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http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics
1. Steinfeld, page 177, # 1.
The corners of a cube are numbered 1, 2, 3, 4 clockwise around the top face of the cube, and 5, 6, 7, 8
clockwise around the bottom face, corner 5 lying under corner 1, corner 6 under corner 2, and so on. A face
center is denoted by the two numbers of the corners between which a face diagonal can be drawn which passes
through that face center (for example, either 13 or 24 would denote the center of the top face).
(a) The structures of several kinds of A B4 molecules are described as follows with the above numbering
system. The A–atom is placed at the center of the cube, and the A— B bond distances are given by the
cube dimensions. However, the B–atoms are not necessarily all equivalent, the actual equivalence being
determined by the symmetry elements remaining in the A B4 structures.
(i) A B4 (1, 2, 3, 4)
(ii) A B4 (1, 3, 5, 7)
(iii) A B4 (1, 3, 6, 8)
(iv) A B4 (1, 5, 16, 18)
(v) A B4 (13, 36, 68, 18)
Give the point-group symbol for each A B4 structure (for example, C3v , D6h , and so on).
(b) Classify the ﬁve molecules of Part (a) as to type of molecular rotator (linear, symmetrical top, and so
(c) Which of the ﬁve molecules of Part (a) will give a pure-rotational spectrum in the far-infrared or mi
crowave region? Which will give a pure-rotational Raman eﬀect?
2. Steinfeld, page 196, # 1.
The harmonic oscillator (mass m) in two dimensions has a potential energy V expressed in polar coordinates r,
θ, of the general form 2V = k1 r2 cos2 θ + k2 r2 sin2 θ, where k1 and k2 are force constants. For the special case
k1 = k2 = k, the oscillator has a single frequency v = (k/m)1/2 /2π, and its Schr¨ dinger equation has solutions
of the form �
ψv,� = Nv,� exp �
2 5.76 Problem Set #4 Spring, 1985 page 2 where v, � are quantum numbers (v = 0, 1, 2, . . . , ∞; ±� = 0, 2, 4, . . . , v for v even, ±� = 1, 3, 5, . . . v for v odd);
Nv,� is a normalization constant; α = 4π2 mv/h; P(r), a polynomial in r only, depends for its form on the values
of v and � and is an even function for even v, odd for odd v.
(a) The energy levels of this two-dimensional oscillator are Ev = (v + 1)hv. What is the degeneracy of the
(b) Find Nv,� for v = 1, � = +1, for which P(r) = α1/2 r.
(c) Show that any two ψ’s of the same v but diﬀerent � are orthogonal.
(d) Find the average value of the angular momentum pθ for any state v, �.
(e) Find the average value of r−2 for the state v = 1, � = +1.
Substitute the results of the above in the equation for the relationship between E and pθ in the plane rotor
and ﬁnd E for the state v = 1, � = +1. Explain the diﬀerence between this value of E and that given by the
equation of Part (a).
3. Steinfeld, page 197, # 2 The carbon suboxide molecule C3 O2 , is believed to be linear and symmetrical. (a) Classify the normal vibrations of C3 O2 according to the symmetry species of point group D∞h .
(b) Draw the modes of vibration of C3 O2 .
(c) How many polarized lines should appear in the vibrational Raman spectrum? How many fundamental
infrared bands should have P–, Q–, and R–branches?
(d) The ground electronic state of C3 O2 is nondegenerate. What is the degeneracy of ψmol when C3 O2 is in
its ground electronic state with v1 = v2 = v3 = v4 = v5 = v6 = 0, v7 = 1, and the rotational quantum
number J = 5? The normal coordinate Q7 is antisymmetric with respect to simultaneous interchange of
all pairs of equivalent nuclei. What is the degeneracy of ψmol for the above state with J = 6 instead of 5?
4. Steinfeld, page 199, # 8.
(a) Let ψa ≡ ψv1 =1 (Q1 )ψv2 =0 (Q2 ) and ψb ≡ ψv1 =0 ψv2 =2 be the normalized harmonic-oscillator wave functions
of a polyatomic molecule corresponding to excited vibrational states of unperturbed energies Ea and Eb .
If these two states are in Fermi resonance, second-order non-degenerate theory can be applied. Assume
that the interaction energies Haa and Hbb are zero and that Hab arises from one or more anharmonic terms in the potential function. In a certain molecule, the levels ψa and ψb are observed to be in Fermi
resonance, the transitions to the perturbed levels being observed at 1400 and 1500 cm−1 , whereas the 5.76 Problem Set #4 Spring, 1985 page 3 level ψv1 =0 ψv2 =1 has an energy of 740 cm−1 above the zero level (see diagram). Deduce the unperturbed
Ea –value from the above data (Hab is to be evaluated from the data, not by integration). (b) The intensity of the Raman line for the transition from the ground state ψ0,0 to the unperturbed state ψa
in the absence of Fermi resonance is proportional to the square of the matrix element
Q 1 � ψa
where ∂α/∂Q1 is a non-zero constant. The corresponding matrix element �
� ∂ Q 2 Q 2 � ψb
is zero because Δv2 = 2. Find the ratio of the intensities of the two Raman lines for the transitions from
state |ψ00 � to the perturbed states |ψA � and |ψB �.
5. 16 O12 C32 S is a linear molecule. The bond lengths are
rCO = 1.16 Å
rCS = 1.56 Å and the observed fundamental vibrational frequencies are
ν1 = 858.9 cm−1 stretch
ν2 = 520.4 cm−1 bend
ν3 = 2062.2 cm−1 stretch.
(a) Obtain kCS , kCO , and kΘ [rCO rCS ]−1 in dynes/cm.
(b) What are the amplitudes for C–O and C–S stretch in ν1 ?
(c) What are the vibrational frequencies for 18 O12 C32 S? ...
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