Improving description of the potential
energy surfaces with self interaction
free density functional theory
I am on the Web:
http://www.cobalt.chem.ucalgary.ca/ps/posters/SIC-Barriers/
S. Patchkovskii and T. Ziegler
Department of Chemistry, University of Calgary,
2500 University Dr. NW, Calgary, Alberta,
T2N 1N4 Canada
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
Introduction
One of the fundamental assumptions of quantum chemistry is that an electron does not interact
with itself. Applied to the density functional theory (DFT), this leads to a simple condition on the
exact (and unknown) exchange-correlation functional: for any one-electron density distribution, the
exchange-correlation (XC) energy must identically cancel the Coulomb self-interaction energy of
the electron cloud.
Although this condition has been well-known since the very first steps in the development of
DFT, satisfying it within model XC functionals has proven difficult. None of the approximate XC
functionals, commonly used in quantum chemistry today, are self-interaction free. The presence of
spurious self-interaction has been postulated as the reason behind some of the qualitative failures
of approximate DFT.
Some time ago, Perdew and Zunger (PZ) proposed a simple correction, which removes the
self-interaction from a given approximate XC functional. Unfortunately, the PZ self-interaction
correction (SIC) is not invariant to unitary transformations between the occupied molecular
orbitals. This, in turn, leads to difficulties in practical implementation of the scheme, so that
relatively few applications of PZ SIC to molecular systems have been reported.
Recently, Krieger, Li, and Iafrate (KLI) developed an accurate approximation to the optimized
effective potential, which allows a straightforward implementation of orbital-dependent
functionals, such as PZ SIC. We have implemented this SIC-KLI-OEP scheme in Amsterdam
Density Functional (ADF) program. Here, we report on the first application of the technique to
“difficult” activation barries.
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
Self-interaction energy in DFT
In Kohn-Sham DFT, the total electronic energy of the system is given by a sum of the kinetic
energy, classical Coulomb energy of the electron charge distribution, and the exchangecorrelation energy:
KS
E tot 

N
 nσi σi
σ  α,β i 1
Kinetic
Energy

1
2
ˆ  σi
 
  

ρr   
ρ
r
 12  1 2 dr1dr2   vext r ρr dr  E xc ρα , ρβ
r12

Energy in the
external
potential
(Classical)
Coulomb energy

(Non-classical)
Exchange-correlation
energy
At the same time, for a one-electron system, the total electronic energy is simply:
E
KS
tot
1  electron  


1
2
  
ˆ    vext r ρr dr
Therefore, for any one-electron density , the exact exchange-correlation functional must
satisfy the following condition:
 

ρ
r
1
1  ρr2   
dr1dr2  E xc ρ ,0  0

2
r12
This condition is NOT satisfied by any popular approximate exchange-correlation functional
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
Perdew-Zunger self-interaction correction
In 1981, Perdew and Zunger* (PZ) suggested a prescription for removing self-interaction from
Kohn-Sham total energy, computed with an approximate XC functional Exc. In the PZ approach,
total enery is defined as:
E tot  E tot 
PZ
KS

σ  α,β
Kohn-Sham
total energy


 1 ρσi r1 ρσi r2   

dr1dr2  E xc ρσi ,0

2 

r12
i 1 

N
(Classical) Coulomb
self-interaction
(Nonclassical) self-exchange
and self-correlation
The PZ correction has some desirable properties, most importantly:
• Correction (term is parentheses) vanishes for the exact functional Exc
• The functional EPZ is exact for any one-electron system
• The XC potential has correct asymptotic behavior at large r
At the same time,
• Total energy is orbital-dependent
• Exchange-correlation potentials are per-orbital
*:
J.P. Perdew and A. Zunger, Phys. Rev. B 1981, 23, 5048
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
Self-consistent implementation of PZ-SIC
The non-trivial orbital dependence of the PZ-SIC energy leads to complications in practical selfconsistent implementation of the correction. Compare the outcomes of the standard variational
minimization of EKS and EPZ:
Kohn-Sham
Perdew-Zunger
E
KS
KS
f̂
tot
σ
KS
f̂
σ
σi   σi σi

1

2
E



ˆ  vc r   vext r   v xc,  r 
PZ
tot
PZ
f̂
f̂
σi
 σi   σi σi
PZ
σi

KS
f̂
σi



 vPZ
r
i


δ E xc ρσi ,0
ρσi r1  
v i r   
    d r1
δ ρσi
r  r1
PZ
All MOs are eigenfunctions of the
same Fock operator
MOs are eigenfunctions of
different Fock operators
The orbital dependence of the fPZ operator makes self-consistent implementation of PZ-SIC
difficult, compared to Kohn-Sham DFT. However, the PZ self-interaction correction can also be
implemented within an optimized effective potential (OEP) scheme, with eigenequations formally
identical to KS DFT:
E
PZ
tot
OEP
f̂
σ
OEP
f̂
σi
 σi   σi σi

KS
f̂
σi
Chosen to
minimize EPZ



 vOEP
r
σ
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
SIC, OEP, and KLI-OEP
Determining the exact OEP is difficult, and involves solving an integral equation on v OEP(r):
 n  v
i
OEP
σ


PZ 
r '  v σi r '
j i
i


 j  r '   j  r 
 j   i

dr '  0
An exact solution of the OEP equation is only possible for small, and highly symmetric systems,
such as atoms. Fortunately, an approximation due to Krieger, Li, and Iafrate* is believed to
approximate the exact OEP closely. The KLI-OEP is given by a density-weighted average of perorbital Perdew-Zunger potentials:

Nσ

ρσi PZ 
- OEP


v KLI
r

i ρ vσi r   xσi
σ
σ

Constants xi are obtained from the requirement, that the orbital densities “feel” the effective
potential just as they would “feel” their own per-orbital potentials:


 
 
KLI- OEP 
PZ 







dr
v
r
ρ
r
d
r

v
r

x
ρ
r
σi
σi
σi
 σ
 σi
KLI-OEP:
• … is exact for perfectly localized systems
• … approximates the exact OEP closely in atomic systems
• … guarantees the correct asymptotic behavior of the potential at r 
*:
J.B. Krieger, Y. Li, and G.J. Iafrate, Phys. Rev. A 1992, 45, 101
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
Implementation in ADF
•
•
•
•
•
•
Numerical implementation, in Amsterdam Density Functional (ADF) program
SIC-KLI-OEP computed on localized MOs (using Boys-Foster localization criterion),
maximizing SIC energy
Both local and gradient-corrected functionals are supported
Frozen cores are supported
All properties are available with SIC
Efficient evaluation of per-orbital Coulomb potentials, using secondary fitting of per-orbital
electron density, avoids the bottleneck of most analytical implementations:
Exact
density





 r    P r  r    A  r    r 

Density matrix




Basis functions
Fitted
density
Fit functions
The per-orbital Coulomb potentials are then computed as a sum of one-centre contributions:



 r 
vc r   v c r    A  r  r dr 
•
•
Computation time  2x-10x compared to KS DFT
Standard ADF fitting basis sets have to be reoptimized, to ensure adequate fits to per-orbital
densities of inner orbitals (core and semi-core).
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
H2+H+H+
E(total), Hartree
-0.45
Exact
SIC-VWN
-0.5
+
SIC-revPBE
-0.55
VWN
revPBE
-0.6
-0.65
0
0.5
1
1.5 2 2.5 3 3.5
R(H-H), Angstrom
4
4.5
For one-electron systems, such as H2+, SIC-DFT is an exact theory (just like Hartree-Fock ab initio)
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
He2+He+He+
E(total), Hartree
-4.85
+
VWN
-4.9
CCSD(T)/cc-pVTZ
SIC-revPBE
-4.95
SIC-VWN
-5
-5.05
0.5
revPBE
1
1.5
2
R(He-He), Angstrom
2.5
3
For many-electron systems, SIC-DFT is no longer exact. However, both local (VWN) and gradientcorrected (revPBE) functionals lead to qualitatively correct dissociation curves, if applied together
with SIC. In this system, both parent functionals fail completely in Kohn-Sham DFT.
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
H + H2H2+H
+
+
DFT†
CCSD(T)*
E
VWN
9.9
SIC-DFT†
revPBE
-2.8
4.7
VWN
5.8
revPBE
11.8
kcal/mol
*:
B.G. Johnson, C.A. Gonzales, P.M.W. Gill, G.A. Pople, Chem. Phys. Lett. 1994, 221,110
†:
Single-point energies at optimized ab initio geometries
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
H + HClH2+Cl
+
+
DFT†
G3*
VWN
SIC-DFT†
revPBE
VWN
revPBE
rE
-3.1
+9.1
-1.6
+1.7
-1.0
E
+5.6
-6.0
+0.1
+3.1
+8.7
*: T.C. Allison, G.C. Lynch, D.G. Truhlar, M.S. Gordon, J. Phys. Chem. 1996, 100, 13575
†:
Single-point energies at optimized ab initio geometries
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
H +N2H2N2H+H2
+
+
MRCI//
CASSCF*
DFT†
VWN
SIC-DFT†
revPBE
VWN
revPBE
rE
-33.6
-34.9
-39.6
-27.2
-30.5
E
+5.9
-11.8
-4.9
+5.0
+9.9
*: D.P. Linder, X. Duan, M. Page, J. Chem. Phys. 1996, 104, 6298
†:
Single-point energies at optimized ab initio geometries
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
CH3 + H2H + CH4
+
+
MP2+
Expt*
DFT†
VWN
SIC-DFT†
revPBE
VWN
revPBE
rE
-3.7
-9.5
-1.4
-16.7
-9.7
E
+11.3
-4.7
+7.5
-0.7
+8.3
*: D.P. Linder, X. Duan, M. Page, J. Chem. Phys. 1996, 104, 6298
†:
Single-point energies at optimized ab initio geometries
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
OHO  OHO
CCSD(T)
//MP2*
E
DFT†
VWN
+4.1
SIC-DFT†
revPBE
-2.4
VWN
+1.5
+0.7
revPBE
+5.1
kcal/mol
*: M.G. Frisch, A.C. Scheiner, H.F. Schaefer III, J. Chem. Phys. 1994, 82, 4194
†:
Single-point energies at optimized ab initio geometries
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
F- + CH3F  CH3F + F+
EB
EIB
EC
DFT†
W1’*
VWN
…
SIC-DFT†
revPBE
VWN
revPBE
EC
+13.7
+25.3
+19.9
+16.5
+12.0
EIB
+13.4
+6.4
+7.3
+18.0
+17.2
*: S. Parthiban, G. de Oliveira, J.M.L. Martin, J. Phys. Chem. A 2001, 105, 895
†:
Single-point energies at optimized ab initio geometries
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
Cl- + CH3Cl  CH3Cl + Cl+
EB
EC
EIB
DFT†
W1’*
VWN
…
SIC-DFT†
revPBE
VWN
revPBE
EC
+10.9
+15.1
+11.6
+11.5
+8.3
EIB
+13.6
+6.0
+7.4
+18.7
+19.6
*: S. Parthiban, G. de Oliveira, J.M.L. Martin, J. Phys. Chem. A 2001, 105, 895
†:
Single-point energies at optimized ab initio geometries
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
C2N4H2  2 HCN + N2
+
B3LYP
+expt*
DFT†
VWN
SIC-DFT†
revPBE
VWN
revPBE
rE
-46.4
-29.5
-79.6
-38.4
-84.4
E
51.8
+12.0
-5.4
+71.9
+50.5
*: J. Baker, M. Muir, J. Andzelm, J. Chem. Phys. 1995, 102, 2063; Experimental rE; theoretical E
†:
Single-point energies at optimized B3LYP geometries
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
Summary of the reaction barriers
DFT
Reaction
“best”
VWN
SIC-DFT
revPBE
VWN
revPBE
H + H2H2+H
+9.9
-2.8
4.7
5.8
11.8
H + HClH2+Cl
+5.6
-6.0
+0.1
+3.1
+8.7
H +N2H2N2H+H2
+5.9
-11.8
-4.9
+5.0
+9.9
CH3 + H2H + CH4
+11.3
-4.7
+7.5
-0.7
+8.3
+4.1
-2.4
+1.5
+0.7
+5.1
F- + CH3F  CH3F + F-
+13.4
+6.4
+7.3
+18.0
+17.2
Cl- + CH3Cl  CH3Cl + Cl-
+13.6
+6.0
+7.4
+18.7
+19.6
C2N4H2  2 HCN + N2
+51.8
+12.0
-5.4
+71.9
+50.5
161%
92%
57%
52%
OHO  OHO
RMS relative error, %
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal
Gotcha!
•
SIC-DFT total energies are not invariant to rotations between occupied
molecular orbitals.
• Localized MOs (Boys-Foster) are do not necessarity minimize EPZ
This may lead to unexpected results:
MP2+
Expt
CH3 + H2H + CH4:
SIC-DFT/All
electron
VWN
revPBE
SIC-DFT/1s frozen
core on carbon
VWN
revPBE
rE
-3.7
-180.3
-192.8
-16.7
-9.7
E
+11.3
-198.5
-200.9
-0.7
+8.3
Frozen core
All-electron
2pz
1s
E(SIC)
-126.6
+3.0
Potential energy surfaces with self-interaction free DFT
kcal/mol
19
+28.8
+28.8
CSC’2001, Montréal
Summary and Outlook
•
•
•
In molecular DFT calculations, self-interaction can be cancelled out with modest effort
Removal of self-interaction leads to qualitatively correct description of “difficult” reaction
barriers
The orbital dependence of the self-interaction correction can lead to unpleasant “gotchas”!
Future developments:
•
•
•
Applications to heavier nuclei
– High-level correlated ab initio too costly
– Other approaches (hybrid DFT, empirical corrections) seem not to help
Other molecular properties which require accurate exchange correlation potentials
– Excitation energies; time-dependent properties
Development of SIC-specific approximate functionals
Acknowledgements. This work has been supported by the National
Sciences and Engineering Research Council of Canada (NSERC), as
well as by the donors of the Petroleum Research Fund, administered by
the American Chemical Society. Dr. Jochen Autschbach is acknowledged
for helpful discussions
Potential energy surfaces with self-interaction free DFT
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CSC’2001, Montréal