1. Ideal diatomic gas: internal degrees of freedom
Polyatomic species can store energy in a variety of ways:
translational motion
rotational motion
vibrational motion
electronic excitation
Each of these modes has its own manifold of energy states, how do we cope?

2. Internal modes: separability of energies
Assume molecular modes are separable
treat each mode independent of all others
i.e. translational independent of vibrational, rotational, electronic, etc, etc
Entirely true for translational modes
Vibrational modes are independent of:
rotational modes under the rigid rotor assumption
electronic modes under the Born-Oppenheimer approximation

3. Internal modes: separability of energies
Thus, a molecule that is moving at high speed is not forced to vibrate rapidly or rotate very fast.
An isolated molecule which has an excess of any one energy mode cannot divest itself of this surplus except at collision with another molecule.
The number of collisions needed to equilibrate modes varies from a few (ten or so) for rotation, to many (hundreds) for vibration.

4. Internal modes: separability of energies
Thus, the total energy of a molecule j:

5. Weak coupling: factorising the energy modes
Admits there is some energy interchange
in order to establish and maintain thermal equilibrium
But allows us to assess each energy mode as if it were the only form of energy present in the molecule
Molecular partition function can be formulated separately for each energy mode (degree of freedom)
Decide later how individual partition functions should be combined together to form the overall molecular partition function

6. Weak coupling: factorising the energy modes
Imagine an assembly of N particles that can store energy in just two weakly coupled modes a and w
Each mode has its own manifold of energy states and associated quantum numbers
A given particle can have:
a-mode energy associated with quantum number k
w-mode energy associated with quantum number r

7. Weak coupling: factorising the energy modes
The overall partition function, qtot:
expanding we would get:

8. Weak coupling: factorising the energy modes
but e(a+b) = ea.eb, therefore:
each term in every row has a common factor of
in the first row, in the second, and so on. Extracting these factors row by row:

9. Weak coupling: factorising the energy modes
the terms in parentheses in each row are identical and form the summation:

10. Weak coupling: factorising the energy modes
If energy modes are separable then we can factorise the partition function and write:

11. Factorising translational energy modes
Total translational energy of molecule j:
which allows us to write:

12. Factorising internal energy modes
Total translational energy of molecule j:
using identical arguments the canonical partition function can be expressed:
but how do we obtain the canonical from the molecular partition function Qtot from qtot? How does indistinguishability exert its influence?

13. Factorising internal energy modes
When are particles distinguishable (having distinct configurations, and when are they indistinguishable?
Localised particles (unique addresses) are always distinguishable
Particles that are not localised are indistinguishable
Swapping translational energy states between such particles does not create distinct new configurations
However, localisation within a molecule can also confer distinguishability

14. Factorising internal energy modes
When molecules i and j, each in distinct rotational and vibrational states, swap these internal states with each other a new configuration is created and both configurations have to be counted into the final sum of states for the whole system. By being identified specifically with individual molecules, the internal states are recognised as being intrinsically distinguishable.
Translational states are intrinsically indistinguishable.

15. Canonical partition function, Q
and thus:
This conclusion assumes weak coupling. If particles enjoy strong coupling (e.g. in liquids and solutions) the argument becomes very complicated!

16. Ideal diatomic gas: Rotational partition function
Assume rigid rotor for which we can write successive rotational energy levels, eJ, in terms of the rotational quantum number, J.
where I is the moment of inertia of the molecule, m is the reduced mass, and B the rotational constant.

17. Ideal diatomic gas: Rotational partition function
Another expression results from using the characteristic rotational temperature, qr,
1st energy increment = 2kqr
2nd energy increment = 4kqr

18. Ideal diatomic gas: Rotational partition function
Rotational energy levels are degenerate and each level has a degeneracy gJ = (2J+1). So:
If no atoms in the atom are too light (i.e. if the moment of inertia is not too small) and if the temperature is not too low (close to 0 K), allowing appreciable numbers of rotational states to be occupied, the rotational energy levels lie sufficiently close to one another to write:

19. Ideal diatomic gas: Rotational partition function
This equation works well for heteronuclear diatomic molecules.
For homonuclear diatomics this equation overcounts the rotational states by a factor of two.

20. Ideal diatomic gas: Rotational partition function
When a symmetrical linear molecule rotates through 180o it produces a configuration which is indistinguishable from the one from which it started.
all homonuclear diatomics
symmetrical linear molecules (e.g. CO2, C2H2)
Include all molecules using a symmetry factor s
s = 2 for homonuclear diatomics, s = 1 for heteronuclear diatomics
s = 2 for H2O, s = 3 for NH3, s = 12 for CH4 and C6H6

21. Rotational properties of molecules at 300 K
qr/K s T/qr qrot H CH
HCl HI N CO CO I

22. Rotational canonical partition function
relates the canonical partition function to the molecular partition function. Consequently, for the rotational canonical partition function we have:

23. Rotational Energy
this can differentiated wrt temperature, since the second term is a constant with no T dependence

24. Rotational heat capacity
this equation applies equally to all linear molecules which have only two degrees of freedom in rotation. Recast for one mole of substance and taking the T derivative yields the molar rotational heat capacity, Crot, m. Thus, when N = NA, the molar rotational energy is Urot,m

25. Rotational entropy
Srot is dependent on (reduced) mass (I = mr2), and there is also a constant in the final term, leading to:

26. Rotational entropy
Typically, qrot at room T is of the order of hundreds for diatomics such as CO and Cl2. Compare this with the almost immeasurably larger value that the translational partition function reaches.

27. Extension to polyatomic molecules
In the most general case, that of a non-linear polyatomic molecule, there are three independent moments of inertia.
Qrot must take account of these three moments
Achieved by recognising three independent characteristic rotational temperatures qr, x, qr, y, qr, z corresponding to the three principal moments of inertia Ix, Iy, Iz
With resulting partition function:

28. Conclusions
Rotational energy levels, although more widely spaced than translational energy levels, are still close enough at most temperatures to allow us to use the continuum approximation and to replace the summation of qrot with an integration.
Providing proper regard is then paid to rotational indistinguishability, by considering symmetry, rotational thermodynamic functions can be calculated.

29. Ideal diatomic gas: Vibrational partition function
Vibrational modes have energy level spacings that are larger by at least an order of magnitude than those in rotational modes, which in turn, are 25—30 orders of magnitude larger than translational modes.
cannot be simplified using the continuum approximation
do not undergo appreciable excitation at room Temp.
at 300 K Qvib ≈ 1 for light molecules

30. The diatomic SHO model
We start by modelling a diatomic molecule on a simple ball and spring basis with two atoms, mass m1 and m2, joined by a spring which has a force constant k.
The classical vibrational frequency, wosc, is given by:
There is a quantum restriction on the available energies:

31. The diatomic SHO model
The value is know as the zero point energy
Vibrational energy levels in diatomic molecules are always non-degenerate.
Degeneracy has to be considered for polyatomic species
Linear: 3N-5 normal modes of vibration
Non-linear: 3N-6 normal modes of vibration

32. Vibrational partition function, qvib
Set e0 = 0, the ground vibrational state as the reference zero for vibrational energy.
Measure all other energies relative to reference ignoring the zero-point energy.
in calculating values of some vibrational thermodynamic functions (e.g. the vibrational contribution to the internal energy, U) the sum of the individual zero-point energies of all normal modes present must be added

33. Vibrational partition function, qvib
The assumption (e0 = 0) allows us to write:
Under this assumption, qvib may be written as:
a simple geometric series which yields qvib in closed form:
where qvib = hw/k = characteristic vibrational temperature

34. Vibrational partition function, qvib
Unlike the situation for rotation, qvib, can be identified with an actual separation between quantised energy levels.
To a very good approximation, since the anharmonicity correction can be neglected for low quantum numbers, the characteristic temperature is characteristic of the gap between the lowest and first excited vibrational states, and with exactly twice the zero-point energy,

35. Ideal diatomic gas: Vibrational partition function
Vibrational energy level spacings are much larger than those for rotation, so typical vibrational temperatures in diatomic molecules are of the order of hundreds to thousands of kelvins rather than the tens of hundreds characteristic of rotation.
Species
qvib/K
qvib
300 K) H2
5987
1.000 HD
5226 D2
4307 N2
3352 CO
3084
Cl2
798
1.075 I2
307
1.556

36. Vibrational partition function, qvib
Light diatomic molecules have:
high force constants
low reduced masses
Thus:
vibrational frequencies (wosc) and characteristic vibrational temperatures (qvib) are high
just one vibrational state (the ground state) accessible at room T
the vibrational partition function qvib ≈ 1

37. Vibrational partition function, qvib
Heavy diatomic molecules have:
rather loose vibrations
Lower characteristic temperature
Thus:
appreciable vibrational excitation resulting in:
population of the first (and to a slight extent higher) excited vibrational energy state
qvib > 1

38. Vibrational partition function, qvib
Situation in polyatomic species is similar complicated only by the existence of 3N-5 or 3N-6 normal modes of vibration.
Some of these normal modes are degenerate
(1), (2), (3), … denoting individual normal modes 1, 2, 3, …etc.
Species
qvib/K
∏(qvib)
300 K)
CO2
3360
1.091
1890
954(2)
NH3
4880(2)
1.001
4780
2330(2)
1360
CHCl3
4330
2.650
1745(2)
1090(2)
938
523
374(2)

39. Vibrational partition function, qvib
As with diatomics, only the heavier species show values of qvib appreciably different from unity.
Typically, qvib is of the order of ~3000 K in many molecules. Consequently, at 300 K we have:
in contrast with qrot (≈ 10) and qtrs (≈ 1030)
For most molecules only the ground state is accessible for vibration

40. High T limiting behaviour of qvib
At high temperature the equation gives a linear dependence of qvib with temperature.
If we expand , we get:
High T limit

41. T dependence of vibrational partition function
As T increases, the linear dependence of qvib upon T becomes increasingly obvious

42. The canonical partition function, Qvib
Using we can find the first differential of lnQ with respect to temperature to give:

43. The vibrational energy, Uvib
This is not nearly as simple as:
linear molecules

44. The vibrational energy, Uvib
This does reduce to the simple form at equipartition (at very high temperatures) to:
(equipartition)
Normally, at room T:

45. The zero-point energy
So far we have chosen the zero-point energy (1/2hw) as the zero reference of our energy scale
Thus we must add 1/2hw to each term in the energy ladder
For each particle we must add this same amount
Thus, for N particles we must add U(0)vib, m = 1/2Nhw

46. Vibrational heat capacity, Cvib
The vibrational heat capacity can be found using:
The Einstein Equation
This equation can be written in a more compact form as:

47. Vibrational heat capacity, Cvib
FE with the argument qvib/T is the Einstein function
The Einstein function

48. The Einstein heat capacity
low T
High T

49. The Einstein function
The Einstein function has applications beyond normal modes of vibration in gas molecules.
It has an important place in the understanding of lattice vibrations on the thermal behaviour of solids
It is central to one of the earliest models for the heat capacity of solids

50. The vibrational entropy, Svib
We know and N = NA for one mole, thus:

51. Variation of vibrational entropy with reduced temperature

52. Electronic partition function
Characteristic electronic temperatures, qel, are of the order of several tens of thousands of kelvins.
Excited electronic states remain unpopulated unless the temperature reaches several thousands of kelvins.
Only the first (ground state) term of the electronic partition function need ever be considered at temperatures in the range from ambient to moderately high.

53. Electronic partition function
It is tempting to decide that qel will not be a significant factor. Once we assign e0 = 0, we might conclude that:
To do so would be unwise!
One must consider degeneracy of the ground electronic state.

54. Electronic partition function
The correct expression to use in place of the previous expression is of course:
Most molecules and stable ions have non-degenerate ground states.
A notable exception is molecular oxygen, O2, which has a ground state degeneracy of 3.

55. Electronic partition function
Atoms frequently have ground states that are degenerate.
Degeneracy of electronic states determined by the value of the total angular momentum quantum number, J.
Taking the symbol G as the general term in the Russell—Saunders spin-orbit coupling approximation, we denote the spectroscopic state of the ground state of an atom as:
spectroscopic atom ground state = (2S+1)GJ

56. Electronic partition function
spectroscopic atom ground state = (2S+1)GJ
where S is the total spin angular momentum quantum number which gives rise to the term multiplicity (2S+1). The degeneracy, g0, of the electronic ground states in atoms is related to J through:
g0 = 2J+1 (atoms)

57. Electronic partition function
For diatomic molecules the term symbols are made up in much the same way as for atoms.
Total orbital angular momentum about the inter-nuclear axis.
Determines the term symbol used for the molecule (S, P, D, etc. corresponding to S, P, D, etc. in atoms).
As with atoms, the term multiplicity (2S+1) is added as a superscript to denote the multiplicity of the molecular term.

58. Electronic partition function
In the case of molecules it is this term multiplicity that represents the degeneracy of the electronic state.
For diatomic molecules we have:
spectroscopic molecular ground state = (2S+1)G
for which the ground-state degeneracy is:
g0 = 2S + 1 (molecules)

59. Electronic partition function
Species
Term Symbol gn
qel/K Li
2S1/2
g0 = 2 C
3P0
g0 = 1 N
4S3/2
g0 = 4 O
3P2
g0 = 5 F
2P3/2
2P1/2
g1 = 2
590 NO
178 O2
3S-g
g0 = 3
1Dg
g1 = 1
11650

60. Electronic partition function
Where the energy gap between the ground and the first excited electronic state is large the electronic partition function simply takes the value g0.
When the ground-state to first excited state gap is not negligible compared with kT (qel/T is not very much less than unity) it is necessary to consider the first excited state.
The electronic partition function becomes:

61. Electronic partition function
For F atom at 1000 K we have:
For NO molecule at 1000 K we have: