1. Molecular Information Content
Béla Viskolcz
Department of Chemical Informatics
University of Szeged

2. Degrees of freedom
Degrees of freedom: the total number of independent variables whose values have to be specified for a complete description of the system.
One atom: In the context of molecular motion these are the spatial coordinates of all the particles. Since we need three coordinates to describe the position of an atom, we say the atom has three degrees of freedom.

3. One atom

4. Diatomic molecules
Diatomic molecule: If the atoms are not bound to one another, there will be no relation among the coordinates of the two atoms. On the other hand, when the two atoms are bound, the displacement of each other is coupled to the other. The result is to give three translational, one vibrational, and two rotational degrees of freedom for the molecule. The vibrational and rotational degrees of freedom are also referred to as internal degrees of freedom.

5. Polyatomic molecules Polyatomic molecule with N atoms:
Linear molecule: The 3N degrees of freedom of the atoms become three translational degrees of freedom and (3N-3) internal degrees of freedom. By analogy with diatomic molecules, we expect two rotational degrees of freedom for any linear polyatomic molecule (e.g. CO2 and C2H2). The remaining (3N-5) internal coordinates must correspond to vibrations.

6. Polyatomic molecules Polyatomic molecule with N atoms:
Bent molecule: A bent molecule loses a vibrational degree of freedom while gaining a rotational degree of freedom. Therefore, a nonlinear polyatomic molecule has three rotational degrees of freedom, hence (3N-6) vibrational degrees of freedom.

7. e.g. vibrational motions for CO2 and H2O:

8. Összefoglalás Degrees of freedom Atom Linear polyatomic Bent
Translation al 3
Vibrational
3N-5
3N-6
Rotational 2

9. Molecular description in space
Atom (origo)
Atom (two atoms in one line) DISTANCE (streching)
Atom (three atoms in one plane) ANGLE (bending)
Atom (four atoms in space – two plane angle dihedral angel (torsion)

10. Introduction Statistical Mechanics Properties of bulk fluid
Properties of individual molecules
Position
Molecular geometry
Intermolecular forces
Properties of bulk fluid
(macroscopic properties)
Pressure
Internal Energy
Heat Capacity
Entropy
Viscosity

11. The partition functions for 5 mode motions are expressed as

13. Potential V(R) for nuclear
motion in a diatomic molecule
Harmonic oscillator potential

14. Energy level

15. Zero-point energy
zero-point energy is the energy at ground state or the energy as the temperature is lowered to absolute zero.
Suppose some energy level of ground state is ε0, and the value of energy at level i is εi, the energy value of level i relative to ground state is
Taking the energy value at ground state as zero, we can denote the partition function as q0.

16. The vibrational energy at ground state is
Born- Oppenheimer
The vibrational energy at ground state is
therefore
the number of distribution in any levels does not depend on the selection of zero-point energy.

17. Since εt,0≈0, εr,0=0, at ordinary temperatures.

18. Translational partition function
Energy level for translation
The partition function

20. Rotational partition function
The rotational energy of a linear molecule is given by εr = J(J+1)h2/8π2I and each J level is 2J+1 degenerate.
define the characteristic rotational temperature

21. Θr<<T at ordinary temperature, The summation can be approximated by an integral
Let J(J+1)=x, hence J(2J+1)dJ=dx, then

22. Vibrational partition function
Vibrational energies for one dimensional oscillator are
Vibration is non-degenerate, g=1. The partition function is

23. take the ground energy level as zero,
For NO, the characteristic vibrational temperature is 2690K. At room temperature Θv/T is about 9; the , indicating that the vibration is almost in the ground state.

24. Thermodynamic energy and partition function

25. Thermodynamic energy and partition function
Substitute this equation into equation (8.48), we have

26. Substitute the factorization of partition function for q
Only qt is the function of volume, therefore

27. If the ground energy is specified to be zero, then

28. It tells us that the thermodynamic energy depends on the zero point energy. Nε0　is the total energy of system when all particles are localized in ground state. It (denoted as U0) can also be thought of as the energy of system at 0K. Then,

29. U0 can be expressed as the sum of different energies

30. The calculation of
(1) The calculation of

31. The calculation of
The degree of freedom of rotation for diatomic or linear molecules is 2, the contribution to the energy of every degree is also ½ RT for a mole substance.

32. The calculation of
Usually, Θv is far greater than T, the quantum effect of vibration is very obvious. When Θv/T>>1,
Showing that the vibration does not have contribution to thermodynamic energy relative to ground state.

33. If the temperature is very high or the Θv is very small, then Θv/T<<1, the exponential function can be expressed as

34. For monatomic gaseous molecules we do not need to consider the rotation and vibration, and the electronic and nuclear motions are supposed to be in their ground states. The molar thermodynamic energy is

35. For diatomic gaseous molecules vibration and rotation must be considered. If only lowest vibrational levels are occupied, the molar thermodynamic energy is

36. If all vibrational energies are equally accessible, the molar thermodynamic energy for vibration is
The molar thermodynamic energy for diatomic molecules is then

37. Heat capacity and partition function
The molar heat capacity, CV,m, can be derived from the partition function.
Replace q with
We can see from above equations that heat capacity does not depends on the selection of zero point of energy.

38. Electrons and nucleus are in ground state

39. The calculation of CV,t, CV,r and CV,v

40. (2) The calculation of CV,r
If the temperature is very low, only the lowest rotation state is occupied and then rotation does not contribute to the heat capacity.

41. The calculation of CV,v

42. Generally, Θv/T>>1, equation becomes
It shows that under general conditions, the contribution to heat capacity of vibration is approximately zero.

43. When temperature is high enough,

44. In gases, all three translational modes are active and their contribution to molar heat capacity is
The number of active rotational modes for most linear molecules at normal temperature is 2
In most cases, vibration has no contribution to the heat capacity,

45. Entropy and partition function
Entropy and microstate
Boltzmann formula
k = ×10-23 J K-1
As the temperature is lowered, the Ω, and hence the S of the system decreases. In the limit T→0, Ω=1, so lnΩ=0, because only one configuration is compatible with E=0. It follows that S→0 as T→0, which is compatible with the third law of thermodynamics.

46. For example
When N approaches infinity,

47. Entropy and partition function
For a non-localized system, the most probable distribution number is
Using Stirling equation ln N!=N ln N - N and Boltzmann distribution expression

48. We have,

49. For localized system
Entropy does not depend on the selection of zero point energy .

50. Factorizing the partition function into different modes of motions and using
We can give

51. For identical particle system, entropies for every mode of motion can be expressed as

52. Calculation of statistical entropy
At normal condition electronic and nuclear motions are in ground state, and in general physical and chemical process the contribution to the entropy by two modes of motion keeps constant. Therefore only translational, rotational and vibrational entropies are involved in computation of statistical entropy.

53. Calculation of statistical entropy

54. For ideal gases, the Sackur–Tetrode equation is used to calculate the molar translational entropy.

55. (2) Calculation of Sr For linear molecules
When all rotational energy levels are accessible
We obtain

56. (3) Calculation of Sv
Substitute
Into the following equation

57. Entropy contribution Electronic: 1 Translational: Rotational:
Vibrational:
Symmetry: S = RT ln2

58. Rotational contribution
„Non linear“
„Linear”

59. Vibrational contribution
nK  0
B. Viskolcz, Sz. N. Fejer and I.G. Csizmadia: J. Phys. Chem.A.3808, 110, (2006).