5.4 Crystals of Deformable Molecules
101
essary for reality [12], it is preferable to formulate the problem using the normal
coordinates for intramolecular vibrations [13]. As well known, there are (3n − 6)
or (3n − 5) intramolecular vibrational modes for a molecule of non-linear or linear shape consisting of n atoms, respectively. Thus the Lagrangian of deformable
molecules can be written as
L = T − V
= −V +
p
1
2
m p
d r p
dt
2
+
1
2
t
ω p i p ω p +
1
2
dξ p
dt
2
, (5.94)
where the inner sum runs over intramolecular modes. Note that the potential energy V
in the above Lagrangian includes not only the lattice energy but also the intramolecular potential energy. The force matrix has the following form,
F =
⎛
⎝
F
TT F
TR F
TI
F
RT F
RR F
RI
F
IT F
IR F
II
⎞
⎠ ,
(5.95)
where the superscript “I” is for intramolecular vibrations. Components of the force
matrix are
F
ab
jα j β =
p
φ
ab
αβ ( p, p
) exp(−i d l q),
(5.96)
with d l defined in Eq. 5.74. Here, α and β are used to distinguish intramolecular
modes if the superscript “a” and “b” are “I”. The force constants are defined as
φ
ab
αβ ( p, p
) =
∂
2 V
∂x α, p ∂x β, p
eq
,
(5.97)
similarly to Eq. 5.59, while understanding x = ξ for a = I.
The consideration on the sum rule similarly to the case of rigid molecules yields
φ
aT
αβ ( p, p) = −
p
= p
φ
aT
αβ ( p, p
),
(5.98)
φ
Ta
αβ ( p, p) = φ
aT
βα ( p, p),
(5.99)
with a = T, R or I. The sum rules for the RR sector remain the same (see the end of
the previous section).
Special consideration is necessary for the “II” sector. Suppose a simple crystal,
a unit cell of which contains two diatomic molecules (n mol = 2). Since a single
molecule has only one intramolecular vibration (stretching), the “II” sector of the
mass and force matrices are (2 × 2)-matrices. That of the mass matrix is
101
essary for reality [12], it is preferable to formulate the problem using the normal
coordinates for intramolecular vibrations [13]. As well known, there are (3n − 6)
or (3n − 5) intramolecular vibrational modes for a molecule of non-linear or linear shape consisting of n atoms, respectively. Thus the Lagrangian of deformable
molecules can be written as
L = T − V
= −V +
p
1
2
m p
d r p
dt
2
+
1
2
t
ω p i p ω p +
1
2
dξ p
dt
2
, (5.94)
where the inner sum runs over intramolecular modes. Note that the potential energy V
in the above Lagrangian includes not only the lattice energy but also the intramolecular potential energy. The force matrix has the following form,
F =
⎛
⎝
F
TT F
TR F
TI
F
RT F
RR F
RI
F
IT F
IR F
II
⎞
⎠ ,
(5.95)
where the superscript “I” is for intramolecular vibrations. Components of the force
matrix are
F
ab
jα j β =
p
φ
ab
αβ ( p, p
) exp(−i d l q),
(5.96)
with d l defined in Eq. 5.74. Here, α and β are used to distinguish intramolecular
modes if the superscript “a” and “b” are “I”. The force constants are defined as
φ
ab
αβ ( p, p
) =
∂
2 V
∂x α, p ∂x β, p
eq
,
(5.97)
similarly to Eq. 5.59, while understanding x = ξ for a = I.
The consideration on the sum rule similarly to the case of rigid molecules yields
φ
aT
αβ ( p, p) = −
p
= p
φ
aT
αβ ( p, p
),
(5.98)
φ
Ta
αβ ( p, p) = φ
aT
βα ( p, p),
(5.99)
with a = T, R or I. The sum rules for the RR sector remain the same (see the end of
the previous section).
Special consideration is necessary for the “II” sector. Suppose a simple crystal,
a unit cell of which contains two diatomic molecules (n mol = 2). Since a single
molecule has only one intramolecular vibration (stretching), the “II” sector of the
mass and force matrices are (2 × 2)-matrices. That of the mass matrix is