Molecular spectroscopy and structure, Nauka - różności, Fizyka medyczna, metody spektroskopowe
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Molecular Spectroscopy and Structure
by
Peter F. Bernath
Departments of Chemistry and Physics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
-2-
15.1 INTRODUCTION
15.2 ROTATIONAL SPECTROSCOPY
15.2.1 Diatomics
15.2.2 Linear Molecules
15.2.3 Symmetric Tops
15.2.4 Asymmetric Tops
15.2.5 Spherical Tops
15.3 VIBRATIONAL SPECTROSCOPY
15.3.1 Diatomics
15.3.2 Linear Molecules
15.3.3 Symmetric Tops
15.3.4 Asymmetric Tops
15.3.5 Spherical Tops
15.3.6 Raman Spectroscopy
15.4 ELECTRONIC SPECTROSCOPY
15.4.1 Diatomics
15.4.2 Polyatomics
15.5 STRUCTURE DETERMINATION
15.6 REFERENCES
-3-
15.1 INTRODUCTION
Our understanding of rotational-vibrational-electronic (rovibronic) spectra of molecules is
based on the non-relativistic Schrödinger equation [1],
(15.1)
The Born-Oppenheimer approximation is used to separate electronic and nuclear motion and then
the nuclear motion is further assumed to be separable into vibrational and rotational motion, leading
to the simple equations
(15.2)
and
(15.3)
For molecules with net electronic spin and net electronic orbital angular momentum, additional terms
such as spin-orbit coupling need to be added to the Hamiltonian of equation (15.1).
The manifold of energy levels described by equation (15.2) are connected by transitions as
determined by selection rules. More generally [2], an absorption line between energy levels E
1
and
E
0
is represented by Beer’s law
(15.4)
where I
0
is the initial radiation intensity,
F
is the cross-section (in m
2
), N
0
-N
1
is the population
density difference (m
-3
) and
R
is the path length (m). The intrinsic line strength of a transition is thus
measured by a cross-section, which is proportional to the square of a transition moment integral, i.e.
(15.5)
where Ôp is a transition moment operator and A is the Einstein A factor for emission. Selection
rules and line strengths are obtained by a detailed examination of equation (15.5).
15.2 ROTATIONAL SPECTROSCOPY
All gas phase molecules have quantized rotational energy levels and pure rotational
transitions are possible. A molecule can, in general, rotate about three geometric axes and can have
three different moments of inertia relative to these axes. The moment of inertia about an axis is
-4-
defined as
(15.6)
where m
i
is the mass of the atom i and r
i
z
is the perpendicular (shortest) distance between this atom
and the axis. The internal axis system of a molecule is chosen to have its origin at the center of mass
and is rotated so that the moment-of-inertia tensor is diagonal [2]. This is the principal-axis system
for a rigid molecule. The three moments of inertia I
x
, I
y
and I
z
can be used to classify molecules into
four different types of “tops”:
1. Linear molecule (including diatomics), I
x
= I
y
; I
z
= 0, e.g. CO, HCCH.
2. Spherical top, I
x
= I
y
= I
z
, e.g. CH
4
, SF
6
.
3. Symmetric top, I
x
= I
y
I
z,
e.g. BF
3
, CH
3
Cl.
4. Asymmetric top, I
x
I
y
I
z,
e.g. H
2
O, CH
3
OH.
The internal molecular axes x, y, and z are labeled according to a certain set of rules based on
molecular symmetry [3]. An additional labeling scheme is also used that is based on the size of the
moments of inertia. In particular, the axis labels A, B, and C are chosen to make the inequality I
A
#
I
B
#
I
C
true. Thus molecular symmetry determines the x, y and z labels but it is the size of the
moments of inertia that set the A, B and C labels. In terms of the A, B and C labels, it is
conventional to classify molecules into five categories:
1. Linear molecules, I
A
= 0, I
B
= I
C
.
2. Spherical tops, I
A
= I
B
= I
C
.
3. Prolate symmetric tops, I
A
<I
B
=I
C
, e.g. CH
3
Cl.
4. Oblate symmetric tops, I
A
=I
B
<I
C
, e.g. BF
3
.
5. Asymmetric tops, I
A
<I
B
<I
C
.
Underlying all of these considerations is the assumption that the concept of a molecular
structure is useful. Floppy species such as Van der Waals molecules or molecules with internal
rotors do not always have a well-defined molecular structure when zero-point motions are
considered. Clearly all molecules have a hypothetical equilibrium structure, but vibrational motion
even in the zero-point level can destroy the concept of a molecular structure with well-defined bond
lengths and bond angles. Fluxional molecules are best handled using concepts based on
permutation-inversion group theory [4].
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15.2.1 Diatomics
For a rigid diatomic molecule in a
1
E
+
electronic state (no net spin or orbital angular
momentum) the rotational energy levels are given by
,
(15.7)
where B is the rotational constant and J, the rotational quantum number, has values 0, 1, 2, ... The
units of (15.7) are determined by the units chosen for B, which are generally cm
-1
, MHz or (rarely)
J (joules).
Various equations for B are:
B (joules) =
(15.8)
B (MHz) =
×10
-6
=
,
(15.9)
B (cm
-1
) =
=
(15.10)
in which I, the moment of inertia, is defined by
(15.11)
and the masses m
A
and m
B
are separated by a distance
r. The reduced mass
:
of the AB molecule
(15.12)
is conventionally calculated using atomic (not nuclear) masses [5]. The use of a single symbol B
for three separate physical quantities (energy, frequency and wavenumber) is clearly confusing but
is the spectroscopic custom. Thus spectroscopists talk about “energy” levels but locate them using
cm
-1
units.
A real molecule is not a rigid rotor because the bond between atoms A and B can stretch at
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