orbital-corrected orbital-free density functional theory (oo-dft) department of chemistry,...
DESCRIPTION
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).TRANSCRIPT
Orbital-Corrected Orbital-Free Density Functional Theory (OO-DFT)
Department of Chemistry, University of British Columbia Yan Alexander Wang
3 August 2007 ● 2007 IMA Summer Program ● University of Minnesota
http://www.chem.ubc.ca/faculty/wang
Principal CoworkerDr. Baojing Zhou
$$$ from NSERC
• 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).
Outline Density Functional Theory (DFT)
Kohn-Sham (KS) DFT Orbital-Free (OF) DFT
Orbital-Corrected OF-DFT (OO-DFT) The formulation Application of OO-DFT: cubic-diamond (CD) Si, fcc
Ag, CD Si vacancy & (100) surface
Linear-Scaling OF-DFT Kinetic energy density functional (KEDF) Local pseudopotential from a bulk environment (BLPS) Applications of OF-DFT: Si, fcc Ag
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).
Orbital-Based Kohn-Sham (KS) Scheme
External potential
Hartree potential
Exchange-correlation potential
the KS KEDF
KS equation:
KS effective potential:
Brillouin-zone sampling Orbital orthonormalization
= the number of electrons
scaling due to:
Kohn & Sham, Phys. Rev. 140, A1133 (1965).
Linear [ ] scaling OF-DFT is significantly faster
allows properties of >1000 atoms of nearly-free-electron-like metals to be accurately (ca. meV/atom) predicted with DFT
Motivation for Linear-Scaling DFT Method: OF-DFT
Migration of two types of grain boundaries at 300 K in crystalline Na containing 6714 atoms simulated by OF-DFT for 1.6 ps.
Watson & Madden, PhysChemComm 1, 1 (1998).
KS-DFT method is not able to simulate systems containing more than 1000 atoms due to the scaling
Condensed matter solution to is DFT.
Linear-Scaling OF-DFT
Two terms in energy pose difficulties without orbitals
OF-DFT avoids bottlenecks present in KS-DFT No orbital orthonormalization Periodic systems: No Brillouin-zone (k-point) integration
Linear scaling is achieved by employing FFT to calculate kinetic, Hartree, and external energies in reciprocal space.
Exchange-correlation energy naturally short-ranged, so
The kinetic energy Valence-(core + nuclear) “external” attraction energy (LPS)
Popular KEDFs The Thomas-Fermi KEDF (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) nor correct linear-response behavior
Wang & Carter, in Theoretical Methods in Condensed Phase Chemistry, edited by S. D. Schwartz (Springer, New York, 2002), p. 117.
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 global LPS further decomposed to obtain an atom-centered BLPS
First-Principles Local Pseudopotential (LPS)
• Wang & Parr, Phys. Rev. A 47, R1591 (1993).• Zhou, Wang & Carter, Phys. Rev. B 69, 125109 (2004).
Pseudopotentials for Si: NLPS vs. ALPS vs. BLPS
The LPSs become much more repulsive near the core ( Bohr) to force the higher angular momentum electrons out of the core region.
The BLPS is significantly
more repulsive than the
ALPS in 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).
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).
OF-DFT/BLPS for the fcc Ag: Approximate KEDFs
WGC KEDF not 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-DFT
due to strongly localized d-electrons new KEDFs needed
Zhou & Carter, J. Chem. Phys. 122, 184108 (2005).
Equation of state (EOS) Density along the diagonal of the (001) plane
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 DFT to large systems (>1000 atoms) (2) Combine the merits of OF-DFT and KS-DFT
Zhou & Wang, J. Chem. Phys. 124, 081107 (2006).
KS-DFT: , but still expensive, SCF iterations OF-DFT: , lack accurate KEDF and LPS LPS: not transferable enough, less accurate than NLPS
without expensive many SCF KS iterations , without the limits of accurate KEDF and LPS able to use NLPS, not only LPS
Essence of OO-DFT from OF-DFT
KS effective potential from
Single non-self-consistent (NSC) KS iteration
: local (LPS) or nonlocal (NLPS)
from OO-DFT
Zhou & Wang, J. Chem. Phys. 124, 081107 (2006).
Total Energies in OO-DFT
We propose the Zhou-Wang- (ZW) functional
: interpolation parameter Value of depends on and Allows systematic error cancellation
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).
Hohenberg-Kohn-Sham (HKS) functional
Rational of the Zhou-Wang- Functional
The Zhou-Wang- (ZW) functional
Exact linear interpolation
Linear-response theory and functional expansion
linear-response kernel
Value of depends on the chemical environment, not very sensitive to the size of the system.
• Finnis, J. Phys.: Condens. Matter 2, 331 (1990).• Zhou & Wang, J. Chem. Phys. 124, 081107 (2005).
IF ???
OF-DFT with less optimal KEDF is completely wrong! better than Optimal : 0.28 for the BLPS; 0.30 for the NLPS The ZW functional ≈ KS-DFT, even for NLPS!
Zhou & Wang, J. Chem. Phys. 124, 081107 (2006).
Test the ZW functional: Total Energies of CD Si
Test the ZW functional: fcc Ag (OO1 vs. OO2)
OO1 = 1st NSC KS iteration OO2 = 2nd NSC KS iteration
1st NSC KS iteration is not enough for chemical accuracy
2nd NSC KS iteration
Total energies:
Zhou & Wang, J. Chem. Phys. 124, 081107 (2006).
OO1 BLPS: better NLPS: better Optimal :
0.39 for the BLPS0.65 for the NLPS
much better
KS-DFT well reproduced Optimal :
0.41 for the BLPS0.58 for the NLPS
≈ KS-DFT
OO2
Zhou & Wang, J. Chem. Phys. 124, 081107 (2006).
Test the ZW functional: Total Energies of fcc Ag
Summary and Conclusions (I) OO-DFT remedies drawbacks of OF-DFT
The ZW functional ≈ KS-DFT !!!
OO-DFT is linear-scaling and can handle large systems (>1000 atoms)
Ab-initio OO-DFT molecular dynamics (OOMD) is coming!
Require only a single or double NSC KS-DFT iterations
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
• 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, IJQC 100, 245 (2004).
Combining CPMD with BOMD• Car & Parrinello, Phys. Rev. Lett. 55, 2471 (1985).• Radeke & Carter, Annu. Rev. Phys. Chem. 48, 243 (1997).
Dynamical Prediction of
• Finnis, J. Phys.: Condens. Matter 2, 331 (1990).• Zhou & Wang, J. Chem. Phys., in press & submitted (2007).
Interpolation scheme
Errors in EHKS and EHarris to ()2
Linear-response kernel
Good Guess for the exact KS density
At 2nd iteration, Pulay’s DIIS method used for density mixing (enough for transition metals with localized d-electrons)
in from OF-DFT
At 1st iteration (enough for nonmetallic main-group materials)
Kerker’s matrix used for density mixing
• Kresse & Furthmüller, Comput. Mater. Sci. 6, 15 (1996).• Kerker, Phys. Rev. B 23, 3082 (1981).
out from OO1
• Zhou & Wang, J. Chem. Phys., in press (2007).• Zhou & Wang, J. Chem. Phys., submitted (2007).
Various Improved Total Energies
• Zhou & Wang, Int. J. Quantum Chem., in press (2007).• Zhou & Wang, J. Chem. Phys., submitted (2007).
Interpolation schemes: ZW & WZ
Corrected HKS & Harris functionals: cHKS & cHarris
Strutinsky shell correction model: SCM
Numerical equivalencies
• Strutinsky, Nucl. Phys. A 122, 1 (1968).• Yannouleas et al., Phys. Rev. B 57, 4872 (1998).• Ullmo et al., Phys. Rev. B 63, 125339 (2001).• Zhou & Wang, J. Chem. Phys., in press (2007).
Test the Dynamically Determined : Total Energies
Cubic diamond (CD)
in from OF-DFT
only 1 iteration
Zhou & Wang, J. Chem. Phys., in press (2007).
OO1:
Test the Dynamically Determined : Total Energies
(fcc)
in from OF-DFT
only 2 iterations
Zhou & Wang, J. Chem. Phys., in press (2007).
OO1
OO2:
Test the Dynamically Determined : CD Si Vacancy
1~2 magnitudes more accurate than conventional schemes.
Zhou & Wang, J. Chem. Phys., submitted (2007).
Atomic densities as initial guess
Test the Dynamically Determined : CD Si (100) Surface
1~2 magnitudes more accurate than conventional schemes.
Atomic densities as initial guess
Zhou & Wang, J. Chem. Phys., submitted (2007).
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)
Alkali metals Alkaline earth metals Poor metals
H Nonmetals Nobel gases
Transition metals Lanthanide series Actinide series
Wang & Carter, in Theoretical Methods in Condensed Phase Chemistry, edited by S. D. Schwartz (Springer, New York, 2002), p. 117.
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)
Alkali metals Alkaline earth metals Poor metals
H Nonmetals Nobel gases
Transition metals Lanthanide series Actinide series
Summary and Conclusions: the O3 Paradigm
• 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).
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)
Alkali metals Alkaline earth metals Poor metals
H Nonmetals Nobel gases
Transition metals Lanthanide series Actinide series
• 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).