Improving description of the potential energy surfaces with self interaction free density functional theory

1. 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 CSC’2001, Montréal

2. One of the fundamental assumptions of quantum chemistry is that an electron does not interactwith 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 firststeps in the development of DFT, satisfying it within model XC functionals has proven difficult. None of the approximate XCfunctionals, 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. Introduction Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

3. Energy in the external potential (Non-classical) Exchange-correlation energy Kinetic Energy (Classical) Coulomb energy 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 Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

4. 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 Kohn-Sham total energy (Classical) Coulomb self-interaction (Nonclassical) self-exchange and self-correlation 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: *: J.P. Perdew and A. Zunger, Phys. Rev. B1981, 23, 5048 Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

5. Kohn-Sham Perdew-Zunger All MOs are eigenfunctions of the same Fock operator MOs are eigenfunctions of different Fock operators 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 Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

6. Determining the exact OEP is difficult, and involves solving an integral equation on vOEP(r): SIC, OEP, and KLI-OEP 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 systems • … guarantees the correct asymptotic behavior of the potential at r  *: J.B. Krieger, Y. Li, and G.J. Iafrate, Phys. Rev. A1992, 45, 101 Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

7. 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: Fitted density Exact density Density matrix Basis functions Fit functions Implementation in ADF 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). Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

8. -0.45 Exact SIC-VWN -0.5 + SIC-revPBE E(total), Hartree -0.55 VWN revPBE -0.6 -0.65 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 R(H-H), Angstrom H2+H+H+ 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 CSC’2001, Montréal

9. + -4.85 VWN -4.9 E(total), Hartree CCSD(T)/cc-pVTZ SIC-revPBE -4.95 SIC-VWN -5 revPBE -5.05 3 0.5 1 1.5 2 2.5 R(He-He), Angstrom He2+He+He+ For many-electron systems, SIC-DFT is no longer exact. However, both local (VWN) and gradient-corrected (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 CSC’2001, Montréal

10. + + H + H2H2+H *: 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 CSC’2001, Montréal

11. + + H + HClH2+Cl *: 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 CSC’2001, Montréal

12. + + H +N2H2N2H+H2 *: 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 CSC’2001, Montréal

13. + + CH3 +H2H+ CH4 *: 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 CSC’2001, Montréal

14. OHO  OHO *: 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 CSC’2001, Montréal

15. + EB EIB EC … F- + CH3F  CH3F + F- *: S. Parthiban, G. de Oliveira, J.M.L. Martin, J. Phys. Chem. A2001, 105, 895 †: Single-point energies at optimized ab initio geometries Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

16. + EB EIB EC … Cl- + CH3Cl  CH3Cl + Cl- *: S. Parthiban, G. de Oliveira, J.M.L. Martin, J. Phys. Chem. A2001, 105, 895 †: Single-point energies at optimized ab initio geometries Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

17. + C2N4H2  2 HCN + N2 *: 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 CSC’2001, Montréal

18. Summary of the reaction barriers Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

19. 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: Gotcha! CH3 +H2H+ CH4: Frozen core All-electron 2pz 1s E(SIC) kcal/mol -126.6 +3.0 +28.8 +28.8 Potential energy surfaces with self-interaction free DFT CSC’2001, Montréal

20. 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 Summary and Outlook 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 CSC’2001, Montréal