Orbital functionals derived from variational
functionals of the Green function
Nils Erik Dahlen
Robert van Leeuwen
Rijksuniversiteit Groningen
Ulf von Barth
Lund University
Outline
1. Deriving orbital functionals from variational energy functionals of the
Green function.
• Luttinger-Ward functional.
• Klein functional.
2. Discuss the difference between different kinds of such functionals.
• Relation between DFT energies and self-consistent Green function results.
• Stability of the functionals with respect to the input density.
4. Illustrate with results for atoms and molecules.
• Total energies from the variational functionals and from self-consistent Green
function.
• Binding energies.
5. Conclusion
∇2
%
#
#
3
Starting
variational
energy
functionals
function
− v(r)point:
G(rt,
r" t" ) = δ(r
− r" )δ(t
− t" ) + ofdthe
r̄ Green
dt̄ Σ(rt,
r̄t̄)G(r̄t̄,
2
The Green function solves the Dyson equation
$
%
2
∇
iω +
− w(r) − vH (r) G(r, r" ; iω) = δ(r − r" )
2
#
3
"
+ d r̄ Σ[G](r, r̄; iω)G(r̄, r ; iω)
The
2
∇
%
# to#which physical
self-energy
Σ[G] is approximated according
"
"
3
− v(r) G(r,
r ; τ ) =toδ(r
r )δ(τ ) + d r̄ dτ̄ Σ(r, r̄; τ − τ̄ )G(r̄,
mechanisms
we believe
be −
important.
2
The Dyson equation should be solved to self-consistency.
Σ[G](rt, r" , t" )
It is essential to use conserving self-energy approximations:
δΦ
Σ=
δG
Second
" order:
! !
!
!
− v(r) G(rt, r t ) = δ(r − r )δ(t − t ) +
#
3
d r̄
#
dt̄ Σ(rt, r̄t̄)G
"
#
#
!
!
3
GW:
− v(r) G(r, r ; τ ) = δ(r − r )δ(τ ) + d r̄ dτ̄ Σ(r, r̄; τ − τ̄ )G
!
!
Σ[G](rt, r , t )
!
"
−1
TheEvariational
functionals
E[G]
are
U0 − Tr {ΣG}
− Tr
ln stationary
−G0 + vH + Σ
LW [G] = Φ −
δE[G]
=0
#
δG
[n] = T0 [n] + wn + U0 + Φ − Tr {G(Σ − vxc ) + ln [1 − G(Σ − vxc )]}
when G is a self-consistent solution of the Dyson equation. Examples:
1: The Luttinger-Ward functional:
!
"
−1
EK [G] = Φ + U0 − Tr 1 − G0
!
"
−1
G − Tr ln −G
!
"
−1
ELW [G] = Φ − U0 − Tr {ΣG} − Tr ln !−G0 + vH + Σ "
−1
#
ELW [G] =KSΦ − U0 − Tr {ΣG} − Tr ln −G0 + vH + Σ
EK [n] = T0 [n] +
wn + U0 + Φ
2: The Klein functional:
!
"
!
"
−1
−1
EK [G] = Φ +G(rt,
U0 r−
" "Tr ! 1 − G0 G" − Tr ln !−G
"
,t )
−1
−1
EK [G] = Φ + U0 − Tr 1 − G0 G − Tr ln −G
Where
$
%
2
∇
−1
"
"
G0 (r, r ; iω ) ≡ iω +
− w(r) δ(r − r" )
2 G(rt, r" , t" )
"
"
t )self-consistent G.
• These functionals give the sameG(rt,
value r
at ,the
#
# to the quality of the input G.
• The LW functional is much less sensitive
dω
Density functionals
δE[G]
the energy =functionals
0
δG
δE[G]
The variational property of
suggests that a crude
approximation to the Green function can
= 0 give a good estimate for the
δG
self-consistent energy.
G(r, r! ; iω) =
$ φi (r)φ∗ (r! )
i
iω − %i Green functions:
We can, e.g., restrict the input to noninteracting
i
G(r, r! ; iω) =
$ φi (r)φ∗ (r! )
φi (r)
i
i
iω − %i
A special case is orbitals obtained from a local potential,
!
"
2
∇
−
+ vKS (r) φi (r) = %i φi (r)
2
This (arbitrary) potential can be considered a functional of the density due
to the 1-1 correspondence
=
0
!
"
2
LW [GHF ] of the density:
∇
Consequently we define
theδn(r)
energy as a Efunctional
−
+ vKS (r) φi (r) = &i φi (r)
2 E KS [n] ≡ E[G ]
KS
"
!
"
−1
−1
E
[G]
=
Φ
+
U
−
Tr
1
−
G
G
−
Tr
ln
−G
0
K
0
v
=
w
+
v
+
v we find:
For a density obtained from an arbitrary
1. The Klein functional
!
KS
H
xc
vKS (r)
KS
δE [n]
#
=0
2
KS
δn(r)
EK
[n] = T0 [n]
+ wn + U0 + Φ
KS
ELW [n]
≡ ELW [GKS ]
For the GWA, this functional is equivalent to DFT-RPA.
For the second!
"
−1
ELW [G] = equivalent
Φ − U0 − Tr
− Tr ln perturbation
−G0 + vH +theory.
Σ
order approximation,
to{ΣG}
second-order
1" "
G(rt,
r
,
t
)
KS
δELW [n]
2. The Luttinger-Ward functional
=0
KS
ELW
[n]
= T0 [n] +
#
δn(r)
wn + U0 + Φ − Tr {G(Σ − vxc ) + ln [1 − G(Σ − vxc )]}
KS
E $ [n] ≡ E[G
]
2 KS
%
∇
−1
"
"
"
G
(r,
r
;
iω
)
≡
iω
+
−
w(r)
δ(r
−
r
)
0
The Kohn-Sham potential is determined
! 2 −1 by
" the condition
!
"
−1
EK [G] = Φ + U0 − Tr 1 − G0 G − Tr ln −G
δE KS [n]
=0
δn(r)δΦ
"
"
G−1
0 (r, r ; iω )
∇
≡ iω +
− w(r) δ(r − r" )
2
1. The Klein functional gives the “ordinary” OEP equation
#
dω
2π
#
∗
!
∗
!
!!
#
#
φi (r)φ
(r )
φ
(r)φ
(r
)
dω
i
!
i
!
i
dr1 dr2 G(r,
GKS (r,
r1r
; iω)Σ(r
; iω)G=
dr1 GKS (r, r1 ; iω)GKS (r1 , r; iω)vxc (r1 )
2=
KS (r2 , r; iω) =
G(r,
r 1;, riω)
; iω)
2π
iω −iω
$i − $i
i
i
2. The LW functional gives the
where
δΦ
Σ
=
equation
δG
" "
% %
#
$
δ(Σ
+
v
)
#
$
H
δ(Σ
+
$
%
#
# vH )
2
0
=
Tr
Ḡ
−
G
∇
" "Ḡ
KS
0
=
Tr
KS
i∂t +
− v(r) G(rt, r t ) = −
δ(r
−G
r" )δ(t
− t" ) + d3 r̄ dt̄ Σ(rt, r̄t̄)G(r̄t̄, r" t" )
δvKSδv
(rKS
2
1 ) (r1 )
$
%
2
∇
iω +
− v(r) G(r, r" ; iω) = δ(r − r" )
2
KS
KS
KS # 3 xc
KS
KS
KS
+ KS
d r̄ Σ[G](r, r̄;xc
iω)G(r̄,KS
r" ; iω)
Ḡ =
G
+
G
(Σ[G
]
−
v
[G
])
Ḡ
Ḡ = G + G (Σ[G ] − v [G ])Ḡ
• The
$
%
#
#
∇2
stationary
of
functionals
give
the
−∂τ +
−points
v(r) G(r,
r" ;these
τ ) = δ(rtwo
− r" )δ(τ
) + d3 r̄ dτ̄do
Σ(r,not
r̄; τ −
τ̄ )G(r̄,
r" ; τ̄ )
2
φi (r)
same energy
φ
(r)
i
• The variational property means that we
do not have to solve
" "
Σ[G](rt, r , t )
these OEP equations to find a good estimate of the total energy
For a given self-energy approximation:
1. Is the OEP energy similar to the self-consistent Green function energy?
2. Does the DFT energy depend strongly on the form of the functional
(Klein, LW,...)?
3. Are the orbital functionals sensitive to the quality of the input
DFT orbitals?
4. Can a method which produces poor total energies give accurate
total energy differences?
Numerical implementation
• We use Slater basis functions
• In the evaluation of the variational energy functionals, the
Green functions are represented on the imaginary frequency
axis.
• The self-consistent Green function is calculated on the
imaginary time-axis.
Compare with self-consistent solutions of the Dyson equation with the
second-order self-energy:
(2)
EK [nLDA ]
(2)
ELW[nLDA ]
(2)
ELW[GHF ]
SCa
-2.9068
-14.6880
-128.9111
-199.9662
-199.1487
-1.1817
-8.0774
-2.8937
-14.5953
-128.8068
-199.8933
-199.0918
-1.1595
-8.0394
-2.8969
-14.6405
-128.8332
-199.9093
199.1025
-1.1658
-8.0526
-2.8969
-14.6409
-128.8339
-199.9097
-199.1020
-1.1659
-8.0528
He
Be
Ne
Mg
Mg2+
H2
LiH
a Dahlen
Ne
Mg
Mg2+
H2
LiH
and van Leeuwen, J. Chem. Phys. (to be published)
• Energies from the LW functional are in good agreement the self-consistent
results.
(2)
(2)
(2)
E
[G
]
E
[G
]
E
SC
LDA
HF
LW
LW
LW
• We have also included the LW results from HF Green functions (in
spectacular agreement with the self-consistent results).
-
Total energies of diatomic molecules
1
Molecule
H2
Li2
LiH
N2
(2)
(2)
(2)
GW
EK
[GLDA ]
GW
ELW
[GLDA ]
GW
ELW
[GHF ]
EK [GLDA ]
ELW [GLDA ]
ELW [GHF ]
-1.210
-15.084
-8.134
-109.720
-1.176
-14.995
-8.077
-109.514
-1.189
-15.035
-8.102
-109.573
-1.182
-15.008
-8.077
-109.739
-1.160
-14.925
-8.039
-109.315
-1.166
-14.964
-8.053
-109.472
•The two orbital functionals (LW and Klein) give significantly different
results both for the second-order approximation and for the GWA.
• The LW results from LDA and HF Green functions are similar.
Molecule
H2
LiH
Li2
N2
(2)
(2)
(2)
GW
EK
[GLDA ]
GW
ELW
[GLDA ]
GW
ELW
[GHF ]
EK [GLDA ]
ELW [GLDA ]
ELW [GHF ]
Expt.
0.172
0.086
0.026
0.349
0.158
0.085
0.029
0.311
0.162
0.081
0.020
0.310
0.183
0.104
0.061
0.546
0.160
0.073
-0.009
0.247
0.166
0.084
0.026
0.369
0.175
0.092
0.039
0.364
1
General conclusions from calculations on atoms, molecules, and
EK [G
function
energy 1996):
the
homogenous
electron
gasGreen
(Hindgren
and Almbladh,
HF ] ≈ ELW [G
HF ] ≈ SC
[G
≈ HF
ELW
] ≈ SC Green function energy
EK] [G
] ≈[GEHF
• HF
LW [GHF ] ≈ SC Green function energy
This is true also for the homogenous electrong gas in the GW
approximation.
ELW [GKS ] ≈ SC Green function energy
• ELW [GKS ] ≈ SC Green function energy
• EK [GKS ]
Differs considerably from SC energy
As shown for the atoms and for molecules and also for the
homogenous
gas. SC energy
EK [GKS ] ≈electron
Differs from
1
1
Binding energies
Molecule
H2
LiH
Li2
N2
(2)
(2)
(2)
GW
EK
[GLDA ]
GW
ELW
[GLDA ]
GW
ELW
[GHF ]
EK [GLDA ]
ELW [GLDA ]
ELW [GHF ]
Expt.
0.172
0.086
0.026
0.349
0.158
0.085
0.029
0.311
0.162
0.081
0.020
0.310
0.183
0.104
0.061
0.546
0.160
0.073
-0.009
0.247
0.166
0.084
0.026
0.369
0.175
0.092
0.039
0.364
• The DFT-RPA functional gives excellent binding energies.
(See also F. Furche, PRB (2001), M. Fuchs and X. Gonze, PRB (2001)
• Self-consistent Green functions from the second-order approximation
give results of comparable quality
The total energy of the H2 molecule with Φ = ΦGW
KS [n]
δELW
=0
δn(r)
1
EK [GHF ] ≈ ELW [GHF ] ≈ SC Green
function
energy ]
E KS
[n] ≡ E[G
KS
KS [n]
δE
ELW [GKS ] ≈ SC Green function energy
=0
δn(r)
2
EK [GKS ]
GW
Differs considerably from
SCΦenergy
Φ=
• The orbital functionals are not sensitive to the choice of input orbitals
• The difference between ELW[GLDA ] and ELW [GHF ] is larger than for the atoms,
indicating that the DFT results are not close to the self-consistent Green
function values
•The DFT-RPA energy converges very slowly to a constant value from above.
Not converged for R≈20.
The total energy of H2calculated from
the second-order approximation.
Good agreement between LW results
and the self-consistent energies.
All functionals diverge for large R .
The binding curve calculated with
orbitals from spin-unrestricted
LDA and HF.
Current and future projects:
• Compare to self-consistent GW (Adrian Stan)
• Calculations using variational functionals of G and W
[Almbladh, von Barth, van Leeuwen, Int. J. Mod. Phys. B (1996)]
• The energy functionals of the Green functions are trivially
generalized to action functionals on the Keldysh contour. This is used
to generate time-dependent OEP from action functionals.
exchange-correlation functionals with memory.
Conclusions
• Variational functionals of the Green functions are useful for
deriving orbital functionals in DFT.
• The orbital functional derived from the LW functional give
results similar to the self-consistent Green function.
• Spectacular agreement between the LW energies from HF Green
functions and self-consistent energies.
• The functionals are stable with respect to the input density.
• The second-order approximation produces better total energies
than the GWA for atoms and small molecules.
• Binding energies are accurately given by DFT-RPA.