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Short Course on Density Functional Theory and Applications
VII. Hybrid, Range-Separated, and One-shot Functionals
Quantum Theory ProjectDept. of Physics and Dept. of Chemistry
Samuel B. Trickey©Sept. 2008
∂
• Given that it has only 3 parameters and performs remarkably well, is B3LYP about as good as as we are going to do?
• Quite some time ago, Ruiz, Salahub, and Vela [J. Phys. Chem. 100, 12265 (1996)]
responded in the negative: “The B3LYP results lie between those of the GGA and MP2.” “[Our results] for the so-called half-and-half potential are in very good agreement with those obtained through second-order Møller-Plesset calculations and with available experimental data. However, the more widely used three-parameter, B3LYP, functional does not perform well; the hybrid methods are not a panacea.”
So Is B3LYP the Answer?
( )( )
LDAxc
fr
ρρ ρ
∂= ∂ �
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• They studied C2H4…………X2 (X=F, Cl, Br, or I) complexes in three orientations →→→→•See their table, next slide.
∂
So Is B3LYP the Answer?
( )( )
LDAxc
fr
ρρ ρ
∂= ∂ �
QTP
Ruiz, Salahub, and Vela [J. Phys. Chem. 100, 12265 (1996)]
• Errors in transition-metal ™ -ligand bond dissociation energies (BDE, in • kJ mol–1) of methyl and carbene complexes of first-row transition metal cations
So Is B3LYP the Answer?
– BDE by CI methods MCPF, CCSD(T) always too small, sometimes significantly (carbenes!)
– Curves for B3LYP and PCI-80(MP2) quite similar in shape [PCI-80: correlation energy (from MP2 or CCSD) scaled by an empirical factor]
– For M-CH 3+, PCI-80 uniformly very good; B3LYP in general too large, second best
– For M-CH 2+, PCI-80 often somewhat too large; B3LYP overall best
Credit: N. Rösch
QTP
Ways Forward? - KLIThere is a relatively fast way to do ExX approximately with fairly high acccuracy, the Krieger-Lee-Iafrate (KLI) approximation. Suppose we had a good Ec func-tional to go with this approximate ExX. Is that a way forward? Here is a sobering table. “F” under method is KLI ExX.
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NONE of the ExX + Eccombinations does better than GGA! Finding an Ec to go with ExX is not easy.
Ways Forward? Local Hybrid FunctionalsHybrid functionals mix exact (single-determinant) exchange, ExKS , with approximate Ex and Ec contributions in fixed proportions:
, , (1 ) xcHybrid xKS x approx c approxE E E Eλ λ= + − +
Why not do the mixing locally, i.e., pointwise? Here is the Jaramillo et al. version of the idea [J. Chem. Phys. 118, 1068 (2003)
( ) ( ) ( ) ( )( ) ( ) ( ){ }( ) ( ) ( ) ( )
, ,
2
1 xcLocHybrid xKS x approx c approx
W
E d n u u u
n
λ λ
τ
= + − +
∇
∫ r r r r r r r
rr
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( ) ( )( ) ( ) ( )
( ) : 1 ; :
8W
n
n
τλ τ
τ∇
= − =rr
r rr r
Recall (Lect. V-3, 8, II-27): ττττW/ ττττ is a so-called “iso-orbital indicator”.
Results – next slide.
Ways Forward? Local Hybrid Functionals (cont’d.)
Dissociation of H2+ for BLYP,
B3LYP, and lh-BLYP XC. 6-311G++(3df,3pd) basis. [J. Chem. Phys. 118, 1068 (2003)
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Ways Forward? Range-separated HybridsRather than mix exact (single-determinant) exchange, ExKS , with approximate Exon a local basis, what about separating the Coulomb potential into a short- and long-ranged part and treating them separately to get a hybrid? [A. Savin et al.,
Internat. J. Quantum Chem. 56, 327 (1995); Chem. Phys. Lett. 275, 151 (1997)]Motivation: approximate Exc functionals are local or “semi-local”, so use them at short range and use exact ExKSat long range.
( ) ( )1 1( ) : +
1 ( )( )ij
i ji j
i ji j
LR iji j
SR iji j
rgg r
rr
r
rg r
r
γγ= ≡ = + =
−
−
r r
Here is the range-separated Coulomb interaction:
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( ) ( ) ( ) 2
, ,
0
1 1ˆ ˆ;2 2
2 erf ; erf :
(( ) )
i ji j
ee LR
i j
LR ij
i
ee SRi j
j
Si j
xx
i j i
R ij
j
r
r r x d
rr
g r g r
x eγ µπ
≠ ≠
′−
−
= =
′= =
∑ ∑
∫
r r
V V
Ways Forward? Range-separated Hybrids (cont’d.)
[ ]
{ }[ ] [ ] [ ]
{ } [ ]
, ,
, ,
0 0 0 , , 0
, , 0
ˆ ˆ ˆ: min
ˆ ˆ ˆˆ ˆ ˆmin min min
ˆ ˆ ˆmin
ˆ ˆ ˆˆ ˆ ˆmin min min min
RS ee LR ee SRn
ee LR ee ee LRn n n
RS ext ee LR ee SR ext
ee LR ee ee LR extn n n
F n
E F n E n E n
E n
ψ
ψ ψ ψ
ψ
ψ ψ ψ
ψ ψ
ψ ψ ψ ψ ψ ψ
ψ ψ
ψ ψ ψ ψ ψ ψ
= + +
= + + + − +
= + = + + +
= + + + − + +
֏
֏ ֏ ֏
֏ ֏
T V V
T V T V T V
T V V
T V T V T V
And here is the range-separated universal functional and its minimum:
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Usual Evcxtf unctional LR version of Evcxt
[ ] ,ˆ ˆmin [ ]RS ee LR SR
nF n F n
ψψ ψ= + +
֏
T V
Ways Forward? Range-separated Hybrids (cont’d.)
[ ]
( )
( ) ( ) ( )
( ) ( )( ) ( )( ) ( ) ( ) }
,
1
1/3,
3 2 3
ˆ ˆ; [ ]
11, , det
!
3 3 3erf(1/ 2 )
2 8
2 4 exp 1/ (4 ) 3 4
e
RS ee LR SR
e N
e
SRxLDA
F n F n
NN
u A A
A A A A A
µ
ϕ ϕ
ππ
Φ = Φ + Φ +
Φ =
= − −
+ − − − +
r r r
r r r r r
… …
T V
One version of a “range-separated hybrid LDA”, makes the following approx-imations [Gerber et al., J. Chem. Phys. 127, 054101 (2007)]
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( ) ( )( ) ( )( ) ( ) ( ) }( )
( ) ( )1/32 1/3
2 4 exp 1/ (4 ) 3 4
2 3
A A A A A
An
µπ
+ − − − +
=
r r r r r
rr
The SR xLDA comes from short-ranged HEG work of Toulouse, Savin, and Fladd[Internat. J. Quantum Chem. 100, 1047 (2004)]. In the calculations shown on the next slide,Gerber et al. used VWN for Ec.
Ways Forward? Range-separated Hybrids (cont’d.)
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HSE= Heyd, Scuseria, Ernzerhof hybrid
[Gerber et al., J. Chem. Phys. 127, 054101 (2007)]
Remark – though this is good work based on a very clever idea, it isn’t obvious that the results are a major improvement on ordinary hybrids.
Ways Forward? Range-separated Hybrids (cont’d.)
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PBE0= PBE-based hybrid [Gerber et al., J. Chem. Phys. 127, 054101 (2007)]
““““The RSHX functional, which has the main feature of providing a correctasymptotic behavior of the exchange potential, has a tendency to improve the description of structural parameters with respect to local and generalized gradient approximations. The band gaps are too strongly opened by the presence of the long-range Hartree-Fock exchange in all but wide-gap systems. In the difficult case of transition metal oxides, the gap is overestimated, whilemagnetic moments and lattice constants are slightly underestimated.”
Ways Forward? Range-separated Hybrids (cont’d.)
From the Abstract of Gerber et al. [J. Chem. Phys. 127, 054101 (2007)]
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Note: “Hartree-Fock exchange” is again a misnomer, even though they work with an orbital-dependent (non-local) potential.
Adiabatic ConnectionNow comes one of the more powerful concepts for understanding functionals and constructing them. The objective is a smooth transformation from the non-interactingKS system to the fully interacting physical system. (This is distinct from the smooth mixingthat is in range-separated and local hybrid functionals.)
,0 ,0 ,0,0 ,0 ,0 ,0
ˆˆ ˆ| | | | | |
E HH Hλ λ λλ
λ λ λ λ λ λλ λ λ λ∂ ∂Ψ ∂Ψ∂= ⟨ Ψ ⟩ + ⟨Ψ Ψ ⟩ + ⟨Ψ ⟩∂ ∂ ∂ ∂
The first ingredient is the Hellmann-Feynman theorem. Suppose the Hamiltonian depends smoothly on a parameter λλλλ. Then, the ground state is
,0 ,0 ,0 ,0 ,0ˆ ; | 1H Eλ λ λ λ λ λΨ Ψ ⟨Ψ Ψ ⟩ == Differentiate w/r to the parameter
QTP
,0 ,0 ,0 ,0
,0 ,0 ,0 ,0 ,0 ,0 ,0
| | | | | |
ˆ ˆ| | | | |
H H
H HE
λ λ λ λ λ λ
λ λλ λ λ λ λ λ λ
λ λ λ λ
λ λ λ
= ⟨ Ψ ⟩ + ⟨Ψ Ψ ⟩ + ⟨Ψ ⟩∂ ∂ ∂ ∂
∂ ∂∂= ⟨Ψ Ψ ⟩ + ⟨Ψ Ψ ⟩ = ⟨Ψ Ψ ⟩∂ ∂ ∂
0 1ˆ ˆ ˆH H Hλ λ= +
1
1 0 10
ˆ| |E E d Hλ λ λ λλ ψ ψ= =− = ⟨ ⟩∫it follows from the Hellmann-Feynman theorem that
The second ingredient is the Pauli coupling constant trick. Given the Hamiltonian
Go back to Levy-Lieb constrained search but for a one-parameter Hamiltonian as in the Pauli trick. Then the universal functional is
Adiabatic Connection (cont’d.)
1
0
1 1 1 , ,
ˆ[ ] min | |
ˆ ˆ0: [ ] min | | | | [ ]
ˆ ˆ1: [ ] min | | | |
n ee
n min,n min,n s
n ee min n ee min n
F n T V
F n T T T n
F n T V T V
λ
λ
λ λ λ
λ λ
ψ
ψ
ψ λ ψ
λ
λ ψ ψ ψ ψ=
Φ
= =
= ⟨ + ⟩
= = ⟨Φ Φ⟩ ≡ ⟨Φ Φ ⟩ ≡
= = ⟨ + ⟩ ≡ ⟨ + ⟩
֏
֏
֏
Therefore1 0 [ ] [ ] [ ] [ ]xc eeF n F n E n E n− = +
By invoking a Lagrange multiplier potential v ( r ) which keeps the density
QTP
By invoking a Lagrange multiplier potential vλλλλ( r ) which keeps the densityUNchanged across the whole range 0 ≤≤≤≤ λλλλ ≤≤≤≤ 1, and using the Hellmann-Feynmantheorem and Pauli trick, one can prove the adiabaticconnection
1
, ,0
ˆ[ ] [ ] | | [ ] [ ]xc min ee min eeE n d n V n E nλ λλ ψ ψ= ⟨ ⟩ −∫
Notice that (a) v-representability is back in the picture, both non-interacting andinteracting and (b) we don’t have to know the potential vλλλλ( r ).
Ways Forward? Adiabatic Connection Functionals
[ ]
0 ,20
0, [ ]
[ ], 2nd -order Goerling-Levy C
0
xKS
GLc
W n E n
WW E n
W
λ
λλ
=
∂′ ≡ =∂
∂ < ∀∂
Some facts are known about the W functional:
The adiabatic connection can be rewritten simply to a suggestive form
[ ]1 1
, ,0 0
ˆ[ ] [ ] | | [ ] [ ] : ,xc min ee min eeE n d n V n E n d W nλ λλ ψ ψ λ λ = ⟨ ⟩ − = ∫ ∫
QTP
A scheme for calibrating approximate interpolation λλλλ=0 →→→→ λλλλ= 1 based on approximate Exc also is known.
[ ] ( )
[ ] ( ) ( )
( ) ( )
1/
2, , 1/ , 1/
31/
, [ ]
2 [ ] [ ]
/
xc
x approx c approx c approx
W n E n
E n E n E n
n n
λ
λ λ
λ
λλ
λ λλ
λ λ−
∂=∂
∂≈ + +∂
=
r
r r
r r[Cohen, Mori-Sánchez, and Yang , J. Chem. Phys. 127, 034101 (2007)]
Ways Forward? Adiabatic Connection Functionals (cont’d.)
[ ] { } { }{ }
{ } ( )2
,, ,
1 ,
ln 1,xc
b nW n a n
c n
b cbE n a
c c
λ ϕλ ϕ
λ ϕ
ϕ
= + +
+= + −
The Mori-Sánchez, Cohen, and Yang [J. Chem. Phys. 124, 091102 (2006)] model adiabatic-connection path is
{ } ( )0 2
ln 1,xc xKS
c cE n E W
cϕ
− +′= +
The choice of BLYP as the interpolating functional, for example, gives
QTP
A selected summary of results [Cohen, Mori-Sánchez, and Yang, J. Chem. Phys. 127, 034101
(2007)] follows. The original paper has many very large tables.
( )0 p
p
BLYPxKS
BLYPp xKS
c
E W Wc
E W
λ
λ
λ
λ
′− − +=
−
Ways Forward? Adiabatic Connection Functionals (cont’d.)
Remark – we are back to the no free lunch theorem. These are intricate functionals that require
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[Cohen, Mori-Sánchez, and Yang , J. Chem. Phys. 127, 034101 (2007)]
intricate functionals that require extensive work to generate and program, yet give comparatively modest improvement over simple hybrids.
Ways Forward? Back to the GGA?!?
Armiento and Mattsson[Phys. Rev. B 72, 085108 (2005)]
produced a GGA that behaves properly for two model systems, the HEG and the jellium surface and, in effect, interpolates between them by measur-ing the local inhomo-geneity. The resulting functional has a structure
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functional has a structure that looks a lot like PBE but is different in content. Results are impressive [J. Chem. Phys. 128, 084714 (2008)]
Consideration of better ways to constrain (hence parameterize) GGAs is an area of active research.
( ){ }(1) (1) (1) (1)
(
212
1)
( )] ( )] ( )]
( )
[ ( )] [ [ [
1[ ( )
( )( ) ( )] [ ] [ ]
2 | |
KS A k A A A k k k
A AHarris A k xc A
ext ee xc
xc Ak
h n n n n
n nE n d d E n
v v v
d V n
φ φ ε φ
ε
= =
′= − +
− ∇ + + +
−′
′−∑ ∫∫ ∫
r r r r
r rr rr
rrr r
r
A Digression: One-shot Functionals
Irrespective of the choice of Exc approximation, it often is desirable to get an estimate of the DFT energy for multiple nuclear configurations {R} of some system, without, doing the full scf calculation. The Harris approximate, non-iterative functional is the best known of several “one-shot” ways to do this. [J. Harris, Phys. Rev. B 31, 1770 (1985)]
Suppose that one has a reasonably good approximate density, nA. Then
The rough physical reasoning is that the KS equation was derived from variational stability. Therefore, a non-self-consistent solution of it should lead to an error reduction which is embodied in the resulting eigenvalues εεεεi
(1) The other terms handle over-counting.
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Ways Forward – Some Commentary
• It is probably fair to say that a majority of the DFT functional development community believes explicitly orbital-dependent functionals, beyond the level of MGGAs, are a necessity.• However, the evolution of increasingly sophisticated hybrids seems to be reaching a state of diminishing returns. • A minority points to M06-L and AM05 as examples to argue that orbital-dependence beyond the level of MGGAs is, in fact, not necessary. The agenda, therefore, of this minority, is to find MGGA and similar functionals with hybrid-level performance. • The use of full ExX is still hampered by the lack of really good approximate Ec
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• The use of full ExX is still hampered by the lack of really good approximate Ec
functionals to accompany ExKS
