[PPT] - Pseudopotentials (Part I): Georg KRESSE Institut f ur PowerPoint Presentation

SLIDE 1

Pseudopotentials (Part I):

Georg KRESSE

Institut f¨ ur Materialphysik and Center for Computational Materials Science Universit¨ at Wien, Sensengasse 8, A-1090 Wien, Austria

ienna imulation ackage b-initio

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 1

SLIDE 2

Overview

the very basics

– periodic boundary conditions – the Bloch theorem – plane waves – pseudopotentials

determining the electronic groundstate
effective forces on the ions

general road-map to the things you will hear in more detail later

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 2

SLIDE 3

Periodic boundary conditions

as almost all plane wave codes VASP uses al-

ways periodic boundary conditions

the interaction between repeated images must

be handled by a sufficiently large vacuum re- gion

sounds

disastrous for the treatment

f

molecules but large molecules can be handled with a comparable or even better performance than by e.g. Gaussian

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 3

SLIDE 4

The Bloch theorem

the translational invariance implies that a good quantum number exists, which is

usually termed k k corresponds to a vector in the Brillouin zone

all electronic states can be indexed by this quantum number

Ψk

in a one-electron theory, one can introduce a second index, corresponding to the one

electron band n ψn

✁

k

✂

r

✄

the Bloch theorem implies that the (single electron) wavefunctions observe the equations ψn

✁

k

✂

r

☎

τ

✄✝✆

ψn

✁

k

✂

r

✄

eikτ

✞

where τ is any translational vector leaving the Hamiltonian invariant

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 4

SLIDE 5

The DFT Hamiltonian

the charge density is determined by integrating over the entire Brillouin zone and

summing over the filled bands ρe

✂

r

✄✝✆

∑∞

n

✟

d3k fn

✁

kψn

✁

k

✂

r

✄

ψ

✠

n

✁

k

✂

r

✄

where the charge density is cell periodic (can be seen by inserting the Bloch theorem) and fn

✁

k

✆ ✂

1

☎

exp

✂

β

✂

εn

✁

k

✡

εFermi

✄ ✄ ✄ ☛

1 are the Fermi-weights

the KS-DFT equations (Schr¨
dinger like) are given by

☞ ✡

¯ h2 2me ∆

☎

V eff

✂

r

✞ ✌

ρe

✂

r

✍ ✄ ✎ ✄ ✏

ψn

✁

k

✂

r

✄✝✆

εn

✁

kψn

✁

k

✂

r

✄

V eff

✂

r

✞ ✌

ρe

✂

r

✍ ✄ ✎ ✄✝✆

e2 ρe

✂

r

✍ ✄ ☎

ρion

✂

r

✍ ✄ ✑

r

✡

r

✍ ✑

d3r

✍ ☎

Vxc

✂

ρe

✂

r

✄ ✄

ρion is the ionic charge distribution

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 5

SLIDE 6

Plane waves

introduce the cell periodic part un

✁

k of the wavefunctions

ψn

✁

k

✂

r

✄✝✆

un

✁

k

✂

r

✄

eikr

✞

un

✁

k

✂

r

✄

is cell periodic (insert into Bloch theorem)

all cell periodic functions are now written as a sum of plane waves

un

✁

k

✂

r

✄✝✆

1 Ω1

✒

2 ∑ G

CGnkeiGr

✞

ψn

✁

k

✂

r

✄ ✆

1 Ω1

✒

2 ∑ G

CGnkei

✓

G

✔

k

✕

r

ρ

✂

r

✄✝✆

∑

G

ρGeiGr

✞

V

✂

r

✄✝✆

∑

G

VGeiGr

✞

in practice only those plane waves

✑

G

☎

k

✑

are included which satisfy ¯ h2 2me

✑

G

☎

k

✑

2

✖

Ecutoff

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 6

SLIDE 7

Fast Fourier transformation

τ1 τ1 π / τ 1 τ2 b1 b2 real space reciprocal space FFT 1 2 3 1 2 3 4 5 −1 −2 −3 −4 N/2 −N/2+1 1 1 x = n / N 1 1 g = n 2 N−1

cut G

Crnk

✗

∑

G

CGnkeiGr CGnk

✗

1 NFFT ∑

r

Crnke

✘

iGr

ψn

✙

k

✚

r

✛ ✗

1 Ω1

✜

2Crnkeikr

✢

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 7

SLIDE 8

Why are plane waves so convenient

historical reason:

many elements exhibit a band-structure that can be interpreted in a free electron picture (metallic s and p elements) the pseudopotential theory was initially developed to cope with these elements (pseudopotential perturbation theory)

practicle reason:

the total energy expressions and the Hamiltonian H are dead simple to implement a working pseudopotential program can be written in a few weeks using a modern rapid prototyping language

computational reason:

because of it’s simplicity the evaluations of Hψ is exceedingly efficient using FFT’s

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 8

SLIDE 9

Computational reason

evaluation of Hψn

✁

k

✂

r

✄ ✡

¯ h2 2me ∆

☎

V

✂

r

✄

ψn

✁

k

✂

r

✄

and using the convention

✣

r

✑

G

☎

k

✤✝✆

1 Ω1

✥

2 ei

✓

G

✔

k

✕

r

✦ ✣

G

☎

k

✑

ψn

✁

k

✤✝✆

CGnk

kinetic energy:

✣

G

☎

k

✑ ✡

¯ h2 2me ∆

✑

ψn

✁

k

✤✝✆

¯ h2

✑

G

☎

k

✑

2 2me CGnk Nplanewaves

local potential:

✣

G

☎

k

✑

V

✑

ψn

✁

k

✤✝✆

1 NFFT ∑

r

VrCrnke

☛

iGr

NFFT logNFFT

if H would be stored as a matrix with Nplanewaves

✧

Nplanewaves components Nplanewaves

✧

Nplanewaves operations would be required

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 9

SLIDE 10

Local part of Hamiltonian

ψr r V ψr r V ψG+k FFT Gcut FFT ψ <k+G V >

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 10

SLIDE 11

Pseudopotential approximation

the number of plane waves would exceed any practicle limits except for H and Li

★

pseudopotentials instead of exact potentials must be applied

three different types of potentials are supported by VASP

– norm-conserving pseudopotentials – ultra-soft pseudopotentials – PAW potentials they will be discussed in more details in later sessions

all three methods have in common that they are presently frozen core methods

i.e. the core electrons are pre-calculated in an atomic environment and kept frozen in the course of the remaining calculations

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 11

SLIDE 12

Pseudopotential: essential idea

wave-function R (a.u.) 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5

s : E= -0.576 R c=1.9 p : E= -0.205 R c=1.9

✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰

3p 2s 1s 3s 2p

✱ ✱ ✱ ✱ ✲ ✲ ✲ ✲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾

Al Al Al

2p 1s 3p 3s

effectiv Al atom PAW Al atom exact potential (interstitial region) pseudopotential

2p and 1s are nodeless !!!! nodal structure is retained

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 12

SLIDE 13

Scattering approach

wave-function R (a.u.) 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5

s : E= -0.576 R c=1.9 p : E= -0.205 R c=1.9

✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

Al Al

solve the Schr¨
dinger equation in the interstitial region only, using energy and

angular momentum dependent boundary conditions at the spheres

∂φl

✚

r

❇

ε

✛

∂r 1 φl

✚

r

❇

ε

✛ ❈ ❈ ❈ ❈

rc

✗

∂ logφl

✚

r

❇

ε

✛

∂r

❈ ❈ ❈ ❈

rc

φl are the regular solutions of the radial Schr¨

dinger equation inside the spheres for

the angular momentum quantum number l and the energy ε

details of the wavefunctions (number of nodes in the spheres) do not enter
pseudopotential: select one specific energy ε in the centre of the valence band and

replace the exact wavefunction φl by a pseudo-wavefunction with ˜ φl

∂ logφl

✚

r

❇

ε

✛

∂r

❈ ❈ ❈ ❈

rc

✗

∂ log ˜ φl

✚

r

❇

ε

✛

∂r

❈ ❈ ❈ ❈

rc

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 13

SLIDE 14

Norm-conserving PP

wave-function R (a.u.) 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5

s : E= -0.576 R c=1.9 p : E= -0.205 R c=1.9

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

❍

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑

Al Al

bviously one would like to get the right distribution of charge between the spheres

and the interstitial region as well

✦

norm-conserving pseudopotentials

rc

φl

✂

r

✞

ε

✄

φ

✠

l

✂

r

✞

ε

✄

4πr2dr

✆

rc

˜ φl

✂

r

✞

ε

✄

˜ φ

✠

l

✂

r

✞

ε

✄

4πr2dr

norm-conservation has another important consequence

the scattering properties are not only correct at the reference energy ε but also in a small energy interval around ε ∂ ∂ε ∂ logφl

✂

r

✞

ε

✄

∂r

▲ ▲ ▲ ▲

rc

✆

∂ ∂ε ∂ log ˜ φl

✂

r

✞

ε

✄

∂r

▲ ▲ ▲ ▲

rc

▼

l

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 14

SLIDE 15

Pseudopotential generation

all-electron calculation for a reference atom
chose energies εl at which the pseudisation is performed

usually these are simply the eigen-energies of the bound valence states, but in principle any energy can be chosen (centre of valence band)

replace the exact wavefunction by a node less pseudo-wavefunction observing the

following four requirements: ˜ φ

✂

rc

✄ ✓

n

✕ ✆

φ

✂

rc

✄ ✓

n

✕

for n

✆ ✞❖◆ ◆ ◆ ✞

2

✂

n

✄

is the n

◆

th derivative 4π

rc

˜ φ

✂

r

✄

2 r2dr

✆

4π

rc

φ

✂

r

✄

2 r2dr

norm-conservation condition

this pseudopotential conserves exactly the scattering properties of the original atom

in the atomic configuration but it is an approximation in another environment

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 15

SLIDE 16

Projector augmented wave method

P.E. Bl¨

chl, Phys. Rev. B50, 17953 (1994); G. Kresse, and J. Joubert, Phys. Rev. B 59, 1758 (1999).
wave function (and energy) are decomposed into three terms:

✑

ψn

✤ ✆ ✑

˜ ψn

✤ ✡

∑

atoms

✑

˜ φlmε

✤

clmε

☎

∑

atoms

✑

φlmε

✤

clmε = + − exact exact onsite pseudo−onsite pseudo (node less) plane waves radial grids radial grids

P P P P P P P P P P P P P P P P P P P P P P P P P P P P ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

no interaction between different spheres and plane waves

❵

efficient

decomposition into three terms holds for

– wave functions – charge densities – kinetic energy – Hartree.- and exchange correlation energy

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 16

SLIDE 17

Determining the electronic groundstate

by iteration – self consistency (old fashioned)
start with a trial density ρe, set up the Schr¨
dinger equation, and solve the

Schr¨

dinger equation to obtain wavefunctions ψn

✂

r

✄ ✡

¯ h2 2me ∇2

☎

V eff

✂

r

✞ ✌

ρe

✂

r

✍ ✄ ✎ ✄

ψn

✂

r

✄✝✆

εnψn

✂

r

✄

n

✆

1

✞❖◆ ◆ ◆ ✞

Ne

❛

2

as a result one obtains a new charge density ρe

✂

r

✄ ✆

∑n

✑

ψn

✂

r

✄ ✑

2 and a new one

electron potential: V eff

✂

r

✞ ✌

ρe

✂

r

✍ ✄ ✎ ✄✝✆

e2 ρe

✂

r

✍ ✄ ☎

ρion

✂

r

✍ ✄ ✑

r

✡

r

✍ ✑

d3r

✍ ☎

Vxc

✂

ρe

✂

r

✄ ✄

new Schr¨

dinger equation

★

iteration

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 17

SLIDE 18

Self-consistency scheme

trial-charge ρin and trial-wavevectors ψn

❜ ❜ ❜ ❜ ❝

set up Hamiltonian H

✂

ρin

✄ ❝

iterative refinements of wavefunctions

✌

ψn

✎ ❝

new charge density ρout

✆

∑n fn

✑

ψn

✂

r

✄ ✑

2

❝

refinement of density ρin

✞

ρout

★

new ρin

❝ ❞ ❞ ❞ ❞ ❞ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❞ ❞ ❞ ❞ ❞

no ∆E

✖

Ebreak calculate forces, update ions

❢ ❝ ❢ ❣

two subproblems

ptimization of

❤

ψn

✐

and ρin

❣

refinement of density: DIIS algorithm

P. Pulay, Chem. Phys. Lett. 73,

393 (1980).

❣

refinement of wavefunctions: blocked Davidson like algorithm

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 18

SLIDE 19

What have all iterative matrix diagonalisation schemes in common ?

ne usually starts with a set of trial vectors (wavefunctions) representing the filled

states and a few empty one electron states

✌

ψn

✑

n

✆

1

✞ ◆ ◆ ✞

Nbands

✎

these are initialised using a random number generator

then the wavefunctions are improved by adding a certain amount of the residual

vector to each the residual vector is defined as

✑

R

✂

ψn

✄ ✤✝✆ ✂

H

✡

εappS

✄ ✑

ψn

✤

εapp

✆ ✣

ψn

✑

H

✑

ψn

✤

Hψn is exactly operation discussed before (efficient)

adding a small amount of the residual vector

ψn

✦

ψn

☎

λR

✂

ψn

✄

is in the spirit of the steepest descent approach (“Jacobi relaxation”)

G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 19

SLIDE 20

We know the groundstate, what now ?

Hellman-Feynmann theorem allows to calculate the ionic forces and the stress tensor

by moving along the ionic forces (steepest descent) the ionic groundstate can be

calculated

we can displace ions from the ionic groundstate, and determine the forces on all other

ions

★

effective inter-atomic force constants and vibrational frequencies

molecular dynamics by using Newtons equation of motion
G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 20

SLIDE 21

Vibrational properties

the vibrational frequencies are calculated using a brute force method:

in the supercell all “selected atoms” are displaced from their groundstate position forces

★

force constants

Γ M K Γ 200 400 600 800 1000 1200 1400 1600

Frequency cm

1

LO SH LA ZO SH ZA

not possible presently

symmetry is not used !!!
works well for molecules, but has to be applied with great care to solids
G. KRESSE, PSEUDOPOTENTIALS (PART I)

Page 21

SLIDE 22

Pseudopotentials (Part II) and PAW

Georg KRESSE

Institut f¨ ur Materialphysik and Center for Computational Materials Science Universit¨ at Wien, Sensengasse 8, A-1090 Wien, Austria

ienna imulation ackage b-initio

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 1

SLIDE 23

Overview

pseudopotential basics
normconserving pseudopotentials

adopted pseudization strategy

G. Kresse, and J. Hafner, J. Phys.: Condens. Matter 6 (1994)
from normconserving to ultrasoft pseudopotentials
the PAW method
G. Kresse, and J. Joubert, Phys. Rev. B 59, 1758 (1999).
where to be careful ?

– local pseudopotentials – simultaneous representation of valence and semi-core states – magnetic calculations

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 2

SLIDE 24

Normconserving pseudopotentials: General strategy

all-electron calculation for a reference atom

(rhfsps)

pseudization of valence wave functions

(rhfsps)

chose local pseudopotential and factorize

(fourpot3)

un-screening of atomic potential to obtained ionic pseudopotential

(fourpot3)

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 3

SLIDE 25

Pseudization of valence wave functions

different schemes have been proposed in literature, but the general strategy is always similar

calculate exact all-electron wave function φ

✁

r

✂

replace exact φ

✁

r

✂

inside pseudization radius by a suitable “soft” pseudo wave function ˜ φ

✁

r

✂

must fulfill some continuity conditions ˜ φ

✁

r

✂ ✄ ☎ ✆ ✝

∑i αiβi

✁

r

✂

r

✞

rc φ

✁

r

✂

r

✟ ✄

rc ˜ φ

✁

rc

✂✡✠

n

☛ ✄

φ

✁

rc

✂✡✠

n

☛

for n

✄ ☞✍✌ ✌ ✌ ☞

2

possibly impose normconservation condition

4π

rc

˜ φ

✁

r

✂

2 r2 dr

✄

4π

rc

φ

✁

r

✂

2 r2 dr

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 4

SLIDE 26

Which expansion set should one use?

many different basis sets have been proposed in the literature

presently the two most prominent ones are – polynomials (Troullier and Martins) ˜ φ

✁

r

✂ ✄

c0

✎

c2r2

✎

c4r4

✎

c6r6

✎

c8r8

✎

c10r10

✎

c12r12 – spherical Bessel-functions (RRKJ—Rappe, Rabe, et. al.) ˜ φ

✁

r

✂ ✄

3

✠

4

☛

∑

i

✏

1 αi jl

✁

q

✑

ir

✂

with q

✑

i such that

jl

✁

q

✑

irc

✂ ✑

jl

✁

q

✑

irc

✂ ✄

φ

✁

rc

✂ ✑

φ

✁

rc

✂ ☞

the last one is the standard scheme for VASP pseudopotentials

the basis set I use is generally minimal (3 or sometimes 4 Bessel-functions)

for PAW and US pseudopotentials only 2 spherical Bessel-functions are required

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 5

SLIDE 27

Why are spherical Bessel-functions so convenient

close analogy between plane waves and spherical Bessel-functions

the required cutoff can be calculated directly from the expansion set

˜ φ

✒

r

✓✕✔

3

✖

4

✗

∑

i

✘

1

αi jl

✒

q

✙

ir

✓

find maximum qi

✚

Ecut

✛

¯ h2 2me max

✁

qi

✂

2

✜

1

✌

5

I always use a minimal basis set

in the original RRKJ scheme, more spherical Bessel-functions were used, and the wave functions were optimized for a selected cutoff

ur tests indicate that this is contra-productive
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 6

SLIDE 28

Factorization

required to speed up the calculations D.M. Bylander, et al., Phys. Rev. B 46, 13756 (1992)

chose local reference potential Vloc
construct a projector such that

✢

p

✣

˜ φ

✤ ✄

1

✣

p

✤

∝

✥

¯ h2 2me ∆

✎

Vloc

✥

ε

✣

˜ φ

✤

the factorized Hamiltonian is given by

H

✄ ✥

¯ h2 2me ∆

✎

Vloc

✎ ✣

p

✤

D

✢

p

✣

with D

✄ ✢

˜ φ

✣ ✦

¯ h2 2me ∆

✥

Vloc

✎

ε

✧ ✣

˜ φ

✤

ne recognizes immediately that:

✢

˜ φ

✣ ✦ ✥

¯ h2 2me ∆

✎

Vloc

✎ ✣

p

✤

D

✢

p

✣ ✧ ✣

˜ φ

✤ ✄

ε

✢

˜ φ

✣

˜ φ

✤

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 7

SLIDE 29

What have we acchived at this point

wave-function R (a.u.) 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5

s : E= -0.576 R c=1.9 p : E= -0.205 R c=1.9

the exact wavefunction has been replaced by it’s pseudo counterpart

and a “pseudo” Hamiltonian has been constructed

✥

¯ h2 2me ∆

✎

VAE

✣

φ

✤ ✄

ε

✣

φ

✤ ✚ ✥

¯ h2 2me ∆

✎

Vloc

✎ ✣

p

✤

D

✢

p

✣ ✣

˜ φ

✤ ✄

ε

✣

˜ φ

✤

at the energy ε, φ and ˜

φ are identical outside of the cutoff radius

φ and ˜

φ have the same norm inside the cutoff radius

★

at the energy ε

✎

δε φ and ˜ φ are identical outside of the cutoff radius

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 8

SLIDE 30

Two reference energies

pseudize at two reference energies:

✩

φi

✣

i

✄

1

☞

2

✪

construct two projectors such that

✢

pi

✣

˜ φ j

✤ ✄

δij for all i

☞

j

✣

pi

✤ ✄

∑

j

αij

✥

¯ h2 2me ∆

✎

Vloc

✥

ε j

✣

˜ φ j

✤

factorized Hamiltonian is given by

H

✄ ✥

¯ h2 2me ∆

✎

Vloc

✎

∑

ij

✣

pi

✤

Dij

✢

pj

✣

Dij

✄ ✢

˜ φi

✣ ✦

¯ h2 2me ∆

✥

Vloc

✎

ε j

✧ ✣

˜ φ j

✤

ne recognizes immediately that:

✢

˜ φi

✣

H

✣

˜ φ j

✤ ✄

ε j

✢

˜ φi

✣

˜ φ j

✤

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 9

SLIDE 31

Two reference energies: practical considerations

the pseudo wavef. must fulfill a generalized normconserv. condition:

4π

rc

˜ φi

✁

r

✂

˜ φ j

✁

r

✂

r2 dr

✄

4π

rc

φi

✁

r

✂

φ j

✁

r

✂

r2 dr

✫

i

☞

j

in the VASP PP generation program only

rc

˜ φi

✁

r

✂

˜ φi

✁

r

✂

r2 dr

✄

rc

φi

✁

r

✂

φi

✁

r

✂

r2 dr is enforced

to correct for this error, augmentation charges would be required, but these are

neglected as a result Dij is not Hermitian Dij

✄ ✢

˜ φi

✣ ✦

¯ h2 2me ∆

✥

Vloc

✎

ε j

✧ ✣

˜ φ j

✤

ff-diagonal elements are averaged to make the matrix D symmetric
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 10

SLIDE 32

US pseudopotentials, very similar to NC pseudopotentials

pseudize at two reference energies
construct two projectors such that

✢

pi

✣

˜ φ j

✤ ✄

δij for all i

☞

j

✣

pi

✤ ✄

∑

j

αij

✥

¯ h2 2me ∆

✎

Vloc

✥

ε j

✣

˜ φ j

✤

the factorized Hamiltonian and overlap operator are given by

H

✄ ✥

¯ h2 2me ∆

✎

Vloc

✎

∑

ij

✣

pi

✤

Dij

✢

pj

✣

S

✄

1

✎

∑

ij

✣

pi

✤

Qij

✢

pj

✣

Dij

✄ ✢

˜ φi

✣ ✦

¯ h2 2me ∆

✥

Vloc

✎

ε j

✧ ✣

˜ φ j

✤ ✎

ε jQij Qij

✄ ✢

φi

✣

φ j

✤ ✥ ✢

˜ φi

✣

˜ φ j

✤

ne can show that:

✢

˜ φi

✣

H

✣

˜ φ j

✤ ✄ ✢

˜ φi

✣

S

✣

˜ φ j

✤

ε j

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 11

SLIDE 33

What does all that mean?

let us look again at the definition of Dij Dij

✄ ✢

˜ φi

✣

¯ h2 2me ∆

✥

Vloc

✎

ε j

✣

˜ φ j

✤ ✎

ε jQij

✄ ✢

˜ φi

✣

¯ h2 2me ∆

✥

Vloc

✣

˜ φ j

✤ ✎

ε j

✁ ✢

˜ φi

✣

˜ φ j

✤ ✎

Qij

✂ ✄ ✥ ✢

˜ φi

✣ ✥

¯ h2 2me ∆

✎

Vloc

✣

˜ φ j

✤ ✎

ε j

✁ ✢

˜ φi

✣

˜ φ j

✤ ✎ ✢

φi

✣

φ j

✤ ✥ ✢

˜ φi

✣

˜ φ j

✤ ✂ ✄ ✥ ✢

˜ φi

✣ ✥

¯ h2 2me ∆

✎

Vloc

✣

˜ φ j

✤ ✎ ✢

φi

✣

ε j

✣

φ j

✤ ✄ ✥ ✢

˜ φi

✣ ✥

¯ h2 2me ∆

✎

Vloc

✣

˜ φ j

✤ ✬ ✭✮ ✯

energy pseudo onsite

✎ ✢

φi

✣ ✥

¯ h2 2me ∆

✎

VAE

✣

φ j

✤ ✬ ✭✮ ✯

energy AE onsite

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 12

SLIDE 34

US-PP: what they really do

character of wave function:

ci

✄ ✢

˜ pi

✣

˜ Ψn

✤ ✢

pi

✣

˜ φ j

✤ ✄

δij

nsite occupancy matrix (or density matrix): ρij

✄ ✢

˜ Ψn

✣

pi

✤ ✢

pj

✣ ✢

˜ Ψn

✤

energy is the sum of three terms

E

✔ ✰

∆

✱

Vloc

✰

ρi j

✲

˜ φi

✳ ✰

∆

✱

Vloc

✳

˜ φ j

✴ ✱

ρi j

✲

φi

✳ ✰

∆

✱

VAE

✳

φ j

✴

= +

pseudo-onsite

pseudo AE-onsite AE

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ✿ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄

US-PP method is in principle an exact frozen core all-electron method
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 13

SLIDE 35

Mixed basis set with an implicit dependency

US-PP’s carry a small rucksack, with two additional sets of basis functions

defined around each atomic site – one for the soft pseudo-wave functions

✩

˜ φi

✪

– one for the AE wave functions

✩

φi

✪

for each atomic sphere the energy is evaluated using these two sets

and the calculated energy is subtracted and added, respectively

the onsite occupancy matrix (density matrix) for these two sets is calculated from

the plane wave coefficients ρij

✄ ✢

˜ Ψn

✣

pi

✤ ✢

pj

✣ ✢

˜ Ψn

✤

PAW inspired formulation
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 14

SLIDE 36

Practical considerations

the US-PP method can not be implemented exactly, since currently no method

exists to handle the rapid variations of the all-electron wave functions

a regular grid does not work, maybe wavelets would be an option

in practice, I therefore adopt a modified prescription for US-PP’s:

exact AE wave functions

✚

norm-conserving wave functions φ

✁

r

✂ ✚

φnorm

❅

conserving

✁

r

✂

augmentation charge: Qij

✁

r

✂ ✄

φnc

i

✁

r

✂

φnc

j

✁

r

✂ ✥

˜ φi

✁

r

✂

˜ φ j

✁

r

✂

these US-PP’s yield exactly the same results as the corresponding NC-PP’s, but at

much lower cutoffs

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 15

SLIDE 37

Why are US-PP’s softer

pseudo wave function is represented as a sum of two spherical Bessel functions

instead of three ˜ φ

✁

r

✂ ✄

2

∑

i

✏

1 αi jl

✁

q

✑

ir

✂

with q

✑

i such that

jl

✁

q

✑

irc

✂ ✑

jl

✁

q

✑

irc

✂ ✄

φ

✁

rc

✂ ✑

φ

✁

rc

✂ ☞

additionally rc can be increased compared to NC potentials

there is no need to represent the charge distribution of the AE wave function, since this is done by the augmentation the pseudo wave functions follow remarkably well the AE wave functions, even for values much smaller than rc

basis sets are roughly a factor 2-3 smaller
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 16

SLIDE 38

Example for this behavior

e.g. Cu (NC) Cu (US)

wave-function R (a.u.) 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

s :E=

0.357

Rc= 2.5 p : E=

0.058

Rc= 2.5 d : E=

0.393

Rc= 2.0

wave-function R (a.u.) 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

s : E=

0.357

Rc= 2.5 p : E=

0.058

Rc= 2.5 d : E=

0.393

Rc= 2.0

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 17

SLIDE 39

The role of the local potential

the local potential needs to describe scattering properties for radial quantum

numbers not included in the projectors – much underestimated problem – resulting errors can be 1-2 % in the lattice constant

the tails of d electrons (transition metals) or p electrons (oxygen) overlap into the

pseudization region they are picked up as high l components (FLAPW)

ideally one would like to use very attractive local potentials

but

★

ghost-state problem

in most cases, compromises must be made
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 18

SLIDE 40

Ghost-state problems

particularly severe for alkali, alkali-earth and early transition metals
the more attractive the local potential and the smaller rc, the more likely it is to

have a ghost-state; Zr example

xl (E) E (Ry)

3
2
1

1 2

4
2

2 4

s p d f g

xl (E) E (Ry)

3
2
1

1 2

4
2

2 4

s p d f g

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 19

SLIDE 41

solution: treat semi-core states as valence

Pseudopotential generation in practice: the PSCTR file

TITEL = US O LULTRA = T use ultrasoft PP ? RWIGS = 1.40 nn distance ICORE = 2 NE = 100 LCOR = .TRUE. QCUT =

1

Description l E TYP RCUT TYP RCUT 15 1.13 23 1.40 15 1.13 23 1.40 1 15 1.13 23 1.55 1 15 1.13 23 1.55 2 0.0 7 1.55 7 1.55

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 20

SLIDE 42

PAW: basic idea

P.E. Bl¨

chl, Phys. Rev. B50, 17953 (1994)
Kohn-Sham equation

E

✄

∑

n

fn

✢

Ψn

✣ ✥

1 2∆

✣

Ψn

✤ ✎

EH

❆

n

✎

nZ

❇ ✎

Exc

❆

n

❇ ✌

frozen core approximation

for the valence electrons, a transformation from the pseudo to the AE wavefunction is defined:

✣

Ψn

✤ ✄ ✣

˜ Ψn

✤ ✎

∑sites

lmε

✁ ✣

φlmε

✤ ✥ ✣

˜ φlmε

✤ ✂ ✢

˜ plmε

✣

˜ Ψn

✤

lm is an index for the angular and magnetic quantum numbers

ε refers to a particular reference energy

˜

plmε projector function ˜ φlmε partial wave

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 21

SLIDE 43

PAW: basic idea

transformation:

✣

Ψn

✤ ✄ ✣

˜ Ψn

✤ ✎

∑

✁ ✣

φlmε

✤ ✥ ✣

˜ φlmε

✤ ✂ ✢

˜ plmε

✣

˜ Ψn

✤

the “character” of an arbitrary pseudo-wavefunction ˜

Ψn at one site can be calculated by multiplication with the projector function at that site clmε

✄ ✢

˜ plmε

✣

˜ Ψn

✤

inside each sphere the wavefunctions can be determined:

✣

˜ Ψn

✤

sphere

✄ ✁ ✛ ✂

∑

lmε

✣

˜ φl

❈

m

❈

ε

❈ ✤

clmε

✣

Ψn

✤

sphere

✄ ✁ ✛ ✂

∑

lmε

✣

φl

❈

m

❈

ε

❈ ✤

clmε

the projector functions must be dual to the pseudo-wavefunction

✢

˜ plmε

✣

˜ φl

❈

m

❈

ε

❈ ✤ ✄

δl

❉

l

❈

δm

❉

m

❈

δε

❉

ε

❈

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 22

SLIDE 44

PAW: addidative augmentation

character of wavefunction:

clmε

✄ ✢

˜ plmε

✣

˜ Ψn

✤

✣

Ψn

✤ ✄ ✣

˜ Ψn

✤ ✥

∑

✣

˜ φlmε

✤

clmε

✎

∑

✣

φlmε

✤

clmε

= +

pseudo-onsite

pseudo AE-onsite AE

❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

❍

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚

same trick works for

– wavefunctions – charge density – kinetic energy – exchange correlation energy – Hartree energy

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 23

SLIDE 45

Derivation of the PAW method is straightforward

for instance, the kinetic energy is given by

Ekin

✄

∑

n

fn

✢

Ψn

✣ ✥

∆

✣

Ψn

✤

by inserting the transformation (i

✄

lmε)

✣

Ψn

✤ ✄ ✣

˜ Ψn

✤ ✎

∑

i

✁ ✣

φi

✤ ✥ ✣

˜ φi

✤ ✂ ✢

˜ pi

✣

˜ Ψn

✤ ✌

into Ekin one obtains: Ekin

✄

˜ E

✥

˜ E1

✎

E1 (assuming completeness)

∑

n

fn

✲

˜ Ψn

✳ ✰

∆

✳

˜ Ψn

✴ ❯ ❱❲ ❳

˜ E

✰

∑

site ∑

✖

i

❨

j

✗

ρi j

✲

˜ φi

✳ ✰

∆

✳

˜ φ j

✴ ❯ ❱❲ ❳

˜ E1

✱

∑

site ∑

✖

i

❨

j

✗

ρi j

✲

φi

✳ ✰

∆

✳

φ j

✴ ❯ ❱❲ ❳

E1

ρij is an on-site density matrix: ρij

✄

∑n fnc

❩

i cj

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 24

SLIDE 46

Hartree energy

the pseudo-wavefunctions do not have the same norm as the AE wavefunctions

inside the spheres

to deal with long range electrostatic interactions between spheres

a soft compensation charge ˆ n is introd. (similar to FLAPW)

= +

AE

pseudo + compens. pseudo+comp. onsite AE-onsite

❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❬ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❫ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

Hartree energy becomes: EH

✄

˜ E

✥

˜ E1

✎

E1 EH

❆

˜ n

✎

ˆ n

❇ ✥

∑sites EH

❆

˜ n1

✎

ˆ n1

❇ ✎

∑sites EH

❆

n1

✎

ˆ n1

❇

˜ n1 pseudo-charge at one site ˆ n1 compensation charge at site

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 25

SLIDE 47

PAW energy functional

P.E. Bl¨

chl, Phys. Rev. B50, 17953 (1994).
total energy becomes a sum of three terms E

✄

˜ E

✎

E1

✥

˜ E1 ˜ E

✄

∑

n

fn

✢

˜ Ψn

✣ ✥

1 2∆

✣

˜ Ψn

✤ ✎

Exc

❆

˜ n

✎

ˆ n

✎

˜ nc

❇ ✎

EH

❆

˜ n

✎

ˆ n

❇ ✎

vH

❆

˜ nZc

❇ ✁

˜ n

✁

r

✂ ✎

ˆ n

✁

r

✂ ✂

d3r

✎

U

✁

R

☞

Zion

✂

˜ E1

✄

∑

sites ∑

✠

i

❉

j

☛

ρij

✢

˜ φi

✣ ✥

1 2∆

✣

˜ φ j

✤ ✎

Exc

❆

˜ n1

✎

ˆ n

✎

˜ nc

❇ ✎

EH

❆

˜ n1

✎

ˆ n

❇ ✎

Ωr

vH

❆

˜ nZc

❇ ❦

˜ n1

✁

r

✂ ✎

ˆ n

✁

r

✂ ❧

d3r E1

✄

∑

sites ∑

✠

i

❉

j

☛

ρij

✢

φi

✣ ✥

1 2∆

✣

φ j

✤ ✎

Exc

❆

n1

✎

nc

❇ ✎

EH

❆

n1

❇ ✎

Ωr

vH

❆

nZc

❇

n1

✁

r

✂

d3r

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 26

SLIDE 48

˜

E is evaluated on a regular grid Kohn Sham functional evaluated in a plane wave basis set with additional compensation charges to account for the incorrect norm of the pseudo-wavefunction (very similar to ultrasoft pseudopotentials) ˜ n

✄

∑n fn ˜ Ψn ˜ Ψ

❩

n

pseudo charge density ˆ n compensation charge

E1 and ˜

E1 are evaluated on radial grids centered around each ion Kohn-Sham energy evaluated for basis sets

✩

˜ ψi

✪

and

✩

ψi

✪

these terms correct for the shape difference between the pseudo and AE wavefunctions

no cross-terms between plane wave part and radial grids exist
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 27

SLIDE 49

General scheme

= +

AE

pseudo + compens. pseudo+comp. onsite AE-onsite

♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ q q q q q q q q q q q q q q q q q q q q q q q q q q q q r r r r r r r r r r r r r r r r r r r r r r r r r r r r s s s s s s s s s s s s s s s s s s s s s s s s s s s s t t t t t t t t t t t t t t t t t t t t t t t t t t t t ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

applies to all quantities

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 28

SLIDE 50

US-PP

D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).
the original derivation of US-PP is somewhat “problematic”

it’s more like accepting things than understanding them

in fact, the equations for US-PP’s can be derived rigidly from the PAW functional

by linearisation of the on-site terms E1 and ˜ E1 around the atomic reference configuration this shows the close relation between both approaches

but it also indicates when US-PP’s might be problematic:

the more the environment differs from the reference state the less accurate US-PP are

ur tests indicate that magnetism is the strongest perturbation

in other cases, US-PP and the PAW yield almost identical results

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 29

SLIDE 51

Construction of PAW potentials

first an AE calculation for a reference atom is performed
the AE wavefunctions of the valence states are pseudised

wave-function R (a.u.) 1 2 3 4 1 2 3

s : E= -0.296 R c=2.6 p1/2: E= -2.537 R c=2.6 d3/2: E= -0.228 R c=2.6

projectors are constructed as

✣

pi

✤ ✄

∑

j

αij

✥

¯ h2 2me ∆

✎

Vloc

✥

ε j

✣

˜ φ j

✤

they must obey

✢

˜ pi

✣

˜ φn

✤ ✄

δi

❉

n

projectors are dual to the pseudo wavefunction

to have a rather complete set of projectors

two partial waves for each quantum channel lm are constructed

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 30

SLIDE 52

PAW — US-PP method: molecules

results for the bond length of several molecules obtained with the US-PP, PAW

and AE approaches

plane wave cutoffs were around 200-400 eV
US-PP and the PAW method give the same results within 0.1-0.3%
well converged relaxed core AE calculations yield identical results
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 31

SLIDE 53

US-PP(data base) US-PP(special) PAW AE H2 1.447 1.447 1.446a Li2 5.127 5.120 5.120a Be2 4.524 4.520 4.521a Na2 5.667 5.663 5.67 a CO 2.163 2.141 (2.127) 2.141 (2.128) 2.129a N2 2.101 2.077 (2.066) 2.076 (2.068) 2.068a F2 2.696 2.640 (2.626) 2.633 (2.621) 2.615a P2 3.576 3.570 3.570 3.572a H2O 1.840 (1.834) 1.839 (1.835) 1.833a α(H2O)(

⑥

) 105.3 (104.8) 105.3 (104.8) 105.0a BF3 2.476 (2.470) 2.476 (2.470) 2.464b SiF4 2.953 (2.948) 2.953 (2.948) 2.949b values in paranthese were obtained with hard potentials at 700 eV

a NUMOL, R.M. Dickson, A.D. Becke, J. Chem. Phys. 99,3898 (1993), b Gaussian94

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 32

SLIDE 54

PAW — AE methods: molecules - energetics

400 eV plane wave cutoff xc PBE PBE rPBEb rPBEb exp meth PAW AEa PAW AEa CO 11.65 11.66 11.15 11.18 11.24 N2 10.39 10.53 10.09 10.09 9.91 NO 7.31 7.45 6.95 7.01 6.63 O2 6.17 6.14-6.24 5.75 5.78 5.22

a S. Kurth, J.P. Perdew, P. Blaha, Int. J. Quantum Chem. 75, 889 (1999). b revised Perdew Burke Ernzerhof functional, B. Hammer, L.B. Hansen, J.K. Norskov, Phys.

Rev. B 59, 7413 (1999).
G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 33

SLIDE 55

PAW — US-PP method: bulk, semiconductors

results for the equilibrium lat- tice constant a, cohesive energy Ecoh (with respect to non spin polarised atoms) and bulk mod- ulus B for several materials cal- culated with the the PAW, US- PP, and the FLAPW approach a

✒

˚ A3

✓

Ecoh(eV) B (MBar) diamond US-PP(current) 3.53

10.15

4.64 PAW(current) 3.53

10.13

4.63 LAPWa 3.54

10.13

4.70 PAWa 3.54

10.16

4.60 silicon US-PP(current) 5.39

5.96

0.95 PAW(current) 5.39

5.96

0.95 LAPWa 5.41

5.92

0.98 PAWa 5.38

6.03

0.98

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 34

SLIDE 56

PAW — US-PP method: bulk, metals

results for the equilibrium lat- tice constant a, cohesive energy Ecoh (with respect to non spin polarised atoms) and bulk mod- ulus B for several materials cal- culated with the the PAW, US- PP, and the FLAPW approach a

✒

˚ A3

✓

Ecoh(eV) B (MBar) bcc V US-PP(current) 2.93

9.41

2.02 PAW(current) 2.93

9.39

2.09 LAPWa 2.94

9.27

2.00 PAWa 2.94

9.39

2.00 fcc Ca US-PP(current) 5.34

2.20

0.0181 PAW(3s3p val) 5.34

2.19

0.0187 PAW(3p val) 5.34

2.20

0.0187 LAPWa 5.33

2.20

0.019 PAWa 5.32

2.24

0.019

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 35

SLIDE 57

PAW — US-PP method: bulk, ionic compounds

results for the equilibrium lat- tice constant a, cohesive energy Ecoh (with respect to non spin polarised atoms) and bulk mod- ulus B for several materials cal- culated with the the PAW, US- PP, and the FLAPW approach a

✒

˚ A3

✓

Ecoh(eV) B (MBar) CaF2 US-PP(current) 5.36

6.32

0.97 PAW(3s3p val) 5.35

6.32

1.01 PAW(3p val) 5.31

6.36

1.00 LAPWa 5.33

6.30

1.10 PAWa 5.34

6.36

1.00

a N.A.W. Holzwarth, et al.; Phys. Rev. B 55, 2005 (1997)

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 36

SLIDE 58

Semi-core states; alkali and alkali earth metals

from a practicle point of view, an accurate treatment of these elements in ionic

compounds is very important: oxides e.g. perovskites

strongly ionized, and small core radii around 2.0 a.u. (1 ˚

A) are desirable

e.g. Ca: one would like to treat 3s, 3p, 4s states as valence states

wave-function R (a.u.) 1 2 3 4 1 2 3

3s : E= -3.437 R c=2.3 3p : E= -2.056 R c=2.3 4s : E= -0.284 R c=2.3

it is very difficult to represent the charge distribution of the 3s and 4s states equally well in a pseudopotential approaches general problem for pseudopo- tentials

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 37

SLIDE 59

Semi core states

in VASP, NC wavefunctions describe the augmentation charges Qij

✁

r

✂ ✄

φnc

i

✁

r

✂

φnc

j

✁

r

✂ ✥

˜ φi

✁

r

✂

˜ φ j

✁

r

✂

it is very difficult to construct accurate NC-PP for 3s and 4s (mutual orthogonality)

wave-function R (a.u.) 1 2 3 4 1 2 3

3s : E= -3.437 R c=2.3 3p : E= -2.056 R c=2.3 4s : E= -0.284 R c=2.3

node in 4s must be included in some way if one succeeds, the augmentation charges be- come quite hard, and require fine regular grids PAW method is the best solution to this prob- lem

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 38

SLIDE 60

PAW — US-PP method: atoms

comparison of GGA PAW, US-PP and scalar relativistic all-electron calcula- tions for O, N, Fe, Co and Ni magnetic energy: ∆Em

✔

EM

✒

gs

✓ ✰

ENM

✒

4s13dn

⑦

1

✓

(in eV) US-PP PAW AE O gs 2s22p4 2s22p4 2s22p4 ∆Em 1.55 1.40 1.41 N gs 2s22p3 2s22p3 2s22p3 ∆Em 3.14 2.88 2.89 Fe gs 3d6

⑧

24s1

⑧

8

3d6

⑧

24s1

⑧

8

3d6

⑧

24s1

⑧

8

∆Em 2.95 2.77 2.76 Co gs 3d7

⑧

74s1

⑧

3

3d7

⑧

74s1

⑧

3

3d7

⑧

74s1

⑧

3

∆Em 1.40 1.32 1.31 Ni gs 3d94s1 3d94s1 3d94s1 ∆Em 0.54 0.52 0.52

for atoms the magnetisation is wrong by 5-10% with US-PP

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 39

SLIDE 61

Bulk properties of Fe

energy differences between different phases of Fe

FLAPWa PAW US-AE US-PP bcc Fe NM 373 372 369 bcc Fe FM

73
73
73
191

fcc Fe NM 78 61 62 62 hcp Fe NM

a FLAPW, L. Stixrude and R.E. Cohen, Geophys. Res. Lett. 22, 125, (1995).

PAW and FLAPW give almost identical results
US-PP overestimates the magnetisation energy by around 5-10 %
calculations for other systems indicate that the accuracy of the PAW method for

magnetic systems is comparable to other AE methods – α-Mn, bulk Cr and Cr surfaces, LaMnO3 (perovskites)

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 40

SLIDE 62

Why do pseudopotentials fail in spin-polarised calculations

non linear core corrections were included in the PP’s !
pseudo-wavefunction for a normconserving pseudopotentials

wave-function R (a.u.) 1 2 3 4 1 2 3 s : E= -0.297 R c=2.2 p1/2: E= -0.094 R c=2.5 d3/2: E= -0.219 R c=2.0

— all electron – – pseudo

the peak in the d-wavefunction is shifted outward to make the PP softer

similar compromises are usually made in US-pseudopotentials
as a result, the valence-core overlap is artifically reduced and the spin

enhancement factor ξ

✁

r

✂

is overestimated ξ

✁

r

✂ ✄

m

✠

r

☛

nvalence

✠

r

☛⑩⑨

ncore

✠

r

☛

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 41

SLIDE 63

Oxides

CeO2 and UO2 CeO2 PAW FLAPW Exp a ( ˚ A3) 5.47 ˚ A 5.47 ˚ A 5.41 ˚ A B 172 GPa i 176 GPa 236 GPa UO2 PAW FLAPW Exp a ( ˚ A3) 5.425 ˚ A 5.46 ˚ A B 200 GPa 209 GPa

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 42

SLIDE 64

Computational costs, efficiency

code complexity of the kernel of course increases with PAW

local pseudopotentials low 2000 lines NC-PP pseudopotential low-medium 7000 lines US-PP medium 10 000 lines PAW medium-high 15 000 lines

parallelisation or error removal becomes progressively difficult
computational efficiency:

Ge 64 atoms, 1 k-point (Γ), Alpha ev6 (500 MHz), 14 electronic cycles type cutoff time total error per atom NC-PP 140 eV 514 sec 400 meV NC-PP 240 eV 1030 sec 10 meV US-PP 140 eV 522 sec 10 meV PAW 140 eV 528 sec 10 meV

the extra costs for PAW or US-PP’s at a fixed cutoff are small PAW method is particularly good for transition metals and oxides

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 43

SLIDE 65

PAW advantages

formal justification is very sound
improved accuracy for:

– magnetic materials – alkali and alkali earth elements, 3d elements (left of PT) – lathanides and actinides

generation of datasets is fairly simple (certainly easier than for US-PP)
AE wavefunction are available
comparison to other methods:

– all test indicate that the accuracy is as good as for other all electron methods (FLAPW, NUMOL, Gaussian) – efficiency for large system should be significantly better than with FLAPW

G. KRESSE, PSEUDOPOTENTIALS (PART II) AND PAW

Page 44