Introduction to the Theory of
Pseudopotentials
Patrick Briddon
Materials Modelling Group
EECE,
University of Newcastle,
UK.
Contents
Pseudopotential 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
Core states are very hard to describe
accurately. They:
• vary rapidly. This makes
– plane wave expansions impossible.
– Gaussian expansions difficult
• Expensive and hard to do.
• oscillate - positions of nodes is
important.
Core states contd.
Core states:
• 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.
Empirical Pseudopotentials
Main idea is to look for a form for the
potential Vps(r) so that the solutions to:
ps
ˆ
T  V r 
 E 
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 Pseudopotentials
We descreen the pseudopotential:
Split charge density into core and
valence contributions:
nr   n r   n r 
c
v
n r   core charge density
c
n r   valence charge density
v
Ionic Pseudopotentials
Then construct the transferrable ionic
pseudopotential:
n r  dr
v
V r   V r   
 Vxc n r 
r  r
v
ps
ion
ps


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
Schemes
Troullier, Martins (1993)
– An improvement on BHS
– refinement to HSC procedure
– widely used today
• Vanderbilt (1990)
– ultrasoft pseudopotentials
– Important for plane waves
– widely used today
Schemes
Troullier, Martins (1993)
– An improvement on BHS
– refinement to HSC procedure
– widely used today
• Vanderbilt (1990)
– ultrasoft pseudopotentials
– Important for plane waves
– widely used today
Schemes
Hartwigsen, 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 derivative
d ln R
dr rc
Non-locality
• Norm conserving pseudopotentials are
non-local (semi-local).
• This means we canot write the action of
potential thus:
V   V r  r 
Non-locality
Instead we have different potentials for
different atomic states :
Vˆ s  Vs r  s r 
Vˆ p  V p r  p r 
This is the action of an operator which
my thus be written as
Non-locality

ps
ˆ
V  Vl r  l l
l 0
or
Vˆ    V r, r  r dr
ps
*
ˆ
V r, r  Vl r, r Ylm  Ylm 
lm
with
Vl r, r  Vl r  r  r 
Kleinman Bylander Form
Problem: Take matrix elements in the basis
set i(r), i=1, N:
ps
ˆ














r
V

r
d
r


r
V
r
,
r

r
d
r
d
r
i
j
i
j


   4r F r F
2
where
i
lm
j*
lm
lm
F r     i r Ylm  ,  d d
i
lm
r Vl r  dr
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
Vl r, r  Vl r Vl r
So that this time
where
Vij   F F
i
lm
j*
lm
lm
F    i r Vl r Ylm  ,  dr
i
lm
N integrals per atom. Improvement crucial
for plane wave calculations to do 100 atoms
Kleinman Bylander Form contd
The Kleinman and Bylander form
Vl r, r  Vl r Vl r
Is called SEPARABLE or sometimes
FULLY NON-LOCAL
They:
1. Developed a standard pptl – e.g. BHS
2. 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:
v

n
r dr
ps
ps
v
Vion r   V r   
 Vxc n r 
r  r


We have subtracted the potential coming
from valence charge density.
Non-Linear Core Corrections
contd
OK for Hartree potential as:
n r  dr
n r  dr
V n   

r  r
r  r
c
v
H
However:
V n   Vxc n r  Vxc n r 
xc
clearly
c
v
n  n   n   n 
c
v 13
c 13
v 13
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
a0
Val
AlAs [1s22s22p6]
3s23p1 0%
2
6
10
2
4s 4p
1
-2%
2
6
10
2
5s 5p
1
-4%
2
6
10
2
GaAs 3s 3p 3d
InAs 4s 4p 4d
ZnSe 3s 3p 3d
4s
-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:
n r dr
tot
V r   V r   
 Vxc n r 
r  r
v
ps
ion
ps


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