課程名稱︰化學鍵 Structure and Chemical Bonding
課程性質︰大學部選修
課程教師︰陸駿逸/林金全
開課系所︰化學系
考試時間︰2005/11/10 13:20~15:10
試題 :
(h 表示 h bar = h/2π dt 表示 對t偏微 d^2x表示對x偏微二次)
Structure and Chemical Bonding
Midterm Exam.
Exam time: 13:20~15:10 2005/11/10
1.(15 points) There are two Schrodinger equations:
the time dependent equation
(1) ih dtΨ(x,t) = -(h/2m) d^2xΨ(x,t) + V(x)Ψ(x,t)
and the time-independent equation
(2) -(h/2m) d^2xΨ(x,t) + V(x)Ψ(x,t) = EΨ(x,t)
(a) Explain the relation between these two equations.
(b) Given a solution of equation(2) ψ(x) and E, construct a solution Ψ(x,t)
of equation(1).
(c) When do you use equation(1) but not equation(2) ?
2.(15 points) When we construct the (approximated) MO by linear combination of
the two atomic orbitals χ1 and χ2, asψ= N (c1χ1 + c2χ2) where N is the
normalization constant, do we solve the Schrodinger equation Hψ=Eψ to
determine c1 and c2? Do our colutions obey the Schrodinger equation?
3.(35 points) Solve the two MO of a diatomic molecule AB. Given that
<χA|H|χA>= -300 kJ/mol, <χB|H|χB>= -400 kJ/mol, <χA|H|χB>=<χB|H|χA>
= -200 kJ/mol, <χA|χB>=<χB|χA>= 0.3, <χA|χA>=<χB|χB>=1.
Consider the linear combination ψ= N (cAχA + cBχB) where N is the
normalization constant, find the suitable cA, cB and the energy E of ψ.
You should get two sets of solutions.
4.(15 points) (a) What is the symmetry origin of the Pauli exclusion principle
for electrons?
(b) Is there an exclusion principle for photons?
(c) If one approximates the many electron wave function by the product of the
one electron wave functions (i.e. orbitals), what is the statement of the
Pauli exclusion principle?
5.(20 points) Construct the MO of the triangular H3. The distance between Ha
and Hb, is the same as the one between Ha and Hc. The distance between Hb and
Hc is a little shorter.
(a) Select a pair of fragments for construction of the MO. Identify a
suitable symmetry element with which one can separate the fragments into
the symmetric and antisymmetric types.
(b) Draw the MO orbitals and the energy levels.
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