Introduction
A diatomic molecule differs from an atom in
having additional degrees of freedom caused by the ability of the molecule
to rotate and vibrate about its center of mass. These motions can be, in
first approximation, considered separately and lead to sets of energy
levels, 
, but with
relatively small values of 
.
Since
= h
, the frequencies of
electromagnetic radiation absorbed or emitted by a transition between two
of these levels will lie in the infrared (frequency (nu)
> 1013Hz, wavelength (lambda) < 20 
m)
or microwave regions ( < 1011Hz,
lambda > 20m) of the
electromagnetic spectrum.
The simplest models for the motion of a diatomic molecule in its electronic ground state are the harmonic oscillator which describes the vibrations of the two atoms along the bond direction and the rigid rotor which approximates the rotation of the molecule in space. Since a diatomic molecule is under investigation, only one bond can vibrate and the rotation of the molecule occurs only about an axis perpendicular to the bond direction and through the center of mass.
Quantum Mechanical Results Related to Molecular Motion
Vibration
Application of Quantum Mechanics to the Simple Harmonic Oscillator (SHO) leads to the energy levels for the vibration of the chemical bond:
where v is a quantum number having only the values v=0,1,2,..., and h is Planck's constant.
is the vibrational frequency (in Hz) given by
where
is the reduced mass of the diatomic molecule, equal to (m1m2)/(m1+m2)
(of the order of 10-27kg for HCl), and k is the force constant
of the bond in units of Nm-1 or dynes cm-1. Note
that the heavier the molecule and/or the weaker the bond, the smaller the
vibrational frequency. The vibrational frequency is often expressed in
units of cm-1, rather than Hz, through the relation
c =
where c is the velocity of light (3 x 1010 cm s-1).
This scaling makes the vibrational frequency of most molecules a
convenient number in units of cm-1, or wavenumbers, i.e.
varies from about 100 to
4,000 cm-1. (The molecule of concern here, HCl, has
ca. 3000 cm-1.) Note from Eq. (1) that the lowest energy level
of a harmonic oscillator has o=
h/2 (the zero point energy)
and adjacent levels are equally spaced with =hc.
Rotation
Quantum Mechanics also provides the rotational eigen-energies of a rigid linear rotor
where J is a quantum numbercan take only the values 0,1,2,..., and I is the moment of inertia of the dumb-bell.
Here I=m1r12+m2r22
= R2 where
is again the reduced mass, and R is the interatomic distance (=r1+r2).
If the rotational constant is defined by
B=h/(8
2I)
for the diatomic molecule. The rotational
constant can be expressed in wavenumber units
= B/c in cm-1
and varies from about 0.05 to 20 cm-1
among diatomics (for HCl, ca.
11cm-1). From Eq. (3), one sees that the spacing between
adjacent levels for a rigid rotor is
(J)=2J'
where J' designates the upper rotational
level. Thus, the rotational levels are not equally spaced, and they are
much closer together than the vibrational levels of the same molecule
because of the relatively small values of 2
relative to that of .
Infra-Red Transitions
Since both rotational and vibrational motions are simultaneously occurring in the diatomic, the energy level scheme for two adjacent vibration levels (spaced, say, 3000 cm-1 apart) where ca. 10 cm-1 in
the two states, would be as shown in Fig. 1 (Transitions between levels
indicated there will be referred to later.). Each vibrational state has a
set of rotational levels. These rotational levels are not exactly the same
since varies slightly with v, the
quantum number labeling the vibrational state. (This latter effect is
called vibration-rotation coupling)
Figure 1: Transitions between the lowest vibrational state, denoted v=0, and the next highrest vibrational state (v=1). Each vibrational state has many rotational levels denoted by a rotational quantum number (J' for the upper vibrational state, and J" for the lower). The spacing between rotational levels of the same vibrational state has been greatly exagerrated for clarity.
Equations (1) and (3) may be combined and if expressed as Term Values of a given state and level of the molecule as labelled by v and J, T(v,J) [in units of cm-1 then
However, these are only a first approximation to the true Term Values since there are small additional terms to be added to account for departures from our simplest models of vibration and rotation:
Here exe
is a small correction for departure from harmonic vibrational motion
(i.e., a non-parabolic potential energy). This correction is termed the anharmonicity
of the bond. Anharmonicity must occur in all chemical bonds since at a
certain energy all chemical bonds break. the term De corrects
for the non-rigidity of the bond which can be expected to stretch as the
molecules rotates, moreso at the high J values (i.e., centrifugal
distortion) the term 
e
is due to the interaction between rotation and vibration; since R changes
during the vibrational motion, so does I and therefore .
Quantum mechanics also stipulates that only
the certain transitions shown in Fig. 1 are "allowed" via
electric dipole radiation. The selection rules for this process are
such that both v and J change by 1 quantum in allowed transitions. (v=+/-1,
J=+/-1) Only molecules with
permanent electric dipole moments may exhibit infrared absorption, v=+1,or
emission, v=-1. J=0
is allowed only for electronic transitions in some diatomic molecules and
non-linear polyatomic molecules. Traditionally, the J
transitions are grouped together as branches (see Fig. 1):
J
= + 1, R branch
J
= - 1, P branch
J
= 0, Q branch (forbidden here)
To a first approximation the length of the
arrows in Fig. 1 (proportional to the frequency
of the transition) differ by 2
among the P branch transitions and among the R branch transitions, but
with 4 between the central P(1)
and R(0) lines. If the forbidden Q branch transition v=0, J"=0 to
v=1, J'=0 (call it o)
were included in the Figure then all spacings would be 2.
If the very small term involving De is neglected in Eq. (4),
the combined rotation-vibration transitions in Fig. 1 can be written
For least squares fitting the P and R equations can be conveniently combined by letting m=(J"+1) in the R branch and (m = -J") in the P branch, yielding
(Note that there is no value corresponding
to o (m=0) for this
molecule.) For example, the transitions in Fig. 1 can be assigned values
of m from -5 to +5 (with 0 missing) and a plot of
(m) versus m will yield e
and e. Then by use
of Eq. (7) one can also find o.
A typical spectrum is given in Fig. 2.
