A Projector Augmented Wave Formulation of the
Optimized Effective Potential Formalism a
N. A. W. Holzwarth and Xiao Xu
Wake Forest University, Winston-Salem, NC, USA
• Motivation and overview
• PAW formalism
• OEP formalism
• Selected atomic examples
• “Frozen” core electrons
• Summary and conclusions
a Supported by NSF grants DMR-0405456, 0427055, and 0705239; computations performed on the Wake
Forest University DEAC cluster. Helpful discussions with L. Kronig, N. Modine, A. Wright, and W. Yang are
also gratefully acknowledged.
CCCC7 July 2009
1
Motivation
• Why Optimized Effective Potential (OEP)?a
– Orbital-dependent exchange-correlation functionals within the Kohn-Sham
framework of density functional theoryb
– Self-interaction-free treatment of electrons
• Why Projector Augmented Wave (PAW)?c
– Efficient pseudopotential-like scheme
– Accurate evaluation of multipole moments in Coulomb and exchange interaction
termsd
a Review
articles: S. Kümmel and L. Kronik, RMP 80, 3-60 (2008); T. Grabo et al in Strong Coulomb
correlations in electronic structure calculations, V. Anisimov, ed. Gordon and Breach (2000), pg. 203-311.
b W. Kohn and L. Sham, PR 140, A1133-A1138 (1965)
c P. Blöchl, PRB 50, 17953-17979 (1994)
d J. Paier et al, JCP 122, 234102 (2005)
CCCC7 July 2009
2
Motivation (continued)
Self-interaction problem
Within Kohn-Sham theory, the total electronic energy is a functional of the electron
density ρ(r) having the form:
Etot (ρ) = EK (ρ) + EN (ρ) + EH (ρ) + Ex (ρ) + Ec (ρ) .
| {z }
| {z }
| {z } | {z } | {z }
Kinetic
Nuclear
Here,
e2
EH (ρ) ≡
2
Z Z
Hartree
Exchange
0
ρ(r)ρ(r
)
3
3 0
,
d rd r
|r − r0 |
(1)
Correlation
(2)
representing the Coulomb interaction between electrons, including the self-interaction.
Only for Ex chosen to have the form of Fock exchange, can this self-interaction be
completely removed from the formalism.
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Motivation (continued)
Fock exchange form
Z
Z
∗
0
∗ 0
Ψ
(r)Ψ
(r)Ψ
(r
)Ψ
e2 X
q
p
p
q (r )
3
3 0
Ex ({Ψp (r)}) = −
δσp σq d r d r
,
2
|r − r0 |
(3)
pq(occ)
where the summation over all occupied states includes the the self-interaction correction
for p ≡ q and σp ≡ σq . For wavefunctions Ψp (r) representing extended states, the
self-interaction energy is not large. Two notable exceptions are:
H 1s
0.8
Fe 3d
EXX
0.6
|rψ(r)|
2
EXX
For a H atom:
LDA
= −0.9 Ry
Etot
EXX
Etot
= −1.0 Ry.
0.4
0.2
LDA
LDA
0
0
1
2
r (bohr)
3
0
1
2
3
For a Fe atom:
²LDA
− ²LDA
= 0.2 Ry
4s
3d
²EXX
− ²EXX
= 0.3 Ry.
4s
3d
r (bohr)
CCCC7 July 2009
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Motivation (continued)
Some self-interaction correction schemes
• SIC-LSD – Perdew and Zunger, PRB 23, 5048-5079 (1981)
• LDA+U – Anisimov et al, PRB 44, 943-954 (1991); Cococcioni and
Gironcoli, PRB 71, 035105 (2005)
⇒ Orbital-dependent density functionals:
Ex ≈ Fock exchange
Ec = consistent correlation functional.
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Scope of present work (so far)
• Choose Ex to be Fock exchange (EXX). The corresponding Kohn-Sham local potential is
Vx (r).
• Choose Ec ≡ 0 for the moment (hoping that an appropriate functional will soon be
developed) (OEP). The corresponding Kohn-Sham local potential Vc (r) would then be added.
• Develop OEP-EXX for all electrons in spherical atoms
• Adapt OEP-EXX formalism to the projector augmented wave (PAW)
formalism in order to carry out calculations for extended systems.
⇒ Formulate and test a frozen core approximation scheme.
⇒ Apply the PAW formalism to valence electrons
CCCC7 July 2009
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All-electron atomic OEP-EXX equations
The basic equations can be derived as a constrained minimization problem to determine
Kohn-Sham radial orbitals {ψp (r)}a and to minimize the total energy of the system with
the help of Lagrange multipliers:b
F ({ψp }, Vx , {gp }, {²p }, {λqp }) =
X
X
(hgp |HKS − ²p |ψp i + cc) −
λpq (hψp |ψq i − δpq ) .
Etot ({ψp }) −
p
(4)
pq
The Kohn-Sham Hamiltonian HKS is given by the kinetic energy operator and potential
terms:
b + VN (r) + VH (r) + Vx (r),
HKS = K
(5)
where
VN (r) ≡
δEN
,
δρ(r)
VH (r) ≡
δEH
,
δρ(r)
(6)
and the local exchange potential Vx (r) is to be determined.
a The
index p stands for atomic shell quantum numbers np lp corresponding to shell occupancy Np .
b Hyman, Stiles, and Zangwill, PRB 62, 15521-15526 (2000)
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All-electron atomic OEP-EXX equations (continued)
At self-consistency, the following equations must be satisfied
simultaneously:
b + VN + VH + Vx
HKS = K
(HKS
HKS |ψp i = ²p |ψp i
hψp |ψq i = δpq (7)
X
1 δEx
λpq |ψq i,
− ²p ) |gp i =
− Vx (r)|ψp i −
∗
Np δψp (r)
q
where λpq
δEx
= hψq | ∗ i − hψq |Vx |ψp i and
δψp (r)
X
Np gp (r)ψp (r) = 0.
(8)
hgp |ψq i = 0.
(9)
p
In practice, the equations are solved iteratively yielding results consistent
with the Green’s function formulation of earlier work.a
a Talman
and Shadwick, PRA 14 36-40 (1976); Engel and Vosko, PRA 47 2800-2811 (1993); T. Grabo et
al in Strong Coulomb correlations in electronic structure calculations, V. Anisimov, ed. Gordon and Breach
(2000), pg. 203-311.
CCCC7 July 2009
8
All-electron atomic OEP-EXX results for carbon
Auxiliary functions gp (r):
2
0.008
2
C (2s 2p )
2s
0.006
gp(r)
0.004
1s
0.002
0
All-electron local exchange potential Vx (r), for
ground state (2s2 2p2 ) and for excited state
(2s1 2p3 ).
-0.002
-0.004
2p
-0.006
1
-0.008
0
1
2
3
4
5
C
7
6
r (bohr)
rVx (bohr.Ry)
0
Constributions to shift equation:
0.01
2
2
C (2s 2p )
Npψp(r)gp(r)
0.005
1s
-1
-2
2s
0
-3
0
1
3
2
2
Vx (2s 2p )
Vx (2s 2p )
0.5
1
1.5
2
2.5
3
r (bohr)
2p
-0.005
-0.01
0
1
2
3
4
5
6
7
r (bohr)
CCCC7 July 2009
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All-electron atomic OEP-EXX results for Vx (r)
0
-2
Be
Li
rVx (bohr.Ry)
-1
B
C
N
Ne O
F
-3
0
-1
Mg
Na
Al
Si
P
S
Cl
Ar
-2
-3
1
2
3
4
5
0
1
2
r (bohr)
3
4
5
r (bohr)
0
rVx (bohr.Ry)
rVx (bohr.Ry)
0
-1
-2
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
-3
0
1
2
3
4
5
r (bohr)
CCCC7 July 2009
10
Frozen core atomic OEP-EXX equations
In order to focus computational effort on the valence properties of materials, it is necessary to develop a frozen
core approximation scheme. For each material (other than H and He), it is possible to partition the states into
core and valence contributions:
8
< v for valence shells (typically partitially occupied)
p→
(10)
: c for core shells
In a similar way, we can partition, the electron wavefunctions
8
< ψ (r) for valence states
v
ψp (r) →
: ψc (r) for core states
and the auxiliary functions
8
< g (r)
v
gp (r) →
: gc (r)
for valence states
for core states
(11)
(12)
and the local exchange potential
Vx (r) = Vxcore (r) + Vxvale (r).
(13)
The challenge is to devise a scheme where the functional variation is allowed only for {ψv (r)} and
Vxvale , consistent with the all-electron equations of the reference configuration.
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11
Frozen core atomic OEP-EXX equations (continued)
Partitioning of the reference configuration to define Vxcore
All-electron auxiliary function equation:
(HKS
X
1 δEx
− ²v ) |gv i =
− Vx (r)|ψv i −
λvq |ψq i
Nv δψv∗ (r)
q
(14)
For the reference configuration, we assume a partitioning of the valence auxiliary function:
gv (r) ≡ gvcore (r) + gvvale (r)
(15)
(HKS −
²v ) |gvvale i
X
1 δExvv
vale
vale
=
−
V
(r)|ψ
i
−
λ
v
x
vq |ψq i,
Nv δψv∗ (r)
q
(16)
(HKS −
²v ) |gvcore i
X
1 δExcv
core
core
−
V
(r)|ψ
i
−
λ
=
v
x
vq |ψq i.
Nv δψv∗ (r)
q
(17)
satisfying the following two equations:
and
The original shift equation becomes
8 X
Nv gvvale (r)ψv (r) = 0
>
>
>
>
v
< X
X
X
Np gp (r)ψp (r) = 0 →
Nc gc (r)ψc (r) +
Nv gvcore (r)ψv (r) = 0
>
>
p
c
v
>
>
: |
{z
}
(18)
Core shift
CCCC7 July 2009
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Frozen core atomic OEP-EXX equations (continued)
Total valence energy in frozen core approximation
cv
vv
vale
Etot
= EK (ρv ) + EN (ρv ) + EH
(ρv ) + EH
(ρv ) + Excv (ρv ) + Exvv ({ψv }),
(19)
Z
where
Excv (ρv ) ≈
d3 r Vxcore (r)ρv (r).
Once Vxcore (r) has been determined for the reference configuration, the self-consistent equations to solve for
excited valence configurations are:
b + VN + VH + V core + V vale
HKS = K
x
x
(HKS −
²v ) |gvvale i
where
HKS |ψv i = ²v |ψv i
X
1 δExvv
vale
=
− Vx (r)|ψv i −
λvale
vw |ψw i,
∗
Nv δψv (r)
w
δExvv
= hψw | ∗
i − hψw |Vxvale |ψv i.
δψv (r)
X
Nv gvvale (r)ψv (r) = 0.
(20)
(21)
λvale
vw
(22)
v
Here, |ψv i, |gvvale i, and Vxvale are updated self-consistently through these equations.
CCCC7 July 2009
13
Frozen core atomic OEP-EXX results for carbon
Frozen core functions gvvale (r) and gvcore (r):
2
0.008
2
C (2s 2p )
0.006
0.004
vale
(r)
2
0.008
gv
All-electron auxiliary functions gv (r):
0
-0.002
2
C (2s 2p )
2s
2s
0.002
2p
-0.004
0.006
-0.006
1s
0.002
-0.008
0
⇒
0
1
2
3
4
5
7
6
r (bohr)
2
0.008
2
C (2s 2p )
-0.002
0.006
-0.004
2p
0.004
2s
1
2
3
4
r (bohr)
5
6
7
core
-0.008
0
(r)
-0.006
gv
gp(r)
0.004
0.002
0
-0.002
2p
-0.004
-0.006
-0.008
0
1
2
3
4
5
6
7
r (bohr)
CCCC7 July 2009
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Frozen core atomic OEP-EXX results for carbon
Frozen core shift components:
0.01
2
2
C (2s 2p )
0.005
Nvψv(r)gv
vale
(r)
2s
All-electron shift components Np gp (r)ψp (r)
0.01
2
2
C (2s 2p )
1s
2p
-0.005
2s
-0.01
0
⇒
0
1
2
4
5
0.01
7
6
2
-0.005
2
C (2s 2p )
Core shift
2p
0.005
(r)
2s
core
-0.01
0
3
r (bohr)
1
2
3
4
r (bohr)
5
6
7
Nvψv(r)gv
Npψp(r)gp(r)
0.005
0
0
2p
-0.005
-0.01
0
1
2
3
4
5
6
7
r (bohr)
CCCC7 July 2009
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Frozen core atomic OEP-EXX results for carbon
Local exchange potential Vx for reference configuration ([He]2s2 2p2 ) and its
decomposition into Vxvale and Vxcore :
1
2
Total
local
exchange
potential
Vxvale + Vxcore for excited configuration ([He]2s1 2p3 ), comparing frozen
core and all-electron results:
1
2
1
C (2s 2p )
Vx
vale
core
0
rVx (bohr.Ry)
rVx (bohr.Ry)
0
Vx
-1
Vx
-2
-3
0
3
C (2s 2p )
Vx
1
1.5
2
2.5
3
+Vx
vale
-1
-2
0.5
core
-3
0
Vx (all electron)
0.5
r (bohr)
1
1.5
2
2.5
3
r (bohr)
CCCC7 July 2009
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Frozen core atomic OEP-EXX results for iron
Frozen core shift components:
6
0.015
0.01
(r)
2
Fe (3d 4s )
4s
Nvψv(r)gv
vale
0.005
0
Valence auxiliary functions:
-0.005
0.02
-0.01
0.01
2
3
4
5
7
6
(r)
core
2
Fe (3d 4s )
Core shift
0.01
0
gv
6
0.015
Nvψv(r)gv
vale
r (bohr)
4s
(r)
1
2
Fe (3d 4s )
-0.015
0
6
3d
3d
-0.01
4s
0.005
-0.02
0
0
1
3
4
5
6
7
r (bohr)
3d
-0.005
2
-0.01
-0.015
0
1
2
3
4
5
6
7
r (bohr)
CCCC7 July 2009
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Frozen core atomic OEP-EXX results for iron
Comparing decomposition: Vx = Vxvale + Vxcore for two choices of core
configuration
rVx (bohr . Ry)
0
Vx
core
Vx
Vx
-1
Vx
vale
-2
-3
0
1
2
3
vale
Vx
(Ar core)
Vx
0
core
(Ne core)
1
r (bohr)
2
3
r (bohr)
CCCC7 July 2009
18
Summary of frozen core errors in excitation energies
vale
∆∆E = Etot
(excited) − Evale
(ground) − Etot (excited) − Etot (ground)
{z tot
} |
{z
}
|
Frozen core
All-electron
∆∆E (Ry)
sp excitations :
0.03
0.02
x
y
x-1
4s 4p => 4s 4p
sd excitations :
y+1
(Ar core)
0.01
(Ne core)
0
0.1
Ca // Zn
K
Ga
Ge
As
Se
Br
(Ar core)
0.03
0.02
x
y
x-1
3s 3p => 3s 3p
0.01
y+1
(He core)
(Ne core)
0
Na
Mg
∆∆E (Ry)
∆∆E (Ry)
0.08
Al
Si
P
S
Cl
0.06
x
0.04
y
x-1
y+1
3d 4s => 3d 4s
0.02
(Ne core)
∆∆E (Ry)
0
0.03
0.02
x
y
x-1
2s 2p => 2s 2p
y+1
Sc
(He core)
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
0.01
0
Li
Be
B
C
N
O
F
CCCC7 July 2009
19
The projector augmented wave method (PAW)
The PAW formalism was developed by P. Blöchl (PRB 50,
17953–17979 (1994)), having similarities to the
soft-pseudopotential formalism of D. Vanderbilt (PRB 41
7892-7895 (1990)). One distinguishing feature of PAW, is
the transformation between all-electron valence
e v (r) :
wavefunctions Ψv (r) and Ψ
e v (r) +
Ψv (r) = Ψ
”
X“
a
a
a
a
e
e
φi (r − R ) − φi (r − R ) hpa
i |Ψv i,
(23)
ai
a
ea
where φa
i , φi , and pi are atom-centered all-electron basis,
pseudo-electron basis, and projector functions, respectively.
Computational work is performed on pseudofunctions
e v (r) after which all-electron functions Ψv (r) can be
Ψ
retreived from PAW transformation.
CCCC7 July 2009
20
Comparison of results from different codes
Li2O
FeO
PWscf
PAW
LAPW
GGA
PWscf
PAW
LAPW
0.3
PWscf
PAW
LAPW
PWscf
PAW
LAPW
0.3
0.8
0.2
0.4
0.1
0.1
0.2
8.5
9
9.5
PWscf
PAW
LAPW
0
7.5 7.75 8 8.25 8.5 8.75
a (bohr)
a (bohr)
Fluorite structure
NaCl structure
GGA
PWscf
PAW
LAPW
0.6
0.2
0
8
LDA
1
GGA
0.4
∆E (eV)
∆E (eV)
0.4
LDA
0.5
LDA
∆E (eV)
0.5
PO4
1.2
0
2.8
2.9
3
3.1
b (bohr)
Tetrahedral molecule
With care, consistent numerical results can be achieved using different formalisms and codes.
PWscf
PAW
LAPW
www.pwscf.org
pwpaw.wfu.edu
www.wien2k.at
soft-pseudopotentials
Projector Augmented Wave
Linearized Augmented Plane Wave
CCCC7 July 2009
21
The PAW method (continued)
Importance of PAW transformation to Hartree and Fock integrals
Given the PAW transformation for a valence wavefunction:
”
X“
a
a
a
a
e
e
e
φi (r − R ) − φi (r − R ) hpa
Ψv (r) = Ψv (r) +
i |ψv i,
(24)
ai
it is also possible to evaluate the valence electron energy of the system in the form:
“
”
X
vale
a
a
e
e
Etot =
Etot
+
Etot − Etot
.
| {z }
a
|
{z
}
pseudo energy
atom-centered corrections
CCCC7 July 2009
(25)
22
Hartree and Fock integrals (continued)
The product of two wavefunctions Pvw (r) ≡ Ψ∗v (r)Ψw (r) can be well-approximated with the form
”
X“
a
a
a
a
e
e
Pvw (r − R ) − Pvw (r − R ) ,
Pvw (r) = Pvw (r) +
(26)
a
where
e ∗ (r)Ψ
e w (r),
Pevw (r) ≡ Ψ
v
and
a
Pvw
(r)
a
− Pevw
(r) ≡
(27)
“
”
X
a
a e
a∗
a
a∗
a
e
e
e
hΨv |pi ihpj |Ψw i φi (r)φj (r) − φi (r)φj (r) .
(28)
ij
Equivalently, we can write:
Pvw (r) = Pevw (r) + Pbvw (r) +
X“
a
Pvw
(r
”
a
a
a
a
e
b
− R ) − Pvw (r − R ) − Pvw (r − R ) ,
a
(29)
a
where the “compensation charge”
Pbvw (r) ≡
X
a
Pbvw
(r − Ra )
(30)
a
is a smooth function localized within the atomic augmentation spheres such that:
8
Z
< V a (r) for |r − Ra | ≤ ra
a
0
a
0
a
0
vw
c
3 0 Pvw (r ) − P̃vw (r ) − P̂vw (r )
d r
=
: 0
|r − r0 |
otherwise
CCCC7 July 2009
.
(31)
23
PAW atomic OEP-EXX equations
Based on the form of the frozen core constrained minimization problem:
F vale ({Ψv }, Vxvale , {gvvale }, {²v }, {λvale
vw }) =
” X
X“
vale
vale
hgv |HKS − ²v |Ψv i + cc −
Etot ({Ψv }) −
λvw (hΨv |Ψw i − δvw ) .,
v
(32)
vw
the PAW contrained minimization expression is given by:
e v }, Ve vale , {[V a ]ij }, {e
Fe({Ψ
gv }, {²v }, {λvw }) =
x
x
” X
“
”
X“
vale e
PAW
PAW e
PAW e
e
Etot ({Ψv }) −
he
gv |H
− ²v O
|Ψv i + cc −
λvw hΨv |O
|Ψw i − δvw .
v
vw
(33)
The PAW Hamiltonian takes the form:
X
a
PAW
a
e
H
=H+
|pa
i iDij hpj |
and
O
PAW
=1+
X
a
a
|pa
i iOij hpj |,
(34)
aij
aij
a ≡ hφa |φa i − hφ
ea i. The smooth Hamiltonian takes the form:
ea |φ
where Oij
j
i
j
i
e =K
b + Ve (r)
H
where
Ve (r) = Vloc (r) + VeH (r) + Vexvale (r);
(35)
Vloc (r) includes effects from the pseudized nuclear and core electron potentials. The atomic matrix elements
a include the valence local exchange potential contributions:
Dij
a,vale a
ea i.
ea |Ve a,vale |φ
[Vxa ]ij ≡ hφa
|φj i − hφ
x
i
j
i |Vx
CCCC7 July 2009
(36)
24
Update equations for OEP-EXX in PAW formalism
a
a
ea
Once basis and projector functions |φa
i i, |φi i, and |pi i and local potential Vloc has been determined for each
atom, the self-consistent PAW equations are:
X
a
a
PAW
vale
e v i = ²v OPAW |Ψ
e vi
b
e
e
+
|pa
H PAW |Ψ
(37)
H
= K + Vloc + VH + Vx
i iDij hpj |
aij
“
”
X
X
1 δExvv
PAW
PAW
a
a
a e
vale
e w i,
e
e
H
− ²v O
|e
gv i =
−Vx (r)|Ψv i−
|pi i[Vx ]ij hpj |Ψv i−
λvw OPAW |Ψ
e ∗ (r)
Nv δ Ψ
v
w
aij
(38)
vv
X
δEx
e w |Ve vale |Ψ
e vi −
e w |pa i[V a ]ij hpa |Ψ
e v i.
where λvw = hψw |
i − hΨ
hΨ
x
x
i
j
e ∗ (r)
δΨ
v
aij
X
e v (r) = 0.
Nv g
ev (r)Ψ
(39)
a e
Nv he
gv |pa
i ihpj |Ψv i = 0.
(40)
v
X
v
e v i, |e
Here, |Ψ
gv i, Vexvale , and {[Vxa ]ij } are updated self-consistently through these equations.
CCCC7 July 2009
25
Recipes for constructing PAW atomic functions
a
{|φai i}, {|φeai i}, {|pai i}, and Vloc
⇒ There is considerable flexibility in choice of functions (within some constraints).
⇒ atompaw code is available from website http://pwpaw.wfu.edu (OEP part not yet included);
interfaces with codes for solids – (pwpaw, socorro (from Sandia National Labs), and abinit (international
code based in Belgium))
General recipe
Based on D. Vanderbilt’s scheme for soft pseudopotentials (other possibilities were programmed by Marc
Torrent in atompaw for abinit)
a
1. Generate all-electron basis set: HKS |φa
i i = ²i |φi i including valence states |ψv i and (optionally) some
continuum states.
ea
2. For every all-electron basis functions |φa
i i, construct corresponding pseudo basis function |φi i such that
ea (r) ≡ φa (r) for r > ra .
φ
c
i
i
ea a
3. Construct projector functions |pa
i i such that hφi |pj i = δij .
(a) Construct local pseudopotential V P S such that V P S (r) ≡ V (r) for r > rca .
(b) Solve the following equations for {|pa
i i}:
“
”
X
PS
P
S
a
ea i.
ea b
e
b
− ²i |φ
K+V
− ²i |φi i =
|pa
i
j ihφj |K + V
j
a :
4. Unscreen local pseudopotential to find Vloc
a
Vloc
(r) = V P S (r) − VeH (r) − Vexvale (r)
CCCC7 July 2009
26
a
Recipes for Vloc
(continued)
In order to determine Vexvale (r) for unscreening the local pseudopotential, the following simplified OEP
relations must be satisfied.
“
”
X
X
1 δExvv
a
a
PS
vale
ew i, (41)
e
b
e
λvw OPAW |ψ
|pi i[Vx ]iv −
K +V
− ²v |e
gv i =
− Vx (r)|ψv i +
e∗ (r)
Nv δ ψ
v
w
i
X
ev (r) = 0.
Nv g
ev (r)ψ
(42)
v
he
gv |pa
ii=0
for all
g
ev (r) ≡ gvvale (r)
for
li = lv .
(43)
r > rca .
(44)
Most promising method:
• Assume
8
< rlv +1+s PM C rn
n=0 n
g
ev (r) =
: g vale (r)
v
r ≤ rca
r > rca ,
where coefficients {Cn } are determined from matching conditions.
• Determine Vexvale (r) from g
ev (r) and other functions.
CCCC7 July 2009
27
Example of basis and projector functions for carbon
Results for rc = 1.3 bohr
4
C 2s
C 2p
3
p2s(r)
p2p(r)
2
1
0
0
~φ (r)
2s
φ2p(r)
~φ (r)
2p
φ2s(r)
0.5
1
1.5
0
0.5
r (bohr)
1
1.5
r (bohr)
CCCC7 July 2009
28
Example of potential functions for carbon
Results for rc = 1.3 bohr
2
rV(r) (bohr.Ry)
0
Vloc
-2
Vx
-4
vale
~
V
-6
-8
~ vale
Vx
V
-10
-12
-14
0
1
2
3
4
5 0
1
r (bohr)
2
3
4
5
r (bohr)
CCCC7 July 2009
29
Summary and Conclusions
⇒ Choosing Ex to have the form of Fock exchange has the advantage of removing
electron self-interaction from the formalism. An orbital dependent correlation
functional Ec can be added to the formalism in a straight-forward way.
⇒ It is possible to define a core electron contribution to the OEP, Vxcore (r), which
approximates the valence-core exchange contributions.
⇒ Tests of the frozen core approximation on atomic excitations show controllable
errors. For increased accuracy, semi-core states can be included with the valence
states.
⇒ The PAW formalism is able to treat the multipole moments of the Coulomb integrals
of the Hartree and Fock functionals.
⇒ The PAW-OEP equations can be derived directly from the frozencore approximation.
a
(r) looks promising.
Full implementation including the calculation of Vloc
CCCC7 July 2009
30