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O rbital-Corrected O rbital-Free D ensity F unctional T heory ( OO-DFT )

Zhou & Wang, J. Chem. Phys. , 124 , 081107 (2006). Zhou & Wang, Int. J. Quantum Chem. , in press (2007). Zhou & Wang, J. Chem. Phys. , in press (2007). Zhou & Wang, J. Chem. Phys. , submitted (2007). Principal Coworker Dr. Baojing Zhou. $$$ from NSERC. Yan Alexander Wang.

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O rbital-Corrected O rbital-Free D ensity F unctional T heory ( OO-DFT )

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  1. Zhou & Wang, J. Chem. Phys., 124, 081107 (2006). • Zhou & Wang, Int. J. Quantum Chem., in press (2007). • Zhou & Wang, J. Chem. Phys., in press (2007). • Zhou & Wang, J. Chem. Phys., submitted (2007). Principal Coworker Dr. Baojing Zhou • $$$ from NSERC Yan Alexander Wang Orbital-Corrected Orbital-Free Density Functional Theory (OO-DFT) http://www.chem.ubc.ca/faculty/wang Department of Chemistry, University of British Columbia 3 August 2007 ● 2007 IMA Summer Program ● University of Minnesota

  2. Density Functional Theory (DFT) • Kohn-Sham (KS) DFT • Orbital-Free (OF) DFT • Linear-Scaling OF-DFT • Kinetic energy density functional (KEDF) • Local pseudopotential from a bulk environment (BLPS) • Applications of OF-DFT: Si, fcc Ag • Orbital-CorrectedOF-DFT(OO-DFT) • The formulation • Application ofOO-DFT: cubic-diamond (CD) Si, fcc Ag, CD Si vacancy & (100) surface Outline

  3. Density-Functional Theory (DFT) • Total energy as a functional of • HK theorems legitimize as the basic variational variable • Thomas-Fermi-Hohenberg-Kohn (TFHK) equation • Drawback: exact unknown Hohenberg & Kohn, Phys. Rev. 136, B864 (1964).

  4. Orbital-Based Kohn-Sham (KS) Scheme • the KS KEDF • KS equation: • KS effective potential: External potential • scaling due to: Hartree potential • Brillouin-zone sampling • Orbital orthonormalization • = the number of electrons Exchange-correlation potential Kohn & Sham, Phys. Rev. 140, A1133 (1965).

  5. Condensed matter solution to is DFT. • KS-DFT methodis not able to simulate systems containing more than 1000atoms due to the scaling • Linear [] scaling OF-DFT is significantly faster • allowsproperties of>1000 atoms of nearly-free-electron-like metals to be accurately(ca.meV/atom)predicted with DFT Migration of two types of grain boundaries at 300 K in crystalline Nacontaining6714 atoms simulated by OF-DFT for 1.6 ps. Motivation for Linear-Scaling DFT Method: OF-DFT Watson & Madden, PhysChemComm 1, 1 (1998).

  6. OF-DFT avoids bottlenecks present in KS-DFT • No orbital orthonormalization • Periodic systems: No Brillouin-zone (k-point) integration • The kinetic energy • Valence-(core + nuclear) “external” attraction energy(LPS) Linear-Scaling OF-DFT • Two terms in energy pose difficulties without orbitals • Linear scaling is achieved by employingFFTto calculate kinetic, Hartree, and external energies in reciprocal space. • Exchange-correlation energy naturally short-ranged, so

  7. Popular KEDFs • The Thomas-FermiKEDF(exact for uniform electron gas) • The von Weizsäcker KEDF(exact for one-orbital system) • KEDF based on the conventional gradient expansion (CGE) • Defects:these do not exhibit the quantum mechanical idempotency property (required for all physically allowed densities) norcorrect linear-response behavior Wang & Carter, in Theoretical Methods in Condensed Phase Chemistry, edited by S. D. Schwartz (Springer, New York, 2002), p. 117.

  8. First-Principles Local Pseudopotential (LPS) • OF-DFT can only use LPSs, , because more accurate non-local pseudopotentials (NLPSs), , involve projections onto orbitals. • Goal: design a LPS that reproduces the KS NLPS density • Strategy: exploit the first HK theorem: • Path : invert the KS equations via the Wang-Parr approach to obtain the KS effective potential • Then, the Hartree and exchange-correlation potentials are removed from the KS effective potential to obtain a global LPS: • Implementation in both atomic (ALPS) and bulk environments (BLPS). For the latter, the globalLPS further decomposed to obtain an atom-centeredBLPS • Wang & Parr, Phys. Rev. A 47, R1591 (1993). • Zhou, Wang & Carter, Phys. Rev. B 69, 125109 (2004).

  9. Pseudopotentials for Si: NLPS vs. ALPS vs. BLPS • The LPSs become much more repulsive near the core ( Bohr) to force the higher angular momentumelectrons out of the core region. • The BLPS is significantly more repulsive than the ALPSin the core region. • Real-space PS for Si • NLPS (green) • ALPS (red) • BLPS (blue) • Coulomb potential (cyan) Si Zhou, Wang & Carter, Phys. Rev. B 69, 125109 (2004).

  10. OF-DFT/BLPS for Si: Total Energies • WGC KEDF improves upon the WT KEDF. • Covalent CD Si: significant deviations using OF-DFT due to localized bonding. • Metallic fcc Si: shape of the EOS from OF-DFT very close to that from KS-DFT. Zhou, Ligneres & Carter, J. Chem. Phys. 122, 044103 (2005).

  11. Density along the diagonal of the (001) plane • Equation of state (EOS) OF-DFT/BLPS for the fcc Ag: Approximate KEDFs • WGC KEDFnot used due to convergence problems • OF density using is better than using WT KEDF • Unacceptable errors due to KEDF exist in the EOS from OF-DFTdue to strongly localized d-electronsnew KEDFs needed Zhou & Carter, J. Chem. Phys. 122, 184108 (2005).

  12. KS-DFT: , but still expensive, SCF iterations • OF-DFT: , lack accurate KEDF and LPS • LPS: not transferable enough, less accurate than NLPS • withoutexpensive many SCF KS iterations • , without the limits of accurate KEDF and LPS • able to use NLPS, not only LPS Orbital-Corrected OF-DFT (OO-DFT) • Most of first-principles Quantum Mechanical methods do not scale linearly • Bottlenecks of Density Functional Theory(DFT)calculations • Goals: (1) Apply DFTtolarge systems (>1000atoms) (2) Combine the merits of OF-DFT and KS-DFT Zhou & Wang, J. Chem. Phys. 124, 081107(2006).

  13. from OF-DFT • Singlenon-self-consistent (NSC) KS iteration • : local (LPS) or nonlocal (NLPS) Essence of OO-DFT • KS effective potential from • from OO-DFT Zhou & Wang, J. Chem. Phys. 124, 081107(2006).

  14. Hohenberg-Kohn-Sham (HKS) functional • We propose the Zhou-Wang- (ZW) functional • : interpolation parameter • Value ofdepends on and • Allows systematic error cancellation Total Energies in OO-DFT • Harris functional • Usually, not always, • Chelikowsky & Louie, Phys. Rev. B 29, 3470 (1984). • Harris, Phys. Rev. B 31, 1770 (1985). • Zhou & Wang, J. Chem. Phys. 124, 081107 (2006).

  15. Value of depends on the chemical environment, not very sensitive to the size of the system. Rational of the Zhou-Wang- Functional • Linear-response theory and functional expansion linear-response kernel • Exact linear interpolation IF ??? • The Zhou-Wang- (ZW) functional • Finnis, J. Phys.: Condens. Matter 2, 331 (1990). • Zhou & Wang, J. Chem. Phys. 124, 081107 (2005).

  16. Test the ZWl functional: Total Energies of CD Si • OF-DFT with less optimal KEDF is completely wrong! • better than • Optimal : 0.28 for the BLPS; 0.30for the NLPS • The ZWfunctional≈KS-DFT, even for NLPS! Zhou & Wang, J. Chem. Phys. 124, 081107(2006).

  17. Total energies: Test the ZWl functional: fcc Ag (OO1 vs. OO2) • 1st NSCKS iteration is not enough for chemical accuracy • 2ndNSC KS iteration • OO1= 1stNSC KS iteration OO2 =2ndNSC KS iteration Zhou & Wang, J. Chem. Phys. 124, 081107(2006).

  18. OO1 • OO2 • BLPS: better • NLPS: better • Optimal : • 0.39for the BLPS • 0.65 for the NLPS • much better • KS-DFT well reproduced • Optimal : • 0.41 for the BLPS • 0.58 for the NLPS • ≈ KS-DFT Test the ZWl functional: Total Energies of fcc Ag Zhou & Wang, J. Chem. Phys. 124, 081107(2006).

  19. Summary and Conclusions (I) • Require only a single or doubleNSCKS-DFT iterations • OO-DFT remediesdrawbacks of OF-DFT • OO-DFT islinear-scalingand can handle large systems (>1000 atoms) • The ZW functional ≈ KS-DFT!!! • Two irrevocable factors for the success of OO-DFT • High-quality from the state-of-the-art OF-DFT • The built-in systematic error cancellation in the ZW functional • Ab-initio OO-DFT molecular dynamics (OOMD) is coming! • Combining CPMD with BOMD • Car & Parrinello, Phys. Rev. Lett. 55, 2471 (1985). • Radeke & Carter, Annu. Rev. Phys. Chem.48, 243 (1997). • Strutinsky, Nucl. Phys. A 122, 1 (1968); Yannouleas et al., Phys. Rev. B 57, 4872 (1998); Ullmo et al., ibid.63, 125339 (2001); Zhou & Wang, J. Chem. Phys., in press (2007). • Benoit et al., Phys. Rev. Lett. 87, 226401 (2001); Zhu & Trickey, IJQC100, 245 (2004).

  20. Dynamical Prediction of  • Errors in EHKS and EHarris to ()2 • Linear-response kernel • Interpolation scheme • Finnis, J. Phys.: Condens. Matter2, 331 (1990). • Zhou & Wang, J. Chem.Phys., in press & submitted (2007).

  21. Good Guess for the exact KS density • At 1st iteration (enough for nonmetallic main-group materials) • infrom OF-DFT • outfrom OO1 • Kerker’smatrix used for density mixing • At 2nd iteration, Pulay’sDIISmethod used for density mixing (enough for transition metals with localized d-electrons) • Zhou & Wang, J. Chem. Phys., in press (2007). • Zhou & Wang, J. Chem. Phys., submitted (2007). • Kresse & Furthmüller, Comput. Mater. Sci.6, 15 (1996). • Kerker, Phys. Rev. B23, 3082 (1981).

  22. Various Improved Total Energies • Corrected HKS & Harris functionals: cHKS & cHarris • Interpolation schemes: ZW& WZ • Strutinsky shell correction model: SCM • Strutinsky, Nucl. Phys. A 122, 1 (1968). • Yannouleas et al., Phys. Rev. B 57, 4872 (1998). • Ullmo et al., Phys. Rev. B63, 125339 (2001). • Zhou & Wang, J. Chem. Phys., in press (2007). • Numerical equivalencies • Zhou & Wang, Int. J. Quantum Chem., in press (2007). • Zhou & Wang, J. Chem. Phys., submitted (2007).

  23. Test theDynamically Determined : Total Energies Cubic diamond (CD) • in from OF-DFT • only 1 iteration • OO1: Zhou & Wang, J. Chem. Phys., in press (2007).

  24. OO2: Test theDynamically Determined : Total Energies (fcc) OO1 • in from OF-DFT • only 2 iterations Zhou & Wang, J. Chem. Phys., in press (2007).

  25. Test theDynamically Determined :CD Si Vacancy • Atomic densities • as initial guess • 1~2 magnitudes more accurate than conventional schemes. Zhou & Wang, J. Chem. Phys., submitted (2007).

  26. Test theDynamically Determined :CD Si (100) Surface • Atomic densities • as initial guess • 1~2 magnitudes more accurate than conventional schemes. Zhou & Wang, J. Chem. Phys., submitted (2007).

  27. Alkali metals • Alkaline earth metals • Poor metals • H • Nonmetals • Nobel gases • Transition metals • Lanthanide series • Actinide series Summary and Conclusions: the O3 Paradigm • O1: OF-DFT (Orbital-Free Density Functional Theory) • O2: OO1(OF-DFT + 1-iteration OO-DFT, a priori & a posteriori) • O3: OO2(OF-DFT + 2-iteration OO-DFT, a priori & a posteriori) Wang & Carter, in Theoretical Methods in Condensed Phase Chemistry, edited by S. D. Schwartz (Springer, New York, 2002), p. 117.

  28. Alkali metals • Alkaline earth metals • Poor metals • H • Nonmetals • Nobel gases • Transition metals • Lanthanide series • Actinide series Summary and Conclusions: the O3 Paradigm • O1: OF-DFT (Orbital-Free Density Functional Theory) • O2: OO1(OF-DFT + 1-iteration OO-DFT, a priori & a posteriori) • O3: OO2(OF-DFT + 2-iteration OO-DFT, a priori & a posteriori) • Zhou & Wang, J. Chem. Phys., 124, 081107 (2006). • Zhou & Wang, Int. J. Quantum Chem., in press (2007). • Zhou & Wang, J. Chem. Phys., in press & submitted (2007).

  29. Alkali metals • Alkaline earth metals • Poor metals • H • Nonmetals • Nobel gases • Transition metals • Lanthanide series • Actinide series Summary and Conclusions: the O3 Paradigm • O1: OF-DFT (Orbital-Free Density Functional Theory) • O2: OO1(OF-DFT + 1-iteration OO-DFT, a priori & a posteriori) • O3: OO2(OF-DFT + 2-iteration OO-DFT, a priori & a posteriori) • Zhou & Wang, J. Chem. Phys., 124, 081107 (2006). • Zhou & Wang, Int. J. Quantum Chem., in press (2007). • Zhou & Wang, J. Chem. Phys., in press & submitted (2007).

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