WPI-AIMR ikutaro@wpi-aimr.tohoku.ac.jp 1
vdw-df vdw-df vdw-df 2
(Semi-)local approximation in density-functional theory - Local density approximation (LDA) Good structural and dynamical properties of solids Too large atomization energy (cohesive energy) - Generalized gradient approximation (GGA) Better structural properties of molecules Better atomization energies for molecules and solids Overestimation of lattice constants (hence underestimation of bulk modulus and vibrational frequencies) Good for describing covalent/ionic bonds Good for hydrogen bonds Lack of long-range dispersion (van der Waals) forces: Weak interactions are described inaccurately 3
van der Waals density functional (vdw-df) Exchange-correlation E xc = E revpbe x + E LDA c + E nl c Nonlocal correlation which accounts for vdw interaction E nl c = 1 2 dr 1 dr 2 n(r 1 )φ(q 1,q 2,r 12 )n(r 2 ) q i = q 0 (n(r i ), n(r i ) ) r 12 = r 1 r 2 Dion et al., Phys. Rev. Lett. 92, 246401 (2004). 4
vdw-df is able to describe vdw interaction Rare gas dimer GGA vdw-df Dion et al., Phys. Rev. Lett. 92, 246401 (2004). 5
vdw-df gives essentially the same results as PBE- GGA for water monomer θhoh doh (A ) θhoh (degree) doh Potential energy curve PBE* 0.975 104.5 vdw-df 0.972 104.6 CCSD(T) 0.958 104.5 Expt. 0.958 104.5 *Perdew-Burke-Ernzerhof GGA IH, J. Chem. Phys. 133, 214503 (2010). 6
vdw-df is able to describe covalent bonding 0.12 0.10 Graphene vdw!df LDA PBE!E tot (ev) 0.08 0.06 0.04 0.02 PBE vdw-df 0.00 0.242 0.244 0.246 0.248 0.25 Lattice constant (nm) IH and M. Otani, Phys. Rev. B 82, 153412 (2010). 7
vdw-df yields more accurate binding energy of ice Ih, but equilibrium volume is slightly overestimated Binding energy (Eb), lattice parameter (c/a), and equilibrium volume (V0), bulk modulus (B0) Eb (ev/h2o) c/a V0 (A 3 /H2O) B0 (GPa) PBE 0.665 1.636 30.23 15.6 vdw-df 0.602 1.632 34.01 10.5 CCSD(T) 0.577 1.627 32.10 15.2 Expt. 0.580 1.628 32.05 12.1 CCSD(T): A. Hermann, P. Schwerdtfeger, Phys. Rev. Lett. 101, 183005 (2008). IH, J. Chem. Phys. 133, 214503 (2010). 8
Problems in vdw-df Equilibrium distance overestimated Too repulsive exchange energy used Rare gas dimer GGA vdw interaction near equilibrium overestimated Too large vdw attraction of vdw-df nonlocal correlation vdw-df Dion et al., Phys. Rev. Lett. 92, 246401 (2004). 9
Band-gap opening at the K point of graphene/ni(111) is not reproduced by vdw-df ARPES experiment vdw-df calculation A. Varykhalov, et al., Phys. Rev. Lett. 101, 157601 (2008) M. Vanin, et al., Phys. Rev. B 81, 081408(R) (2010) 10
Adsorption energy is calculated accurately, while adsorption distance is generally overestimated by vdw-df Expt. GGA vdw-df DFT-D Expt. Z C 0.42 0.37 0.24 0.234 Δφ (ev) -0.15-0.37-0.97-0.90 K. Toyoda, IH, K. Lee, S. Yanagisawa, Y. Morikawa, J. Chem. Phys. 132, 134703 (2010) 11
Problems in vdw-df Equilibrium distance overestimated Too repulsive exchange energy used Rare gas dimer GGA vdw interaction near equilibrium overestimated Too large vdw attraction of vdw-df nonlocal correlation near the equilibrium vdw-df E vdw-df xc = E revpbe x + E LDA c + E nl c Dion et al., Phys. Rev. Lett. 92, 246401 (2004). 12
The revpbe exchange reasonably reproduce the Hartree- Fock exchange, but too repulsive at small separation Dion et al., Phys. Rev. Lett. 92, 246401 (2004). 13
vdw-df vdw-df vdw-df vdw-df Self-consistent vdw-df potential STATE 14
vdw-df Lee vdw-df2 vdw-df Self-consistent vdw-df potential STATE 15
Improved vdw-df: vdw-df2 C09x Exchange functional of Cooper (C09) Phys. Rev B 81, 161104(R) (2010) Nonlocal correlation of Lee et al (vdw-df2) Phys. Rev. B 82, 081101(R) (2010). for improved description of interface geometry and electronic structure IH and M. Otani, Phys. Rev. B 82, 153412 (2010). 16
Exchange enhancement factor E GGA x = drne unif x (n)f x (s) 2.4 2.2 2.0 PBE revpbe C09 Reproduce revpbe at a large reduced gradient F x 1.8 1.6 1.4 1.2 1.0 Small Fx at a small reduced gradient to reduce repulsion at short distance 0 2 4 6 8 10 s C09 [Cooper, Phys. Rev. B 81, 161104(R) (2010)] 17
Application of improved vdw-df to adsorption systems Graphene/metal C60/metal 18
C09 exchange and vdw-df2 correlation improves the adsorption distance of graphene 300 Graphene/Ni(111) 200 PBE vdw-df Binding energy (mev) 100 0 Expt. vdw-df2 C09 x -100-200 0.2 0.3 0.4 0.5 Adsorption distance (nm) IH and M. Otani, Phys. Rev. B 82, 153412 (2010). 19
Band-gap opening at the K point on Ni (111) is reproduced by vdw-df2 C09x vdw-df2 C09x band structure ARPES experiment IH and M. Otani, Phys. Rev. B 82, 153412 (2010). A. Varykhalov, et al., Phys. Rev. Lett. 101, 157601 (2008) 20
C09 exchange and vdw-df2 correlation improves the adsorption distance of graphene Graphene/Pt(111) Binding energy (mev) 200 100 0 Expt. vdw-df vdw-df2 vdw-df C09 x vdw-df2 C09 x -100 0.2 0.3 0.4 0.5 Adsorption distance (nm) 21
Electronic structure of graphene varies depending on the nature of the substrate Band structures of graphene on metal surfaces calculated at the equilibrium distances by vdw-df2c09x Ni E-EF (ev) 5 0 Cu High EF -5-10 Low -15 n-type doping EF E-EF (ev) -20 5 Pd 0 Au -5 EF -10 p-type doping -15-20 Γ n-type doping Κ ΜΓ Κ IH and M. Otani, Phys. Rev. B 82, 153412 (2010). Μ 22
C60/Au(111): accurate binding energy with vdw-df2 C09x 1 Binding energy (ev) 0-1 PBE Distance vdw-df2 C09 x -2 Expt. 0.1 0.2 0.3 0.4 0.5 0.6 Distance (nm) IH and M. Tsukada, Phys. Rev. B 83, 245437 (2011). 23
C09 exchange and vdw-df2 correlation improves the interlayer distance of graphite Graphite c Interlayer binding energy (mev) 75 50 25 0-25 -50-75 LDA PBE vdw-df vdw-df2 vdw-df C09 x vdw-df2 C09 x Expt. 0.55 0.60 0.65 0.70 0.75 Lattice constant c (nm) LDA PBE vdw-df vdw-df2 C09x Expt. c (nm) 0.664 0.853 0.714 0.652 0.6672 24
vdw-df2 C09x predicts accurate geometry of solid Hexagonal Boron nitride c Interlayer binding energy (mev) 75 50 25 0-25 -50 Experimental interlayer distance LDA PBE vdw-df vdw-df2 C09 z -75 0.55 0.60 0.65 0.70 0.75 Lattice constant c (nm) LDA PBE vdw-df vdw-df2 C09x Expt. c (nm) 0.646 0.848 0.708 0.642 0.6603 25
vdw-df vdw-df2 C09x / 26
STATE self-consistent vdw-df potential [Public domain (Quantum-espresso SIESTA)] vdw-df2 C09x / / 27