3.3 Properties of χ (2)
65
which can be converted each other using Eqs. (3.42) and (3.44) as
x e x + y e y + z e z =
e x e y e z
⎛
⎝
x
y
z
⎞
⎠ =
e ξ e η e ζ
D
T
· D
⎛
⎝
ξ
η
ζ
⎞
⎠
=
e ξ e η e ζ
⎛
⎝
ξ
η
ζ
⎞
⎠ = ξ e ξ + η e η + ζ e ζ .
D in Eq. (3.44) is called the rotational matrix, 5 and the transformation of Eq. (3.44)
holds for any vector (first-rank tensor).
Now let us express Eq. (3.41) with the α (2) elements in the molecule-fixed coordinates. The molecular hyperpolarizability α (2) is represented in two ways, α
(2),space
pqr
in the space-fixed coordinates or α
(2),mol
p q r in the molecule-fixed coordinates. As α (2)
is a third-rank tensor, the two representations of α (2) elements are related by
α
(2),space
pqr
=
ξ ∼ζ
p
ξ ∼ζ
q
ξ ∼ζ
r
D pp D qq D rr α
(2),mol
p q r .
(3.45)
Therefore, χ
(2)
pqr in Eq. (3.41) is represented by
χ
(2)
pqr =
molecule
l
α
(2)
l,pqr =
molecule
l
ξ ∼ζ
p
ξ ∼ζ
q
ξ ∼ζ
r
D l,pp D l,qq D l,rr α
(2),mol
p q r ,
(3.46)
using the rotational matrix D l specified for the l-th molecule. The average α
(2)
pqr in
Eq. (3.41) is accordingly expressed by
α
(2)
pqr =
1
N
N
l=1
ξ ∼ζ
p
ξ ∼ζ
q
ξ ∼ζ
r
D l,pp D l,qq D l,rr α
(2),mol
p q r
=
p
q
r
D pp D qq D rr α
(2),mol
p q r .
(3.47)
5 The rotation matrix is often introduced with D −1 = D T , which satisfies
⎛
⎝
e ξ
e η
e ζ
⎞
⎠ = D
T
⎛
⎝
e x
e y
e z
⎞
⎠ and
⎛
⎝
ξ
η
ζ
⎞
⎠ = D
T
⎛
⎝
x
y
z
⎞
⎠ .
Do not confuse the two definitions, which are transpose each other.