ファン・デル・ワールス密度汎関数の開発と応用
濱田幾太郎
東北大学

原子分子材料科学高等研究機構（WPI-AIMR）

ikutaro@wpi-aimr.tohoku.ac.jp

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講演内容
• イントロダクション• 研究の背景• ファン・デル・ワールス密度汎関数（vdW-DF）の紹介• 計算手法とその応用例• 研究の動機となったvdW-DFの問題点• 本研究課題での研究内容• vdW-DFの応用計算例の紹介（吸着系を中心に）• まとめと展望

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(Semi-)local approximationin 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
revPBE
LDAnl
Exc = Ex+ Ec+ Ec

Nonlocal correlation which accounts for vdW interaction

1nl
Ec =2!!

dr1 dr2 n(r1 )φ(q1 , q2 , r12 )n(r2 )qi = q0 (n(ri ), |∇n(ri )|)r12 = |r1 − r2 |Dion et al., Phys. Rev. Lett. 92, 246401 (2004).

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vdW-DF is able to describe vdW interaction
Rare gas dimer

GGA

vdW-DF

Dion et al., Phys. Rev. Lett. 92, 246401 (2004).

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vdW-DF gives essentially the same results as PBEGGA for water monomer
dOH
(Å)

θHOH
(degree)

PBE*

0.975

104.5

vdW-DF

0.972

104.6

CCSD(T)

0.958

104.5

Expt.

0.958

104.5

θHOH
dOH
Potential energy curve*Perdew-Burke-Ernzerhof GGAIH, J. Chem. Phys. 133, 214503 (2010).

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vdW-DF is able to describe covalent bonding
Graphene
0.12

vdW!DFLDA
PBE

0.10

!Etot (eV)

0.08
0.06
0.04

PBE

vdW-DF

0.02
0.00
0.242

0.244
0.246
0.248
Lattice constant (nm)0.25

IH and M. Otani, Phys. Rev. B 82, 153412 (2010).

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vdW-DF yields more accurate binding energy of ice Ih,but equilibrium volume is slightly overestimatedBinding energy (Eb), lattice parameter (c/a), andequilibrium volume (V0), bulk modulus (B0)c/aV0(Å3/H2O)

B0(GPa)

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

Eb
(eV/H2O)PBE

CCSD(T): A. Hermann, P. Schwerdtfeger, Phys. Rev. Lett. 101, 183005 (2008).

IH, J. Chem. Phys. 133, 214503 (2010).

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Problems in vdW-DF• Equilibrium distanceoverestimatedRare gas dimer• Too repulsive exchange
energy used

GGA
• vdW interaction nearequilibrium overestimated• Too large vdW attraction ofvdW-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

A. Varykhalov, et al., Phys. Rev. Lett. 101, 157601 (2008)

vdW-DF calculation

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.

0.42

0.37

0.240.234

Δφ (eV) -0.15-0.37-0.97-0.90

ZC11K. Toyoda, IH, K. Lee, S. Yanagisawa, Y. Morikawa, J. Chem. Phys. 132, 134703 (2010)

Problems in vdW-DF• Equilibrium distanceoverestimatedRare gas dimer• Too repulsive exchange
energy usedGGA• vdW interaction nearequilibrium overestimated• Too large vdW attraction ofvdW-DF nonlocal correlationnear the equilibrium

vdW-DF

vdW-DFExc= ExrevPBE + EcLDA + Ecnl

Dion et al., Phys. Rev. Lett. 92, 246401 (2004). 12

The revPBE exchange reasonably reproduce the HartreeFock exchange, but too repulsive at small separation

Dion et al., Phys. Rev. Lett. 92, 246401 (2004).

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研究課題
• vdW-DFの枠内での精度良い計算に向けて• vdW-DFのための高精度な交換エネルギー汎関数の開発• vdW-DFの相関項の改善• vdW-DFのより効率良い計算に向けて• Self-consistent vdW-DF potentialのSTATEへの実装

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研究課題
• vdW-DFの枠内での精度良い計算に向けて• 新しい交換エネルギーの応用と検証• 相関エネルギーはLeeらのvdW-DF2で既に改善• vdW-DFのより効率良い計算に向けて• Self-consistent vdW-DF potentialのSTATEへの実装

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Improved vdW-DF: vdW-DF2C09x• 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 electronicstructure
IH and M. Otani, Phys. Rev. B 82, 153412 (2010).
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Exchange enhancement factor
ExGGA =

2.2

drneunifx (n)Fx (s)Reproduce revPBE at a large
reduced gradientPBE
revPBE
C09

2.4!

Fx

2.0
1.8
1.6
Small Fx at a small reduced gradient
to reduce repulsion at short distance

1.4
1.2
1.00246810s

C09 [Cooper, Phys. Rev. B 81, 161104(R) (2010)]

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Application of improved vdW-DF toadsorption systems•Graphene/metal•C60/metal
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C09 exchange and vdW-DF2 correlation
improves the adsorption distance of graphene
Graphene/Ni(111)

Binding energy (meV)300
PBE
vdW-DF200

vdW-DF2

C09x

1000Expt.
-100

-200

0.2

0.3

0.4

0.5

Adsorption distance (nm)
IH and M. Otani, Phys. Rev. B 82, 153412 (2010).

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Band-gap opening at the K point on Ni (111) isreproduced by vdW-DF2C09x
vdW-DF2C09x band structure

IH and M. Otani, Phys. Rev. B 82, 153412 (2010).

ARPES experiment

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)
vdW-DF
vdW-DF2
C09
vdW-DF x
C09
vdW-DF2 x

Binding energy (meV)

200

1000Expt.

-100
0.2

0.3

0.4

0.5

Adsorption distance (nm)21Electronic structure of graphene varies depending on thenature of the substrate
Band structures of graphene on metal surfaces
calculated at the equilibrium distances by vdW-DF2C09x

Ni

E-EF (eV)

50Cu HighEF-5-10

Low-15n-type doping

EFE-EF (eV)-20
5Pd0Au-5EF-10

p-type doping -15-20

Γn-type dopingΚΜΓΚ

IH and M. Otani, Phys. Rev. B 82, 153412 (2010).

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C60/Au(111): accurate binding energy withvdW-DF2C09x

Binding energy (eV)1Distance

0-1
PBE
C09
vdW-DF2 x-2

Expt.
0.1

0.2

0.3

0.4

Distance (nm)

IH and M. Tsukada, Phys. Rev. B 83, 245437 (2011).

0.50.623

C09 exchange and vdW-DF2 correlation
improves the interlayer distance of graphite

cInterlayer binding energy (meV)Graphite
LDAPBE
vdW-DF
vdW-DF2
C09
vdW-DF x
C09
vdW-DF2 x
Expt.

7550250-25-50-75
0.55

c (nm)

0.60

0.70
0.65
Lattice constant c (nm)0.75

LDAPBE

vdW-DF

vdW-DF2C09x

Expt.

0.664

0.853

0.714

0.652

0.667224

C09x
vdW-DF2
predicts accurate geometry of solid

Hexagonal Boron nitridecInterlayer binding energy (meV)75LDAPBE
vdW-DF
C09
vdW-DF2 zExperimental
interlayer distance50250-25-50-750.55

c (nm)

0.60

0.65

0.70

Lattice constant c (nm)

LDA

PBE

vdW-DF

vdW-DF2C09x

Expt.

0.646

0.848

0.708

0.642

0.6603

0.75

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まとめ
• vdW-DFの問題点と改善、応用• vdW-DF2C09xの提案• 分子/金属界面だけではなく、固体も精度良く計算可能に

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今後の展望• STATEへのself-consistent vdW-DF potentialの実装[Public domainの幾つかのコードでは既に使用可能に(Quantum-espresso、SIESTA)]• vdW-DF2C09xの様々な系への適用• 水、氷、水/金属界面• 有機/金属界面• 分子性結晶への適用計算

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