Kohn-Sham density functional theory for strong-interacting

Transcription

Kohn-Sham density functional theory
for strong-interacting systems
Paola Gori-Giorgi
Theoretical Chemistry, Vrije Universiteit Amsterdam (NL)
Francesc Malet
André Mirtschink
Outline
•
The exact strong-interacting limit (SCE) of
DFT as an approximation for the exchangecorrelation energy
•
Self-consistent KS SCE results for model
quantum wires and quantum dots
•
Generalization to open systems with
fluctuating particle number
•
•
Chemistry
Monge-Kantorovich formulation of the SCE
functional
Kohn-Sham DFT and strong correlation
Example: two electrons (singlet) in 1D harmonic confinement
L: effective length
weak correlation (small L)
.e
.e
Strong correlation (large L)
.e
-
-
-
.e
-
Kohn-Sham DFT and strong correlation
Kohn-Sham potential:
“Bump” in the exact KS
potential: localization
- Buijse, Baerends & Snijders
Phys. Rev. A 40, 4190 (1989)
- Helbig, Tokatly & Rubio,
J. Chem. Phys. 131, 224105 (2009)
The “bump” is missing
in the KS LDA potential:
delocalized density
Stretched bond, e.g. H2 molecule
LDA
exact
- Buijse, Baerends & Snijders Phys. Rev. A 40, 4190 (1989)
- Helbig, Tokatly & Rubio, J. Chem. Phys. 131, 224105 (2009)
Stretched bond, e.g. H2 molecule
LDA
exact
exact KS potential localizes more:
“bump” in the midpoint
- Buijse, Baerends & Snijders Phys. Rev. A 40, 4190 (1989)
- Helbig, Tokatly & Rubio, J. Chem. Phys. 131, 224105 (2009)
Unrestricted KS & strong correlation...
•
Like UHF, UKS can mimic strong correlation
(not always!)
•
Often better energies, but wrong
characterizations of several properties
•
•
Formally not well founded
•
Crucial for: transition metals, Mott
insulators, bond breaking, nanostructures,...
Huge literature: controversial, wrong
interpretation, etc...
Exchange-correlation from the
strong-interaction limit of DFT:
The “bumps” in the KS potential
from the right physics
(no spin/spatial-symmetry
breaking)
Hohenberg-Kohn functional
Kohn-Sham DFT:
Kohn-Sham DFT:
Ts [ρ]
(known)
(known) (unknown)
Kohn-Sham DFT:
Kohn-Sham equations:
Kohn-Sham DFT:
Our approach:
Kohn-Sham DFT:
Our approach:
Ts [ρ]
Strictly-correlated-electrons (SCE) DFT:
- M. Seidl, PRA 60, 4387 (1999)
- M. Seidl, P. Gori-Giorgi and A. Savin, PRA 75,
042511 (2007)
- P.Gori-Giorgi, G.Vignale and M. Seidl, JCTC 5, 743 (2009)
- P. Gori-Giorgi, M. Seidl, and G. Vignale, PRL 103,
166402 (2009).
Vee [ρ]
SCE
e-e interaction energy of
system with zero kinetic
energy and density ρ
(SCE reference system)
Kohn-Sham DFT:
Our approach:
Vee [ρ]
SCE
Strictly-interacting-electrons (SCE) DFT
.r 'e≡ f (r)
-
2
2
The SCE reference system:
. er
! For a given smooth density ρ(r):
-
.re' ≡ f (r)
.r 'e≡ f (r)
.e
(1)
-
3
3
-
-
5
r4' ≡ f4(r)
KS-SCE approach:
5
! A local one-body potential can be
defined:
(2)
1-Integrate (1) to obtain the co-motion functions fi (r)
2-Integrate (2) to obtain the potential vSCE(r)
3-Approximate the Hxc potential of KS DFT with vSCE(r)
SCE
Vee
[
]=
⇥
(r)
dr
N
SCE
Vee
[⇢]
= vSCE (r)
⇢(r)
N
N
i=1 j=i+1
1
|fi (r)
fj (r)|
SCE
Vee
[
SCE
Vee
[⇢]
⇢(r)
]=
⇥
(r)
dr
N
N
N
1
i=1 j=i+1
|fi (r)
fj (r)|
N
= vSCE (r)
vSCE (r) =
i=2
r
|r
fi (r)
fi (r)|3
shortcut to the functional derivative
F. Malet. et al., PRB 87 115146 (2013)
The self-consistent KS SCE total
energy is a rigorous lower bound
to the exact energy
The self-consistent KS SCE total
energy is a rigorous lower bound
to the exact energy
The proof is trivial
(minimum of a sum is always larger than
the sum of the minima; self-consistency
lowers the energy)
First tests: quasi 1D systems
~
b =
2m !?
1
2
V? (y, z) = m !?
(y 2 + z 2 )
2
N
H1D =
wb (x) =
N
2
N
XX
~2 X @ 2
+
wb (|xi
2
2m i=1 @xi
i=1 j>i
p
e2⇤ x2 /4b2
⇡
e
2 b
erfc
⇣x⌘
2b
xj |) +
N
X
vext (xi )
i=1
10
wb(x), b=0.1
wb(x), b=0.2
1/x
8
6
4
2
0
0
0.5
Bednarek et al. PRB 68, 045328 (2003)
LDA: Casula, Sorella & Senatore PRB 74, 245427 (2006)
1
1.5
x
2
2.5
Electrons confined in (quasi)1D
Co-motion functions:
0
vSCE
[⇢](x) =
N
X
wb0 (|x
fi (x)|)sgn(x
i=2
Seidl, Phys. Rev. A 60, 4387 (1999)
Buttazzo, De Pascale, & Gori-Giorgi, Phys. Rev. A 85, 062502 (2012)
Malet & Gori-Giorgi, Phys. Rev. Lett., 109, 246402 (2012)
fi (x))
Applications to 1D systems
1D harmonic confinement:
•
L: effective length
Qualitatively good resuls in both the
weak and strong correlation regimes
F. Malet and P. Gori-Giorgi, PRL 109 246402 (2012); F. Malet. et al., PRB 87 115146 (2013)
•
Qualitatively good resuls in both the
weak and strong correlation regimes
•
Tends to the exact result in the very
strongly-interacting limit
L: effective length
L: effective length
L: effective length
KS SCE is the first KS DFT approach able to
capture the density peak splitting in quasi
1D without introducing magnetic order
Previous attempts include self-interaction corrections (SIC) and GGA:
S. H. Abedinpour, M. Polini, G. Xianlong, and M. P. Tosi, Eur. Phys. J. B 56, 127 (2007)
D. Vieira and K. Capelle, J. Chem. Theory Comput. 6, 3319 (2010)
D. Vieira, Phys. Rev. B 86, 075132 (2012)
Total Energies
Total Energies
Total Energies
CI matrix 105 - 106
Total Energies
CI matrix 105 - 106
KS SCE is a rigorous lower bound to the exact ground-state energy
Ionization potentials from minus HOMO
Generalization to open systems:
The Derivative Discontinuity at
integer particle numbers
What exact spin-restricted KS should do
N+electron&
open&shell&
N&
N+η&
exact&EN+1& &EN&&
Δxc=I& &A&&
exact&EN& &EN+1&&
N+electron&
closed&shell&
N&
N+η&
exact&EN+1& &EN&&
Δxc&
exact&EN& &EN+1&&
What exact spin-restricted KS should do
N+electron&
open&shell&
N&
N+η&
What approximate
functionlals do (UKS)
exact&EN+1& &EN&&
Δxc=I& &A&&
exact&EN& &EN+1&&
N+electron&
closed&shell&
N&
N+η&
exact&EN+1& &EN&&
Δxc&
exact&EN& &EN+1&&
Vydrov, Scuseria & Perdew, JCP 126, 154109 (2007)
SCE for fractional particle numbers
total%%M=5%electrons%
Q=2.5%
Q=2.5%
Z
fi+1 (x)
⇢(y)dy = 1
fi (x)
example: N=2.5
Z
fi+1 (x)
⇢(y)dy = 1
fi (x)
co-motion functions for N + ⌘ electrons
get the SCE potential by integrating
0
vSCE
[⇢](x) =
N
X
wb0 (|x
fi (x)|)sgn(x
i=2
with boundary condition
vSCE [⇢](x ! 1) = 0
fi (x))
Self-consistent KS HOMO eigenvalue
quasi 1D
N
A. Mirtschink, M. Seidl, and P. Gori-Giorgi, arXiv:1305.3982, accepted in PRL
Self-consistent KS HOMO eigenvalue
quasi 1D
3D (Hooke’s atom)
0.0007
exact"
0.0006
0.0005
ω=10*5"
0.0004
KS"SCE"
0.0003
0.0002
0.0001
0.0000
N
0.5
0.5"
1.0
1.0"
Q"
N
1.5
1.5"
2.0
2.0"
A. Mirtschink, M. Seidl, and P. Gori-Giorgi, arXiv:1305.3982, accepted in PRL
L: effective length
•
KS-SCE allows to treat large
strongly-correlated quasi 1D
systems (vs CI, limited to 6-8
particles, QMC and DMRG ~ 100
electrons).
•
Computational time similar to
KS LDA calculations
•
Applications in Physics (model
semiconductor quantum wires);
nanotransport, quantum
computation, ...
Electrons confined in two dimensions
N
Ĥ =
i=1
N
N
2
⇤2i
1
+
+
2
|r
rj |
2
i=1 j=i+1 i
N
ri2
i=1
Self-consistent KS SCE density
N = 6 ! = 0.005
C. B. Mendl (Munich Technical University)
F. Malet
What about Chemistry?
1D as a test lab for 3D chemistry
1
ws (x) = p
x2 + a2
1D model for H2 :
RH-H
Total energies
HF)
KS)LDA)
CI)
0th$order)KS$SCE)+)ZPE)
0th$order)KS$SCE)
Negative ions (3D)
N =2
vext (r) =
Z
r
r2
Z ! Zcrit ⇡ 0.9110289
Monotonicity of density vs. Z from John Morgan’s program
0.14
Z=.9110289
Z=.92
0.12
Z=.93
Z=.94
0.1
Z=.95
Z=1
0.08
Z=2
0.06
0.04
0.02
0
0
2
4
6
r (a0)
8
10
the system looses 1 electron
accurate results
(C. Umrigar & J. Morgan)
Negative ions (3D)
N =2
vext (r) =
Z
r
Z ! Zcrit ⇡ 0.9110289
KS SCE self-consistent:
Zcrit ⇡ 0.73
A. Mirtschink
Negative ions (3D)
N =2
vext (r) =
Z
r
Z ! Zcrit ⇡ 0.9110289
KS SCE +local correction (self-consistent):
Zcrit ⇡ 0.94
A. Mirtschink
How to treat general 3D
systems?
Monge-Kantorovich formulation
Buttazzo, De Pascale, and Gori-Giorgi, Phys. Rev. A. 85, 062502 (2012)
Cotar, Friesecke, and Kluppelberg, Comm. Pure Appl. Math. 66, 548 (2013)
see also talk by Gero Friesecke!
Optimal transport formulation
(Monge-Kantorovich)
1 (r)dr
1 (r)
=
2 (r)dr
2 (r)
cost function (work necessary to move a unit mass)
c(r1 , r2 ) = |r2
r1 |
Optimal transport formulation
(Monge-Kantorovich)
1 (r)dr
1 (r)
=
2 (r)dr
2 (r)
cost function (work necessary to move a unit mass)
c(r1 , r2 ) = |r2
r1 |
minimize total cost ) min
f
Z
optimal map
c(r, f (r))⇢1 (r)dr
min
P
(Z
c(x1 , ..., xN )P (dx1 , ...dxN ) :
c(x1 , ..., xN ) =
N
X1
N
X
i=1 j=i+1
⇡j# P
1
|xi
xj |
⇢
for j = 1, ..., N
=
N
)
min
P
(Z
c(x1 , ..., xN )P (dx1 , ...dxN ) :
c(x1 , ..., xN ) =
N
X1
N
X
i=1 j=i+1
1
x
⇡j# P
⇢
for j = 1, ..., N
=
N
1
|xi
xj |
x2
x
here: the further,
the better!
)
Things you can prove from Optimal Transport
Existence of P (r1 , r2 , . . . , rN ) (= |
SCE (r1 , r2 , . . . rN )|
2
)
vSCE (r) exists and it is bounded (Kantorovich potential)
Not possible (so far) to prove general existence of f i (r)
Dual Kantorovich problem
Dual Kantorovich problem
VSCE [⇢] = min h |V̂ee | i
!⇢
VSCE [⇢] = max
u
8
<Z
:
⇢(r)u(r)dr :
N
X
i=1
u(ri ) 
N X
N
X
i=1 j=i+1
vSCE (r) = u(r) + C
Kantorovich potential
1
|ri
9
=
rj | ;
Kantorovich formulation:
key to treat the 3D case?
Conclusions and outlook
! Strongly-interacting limit of DFT: approximations for the
xc functional able to describe strong correlation within the
restricted KS scheme (no artificial symmetry breaking)
! One can treat strongly-correlated quasi 1D systems with large
electron numbers
Perspectives:
" Improve the algorithms for 2D and 3D systems
" Corrections to the KS SCE functional:
" Inclusion of spin states
" Approximate SCE forms for new non-local functionals
" Nanotransport
Acknowledgments
Francesc Malet (VU Amsterdam)
André Mirtschink (VU Amsterdam
Klaas Giesbertz (VU Amsterdam)
Michael Seidl (University of Regensburg)
Giovanni Vignale (University of Missouri)
Andreas Savin (CNRS, University Paris VI)
Giuseppe Buttazzo (University of Pisa)
Luigi de Pascale (University of Pisa)
Stephanie Reimann (Lund University)
Jonas Cremon (Lund University)
Christian Mendl (Technical University Munich)
Acknowledgments
$$$
Acknowledgments
$$$
Thank you for your
attention!

Kohn-Sham density functional theory for strong-interacting

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