Tutorial 3:van der Waals Interactions

Jan Hermann Alexandre TkatchenkoFritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin

Hands-On Summer SchoolLos Angeles, July 24, 2014

Pairwise approaches to van der Waals

Interatomic methods Non-local functionals

• Grimme'sDFT-Dn1

• Tkatchenko–Scheffler2 C� functionals of density

+ Fast,simple − Atomic parameters input − Problems with metals and

charge-transfer character

• Langreth–Lundqvist3

• Vydrov–vanVoorhis4

+ Pure density functionals − Questionable for

molecules,metals

1Grimme, J. Comp. Chem. (2004) 2Tkatchenko, Scheffler, Phys. Rev. Lett. (2009)3Dion, Phys. Rev. Lett. (2004) 4Vydrov, van Voorhis, Phys. Rev. Lett (2004)

? n(r�)Φ[n](r�, r�)n(r�)dr�dr�∑i< j

f (Ri j)C6,i j[n]R6i j

How to go beyond pairwise van der Waals

Many-body dispersion1 EXX+RPA@DFT2

• χ� from localized atomic response functions

• Basedondipoleharmonicoscillators

+ Can be solved analytically − Open problem with metals

• χ� from delocalized molecular orbitals

+ Works for all sorts of systems − Costly numerical solution

1Tkatchenko, DiStasio, Car, Scheffler, Phys. Rev. Lett. (2012)2Zhu, Toulouse, Savin, Ángyán, J. Chem. Phys. (2010) and refs therein

• Random-phaseapproximation(RPA)framework

Ec = −��π

=ô

�dω

ô

9n=�

�n Tr[(χ�V )n]

BenzeneisabuildingelementofmanyvdWsystems

Frombenzenemoleculetographite

• Benzenemolecule• Benzenedimer• Benzenecrystal• Benzenechain• Benzeneongraphene• Graphenebilayer• Graphenemultilayer

3

2 α̃ = α αT<Rα̃

1

How many-body dispersion works

Self-consistentscreening• Responsefunctionsofnear

atoms interact

• Eachatomreplacedbyoscillator with α from Tkatchenko–Scheffler

Random-phase approx• Givesthelong-rangevan

der Waals energy• Solvedanalytically

Ec = −12π

∫∞

0dω

∞

∑n=2

1n Tr[(α̃T>R)n]

α = αfreeV[n]Vfree

Non-empirical vdW methods provide more info

• noinformationbeyondenergy

• polarizability• anisotropy• dipolefluctuationmodes

Fromempirical... ...tofirstprinciples

Structureisthestartingpointofmostcalculations

• Localminimaonthepotential energy surface

• Transitionstates• Equilibriumstructure

(globalminimum)

Benzenedimer• TestcaseforvanderWaals

for decades• StrongvdWinteractions• Quadrupole–quadrupole

interactions• Stronganisotropy

Many-body effects get interesting in crystals

• Many-bodyeffectsgetamplified• Multipledirectionsofinteraction• Anisotropy

Emergenceoflong-rangefluctuationsinlow-dimensional systems• Ittakeslongerdistancesbeforemany-bodyeffectsgetsaturated• 2Dsystems:physisorption,interfaces• 1Dsystems:self-assembly,quantumwires