Modeling of Nanostructures and Materials
Summer Semester 2014 Lecture
Jacek A. Majewski
Chair of Condensed Matter Physics Institute of Theoretical PhysicsFaculty 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"
!22M
!#
!! ! 2m
2"
i2#
i+ 1 2 |
RZ!
!Z"e2!
! !
R!
|
#
! ,!! |
R!
Z"e2!
! !
ri|
#
i,!+ 1 2 | !r
e2 i! !
rj|
#
i, jMaterials Science:
Examples of Schrödinger Equation?
Ab-initio (first principles) Method –
ONLY Atomic Numbers {
Z
i} as input parameters Materials are composed of nuclei and electronsthe 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 !
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[ !! ] == !! !
22m # #
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$$ #
%% && '' (( )) !!
i( r') !
!! !
22m
"
"
"
2++ !!
ext( !r )++!!
H( !r )++!!
x( !r )++!!
c( !r )
##
$$ %% &&
'' ((!!
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)
#
$ % &
' ( !
"= #
"!
"Solution of the Kohn-Sham Equations
Direct methods on a mesh in r-spaceExpansion 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
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
2r
Si: (pseudo core) 3s2 3p2
4 (valence) electrons
V
ext(r) = V
ps(r)
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 ##
!!
" "
!! =={ˆat ( 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
"
vVps 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 2V 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
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 2ps==
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
”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
= 0,1,2Norm 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 pseudopotentialProjection 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
-” componentsExchange-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)
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')
$$%%
%%
&&
''((
((cnk!(!
G) == !!nk!cnk!(! G)
!! !
22m
" !
"
2++ !!
H( ! r ) ++ !!
xc( !
r ) ++ ˆ V
ionps##
$$ %%
%%
&&
'' ((
(( !!
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
(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
!! !
22m
"
!
!
2++ !!
H( r ) ++ ! !!
xc( r ) ++ ˆ ! V
nonlocps
( r, ! r ') !
!!
""
## ##
$$
%% &&
&& !!
nk!( r ) == ! !!
nk!!!
nk!( r ) !
!"
lV
ionps,!!( !r,!r') == !!
l ,ionps,""!
lm! ( r )Y
lm*( !r )Y
lm( !r') V
ionps,!!( !r,!r') == !!
l0 ,ionps,!!( r ) ++ [""
l ,ionps,!!!
lm! ( r ) "" ""
l0 ,ionps,!!( r )]Y
lm*( !r )Y
lm( !r')
ps loc l
lm
ˆ! == ! ( r ) ++ ! ! ! ( r ) lm lm
Semilocal Pseudopotentials
Norm conserving pseudopotential – semilocal PP
local in r, nonlocal in angles
Expensive calculations of !! ps ( ! k ++ !
G, ! k ++ !
G')
!"
lis short range
Kohn-Sham equations in pseudopotential formalism V
ps( !r,!r') == !!
locps( !r )++!!
nonlocps( !r,!r')
!!
loc( !
r ) == !!
H( !
r ) ++ !!
xc( !
r ) ++ !!
locps( ! r )
!!!2 2m
"!
"2++!!H(!r)++!!xc(!r)++!!locps(!r)++!!nonlocps (!r,!r')
##
$$%%
%%
&&
''((
((!!nk!(!r) == !!nk!!!nk!(!r)
!! !
22m
" !
"
2++ !!
loc( !
r ) ++ !!
nonlocps( ! r, !
## r ')
$$ %%
%%
&&
'' ((
(( !!
nk!( !
r ) == !!
nk!!!
nk!( ! r )
!! ( r ) == ! !! *
i( r ) !
i==1
!
N! !!
i( r ) !
Local and nonlocal parts of pseudopotential
V
ionps,!!( !r,!r') == !!
locps,""( r ) ++ !!
l""( r )
!
lm! Y
lm*( !r )Y
lm( !r') Pseudopotential for atomic species !
Local
Non-local
V
ps( !r,!r') == V
ionps,!!( !r !! !X
!!, !r'!! !X
!!)
"
!!"
V
ps( !r,!r') == !!
locps,""(| r !! ! X
!!|)
"
!!" ++
+
"
!!" ""
l!!(| !r !! !X
!!|)
"
lm" Y
lm*( !r !! !X
!!)Y
lm( !r'!! !X
!!) Pseudopotential
for a collection of atoms
local part
non-local part
l lm l lm K Bps loc
lm l lm
lm
!" # !" # ˆV " ( r )
# | " | #
!!
== ++ " "
Kleinman-Bylander Separable Pseudopotentials
Fully non-local separable PP
T ++ ˆV !
ps!! !!
i(( )) !!i == 0
Exact for the reference atomic energies Approximate for all other energies
Much easier calculations ( in comparison to semi-local PP) of Fourier components
cnk!(! Eigenvalue problem – system of equations for G) expansion coefficients
Fourier transform
!!val(! G) == 1
!
!0 cnk!
n! k!
"
G'"
(G ++! G')c! n!*k(!
Self-consistent
G)problem
!!loc(! G !!!
G') ==!!H(! G !!!
G') ++!!xc(! G !!!
G') ++!!locps(! G !!!
G')
G'!
!
!
!2m2(k ++! G)!2!!G,! ! G'++!!loc(!G ""!
G') ++ !!nonlocps (! k ++!
G,! k ++!
## G')
$$%%
%%
&&
''((
((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
maxG !
max: 1 2 ( !
k ++ !
G
max)
2== E
cutoffN 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 )
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 !!
occsuch 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
2operations 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) == !!
*µµµµ 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,! vloc
Z e
locZ 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') !
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
2Therefore, 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!