Ab initio Wannier-function-based correlated calculations of Born effective charges of crystalline Li$_{2}$O and LiCl

0.19 MB·English

by  Priya Sony

Ab initio Wannier-fun
tion-based
orrelated
al
ulations of Born
2
e(cid:27)e
tive
harges of
rystalline Li O and LiCl
∗
Priya Sony and Alok Shukla
∗
8
Physi
s Department, Indian Institute of Te
hnology, Powai, Mumbai 400076, INDIA
0
0
2 Abstra
t

n
In thispaper we have usedour re
ently developed ab initio Wannier-fun
tion-based methodology
a
J
2
to perform extensive Hartree-Fo
k and
orrelated
al
ulations on Li O and LiCl to
ompute their
2
2
Born e(cid:27)e
tive
harges. Results thus obtained are in very good agreement with the experiments. In

]
ci parti
ular, for the
ase of Li2O, we resolve a
ontroversy originating in the experiment of Osaka
s
- and Shindo [Solid State Commun. 51 (1984) 421℄ who had predi
ted the e(cid:27)e
tive
harge of Li
l
r
t
m ions to be in the range 0.58(cid:22)0.61, a value mu
h smaller
ompared to its nominal value of unity,
.
t thereby, suggesting that the bonding in the material
ould be partially
ovalent. We demonstrate
a
m
thate(cid:27)e
tive
harge
omputed byOsakaandShindo istheSzigeti
harge, andon
etheBorn
harge
-
d
n is
omputed, it is in ex
ellent agreement with our
omputed value. Mulliken population analysis of
o
2
c Li O also
on(cid:28)rms ioni
nature of the bonding in the substan
e.
[

1
PACS numbers: 77.22.-d, 71.10.-w, 71.15.-m
v
8
0
3
3
.
1
0
8
0
:
v
i
X
r
a
1
2 I INTRODUCTION
I. INTRODUCTION
Ab initio
al
ulationof diele
tri
response properties of materialsare routinely performed
1
using methods based upon density-fun
tional theory (DFT). Most of these
al
ulations
presently are based upon the so-
alled 'modern theory of polarization' whi
h is based upon
2,3
a Berry-phase (BP) interpretation of ma
ros
opi
polarization of solids. The pra
ti
al
implementation of the BP approa
h within various versions of the DFT, su
h as the lo
al-
density approximation (LDA), is straightforward be
ause of their mean-(cid:28)eld nature. How-
ever, it is desirable to go beyond the mean-(cid:28)eld level, so as to
ompute the in(cid:29)uen
e of
ele
tron
orrelation e(cid:27)e
ts on polarization properties su
h as the Born
harge of
rystals.
Re
ently, we have proposed an approa
h whi
h allows
omputation of various polarization
4,5
properties of
rystalline insulators within a many-body framework. The approa
h utilizes
a Wannier-fun
tion based real-spa
e methodology,
oupled with a (cid:28)nite-(cid:28)eld approa
h, to
perform
orrelated
al
ulations using a Bethe-Goldstone-like many-body hierar
hy. Its su
-

essful implementationwas demonstrated by performingab initio many-body
al
ulationsof
4 5
Born
harges, and opti
al diele
tri

onstants, of various insulating
rystals. We note that
an ab initio wave-fun
tional-based real-spa
e
orrelated approa
h has re
ently been su

ess-
6
fully demonstrated by Hozoi et al.(author?) to
ompute the quasi-parti
le band stru
ture
of
rystalline MgO.
4
In this paper we apply our re
ently-developed approa
h, to perform ab initio
orrelated
2 2

al
ulations of Born
harges of
rystalline Li O and LiCl. Li O is a te
hnologi
ally impor-
7
tant material with possible appli
ations in thermonu
lear rea
tors, as also in solid-state
8
batteries. Based upon their infrared re(cid:29)e
tivity and Raman s
attering based experiment
2 7
on Li O, Osaka and Shindo reported the value of e(cid:27)e
tive
harge of Li ions to be in the
range 0.58(cid:22)0.61, a value mu
h smaller
ompared to the nominal value of unity expe
ted
in an ioni
material. Therefore, they spe
ulated whether the nature of
hemi
al bond in
7
the substan
e is partly
ovalent. In this work we present a
areful investigation of this
subje
t, and resolve the
ontroversy by showing that the e(cid:27)e
tive
harge reported by Os-
7
aka and Shindo was the Szigeti
harge and not the Born
harge. On
e the Born
harge
is
omputed from the Szigeti
harge, its experimental value 0.95 is in ex
ellent agreement
with our theoreti
al value of 0.91, and mu
h
loser to the nominal value. We also present a
plot of the Wannier fun
tions in the substan
e as also the Mulliken population analysis to
3
2

on(cid:28)rm the ioni
bonding in Li O. As far as LiCl is
on
erned, earlier we
omputed its Born
9

harge at the Hartree-Fo
k level using our Wannier fun
tion methodology, and found its
value to be signi(cid:28)
antly smaller as
ompared to the experiments. Therefore, in this work,
we perform
orrelated
al
ulations of the Born
harge of LiCl, and, indeed, obtain mu
h
better agreement with the experiments.
Remainder of this paper is organized as follows. In se
tion II we brie(cid:29)y des
ribe the
theoreti
al aspe
ts of our Wannier-fun
tion-based methodology. Next in se
tion III we
present and dis
uss the results of our
al
ulations. Finally, in se
tion IV we present our

on
lusions.
II. METHODOLOGY
Z∗ (i) i
The Born e(cid:27)e
tive
harge tensor, αβ , asso
iated with the atoms of the -th sublatti
e,
10
is de(cid:28)ned as (el)
∂P (cid:12)
Z∗ (i) = Z + (Ω/e) α (cid:12) ,
αβ i ∂u (cid:12) (1)
iβ (cid:12)
=
(cid:12)E 0
Z Ω
i
where is the
harge asso
iated with the nu
lei (or the
ore) of the sublatti
e, is the
(el)
e P α
α
volumeoftheunit
ell, ismagnitudeoftheele
troni

harge, and isthe -thCartesian

omponent of the ele
troni
part of the ma
ros
opi
polarization indu
ed as a result of
β ∆u ∆u
iβ iβ
the displa
ement of the sublatti
e in the -th Cartesian dire
tion, . For small ,
∂Pα(el)(cid:12) = ∆Pα(el) ∆P(el)
one assumes ∂uiβ (cid:12) = ∆uiβ , with α
omputed as the
hange in the ele
troni
(cid:12)E 0
∆u
iβ
polarization per unit
ell due to the displa
ement . For a given latti
e
on(cid:28)guration
(el)
λ P (λ)
α
denoted by parameter , we demonstrated that , for a many-ele
tron system,
an be
4

omputed as the expe
tation value
q
P(el)(λ) = e Ψ(λ) X(el) Ψ(λ) ,
α NΩh 0 | α | 0 i (2)
q = e N ( )
e
( − ) is the ele
troni

harge, → ∞ , represents the total number of unit
ells in
X(el) = Ne x(k) α
the
rystal, α Pk=1 α is the -th
omponent of the many-ele
tron position operator
(λ)
N Ψ
for the e ele
trons of the
rystal, and | 0 i represents the
orrelated ground-state wave-
fun
tionof the in(cid:28)nite solid, expressed interms of Wannierfun
tions. Forone-ele
tronmod-
4
els su
h as Kohn-Sham theory, or the Hartree-Fo
k method, we showed that Eq. (2) leads
2
to expressions
onsistent with the Berry-phase-based theory of King-Smithand Vanderbilt ,
3
and Resta .
4 II METHODOLOGY
Cal
ulating the expe
tation value of Eq. (2)
an be tedious for a general
orrelated
wave fun
tion. We avoid this by using the (cid:28)nite-(cid:28)eld approa
h whereby the expe
tation
values of various
omponents of dipole operator
an be obtained by performing both the
HF and
orrelated
al
ulations in presen
e of a small ele
tri
(cid:28)eld in that dire
tion, and
then
omputing the (cid:28)rst derivative of the
omputed total energy with respe
t to the applied
4,5
ele
tri
(cid:28)eld. This approa
h of
omputing expe
tation values derives its legitima
y from
the generalized Hellman-Feynman theorem, and is
alled (cid:16)(cid:28)nite-(cid:28)eld method(cid:17) in quantum
11

hemistry literature. Therefore, the present set of HF and
orrelated
al
ulations are dis-
12,13,14,15,16
tin
t from our previous ones in that the present
al
ulations have been performed
in presen
e of an external ele
tri
(cid:28)eld. For details related to our Wannier-fun
tion-based
HF methodology, in the presen
e of external ele
tri
(cid:28)elds, we refer the reader to our earlier
4,5
papers.
E
corr
Correlation
ontributionstotheenergy perunit
ell , even inthepresen
e ofexternal
17
ele
tri
(cid:28)elds, were
omputed a

ording to the in
remental s
heme of Stoll
1
Ecorr = Xǫi + X∆ǫij +
2!
i i6=j
1
X ∆ǫijk + ,
3! ··· (3)
i6=j6=k
ǫ , ∆ǫ , ∆ǫ ,... ,...
i ij ijk
where et
. are respe
tively the one-, two- and three-body
orrelation
in
rements obtained by
onsidering simultaneous virtual ex
itations from one, two, or three
i, j, k,...
15
o

upied Wannier fun
tions, and label the Wannier fun
tions involved. Be
ause
n
of the translational symmetry, one of these Wannier fun
tions is required to be in the
referen
eunit
ell,whiletheremainingones
anbeanywhereelseinthesolid. Inourprevious
studies performed on ioni
systems, we demonstrated that the in
rements
an be trun
ated
n = 2)
to two-body e(cid:27)e
ts ( , with the Wannier fun
tions not being farther than the third-
15,16
nearest neighbors. We follow the same trun
ation s
heme here as well, with the method
of
orrelation
al
ulation being the full-
on(cid:28)guration-intera
tion (FCI) as in our previous
4,5,15,16
works, where other te
hni
al details related to the approa
h are also des
ribed.
2
A Li O 5
III. RESULTS AND DISCUSSION
2
In this se
tion, (cid:28)rst we present the results of our
al
ulations of Born
harges of Li O,
followed by those of LiCl.
2
A. Li O
2
The primitive
ell of Li O
onsists of a three-atom basis (two Li and one O), with ea
h
atom belonging to an f

latti
e. In our
al
ulations anion was pla
ed at the origin of
( a/4 a/4, a/4)
the primitive
ell, while the
ations were lo
ated at the positions ± ± ± . This
stru
ture is usuallyreferred to as the anti-(cid:29)uoritestru
ture, to whi
h other related materials
2 2
su
h as Na O and K O also belong. Based upon intuition, one would
on
lude that the
2s
bonding in the substan
e will be of ioni
type, with the valen
e ele
trons of the two
lithium atoms of the formula unit being transferred to the oxygen. Thus the two Li atoms
willbein
ationi
statewhiletheoxygenwillbeindianioni
state. However, theargumentfor
−2
partial
ovalen
e inthe system goesas follows. It iswell-known that the free O ions do not
exist in nature. Therefore, in lithium oxide it will be stabilized only be
ause of the
rystal-
−2
(cid:28)eld e(cid:27)e
ts. Still, one
ould argue, that O ion in solid state will be very di(cid:27)use, leading
to partial
ovalen
y. In light of these arguments, it is interesting to examine the results of
7 2
the experiments of Osaka and Shindo performed on the single
rystals of Li O, where they
measured the frequen
ies of fundamental opti
al and Raman a
tive modes using infrared
and Raman spe
tros
opy. Using the measured frequen
ies and several other parameters, the
authors (cid:28)tted their data both to rigid-ion and shell models of latti
e dynami
s, to obtain
7
values for Li e(cid:27)e
tive
harges of 0.58 and 0.61, respe
tively. Noting that the obtained
e(cid:27)e
tive
harge is quite di(cid:27)erent from the nominal value of unity, they argued that the
2 7
result suggests that Li O has partially
ovalent
hara
ter. Next, we argue, however, that
18
the e(cid:27)e
tive
harge reported by Osaka and Shindo is a
tually the Szigeti
harge, and not
the Born
harge. The shell-model value of 0.61 of the e(cid:27)e
tive
harge of Li was
omputed
7 19
by Osaka and Shindo using the expression for the (cid:29)uorite stru
ture derived by Axe
3√ǫ
Z = [(ν2 ν2 )µπr3]1/2 ∞,
S LO − TO 0 ǫ (4)
∞+2
ν /ν r = a
where LO TO are the longitudinal/transverse opti
al phonon frequen
ies, 0 2, where
a µ ǫ
∞
is the latti
e
onstant, is the e(cid:27)e
tive mass of the three ion system, while is the
6 III RESULTS AND DISCUSSION
high-frequen
y diele
tri

onstant of the material. The expression in Eq. 4 is nothing but
Z
18 T
the Szigeti
harge of the ion from whi
h the Born
harge ( )
an be
omputed using the
18
relation
(ǫ +2)
∞
Z = Z
T S
3 (5)
ǫ = 2.68 Z ( ) = 0.61
∞ S 7
If we use the values and Li obtained by Osaka and Shindo in
0.95
2
Eq. 5, we obtain the experimental value of Born
harge of Li ions in Li O to be whi
h
is very
lose to the nominal value of unity. From this, the experimental value of the Born
1.90

harge of the O ions
an be dedu
ed to be . These values of the Born
harges of Li and
2
O ions imply, in the most unequivo
al manner, an ioni
stru
ture for Li O
onsistent with
the intuition, without any possibility of
ovalen
y.
Next, we report the results of our ab initio
orrelated
al
ulations of Born
harges of
2
bulk Li O. These
al
ulations were performed using the lobe-type Gaussian basis fun
tions
whi
h, as demonstrated in the past, represent Cartesian Gaussian basis fun
tions to a very
12,13,14 2 14
good a

ura
y. As in our previous HF study of the ground state properties of Li O,
20
our
al
ulationswere initiated by a smallerbasis set reported by Dovesi et al.
onsisting of
(2s,1p) (4s,3p)
fun
tions forlithium,and fun
tionsforoxygen. However, we alsoaugmented
d
20
our basis set by a single -type exponent of 0.8, re
ommended also by Dovesi et al., so as
to
he
k the in(cid:29)uen
e of higher angular momentum fun
tions on our results. Hen
eforth,
(s,p) (s,p,d)
we refer to these two basis sets as and basis sets, respe
tively. For the purpose
T = 0
of the
al
ulations, f

geometry along with the experimentally extrapolated latti
e
a = 4.573
8

onstant value of Å was assumed. For our (cid:28)nite-(cid:28)eld approa
h to Born
harge
0.001
des
ribed earlier, ele
tri
(cid:28)elds of strength ± atomi
units (a.u.), along with an oxygen
∆u = 0.01a x
latti
e displa
ement , both in the −dire
tion, were used. This leads to the
Z∗
determination of xx
omponent of the Born
harge
orresponding to the oxygen latti
e.
1s
During the
orrelated
al
ulations, Wannier fun
tions
orresponding to fun
tions of both
2s 2p
Li and O were held frozen, while virtual ex
itations were
arried out from the and
Z∗ =
orbitals of oxygen anion. At the HF level we veri(cid:28)ed that the symmetry relation xx
Z∗ = Z∗
yy zz
holds, along with the fa
t that the o(cid:27)-diagonal elements of the Born
harge
Z∗ ( ) + 2Z∗ ( ) = 0
tensor are negligible. Additionally, the opti
al sum rule αβ O αβ Li , was also
veri(cid:28)ed.
2
Our
al
ulated results for the Born
harge of oxygen in Li O are presented in table I. It is
obvious from the table that irrespe
tive of the basis set used, most of the
ontribution to the
2
A Li O 7
2
Table I: Correlated values of Born Charge of Li O obtained in our
al
ulations. Column with
headingHFreferstoresultsobtainedattheHartree-Fo
klevel. Heading(cid:16)one-body(cid:17) referstoresults
obtained after in
luding the
orre
tions due to (cid:16)one-body(cid:17) ex
itations from ea
h Wannier fun
tion
of the unit
ell, to the HF value. Two-body (O) implies results in
lude additional
orre
tions
due to simultaneous ex
itations from two distin
t Wannier fun
tions lo
ated on the anion in the
referen
e unit
ell. Two-body (1NN) and two-body (2NN)
orrespond to two-body
orrelation
e(cid:27)e
ts involving 1st and 2nd, nearest neighbor Wannier fun
tions, respe
tively.
Basis Set Oxygen Born Charge Experiment
a
HF One-body Two-Body (O) Two-body (1NN) Two-body (2NN) 1.90
(s,p)
1.794 1.783 1.763 1.781 1.777
(s,p,d)
1.851 1.840 1.814 (cid:22) (cid:22)-
a
7
As dedu
ed from the Szigeti
harge reported in referen
e.
Born
harge, as expe
ted, is obtained at the HF level, with relatively smaller
ontribution
of ele
tron-
orrelation e(cid:27)e
ts. Correlation e(cid:27)e
ts do diminish the value of the Born
harge
as
ompared to its HF value, however, rather insigni(cid:28)
antly. Comparatively, speaking the
d
in
lusionof one −type basis fun
tion on oxygen has greater in(cid:29)uen
e on the Born
harge of
(s,p)
oxygen, and its value is in
reased as
ompared to the one obtained using the basis set.
(s,p,d)
With the basis set, we found negligible in(cid:29)uen
e on the Born
harge from the two-
body
ontributions outside of the referen
e unit
ell. Thus our
al
ulations, as per opti
al
sum rule, predi
t a value of Born
harge of Li to be
lose to 0.91, in very good agreement
7
with the experimental results, with the error being less than 5%. In order to obtain a
2p
pi
torial view of ele
troni
stru
ture in the material, in Fig. 1 we plot one of the oxygen
(s,p,d) (1,1,1)
valen
eWannierfun
tions,
omputedusingthe basisset, alongthe dire
tion.
To simulate the e(cid:27)e
t of one of the phonon modes, we displa
ed the oxygen atom of the
(0.01a,0.01a,0.01a) a
primitive
ell to the position , where is the latti
e
onstant, leaving
the Li atoms undisturbed. From the (cid:28)gure it is obvious that the Wannier fun
tion is highly
lo
alized around the oxygen site, with small value in the interatomi
region, implying the
ioni

hara
ter of the material.
Another quantity whi
h
ould help us determine whether the system is ioni
or
ovalent
(s,p)
is the Mulliken population asso
iated with the atoms of the primitive
ell. With the
2.005 0.984
basisset, theLiandOMullikenpopulationswere foundtobe− and ,respe
tively,
8 III RESULTS AND DISCUSSION
2p (1,1,1)
Figure 1: One of the valen
e Wannier fun
tions of oxygen plotted along the dire
tion.
In order to simulate the e(cid:27)e
t of opti
al phonons,
al
ulation was performed with oxygen shifted to
(0.01a,0.01a,0.01a)
the position , and the Li positions un
hanged. Nodes in the Wannier fun
tion
( a/4, a/4, a/4) 1s
at positions ± ± ± are due to its orthogonality to the Li Wannier fun
tions lo
ated
there.
1
0.5
)
r
(p 0
2
φ
-0.5
-1
-5 -4 -3 -2 -1 0 1 2 3 4 5
r (a.u.)
(s,p,d) 2.004 0.984
while with the basis set the populations were − and . It is obvious that
the
omputed Mulliken populations have
onverged with respe
t to the basis set, and are
very
lose to the nominal ioni
ities of the two atoms. This further
on(cid:28)rms the ioni
pi
ture
of the bonding in this substan
e.
B. LiCl
21
These
al
ulations were performed using the basis set proposed by Pren
ipe et al. in
their ab initio HF study of alkali halides. The experimental f

geometry with latti
e

onstant of 5.07 Å was assumed. The unit
ell employed during the
al
ulations
onsisted of
(a/2,0,0)
anion pla
ed at the origin, and the
ation at the position . During the many-body
3s 3p

al
ulations, only the valen
e Wannier fun
tions
omposed of , and orbitals lo
alized
on Cl, were
orrelated. In order to
al
ulate the Born
harge, the anion was displa
ed in the
x 0.01a
-dire
tion by an amount . For the (cid:28)nite-(cid:28)eld
omputation of the dipole expe
tation
B LiCl 9
Table II: Correlated values of Born Charge of LiCl obtained in our
al
ulations. Rest of the
information is same as in the
aption of table I.
Basis Set Chlorine Born Charge Experiment
a
HF One-body Two-Body (O) Two-body (1NN) 1.231
(s,p)
1.019 1.020 1.037 1.013
(s,p,d)
1.098 1.128 1.178 (cid:22)
a
22
Computed from the experimental data reported in referen
e.
0.01 x
value, an external ele
tri
(cid:28)eld of strength ± a.u., also along the -axis, was applied.
(s,p)
The results of our
al
ulations are presented in table II. With our basis set the

orrelatione(cid:27)e
tswerein
ludeduptotheleveloftwo-bodyin
rementsinvolvingtheWannier
(s,p,d)
fun
tions in the nearest-neighboring
ells. However, with the augmented basis set,
in
lusionof two-body in
rements beyond the referen
e unit
ell, didnot make any signi(cid:28)
ant
2
di(cid:27)eren
e to the results. Similar to the
ase of, Li O, we note that: (a) in
lusion of one
d
-type fun
tion improves the results both at the HF and the
orrelated levels, and (b) the
in
lusionof ele
tron-
orrelatione(cid:27)e
ts generally improves the results as
ompared to the HF
values. The (cid:28)nal value of Born
harge of LiCl of 1.178 obtained from these
al
ulations, is
again in very good agreement with the experimental value, with an error smaller than 5%.
IV. CONCLUSIONS
In
on
lusion, we performed Wannier-fun
tion-based ab initio
orrelated
al
ulations on
2

rystalline Li O and LiCl to
ompute their Born e(cid:27)e
tive
harges, using a methodology re-

ently developed inour group. Cal
ulationswere performed using polarizablebasis sets, and
the results obtained were in very good agreement with the experimental ones. Additionally,
we also resolved an ambiguityasso
iated withrather smallvalue of experimentallymeasured
2
e(cid:27)e
tive
harge of Li O, whi
h had raised the possibility of partial
ovalen
e in its
hemi
al
7
stru
ture, by demonstrating that the measured
harge was the Szigeti
harge, and not the
Born
harge of the
ompound. The Born
harge obtained from the Szigeti
harge was found
to be in good agreement with its nominal ioni
value. By further
omputing the Mulliken

harges and exploring the Wannier fun
tions, we demonstrated that the results were
on-
2
sistent with a ioni
pi
ture of Li O. The
omputational e(cid:27)ort involved in our
al
ulations
10 IV CONCLUSIONS
is signi(cid:28)
ant heavier than in the
al
ulations performed using the density-fun
tional theory.
This is be
ause our approa
h is many-body in nature, requiring
al
ulations beyond the
mean-(cid:28)eld. At present we are exploring the possibility of using our approa
h to
ompute
the Born
harges offerroele
tri
,aswellas
ovalentmaterials,andwe willpublishourresults
in future as and when they be
ome available.
A
knowledgments
This work was supported by grant no. SP/S2/M-10/2000 from Department of S
ien
e
and Te
hnology (DST), Government of India.
∗
Present Address: Group of Atomisti
Modeling and Design of Materials, University of Leoben,
Franz-Josef-Strasse 18, A-8700 Leoben, Austria
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