Efficient methods for computing exchange-correlation potentials for
orbital-dependent functionalsViktor N. Staroverov
Department of Chemistry, The University of Western Ontario, London, Ontario, Canada
IWCSE 2013, Taiwan National University, Taipei, October 14‒17, 2013
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Orbital-dependent functionals
𝐸XC [𝜌 ]=∫ 𝑓 ( {𝜙𝑖 }) 𝑑 𝐫
• More flexible than LDA and GGAs (can satisfy more exact constraints)
• Needed for accurate description of molecular properties
Kohn-Sham orbitals
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Examples
• Exact exchange
• Hybrids (B3LYP, PBE0, etc.)
• Meta-GGAs (TPSS, M06, etc.)
𝐸Xexact [𝜌 ]=− 1
4 ∑𝑖 , 𝑗=1
𝑁
∫𝑑 𝐫∫𝑑 𝐫 ′ 𝜑𝑖 (𝐫 )𝜑 𝑗∗ (𝐫 )𝜑𝑖
∗ (𝐫 ′ )𝜑 𝑗 (𝐫 ′ ) |𝐫−𝐫′|
same expression as in the Hartree‒Fock theory
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The challenge
𝑣 XC (𝐫 )=𝛿 𝐸X C
❑ [{𝜙𝑖 }] 𝛿𝜌 (𝐫 )
=?
Kohn‒Sham potentials corresponding to orbital-dependent functionals
cannot be evaluated in closed form
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Optimized effective potential (OEP)method
𝛿𝐸 total❑
𝛿𝑣 XC (𝐫) =0
Find as the solution to the minimization problem
OEP = functional derivative of the functional
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Computing the OEP
Expand the Kohn‒Sham orbitals:
Expand the OEP:
𝑣 X C (𝐫 )=∑𝑘=1
𝑚
𝑏𝑘 𝑓 𝑘(𝐫)
𝜙𝑖 (𝐫 )=∑𝑘=1
𝑛
𝑐𝑘𝑖 𝜒𝑘(𝐫 )
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)
�̂�XC𝜙 𝑖 (𝐫 )=𝛿 𝐸X C
❑ [{𝜙𝑖 }] 𝛿𝜙𝑖
∗(𝐫 )
Define operator such that
The OAP is a weighted average:
𝑣 XC (𝐫 )=∑𝑖=1
𝑁
𝜙𝑖∗ (𝐫 ) �̂�XC𝜙 𝑖 (𝐫 )
∑𝑖=1
𝑁
𝜙𝑖∗(𝐫 )𝜙𝑖 (𝐫 )
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Example: Slater potential
Fock exchange operator:
Slater potential:
�̂� 𝜙 𝑖 (𝐫 ) ≡𝛿𝐸 X
exact
𝛿𝜙𝑖∗(𝐫 )
𝑣 S (𝐫 )= 1𝜌 (𝐫 ) ∑𝑖=1
𝑁
𝜙𝑖∗(𝐫) �̂� 𝜙 𝑖(𝐫)
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Calculation of orbital-averaged potentials
• by definition (hard, functional specific)
• by inverting the Kohn‒Sham equations (easy, general)
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Kohn‒Sham inversion
𝜏𝐿
𝜌 +𝑣+𝑣H +𝑣XC= 1𝜌∑𝑖=1
𝑁
𝜖 𝑖|𝜙𝑖|2
[− 12∇2+𝑣+𝑣H +𝑣XC ]𝜙𝑖=𝜖 𝑖𝜙 𝑖
Kohn‒Sham 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 Kohn‒Sham inversion
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A. P. Gaiduk,I. G. Ryabinkin, VNS,JCTC 9, 3959 (2013)
Removal of oscillations
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Kohn‒Sham inversion for orbital-specific potentials
𝜏𝐿
𝜌 +𝑣+𝑣H +𝑣XC= 1𝜌∑𝑖=1
𝑁
𝜖 𝑖|𝜙𝑖|2
[− 12∇2+𝑣+𝑣H +�̂�XC ]𝜙𝑖=𝜖 𝑖𝜙 𝑖
Generalized Kohn‒Sham equations:
same manipulations
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Example: Slater potential through Kohn‒Sham inversion
𝑣 S (𝐫 )=
14∇
2
𝜌 (𝐫 ) −𝜏 (𝐫 )+∑𝑖=1
𝑁
𝜖 𝑖∨𝜙 𝑖❑(𝐫 )|2
𝜌 (𝐫 )−𝑣 (𝐫 ) −𝑣H (𝐫)
𝜏=12∑𝑖=1
𝑁
¿∇𝜙 𝑖∨¿2=𝜏𝐿+14∇2𝜌 ¿
where
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Slater potential via Kohn‒Sham inversion
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OAPs constructed by Kohn‒Sham inversion
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Correlation potentials via Kohn‒Sham inversion
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Kohn‒Sham inversion for a fixed set of Hartree‒Fock orbitals
𝑣 XOEP ≈𝑣 X
model=−𝜏 𝐿
HF+∑𝑖=1
𝑁
𝜖 𝑖¿𝜙 𝑖HF |2
𝜌HF −𝑣−𝑣HHF
Slater potential:
𝑣 SHF=
−𝜏𝐿HF +∑
𝑖=1
𝑁
𝜖 𝑖HF ¿𝜙 𝑖
HF |2
𝜌HF −𝑣−𝑣HHF
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)
𝜙𝑖=𝜙𝑖HF
𝑣 XECOAP=𝑣S
HF + 1𝜌HF ∑
𝑖=1
𝑁
(𝜖 𝑖−𝜖¿¿ 𝑖HF)|𝜙 𝑖HF|2 ¿
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ECOAP KLI LHF
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Calculated exact-exchange (EXX) energies
, mEh
KLI ELP=LHF=CEDA ECOAPm.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. Hartree‒Fock exchange-correlation (HFXC) potential
An HFXC potential is the which reproduces a HF density within the Kohn‒Sham scheme:
𝜌 (𝐫 )=∑𝑖=1
𝑁
|𝜙𝑖 (𝐫 )|2=¿∑
𝑖=1
𝑁
|𝜙 𝑖HF (𝐫 )|2
=𝜌HF (𝐫 )¿
[− 12∇2+𝑣 (𝐫 )+𝑣H (𝐫 )+𝑣XC (𝐫 )]𝜙 𝑖(𝐫 )=𝜖𝑖𝜙𝑖 (𝐫)
That is, is such that
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Inverting the Kohn–Sham equations
𝜏𝐿
𝜌 +𝑣+𝑣H +𝑣XC= 1𝜌∑
𝑖=1
𝑁
𝜖 𝑖|𝜙𝑖|2
[− 12∇2+𝑣+𝑣H +𝑣XC ]𝜙𝑖=𝜖 𝑖𝜙 𝑖
Kohn‒Sham equations:
local ionizationpotential
multiply by ,sum over i,divide by
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Inverting the Hartree–Fock equations
𝜏 𝐿HF
𝜌HF +𝑣+𝑣H +𝑣SHF= 1
𝜌HF ∑𝑖=1
𝑁
𝜖𝑖HF|𝜙 𝑖
HF|2
Hartree‒Fock equations:
Slater potential builtwith HF orbitals
[− 12∇2+𝑣+𝑣H +𝐾 ]𝜙𝑖
HF=𝜖 𝑖HF 𝜙𝑖
HF
same manipulations
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Closed-form expression for the HFXC potential
𝑣 XCHF=𝑣S
HF + 1𝜌∑
𝑖=1
𝑁
𝜖 𝑖∨𝜙 𝑖 |2 − 1
𝜌HF ∑𝑖=1
𝑁
𝜖 𝑖HF|𝜙𝑖
HF|2+ 𝜏
HF
𝜌HF − 𝜏𝜌
, but , , and
We treat this expression as a model potential within the Kohn‒Sham SCF scheme.
Here
Computational cost: same as KLI and Becke‒Johnson (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
where𝐸 vir= ∫ 𝑣X (𝐫 ) [3 𝜌 (𝐫 )+𝐫 ⋅∇ 𝜌 (𝐫 ) ]𝑑𝐫
For exact OEPs,
𝐸 vir −𝐸EXX=0 ,
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HFXC potentials in finite basis sets
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Hierarchy of approximations to the EXX potential
𝑣 X❑=𝑣S
HF + 1𝜌HF ∑
𝑖=1
𝑁
(𝜖 𝑖−𝜖𝑖HF )|𝜙𝑖
HF|2+ 𝜏
HF −𝜏𝜌HF
OAP
ECOAP
HFXC
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Summary
• Orbital-averaged potentials (e.g., Slater) can be constructed by Kohn‒Sham inversion
• Hierarchy or approximations to the OEP: OAP (Slater) < ECOAP < HFXC
• ECOAP Slater potential KLI LHF
• HFXC potential OEP
• Same applies to all occupied-orbital functionals
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Acknowledgments
• Eberhard Engel• Leeor Kronik
for OEP benchmarks