1. 1 Curing NMR with SIC-DFT CSTC’2001, Ottawa Curing difficult cases in magnetic properties prediction with SIC-DFT S. Patchkovskii, J. Autschbach, and T. Ziegler Department of Chemistry, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, T2N 1N4 Canada I am on the Web: http://www.cobalt.chem.ucalgary.ca/ps/posters/SIC-NMR/

2. 2 Curing NMR with SIC-DFT 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.

3. 3 Curing NMR with SIC-DFT CSTC’2001, Ottawa Self-interaction energy in DFT Kinetic Energy (Classical) Coulomb energy Energy in the external potential (Non-classical) Exchange-correlation energy At the same time, for a one-electron system, the total electronic energy is simply: This 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 energy, classical Coulomb energy of the electron charge distribution, and the exchange- correlation energy: Therefore, for any one-electron density , the exact exchange-correlation functional must satisfy the following condition:

4. 4 Curing NMR with SIC-DFT CSTC’2001, Ottawa Perdew-Zunger self-interaction correction The PZ correction has some desirable properties, most importantly: Correction (term is parentheses) vanishes for the exact functional E xcCorrection (term is parentheses) vanishes for the exact functional E xc The functional E PZ is exact for any one-electron systemThe functional E PZ 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, Total energy is orbital-dependentTotal energy is orbital-dependent Exchange-correlation potentials are per-orbitalExchange-correlation potentials are per-orbital Kohn-Sham total energy (Classical) Coulomb self-interaction (Nonclassical) self-exchange and self-correlation In 1981, Perdew and Zunger * (PZ) suggested a prescription for removing self-interaction from Kohn-Sham total energy, computed with an approximate XC functional E xc. In the PZ approach, total enery is defined as: * : J.P. Perdew and A. Zunger, Phys. Rev. B 1981, 23, 5048

5. 5 Curing NMR with SIC-DFT CSTC’2001, Ottawa 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 E KS and E PZ :Kohn-ShamPerdew-Zunger All MOs are eigenfunctions of the same Fock operator MOs are eigenfunctions of different Fock operators The orbital dependence of the f PZ 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 E PZ

6. 6 Curing NMR with SIC-DFT CSTC’2001, Ottawa SIC, OEP, and KLI-OEP Determining the exact OEP is difficult, and involves solving an integral equation on v OEP (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: KLI-OEP: … is exact for perfectly localized systems… is exact for perfectly localized systems … approximates the exact OEP closely in atomic and molecular systems… approximates 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   * : J.B. Krieger, Y. Li, and G.J. Iafrate, Phys. Rev. A 1992, 45, 101 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:

7. 7 Curing NMR with SIC-DFT CSTC’2001, Ottawa Implementation 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), maximizing SIC energySIC-KLI-OEP computed on localized MOs (using Boys-Foster localization criterion), maximizing 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 SICAll 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:Efficient evaluation of per-orbital Coulomb potentials, using secondary fitting of per-orbital electron density, avoids the bottleneck of most analytical implementations: Computation time  2x-10x compared to KS DFTComputation 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).Standard ADF fitting basis sets have to be reoptimized, to ensure adequate fits to per-orbital densities of inner orbitals (core and semi-core). The per-orbital Coulomb potentials are then computed as a sum of one-centre contributions: FitteddensityExactdensity Fit functions Basis functions Density matrix

8. 8 Curing NMR with SIC-DFT CSTC’2001, Ottawa NMR chemical shifts: 13 C RMS Error VWN9.2 BP866.6 SIC- VWN 7.1 SIC- BP86 6.6

9. 9 Curing NMR with SIC-DFT CSTC’2001, Ottawa NMR chemical shifts: 29 Si RMS Error VWN13.9 revPBE10.0 SIC- VWN 12.4 SIC- revPBE 12.0

10. 10 Curing NMR with SIC-DFT CSTC’2001, Ottawa -50 0 50 100 150 200 250 300 350 -300-200-1000100200300400 Error in calculated chemical shift, ppm Experimental 14 N, 15 N chemical shift, ppm O 2 N-N * O O 2 N * -NO (CH 3 ) 2 N-N * O VWN BP86 VWN-SIC NMR chemical shifts: 14 N, 15 N RMS Error VWN86.3 BP8669.2 SIC- VWN 21.3 SIC- BP86 17.0

11. 11 Curing NMR with SIC-DFT CSTC’2001, Ottawa -100 0 100 200 300 400 02004006008001000120014001600 Error in calculated chemical shift, ppm Experimental 17 O chemical shift, ppm H 2 CO OF 2 O 2 N-NO * O-O-O * VWN BP86 VWN-SIC NMR chemical shifts: 17 O RMS Error * VWN135.3 BP86105.3 SIC- VWN 61.2 SIC- BP86 45.6 * Excluding O 3

12. 12 Curing NMR with SIC-DFT CSTC’2001, Ottawa NMR chemical shifts: 19 F RMS Error VWN27.3 BP8620.7 SIC- VWN 14.5 SIC- BP86 12.3

13. 13 Curing NMR with SIC-DFT CSTC’2001, Ottawa NMR chemical shifts: 31 P RMS Error VWN63.8 revPBE49.0 SIC- VWN 34.8 SIC- revPBE 21.3 B3-LYP * 27.1 MP2 * 23.7 * Excludes PBr 3

14. 14 Curing NMR with SIC-DFT CSTC’2001, Ottawa 31 P: PX 3 (X=F,Cl,Br) 100 150 200 250 300 350 400 PF 3 PCl 3 PBr 3 31 P chemical shift, ppm expt VWN revPBE SIC-VWN SIC-revPBE

15. 15 Curing NMR with SIC-DFT CSTC’2001, Ottawa SIC-DFT: Uniform description of the chemical shifts RMS error/total shift range, percent

16. 16 Curing NMR with SIC-DFT CSTC’2001, Ottawa Chemical shift tensors in SIC-DFT Expt.VWNBP86 SIC- VWN  iso 201202202200 1111272290285276 2222246256253245 333384626780 Expt.VWNBP86 SIC- VWN  iso 171289 1111275327303282 2222108938996 3333-347-385-368-351

17. 17 Curing NMR with SIC-DFT 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 ( 17 O, 15 N, 31 P) 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.