2.2.3 Wilson–Decius–Cross Method
As stated in the previous section within the harmonic oscillator approximation, a band
on the vibrational spectrum corresponding to t = 0 ! t = 1 transition appears at the
wavenumber corresponding to classical vibrational frequency. This can be generalized; finding classical vibrational frequencies of a polyatomic molecule will give us its
approximate vibrational spectrum. The detailed derivation of relevant formulas is
given elsewhere [4, 5] and will not be repeated here. We merely present the proper
practice and report formulas necessary for future discussion.
Finding classical vibrational frequencies of a polyatomic molecule consists in
solving Lagrange equations of motion. This requires the knowledge of the vibrational kinetic and potential energies of a molecule. The most straightforward
coordinate system to treat molecular vibrations is the Cartesian system, the proper
choice of which is the following: Its origin is fixed at the center of mass of a
molecule, and it rotates with the molecule. The instantaneous position of the Ath
atom is r A ¼ x A ; y A ; z A
ð
Þ , while its equilibrium position is r A;e ¼ x A;e ; y A;e ; z A;e
À
Á
.
First it should be noted that change of any of the 3N Cartesian coordinates from
its equilibrium position results in translation and possibly rotation of a molecule,
since
P N
A¼1 M A r A 6 ¼ 0 and, where M A denotes mass of the Ath atom. To a good
approximation kinetic energy T of a molecule is given by
T ¼
1
2
X N
A¼1
M A _
q A
j j
2
ð2:16Þ
where q A ¼ r A À r A;e ¼ x A À x A;e ; y A À y A;e ; z A À z A;e
À
Á ¼ Dx A ; Dy A ; Dz A
ð
Þ , and the
dot denotes time derivative, i.e., velocity of the Ath atom, provided relations
X N
A¼1
M A r A ¼ 0
ð2:17Þ
and
X N
A¼1
M A r A;e  r A ¼ 0
ð2:18Þ
are satisfied. Equations (2.17) and (2.18) are called first and second Sayvetz conditions, respectively (sometimes Eckart conditions) and denote that whenever
atomic vibration is to generate translation (generate a change in a position of a mass
center) and/or rotation (generate zero-order vibrational angular momentum; this can
be easily seen after differentiation of Eq. (2.18) with respect to time) the coordinate
system adjusts in such a way that both components of the motion are eliminated.
These conditions were introduced to eliminate to the best possible extent coupling
between translation and vibrations as well as translation and rotation in a general
2 Scaling Procedures in Vibrational Spectroscopy
59
As stated in the previous section within the harmonic oscillator approximation, a band
on the vibrational spectrum corresponding to t = 0 ! t = 1 transition appears at the
wavenumber corresponding to classical vibrational frequency. This can be generalized; finding classical vibrational frequencies of a polyatomic molecule will give us its
approximate vibrational spectrum. The detailed derivation of relevant formulas is
given elsewhere [4, 5] and will not be repeated here. We merely present the proper
practice and report formulas necessary for future discussion.
Finding classical vibrational frequencies of a polyatomic molecule consists in
solving Lagrange equations of motion. This requires the knowledge of the vibrational kinetic and potential energies of a molecule. The most straightforward
coordinate system to treat molecular vibrations is the Cartesian system, the proper
choice of which is the following: Its origin is fixed at the center of mass of a
molecule, and it rotates with the molecule. The instantaneous position of the Ath
atom is r A ¼ x A ; y A ; z A
ð
Þ , while its equilibrium position is r A;e ¼ x A;e ; y A;e ; z A;e
À
Á
.
First it should be noted that change of any of the 3N Cartesian coordinates from
its equilibrium position results in translation and possibly rotation of a molecule,
since
P N
A¼1 M A r A 6 ¼ 0 and, where M A denotes mass of the Ath atom. To a good
approximation kinetic energy T of a molecule is given by
T ¼
1
2
X N
A¼1
M A _
q A
j j
2
ð2:16Þ
where q A ¼ r A À r A;e ¼ x A À x A;e ; y A À y A;e ; z A À z A;e
À
Á ¼ Dx A ; Dy A ; Dz A
ð
Þ , and the
dot denotes time derivative, i.e., velocity of the Ath atom, provided relations
X N
A¼1
M A r A ¼ 0
ð2:17Þ
and
X N
A¼1
M A r A;e  r A ¼ 0
ð2:18Þ
are satisfied. Equations (2.17) and (2.18) are called first and second Sayvetz conditions, respectively (sometimes Eckart conditions) and denote that whenever
atomic vibration is to generate translation (generate a change in a position of a mass
center) and/or rotation (generate zero-order vibrational angular momentum; this can
be easily seen after differentiation of Eq. (2.18) with respect to time) the coordinate
system adjusts in such a way that both components of the motion are eliminated.
These conditions were introduced to eliminate to the best possible extent coupling
between translation and vibrations as well as translation and rotation in a general
2 Scaling Procedures in Vibrational Spectroscopy
59