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NSF NIH ONR DOE
Density Functional Theory and Non-covalent Interactions
Weitao Yang, Duke University
ARO Workshop Univ of Delawere August, 2012
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Outline
● Empirical Corrections for vdw Interactions ● Revealing Non-Covalent Interactions ● Isodesmic reaction energies of n-alkanes
to form ethane ● Eaxct conditions on functionals and their
violation
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LDA and GGA do not describe vdw interaction
For rare gas diatomic molecules: • LDA severely over-binds for rare gas diatomic molecules. • GGA only displays repulsive potential curves with no
minimum. Kristyan and Pulay, Chem. Phys. Lett. 229, 175 (1994) Perez-Jorda and Becke, Chem. Phys. Lett. 233, 134 (1995)
• GGA does not make any improvement over LDA systemmatically.
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The GGA exchange functional
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small density and large density gradient play a very important role in GGA Zhang, Pan and Yang, JCP, 107, 7921 (1997) Ne Dimer
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Empirical Correction for vdw Interaction
● with Hartree-Fock Theory
R. Ahlrichs, R. Penco, and G. Scoles, Chem. Phys. 19, 119 (1977)
● with tight-binding theory (SCC-DFTB)
Elstner et al., J. Chem. Phys. 114, 5149 (2001)
H.Liu, M. Elstner, E. Kaxiras, T. Frauenheim, J. Hermans, and W. Yang, Proteins: Struct., Funct., Genet. 44, 484 (2001)
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● with density functional theory
Qin Wu and Weitao Yang, JCP, 116, 515(2002)
Dampping functions
Combination rules
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General improvement. Accuracy is functional dependent
Solid line: accurate values. Circles: B3LYP. Triangles: BLYP. Stars: BPW91. Pluses: PW91xc.
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Empirical Correction: DFT-D
S. Grimme, J Comput Chem 25, 1463(2004)
• Functional-dependent parameterization
• a global scaling factor for the atomic coefficients Optimum scale factors of 1.4, 1.3, and 0.7 for BLYP, BP86, and PBE
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DFT-D3: ab initio parametrization of DFT-D for the 94 elements H-Pu
Stefan Grimme, Jens Antony, Stephan Ehrlich, and Helge Krieg, J. Chem. Phys. 132, 154104 (2010)
• Atom-pairwise specific dispersion coefficients and cutoff radii that are both computed from first principles.
• The coefficients for new eighth-order dispersion terms are computed.
• System geometry dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination numbers CN.
• Two global parameters for each density functional
• Three-body nonadditivity terms are considered
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Main ingredients
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Beyond DFT-D: calculating C6
● Empirical corrections, such as DFT-D – ● No electronic structure calculation is involved
with the caculations of C6. ● Very low cost, like MM force field ● It can be viewed as a frozen density
approximation
● Calculating C6, C8 from electronic structure information.
● Becke and Johnson ● Scheffler and coworkers ● …
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Can we use the density to detect non-covalent interactions in real space ?
C1
H
H
HH
C1
H
H HH
Covalent bonds Dispersion?
Non Covalent Interaction in Real Space?
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Erin Johnson (Duke, now UC-Merced)
Revealing Non-covalent interactions
Julia Conteras-Garcia
Paula Mori-Sanchez (Duke, now Univ. Autonoma Madrid)
Aron J. Cohen (Duke, now Cambridge)
Shahar Keinan
Johnson, Contreras, Keinan, Mori-Sanchez, Cohen , and WY, JACS, 2010. Contreras , Johnson, Keinan,Chaudret, Piquemal, Beratan, and WY J. Chem. Theory Comput. 2011
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and
The features of the density between molecules
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AO tails
covalent bonds
nuclei
Methane
C1
H
H HH
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Methane and Water
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Two Methane Molecules
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non-covalent interactions
Two Water and Two Methane Molecules
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Attractive and repulsive interactions can be distinguished by the sign of the second Hessian eigenvalue (λΗ).
This value corresponds to the variation of the density along the axes of maximal curvature.
λH = 0
density accumulation λH < 0
λH > 0 density depletion
Hessian of Density
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Plot gradient isosurfaces in real space and color by sign(λΗ)ρ
Strong, attractive interactions:
H-bonding
Strong, repulsive interactions:
steric crowding
ρ ≈ 0 λH ≈ 0
ρ > 0 λH > 0
ρ > 0 λH < 0
increasing ρ increasing ρ
Weak interactions:
van der Waals, including dispersion
Seeing the types and strengths of interactions
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H-bonding
Steric repulsion Dispersion
Seeing the types and strengths of interactions
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Seeing the types and strengths of interactions
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Graphite
Diamond
Bulk matter
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Use promolecular densities
pro atomi
iρ ρ= ∑
For Large Systems
• Any reasonable electron density can be used,
• only requires input atomic coordinates,
• shows continuous surfaces rather than pair-wise contacts.
This offers exciting possibilities for
• analysis of interactions in, and between, biomolecules,
• design of ligands and catalysts,
• self-assembled materials.
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c) d)
SCF PROMOLECULAR
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Beta sheet of protein
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Alpha helix of protein
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Double helix of DNA
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A
B C
A V5X ligand in the active site of HDAC8 protein
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•Non-covalent interactions have a unique signature and their presence can be revealed solely from the electron density.
•Non-covalent interactions are highly non-local and manifest in real space as low-gradient isosurfaces with low densities. •This approach provides a rapid and rich representation of van der Waals interactions, hydrogen-bonds, and steric repulsion, requiring only the atomic coordinates as input.
•Programs freely available: http://www.chem.duke.edu/~yang/Software/softwareNCI.html
Summary on NCI
Johnson, Contreras, Keinan, Mori-Sanchez, Cohen , and WY, JACS, 2010. Contreras , Johnson, Keinan,Chaudret, Piquemal, Beratan, and WY J. Chem. Theory Comput. 2011
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Quantitative energetic information on hydrogen bonding
● NCI is shown to allow quantitative treatment of hydrogen-bond energetics.
● The evolution of NCI regions along a PES follows a well-behaved pattern which, upon integration of the electron density, is capable of mimicking conventional hydrogen-bond interatomic potentials.
Contreras-García, Weitao Yang and Erin R. Johnson, J. Phys. Chem. A, 2011
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Quantitative energetic information on hydrogen bonding
Contreras-García, Weitao Yang and Erin R. Johnson, J. Phys. Chem. A, 2011
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A Challenge
● Vdw interactions are nonlocal ● How to extract quantitative energetic
information for vdw interaction, from local information in density and reduced gradient
● GGA like? (Truhlar’s talk)
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Isodesmic reaction energies of n-alkanes to form ethane
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Many Studies ● Several explanations have been proposed, involving
both the exchange or correlation nature of the problem and its distance range (i.e., medium-range or long-range interactions).
● More complex functional forms, involving range-separation and dispersion, or excited-state information.
● Using MP2-based correlation energy terms (double-hybrid functionals), can recover good agreement with experimental results for these isodesmic reaction energies.
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Johnson, Contreras-García, WY, JCTC, in press, 2012
Experimental reaction energy of 0.04 kcal/mol for butane, increasing to 0.23 kcal/mol for n-octane. Therefore, this reaction is essentially thermoneutral.
New Isodesmic Reactions
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eq 1
eq 7
eq 6
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• The reaction energy differences originate from localized interactions between contiguous CH2 units in the n-alkane, i.e., from 1,3 interactions.
• The reduced density gradient has smaller values for noncovalent 1,3 interactions in n-alkane reactants compared to ethane products.
• The gradient contribution to the exchange energy is consequently reduced, giving a constant energy bias against each propane unit in an n-alkane.
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Density Functional Theory • DFT is exact and should give agreement with experiment or
high-level ab initio calculations in all situations. • Approximate functionals perform well in many systems but
can fail dramatically in other situations.
• This can be traced back to errors of DFA (density functional approximation)
• The understanding of these errors will hopefully lead to new and improved functionals.
• The same challenges for other approximate QM methods.
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A large class of problems • Wrong dissociation limit for molecules and ions • Over-binding of charge transfer complex • too low reaction barriers • Overestimation of polarizabilities and hyperpolarizabilities • Overestimation of molecular conductance in molecular electronics • Incorrect long-range behavior of the exchange-correlation potential • Charge-transfer excited states • Band gaps too small • Diels-Alder reactions, highly branched alkanes, dimerization of
aluminum complexes
Fractional Charges
Delocalization Error
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DFT for fractional number of electrons from grand ensembles, Perdew, Parr, Levy, and Balduz, PRL. 1982
Yang, Zhang and Ayers, PRL, 2000 – pure states in molecular dissociation
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E(N)
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Delocalization Error Define the Delocalization Error as the violation of the linearity condition for fractional charges
Cohen, Mori-Sanchez and Yang, 2008 Science
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Delocalization Error in GGA, LDA, B3LYP
• Energy of dissociation of molecular ion: too low • Charge transfer complex energy: too low • Transition state energy: too low • Charge transfer excitation energy: too low • Band gap: too low • Molecular conductance: too high • (Hyper)polarizability for long molecules: too high • Diels-Alder reaction products, highly branched
alkanes, dimerization of aluminum complexes: too high
Too low energy for fractional charge systems
Cohen, Mori-Sanchez and Yang, 2008 Science
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Another large class of problems • huge error dissociation of chemical bonds • transition metal dimmers • some magnetic properties • strongly correlated systems • Mott insulators, high superconductors • degeneracy and near degeneracy
cT
Fractional Spins
Static Correlation Error
Aron Cohen, Paula Mori-Sanchez and WY, JCP, 2008
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Static Correlation Error
Define the Static Correlation Error as the violation of the constancy condition for fractional spins
Cohen, Mori-Sanchez and Yang, 2008, JCP; 2008 Science
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Exact conditions on DFT
, ,[ ]i N i N i NiE c E Eρ ρ = = ∑
[ ]1 1(1 ) (1 )N N N NE E Eδ ρ δρ δ δ+ +− + = − +
, 1, 1(1 ) (1 )i N i j N j N Ni jE c d E Eδ ρ δ ρ δ δ+ +
− + = − + ∑ ∑• !! The exact XC functional cannot be an explicit and differentiable functional of the electron density/density matrix, either local or nonlocal.
•Valid for density functionals, and also for 1-body density matrix functionals, 2-RDM theory, and other many-body theories.
Fractional Charge: 1982: Perdew, Levy, Parr and Baldus
Fractional Spins: 2000, PRL, WY, Zhang and Ayers; 2008, JCP, Cohen, Moris-Sanchez, and WY
Fractional Charges and Spins: 2009: PRL, Moris-Sanchez, Cohen and WY
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: , ,Falt plan for Hydrogen H H and H+ −−
H+
H(↓)
H-
H(↑)
Fractional Charges and Spins: PRL 2009, Cohen, Moris-Sanchez, and WY
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B3LYP
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Analytical Evaluation of Fukui Functions and New Conditions on the Exact Functional
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WY, Aron J. Cohen, Frank De Proft and Paul Geerlings JCP, 2012. Work initiated during my visit to Vrije Universiteit Brussel (VUB), as the International Solvay Chair in Chemistry
Fukui funcitons Parr and Yang, 1984
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Hardness Conditions-New Exact Conditions
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Fukui Function Conditions-New Exact Conditions
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The fukui function and hardness conditions
Pointwise (local) constraints on the exact functionals
Hardness conditions
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Improving band gap prediction in density functional theory from
molecules to solids PRL, 2011, Xiao Zheng, Aron J. Cohen, Paula Mori-
Sanchez, Xiangqian Hu, and Weitao Yang
Paula Mori-Sanchez (Univ. Autonoma Madrid)
Xiao Zheng (USTC)
Aron J. Cohen (Cambridge) Xiangqiang Hu
Understanding what is required of the functiona helps improving approximationsl
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Energy versus fractional electron number
The SC significantly restores linearity condition for energy
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Band gaps of H-passivated Si nanocrystals Computational details Systems under study: H-passivated spherical Si nanocrystals. The largest
system is Si191H148 with a diameter of 20 angstrom
Geometries optimized by B3LYP with Lanl2dz ECP basis set, except that the largest system Si191H148 is optimized with semiempirical PM3 method
Diffusive basis functions are important to obtain accurate HOMO-LUMO gaps: 6-31G for H atoms; an sp-shell with an exponent of 0.0237 au and a d-shell with an exponent of 0.296 au added to Lanl2dz basis for Si atoms
Si191H148
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Band gaps of H-passivated Si nanocrystals
The S-MLDA HOMO-LUMO gaps agree well with the GW gaps
S-MLDA
LDA