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Pseudopotentials: CASTEP Workshop: Frankfurt 2012 1 / 33

Pseudopotentials

Keith RefsonSTFC Rutherford Appleton Laboratory

September 2012

Introduction

Introduction

❖ Synopsis

❖ Why Pseudopotentials?

Pseudopotential Theory

Pseudopotential inpractice

Where to obtain potentials

Conclusions


Synopsis

Introduction

❖ Synopsis





Conclusions


● Aim to de-mystify a commonly misunderstood (and feared) subject● Introduce essential theory of pseudopotentials● Describe and contrast methods of pseudopotential construction and

relationships between norm-conserving, ultrasoft and PAW varieties.● Generating and testing pseudopotentials for CASTEP the OTF generator.● Accuracy and testing of pseudopotentials.● Sources of pseudopotentials and standard libraries.

Recommended Reading and Further Study

● Jorge Kohanoff Electronic Structure Calculations for Solids and Molecules,Theory and Computational Methods, Cambridge, ISBN-13: 9780521815918

● Richard M. Martin Electronic Structure: Basic Theory and PracticalMethods: Basic Theory and Practical Density Functional Approaches Vol 1Cambridge University Press, ISBN: 0521782856

Why Pseudopotentials?

Introduction

❖ Synopsis





Conclusions


1s

3s

2p

2s

1s2s2p3s

● Chemical bonding arises from va-lence electron overlap

● Core states insensitive to chemicalenvironment.

● Frozen-core approximation elimi-nates core states from plane-wavecalculation

● Smoothed valence states requiremuch lower plane-wave cutoff.


Introduction


❖ PseudopotentialOperator


❖ Ab initioPseudopotentialConstruction

❖ The HSC criteria

❖ PSP form inside rc

❖ Optimisedpseudopotentials

❖ Non-linear corecorrections

❖ Separable form

❖ Separable form (II)

❖ UltrasoftPseudopotentials



❖ More Projectors



Conclusions


Pseudopotential Operator

Introduction









❖ Separable form





❖ More Projectors



Conclusions


All-electron Kohn-Sham equations:(

T + Vext + VH + VXC

)

|ψi〉 = ǫAEi |ψi〉 ; i ∈ {c, v}

with

Vext = Vcoul = −∑

I

ZI

|r −RI |

are replaced with pseudopotential Kohn-Sham equations:(

T + VPS + V(v)

H + VXC

)

|φi〉 = ǫPSi |φi〉 ; i ∈ {v}

for valence states only and with

VPS =∑

I

{

Vloc(|r −RI |) +

lmax∑

l=0

l∑

m=−1

|Ylm〉Vl(|r −RI |) 〈Ylm|

}

The nonlocal part of the pseudopotential is written as

V NLI =

lmax∑

l=0

l∑

m=−1


Pseudopotential Operator

Introduction









❖ Separable form





❖ More Projectors



Conclusions


● Historically, pseudopotential approximation was derived from OPW(orthogonalised plane wave) theory. This shows that V PS is energydependent operator (see §11.2 in R. M. Martin).

● Modern approach is from PAW theory, which derives HPS from H accordingto well defined approximations.

● If energy-dependence is small, pseudopotential can be used for more thanone chemical environment and is called transferrable.

● We require that ǫPS = ǫAE and φPS(r) = ψAE(r) if r > rc● Vloc(r) and Vl(r) are not unique. Can make use of functional freedom to

enhance transferrability and computational efficiency with a plane-wavebasis.

● (N.B. We use VH = V cH + V v

H but VXC 6= V cXC + V v

XC as V XC(n) is not a linearfunction of n.)

Ab initio Pseudopotential Construction

Introduction









❖ Separable form





❖ More Projectors



Conclusions


0 1 2 3 4 5r (a.u.)

-0.4

-0.2

0

0.2

0.4

0.6

r ψ

(r)

ψ4s

AE

ψ4p

AE

ψ4d

AE

ψ4s

NL

ψ4p

NL

ψ4d

NL

Valence wavefunctions for Ge

0 1 2 3 4 5r (a.u.)

-50

-40

-30

-20

-10

0

10

20

30

40

50

Vio

n (R

y)

Vs

Vp

Vd

2Zeff

/r

Vloc

Ionic pseudopotential for Ge

✓

✒

✏

✑

● PSPs constructed with atomic DFT code.● Solve K-S atom for ψAE

i,l in some choice ofreference configuration.

● Construct φPSi,l nodeless for r < rc and

φPSi,l(r > rc) = ψAE

i,l(r > rc)∫

rc

0

∣

∣

∣φPSi,l

∣

∣

∣

2

r2dr =

∫

rc

0

∣

∣

∣ψAEi,l

∣

∣

∣

2

r2dr

● invert radial Schrödinger equation to find Vl

under condition ǫPSi = ǫAE

i

● unscreen with valence charge density

V PSl (r) = Vl(r)− VH(r)− VXC(r)

● Plane-wave PS operator is

V NLI =

∑

l,m

|Ylm〉 δV PSl (|r −RI |) 〈Ylm|

With δV PSl (r) = V PS

l (r)− Vloc(r)● Use relativistic ψAE

i,l but invert non-relativisticS.E. to include relativity.

The HSC criteria

Introduction









❖ Separable form





❖ More Projectors



Conclusions


Hamman, Schlüter and Chang (PRL, 43, 1494 (1979): 4 criteria for φPS

1. PS and AE orbitals should be identical φPSi,l(r > rc) = ψAE

i,l(r > rc)

2. PS and AE eigenvalues should be identical ǫPSi = ǫAE

i

3. Norm conservation: total charge of PS and AE orbitals should be equal∫

rc

0

∣

∣

∣φPSi,l

∣

∣

∣

2

r2dr =

∫

rc

0

∣

∣

∣ψAEi,l

∣

∣

∣

2

r2dr

4. log derivative of PS and AE orbitals and energy derivatives must agree

d

drlog φPS(r)

∣

∣

∣

∣

r>rc

=d

drlogψAE(r)

∣

∣

∣

∣

r>rc

d

dǫ

d

drlog φPS(r)

∣

∣

∣

∣

r>rc

=d

dǫ

d

drlogψAE(r)

∣

∣

∣

∣

r>rc

Identity from Friedel sum rule

−1

2

[

d

dǫ

d

drlog φ(r)

]

R

=

∫

R

0

r2|φ(r)|2dr

⇒ norm-conserving pseudopotentials are transferrable (to 1st order)HSC scheme gave first pseudopotentials suitable for total energy calculations.

PSP form inside rc

Introduction









❖ Separable form





❖ More Projectors



Conclusions


Considerable freedom of form of φPSl (r); r < rc.

Several generation schemes aimed at making softer pseudopotentials, i.e.requiring lower plane-wave cutoff energy.

Hamman Schlüter Chiang Chose parameterised functional form of Vl andsolved for coefficients.Bachalet, Hamman, Schlüter: Pseudopotentials that work: From H to Pu(PRB 26, 4199 (1982)

KerkerφPSl (r) = rl+1ep(r); p(r) = αr4 + βr3 + γr2 + δ

with coefficients determined from 4 HSC conditions.Troullier and Martins 12th order even polynomial gives smoother form:

φPSl (r) = rl+1ep(r); p(r) =

6∑

i=0

c2ir2i

ci fitted from HSC conditions plus high-order derivative matching.

TM widely adopted as they require lower plane-wave cutoff, and sill inwidespread use today.

Optimised pseudopotentials

Introduction









❖ Separable form





❖ More Projectors



Conclusions


● Use variational freedom of φ(r) to achieve low plane-wave cutoff.● Original RRKJ form (PRB 41, 1227 (1990))

φPSl (r) =

n∑

i=1

cijl(qi, r)

wherejl(qi, r)

are spherical bessel functions.● qi are fixed by derivative matching at rc.● ci are adjusted to minimize KE residual above qc

∆EK = −

∫

∞

0

φ∗

l (r)∇2φl(r)d

3r −

∫ qc

0

q2|φl(q)|2dq

● qc is additional parameter, chosen so ∆Ek is reasonably small.● Refinements [J.S. Lin, PRB 47, 4174(1993)]: use only 4 spherical Bessel

functions● Refinements [M.H.Lee, PhD Thesis 1996]: Use 3 spherical Bessel functions

and tune qc and transferrability

Non-linear core corrections

Introduction









❖ Separable form





❖ More Projectors



Conclusions


✞

✝

☎

✆

VX,LDA(n) ∼ n1

3

VXC(nc + nv) 6= VXC(nc) + VXC(nv)

if nc(r) and nv(r) overlap.Use pseudized “partial” core density tocompute VXC(nv(r) + npcc(r))PW code must use same npcc(r)) as usedto unscreen PSP.

0 1 2 3r (a.u.)

-0.5

0

0.5

1

1.5

2

r ψ

(r)

ψ3s

AE

ψ3p

AE

ψ3d

AE

ψ4s

AE

0 1 2 3r (a.u.)

0

10

20

30

40

4πr

2 ρ(r)

core density partial core valence density

Separable form (II)

Introduction









❖ Separable form





❖ More Projectors



Conclusions


● Kleinman and Bylander [PRL 48, 1425(1982)] substitution

|βlm〉 =∣

∣

∣δVlφ

PSlm

⟩

and Blm = 1/⟨

φPSlm

∣

∣

∣δVl

∣

∣

∣φPSlm

⟩

where φPSlm are pseudo-atomic orbitals.

● straightforward to show that

V KB∣

∣

∣φPSlm

⟩

= V NL∣

∣

∣φPSlm

⟩

● N.B. K-B projectors change drastically with different choice of local potential.● Risk of Kohn-Sham states of V KB in solid which are not solutions of V NL ⇒

ghost states● Vanderbilt [PRB 41, 7892 (1990)] generalised the form to

V NLI =

∑

jk

Bjk |βj〉〈βk|

and showed how to directly generate potential in separable form● Third method is Gauss Hermite integration (Goedecker & Hutter)

Ultrasoft Pseudopotentials

Introduction









❖ Separable form





❖ More Projectors



Conclusions


✗

✖

✔

✕

● Norm conservation ⇒ nodeless 2p, 3d, 4fstates inevitably hard

● Vanderbilt [PRB 41,7892(1990)] relaxnorm-conservation.

V NLI =

∑

jk

Djk |βj〉〈βk|

withDjk = Bjk + ǫjqjk

and

Qjk(r) = ψ∗,AEj (r)ψAE

k (r)−φ∗,PSj (r)φPS

k (r)

qjk =⟨

ψAEj |ψAE

k

⟩

−⟨

φPSj |φPS

k

⟩

=

∫ rc

0

Qjk(r)dr

● Qjk(r) are augmentation functions0 1 2

r (Bohr)

Fe 3d USP


Introduction









❖ Separable form





❖ More Projectors



Conclusions


With overlap operator S defined as

S = 1 +∑

jk

qjk |βj〉〈βk|

orthonormality of ψAE ⇒ S-orthonormality of ψPS

⟨

ψAEj |ψAE

k

⟩

=⟨

φPSj

∣

∣

∣S∣

∣

∣φPSk

⟩

= δjk

The density aquires additional augmentation term

n(r) =∑

i

|φi(r)|2 +

∑

jk

ρjkQjk(r); ρjk =∑

i

〈φi|βj〉〈βk|φi〉

The K-S equations are transformed into generalised eigenvalue equations

Hφi = ǫiSφi

What gain does this additional complexity give?


Introduction









❖ Separable form





❖ More Projectors



Conclusions


● φPS can be made much smoother by dropping norm-conservation.● Charge density restored by augmentation.● Transferrability restored by use of 2 or 3 projectors for each l● Quantities qjk, Bjk are just numbers and |βj(r)〉 required to construct S are

similar to norm-conserving projectors.● Only functions Qjk(r) have fine r-dependence, and they only appear when

constructing augmented charge density.● Everything except Qjk(r) easily transferred from atomic to grid-based plane

wave code.● Vanderbilt USP ⇒ pseudize Qjk(r) at some rinner ≈ rc/2, preserving norm

and higher moments of charge density.● Blöchl PAW ⇒ add radial grids around each atom to represent Qjk(r) and

naug(r)● In PW code add 2nd, denser FFT grid for naug(r) (and VH(r) - specified by

parameter fine_grid_scale.● Set fine_grid_scale= 2..4 depending on rc and rinner; good guess is

rc/rinner

More Projectors

Introduction









❖ Separable form





❖ More Projectors



Conclusions


● logarithmic derivative ( ddr

log φ(r)) vs energy plots are guide totransferrability.

● 2 projectors ⇒ superior transferrability.

Pseudopotential in practice

Introduction



❖ Comparison of accuracy

❖ Semicore states


Conclusions


Comparison of accuracy

Introduction




❖ Semicore states


Conclusions


LDA lattice parameters of rutile TiO2.a/A c/A u

PW-LDA (OTF1) 4.550 (-0.18%) 2.919 (-0.03%) 0.3039PW-LDA (OTF2)(SC) 4.549 (-0.20%) 2.919 (-0.03%) 0.3039PW-LDA (OTF3)(LC) 4.597 (+0.86%) 2.902 (-0.62%) 0.3023PW-LDA (TM) 4.536 (-0.48%) 2.915 (-0.17%) 0.304PW-LDA (RRKJ-OPT-DNL) 4.563 (+0.11%) 2.932 (+0.41%) 0.3040LCAO-LDA (CRYSTAL) 4.548 (-0.22%) 2.944 (+0.82%) 0.305PW-LDA (PAW) 4.557 (-0.22%) 2.928 (+0.27%) 0.304FP-LAPW-LDA 4.558 2.920 0.3039Expt 4.582 (+0.53%) 2.953 (+1.13%) 0.305

Semicore states

Introduction




❖ Semicore states


Conclusions


● Frozen core approx. poor for semi-corestates in some elements, e.g. Ti 3s and 3p.

● Consequences can include inaccurate lat-tice parameters, dielectric properties.

● Redefine valence to include 3s and 3pstates.

● PROBLEM if only one δVl per angular mo-mentum channel

● Vs fixed at energy of 3s ⇒ transferrabilityproblem gives inaccurate 4s.

● One solution is designed nonlocal method ofRamer and Rappe [PRB 59, 12471 (1999)]implemented in OPIUM

● Another solution is Vanderbilt method of us-ing projectors at 2 or more energies.

0 1 2 3 4 5r (a.u.)

-0.5

0

0.5

1

1.5

2

r ψ

(r)

ψ3s

AE

ψ3p

AE

ψ3d

AE

ψ4s

AE

ψ3s

NL

ψ3p

NL

ψ3d

NL

ψ4s

NL

Ti 3s and 3p valence

0 1

-0.5

0

0.5

1

1.5

2

r ψ

(r)

Ti 3s and 3p valence + DNL


Introduction




❖ Pseudopotentiallibraries for CASTEP

❖ Library Documentation


❖ GeneratingPseudopotentials

❖ CASTEP OTFPseudopotentials

❖ Testing: atomic tests

❖ Testing: solid-state tests

❖ What can go wrong


Conclusions


Pseudopotential libraries for CASTEP

Introduction













Conclusions


● Norm-conserving library (1990s) xx_00.recpotLDA-only. Comprehensive coverage of periodic table (except lanthanides/actinides).Moderate accuracy, with some poor, but well documented. Supplied along with commercialand academic CASTEP.

● New norm-conserving library (2010-) xx_OP_00PBE.recpotLDA and PBE-GGA. Sporadic converage of elements. Higher accuracy and transferrability.Supplied along with commercial and academic CASTEP.

● Rappe and Bennett library http://opium.sourceforge.net.Norm-conserving with DNL. Good accuracy. Reasonable converage of elements. LDA andPBE-GGA. .recpot version downloadable fromhttp://ccpforge.cse.rl.ac.uk/gf/project/castep/

● Vanderbilt USP library (1995-) xx_00.uspLDA and PBE-GGA. Comprehensive coverage of periodic table. Mostly reasonableaccuracy with occasional exceptions (Fe_00.uspcc). Supplied along with CASTEP(commercial and academic).

● OTF USP library (built in default). All XC functionals. Comprehensive coverage of elementsat Near all-electron accuracy. Suitabe for NMR and EELS.

● Accelrys OTF USP library (otfg.cell) All XC functionals. Comprehensive coverage ofelements at Near all-electron accuracy. Suitabe for NMR and EELS.

http://opium.sourceforge.net

http://ccpforge.cse.rl.ac.uk/gf/project/castep/

Library Documentation

Introduction













Conclusions


Most .usp and .recpot files in library contain documentation of cutoff andvalidation tests.

Convergence t e s t−−−−−−−−−−−−−−−−

Si2 dimer , or thorombic c e l l , a=6.05 , b=5.95 , c=6.00 Angstrom , LDAF r a c t i o n a l coord ina tes :(0.68402 0.69789 0.69079) and (0.47295 0.47852 0.47599)

=============================================================Ecut E to t dE Force on atom 1(eV) (eV) (eV / atom ) (eV /A)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

80 (COARSE) −208.846 0.163 −0.32739 −0.49454 −0.39083120 (MEDIUM) −209.065 0.054 −0.14034 −0.18670 −0.06503160 ( FINE ) −209.102 0.035 −0.25120 −0.26779 −0.28774180 (PRECISE) −209.148 0.012 −0.25123 −0.26911 −0.29407280 −209.169 0.001 −0.26726 −0.23212 −0.25473320 −209.171 0.000 −0.26858 −0.23360 −0.25605400 −209.172 0.000 −0.26189 −0.23120 −0.24662450 −209.172 0.000 −0.25998 −0.23120 −0.24575800 −209.172 −0.26368 −0.23284 −0.25065=============================================================

Library Documentation

Introduction













Conclusions


Most .usp and .recpot files in library contain documentation of cutoff andvalidation tests.

V a l i d a t i o n t e s t−−−−−−−−−−−−−−−

#1 Cr3Si (SG 223) , exp . l a t t i c e constant a=4.555 , CASTEP wi thGGA ( FINE c u t o f f ) g ives 4.525 (−0.7%)

#2 Cubic Si , exp . l a t t i c e constant a=5.429 , CASTEP (GGA, PRECISE)gives 5.440 (+0.2%)

#3 MoSi2 (SG 139) , exp . l a t t i c e constants a=3.2 , c =7.85 ,CASTEP (GGA, PRECISE) gives 3.195 (−0.1%) and 7.791 (−0.8%).

#4 SiO2 ( alpha quartz , SG 154) , exp . l a t t i c e constants area=4.91 , c =5.402 , CASTEP (GGA, PRECISE) gives 4.987 (+1.5%) and5.459 (+1.1%)

#5 SrAu2Si2 , exp . l a t t i c e constants a=4.37 , c =10.14 , CASTEP(GGA, FINE ) g ives 4.437 (+1.5%) and 10.074 (−0.7%)

N.B. Really ought to be comparing against all-electron calculations not expt. Aim is to testpseudopotential, not XC functional.

Generating Pseudopotentials

Introduction













Conclusions


Many pseudopotential codes available:

● fhi98pp (http://th.fhi-berlin.mpg.de/th/fhi98md/fhi98PP/)TM, Hamman

● OPIUM (http://opium.sourceforge.net) RRJK, TM, Kerker● Vanderbilt USP code

(http://www.physics.rutgers.edu/~dhv/uspp/) USP● CASTEP - USP, TM, Optimised

http://th.fhi-berlin.mpg.de/th/fhi98md/fhi98PP/

http://opium.sourceforge.net

http://www.physics.rutgers.edu/~dhv/uspp/

CASTEP OTF Pseudopotentials

Introduction













Conclusions


● Vanderbilt ultrasoft - scheme used is very similar to Vanderbilt’s own and superior to VASPs.● Optimized (RRKJ) norm-conserving● Troullier-Martins norm-conserving● Specified by special descriptive string in .cell file

%BLOCK species_potO 2|1.3|16.537|18.375|20.212|20UU:21UU(qc=7.5)%ENDBLOCK species_pot

● local potential is d● rc = 1.3● 20UU means use 2 2s ultrasoft projectors● Optimize orbitals with qc = 7.5● Other properties such as rinner set as default.

%BLOCK species_potO 2|1.3|16.537|18.375|20.212|20N:21N(tm)%ENDBLOCK species_pot

generates a Troullier-Martins O with one s and one p projector.

%BLOCK species_potO 2|1.3|16.537|18.375|20.212|20UU:21UU{1s1}(qc=7.5)%ENDBLOCK species_pot

changes the reference configuration - in this case making a potential with a core hole.

Testing: atomic tests

Introduction













Conclusions


Square brackets “[]” turns testing on

O 2|1.3|16.537|18.375|20.212|20UU:21UU(qc=7.5)[]

will write grace plot files for betas, potentials and cutoff energy characteristics. Test correctnessby comparing AE and PS eigenvalues

---------------------------------------AE eigenvalue nl 10 = -18.40702924AE eigenvalue nl 20 = -0.54031716AE eigenvalue nl 21 = -0.01836880---------------------------------------PS eigenvalue nl 20 = -0.54026290PS eigenvalue nl 21 = -0.01829559---------------------------------------

Test transferrability with test at configuration different from reference.

O 2|1.3|16.537|18.375|20.212|20UU:21UU(qc=7.5)[2p4.75]

Check that cutoff energy curve allows convergence to (say) 0.001 eV/atom. Otherwise you mayneed to increase qc.

Testing: solid-state tests

Introduction













Conclusions


● Lattice parameters useful but not sufficient test of transferrability- onlyverifies minimum of binding curve, not energy.

● Γ-point phonon frequencies of elemental solid or simple binary are goodtest.

● Equation of state (compression curve) and fitted Birch-Murnaghanparameters also good.

● Stringent (but expensive) test is cohesive energy of solid.

What can go wrong

Introduction













Conclusions


● Generation is by a recipe - not guaranteed to always work.● Failure to meet basic PSP eigenfunction/value matching● Cannot generate PSP because of numerical ill-conditioning● Ghost states● Bad choice of local component● Core overlap in solid (eg CO2−

3 or CO molecule)● Semicore states may need to be treated as valence (at least 1st half of 3d

transition metal series).

Comparison of accuracy

Introduction













Conclusions


LDA lattice parameters of rutile TiO2.a/A c/A u

PW-LDA (OTF1) USP/NAE 4.550 (-0.18%) 2.919 (-0.03%) 0.3039PW-LDA (OTF2) USP 4.549 (-0.20%) 2.919 (-0.03%) 0.3039PW-LDA (VAN) USP 4.551 (-0.15%) 2.921 (+0.03%) 0.3039PW-LDA (TM) 4.536 (-0.48%) 2.915 (-0.17%) 0.304PW-LDA (R&B) NC 4.563 (+0.11%) 2.932 (+0.41%) 0.3040PW-LDA (Old-NC ) NC 4.596 (+0.83%) 2.983 (+2.16%) 0.3041PW-LDA (OP-NC) NC 4.526 (-0.70%) 2.908 (-0.41%) 0.3041PW-LDA (PAW) PAW 4.557 (-0.22%) 2.928 (+0.27%) 0.304FP-LAPW-LDA LAPW 4.558 2.920 0.3039Expt 4.582 (+0.53%) 2.953 (+1.13%) 0.305

● Comparison should be against all-electron, not experiment.● USPs are systematically more accurate than norm-conserving.● Rappe and Bennett DNL Optimized norm-conserving achive 0.5% lattice param accuracy.● Old CASTEP NC library falls short of modern standards of accuracy.● Other elements and tests may reveal other problems.

Conclusions

Introduction




Conclusions

❖ Conclusion


Conclusion

Introduction




Conclusions

❖ Conclusion


● Pseudopotentials based on well-founded theory.● Generation schemes involve some numerical “cookery” and sometimes fail

to work (well or at all).● Transferrability is measurable and pseudopotential accuracy can be

systematically improved.● Ultrasoft pseutopotentials can be highly accurate. (0.25% in lattice

parameter)● Even norm conserving schemes can approach this (sometimes)● CASTEP’s OTF generator gives you great power to make your own custom

pseudopotentials● With great power comes great responsibility.... TEST CAREFULLY

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