Introduction to the Theory of Pseudopotentials

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Introduction to the Theory of Pseudopotentials

• Patrick Briddon

Materials Modelling Group EECE, University of
Newcastle, UK.
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Contents

• Pseudopotential theory.
• The concept
• Transferability
• Norm conservation
• Non-locality
• Separable form
• Non-linear core corrections

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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.

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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.

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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.

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Empirical Pseudopotentials
Main idea is to look for a form for the potential
Vps(r) so that the solutions to
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.
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Transferability

• Problem these pseudopotentials cannot be
  transferred from one system to another.
• e.g. diamond pseudopotential no good for
  graphite, C60 or CH4.

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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.

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Ionic Pseudopotentials
We descreen the pseudopotential Split charge
density into core and valence contributions
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Ionic Pseudopotentials
Then construct the transferrable ionic
pseudopotential
We have subtracted the potential from the valence
density. The remaining ionic pseudopotential is
more transferrable.
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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)

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Norm Conservation

• A key idea introduced in 1980s.
• Peviously defined a cutoff radius rc
• if r gt rc, Vps Vtrue.
• Now require ?ps ?true if r gt rc.
• Typically match ?ps and first two (HSC) or four
  (TM) derivatives at rc

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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.

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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.

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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

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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

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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

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Schemes

• Hartwigsen, Goedecker, Hutter (1998)
• Separable
• Extended norm conservation
• The AIMPRO standard choice
• BUT ...
• ALL LOOK COMPLETELY DIFFERENT!

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Accuracy

• Look at atoms in different reference
  configuation.
• E.g. C2s22p2 and C2s12p3.
• ?E 8.23 eV (all electron)
• ?E 8.25 eV (pseudopotential)

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Silicon Pseudopotential
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Silicon Pseudopotential

• Some things to note
• Asymptotic behaviour correct, rgtrc
• Non-singular at origin (i.e. NOT 1/r)
• Very different s, p, d forms

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Pseudo and All electron Wavefunctions (Si)
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Silicon Wavefunctions

• Some things to note
• Nodeless pseudo wavefunction, rgtrc
• Agree for rgtrc. Cutoff is around 2.
• Smooth not rapidly varying

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Log derivative
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Non-locality

• Norm conserving pseudopotentials are non-local
  (semi-local).
• This means we canot write the action of potential
  thus

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Non-locality

• Instead we have different potentials for
  different atomic states

This is the action of an operator which my thus
be written as
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Non-locality
or
with
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Kleinman Bylander Form
Problem Take matrix elements in the basis set
?i(r), i1, N
where
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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?

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Kleinman Bylander Form contd
Kleinman and Bylander wrote
So that this time
where
N integrals per atom. Improvement crucial for
plane wave calculations to do 100 atoms
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Kleinman Bylander Form contd
The Kleinman and Bylander form
Is called SEPARABLE or sometimes FULLY
NON-LOCAL

• They
• Developed a standard pptl e.g. BHS
• Modified it to make it separable.

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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.
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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.
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Non-Linear Core Corrections
An issue arises when constructing ionic
pseudopotentials
We have subtracted the potential coming from
valence charge density.
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Non-Linear Core Corrections contd
OK for Hartree potential as
However
clearly
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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.
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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?
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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?
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Effect of large core
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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
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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
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Summary

• The concept of a pseudopotential
• A norm conserving pseudopotential
• A non-local pseudopotential
• A separable pseudopotential.
• A nonlinear core correction.

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Reading...

• Kerker paper
• BHS paper
• Troullier Martin paper
• HGH papers
• Louie-Froyen-Cohen paper