efficient methods for computing exchange-correlation potentials for orbital-dependent functionals...
TRANSCRIPT
Efficient methods for computing exchange-correlation potentials for
orbital-dependent functionals
Viktor N. Staroverov
Department of Chemistry, The University of Western Ontario, London, Ontario, Canada
IWCSE 2013, Taiwan National University, Taipei, October 14โ17, 2013
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Orbital-dependent functionals
๐ธXC [๐ ]=โซ ๐ ( {๐๐ }) ๐ ๐ซ
โข More flexible than LDA and GGAs (can satisfy more exact constraints)
โข Needed for accurate description of molecular properties
Kohn-Sham orbitals
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Examples
โข Exact exchange
โข Hybrids (B3LYP, PBE0, etc.)
โข Meta-GGAs (TPSS, M06, etc.)
๐ธXexact [๐ ]=โ
14โ๐ , ๐=1
๐
โซ๐ ๐ซโซ๐ ๐ซ โฒ ๐๐ (๐ซ )๐ ๐โ (๐ซ )๐๐
โ (๐ซ โฒ)๐ ๐ (๐ซ โฒ ) |๐ซโ๐ซโฒ|
same expression as in the HartreeโFock theory
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The challenge
๐ฃ XC (๐ซ )=๐ฟ ๐ธX C
โ [{๐๐ }] ๐ฟ๐ (๐ซ)
=?
KohnโSham potentials corresponding to orbital-dependent functionals
cannot be evaluated in closed form
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Optimized effective potential (OEP)method
๐ฟ๐ธ totalโ
๐ฟ๐ฃ XC (๐ซ) =0
Find as the solution to the minimization problem
OEP = functional derivative of the functional
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Computing the OEP
Expand the KohnโSham orbitals:
Expand the OEP:
๐ฃ X C (๐ซ )=โ๐=1
๐
๐๐ ๐ ๐(๐ซ)
๐๐ (๐ซ )=โ๐=1
๐
๐๐๐ ๐๐(๐ซ )
Minimize the total energy with respect to {} and {}
orbital basis functions
auxiliary basis functions
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Attempts to obtain OEP-X in finite basis sets
size
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I. First approximation to the OEP:An orbital-averaged potential (OAP)
๏ฟฝฬ๏ฟฝXC๐ ๐ (๐ซ )=๐ฟ ๐ธX C
โ [{๐๐ }]
๐ฟ๐๐โ(๐ซ )
Define operator such that
The OAP is a weighted average:
๐ฃ XC (๐ซ )=โ๐=1
๐
๐๐โ (๐ซ ) ๏ฟฝฬ๏ฟฝXC๐ ๐ (๐ซ )
โ๐=1
๐
๐๐โ(๐ซ )๐๐ (๐ซ )
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Example: Slater potential
Fock exchange operator:
Slater potential:
๏ฟฝฬ๏ฟฝ ๐ ๐ (๐ซ ) โก๐ฟ๐ธ X
exact
๐ฟ๐๐โ(๐ซ )
๐ฃ S (๐ซ )= 1๐ (๐ซ ) โ๐=1
๐
๐๐โ(๐ซ) ๏ฟฝฬ๏ฟฝ ๐ ๐(๐ซ)
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Calculation of orbital-averaged potentials
โข by definition (hard, functional specific)
โข by inverting the KohnโSham equations (easy, general)
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KohnโSham inversion
๐๐ฟ
๐+๐ฃ+๐ฃH +๐ฃXC= 1
๐โ๐=1
๐
๐ ๐|๐๐|2
[โ 12โ2+๐ฃ+๐ฃH +๐ฃXC ]๐๐=๐ ๐๐ ๐
KohnโSham equations:
multiply by ,sum over i,divide by
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LDA-X potential via Kohn-Sham inversion
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PBE-XC potential via KohnโSham inversion
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A. P. Gaiduk,I. G. Ryabinkin, VNS,JCTC 9, 3959 (2013)
Removal of oscillations
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KohnโSham inversion for orbital-specific potentials
๐๐ฟ
๐+๐ฃ+๐ฃH +๐ฃXC= 1
๐โ๐=1
๐
๐ ๐|๐๐|2
[โ 12โ2+๐ฃ+๐ฃH +๏ฟฝฬ๏ฟฝXC ]๐๐=๐ ๐๐ ๐
Generalized KohnโSham equations:
same manipulations
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Example: Slater potential through KohnโSham inversion
๐ฃ S (๐ซ )=
14โ
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๐ (๐ซ ) โ๐ (๐ซ)+โ๐=1
๐
๐ ๐โจ๐ ๐โ(๐ซ )|2
๐ (๐ซ )โ๐ฃ (๐ซ ) โ๐ฃH (๐ซ)
๐=12โ๐=1
๐
ยฟโ๐ ๐โจยฟ2=๐๐ฟ+14โ2๐ ยฟ
where
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Slater potential via KohnโSham inversion
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OAPs constructed by KohnโSham inversion
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Correlation potentials via KohnโSham inversion
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KohnโSham inversion for a fixed set of HartreeโFock orbitals
๐ฃ XOEP โ๐ฃ X
model=โ๐ ๐ฟ
HF+โ๐=1
๐
๐ ๐ยฟ๐ ๐HF |2
๐HF โ๐ฃโ๐ฃHHF
Slater potential:
๐ฃ SHF=
โ๐๐ฟHF +โ
๐=1
๐
๐ ๐HF ยฟ๐ ๐
HF |2
๐HF โ๐ฃโ๐ฃHHF
But if , then
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Dependence of KS inversion on orbital energies
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II. Assumption that the OEP and HF orbitals are the same
The assumption
leads to the eigenvalue-consistent orbital-averaged potential (ECOAP)
๐๐=๐๐HF
๐ฃ XECOAP=๐ฃS
HF + 1๐HF โ
๐=1
๐
(๐ ๐โ๐ยฟยฟ ๐HF)|๐ ๐HF|2 ยฟ
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ECOAP KLI LHF
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Calculated exact-exchange (EXX) energies
, mEh
KLI ELP=LHF=CEDA ECOAP
m.a.v. 2.88 2.84 2.47
Sample: 12 atoms from He to BaBasis set: UGBS
A. A. Kananenka, S. V. Kohut, A. P. Gaiduk, I. G. Ryabinkin, VNS, JCP 139, 074112 (2013)
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III. HartreeโFock exchange-correlation (HFXC) potential
An HFXC potential is the which reproduces a HF density within the KohnโSham scheme:
๐ (๐ซ )=โ๐=1
๐
|๐๐ (๐ซ )|2=ยฟโ
๐=1
๐
|๐ ๐HF (๐ซ )|2
=๐HF (๐ซ )ยฟ
[โ 12โ2+๐ฃ (๐ซ )+๐ฃH (๐ซ )+๐ฃXC (๐ซ )]๐ ๐(๐ซ )=๐๐๐๐ (๐ซ )
That is, is such that
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Inverting the KohnโSham equations
๐๐ฟ
๐+๐ฃ+๐ฃH +๐ฃXC= 1
๐โ๐=1
๐
๐ ๐|๐๐|2
[โ 12โ2+๐ฃ+๐ฃH +๐ฃXC ]๐๐=๐ ๐๐ ๐
KohnโSham equations:
local ionizationpotential
multiply by ,sum over i,divide by
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Inverting the HartreeโFock equations
๐ ๐ฟHF
๐HF +๐ฃ+๐ฃH +๐ฃSHF= 1
๐HF โ๐=1
๐
๐๐HF|๐ ๐
HF|2
HartreeโFock equations:
Slater potential builtwith HF orbitals
[โ 12โ2+๐ฃ+๐ฃH +๐พ ]๐๐
HF=๐ ๐HF ๐๐
HF
same manipulations
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Closed-form expression for the HFXC potential
๐ฃ XCHF=๐ฃS
HF + 1๐โ๐=1
๐
๐ ๐โจ๐ ๐ |2 โ
1๐HF โ
๐=1
๐
๐ ๐HF|๐๐
HF|2+ ๐
HF
๐HF โ๐๐
, but , , and
We treat this expression as a model potential within the KohnโSham SCF scheme.
Here
Computational cost: same as KLI and BeckeโJohnson (BJ)
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HFXC potentials are practically exact OEPs!
Numerical OEP: Engel et al.
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HFXC potentials can be easily computed for molecules
Numerical OEP: Makmal et al.
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Energies from exchange potentials
, mEh
KLI LHF BJ Basis-set OEP HFXC
m.a.v. 1.74 1.66 5.30 0.12 0.05
Sample: 12 atoms from Li to CdBasis set for LHF, BJ, OEP (aux=orb), HFXC: UGBS KLI and true OEP values are from Engel et al.
I. G. Ryabinkin, A. A. Kananenka, VNS, PRL 139, 013001 (2013)
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Virial energy discrepancies
, mEh
KLI LHF BJ Basis-set OEP HFXC
m.a.v. 438.0 629.2 1234.1 1.76 2.76
where
๐ธ vir= โซ ๐ฃX (๐ซ ) [3 ๐ (๐ซ )+๐ซ โ โ ๐ (๐ซ) ]๐๐ซ
For exact OEPs,
๐ธ vir โ๐ธEXX=0 ,
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HFXC potentials in finite basis sets
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Hierarchy of approximations to the EXX potential
๐ฃ Xโ=๐ฃS
HF + 1๐HF โ
๐=1
๐
(๐ ๐โ๐๐HF )|๐๐
HF|2+ ๐
HF โ๐๐HF
OAP
ECOAP
HFXC
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Summary
โข Orbital-averaged potentials (e.g., Slater) can be constructed by KohnโSham inversion
โข Hierarchy or approximations to the OEP: OAP (Slater) < ECOAP < HFXC
โข ECOAP Slater potential KLI LHF
โข HFXC potential OEP
โข Same applies to all occupied-orbital functionals
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Acknowledgments
โข Eberhard Engelโข Leeor Kronik
for OEP benchmarks