1. First principles computation of second-order
elastic constants and equations of state for
tetragonal BaT iO3
Ghous Narejo and W. F. Perger
Electrical Engineering Department
Michigan Tech University
Abstract
First-principles computational techniques are employed for computing the second-
order elastic constants (SOEC) and equations of state for the tetragonal phase of
BaT iO3 . The bulk modulus is computed by two independent methods. A variety
of potentials and basis sets are used. The impact of the potentials, the basis sets
and the sensitivity of the crystalline geometry optimization on the computational
values is discussed.
Key words: Elastic constant, Bulk modulus
1 Introduction
The perovskites are an important class of materials which have potential to
find applications in the fields of memory, logic design and switching. However,
the theoretical exploration of these materials is not on par with their tech-
nological importance. Crystalline BaT iO3 possesses a perovskite geometry
which may occur in cubic, tetragonal, rhombohedral and orthorhombic crys-
talline phase. There is a general lack of the data on the mechanical properties
of all phases of BaT iO3 . However, there have been attempts [?] to understand
the electronic and mechanical properties of this material. The computations of
elastic constants and bulk modulus were done by Piskunov et al [?] by ab-inito
computational method for cubic BaT iO3 .
Email address: wfp@mtu.edu (W. F. Perger).
URL: http://www.ee.mtu.edu/faculty/wfp.html (W. F. Perger).
Preprint submitted to Elsevier 7 February 2011

2. At room temperature, BaT iO3 has tetragonal crystalline geometry. In the
past, the elastic constants and bulk modulus of tetragonal BaT iO3 could not
be computed due to the complexity of these computations. In this paper, the
computations of elastic constants and the bulk modulus of tetragonal BaT iO3
are done with ab-initio Hartee Fock (HF), density functional theory (DFT)
and hybrid potentials. The ab-initio computational techniques are employed
effectively due to recent advancements in basis sets, geometry optimization,
and computational power.
In this work, the computations of the tetragonal phase of BaT iO3 are made
with the CRYSTAL09 code using a variety of basis sets. The CRYSTAL09
code has the unique characteristic to employ the Hartree-Fock (HF), exchange
correlation potentials and the effective combination of the former with the
latter. In addition, the availability of a variety of basis sets and geometry
optimization techniques further enhances the efficiency of these computations.
An important factor that affects the efficiency of computation is obtaining the
optimized geometry of the crystalline structure. The geometry optimization
of the tetragonal phase of BaT iO3 is an essential step during computations
of elastic constants and bulk modulus. The total geometry optimization is
done to optimize the atomic positions and lattice parameters of the unit cell
of tetragonal BaT iO3 . These special features of the geometry optimization in
the CRYSTAL09 code [?] makes it an efficient program to achieve the optimum
computational results.
The computations of the bulk modulus are done independently with ELAST-
CON and EOS programs. These programs employ the geometry optimization
techniques available in the CRYSTAL09 code. The ELASTCON algorithm
determines the number of crystalline deformations based upon the crystalline
geometry of tetragonal BaT iO3 . Geometry optimization of the crystalline
structure is carried out after each deformation. The analytic first derivative,
numerical second derivative and Levenberg Marquardt (LM) curve fitting is
done in sequence to compute elastic constants for the tetragonal BaT iO3 . A
detailed discussion of the ELASTCON program can be seen in ref. [?].
The equation of state algorithm is utilized for computation of bulk modulus
from a pressure vs. volume curve. The range of volumes around the optimized
equilibrium may be selected by the input parameters available in the EOS
program. The computation of total energy is then done for this range of vol-
umes. The geometry optimization is done for each volume while keeping the
volume constant. The computation of elastic constants and bulk modulus of
tetragonal BaT iO3 is complicated due to its crystalline geometry and the po-
sition of the atoms in the perovskite crystalline structure. With the lowering of
crystalline symmetry the number of elastic constants increases and geometry
optimization steps are increased.
2

3. The crystalline geometry of the tetragonal BaT iO3 must be fully optimized
during the computation of the elastic constants and bulk modulus. This is
possible only if the tetragonal crystalline system is relaxed during each in-
dividual computation. The task of crystalline geometry optimization of the
perovskite BaT iO3 has remained a challenge as the crystalline system un-
dergoes a sudden decrease in energy when it is deformed. The ELASTCON
algorithm has detected this sudden decrease in the strain vs. energy computa-
tions as reported earlier [?,?,?]. Detailed energy vs. strain computations were
carried out to explore the sudden decrease in energy. The ELASTCON algo-
rithm issues a warning message if there is an abrupt and unexpected change
in the energy during optimization which helps to understand the optimization
process further.
2 ELASTCON and EOS programs in CRYSTAL09 code
The computation of elastic constants and bulk modulus is an automated pro-
cess in ELASTCON program and is determined from the crystalline symmetry
of tetragonal BaT iO3 . Deformations are applied depending upon the tetrago-
nal crystalline geometry. The analytic first derivative and the numerical second
derivative of total energy are computed for each deformation. The Levenberg
Marquardt (LM) curve fitting [?] is used to compute the elastic constants and
bulk modulus.
The linear deformation of solids is expressed by Hooke’s law as
σij = Cijkl kl (1)
kl
where i, j, k, l = 1, 2, 3, σi,j,k,l , k,l and Ci,j,k,l are stress, strain and second-order
elastic constant tensors.
The second-order elastic constants may be computed with different tech-
niques. The molecular dynamics and ab-initio computational techniques are
two prominent methods to compute the elastic constants and bulk modulus.
The ab-initio computational techniques compute the second-order elastic con-
stant (SOEC) from the total energy. The elastic constants can be computed
from the Taylor series expansion of the total energy with respect to the applied
strains, as shown in Eq. (??). The Taylor series terms up to the second-order
may be utilized for the estimation of the elastic constants and bulk modulus
if the strains are very small and the higher order terms have negligible effects
3

4. on the computational results.
V
E (V, ) = E(V0 ) + σα α + Cαβ α β + ..... (2)
α 2 αβ
The terms α, β = 1, .., 6 express the elastic constants in Voigt notation and
V0 is the equilibrium volume. In Eq. (??), the non-zero stress term may be
ignored if the crystalline geometry of the system is fully optimized. The third
term in Eq. (??) can be rewritten to express the elastic constant as the second
derivative of the total energy with respect to the applied strain in a crystalline
direction
1 ∂ 2E
Cαβ = . (3)
V ∂ α∂ β 0
In Eq. (??), the terms Cαβ , E, and V express the elastic constant tensor,
energy and volume of the crystalline structure, respectively.
The ab-initio computations are done by calculating the analytic first deriva-
tive of the total energy and numerical second derivative with respect to the
applied strain. The appropriate number of strains are applied in a systematic
manner, the elastic constants are calculated, and the compliance coefficients
are computed from Eq. (??). The compliance coefficients are then utilized for
the computation of the bulk modulus as shown in Eq. (??).
[S] = [C]−1 (4)
B = 1/(S11 + S22 + S33 + 2(S12 + S13 + S23 )). (5)
The terms Sij and B in Eq. (??) express the compliance tensor elements and
bulk modulus respectively.
The EOS algorithm in CRYSTAL09 computes the energy for a range of
volumes around the optimized equilibrium volume. The Crystal09 code is
equipped with a wide variety of equations of state such as Birch Murnaghan,
3rd order Birch Murnaghan, logarithmic, Vinet and polynomial. In this pa-
per, the 3rd order Birch Murnaghan equation of state algorithm was utilized
for computing the bulk modulus from the energy vs. volume computations as
expressed in Eq. (??):
B −1
1 Vo V 1
E(V ) = Bo Vo + − + Eo . (6)
B (B − 1) V B Vo B − 1
In Eq. (??), V0 represents the volume at the lowest energy, B0 is the bulk
4

5. modulus at pressure P = 0, B is the derivative of bulk modulus B at P = 0
and E0 is the minimum energy. The optimization of crystalline geometry at
each step is done during energy-volume (E-V) calculations. The bulk modulus
results are obtained with Levenberg-Marquardt curve fitting of the E vs. V
computations.
3 Choice of basis sets
The selection of basis sets [?,?] vs. [?,?] affects the computations of the elastic
constants and bulk modulus. The selection of adequate basis sets is an impor-
tant factor in the computational physics for examining the complex material
physics of the perovskite BaT iO3 . We have chosen 6-31d1 and 8-411d11 basis
sets for Oxygen and HAYWSC ECP basis set for Ba and Ti atoms from the
Crystal06 basis set library [?]. We have named the combination of basis sets
8-411d11 and HAYWSC ECP as basis set 1 and 6-31d1 and HAYWSC ECP
as basis set 2. The computational efficiency of the results with basis set 1 is
also reported earlier for cubic BaT iO3 in ref. [?].
The variation in the values of elastic constants and bulk modulus confirm the
fact that the improper choice of basis sets can severely affect the computations.
The efficient vs. inefficient usage of basis sets hint at a trade-off between
the efficiency of the results and the computational time. Our approach is to
check the credibility of the results by employing the basis sets in two different
algorithms and comparing the results attained by each of these basis sets.
4 Geometry optimization
For temperatures above 393◦ K, BaT iO3 has a stable cubic crystalline struc-
ture. However, as the temperature is lowered, the BaT iO3 crystalline structure
changes into tetragonal, orthorhombic and rhombohedral phases at 393◦ K,
278◦ K and 183◦ K [?]. This paper reports the computations of the elastic con-
stants and bulk modulus of the tetragonal P 4mm BaT iO3 . The experimental
values [?] of the lattice constants and the atomic positions for tetragonal
BaT iO3 are used as an input to the Crystal09 code.
Geometry optimization is an essential step for the computations of the elastic
constants and bulk modulus. It relaxes the crystalline structure and removes
the effect of the strain due to the non-optimized crystalline geometry. The
effect of the non-optimized crystalline geometry would appear as the second
term on the right hand side of Eq. (??) not being zero if the geometry opti-
mization is not employed in the computations. The geometry optimization of
5

6. the atomic positions and cell parameters of a tetragonal BaT iO3 crystalline
system was done with Hartree Fock, density functional theory (DFT) poten-
tials and hybrid mixing of the former with the latter.
In the ELASTCON program [?], the computation of the bulk modulus requires
a series of deformations along the crystalline axes and optimization of the in-
ternal co-ordinates of atomic positions and cell parameters for each of these
deformations. This step is followed by determining analytic first-derivatives
and numerical second-derivatives of the total energy with respect to applied
deformations resulting in elastic constants. The elastic constants are then em-
ployed for computation of the bulk modulus as in Eq. (??).
In the EOS algorithm [?], a series of volumes is chosen around the equilibrium
and optimization of the internal co-ordinates is performed holding the volume
constant. The energy vs. volume curve is fitted to an equation of state and the
bulk modulus is computed. (See ref. [?] for the detailed discussion about the
types of geometry optimization implemented in Crystal06 code). The values of
the optimized lattice constants along with the energy and volume are reported
in Tables ?? and ??.
5 Selection of Hartree Fock, DFT and hybrid potentials
The computations of elastic constants and bulk modulus were performed with
the potentials HF, DFT and hybrid mixing of the former with the latter. The
DFT and hybrid exchange correlation potentials are employed due to the lack
of correlation of HF as reported in [?]. Moreover, the aim of employing HF,
DFT and hybrid potentials is to verify and confirm the computational merit of
these results. The computational results of the optimized lattice constants ob-
tained with basis set 1 and exchange correlation potential PWGGA and hybrid
exchange correlation potentials B3LYP and B3PW have shown an improve-
ment with respect to experiment. Therefore the employment of HF, DFT and
hybrid exchange correlation potentials have essentially reinforced the compu-
tational merit of PWGGA and hybrid potentials. The computational values
of the optimized lattice constants, elastic constants and bulk modulus results
obtained with HF, DFT and hybrid potentials are reported in Tables ?? -??.
6

7. Table 1
The values of relaxed lattice constants (in ˚), ambient volume (in ˚3 ),
A A
and total energy, E (in a.u.), for tetragonal BaT iO3 . The computations
were done by using Hartree-Fock, DFT-BLYP, DFT-B3LYP and DFT-
PWGGA potentials. The basis sets employed for Ba, Ti and O atoms are
collectively named as basis set 1 (see section ??).
a c c/a Vol. E
HF 3.96 4.26 1.07 67.17 -307.5388
LDA 3.93 3.93 1.00 61.12 -307.8828
PWGGA 4.00 4.03 1.00 64.75 -309.5928
BLYP 4.05 4.14 1.02 68.12 -309.4749
B3LYP 4.01 4.10 1.02 66.15 -309.4084
B3PW 3.98 4.02 1.01 63.75 -309.5360
Exp. [?,?] 3.99 4.03 1.01 64.16 -
Table 2
The values of relaxed lattice constants (in ˚), ambient volume (in ˚3 ),
A A
and total energy, E (in a.u.), for tetragonal BaT iO3 . The values are com-
puted with Hartree-Fock, DFT-PWGGA, DFT-BLYP and DFT-B3LYP
potentials. The basis set used for Ba, Ti and O atoms are collectively
named as basis set 2 (see section ??).
a c c/a Vol. E
HF 3.98 3.98 1.00 63.23 -307.4627
PWGGA 3.95 3.95 1.00 61.94 -309.4862
BLYP 4.00 4.00 1.00 64.06 -309.3636
B3LYP 3.97 3.97 1.00 62.66 -309.3082
Exp. [?,?] 3.99 4.03 1.01 64.16 -
7

8. Table 3
The elastic constants and bulk modulus computational results using the
HF and DFT LDA, PWGGA, BLYP, and B3LYP and B3PW potentials
with basis set 1 (see section ??). All values are in GPa.
C11 C12 C13 C33 C44 C66 B
HF 308.16 127.89 88.00 66.09 120.55 162.87 61.64
LDA 443.63 93.05 93.08 443.54 224.297 224.31 209.91
PWGGA 353.79 76.01 56.46 227.22 174.97 193.53 138.64
BLYP 284.53 82.10 58.05 146.28 73.33 160.04 109.82
B3LYP 315.30 94.53 64.13 166.22 76.47 176.58 123.31
B3PW 370.94 88.85 61.25 228.04 147.91 203.69 145.11
Exp.[?] 211+-6 107+-5 114+-8 160 +-11 56.2+-1.7 127+-4 125-141[?]
[?] 242.7 128.3 122.6 147.9 54.9 120.1 -
[?] 275.1 178.9 151.55 164.8 54.3 113.1 -
[?] 222.9 - 147.0 240.0 61.7 133.7 -
Table 4
The elastic constants and bulk modulus computations using the HF, DFT
LDA, PWGGA, BLYP, B3LYP and B3PW potentials with basis set 2
(see section ??). All values are in GPa.
C11 C12 C13 C33 C44 C66 B
HF 408.03 154.19 154.19 408.02 184.72 184.72 238.80
PWGGA 444.52 107.43 107.42 444.52 107.42 213.71 219.79
BLYP 381.51 111.14 111.15 381.52 184.79 184.79 201.27
B3LYP 414.21 122.55 122.54 414.21 200.01 200.01 219.77
Exp.[?] 211+-6 107+-5 114+-8 160 +-11 56.2+-1.7 127+-4 125-141[?]
[?] 242.7 128.3 122.6 147.9 54.9 120.1 -
[?] 275.1 178.9 151.55 164.8 54.3 113.1 -
[?] 222.9 - 147.0 240.0 61.7 133.7 -
8

9. 6 Computation of bulk modulus by equation of state (EOS)
The EOS algorithm utilizes the systematic changes [?] in the volume around
the optimized equilibrium state of a crystalline structure. The EOS calcula-
tions are carried out by selecting a range of volumes around a minimum total
energy at an equilibrium state of the tetragonal BaT iO3 crystalline structure.
The EOS algorithm has permits selection of a range of volumes and a number
of volumes within that range. At each of the volumes in the range, the con-
stant volume optimization is carried out. The energy vs. volume results are
curve-fitted to an EOS such as Murnaghan EOS [?] as shown in Eq. (??).
The Crystal09 code [?] can accomplish an optimization of the internal co-
ordinates and lattice parameters while keeping the volume constant (see ref. [?]
for the detailed description of CVOLOPT option of geometry optimization).
Table ?? and ?? show the results of EOS algorithm for the calculation of a
series of total energies at the chosen volumes and fitted to Eq. (??) using a
Levenburg-Marquardt [?] curve fitting algorithm.
7 Discussion of computational results
[?] We have observed the deviation in values of optimized lattice constants a
and c of tetragonal BaT iO3 as compared with the experimental values [?,?].
The main sources of these deviations are basis sets and potentials and the
sensitivity of each due to the displacive nature of tetragonal BaT iO3 [?]. It
is generally observed that local DFT (LDFT) exchange correlation potentials
underestimate and nonlocal DFT (NLDFT) exchange correlation potentials
overestimate the lattice constants. The comparison of lattice constants a, c
and the ratio c/a with experimental values is shown in Tables ?? and ??.
The values of optimized lattice constants have shown a better agreement with
the experimental values for basis set 1. The computational values of a and c are
slightly higher than the experimental values for DFT-PWGGA, DFT-BLYP,
DFT-B3LYP and DFT-B3PW exchange correlation potentials employed with
basis set 1. However, the optimized lattice constants a and c computed with HF
and DFT-LDA exchange correlation potential employed with basis set 1 show a
different trend. The optimized lattice constants a and c, computed with DFT-
LDA exchange correlation potential, are lower than the experimental lattice
constants. On the other hand, the optimized lattice constant a has decreased
and c has increased from the experimental value for the computations done
with HF potential. Surprisingly, the optimized values of lattice constants a and
c computed with basis set 2 show large deviations from the experimental values
of 3.99 and 4.03. The values of the optimized lattice constants a and c are equal
9

10. for Hartree Fock, exchange correlation and hybrid potentials employing basis
set 2 as shown in Table ??.
The computational values of bulk modulus, done with ELASTCON and EOS
methods employing HF, DFT and hybrid potentials with basis set 1, are re-
ported in Tables ?? and ??. The DFT-LDA have shown higher bulk mod-
ulus values due to underestimation of the optimized lattice constant values.
Whereas, the computations done with DFT-PWGGA and DFT-BLYP have
shown lower values of bulk modulus due to the overestimation of optimized
lattice constants. The computations done with hybrid potentials have shown
the optimum values of bulk modulus as a consequence of the fact that the op-
timized lattice constants are comparatively close to the experimental values.
In contrast with these results, the computational values of bulk modulus are
higher with the basis set 2 as reported in Tables ?? and ??. The large increase
in computational values of bulk modulus for the basis set 2 is attributed to
the deviation in optimized lattice constants from the experimental values.
The order -disorder nature [?] of perovskite BaT iO3 is another important
factor that makes the computation of optimum values of the bulk modulus
problematic, requiring accurate basis sets and potentials. It has been observed
that the computation of the bulk modulus for tetragonal BaT iO3 is coupled
intimately with the geometry optimization of its complex crystalline structure.
Slight deviations in the values of optimized lattice constants a and c have
shown large effects on the computational values of bulk moduli.
The bulk moduli obtained with PWGGA and hybrid potentials employing
basis set 1 are computationally accurate. This analysis is based upon the
computational results obtained with these potentials and basis sets for cu-
bic BaT iO3 [?] and rutile T iO2 [?] crystalline systems. Moreover, the values
of bulk modulus computed with ELASTCON and EOS algorithms are suffi-
ciently close to each other for the same potential and basis set. These facts
point to the relative merit of the approach for the selection of a variety of
basis sets, potentials and algorithms. Therefore, the computational values of
elastic constants computed with hybrid potentials are reliable if we ignore the
variations in computations originating due to the order-disorder nature of the
tetragonal BaT iO3 .
10

11. Table 5
Equation of state results for tetragonal BaT iO3 with the Birch Mur-
naghan 3rd order equation. The energy-volume curve was fitted with
eleven points and the range of volume around equilibrium was chosen as
±10%. The basis sets employed for for Ba, Ti and O atoms are collectively
named as basis set 1 (see section ??).
˚
B(GPa) Vo (A3 ) E0 (a.u.)
HF 61.71 67.41 -307.5386
LDA 197.72 61.16 -307.8827
PWGGA 144.34 64.75 -309.5928
BLYP 96.38 68.09 -309.4749
B3LYP 103.51 66.07 -309.4083
B3PW 149.14 63.82 -309.5360
Exp.[?] 125-141 63.75 -
Table 6
Equation of state data for BaT iO3 using the 3rd-order Birch Murnaghan
equation. Eleven points in the energy-volume curve were used and the
range of volumes used around equilibrium was ±10%. The basis sets used
for Ba, Ti and O atoms are collectively named as basis set 2 (see section
??).
˚
B(GPa) Vo (A3 ) E0 (a.u.)
HF 202.41 64.23 -307.4678
LDA 256.96 59.07 -307.7672
PWGGA 220.55 61.93 -309.4862
BLYP 202.76 64.09 -309.3636
B3LYP 229.02 62.77 -309.3082
B3PW 229.68 61.41 -309.4395
Exp. [?] 125-141 63.75 -
11

12. Table 7
Equation of state data for BaT iO3 using the 3rd-order Birch Murnaghan
equation. Eleven points in the energy-volume curve were used and the
range of volumes used around equilibrium was ±10%. The basis sets used
for Ba, Ti and O atoms are collectively named as basis set 2 (see section
??).
B(GP a) B(GP a)
HF 61.71 61.64
LDA 197.72 209.91
PWGGA 144.34 138.64
BLYP 96.38 109.82
B3LYP 103.51 123.31
B3PW 149.14 145.11
Exp. [] 125-141
Table 8
Equation of state data for BaT iO3 using the 3rd-order Birch Murnaghan
equation. Eleven points in the energy-volume curve were used and the
range of volumes used around equilibrium was ±10%. The basis sets used
for Ba, Ti and O atoms are collectively named as basis set 2 (see section
??).
B(GP a) B(GP a)
HF 202.41 238.80
PWGGA 220.55 201.27
BLYP 202.76 219.77
B3LYP 229.02 219.77
Exp. [?] 125-141
12

13. 8 Conclusions
The second order elastic constants and equations of state computations were
successfully for tetragonal BaT iO3 . We have observed a close agreement in
the values of the bulk modulus computed with ELASTCON and EOS meth-
ods. There have been attempts [?,?,?] to compute the different properties of
tetragonal BaT iO3 . However, the experimental and computational values of
elastic constants and bulk modulus of tetragonal BaT iO3 are not available in
literature. We have further verified the computational accuracy of the results
by implementing different algorithms, potentials and basis sets on the rutile
T iO2 [?] and cubic BaT iO3 [?]. The computational results obtained with these
crystalline systems have provided additional evidence for the accuracy of the
computational results for tetragonal BaT iO3 . It has been observed that the
crystalline structure and position of atoms in tetragonal phase [?,?] has in-
creased the complexity of our computations. In our opinion the agreement in
computational values of bulk modulus provided an opportunity to verify the
computational accuracy of elastic constants of tetragonal BaT iO3 .
Acknowledgements
One of the authors (WFP) gratefully acknowledges the support of the Office
of Naval Research Grant N00014-01-1-0802 through the MURI program.
13