2004. 1.16
1.True of False
Identify each statement as either true or false. If a statement is true
uncer special circumstances, label it as false. (4%*3=12%)
a)The total electronic angular momentum of a diatomic molecule is a good
quantum number.
b)The orbital angular momentum of the electrons of a diatomic molecule is a
good quantum number.
c)The component on the bond axis of the orbital angular momentum of an
electron in a diatomic molecule is a good quantum number.
2.Neon excited dimer
Excited states of diatomic neon exist, although the ground state has bond
order zero. Give the electron configuration and term symbol for two
different states that might exist. (10%)
3.1,3-Butadiene
Consider the Huckel molecular orbitals for butadiene given as follows:
Ψ1 = 0.372ψ1 + 0.602ψ2 + 0.602ψ3 + 0.372ψ4
Ψ2 = 0.602ψ1 + 0.372ψ2 - 0.372ψ3 - 0.602ψ4
Ψ3 = 0.602ψ1 - 0.372ψ2 - 0.372ψ3 + 0.602ψ4
Ψ4 = 0.372ψ1 - 0.602ψ2 + 0.602ψ3 - 0.372ψ4
Each ψi is an atomic pz orbital on carbon atom i, so there is a nodal plane
in the xy plane for each molecular orbital. (20%)
a)Sketch other nodes in these orbitals as we move from atom 1 to atom 4.
b)The contribution to the electronic energy of a single π bond is 2β.
What is the contribution for two conjugated bonds in butadiene?
c)What is the extra stabilization in butadiene due to the conjugation(in
terms of β)?
d)The π bond order between carbon atoms μ and ν is given by
occ
p = 2Σ c c , calculate the π bond order for C1C2 and C2C3
μν i μi νi
bonds.
4.Square Cyclobutadiene (18%)
a)Set up the Huckel Hamiltonian matrix for the square cyclobutadiene, then
solve for the roots of the secular determinant and corresponding molecular
orbitals in terms of the atomic orbital p1,p2,p3,and p4. Make sure these
orbitals can be classified according to symmetry operations σx and σy.
Then make a schematic plot of molecular orbitals and the corresponding
molecular orbital energy diagram.
b)Instead of using σx and σy to do symmetry classification, construct
molecular orbitals as eigenfunctions of symmetry operations σ(45度) and
σ(135度). What is the relationship between these molecular orbitals and
those obtained in the previous problem (a)?
a.。──。 b.。──。
│_│_│__σx │＼／│
│ │ │ │／＼│
。──。 。──。
│ ／ ＼
σy σ(45度) σ(135度)
c)Filling in four electrons into the system, determine the degeneracy of the
ground state. If the electron-electron interation is taken into account,
these degenerate stated will be splitted into several groups, what will
be the relative ordering of energies for these groups according to the
Hund's rule.
5.Vibrational Spectroscopy
The fundamental vibration frequency of gaseous 14N16O is 1904 cm-1. (20%)
a)Calculate the force constant, using the simple harmonic oscillator model.
(Hints:The fundamental vibraqtional frequency of oscillator is given by
ν=1/2 π√(k/μ) , where μ= mNmO/(mN+mO) is the reduced mass.
~
Infrared spectroscopist usually express ν in wavenumbers ν =ν/c)
b)Calculate the fundamental vibrational frequency of gaseous 15N16O.
c)When 14N16O is bound to hemoglobin, an absorption band at 1615 cm-1 is
observed and is believed to correspond to the bound NO species. Assuming
that when an NO binds to hemoglobin, the oxygen is so anchored that its
effective mass becomes infinite, estimate the vibration frequency of bound
14N16O from the vibration frequency of gaseous NO.
d)Using another model, that binding of NO to hemoglobin does not change the
reduced mass of the NO vibrator, calculate the force constant of bound NO.
e)Estimate the vibration of 15N16O bound to hemoglobin using, respectively,
the assumption made in parts (c) and (d).
6.Microwave Spectroscopy
A microwave spectrometer capable of operating only between 60 and 90 cm-1
was used to observe the rotational spectra of HI and DI. Absorptions were
measured as follows
HI(cm-1) DI(cm-1) Useful formula
64.275 65.070 ~ ~
77.130 71.577 F(J)=Er/hc=BJ(J+1), B = h/8π^2 Ic
89.985 78.094 I = μ r^2
84.591
~
Find B, I, r for each molecule, and determine the J values between which
transitions occur for the first line listed above for each molecule. Do
your results support the usual assumption that bond length is unchanged by
isotopic substitution? (h=6.626×10^-34 Js, c=2.998×10^8 ms-1, mI=126.9)
(10%)
7.IR and Raman Selectrion Rule
Cyanogen is a symmetical linear molecule. The seven normal modes of
cyanogen, C2N2, are shown in the following figure:
← →← → + ← ← → → + →← ← → +
ν1 。─。─。─。Σg ν2 。─。─。─。Σg ν3 。─。─。─。Σu
↑ ↑ ↑ ↑
ν4 。─。─。─。Πg ν5 。─。─。─。Πu
↓ ↓ ↓ ↓
a)Which are doubly degenerate vibrations?
b)Briefly explain the underlying physical mechanisms that lead to infrared
absorption and Raman scattering by a vibrational mode.
c)Which vibrations are infrared active? Which vibrations are Raman active?
(12%)
Total:102%
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