444
10 The Statistical Basis of Bioenergetics
10.6.7 Asymmetric Molecules
Asymmetric molecules have different moments of inertia around all three principle
axes. A prominent example is water H–O–H. Another is nitrogen dioxide O–N=O,
a noxious red gas and an air pollutant from coal and hydrocarbon burning.
Energy levels and their degeneracies are found by solving the Schrödinger
equation for the rotor. The solutions are expressible in terms of functions of the
Euler angles α, β, γ ) which determine the orientation of the molecule in space. The
square of the total angular momentum J 2 ≡ J 2
a + J 2
b + J 2
c will be conserved, so
its possible values are fixed to be J (J + 1) ¯
h 2 , and we can use J as one of the
labels of the energy states. For each J , there will be a set of 2J + 1 distinct energy
states, whose degeneracy depends on the shape of the molecule. The solutions can
be usefully expanded in terms of the Wigner D
∗j
m,m (α, β, γ ) functions, or Schwinger
rotation states. 23
With a little algebra, the rotational Hamiltonian (energy operator),
H rot = A J
2
a + B J
2
b + C J
2
c
can be put into the form
H rot =
1
2
(A + C) J
2
+
1
2
(A − C)h r (κ, J, τ )
(10.54)
where
h r (κ, J, τ ) ≡ J
2
a + κ J
2
b − J
2
c ,
is called the reduced rotational energy which depends on an asymmetry parameter
κ ≡ (2B − A − C)/(A − C). Note: κ = −1 for a prolate rotor (A = B > C), 1 for
an oblate rotor (A > B = C), and in the limiting case of a linear molecule, I A → 0
so A → ∞ and κ → −1. The index τ runs from −J to J labels the possible
energies for a given J . The measured values taken by h r called r , are the roots of
a secular equation 24 of degree 2J + 1. Solving the secular equation for J ≥ 2 can
be a formidable task, so we have no general expression for the molecular partition
function in this case. Applying the symmetry group for the molecule helps reduce
23 See P.D. Jarvis and L.A. Yates, The molecular asymmetric rigid rotor Hamiltonian as an exactly
solvable model, arXiv:0803.256v1 (2008).
24 A secular equation is a matrix determinant relation of the form det |H − I | = 0, where H is
a hermitian matrix, I a unit matrix, and is one of the unknown ‘eigenvalues’. Observables in
quantum theory correspond to hermitian matrices, i.e. matrices satisfying H T ∗ = H , where T is
the transpose operation and ∗ is the complex conjugation.
10 The Statistical Basis of Bioenergetics
10.6.7 Asymmetric Molecules
Asymmetric molecules have different moments of inertia around all three principle
axes. A prominent example is water H–O–H. Another is nitrogen dioxide O–N=O,
a noxious red gas and an air pollutant from coal and hydrocarbon burning.
Energy levels and their degeneracies are found by solving the Schrödinger
equation for the rotor. The solutions are expressible in terms of functions of the
Euler angles α, β, γ ) which determine the orientation of the molecule in space. The
square of the total angular momentum J 2 ≡ J 2
a + J 2
b + J 2
c will be conserved, so
its possible values are fixed to be J (J + 1) ¯
h 2 , and we can use J as one of the
labels of the energy states. For each J , there will be a set of 2J + 1 distinct energy
states, whose degeneracy depends on the shape of the molecule. The solutions can
be usefully expanded in terms of the Wigner D
∗j
m,m (α, β, γ ) functions, or Schwinger
rotation states. 23
With a little algebra, the rotational Hamiltonian (energy operator),
H rot = A J
2
a + B J
2
b + C J
2
c
can be put into the form
H rot =
1
2
(A + C) J
2
+
1
2
(A − C)h r (κ, J, τ )
(10.54)
where
h r (κ, J, τ ) ≡ J
2
a + κ J
2
b − J
2
c ,
is called the reduced rotational energy which depends on an asymmetry parameter
κ ≡ (2B − A − C)/(A − C). Note: κ = −1 for a prolate rotor (A = B > C), 1 for
an oblate rotor (A > B = C), and in the limiting case of a linear molecule, I A → 0
so A → ∞ and κ → −1. The index τ runs from −J to J labels the possible
energies for a given J . The measured values taken by h r called r , are the roots of
a secular equation 24 of degree 2J + 1. Solving the secular equation for J ≥ 2 can
be a formidable task, so we have no general expression for the molecular partition
function in this case. Applying the symmetry group for the molecule helps reduce
23 See P.D. Jarvis and L.A. Yates, The molecular asymmetric rigid rotor Hamiltonian as an exactly
solvable model, arXiv:0803.256v1 (2008).
24 A secular equation is a matrix determinant relation of the form det |H − I | = 0, where H is
a hermitian matrix, I a unit matrix, and is one of the unknown ‘eigenvalues’. Observables in
quantum theory correspond to hermitian matrices, i.e. matrices satisfying H T ∗ = H , where T is
the transpose operation and ∗ is the complex conjugation.