Formulae of molecules and atoms (radio spectra)
Meaning of quantum numbers and related symbols
(Most contents from NIST diatomic spectral database documents)
I or Ii – Angular momentum quantum number of nuclear spin for one (or ith) nucleus
S – Resultant angular momentum quantum number of electron spins
Σ – Projection of S on the molecular axis (for Hund’s case a only)
Λ – Absolute value of the projection of the resultant orbital electronic angular
momentum on the molecular axis (for Hund’s case a and b only)
Ω – Absolute value of the projection of the total electronic angular momentum on the
molecular axis (= Σ+Λ)
N – Rotational angular momentum quantum number, excluding electronic and nuclear
spins
J – Resultant total angular momentum quantum number, excluding nuclear spins
mJ – Projection of J along any given axis
Fi – Resultant angular momentum quantum number including nuclear spin for i nuclei
Itot – Resultant angular momentum quantum number of two or more nuclear spins that
couple together before being coupled into total rotational angular momentum F.
F – Resultant total angular momentum quantum number
mF – Projection of F along any given axis
X – Quantum number employed when Fi’s are not good quantum numbers. This value
simply numbers the levels from lowest to highest energy for the same F quantum
number
v – Vibrational quantum number
νi – vibrational quantum numbers for the ith vibrational mode for molecules with
more than 2 nucei. Vibrational modes are usually sorted with increasing energy (with
decreasing degree of symmetry) and designated as ν1, ν2, ν3,… sequentially.
l – Quantum number for the projection of degenerate 2-dimensional bending vibration
angular momentum on molecular axis (for linear molecules only)
Σ, Π, Δ -- Electronic state designation for which Λ = 0, 1, 2, respectively. The
symbols X, A, B, E, F, … precede the state designation where X refers to the ground
state, A refers to the first excited electronic level, ...
1
General Formulas
Radiative transfer
The general radiative transfer equation for a ray is
dI
  I   ,
ds
where Iν is the radiation intensity in ergs-1cm-2Hz-1sr-1, s is the radiation transfer route
length, κν is the absorption coefficient (opacity) in cm-1, εν is the emission coefficient
(emissivity) in ergs-1cm-3Hz-1sr-1. It is convenient to define an optical depth τν as
d   ds
s0
or
    ds (with s  s0 ) .
s
Note that both τν and s are positive quantities, while the minus sign in the first
equation means the two quantities increase in opposite directions, with τν =0 at s0. The
radiation is going along increasing direction of s. Then, the radiation transfer equation
becomes
dI
 I  S ,
d 
with the source function
S   /  .
In thermodynamic equilibrium (TE), Iν= Sν= Bν(Tk), with Tk being the kinematical
temperature of gas (e.g., in the interior of a star). Here, the Plank function is
B T  
2h 3
1
2
h / kT
c e
1
For molecular gas not in TE but in local thermodynamic equilibrium (LTE), although
the radiation is not in equilibrium with the gas, we usually still can define an single
excitation temperature Tex for all energy levels of a molecule (according to Boltzmann
distribution, see below), so that Sν= Bν(Tex) can be fulfilled for all transitions. Even
when the molecular gas is not in LTE, this definition of Tex still can be applied, but Tex
may be different for different transitions of the same molecule.
The formal solution of the radiative transfer equation is
I s   I 0e  0   s  
 0 
S e

 

d  I 0e   B T  1  e 
   s 

s
In the second equation, we have replaced the optical depth difference τ(0)- τ(s) by a
new optical depth τ that is defined to be zero at s=0 and increases in the same
direction as s. If we subtract the back ground intensity Iν(0)= Bν(Tbg) from the
measured intensity Iν(s), we get
2



I s   I s   I 0  B(Tex )  B(Tbg ) 1  e  ,
or in terms of brightness temperature



TR  J (Tex )  J (Tbg ) 1  e  .
Here, TR is the brightness temperature directly measured by radio telescope using the
beam switch method (so that the back ground brightness has been subtracted). J(T) is
Relaigh-Jeans temperature:
h
1
2

B T 
k e h / kT  1 2k
Einstain coefficients Aul, Bul and Blu
The three Einstein coefficients that describe the interaction between molecule and
J T  
radiation are the spontaneous transition probability Aul in s-1, the induced emission or
absorption probability Bul and Bul in s-1erg-1cm2sr. For gaseous molecule with
radiative transitions between an upper level “u” and a lower level “l”, the three
coefficients are defined by
Aul 
8 2 2 e 2 g l
2h 3
f
,
A

B
, g u Bul  g l Blu
lu
ul
ul
me c 3 g u
c2
Here, flu is oscillator strength, e is electron charge, me is electron mass, c is the speed
of light, ν is the transition frequency from upper, gu, gl are statistic weights. Therefore,
for gas number density nu and nl in the two levels, the spontaneous photon emission
rate is nuAul (s-1) while the induced photon emission or absorption rates in a radiation
field with mean radiation intensity of J(νul) (erg/s/Hz/cm2/sr) are nuBulJ(νul) and
nlBluJ(νul) (s-1), respectively. (NOTE1: The definition of Bul and Blu may be slightly
different by using radiation energy density U(νul) instead of the mean intensity J(νul)
according to the relationship U  ul   4J  ul  / c .) (NOTE2: If the levels are
degenerate, the above emission rates are only for transition from one initial sublevel
to all final sublevels. Therefore, to calculate the total emission rates, one must
multiply them by the statistic weight of the initial level.)
The Einstein coefficient Aul is related to the electric or magnetic dipole moment μul
of the transition as
Aul 
64 4 3
2
 ul  ul .
3
3hc
For rotational transitions, the dipole moment is usually expressed as
 ul
2
 x
x
2
S ul
,
gu
in which μx is the permanent dipole moment along axis x (the symmetry axis of a
molecule along which the permanent dipole moment is non-zero), xSul is the so called
3
line strength or transition strength for the transition from level “u” to “l”. The line
strength is only determined by the direction part of the molecular wave function and
so is only related to the symmetry property of the molecule. Especially, the line
strength has no relation to molecular weight, size, composition etc. The formulas of
the line strength for molecules with different symmetry types are given in detail in
latter sections.
Absorption coefficients in terms of Einstein coefficients
Absorption coefficient per unit volumn of molecular gas is
h
nl Blu  nu Bul   
 
4
h

nl Blu 1  e  h / kTex   
.
4
g
c2

n u Aul 1  e  h / kTex   
2 l
8
gl



c2




nu Aul e h / kTex  1   
8
From here on, the subscript “ul” of frequency has been omitted for clarity.
Substituting Aul into this formula, one gets
 
2
8 3
8 3 nu 2 x
2
nu ul e h / kTex  1    
 x  Sul e h / kTex  1   
3hc
3hc g u




Boltzmann distribution
In local thermal equilibrium (LTE), the molecular energy level population distribution
nj obeys the Boltzmann distribution law
nj
gj

N  E j / kTk
e
,
Q
in which N is the total number density, Tk is the kinematical temperature, Ej is the
energy of the level j, and Q is the partition function which is defined as
Q   g je
 E j / kTk
.
j
One must be careful about the calculation of Q when fine structure or hyperfine
structure lines are considered. For those molecules with non-zero electron and/or
nuclear spins, the (hyper-) fine-structure splitting increases the total number of levels
and makes the Q value much larger. Therefore, with the same number density N, the
population nj of each (hyper-) fine level becomes smaller, and the individual (hyper-)
fine line intensity also becomes weaker. This consideration is also valid for those
asymmetric top molecules that have many energy levels due to the lack of symmetry.
Relative intensity of optically thin rotational lines
In the case of optically thin case, we can omit the opacity effect and use the
4
spontaneous photon emission rate to roughly estimate the relative intensity of
different rotational lines. For a homogeneous and optically thin molecular cloud, we
can omit the opacity effect so that the integrated emission line intensity just equals to
64 3 3  x x Sul N  Eu / kT
h
c


e
 I dV   4 nu Aul   d 
3c 2
Q
Here, the formulas of the Einstein A coefficient and the boltzmann distribution law
2
have been applied. With the formulas for pure rotational transitions given in latter
sections: E j  hBJ J  1 and J  J 1  2BJ , one has
2x
I int
The
512 3 B 3  x S J , J 1 N 3 hBJ  J 1/ kT
  I dV 
J e
3c 2
Q
strongest
line
occurs
at
J  1 / 16  3kT / 2hB  1 / 4 
0.0625  31.255  T (K) / B(GHz )  0.25 . Take the example of CO in the ground
vibration state, B=57635.968 MHz from CDMS database. Therefore, the strongest
optically thin is at J  0.0625  0.542  T (K)  0.25 . For a temperature of T = 9.375
K, the strongest line is at around J=2. For T = 300 K, the strongest line is around
J=12.5. These results qualitatively agree with the line intensity plots from CDMS
database (see in the figures below).
028503 CO, v=0
028503 CO, v=0
Rotation diagram analysis
Although further extension of above discussions of relative intensity of optically thin
lines just results in the formula for rotation diagram analysis (See for example the
derivation by Cummins et al., 1986, ApJS, 60, 819), here I would like to restart from
the formal solution of radiative transfer equation to derive it. From the formal solution
of RT equation, the observed radio line intensity after the removal of back ground
emission is


TR  Tex  Tbg  1  e .
Here, Rayleigh-Jeans approximation has been applied to Tex and Tbg. In the optically
5
thin case,
TR  Tex  Tbg 
From above discussions of absorption coefficient, one has the optical thickness of a
homogeneous cloud as
n
8 3 2 x
    dl 
 x  Sul e h / kTex  1    u dl
3hc
gu
0
0
.
2
3
x
8  x  S ul N
h / kTex
 Eu / kTex

   e
1 e
3hc
Q

L


L

Here, L is the size of the cloud, N is the total column number density of the molecules
and Q is the partition function. Substituting it into previous formula and integrate over
frequency, one gets the integrated line temperature
8 3  x  x Sul N h / kTex
e
 1 e  Eu / kTex
3hc
Q
By using the relation d /  dV / c and applying Rayleigh-Jeans approximation, we
have the integrated line intensity
 TR d  Tex  Tbg 
W   TR dV 

2

2
8 3 x x Sul Tex  Tbg N  Eu / kTex
e
.
3k
Tex Q
Then we obtain the standard formula for rotation analysis:


 Tex  Tbg N  Eu / k
3kW
 
lg  3 2 x   lg 
lg e
 Tex Q  Tex
 8  x Sul 
(See the similar derivation by Kawaguchi et al., 1995, PASJ, 47, 853) Note that this
formula is applicable only for energy levels (in unit of K) far lower than kinematical
temperature of the gas. Otherwise, non-LTE effects may emerge.
Energy level balance equations (Rate equations)
In above discussions, TE or LTE has been assumed. However, the actual molecular
gas may not fulfill that condition. For example, when external radiation field plays an
important role in the excitation of energy levels, LTE is usually not met. In this case,
one needs to compute the level populations by the level balance equations:


ni   Aij   Bij Jij  Cij  
j ( i )
 j ( i )

n A
j ( i )
j
ji

 n B
j ( i )
j
J  C ji 
ji ji
Here, Aij and Bij are Einstein constants, Cij is collision rate in unit of s-1, Jνij is the
mean radiation intensity in unit of Wm-2Hz-1Sr-1 that must be determined by solving
radiative transfer equation. This set of equation must be closed by the equation of
particle conservation
6
n
i
N
Here N is total number density of the species. Once the mean radiation intensity is
known, these linear equations of level population can be easily solved by standard
method.
Format of molecular energy level specification
Similar as atomic energy level formulas, molecular formula also take the form


X 2 S 1  ,  i , v, N , J , , , F1 , F2 ,..., I tot , F  1

Here the terms before the parenthesis is electronic state specification. X means it is in
the ground electronic state (n=0). X may be replaced by A, B, C, … for excited
electronic states with n=1, 2, 3, … However, molecules in electronically excited states
are rare, because the huge excitation energy usually will get the molecule dissociated.
In the second electronic term, Λ represents projection of resultant electronic orbital
angular momentum along the major symmetry axis of the molecule. Upper case Greek
letters Σ, Π, Δ, … are used for |Λ| = 0, 1, 2, … respectively. However, for strongly
asymmetric molecules for which there is no obvious symmetry axis, the Greek letters
may be replace by upper case alphabets A, B, E, F, … to indicate the different
symmetry properties. S is the total electron spin quantum, hence 2S+1 is also called
multiplicity of each electronic level. Ω represents the resultant projection of total
electronic angular momentum along the major symmetry axis of the molecule (mainly
composed of projection of electronic spin Σ and that orbital angular momentum Λ, but
in the case of bending linear molecules, it also include the projection of the 3D
bending angular momentum l).
The quantum numbers (Qn) in the parenthesis are related to vibration and
rotation of the molecule. Those Qns not related to spin are usually called primary Qns,
while those involving spins are called secondary Qns. The Qn v is for vibrational
states of diatomic molecules, while νil is used for linear chains where i = 1, 2, 3, … for
different vibrational frequency modes ν1, ν2, ν3, …. The superscript l is for the
projection of the 3D vibrational angular momentum along the symmetry axis. N is the
Qn for rotational angular momentum, J is the Qn for resultant total angular
momentum (including electronic components), Ω is the Qn for resultant total angular
momentum in the Hund’s case (a) coupling, Λ is the Qn for projection of resultant
total angular momentum along symmetry axis. The rest are secondary Qns: F1,
F2, …Fi, … are for resultant angular momentum when the ith non-zero nuclear spin is
taken into accound; Itot is the resultant of nuclear spins of two identical nuclei that
usually couple first before being coupled to the total angular momentum; F is for the
ultimate total angular momentum (the only quantity that strictly conserves).
7
Atoms
(Most contents from chapter 5 of Townes C.H. and Schawlow A.L., Microwave
Spectroscopy, New York: McGraw-Hill, 1955)
Quantum numbers
Usually, only hyperfine structure transitions of atoms appear in millimeter wavelength.
Quantum numbers for an atomic level with nuclear spin are L, J, mJ, I, F, mF.
Energy level
There are mainly two kinds of hyperfine structure splitting due to electric quadrupole
momentum and magnetic dipole momentum respectively. The formula of quadrupole
hyperfine structure level is
eq J Q
1
34 cc  1  I I  1J  1
WQ 
2 I 2 I  12 J  1
With
c  F F  1  I I  1  J J  1
Here, eqJQ is the quadrupole coupling constant.
The formula of magnetic dipole hyperfine structure level is
W
a
F F  1  I I  1  J J  1
2
Here, a is the so-called interval factor, a constant for a given electronic state and
nucleus.
With external magnetic field, ……(Zeeman effects)
Statistic weight
For the pure hyperfine structure level: SF = 2F+1
Selection rules
Only pure hyperfine transitions fall in microwave. The selection rule for these pure
hyperfine transitions is
L  0; J  0; mJ  0; F  1; mF  0,1
Magnetic dipole moment for the transitions
The pure hyperfine transitions in microwave are usually only strong in the ground
electronic state 2S1/2 of atoms, because most atoms in space are populated in this
ground level and the hyperfine structure splitting is the largest in this state. For pure
hyperfine transitions in the 2S1/2 state with external magnetic filed, the magnetic
8
dipole moment for transition F  I  12  F  I  12 is:


 2 4 I  12 2  m F 2
,
for m F  0
 0
2

2


2
I

1
 ij  
3
1
 2 2I  2  m F I  2  m F  , for m  1
F
 0
2 I  12
Here, mF is the projection of total angular momentum F along the direction of external
magnetic filed B, μ0 = he/4πmc is the Bohr magneton (the statistic weight for each
non-degenerate magnetic level is 1). Note that the transitions with mF  0 are
always polarized with their magnetic vector parallel to magnetic filed B, while
transitions with mF  1 are always polarized with their magnetic vector
perpendicular to magnetic filed B.) The line strength in this case is


 4 I  12 2  mF 2
,
for mF  0

2



2
I

1
S 
3
1
 2I  2  mF I  2  mF  , for mF  1

2 I  12
For pure hyperfine transitions without external magnetic field, the magnetic dipole
moment averaged over all transitions with the same ΔmF and summed over 3
directions for absorption transition F  I  12  F  I  12 is
 ij
2
av
 0
2
4I  1
2I  1
The line strength in this case is
8 I I  1
2I  1
Inversely, the magnetic dipole moment for emission transition F  I  12  F  I  12
S
should be
ij
2
av
 0
2
S
4I
2
 0
,
gF
2I  1
Here, gF = 2F+1 = 2(I+1) is the statistic weight of each level.
Miscellaneous
ss  1 2h

Angular momentum of an electron:

Magnetic moment of an electron: 1   / 2  0 (μ0 is the Bohr magneton,

  2e 2 / hc  1 / 137 is the fine structure constant.)
Angular momentum of a nucleus: I 2h

The most famous example of pure hyperfine structure transition is the 21cm H
emission line (1.420405 GHz) from the interstellar space of the Galaxy.
9
Diatomic molecules
(Most contents from chapter 1 of Townes C.H. and Schawlow A.L., Microwave
Spectroscopy, New York: McGraw-Hill, 1955)
Quantum numbers (my opinion)
For simple diatomic molecules: J(=N), v (e.g., X1Σ state molecules)
with only fine structure splitting: N, v, J (e.g., X3Σ state molecules)
with only hyperfine structure splitting: J(=N), v, F1, F
with both fine and hyperfine structure splitting: N, v, J, F1, F
for Hund’s case a coupling (L>0,S>0): Ω, Λ, J (e.g., X2Π state molecules)
for Hund’s case a with hyperfine structure: Ω, Λ, J, F1, F (e.g., OH)
Here, N, v, Ω, Λ are called principal quantum numbers, J, F1, F are called spin
involved quantum numbers. For molecules with identical nuclei, the sign of Λ should
reflect parity. F1 is needed only when both nuclei have non-zero spin. In the case
where J=N, the convention is to use J and omit N.
Energy level
For a rigid rotor with neither electronic nor nuclear spin, level energy
W
h

J  J  1  BJ J  1 ,
h 8 2 I
where h is Plank’s constant, I is the momentum of inertia, B is rotation constant, J is
the rotational quantum number.


For a non-rigid rotor, if we adopt Morse potential U r   D 1  e a r re  (where D
2
is the dissociation energy of the molecule, re is equilibrium distance between nuclei, a
is a constant) for low vibration states with r ~ re, the level energy containing
centrifugal stretching is
W J ,v
  e v  12    e e v  12   Be J J  1  De J 2 J  1   e v  12 J J  1
h
Here, J and v are rotation and vibration quantum numbers, subscript e means
2
2
quantities when atoms are in equilibrium position, and
a
e 
2
he
4B
h
h3
, e 
, Be  2 , De 
 e2
2 6
6 3

4D
8 I e
128  e re
e
3
2D
10
e 
 1
1


2
2
16 2 re D  are a 2 re
3h 2 e

 B 3 6B 2
6 e e  e

e
e

Here, μ=m1m2/(m1+m2) is the reduced mass the two nuclei m1 and m2.
A more appropriate solution for any potential function that can be expanded as a
series of power of (r-re) in the neighborhood of the potential minimum is the
Dunham’s solution:
Wv , J
l
j
 Fv , J   Ylj v  12  J j J  1
h
l, j
Here, l and j are summation indices, Ylj are coefficients depending on molecular
constants. In the case of Be/ωe<<1 (say, for molecules not very light, e.g., the
molecule H2 is a bad case), one has the approximate relations:
Y10   e   1 / 2 , Y20   e  e   1 , Y30   e y e   3 / 2 , Y40   e z e   2 ,
Y01  Be   1 , Y11   e   3 / 2 , Y21   e    2 ,
Y02   De    2 , Y12   e   5 / 2 ,
Y03  H e   3
(Note: Part of the scaling relation is from the documents of NIST diatomic spectral
database.) The above scaling relations with the reduced mass μ of the molecule are
very useful to approximately calculate the energy levels of rare isotopic diatomic
molecules from their main isotope, because for many molecules, the only significant
difference among isotopomers is nothing but the reduced mass.
When electronic orbital angular momentum and/or spin exist, the energy level
structure is determined by the angular momentum coupling schemes. There are four
typical coupling schemes: the Hund’s cases (a,b,c,d) (see Appendix B for details). The
most useful one is the Hund’s case (b) coupling, the case (a) occurs sometime, while
the cases (c) and (d) are very rare.
For Hund’s case (a) coupling, the energy level is


W / h  Bv J J  1   2  A 2
Here Bv is the rotation constant corresponding to vibrational state v, A is also a
rotation constant that is usually much larger than Bv. Ω is the quantum number for the
projection of J along the molecular axis.
For Hund’s case (b) coupling, the energy level is
….
For Hund’s case (c) coupling is identical to that of case (a). For Hund’s case (d), the
energy level is
W O  / h  BvOO  1
11
For fine and hyper fine splitting, the energy levels are… (????)
What is the role of parity in the energy level formula? (For example, the energy levels
of SiC are different for different parities.)
With external magnetic field, ……(Zeeman effects)
Statistic weight
For pure rotational levels: gJ = 2J+1
The only angular momentum not included in the quantum number J is nuclear spin.
For rotational levels with non-zero nuclear spins (hyperfine structure splitting), the
statistic weight for a pure hyperfine splitting level is
g F  2F  1 ,
while that for a pure rotational level (not differentiating hyperfine components) is:
g I , J   2 F  1  2 I  12 J  1 .
F
It’s not difficult to verify the second equality using the quantum angular momentum
coupling rule: F  J  I ,, J  I .
Selection rules
Selection rules for electric dipole transitions in the absence of external field: (from
NIST diatomic spectra database documents)
For X1Σ molecules: J  1; F  0,1; v  0,1,2,...
For X3Σ molecules: N  1; J  0,1; F  0,1 (e.g., O2, SO)
Actually N  3 is also allowed for intermediate coupling case,
for magnetic dipole transitions: N  0,2; J  0,1
2
For X Π molecules:   0; J  0,1;  -; F  0,1 (Hund’s case a, e.g., NO)
Electric dipole moment for transitions
For the case of L=0, S=0 (say, 1Σ state),
ij 2
 2 J 1
  d 2 J  1 , for J  J  1

J 1
 d 2
, for J  1  J
2J  3

Here, μd is the permanent electric dipole moment of the molecule. The line strength
thus is
S  J  1  J up
For the case of L>0 (nonzero electronic orbit momentum), the molecule can be treated
in analogue of symmetric top, so that the rotational line strength for Hund’s case (a)
12
coupling is


S  J up  2 / J up ,
2
in which Λ is the projection of L along the molecular axis. For the Hund’s case (b), the
line strength is


….??????? S  J up  2 / J up ??????
2
In the case of fine structure splitting (S>0), the total of line strength of all fine
structure lines just equals to the rotational line strength given by above formulas with
the quantum number J replaced by N, while the relative intensity (RI) of fine structure
lines should be determined by the quantum angular momentum coupling formula (see
details on this in Appendix A). Eventually, the line strength for a single fine structure
transition is simply S(N,J) = S(N) * RI.
13
Linear molecules
(Most contents from chapter 2 of Townes C.H. and Schawlow A.L., Microwave
Spectroscopy, New York: McGraw-Hill, 1955)
Quantum numbers
For the case with L=0,S=0,Ii=0: J, ν1,…, l (J=N)
with L=0,S>0,Ii=0: N, ν1,…, l, J
with L=0,S=0,Ii>0: J, ν1,…, l, F1,…, Itot, F (J=N)
with L>0,S>0,Ii>0: N, ν1,…, l, J, F1,…, Itot, F (Hund’s case b)
with L>0,S>0,Ii=0: Ω, Λ, J (Hund’s case a, e.g., X2Π state molecules)
with L>0,S>0,Ii=0: Ω, Λ, J, F1,…, Itot, F (Hund’s case a)


Here, N, ν1,…, l, Ω, Λ are called principal quantum numbers, J, F1,…, Itot, F are
called spin involved quantum numbers. For molecules with identical nuclei, the
sign of Λ may reflect parity. F1,… are needed only when any nuclei have
non-zero spin. Itot is needed when two spins couple with each other before being
coupled to the total angular momentum.
There are two ways to specify the vibration quantum numbers ν1,… values: 1)
list all ν1l values, with non-zero l as superscript: ν1, ν2l, ν3,… (suitable for small
molecules, e.g., 0110 state of CO2); 2) only specify the non-zero ν1l values
(suitable for large molecules, e.g., ν7=11,N=23,J=47/2 of C4H).
Energy level
With centrifugal stretching effects, energy level of a linear molecule is:
d 
W

 BJ  J  1 with B  Be    i  vi  i   DJ J  1,
h
2

i
where αi is rotational constants that reflect the change of B from its equilibrium value
Be due to ith vibration mode. di is the degeneracy of the ith vibration mode, or the
number of degenerate modes with the same αi, D is the centrifugal stretching
coefficient.
For a degenerate excited bending mode, with the l-doubling considered, the energy
level for the case of |l|=1 is
q 
W

  k v  1   Bv  l  J J  1  Bv l 2  Dv J J  1  l 2
h
2


with
14

d 

Bv  Be    i  vi  i 
2

i
Here, νk is the zero energy of the ground level of the degenerate vibration mode, l is
the bending vibration angular momentum quantum number, ql is the l-doubling
coefficient. For higher excited bending modes with |l|>1, the l-doubling is usually
very small (not important for mm wavelength).
With external magnetic field, ……(Zeeman effects)
Statistic weight
For pure rotational levels: gJ = 2J+1
As long as the molecule is not in the degenerate excited bending mode, the only
angular momentum not included in the quantum number J is nuclear spin. For
rotational levels with non-zero nuclear spins (hyperfine structure splitting), the spin
statistics is (2I+1). Therefore, the statistic weight of a hyperfine is: gF = (2I+1)(2J+1)
=2F+1.
Selection rules
Selection rules for electric dipole transitions in the absence of external field:
J  0,1; l  0 with J  0 while J  0 or l  0 not allowed 
Electric dipole moment for transitions
 2 J  12  l 2
 J , l  J  1, l1
, for  1
 d

J  12 J  1
 J , l 2  J  1, l 2
 ij 2  
2
l
  2
, for J , l1  J , l 2
d

J  12 J  1
Here, μd is the permanent dipole moment of the molecule and l1 and l2 represent the
two l-doubled sublevels with lower and higher energy, respectively. One can easily
see that for ground vibrational level with l=0, the formula is identical to that of the
diatomic molecule. The line strength thus is
 2 2
 J , l1  J  1, l1
 J up  l / J up , for 
S 
 J , l2  J  1, l2
l 2 / J  1,
for J , l1  J , l2



One can see the similarity of above formula with the dipole moment of symmetric top.
Actually the excited bending mode of linear molecule can be considered as an
asymmetric top with slight asymmetry. The quantum numbers of the excited bending
mode can be analogously written in the form of asymmetrical top as:
For the higher l-doubling level: J,l => J|l|,J+1-|l|, and
For the lower l-doubling level: J,l => J|l|,J-|l|
15
Then, in the slight asymmetry limit, the dipole formula is the same as that of a
symmetric top.
16
Symmetric top
(Most contents from chapter 3 of Townes C.H. and Schawlow A.L., Microwave
Spectroscopy, New York: McGraw-Hill, 1955)
Energy level
For a rigid symmetric top, the energy level is
W J , K 
 BJ J  1  C  B K 2
h
Here C is the rotation constant along the symmetry axis, B perpendicular to the axis.
With centrifugal stretching considered, the energy level is
W J , K 
2
 BJ  J  1  C  B K 2  D J J 2  J  1  D JK J  J  1K 2  DK K 4
h
For a rigid symmetric top with l-type doubling, the energy level is
W J , K 
 BJ  J  1  C  B K 2  2 KCl
h
Here, |ς| ≤ 1 is a coefficient describing how well the bending motion is perpendicular
to the molecular axis C, with a value of 1 for the entirely perpendicular case.
With external magnetic field, ……(Zeeman effects)
Statistic weight
Usually, a symmetric top has at least 3-fold of symmetry (except the rear case of
accidental symmetric top made from linear molecules). The statistic weight of a pure
rotational energy level is gJ = 2J+1. With hyperfine structure splitting, the statistic
weight then becomes gF = 2F+1. For a molecular energy level with hyperfine structure
splitting but with the hyperfine structure transitions unresolved, the statistic weight of
such an unresolved rotational level is composed of two independent parts: pure
rotational levels weight gJ and the spin statistics weight S(J,K). Then, the total statistic
weight of the unresolved rotational level is the multiplication of the two parts:
g J , I , K  g J S I , K 
The spin statistic weight for any given rotational level J of a 3-fold symmetric top is
17
 13 2 I
1
 3 2 I
 1 2 I
3
S I , K   

 1 2 I
3


 14 I 2  4 I  3, for K  3n, K  0;
 14 I 2  4 I ,
for K  3n;
for K  0; J even lower inversion level or
 12 I  1I ,
J odd upper inversion level,
Fermi - Dirac statistics
for K  0; J odd lower inversion level or
 12 I  3I  1,
J even upper inversion leve,
Fermi - Dirac statistics
in which I is total nuclear spin quantum number, n is a non-negative integer. (A
nucleus with odd mass number usually has half integer spin and so obeys
Fermi-Dirac statistics, say, exchange of any pair of such identical fermion nuclei
results in the reverse of the sign of the molecular wave function. In this case, one says
the total wave function of the molecule is anti-symmetric w.r.t. the exchange of the
identical nuclei. Oppositely, a nucleus with even mass number usually has integer spin
and so obeys Bose-Einstein statistics, say, exchange of any pair of such identical
boson nuclei will result in completely the same molecular wave function. One says
that the total wave function of the molecule is symmetric w.r.t. the exchange of
identical nuclei.)
Similarly, for a 4-fold symmetric top molecule, the spin statistic weight is


I  12 I  1 2 I 2  I  1 ,


 I 2 I  1 2 I 2  3I  2 ,


2



S I , K   I I  12 I  1
 12 I  12 I  1 2 I 2  3I  2 ,
1
 2 I I  12 I  12 I  1,
 1 I 2 I  1 2 I 2  I  1 ,
2
 12 I I  12 I  12 I  3,






for K  4n, K  0, Bose - Einstein statistics or
K  4n  2, Fermi - Dirac statistics ;
for K  4n, K  0, Fermi - Dirac statistics or
K  4n  2, Bose - Einstein statistics ;
for K  4n  1;
for K  0; J even , Bose - Einstein statistics ;
for K  0; J odd, Bose - Einstein statistics ;
for K  0; J even, Fermi - Dirac statistics ;
for K  0; J odd, Fermi - Dirac statistics .
Selection rules
Selection rules for electric dipole transitions in the absence of external field:
J  0,1; K  0;  Electric dipole moment for transitions
18
 2  J  12  K 2
, for J , K  J  1, K; (  -)
 d



J

1
2
J

1

2
 2 K
2

,
for J , K  J , K ; (  -)
ij   d


J
J

1

 2 J2 K2
,
for J , K  J  1, K; (  -)
 d
J 2 J  1

Here, μd is the permanent dipole moment of the molecule. The J,K – J,K transition is
the inversion transition of molecules like NH3. One can easily see that for the limit
case of K=0, the formula is identical to that of the diatomic molecule. The line
strength thus is
 J up 2  K 2
, for J , K  J  1, K; (  -)

 J up
?????
S  2
 K 2 J  1 , for J , K  J , K ; (  -)
 J J  1
One can easily see that the line strength is identical for both emission and absorption.
When (hyper-)fine structure splitting exists, the sum of line strength of all (hyper-)fine
lines should be just equal to the rotational line strength calculated by above formulas.
The relative intensity (RI) of each (hyper-)fine lines should be computed according to
quantum angular momentum coupling formulation (see more details in Appendix A).
19
Asymmetric top
(Most contents from chapter 4 of Townes C.H. and Schawlow A.L., Microwave
Spectroscopy, New York: McGraw-Hill, 1955)
Energy level
For a rigid asymmetric top that is merely slightly asymmetric, the level energy is
For prolate top:
For oblate top:
B  C  
W B  C 


J  J  1   A 
 p
h
2
2 

 A  B  
W A  B


J J  1   C 
 o
h
2
2 

in which A, B and C are rotation constants along the three molecular axes and the
convention of A>B>C is assumed, and
 p  K 2  c1b p  c2 b p 2  c3b p 3  
o  K 2  c1bo  c2 bo 2  c3bo 3  
Here,
B C
B  C   2 A
B A
bo    1 /   3 
B  A  2C
b p    1 /   3 
are asymmetry parameters for prolate and oblate tops, respectively, with

2B   A  C 
, for A  B  C
AC
being King’s asymmetry parameter.
For a highly asymmetric rigid top, the energy level is
A  C  E
W A  C 

J  J  1 
K 1 , K1
h
2
2
with E K 1 ,K1 only a function of κ. A list of E K 1 ,K1 can be found in G. Erlandsson,
(1954, Arkiv For Fysik, 8, 189). There is a useful relationship: Emn     Enm    .
For a non-rigid asymmetric top, the energy level including centrifugal stretching
effects is
W W0
2

 A1W0  A2W0 J  J  1 
h
h
2
2
4
2
A3 J 2  J  1  A4 J  J  1 Pz av  A5 Pz av  A6W0 Pz av
 
 
 
Here, Ai’s are constants, W0 is the rigid asymmetric top level energy given above,
P 
2
z
av
 W0 / 1 / I zz 
20
with Izz being the moment of inertia matrix element. In the limit cases of symmetric
top or near symmetric top, one has
P 
2
z
av
 K 2 and
P 
4
z
av
 K4
Statistic weight
For pure rotational levels gJ = 2J+1. If hyperfine structure splitting exists, each
hyperfine structure level has statistic weight of gF = 2F+1.
Selection rules
Selection rules for electric dipole transitions in the absence of external field:
J  0,1
Electric dipole moment for transitions
x
ij 2   x 2
S J k 1,k 2 J k 3 k 4  
2 J  1
, for J k1,k 2  J k 3,k 4
Here, μx is the permanent dipole moment of the molecule while x is the axis along
which the dipole moment is non-zero and so is responsible for allowed transitions.
The calculation of the term
x
S J k 1, k 2 J k 3 k 4   is complex. A table of the values of this
term is given for different x, J, K1, K2, κ and transition types P, Q, and R in the
Appendix V of the book by Townes and Schawlow (1955). The transition types P, Q
and R are corresponding to ΔJ = -1, 0, and 1, respectively. Here are several useful
relations:
x
x
S J K 1K 2 J K 3 K 4    x S J K 3 K 4 J K 1K 2  
S J K 1K 2 J K 3 K 4    x ' S J K 4 K 3 J K 2 K 1   
Here x and x’ are the ones with the largest and smallest rotation constant (or
momentum of inertia) among the three molecular axes. So the second formula means
the change from prolate top base functions to oblate top ones, or vise versa.
For slightly symmetric tops (with parameter κ close to 1 or -1), the line strength is
very close to that of symmetric top by taking Ka as K. One of the good examples of
such slightly asymmetric top is SiCC and its isotopologues.
21
Appendix A
Relative intensity (RI) of (hyper-)fine structure lines
(from notes by Dr. Dinh-Van-Trung)
The relative intensity of fine or hyperfine structure lines is determined by the quantum
mechanic angular momentum coupling theory. First of all, let’s consider a pure
rotational transition from any level J to any level J’, the electric dipole moment of the
transition is

 J ' ||  1|| J
2
J ,J '
2
 J 0   J ' , J  1 .
2
M
Here, M is the projection quantum number of J along any axis. Therefore, only
transition from J to J’=J-1 has nonzero dipole (allowed). The dipole moment of a
transition from one degenerate sublevel of J to all degenerate sublevels of J’ is
2
J ,J ' 

2
J ,J '
M
2J 1

J
0 2 .
2J 1
Hence, line strength for a pure rotation transition is
S  J up
Now, let’s consider a Hund’s case (b) coupling of rotational angular momentum N,
electronic spin S and nuclear spin I. N couples with S to produce angular momentum J,
then J couples with I to produce F. This can be expressed using the Dirak bracket as
FIJSN ~ F  I  S  N .
Then, the transition dipole moment that is used in the formula of Einstein A
coefficient is
x 
2

F ' M F ' IJ ' SN '  x FM F IJSN
1
2
F 'M F ' x
2
J ' F ' I 
2
1
 2 F '1
 J ' SN '  x JSN
 F J 1
2
2
J ' F ' I  N ' J ' S 
1
 2 F '12 J  12 J '1
 
 N ' x N
 F J 1  J N 1 
2
2
J ' F ' I  N ' J ' S 
2
 2 F '12 J  12 J '1
 
 N 0
F
J
1
J
N
1

 

2
Thus the line strength S can be derived from the relationship:
 x   0 2 S / 2 J  1
2
Here the symbol {} is the so-called 6-j symbol. The constant “1” in the right bottom
22
corner means that we are calculating electric dipole transitions. The formulas to
calculate 6-j symbols are
b
c 
a

ss  1s  2a  1s  2a 
s

   1 

 2b  12b2b  12c  12c2c  1
1 c  1 b  1
b
c
a
s  2s  1s  2a s  2b s  2c  1 

   1 

 2b2b  12b  22c  12c2c  1
1 c  1 b
1/ 2
1/ 2
b
c 
a
s  s  2b  1s  2b s  2c  1s  2c  2  

   1 

 2b  12b  22b  32c  12c2c  1 
1 c  1 b  1
1/ 2
a b c 
2bb  1  cc  1  aa  1
s 1

   1
2b2b  12b  22c2c  12c  21/ 2
1 c b
where s = a+b+c. Following property of the 6-j symbol is important for its
implimentation: exchange of any two columns or exchange of two rows of any two
columns doesn’t change the value. There are more 6-j formulas for other types of
transitions (e.g., quadrupole transtions).
23
Appendix B
The four Hund’s cases for electronic angular momentum coupling
The angular momenta need to couple with each other because they exert torque on
each other, so that some of the momenta have fast precession around other momenta
or axes. The fast precession of momenta makes the momenta not conservative any
more. But the projection of the fast precession momenta along their precession axes
might be still conservative and so can be used to tag the energy levels of the molecule.
For a linear molecule with electronic orbital momenta li and spins si, there are four
types of most possible coupling cases: the Hund’s cases (a,b,c,d). In the four Hund’s
cases, all individual orbital momenta and spins first couple with each other to form a
total orbital momentum L and total spin S. Hund’s case (b) is the case found in most
astro-molecules. Hund’s case (a) is much rarer, while Hund’s case (c) and (d) are not
found yet.
Hund’s case (a) coupling: The coupling between the momenta L, S and the
molecular axis is strong. Their projections on the axis are Λ and Σ, respectively. Λ and
Σ add up to form Ω. Then Ω couples with couples with rotational angular momentum
R to form the resultant angular momentum J. The coupling flow chart:
(1) L+axis=>Λ, S+axis=>Σ
(2) Λ+Σ=>Ω
(3) Ω+R=>J
Hund’s case (b) coupling: In this most possible case, the coupling between the
momentum L and the molecular axis is still strong, but the coupling of S with the
molecular axis is weak. So the projection of L along the axis, Λ, couples with
rotational angular momentum R to form the resultant rotational angular momentum N.
N directly couple with S to form the resultant angular momentum J. The coupling
flow chart:
(1) L+axis=>Λ
(2) Λ+R=>N
(3) N+S=>J
Hund’s case (c) coupling: In this case, the direct coupling of momenta with
molecular axis is weak, L and S first directly couple to resultant electronic angular
momentum Jc, then Jc couple with axis with projection Ω. Ω further couples with R to
form the resultant angular momentum J. The coupling flow chart:
(1) L+S=>Jc
(2) Jc+axis=>Ω
(3) Ω+R=>J
24
Hund’s case (d) coupling: In this case, the coupling of momenta with molecular axis
is always very weak, so that L directly couples with R to form resultant angular N,
then N directly couples with S to form the resultant angular momentum J. The
coupling flow chart:
(1) L+R=>N
(2) N+S=>J
25