Lecture on First-principles Computations (18):

van der Waals Interactions

任新国 (Xinguo Ren)

中国科学技术大学量子信息重点实验室

Hefei, 2018.11.16

Key Laboratory of Quantum Information, USTC

Van der Waals (vdW) interactions

RAB

A B

++ +

++ -

-

-

--

-

-

+

++ -

-

-

+

+

+

-

-+

-

-

+

+

+

-

-+

RAB

A B

t=t1

t=t2

𝐸𝑑𝑖𝑠𝑝 = −𝐶6

𝑅𝐴𝐵6 −

𝐶8

𝑅𝐴𝐵8 −

𝐶10

𝑅𝐴𝐵10 +⋯

Interaction between fluctuating multipoles (dispersion forces)

𝐶6𝐴𝐵 =

3

πන𝑑ωα𝐴 𝑖ω α𝐵 𝑖ω

Electric dipole polarizability

vdW interactions are ubiquitous

Describing vdW interactions within DFT is a challenge

LDA, GGAs, and hybrid functions don't capture the 1/R6 vdW tail!

Kr2

Adenine (腺嘌呤)

Thymine(胸腺嘧啶)

H

N

OC

Unit:KJ/mol

1eV=96.7KJ/mol

The DFT-D method

𝐸DFT−D = 𝐸DFT + 𝐸disp

DFT under conventional local/semi-local approximations, no long-range vdW tail.

Dispersion force

𝐸disp = −

𝐴𝐵

𝑓𝑑𝑎𝑚𝑝 𝑅𝐴𝐵 , 𝐴, 𝐵𝐶6𝐴𝐵

𝑅𝐴𝐵6

(all atom pairs)

Damping fuction

𝑓𝑑𝑎𝑚𝑝 𝑅𝐴𝐵 , 𝐴, 𝐵 =1

1 + 𝑒−γ Τ𝑅𝐴𝐵 𝑅𝐴0+𝑅𝐵

0 −1

vdW radius

The damping function

𝑓𝑑𝑎𝑚𝑝 𝑅𝐴𝐵, 𝐴, 𝐵 =1

1 + 𝑒−𝛾 Τ𝑅𝐴𝐵 𝑅𝐴0+𝑅𝐵

0 −1

𝐸𝑑𝑖𝑠𝑝 = −

𝐴𝐵

𝑓𝑑𝑎𝑚𝑝 𝑅𝐴𝐵, 𝐴, 𝐵𝐶6𝐴𝐵

𝑅𝐴𝐵6

Grimme

Becke-Johnson

𝐸disp = −

𝐴𝐵

𝐶6𝐴𝐵

𝑅𝐴𝐵6 + 𝑐𝑜𝑛𝑠𝑡

DFT-D2 (Grimme's method)

S. Grimme, J. Comput. Chem. 27, 1787 (2006)

𝐸disp = −𝑠6

𝐴𝐵

𝑓𝑑𝑎𝑚𝑝 𝑅𝐴𝐵, 𝐴, 𝐵𝐶6𝐴𝐵

𝑅𝐴𝐵6

Global scaling factor, depending on the DFT functional

𝑓𝑑𝑎𝑚𝑝 𝑅𝐴𝐵 , 𝐴, 𝐵 =1

1 + 𝑒−𝛾 Τ𝑅𝐴𝐵 𝑅𝐴0+𝑅𝐵

0 −1

𝐶6𝐴𝐵 = 𝐶6

𝐴𝐴𝐶6𝐵𝐵

𝐶6𝐴𝐴 = 0.05𝑁𝐼𝑝

𝐴α𝐴

2, 10, 18, 36, ...

Atomic ionizationenergy

Atomic dipolepolarizability

(Å)(Jnm6/mol)

Becke-Johnson schemeA. D. Becke and E. R. Johnson, J. Chem. Phys. 127, 154108 (2007)

The exchange hole:

Electron plus its exchange holedefine a non-zero dipole moment

Δ𝐸𝑎𝑣: average excitation energy

ℎ𝑋𝜎 𝒓1, 𝒓2 =1

𝑛𝜎(𝒓1)

𝑖,𝑗

𝑜𝑐𝑐.

𝜓𝑖𝜎(𝒓𝟏)𝜓𝑗𝜎(𝒓1)𝜓𝑖𝜎(𝒓2)𝜓𝑗𝜎(𝒓𝟐)

𝑑𝑋𝜎 𝒓1 = නℎ𝑋𝜎 𝒓1, 𝒓2 𝒓2𝑑3𝑟2 − 𝒓1

𝑑𝑋𝜎𝑒−(𝒓, Ω)

ℎ−(𝒓 − 𝑑𝑋𝜎, Ω)

Nucleus

𝑀𝑙𝜎 = − 𝒓𝑙 − (𝒓 − 𝑑𝑥𝜎𝑙 (𝒓))𝑙

𝑀𝑙2 =

𝜎

න𝑛𝜎(𝒓)𝑀𝑙𝜎2 𝑑3𝑟

𝐶6 =2

3

𝑀12𝐴𝑀1

2𝐵

Δ𝐸𝑎𝑣

𝐶8 =𝑀1

2𝐴𝑀2

2𝐵+ 𝑀2

2𝐴𝑀1

2𝐵

Δ𝐸𝑎𝑣

TS-vdW scheme

A. Tkatchenko and M. Scheffler, PRL 102, 073005 (2009)

The C6 coefficients depend on the chemical environment!

𝐶6𝐴𝐵 =

3

πන𝑑ωα𝐴 𝑖ω α𝐵 𝑖ω

α𝐴 ω ≈α𝐴0

1 − Τω η𝐴2

Plasmon-pole approximation

𝐶6𝐴𝐵 =

2C6𝐴𝐴𝐶6

𝐵𝐵

α𝐵0

α𝐴0 𝐶6

𝐴𝐴 +α𝐴0

α𝐵0 𝐶6

𝐵𝐵

𝐶6𝐴𝐴 =

𝑉𝐴eff

𝑉𝐴free

2

𝐶6,𝑓𝑟𝑒𝑒𝐴𝐴

Free C6 coefficient 𝑉𝐴eff

𝑉𝐴free

=𝑑3 𝑟𝑟3𝑤𝐴 𝐫 𝑛 𝐫

𝑑3 𝑟𝑟3𝑛𝐴free 𝐫

𝑤𝐴 𝐫 = ൘𝑛𝐴free 𝐫

𝐵

𝑛𝐵free 𝐫

(Hirshfeld partitioning)

𝐶6𝐴𝐵 =

3

2

η𝐴0η𝐵

0

(η𝐴0+η𝐵

0 )α𝐴0α𝐵

0

DFT-D3S. Grimme et al., J. Chem. Phys. 132, 154104 (2010)

𝐸𝑑𝑖𝑠𝑝 = 𝐸𝑑𝑖𝑠𝑝2

+ 𝐸𝑑𝑖𝑠𝑝3

Pair-wise two-body termsThree-body terms

𝐸disp2

= −

𝐴𝐵

𝑛=6,8

𝑠𝑛 𝑓𝑑𝑎𝑚𝑝𝑛 𝑅𝐴𝐵, 𝐴, 𝐵

𝐶𝑛𝐴𝐵

𝑅𝐴𝐵𝑛

Depends on the chemical environment, e.g., the coordination numbers.

𝐸𝑑𝑖𝑠𝑝3

= −

𝐴𝐵𝐶

𝑓𝑑𝑎𝑚𝑝,3ሜ𝑅𝐴𝐵𝐶

𝐶9𝐴𝐵𝐶 3cosθ𝐴cosθ𝐵cosθ𝐶 − 1

𝑅𝐴𝐵3 𝑅𝐵𝐶

3 𝑅𝐴𝐶3

𝐶9𝐴𝐵𝐶 = 𝐶6

𝐴𝐵𝐶6𝐵𝐶𝐶6

𝐴𝐶Axilrod-Teller-Muto term

Remarks on the DFT-D type method

+ Very efficient (tiny extra computational cost)

+ Usefully accurate for many applications

- Non-additive effects not captured

- Empiricism of different levels

- “Double-counting” in the overlapping density regime

- Not true density functional (relying on geometries)

l

van der Waals density functional (vdW-DF)

M. Dion, H. Rydberg, E. Schroeder, D. C. Langreth,and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).

𝐸𝑥𝑐[𝑛] = 𝐸𝑥𝐺𝐺𝐴[𝑛] + 𝐸𝑐

𝐿𝐷𝐴[𝑛] + 𝐸𝑐𝑛𝑜𝑛−𝑙𝑜𝑐𝑎𝑙[𝑛]

𝐸𝑥GGA as close to Hartree-Fock as possible, choose 𝐸𝑥

revPBE

2ℎ𝑦

𝑑

van der Waals density functional (vdW-DF)M. Dion, H. Rydberg, E. Schroeder, D. C. Langreth,and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).

𝐸𝑥𝑐[𝑛] = 𝐸𝑥GGA[𝑛] + 𝐸𝑐

LDA[𝑛] + 𝐸𝑐non−local[𝑛]

𝐸𝑐𝑛𝑜𝑛−𝑙𝑜𝑐𝑎𝑙 =

1

2න𝑑3 𝑟𝑑3𝑟′𝑛 𝒓 𝜙 𝒓, 𝒓′ 𝑛 𝒓′

𝜙 𝒓, 𝒓′ = 𝜙(𝑞0 𝒓 , 𝑞0 𝒓′ , 𝒓 − 𝒓′ ), 𝑞0 𝒓 =𝜖𝑥𝑐0 𝒓

𝜖𝑥LDA 𝒓

𝑘𝐹(𝒓)

𝜈 𝑦 =𝑦2

2ℎ𝑦𝑑

, 𝑣′ 𝑦 =𝑦2

2ℎ𝑦𝑑′

𝑑 = 𝑟 − 𝑟′ 𝑞0 𝒓 , 𝑑′ = 𝑟 − 𝑟′ 𝑞0 𝒓′

𝑑 = 𝐷 1 + 𝛿 , 𝑑′ = 𝐷(1 − 𝛿)

ℎ 𝑦 = 1 − 𝑒−4𝜋𝑦2/9

van der Waals density functional (vdW-DF)M. Dion, H. Rydberg, E. Schroeder, D. C. Langreth,and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).

𝐸𝑥𝑐[𝑛] = 𝐸𝑥GGA[𝑛] + 𝐸𝑐

LDA[𝑛] + 𝐸𝑐non−local[𝑛]

𝐸𝑐𝑛𝑜𝑛−𝑙𝑜𝑐𝑎𝑙 =

1

2න𝑑3 𝑟𝑑3𝑟′𝑛 𝒓 𝜙 𝒓, 𝒓′ 𝑛 𝒓′

𝜙 𝒓, 𝒓′ = 𝜙(𝑞0 𝒓 , 𝑞0 𝒓′ , 𝒓 − 𝒓′ ), 𝑞0 𝒓 =𝜖𝑥𝑐0 𝒓

𝜖𝑥LDA 𝒓

𝑘𝐹(𝒓)

Interaction energy between benzene dimer

Vydrov-Van Voorhis functional (VV09)

Phys. Rev. Lett. 103, 063004 (2009)

Vydrov-Van Voorhis functional (VV09)

𝐸𝑐VV09 = 𝐸𝑐

nl−VV + 𝐸𝑐LSDA

𝐸cnl−VV =

3ℏ

64𝜋2ඵ𝑑𝑟𝑑𝑟′

𝜔𝑝2 𝒓 𝜔𝑝

2 𝒓′ 𝐷(𝐾)

𝜔0 𝒓 𝜔0 𝒓′ [𝜔0 𝒓 +𝜔0 𝒓′ ] 𝒓 − 𝒓′ 6

𝜔𝑝 =4𝜋2𝑛

𝑚: plasmon frequency 𝜔0

2 = 𝜔𝑔2 + 𝜔𝑝

2/3 𝜔𝑔2 𝑟 = 𝐶

ℏ2

𝑚2

𝛻𝑛(𝒓)

𝑛(𝒓)

4

𝐶: 0.0089 (fitting parameter)

𝐾 𝒓, 𝒓′ =𝒓 − 𝒓′

2

𝜅 𝒓 𝜅(𝒓′)

𝜅 𝒓 + 𝜅(𝒓)

1/2

𝐷 𝐾 =4

3𝐾2𝐴𝐵 − 𝐵2

𝐴 =2𝐾

𝜋𝑒−𝐾

2, 𝐵 = erf 𝐾 − 𝐴

𝜅 𝒓 =𝑘𝑠2(𝒓)𝜙2(𝒓)

𝑘𝑠 𝑟 = 4𝑘𝐹/𝜋𝑎0 : Thomas-Fermi screening parameter.

𝜙 𝜁 = 1 + 𝜁 2/3 + 1 − 𝜁 2/3 /2 : spin scaling factor

Bohr radius

Comparison: the S66 test set

S66: 66 vdW-bonded dimolecular complex set

Comparison of different methods

G. Grimme, WIRES Comput. Mol. Sci. 1, 212 (2011)

DCACP: Dispersion-corrected atom-centered potentials