### PPT presentation

```Efficient methods for computing
exchange-correlation potentials for
orbital-dependent functionals
Viktor N. Staroverov
Department of Chemistry, The University of Western Ontario,
IWCSE 2013, Taiwan National University, Taipei, October 14‒17, 2013
Orbital-dependent functionals
XC  =

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
Xexact
1
=−
4

,=1
∗
∗ ′
′

′
− ′
same expression as in the Hartree‒Fock theory
• Hybrids (B3LYP, PBE0, etc.)
• Meta-GGAs (TPSS, M06, etc.)
3
The challenge
Kohn‒Sham potentials corresponding to orbitaldependent functionals
XC
XC [{ }]
=
=?
()
cannot be evaluated in closed form
4
Optimized effective potential (OEP)
method
Find XC () as the solution to the minimization
problem
total
=0
XC ()
OEP = functional derivative of the functional
5
Computing the OEP
Expand the Kohn‒Sham orbitals:

=
()
=1
orbital basis functions
Expand the OEP:

XC  =
()
=1
auxiliary basis functions
Minimize the total energy with respect to { } and { }
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 XC such that
XC [{ }]
XC   =
∗ ()
The OAP is a weighted average:
XC  =

∗

=1   XC

∗

=1  ()

8
Example: Slater potential
Fock exchange operator:
Xexact
≡
∗ ()
Slater potential:
1
S  =

∗ () ()
=1
9
Calculation of orbital-averaged
potentials
• by definition (hard, functional specific)
• by inverting the Kohn‒Sham equations
(easy, general)
10
Kohn‒Sham inversion
Kohn‒Sham equations:
1 2
− ∇ +  + H + XC  =
2
multiply by ∗ ,
sum over i,
divide by

1
+  + H + XC =

2
=1
11
LDA-X potential via Kohn-Sham inversion
12
PBE-XC potential via Kohn‒Sham inversion
13
Removal of
oscillations
A. P. Gaiduk,
I. G. Ryabinkin, VNS,
JCTC 9, 3959 (2013)
14
Kohn‒Sham inversion for orbitalspecific potentials
Generalized Kohn‒Sham equations:
1 2
− ∇ +  + H + XC  =
2
same manipulations

1
+  + H + XC =

2
=1
15
Example: Slater potential through
Kohn‒Sham inversion
1 2
− () +
S  = 4

=1  |
()|2
−   − H ()
where
1
=
2

|
=1
|2
1 2
=  +
4
16
Slater potential via Kohn‒Sham inversion
17
OAPs constructed by Kohn‒Sham inversion
18
Correlation potentials via Kohn‒Sham inversion
19
Kohn‒Sham inversion for a fixed set
of Hartree‒Fock orbitals
Slater potential:
SHF =
−HF +

HF HF |2

=1  |
HF
−  − HHF

HF |2

|
=1
HF
−  − HHF
But if OEP ≈ HF , then
XOEP ≈ Xmodel =
−HF +
20
Dependence of KS inversion on orbital energies
21
II. Assumption that the OEP and HF
orbitals are the same
The assumption
= HF
leads to the eigenvalue-consistent orbitalaveraged potential (ECOAP)
XECOAP
=
SHF
1
+ HF

( −
HF )
HF 2

=1
22
ECOAP ≈ KLI ≈ LHF
23
Calculated exact-exchange (EXX) energies
EXX − OEP , 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)
24
III. Hartree‒Fock exchangecorrelation (HFXC) potential
An HFXC potential is the XC () which reproduces
a HF density within the Kohn‒Sham scheme:
1 2
− ∇ +   + H  + XC   () =   ()
2
That is, XC () is such that

=

=1
2
HF
=

2
= HF ()
=1
25
Inverting the Kohn–Sham equations
Kohn‒Sham equations:
1 2
− ∇ +  + H + XC  =
2
multiply by ∗ ,
sum over i,
divide by

1
+  + H + XC =

2
=1
local ionization
potential
26
Inverting the Hartree–Fock equations
Hartree‒Fock equations:
1 2
− ∇ +  + H +  HF = HF HF
2
same manipulations
HF
HF
+  + H +
SHF
1
= HF

HF HF
2
=1
Slater potential built
with HF orbitals
27
Closed-form expression for
the HFXC potential
HF
XC
1
HF
= S +

=1
1
2
| | − HF

=1
HF HF 2

HF
+ HF −

Here
= HF , but  ≠ HF ,  ≠ HF , and  ≠  HF
We treat this expression as a model potential within the
Kohn‒Sham SCF scheme.
Computational cost: same as KLI and Becke‒Johnson (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
EXX − OEP , 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,
vir − EXX = 0,
where
vir = ∫ X  3  +  ⋅ ∇()
vir − EXX , 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
X
1
HF
= S + HF

− HF HF
=1
HF −

2
+
HF
OAP
ECOAP
HFXC
36
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
37
Acknowledgments
• Eberhard Engel
• Leeor Kronik
for OEP
benchmarks
38
```