Transcript Improving description of the potential energy surfaces

Curing difficult cases in magnetic
properties prediction with SIC-DFT
I am on the Web:
http://www.cobalt.chem.ucalgary.ca/ps/posters/SIC-NMR/
S. Patchkovskii, J. Autschbach, and T. Ziegler
Department of Chemistry, University of Calgary,
2500 University Dr. NW, Calgary, Alberta,
T2N 1N4 Canada
Curing NMR with SIC-DFT
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CSTC’2001, Ottawa
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 applications of the technique to
magnetic resonance parameters.
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CSTC’2001, Ottawa
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:
E tot 
KS

N
1 ˆ
n


 σi σi 2  σi
σ  α,β i 1
Kinetic
Energy
 
  

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
Curing NMR with SIC-DFT
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CSTC’2001, Ottawa
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
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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
fˆ
fˆ
KS
tot
KS
σ
KS
σ
σi   σiσi

1

2
E



ˆ  vc r   vext r   v xc,  r 
PZ
tot
fˆ
fˆ
PZ
σi
 σi   σi σi
PZ
σi
 fˆ
KS
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
Curing NMR with SIC-DFT
fˆ
fˆ
OEP
σ
OEP
σi
 σi   σi σi
 fˆ
KS
σi
Chosen to
minimize EPZ



 vOEP
r
σ
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CSTC’2001, Ottawa
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 and molecular 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
Curing NMR with SIC-DFT
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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).
Curing NMR with SIC-DFT
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CSTC’2001, Ottawa
Error in calculated chemical shift, ppm
NMR chemical shifts: 13C
25
VWN
20
BP86
VWN-SIC
15
10
CH3NC
H2CO
*
RMS Error
CHF3
VWN
9.2
0
BP86
6.6
-5
SICVWN
7.1
SICBP86
6.6
5
-10
-15
-20
py->O, C 4
*
*
CF3C N
*
C(C O)2
-25
0
Curing NMR with SIC-DFT
50
100
150
200
Experimental 13
C chemical shift, ppm
8
250
CSTC’2001, Ottawa
Error in calculated chemical shift, ppm
NMR chemical shifts: 29Si
50
VWN
revPBE
SIC-VWN
SIC-revPBE
40
30
RMS Error
20
VWN
13.9
revPBE
10.0
SICVWN
12.4
SICrevPBE
12.0
10
0
-10
-20
-200
-150
-100
-50
0
Experimental29Si chemical shift, ppm
Curing NMR with SIC-DFT
9
50
CSTC’2001, Ottawa
Error in calculated chemical shift, ppm
NMR chemical shifts: 14N,15N
350
300
VWN
BP86
VWN-SIC
*
O2 N-N O
250
RMS Error
(CH3)2 N-N *O
200
150
*
O2N -NO
100
50
0
VWN
86.3
BP86
69.2
SICVWN
21.3
SICBP86
17.0
-50
-300
Curing NMR with SIC-DFT
-200
-100
0
100
200
300
14 15
Experimental N, N chemical shift, ppm
10
400
CSTC’2001, Ottawa
Error in calculated chemical shift, ppm
NMR chemical shifts: 17O
400
VWN
BP86
VWN-SIC
OF 2
300
RMS Error*
O2N-NO *
H2CO
200
VWN
135.3
BP86
105.3
SICVWN
61.2
SICBP86
45.6
100
0
-100
O-O-O
0
*
*Excluding
O3
200 400 600 800 1000 1200 1400 1600
Experimental 17O chemical shift, ppm
Curing NMR with SIC-DFT
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CSTC’2001, Ottawa
Error in calculated chemical shift, ppm
NMR chemical shifts: 19F
60
VWN
BP86
VWN-SIC
F2
40
RMS Error
20
NF3
VWN
27.3
BP86
20.7
SICVWN
14.5
SICBP86
12.3
0
-20
HF
-40
-100
Curing NMR with SIC-DFT
0
100
200
300
400
500
Experimental 19F chemical shift, ppm
12
600
CSTC’2001, Ottawa
NMR chemical shifts: 31P
Error in calculated chemical shift, ppm
200
150
VWN
revPBE
SIC-VWN
SIC-revPBE
PBr3
RMS Error
100
PH3
0
-50
-100
-300
PH4
+
-200
-100
0
100
200
31
Experimental P chemical shift, ppm
Curing NMR with SIC-DFT
63.8
revPBE
49.0
SICVWN
34.8
SICrevPBE
21.3
B3-LYP*
27.1
MP2*
23.7
PN
PF6-
50
VWN
13
300
*Excludes
PBr3
CSTC’2001, Ottawa
31P:
31P
chemical shift, ppm
400
350
PX3 (X=F,Cl,Br)
VWN
revPBE
SIC-VWN
SIC-revPBE
expt
300
250
200
150
100
PF3
Curing NMR with SIC-DFT
PCl3
14
PBr3
CSTC’2001, Ottawa
RMS error/total shift range, percent
SIC-DFT: Uniform description of the
chemical shifts
14
12
10
LDA
GGA
SIC-LDA
SIC-GGA
8
6
4
2
0
C
Curing NMR with SIC-DFT
N
O
F
15
Si
P
CSTC’2001, Ottawa
Chemical shift tensors in SIC-DFT
Expt.
BP86
SICVWN
iso
201
202
202
200
1
272
290
285
276
2
246
256
253
245
3
84
62
67
80
VWN
BP86
SICVWN
Expt.
Curing NMR with SIC-DFT
VWN
iso
17
12
8
9
1
275
327
303
282
2
108
93
89
96
3
-347
-385
-368
-351
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CSTC’2001, Ottawa
Summary and Outlook
•
•
In molecular DFT calculations, self-interaction can be cancelled out with modest
effort
Removal of self-interaction greatly improves the description of the NMR
chemical shifts for “difficult” nuclei (17O,15N,31P)
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.
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