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