Assumption: statistically equal distribution in final states
  Involves neither intermediate nor transition state 0th order approximation

cf.) Photodissociation of diatomic molecules (HCl, HI, etc.):
  (excess energy) distributes only in translation

	ex.)	Photodissociation of HCN at 215 nm (46512 ) :
		HCN + (46512 )	H + CN( = 0) + 3211	[channel-0]
			H + CN( = 1) + 1151	[channel-1]
		rel. translation + CN rotation

Probability of a specific rovibrational state
      (3.1.1)

	ex.)	Rotational distribution of CN via channel-1
		(max = 24, fig. 3.1)

Fig. 3.1 Rotational distribution of CN formed via HCN + (channel-1)

Probability of a specific vibrational state (indistinctive of rotational states) Rotational sum:
  ,       (3.1.2)
  (summation integration)
      (3.1.3)

ex.)

Branching ratio of channel-0 : channel-1 = (3211)3/2 : (1151)3/2 4.66 : 1

Problem-3.1 [OPTION] Derive an equation similar to 3.1.3 for the case that an atom and a non-linear molecule (3-D rotation) are formed.

[Rotational Sum]

Rotational sum for the case that two fragments are formed
(i.e., Branching probability for the specific vibrational state) :

      (3.1.4)

      (3.1.5)

: Energy partitioned into translation and rotation ( : Total excess energy)
: Translational degree of freedom (= 3), : Number of rotators,
: Total rotational degree of freedom ()
for integer : ,  
for half integer : ,   , ...

[Coefficients for rotational sum (eq. 3.1.5) for specific cases]

Problem-3.2 1) Calculate the prior vibrational energy distribution of OH formed via the reaction O() + H2 OH + H ( = -182.2 ) at 298 K.   Considering the thermal energy of the relative translation, , and that of rotation, , (note that = 2 for H2), the total excess energy at temperature is;   .   The vibrational frequency of OH is 3568 . 2) [OPTION] Calculate the prior rotational distribution OH in its vibrational ground state ( = 0) formed in the above reaction.   The rotational constant of OH is 18.51 .

Branching probability for a reaction channel (vibrational sum)

	ex.)	O() + HD	H + OD	channel-a
			D + OH	channel-b
		[O() + H2 OH + H] = -182.2 Vibrational frequencies () - OH: 3568, OD: 2632, H2: 4162, HD: 3633 Rotational constants () - OH: 18.51, OD: 9.87 (after ZPE & thermal corr. @ 298 K) (a) = 15952, (b) = 15484 () max(a) = 6 [15792 ], max(b) = 4 [14272 ]
		Branching ratio = = 1 : 1.017

	cf.)	summation integration
		(3.2.1) note)

[Vibrational sum]

General formula by replacement of summation by integration :

      (3.2.2)

  ,   : vibrational degree of freedom,   _i : vibrational frequency
  : excess energy measured from the classical origin

	ex.)	O() + CH4 OH + CH3 ( = -182.3 )   - Prior vibrational distribution of OH (298 K)
			Motion other than OH-vibration (3.2.2), OH = 3568 ZPE(CH3) = 76.1 = (182.3 + 76.1) [] + 3RT = 22222 cf.) = 182.3 [] + 3 = 15860 max = 4 [14272 ], = (3 + 2 + 1) / 2 = 3, = 6, + = 9 Vibrational distribution (fig. 3.2) = 0     1 = 1     0.207 = 2     0.0306 = 3     0.00270 = 4     9.60 10-5

Fig. 3.2 Vibrational distribution of OH formed via O() + CH₄

Problem-3.3 Calculate the prior vibrational distribution of OH formed by O() + C2H6 OH + C2H5 ( = -210.7 , ZPE(C2H5) = 148.5 ) similarly to above, and compare it with the cases for O() + H2 and O() + CH4.

[Atoms]     (except for the excited states of rare gas atoms)

(spectrum) Term :

  : electron spin q. n.
  : electron orbital angular momentum q. n.
  : total angular momentum q. n.
  [] : symbolic representation of   - S, P, D, F, G, H, ... for = 0, 1, 2, 3, 4, 5, ...

Total degeneracy :
Degeneracy of the fine-structure state:

  ex.)

term						fine str.   term
	2	0	5	5	2
	0	1/2	2	2	1/2
	1	1/2	6	2	1/2
				4	3/2
	1	1	9	1	0
				3	1
				5	2

	ex.)				branching ratio
		HCl +	H + Cl()	(a)	4 trans(a)
			H + Cl()	(b)	2 trans(b)

[Linear Molecules]

(spectrum) Term :

  : projection of to the molecular axis
  : projection of to the molecular axis
  []: symbolic representation of   - , , , , ... (for = 0, 1, 2, 3, ...)
  (+-): parity (+ or -, only for S states)

Total degeneracy :
Degeneracy of the fine-structure state :

  ex.)

term					+	fine str.   term
	2	0	2	2	2
	0	1/2	2	1	1/2
	1	1/2	4	2	1/2
				2	3/2
	1	1	6	2	0
				2	1
				2	2

	ex.)				branching ratio
		NH3	H2 + NH()	(X)	3 vib-sum(X)
			H2 + NH()	(a)	2 vib-sum(a)

[Non-Linear Molecules]

(spectrum) Term :

  : symmetry species of the electronic state ... , ', ₂, ₁, , , etc.

Total degeneracy :
Degeneracy of the fine-structure state :

non-degenerates (A, B) and E of spherical group : no angular momentum
E of cylindrical group, T(F), G, H, ... of spherical group : (non-integer) angular momentum

  ex.)

term				fine str.   term
	0	3	3
	1/2	2	1
	1/2	4	2
			2

	ex.)				branching ratio
		CH3OH	H + CH3O()	(X)	4 vib-sum(X)
			H + CH3O()	(A)	2 vib-sum(A)

[Number of optical isomers]

	ex.)				branching ratio
		CFCl2Br +	Cl() + CFClBr	(a)	2 vib-sum(a)
			Br() + CFCl2	(b)	2 vib-sum(b)

channel-a two optical isomers for CFClBr (= two reaction pathway = inversion doubling)

[Rotational distribution and nuclear spin statistics]

  - Nuclear spin :
  - Resultant total nuclear spin of a molecule :
  - Bose/Fermi particle-Bose/Fermi statistics (symmetric/assymmetric to permutation)

ex.)

H2( = 0) - Rotational state distribution - H nucleus : = 1/2 (= proton / Fermi particle)     tot should be Asymmetric - elec () : Sym.,   vib ( = 0) : Sym.

	ortho-H2		para-H2
	1 ()		0 ()
n.s	Sym.		Asym.
rot	Asym.		Sym.
	1, 3, ... (odd)		0, 2, ... (even)
	3 ()		1 ()

ex.)

N2 or D2( = 0) - Rotational state distribution - N or D nucleus : = 1 (Bose particle)     tot should be Symmetric

	ortho-N2/D2		para-N2/D2
	0 () or 2 ()		1 ()
n.s	Sym.		Asym.
rot	Sym.		Asym.
	0, 2, ... (even)		1, 3, ... (odd)
	(1+5) ()		3 ()

  - = 2 either in sym. or asym. rotational state, rotational density of states is 1/2

Problem-3.4   Calculate the prior branching fractions for the reaction of O() with CH4, / O() + CH4 CH3 + OH   -182.7 channel-a CH3O + H   -129.1 channel-b CH2OH + H   -173.2 channel-c at room temperature (298 K). Two experimental measurements on H-atom reported the sum of branching fractions for channel-b and c as 14% and 25%. Compare these with the prior branching fractions. Molecular constants   chemical species H OH CH3 CH3O CH2OH   mass / amu 1.008   17.003   15.024 31.019   31.019     symmetry -   rot. sym. num.; 1 1 6 3 1   num. opt. isom.; OPT 1 1 1 1 2   electronic state       deg. of electron. state;   2 4 2 4 2   rot. deg. of freedom - 2 3 3 3   rot. consts. /     (= ) 5.169 6.555     18.51 9.578 0.9317 1.0061     4.742 (= ) 0.8879   vib. deg. of freedom - 1 6 9 9   vib. freqencies* /     1 3568 3004 2840 3650     2 606 1362 3019     3   3161 (2)   1047 2915     4 1396 (2)   2774 (2)   1459     5 1487 (2) 1334     6 652 (2) 1183     7 1048     8 607     9 420     * Numbers in parentheses are degeneracy of the vibrational modes.     1 = 11.96266 , = 8.31451