2 Principles and Characteristics of NIR Spectroscopy
21
state m to another state n by absorbing or emitting IR light, it is necessary that the
following definite integral or at least one of (μ y ) mn and (μ z ) mn which are expressed by
a similar equation to (2.2) is not 0, where μ x denotes an x-component of the electric
dipole moment; ψ denotes the eigenfunction of the molecule in its vibrational state;
and Q denotes a displacement along a normal coordinate (i.e., a normal vibration
expressed as a single coordinate). Now, let us consider only (μ x ) mn . A distribution of
electrons in the ground state changes as the coordinate expressing a vibration varies,
and thus, the electric dipole moment is a function of the normal coordinate Q. Hence,
μ x can be expanded as follows.
Expressed by a displacement of atoms during the vibration, Q has a small value.
This allows to omit Q
2 and the subsequent terms in the equation above. Substituting
the terms up to Q of Eq. (2.3) in Eq. (2.2), is obtained. Due to the orthogonality of
the eigenfunction, the first term of this equation is 0 except when m = n holds. The
first term denotes the magnitude of the permanent dipole of the molecule. For the
second term to have a value other than 0, both (∂μ x /∂ Q) 0 = 0 and
ψ n Q ψ m d Q =
0 must be satisfied. These two conditions lead to the two selection rules. The nature
of the eigenfunction allows the integral to have the value other than 0 only when
n = m ± 1 holds. Considering Q
2 and the subsequent terms of Eq. (2.3) as well, we
can prove that even when n = m ± 1 fails to hold, (μ x ) mn has a value, even though
small, other than 0. The first selection rule regarding IR absorption is thus proved.
The other selection rule, which is based upon the symmetry of a molecule, comes
from (∂μ x /∂ Q) 0 = 0. The relationship (∂μ x /∂ Q) 0 = 0 indicates that IR absorption
takes place only when a certain vibration changes the electric dipole moment. The
vibration is IR active when (∂μ x /∂ Q) 0 = 0 holds, but is IR inactive when (∂μ x /∂
Q) 0 = 0 holds.
Most molecules are in the ground vibrational state at room temperature, and
thus, a transition from the state ν” = 0 to the state ν” = 1 (first excited state) is
possible. Absorption corresponding to this transition is called the fundamental.
Although most bands which are observed in an IR absorption spectrum arise from
the fundamental, in some cases, also in the IR spectrum one can observe bands which
correspond to transitions from the state ν” = 0 to the state ν” = 2, 3,… They are
called first, second, overtones. Bands due to combinations are also observed in the
IR spectra. However, since overtones and combinations are forbidden with harmonic
oscillator approximation, overtone and combination bands are very weak even in
real molecules. Because of anharmonicity, although the intensities are weak, the
forbidden bands appear.
2.2.2 Molecular Vibrations
One must learn molecular vibrations to understand all kinds of vibrational spectroscopy; IR, NIR, FIR/terahertz, and Raman spectroscopy. Vibrations of a polyatomic molecule are, in general, complex, however, according to harmonic oscillator
approximation (i.e., an approximation on the assumption that the restoring force
21
state m to another state n by absorbing or emitting IR light, it is necessary that the
following definite integral or at least one of (μ y ) mn and (μ z ) mn which are expressed by
a similar equation to (2.2) is not 0, where μ x denotes an x-component of the electric
dipole moment; ψ denotes the eigenfunction of the molecule in its vibrational state;
and Q denotes a displacement along a normal coordinate (i.e., a normal vibration
expressed as a single coordinate). Now, let us consider only (μ x ) mn . A distribution of
electrons in the ground state changes as the coordinate expressing a vibration varies,
and thus, the electric dipole moment is a function of the normal coordinate Q. Hence,
μ x can be expanded as follows.
Expressed by a displacement of atoms during the vibration, Q has a small value.
This allows to omit Q
2 and the subsequent terms in the equation above. Substituting
the terms up to Q of Eq. (2.3) in Eq. (2.2), is obtained. Due to the orthogonality of
the eigenfunction, the first term of this equation is 0 except when m = n holds. The
first term denotes the magnitude of the permanent dipole of the molecule. For the
second term to have a value other than 0, both (∂μ x /∂ Q) 0 = 0 and
ψ n Q ψ m d Q =
0 must be satisfied. These two conditions lead to the two selection rules. The nature
of the eigenfunction allows the integral to have the value other than 0 only when
n = m ± 1 holds. Considering Q
2 and the subsequent terms of Eq. (2.3) as well, we
can prove that even when n = m ± 1 fails to hold, (μ x ) mn has a value, even though
small, other than 0. The first selection rule regarding IR absorption is thus proved.
The other selection rule, which is based upon the symmetry of a molecule, comes
from (∂μ x /∂ Q) 0 = 0. The relationship (∂μ x /∂ Q) 0 = 0 indicates that IR absorption
takes place only when a certain vibration changes the electric dipole moment. The
vibration is IR active when (∂μ x /∂ Q) 0 = 0 holds, but is IR inactive when (∂μ x /∂
Q) 0 = 0 holds.
Most molecules are in the ground vibrational state at room temperature, and
thus, a transition from the state ν” = 0 to the state ν” = 1 (first excited state) is
possible. Absorption corresponding to this transition is called the fundamental.
Although most bands which are observed in an IR absorption spectrum arise from
the fundamental, in some cases, also in the IR spectrum one can observe bands which
correspond to transitions from the state ν” = 0 to the state ν” = 2, 3,… They are
called first, second, overtones. Bands due to combinations are also observed in the
IR spectra. However, since overtones and combinations are forbidden with harmonic
oscillator approximation, overtone and combination bands are very weak even in
real molecules. Because of anharmonicity, although the intensities are weak, the
forbidden bands appear.
2.2.2 Molecular Vibrations
One must learn molecular vibrations to understand all kinds of vibrational spectroscopy; IR, NIR, FIR/terahertz, and Raman spectroscopy. Vibrations of a polyatomic molecule are, in general, complex, however, according to harmonic oscillator
approximation (i.e., an approximation on the assumption that the restoring force