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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
Orbital-dependent functionals
πΈXC π =
π ππ
ππ«
Kohn-Sham orbitals
• More flexible than LDA and GGAs (can satisfy
more exact constraints)
• Needed for accurate description of molecular
properties
2
Examples
• Exact exchange
πΈXexact
1
π =−
4
π
ππ«
π,π=1
∗
∗ ′
′
π
π«
π
π«
π
π«
π
π«
π
π
π
π
ππ« ′
π« − π«′
same expression as in the HartreeβFock theory
• Hybrids (B3LYP, PBE0, etc.)
• Meta-GGAs (TPSS, M06, etc.)
3
The challenge
KohnβSham potentials corresponding to orbitaldependent functionals
π£XC
πΏπΈXC [{ππ }]
π« =
=?
πΏπ(π«)
cannot be evaluated in closed form
4
Optimized effective potential (OEP)
method
Find π£XC (π«) as the solution to the minimization
problem
πΏπΈtotal
=0
πΏπ£XC (π«)
OEP = functional derivative of the functional
5
Computing the OEP
Expand the KohnβSham orbitals:
π
ππ π« =
πππ ππ (π«)
π=1
orbital basis functions
Expand the OEP:
π
π£XC π« =
ππ ππ (π«)
π=1
auxiliary basis functions
Minimize the total energy with respect to {πππ } and {ππ }
6
Attempts to obtain OEP-X in finite basis sets
size
7
I. First approximation to the OEP:
An orbital-averaged potential (OAP)
Define operator π’XC such that
πΏπΈXC [{ππ }]
π’XC ππ π« =
πΏππ∗ (π«)
The OAP is a weighted average:
π£XC π« =
π
∗
π
π=1 π π« π’XC ππ
π
∗
π
π=1 π (π«)ππ π«
π«
8
Example: Slater potential
Fock exchange operator:
πΏπΈXexact
πΎππ π« ≡
πΏππ∗ (π«)
Slater potential:
1
π£S π« =
π π«
π
ππ∗ (π«)πΎππ (π«)
π=1
9
Calculation of orbital-averaged
potentials
• by definition (hard, functional specific)
• by inverting the KohnβSham equations
(easy, general)
10
KohnβSham inversion
KohnβSham equations:
1 2
− ∇ + π£ + π£H + π£XC ππ = ππ ππ
2
multiply by ππ∗ ,
sum over i,
divide by π
ππΏ
1
+ π£ + π£H + π£XC =
π
π
π
ππ ππ
2
π=1
11
LDA-X potential via Kohn-Sham inversion
12
PBE-XC potential via KohnβSham inversion
13
Removal of
oscillations
A. P. Gaiduk,
I. G. Ryabinkin, VNS,
JCTC 9, 3959 (2013)
14
KohnβSham inversion for orbitalspecific potentials
Generalized KohnβSham equations:
1 2
− ∇ + π£ + π£H + π’XC ππ = ππ ππ
2
same manipulations
ππΏ
1
+ π£ + π£H + π£XC =
π
π
π
ππ ππ
2
π=1
15
Example: Slater potential through
KohnβSham inversion
1 2
π» π π« − π(π«) +
π£S π« = 4
π π«
π
π=1 ππ |ππ
(π«)|2
− π£ π« − π£H (π«)
where
1
π=
2
π
|π»ππ
π=1
|2
1 2
= ππΏ + π» π
4
16
Slater potential via KohnβSham inversion
17
OAPs constructed by KohnβSham inversion
18
Correlation potentials via KohnβSham inversion
19
KohnβSham inversion for a fixed set
of HartreeβFock orbitals
Slater potential:
π£SHF =
−ππΏHF +
π
HF HF |2
π
π=1 π |ππ
πHF
− π£ − π£HHF
π
HF |2
π
|π
π=1 π π
πHF
− π£ − π£HHF
But if ππOEP ≈ ππHF , then
π£XOEP ≈ π£Xmodel =
−ππΏHF +
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Dependence of KS inversion on orbital energies
21
II. Assumption that the OEP and HF
orbitals are the same
The assumption
ππ = ππHF
leads to the eigenvalue-consistent orbitalaveraged potential (ECOAP)
π£XECOAP
=
π£SHF
1
+ HF
π
π
(ππ −
ππHF )
HF 2
ππ
π=1
22
ECOAP ≈ KLI ≈ LHF
23
Calculated exact-exchange (EXX) energies
πΈEXX − πΈOEP , mEh
m.a.v.
KLI
ELP=LHF=CEDA
ECOAP
2.88
2.84
2.47
Sample: 12 atoms from He to Ba
Basis 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 exchangecorrelation (HFXC) potential
An HFXC potential is the π£XC (π«) which reproduces
a HF density within the KohnβSham scheme:
1 2
− ∇ + π£ π« + π£H π« + π£XC π« ππ (π«) = ππ ππ (π«)
2
That is, π£XC (π«) is such that
π
π π« =
π
ππ π«
π=1
2
ππHF
=
π«
2
= πHF (π«)
π=1
25
Inverting the Kohn–Sham equations
KohnβSham equations:
1 2
− ∇ + π£ + π£H + π£XC ππ = ππ ππ
2
multiply by ππ∗ ,
sum over i,
divide by π
ππΏ
1
+ π£ + π£H + π£XC =
π
π
π
ππ ππ
2
π=1
local ionization
potential
26
Inverting the Hartree–Fock equations
HartreeβFock equations:
1 2
− ∇ + π£ + π£H + πΎ ππHF = ππHF ππHF
2
same manipulations
ππΏHF
πHF
+ π£ + π£H +
π£SHF
1
= HF
π
π
ππHF ππHF
2
π=1
Slater potential built
with HF orbitals
27
Closed-form expression for
the HFXC potential
HF
π£XC
1
HF
= π£S +
π
π
π=1
1
2
ππ |ππ | − HF
π
π
π=1
HF HF 2
ππ ππ
π HF π
+ HF −
π
π
Here
π = πHF , but ππ ≠ ππHF , ππ ≠ ππHF , and π ≠ π HF
We treat this expression as a model potential within the
KohnβSham SCF scheme.
Computational cost: same as KLI and BeckeβJohnson (BJ)
28
HFXC potentials are practically exact OEPs!
Numerical OEP: Engel et al.
29
30
31
HFXC potentials can be easily computed
for molecules
Numerical OEP: Makmal et al.
32
Energies from exchange potentials
πΈEXX − πΈOEP , mEh
m.a.v.
KLI
LHF
BJ
1.74
1.66
5.30
Basisset OEP
0.12
HFXC
0.05
Sample: 12 atoms from Li to Cd
Basis 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)
33
Virial energy discrepancies
For exact OEPs,
πΈvir − πΈEXX = 0,
where
πΈvir = ∫ π£X π« 3π π« + π« ⋅ ∇π(π«) ππ«
πΈvir − πΈEXX , mEh
KLI
m.a.v. 438.0
LHF
BJ
629.2 1234.1
Basis-set
HFXC
OEP
1.76
2.76
34
HFXC potentials in finite basis sets
35
Hierarchy of approximations to
the EXX potential
π£X
1
HF
= π£S + HF
π
π
ππ − ππHF ππHF
π=1
HF − π
π
2
+
πHF
OAP
ECOAP
HFXC
36
Summary
• Orbital-averaged potentials (e.g., Slater) can be
constructed by KohnβSham inversion
• Hierarchy or approximations to the OEP:
OAP (Slater) < ECOAP < HFXC
• ECOAP Slater potential ο» KLI ο» LHF
• HFXC potential ο» OEP
• Same applies to all occupied-orbital functionals
37
Acknowledgments
• Eberhard Engel
• Leeor Kronik
for OEP
benchmarks
38
