Transcript knc_sym
Symmetry, degeneracy, and
+
statistical weights of H3
Kyle N. Crabtree
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA
Benjamin J. McCall
University of Illinois, Urbana, IL
Motivation
H5+
Interstellar Chemistry
Chemical Physics
Oka (2004) J. Mol. Spectrosc. 228, 635
H3+ + H2 H2 + H3+
Factor of 2?
4x3=
4x1=
2x2x3=
2x2x1=
Factors of 2!
• Nuclear spin:
gI = 2I + 1
o (I = 3/2) 4
p (I = 1/2) 2
PI symmetry
4A1 + 2E
4x1=4
2x2=4
Rotation Group
(D1/2)3 = D3/2 + 2D1/2
4x1=4
2x2=4
• Rotation:
Symmetric top states: |J,k> and |J,-k> are degenerate (k > 0)
Factor of 2 for all K >0 states? (K = |k|)
Ortho-H3+: J = even, K = 0 states forbidden in vib. ground state!
• Vibration:
n2 band: E symmetry (factor of 2?)
Vibrational angular momentum l = {v2, v2-2, … -v2}
Rovibrational projection g = k-l; G = |g|
H3+ CNP symmetry
Nuclear permutation (12)
2
(12)
=
1
(12)(23) = (123)
3
=
Hamiltonian invariant to
permutation of identical
nuclei
Pauli Exclusion Principle
Gtot = Gel × Grv × Gns = A2
C3v(M)
E
3(12)
2(123)
A1
A2
E
1
1
2
1
-1
0
1
1
-1
G1 × G2 A1
A1
A1
A2
A2
E
E
A2
A2
A1
E
E
E
E
A1 + A 2 + E
Nuclear spin symmetry
AM of H3+: 3/2 or 1/2
projection
Angular momentum coupling
Clebsch-Gordan coefficients
AM of “embedded” H2
proton AM projections (±½)
ortho-H3+
para-H3+
|3/2,3/2,1>
|1/2,1/2,1>
|3/2,1/2,1>
|1/2,1/2,0>
|3/2,-1/2,1>
|1/2,-1/2,1>
|3/2,-3/2,1>
|1/2,-1/2,0>
Consider (123) operation
NS transformation matrices
(123)
331
311
3-11
3-31
111
110
1-11
1-10
331
311
3-11
3-31
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
111
110
1-11
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-1/2
0
0
√3/2
1-10
0
0
0
0
-√3/2
-1/2
-1/2 √3/2
-√3/2 -1/2
0
0
0
|3/2,-1/2,1> = 3-11
0
NS transformation matrices
I=3/2 (4)
E
(12)
(23)
(13)
(123)
(132)
1
1
1
1
1
1
I=1/2 (2)
1 0
0 1
1 0
0 −1
C3v(M)
E
3(12)
2(123)
A1
1
1
1
−12
−√3
2
A2
1
-1
1
−√3
2
1
2
E
2
0
-1
−12
√3
2
1
2
√3
2
−12
Character is the trace of the
transformation matrix
√3
2
−12
Trace
E
3(12)
2(123)
I = 3/2
1
1
1
−12 −√3
2
I = 1/2
2
0
-1
−√3
2
√3
2
−12
Rotation-vibration symmetry
|J,0,0>
|J,k,l>
|J,-k,-l>
E
1
1 0
0 1
(12)
(-1)J
(23)
(-1)J
0
−1 𝐽𝑒 4𝜋𝑖𝑔/3
−1 𝐽𝑒 −4𝜋𝑖𝑔/3
0
(13)
(-1)J
0
−1 𝐽𝑒 −4𝜋𝑖𝑔/3
−1 𝐽𝑒 4𝜋𝑖𝑔/3
0
(123)
1
• Symmetric top states x
vibrational angular
momentum projection:
|J,k,m> x |l>
(2J+1) |J,k,l>
• Results depend on value
of J and g (k-l):
Case
J = even
J = odd
k=0=l
A1
A2
g=3n
A1 + A 2
g=3n±1
E
(132)
1
0
−1
𝐽
𝑒 2𝜋𝑖𝑔/3
0
𝑒 −2𝜋𝑖𝑔/3
0
−1
0
𝐽
0
𝑒 −2𝜋𝑖𝑔/3
0
𝑒 2𝜋𝑖𝑔/3
Total nuclear wavefunctions: o-H3+
• Direct product of rot-vib and spin functions
|J,k,l> x |I3,m3,I2> |J,k,l,I3,m3,I2>
• Symmetrize Results for o-H3+ (|I3,m3,I2> =
|3/2, m3,1> omitted)
(2J+1)×4
• Pauli exclusion principle
Case
J even
J odd
k=0=l
|J,0,0>: A1, (2J+1)×4
|J,0,0>: A2, (2J+1)×4
g=3n
2-1/2(|J,k,l>+|J,-k,-l>): A1, (2J+1)×4
2-1/2(|J,k,l>-|J,-k,-l>): A2, (2J+1)×4
2-1/2(|J,k,l>+|J,-k,-l>): A2, (2J+1)×4
2-1/2(|J,k,l>-|J,-k,-l>): A1, (2J+1)×4
Total nuclear wavefunctions: p-H3+
A1
E
(2J+1)×2
A1
E
Statistical weights
J
6n
6n+1
6n+2
6n+3
6n+4
6n+5
4x3=
4x1=
2x2x3=
2x2x1=
Grv High-J limit
n(2A1 + 2A2 + 4E) + A1
n(2A1 + 2A2 + 4E) + A2 + E
n(2A1 + 2A2 + 4E) + A1 + 2E
n(2A1 + 2A2 + 4E) + A1 + 2A2 + 2E
n(2A1 + 2A2 + 4E) + 2A1 + A2 + 3E
n(2A1 + 2A2 + 4E) + A1 + 2A2 + 4E
4 o-H3+ spin functions combine with each A2
2 p-H3+ spin functions combine with each E
4x2
2x4
1:1
Factor of 2!
Proton hop reactions
p-H3+ + X
+X
1991 A&A 242, 235
X+ +
o-H2 + HX+
p-H2 + HX+
X+ +
X+ +
Proton hop reactions
p-H3+ + X
o-H2 + HX+
p-H2 + HX+
• High-temperature angular momentum algebra:
50:50 o-H2:p-H2
• p-H3+ total nuclear wavefunction for (J,G)=(1,1)
1/
1/ ,±1/ ,1>
|1,1,0,
2
2
2
-i/2|1,1,0,1/2, ±1/2,0>
+1/2|1,-1,0,1/2, ±1/2,1>
+i/2|1,-1,0,1/2, ±1/2,0>
50% “embedded” o-H2
50% “embedded” p-H2