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i

q =−
ln i

q    /
ln V    .
2
The Grüneisen parameter is often used to characterize fre-
quency changes under applied strain. We found that most
mode-Grüneisen parameters are positive and dispersed
around 1.0 similar to those of corundum.
51 However, the
owest optical mode gives a negative mode-Grüneisen pa-
rameter of −0.20, a feature found in many bulk amorphous
materials.
52–54
We now proceed to the calculation of Gibb’s free energy
G
P,T or G
V,T for
-Al2O3. Once the phonon spectrum
s calculated, it is possible to estimate all other thermody-
namic functions.
48
Within the Born-Oppenheimer approxi-
mation, the Helmholtz free energy of a solid can be decom-
posed into two parts:
F
V,T = Fel

V,T + Fvib

V,T,
3
where V is the volume of the unit cell, T is the temperature,
Fel

V,T and Fvib

V,T are free-energy contributions due to
electron motion and nuclear vibration, respectively. For in-
sulators at ambient temperature, the thermal excitation en-
ergy and entropy contribution to electron free energy
Fel

V,T are negligible, and we may write:
Fel

V,T = Uel

V,T − TSel

V,T
Uel

V,0 = Eel

V,

4
where Eel

V is the ground-state total energy for the electrons
within the framework of density-functional theory. Within
he QHA, the nuclear vibrational free energy is given by,
Fvib

V,T =  q
BZ
 i
3N

1
2 i

V,q
 + kBT ln
1− e−i

V,q
/kBT


5
where the first term in the summation is the zero-point en-
ergy, q is a wave vector in the BZ, i is the phonon branch
ndex, i

V,q is the phonon frequency of the i
th
branch at
wave vector q for volume V. The vibrational entropy
Svib

V,T can be calculated according to,