitly some information about B & H– Use this to construct an approximation that is better

than ignoring the prior steps– Provided that the Hessian (or its inverse) is not too

rapidly changing, this is far more efficient– This is the conventional blue-ribbon method

Curvature condition

Suppose we have a gradient at some position x0 , expand this as a Taylor series:G(x) = G(x0 ) – B(x-x0 )

For two positions x1 , x2G(x2 )-G(x1 ) = - B(x2 -x1 )

Defines = x2 -x1 ; y = G(x1 ) –G(x2 )

Curvature ConditionBs = y

So What

There are many possible values of B which satisfy the Curvature Condition – how do we solve?

Key point:– Choose a minimum norm solution that leads to the least

change in B relative to some prior estimate– This involves the smallest possible change in B– Can be considered as the change which is most

conservative in some sense

Curvature ConditionBs = y

Examples

Let Bk be the prior estimate

SR1

(Not implemented: useful for transition states)

BFGS (PORT in Wien2k)

kT

kkk

Tkkkkkk

kk ssBysBysByBB

)())((

1

kTk

Tkk

kkTk

kTkkk

kk syyy

sBsBssBBB

))((1

Breakdown of the Curvature Condition

For minimization, B must be positive definite otherwise the problem will find a maximum and/or fail

Bs = y ; sTBs = sTy

If sTy < 0, B cannot be positive

Example– E=0.6-x2+0.25x4 on rightBlue, E ; Red,

Hence must trap (damp)Reported as a warning in Wien2k

-0.5

0

0.5

1

0 0.5 1 1.5 2

kkkkkk sByy )1(

otherwise )(

)8.0(

2.0 1

kTkkk

Tk

kkTk

kkTkk

Tkk

yssBssBs

sBsys

Fails OK

E

d2E/dx2

How far to go?

Two main methods– Line search: safe but slow– Trust region: most common method

Wolfe Condition– QN methods converge so long as steps are not too

small, need to obtain adequate curvature information– Too small steps will be dominated by noise– Too small steps susceptible to numerical noise from too

few k-points, iterative diagonalization….

General Form used

Total magnitude of step limited by a “Trust Region”; solve quadratic form to minimize the energy with this as a constraint

Broyden’s Methods

Comparable Form for non-linear equations

(r) = F((r)) ; SCF Mapping

sk = k+1 (r)-k (r) ; yk = F(k+1 (r))- F(k (r))

Broyden’s “Good Method”

Broyden’s “Bad Method”k

Tk

Tkkkk

kk ssssByBB )(

1

kTk

Tkkkk

kk yyyyHsHH )(

1

kTk

Tkkkk

kk yssyHsHH )(

1

Why Good & Bad?

Originally the “Good” method worked for Broyden (PRB…) and the “Bad” method did not

For many years the “Bad” method was never used…..except in mixing as DFT developers probably never read the literature

Subtle differenceGood Broyden: finds optimal density with current model

(greedy algorithm)Bad Broyden: finds most conservative density to

minimize residue (least greedy algorithm)

What is a greedy algorithm?

A greedy algorithm takes decisions on the basis of information at hand without worrying about the consequences. In many cases “greed is good”, but not always.

Example: make 41c with 25c, 10c, 4c coins

Optimum solution: 25+4x4

Greedy solution: start with 41c, use largest reduction– 25c Remainder 16– 10c Remainder 6– 4c Remainder 2

An issue for fixed-point problems

Up to now, all the methods are somewhat causal, appropriate for a minimization where the energy is decreasing

This does not need to be the case for a fixed-point problem as the gradient is not available, instead a residual

There is nothing a-priori to say that point 2 should be better than point 1

Hence….

Multisecant Approach

Consider a number of values:

S = (s0 ,s1 ,….sn ) ; Y=(y0 ,y1 ,….yn )

Expand to a simultaneous solution:

BS = Y ; or HY=S

Minimum-Norm Solution (MSEC)

Take Hk =I

Tkk

Tkkkkkk YYYYHSHH 1

1 ))((

Regularization

What is (Yk

TYk )-1 ?UsePenrose-Moore pseudo inverse

Use a constant – Works “OK” in MSEC1/MSEC3

Generalized Cross-Validation (MSR1) used to find

11)( Tk

Tkk

Tk YYIYY

Why Regularize

At point 3, with information from 1 & 2

Small eigenvalues along the poorly defined direction are not well known

Regularization damps the effect of small eigenvalues (similar to Wiener filter)

Unregularized problems tend to be unstable and/or oscillate badly

Unpredicted Step

General Form

Control the magnitude of the unpredicted step (Greed) – used to be called “mixing factor” but I urge you not to use this term!

Trust region automatically adjusts it

))((

))(( 1

FH

YIYYYSIH Tkk

Tkkk

))(}()({

))(()(1

1

FYIYYYI

FYIYYST

kkT

kk

Tkk

Tkk Predicted

Unpredicted

Scaling

Wien2k has several different basis set components– Radial mesh within RMT– Plane wave outside– Density Matrix/Orbital Potentials– Atomic Positions/Gradients (MSR1a)

Need to approximately scale – effects the regularization

Use L2 normalization (sum of squares) for all residuals

Natural scaling for positions (mau) and gradients (mRyd/au)

Trap small gradients to prevent runaway

May not be optimal…research topic

MSECa

Nominally “simple”

sk = (k (r)-k

(r), xk -xk-1 )

yk = (F(k (r))- F(k

(r)), Gk-1 (x1 )-Gk (x2 ))

Insert directly into multisecant form… but it does not really work

MSR1a

What works

kk

kkTk

Tkk

Tkkkk

SY

YSA

AYAYSIH

11 ))((

Good Broyden, Otherwise Bad Broyden

Regularize (pseudo-inverse)

Note: this is a greedier algorithm

Some development continuing…

kTk YA

PORT

Free

Reverse Communication

Conventional BFGS

Adaptive Trust-Region

Very Stable

Traps overlapping RMT

Sensitive to numerical errors (i.e. can annoy PB)

Total energies and atomic forces (Yu et al.; Kohler et al.)

Total Energy:

Electrostatic energy

Kinetic energy

XC-energy

Force on atom

Hellmann-Feynman-force

Pulay corrections

Core

Valence

expensive, contains a summation of matrix elements over all occupied states

KiKKKi

KKii

ikivaleffval

effcorecore

mm

esm

rHF

HKKidSrrK

KcKcnrdrrVF

rdrVrF

rYrr

rVZF

)()()()(

)()()()(

)()(

)ˆ()(lim

*2

,

*

,

1

1

1

1

0

)()(][

)()(][

)(21)()(2

1][

3

3

3

rrrdE

rVrrdnT

rVZrVrrdU

xcxc

effi ii

eses

valcoreHF

tot FFFRd

dEF

Optimization of internal parameters using “forces”

Forces only for “free” structural parameters:

NaCl: (0,0,0), (0.5,0.5,0.5) : all positions fixed by symmetry

TiO2 : Ti (0,0,0), O (u,u,0): one free parameter (u,x,y,z)

Forces are only calculated when using “-fc”:

run_lapw –fc 1.0 (mRy/bohr)

grep :for002 case.scf

200.

-130.

140.

135 only FHF + Fcore

120

122 forces converging

121

changes “TOT” to “FOR” in case.in2

-12.3 FHF + Fcore + Fval , only this last number is correct

Forces are useful for

structural optimization (of internal parameters)

phonons

Structural optimization of internal parameters

/home/pblaha/tio2> min_lapw -h

OPTIONS:

-p -> does a k-point parallel calculation

-it -> use iterative diagonalization

-sp -> does a spin-polarized calculation (runsp_lapw)

-NI -> without initialization of input-files (continue after a “crash”)

-i NUMBER -> max. NUMBER (50) of structure changes

-j JOB -> job-file JOB (run_lapw -I -fc 1. -i 40)

CONTROL FILES:

.minstop stop after next structure change

tio2.inM (generated automatically by “pairhess” at first call of min_lapw)

PORT 2.0 #(NEW1, NOSE, MOLD, tolf (a4,f5.2))

0.0 1.0 1.0 1.0 # Atom1 (0 will constrain a coordinate)

1.0 1.0 1.0 1.0 # Atom2 (NEW1: 1,2,3:delta_i, 4:eta (1=MOLD, damping))

monitor minimization in file case.scf_mini

contains last iteration of each geometry step

each step N is saved as case_N.scf (overwritten with next min_lapw !)

Optimization of atomic posistions (E-minimization via forces)

• damped Newton mechanics scheme (NEW1: with variable step)

• quite efficient quasi-Newton (PORT) scheme• minimizes E (using forces as gradients) • If minimizations gets stuck or oscillates: (because E and Fi are inconsistent):

• touch .minstop; min –nohess (or rm case.tmpM .min_hess)• improve scf-convergence (-ec), Rkmax, k-mesh, …• change to NEW1 scheme (LDM not so sure…)

W impurity in Bi (2x2x2 supercell: Bi15 W)

0 2 4 6 8 10 12 14

-40

-20

0

20

40

60

for01 for04x for04z for06x for06z

forc

es (m

Ry/

a 0)

time step

0 2 4 6 8 10 12 14

-679412.54

-679412.52

-679412.50

-679412.48

-679412.46

-679412.44

Ene

rgy

(Ry)

tim e step

0 2 4 6 8 10 12 14

-0.04

-0.02

0.00

0.02

0.04

pos01 pos04x pos04z pos06

posi

tion

time step

0 2 4 6 8 10 12 14

-4

-2

0

2

4

6

8

EFG

(1021

V/m

2 )

time step

Energy

Forces

PositionsEFG

Some practical details (PORT)

Setup the problem “adequately”

Run “x pairhess” and copy .min_pair .minrestart (done automatically)

Check the frequencies – do they look reasonable (do not be too precise)– Yes – go ahead– No – edit case.inpair

Determine how to minimize (sp, not, LDA+U?)

Either use in-line commands or a separate executable (e.g. Job.csh)

Run min_lapw

case.outputM

grep -e :D *tM:DD7DOG Newton Step, radius 0.561E-01:DD7DOG Dogleg Step 0.263 Radius 0.561E-01:DD7DOG Relaxed Newton 0.881 Radius 0.561E-01:DD7DOG Relaxed Newton 0.777 Radius 0.150E+00:DD7DOG Relaxed Newton 0.730 Radius 0.150E+00:DD7DOG Relaxed Newton 0.135 Radius 0.150E+00:DD7DOG Newton Step, radius 0.879E-01:DD7DOG Newton Step, radius 0.594E-01:DD7DOG Newton Step, radius 0.434E-01

Pairhess (MgO + H2 O)x pairhesscase.inM present and used for constrainsAverage Hessian Eigenvalue 140.0 mRyd/au^2, Frequency 489.80 cm-1Min & Max of Eigenvalues, mRyd/au^2 16.0 520.1Min & Max frequencies, cm-1 79 2353Check .minpair, the estimate, and output in /home/ldm/Nat/Nat.outputpairPairHess END

tail *.outputpair********* Eigenmode Listing ********* Eigenvalue 1 16.0 mRyd/au^2 79 cm-1Eigenvalue 2 35.0 mRyd/au^2 96 cm-1Eigenvalue 3 38.0 mRyd/au^2 99 cm-1….Eigenvalue 22 377.3 mRyd/au^2 1698 cm-1Eigenvalue 23 478.4 mRyd/au^2 2031 cm-1Eigenvalue 24 520.1 mRyd/au^2 2353 cm-1

After initial refinement

Increase (as appropriate)– RKMAX– Change RMT (use clminter)– Change number of k-points

cp .min_hess .minrestart– Re-initializes approximate Hessian (can make it

twice as fast)

Redo min_lapw as before

Post Analysis (eigenhess)Eigenhess code to analyze .min_hess data

It currently:1) If you change case.inM, it will rewrite an appropriate .minpair

(copy to .minrestart and .min_hess then run min -I)2) It will output the eigenvectors/values of the BFGS approximation3) Will create a file case.struct.cif with pseudo-thermal ellipsoids

(a1g symmetry allowed modes) if CIF is in case.incon4) Will add to this cif files the first 9 energy eigenvectors of the

thermal displacements if CIF is in case.incon5) Will output the current BFGS Hessian in case.hess6) Will output in case.struct.xyz the first 9 energy eigenvectors

if XYZ is in case.incon. The number of cells output in this caseis determined by three integers i1,i2,i3 where each is the numberof repeats along the respective axes (default none)

7) For XYZ, will shift by 1/2,1/2,1/2 if SHIFT is specified, similarlywith XSHIFT, YSHIFT and ZSHIFT

8) For XYZ, will print in case.v.xyz the vibrations9) If ALLM is specified, will output all the eigenmodes (files may be large)Note: the xyz files can be read by Jmol and some other viewers

Practical Details (MSEC)

In 99% of cases, do not adjust anything

At most, reduce Greed to 0.1 or 0.05 – not further– Too small a Greed, algorithm will starve to

death– Too large a Greed, algorithm can go unstable– But…..this is controlled within the code

Things to look at

:DIS : CHARGE DISTANCE ( 0.0018343 for atom 1 spin 1) 0.0003262:PLANE: INTERSTITIAL TOTAL 9.00853 DISTAN 1.013E-01 % :CHARG: CLM CHARGE /ATOM 18.62575 DISTAN 2.758E-03 % :DIRA : |BROYD|= 1.217E-02 |PRATT|= 1.252E-03 ANGLE= 86.9 DEGREES:DIRP : |BROYD|= 1.775E-02 |PRATT|= 1.554E-03 ANGLE= 87.2 DEGREES:DIRB : |BROYD|= 1.822E-02 |PRATT|= 2.518E-03 ANGLE= 85.9 DEGREES:FRMS (mRyd/au) 1.606 :DRMS (au) 1.944E-03 :MAX (au) 4.490E-03:MIX : MSR1a REGULARIZATION: 3.50E-04 GREED: 0.121:ENE : ********** TOTAL ENERGY IN Ry = -135343.41926326

Practical Details (MSR1a)

Preconverge– PB, well– LDM, not so well

Edit case.inm (MSR1a in first line)

Run as normal (but check)

Perhaps use (see README.doc)echo 0.05 > .prattecho 0.10 > .msec

Performance

Typical convergence for an insulator

y = 0.4196e-0.0404x

1.E-04

1.E-03

1.E-02

1.E-01

1.E+000 50 100 150 200

Iteration

E-E l

ow

:WARN

Printed when:– Algorithm reduces the step as it is considered to be too

greedy– When the step is reduced as the RMT’s would overlap

Only the second is a concern, and then only if it occurs many times

Note: the algorithm (PORT too) can stagnate if the RMTs are too large (they are not as clever as you…)

Also, MSR1a is sensitive to how well posed is the physics (research ongoing), a good debugger…

Convergence of QN methods

Depends upon clustering of eigenvectors

Rationalization

E = -Gx + 1/2xTBx ; pick direction dE(d) = -(G.d) + 2/2{dTBd}

= (G.d)/{dTBd}

If B has eigenvalues/vectors i , vid= i vi ; dTBd = i i ; dominated by large iQN methods optimize largest eigenvalues first, then

smallestSimilar eigenvalues are optimized in parallelHolds (probably) for MSEC as well (hard to prove)

For a DFT problem

Energy minimization– Elastic Band Structure (similar to phonons, but no

mass)– Similar to phonons, only a limited number of bands

(depends on number of atom-atom bond types, not so much on number of atoms)

Fixed-Point Density Problem– Vertical Dielectric Band Structure– Will scale as number of distinctly different electronic

environments – similar to number of different atom types

Note:

If the “bonding” changes during optimization, H changes so the prior information is not so good

Metals have a richer Quasi-DBS, so may converge more slowly

Often need more k-points (math of this is not well developed)

Soft-modes converge last…be careful

If something goes wrong…

In most cases this is because:– The problem is poorly posed– Too few k-points– The structure is very far from equilibrium

So– Increase k-points– Maybe increase T (TEMPS), sometimes does

not work– Think

What not to do

Do not reduce the Greed below 0.05 (probably do not change it, or maybe 0.1)

MSEC/MSR1 are not the same as what is in other mixing algorithms in the literature, they are quite different

Supercells

(0,0,0) P 8 atoms (0,0,0) (.5,0,0) (.5,.5,0) (.5,.5,.5)(0,.5,0) (.5,0,.5)(0,0,.5) (0,.5,.5)

B 4 atoms yes yes no noF 2 atoms yes no no yes

4x4x4 supercells: P (64), B (32), F (16) atoms

supercells (1

2 atoms)

2x2x2 = 8 atoms

22 x

Supercells

Program „supercell“:

start with „small“ struct file

specify number of repetitions in x,y,z (only integers, e.g. 2x2x1)

specify P, B or F lattice

add „vacuum“ for surface slabs (only (001) indexed surfaces)

shift all atoms in cell

You must break symmetry!!!

replace (impurities, vacancies) or

displace (phonons) or

label (core-holes, specific magnetic order; change “Fe” to “Fe1”; this tells the symmetry-programs that Fe1 is NOT a Fe atom!!)

at least 1 atom

At present „supercell“ works only along unit-cell axes!!!

Structeditor (by R.Laskowski)

requires octave (matlab) and opendx (visualization)

allows complex operations on struct-files

Other methods

Cryscon (Shape Software), shareware, allows general transformation

Many, many other codes will handle CIF files, and then convert back using cif2struct

Excel (simple manipulations)

Note: x struct2cif converts the other direction (adequately)

Needed if you publish a structure (proper is to deposit this type of information; now required by Surface Science)

Calculating surfaces

n

Choose the functional (LDA, PBE, WC)n

Find the DFT equilibrium volume for the bulk, with similar RMT, RKMAX (may have to redo later)

n

Create the structure – supercell or other (e.g. Cryscon from Shape Software)

n

Look at it – you probably did something wrong!n

By hand (or using bond-valence sums) adjustn

Use the same density of k-points in reciprocal spacen

Avoid 1 k-point at gamma – often not very accuraten

Use a “big enough” cell

TiO2 c(2x2)

Electronic Entropy cannot be neglectedShifts O2 chemical potential by >1eVDominant stabilization term

PRL 100, 86102 (2008)

Comment: Constraints

n

Personally, I consider this a very bad idea unless there is a really good reason– Wien2k exploits symmetry, so use centro-symmetric

cells and do not terminate artificially– It is hard to ensure that this does not introduce more

artifacts– The problem now becomes a constrained optimization,

and these converge worse so in fact it will probably take longer even though there may be less movable atoms

Be careful

n

All software has bugs – by definitionn

Many surface calculations are at the limit of what can be done

n

Few computer codes will tell you “don’t do that you stupid #!?” – GIGO

n

Never assume theory (or experiment) is correctn

Think about what you are doing as computer experiments

n

Think science, not just typing at a terminal

The DFT is converged…publish?Alas, No

Some solutions are more equal than others

n

How do we know the model is correct?

Frequently models are proposed for surface structures, not rigorous crystallographic solutions

All models will refine to an improved fit, but this does not prove they are right

Only a few techniques can search configurational space to find the best fit

It is rarely plausible to test with DFT calculations all possible models

How accurate are DFT results?

n

DFT is an approximate theory, and the functionals used are far from perfect– How good are the atomic positions?– How good is the surface charge density?– How good is the relative positions of bands?– How good is the energy – exchange correllation

terms?

Errors in Energies

n

Calibrate your DFT– Benchmark energies for a know case using

more than one functional– Calculate energies for the surface using the

same functionals– Error in energies can be estimated– Typical numbers are 0.05-0.1 eV/1x1 surface

cell

Perovskite Surfaces

~43,000 articles on perovskites– ~21,000 on strontium titanate

637 articles on perovskite surface structures– 571 articles say what reconstruction is present

Almost none provide crystallographic solutions to surface structures

58

(001) (110) (111)

2x12x22x2 c4x2 c6x2 √5x√5√13x√13

3x14x15x16x11x4

3x34x45x56x69x9

SrTiO3 reconstructions

2x1

c4x2

Previously solvedTodaySoon

SrTiO3 (110): 900C in O2

40 nm

001

1x1

60

N. Erdman, PhD Thesis, 2002

Dark Field

1000 °C in flowing O2

DP’s from Arun

Subramanian

_(110)

(001)

1x1

61

3

1

Direct Methods

In diffraction experiments we measure intensities (phase information is lost):

If the phases were known we have scattering potential maps (electron diffraction)

Use Genetic Algorithm to search phase space and assign phases to beams Feasible

Solution

FT

FT-1

Fourier spaceconstraints (S2 )

Real spaceconstraints (S1 )

Observed Intensities(assigned phases)

(Genetic Algorithm)

Recovery Criterion

YES

NO

2)()( kFkI

))(exp()(.. kikFFS

L. D. Marks, W. Sinkler, and E. Landree, Acta Crystallogr. Sect. A 55, 601 (1999); L. D. Marks, Phys. Rev. B 60, 2771 (1999); L. D. Marks et al., Surf. Rev. Lett. 5, 1087 (1998).

Direct Methods Solution

63

Atomic Positions Refined

64

SrTiO3 (110) 3x1

TiO2 overall surface stoichiometry– Ti5 O7 atop O2 termination– Ti5 O13 atop SrTiO termination

Surface composed of corner sharing TiO4 tetrahedra– Arranged in rings of 6 or 8 tetrahedra– 4 corner share with bulk octahedra– 1 edge shares with bulk octahedron

Enterkin et. al., Nature Materials, 2010

Blue polyhedra are surface polyhedra, gold are bulk octahedra, 

orange spheres Sr, blue spheres Ti, red spheres O

SrTiO3 (110) nx1

∞x1

66

2x1

3x1

4x1

5x1

6x1

Expansion to other nx1 by changing the number of TiO4 surface tetrahedra per ring

Intergrowths

STM with simulations (DFT)

Enterkin et. al., Nature Materials, 2010

4x1

3x1 3x1

6x15x16x1

4x14x15x1

Pauling’s Rules

1. A coordinated polyhedron of anions is formed about each cation, the cation-anion distance determined by the sum of ionic radii and the coordination number (C.N.) by the radius ratio.

2. An ionic structure will be stable to the extent that the sum of the strengths of the electrostatic bonds that reach an anion equal the charge on that anion.

The principles determining the structure of complex ionic crystals. J. Am. Chem. Soc. 51 (4): 1010–1026. 1929.

STO (110) Bond Valence SumsBond Valence Sr Half-O2 x1 6x1 5x1 4x1 3x1 2x1

Surf Ti(Sr) 1.88 3.83 3.41 3.87 3.93 3.99 4.06 4.02

Surf O 1.49- 1.94- 1.21- 1.88- 1.95- 1.99- 2.05- 2.07-

Sub-surf O 2.61- 1.88- 2.07- 2.02- 2.04- 2.05- 2.05- 1.94-Surface 

Energy(eV per 

1x1 

unit)

X: TiO2

at Surface

SrO monolayerat Surface

TPSSh

Bulk: Sr2.1+Ti 4.1+O32.1-

Oxide Surfaces

God made the bulk; the surface was invented by the devil

Wolfgang Pauli

Maybe not so hard for oxide surfaces…but be careful!

Calculations of Phonons: The Direct Method

WIEN2k + Phonon

http://wolf.ifj.edu.pl/phonon/

Copyright by K.Parlinski

alternatively use D.Alfe`s PHON code +W2P-interface from G.Madsen(see www.wien2k.at/unsupported)

Supercell dynamical matrix. Exact wave vectors.

Conventional dynamical matrix:

Supercell dynamical matrix:

These two matrices are equal if

• interaction range is confined to interior of supercell (supercell is big enough)• wave vector is commensurate with the supercell and fulfils the condition (independent of interaction range):

At wave vectors ks the phonon frequencies are “exact”,provided the supercell contains the complete list ofneighbors. Wave vectors ks are commensurate with the supercell size.

1x1x1 2x2x2 3x3x3Exact wave vectors

X M

Exact: Exact:

X, M, R

Exact:

Phonon dispersions + density of states

Total + Germanium Total + Oxygen

GeO2 P4_2/mnm

Wave vector

Frequency

Thermodynamic functions of phonon vibrations

Internal energy:

Free energy:

Entropy:

Heat capacity Cv :

Thermal displacements:

PHONON-I

PHONON

by K.Parlinski (Crakow)

runs under MS-windows

uses a „direct“ method to calculate Force- constants with the help of an ab initio program

with these Force- constants phonons at arbitrary k-points can be obtained

Define your spacegroup

Define all atoms

http://wolf.ifj.edu.pl/phonon/

Phonons:

selects symmetry adapted atomic displacements (4 displacements in cubic perovskites)

(Displacement pattern for cubic perovskite)

select a supercell: (eg. 2x2x2 atom P-type cell)

calculate all forces for these displacements with high accuracy(WIEN2k)

force constants between all atoms in the supercell

dynamical matrix for arbitrary q-vectors

phonon-dispersion (“bandstructure”) using PHONON (K.Parlinski)

PHONON-II

Define an interaction range (supercell)

create displacement file

transfer case.d45 to Unix

Calculate forces for all required displacements

init_phonon_lapw

for each displacement a case_XX.struct file is generated in an extra directory

runs nn and lets you define RMT values like:

1.85 1-16

• init_lapw: either without symmetry (and then copies this setup to all case_XX) or with symmetry (must run init_lapw for all case_XX) (Do NOT use SGROUP)

• run_phonon: run_lapw –fc 0.1 –i 40 for each case_XX

PHONON-III

analyze_phonon_lapw

reads the forces of the scf runs

generates „Hellman-Feynman“ file case.dat and a „symmetrized HF- file case.dsy (when you have displacements in both directions)

check quality of forces:

sum Fx should be small (0)

abs(Fx ) should be similar for +/- displacements

transfer case.dat (dsy) to Windows

Import HF files to PHONON

Calculate force constants

Calculate phonons, analyze phonons eigenmodes, thermodynamic functions

Applications:

phonon frequencies (compare with IR, raman, neutrons)

identify dynamically unstable structures, describe phase transitions, find more stable (low T) phases.