Introduction to the Theory of Pseudopotentials

Patrick Briddon

Materials Modelling GroupEECE,

University of Newcastle,UK.

ContentsPseudopotential theory.

– The concept– Transferability – Norm conservation– Non-locality– Separable form– Non-linear core corrections

Pseudopotentials• A second main plank of modern

calculations

• Key idea - only valence electrons involved in chemical reactions

• e.g. Si = [1s22s22p6 ]3s23p2

• Chemical bonding controlled by overlap of 3s23p2 electrons with neighbouring atoms.

• Idea: avoid calculating the core states altogether.

The problem with core states

• vary rapidly. This makes

– plane wave expansions impossible.

– Gaussian expansions difficult• Expensive and hard to do.

• oscillate - positions of nodes is important.

Core states are very hard to describe accurately. They:

Core states contd.

• make the valence states oscillate.

• require relativistic treatment.

• make the energy very large. This makes calculations of small changes (e.g. binding energies) very hard.

Core states:

Empirical Pseudopotentials

Main idea is to look for a form for the potential Vps(r) so that the solutions to:

EpsVT rˆ

for a reference system agree with expt. E.g. get band structures of bulk Si, Ge.

Then, use the potentials to look at SiGe or SiGe microstructures.

Transferability

• Problem: these pseudopotentials cannot be transferred from one system to another.

• e.g. diamond pseudopotential no good for graphite, C60 or CH4.

Transferability

Why is this?

• the valence charge density is very different in different chemical situations - only the core is frozen.

• We should not try to transfer the potential from the valence shell.

Ionic PseudopotentialsWe descreen the pseudopotential: Split charge density into core and valence contributions:

rrr vc nnn

density charge valencen

density charge corev

c

r

rn

Ionic Pseudopotentials

rrr

rrrr v

xc

v

nVdn

VV

pspsion

Then construct the transferrable ionic pseudopotential:

We have subtracted the potential from the valence density. The remaining ionic pseudopotential is more transferrable.

Ab Initio pseudopotentials

• This approach allows us to generate pseudopotentials from atomic calculations.

• These should transfer to solid state or molecular environment.

• ab initio approach possible.

• Look at some schemes for this.

• “Pseudopotentials that work from H to Pu” by Bachelet, Hamann and Schluter (1982)

Norm Conservation

• A key idea introduced in 1980s.

• Peviously defined a cutoff radius rc:

– if r > rc, Vps = Vtrue.

• Now require ps = true if r > rc.

• Typically match ps and first two (HSC) or four (TM) derivatives at rc

Cutoff Radius

• rc is a quality parameter NOT an adjustable parameter.

• We do not “fit” it!

• Small rc means ps = true for greater range of r more accurate.

Cutoff Radius

• BUT, small r will lead to rapidly varying ps (eventually it will have nodes).

• Use biggest rc that leaves results unchanged.

• Generall somewhere between outermost maximum and node.

Schemes• Kerker (1980)

– not widely used

• Hamann, Schlüter, Chiang, 1982– basis of much future work

• Bachelet, Hamann, Schlüter, 1982– fitted HSC procedure for all elements

SchemesTroullier, Martins (1993)

– An improvement on BHS– refinement to HSC procedure– widely used today

• Vanderbilt (1990)– ultrasoft pseudopotentials– Important for plane waves– widely used today

SchemesTroullier, Martins (1993)

– An improvement on BHS– refinement to HSC procedure– widely used today

• Vanderbilt (1990)– ultrasoft pseudopotentials– Important for plane waves– widely used today

SchemesHartwigsen, Goedecker, Hutter (1998)

– Separable– Extended norm conservation– The AIMPRO standard choice

BUT ...

ALL LOOK COMPLETELY DIFFERENT!

Accuracy

• Look at atoms in different reference configuation.

• E.g. C[2s22p2] and C[2s12p3].

E = 8.23 eV (all electron)E = 8.25 eV (pseudopotential)

Silicon Pseudopotential

Silicon Pseudopotential

Some things to note:

• Asymptotic behaviour correct, r>rc

• Non-singular at origin (i.e. NOT 1/r)

• Very different s, p, d forms

Pseudo and All electron Wavefunctions (Si)

Silicon Wavefunctions

Some things to note:

• Nodeless pseudo wavefunction, r>rc

• Agree for r>rc. Cutoff is around 2.

• Smooth – not rapidly varying

Log derivativecr

drRd ln

Non-locality

• Norm conserving pseudopotentials are non-local (semi-local).

• This means we canot write the action of potential thus:

rr VV

Non-locality

Instead we have different potentials for different atomic states :

rr

rr

ppp

sss

VV

VV

ˆ

ˆ

This is the action of an operator which my thus be written as

Non-locality

llrVVl

lps

0

ˆ

or

*,,ˆlmlm

lml

ps YYrrVV rr

with rrrVrrV ll ,

rrrr d,ˆ VV

Kleinman Bylander Form

Problem: Take matrix elements in the basis set i(r), i=1, N:

lml

jlm

ilm

jps

iji

drrVrFrFr

ddVdV

*24

,ˆ

rrrrrrrrr

ddYrF lmiilm , r

where

Kleinman-Bylander Form

• Problem is: There are N2 integrals per atom is the basis set is not localised.

• A disaster for plane waves.

• Not the best for Gaussians

• Recall there is no such things as “the pseudopotential”.

• Can we chose a form that helps us out?

Kleinman Bylander Form contd

Kleinman and Bylander wrote

So that this time

rVrVrrV lll ,

lm

jlm

ilmij FFV *

rr dYrVF lmliilm ,

where

N integrals per atom. Improvement crucial for plane wave calculations to do 100 atoms

Kleinman Bylander Form contd

The Kleinman and Bylander form

Is called SEPARABLE or sometimes FULLY NON-LOCAL

rVrVrrV lll ,

They:1. Developed a standard pptl – e.g. BHS2. Modified it to make it separable.

The HGH pseudopotentials

HGH pseudopotentials are also fully separable.

They proposed a scheme to generate in this way directly (i.e. Not a two stage process).

Thus they avoided issues with “ghost states” that were initially encountered when trying to modifuy a previously generated pptl.

The HGH pseudopotentials

HGH pseudopotentials are also fully separable.

They proposed a scheme to generate in this way directly (i.e. Not a two stage process).

Thus they avoided issues with “ghost states” that were initially encountered when trying to modifuy a previously generated pptl.

Non-Linear Core Corrections

An issue arises when constructing ionic pseudopotentials:

rrr

rrrr v

xc

v

nVdn

VV

psps

ion

We have subtracted the potential coming from valence charge density.

Non-Linear Core Corrections contd

OK for Hartree potential as:

rrrr

rrrr dndn

nVvc

H

However:

rr vxc

cxc

xc nVnVnV

313131 vcvc nnnn clearly

Non-Linear Core Corrections contd

This is true if valence and core densities do not overlap spatially.

i.e. Core states vanish before valence states significant.

Problem: this just does not always happen.

NLCC contd

Is a problem when it is difficult to decide what is a core electron and what is a valence electron.

e.g. Cu: 1s22p22p63s23p64s23d10

The issue is the 3d electrons – a filled shell. Largely do not participate in bonding. Are they core ot not?

NLCC contd

What about

e.g. Zn: 1s22p22p63s23p64s13d10

The same question. What happens if we look at ZnSe using “3d in the core”?

What about ZnO?

Effect of large core

core Val a0

AlAs [1s22s22p6] 3s23p1 0%

GaAs 3s23p63d10 4s24p1 -2%

InAs 4s24p64d10 5s25p1 -4%

ZnSe 3s23p63d10 4s2 -10%

Non-Linear Core Corrections contd

A solution is to use a NLCC

Descreen with the potential from the total density, not just the valence density:

rrr

rrrr tot

xc

v

nVdn

VV

psps

ion

Non-Linear Core Corrections contd

Fixes lattice constant completely for GaAs, InAs. Good for GaN, ZnSe.

Band structure still be affected. CARE.

NLCC will not work if the states change shape when moving from atom to solid. Other properties ma

Summary

• The concept of a pseudopotential

• A norm conserving pseudopotential

• A non-local pseudopotential

• A separable pseudopotential.

• A nonlinear core correction.

Reading...

• Kerker paper

• BHS paper

•Troullier Martin paper

• HGH papers

• Louie-Froyen-Cohen paper