4.2 Examples of χ (2) Tensor and Orientation
87
We notice in Eq. (4.9) that the sign of Im[α
(2)
yyz ] is determined by the sign of
cos θ , as the other factors are always positive. By its definition, θ is the tilt angle
of the local O-H 1 bond (ζ axis) from the surface normal (z axis) in the space-fixed
coordinate (see Fig. 3.3). Therefore, if the O-H 1 bond points upward (cos θ > 0),
Im[α
(2)
yyz ] spectrum is positive (Fig. 4.1a). On the other hand, if the O-H 1 bond points
downward (cos θ < 0), Im[α
(2)
yyz ] spectrum is negative (Fig. 4.1b). This qualitative
relation between the O-H orientation and the sign of Im[α
(2)
yyz ] is useful to interpret
the sign of O-H band of Im[χ
(2)
yyz ] spectrum in the SSP polarization.
4.2.2 C-H Stretching
The C-H stretching vibration usually appears in 2800–3000 cm −1 region. The C-H
vibrations are widely seen in organic molecules, particularly in alkyl moieties such
as methyl (CH 3 -) and methylene (-CH 2 -) groups. The relation between molecular
orientation and χ (2) tensor elements has been formulated in details [7–9, 23]. Here
we briefly present essential formulations in the case of methyl C-H symmetric
stretching mode. Further discussion of the C-H bands including other modes will
be given in Chap. 10 with the help of MD simulation.
To make the qualitative relation clear, we assume the (pseudo) C 3v symmetry for
the methyl group. The symmetry allows us to simplify the vibrational analysis of
the local methyl C-H vibrations. (Note that such symmetry may not rigorously hold
for actual methyl groups, such as in methanol or ethanol [10, 24, 25].) Therefore,
we treat the C-H mode of acetonitrile as an ideal C 3v molecule. Table 4.2 displays
the calculated derivatives of dipole moment and polarizability of acetonitrile
with respect to the methyl C-H symmetric stretching mode in the molecule-fixed
coordinates [19]. The dipole derivative ∂μ r /∂q 1 has a non-zero element along the
r = ζ axis (molecular principal axis) with respect to the C-H symmetric stretching,
and the polarizability derivative ∂α p q /∂q 1 has non-zero diagonal elements for
(p q ) = (ξξ) = (ηη) and (p q ) = (ζ ζ ). We further notice that the dipole
derivative ∂μ ζ /∂q 1 is negative, and the ratio of the two independent elements of
the polarizability derivative,
R =
∂α ξξ /∂q 1
∂α ζ ζ /∂q 1
=
∂α ηη /∂q 1
∂α ζ ζ /∂q 1
(4.10)
is estimated to be 0.80 in Table 4.2.
Sign of Im[χ (2) ] First, we provide qualitative discussion about the molecular
orientation and the sign of the α
(2),space
yyz
element for the SSP polarization. The sign
of Im[α
(2),space
yyz
] is determined by the product of (∂α yy /∂q 1 ) and (∂μ z /∂q 1 ) in the
space-fixed coordinates. The latter is determined by the molecular orientation as
∂μ z
∂q 1
=
ξ ∼ζ
r
D zr
∂μ r
∂q 1
= −0.005970 · cos θ,
87
We notice in Eq. (4.9) that the sign of Im[α
(2)
yyz ] is determined by the sign of
cos θ , as the other factors are always positive. By its definition, θ is the tilt angle
of the local O-H 1 bond (ζ axis) from the surface normal (z axis) in the space-fixed
coordinate (see Fig. 3.3). Therefore, if the O-H 1 bond points upward (cos θ > 0),
Im[α
(2)
yyz ] spectrum is positive (Fig. 4.1a). On the other hand, if the O-H 1 bond points
downward (cos θ < 0), Im[α
(2)
yyz ] spectrum is negative (Fig. 4.1b). This qualitative
relation between the O-H orientation and the sign of Im[α
(2)
yyz ] is useful to interpret
the sign of O-H band of Im[χ
(2)
yyz ] spectrum in the SSP polarization.
4.2.2 C-H Stretching
The C-H stretching vibration usually appears in 2800–3000 cm −1 region. The C-H
vibrations are widely seen in organic molecules, particularly in alkyl moieties such
as methyl (CH 3 -) and methylene (-CH 2 -) groups. The relation between molecular
orientation and χ (2) tensor elements has been formulated in details [7–9, 23]. Here
we briefly present essential formulations in the case of methyl C-H symmetric
stretching mode. Further discussion of the C-H bands including other modes will
be given in Chap. 10 with the help of MD simulation.
To make the qualitative relation clear, we assume the (pseudo) C 3v symmetry for
the methyl group. The symmetry allows us to simplify the vibrational analysis of
the local methyl C-H vibrations. (Note that such symmetry may not rigorously hold
for actual methyl groups, such as in methanol or ethanol [10, 24, 25].) Therefore,
we treat the C-H mode of acetonitrile as an ideal C 3v molecule. Table 4.2 displays
the calculated derivatives of dipole moment and polarizability of acetonitrile
with respect to the methyl C-H symmetric stretching mode in the molecule-fixed
coordinates [19]. The dipole derivative ∂μ r /∂q 1 has a non-zero element along the
r = ζ axis (molecular principal axis) with respect to the C-H symmetric stretching,
and the polarizability derivative ∂α p q /∂q 1 has non-zero diagonal elements for
(p q ) = (ξξ) = (ηη) and (p q ) = (ζ ζ ). We further notice that the dipole
derivative ∂μ ζ /∂q 1 is negative, and the ratio of the two independent elements of
the polarizability derivative,
R =
∂α ξξ /∂q 1
∂α ζ ζ /∂q 1
=
∂α ηη /∂q 1
∂α ζ ζ /∂q 1
(4.10)
is estimated to be 0.80 in Table 4.2.
Sign of Im[χ (2) ] First, we provide qualitative discussion about the molecular
orientation and the sign of the α
(2),space
yyz
element for the SSP polarization. The sign
of Im[α
(2),space
yyz
] is determined by the product of (∂α yy /∂q 1 ) and (∂μ z /∂q 1 ) in the
space-fixed coordinates. The latter is determined by the molecular orientation as
∂μ z
∂q 1
=
ξ ∼ζ
r
D zr
∂μ r
∂q 1
= −0.005970 · cos θ,