Pseudopotentials (Part I):
Georg KRESSE
Institut für Materialphysik and Center for Computational Materials Science
Universität Wien, Sensengasse 8, A-1090 Wien, Austria
b-initio
ackage
ienna
G. K RESSE , P SEUDOPOTENTIALS (PART I)
imulation
Page 1
Overview
the very basics
– periodic boundary conditions
– the Bloch theorem
– plane waves
– pseudopotentials
determining the electronic groundstate
effective forces on the ions
general road-map to the things you will hear in more detail later
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 2
Periodic boundary conditions
as almost all plane wave codes VASP uses always periodic boundary conditions
the interaction between repeated images must
be handled by a sufficiently large vacuum region
sounds disastrous for the treatment of
molecules
but large molecules can be handled with a
comparable or even better performance than
by e.g. Gaussian
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 3
The Bloch theorem
the translational invariance implies that a good quantum number exists, which is
usually termed k
k corresponds to a vector in the Brillouin zone
all electronic states can be indexed by this quantum number
Ψk
in a one-electron theory, one can introduce a second index, corresponding to the one
electron band n
ψn k r
the Bloch theorem implies that the (single electron) wavefunctions observe the
equations
ψn k r eikτ
τ
ψn k r
where τ is any translational vector leaving the Hamiltonian invariant
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 4
The DFT Hamiltonian
the charge density is determined by integrating over the entire Brillouin zone and
summing over the filled bands
3
∑∞
n d k f n k ψn k r ψ n k r
ρe r
where the charge density is cell periodic (can be seen by inserting the Bloch theorem)
1 are the Fermi-weights
1 exp β εn k εFermi
and fn k
the KS-DFT equations (Schrödinger like) are given by
ε n k ψn k r
Vxc ρe r
ρion r 3
d r
r r
ρe r
e2
V eff r ρe r
ψn k r
V eff r ρe r
h̄2
2me ∆
ρion is the ionic charge distribution
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 5
Plane waves
introduce the cell periodic part un k of the wavefunctions
ψn k r
un k r eikr
un k r is cell periodic (insert into Bloch theorem)
all cell periodic functions are now written as a sum of plane waves
kr
iG
C
e
Gnk
∑
2
1
Ω1
ψn k r
un k r
1
iGr
C
e
Gnk
∑
Ω1 2 G
G
iGr
ρ
e
G
∑
ρr
G
∑ VG eiGr
V r
G
in practice only those plane waves G
k are included which satisfy
k2
h̄2
G
2me
Ecutoff
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 6
Fast Fourier transformation
FFT
real space
τ
reciprocal space
2
G
cut
b2
0
1
2
3
N−1 0
τ
1
0
1
2
b1
g = n 2π / τ
1
1
1
1
ikr
C
e
rnk
Ω1 2
G. K RESSE , P SEUDOPOTENTIALS (PART I)
r
ψn k r
NFFT
iGr
G
∑ Crnk e
1
∑ CGnk eiGr CGnk
5 −4 −3 −2 −1 0
N/2 −N/2+1
x =n /N τ
1
1
1
Crnk
3 4
Page 7
Why are plane waves so convenient
historical reason:
many elements exhibit a band-structure that can be interpreted in a free electron
picture (metallic s and p elements)
the pseudopotential theory was initially developed to cope with these elements
(pseudopotential perturbation theory)
practicle reason:
the total energy expressions and the Hamiltonian H are dead simple to implement
a working pseudopotential program can be written in a few weeks using a modern
rapid prototyping language
computational reason:
because of it’s simplicity the evaluations of Hψ is exceedingly efficient using FFT’s
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 8
Computational reason
evaluation of Hψn k r
kr
G
!
k ψn k
Ω1 2
ei G
1
k
and using the convention r G
ψn k r
V r
h̄2
∆
2me
CGnk
kinetic energy:
h̄2 G k 2
CGnk
2me
k
G
h̄2
∆ ψn k
2me
Nplanewaves
local potential:
VrCrnk e
NFFT ∑
k V ψn k
G
1
iGr
NFFT log NFFT
r
"
if H would be stored as a matrix with Nplanewaves Nplanewaves components
Nplanewaves Nplanewaves operations would be required
"
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 9
Local part of Hamiltonian
Gcut
FFT
ψ
ψ
G+k
V
r
r
FFT
<k+G V ψ >
ψ V
r r
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 10
Pseudopotential approximation
the number of plane waves would exceed any practicle limits except for H and Li
#
pseudopotentials instead of exact potentials must be applied
three different types of potentials are supported by VASP
– norm-conserving pseudopotentials
– ultra-soft pseudopotentials
– PAW potentials
they will be discussed in more details in later sessions
all three methods have in common that they are presently frozen core methods
i.e. the core electrons are pre-calculated in an atomic environment and kept frozen in
the course of the remaining calculations
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 11
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3s
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3s
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Page 12
G. K RESSE , P SEUDOPOTENTIALS (PART I)
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nodal structure
is retained
2p and 1s are
nodeless !!!!
2p
2s
1s
%$ %$
%$ %$
%$ %$
%$ %$
PAW Al atom
: E= -0.205 R c=1.9
%$ %$ %$
%$ %$ %$
%$ %$ %$
%$ %$ $%
effectiv Al atom
4
3
2
R (a.u.)
1
0
0.0
p
0.5
Al
'& '& '&
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%$ %$ %$
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Al
1.0
: E= -0.576 R c=1.9
1.5
s
Al
pseudopotential
exact potential (interstitial region)
2.0
wave-function
Pseudopotential: essential idea
2.5
2.5
A@ A@ A@ A@ A@ A@ A@ A@
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A@ A@ A@ A@ A@ A@ A@ A@
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Al
wave-function
=< =< =< =< =< =< =< =<
=< =< =< =< =< =< =< =<
=< =< =< =< =< =< =< =<
=< =< =< =< =< =< =< =<
;: ;: ;: ;: ;: ;: ;: ;:
;: ;: ;: ;: ;: ;: ;: ;:
;: ;: ;: ;: ;: ;: ;: ;:
;: ;: ;: ;: ;: ;: ;: ;:
Scattering approach
Al
2.0
1.5
s
: E= -0.576 R c=1.9
p
: E= -0.205 R c=1.9
1.0
0.5
0.0
0
1
2
R (a.u.)
3
4
solve the Schrödinger equation in the interstitial region only, using energy and
angular momentum dependent boundary conditions at the spheres
C C C C
rc
∂ log φl r ε
∂r
B
B
C C C C
B
1
∂ φl r ε
∂r
φl r ε
rc
φl are the regular solutions of the radial Schrödinger equation inside the spheres for
the angular momentum quantum number l and the energy ε
details of the wavefunctions (number of nodes in the spheres) do not enter
pseudopotential: select one specific energy ε in the centre of the valence band and
replace the exact wavefunction φl by a pseudo-wavefunction with φ̃l
C C C C
rc
∂ log φ̃l r ε
∂r
B
B
C C C C
∂ log φl r ε
∂r
G. K RESSE , P SEUDOPOTENTIALS (PART I)
rc
Page 13
2.5
wave-function
KJ KJ KJ KJ KJ KJ KJ KJ
KJ KJ KJ KJ KJ KJ KJ KJ
KJ KJ KJ KJ KJ KJ KJ KJ
KJ KJ KJ KJ KJ KJ KJ KJ
IH IH IH IH IH IH IH IH
IH IH IH IH IH IH IH IH
IH IH IH IH IH IH IH IH
IH IH IH IH IH IH IH IH
Al
GF GF GF GF GF GF GF GF
GF GF GF GF GF GF GF GF
GF GF GF GF GF GF GF GF
GF GF GF GF GF GF GF GF
ED ED ED ED ED ED ED ED
ED ED ED ED ED ED ED ED
ED ED ED ED ED ED ED ED
ED ED ED ED ED ED ED ED
Norm-conserving PP
Al
2.0
1.5
s
: E= -0.576 R c=1.9
p
: E= -0.205 R c=1.9
1.0
0.5
0.0
0
1
2
R (a.u.)
3
4
!
obviously one would like to get the right distribution of charge between the spheres
and the interstitial region as well
norm-conserving pseudopotentials
rc
0
φ̃l r ε φ̃l r ε 4πr2 dr
0
φl r ε φl r ε 4πr dr
rc
2
norm-conservation has another important consequence
the scattering properties are not only correct at the reference energy ε but also in a
small energy interval around ε
G. K RESSE , P SEUDOPOTENTIALS (PART I)
l
M
L L L L
rc
∂ ∂ log φ̃l r ε
∂ε
∂r
L L L L
∂ ∂ log φl r ε
∂ε
∂r
rc
Page 14
Pseudopotential generation
all-electron calculation for a reference atom
chose energies εl at which the pseudisation is performed
usually these are simply the eigen-energies of the bound valence states, but in
principle any energy can be chosen (centre of valence band)
replace the exact wavefunction by a node less pseudo-wavefunction observing the
following four requirements:
n
n is the n th derivative
N
N
2
rc
4π
norm-conservation condition
φ r 2 r2 dr
φ̃ r r dr
2 2
0
0
ONN
for n
rc
4π
φ rc
n
φ̃ rc
0
this pseudopotential conserves exactly the scattering properties of the original atom
in the atomic configuration
but it is an approximation in another environment
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 15
Projector augmented wave method
P.E. Blöchl, Phys. Rev. B50, 17953 (1994); G. Kresse, and J. Joubert, Phys. Rev. B 59, 1758 (1999).
wave function (and energy) are decomposed into three terms:
atoms
atoms
YX YX YX
YX YX YX
YX YX YX
YX YX XY
[Z [Z [Z
Z[ Z[ Z[
[Z [Z [Z
[Z [Z [Z
YX YX YX
YX YX YX
YX YX YX
YX YX XY
SR SR SR
SR SR SR
SR SR SR
SR SR SR
QP QP QP
QP QP QP
QP QP QP
QP QP PQ
[Z
Z[
[Z
[Z
SR
SR
SR
SR
YX
YX
YX
YX
QP
QP
QP
QP
[Z [Z [Z
Z[ Z[ Z[
[Z [Z [Z
Z[ [Z [Z ]\ ]\ ]\
]\ ]\ ]\
]\ ]\ ]\
]\ ]\ \]
SR SR SR
SR SR SR
SR SR SR
RS SR SR UT UT UT
UT UT UT
UT UT UT
UT UT TU
_^ _^ _^
_^ _^ _^
_^ _^ _^
_^ _^ _^
]\ ]\ ]\
]\ ]\ ]\
]\ ]\ ]\
]\ ]\ \]
WV WV WV
WV WV WV
WV WV WV
WV WV WV
UT UT UT
UT UT UT
UT UT UT
UT UT TU
_^ _^
_^ _^
_^ _^
_^ _^
]\ \]
]\ ]\
]\ ]\
]\ ]\
WV WV
WV WV
WV WV
WV WV
UT TU
UT UT
UT UT
UT UT
`
– kinetic energy
– charge densities
– Hartree.- and exchange correlation energy
_^ _^
_^ _^
_^ _^
_^ _^
WV WV
WV WV
WV WV
WV WV
– wave functions
Page 16
G. K RESSE , P SEUDOPOTENTIALS (PART I)
efficient
no interaction between different spheres and plane waves
QP QP QP
QP QP QP
QP QP QP
QP QP PQ
pseudo (node less) pseudo−onsite exact onsite
plane waves
radial grids
radial grids
exact
+
−
=
φlmε clmε
∑
φ̃lmε clmε
∑
ψ̃n
ψn
decomposition into three terms holds for
Determining the electronic groundstate
by iteration – self consistency (old fashioned)
start with a trial density ρe , set up the Schrödinger equation, and solve the
Schrödinger equation to obtain wavefunctions ψn r
Ne 2
ONN
a
2
and a new one
∑n ψn r
Vxc ρe r
ρion r 3
d r
r r
e
ρe r
r ρe r
V
2
1
as a result one obtains a new charge density ρe r
electron potential:
eff
n
N
ε n ψn r
ψn r
V eff r ρe r
h̄2 2
∇
2me
#
new Schrödinger equation
iteration
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 17
Self-consistency scheme
b b
b b
trial-charge ρin and trial-wavevectors ψn
f f
c
set up Hamiltonian H ρin
g
c
iterative refinements of wavefunctions ψn
g
c
new charge density ρout
2
∑n fn ψn r
c
refinement of density ρin ρout
#
new ρin
refinement of density:
DIIS algorithm
P. Pulay, Chem. Phys. Lett. 73,
393 (1980).
g
refinement of wavefunctions:
blocked Davidson like algorithm
d
ce
e
d
ee dd
e d
Ebreak
e
e
d
∆E
d e
dd ee
no
i
h
two subproblems
optimization of ψn and ρin
d c
calculate forces, update ions
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 18
What have all iterative matrix diagonalisation schemes in common ?
one usually starts with a set of trial vectors (wavefunctions) representing the filled
states and a few empty one electron states
1
Nbands
NN ψn n
these are initialised using a random number generator
then the wavefunctions are improved by adding a certain amount of the residual
vector to each
the residual vector is defined as
ψn H ψ n
εapp
εapp S ψn
H
R ψn
Hψn is exactly operation discussed before (efficient)
adding a small amount of the residual vector
!
λR ψn
ψn
ψn
is in the spirit of the steepest descent approach (“Jacobi relaxation”)
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 19
We know the groundstate, what now ?
Hellman-Feynmann theorem allows to calculate the ionic forces and the stress tensor
by moving along the ionic forces (steepest descent) the ionic groundstate can be
calculated
we can displace ions from the ionic groundstate, and determine the forces on all other
ions
#
effective inter-atomic force constants and vibrational frequencies
molecular dynamics by using Newtons equation of motion
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 20
Vibrational properties
Frequency cm
-1
the vibrational frequencies are calculated using a brute force method:
in the supercell all “selected
atoms” are displaced from
LO
1600
their groundstate position
SH
1400
1200
1000
800
600
400
200
0
Γ
LA
ZO
SH
ZA
M
K
Γ
not possible presently
#
forces
force constants
symmetry is not used !!!
works well for molecules, but has to be applied with great care to solids
G. K RESSE , P SEUDOPOTENTIALS (PART I)
Page 21
Pseudopotentials (Part II) and PAW
Georg KRESSE
Institut für Materialphysik and Center for Computational Materials Science
Universität Wien, Sensengasse 8, A-1090 Wien, Austria
b-initio
ackage
ienna
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
imulation
Page 1
Overview
pseudopotential basics
normconserving pseudopotentials
adopted pseudization strategy
G. Kresse, and J. Hafner, J. Phys.: Condens. Matter 6 (1994)
from normconserving to ultrasoft pseudopotentials
the PAW method
G. Kresse, and J. Joubert, Phys. Rev. B 59, 1758 (1999).
where to be careful ?
– local pseudopotentials
– simultaneous representation of valence and semi-core states
– magnetic calculations
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 2
Normconserving pseudopotentials: General strategy
all-electron calculation for a reference atom
(rhfsps)
pseudization of valence wave functions
(rhfsps)
chose local pseudopotential and factorize
(fourpot3)
un-screening of atomic potential to obtained ionic pseudopotential
(fourpot3)
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 3
Pseudization of valence wave functions
different schemes have been proposed in literature, but the general strategy is always
similar
calculate exact all-electron wave function φ r
replace exact φ r inside pseudization radius by
a suitable “soft” pseudo wave function φ̃ r must fulfill some continuity conditions
rc
n
2
0
rc
for n
φ rc
r
φr
n
φ̃ rc
r
∑i αi βi r
φ̃ r
possibly impose normconservation condition
rc
rc
4π
0
φ r 2 r2 dr
φ̃ r r dr
4π
2 2
0
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 4
Which expansion set should one use?
many different basis sets have been proposed in the literature
presently the two most prominent ones are
– polynomials (Troullier and Martins)
c10 r10
c8 r 8
c6 r 6
c4 r 4
c2 r 2
c0
φ̃ r
c12 r12
– spherical Bessel-functions (RRKJ—Rappe, Rabe, et. al.)
i 1
φ rc
φ rc
with qi such that
qi r
j l q i rc
j l q i rc
∑ αi j l
φ̃ r
34
the last one is the standard scheme for VASP pseudopotentials
the basis set I use is generally minimal (3 or sometimes 4 Bessel-functions)
for PAW and US pseudopotentials only 2 spherical Bessel-functions are required
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 5
Why are spherical Bessel-functions so convenient
close analogy between plane waves and spherical Bessel-functions
the required cutoff can be calculated directly from the expansion set
34
∑ αi jl
φ̃ r
qi r
i 1
2
15
qi
h̄2
2me max
Ecut
find maximum qi
I always use a minimal basis set
in the original RRKJ scheme, more spherical Bessel-functions were used, and the
wave functions were optimized for a selected cutoff
our tests indicate that this is contra-productive
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 6
Factorization
required to speed up the calculations D.M. Bylander, et al., Phys. Rev. B 46, 13756 (1992)
chose local reference potential Vloc
p φ̃
construct a projector such that
1
Vloc
ε
φ̃
p ∝
h̄2
∆
2me
the factorized Hamiltonian is given by
H
Vloc
Vloc
pD p
"
ε φ̃ φ̃
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
φ̃
!
h̄2
2me ∆
one recognizes immediately that: φ̃
ε φ̃
"
!
h̄2
2me ∆
φ̃
pD p
D
with
Vloc
h̄2
∆
2me
Page 7
wave-function
2.5
What have we acchived at this point
2.0
1.5
s
: E= -0.576 R c=1.9
p
: E= -0.205 R c=1.9
1.0
0.5
0.0
1
0
2
R (a.u.)
3
4
the exact wavefunction has been replaced by it’s pseudo counterpart
and a “pseudo” Hamiltonian has been constructed
φ̃
pD p
εφ
ε φ̃
Vloc
φ
VAE
h̄2
∆
2me
h̄2
∆
2me
at the energy ε, φ and φ̃ are identical outside of the cutoff radius
#
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
at the energy ε
φ and φ̃ have the same norm inside the cutoff radius
φ̃ are identical outside of the cutoff radius
δε φ and
Page 8
Two reference energies
pseudize at two reference energies: φi i
%
δi j
pi φ̃ j
φ̃ j
j
εj
Vloc
∑ αi j
pi
h̄2
∆
2me
for all i j
$
construct two projectors such that
12
factorized Hamiltonian is given by
p i Di j p j
∑
ij
φ̃ j
εj
"
!
ε j φ̃i φ̃ j
one recognizes immediately that: φ̃i H φ̃ j
Vloc
h̄2
2me ∆
φ̃i
Di j
Vloc
H
h̄2
∆
2me
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 9
Two reference energies: practical considerations
the pseudo wavef. must fulfill a generalized normconserv. condition:
rc
&
0
i j
φi r φ j r r2 dr
0
4π
φ̃i r φ̃ j r r dr
4π
rc
2
in the VASP PP generation program only
rc
0
0
φi r φi r r2 dr
φ̃i r φ̃i r r dr
rc
2
is enforced
to correct for this error, augmentation charges would be required, but these are
neglected
as a result Di j is not Hermitian
φ̃ j
εj
Vloc
"
!
h̄2
2me ∆
φ̃i
Di j
off-diagonal elements are averaged to make the matrix D symmetric
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 10
US pseudopotentials, very similar to NC pseudopotentials
pseudize at two reference energies
εj
Vloc
φ̃ j
j
for all i j
∑ αi j
pi
h̄2
∆
2me
δi j
pi φ̃ j
construct two projectors such that
the factorized Hamiltonian and overlap operator are given by
S
1
φ̃i φ̃ j
φi φ j
Qi j
ε j Qi j
"
!
φ̃ j
εj
Vloc
p i Qi j p j
ij
h̄2
2me ∆
∑
p i Di j p j
∑
ij
φ̃i
Vloc
Di j
H
h̄2
∆
2me
φ̃i S φ̃ j ε j
one can show that: φ̃i H φ̃ j
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 11
What does all that mean?
let us look again at the definition of Di j
ε j Qi j
φ̃ j
Qi j
ε j φ̃i φ̃ j
Vloc φ̃ j
εj
h̄2
∆
φ̃i
2me
Vloc
φ̃i
Di j
h̄2
∆
2me
φ̃i φ̃ j
VAE φ j
h̄2
∆
2me
φi
Vloc φ̃ j
φ̃i
h̄2
∆
2me
φi ε j φ j
Vloc φ̃ j
φi φ j
φ̃i
h̄2
∆
2me
ε j φ̃i φ̃ j
Vloc φ̃ j
φ̃i
h̄2
∆
2me
*
'
*
'
()
()
energy pseudo onsite
energy AE onsite
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 12
US-PP: what they really do
,
/
.
/
.
-
+ .
,
,
+ .
+
+
,
-
98 98 98
98 98 98
98 98 98
98 98 89
10 10 10
10 10 10
10 10 10
10 10 01
;: ;: ;:
;: ;: ;:
;: ;: ;:
;: ;: ;:
98 98 98
98 98 98
98 98 98
98 98 89
32 32 32
32 32 32
32 32 32
32 32 32
10 10 10
10 10 10
10 10 10
10 10 01
;:
;:
;:
;:
32
32
32
32
98
98
98
98
10
10
10
10
;: ;: ;:
;: ;: ;:
;: ;: ;:
:; ;: ;: =< =< =<
=< =< =<
=< =< =<
=< =< <=
32 32 32
32 32 32
32 32 32
23 32 32 54 54 54
54 54 54
54 54 54
54 54 45
?> ?> ?>
?> ?> ?>
?> ?> ?>
?> ?> ?>
=< =< =<
=< =< =<
=< =< =<
=< =< <=
76 76 76
76 76 76
76 76 76
76 76 76
54 54 54
54 54 54
54 54 54
54 54 45
?> ?>
?> ?>
?> ?>
?> ?>
=< <=
=< =<
=< =<
=< =<
76 76
76 76
76 76
76 76
54 45
54 54
54 54
54 54
?> ?>
?> ?>
?> ?>
?> ?>
76 76
76 76
76 76
76 76
Page 13
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
AE-onsite
pseudo-onsite
pseudo
AE
+
=
VAE φ j
∆
ρi j φ i
Vloc φ̃ j
∆
ρi j φ̃i
Vloc
∆
E
Ψ̃n pi p j Ψ̃n
onsite occupancy matrix (or density matrix): ρi j
δi j
pi φ̃ j
p̃i Ψ̃n
character of wave function: ci
energy is the sum of three terms
US-PP method is in principle an exact frozen core all-electron method
Mixed basis set with an implicit dependency
US-PP’s carry a small rucksack, with two additional sets of basis functions
defined around each atomic site
%
$
– one for the soft pseudo-wave functions φ̃i
%
$
– one for the AE wave functions φi
for each atomic sphere the energy is evaluated using these two sets
and the calculated energy is subtracted and added, respectively
the onsite occupancy matrix (density matrix) for these two sets is calculated from
the plane wave coefficients
Ψ̃n pi p j Ψ̃n
ρi j
PAW inspired formulation
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 14
Practical considerations
the US-PP method can not be implemented exactly, since currently no method
exists to handle the rapid variations of the all-electron wave functions
a regular grid does not work, maybe wavelets would be an option
in practice, I therefore adopt a modified prescription for US-PP’s:
exact AE wave functions
conserving
r
φ̃i r φ̃ j r
nc
φnc
i r φj r
augmentation charge: Qi j r
φnorm
@
φr
norm-conserving wave functions
these US-PP’s yield exactly the same results as the corresponding NC-PP’s, but at
much lower cutoffs
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 15
Why are US-PP’s softer
pseudo wave function is represented as a sum of two spherical Bessel functions
instead of three
φ rc
φ rc
i 1
j l q i rc
j l q i rc
with qi such that
qi r
∑ αi j l
φ̃ r
2
additionally rc can be increased compared to NC potentials
there is no need to represent the charge distribution of the AE wave function,
since this is done by the augmentation
the pseudo wave functions follow remarkably well the AE wave functions, even
for values much smaller than rc
basis sets are roughly a factor 2-3 smaller
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 16
Example for this behavior
Cu (US)
3.5
3.5
3.0
3.0
2.5
s
wave-function
wave-function
e.g. Cu (NC)
:E=-0.357 Rc=2.5
2.0
1.5
p : E=-0.058 Rc=2.5
1.0
s
: E=-0.357 Rc=2.5
2.0
1.5
p : E=-0.058 Rc=2.5
1.0
0.5
0.0
0
2.5
0.5
d : E=-0.393 Rc=2.0
1
2
R (a.u.)
3
4
0.0
0
d : E=-0.393 Rc=2.0
1
2
3
4
R (a.u.)
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 17
The role of the local potential
the local potential needs to describe scattering properties for radial quantum
numbers not included in the projectors
– much underestimated problem
– resulting errors can be 1-2 % in the lattice constant
the tails of d electrons (transition metals) or p electrons (oxygen) overlap into the
pseudization region
they are picked up as high l components (FLAPW)
ideally one would like to use very attractive local potentials
#
but
ghost-state problem
in most cases, compromises must be made
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 18
Ghost-state problems
particularly severe for alkali, alkali-earth and early transition metals
the more attractive the local potential and the smaller rc , the more likely it is to
have a ghost-state; Zr example
4
4
2
g
p f
0
g
f
s
xl (E)
xl (E)
2
s
d
-2
0
d
-2
p
-4
-4
-3
-2
-1
0
E (Ry)
1
2
-3
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
-2
-1
0
E (Ry)
1
2
Page 19
solution: treat semi-core states as valence
Pseudopotential generation in practice: the PSCTR file
TITEL = US O
LULTRA =
T
use ultrasoft PP ?
RWIGS =
1.40 nn distance
ICORE
NE
LCOR
QCUT
l
0
0
1
1
2
=
=
=
=
2
100
.TRUE.
-1
Description
E
TYP
0
15
0
15
0
15
0
15
0.0
7
RCUT
1.13
1.13
1.13
1.13
1.55
TYP
23
23
23
23
7
RCUT
1.40
1.40
1.55
1.55
1.55
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 20
PAW: basic idea
P.E. Blöchl, Phys. Rev. B50, 17953 (1994)
Kohn-Sham equation
B
Exc n
A
nZ
B
EH n
A
n
Ψn
∑ fn
E
1
∆ Ψn
2
frozen core approximation
for the valence electrons, a transformation from the pseudo to the AE
wavefunction is defined:
p̃lmε Ψ̃n
φ̃lmε
∑sites
lmε φlmε
Ψ̃n
Ψn
lm is an index for the angular and magnetic quantum numbers
ε refers to a particular reference energy
p̃lmε projector function
φ̃lmε partial wave
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 21
PAW: basic idea
p̃lmε Ψ̃n
φ̃lmε
∑ φlmε
Ψ̃n
Ψn
transformation:
the “character” of an arbitrary pseudo-wavefunction Ψ̃n at one site can be
calculated by multiplication with the projector function at that site
p̃lmε Ψ̃n
clmε
inside each sphere the wavefunctions can be determined:
sphere
clmε
C
∑ φl m ε
C
clmε
C
∑ φ̃l m ε
Ψ̃n
lmε
sphere
C
C
C
Ψn
lmε
the projector functions must be dual to the pseudo-wavefunction
C D
C D
δl l δm m δε ε
C D
C
C
C
p̃lmε φ̃l m ε
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 22
PAW: addidative augmentation
PO PO PO PO PO PO
PO PO PO PO PO PO
PO PO PO PO PO PO
PO PO PO PO PO PO
NM NM NM NM NM NM NM
NM NM NM NM NM NM NM
NM NM NM NM NM NM NM
NM NM MN NM NM NM NM
HG HG HG HG HG HG
HG HG HG HG HG HG
HG HG HG HG HG HG
HG HG HG HG HG HG
FE FE FE FE FE FE FE
FE FE FE FE FE FE FE
FE FE FE FE FE FE FE
FE FE EF FE FE FE FE
TS TS TS TS
TS TS TS
TS TS TS
TS TS TS
RQ RQ RQ
RQ RQ RQ
RQ RQ RQ
RQ RQ QR
PO PO TS
PO PO TS
PO PO TS
OP PO RQ RQ RQ
RQ RQ RQ
RQ RQ RQ
RQ RQ QR
LK LK LK LK
LK LK LK
LK LK LK
LK LK LK
JI JI JI
JI JI JI
JI JI JI
JI JI IJ
HG HG LK
HG HG LK
HG HG LK
GH HG JI JI JI
JI JI JI
JI JI JI
JI JI IJ
RQ
RQ
RQ
RQ
JI
JI
JI
JI
TS TS TS
TS TS TS
TS TS TS
TS TS ST
LK LK LK
LK LK LK
LK LK LK
LK LK KL
Page 23
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
– kinetic energy
– Hartree energy
– charge density
– exchange correlation energy
– wavefunctions
AE-onsite
pseudo-onsite
pseudo
AE
+
=
∑ φlmε clmε
∑ φ̃lmε clmε
Ψ̃n
Ψn
p̃lmε Ψ̃n
character of wavefunction: clmε
same trick works for
Derivation of the PAW method is straightforward
for instance, the kinetic energy is given by
∆ Ψn
φi
φ̃i
Ψn
∑ fn
Ekin
n
by inserting the transformation (i
p̃i Ψ̃n
∑
Ψ̃n
Ψn
lmε)
i
Ẽ 1
/
-
.
∆ φj
Y
site i j
Y
site i j
φi
+ .
∑ ∑ ρi j
,/
∆ φ̃ j
.
φ̃i
+ .
∑ ∑ ρi j
-
/
n
E 1 (assuming completeness)
Ẽ
+
.
∆ Ψ̃n
+ .
Ψ̃n
-
∑ fn
into Ekin one obtains: Ekin
X
U
VW
X
U
VW
VW
Ẽ 1
E1
∑n f n ci c j
Z
ρi j is an on-site density matrix: ρi j
X
U
Ẽ
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 24
Hartree energy
the pseudo-wavefunctions do not have the same norm as the AE wavefunctions
inside the spheres
to deal with long range electrostatic interactions between spheres
a soft compensation charge n̂ is introd. (similar to FLAPW)
ji ji ji ji
ji ji ji
ji ji ji
ji ji ji
hg hg hg
hg hg hg
hg hg hg
hg hg gh
hg
hg
hg
hg
`_
`_
`_
`_
ji ji ji
ji ji ji
ji ji ji
ji ji ij
ba ba ba
ba ba ba
ba ba ba
ba ba ab
B
A
B
A
B
A
Page 25
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
n̂1
∑sites EH n1
fe fe ji
fe fe ji
fe fe ji
ef fe hg hg hg
hg hg hg
hg hg hg
hg hg gh
ba ba ba ba
ba ba ba
ba ba ba
ba ba ba
`_ `_ `_
`_ `_ `_
`_ `_ `_
`_ `_ _`
^] ^] ba
^] ^] ba
^] ^] ba
]^ ]^ `_ `_ `_
`_ `_ `_
`_ `_ `_
`_ `_ _`
n̂1 compensation charge at site
ñ1 pseudo-charge at one site
E1
Ẽ 1
Ẽ
Hartree energy becomes: EH
n̂1
∑sites EH ñ1
n̂
EH ñ
fe fe fe fe fe fe
fe fe fe fe fe fe
fe fe fe fe fe fe
fe fe fe fe fe fe
dc dc dc dc dc dc dc
dc dc dc dc dc dc dc
dc dc dc dc dc dc dc
cd cd cd cd cd cd cd
^] ^] ^] ^] ^] ^]
^] ^] ^] ^] ^] ^]
^] ^] ^] ^] ^] ^]
]^ ]^ ]^ ]^ ]^ ]^
\[ \[ \[ \[ \[ \[ \[
[\ [\ [\ [\ [\ [\ [\
\[ \[ \[ \[ \[ \[ \[
\[ \[ [\ \[ \[ \[ \[
pseudo + compens. pseudo+comp. onsite AE-onsite
AE
+
=
PAW energy functional
P.E. Blöchl, Phys. Rev. B50, 17953 (1994).
d3r
A
ñc
B
n̂
Exc ñ1
U R Zion
D
i j
sites
1
∆ φ̃ j
2
ñc
n̂ r
A
φ̃i
∑ ∑ ρi j
B
vH ñZc ñ r
B
Ẽ
1
n̂
Ẽ 1
B
n̂
A
EH ñ
E1
n
Exc ñ
A
1
∆ Ψ̃n
2
∑ fn Ψ̃n
Ẽ
Ẽ
total energy becomes a sum of three terms E
l
k
A
nc
B
Exc n1
D
i j
sites
1
∆ φj
2
d3r
n̂ r
B
∑
∑ ρi j φi
vH ñZc ñ1 r
A
A
E1
Ωr
B
n̂
EH ñ1
B
Ωr
vH nZc n1 r d 3 r
A
A
B
E H n1
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 26
Ẽ is evaluated on a regular grid
Kohn Sham functional evaluated in a plane wave basis set
with additional compensation charges to account for the incorrect norm of the
pseudo-wavefunction (very similar to ultrasoft pseudopotentials)
n̂
∑n fn Ψ̃n Ψ̃n
Z
ñ
pseudo charge density
compensation charge
E 1 and Ẽ 1 are evaluated on radial grids centered around each ion
%
$
%
$
Kohn-Sham energy evaluated for basis sets ψ̃i and ψi
these terms correct for the shape difference between the pseudo and AE
wavefunctions
no cross-terms between plane wave part and radial grids exist
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 27
General scheme
xw xw xw xw xw xw
xw xw xw xw xw xw
xw xw xw xw xw xw
xw xw xw xw xw xw
vu vu vu vu vu vu vu
vu vu vu vu vu vu vu
vu vu vu vu vu vu vu
uv uv uv uv uv uv uv
po po po po po po
po po po po po po
po po po po po po
op op op op op op
nm nm nm nm nm nm nm
mn mn mn mn mn mn mn
nm nm nm nm nm nm nm
nm nm mn nm nm nm nm
|{ |{ |{ |{
|{ |{ |{
|{ |{ |{
|{ |{ |{
zy zy zy
zy zy zy
zy zy zy
zy zy yz
zy
zy
zy
zy
rq
rq
rq
rq
|{ |{ |{
|{ |{ |{
|{ |{ |{
|{ |{ {|
ts ts ts
ts ts ts
ts ts ts
ts ts st
Page 28
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
xw xw |{
xw xw |{
xw xw |{
wx xw zy zy zy
zy zy zy
zy zy zy
zy zy yz
ts ts ts ts
ts ts ts
ts ts ts
ts ts ts
rq rq rq
rq rq rq
rq rq rq
rq rq qr
po po ts
po po ts
po po ts
op op rq rq rq
rq rq rq
rq rq rq
rq rq qr
pseudo + compens. pseudo+comp. onsite AE-onsite
AE
+
=
applies to all quantities
US-PP
D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).
the original derivation of US-PP is somewhat “problematic”
it’s more like accepting things than understanding them
in fact, the equations for US-PP’s can be derived rigidly from the PAW functional
by linearisation of the on-site terms E 1 and Ẽ 1 around the atomic reference
configuration
this shows the close relation between both approaches
but it also indicates when US-PP’s might be problematic:
the more the environment differs from the reference state the less accurate US-PP
are
our tests indicate that magnetism is the strongest perturbation
in other cases, US-PP and the PAW yield almost identical results
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 29
Construction of PAW potentials
first an AE calculation for a reference atom is performed
the AE wavefunctions of the valence states are pseudised
projectors are constructed as
s
: E= -0.296 R c=2.6
j
p1/2: E= -2.537 R c=2.6
1
0
1
2
R (a.u.)
3
4
φ̃ j
δi n
D
p̃i φ̃n
0
εj
they must obey
d3/2: E= -0.228 R c=2.6
Vloc
∑ αi j
h̄2
∆
2me
pi
2
wave-function
3
projectors are dual to the pseudo wavefunction
to have a rather complete set of projectors two partial waves for each quantum
channel lm are constructed
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 30
PAW — US-PP method: molecules
results for the bond length of several molecules obtained with the US-PP, PAW
and AE approaches
plane wave cutoffs were around 200-400 eV
US-PP and the PAW method give the same results within 0.1-0.3%
well converged relaxed core AE calculations yield identical results
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 31
US-PP(data base)
US-PP(special)
PAW
AE
H2
1.447
1.447
1.446a
Li2
5.127
5.120
5.120a
Be2
4.524
4.520
4.521a
Na2
5.667
5.663
5.67 a
2.163
2.141 (2.127)
2.141 (2.128)
2.129a
N2
2.101
2.077 (2.066)
2.076 (2.068)
2.068a
F2
2.696
2.640 (2.626)
2.633 (2.621)
2.615a
P2
3.576
3.570
3.570
3.572a
H2 O
1.840 (1.834)
1.839 (1.835)
1.833a
α(H2 O)( )
105.3 (104.8)
105.3 (104.8)
105.0a
BF3
2.476 (2.470)
2.476 (2.470)
2.464b
SiF4
2.953 (2.948)
2.953 (2.948)
2.949b
}
CO
values in paranthese were obtained with hard potentials at 700 eV
a
NUMOL, R.M. Dickson, A.D. Becke, J. Chem. Phys. 99,3898 (1993), b Gaussian94
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 32
PAW — AE methods: molecules - energetics
400 eV plane wave cutoff
xc
PBE
PBE
rPBEb
rPBEb
meth
PAW
AEa
PAW
AEa
CO
11.65
11.66
11.15
11.18
11.24
N2
10.39
10.53
10.09
10.09
9.91
NO
7.31
7.45
6.95
7.01
6.63
O2
6.17
6.14-6.24
5.75
5.78
5.22
exp
a
S. Kurth, J.P. Perdew, P. Blaha, Int. J. Quantum Chem. 75, 889 (1999).
b revised Perdew Burke Ernzerhof functional, B. Hammer, L.B. Hansen, J.K. Norskov, Phys.
Rev. B 59, 7413 (1999).
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 33
PAW — US-PP method: bulk, semiconductors
a Å3
Ecoh (eV)
B (MBar)
diamond
results for the equilibrium lattice constant a, cohesive energy
Ecoh (with respect to non spin
polarised atoms) and bulk modulus B for several materials calculated with the the PAW, USPP, and the FLAPW approach
US-PP(current)
3.53
-10.15
4.64
PAW(current)
3.53
-10.13
4.63
LAPWa
3.54
-10.13
4.70
PAWa
3.54
-10.16
4.60
US-PP(current)
5.39
-5.96
0.95
PAW(current)
5.39
-5.96
0.95
LAPWa
5.41
-5.92
0.98
PAWa
5.38
-6.03
0.98
silicon
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 34
PAW — US-PP method: bulk, metals
a Å3
Ecoh (eV)
B (MBar)
bcc V
results for the equilibrium lattice constant a, cohesive energy
Ecoh (with respect to non spin
polarised atoms) and bulk modulus B for several materials calculated with the the PAW, USPP, and the FLAPW approach
US-PP(current)
2.93
-9.41
2.02
PAW(current)
2.93
-9.39
2.09
LAPWa
2.94
-9.27
2.00
PAWa
2.94
-9.39
2.00
US-PP(current)
5.34
-2.20
0.0181
PAW(3s3p val)
5.34
-2.19
0.0187
PAW(3p val)
5.34
-2.20
0.0187
LAPWa
5.33
-2.20
0.019
PAWa
5.32
-2.24
0.019
fcc Ca
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 35
PAW — US-PP method: bulk, ionic compounds
a Å3
results for the equilibrium lattice constant a, cohesive energy
Ecoh (with respect to non spin
polarised atoms) and bulk modulus B for several materials calculated with the the PAW, USPP, and the FLAPW approach
Ecoh (eV)
B (MBar)
CaF2
a
US-PP(current)
5.36
-6.32
0.97
PAW(3s3p val)
5.35
-6.32
1.01
PAW(3p val)
5.31
-6.36
1.00
LAPWa
5.33
-6.30
1.10
PAWa
5.34
-6.36
1.00
N.A.W. Holzwarth, et al.; Phys. Rev. B 55, 2005 (1997)
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 36
Semi-core states; alkali and alkali earth metals
from a practicle point of view, an accurate treatment of these elements in ionic
compounds is very important: oxides e.g. perovskites
strongly ionized, and small core radii around 2.0 a.u. (1 Å) are desirable
e.g. Ca: one would like to treat 3s, 3p, 4s states as valence states
wave-function
3
3s : E= -3.437 R c=2.3
2
3p : E= -2.056 R c=2.3
it is very difficult to represent
the charge distribution of the 3s
and 4s states equally well in a
pseudopotential approaches
1
4s : E= -0.284 R c=2.3
0
0
1
2
R (a.u.)
3
4
general problem for pseudopotentials
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 37
Semi core states
in VASP, NC wavefunctions describe the augmentation charges
φ̃i r φ̃ j r
nc
φnc
i r φj r
Qi j r
it is very difficult to construct accurate NC-PP for 3s and 4s (mutual orthogonality)
wave-function
3
3s : E= -3.437 R c=2.3
2
3p : E= -2.056 R c=2.3
1
4s : E= -0.284 R c=2.3
0
0
1
2
R (a.u.)
3
node in 4s must be included in some way
if one succeeds, the augmentation charges become quite hard, and require fine regular grids
PAW method is the best solution to this problem
4
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 38
PAW — US-PP method: atoms
O
N
∆Em
1.55
1.40
1.41
gs
2s2 2p3
2s2 2p3
2s2 2p3
∆Em
3.14
2.88
2.89
gs
3d 6 2 4s1 8
3d 6 2 4s1 8
3d 6 2 4s1 8
∆Em
2.95
2.77
2.76
gs
3d 7 7 4s1 3
3d 7 7 4s1 3
3d 7 7 4s1 3
∆Em
1.40
1.32
1.31
gs
3d 9 4s1
3d 9 4s1
3d 9 4s1
∆Em
0.54
0.52
0.52

2s2 2p4

2s2 2p4

2s2 2p4

gs
AE

+
PAW

Fe
~
comparison of GGA PAW, US-PP and
scalar relativistic all-electron calculations for O, N, Fe, Co and Ni
magnetic energy:
EM gs
ENM 4s1 3d n 1 (in
∆Em
eV)
US-PP





Ni

Co
for atoms the magnetisation is wrong by 5-10% with US-PP
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 39
Bulk properties of Fe
energy differences between different phases of Fe
FLAPWa
bcc Fe NM
US-AE
US-PP
373
372
369
bcc Fe FM
-73
-73
-73
-191
fcc Fe NM
78
61
62
62
0
0
0
hcp Fe NM
a
PAW
FLAPW, L. Stixrude and R.E. Cohen, Geophys. Res. Lett. 22, 125, (1995).
PAW and FLAPW give almost identical results
US-PP overestimates the magnetisation energy by around 5-10 %
calculations for other systems indicate that the accuracy of the PAW method for
magnetic systems is comparable to other AE methods
– α-Mn, bulk Cr and Cr surfaces, LaMnO3 (perovskites)
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 40
Why do pseudopotentials fail in spin-polarised calculations
non linear core corrections were included in the PP’s !
pseudo-wavefunction for a normconserving pseudopotentials
wave-function
3
— all electron
s
: E= -0.297 R c=2.2
– – pseudo
2
the peak in the d-wavefunction is
shifted outward to make the PP
softer
p1/2: E= -0.094 R c=2.5
1
d3/2: E= -0.219 R c=2.0
0
0
1
2
3
4
R (a.u.)
similar compromises are usually made in US-pseudopotentials
as a result, the valence-core overlap is artifically reduced and the spin
enhancement factor ξ r is overestimated
mr
nvalence r ncore r
€
ξr
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 41
Oxides
CeO2 and UO2
CeO2
PAW
FLAPW
Exp
5.47 Å
5.47 Å
5.41 Å
172 GPa i
176 GPa
236 GPa
UO2
PAW
FLAPW
Exp
a (Å3 )
5.425 Å
5.46 Å
B
200 GPa
209 GPa
a (Å3 )
B
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 42
Computational costs, efficiency
code complexity of the kernel of course increases with PAW
local pseudopotentials
low
2000 lines
NC-PP pseudopotential
low-medium
7000 lines
US-PP
medium
10 000 lines
PAW
medium-high
15 000 lines
parallelisation or error removal becomes progressively difficult
computational efficiency:
Ge 64 atoms, 1 k-point (Γ), Alpha ev6 (500 MHz), 14 electronic cycles
type
cutoff
time
total error per atom
NC-PP
140 eV
514 sec
400 meV
NC-PP
240 eV
1030 sec
10 meV
US-PP
140 eV
522 sec
10 meV
PAW
140 eV
528 sec
10 meV
the extra costs for PAW or US-PP’s at a fixed cutoff are small
PAW method is particularly good for transition metals and oxides
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 43
PAW advantages
formal justification is very sound
improved accuracy for:
– magnetic materials
– alkali and alkali earth elements, 3d elements (left of PT)
– lathanides and actinides
generation of datasets is fairly simple (certainly easier than for US-PP)
AE wavefunction are available
comparison to other methods:
– all test indicate that the accuracy is as good as for other all electron methods (FLAPW,
NUMOL, Gaussian)
– efficiency for large system should be significantly better than with FLAPW
G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW
Page 44