Curing NMR with SIC-DFT CSTC’2001, Ottawa1

Curing difficult cases in magnetic Curing difficult cases in magnetic properties prediction with SIC-DFTproperties prediction with SIC-DFT

S. Patchkovskii, J. Autschbach, and T. ZieglerS. Patchkovskii, J. Autschbach, and T. Ziegler

Department of Chemistry, University of Calgary, Department of Chemistry, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, 2500 University Dr. NW, Calgary, Alberta,

T2N 1N4 CanadaT2N 1N4 Canada

I am on the Web: I am on the Web: http://www.cobalt.chem.ucalgary.ca/ps/posters/SIC-NMR/http://www.cobalt.chem.ucalgary.ca/ps/posters/SIC-NMR/

Curing NMR with SIC-DFT CSTC’2001, Ottawa2

IntroductionIntroductionOne of the fundamental assumptions of quantum chemistry is that an electron does not interactOne 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 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 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 exchange-correlation (XC) energy must identically cancel the Coulomb self-interaction energy of the electron cloud.the electron cloud.

Although this condition has been well-known since the very firstAlthough this condition has been well-known since the very first steps in the development of steps in the development of DFT, satisfying it within model XCDFT, satisfying it within model XC f functionals has proven difficult. None of the approximate XCunctionals has proven difficult. None of the approximate XC functionals, commonly used in quantum chemistry today, are self-interaction free. The presence of 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 spurious self-interaction has been postulated as the reason behind some of the qualitative failures of approximate DFT. approximate DFT.

Some time ago, Perdew and Zunger (PZ) proposed a simple correction, which removes the 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 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 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 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.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 Recently, Krieger, Li, and Iafrate (KLI) developed an accurate approximation to the optimized effective potential, which allows a straightforward implementation of orbital-dependent functionals, 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 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 Functional (ADF) program. Here, we report on the applications of the technique to magnetic resonance parameters.resonance parameters.

Curing NMR with SIC-DFT CSTC’2001, Ottawa3

Self-interaction energy in DFTSelf-interaction energy in DFT

βαxcext

N

iσiσiσi

α,βσ

ρρrdrρrrdrdr

rρrρn ,EvˆE 21

12

212

1

12

1KStot

Kinetic Kinetic EnergyEnergy

(Classical) Coulomb (Classical) Coulomb energyenergy

Energy in the Energy in the external external potentialpotential

(Non-classical) (Non-classical) Exchange-correlation Exchange-correlation

energyenergy

At the same time, for a one-electron system, the total electronic energy is simply: At the same time, for a one-electron system, the total electronic energy is simply:

00,E2112

212

1 ρrdrdr

rρrρxc

This condition is NOT satisfied by any popular approximate exchange-correlation functionalThis condition is NOT satisfied by any popular approximate exchange-correlation functional

In Kohn-Sham DFT, the total electronic energy of the system is given by a sum of the kinetic 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-energy, classical Coulomb energy of the electron charge distribution, and the exchange-correlation energy:correlation energy:

rdrρrext

vˆelectron1E 2

1KStot

Therefore, for any one-electron density Therefore, for any one-electron density , the exact exchange-correlation functional must satisfy , the exact exchange-correlation functional must satisfy the following condition:the following condition:

Curing NMR with SIC-DFT CSTC’2001, Ottawa4

Perdew-Zunger self-interaction correctionPerdew-Zunger self-interaction correction

The PZ correction has some desirable properties, most importantly:The PZ correction has some desirable properties, most importantly:

• Correction (term is parentheses) vanishes for the exact functional ECorrection (term is parentheses) vanishes for the exact functional Excxc

• The functional EThe functional EPZPZ is exact for any one-electron system is exact for any one-electron system

• The XC potential has correct asymptotic behavior at large rThe XC potential has correct asymptotic behavior at large r

At the same time, At the same time,

• Total energy isTotal energy is orbital-dependent orbital-dependent

• Exchange-correlation potentials are Exchange-correlation potentials are per-orbitalper-orbital

N

iσixc

σiσi

α,βσ

ρrdrdr

rρrρ

121

12

212

1KStot

PZtot 0,EEE

Kohn-Sham Kohn-Sham total energytotal energy

(Classical) Coulomb (Classical) Coulomb self-interactionself-interaction

(Nonclassical) self-exchange (Nonclassical) self-exchange and self-correlationand self-correlation

In 1981, Perdew and ZungerIn 1981, Perdew and Zunger** (PZ) suggested a prescription for removing self-interaction from (PZ) suggested a prescription for removing self-interaction from Kohn-Sham total energy, computed with an approximate XC functional EKohn-Sham total energy, computed with an approximate XC functional Exc.xc. In the PZ approach, In the PZ approach, total enery is defined as:total enery is defined as:

**: J.P. Perdew and A. Zunger, : J.P. Perdew and A. Zunger, Phys. Rev. BPhys. Rev. B 19811981, , 2323, 5048, 5048

Curing NMR with SIC-DFT CSTC’2001, Ottawa5

Self-consistent implementation of PZ-SICThe 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:

EKS

tot

rrrσ

xc,extc2

1KSvvvˆf̂

σiσiσiσ f̂

KS

EPZ

tot

riσiσi

PZKSPZvf̂f̂

σiσiσiσi f̂

PZ

1

1

1σi

σi

σi rrr

rρ

ρ

ρr

i

dδ

0,Eδv xcPZ

Kohn-ShamKohn-Sham Perdew-ZungerPerdew-Zunger

All MOs are eigenfunctions of the All MOs are eigenfunctions of the samesame Fock operator Fock operator

MOs are eigenfunctions of MOs are eigenfunctions of differentdifferent Fock operators 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:

σiσiσiσ f̂

OEP

Chosen to minimize EPZ

EPZ

tot

rσσiσi

OEPKSOEPvf̂f̂

Curing NMR with SIC-DFT CSTC’2001, Ottawa6

SIC, OEP, and KLI-OEPSIC, OEP, and KLI-OEPDetermining the exact OEP is difficult, and involves solving an integral equation on vDetermining the exact OEP is difficult, and involves solving an integral equation on v OEPOEP(r):(r):

i ij

rri rdrrn

ij

jj

σiσ0''v'v

'PZOEP

An exact solution of the OEP equation is only possible for small, and highly symmetric systems, 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 Iafratesuch as atoms. Fortunately, an approximation due to Krieger, Li, and Iafrate** is believed to is believed to approximate the exact OEP closely. The KLI-OEP is given by a density-weighted average of per-approximate the exact OEP closely. The KLI-OEP is given by a density-weighted average of per-orbital Perdew-Zunger potentials:orbital Perdew-Zunger potentials:

σ

σ

N

iσiσi

σ

σi xrρ

ρr

PZOEP-KLI vv

KLI-OEP:KLI-OEP:• … … is exact for perfectly localized systemsis exact for perfectly localized systems• … … approximates the exact OEP closely in atomic and molecular systemsapproximates the exact OEP closely in atomic and molecular systems• … … guarantees the correct asymptotic behavior of the potential at r guarantees the correct asymptotic behavior of the potential at r

rdrρxrrdrρr σiσiσiσiσ

PZOEP-KLI vv

**: J.B. Krieger, Y. Li, and G.J. Iafrate, : J.B. Krieger, Y. Li, and G.J. Iafrate, Phys. Rev. APhys. Rev. A 19921992, , 4545, 101, 101

Constants xConstants xii are obtained from the requirement, that the orbital densities “feel” the effective are obtained from the requirement, that the orbital densities “feel” the effective potential just as they would “feel” their own per-orbital potentials:potential just as they would “feel” their own per-orbital potentials:

Curing NMR with SIC-DFT CSTC’2001, Ottawa7

Implementation in ADFImplementation in ADF• Numerical implementation, in Amsterdam Density Functional (ADF) programNumerical implementation, in Amsterdam Density Functional (ADF) program

• SIC-KLI-OEP computed on localized MOs (using Boys-Foster localization criterion), SIC-KLI-OEP computed on localized MOs (using Boys-Foster localization criterion), maximizing SIC energymaximizing SIC energy

• Both local and gradient-corrected functionals are supportedBoth local and gradient-corrected functionals are supported

• Frozen cores are supportedFrozen cores are supported

• All properties are available with SIC All properties are available with SIC

• Efficient evaluation of per-orbital Coulomb potentials, using secondary fitting of per-orbital Efficient evaluation of per-orbital Coulomb potentials, using secondary fitting of per-orbital electron density, avoids the bottleneck of most analytical implementations:electron density, avoids the bottleneck of most analytical implementations:

rd

rr

rrr Avv cc

• Computation time Computation time 2x-10x compared to KS DFT 2x-10x compared to KS DFT

• Standard ADF fitting basis sets have to be reoptimized, to ensure adequate fits to per-orbital Standard ADF fitting basis sets have to be reoptimized, to ensure adequate fits to per-orbital densities of inner orbitals (core and semi-core).densities of inner orbitals (core and semi-core).

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

rrrrr AP

FittedFitted

densitydensity

ExactExact

densitydensity

Fit functionsFit functionsBasis functionsBasis functionsDensity matrixDensity matrix

Curing NMR with SIC-DFT CSTC’2001, Ottawa8

-25-25

-20-20

-15-15

-10-10

-5-5

00

55

1010

1515

2020

2525

00 5050 100100 150150 200200 250250

Err

or in

cal

cula

ted

chem

ical

shi

ft,

ppm

Err

or in

cal

cula

ted

chem

ical

shi

ft,

ppm

Experimental Experimental 1313C chemical shift, ppmC chemical shift, ppm

CHFCHF33

CHCH33NCNC **HH22COCO

CFCF33CC**NN

C(CC(C **O)O)22

py->O, Cpy->O, C44**

VWNVWNBP86BP86

VWN-SICVWN-SIC

NMR chemical shifts: NMR chemical shifts: 1313CC

RMS ErrorRMS Error

VWNVWN 9.29.2

BP86BP86 6.66.6

SIC-SIC-VWNVWN

7.17.1

SIC-SIC-BP86BP86

6.66.6

Curing NMR with SIC-DFT CSTC’2001, Ottawa9

NMR chemical shifts: NMR chemical shifts: 2929SiSi

RMS ErrorRMS Error

VWNVWN 13.913.9

revPBErevPBE 10.010.0

SIC-SIC-VWNVWN

12.412.4

SIC-SIC-revPBErevPBE

12.012.0

-20-20

-10-10

00

1010

2020

3030

4040

5050

-200-200 -150-150 -100-100 -50-50 00 5050

Err

or in

cal

cula

ted

chem

ical

shi

ft,

ppm

Err

or in

cal

cula

ted

chem

ical

shi

ft,

ppm

Experimental Experimental 2929Si chemical shift, ppmSi chemical shift, ppm

VWNVWNrevPBErevPBE

SIC-VWNSIC-VWNSIC-revPBESIC-revPBE

Curing NMR with SIC-DFT CSTC’2001, Ottawa10

-50-50

00

5050

100100

150150

200200

250250

300300

350350

-300-300 -200-200 -100-100 00 100100 200200 300300 400400

Err

or in

cal

cula

ted

chem

ical

shi

ft, p

pmE

rror

in c

alcu

late

d ch

emic

al s

hift,

ppm

Experimental Experimental 1414N,N,1515N chemical shift, ppmN chemical shift, ppm

OO22N-NN-N **OO

OO22NN **-NO-NO

(CH(CH33))22N-NN-N **OO

VWNVWNBP86BP86

VWN-SICVWN-SIC

NMR chemical shifts: NMR chemical shifts: 1414N,N,1515NN

RMS ErrorRMS Error

VWNVWN 86.386.3

BP86BP86 69.269.2

SIC-SIC-VWNVWN

21.321.3

SIC-SIC-BP86BP86

17.017.0

Curing NMR with SIC-DFT CSTC’2001, Ottawa11

-100-100

00

100100

200200

300300

400400

00 200200 400400 600600 800800 10001000 12001200 14001400 16001600

Err

or

in c

alc

ulat

ed

chem

ical

shi

ft, p

pm

Err

or

in c

alc

ulat

ed

chem

ical

shi

ft, p

pm

Experimental Experimental 1717O chemical shift, ppmO chemical shift, ppm

HH22COCO

OFOF22

OO22N-NON-NO **

O-O-OO-O-O **

VWNVWNBP86BP86

VWN-SICVWN-SIC

NMR chemical shifts: NMR chemical shifts: 1717OO

RMS ErrorRMS Error**

VWNVWN 135.3135.3

BP86BP86 105.3105.3

SIC-SIC-VWNVWN

61.261.2

SIC-SIC-BP86BP86

45.645.6

**Excluding OExcluding O33

Curing NMR with SIC-DFT CSTC’2001, Ottawa12

-40-40

-20-20

00

2020

4040

6060

-100-100 00 100100 200200 300300 400400 500500 600600

Err

or

in c

alc

ulat

ed

chem

ical

shi

ft, p

pm

Err

or

in c

alc

ulat

ed

chem

ical

shi

ft, p

pm

Experimental Experimental 1919F chemical shift, ppmF chemical shift, ppm

FF22

NFNF33

HFHF

VWNVWNBP86BP86

VWN-SICVWN-SIC

NMR chemical shifts: NMR chemical shifts: 1919FF

RMS ErrorRMS Error

VWNVWN 27.327.3

BP86BP86 20.720.7

SIC-SIC-VWNVWN

14.514.5

SIC-SIC-BP86BP86

12.312.3

Curing NMR with SIC-DFT CSTC’2001, Ottawa13

NMR chemical shifts: NMR chemical shifts: 3131PP

RMS ErrorRMS Error

VWNVWN 63.863.8

revPBErevPBE 49.049.0

SIC-SIC-VWNVWN

34.834.8

SIC-SIC-revPBErevPBE

21.321.3

B3-LYPB3-LYP** 27.127.1

MP2MP2** 23.723.7

**Excludes PBrExcludes PBr33-100-100

-50-50

00

5050

100100

150150

200200

-300-300 -200-200 -100-100 00 100100 200200 300300

Err

or

in c

alc

ulat

ed

chem

ical

shi

ft, p

pm

Err

or

in c

alc

ulat

ed

chem

ical

shi

ft, p

pm

Experimental Experimental 3131P chemical shift, ppmP chemical shift, ppm

VWNVWNrevPBErevPBE

SIC-VWNSIC-VWNSIC-revPBESIC-revPBE

PNPN

PBrPBr33

PHPH33

PHPH44++

PFPF66--

Curing NMR with SIC-DFT CSTC’2001, Ottawa14

31P: PX3 (X=F,Cl,Br)

100

150

200

250

300

350

400

PF3 PCl3 PBr3

31P

che

mic

al s

hift

, pp

m

expt

VWNrevPBE

SIC-VWNSIC-revPBE

Curing NMR with SIC-DFT CSTC’2001, Ottawa15

SIC-DFT: Uniform description of the SIC-DFT: Uniform description of the chemical shiftschemical shifts

0

2

4

6

8

10

12

14

C N O F Si P

LDAGGASIC-LDASIC-GGA

RM

S e

rror

/tot

al s

hift

ran

ge, p

erce

ntR

MS

err

or/t

otal

shi

ft r

ange

, per

cent

Curing NMR with SIC-DFT CSTC’2001, Ottawa16

Chemical shift tensors in SIC-DFTChemical shift tensors in SIC-DFT

Expt.Expt. VWNVWN BP86BP86 SIC-SIC-VWNVWN

isoiso 201201 202202 202202 200200

11 272272 290290 285285 276276

22 246246 256256 253253 245245

33 8484 6262 6767 8080

Expt.Expt. VWNVWN BP86BP86 SIC-SIC-VWNVWN

isoiso 1717 1212 88 99

11 275275 327327 303303 282282

22 108108 9393 8989 9696

33 -347-347 -385-385 -368-368 -351-351

Curing NMR with SIC-DFT CSTC’2001, Ottawa17

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.