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1
Nonlocal density functional theory for chemicalreactions
Benjamin G. Janesko
Texas Christian University
2013.02.14
2
Chemistry in Texas
Several large programs: Rice, University of Texas, Texas A&M
Computational chemistry: University of North Texas, TexasTech, Southern Methodist University
Local research support: Welch foundation, collaborations withTexas A&M Qatar
3
Chemistry at TCU
∼ 10, 000undergraduates
11 research-activechemistry faculty
∼ 25 Ph. D. students
Strengths: Organic,bio-inorganic,macromolecularchemistry
Computation isincreasingly important
4
Group research overview
1 Use electronic structure theory, particularly density functionaltheory (DFT), to model molecular structure and reactivity
2 Refine DFT for difficult problems like reactions on catalystsurfaces
3 Develop new DFT approximations for these problems
5
Explanation #1 of density functional theory
E
Freshman chemistry molecular orbital theory, plus approximateelectron-electron interactions
Different DFT methods use different approximations
∼ 2− 6 kcal/mol accuracy for thermochemistry & kinetics ofmedium-sized molecules
Qualitatively correct trends in reactivity
6
Outline
1 DFT for alkyl cross-coupling
2 Other DFT applications
3 New functionals for surface chemistry
4 “Rung 3.5” density functionals
5 Summary
7
Cross-coupling
Noble-metal-catalyzed C-C bond formation
Suzuki, Negishi, Hiyama, Heck, etc.
8
Alkyl Suzuki coupling
Alkyl reactants are challenging
Strong Csp3-X bonds impede initial oxidative additionβ-hydride elimination from coordinatively unsaturated Pd(II)Electron withdrawing substitutent CN is essential
Choice of ligand and reaction conditions is important
SPhos, RuPhos, JohnPhos, etc.
Few “design rules” optimizing ligands for particular reactants
A. He, J. R. Falck, JACS 132, 2524 (2010)
9
Initial oxidative addition
10
Anion accelerates & controls stereoselectivity
B. Pudasaini, B. G. Janesko, Organometallics 31, 4610 (2012)
11
Cyano deactivates β-hydride elimination
B. Pudasaini, B. G. Janesko, Organometallics 31, 4610 (2012)
12
Ligand trans influence and sterics control selectivity
B. Pudasaini, B. G. Janesko, Organometallics 31, 4610 (2012)
13
Ligand trans influence and sterics control selectivity
B. Pudasaini, B. G. Janesko, Organometallics 31, 4610 (2012)
14
Ligand protection in another alkyl cross-coupling
Dialkylbiaryl ligand protects coordinatively unsaturated Pd(II )
intermediate from undesirable side reactions.
B. Pudasaini, B. G. Janesko, Organometallics 30, 4564 (2011)
15
Outline
1 DFT for alkyl cross-coupling
2 Other DFT applications
3 New functionals for surface chemistry
4 “Rung 3.5” density functionals
5 Summary
16
Ionic liquid solvents for biofuels
B. G. Janesko, Phys. Chem. Chem. Phys. 13, 11393 (2011)
17
Biofuel production in ionic liquids
Solvent Barrier ε
None 20.7 1.0Dichloroethane 23.7 10.1(BMIM)(PF6) 25.7 11.4Water 28.1 78.4
Rate-limiting transition barrier to base-catalyzed lignin hydrolysis
B. G. Janesko, in preparation
18
Conducting polymers for organic photovoltaics
P. Sista, B. Xue, M. Wilson, N. Holmes, R. S. Kularatne, H. Nguyen, P. C. Dastoor, W. Belcher, K. Poole, B. G.
Janesko, M C. Biewer, M. C. Stefan; Macromolecules 45, 772 (2012)
19
Frustrated Lewis pair nanoribbons
B. G. Janesko, J. Phys. Chem. C 116, 16476 (2012)
20
Organocatalysts for organophosphorus production
-10
-5
0
5
10
15
20
25
30
35
40
ΔE
vs.
fre
e r
ea
cta
nts
(k
cal/
mo
l)
Formic acid Thioformic acidDimer
M. Bridle, B. G. Janesko, J.-L. Montchamp, in preparation
21
Why bother designing new DFT methods?
22
Outline
1 DFT for alkyl cross-coupling
2 Other DFT applications
3 New functionals for surface chemistry
4 “Rung 3.5” density functionals
5 Summary
23
Explanation #2 of density functional theory
Many-body problems are hard
Two moons interacting with a planet and each otherTwo electrons interacting with a nucleus and each other
Analytic solutions generally unavailable
Numerical solutions scale as eN or N! for N particles
Molecular orbital theory ignores electron-electron interactions
DFT does something clever
24
Explanation #2 of density functional theory
N noninteracting Fermions in orbitals {φi (~r)}, with electronprobability density
ρ(~r) =N∑i=1
|φi (~r)|2
can exactly model the ground state of N interactingelectrons (Hohenberg & Kohn)
Electron interactions treated by self-Coulomb repulsion
1
2
∫d3~r1
∫d3~r2
ρ(~r1)ρ(~r2)
|~r1 −~r2|
and an “exchange-correlation” density functional (function ofa function) EXC [ρ(~r)]
25
Approximate exchange-correlation functionals
Exact EXC [ρ] requires solving the many-body problem
Not possible in real systems
Real calculations use an alphabet soup of approximate XCfunctionals
While “DFT” is exact, approximate XC functionals are not
Each approximation has costs and benefits
26
Costs and benefits of approximate functionals
Semilocal DFT
EXC [ρ] =
∫d3~r eXC (ρ(~r),∇ρ(~r), . . .)
Inexpensive enough for solids
Over-delocalize electrons (overbinding)
Nonlocal Hybrid
E exX = −1
2
N∑i ,j=1
∫d3~r
∫d3~r ′
φ∗i (~r)φj(~r′)φ∗j (~r)φi (~r
′)
|~r −~r ′|
Empirical admixture of E exX fixes overbinding
All molecular calculations from Part 1 used hybrid DFT
Long range of integrand expensive in metals
27
DFT for heterogeneous catalysis
Semilocal DFT used for ∼ 20 years
Successes:
Computational design of catalytic alloys
Limitations:
Science 2005: Two semilocal DFT methods give largediscrepancy in N2 bond breaking on metal surface0.6 eV barrier height discrepancy10 order of magnitude reaction rate discrepancy!
28
A solution from DFT for molecules
Screen out long-range exact exchange
ESRX = −1
2
∑ij
∫d3~r
∫d3~r ′
φ∗i (~r)φj(~r′)φ∗j (~r)φi (~r
′)
|~r −~r ′|
× erfc(ω|~r −~r ′|
)HSE06 and HISS screened hybrids are readily applied tometals and surfaces
Can they improve semilocal DFT’s reaction barriers?
29
Tests on a model Si cluster
+
-1.60
-1.20
-0.80
-0.40
0.00
0.40
0.80
1.20
Re
lati
ve E
ne
rgy
(eV
)
Adsorbed NH3
Product
TS
Reactants
HISS
PBE revPBE HSE06 CCSD(T)/CBS
NH3 dissociation on Si9H12 cluster
Screened hybrids better reproduce accurateCCSD(T)/CBS-extrapolated benchmarks
R. Sniatynsky, B. G. Janesko, F. El-Mellouhi, E. N. Brothers, J. Phys. Chem. C 116, 26396 (2012)
30
Tests on a realistic Si surface
HISSHSE06
PBE
On-DimerPathway
Inter-DimerPathway
+
NH3 Adsorbed
Inter-DimerTransition State
On-DimerTransition State
On-DimerProduct
Inter-DimerProduct
Accurate benchmarks are computationally infeasible
Screened hybrids increase barrier, improve selectivity
R. Sniatynsky, B. G. Janesko, F. El-Mellouhi, E. N. Brothers, J. Phys. Chem. C 116, 26396 (2012)
31
Outline
1 DFT for alkyl cross-coupling
2 Other DFT applications
3 New functionals for surface chemistry
4 “Rung 3.5” density functionals
5 Summary
32
Rationalizing DFT’s alphabet soup
LSDA GGA
Chemical accuracyheaven
Hartree world
Meta-GGA
Hybrid
Fifth-rung Small and medium-sized molecules
Rung 3.5 functionals
Large molecules
Solids and surfaces
“Jacob’s Ladder” ofapproximate XCfunctionals
First 3 rungs often aren’taccurate enough forchemistry
4th rung often expensivefor solids & surfaces
We seek a compromise
33
Two problems with standard hybrid DFT
Metal surface, 5-10% screened nonlocal exchange
A
B
C
DCovalent bonds,~25% nonlocal exchange
Transition states,~50% nonlocal exchange
Long-range exact exchange is expensive
Fixed by screened hybrids
Optimum fraction of exact exchange varies for differentsystems & properties
34
A closer look at exchange
Exact exchange comes from the noninteracting system’snonlocal density matrix
E exX = −1
2
∫d3~r
∫d3~r ′|γ(~r ,~r ′)|2
|~r −~r ′|
γ(~r ,~r ′) =N∑i=1
φi (~r)φ∗i (~r ′)
Semilocal functionals use a semilocal model constructedfrom information about ~r
ESLX = −1
2
∫d3~r
∫d3~r ′|γSL(ρ(~r),∇ρ(~r), . . . ,~r −~r ′)|2
|~r −~r ′|
Hybrid exchange aE exX + (1− a)ESL
X models nondynamicalcorrelation (chemical bonding)
35
The “Rung 3.5” compromise
eX (~r) = −1
2
∫d3~r ′
|γ(~r ,~r ′)|2
|~r −~r ′|
eSLX (~r) = −1
2
∫d3~r ′
|γSL(ρ(~r),~r ′ −~r)|2
|~r −~r ′|
eΠX (~r) = −1
2
∫d3~r ′
γ(~r ,~r ′)γSL(ρ(~r),~r ′ −~r)
|~r −~r ′|
More delocalized than semilocal DFT
Less delocalized than |γ(~r ,~r ′)|2 (hybrids)
36
Benefit #1: γSL screens large |~r −~r ′|
eΠX (~r) = −1
2
∫d3~r ′
γ(~r ,~r ′)γSL(ρ(~r),~r ′ −~r)
|~r −~r ′|
-9
-8
-7
-6
-5
-4
-3
-2
10 12 14 16 18
Log1
0 a
bso
lute
en
erg
y e
rro
r (H
artr
ee
)
Range of nonlocal interactions (Angstrom)
PBE
HSE06
Pi-LDA
HSE-2X
B3LYP
B. G. Janesko, in preparation
37
Benefit #2: γ includes useful nonlocal information
G3/99 Reaction ReactionMethod ∆Ho
f Energies Barriers
PBE 17.9 3.5 8.9Π1PBE 9.5 2.2 7.6PBE0 5.0 2.2 3.9
6-311++G(2d,2p) MAE (kcal/mol) in G399 and NHTBH38/04data sets
A. Aguero, B. G. Janesko, J. Chem. Phys. 136, 024111 (2012)
38
Band gaps
LPPyB PA PITN PEDOT MEH-PPV PTh PCDTBT CN-PPV PFu PPV PPy0
0.5
1
1.5
2
2.5
Bandgap (
eV) Ref
B3LYP
HSE06
Π-LSDA
PBE+Π(s)
M06-L
TPSS
PBE
B. G. Janesko, J. Chem. Phys. 134, 184105 (2011)
39
Excitation energies & polarizabilities
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
LDA PBE PiLDA HSE PBE0
Fir
st e
xci
tati
on
en
erg
y (
eV
)
270
275
280
285
290
295
300
305
310
315
LDA PBE PiLDA HSE PBE0
Po
lari
zab
ilit
y (
au
)
G. Scalmani, M. J. Frisch, B. G. Janesko, in preparation
40
Potential for surface chemistry
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
E (e
V)
Reference
PBE
Π1-PBE
PBE+Π(s)
+
41
Nonempirical, nonlocal DFT
Lieb-Oxford bound: E exXC ≥ λELDA
X
Variational bound: E exXC ≤ E ex
X
Cauchy-Schwarz: eexX (~r) ≤ (eΠX (~r))2
eSLX (~r)
Odashima & Capelle 2009
EXC ' βE exX + (1− β)λELDA
X
42
More flexible approximations
Odashima, Capelle, Haunschild, Perdew, Scuseria 2012
EXC =
∫d3~r eXC (~r)
eXC (~r) ' β(~r)eexX (~r) + (1− β(~r))λeLDAX (~r)
Our Rung 3.5 version
eXC (~r) ' β(~r)(eΠ
X (~r))2
eSLX (~r)+ (1− β(~r))λeLDAX (~r)
43
Nomempirical Rung 3.5 functionals
Method Params ∆Hof Barriers Bonds
Semilocal PBE 0 17.9 8.9 2.83Π-PBE 1 19.5 9.5 2.42ΠOC 0 14.3 7.4 2.62PBE0 hybrid 1 4.2 3.9 1.37
One of the most accurate nonempirical DFT methods
B. G. Janesko, J. Chem. Phys. 137, 224110 (2012)
44
Outline
1 DFT for alkyl cross-coupling
2 Other DFT applications
3 New functionals for surface chemistry
4 “Rung 3.5” density functionals
5 Summary
45
Summary
DFT calculations can quantify and extend chemical intuition,helping design new catalysts and new materials
Screened hybrids fix some of standard DFT’s limitations forsurface chemistry
“Rung 3.5” DFT functionals provide potential for furtherimprovements
46
Acknowledgements
Group: Bimal Pudasaini, John Determan, Mark Bridle
Undergraduates: Austin Aguero, Jessie Girgis, Katelyn Poole
Collaborators: Ed Brothers (Texas A&M Qatar), Jean-LucMontchamp (TCU), Juan Peralta (Central MichiganUniversity), Mihaela Stefan (UT Dallas), Gaussian, Inc.
TCU startup and undergraduate research funds
Qatar National Research Foundation, NSF REU
47
Extra slides
48
Oxidative addition, no Lewis base
49
β-elimination from OA product
50
Transmetalation
51
β-elimination from TM product
52
γSL models
γLDA1(ρ(~r), |~r −~r ′|) = ρ(~r) exp(−by2
)γGGA(ρ,∇ρ,u) = γLDA2(ρ, u) +
1
2u · ∇ρ exp
(f (s)y2
)∫
d3~rFPBEX (s)
(w(~r)eΠLDA1
X [ρ](~r) + (1− w(r))eLDAX (ρ(~r)))
53
Exchange decay in hydrogen chain
-6
-4
-2
0
2
4
6
12 16 20 24 28 32
Tota
l en
erg
y e
rro
r (1
0^(
-7)
Har
tre
e)
Range of nonlocal interactions (Angstrom)
54
Models for γSL
Simple:
γLDA(ρ(~r), |~r −~r ′|) = ρ(~r)e−by2
y = ρ(~r)1/3|~r −~r ′|
More complicated:
γGGA(ρ,∇ρ,~r −~r ′) = γLDA(ρ, |~r −~r ′|)
+1
2∇ρ · (~r ′ −~r ′)e−f (s)y2
s =|∇ρ|ρ4/3
55
Practical Rung 3.5 implementation
Practical DFT calculations expand the orbitals and densitymatrix in a finite basis set fixed throughout the calculation
φi (~r) =M∑µ=1
ciµχµ(~r)
γ(~r ,~r ′) =∑µν
Pµνχµ(~r)χν(~r ′)
56
Practical Rung 3.5 implementation
Rung 3.5 calculations are more practical if we expand themodel density matrix in a different finite basis
γSL(ρ(~r),~r ′ −~r) =M′∑η=1
cη(ρ(~r)) exp(−αη|~r ′ −~r |2
)eΠX (~r) =
∫d3~r ′
γ(~r ,~r ′)γSL(ρ(~r),~r −~r ′)|~r −~r ′|
=∑µνη
Pµνcη(ρ(~r))χµ(~r)Aνη(~r)
Aνη(~r) =
∫d3~r ′
χν(~r ′) exp(−αη|~r ′ −~r |2
)|~r ′ −~r |
G. Scalmani, M. J. Frisch, B. G. Janesko, in preparation