[試題] 101上 金必耀 化學數學一 期末考

[wenx1002 (wenx1002)]

課程名稱︰化學數學一
課程性質︰化學系必修
課程教師︰金必耀
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2013/ 1/ 8
考試時限（分鐘）： 130 min
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

(自行攜帶Character tables)

1.(10%)

Show that if a set of matrics A,B... represents a group, then applying the

same sililarity transformation to all elements of the group:

     -1          -1
A'= S  AS , B'= S  BS,... produces another representation of the same group.



2.(15%)

(a) Identify the point groups of the following molecules: cyclopropane, bezene

, and ethane in the straggered, eclipsed, and gauche conformations.

(b) Locate the four mirror planes of cyclopropane.

(c) How many of the mirror planes of benzene can you find?

(d) How many mirror planes does ethane have in the staggered, eclipsed, and the

gauche conformations of ethane as vertical, horizontal or diagonal mirrors.


3.(10%)

If f transforms as A2 in C2v and g transforms as B1, what is the symmetry

species in C2v of the product fg ?


4.(10%)

Formaldehyde, CH2O, has C2v symmetry. The ground atate has symmetry species A1

, and low-lying excited states have symmetry species A2, B2, and A1. To which

excited states are transitions allowed? What are the directions of polarization

of the allowed transitions?


5.(15%)

Assume tetramethylethane to be planar and all resonance integrals are the same.

Find out all Huckel molecular orbitals and eigenvalues of this molecule by

taking suitable symmetry adaptation first. Predict the delocalization energy

and spin multiplicity for its ground state.


                              CH2   CH2
                              \    /
                               C一C
                              /    \
                              CH2   CH2

6.(20%)

Boron hydride, BH3, is a planar, D3h, molecule. We take the z-axis to be along

the principal, C3, axis and place the x-axis along B-H1 .

(a) Find the reducible matrix representations for the reducible representation

Γ and the corresponding character for the basis of three H(1s) orbitals, s1,

s2, s3, in BH3.

(b) Identify the irreducible representations for the reducible representation

Γ. Then find the basis, Φ1,Φ2,Φ3 for these irreducible representations

by applying the SALC for the three H(1s) orbitals.

(c) Ignoring the 1s and 2s orbitals of centrall boron atom, identify the

irreducible representations to which three 2p orbitals, px, py, pz, belong.

(d) Sketch the Hamiltonian matrix of BH3 spanned by these six orbitals, i.e.

Φ1,Φ2,Φ3,px,py, and pz. Identify nonvanishing matrix elements

Hij= <Ψi|H|Ψj>, where Ψi and Ψj are one of these six symmetry-adapted

orbitals.

(e)(Extra credit) Solve the Hamiltonian matrix, and draw the molecular orbital

energy diagram(MOED) for BH3. Sketch molecular orbital for each energy level

in the MOED qualitatively.


7.(10%)

In D3h, if Ψi has symmetry species E',Ψj has symmetry species E'', what is

the set of characters corresponding to the direct product representation for

<Ψi|H|Ψj>? Of what irreducible representation is it composed?


8.(10%)

Consider a cubane molecule, C8H8. What irreducible reprecentations are spanned

by the set of CH bonding orbitals? And by the set of 12 C-C bonding orbitals?


--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 140.112.4.195
推 firepeter   :原PO大帥哥
推 williamwei82:原PO大帥哥
推 danhao      :原PO大帥哥
推 aochiu1992  :原PO大帥哥
推 icheee      :原PO大帥哥                           超快我剛醒耶
推 jew710934   :原PO大帥哥
推 noreg0108807:原PO大師哥
推 franc40810  :原PO大帥哥
推 darren8221  :原P0大帥哥
推 oray914     :原PO大帥哥                           七樓有事嗎??
→ ruby40861   :原PO大帥哥                                神速耶~
→ potter99168 :原P0犬師哥
推 edda20047   :原PO大帥哥
推 jane820113  :原po大帥哥                       明天明明還有普生
推 taasnsg     :原PO大帥哥
推 vanchou     :原PO大帥哥