E for small wavenumbers however this large

i.e. for small wavenumbers. However, this large computational cost in DFT can be
avoided by calculating phonon dispersion relations using Density Functional
Perturbation Theory, as described in [ 2 ].
5.1.2 Molecular Dynamics
Alternatively, the phonon frequencies can also be obtained from molecular
dynamics runs [ 3 ]. The relative displacements u
l j
in Eq. 5.8 can be Fourier-
transformed, leading to
±
k j
ð t Þ ¼
1
N
cell
X
j
X
l
exp ð² i k R
l
Þ u
l j
/
X
m
exp ð² i x ð k ; m Þ t Þ ; ð 5 : 12 Þ
where N
cell
is the number of primitive cells in the supercell. Fourier-transforming
equation 5.12 to frequency space gives
±
k
ð x Þ /
X
m
d ð x ² x
k ; m
Þ : ð 5 : 13 Þ
The spectral analysis of ±
k
ð x Þ ; i.e. finding sharp peaks in the power spectrum
P
k j
³ j ±
k j
ð x Þj
2
ð 5 : 14 Þ
gives the phonon frequencies.
The advantage of this method is that it can be used for more complicated
systems, where explicit calculation of the full dynamical matrix would be extre-
mely expensive. Furthermore, we can calculate the temperature dependence of the
phonon spectrum by simply performing molecular dynamics simulations at dif-
ferent temperatures. The temperature dependence of the phonon spectrum is due to
anharmonic effects, i.e., at larger displacements when terms higher than second
order contribute to the potential energy in Eq. 5.4 .
5.1.3 Thermodynamics
The quantum mechanical solution of a system of harmonic oscillators [ 1 ] states
that the allowed energies of a phonon mode labelled by k and m are
E
k m
¼
1
2
þ n
³ ´
x ð k ; m Þ ; ð 5 : 15 Þ
where ± h is the reduced Planck constant, and n is a non-negative integer. The
canonical partition function of a system can be calculated as
5.1 Lattice Dynamics 53

Z ¼
X
j
exp ð² b E
j
Þ ; ð 5 : 16 Þ
where E
j
is the energy of the j th state and b ¼
1
k
B
T
: Substituting 5.15 into this
expression, we obtain
Z
vib :
¼
Y
k ; m
X
1
n
k m
¼ 0
exp ² b
1
2
þ n
k m
³ ´
± h x ð k ; m Þ
³ ´
" #
ð 5 : 17 Þ
which can be simplified by using
X
1
n ¼ 0
exp ð² nx Þ ¼
1
1 ² exp ð² x Þ
ð 5 : 18 Þ
to
Z
vib :
¼
Y
k ; m
exp ð² b ± h x ð k ; m Þ = 2 Þ
1 ² exp ð² b ± h x ð k ; m ÞÞ
: ð 5 : 19 Þ
In the case of a crystal, the total partition function is
Z ¼ exp ð² b/
0
Þ Z
vib :
: ð 5 : 20 Þ
The partition function can be used to obtain all thermodynamic quantities. For
example, the free-energy can be obtained as
F ¼ ² k
B
T ln Z ð 5 : 21 Þ
¼ /
0
þ k
B
T
X
k ; m
ln 2 sinh ð b ± h x ð k ; m Þ = 2 Þ ½ ; ð 5 : 22 Þ
and the internal energy is
U ¼
1
Z
o Z
o b
ð 5 : 23 Þ
¼ /
0
þ
X
k ; m
± h x ð k ; m Þ
1
2
þ
1
exp ð² b ± h x ð k ; m ÞÞ ² 1
³ ´
: ð 5 : 24 Þ
This result leads us to a rather crude method for approximating the real temper-
ature in the case of a classical molecular dynamics run [ 4 ]. We equate the kinetic
energy E
kin.