Canonical Ensemble

• Under free and random energy exchange among molecules in the system (normal gas-state)
• The total energy of the system is defined by temperature, .

<Boltzmann Statistics>

Probability of finding a molecule in the state (of energy and multiplicity )  

ex.)	Distribution of the ground triplet states of atomic oxygen; ( 0, = 5), ( = 158.5 , = 3), and ( = 226.5 , = 1); at 298 K (1 = 1.4388 K)

<Partition Function>

Probability of finding molecule A in the system (e.g., A = HCN and not HNC, 1/2 H₂ + 1/2 N₂ + C, nor etc.) = Sum of the probability of finding for all states of A   [partition function]     (1.1.1)

ex.)	Distribution of the triplet ground states ( = + + ) and the excited () state [ = 15867.7 , = 5] of atomic oxygen at 298 K:

ex.)	Vibrational partition function of a harmonic oscillator ():
		(1.1.2)

<Chemical Equilibrium>

Ratio of the distributions of molecule B to A:
  (When the same zero energy point was used to calculate both and )   or
    (1.1.3)   (When and are calculated from an each ground sate)

<Transition-State Theory>

Rate coefficient (A [TS*] B)     (1.1.4)

• Probability of passing through the states in the transition state obey Boltzmann distribution

• Isolated molecules (molecules after photolysis, molecules just after reaction, or etc.�j
• The system is defined by energy .
• The prior distribution

<Multiplicity>

• Degeneracy of multiple solutions to a (time-independent) Schrodinger equation with the same eigenvalue (energy).
• Ratio of the probabilities of finding a system in the state A (multiplicity ) to that in the state B (multiplicity ; at the same energy as A) = :

ex.)	Atomic orbitals of H: = (: principal q.n.), = (: azimuthal q.n.)
	K-shell ( = 1): 1s ( = 0, = 1) only   = = 1
	L-shell ( = 2): 2s ( = 0, = 1), 2p ( = 1, = 3)   = = 4
	M-shell ( = 3): 3s ( = 0, = 1), 3p ( = 1, = 3), 3d ( = 2, = 5)   = = 9

ex.)	Electronic states of H-atom:		total multiplicity
	ground state:	() [ = = 2 1 = 2]	(K-shell) 2
	excited states:	() [ = 2 3 = 6]	(L-shell) 8
		() [ = 2 1 = 2]

ex.)	Bending vibration () of CO2 ... 2-D harmonic oscillator
	,   ;   (: vibrational q.n.)

<Density of states>

• = Number of states per unit energy [states / ]
• Probability ratio between molecule A (density of states = ) and B () = :

ex.)	Photodecomposition of HCl at 248 nm (40320 ):
	HCl + (40320 )	H + Cl() + (4240 )	(a)
		H + Cl() + (3359 )	(b)
	Statistical branching ratio, :
	= [Cl()] (4240 ) : [Cl()] (3359 )
	=

Problem-1   Calculate the statistical branching ratio between H + I() [channel-a] and H + I() [channel-b] upon the photolysis of HI at 266 nm.   Use the bond dissociation energy of HI [ H + I() ] = 298 and excitation energy of I() form I() (ground state) = 0.943 eV.