Page 288 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry
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Encyclopedia of Physical Science and Technology EN009N-447 July 19, 2001 23:3
Microwave Molecular Spectroscopy 805
TABLE I Types of Molecular Rotors a TABLE II Expressions for the Moments of Inertia of Some
Simple Molecules
Spherical top I a = I b = I c All three moments of inertia
Molecules Moments of inertia
molecules are equal, e.g., CH 4
Linear molecules I a = 0, I b = I c Axis a along the internuclear
m X m Y 2
axis, b and c perpendicular Diatomic, XY I = d XY
m X + m Y
to this axis, e.g., OCS, HCl
1
2
2
Symmetric-top Linear, XYZ I = m X m Y d YX + m Y m Z d YZ
M
molecules
Prolate top I a < I b = I c Axis of least moment of + m X m Z (d YX + d YZ ) 2
inertia, a, along the θ
b 2 2
symmetry axis, e.g., CH 3 F Bent, XY 2 I x = 2m Y d XY sin 2
Oblate top I a = I b < I c Axis of largest moment of 2m X m Y 2 2 θ
I y = d XY cos
inertia, c, along the M 2
symmetry axis, e.g., BCl 3 I z = I x + I y
Asymmetric-top I a = I b = I c All three moments of inertia c 2 2 θ m X m Y 2
Pyramidal, XY 3 I x = I y = 2m Y d XY sin + d XY
molecules different, e.g., SO 2 2 M
θ
× 3 − 4 sin 2
a 2
For a symmetric top, the molecule is designated a prolate or oblate
rotor, depending on which inertia axis corresponds to the molecular θ
I z = 4m Y d 2 sin 2
symmetry axis. Most molecules belong to the asymmetric rotor case. XY 2
a The d ij is the bond distance between atoms i and j; M is the total
mass of the appropriate molecule; and m i is the mass of the ith atom.
principal moments of inertia. The various cases are sum- b The x axis corresponds to the C 2 axis, with the z axis perpendicular
marized in Table I. It may be noted that molecules with a to the xy plane and with θ as the Y X Y bond angle.
threefold or higher axis of symmetry are symmetric tops, c The z axis is the C 3 symmetry axis and θ is the Y X Y bond angle.
and this symmetry axis is a principal intertial axis. Also, Note that the acute angle β between the X Y bond and the symmetry
√
any two perpendicular axes that are perpendicular to the axis is related to the bond angle by sin(θ/2) = ( 3/2) sin β.
symmetry axis are principal axes, and the corresponding
moments of intertia are equal. As indicated in the table,
where x , y , and z are the coordinates of the ith atoms
i
i
i
there are two types of symmetric rotors. For the prolate
of mass m i relative to the arbitrary coordinate system and
case, the molecule is elongated like a football, while for
M is the total mass of the molecule. The center-of-mass
the oblate case, the molecule is flattened like a disk. Most
coordinates of the atoms are computed from
molecules are asymmetric tops, and if the molecule has
some symmetry, one or more of the principal axes may
x i = x − ¯ x, y i = y − ¯ y, z i = z − ¯ z. (11)
i
i
i
be selected. If a molecule has a twofold axis of symmetry,
then this axis must be a principal axis. If a symmetry plane The elements of the moment of inertia tensor are evaluated
is present, then two principal axes must lie in this plane and from these coordinates by means of the expressions
the third must be perpendicular to this plane. It often oc-
2
2
curs that two moments of inertia are accidentally close to I xx = m i y + z , I xy =− m i x i y i ,
i
i
each other and the slightly asymmetric top approximates
2 2
one of the symmetric tops. In such cases, it is referred to I yy = m i x + z , I xz =− m i x i z i , (12)
i
i
as a near-prolate or near-oblate asymmetric top. 2 2
Expressions for the principal moments of inertia of I zz = m i x + y , I yz =− m i y i z i .
i
i
some simple molecules are collected in Table II. To evalu-
The I xx ,... are the moments of inertia and the I xy ,...
ate the moments of inertia in the general case, an arbitrary
are called the products of inertia. The inertia matrix I is
but convenient coordinate system may be chosen in the
symmetric, and diagonalization of this matrix by standard
molecule. The center of mass (¯ x, ¯ y, ¯ z)isgivenby
techniques (such as Jacobi’s rotation method),
m i x i
¯ x = , I a 0 0
M
¯
RIR = 0 0 , (13)
I b
m i y i
¯ y = , (10) 0 0 I c
M
m i z i provides theprincipal moments of inertia, and thetransfor-
¯ z = ,
M mation matrix R provides the orientation of the principal
