推 a3225737 :太精美了= = 06/22 22:47
→ firepeter :被你稱讚是我的榮幸 06/22 23:04
推 a3225737 : = = 06/23 00:15
課程名稱︰化學數學二
課程性質︰化學系選修
課程教師︰陳振中
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2011/6/22
考試時限（分鐘）：120min
是否需發放獎勵金：是
試題 :
Mathematics for Chemistry(II) Final Exam 2011.6.22
Section A (70%)
1.(12%) Write down the characters of the representations of the following direct
products, and determine the irreducible representations which comprise them for
group D4h:
A1u x A1u A1g x B1g B2u x Eg Eg x Eu
2.(20%) Consider the group C3v. Let the C3 axis coincide with the z-axis. Write
out the complete matrices for all irreducible representations for this group.
Derive from these the character table.
3.(16%) Derive the following character table and determine the irreps
transformed by the five d-orbitals.
Character table for D3h group
──────────────────────────────────────
｜ ｜ E ｜2C3 ｜3C2'｜σh ｜2S3 ｜3σv｜linear, rotations ｜ quadratic ｜
──────────────────────────────────────
｜A1'｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜
──────────────────────────────────────
｜A2'｜ ｜ ｜ ｜ ｜ ｜ ｜ Rz ｜ ｜
──────────────────────────────────────
｜E' ｜ ｜ ｜ ｜ ｜ ｜ ｜ (x.y) ｜ ｜
──────────────────────────────────────
｜A1"｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜
──────────────────────────────────────
｜A2"｜ ｜ ｜ ｜ ｜ ｜ ｜ z ｜ ｜
──────────────────────────────────────
｜E" ｜ ｜ ｜ ｜ ｜ ｜ ｜ (Rx,Ry) ｜ ｜
──────────────────────────────────────
4.(22%) Suppose we have a planar molecule as shown below
p1 p2
=======
｜ ｜
←── p3｜=====｜p4 ● z
x ｜ ｜
=======
p5 p6
｜
｜y
↓
(a) Using the Huckel MO approximation, determine the MOs of the symmetry B1u
and their energies.
(b) Some of the Huckel molecular orbitals are schematically illustrated, where
positive and negative coefficients of the orbitals are represented by unshaded
and shaded circles, respectively. Note that only the upper lobes of the p
orbitals are shown.
‧=====‧ ●=====● 。=====‧ ‧=====‧ ○=====● 。=====‧
｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜
●=====● ｜=====｜ ○=====● ○=====○ ｜=====｜ ●=====○
｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜ ｜
‧=====‧ ○=====○ 。=====‧ ‧=====‧ ●=====○ 。=====‧
ψ1 ψ2 ψ3 ψ4 ψ5 ψ6
Classify each of these orbitals relative to the point group of the molecule
(i.e. determine the corresponding irrep of each orbital).
Section B (30%)
1.(10%) It has been said that the first point, i.e. t=0, of time domain data
will determine the value of the overall intrgral of the frequency domain
spectrum. Prove the statement using Fourier transformation theory.
2.(20%) Consider the hypothetical dicyclobutadiene organometallic sandwich
compound (C4H4)2M of D4h symmetry:
_________
ψ3/ /ψ2
/ / (PS：每個ψ都是一個p軌域...實在是太難畫了 所以就這樣囉)
/ / (PS：每個p軌域靠近M的那端為+ 遠離M的那端為-)
/________/ ↑
ψ4 ψ1 ｜
z｜ y ↗
● M ／
_________
ψ1'/ /ψ4' ＼
/ / x ↘
/ /
/________/
ψ2' ψ3'
(i) Construct all the symmetry adapted orbitals from the ligand pπ orbitals.
You must normalized all the SALCs.
(ii)Mathch all the metals s, p, and d orbitals with all the ligand symmetry
orbitals to obtain bonding, antibonding, and non-bonding molecular orbitals.
Indicate all the bond types(σ,π,δ,nonbonding)
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