Dispersion forces from van der
Waals density functionals
Jose M. Soler
Departamento de Física de la Materia Condensada
Universidad Autónoma de Madrid
Collaborators
Mariví Fernández-Serra, Stony Brook
Simone S. Alexandre, Belo Horizonte
Guillermo Román-Pérez, UAM
Michelle Fritz, UAM
Mohammed Moaied, UAM
Eduardo Anglada, UAM
Jue Wang, Stony Brook
Betul Pamuk, Stony Brook
Peter Stephens, Stony Brook
Phil B. Allen, Stony Brook
Rafael Ramírez, ICMM-CSIC
Carlos Herrero, ICMM-CSIC
Emilio Artacho, Nanogune & Cambridge
Outline
•  Differences between vdW-DF versions
•  Efficient implementation of vdW-DFs
•  VdW effects in liquid water
•  Nuclear quantum effects in ice
VdW XC density functional
Dion, Rydberg, Schröder, Langreth, and Lundqvist,
PRL 92, 246401 (2004)
GGA-revPBE
x
Z ab
LDA
c
nl
c
E xc = E
+E
+E
1
nl
E c = ∫∫ dr1 dr2 n(r1 ) n(r2 ) Φ(q1,q2 ,r12 )
2
⎡ ε LDA (n) Z ⎛ k ⎞ 2 ⎤
q(n,∇n) = ⎢1+ cLDA
− ab ⎜ G ⎟ ⎥ kF
9 ⎝ 2k F ⎠ ⎥⎦
⎢⎣ ε x (n)
∇n
2 1/ 3
= −0.8491
k F = ( 3π n )
kG =
n
Non-local correlation kernel
q1 + q2
r12
2
q1 − q2
δ=
q1 + q2
D=
€
n = const ⇒ δ = 0 ⇒
2
4
π
D
Φ(D) dD = 0 ⇒
∫
E cnl = 0
12(4 π /9) 3 1
r12 →∞ ⇒ D →∞ ⇒ Φ →− 2 2 2 2 6
q1 q2 (q1 + q2 ) r12
Universal ‘seamless’ functional
vdW-DF, version 2
K. Lee et al, PRB 82, 081101 (2010)
vdW-DF2
Exch: revPBEx  rPW86x
Corr: -Zab=0.848  1.887
Other proposals:
- J.Klimes, D.R.Bowler, and A.Michaelides, JPCM 22, 22201 (2010)
-  J.Wellendorff et al, Topics Catal. 54, 1143 (2011)
vdW-DF-VV10
O. A. Vydrov & T. Van Voorhis, JCP 133, 244103 (2010)
E xc = E xGGA-rPW86 + E cGGA-PBE + E cnl + βN
1
nl
E c = ∫∫ dr1 dr2 n1 n 2 Φ(n1,∇n1,n 2 ,∇n 2 ,r12 )
2
3/2
Φ=−
g = q 2 r122 + κ
g1g2 (g1 + g2 )
1/ 4
⎛ 4 3
⎞
4
q = ⎜ k F + CkG ⎟
⎝ 9π
⎠
C = 0.0093
b = 5.9
1/ 2
⎛ 3π ⎞
κ = b⎜
k F ⎟
⎝ 4
⎠
1 ⎛ 3 ⎞
β = ⎜ 2 ⎟
32 ⎝ b ⎠
C and b are fitted empirically
€
3/4
Nonlocal kernel
vdW-DF2
vdW-DF1
vdW-VV
Asymptotic behavior
1
1
r12 →∞ ⇒ Φ →− 2 2 2 2 6
q1 q2 (q1 + q2 ) r12
vdW-DF2
€
vdW-VV
vdW-DF1
Oxigen atom density
AE
PS
PS
AE
Oxigen atom density
in (kF,kG) space
AE
PS
Density × kernel
1
E =
2
nl
c
€
∫∫ dr dr
AE
1
2
n1 n 2 Φ(n1,∇n1,n 2 ,∇n 2 ,r12 )
PS
Water molecule
DRSLL
Enl=6.8
LMKLL
Enl=2.8
VV
Enl=3.1
Liquid water
molecule
liquid
Simple and complex systems
graphite
Pt2S8C8H12I
The double integral problem
• Φ(q1,q2,r12) decays as r12-6
• Ecnl= ∫∫ d3r1 d3r2 n(r1) n(r2) Φ(q1,q2,r12)
can be truncated for r12 > rc ~ 15Å
• In principle O(N) calculation for systems
larger than 2rc ~ 30Å
• But... with Δx ~ 0.15Å (Ec=120Ry) there
are ~(2×106)2 = 4×1012 integration points
• Consequently, direct evaluation of vdW
functional is much more expensive than
LDA/GGA
Factoring Φ(q1,q2,r12)
Φ(q1,q2 ,r12 ) ≅ ∑ pα (q1 ) pβ (q2 )Φαβ (r12 )
α ,β
nl
c
∫∫ dr dr n(r )n(r )Φ(q ,q ,r )
≅ ∑ ∫∫ dr dr θ (r ) θ (r )Φ (r
E =
1
2
1
1
2
2
α
1
1
β
2
2
12
αβ
α ,β
= ∑ ∫ dk θα (k) θ β (k) Φαβ (k)
α ,β
θα (r) ≡ ρ (r) pα [q( n(r),∇n(r))]
12
)
Interpolation as an expansion
f
f1
f3
f2
f4
x
=
f2p2 f3p3
f1p1
x1
x2
f4p4
x3 x4
General recipe: fj=δij ⇒ f(x)=pi(x)
Factoring by interpolation
Nonlocal correlation potential
E cnl ≡ ΔV 2 ∑ ∑θαiθ βj Φαβ (rij )
α , β i, j
∇ρi ≅ ∑ gij n j
j
v
nlc
i
⎛ ∂θ
⎞
∂
θ
1 ∂E
αj
αi
⎜
⎟⎟
≡
= 2ΔV ∑⎜ uαi
+ ∑ gij uαj
ΔV ∂ρi
∂n i
∂∇n j ⎠
α ⎝
j
nl
c
uαi = ΔV ∑ ∑θ βj Φαβ (rij )
β
j
3
d
∫ r2θ β (r2 )Φαβ (r12 ) =
3
ikr1
d
k
e
θ β (k)Φαβ (k)
∫
Kernel cusp
n1,n 2 →0 ⇒ q1,q2 →0 ⇒ Φ(q1r12 ,q2 r12 ) →∞
Φ(d1,d2 ) = Φsoft (d1,d2 ) + Φcusp (d1,d2 )
1
E ≈ ∫∫ dr1dr2 n(r1 )n(r2 )Φsoft (q1,q2 ,r12 ) +
2
∞
1
2
2
d
r
n
(r
)
4
π
r
drΦcusp (d1,d1 )
∫
1
1 ∫0
2
nl
c
A better solution (Gygi et al):
€
F(q1,q2 ,r12 ) ≡ q1q2Φ(q1,q2 ,r12 ) ≅ ∑ pα (q1 ) pβ (q2 ) Fαβ (r12 )
α ,β
∫∫ dr dr n(r )n(r )Φ(q ,q ,r )
≅ ∑ ∫∫ dr dr θ (r ) θ (r ) F (r )
E cnl =
1
2
1
1
2 α
α ,β
n(r)
θα (r) ≡
pα (q(r))
q(r)
2
1
1
β
2
2
αβ
12
12
Implementation of vdW-DF-VV10
1
E = ∫∫ dr1 dr2 n1 n 2 Φ(n1,∇n1,n 2 ,∇n 2 ,r12 )
2
n1n 2Φ ≡ F(k F1,kG1,k F 2 ,kG 2 ,r12 ) ≅
nl
c
∑ ∑ p (k
α
F1
) pβ (kG1 ) pγ (kF 2 ) pδ (kG 2 ) Fαβγδ (r12 )
α , β γ ,δ
1
E ≅ ∑ ∑ ∫∫ dr1dr2 θαβ (r1 ) θ γδ (r2 ) Fαβγδ (r12 )
2 α ,β γ ,δ
nl
c
θαβ (r) ≡ pα ( k F (r)) pβ ( kG (r))
Alternative implementation (Sabatini & de Gironcoli):
€
ΦVV (n1,∇n1,n 2 ,∇n 2 ,r12 ) ≈ Φapprox (q1,q2 ,r12 )
O( N log(N) ) algorithm
do, for each grid point i
find ni and ∇ni
Input: ni on a regular grid
find qi=q(ni ,∇ni )
find θαi = ni pα(qi )
Output: Exc , vixc on the grid
∀α
No need of supercells in solids
end do
Fourier-transform θαi → θαk
No cutoff radius of interaction
∀α
do, for each reciprocal vector k
find uαk = ∑β Φαβ(k) θβk
∀α
end do
Inverse-Fourier-transform uαk → uαi
∀α
do, for each grid point i
find ni , ∇ni , and qi
find θαi , ∂θαi /∂ni , and ∂θαi /∂ ∇ni
find vi
end do
∀α
vdW effects in water
J.Wang et al, JCP 134, 024516 (2011)
Known problems:
•  GGA water is too structured and rigid
•  Equilibrium density is too low
•  Diffusion coefficient is too low
=> Simulations must be performed at too high
pressures and temperatures, to reproduce
experiments
Pressure-density curves
Why vdW interactions
increase density?
•  Shorter bond lenghts?
•  Larger average coordinations?
•  Other?
Dimer interaction energies
H bonded
vdW bonded
vdW-DF RDF
Pair distribution functions
GGA
vdW-DF
H-bond network
GGA
vdW-DF
3.42 H-bonds/molecule
3.15 H-bonds/molecule
Angular distribution function
First shell
r = 2.8 Ang
‘Interstitial shell’
r = 3.5 Ang
Interstitial sites
Interstitial sites
2.9 Å
Diffusivities
Liquid water. Conclusions
•  The density and diffusivity of water
greatly improve with vdW-DF
•  vdW interactions increase the density by
populating ‘interstitial sites’
•  This weakens the H-bond network and
increases the diffusivity
Liquid water. Still unsatisfactory.
•  Defficiencies:
•  Density too large
•  Bulk modulus too large
•  g(r) not yet quite satisfactory
•  Missing effects:
•  Better functional?
•  Nuclear quantum effects
Vydrov-VanVoorhis functional
GGAs
vdW-VV
vdW-PBE
Vydrov-VanVoorhis functional
Quantum nuclei:
Inverse isotopic effect
B. Pamuk et al, PRL 108, 193003 (2012)
K.Röttger et al, Acta Cryst. B50, 644 (1994)
D2 O
H2O
Also present in liquid water!
Normal isotopic effect
U
Heavier
Lighter
r
Inverse isotopic effect
O
H
O
Quasi-harmonic approximation
V
∂ (ω i /2) 1
ω i
δV = ∑
= ∑
γi
B0 i
∂V
B0 i 2
€
∂ ln ω i
γi = −
∂ lnV
Path integral simulations
J. M. McMahon, M. A. Morales, and D. M. Ceperley, APS 2012
QHA IN THE LIQUID LOCAL MINIMA
QHA in quenched snapshots
Classical limit
overestimates
the volume
Heavy water (at
290K) has 0.2%
more volume per
molecule than light
water.
Quantum effects.
Conclusions
•  Inverse isotopic effect in solid and liquid water
•  Unlike TIP4P/F, DFT reproduces it in ice
•  Caused by stretching modes
•  Depends on very subtle competition between O-H
stretching and O-O librations
•  Indications that DFT reproduces it in the liquid
•  Indications of that quantum effects strengthen the
hydrogen-bond network in the liquid