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Modeling of Nanostructures and Materials

Summer Semester 2014 Lecture

Jacek A. Majewski

Chair of Condensed Matter Physics Institute of Theoretical Physics

Faculty of Physics, Universityof Warsaw

E-mail: Jacek.Majewski@fuw.edu.pl

(Jacek A. Majewski) Nevill Gonzalez Szwacki

Modeling of Nanostructures and Materials

Lecture 3 March 10, 2014

Kohn-Sham Method with

Plane-waves and pseudopotentials

!!

Generation of norm conserving pseudopotentials (PPs)

!!

Separable (Kleinman-Bylander) PPs

!!

Unscreening of PPs (

"" ionic PPs)

!!

Practical aspects of the calculations

H ! = ! E

H = !

!

2

"

!2

2M

!

#

!

! ! 2m

2

"

i2

#

i

+ 1 2 |

RZ

!

!Z"e2

!

! !

R!

|

#

! ,!

! |

R

!

Z"e2

!

! !

ri

|

#

i,!

+ 1 2 | !r

e2 i

! !

rj

|

#

i, j

Materials Science:

Examples of Schrödinger Equation?

Ab-initio (first principles) Method –

ONLY Atomic Numbers {

Z

i} as input parameters Materials are composed of nuclei and electrons

the interactions are known

{Z!,M!,

R!!}

{!r

i

}

Kinetic energy of nuclei

Kinetic energy of electrons

Nucleus-Nucleus interaction

Electron-Nucleus interaction

Electron-Electron interaction

Interacting particles Independent particles

!(!x1, !x2,…, !x1023) !1(x!

1)!2(x!

2)!3(x!

1023)

Idea: consider electrons as independent particles moving in an effective potential

Density Functional Theory (DFT) in Kohn-Sham realization

This reduction is rigorously possible !

(2)

DFT- The Kohn- Sham Method

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965) System of interacting

electrons with density

!(!r)

System of non-interacting electrons with the same density

!( !r)

“Real” system “Fictitious” or Kohn-Sham reference system

!! ( r ) !

T [ !] T [ !]

S

!!

S

( !r )== !!( !r )

E[ !! ] == d!r!! !!

ext

( !r )!!( !r ) ++T

S

[ !! ] ++U [ !! ] ++ E

x

[ !! ] ++ E

c

[ !! ]

!! ( !

r ) == !! *

i

( ! r )

i==1

!

N

! !!

i

( r ) !

T

s

[ !! ] == !! !

2

2m # #

i==1N

"" d"r !!

i*

( !r ) !

$

$

2

!!

i

( !r )

unknown!!!

E

x

[ !! ] == !! 1

2 d rd !

""""

#

i

# r' ! !!

i*

( r ) ! !!

j

| ( r !! ! r ) ! !! ! r' |

*j

( r') !

#

j

$$ #

%% && '' (( )) !!

i

( r') !

!! !

2

2m

"

"

"

2

++ !!

ext

( !r )++!!

H

( !r )++!!

x

( !r )++!!

c

( !r )

##

$$ %% &&

'' ((!!

i

( !r ) == !!

i

""

i

( !r ) The Kohn- Sham Method –

The Kohn-Sham Equations

!! ( r ) == ! !! *

i

( ! r )

i==1

!

N

! !!

i

( ! r )

!E [ "]x

!" c

!E [ "]

!"

!!H(! r ) ==!!U

!!"" == d r'!

!!

|!r ""!!(!r')! r' |

!!ext(!

r ) == ""e2 Zs

|! r ""!

##s""! Rn|

$

s,n

$

Schrödinger-like equations with local potential

These equation are nonlinear and must be solved iteratively (self-consistently)

The Kohn- Sham Method – ‚Aufbau‘ principle

HOMO LUMO

! 1

! 2

! N N

! ++1 unoccupied occupied How to calculate one particle density?

DFT: Implementations of the Kohn-Sham Method

Fully relativistic Semi-relativistic Non-relativistic

Non-periodic periodic

All-electron full potential All-electron muffin-tin All-electron PAW Pseudopotential

Non-spin-polarized Spin polarized

Beyond LDA

Generalized Gradient Approximation (GGA) Local Density Approximation (LDA)

GW (quasi-particles) EXX (exact exchange) sX-LDA

Time dependent DFT LDA+U

Atomic orbitals Plane Waves Augmentation Fully numerical (real space)

Gaussians(GTO) Slater type (STO) Numerical Plane waves (FPLAPW) Spherical waves (LMTO, ASW)

! 1 2 "

2

+ V

ext

( !r) + V

xc

( !r)

#

$ % &

' ( !

"

= #

"

!

"

(3)

Solution of the Kohn-Sham Equations

Direct methods on a mesh in r-space

Expansion of the Kohn-Sham orbitals in a basis

Eigenvalueproblem

Bandstructure

{!!

""k!

(!r)}

!!

nk!

(!r) == ! c

!!

(n, ! k)!!

""k!

(!r)

!

!!

!

!!'

!

!

!!""!

k| ""!2 2m

#!

#2++!!KS(!

r ) |!!""'k! ""!!n(!

k) !!""k!|!!""'k!

$$

%%

&&

&&

'' ((

))))c!!'(n,! k) == 0

[H!!!!'(! k) !!""n(!

k)S!!!!'(! k)]c!!'(n,!

k) == 0

det[H

!!!! '

( ! k) !! !!

n

( !

k)S

!!!! '

( ! k)] == 0

!!

n

( ! k)

Hamiltonian

matrix elements Overlap integrals

Kohn-Sham

plane-waves formalism

Plane-wave formalism

!!

!p

( !r ) == 1

V exp( i!p!! !r )

!! n K !!S ( !r ) == C ! !pn !! !p ( !r )

" P

"

Plane-waves

Plane-waves constitute orthonormal system

!!

!p

| !!

!p'

== !!

!p!p'

Problem: !p

continuous variable

For periodic systems, one can introduce discrete values

Plane-wave formalism

{ !

G} G !

n

== n

1

!

b

1

++ n

2

! b

2

++ n

3

!

b

3

-

reciprocal lattice vectors Periodic systems

!a

i

{{ }} primitive translations characterizing periodicity

k ! ! ! BZ Wave vectors characterize Bloch States

!p == !k ++ !G

!!

G!

( !r ) == 1

V exp( i !

G !! !r ) !!

G!

| !!

G'!

== !!

G!G'!

Discrete, orthonormal set of basis functions

(4)

Plane-wave formalism

!!

nk!

( r ) == 1 !

!

! e

ik""!r!

u

nk!

( ! r ) == 1

!

! c

nk!

( !

G)e

i(k ++! G)""!r! G!

#

#

Expansion coefficients

!!

Basis – plane waves kG!!

(!r) == e

i(k ++! G)!!! r!

{!

G} !

Gn==n1! b1++n2!

b2++n3! b3 - reciprocal lattice vectors

{ ! G : 1 2 ( !

k ++ !

G)

2

!! E

cutoff

}

In practical calculations

Kinetic Energy Cutoff

Concept of pseudopotential

•! When solving the K-S equations for the full system of nuclei and electrons we find that

:

– ! Close to the nuclei, the potential is dominated by strong Coulomb interactions

–! In interstitial regions the potential is much weaker and reflects the symmetry of the crystal

•! A typical wavefunction would look like:"

Typical electronic potential and wave function

Strong oscillations in the core region

The Pseudopotential Concept

•! IDEA: group all the electrons around the nuclear core into an effective ionic core, where all the strong oscillations close to the nuclei are damped, and leave out only the valence electrons that contribute to the bonding of the solid. Core electrons are left basically unchanged going from the atom to the solid!

Si: 1s2 2s2 2p6 3s2 3p2

14 electrons

V

ext

(r) = !14e

2

r

Si: (pseudo core) 3s2 3p2

4 (valence) electrons

V

ext

(r) = V

ps

(r)

(5)

PSEUDOPOTENTIALS - Basic Idea

The basic idea of the pseudopotential theory:

Core electrons are localized and therefore chemically inactive (inert)

Valence electrons determine chemical properties of atoms and SOLIDS

Nucleus electrons Core

Valence electrons

Describe valence states by smooth wavefunctions

Construction of pseudopotentials from atomic wavefunctions -!core states - valence states

How to get smooth pseudo-valence- wavefunctions from atomic valence wavefunctions ?

ˆat c == c c

H !! E !!

ˆat v == v v

H !! E !! !!c !!v

!!v

==

| 0

v c

!! !!

Orthogonality condition leads to oscillations in !!v

IDEA: Project out oscillations from !!v

!!v

PSEUDOPOTENTIALS – Philips-Kleinman Method

Phillips & Kleinman, Phys. Rev. 116, 287 (1959)

PSEUDOPOTENTIALS – Philips-Kleinman Method

== ++ ==

== !! ++ ==

== ++ !!

"

"

"

" " "

"

"

ˆ

( )

at v v v c c cv

c

v v v c cv c c cv

c c

v v c v c cv

c

H E E

E E E

E E E

## $$ $$ %%

## $$ %% $$ %%

## $$ %%

!!v ==!!v ++ !!c

!

c

!

!!c|!!v

""cv

! "### ##$

!!

" "

!! ==

ˆat v ( c v) c c| v v v

c

H ## E E $$ $$ ## E ##

!!

" "

!! ==

at ( c v) c c} v v v c

H E E ## ## $$ E $$

== ++

" "

!!

ˆps ˆat ( c) c c

c

H H E E ## ##

First, we define pseudo-wave-function for valence electrons

Second, we act with atomic Hamiltonian on the pseudo-wavefunction

The pseudo-wave-function fulfills Schrödinger-like equation with Hamiltonian that is dependent on energy and contains additional repulsive, nonlocal potential.

== ++

" "

!!

ˆps ˆcore ( c) c c c

V V E E ## ##

PSEUDOPOTENTIALS – Philips-Kleinman Method

Effective potential acting on the pseudo-valence electrons ˆps v == v v

H !! E !!

ˆat v == v v

H !! E !!

Energies are identical

Atom Pseudoatom

ˆps

V is weaker than Vˆcore

rc

!v

"

v

Vps Z es2

r 0

Z

s – No. of valence electrons in neutral atom

==!! 2

core( ) Z es

V r

r

!! !!

(0 r rc)

>> c r r

v== v

!! " "

ps==!!Z es 2

V r

Within the core region The potential

and atomic valence wavefunctions are substituted by pseudopotential and knot free pseudowavefunction

Outside the core region Vps

!!v

Construction procedure keeps the energies of atomic and pseudoatomic states unchanged.

r

After paper of Philips & Kleinman, various models of pseudopotentials have been developed. Main weakness: many parameters involved

(6)

Parameter Free (Ab-initio) Pseudopotentials – Norm Conserving Pseudopotentials

Density Functional Theory for Atoms – Kohn-Sham equations for atoms Spherical symmetry of atoms is assumed

!

!at(!

r ) == Rl(r)Ylm(!

!!) ==

"

lm

"

ulr(r)Ylm(!

!!)

"

lm

"

!! ""

## ++ ==

$$ %%

$$ %%

&& ''

2 2 ,

1 ( ) ( ) ( )

2 l KSat l l l

d r u r u r

dr (( )) Atomic units: ! == == ==e m 1

== !! ++ ++ ++ ++ ++ ++

, 2

( ) ( 1) ([ ]) ([ ])

2

at s

l KS r Z l l H c v xc c v

r r

"" "" ## ## "" ## ##

== ++

at c v

!! !! !!

Effective Kohn-Sham potential contains all electronic interactions For each “l”, one-dimensional Kohn-Sham equation

Atomic

density Core density

Valence density

Construction of Norm Conserving Pseudopotentials

Construction of pseudo-wavefunctions from atomic solutions u rlps( )

l( )

!! "" u r

## ++ ==

$$ %%

$$ %%

&& ''

2 2

1 ( ) ( ) ( )

2 lps lps l lps

d r u r u r

dr (( ))

The pseudo-wavefunctions have to fulfill certain conditions u rlps( ) for

== >>

( ) ( )

ps l cl

u rl u r r r

!!

0rcl

| R r r dr

lps

( ) |

2 2

== !!

0rcl

| ( ) | R r r dr

l 2 2

ps==

l

!!l !!

== ==

( ) == ( )

( ) ( )

cl cl

ps l

l

ps l

l

r r r r

d R r d R r

dr dr

R r R r

Pseudo-wavefunctions and atomic wavefunctions lead to identical charge in the core region r r!! cl NORM CONSERVATION

Identical logarithmic derivatives at cutoff radii

2

1

3 4

These conditions do not determine the pseudo-wavefunctions uniquely Different types of ab-initio pseudopotentials

Construction of Norm Concerving Pseudopotentials

Troullier-Martins-Pseudopotentials N. Troullier & J. L. Martins, Phys. Rev. B 43, 8861 (1991) Older pseudopotentials

BHS pseudopotential G.B. Bachelet, D.R. Haman, and M. Schlüter, Phys. Rev. B 26, 4199 (1982)

Kerker pseudopotential G.P.Kerker, J. Phys. C 13, L189 (1980)

The pseudo-wavefunction in the core region (r < rcl)

== ++1 2

( ) exp[ ( )]

ps l

l l

u r r p r pl - polynomial of 6th order

Coefficients of the polynomial are determined from:

a)! Conditions 1-4

b)! Continuity of the first, second, third, and fourth derivative of ulps in rcl c)! Second derivative of ionic pseudopotential should vanish inr

Very good convergence properties !

When pseudo-wavefunctions are established, then proceed to the next step of pseudopotential construction.

lps

u

Construction of Norm Conserving Pseudopotentials

Inversion of the Kohn-Sham equations Atomic pseudopot.

== !! ++ ++

2 2

, 2

( 1) 1 ( )

( ) 2 2 ( )

lps

ps l

l atom ps

l

d u r

l l dr

r r u r

"" ##

Atomic pseudopotential contains interaction between valence electrons.

This interaction should be subtracted.

Note, are knot free u rlps( )

Unscreening procedure !!

, ( ) , ( )

ps ps

l atom r Vl ion r

"" IONIC PSEUDOPOTENTIAL

== !! !!

,

( )

,

( ) ([ ]; ) ([ ]; )

ps ps ps ps

H xc

l ion l atom val val

V r "" r "" ## r "" ## r

, ( )

l ionps

V r - For each angular momentum quantum number “

l

(7)

Norm Conserving Pseudopotentials

0 1 2 3

r[a.u.]

V(r)[Ry]ps

s

Si

p

d 0 -2 -4 -6 -8 -10

-Z ev 2

r

0 1 2 3 4

0.8

0 0.4

-0.4

Si

s d

p

Radial wave function

r[a.u.]

Comparison of the pseudo-wavefunction (solid lines) and the corresponding all-electron wavefunctions (dashed lines)

Components of the ionic pseudo-potential for angular momentum

l

= 0,1,2

Norm Conserving Pseudopotentials

Vˆionps== Vl,ionps (r) ˆPl

!

l

!

== Vl,ionps (r) l l

!

l

!

r | l == Y! lm(!ˆr)

r | ˆ! Vionps|!

r ' == Vionps(! r,!

r ')

Vˆionps==

n,s

!

!

Vl,ionps,(s)(|!r "" !Rn""!!!s|) ˆPl

!

l

!

,ps ( )

l ion

V r - different component for each “

l

Non-local pseudopotential

Projection operator

In Solids:

Extensions

Relativistic effects are extremely important for core electrons Dirac equation for atoms

Schrödinger-like equation for pseudo-valence wavefunctions

l

” component of the ionic pseudopotential is obtained through the averaging over “

j

+” and “

j

-” components

Exchange-correlation functional is nonlinear in !! !!== c++!!v

xc[ c++ v]!! xc[ ]c ++ xc[ ]v

"" ## ## "" ## "" ##

Equality was assumed, for simplicity, for the unscreening procedure Nonlinear core correction

Norm Conserving Pseudopotentials

Nonlinear Core Correction

Unscreening using smooth function of core density

Louie et al.,Phys. Rev. B 26, 1738 (1982)

(8)

Nonlinear Core Correction – How it works?

Atomic energy splitting for Fe atom

High sensitivity to cutoff

Use of full-core improves accuracy but usually costs are too high

Pseudopotential Smoothness & Accuracy

Iron wavefunctions

and pseudo-wavefunctions Move outward cutoff radius to get smoother pseudo-wavefunctions Acceptable basis size Penalty:

decreased transferability

Small cutoff = sharp function expensive to expand in PWs Pseudopotential quality is measured by its transferability,

i.e., ability of the PP to match AE values when put in different chemical environments

Plane-wave formalism

Kohn-Sham equations in momentum space

G'!

!

!

"2m2(k ++! G)!2!!G,!! G'++!!H(!

G ""! G') ++ !!xc(!

G ""! G') ++Vps(!

k ++! G,!

k ++!

## G')

$$%%

%%

&&

''((

((cnk!(!

G) == !!nk!cnk!(! G)

!! !

2

2m

" !

"

2

++ !!

H

( ! r ) ++ !!

xc

( !

r ) ++ ˆ V

ionps

##

$$ %%

%%

&&

'' ((

(( !!

nk!

( !

r ) == !!

nk!

!!

nk!

( ! r )

cnk!(

!

Eigenvalue problem – system of equations G)

for expansion coefficients

Fourier transformation

Expansion with respect to basis = Fourier series

(KS-Eq.)

Plane-wave formalism

External potential substituted by

Pseudopotential

(9)

(KS-Eq.)

Pseudopotential in Kohn-Sham Method

External potential substituted by Pseudopotential

non-local pseudopotential !!

!r | ˆV

ionps

| !r' ==V

ionps

(!r,!r')

External potential (pseudopotential) is non-local !

Is it compatible with derivation of Kohn-Sham equations?

Not really, but generalization of the formalism possible

!! !

2

2m

"

!

!

2

++ !!

H

( r ) ++ ! !!

xc

( r ) ++ ˆ ! V

nonlocps

( r, ! r ') !

!!

""

## ##

$$

%% &&

&& !!

nk!

( r ) == ! !!

nk!

!!

nk!

( r ) !

!"

l

V

ionps,!!

( !r,!r') == !!

l ,ionps,""

!

lm

! ( r )Y

lm*

( !r )Y

lm

( !r') V

ionps,!!

( !r,!r') == !!

l0 ,ionps,!!

( r ) ++ [""

l ,ionps,!!

!

lm

! ( r ) "" ""

l0 ,ionps,!!

( r )]Y

lm*

( !r )Y

lm

( !r')

ps loc l

lm

ˆ! == ! ( r ) ++ ! ! ! ( r ) lm lm

Semilocal Pseudopotentials

Norm conserving pseudopotential – semilocal PP

local in r, nonlocal in angles

Expensive calculations of !! ps ( ! k ++ !

G, ! k ++ !

G')

!"

l

is short range

Kohn-Sham equations in pseudopotential formalism V

ps

( !r,!r') == !!

locps

( !r )++!!

nonlocps

( !r,!r')

!!

loc

( !

r ) == !!

H

( !

r ) ++ !!

xc

( !

r ) ++ !!

locps

( ! r )

!!!2 2m

"!

"2++!!H(!r)++!!xc(!r)++!!locps(!r)++!!nonlocps (!r,!r')

##

$$%%

%%

&&

''((

((!!nk!(!r) == !!nk!!!nk!(!r)

!! !

2

2m

" !

"

2

++ !!

loc

( !

r ) ++ !!

nonlocps

( ! r, !

## r ')

$$ %%

%%

&&

'' ((

(( !!

nk!

( !

r ) == !!

nk!

!!

nk!

( ! r )

!! ( r ) == ! !! *

i

( r ) !

i==1

!

N

! !!

i

( r ) !

Local and nonlocal parts of pseudopotential

V

ionps,!!

( !r,!r') == !!

locps,""

( r ) ++ !!

l""

( r )

!

lm

! Y

lm*

( !r )Y

lm

( !r') Pseudopotential for atomic species !

Local

Non-local

V

ps

( !r,!r') == V

ionps,!!

( !r !! !X

!!

, !r'!! !X

!!

)

"

!!

"

V

ps

( !r,!r') == !!

locps,""

(| r !! ! X

!!

|)

"

!!

" ++

+

"

!!

" ""

l!!

(| !r !! !X

!!

|)

"

lm

" Y

lm*

( !r !! !X

!!

)Y

lm

( !r'!! !X

!!

) Pseudopotential

for a collection of atoms

local part

non-local part

(10)

l lm l lm K Bps loc

lm l lm

lm

!" # !" # ˆV " ( r )

# | " | #

!!

== ++ " "

Kleinman-Bylander Separable Pseudopotentials

Fully non-local separable PP

T ++ ˆV !

ps

!! !!

i

(( )) !!

i

== 0

Exact for the reference atomic energies Approximate for all other energies

Much easier calculations ( in comparison to semi-local PP) of Fourier components

cnk!(! Eigenvalue problem – system of equations for G) expansion coefficients

Fourier transform

!!val(! G) == 1

!

!0 cnk!

n! k!

"

G'

"

(G ++! G')c! n!

*k(!

Self-consistent

G)

problem

!!loc(! G !!!

G') ==!!H(! G !!!

G') ++!!xc(! G !!!

G') ++!!locps(! G !!!

G')

G'!

!

!

!2m2(k ++! G)!2!!G,! ! G'++!!loc(!

G ""!

G') ++ !!nonlocps (! k ++!

G,! k ++!

## G')

$$%%

%%

&&

''((

((cnk!(!

G') == !!nk!cnk!(! G)

!!

nk!(

!

r ) == 1

!

! cnk!(

!

G)ei(k ++! G)""! r! G!

#

#

Kohn-Sham equations in pseudopotential plane-wave formalism

Kohn-Sham Eqs.

in real space

Pseudopotential plane-wave formalism – practical aspects

{ ! G : 1 2 ( !

k ++ !

G)

2

!! E

cutoff

} Number of plane waves

in the wavefunction expansion N

Number of needed Fourier coefficients of the local potential? !!

loc

( !

G !! ! G')

max | ! G !! !

G' |== 2 ! G

max

G !

max

: 1 2 ( !

k ++ !

G

max

)

2

== E

cutoff

N 8

!!( !

G ) -- 8 N Fourier coefficients required

Pseudopotential plane-wave formalism – practical aspects

G'!

!

!

!2m2(k ++! G)!2!!G,! ! G'++!!loc(!

G ""!

G') ++ !!nonlocps (! k ++!

G,! k ++!

## G')

$$%%

%%

&&

''((

((cµµ(!

G') == !!µµcµµ(! G)

Solution of eigenvalue problem gives N eigenvalues and eigenfunctions

For self-consistent solution of the problem, it is necessary to known only occupied states

Number of occupied states << N

Traditional methods (based on the solution of eigenvalue problem) only practicable for moderate N (say N~2000) For N > 2000 (large supercells), reformulation of the problem is required

µµ !! ( n !

k )

(11)

Pseudopotential plane-wave formalism – practical aspects – Iterative methods

g

(µµ)

( ! G ) ==

!

!

G'

! ( H( G, ! G') "" !! !

µµ

""

G ,!G'!

)c

µµ

( ! G') Gradient

We are looking for wavefunctions mutually orthogonal

µ N !!

occ

such that the gradients vanish Searching procedure ?

( n ) ( n ) ( µ )

µ µ

c

++1

== c ++ !g ( n ) e.g. steepest descent

conjugate gradient Davidson method

Required: effective method to calculate gradient

µµ !! ( n!

k )

µµ | µµ' == c

µµ*

( ! G )c

µµ'

( !

!

G )

!

G

! == !!

µµµµ'

g

( µµ )

( ! G ) == 0

Pseudopotential plane-wave formalism – practical aspects

Calculation of from the formula

!!val(! G) == 1

!

!0 cnk!

n! k!

"

G'

"

(G ++! G')c! n!

*k(! G)

requires ~ N

2

operations Inefficient !!

Better solution

Introduce mesh in r-space with 8N points Fourier transform wavefunction in G-space into wave function in real space

{ !r

i

}

Use Fast Fourier Transform – it costs ~ NlogN operations Calculate

Use inverse FFT to obtain Total cost ~ NlogN

!!

µµ

( !r

i

) c

µµ

( !

G

j

) ! ! !!

µµ

( !r

i

)

!!( !r

i

) == !!

*µµ

µµ occ

!

! ( !r

i

)!!

µµ

( !r

i

)

!!( !

G ) !! ( !r

i

) ! ! !! ( ! G

j

)

!! ( ! G )

Use FFT to obtain local pseudopotential plus Hartree on mesh

Pseudopotential plane-wave formalism – practical aspects: local potential

Calculation of Hartree potential is very cheap:

How to deal with other parts of local potential?

Separate local pseudopotential into long-range and

short range part

ps,! v ps,! v

loc

Z e

loc

Z e

" ( r ) == !! r

2

++ ( " ( r ) ++ r

2

)

loc,SRps,!

" ( r ) Calculate Fourier coefficient of Coulomb potential analytically and of the short range one numerically Calculate

!!

H

( ! G )~ !!(

G ) ! G

2

!!

locps

( ! G

j

)

{ !r

i

} !!

locps

( !

G

j

) ++ !!

H

( !

G

j

) ! ! !!( !r

i

)

Pseudopotential plane-wave formalism – practical aspects: local potential

Calculate exchange-correlation potential on the mesh { !r

i

} using values of (LDA, GGA approximation) Compute

FFT to get

Very simple calculation of

gkin(µµ)(!

G) == !

!

G'

!

!2m2(k ++! G)!2!!G,! ! G'

""

##$$

$$

%%

&&

''''cnk!(! G')

No problem!

!!

xc

( !r

i

) !! ( !r

i

)

g

loc( µµ )

( !r

i

) :== [!!

xc

( !r

i

) ++ !!

locps

( !r

i

)]!!

µµ

( !r

i

)

gloc( µµ )(!ri)

g

loc

( !

G

j

)

g

loc(µµ)

( !

G ) == !!

loc

( !

!

G !!

"

G'

" G')c !

µµ

( G') !

(12)

Pseudopotential plane-wave formalism – practical aspects: nonlocal potential

g

nonloc(µµ)

( !

G) == !!

nonlocps

( ! k ++ !

G, ! k ++ !

!

G')

!

G'

! c

n!

k

( ! G') Calculation of gradient corresponding to nonlocal semilocal pseudopotential is very costly

Calculation of Fourier coefficients always of the order of N

2

Therefore, separable nonlocal pseudopotential of the Bylander-Kleinman form

Local part maybe identical in K-B and semilocal PP

!!

nonlocK !!B

( !r,!r') ==

"

lm

" f

lm,!!*

( !r !! !X

!!

) f

lm,!!

( !r'!! !X

!!

)

"

!!

"

Pseudopotential plane-wave formalism – practical aspects: nonlocal potential

The knowledge of is sufficient to calculate

Computational cost proportional to N, but with rather large prefactor

!!

nonlocK !!B

( !r,!r') ==

"

lm

" f

lm,!!*

( !r !! !X

!!

) f

lm,!!

( !r'!! !X

!!

)

"

!!

"

f

lm,!!

( ! G )

g nonloc ( !! ) ( ! G )

Pseudopotential plane-wave formalism – practical aspects norm conserving PPs

Computational schemes with norm conserving PPs

Computational burden ~ N log N Good transferability of the PPs

Atoms from the first row of Periodic Table require kinetic energy cutoff of ~60-70 Ry

Too many plane-waves required in many applications atoms from the first row of Periodic Table

semicore d-states

Ultra soft pseudoptentials

Even one atom of this type requires large cutoff

Features of the Pseudopotential Method

Pseudopotential is approximation to all-electron case, but!

Very accurate

•Comparable accuracy to AE in most cases

•Simpler formalism Low computational cost

•Perform calculations on ‘real-life’ materials Allows full advantage of plane-wave basis sets

•Systematic convergence

•Easy to perform dynamics

Cytaty

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