density functional theory in the solid state -...

Density functional theory in the solid state

Ari P Seitsonen

IMPMC, CNRS & Universités 6 et 7 Paris, IPGP

École thematique calcul/RMN

Density functional theoryBloch theorem / supercells

Plane wave basis setPseudo potentials

MotivationHistoryKohn-Sham method

Summary

1 Density functional theoryMotivationHistoryKohn-Sham method

2 Bloch theorem / supercells

3 Plane wave basis set

4 Pseudo potentials

DFT in the solid state École thematique calcul/RMN 2 / 99




Motivation: Why use DFT?

Explicit inclusion of electronic structurePredictable accuracy (unlike fitted/empirical approaches)Knowledge of the electron structure can be used for theanalysis; many observables can be obtained directly

Preferable scaling compared to many quantum chemistrymethods





History of DFT — I

There were already methods in the early 20th centuryThomas-Fermi-methodHartree-Fock-method





History of DFT — II

Walter Kohn





History of DFT — III: Foundations





Hohenberg-Kohn theorems: Theorem I

Given a potential, one obtains the wave functions viaSchrödinger equation:

V (r) ⇒ ψi (r)

The density is the probability distribution of the wavefunctions:

n (r) =∑

i

|ψi (r)|2

ThusV (r) ⇒ ψi (r) ⇒ n (r)





Hohenberg-Kohn theorems: Theorem I

TheoremThe potential, and hence also the total energy, is a uniquefunctional of the electron density n(r)

ThusV (r) ⇒ ψi (r) ⇒ n (r) ⇒ V (r)

The electron density can be used to determine all properties ofa system





Hohenberg-Kohn theorems: Theorem II

TheoremThe total energy is variational: In the ground state the totalenergy is minimised

ThusE [n] ≥ E [nGS]





History of DFT — IV: Foundations





History of DFT — V: The reward

. . . in 1998:





Kohn-Sham method: Total energy

Let us write the total energy as:

Etot[n] = Ekin[n]+Eext[n]+EH[n]+Exc[n]

Ekin[n] = QM kinetic energy of electronsEext[n] = energy due to external potential (usually ions)EH[n] = classical Hartree repultion (e− − e−)Exc[n] = exchange-correlation energy





Kohn-Sham method: Noninteracting electrons

To solve the many-body Schrödinger equation as such is anunformidable task

Let us write the many-body wave function as a determinantof single-particle equationsThen kinetic energy of electrons becomes

Ekin,s =∑

i

−12

fi⟨ψi (r) | ∇2 | ψi (r)

⟩

fi = occupation of orbital i (with spin-degeneracy included)





Kohn-Sham method: External energy

Energy due to external potential; usually Vext =∑

I − ZI|r−RI |

Eext =

∫rn (r) Vext (r) dr

n (r) =∑

i

fi |ψi (r)|2





Kohn-Sham method: Hartree energy

Classical electron-electron repulsion

EH =12

∫r

∫r′

n (r) n (r′)|r − r′| dr′ dr

=12

∫rn (r) VH (r) dr

VH (r) =

∫r′

n (r′)|r − r′| dr′





Kohn-Sham method: Exchange-correlation energy

The remaining component: Many-body complicationscombined

=⇒ Will be discussed later





Total energy expression

Kohn-Sham (total1) energy:

EKS[n] =∑

i

−12

fi⟨ψi | ∇2 | ψi

⟩+


+12

∫r

∫r′

n (r) n (r′)|r − r′| dr′ dr + Exc

1without ion-ion interactionDFT in the solid state École thematique calcul/RMN 17 / 99




Kohn-Sham equations

Vary the Kohn-Sham energy EKS with respect to ψ∗j (r′′): δEKS

δψ∗j (r′′)

⇒ Kohn-Sham equations

−1

2∇2 + VKS (r)

ψi (r) = εiψi (r)

n (r) =∑

i

fi |ψi (r)|2

VKS (r) = Vext (r) + VH (r) + Vxc (r)

Vxc (r) = δExcδn(r)





Kohn-Sham equations: Notes

−1

2∇2 + VKS (r)

ψi (r) = εiψi (r) ; n (r) =

∑i

fi |ψi (r)|2

Equation looking like Schrödinger equationThe Kohn-Sham potential, however, depends on densityThe equations are coupled and highly non-linear⇒ Self-consistent solution requiredεi and ψi are in principle only help variables (only εHOMOhas a meaning)The potential VKS is localThe scheme is in principle exact





Kohn-Sham equations: Self-consistency

1 Generate a starting density ninit

2 Generate the Kohn-Sham potential ⇒ V initKS

3 Solve the Kohn-Sham equations ⇒ ψiniti

4 New density n1

5 Kohn-Sham potential V 1KS

6 Kohn-Sham orbitals ⇒ ψ1i

7 Density n2

8 . . .

. . . until self-consistency is achieved (to required precision)





Kohn-Sham equations: Self-consistency

Usually the density coming out from the wave functions ismixed with the previous ones, in order to improveconvergenceIn metals fractional occupations numbers are necessaryThe required accuracy in self-consistency depends on theobservable and the expected





Kohn-Sham energy: Alternative expression

Take the Kohn-Sham equation, multiply from the left withfiψ∗

i and integrate:

−12

fi∫

rψi (r)∇2ψi (r) dr + fi

∫rVKS (r) |ψi (r)|2 dr = fiεi

Sum over i and substitute into the expression forKohn-Sham energy:

EKS[n] =∑

i

fiεi − EH + Exc −∫

rn (r) Vxcdr





Exchange-correlation functional

The Kohn-Sham scheme is in principle exact — however,the exchange-correlation energy functional is not knownexplicitlyExchange is known exactly, however its evaluation isusually very time-consuming, and often the accuracy is notimproved over simpler approximations (due toerror-cancellations in the latter group)Many exact properties, like high/low-density limits, scalingrules etc are knownFamous approximations:

Local density approximation, LDAGeneralised gradient approximations, GGAHybrid functionals. . .





Exchange-correlation functional: LDA

Local density approximation:Use the exchange-correlation energy functional forhomogeneous electron gas at each point of space:Exc

∫r n (r) eheg

xc [n(r)]drWorks surprisingly well even in inhomogeneous electronsystems, thanks fulfillment of certain sum rules

Energy differences over-bound: Cohesion, dissociation,adsorption energiesLattice constants somewhat (1-3 %) too small, bulk moduliitoo largeAsymptotic potential decays exponentially outside chargedistribution, leads to too weak binding energies for theelectrons, eg. ionisation potentialsContains self-interaction in single-particle case





Exchange-correlation functional: GGA

Generalised gradient approximation:The gradient expansion of the exchange-correlation energydoes not improve results; sometimes leads to divergenciesThus a more general approach is taken, and there is roomfor several forms of GGA: Exc

∫r n (r) exc[n(r), |∇n(r)|2]dr

Works reasonably well, again fulfilling certain sum rulesEnergy differences are improvedLattice constants somewhat, 1-3 % too large, bulk moduliitoo smallContains self-interaction in single-particle caseAgain exponential asymptotic decay in potential⇒ Negative ions normally not boundUsually the best compromise between speed and accuracyin large systems





Exchange-correlation functional: Hybrid functionals

Hybrid functionalsInclude partially the exact (Hartree-Fock) exchange:Exc αEHF + (1 − α)EGGA

x + EGGAc ; again many variants

Works in general wellEnergy differences are still improvedVibrational properties slightlyImproved magnetic moments in some systemsPartial improvement in asymptotic formUsually the best accuracy if the computation burden can behandledCalculations for crystals appearing





Exchange-correlation functional: Observations

The accuracy can not be systematically improved!van der Waals interactions still a problem (tailoredapproximations in sight)Most widely used parametrisations:

GGAPerdew-Burke-Ernzerhof, PBE (1996); among phycisistsBeck–Lee-Yang-Parr, BLYP (early 1990’s); among chemists

Hybrid functionalsPBE0; among phycisistsB3LYP; among chemists




Summary

1 Density functional theory



4 Pseudo potentials




Periodic systems

In realistic systems there are ≈ 1020 atoms in cubicmillimetre — unformidable to treat by any numericalmethodAt this scale the systems are often repeating (crystals). . . or the observable is localised and the system can bemade periodicChoices: Periodic boundary conditions or isolated(saturated) cluster




Periodic systems




Periodic systems

Is it possible to replace the summation over translations L witha modulation?

Bloch’s theoremFor a periodic potential V (r + L) = V (r) the eigenfunctionscan be written in the form

ψi (r) = eik·ruik (r) ,

uik (r + L) = uik (r)




Periodic systems: Reciprocal space

Reciprocal lattice vectors:

b1 = 2πa2 × a3

a1 · a2 × a3

b2 = 2πa3 × a1

a2 · a3 × a1

b3 = 2πa1 × a2

a3 · a1 × a2




Periodic systems: Brillouin zone

First Brillouin zone: Part of space closer to the origin thanto any integer multiple of the reciprocal lattice vectors,K′ = n1b1 + n2b2 + n3b3




Integration over reciprocal space

Thus the summation over infinite number of translationsbecomes an integral over the first Brillouin zone:

∞∑L

⇒∫

k∈1.BZdk

In practise the integral is replaced by a weighted sum ofdiscrete points: ∫

kdk ≈

∑k

wk

Thus eg.n (r) =

∑k

wk∑

i

fik |ψik (r)|2




Periodic systems: Dispersion




Band structure: Example Pb/Cu(111)

Photoemission vs DFT calculations for a free-standing layer

Felix Baumberger, Anna Tamai, Matthias Muntwiler, Thomas Greber and Jürg

Osterwalder; Surface Science 532-535 (2003) 82-86

doi:10 1016/S0039-6028(03)00129-8DFT in the solid state École thematique calcul/RMN 36 / 99



Monkhorst-Pack algorithmApproximate the integral with an equidistance grid of kvectors with identical weight:

n =2p − q − 1

2q, p = 1 . . .q

kijk = n1b1 + n2b2 + n3b3




Symmetry operations

If the atoms are related by symmetry operation S(Sψ (r) = ψ (Sr)) the integration over the whole 1stBrillouin zone can be reduced into the irreducible Brillouinzone, IBZ

Sψik (r) = ψik (Sr) = eik·Sruik (Sr) = eik′·ruik′ (r) , k′ = S−1k

∫k

dk ≈∑

k∈BZ

wk =∑

k∈IBZ

∑S

w ′Sk




Irreducible Brillouin zone: Examples




Doubling the unit cell




Doubling the unit cell (super-cells)

If one doubles the unit cell in one direction, it is enough totake only half of the k points in the corresponding directionin the reciprocal spaceAnd has to be careful when comparing energies in cellswith different size unless either equivalent sampling of kpoints is used or one is converged in the total energy inboth cases




Basics of plane wave basis setOperatorsEnergy terms in plane wave basis set

Summary



3 Plane wave basis setBasics of plane wave basis setOperatorsEnergy terms in plane wave basis set

4 Pseudo potentials





Kohn–Sham method

The ground state energy is obtained as the solution of aconstrained minimisation of the Kohn-Sham energy:

minΦ

EKS[Φi(r)]

∫Φ

i (r)Φj(r)dr = δij





Expansion using a basis set

For practical purposes it is necessary to expand theKohn-Sham orbitals using a set of basis functionsBasis set ϕα(r)M

α=1

Usually a linear expansion

ψi(r) =M∑α=1

cαiϕα(r)





Plane waves

PhilosophyAssemblies of atoms are slight distortions to free electrons

ϕα(r) =1√Ω

eiGα·r

(. . . = cos(Gα · r) + i sin(Gα · r))+ orthogonal+ independent of atomic positions+ no BSSE± naturally periodic– many functions needed





Computational box

Box matrix : h = [a1,a2,a3]

Box volume : Ω = det hDFT in the solid state École thematique calcul/RMN 46 / 99




Lattice vectors

Direct lattice h = [a1,a2,a3]

Translations in direct lattice: L = i · a1 + j · a2 + k · a3

Reciprocal lattice 2π(ht)−1 = [b1,b2,b3]

bi · aj = 2πδij

Reciprocal lattice vectors : G = i · b1 + j · b2 + k · b3





Expansion of Kohn-Sham orbitals

Plane wave expansion

ψik(r) =∑

G

cik(G)ei(k+G)·r

To be solved: Coefficients cik(G)

Different routes:Direct optimisation of total energyIterative diagonalisation/minimisation





Dependence on position

Translation:φ(r) −→ φ(r − RI)

φ(r − RI) =∑

G

φ(G)eiG·(r−RI)

=∑

G

(φ(G)eiG·r

)e−iG·RI

Structure Factor:SI(G) = e−iG·RI

Derivatives:

∂φ(r − RI)

∂RI,s= −i

∑G

Gs

(φ(G)eiG·r

)SI(G)





Atom-centred functions

Often there appear atom-centred functions

φl(|r − RI |)Ylm

(r − RI

)They can be efficiently calculated as

φl(|r − RI |)Ylm

(r − RI

)=

∑g

SI (g)φl(g)Ylm(g)

where φ(g) is the Bessel transform

φ(g) = 4π(−i)l∫ ∞

0r2φ(r) jl(gr) dr

jl = spherical Bessel functions of the first kind





Plane waves: Kinetic energy

Kinetic energy operator in the plane wave basis:

−12∇2ϕG(r) = −1

2(iG)2 1√

ΩeiG·r =

12

G2ϕG(r)

Thus the operator is diagonal in the plane wave basis set

Ekin(G) =12

G2





Cutoff: Finite basis set

Choose all basis functions intothe basis set that fulfill

12

G2 ≤ Ecut

— a cut-off sphere

NPW ≈ 12π2 ΩE3/2

cut [a.u.]

Basis set size depends on volume of box and cutoff only— and is variational!





Plane waves: Fast Fourier Transform

The information contained in ψ(G) and ψ(r) are equivalent

ψ(G) ←→ ψ(r)

Transform from ψ(G) to ψ(r) and back is done using fastFourier transforms (FFT’s)Along one direction the number of operations ∝ N log[N]

3D-transform = three subsequent 1D-transformsInformation can be handled always in the most appropriatespace





Plane waves: Integrals

Parseval’s theorem

Ω∑

G

A(G)B(G) =Ω

N

∑i

A(ri)B(ri)

Proof.

I =

∫Ω

A(r)B(r)dr

=∑GG′

A(G)B(G)

∫exp[−iG · r] exp[iG′ · r]dr

=∑GG′

A(G)B(G) Ω δGG′ = Ω∑

G

A(G)B(G)





Plane waves: Electron density

n(r) =∑

ik

wkfik|ψik(r)|2 =1Ω

∑ik

wkfik∑G,G′

cik(G)cik(G′)ei(G−G′)·r

n(r) =2Gmax∑

G=−2Gmax

n(G)eiG·r

The electron density can be expanded exactly in a plane wavebasis with a cut-off four times the basis set cutoff.

NPW(4Ecut) = 8NPW(Ecut)





Plane waves: Operators

The Kohn-Sham equations written in reciprocal space:−1

2∇2 + VKS(G,G′)

ψik (G) = εiψik (G)

However, it is better to do it like Car and Parrinello (1985)suggested: Always use the appropriate space (via FFT)There one needs to apply an operator on a wave function:∑

G′O(G,G′)ψ

(G′) =

∑G′

c(G′) 〈G|O|G′〉

Matrix representation of operators in: O(G,G′) = 〈G|O|G′〉Eg. Kinetic energy operator

TG,G′ = 〈G| − 12∇2|G′〉 =

12

G2δG,G′





Plane waves: Local operators

〈G′|O(r)|G”〉 =1Ω

∑G

O(G)

∫e−iG′·reiG·reiG”·rdr

=1Ω

∑G

O(G)

∫ei(G−G′+G”)·rdr

=1Ω

O(G′ − G”)

Local operators can be expanded in plane waves with a cutofffour times the basis set cutoff





Plane waves: Applying operators

B(G) =∑G′

O(G,G′)A(G′)

Local operators: Convolution

B(G) =∑G′

1Ω

O(G − G′)A(G′) = (O ∗ A)(G)

Convolution in frequency space transforms to product inreal space:

B(ri) = O(ri)A(ri)





Kohn–Sham energy

EKS = Ekin + EES + Epp + Exc

Ekin Kinetic energyEES Electrostatic energy (sum of electron-electron

interaction + nuclear core-electron interaction +ion-ion interaction)

Epp Pseudo potential energy not included in EES

Exc Exchange–correlation energy





Kinetic energy

Ekin =∑

ik

wkfik〈ψik|−12∇2|ψik〉

=∑

ik

wkfik∑GG′

c∗ik(G)cik(G′)〈k + G|−1

2∇2|k + G′〉

=∑

ik

wkfik∑GG′

c∗ik(G)cik(G′) Ω

12|k + G|2 δG,G′

= Ω∑

ik

wkfik∑

G

12|k + G|2 |cik(G)|2





Periodic Systems

Hartree-like terms are most efficiently evaluated inreciprocal space via the

Poisson equation

∇2VH(r) = −4πntot(r)

VH(G) = 4πn(G)

G2

VH(G) is a local operator with same cutoff as ntot





Electrostatic energy

EES =12

∫∫n(r)n(r′)|r − r′| dr′dr+

∑I

∫n(r)V I

core(r)dr+12

∑I =J

ZIZJ

|RI − rJ |

The isolated terms do not converge; the sum only forneutral systemsGaussian charge distributions a’la Ewald summation:

nIc(r) = − ZI(

RcI

)3π−3/2 exp

[−

(r − RI

RcI

)2]

Electrostatic potential due to nIc:

V Icore(r) =

∫nI

c(r′)|r − r′|dr′ = − ZI

|r − RI |erf[ |r − RI |

RcI

]DFT in the solid state École thematique calcul/RMN 62 / 99




Electrostatic energy: Long-ranged forces

Plane wave expansion of ntot

ntot(G) = n(G) +∑

I

nIc(G)SI(G)

= n(G) − 1Ω

∑I

ZI√4π

exp[−1

2G2Rc

I2]

SI(G)

Criterion for parameter RcI : PW expansion of nI

c has to beconverged with density cut-off

Poisson equation

VH(G) = 4πntot(G)

G2







EES = 2πΩ∑G=0

|ntot(G)|2G2 + Eovrl − Eself

Eovrl =∑′

I,J

∑L

ZIZJ

|RI − rJ − L|erfc

⎡⎣ |RI − rJ − L|√

RcI2 + Rc

J2

⎤⎦

Eself =∑

I

1√2π

Z 2I

RcI

Sums expand over all atoms in the simulation cell, all directlattice vectors L; the prime in the first sum indicates thatI < J is imposed for L = 0.





Exchange-correlation energy

Exc =

∫rn(r)εxc(r)dr = Ω

∑G

εxc(G)n(G)

εxc(G) is not local in G space; the calculation in real spacerequires very accurate integration scheme.If the function εxc(r) requires the gradients of the density,they are calculated using reciprocal space, otherwise thecalculation is done in real space (for LDA and GGA; hybridfunctionals are more intensive)





Plane waves: Basic self-consistent cycle

n(r) FFT−→ n(G)Vxc[n](r)

VES(G)

VKS(r) = Vxc[n](r) + VES(r) FFT←− VES(G)

ψik(r) Ni×FFT←− ψik(G)

VKS(r)ψik(r) Ni×FFT−→ [VKSψik] (G)update ψik(G)

ψ′ik(r) Ni×FFT←− ψ′

ik(G)

n′(r) =∑

ik wkfik∣∣ψ′

ik(r)∣∣2





Plane waves: Calculation of forces

With the plane wave basis set one can apply the

Hellmann-Feynman theorem

(FI =) − ddRI

〈Ψ | HKS | Ψ〉 = −〈Ψ | ∂

∂RIHKS | Ψ〉

All the terms where RI appear explicitly are in reciprocalspace, and are thus very simple to evaluate:

∂

∂RIe−iG·RI = −iGe−iG·RI





Plane waves: Summary

Plane waves are delocalised, periodic basis functionsPlenty of them are needed, however the operations aresimpleThe quality of basis set adjusted using a single parametre,the cut-off energyFast Fourier-transform used to efficiently switch betweenreal and reciprocal spaceForces and Hartree term/Poisson equation are trivialThe system has to be neutral ! Usual approach for chargedstates: Homogeneous neutralising backgroundThe energies must only be compared with the same Ecut




Introduction to pseudo potentialsGeneration of pseudo potentials

Summary




4 Pseudo potentialsIntroduction to pseudo potentialsGeneration of pseudo potentials





Why use pseudo potentials?

Reduction of basis set sizeeffective speedup of calculationReduction of number of electronsreduces the number of degrees of freedomFor example in Pt: 10 instead of 78Unnecessary “Why bother? They are inert anyway...”Inclusion of relativistic effectsrelativistic effects can be included "partially" into effectivepotentials





Why pp? Estimate for number of plane waves

plane wave cutoff ↔ most localized function

1s Slater type function ≈ exp[−Zr ]Z: effective nuclear charge

φ1s(G) ≈ 16πZ 5/2

G2 + Z 2

Cutoff Plane wavesH 1 1Li 4 8C 9 27Si 27 140Ge 76 663





Pseudo potential

What is it?

Replacement of the all-electron, −Z/r problem with aHamiltonian containing an effective potentialIt should reproduce the necessary physical properties ofthe full problem at the reference stateThe potential should be transferable, ie. also be accuratein different environments

The construction consists of two steps of approximationsFrozen core approximationPseudisation





Frozen core approximation

Core electrons are chemically inertCore/valence separation is sometimes not clearCore wavefunctions are from atomic reference calculationCore electrons of different atoms do not overlap





Remaining problems

Valence wavefunctions must be orthogonal to core states→ nodal structures → high plane wave cutoffThus pseudo potential should produce node-less functionsand include Pauli repulsionPseudo potential replaces Hartree and XC potential due tothe core electronsXC functionals are not linear: approximation

EXC(nc + nv) = EXC(nc) + EXC(nv)

This assumes that core and valence electrons do notoverlap. This restriction can be overcome with the"non–linear core correction" (NLCC) discussed later





Atomic pseudo potentials: Basic idea

1 Create a pseudo potential for each atomic speciesseparately, in a reference state (usually a neutral or slightlyionised atom)

2 Use these in the Kohn-Sham equations for the valenceonly

(T + V (r, r′) + VH(nv) + VXC(nv)

)Φv

i (r) = εiΦvi (r)

GoalPseudo potential V (r, r′) has to be chosen such that the mainproperties of the all-electron atom are imitated as well aspossible





Pseudisation of valence wave functions





Generation of pseudo potentials: General recipe

1 Atomic all–electron calculation (reference state)⇒ Φv

l (r) and εl .2 Pseudize Φv

l ⇒ ΦPSl

3 Calculate potential from

[T + Vl(r)] ΦPSl (r) = εlΦ

PSl (r)

4 Calculate pseudo potential by unscreening of Vl(r)

V PSl (r) = Vl(r) − VH(nPS) − VXC(nPS)V PS

l is dependent on l ! Otherwise poor transferability





Semi-local pseudo potential

Semi-local form

V PS(r, r′) =∑

l

V PSl (r) | Ylm(r)〉〈Ylm(r′) |

where Ylm are spherical harmonics

This form leads, however, to large computing requirements inplane wave basis sets





Norm-conserving pseudo potentials

Hamann-Schlüter-Chiang-Recipe (HSC) conditionsD R Hamann, M Schlüter & C Chiang, Phys Rev Lett 43, 1494 (1979)

1 Real and pseudo valence eigenvalues agree for a chosenprototype atomic configuration: εl = εl

2 Real and pseudo atomic wave functions agree beyond achosen core radius rc : Ψl(r) = Φl(r) for r ≥ rc

3 Integral from 0 to R of the real and pseudo chargedensities agree for R ≥ rc (norm conservation):〈Φl |Φl〉R = 〈Ψl |Ψl〉R for R ≥ rc , 〈Φ|Φ〉R =

∫ R0 r2|φ(r)|2dr

4 The logarithmic derivatives of the real and pseudo wavefunction and their first energy derivatives agree for r ≥ rc .

Points 3) and 4) are related via−1

2

[(rΦ)2 d

dεddr lnΦ

]R =

∫ R0 r2|Φ|2dr





Recipes for norm-conserving pseudo potentials

Bachelet-Hamann-Schüter (BHS) form

G B Bachelet et al., Phys Rev B 26, 4199 (1982)

Recipe and analytic form of V PSl

Kerker recipe G P Kerker, J Phys C 13, L189 (1980)

analytic pseudisation functionD. Vanderbilt, Phys Rev B 32, 8412 (1985)

Kinetic energy optimized pseudo potentials

A M Rappe et al., Phys Rev B 41, 1227 (1990)J S Lin et al., Phys Rev B 47, 4174 (1993)





Troullier–Martins recipe

N Troullier and J L Martins, Phys Rev B 43, 1993 (1991)

ΦPSl (r) = r l+1ep(r) r ≤ rc

p(r) = c0 + c2r2 + c4r4 + c6r6 + c8r8 + c10r10 + c12r12

determine cn fromnorm–conservationsmoothness at rc (for m = 0 . . .4)dmΦdrm

∣∣∣r=rc−

= dmΦdrm

∣∣∣r=rc+

dΦdr

∣∣r=0 = 0





Separation of local and nonlocal parts

V PS(r, r′) =∞∑

L=0

V PSL (r)|YL〉〈YL|

Approximation: All potentials with L > Lmax are equal to V PSloc

V PS(r, r′) =Lmax∑L=0

V PSL (r)|YL〉〈YL| +

∞∑L=Lmax+1

V PSloc(r)|YL〉〈YL|

=Lmax∑L=0

(V PS

L (r) − V PSloc(r)

) |YL〉〈YL| +∞∑

L=0

V PSloc(r)|YL〉〈YL|

=Lmax∑L=0

(V PS

L (r) − V PSloc(r)

) |YL〉〈YL| + V PSloc(r)





Semi-local pseudo potential

Final semi-local form

V PS(r, r′) = V PSloc(r) +

Lmax∑L=0

∆V PSL (r)|YL〉〈YL|

Local pseudo potential V PSloc

Non-local pseudo potential ∆V PSL = V PS

L − V PSloc

Any L quantum number can have a non-local part









Kleinman–Bylander form

Write the pseudo potential terms as

Kleinman-Bylander operator

V KBPS,nl

(r, r′

)=

∑L

|∆VLφL〉ωL〈∆VLφL|

ωL = 〈φL | ∆VL | φL〉

φL is the pseudo wave function

EPS,nl =∑

ik

wkfik∑

L

〈ψik | ∆VLφL〉ωL〈∆VLφL | ψik〉

Now the expression is fully non-local f (r′) f (r) and thuscomputationally efficient also in plane wave basis set!





Ghost states

Problem: In Kleinman–Bylander form, the node-less wavefunction is no longer the solution with the lowestenergy

Solution: Carefully tune the local part of the pseudopotential until the ghost states disappear

How to detect ghost states: Look for following propertiesDeviations of the logarithmic derivatives of theenergy of the KB–pseudo potential from thoseof the respective semi-local pseudo potentialor all–electron potential.Comparison of the atomic bound state spectrafor the semi-local and KB–pseudo potentials.Ghost states below the valence states areidentified by a rigorous criteria by Gonze et al





Ultra–soft pseudo potentials and PAW method

Many elements require high cutoff for plane wavecalculations

First row elements: O, FTransition metals: Cu, Znf elements: Ce

relax norm-conservation condition∫nPS(r)dr +

∫Q(r)dr = 1





Ultra–soft pseudo potentials and PAW method

Augmentation functions Q(r) depend on environment.No full un-screening possible, Q(r) has to be recalculatedfor each atom and atomic position.Complicated orthogonalisation and force calculations.Allows for larger rc , reduces cutoff for all elements to about30 Ry





Non-linear core correction (NLCC)

For many atoms (eg. alkali atoms, transition metals) corestates overlap with valence states; thus linearisation of XCenergy breaks downPossible cures

Add additional states to valenceadds more electronsneeds higher cutoff

Add core charge to valence charge in XC energy ⇒non–linear core correction (NLCC)S.G. Louie et al., Phys. Rev. B, 26 1738 (1982)






Exc = Exc(n + ncore) where ncore(r) = ncore(r) if r > r0






The total core charge of the system depends on the atomicpositions.

ncore(G) =∑

I

nIcore(G)SI(G)

This leads to additional terms in the derivatives wrt to nuclearpositions and the box matrix (for the pressure).

∂Exc

∂RI,s= −Ω

∑G

iGsV xc(G)nI

core(G)SI(G)





Specification of pseudo potentials

The pseudo potential recipe used and for each l value rcand the atomic reference stateThe definition of the local potential and which angularmomentum state have a non–local partFor Gauss–Hermit integration: the number of integrationpointsWas the Kleinman–Bylander scheme used ?NLCC: definition of smooth core charge and rloc





Testing of pseudo potentials

Calculation of other atomic valence configurations(occupations), comparison with all-electron resultsCalculation of transferability functions, logarithmicderivatives, hardnessCalculation of small molecules, comparison with allelectron calculations (geometry, harmonic frequencies,dipole moments)Calculation of other test systemsCheck of basis set convergence (cutoff requirements)





Convergence test for pseudo potentials in O2





Pseudo potentials: Summary

Pseudo potential are necessary when using plane wave basissets in order to keep the number of the basis functionmanageable

Pseudo potentials are generated at the reference state;transferability is the quantity describing the accuracy of theproperties at other conditions

The mostly used scheme in plane wave calculations is theTroullier-Martins pseudo potentials in the fully non-local,Kleinman-Bylander form

Once created, a pseudo potential must be tested, tested,tested!!!


Plane wave/pseudo potential calculations insolid state — II

Ari P Seitsonen



Different basis setPractical calculations — What to control

Quantum Espresso/PWSCF

Localised basis setsComparison: Localised basis sets vs plane waves

Summary

1 Different basis setLocalised basis setsComparison: Localised basis sets vs plane waves

2 Practical calculations — What to control

3 Quantum Espresso/PWSCF

PWSCF École thematique calcul/RMN 2 / 34




Expansion using a basis set

For practical purposes it is necessary to expand theKohn-Sham orbitals using a set of basis functionsBasis set ϕα(r)M

α=1

Usually a linear expansion

ψi(r) =M∑α=1

cαiϕα(r)





Different basis sets

Plane wavesquantum espresso/PWSCF, CPMD, abinit, paratec, VASP,CASTEP, . . .

Gaussian functionsTurboMole, MolPro, GaussianXX, . . .

Slater functionsADF

Numerical/δ basisDMol3, numol, . . .

MixturesCP2k, Wien2k, . . .





Atomic orbital basis sets

PhilosophyMolecules are assemblies of slightly distorted atoms

ϕα(r) = ϕα(r)Ylm(Θ, φ)

ϕα(r) =

exp[−αr2] Gaussianexp[−αr ] Slater

ϕα(r; RI) : basis functions are attached to nuclear positions





Advantages / disadvantages of AO basis sets

+ according to chemical insight+ small basis sets give already good results– non-orthogonal– depend on atomic position– basis set superposition errors (BSSE)





Properties of plane waves

ϕG(r) =1√Ω

exp[iG · r]

Plane waves are periodic wrt. box hPlane waves are orthonormal

〈ϕG′ |ϕG〉 = δG′,G

Plane waves are complete

ψ(r) = ψ(r + L) =1√Ω

∑G

ψ(G) exp[iG · r]





Comparison to AO basis set

Plane Waves:

12

G2 < Ecut

12

G′2 < Ecut

12

(G + G′)2

<(√

Ecut +√

Ecut

)2= 4Ecut

Atomic orbitals: Every product results in a new function

ϕα(r − A)ϕβ(r − B) = ϕγ(r − C)

Linear dependence for plane waves vs. quadratic dependencefor AO basis sets.




Starting a new calculationUseful toolsExamples of convergence tests

Summary

1 Different basis set

2 Practical calculations — What to controlStarting a new calculationUseful toolsExamples of convergence tests

3 Quantum Espresso/PWSCF





How to start a new calculation





Initial considerations

What is the best method/code available?(classical/approximate/DFT potential)Size of system? (computational resources available)Static or dynamics calculation?





Procedure for DFT calculations

Choice of the system (number of atoms, unit cell, ...; canbe modified later)Test for static computational parametres

description of nuclei (CPMD: pseudo potentials)basis set (CPMD: cut-off energy only)computational unit cell for isolated systems (molecules)exchange-correlation functional





Choice for computational parametres: Unit cell

In plane wave calculations one always has to define acomputational unit cell, even for isolated objects such asatoms or molecules

It should be large enough so that the electronic wavefunctions and density vanish at the boundary of the box, orthat the periodic images of the molecule do not interact witheach otherTest simply by increasing the size of the box systematicallyuntil the main parametres of interest (energies, vibrationalfrequencies, Kohn-Sham eigenvalues, . . . ) do not changeanymore

In naturally periodic systems like solids, liquids, slabgeometries (surfaces, 2-dimensional periodicity), polymers(1-dimensional periodicity) one has to test for theconvergence with respect to k point sampling





Choice for computational parametres: XC functional

Does the system size allow for the use of hybridfunctionals?If not, would LDA provide better accuracy than GGA’s?This is normally the case only in solids or other denselypacked materials, for the bulk properties. Also usedsometimes in cases where GGA’s fail qualitatively (egadsorbtion of molecules with phenyl ring on transitionmetal surfaces)If GGA, which one of them? BLYP (and BP86) is favouredby chemists, PBE by physicistsOther functionals: Meta-GGA’s, asymptotically correctedfunctionals, self-interaction corrected functionals, . . .





Choice for static computational parametres: PP

Considerations when selecting a pseudo potentialWhich valence configuration? Semi-core (= state inbetween “clear” core and valence states; normally−20. . .−50 eV below the vacuum level) in core or valence?Do the core and valence electron density over-lapconsiderably? If yes, is NLCC necessary?Type of pseudo potential? Troullier-Martins, Vanderbilt,other norm-conserving one, . . .Kleinman-Bylander scheme or Gauß-Hermite integration?Which lmax, lloc? The former is more an issue of computingtime, the second crucial to avoidCould the core radii be larger (to reduce basisset/computing time)?





Choice for computational parametres: PP testing

Select a (small) representative system, preferably severalof them; for example of different bonding or hybridisationtypes (eg purely covalent C-C, partially ionic/polarisationC-N or C-H, ionic like H-FStart with a large basis set (please see next topic) in orderto be sure of convergenceCompare to all-electron or other, reliable pseudo potentialcalculation employing the same XC functional, if possible;comparisons with experiments have a second priority, dueto the known, systematic errors in the XC functionalsQuantities to compare: Binding energies, bond lengths,angles, vibrational frequencies, . . .The differences to proper all-electron calculations shouldbe of the order of 0.01 Å or less in bond lengths, < 50 meVfor binding energies, vibrational frequencies < 20 cm−1





Choice for computational parametres: PP parametres

If you need to optimise the parametres of the pseudo potentiallike the core radii, NLCC radius etc, you have to change themand re-run these tests; again until the property of interest doesnot change anymore significantly (Please remember: It isalways nicer to be on the safe side. . . )





Choice for computational parametres: Basis set

With plane wave basis set only one parametre: Cut-offenergy Ecut; units Rydberg atomic units (historical /convenience reasons: Ecut [Ha] = 1

2 |Gmax|2 = 12Ecut [Ry] ⇒

|Gmax| [1/Bohr] =√

Ecut [Ry] )You can start from a low value of Ecut for fast convergence,increase it in steps of 5, 10 or 20 Ry until the properties ofinterest converge; please notice that the total energy of thesystemTypical values

2p elements, transition metals, Troullier-Martins pp’s:50-90 Ry3p, 4p elements, alkali metals with NLCC, transition metalsof the 5p row etc, Troullier-Martins pp’s: 20-40 RyVanderbilt pp’s: 20-35 Ry





Useful tools

units command: Conversion between different units:localhost (~) : units2084 units, 71 prefixes, 32 nonlinear units

You have: 32*(15.9994+2*1.0067) atomicmassunit / (9.885 angstrom)^3You want: g cm^-3

* 0.99094647/ 1.0091362

You have:

openbabel converts between different graphics fileformatsxmgrace or gnuplot for 2-dimensional graphicsxcrysden understand input and output of PWSCFgraphics programs for 3-dimensional plots and atomicconfigurations, movies: gOpenMol vmd, xmakemol, . . .





Convergence tests: Cut-off energy





Convergence tests: k point set





Convergence tests: Cut-off energy





Convergence tests: k point set




IntroductionBasic usageExample

Summary

1 Different basis set

2 Practical calculations — What to control

3 Quantum Espresso/PWSCFIntroductionBasic usageExample





Quantum Espresso http://www.quantum-espresso.org/





PWSCF http://www.pwscf.org/





Quantum Espresso

Historically started from “original” Car-Parrinello codeJoined recently from PWSCF, CPV and FPMD





Running the executable

Machine-dependent!Serial run:./pw.x -input my-input-file > my-output-file &

Parallel run, linuxmpirun -np $NSLOTS ./pw.x -npool 4 -input my-input-file > my-output-file &

Parallel run, LoadLeveler:./pw.x -npool 4 -input my-input-file > my-output-file &





Example: fcc-Cu&control

calculation=’scf’restart_mode=’from_scratch’,pseudo_dir = ’/home/seitsonen/usr/espresso/PP_LIBRARY/’,wfcdir=’/tmp/’prefix=’fcc-Cu’tstress = .true.tprnfor = .true.

/&system

ibrav = 2, a = 4.00, b = 4.00, c = 4.00,nat= 1, ntyp= 1,nbnd = 9ecutwfc = 140

! ecutrho = 0occupations=’smearing’, smearing=’fermi-dirac’, degauss=.00367490107593722133

/&electrons

diagonalization=’david’conv_thr = 1.0e-9mixing_beta = 0.8

/ATOMIC_SPECIESCu 63.5500 Cu_pbe-20071125.UPFATOMIC_POSITIONS angstromCu 0.0 0.0 0.0K_POINTS (automatic)16 16 16 0 0 0





Help on input

Input is explained in file Doc/INPUT_*

One can ask for help on the Mailing list

Different values for ’ibrav/celldm’ are explained at the endof Doc/INPUT_PWPlease be aware on units for lattice vectors andcoordinates: Bohr, Ångström, a0, lattice vectors, . . .


Density functional theory in the solid state — III

Ari P Seitsonen



RepetitionStructural optimisation

Input for GIPAW module

DFTPlane wave basis setPseudo potentials

Summary

1 RepetitionDFTPlane wave basis setPseudo potentials

2 Structural optimisation

3 Input for GIPAW module





Kohn-Sham schemeKohn-Sham equations

−12∇2 + VKS (r)

ψi (r) = εiψi (r)

n (r) =∑

i

fi |ψi (r)|2

VKS (r) = Vext (r) + VH (r) + Vxc (r)

Vxc (r) = δExcδn(r)

Total energy:

Etot[n] =∑

i

−12

fi⟨ψi | ∇2 | ψi

⟩+


+12

∫r

∫r′

n (r) n (r′)|r − r′| dr′ dr + Exc + Eion−ion





Kohn-Sham equations: Notes

−1

2∇2 + VKS (r)

ψi (r) = εiψi (r) ; n (r) =

∑i

fi |ψi (r)|2

Many-body problem casted into a system of single-particle,coupled equationsSelf-consistent solution requiredεi and ψi are in principle only help variables (only εHOMOhas a meaning)The potential VKS is localThe scheme is in principle exact





Exchange-correlation functional

The Kohn-Sham scheme is in principle exact — however,the exchange-correlation energy functional is not knownexplicitlyExchange is known exactly, however its evaluation isusually very time-consuming, and often the accuracy is notimproved over simpler approximations (due toerror-cancellations in the latter group)Many exact properties, like high/low-density limits, scalingrules etc are knownFamous approximations:

Local density approximation, LDAGeneralised gradient approximations, GGAHybrid functionals. . .





Exchange-correlation functional: Observations

The accuracy can not be systematically improved!van der Waals interactions still a problem (tailoredapproximations in sight)Most widely used parametrisations:

GGAPerdew-Burke-Ernzerhof, PBE (1996); among phycisistsBeck–Lee-Yang-Parr, BLYP (early 1990’s); among chemists

Hybrid functionalsPBE0; among phycisistsB3LYP; among chemists





Integration over reciprocal space

Bloch’s Theorem: The summation over infinite number oftranslations becomes an integral over the first Brillouinzone: ∞∑

L

⇒∫

k∈1.BZdk

In practise the integral is replaced by a weighted sum ofdiscrete points: ∫

kdk ≈

∑k

wk

Eg.n (r) =

∑S

∑k∈1. BZ

wk∑

i

fik |ψik (Sr)|2





Expansion of Kohn-Sham orbitals

Plane wave expansion

ψik(r) =∑

G

cik(G)ei(k+G)·r

To be solved: Coefficients cik(G)

+ orthogonal+ independent of atomic positions+ no BSSE± naturally periodic– many functions needed





AdvantagesTranslations

φ(r − RI) =∑

G

(φ(G)eiG·r

)e−iG·RI

∂φ(r − RI)

∂RI,s= −i

∑G

Gs

(φ(G)eiG·r

)e−iG·RI

Poisson equation

∇2VH(r) = −4πn(r)

VH(G) = 4πn(G)

G2

Diagonal terms

Ekin = Ω∑

ik

wkfik∑

G

12|k + G|2 |cik(G)|2





Cutoff: Finite basis set

Choose all basis functions intothe basis set that fulfill

12

G2 ≤ Ecut

— a cut-off sphere

NPW ≈ 12π2 ΩE3/2

cut [a.u.]

Basis set size depends on volume of box and cutoff only— and is variational!





Plane waves: Basic self-consistent cycle

n(r) FFT−→ n(G)Vxc[n](r)

VES(G)

VKS(r) = Vxc[n](r) + VES(r) FFT←− VES(G)

ψik(r) Ni×FFT←− ψik(G)

VKS(r)ψik(r) Ni×FFT−→ [VKSψik] (G)update ψik(G)

ψ′ik(r) Ni×FFT←− ψ′

ik(G)

n′(r) =∑

ik wkfik∣∣ψ′

ik(r)∣∣2





Pseudo potentials

Idea: Replace action of core electrons on valenceelectrons with a pseudo Hamiltonian yielding thecorresponding quantitiesPseudo potentials are generated in atomic, referenceconfigurations; transferability characterises their accuracyThe pseudo wave functions are node-less, soft functionsThey are usually l-dependent and decomposed viaKleynman-Bylander schemePopular choices: Norm-conserving, Vanderbilt (ultra-soft),PAW





Example: Si




Optimisation of atomic structureOptimisation of crystal lattice

Summary

1 Repetition

2 Structural optimisationOptimisation of atomic structureOptimisation of crystal lattice






Structural optimisation

In the beginning of the calculation the atomic coordinatesare usually not accurate (= not at equilibrium of the DFTmethod), coming from

ExperimentsClassical simulationsEducated guess (“chemical intuition”)

Task: Optimise the atomic coordinates and maybe thecrystal parametres= Minimise the total energy with respect to ionic positionsOther type of calculations: Ab initio molecular dynamics(finite ionic T)





Optimisation of atomic geometry

In principle easy: “Only” (3N-6) degrees of freedom (DoF)However, non-linear and coupled dependency on relativepositionsThe optimisation is much easier if one has the gradientsavailable

Forces acting on the ions

FI = −∇Etot = −∂Etot

∂RI






Forces in plane wave basis set:

Hellmann-Feynman theorem

FI = − ddRI

〈Ψ | HKS | Ψ〉 = −〈Ψ | ∂

∂RIHKS | Ψ〉

All the terms with explicit RI explicitly are in reciprocalspace, and are thus very simple to evaluateFor example a local, external potential:

Eloc =

∫rn (r) Vloc (r − RI) dr =

∑G

n∗(G) Vloc (G) e−iG·RI

FI = −∂Eloc

∂RI=

∑G

iG n∗(G) Vloc (G) e−iG·RI






In principle easy: “Only” (3N-6) degrees of freedom (DoF)However, non-linear and coupled dependency on relativepositionsThe optimisation is much easier if one has the gradientsavailable

Forces acting on the ions

FI = −∇Etot = −∂Etot

∂RI

Holds in principle only at Born-Oppenheimer surface (atself-consistency)





Mathematical formulation

At least close to minimum: Quadratic problem

E = Emin +12

(RI − RminI ) · B · (RI − Rmin

I )

with Hessian BIα,Jβ = ∂2E∂RIα∂RJβ

gradient GIα = ∂E∂RIα

= B · (RI − RminI )





Optimisation of atomic structure

Newton’s algorithm

Evaluate B and then B−1

Start from a point R(1)I

Calculate gradien G(1)I = GI(R

(1)I )

After movement: R(2) = R(1)I − B−1 · G(1)

I

Since GI(RI) = B · (RI − RminI ), R(2) = Rmin

I

However, evaluation of B would take too long (and system innot harmonic) ⇒ Not a practical schemeQuasi-Newton methods; approximations for B can be used forpre-conditioning





Optimisation of atomic structure: Algorithms

Steepest descentsConjugate gradientsBroyden-Fletcher-Goldfarb-Shanno (BFGS) methodDamped (Langevin) dynamicsSimulated annealing (molecular dynamics)





Steepest descents, SD

Steepest descents

At point R(k)I , calculate gradient G(k)

I = GI(R(k)I )

R(k+1)I = R(k)

I − αG(k)I

alpha can be either fixed. . . or searched via line minimisation: Move ions along G(k)

I ,calculate new energy, search for a minimum via harmonicapproximation and move the ions with the final, optimal αCan be very slow eg. in steep valleys





Direct inversion in iterative sub-space, DIIS

Direct inversion in iterative sub-space, DIIS

Obtain a new point R(k)I = R(k−1)

I − αG(k−1)I , calculate

gradient G(k)I = GI(R

(k)I )

Find GoptI =

∑i≤k αiG

(i)I with

∑α = 1

RoptI =

∑i≤k αiR

(k)I

Update R(k+1)I = Ropt

I − λGoptI





Broyden-Fletcher-Goldfarb-Shanno, BFGS

Broyden-Fletcher-Goldfarb-Shanno, BFGS

Solve B(k) · s(k) = −G(k)I for s(k)

Perform line search along this direction to obtain α(k)

Obtain the new position from

R(k+1)I = R(k)

I + α(k)s(k)

y(k) =G(k+1)

I − G(k)I

α(k)

B(k+1) = B(k) +y(k) · y(k)

y(k)s(k)− (B(k) · s(k)) · (B(k) · s(k))

s(k) · B(k) · s(k)





Conjugate gradient, CG

Conjugate gradient, CG

Calculate the gradient at current point, G(k)I = GI(R

(k)I )

Conjugate this gradient as

s(k) = G(k)I + γs(k−1) , γ =

(G(k)I − G(k−1)

I ) · G(k)I

G(k−1)I · G(k)

I

Perform line minimisation along s(k) to obtain R(k+1)I





Damped dynamics

Damped dynamicsUpdate the positions dynamicallyForces accelerate the ions, velocities are used in a frictionterm to stop them (large forces — fast acceleration, fastmovement — large friction, “cooling”)Equation of motion

R(k)I = −G(k)

I /MI + γs(k−1)





Quenching dynamics

Quenching/projected dynamicsUpdate the positions dynamicallyForces accelerate the ions, quench if the forces andacceleration are opposite (“overshooting”)Equation of motion

R(k)I = (R(k)

I − R(k−1)I )/(∆t)

V(k)I = R(k)

I · G(k)I

|G(k)I |

G(k)I

|G(k)I |

if R(k)I · G(k)

I < 0, otherwise 0

R(k+1)I = R(k)

I + αV(k)I − (∆t)2G(k)

I





Simulated annealing

Simulated annealing

Perform molecular dynamics (Newton’s equation ofmotion), reduce the temperature periodicallyAt minimum the forces vanish and the atoms stopIs the “natural” way, however the annealing rates are muchfaster than in eg. in experiments/preparation






Which algorithm to use?Few degrees of freedom (or robustness required):Conjugate gradientsMedium number of DoF: DIIS, BFGSLarge number of DoF: Damped dynamics or simulatedannealing






Extrapolation of potential and/or wave function from previousionic steps

Helps to assign a new initial guess for the electronicstructure in the new geometryThus the self-consistent solution is found faster





Optimisation of crystal structure

It is straightforward to calculate the stress tensor in planewave basis set:

Πuv = − 1Ω

∑s

∂Etot

∂hushT

sv

Eg. kinetic energy

∂Ekin

∂huv= −

∑ik

wkfik∑

G

∑s

GuGs(hT )−1sv |cik (G)|2

Π acts as “gradient” in the update of the lattice vectorsOne has to be careful with discontinuities/jumps in thebasis set when changing the lattice vectors: High accuracyneeded already for stress tensor




Summary

1 Repetition

2 Structural optimisation





Parametres affecting scientific output

Exchange-correlation functional (read from pp files)Pseudo potentialsBasis set — Ecut [ecutwfc]k point sampling [section K_POINTS]Convergence: Self-consistency [conv_thr]Spin-polarisation [nspin,multiplicity]Lattice vectors, ionic positions




Parametres affecting verbosity/control

Algorithm for optimisation of electronic structure[diagonalization]Mixing [mixing_*]Amount of output [verbosity]Restarting [restart_mode]Restarting a parallel calculation [wf_collect]Print-outs [tstress,tprnfor]Organisation of files [outdir,wfcdir]Output filenames [prefix]Computing time [max_seconds]Amount of hard disc/memory [disk_io]Pseudo potentials [pseudo_dir]Symmetries [nosym]




GIPAW

Chris J Pickard and Francesco Mauri, PRB 63, 245 101 (2001)

Two tasks:1 Gauge-invariance2 Reconstruction of core wave functions

PAW-like transformation

TB = 1+∑I,n

ei/(2c)r·RI×B[∣∣φI,n

⟩ − ∣∣∣φI,n

⟩] ⟨pI,n

∣∣ e−i/(2c)r·RI×B

Slowly-varying field: eiq·x

Macroscopic shape:

B(1)ind (G = 0) =

8π3χB




GIPAW in PWSCF

First perform a self-consistent run with ’pw.x’Then you can use ’gipaw.x’: The latter uses informationgenerated by ’pw.x’Converge the electronic structure accuratelyThe symmetry operations have to map the coordinate axisinto each other; otherwise the symmetries must beswitched off!




Example: Hegaxonal lattice, GIPAW-NMR&control

calculation = ’scf’ <<<<<<<<<<<<<<<<<<<<<<<<<<<<! restart_mode = ’from_scratch’

restart_mode = ’restart’prefix = ’b2o3I’tstress = .false.tprnfor = .true.pseudo_dir = ’./’outdir = ’./scratch/’

/&system

ibrav = 4celldm(1) = 8.07450432109878800047celldm(3) = 1.92343173431735265832nat = 15ntyp = 2nspin = 1ecutwfc = 60.0nosym = .true. <<<<<<<<<<<<<<<<<<<<<<<<<<<<

/&electrons

diagonalization = ’cg’mixing_mode = ’plain’mixing_beta = 0.7conv_thr = 1.0d-12 <<<<<<<<<<<<<<<<<<<<<<<<<<<<

/ATOMIC_SPECIESB 10.81 B_pbe-20070929.cpi.UPFO 15.9994 O_pbe-20070929.cpi.UPF

ATOMIC_POSITIONS crystalO 0 555685095948 0 398122479874 -0 003161558426




Relaxation of ionic positions

&controlcalculation = ’relax’

...etot_conv_thr = 1.0D-7forc_conv_thr = 1.0D-5

.../...&ions

ion_dynamic = ’bfgs’pot_extrapolation = first_orderwfc_extrapolation = first_order

/...




Relaxation of ionic positions

&inputgipawjob = ’nmr’prefix = ’b2o3I’tmp_dir = ’./scratch/’iverbosity = 1q_gipaw = 0.01read_recon_in_paratec_fmt = .false.file_reconstruction ( 1 ) = "B_AEPS.DAT"file_reconstruction ( 2 ) = "O_AEPS.DAT"use_nmr_macroscopic_shape = .TRUE.

/


density functional theory in the solid state -...

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