Calculating dispersion interactions using maximally localized Wannier functions.

Abstract

We investigate a recently developed approach [P. L. Silvestrelli, Phys. Rev. Lett. 100, 053002 (2008); J. Phys. Chem. A 113, 5224 (2009)] that uses maximally localized Wannier functions to evaluate the van der Waals contribution to the total energy of a system calculated with density-functional theory. We test it on a set of atomic and molecular dimers of increasing complexity (argon, methane, ethene, benzene, phthalocyanine, and copper phthalocyanine) and demonstrate that the method, as originally proposed, has a number of shortcomings that hamper its predictive power. In order to overcome these problems, we have developed and implemented a number of improvements to the method and show that these modifications give rise to calculated binding energies and equilibrium geometries that are in closer agreement to results of quantum-chemical coupled-cluster calculations.

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THE JOURNAL OF CHEMICAL PHYSICS 135, 154105 (2011)
Calculating dispersion interactions using maximally localized
Wannier functions
Lampros Andrinopoulos,a)Nicholas D. M. Hine, and Arash A. Mostofi
The Thomas Young Centre for Theory and Simulation of Materials, Imperial College London,
London SW7 2AZ, United Kingdom
(Received 11 March 2011; accepted 19 September 2011; published online 17 October 2011)
We investigate a recently developed approach [P. L. Silvestrelli, Phys. Rev. Lett. 100, 053002 (2008);
J. Phys. Chem. A 113, 5224 (2009)] that uses maximally localized Wannier functions to evaluate the
van der Waals contribution to the total energy of a system calculated with density-functional theory.
We test it on a set of atomic and molecular dimers of increasing complexity (argon, methane, ethene,
benzene, phthalocyanine, and copper phthalocyanine) and demonstrate that the method, as originally
proposed, has a number of shortcomings that hamper its predictive power. In order to overcome these
problems, we have developed and implemented a number of improvements to the method and show
that these modifications give rise to calculated binding energies and equilibrium geometries that are
in closer agreement to results of quantum-chemical coupled-cluster calculations. © 2011 American
Institute of Physics. [doi:10.1063/1.3647912]
I. INTRODUCTION
Local and semi-local exchange-correlation functionals
used in density-functional theory3,4(DFT) cannot account
for the effect of long-ranged dispersion, or van der Waals
(vdW), interactions. Dispersion interactions are crucial for
weakly bound systems, particularly where no covalent or
ionic bonding is present, and often dominate intermolecular
binding energies and equilibrium geometries. Incorporating
vdW interactions in DFT remains a challenging task and a
wide variety of methods have been developed, approach-
ing the problem from many different perspectives.5–13In
this work we focus on the method recently proposed by
Silvestrelli,1,2which has been applied to various systems14–17
and implemented in a number of modern electronic structure
codes.18,19This approach uses maximally localized Wannier
functions20(MLWFs) as a means of decomposing the
electronic density of the system into a set of localized but
overlapping fragments, which may then be used to calculate
a vdW correction to the DFT total energy by considering
pairwise interactions between density fragments as derived
by Andersson, Langreth, and Lundqvist7(ALL).
In this paper, we explore the parameters and approxima-
tions involved in Silvestrelli’s method and improve its results
where possible by modifying various aspects of the method.
We apply the method and our proposed modifications to a se-
ries of test systems, then to two more challenging systems, a
phthalocyanine and a copper phthalocyanine dimer. We thus
demonstrate that although this method can offer an easily im-
plementable and computationally efficient way of calculating
the dispersion correction to the energy with the possibility
of improved accuracy (once some modifications are applied
to it), it is largely dependent on a number of parameters and
choices one can make.
a)Author to whom correspondence should be addressed. Electronic mail:
l.andrinopoulos09@imperial.ac.uk.
The remainder of the paper is organized as follows: in
Sec. II we recap the necessary background theory relating to
MLWFs and Silvestrelli’s method; in Sec. III we highlight
some of the problems with the method as it stands, and de-
scribe our improvements; in Sec. IV we then present and dis-
cuss results for vdW-corrected total energies and equilibrium
geometries obtained by applying these methods to a series of
dimer systems and compare to quantum chemical coupled-
cluster and semi-empirical vdW (DFT + D) approaches; fi-
nally, in Sec. V we draw our conclusions.
II. THEORETICAL BACKGROUND
A. Maximally localized Wannier functions
Wannier functions21are orthogonal localized functions
that span the same space as the eigenstates of a single parti-
cle Hamiltonian. Consider the set of Noccoccupied (valence)
eigenstates {|um?} of a molecule. The total energy is invariant
with respect to unitary transformations among the eigenstates
|wn? =
Nocc
?
m=1
Umn|um?.
(1)
If the unitary matrix U is chosen such that the resulting Nocc
orbitals {wn(r)} minimize their total quadratic spread, given
by
?
then they are said to be maximally localized Wannier
functions.20Each MLWF is characterized by a value for its
quadratic spread, S2
In the construction of MLWFs it is sometimes useful to
consider not only the valence manifold but also a range of un-
occupied eigenstates above the Fermi level—often those con-
stituting the anti-bonding counterparts to the valence states.
? =
n
??wn|r2|wn? − ?wn|r|wn?2?=
?
n
??r2?n− ¯ r2
n
?,
(2)
n, and its centre, ¯ rn.
0021-9606/2011/135(15)/154105/13/$30.00 © 2011 American Institute of Physics
135, 154105-1
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154105-2Andrinopoulos, Hine, and MostofiJ. Chem. Phys. 135, 154105 (2011)
This not only allows the MLWFs to be more localized22,23but
can also restore symmetries that would otherwise be broken
arbitrarily through the construction of MLWFs for the valence
manifold only.
In order to do so, one defines an outer energy win-
dow, Ewin, consisting of Nwin ≥ Nocc states, from which
one may extract an optimal Ndis-dimensional subspace (Nwin
≥ Ndis≥ Nocc) using the disentanglement approach described
in Ref. 24,
??uopt
m
?=
Nwin
?
p=1
Udis
pm|up?,
(3)
where Udisis a rectangular Nwin× Ndisunitary matrix. Ndis
MLWFs may then be localized by suitable rotation of the op-
timal subspace in the usual manner:
??wdis
n
?=
Ndis
?
m=1
Umn
??uopt
m
?,
(4)
or in terms of the Bloch states:
??wdis
n
?=
Ndis
?
m=1
Nwin
?
p=1
UmnUdis
pm|up?.
(5)
Furthermore, an inner, or frozen, energy window may be de-
fined if one wishes to make certain that a range of low-lying
eigenstates are included unchanged in the optimal subspace,
for example, the occupied states. Algorithms for determining
MLWFs from the eigenstates obtained from electronic struc-
ture calculations are implemented within the WANNIER90
software package.25
The single-particle density operator is given by
ˆ ρ =
Nocc
?
n=1
|un??un|.
(6)
It can also be written in terms of the Nocc fully occupied
valence MLWFs, |wn? or equivalently in terms of a larger
set of Ndisdisentangled MLWFs, |wdis
pied subspace, which can be guaranteed by using a suitable
frozen/inner window in the disentanglement procedure, and
that have occupancies fw
n?, that span the occu-
kl,
ˆ ρ =
Nocc
?
n=1
|wn??wn|,
(7)
=
Ndis
?
k,l=1
fw
kl
??wdis
k
??wdis
l
??,
(8)
where we have substituted Eq. (1) and Eq. (5), respectively,
into Eq. (6), and where the occupancies are given by
fw
kl=
Nocc
?
p=1
Ndis
?
m,s=1
UmlUdis
pmU∗
skU∗dis
ps.
(9)
We can write the density as a sum of diagonal (l = k) and
off-diagonal (l ?= k) terms,
?
≡ ρD(r) + ρOD(r).
It is important to note that in this form, ρD(r) alone in-
tegrates to the number of valence electrons Ne, because the
mutual orthogonality of the MLWFs ensures?ρOD(r)dr = 0.
manifold of occupied states only (Ndis= Nocc), the occupancy
matrix is simply the identity matrix, fkl= δkl, and the charge
density in terms of the MLWFs is simply given by
ρ(r) =
Ndis
l=1
fw
ll
??wdis
l(r)??2+
Ndis
?
l?=m
fw
lmw∗dis
l
(r)wdis
m(r),
(10)
In the case of considering MLWFs obtained from the
ρ(r) =
Nocc
?
n=1
|wn(r)|2.
(11)
It is worth noting that in the case of spin-degenerate systems,
the occupancies must be scaled by a factor of 2.
We have adapted the WANNIER90 code to calculate the
occupation matrices, and can choose to make a diagonal ap-
proximation to the density by retaining only the first term
of Eq. (10). The effect of approximating the true density
with the diagonal approximation will be discussed later in
Sec. IV I in the context of the improvements, described in
Sec. III, to Silvestrelli’s method.
B. Silvestrelli’s method
Silvestrelli’s approach1,2is based on the Andersson,
Langreth, and Lundqvist7expression for the vdW energy
in terms of pairwise interactions between density fragments
ρn(r) and ρl(r?), separated by a distance rnl,
EvdW= −
?
n>l
gnl(rnl)C6nl
r6
nl
,
(12)
where gnl(rnl) is a damping function2which screens the un-
physical divergence of Eq. (12) at short range, and
?
in atomic units. It should be noted that these expressions are
only strictly valid in the limit of non-overlapping density frag-
ments. There are various forms for the damping function26,27
that might have a slight short-range effect but should not af-
fect the long-range behaviour of the vdW energies. Here we
chose to use the damping function as proposed in the original
paper by Silvestrelli.1
Now, in accord with Eq. (11), the MLWFs obtained from
the valence orbitals of a system provide a localized decom-
position of the electronic charge density, such that ρn(r)
= |wn(r)|2, so that Eq. (13) becomes
3
32π3/2
|r|≤rc
C6nl=
3
4(4π)3/2
V
dr
?
V?dr?
√ρn(r)ρl(r?)
√ρn(r) +√ρl(r?),
(13)
C6nl=
?
dr
?
|r?|≤r?c
dr?|wn(r)||wl(r?)|
|wn(r)| + |wl(r?)|,
(14)
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154105-3Dispersion interactions using MLWFsJ. Chem. Phys. 135, 154105 (2011)
where rcis a suitably chosen cutoff radius obtained by equat-
ing the length scale for density change to the electron gas
screening length;2we will revisit this point later.
In order to make the calculation of the integrals more
tractable, the charge density is approximated by replacing
each MLWF wn(r) with a hydrogenic s-orbital that has the
same centre ¯ rn and spread Sn as the MLWF, and whose
analytic form is given by
wH
n(r) =
33/4
√πS3/2
n
e−√3|r−¯ rn|/Sn,
(15)
which, on substitution into Eq. (14) and after some algebra,
gives
C6nl=S3/2
n S3
2 · 35/4F(Sn,Sl),
l
(16)
where
F(Sn,Sl) =
β = (Sn/Sl)3/2, xc=√3rc/Sn, and yc=√3r?
evaluating Eq. (14) using the true MLWFs requires a com-
putationally demanding six-dimensional numerical integra-
tion, Eq. (17) may be evaluated easily since it is only a
two-dimensional integral that depends solely on the MLWF
spreads and centres, not their detailed shapes or orientations.
We note that in the case of a spin-degenerate system,
since every MLWF is doubly occupied, the density of each
fragment must be multiplied by a factor of 2 and, therefore,
the C6nlintegral in Eq. (14) must be scaled by a factor of√2.
?xc
0
dx
?yc
0
dyx2y2e−xe−y
e−x/β + e−y,
(17)
c/Sl. Whereas
III. IMPROVEMENTS TO SILVESTRELLI’S METHOD
The approximations that go into the method described in
Sec. II B will clearly not always hold, and the need to examine
them is clear. In this section, we introduce our enhancements
to the method that address possible drawbacks.
A. Partly occupied Wannier functions
Using a manifold of eigenstates that includes but is larger
than the subspace spanned by just the valence states results in
partly occupied MLWFs that are generally more localized and
that better reflect the symmetries of the system, as opposed to
MLWFs obtained by rotation of the valence subspace only,
which arbitrarily break the symmetry (we will demonstrate
examples of this phenomenon in Sec. IV).
In order to account for the partial occupancy of the
MLWFs, we make a slight modification to Silvestrelli’s ap-
proach, explicitly introducing occupancies in the definition of
theC6nlintegral;sinceinthediagonalapproximation,theden-
sityofeachfragmentisnowgivenbyρn(r) = fw
expression for F(Sn, Sl) in Eq. (17) becomes
?xc
nn|wn(r)|2,the
F(Sn,Sl) =
0
dx
?yc
0
dy
x2y2e−xe−y
nn) + e−y/?fw
e−x/(β?fw
ll
,
(18)
FIG. 1. Partly occupied p-like orbital on ethene molecule. In the method
describedhere,eachofthetwolobes,upper(red)andlower(blue),isreplaced
by an s-orbital and considered a separate fragment.
where the fw
this seemingly simple idea can give rise to a marked improve-
ment in the accuracy of the method.
nnare given by Eq. (9). We will see in Sec. IV that
B. Modification to describe p-like states
MLWFs describing only the valence manifold often take
the form of well-localized functions centred on a bond be-
tween two atoms, and are thus reasonably well described by
the approximation of replacing them with a suitable s-orbital.
When anti-bonding states are included in the construction of
the MLWFs, the resulting orbitals have more atomic-orbital
character. This is demonstrated by the atom-centred p-like
MLWF shown in Fig. 1. It is clear that the density associ-
ated with such an MLWF will not be very well represented
by a single s-like function at its centre. In order to approx-
imate p-like orbitals appropriately when calculating C6, one
could imagine using a suitably oriented analytic expression
for a hydrogenic p-orbital, for example, a canonical pz-orbital
given by
pz(r) =305/4r cosθ
√32πS5/2e−√30r/2S,
(19)
which has been normalized such that its quadratic spread is
?pz|(r − ¯ r)2|pz? = S2. As a consequence of the explicit an-
gular dependence, using this function in Eq. (14) would give
rise to four-dimensional integrals for which analytic solutions
are not readily available. Numerical evaluation of these inte-
grals, for realistic systems, would be prohibitively computa-
tionally expensive. We solve this problem by identifying the
p-like MLWFs in the system and replacing them with the hy-
drogenic form given in Eq. (19). Then, we further approxi-
mate each lobe (lower and upper) of this p-like orbital with
two separate hydrogenic s-orbitals of the form of Eq. (15). In
order to do so, for each of the upper (+) and lower (−) lobes
of the orbital, it is necessary to know the spread S±and centre
¯ r±, given by
?∞
S2
±=
0
?π/2
0
?2π
0
r4p2
z(r)sinθdrdθdφ,
(20)
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154105-4Andrinopoulos, Hine, and MostofiJ. Chem. Phys. 135, 154105 (2011)
¯ r±= ¯ r ±
?∞
0
?π/2
0
?2π
0
r3cosθ p2
z(r)sinθdrdθdφ ˆ z,
(21)
which, after some algebra, simplifies to
S±=
7S
8√2,
(22)
¯ r±= ¯ r ±
15S
8√30
ˆ z,
(23)
where ¯ r and S are the original centre and spread, respectively,
of the true MLWF. These expressions may be easily general-
ized to arbitrary orientations of the symmetry axis of a p-like
state by rotating the offset vectors (¯ r±− ¯ r) accordingly.
Thus, we have developed a formalism whereby the
charge density due to MLWFs with p-like character can
be represented by a pair of s-like hydrogenic orbitals with
appropriate centres and spreads. In Sec. IV we will show
how this works in practice for calculating vdW energy
corrections.
In the relatively simple systems studied in this paper, the
p-like orbitals are easily distinguished from other orbitals by
their partial occupancies, given by Eq. (9), which are typically
closer to 0.5 rather than 1. Alternatively, and especially for
structurally more complex systems, the shape of each MLWF
could be characterized using the efficient method described
in Appendix A of Ref. 28 as another means of automating the
procedure of identifying p-like functions.
C. Symmetry considerations
Minimizing the total spread ? with respect to the
elements of the unitary matrix U, and thus producing
MLWFs, has the effect of picking from the space of all pos-
sible unitary matrices one which produces the most local-
ized Wannier functions accessible through optimization from
a chosen initial guess. This is often enough to uniquely de-
termine the MLWFs. In some cases, however, it does not
give rise to a unique choice, even if the optimization proce-
dure is perfect. For example, the atomic positions and elec-
tron density of the system may possess certain symmetry el-
ements, such as rotations about a particular axis. Then there
will exist a number of equally valid and degenerate represen-
tations of the MLWFs and their centres, which give the same
spread, and are related by symmetry. The minimization pro-
cedure breaks the symmetry by choosing one of these rep-
resentations; in other words there will be a degree of arbi-
trariness in the final MLWFs. It is clear from Eq. (12) that
any degree of non-uniqueness of the centres will cause an
undesirable variability of the vdW energy calculated in Sil-
vestrelli’s method. This is indeed what we observe in some of
the examples below. Moving away from a description of the
MLWFs using the valence states only, and towards using
partly occupied MLWFs that include anti-bonding states and
which retain the symmetries of the system, enables us to over-
come these problems, as we demonstrate below.
IV. APPLICATIONS
A. Calculation details
For the application of Silvestrelli’s method to the
following dimer systems we used the Quantum Espresso
(QE) package18to perform the ground-state DFT calcu-
lations, and WANNIER90 (Ref. 25) to obtain the centres
and spreads of the MLWFs. Our results are compared to
both the semi-empirical DFT + D method29,30as imple-
mented in QE, which is expected to give good asymptotic
behaviour, and a wavefunction-based coupled-cluster ap-
proach, CCSD(T), which is considered the “gold-standard”
of quantum chemistry.
The Perdew-Burke-Ernzerhof
generalized-gradientapproximation
correlation, except in the case of argon where the revPBE
(Ref. 32) functional was used; norm-conserving pseudopo-
tentials, and ?-point sampling of the Brillouin zone were
used throughout. We note that we have chosen to use revPBE
for the argon system since PBE produces significant binding
in rare gas dimers as it overestimates the long-range part of
the exchange contribution.12,33,34For all the other systems
we studied in this paper, however, PBE does not cause
spurious binding and would therefore normally be considered
an appropriate functional. A plane-wave basis set cut-off
energy of 80 Ry was used in all calculations with QE except
for the case of the phthalocyanine and copper phthalocyanine
where a 50 Ry energy cutoff was used. For the dimers
of argon, methane, ethene, phthalocyanine, and copper
phthalocyanine, cubic simulation cells of length 15.87 Å,
15.87 Å, 21.16 Å, and 23.81 Å, respectively, were used. For
the dimers of benzene, a hexagonal cell with a = 15.87 Å
and c = 31.75 Å was used.
For all the systems, the choice of energy windows
when using the disentanglement procedure in WANNIER90
for our modified method was as follows: inner (frozen)
energy windows were chosen to include all the valence
states; outer energy windows ranged from the lowest
eigenvalue of the system, ?0, to a maximum of Ewin
= ?LUMO+ α(?HOMO− ?0), where ?HOMOis the energy of the
highest occupied valence Kohn-Sham (KS) state and ?LUMO
is the energy of the lowest unoccupied KS state. The factor α
= 0.4 was chosen to scale down the valence energy band-
width, used to estimate the energy difference required above
the LUMO when including anti-bonding states. We discuss
the sensitivity of the method to this factor in Sec. IV J.
(PBE)
for
(Ref.31)
and exchange
B. Argon
We will first investigate the severity of the aforemen-
tioned issues relating to symmetry, by considering the case
of an argon dimer. Optimization of the MLWFs describing
a single argon atom produces four doubly occupied MLWFs
arranged tetrahedrally around the atom. Due to spherical sym-
metry, the orientation of these MLWFs with respect to a
given coordinate system is arbitrary for an isolated atom and
the final MLWFs obtained will depend on the initial guess
used. In the dimer, this arbitrariness is removed, at least in
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Page 5

154105-5 Dispersion interactions using MLWFs J. Chem. Phys. 135, 154105 (2011)
FIG. 2. Illustration of three of the many possible configurations of MLWF
centres (small pink spheres) for the two argon atoms (large blue spheres) in
the fragment method.
principle, since the spherical symmetry is broken by the pres-
ence of the other atom at a specific orientation. At large sepa-
rations, this is not in practice necessarily the case: the electron
density overlap between the Ar atoms is vanishingly small,
since the wavefunctions decay exponentially away from the
atom. Therefore, to within attainable numerical precision, the
orientation of the MLWFs on each atom is uncorrelated with
the orientation of the other atom: the MLWFs can be freely
rotated with respect to the atom without affecting the total
spread. Note, however, that since the vdW energy only de-
cays as R−6, its value is influenced by the orientation of the
MLWF centres (and hence their separation) out to distances
beyond which the calculated spread (and thus the optimized
MLWF orientation) has ceased to be sensitive to separation.
This dependence can be investigated in a two-atom sys-
tem by fixing the relative orientations of the MLWF centres
between the two atoms in the dimer. This is achieved by first
calculating the MLWF centres for a single atom of argon
and then translating and rotating these centres to the second
argon atom with various choices of alignment. We will
refer to this approach as the fragment method. In this
method, we calculate the dispersion correction to the en-
ergy for a dimer system using various possible arrange-
ments of MLWF centres on the other atom. Three pos-
sible high-symmetry choices are shown in Fig. 2. For
each of these orientations, Fig. 3 (top) shows the bind-
ing energy of the Ar dimer as the separation of the atoms
varies. We see that there is considerable displacement of the
curves, and the binding energy and the equilibrium sepa-
ration change according to the alignment chosen by up to
0.04 kcal/mol and 0.08 Å, respectively.
In contrast to this fragment approach, in Fig. 3 (bottom)
we show the binding energy as calculated with the normal ap-
proachofusingtheoptimizedMLWFsoftheentiredimersys-
tem. However, here we have used varying initial guesses cor-
responding to the set of possible alignments shown in Fig. 2.
3.9 4.04.14.2 4.34.4
Interdimer distance (Å)
-0.30
-0.28
-0.26
-0.24
-0.22
-0.20
Binding energy (kcal/mol)
Anti-aligned 1 (fragment)
Anti-aligned 2 (fragment)
Aligned (fragment)
3.94.04.14.2 4.3 4.4
Interdimer distance (Å)
-0.30
-0.28
-0.26
-0.24
-0.22
-0.20
Binding energy (kcal/mol)
Random
Aligned
Anti-aligned
Continuous
FIG. 3. Binding energy versus interatomic separation for the argon dimer,
for varying relative orientations of the MLWF centres surrounding each atom
(see Fig. 2). (Top panel) Results obtained using the fragment method, in
which the MLWF centres are calculated for a lone Ar atom and then trans-
lated and rotated to the second Ar atom. (Bottom panel) Results obtained
using the true MLWF centres with various initial guesses for their positions.
The curve labelled “continuous” is obtained by using the MLWF centres from
a configuration at small separation as the initial guess for the centres at larger
separations. In this way, the discontinuities in the curve are avoided and a
unique curve is obtained (see text for details).
We see that at small separations, the MLWF centres always
converge to the same positions, regardless of the initial guess,
and the binding energy curve is nearly independent of the
choice of initial guess (∼10−3kcal/mol variation).
At larger separation, however, the spread minimization
is insufficiently sensitive to the relative orientation of the
MLWFs on different atoms, and does not necessarily alter it
from the initial guess, resulting in several different possible
results depending on the initial orientation of the centres. If
a random initial guess is chosen, then the energy varies dis-
continuously, as a function of separation, within the bounds
imposed by the limiting cases described using the fragment
method.ThisisbecausetheMLWFcentresconverge todiffer-
ent orientations depending on their starting positions (curve
labelled “random” in Fig. 3 (bottom)).
In order to avoid this problem of non-uniqueness of bind-
ing energy curves, a random initial guess is used first for
a configuration at small separation, in the knowledge that
the result will be independent of the guess used. Then the
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