Chapter 5.

Molecular Spectroscopy: Basic Physics

Notes:
• Most of the material presented in this chapter is taken from Stahler and Palla (2004),
Chap. 5.

5.1 Interstellar Molecules

The first thing that needs to be explained when considering the existence of interstellar
molecules is the fact that they are there at all. Indeed, the presence of ultraviolet photons
in the interstellar radiation field would lead us to believe that any molecules previously
created would be destroyed by these high-energy photons. In the present day Universe,
the solution to this problem resides in the fact that wherever we find molecular gas, we
also find interstellar dust. We know that dust grains are extremely efficient in absorbing
radiation at optical and ultraviolet wavelengths, and, therefore, provide the amount of
shielding necessary to protect molecules against harmful radiation.
Another problem consists in explaining how atoms can combine to form both simple and
more complicated molecules. One might assume that normal neutral-neutral interaction in
the gas phase of the interstellar medium would suffice to explain the current observations
of the many different molecular species detected so far (well over 100) in the dense and
cold parts of molecular clouds. As it turns out, there are a few reasons that render this
hypothesis unlikely. First, it is often the case that an energetic activation barrier needs
to be conquered before a neutral-neutral chemical reaction can take place. Since the
corresponding energy level is on the order of !E kB ! 100 K , this type of reaction is
unlikely to take place in such regions of molecular clouds.
Another problem arises from the fact that most chemical reactions are exothermic, i.e.,
they liberate energy upon the creation of a molecule. This characteristic, combined with
the requirement that energy and momentum be conserved during a chemical reaction,
makes it impossible for a two-body interaction to take place. Technically, there is still the
possibility that the molecule resulting from a two-body interaction would be initially in
an excited state and subsequently emit a photon, which would make possible such a
chemical reaction while ensuring that the aforementioned conservation laws are met. This
process know as radiative association is, however, very unlikely to take place in
molecular clouds. It appears that the only way out for the existence of this kind of
reaction resides with the presence of a third body, which would then easily ensure the
conservation of momentum and energy. But such three-body interactions (between three
hydrogen atoms, for example) would require much higher densities than encountered in
the dense and cold parts of molecular clouds. We must, therefore, also discard this
possibility as the main pathway for molecule production (although we will see later that
we cannot completely neglect them).
We now consider another factor that enters the equation when one calculates the rate at
which any chemical reaction, of any kind, between two particles can take place. Although
we have just dismissed the possibility of neutral-neutral reactions as a viable explanation
for the existence of the many molecular species detected in dense molecular clouds, we
do this calculation for comparison purposes at a later time. Let us take, for example, two

57
neutral species A and B that coexist in some region of a cloud with respective densities
nA and nB . Let us now calculate the collision rate of a given particle B collides with
another A such that the following reaction could, in principle, take place

A + B ! C, (5.1)

where C is the molecule that results from this interaction. If ! is the scattering cross-
section characterizing the collision, then the volume covered or traveled by the particle B
as seen by A per unit time is vr ! , where vr is the relative velocity between the two. It
follows that the probability that the particle B meets any particle A is

! c = nA vr " , (5.2)

which is, in effect, the collision rate for this type of encounter. If we assume equipartition
of energy, the relative velocity between two particles in a gas of temperature T is
approximately

12
"k T%
vr ! $ B ' , (5.3)
# µ &

where µ is the reduced mass for the two particles considered. For a typical molecular
cloud with T = 10 K , the relative velocity is on the order of 0.1 km/s. Also, the cross-
section ! for neutral atoms or molecules collisions is approximately 10 !15 cm !2 . We,
therefore, find from equation (5.2) that the collision rate is

! c " 10 #11 nA s #1 , (5.4)

and nc the total number of collisions per unit volume and unit time is

nc ! 10 "11 nA nB cm "3s "1 . (5.5)

Furthermore, if the reaction rate coefficient is denoted by k (in cm 3s !1 ), then the

average number of reactions taking place per unit time is

N r = k nA nB s !1. (5.6)

Comparison with equation (5.2) reveals that the reaction rate coefficient already
incorporates the collision rate within it. We note for the moment that the value obtained
in equation (5.4) for the collision rate leading to neutral-neutral reactions is a function of
the relative velocity between the two particles involved. One could always postulate that,
for some reason, the relative velocity is increased by a large factor (e.g., an order of
magnitude or two), if this were needed to explain observations. A problem with this
scenario is that it would be very hard for two particles colliding at such high speeds to

58
“stick” together and enter into a chemical reaction. They would most likely bounce off
each other instead.
It was eventually discovered that the way out of the problem of accounting for the
observed population of molecular species in dense molecular clouds is to involve charged
atoms or molecules in chemical reactions. These are called ion-molecule reactions. This
is due to the fact that when an ion approaches a neutral particle it will induce an electric
dipole moment into that neutral entity that will greatly increase the scattering cross-
section. More precisely, it can be shown that

! " vr#1 , (5.7)

and that the mean collision rate of any ion A + with a neutral B leading to a potential
reaction of the type

A+ + B ! C+ + D (5.8)

is

! c = nA vr" . (5.9)

The difference with a neutral-neutral collision is quantified by the fact that

vr! " 10 #9 cm 3s #1 , (5.10)

which is the equivalent of a neutral-neutral collision with a relative velocity of 10 km/s

(see equation (5.4))! For similar volume densities as for a neutral-neutral interaction, the
collision rates are increased by two orders of magnitudes. Furthermore, ion-molecule
reactions do not involve any activation barrier, in general. These facts are sufficient to
account for the molecular abundances measured in molecular clouds, at least generally
speaking.
We should note that a certain level of ionization, if very low (i.e., 10 !6 to 10 !9 ), is
always expected to be present in molecular clouds, even in cases where there are no
stellar or interstellar radiation fields in the vicinity. This is because of the omnipresence
of cosmic rays that can penetrate deep into molecular clouds to bring about some level of
ionization.
Perhaps the most fundamental ion-molecule reaction is

H 2 + H +2 ! H +3 + H. (5.11)

The importance of this reaction stems from the high chemical reactivity of the H +3 ion
and its ensuing role for the formation of many other molecules. For example, we can
show how the combination of this reactivity with the existence of atomic carbon and
oxygen, which are produced abundantly in stellar interiors and injected in the interstellar

59
medium at the end of their lives, leads to the formation of carbon monoxide (CO). More
precisely, we have

C + H +3 ! CH + + H 2
CH + + H 2 ! CH +2 + H (5.12)
CH +2 + H 2 ! CH +3 + H

for part of the so-called carbon chemistry, and a similar network

O + H +3 ! OH + + H 2
OH + + H 2 ! OH +2 + H (5.13)
OH + H 2 ! OH + H
+
2
+
3

for the oxygen chemistry. Combining the last reaction of these networks with O and C,
respectively, yields the formation of the very important HCO + molecular ion

O + CH +3 ! HCO + + H 2
(5.14)
C + OH +3 ! HCO + + H 2 .

Finally, since most of the resulting ions will be destructed through dissociative
recombination with an electron e ! we get

HCO + + e ! " CO + H. (5.15)

Carbon monoxide is very stable molecule (it has a triple-bond between the two atom) and
is the second most abundant molecule after molecular hydrogen.
Evidently our main reaction given in equation (5.11) assumes the presence of molecular
hydrogen. As it turns out, this is not a trivial assumption. This is because the chemical
reaction involving two hydrogen atoms to form a hydrogen molecule is exothermic (see
above). As was mentioned earlier, because of the need for conservations of energy and
momentum in the process of creating the molecule the energy released in the creation of
the molecule must be transmitted to a third body. A solution to this problem can be found
if one considers the role played by dust grains for the formation of molecules, especially
H2 .
Because a hydrogen atom has an unpaired electron it will easily stick (through the weak
Van der Walls force) to the surface of a grain after a collision with it. It will then
quantum tunnel through the surface of the grain until it reaches a lattice defect where the
binding energy is high enough to trap it in place. Evidently, the probability that two
atoms meet at the location of such a defect on the surface of grain and form a molecule
can be relatively high if enough atoms collide with the dust grain. It is, in fact, believed
that two atoms will quickly find each other in such a way at the surface of a grain. If the
collision rate of a grain with hydrogen atoms is given by (see equation (5.2))

60
 ! c = nH vr " d , (5.16)

where ! d = " ad2 is the cross-section of a (spherical) dust grain and nH the density of
atomic hydrogen. Then the rate of H 2 formation is given by

1
R H2 = ! H nd" c
2
(5.17)
1
= ! H nd nH vr # d ,
2

where ! H ! 0.3 is the sticking probability of a hydrogen atom to the surface of the grain.
The factor or 1 2 in equation (5.17) is due to the fact that two atoms are needed to create
one molecule. Once the molecule has formed the energy release in the process is easily
absorbed by the dust grain, which is much more massive than the atoms, and the
molecule is ejected from the surface of the grain. Moreover, since they ejected hydrogen
molecules do not have any unpaired electron (unlike the hydrogen atom) they will not
exhibit the tendency to easily stick to a dust grain when colliding at a later time. Dust
grains are thus essential for the formation of the most abundant molecule in molecular
clouds, and allow for the creation of the H +3 molecule (through the reaction given in
equation (5.11)) that is so important for the chemistry and the formation of many other
molecules.

5.2 Molecular Spectroscopy

Molecules being composed of nuclei and electrons along with their interwoven
interactions are, needless to say, impossible systems to completely solve analytically. It
follows that a series of approximations are usually applied when trying to explore their
spectroscopy.
We start with the classical molecular Hamiltonian, which is the sum of the kinetic and
potential energies

T=
1 l
(
! mr X! 2r + Y! r2 + Z! 2r
2 r =1
)
(5.18)
l
C C e2
V= ! r s ,
r < s =1 Rrs

where, for the r th particle, mr is the mass, the position is specified by X r , Yr , and Z r and
measured from some arbitrary space-fixed coordinate system, Cr e is the charge, and Rrs

Rrs = ( Xr ! X s )2 + ( Yr ! Ys )2 + ( Zr ! Z s )2 (5.19)

61
is the distance to particle s . The kinetic energy can be decomposed into two components:
one ( TCM ) due to the motion of the centre of mass of the system, and another ( Trve )
arising from the motions of the nuclei and electrons relative to the centre of mass. To
separate these two terms, we express Xr ,Yr , and Z r the position components of the r th
particle relative to the centre of mass located at X 0 , Y0 , and Z 0 such that

Xr = Xr + X0
Yr = Yr + Y0 (5.20)
Zr = Zr + Z0 .

Once the ensuing calculations are performed we find the following for the Hamiltonian

H rve = Trve + V
1 l
2 r=2
2 ! (
= ! mr Xr + Yr + Z r +
! 2 ! 2 1 l
)
! mr ms ( X! r X! s + Y!rY!s + Z! r Z! s )
2m1 r, s = 2
(5.21)

l
Cr C s e2
+! ,
r < s =1 Rrs

The index “rve” stands for rovibronic (rotation-vibration-electronic), and is associated

with the internal kinetic energy of the molecule, as opposed to the external or
translational kinetic energy. The rovibronic Hamiltonian is the quantity that we need to
consider to determine the spectroscopy of a molecule.
To convert the classical rovibronic Hamiltonian H rve into the needed quantum
mechanical version of the Hamiltonian, we must first express equation (5.21) as a
function of the coordinates Xr ,Yr , and Z r and momenta PXr , PYr , and PZr . We could then
replace these quantities with the corresponding quantum mechanical operators and obtain
an expression for the quantum mechanical Hamiltonian. To accomplish our first
aforementioned task, we make use of the Lagrangian definition for the momenta, i.e.,

!Lrve
PXr = , …, (5.22)
!X! r

where Lrve ! Trve " V is the (rovibronic) Lagrangian. In this case, since the potential
energy is not a function of the velocities, then equation (5.22) simplifies to

!Trve
PXr = , (5.23)
!X! r

and so on. The equation resulting from this is

62
 1 2 !2
l l l
Cr C s e2
Ĥ rve = !! $ 2
"r + $ "r # "s + $ R , (5.24)
r = 2 2mr 2M r, s = 2 r < s =1 rs

with

" " "

!r = e X + eY + eZ , (5.25)
"Xr "Yr "Z r

as a representation in the usual coordinate space.

5.2.1 The Fine Structure and Hyperfine Structure Hamiltonians

The rovibronic Hamiltonian of equation (5.24) does not take into account some
interactions due to the intrinsic magnetic moment (i.e., spin) of the electrons, and the
intrinsic magnetic and electric moments of the nuclei. More precisely, if we only consider
the individual electron spins ŝ i , and the so-called electron spin-spin and spin-orbit
couplings, then the electron fine structure Hamiltonian Ĥ es must be introduced

1 ) %,
2
" e % " P̂i
Ĥ es ! ! $
# m c '&
+
ij +
R̂(
1 R 3 i j $# 2 j '& . / ŝi
! R̂ ( )
! P̂
.-
e j 0i *
C2 e ) " P̂ P̂ % ,
+
e
2 1 3 +
me c 2 ,i Ri2 +*
( )
R̂ i ! R̂2 ( $ i ! 2 ' . / ŝ i
# 2me m2 & .-
(5.26)
" e %
2
)1 " %
+$
# me c '&
1+ R
3
( ) (
ŝ / ŝ j ! 2 )* ŝ i / R̂ i ! R̂ j ,- )* ŝ j / R̂ i ! R̂ j
3 $ i ),-'&
* ij #
j >i + Rij
83
!
3
(
4 R̂ i ! R̂ j ŝ i / ŝ j )( ),.- ,
where ! labels the nuclei, and i and j label the electrons. In equation (5.26), the first
term (a spin-orbit interaction) corresponds to the coupling of the spin of each electron to
the magnetic field it feels (in its reference frame) because of the presence of the electric
Coulomb fields due to the other electrons (in their respective reference frames). The
second term is also a spin-orbit interaction but this time with the Coulomb field of the
nuclei, and the last term corresponds the spin-spin couplings between the intrinsic
magnetic moments of each pair of electrons. The internal molecular Hamiltonian then
becomes Ĥ int = Ĥ rve + Ĥ es , and the total angular momentum must include both the orbital
and electron spin momenta. It is usual to denote the orbital angular momentum with N̂
(instead of Ĵ ), and the total angular momentum with Ĵ . We then write

Ĵ = N̂ + Ŝ, (5.27)

63
where Ŝ is the total electron spin operator. The associated quantum numbers J and m
refer to the sum of the orbital and spin angular momenta, and are those with which the
molecular eigenfunctions can be labeled.
On the other hand, if the interactions of the intrinsic magnetic and electric moments of
the nuclei are taken into account, then the total angular momentum is denoted by F̂ with

F̂ = Ĵ + Î = N̂ + Ŝ + Î, (5.28)

where Î is the total nuclear spin angular momentum. The corresponding nuclear
hyperfine structure Hamiltonian Ĥ hfs will include nuclei spin interactions similar in
form to those of equation (5.26), as well as terms due to the nuclei electric quadrupole
fields. The internal molecular Hamiltonian then becomes

Ĥ int = Ĥ rve + Ĥ es + Ĥ hfs , (5.29)

and the good quantum numbers with which the associated eigenfunctions can be labeled
are those corresponding to F̂ 2 and F̂Z , i.e., F and mF ( mF = 0,1, 2, …, F ).

5.2.2 The Born-Oppenheimer Approximation

Coming back to equation (5.24) for the rovibronic Hamiltonian, we can simplify things
greatly with the realization that the motion of the electrons can be separated from that of
the nuclei. This is expected because of the significantly lower mass of the electrons. We
can therefore imagine that as the nuclei are moving around, the electron will adjust
themselves on a much shorter time scale (almost instantaneously as far as the nuclei are
concerned). This is the idea at the centre of the so-called Born-Oppenheimer
approximation.
To proceed with this approximation, we introduce a new coordinate system that has its
origin at the nuclear centre of mass instead of the molecular centre of mass. This
difference in the location of the origin is important as it relates the motion of the electrons
to the position of the nuclei. The Hamiltonian then becomes

Ĥ rve = Tˆe + TˆN + V ( R N , re ) , (5.30)

where Tˆe and TˆN are the separated kinetic energies of the electrons and nuclei,
respectively. The potential energy can also be advantageously written as

V ( R N , re ) = Vee ( re ) + VNN ( R N ) + VNe ( R N , re ) , (5.31)

where the different terms on the right hand side are for the separate summations of
electron-electron, nucleus-nucleus, and nucleus-electron electrostatic potential energies,
respectively. This Born-Oppenheimer approximation also implies that it is possible to

64
express the rovibronic wave function ! 0rve,nj as the product of the electronic wave
function ! elec,n and the rotation-vibration wave function ! rv,nj

! 0rve,nj = ! elec,n ( R N , re ) ! rv,nj ( R N ) , (5.32)

where the indices n and j relate to the particular electronic and rotation-vibration states,
respectively. Obviously, we must realize that although the nuclei coordinates R N are
assumed constant in ! elec,n , they are certainly allowed to vary in ! rv,nj . The Schrödinger
equation is then

( ) ( )
Ĥ rve ! 0rve,nj ( R N , re ) = "# Tˆe + Vee + VNe + TˆN + VNN $% ! elec,n ( R N , re ) ! rv,nj ( R N )

(= Tˆ + V + V !
e ee Ne )
(R , r ) ! (R )
elec,n N e rv,nj N

+! elec,n ( R N , r ) (Tˆ
e N )
+ VNN ! rv,nj ( R N )
(5.33)
= Velec,n ! elec,n ( R N , re ) ! rv,nj ( R N )

(
+! elec,n ( R N , re ) TˆN + VNN ! rv,nj ( R N ))
= ! elec,n ( R N , r ) (Tˆ
e N )
+ VNN + Velec,n ! rv,nj ( R N ) ,

or

Ĥ rve ! 0rve,nj ( R N , re ) = Erve,nj

0
! elec,n ( R N , re ) ! rv,nj ( R N )
(5.34)
= Erve,nj
0
! 0rve,nj ( R N , re ) ,

with Erve,nj
0
the eigenvalue when the molecule is in the electronic state n and rotation-
vibration state j . It is therefore apparent that the potential energy for the nuclear rotation-
vibration Hamiltonian is (VNN + Velec,n ) and includes a contribution from the electronic
state through the presence of Velec,n . It is customary to rewrite things so that the zero
energy for the rotation-vibration equation, in a given electronic state, is the minimum
value of (VNN + Velec,n ) , which is usually called the electronic energy Eelec,n . Our
molecular problem can then be rewritten with two Schrödinger equations: one
determining the electronic states and another the nuclear (rotation-vibration) states

(Tˆ + V
e ee )
+ VNe ! elec,n ( R N , re ) = Velec,n ! elec,n ( R N , re )
(5.35)
(Tˆ N )
+ VN,n ! rv,nj ( R N ) = Erv,nj ! rv,nj ( R N ) ,

with

65
 VN,n = VNN + Velec,n ! Eelec,n
(5.36)
Erv,nj = Erve,nj
0
! Eelec,n .

5.2.3 Separation of Vibration and Rotation Motions

A further simplification of the problem is effected by introducing yet another coordinate
system. This new system also has its origin at the nuclear centre of mass, but it is fixed to
nuclei in such a way as to follow the rotational motion of the molecule. In comparison,
the coordinate systems considered until now have been fixed in “space”.
Once this substitution is made in the rotation-vibration Hamiltonian, and many other
approximations are made (e.g., ignoring terms due to Coriolis interactions, anharmonic
vibrational motions, etc.), it is possible to separate the two types of motion. The resulting
Hamiltonian corresponding to the second of equations (5.35) is

1 3N $ 6 2
Ĥ rv0 =
1
" !! ! 2 " P̂k + #kQk2 ,
2 !
µ e
Ĵ 2
+
k =1
( ) (5.37)

where µ!!
e
are the reciprocal of the principal moments of inertia of the molecule, Ĵ! are
the components of the total angular momentum, and Qk , !k , and P̂k are, respectively, the
normal modes of vibrations, their eigenvalues, and their associated linear momenta.
A solution to the vibration part of the Hamiltonian (that of a harmonic oscillator)

1 3N " 6 2
0
Ĥ vib = #
2 k =1
(
P̂k + !kQk2 , ) (5.38)

yields a set of wave functions !" (Q ) , which depend on the normal modes of vibration
(one for each non-degenerate mode or set of degenerate modes), with associated energy
levels that are functions of the vibrational quantum number ! . For a non-degenerate
mode the energy levels are given by

" 1%
Evib = $ ! + ' ! 2( , (5.39)
# 2&

where ! = 0,1,2, … and ! " # 1 2 ! . The total energy of vibration for a molecule will
consist of the summation over all the energies (of the type given by equation (5.39)) over
all normal modes of vibration.
The rotational Hamiltonian (that of a rigid rotator) is usually written

1
0
Ĥ rot = "
2 !
µ!!
e
Ĵ!2
(5.40)
= Ae Ĵ a2 + Be Ĵ b2 + Ce Ĵ c2 ,

66
where

1 1 1
Ae = e
, Be = e , and Ce = e . (5.41)
2I aa 2I bb 2I cc

The principal axes a, b, and c are always defined such that Ae ! Be ! Ce , and whether
the molecular fixed z-axis (fixed to the nuclei of the molecule when not vibrating) is
identified with the a, b, and c index the situation is usually defined as type I, II, or III.
The quantity Be is the rotational constant.
It is convenient to discern between the following types of rotators

1. Spherical tops e
I aa = I bbe = I cce CH 4
2. Symmetric tops, e
a) prolate I aa < I bbe = I cce CH 3D
e
b) oblate I aa = I bbe < I cce H +3 , NH 3
3. Linear molecules e
I aa = 0, I bbe = I cce CO, HCN
4. Asymmetric tops e
I aa < I bbe < I cce H 2O

The solution to Schrödinger equation for the Hamiltonian of equation (5.40) will differ
depending on what type of molecule is being considered. Examples of configuration for
the rotational energy levels for molecules of the different types (i.e., spherical top,
symmetrical top, linear, and asymmetric top molecules) are shown in Figure 5.1. For the
purpose of our discussion it will be sufficient to limit ourselves to the case of linear
molecules, it is then easy to show that the solution can be written as

Be Ĵ 2 ! J (" ,# , $ ) = Erot ! J (" ,# , $ ) , (5.42)

and

Erot = Be J ( J + 1) ! 2 , (5.43)

where ! J (" ,# , $ ) is the wave function associated to the angular momentum quantum
number J . The quantities ! , " , and # are the Euler angles that relate the position of the
molecule-fixed and space-fixed set of axes at all times.
Finally, referring to equations (5.35), (5.39), and (5.40) we find that the total energy of a
molecule (within our set of approximations) can be written as

Erv,nj = Eelec + Evib + Erot

" 1% (5.44)
= Eelec,n + $ ! + ' ! 2( + Be J ( J + 1) ! 2 .
# 2&

67
Figure 5.1 - Rotational energy levels for molecules of the different types (i.e., spherical
top ( CH 4 ), symmetric top ( CH 3D ), linear (CO), and asymmetric top ( H 2O ) molecules).
It is also important to note that these energies scale in such a way that

Eelec ! Evib ! Erot . (5.45)

It follows that the gas in the dense and cold parts of molecular clouds will mainly radiate
through rotational transitions. The other types of transitions will happen in hotter regions,
such as regions located close to protostars or within shock fronts.

5.2.4 Selection Rules

Whenever a molecule possesses a finite electric dipole moment µ̂ , it will regulate its
interaction with any ambient electric field. More precisely, to adequately account for this
interaction the molecular Hamiltonian must be augmented by the following perturbation
term

Ĥ dip = !µ̂ " E, (5.46)

The electric dipole is defined as

µ̂ = ! Ci eR̂ i , (5.47)
i

68
where Ci e and R̂ i are the charge and the space-fixed position of particle i , respectively
(the summation is over all electrons and nuclei).
Selection rules for transitions within the Born-Oppenheimer, harmonic oscillator, and
rigid rotators approximations can be established by a careful quantum mechanical
analysis that specifies which states (i.e., wave functions) will be coupled by the
perturbation Hamiltonian of equation (5.46). There is no simple way to state what
electronic transitions will be allowed of not. Fortunately, such rules can be established for
vibrational, rotational, and rovibrational transitions for the different types of molecules
(i.e., rotators).
In general, vibrational transitions for a given normal mode (of index ! r ) are unrestricted.
That is,

!" r = " r# $ " r## = ±1, ± 2, ± 3, …, (5.48)

where " r! and " r!! are the initial and final states, respectively. However, the so-called
infrared active transitions (they happen in the infrared) with !" r = ±1 will normally be
stronger. Other types of transitions, such as overtone (from the vibrational ground state
to one state r with !" r # 2 ) and combination tones (from the vibrational ground state
to more than one state r with !" r # 0 ) transitions are also possible.
Rotational transitions, whether pure (i.e., !" r = 0 and no change in electronic state) or
combined with vibrational transitions, are more complicated to generalize. But a rule on
the angular momentum quantum number J can be stated with

!J = J " # J "" = 0, ± 1 ( J " = J "" = 0 is forbidden ) . (5.49)

where J ! and J !! are the initial and final states, respectively. The reader should be aware,
however, that the simplicity of the rule given in equation (5.49) is somewhat misleading,
as other quantum numbers come into play for rotational transitions. Furthermore, these
vary with the type of rigid rotator considered. In particular, pure rotational transitions
(i.e., !" r = 0 and no change in electronic state) for symmetric top and linear molecules
are only allowed when

!J = ±1 (5.50)

because of energy and angular momentum conservations considerations. A transition

with !J = 0 will only be allowed if !" r # 0 , as long as J ! = J !! " 0 ; these are
rovibrational transitions.
A molecule will have no electric dipole whenever it is symmetric (e.g., H 2 , H +3 , and CH 4
in their electronic ground states). In such cases, interaction of the molecule with an
ambient electric field will be regulated through its electric quadrupole moment Q̂!" (it
is a second rank tensor) through the corresponding perturbation Hamiltonian

69
 1 $E
Ĥ quad = ! %
6 " ,#
Q̂"# " .
$x#
(5.51)

The electric quadrupole moment is defined as

(
Q̂!" = % Ci e 3 X̂! ,i X̂" ,i # R̂i2$!" ,
i
) (5.52)

where Ci e , X̂! ,i , and R̂i are the charge, and the ! th component and the modulus of the
space-fixed position of particle i , respectively (the summation is over all electrons and
nuclei). The main difference in the selection rules, as compared to electric dipole
transitions, is restricted to the rotational transitions. More precisely, we find that

!J = J " # J "" = 0, ± 1, ± 2 ( J " = J "" = 0 is forbidden ) , (5.53)

where J ! and J !! are the initial and final states, respectively. The other rules and
comments described earlier for the electric dipole interaction also apply for the electric
quadrupole transitions. However, it should always be kept in mind that other
considerations can restrict these selection rules (e.g., conservation of energy and
momenta, and Pauli’s exclusion principle).

5.3 The Hydrogen Molecule

It is rather ironic that perhaps the two most important molecules in molecular clouds, i.e.,
H 2 and H +3 , are symmetric. It follows that they cannot be detected through the strong
electric dipole transitions (they do not possess a dipole), but can only be observed via the
relatively weaker electric quadrupole counterparts. Furthermore, this fact, combined with
the low temperatures characterizing the dense and cold parts of molecular clouds, implies
that these molecules will not be detectable in these regions (i.e., the quadrupole
transitions will be too weak). Astronomers rely instead on other molecular species as
tracers for these molecules (e.g., CO for H 2 ). On the other hand, H 2 and H +3 can be
observed in other types of environment (e.g., shocks), which are energetic enough to
induce electronic and vibrational transitions.
Molecular hydrogen will then be subjected to the selection rules specified by equations
(5.48) and (5.53) with the additional restrictions imposed on the nuclear spin states by the
Pauli exclusion principle. This is because the two nuclei are identical particles (i.e.,
protons), and their associated spin wave functions will impose certain symmetries on the
rovibrational wave functions to allow for quadrupole radiative transitions. More
precisely, the selection rules for the angular momentum quantum number is not given by
equation (5.53) but

!J = J " # J "" = 0, ± 2 ( J " = J "" = 0 is forbidden ) . (5.54)

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Figure 5.2 – Rotational levels of H 2 for the first two vibrational states. The J = 2 ! 0
pure rotational electric quadrupole transition in the vibrational ground state is shown, as
well as the 1 ! 0 S (1) rovibrational transition.
Since H 2 is the molecule with the smallest moment of inertia, the energy spacing
between its rotational transitions will be the greatest (see equations (5.40) to (5.43)). For
example, the J = 2 ! 0 pure rotational electric quadrupole transition in the vibrational
ground state has an energy h! kB = 510 K , which is too high for cold molecular clouds.
It is customary to denote a general rovibrational transition between the initial and final
states (" !, J ! ) and (" !!, J !! ) with " ! # " !! O ( J !! ) , Q ( J !! ) , S ( J !! ) , depending whether
!J = 2, 0, or " 2 , respectively. For example, the well-known 1 ! 0 S (1) transition that
occurs at 2.12 µm is shown in Figure 5.2. As can also be seen from this figure, the
energy separation between vibrational states is on the order of almost 10 4 K . The
corresponding separation for electronic states is even higher at approximately 10 5 K .

5.4 Carbon Monoxide

Carbon monoxide is not symmetric and, therefore, has a finite electric dipole moment
that allows for the detection of strong pure rotational transitions with !J = ±1 (see
equation (5.50)). Furthermore, this molecule also has a sizeable principle moment of
inertia about axes perpendicular to the molecular axis. This implies that the spacing
between adjacent energy levels will be accordingly small. Indeed, the energy associated
with the fundamental J = 1 ! 0 ( ! = 0 ) pure rotational transition is !E10 kB = 5.5 K ;
this transition occurs at a wavelength of 2.6 mm or a frequency of 115 GHz. The next
transition J = 2 ! 1 at 230 GHz has a corresponding energy of 16 K. We have here a
molecule that lends itself perfectly to the observational study of the cold regions of

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molecular clouds. The J = 1 ! 0 transition, as well as the positions of rotational energy
levels, is shown in Figure 5.1.

5.4.1 The Critical Density

Molecules located deep inside molecular clouds, away from any substantial sources of
radiation, will be excited through collisions with mainly H 2 . How much a given CO
rotational level, in this case, will be populated depends on the number of, and the energy
involved in, collisions. Referring to equations (5.2) and (5.3) for the neutral-neutral
collision rate we realize that the level of excitation will depend on both the density of H 2
and the kinetic temperature Tkin of the gas.
We define the critical density of the gas ncrit for a well-defined transition the density at
which, for a given temperature Tkin , the collision rate is equal to the Einstein coefficient
for spontaneous emission Aul between the upper (u) and lower (l) levels defining the
transition. That is, there must be enough collisions to maintain a consistent population for
the upper level of the transition. For example, for the J = 1 ! 0 transition of CO at
Tkin = 10 K we have A10 = 7.5 ! 10 "8 s "1 . Equating the collision rate given by equation
(5.4) to this coefficient yields nH2 ! 7.5 " 10 3 cm #3 . A more careful evaluation of the
collision rate for these conditions would yield the more correct value of
nH2 = 3 ! 10 3 cm "3 .

Figure 5.3 – Pure rotational transitions of carbon monoxide within the ground vibrational
state ! = 0 .

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As the density of the gas increases beyond the critical density for the J = 1 ! 0 , other
levels lying at higher energies will also become significantly populated as long as the
density reaches their associated critical densities. The populations of the different levels
are then given by Boltzmann’s relation

Nj N tot ! E j
=
kBTex
e , (5.55)
gj U (Tex )

where N j is the number of molecules in the state of energy E j and degeneracy g j , N tot
is the total number of molecules, and U (Tex ) is the partition function at the excitation
temperature Tex . When the levels are all populated through collisions as is assumed here,
then Tex = Tkin and the molecules are in local thermodynamic equilibrium (LTE).

5.4.2 Vibrational transitions

In regions that are hot enough (e.g., in the surroundings of young stars) vibrational
transitions such as those discussed in Section 5.2.4 can arise, and will generally be
accompanied with rotational transitions (hence rovibrational transitions). For these it is
possible that !J = 0, ±1 . Depending on the value of !J , we have the P-branch
( !J = "1 ), the Q-branch ( !J = 0 ), and the R-branch ( !J = 1 ).

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