Curing difficult cases in magnetic properties prediction with SICDFT

1
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
2
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.

3
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 exchange-correlation
energy
At the same time, for a one-electron system, the
total electronic energy is simply
Therefore, for any one-electron density ?, the
exact exchange-correlation functional must
satisfy the following condition
This condition is NOT satisfied by any popular
approximate exchange-correlation functional
4
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

• 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
5
Self-consistent implementation of PZ-SIC
The non-trivial orbital dependence of the PZ-SIC
energy leads to complications in practical
self-consistent implementation of the correction.
Compare the outcomes of the standard variational
minimization of EKS and EPZ
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
Chosen to minimize EPZ
6
SIC, OEP, and KLI-OEP

• Determining the exact OEP is difficult, and
  involves solving an integral equation on vOEP(r)

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
per-orbital Perdew-Zunger potentials
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
• 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
7
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

The per-orbital Coulomb potentials are then
computed as a sum of one-centre contributions

• 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).

8
NMR chemical shifts 13C
9
NMR chemical shifts 29Si
10
NMR chemical shifts 14N,15N
VWN
350
BP86
VWN-SIC
300

O
N-N
O
2
250

(CH
)
N-N
O
200
3
2
150
Error in calculated chemical shift, ppm

O
N
-NO
2
100
50
0
-50
-300
-200
-100
0
100
200
300
400
14
15
Experimental
N,
N chemical shift, ppm
11
NMR chemical shifts 17O
VWN
400
BP86
VWN-SIC
OF
2
300

O
N-NO
H
CO
2
200
2
Error in calculated chemical shift, ppm
100
0
-100

O-O-O
Excluding O3
0
200
400
600
800
1000
1200
1400
1600
17
Experimental
O chemical shift, ppm
12
NMR chemical shifts 19F
13
NMR chemical shifts 31P
Excludes PBr3
14
31P PX3 (XF,Cl,Br)
15
SIC-DFT Uniform description of the chemical
shifts
16
Chemical shift tensors in SIC-DFT
17
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.