Physics 250-06 “Advanced Electronic Structure”

Pseudopotentials

Contents:

1. Plane Wave Representation

2. Solution for Weak Periodic Potential

3. Solution for a single scattering problem

4. Pseudopotential Approximation

Solving Schroedingers Equation via Fourier Transoforms.

Periodic array of potential wells placed at distance a between the wells. Basic property of the potential V(x+l)=V(x), and of the Hamiltonian H(x+l)=H(x), i.eperiodicity.

( ) ( ) 0H E x

Periodic boundary condition is imposed

( ) ( )x L x

Properties of solutions: The solutions are traveling (or Bloch waves). There are infinite number of solutions which can be labeled by wave vector K, i.e.

( ) ( )iKtK Kx t e x

where t=m*l, m is any integer. Vectors k can take the values following from the periodic boundary condition:

1

2

2

iKLe

KL n

nK

L

when n is any integer.

Non trivial values of K are those for which n runs from 0 to N, where N is number of wells considered, i.e. L=N*l. Important properties of the Bloch waves: Orthogonality

* ( ' ) *' '0 0

* ( ' )' '0

2( ')( ' )

'0

( ) ( ) ( ) ( )

( ) ( )

:

L li K K tK K K K

t

l i K K tK K K K

t

N i n n mi K K t Nnn

t m

x x dx e x x dx

x x dx e

because e e

Example of periodic potential

0 1

2( ) cos( )V x V V x

l

This potential is periodic function as for x=l*m, m=0,…so that it can be a model to

analyze the solutions of Schroedingers equation in periodic potential.

For V1=0 the solutions can be written as plane waves

( ) iKxK x e

The eigenvalues E are

2 20 0| | ( )iKx iKx iKx iKx

KE e H e e V e dx K V

Consider now the equation

20 1

2( cos( ) ) ( ) 0K KV V x E x

l

Let us use a single plane wave as a basis:

( ) iKxK Kx c e

The expansion coefficient is simply equal 1 (assuming plane waves are normalized to 1 within the volume). Let us find the eigenvalues. The matrix eigenvalue problem is collapsed to

20 1

20

| | 0

2( cos( ))

iKx iKxK K

iKx iKxK

K

e H E e c

E e V V x e dxl

E K V

which means that there is no correction to the free electrons eigenstates linear to V1.

We need to improve our basis! With the plane waves it is pretty easy.For this recall first what is

2 nK

Nl

where n is any integer. Let us agree on n running from 0 to N and this will be called k. Then,

'

2 2' n nK n n k G

Nl l

where n is an integer from 0 to N and n’ is any integer. In other words,

K k G so that

2'*

1

iKx ikx iGx

i n mliGt l

e e e

e e

and note also that the latter is periodic function on the lattice ( )iG x t iGxe e

We thus separated the space of all wave vectors K on the subspace of so called irreducible wave vectors k and the subspace of so called reciprocal lattice vectors G. (concept of Brillouin Zone)This can always be done as soon as we introduce a period set by length l, and the periodic boundary condition set by length L=Nl. Now we have a powerful basis set since we can expand

( )( ) k i k G xk G

G

x c e

In other words, for each given k, we use the subspace G which delivers us basis functions. We use those basis functions to represent the wave function for given wave vector k.

Let us return back to our example. Restrict the expansion by two nearest G, i.e. pick n’=-1,0,1, or G=-2pi/l,0,+2pi/l. We have

0 0

0 0

( ) ( )( )0( ) i k G x i k G xk i k G x k k ikx k

k G G GG

x c e c e c e c e

where G0=2pi/l.Compute matrix eigenvalue problem in this basis

0 0 0 0 0

0 0

0 0

0 0

0

( ) ( ) ( ) ( ) ( )0

( ) ( )0

( ) (

| ( ) | | ( ) | | ( ) | 0

| ( ) | | ( ) | | ( ) | 0

| ( ) |

i k G x i k G x i k G x i k G x i k G xk ikx k kG G

i k G x i k G xikx k ikx ikx k ikx kG G

i k G x i k

e H x E e c e H x E e c e H x E e c

e H x E e c e H x E e c e H x E e c

e H x E e

0 0 0 0

0 0

) ( ) ( ) ( )0| ( ) | | ( ) | 0G x i k G x i k G x i k G xk ikx k k

G Gc e H x E e c e H x E e c

Now we realize that plane waves with different K’s or different G’s are orthogonal and therefore

0 0 0 0 0

0 0

0 0

0 0

0 0

0

( ) ( ) ( ) ( ) ( )0

( ) ( )0

( ) ( )

| ( ) | | ( ) | | ( ) | 0

| ( ) | | ( ) | | ( ) | 0

| ( ) |

i k G x i k G x i k G x i k G x i k G xk ikx k kG G

i k G x i k G xikx k ikx ikx k ikx kG G

i k G x i k G x kG

e H x E e c e H x e c e H x e c

e H x e c e H x E e c e H x e c

e H x e c

0 0 0

0

( ) ( ) ( )0| ( ) | | ( ) | 0i k G x i k G x i k G xikx k k

Ge H x e c e H x E e c

We also realize that diagonal element2

0| ( ) |iKx iKxe H x e K V

does not depend on the oscillating part of the potential! We realize that off diagonal elements depend only on oscillating part of the potential.because

0 00

2cos( ) cos( ) 2( )iG x iG xx G x e e

l

The matrix elements are trivially evaluated

0 0 0 0( ) ( )1 1| ( ) | 2 ( ) 2i k G x i k G x iG x iG xikx ikxe H x e V e e e e dx V

0

0 0

( )1

( ) ( )

| ( ) | 2

| ( ) | 0

i k G x ikx

i k G x i k G x

e H x e V

e H x e

We finally obtain0

0 0

0

20 0 1 0

21 0 0 1

21 0 0 0

(( ) ) 2 0

2 ( ) 2 0

2 (( ) ) 0

k kG

k k kG G

k kG

k G V E c V c

V c k V E c V c

V c k G V E c

The roots can be found by looking at the determinant

20 0 1

21 0 1

21 0 0

( ) 2 0

2 2

0 2 ( )

k G V E V

V k V E V

V k G V E

Opening the determinant produces

2 22 2 2 2 20 0 0 0 0 0(( ) )*( )*( ) ) 4 (( ) ) 4 (( ) ) 0k G E k E k G E V k G E V k G E

This quation delivers us three roots E1(k), E2(k), E3(k).

Solution for Periodic Potential

Sphere S

Scattering by a single potentialScattering by a single potential

0

( ) ( ),

( ) ( ) ,

sph

sph

V r V r r S

V r V S V r S

Consider scattering by a potential assumed to be spherically symmetric inside a sphere

0V

( )V re

Sphere S

E

Solve radial Schroedinger equation inside the sphere

ˆ( , ) ( )ll lmr E i Y r

2( ( ) ) ( , ) 0rl sph lV r E r E

Solve Helmholtz equation outsidethe sphere

2 20

20

( ) ( , ) 0rl lV E r

E V

2 ˆ( , ) ( )ll lmr i Y r

SS

Solution of Helmholtz equation outsidethe sphere

2( , ) ( ) ( )l l l l lr a j r b h r

where coefficients provide smoothmatching with

,l la b( , )l r E

{ , }

{ , }

{ , } ' '

l l l

l l l

a W j

b W h

W f g f g g f

A single L-partial wave solves single scattering problem

ˆ( , ) ( , ) ( ),

ˆ( , ) { ( ) ( )} ( ),

lL l L

lL l l l l L

r E r E i Y r r S

r E a j r b h r i Y r r S

Any linear combination solves it as well

( ) ( , )L LL

r A r E

SS

A potential information is hidden inside the coefficients providingsmooth matching

2( , ) ( ) ( )l l l l lr a j r b h r

{ , }

{ , }

{ , } ' '

l l l

l l l

a W j

b W h

W f g f g g f

One can replace real potential by a pseudopotentialwhich would provide the same scattering property!

Replace real potential for each L by its by its pseudopotential

2( ( ) ) ( , ) 0rl l lV r E r E

( ) ( )sph lV r V r

( )lV r should produce such solution that

( , ) ( ) ( )l l l l lS E a j S b h S

{ , }, { , }l l l l l la W j b W h

In other words, (i) Solve Equation for , find(ii) Adjust so that it produces pseudowith the same scattering properties.

( )sphV r ,l la b( )lV r ( , )l r E

The Pseudopotential Approximation for Solids

2

( )

2' '

( ( ) ) ( ) 0

( )

( )

{[( ) ] } 0

kj kj

iGrG

G

i k G rkj k Gj

G

kj GG G G k GjG

V r E r

V r V e

r A e

k G E V A

Unfortunately a plane wave basis set is usually very poorly suited to expanding the electronic wavefunctions because a very large number are required to accurately describe the rapidly oscillating wavefunctions of electrons in the core region. Also we have a divergency problem in 3D ionic potential

Solving solid state problem using plane wave basis set

2 2

2

( ') ' 4 [ ( )]( )

| | | ' |iGrN core N core

ionR R G

Z e r dr Z e GV r e

r R r r R G

It is well known that most physical properties of solids are dependent on the valence electrons to a much greater degree than that of the tightly bound core electrons. It is for this reason that the pseudopotential approximation is introduced.

This approximation uses this fact to remove the core electrons and the strong nuclear potential and replace them with a weaker pseudopotential which acts on a set of pseudowavefunctions rather than the true valence wavefunctions.So, for each atom in periodic table ionic potential can be replaced by pseudopotential which would produce the same scattering process:

2 ( ') '( ) ( )

| | | ' |pseudoN core

ion LR R

Z e r drV r V r

r R r r R

A pseudopotential is not unique, therefore several methods of generation exist. However they must obey several criteria.

The core charge produced by the pseudo wavefunctions must be the same as that produced by the atomic wavefunctions. This ensures that the pseudo atom produces the same scattering properties as the ionic core.

Pseudo-electron eigenvalues must be the same as the valence eigenvalues obtained from the real wavefunctions.

Pseudopotentials of this type are known as non-local norm-conserving pseudopotentials and are the most transferable since they are capable of describing the scattering properties of an ion in a variety of atomic environments.

Mathematically, the angular dependent pseudopotential can be written as: |)(|)(ˆ lrVlrVl

l Here = |l is an angular momentum projection operator. In computation, this can be evaluated

riGl

riG elrVle .2,1 ||)(||

Analytical formula exists to write down the matrix elements. For large systems he Kleinman-Bylander implementation of the nonlocal potential is used. Basically, we first take one l (s, or, p, or d) as the local potential, )(rVloc

Then we can define a nonlocal part as )()()( rVrVrV locll

Then the above nonlocal pseudopotential is approximated as:

ml lll

lmlllmllloc

ll V

YVYVrVlrVlrV

, ||

||)(|)(|)(ˆ

lmY

l are the atomic pseudowavefunctions for angular l.

The sphere of Gc2 being inside the Fourier box of n1*n2*n3 grid.  

1

2

2

( )

2

( )

( ) | ( ) |

( )

c

c

c

Gi k G r

kj k GjG

GocciGr

kj Gkj G

GiGr

GG

r A e

r r e

V r V e

Plane Wave Grids

Gygi’s adaptive grid: ideas from the theory of gravity

Francois Gygi (UC Davis)