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).