(c) 2000-2003 by Akira Miyoshi. All rights reserved.

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Reaction Dynamics 2002 - section-2

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-2.11) 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.) (N_{2})=2,
(HCl)=1,
(CO_{2})=2,
(H_{2}O)=2,
(NH_{3})=3,
(C_{2}H_{4})=4,
(CH_{4})=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
[H_{3}C-CC-CH_{3}]

eigenvalue: (), degeneracy: (2.2.3)

() except for (2.2.4)

density of states: (2.2.5)

**[2-D Rotator]**

e.g. linear molecules [N_{2}, CO_{2}]

eigenvalue: (), degeneracy: (2.2.6)

density of states: (2.2.7)

**[3-D Rotator]**

e.g. non-linear molecules

a) Spherical top ...
[CH_{4}, SF_{6}]:

eigenvalue: (), degeneracy: (2.2.8)

density of states: (2.2.9)

b) Symmetric top ... or
[NH_{3},
CH_{3}]: (approximation)

Asymmetric top ...
(H_{2}CO, 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 H_{2}O.

Fig. 2.1b Comparison of by eq.
2.3.5 (solid line) and exact
(dots) for CH_{3}OH.

Fig. 2.2

Problem-2.2Estimate 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. |