Classical Approximation / Limit

• (microcanonical)
• (canonical)

[1-D (One-Dimensional) Translation]

Motion of a particle (mass ) in a 1-D box of length :

  eigenvalue:   ( = 1, 2, 3, ...)     (2.1.1)

  density of states:     (2.1.2)

[3-D Translation]

Motion of a particle (mass ) in a 3-D box ()
  [Independent in , , and directions]

  eigenvalue:   ()     (2.1.3)

  density of states:     (2.1.4)

= volume:

  density of states per unit volume: (2.1.5)

[3-D Relative Translation]

Relative translation of particle A and B: (mass) (reduced mass)

      (2.1.6)

      (2.1.7)

note:

  1-D translation activated complex in the transition-state theory
  2-D translation diffusion on a surface of an adsorbed molecule
  3-D translation reaction producing two molecules

Problem-2.1 1) Derive the expression for the density of states per unit area of 2-D translation. 2) Derive the 3-D translational energy distribution for canonical ensemble (Maxwell-Boltzmann distribution) at temperature T from equation (2.1.5).

[Rotational Motion]

  moment of inertia:     (2.2.1)
    (: mass of th atom, : distance of th atom from rotation axis)

  rotational constant:     (2.2.2)

  <Rotational symmetry number>
    = number of indistinctive rotational conformation
      (same atoms are indistinctive)
    ex.) (N₂)=2, (HCl)=1, (CO₂)=2, (H₂O)=2, (NH₃)=3, (C₂H₄)=4, (CH₄)=12

  <Rigid rotor approximation>
    is constant (i.e., independent on the speed of rotation, the vibrational states, or etc.)

[1-D Rotator]
  e.g. intramolecular rotations [H₃C-CC-CH₃]

  eigenvalue: (), degeneracy:     (2.2.3)

    () except for     (2.2.4)

  density of states:     (2.2.5)

[2-D Rotator]
  e.g. linear molecules [N₂, CO₂]

  eigenvalue: (), degeneracy:     (2.2.6)

  density of states:     (2.2.7)

[3-D Rotator]
  e.g. non-linear molecules

a) Spherical top ... [CH₄, SF₆]:

  eigenvalue: (), degeneracy:     (2.2.8)

  density of states:     (2.2.9)

b) Symmetric top ... or [NH₃, CH₃]: (approximation)
      Asymmetric top ... (H₂CO, HCO): (approximation)

  density of states: ,     (2.2.10)

[Summary]
  rotational degree of freedom (dimension of rotation)

  ,       (2.2.11)

[Vibrational Motion]

    ( : force constant, : reduced mass)     (2.3.1)

  <Harmonic Oscillator Approximation>
    is constant (i.e., independent of deviation )

  Solution to the classical mechanics:
  ,     ( : frequency [])     (2.3.2)

[One Vibrator]

  eigenvalue: ,       (2.3.3)

  density of states:     (2.3.4)

[ Vibrators]

Number of vibrators: (non-linear molecule), (linear molecule)

  two vibrators:

  three vibrators:
  .....

  vibrators:     (2.3.5)

note:

1) Density of states is crude approximation (fig. 2.1) for vibration since 200-3000 .
2) in eq. 2.3.5, is the energy from the classical origin (see fig. 2.2).
3) ,

Fig. 2.1a Comparison of by eq. 2.3.5 (solid line) and exact (dots) for H₂O.

Fig. 2.1b Comparison of by eq. 2.3.5 (solid line) and exact (dots) for CH₃OH.

Fig. 2.2

Problem-2.2 Estimate the vibrational density of states for ethane [(C-C) = 377 ] at around dissociation threshold by eq. 2.3.5.   Use the following vibrational frequencies [Unit: ; Values in parentheses are degeneracy of the vibration.]     2954, 1388, 995, 289, 2896, 1379, 2969(2), 1468(2), 1190(2), 2985(2), 1469(2), 822(2) Note that the in eq. 2.3.5 is the energy from classical origin.