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 = [1s 2 2s 2 2p 6 ]3s 2 3p 2 Chemical bonding controlled by overlap of 3s 2 3p 2 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 V ps (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.

Transferability Problem: these pseudopotentials cannot be transferred from one system to another. e.g. diamond pseudopotential no good for graphite, C 60 or CH 4.

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:

Ionic Pseudopotentials 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 r c: – if r > r c, V ps = V true. Now require  ps =  true if r > r c. Typically match  ps and first two (HSC) or four (TM) derivatives at r c

Cutoff Radius r c is a quality parameter NOT an adjustable parameter. We do not “fit” it! Small r c 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 r c 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[2s 2 2p 2 ] and C[2s 1 2p 3 ].  E = 8.23 eV (all electron)  E = 8.25 eV (pseudopotential)

Silicon Pseudopotential

Some things to note: Asymptotic behaviour correct, r>r c 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>r c Agree for r>r c. Cutoff is around 2. Smooth – not rapidly varying

Log derivative

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

Non-locality Instead we have different potentials for different atomic states : This is the action of an operator which my thus be written as

Non-locality or with

Kleinman Bylander Form Problem: Take matrix elements in the basis set  i (r), i=1, N: where

Kleinman-Bylander Form Problem is: There are N 2 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 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 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: We have subtracted the potential coming from valence charge density.

Non-Linear Core Corrections contd OK for Hartree potential as: However: 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: 1s 2 2p 2 2p 6 3s 2 3p 6 4s 2 3d 10 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: 1s 2 2p 2 2p 6 3s 2 3p 6 4s 1 3d 10 The same question. What happens if we look at ZnSe using “3d in the core”? What about ZnO?

Effect of large core

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:

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