Supporting Information
Effects of an Electric Field on Interaction of
Aromatic Systems
Il Seung Youn,†, ‡ Woo Jong Cho,*,‡ and Kwang S. Kim*,‡
†
Center for Superfunctional Materials, Department of Chemistry, Pohang University of Science and Technology,
Pohang 790-784, Korea, and ‡Department of Chemistry, Ulsan National Institute of Science and Technology
(UNIST), Ulsan 689-798, Korea
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Derivation of Equation (5)
When the Ar-X complex is under an external electric field E, mutual polarization of Ar and X
results in a non-arithmetic increase of the total polarizability: Ar-X > Ar + X. Selfconsistent dipole moment of each fragment taking into account the mutual polarization effect
is written as
Ar   Ar (E  EX )
X   X (E  EAr ) ,
where EX and EAr are dipole field from the polarized X and Ar, respectively. In the case of
benzene, six carbon atoms in the ring are equally responsible for electric dipole potential on
Ar. From the electric dipole potential, we can derive the following:
X r
V X cos 


2
2
6
6 R
6(d  r 2 )3/2

EX
 
1 V
1
3
2r 2

  X  2 2 3/2 
2
2 5/2 
6
6 r
6  (d  r )
2 (d  r ) 

 X  2r 2  d 2 
 f
 X ,

2
2 5/2 
6  (d  r ) 
6
where the distance between a carbon and an argon atoms R = (d2 + r2)1/2, r is the separation
between the argon atom and center of the ring, d is the carbon-carbon distance, and f is the
term in the square bracket. Likewise, EAr = Ar f as well. Solving for Ar and X, we obtain
Ar 
 Ar   X Ar f
E
1   X Ar f 2
X 
 X   X Ar f
E.
1   X Ar f 2
If the Ar and X are separated far enough so that their electron density overlap is not
significant,
classical
X-Ar ~ X  Ar   X-Ar
E.
Thus, the classical estimate of total polarizability is as follows:
classical
 X
Ar
 X   Ar  2 X Ar f
.
1   X Ar f 2
2
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Unphysical Behavior of the vdW-DF Series
From the discussions above, it is physically nonsense to have Bz-Ar < Bz + Ar. However,
both vdW-DF and vdW-DF2 functionals yield unphysical result when the conventional
computational setting as used in the manuscript is applied (Figure S1, red and sky blue
curves). The unphysical behavior persisted even when the non-local correlation is removed
from vdW-DF2 to yield the rPW86 functional (Figure S1, ivory curve). Interestingly, the
physics of mutual polarization is recovered when a fast Fourier transform (FFT) grid twice as
dense in each direction is used. This is realized by replacing PREC=Normal with
PREC=Accurate option in VASP.
Figure S1. Interaction energy response (ΔΔEBz-Ar) to the external electric field at r = 3.4 Å
obtained with revPBE-vdW-DF, vdW-DF2 and rPW86 (exchange functional for vdW-DF2)
comparing to CCSD(T)/CBS (‘accurate’ stands for PREC=accurate).
To find the exact cause of the problem, we evaluated Bz-Ar, Bz and Ar with various
exchange-correlation (xc) functionals using VASP and Orca as shown in Table S1. The main
reason that we adopted VASP in this study is that it has three most popular (vdW-DF, Grimme
D3 and Tkatchenko-Scheffler) dispersion correction schemes implemented on the same
platform. In particular, to our knowledge, VASP is the only software in which vdW-DF and
vdW-DF2 are implemented. Orca was chosen for the cross-checking purpose to eliminate the
need for the Fourier transform of electron density and wave functions to the reciprocal space.
It is also one of the few quantum chemistry programs where dispersion correction based on
non-local correlation kernel (VV10) is made available. Since VV10 belongs to the same
family with the vdW-DF series, we expect that a numerical instability, if any, of such nonlocal functionals can be tested in the absence of complications from using plane wave basis
set.
The two softwares show inconsistent results for the rPW86 functional: αBz-Ar < αBz +
αAr in VASP as shown in the manuscript and αBz-Ar > αBz + αAr in Orca, in accordance with
what is expected. The spurious ‘reduction’ of polarizability in VASP turns out to arise from
the unphysical overestimation of Bz and Ar when each of them is calculated separately.
Using a denser (×2) FFT grid solves the problem for the vdW-DF2 functional, but not for
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vdW-DF functional. Nevertheless, the quality of electron density when it is transformed back
to the real space is one of the critical factors affecting the accuracy. Another indirect evidence
that the rPW86-based non-local functionals indeed respect the physical reality is that we do
not find any artifact for the VV10 functional when the calculation is carried out in the real
space using Orca.
Table S1. Numerically computed polarizabilities Bz-Ar, Bz and Ar (in Å3) for Bz-Ar
distance of 3.4 Å using several different methods
Bz-Ar
Bz
Ar
10.91
11.46
2.71
9.20
11.69
2.73
9.18
6.82
1.80
9.04
11.12
2.66
Orca
Settings
700 eV cutoff,
normal FFT grid
700 eV cutoff,
normal FFT grid
700 eV cutoff,
2×normal FFT grid
700 eV cutoff,
normal FFT grid
aug-cc-pVQZ
9.08
6.76
1.76
revPBE
Orca
aug-cc-pVQZ
9.05
6.80
1.74
VV10
Orca
aug-cc-pVQZ
9.15
6.76
1.77
8.80
6.77
1.60
xc functional
program
vdW-DF
VASP
vdW-DF2
rPW86
VASP
VASP
CCSD(T)/CBS
Therefore, neither of the semi-local and non-local components of the vdW-DF series
is flawed per se, but their implementation in VASP is sometimes incompatible with the finitefield calculation, most probably due to their sensitivity towards the subtle error in the real
space electron density.
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