复旦大学2008年物理化学AI期末考试

2007～2008学年第一学期期末考试试卷

□A卷

课程名称： Physical Chemistry 课程代码：____________________

开课院系：__ Chemistry _ 考试形式：闭 卷 成绩：

姓 名： 学 号： 专 业： No. Score No. Score This is a closed book exam. Use of a calculator and an English dictionary is permitted. Show all of your work and check your units carefully. Don’t give help to, or get help from, others. Thanks for your cooperation. GOOD LUCK！ Some useful constants and results are:

h6.631034Js

1 9 2 10 3 11 4 12 5 13 6 14 7 15 8 16 me9.111031kg e1.601019Cc2.99108ms1

1kJ/mol1.036102eV

RH109677.581cm11.3811023JK1

h21cm11.9861023J 1Hartree219470cm1 NA6.021023 Ehhc/

h mvxph Eth Mn

111 R22n1n22xrsincos yrsinsin zcos

1112 2r2sin222rsinrrrrsinMI

dxdydzr2sindrdd

(x)k

n2h22nx En sinll8ml2h282IJ(J1)

EEvv1/2h1(atomic unit) 22n

1v2EJ*d1 *d1 *d*d0

ˆzs11ˆz  s22lˆzi

2axdxsin11xsin(2ax) 24a

c2a2b22abcos

x2x1xsinbxdxsin(2bx)cos(2bx) 344b8b21h2k2l2222 2dabci(R)j(R)gij

R a1*(R)aR gRRelative atomic mass: Ag-108 O-16 1. A system is in one of three states described by the wavefunctions 1,2 and

3. The wavefunctions are linear combinations of the orthonormal

eigenfunctions 1， 2, 3 of the Hamiltonian operator, which have eigenvalues of b, 4b, and 9b, respectively. The average of a large number of measurements on identically prepared systems in the same state finds

E57b/16. Which wavefunction correctly describes the state of the system? a) 111111111123; b) 2123; 4422441111 c) 3123

244Briefly give your reason by calculating. （8 points）

2. When considering a one dimensional quantum mechanical system, three chemists

propose three different wave functions (in each N and a are constants): a) Chemist A proposes (x)Ntan(ax) for 0x; (3 points) b) Chemist B proposes (x)Nx1/2exp(ax) for 0x; (3 points) c) Chemist C proposes (x)Nsin(ax) for 0x （3 points） Which chemist is mostly likely to be correct? Justify your answer.

3. Consider the wavefunction NC, where functions and N and C are constants. a) Normalize ; (4 points)

b) Find sZ for the wavefunction . (4 points)

and  are spin

4. The motion of a π electron in benzene can be treated a particle on a ring. If the

particle has mass m and the ring is of radius a, the Schrödinger equation can be written:

22E and the eigenfunctions are 2ma22a). What are the eigenvalues of this equation? (3 points)

12expim

b). When the benzene molecular ion C6H6+ is placed in a magnetic field of strength B oriented in the z direction, a perturbation potential VBlˆz is added to the Hamiltonian of each electron ( lˆz is the orbital angular momentum operator,  is a constant). Evaluate the perturbation energy E' for this effect. （4 points）

5. Consider the following Gaussian trial function for the He atom:

expr12r22

in this function,  is a variational parameter.

a). Write down the complete expression of for determining the trial energy in a variational calculation. Be sure to indicate all of the terms in the Hamiltonian. （4 points）

b) Suppose the resulting expression for the trial energy (in hartree units) is Etrial1111 42What is the optimal value of  ? What is the optimized energy? （4 points） c) Which is the better trial wavefunction, the Gaussian one presented in part (a) or the product of 1s functions: expZ'r1r2 which gives an optimized energy of -2.838 (also in hartree units)? Explain. (3 points)

6. Classify each of the following operators as linear or non-linear. Justify your

answer.

ˆ, that acts on  to generate the exponential of : a) The operator, Fˆe （3 points） Fˆ, that acts on  by integrating it over the range from 0 to b) The operator, Fˆ(x)dx. （3 points） x: F0x

7. In the HF molecule the  bond is formed primarily from the 1s orbital of

hydrogen and the 2pz orbital on the fluorine. To construct this molecular orbital, we must form a linear combination of atomic orbitals

HFc11sHc22pzF

Letting, f11sH and f22pzF, and taking a bunch of integral we obtain

the following matrix elements:

H1113.6eV, H2217.4eV, H12H215.0eV S11S221, S12S210.1

Note, that H11 represents the energy of the 1s electron of a hydrogen atom,

relative to a free electron and H22 represents the energy of the 2p orbital of a fluorine atom, relative to a free electron, thus these numbers are the negratives of ionization potential of these atom.

Using these values:

a) Set up the 22 secular equation and solve it for the two orbital energies E1 and E2. （4 points）

b) Substitute the orbital energy back into the secular equations, and find the ratio of the coefficients c1 and c2. （4 points） c) Normalize the obtained orbitals. （3 points）

d) When an electron is placed into the bonding orbital, is it more localized on F or on H? Give the lowest electron configuration for the HF molecule and predict the dipole moment of this molecule. （3 points）

8. Consider the molecules cis-ClHC=CHCl（顺式）, trans-ClHC=CHCl（反式）,

benzene and CHClFBr.

a) Classify them according to their point group symmetry; （4 points） b) Which of the molecules may possess a permanent electric dipole moment? （4

points）

c) Which of the molecules may possess an optical rotation (activity)? （4 points） Answer:

point group symmetry electric dipole momen optical rotation

cis-ClHC=CHCl trans-ClHC=CHCl benzene CHClFBr

9. Consider the H2O molecule and the basis of the valence orbitals H1s(A), H1s(B),

O2s and the three O2p-orbitals. a) Which point group is H2O? （2 points）

b) Taking two 1s orbitals of hydrogen atom as bases, find the characters of the representation; （4 points）

c) Determine whether it is an irreducible or reducible representations, if it is a reducible representation, reduce it to irreducible representations; (3 points) d) Find the symmetry-adapted linear combinations of H2O using projection operators; Are these orbitals orthogonal? (5 points)

e) Write out the forms of the two valence molecular orbitals of H2O using the LCAO-MO method. (2 points)

f) Determine the resulting representations for the product of A1A2B1B2 in the point group of H2O.(2 points)

E C2 1 1 -1 -1 v 1 -1 1 -1 v' 1 -1 -1 1 A1 A2 1 1 1 1 z Rz x2，y2，z2 xy B1 B2

x，Ry y，Rx xz yz 10. In class we worked out the possible term（光谱支项） symbols for two equivalent

p electrons. Now do the same thing for the following system.

a) s2; b)p5; c)s1p1; d)d1 (9 points)

. 11. Solid silver exists as a face-centered cubic crystal with a4.09Aa) What is the density of silver? Assume that each silver atom has a mass of 108

amu. （4 points）

b) What is the distances for the (100),(110), and（111）planes.(6 points)

12. Write down the d4, d5, d6, d7 high and low spin configurations in (a)

octahedral and (b) tetrahedral ligand field; work out which configurations show Jahn-Teller distortion.(8 points)