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课程名称︰化学数学二 课程性质︰化学系选修 课程教师︰陈振中 开课学院:理学院 开课系所︰化学系 考试日期(年月日)︰2011/6/22 考试时限(分钟):120min 是否需发放奖励金:是 试题 : 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:每个ψ都是一个p轨域...实在是太难画了 所以就这样罗) / / (PS:每个p轨域靠近M的那端为+ 远离M的那端为-) /________/ ↑ 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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