1. Università di Torino
Dottorato in Scienza ed Alta Tecnologia
Indirizzo Scienze della Terra
Ab initio modelling:
Mechanical and Thermodynamic
properties of
Calcium Carbonates Polymorphs
Supervisor: Dott. Mauro Prencipe
PhD Student: Crina Georgeta Ungureanu
2. How the ab initio quantum mechanical calculation method can be applied on the
study of equilibrium and HP – HT phases?
Quantum calculation have been carried out at pressures between:
0 and 5 GPa (Calcite),
0 to 30 GPa (Aragonite)
and temperatures between 298 and 700 K.
Our Aim
Calcium carbonates :
(trigonal) calcite, (orthorhombic) aragonite, vaterite
and (monoclinic) calcite II systems.
This allowed an exclusive investigation of the Mechanical and Thermodynamic
Properties of Calcium Carbonate Polymorphs, which represent the underlying
reference for many studies in different fields as :
Mineralogy, Geophysics and Petrography.
4. Ab initio → “from first principles” : no experimental data are used
and computations are based on quantum mechanics.
5. Hartree – Fock Self Consistent field SCF method: Born Oppenheimer Approximation
∑∑∑∑∑ −
−
−
+
−
++=
in ni
i
nn nnij ji
ji
n
n
i i
i
rR
eZ
rr
e
RR
eZZ
m
p
M
p
H
2
'
2222
'2
1
2
1
22
Nuclear kinetic energy
Potential energy of e-e interaction
Potential nuclei nuclei interaction
Potential energy of electrons
nuclei interaction
Electronic kinetic energy
Hohenberg-Kohn theorems:
Hψ = Eψ→ ))r((EE ρ= ))r((E ρ minima ( ) ( )
2
1
N
i
i
ρ ψ
=
= ∑r r
( ) ( ) ( ) ( ) ( )KS
S ext H xcE T E E Eρ ρ ρ ρ ρ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦r r r r r
Each issue is returned to EXC.
Kohn-Sham implementation:
2
2
( ) ( ) ( )
2
eff i i iV r r E r
m
⎧ ⎫
− ∇ + Ψ = Ψ⎨ ⎬
⎩ ⎭
h
classical electronic
Coulomb potential
ion Coulomb
potential
( ) ( ) ( ) ( )eff ion H XCV r V r V r V r= + +
exchange-correlation potential
of electron gas (LDA,GGA)
DFT
6. The calculation of the vibrational frequencies at Г point for all phases of calcium carbonate
studied, is performed within the harmonic approximation:
vibrational frequencies at central Г point
(each volume , pressure):
diagonalizing the mass-weighted Hessian W
special importance of Г point: IR and RAMAN spectra refer to this point.
Frequency harmonic calculations at Γ point: FREQCALC
V (x): Potential energy surface of a system
as a function of coordinates of the nuclei.
( )
0
2
2
1
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂∂
∂
=
ji
ij
uu
xV
H
ji
ij
ij
MM
H
W =mass-weighted Hessian matrix W: Mi, Mj: masses of atoms i and j.
ui,j: displacement coordinate with respect to equilibrium
7. 1
1
),(
−
=
kT
hjj j
e
Tn ν
ν Bose-Einstein
statistics
Thermal vs Elastic
Properties of a SolidV
V
V
j
j
j
j
∂
∂
−=
∂
∂
−=
ν
ν
ν
γ
ln
ln
Grüneisen parameters
static pressure: PST (V)
zero point pressure: PZP (V)
thermal pressure: PTH (V, T).
From vibrational Frequencies to
Thermo Elastic and Thermodynamic Properties
∑∑ −
++⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
j
y
j
j
j
jj
T
ST
j
e
y
V
kT
y
V
kT
V
E
P
12
γγ
Equation of State (Anderson 1995)
EoS works in Quasi – Harmonic Approximation
8. Bulk modulus, thermal expansion, specific heat and entropy
were obtained in the limit of the quasi-harmonic approximation (Anderson, 1995).
For each normal mode, the Grüneisen’s parameter γj was determined through the
analytic first derivative at each volume (pressure) of the 2nd order polynomial resulting
from the fitting of the numerical νj (V) curves.
Static pressures PST were determined by numerical derivative, with respect to V, of the
4th order polynomial interpolating the EST(V) curve; zero points (PZP) and thermal (PTH)
pressures were obtained by direct application of equations as follow:
PT T
V
V
TV ⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
1
),(α
TT T
P
K
TV ⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
1
),(α
Num. Deriv BM3 EOSThermal Expansion
TTTvp KVTCC 2
α+=∑ −
Ν=
j
X
X
j
V j
j
e
eX
kC 2
2
0
)1(
Heat Capacity
∑ ⎥
⎦
⎤
⎢
⎣
⎡
−−=
−
)1ln(),(),(
/
0
kThj
jj
j
e
kT
h
TnkNTVS
νν
ν
Entropy
9. Applications of ab initio quantum mechanical calculation
CRYSTAL code
Thermo- elastic, Equation of State and Thermodynamic Properties of
CaCO3 polymorphs
Why Calcium carbonates?
Calcium Carbonate is relatively well - defined system and, as
such can be used as a test of the whole algorithm which has
been refined in the present work.
Calcium carbonate crystallizes in nature as:
Calcite R (3+i)c
Aragonite Pmcn
Vaterite Pbnm.
cR
−
3
10. Calcite ( CaCO3 )cR
−
3
The calcite uniaxial (-) structure has been
determined by Bragg in 1914 and has
been described traditionally using the
NaCl structure as a starting point.
Hexagonal frame: a = 4.989 and c = 17.061 Z = 6 CaCO3
Rhombohedral frame: a = 6.375, α = 46°05’ Z = 2 CaCO3.
Calcite
Most calcites are relatively pure. Common
impurities are:
magnesium, ferrous iron and manganese.
11. Aragonite Phase relation of Calcium carbonates
The calcite structure is characterized by
alternate layers of Ca atoms and CO3 groups
along the z axis, and between successive layers
the CO3 groups have opposite orientations.
Aragonite
Aragonite occurs as the inorganic constituent of many invertebrate skeletons and
sediments derived from them. It also occurs as a primary phase in high – pressure
metamorphic rocks.
The aragonite structure: layers of
calcium atoms parallel to (001)
along the z axis and separated by
CO3 layers. The C atoms do not
stack on top of each other along the
z axis and are displaced relative to
each other in the y direction.
12. Monoclinic system: Calcite II
A, aragonite; I-V, polymorphic forms of calcite;
+ H, J + P, I+W, K+S+M, dashed lines,
metastable equilibrium.
This high-pressure (15 kbar ) form of
CaCO3 is monoclinic, space group P21/c
a= 6.334 ± 0.002,
b= 4.948 ± 0.005,
c= 8.033 ± 0.025,
β= 107° 54' and Z = 4.
Other Calcium Carbonate Polymorphs
Orthorombic system: Vaterite
Orthorhombic structure with space group
Pbnm after Meyer (1959).
Hexagonal structure with space group
P63/mmc after Kamhi (1963).
Vaterite (μ - Calcite)
Vaterite is thermodynamically the least stable
CaCO3 polymorph, and it rapidly transforms
into calcite or aragonite in aqueous solution
(Fricke et al. 2006).
13. The vibrational spectrum of CaCO3 Aragonite.
An ab initio study with the CRYSTAL code
BSC BSD Exp
ΔE 0.054
a 5.02311 5.00955 4.96183(1)
b 8.03418 8.02478 7.96914(2)
c 5.85903 5.86119 5.74285(2)
V 236.451 235.621 227.081(1)
Cay 0.41515 0.41527 0.41502(2)
Caz 0.76009 0.75985 0.75985(4)
Cy 0.76258 0.76241 0.7691(1)
Cz -0.08152 -0.08134 -0.0823(1)
O1y 0.92273 0.92163 0.92238(8)
O1z -0.09018 -0.08999 -0.09453(8)
O2x 0.47306 0.47212 0.47499(7)
O2y 0.68097 0.6813 0.68013(5)
O2z -0.08468 -0.08423 -0.08725(7)
dC-O1 1.2877 1.2787 1.2235
dC-O2 1.2983 1.2892 1.3228
dCa-O 2.4306 2.5539 2.5196
Geometrical parameters of aragonite;
calculated values obtained with the
different basis sets (BSC-D) at the
B3LYP level are compared with
experimental results.
a, b and c lattice parameters (Å),
V is the volume of the cell in Å3.
Cai, Ci, O1i and O2i fractional
coordinates of the irreducible atoms in
the primitive cell,
di−j indicate the shortest interatomic
distances (Å).
ΔE (in m Hartree) BSC- BSD
(- 3766.163915688 Hartree)→BSD
better than BSC.
14. mmm orthorhombic cell of aragonite:
Z = 4 CaCO3
N = 20 atoms;
60 vibrational modes can be classified according to the
Irreducible Representations of the mmm point group as follows:
Γtot = 9Ag + 6Au + 6B1g + 9B1u + 6B2g + 6B2u + 9B3g + 6B3u.
24 external vibrations (E): 68,2 – 293,4 cm-1
33 internal vibrational modes (I): 692,9 – 1599 cm-1
Where: 9A1g + 6B1g + 6B2g + 9B3g Raman active
8B1u + 5B2u + 5B3u IR active
6A1u inactive
1B1u+1B2u+1B3u acoustic.
The Raman and IR vibrational modes calculated using BSD basis set and the B3LYP
Hamiltonian and were compared with experimental data: the RMS = 4.1 cm-1.
Two exceptions are noted:
• lowest frequencies < 300 cm-1
• highest >1080 cm-1, for which calculation overestimates the experimental values
as for Raman and IR frequencies (LO-TO) between 9.9 and 12.9 cm-1.
15. Ab-initio quantum-mechanical calculation on aragonite at high pressure.
Elastic and Thermodynamic properties and comparison with experimental data
Structure and vibrational frequencies (at
the Г point) of aragonite have been
calculated from first principles, by using
the hybrid Hartree-Fock/DFT B3LYP
Hamiltonian, at different unit cell
volumes, in the 185-242 Å3 range.
Starting with the equilibrium geometry of the
orthorhombic aragonite primitive cell, new
geometries have been obtained for reduced
volumes of the unit cell which correspond to static
pressures (1st term of EoS formula) ranging from
Pst = 0 GPa at V = 236.44 Å3 (equilibrium
geometry) to Pst = 28.62 GPa at V = 185.46 Å3.
0 400 800 1200 1600
0
5
10
15
gj
n(cm
-1
)
g
j
M ean g=1.637
Calculated Grüneisen’s parameter values (γj),
at equilibrium volume, as a function of frequency (νj).
γj calculated γj experimental
νj(cm-1) γj νj(cm-1) γj
155.3 1.30 155 1.20
180.5 2.35 180 2.60
211.2 1.48 209 1.20
696.1 0.15 702 0.14
707.8 0.21 710 0.20
1092.1 0.16 1084 0.16
16. 0 500 1000 1500
0.01
0.1
1
Thermalpressure(GPa)
300K
400K
500K
600K
700K
ν(cm
-1
)
Calculated thermal pressure contribution of phonons, as
a function of the normal modes frequency
210 215 220 225
0
2
4
6
8
Pressure(GPa)
Volume(A
3
)
EXP
CALC
Calculated (scaled volume) isothermal curve at 298 K:
comparison with experimental values (Martinez et al.,
1996).
Excellent agreement between the calculated total pressure curve at 298 K, as a function
of the scaled volume, and the experimental one (Martinez et al. 1996).
17. 300 400 500 600 700 800 900 1000
0
10
20
30
40
50
60
70
80
KBM3
KNumDeriv
EXP
BulkModulus(GPa)
Temperature(K)
P=0GPa
Calculated BM3 and Num_Deriv bulk
modulus as a function of T (298÷973 K)
at P = 0 GPa.
Comparison with experimental data.
0 1 2 3 4 5
0
20
40
60
80
100
BulkModulus(GPa) Pressure(GPa)
KBM3
KNumDerix
EXP
T=298K
Calculated BM3 and NumDeriv bulk
modulus as a function of P (0÷5 GPa) at
T = 298 K .
Comparison with experimental results.
18. 400 600 800 1000
0
2x10
-5
4x10
-5
6x10
-5
8x10
-5
1x10
-4
a(K
-1
)
Temperature (K)
a
BM3
a
NumDeriv
a
EXP
Calculated BM3 and NumDeriv thermal
expansion values as a function of
T (273÷973 K) at P=0 GPa.
Comparison with experimental data.
T (K) BM3 (K-1
)
NumDeriv
(K-1
)
EXP (K-1
)
298 6.06E-05 5.88E-05 6.36E-05
373 6.40E-05 6.33E-05 6.76E-05
473 6.97E-05 6.99E-05 7.01E-05
573 7.52E-05 7.94E-05 7.14E-05
673 8.06E-05
773 8.59E-05
873 9.11E-05
973 9.60E-05
Better agreement between Thermal
expansion BM3 and experimental
values.
19. 0 50 100 150 200 250 300
0
20
40
60
80
100
Entrophy(J*mole
-1
*K
-1
)
Temperature(K)
SCALC
SEXP
Calculated Entropy values as a function of
T (20-300K) at P = 0 GPa.
Comparison with experimental data.0 50 100 150 200 250 300
0
20
40
60
80
100
HeatCapacity(J*mole
-1
*K
-1
)
Temperature(K)
CpBM3
CpNumDeriv
CpEXP
Cv
Calculated Heat capacity at constant
volume (Cv) and two at constant pressure
Cp BM3 and CpNumDeriv as a function of
T (20÷300 K) at P=0 GPa.
Comparison with experimental values.
KVTCC VP
2
α+=
20. Using the α, KT and CV values, it was possible to calculate the following Grüneisen ratio
= 1.898
at P = 0 GPa and T = 298 K, which appears to be in good agreement
(over-estimation of 3%) with respect to the experimental room temperature value of 1.843.
VT CVKαγ =
The Gruneisen ratio is a very important thermodynamic parameter
which helps to quantify the
relationship between the thermal and elastic properties of a solid.
The pressure (or volume) and temperature variation of γj contribute
significantly to the evaluation of EoS parameters (ex: Pthermal) and thus on
exact (P,T) point computation of many thermodynamic functions.
Using BM3 method, which uses a wider sampling of (P, T) space, we obtain
more stable and reliable (if compared with experimental ones) values of the
thermodynamic functions. (*)
Ungureanu C.G., Prencipe M., Cossio R. “Ab initio quantum-mechanical
calculation of CaCO3 aragonite at high pressure: thermodynamic properties
and comparison with experimental data” Eur. J. Min. 2010, 22, 693–701.
21. Thermodynamic properties of Calcite at high pressure:
an ab initio quantum - mechanical calculation
Optimized geometries, static
energies and zone-centre
vibrational frequencies (νj, j =
1,3n-3; n = 10 atoms in the unit
cell) were determined for a set of
10 different cell volumes (V)
ranging from 356.0839 to 385.7618
A3) at B3LYP and WC1LYP level.
0 200 400 600 800 1000 1200 1400 1600
0.00
0.05
0.10
0.15
0.20
0.25
ThermalPressure(GPa)
ν(cm
-1
)
300K
400K
500K
600K
700K
Thermal pressure contribution Pth of phonons,
calculated as a function of
the normal modes frequency,
at five different temperatures.
Raman and IR vibrational
frequencies of CaCO3 calcite (0 ÷
5 GPa pressure range) has been
calculated at Г point with the
periodic ab initio CRYSTAL09
code using full geometry
optimization at constant volume,
high quality localized basis set and
WC1LYP hybrid Hamiltonian
function.
22. 0 1 2 3 4 5
0
20
40
60
80
100
K(GPa)
Pressure(GPa)
KCALC
KEXP
WC1LYP
Calculated bulk modulus K
at WC1LYP level as a function of P
(0÷5 GPa) at T = 298 K. Comparison
with experimental values (Salje &
Viswanathan, 1976).
300 400 500 600 700
0
1x10
-5
2x10
-5
3x10
-5
α(K-1)
Temperature(K)
αCALC
αEXP
WC1LYP
Calculated thermal expansion α at
WC1LYP level as a function of T (298 -
671 K) at P = 0 GPa: comparison with
experimental values (Salje &
Viswanathan, 1976).
Ungureanu C.G., Prencipe M., Cossio R. “Mechanical, Elastic and
Thermodynamic properties of CaCO3 calcite at high pressure: an ab initio
quantum-mechanical calculation”, Eur. J. Min. , submitted
23. 0 100 200 300 400 500 600 700 800 900 1000
0
50
100
150
200
250
Diff.RMS
=0.31J/mole*K
S(J/mole*K)
Temperature(K)
SCALC
SEXP
WC1LYP
Calculated Entropy values S as a
function of T (20-1000K) at P=0 GPa.
Comparison with experimental (20-
300K; 350 ÷ 750K; 298 and 1000 K).
0 100 200 300 400 500 600 700 800 900 1000
0
50
100
150
Cp(J/moleK)
Temperature(K)
WC1LYP
CpCALC
CpEXP
Diff.RMS
=1.07 J/moleK
Calculated Heat capacity at constant pressure
(CP) at WC1LYP level as a function of T (20 ÷
1000 K) at P = 0 GPa.
Comparison with experimental (20 - 300K;
350 ÷ 750 K. 298 and 1000 K).
Although we note small difference between
thermodynamic properties calculated at
B3LYP and at WC1LYP level, a general
better agreement between WC1LYP
calculated values and experimental ones
can be reported.
*S at T = 298K, was calculated taking into
account the phonon dispersion.
24. Vaterite (μ - Calcite). Ab initio quantum mechanical calculation of
normal vibrational modes at Γ point
(Å) BSD Exp.
a 6.69422 7.15
b 8.50694 8.48
c 4.52341 4.13
V ( Å3) 257.597 250.4142
Cax 0 0
Cay 0 0
Caz 0 0
Cx -0.3693 0.67
Cy 0.25 0.25
Cz 0.25 0.157
O1x -0.45677 0.471
O1y 0.25 0.25
O1z 0.27 0.157
O2x 0.45677 0
O2y 0.25 0.118
O2z 0.27 0.67
dCa-O 2.3841
dC-O1 1.2794
dC-O2 1.2917
The equilibrium geometry
of the orthorhombic vaterite Pnma
and vibrational frequencies calculation
(CRYSTAL code, using hybrid Hartree-
Fock/DFT B3LYP Hamiltonian and BSD).
The cell parameters (Å) obtained are:
a = 6.69422, b = 8.50694, c = 4.52341
α=β = γ = 90° V = 257.597 Å3
and
E = -3.766173E+03 Ha, at B3LYP level.
25. Dielectric properties
The Born effective charge and
dielectric tensors were obtained with
BSD and B3LYP calculation, using the
CPHF method which is available in the
new test version CRYSTAL09.
Components Calculated
xx 2.1615
yy 2.5534
zz 2.388
The orthorhombic cell of vaterite Pnma contains 4 CaCO3 formula units, for a total of
N = 20 atoms; its 60 vibrational modes can be classified according to the Irreducible
Representations of the mmm point group as follows:
Γtot = 7Ag + 8Au + 5B1g + 9B1u + 5B2g + 9B2u + 7B3g + 7B3u.
Where: 7A1g + 5B1g + 5B2g + 7B3g Raman active
9B1u + 9B2u + 7B3u IR active
8A1u inactive
1B1u+1B2u+1B3u translations.
The spectrum of vaterite is divided in two frequency regions
separated by a relatively large gap (320 cm-1)
from 352 cm-1 (B2g) to 672 cm-1 (B1u) "external" modes (E),
frequencies larger than 672 cm-1 correspond to the "internal" modes (I).
26. Calcite II vibrational frequencies
calculation at Γ point
B3LYP WC1LYP Exp.
E (ha) -3.76E+03 -3.77E+03
(Merrill &
Bassett 1975)
a 6.46320917 6.37962599 6.334
b 5.03907252 5.00426778 4.984
c 8.19823057 8.12532514 8.033
β (°) 108.026772 107.567684 107.9
ρ(g/cm-
3) 2.615 2.684 2.77
V(A3) 253.898004 247.305581 241.3
Cax 0.25 0.25 0.234(2)
Cay -0.25 -0.25 0.738(2)
Caz 0.25 0.25 0.217(1)
Cx 0.25 0.25 0.260(8)
Cy 0.25 0.25 0.253(9)
Cz -0.5 -0.5 0.504(3)
O1x 0.38 0.38 0.380(7)
O1y 0.12 0.12 0.156(3)
O1z -0.37 -0.37 0.637(3)
O2x 0.12 0.12 0.134(7)
O2y 0.12 0.12 0.088(4)
O2z 0.37 0.37 0.381(3)
O3x 0.25 0.25 0.221(6)
O3y -0.5 -0.5 0.490(4)
O3z -0.5 0.5 0.467(3)
Geometrical parameters of CaCO3;
calculated values obtained with two
different functionals (B3LYP and
WC1LYP) are compared with
experimental results.
CalciteII is monoclinic, P21/c
a = 6.334 ± 0.020%, b = 4.948 ±
0.015%, c = 8.033 ± 0.025%, and
β = 107.9°.
According to Irreducible
Representations of the mmm point
group as follows:
Γtot = 15Ag + 15Au + 15Bg + 15Bu
Where: 15Ag + 15Bg Raman active
15Bu IR active
15A1u inactive
1B1u+1B2u+1B3u
translation.
27. •We analysed these polymorphs of calcium carbonates and
• obtained appropriate results for the geometry of crystallographic cell and for the
vibrational normal modes,
• using ab initio quantum mechanical calculation
We can conclude that:
CRYSTAL code may be considered one of the best modelization method dedicated
to the study of the solid phases (especially 3D crystalline
structures)
not only at the equilibrium, but even more under HP- HT conditions, which plays
an important role in
mineralogy, geophysics and sedimentary petrology.
The vibration frequencies obtained with ab initio quantum mechanical calculation,
in comparison with
Debye 1911 - the frequencies in a crystal are distributed as though the solid were
an isotropic elastic continuum rather than an aggregate of particles, and Einstein
1907 - all vibrational modes have the same frequency models,
provide the appropriate underlying in the determination of Mechanical and
Thermodinamical Properties of the crystalline systems as calcium carbonate type.
Conclusions
