NSF NIH ONR DOE

Density Functional Theory and Non-covalent Interactions

Weitao Yang, Duke University

ARO Workshop Univ of Delawere August, 2012

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

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.

The GGA exchange functional

small density and large density gradient play a very important role in GGA Zhang, Pan and Yang, JCP, 107, 7921 (1997) Ne Dimer

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)

● with density functional theory

Qin Wu and Weitao Yang, JCP, 116, 515(2002)

Dampping functions

Combination rules

General improvement. Accuracy is functional dependent

Solid line: accurate values. Circles: B3LYP. Triangles: BLYP. Stars: BPW91. Pluses: PW91xc.

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

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

Main ingredients

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 ● …

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?

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

and

The features of the density between molecules

AO tails

covalent bonds

nuclei

Methane

C1

H

H HH

Methane and Water

Two Methane Molecules

non-covalent interactions

Two Water and Two Methane Molecules

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

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

H-bonding

Steric repulsion Dispersion

Seeing the types and strengths of interactions

Seeing the types and strengths of interactions

Graphite

Diamond

Bulk matter

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.

c) d)

SCF PROMOLECULAR

Beta sheet of protein

Alpha helix of protein

Double helix of DNA

A

B C

A V5X ligand in the active site of HDAC8 protein

•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

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

Quantitative energetic information on hydrogen bonding

Contreras-García, Weitao Yang and Erin R. Johnson, J. Phys. Chem. A, 2011

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)

Isodesmic reaction energies of n-alkanes to form ethane

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.

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

eq 1

eq 7

eq 6

• 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.

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.

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

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

E(N)

Delocalization Error Define the Delocalization Error as the violation of the linearity condition for fractional charges

Cohen, Mori-Sanchez and Yang, 2008 Science

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

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

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

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

: , ,Falt plan for Hydrogen H H and H+ −−

H+

H(↓)

H-

H(↑)

Fractional Charges and Spins: PRL 2009, Cohen, Moris-Sanchez, and WY

B3LYP

Analytical Evaluation of Fukui Functions and New Conditions on the Exact Functional

58

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

60

Hardness Conditions-New Exact Conditions

61

Fukui Function Conditions-New Exact Conditions

The fukui function and hardness conditions

Pointwise (local) constraints on the exact functionals

Hardness conditions

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

64

Energy versus fractional electron number

The SC significantly restores linearity condition for energy

65

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

66

Band gaps of H-passivated Si nanocrystals

The S-MLDA HOMO-LUMO gaps agree well with the GW gaps

S-MLDA

LDA