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 14β17, 2013
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Orbital-dependent functionals
πΈXC [π ]=β« π ( {ππ }) π π«
β’ More flexible than LDA and GGAs (can satisfy more exact constraints)
β’ Needed for accurate description of molecular properties
Kohn-Sham orbitals
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Examples
β’ Exact exchange
β’ Hybrids (B3LYP, PBE0, etc.)
β’ Meta-GGAs (TPSS, M06, etc.)
πΈXexact [π ]=β
14βπ , π=1
π
β«π π«β«π π« β² ππ (π« )π πβ (π« )ππ
β (π« β²)π π (π« β² ) |π«βπ«β²|
same expression as in the HartreeβFock theory
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The challenge
π£ XC (π« )=πΏ πΈX C
β [{ππ }] πΏπ (π«)
=?
KohnβSham potentials corresponding to orbital-dependent functionals
cannot be evaluated in closed form
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Optimized effective potential (OEP)method
πΏπΈ totalβ
πΏπ£ XC (π«) =0
Find as the solution to the minimization problem
OEP = functional derivative of the functional
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Computing the OEP
Expand the KohnβSham orbitals:
Expand the OEP:
π£ X C (π« )=βπ=1
π
ππ π π(π«)
ππ (π« )=βπ=1
π
πππ ππ(π« )
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)
οΏ½ΜοΏ½XCπ π (π« )=πΏ πΈX C
β [{ππ }]
πΏππβ(π« )
Define operator such that
The OAP is a weighted average:
π£ XC (π« )=βπ=1
π
ππβ (π« ) οΏ½ΜοΏ½XCπ π (π« )
βπ=1
π
ππβ(π« )ππ (π« )
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Example: Slater potential
Fock exchange operator:
Slater potential:
οΏ½ΜοΏ½ π π (π« ) β‘πΏπΈ X
exact
πΏππβ(π« )
π£ S (π« )= 1π (π« ) βπ=1
π
ππβ(π«) οΏ½ΜοΏ½ π π(π«)
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Calculation of orbital-averaged potentials
β’ by definition (hard, functional specific)
β’ by inverting the KohnβSham equations (easy, general)
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KohnβSham inversion
ππΏ
π+π£+π£H +π£XC= 1
πβπ=1
π
π π|ππ|2
[β 12β2+π£+π£H +π£XC ]ππ=π ππ π
KohnβSham 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 KohnβSham inversion
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A. P. Gaiduk,I. G. Ryabinkin, VNS,JCTC 9, 3959 (2013)
Removal of oscillations
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KohnβSham inversion for orbital-specific potentials
ππΏ
π+π£+π£H +π£XC= 1
πβπ=1
π
π π|ππ|2
[β 12β2+π£+π£H +οΏ½ΜοΏ½XC ]ππ=π ππ π
Generalized KohnβSham equations:
same manipulations
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Example: Slater potential through KohnβSham inversion
π£ S (π« )=
14β
2
π (π« ) βπ (π«)+βπ=1
π
π πβ¨π πβ(π« )|2
π (π« )βπ£ (π« ) βπ£H (π«)
π=12βπ=1
π
ΒΏβπ πβ¨ΒΏ2=ππΏ+14β2π ΒΏ
where
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Slater potential via KohnβSham inversion
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OAPs constructed by KohnβSham inversion
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Correlation potentials via KohnβSham inversion
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KohnβSham inversion for a fixed set of HartreeβFock orbitals
π£ XOEP βπ£ X
model=βπ πΏ
HF+βπ=1
π
π πΒΏπ πHF |2
πHF βπ£βπ£HHF
Slater potential:
π£ SHF=
βππΏHF +β
π=1
π
π πHF ΒΏπ π
HF |2
πHF βπ£βπ£HHF
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)
ππ=ππHF
π£ XECOAP=π£S
HF + 1πHF β
π=1
π
(π πβπΒΏΒΏ πHF)|π πHF|2 ΒΏ
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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. HartreeβFock exchange-correlation (HFXC) potential
An HFXC potential is the which reproduces a HF density within the KohnβSham scheme:
π (π« )=βπ=1
π
|ππ (π« )|2=ΒΏβ
π=1
π
|π πHF (π« )|2
=πHF (π« )ΒΏ
[β 12β2+π£ (π« )+π£H (π« )+π£XC (π« )]π π(π« )=ππππ (π« )
That is, is such that
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Inverting the KohnβSham equations
ππΏ
π+π£+π£H +π£XC= 1
πβπ=1
π
π π|ππ|2
[β 12β2+π£+π£H +π£XC ]ππ=π ππ π
KohnβSham equations:
local ionizationpotential
multiply by ,sum over i,divide by
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Inverting the HartreeβFock equations
π πΏHF
πHF +π£+π£H +π£SHF= 1
πHF βπ=1
π
ππHF|π π
HF|2
HartreeβFock equations:
Slater potential builtwith HF orbitals
[β 12β2+π£+π£H +πΎ ]ππ
HF=π πHF ππ
HF
same manipulations
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Closed-form expression for the HFXC potential
π£ XCHF=π£S
HF + 1πβπ=1
π
π πβ¨π π |2 β
1πHF β
π=1
π
π πHF|ππ
HF|2+ π
HF
πHF βππ
, but , , and
We treat this expression as a model potential within the KohnβSham SCF scheme.
Here
Computational cost: same as KLI and BeckeβJohnson (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
πΈ vir= β« π£X (π« ) [3 π (π« )+π« β β π (π«) ]ππ«
For exact OEPs,
πΈ vir βπΈEXX=0 ,
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HFXC potentials in finite basis sets
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Hierarchy of approximations to the EXX potential
π£ Xβ=π£S
HF + 1πHF β
π=1
π
(π πβππHF )|ππ
HF|2+ π
HF βππHF
OAP
ECOAP
HFXC
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Summary
β’ Orbital-averaged potentials (e.g., Slater) can be constructed by KohnβSham inversion
β’ Hierarchy or approximations to the OEP: OAP (Slater) < ECOAP < HFXC
β’ ECOAP Slater potential KLI LHF
β’ HFXC potential OEP
β’ Same applies to all occupied-orbital functionals
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Acknowledgments
β’ Eberhard Engelβ’ Leeor Kronik
for OEP benchmarks