复旦大学2008年物理化学AI期末考试

复旦大学化学系

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：

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:

Js h 341063.6-?=

kg m e 311011.9-?= C

e 191060.1-?=18109989.2-?=ms c

eV mol kJ 210036.1/1-?=

123110381.1581.109677---?==JK cm R H

J cm 23110986.11--?= 12194701-=cm Hartree 231002.6?=A N

λν/hc h E ==

mv

h

=

λ h p x ≥?? h t E ≥?? π

2h

n M =

???

? ??-=λ2221111n n R φθcos sin r x = φθsin sin r y = θcos =z

φθθθθ

θ2

2

22

222

sin 1sin sin 11??+????+????=?r r r r r r φθθd drd r dxdydz sin 2=

ωI M =

l x

n l x πψsin

2)(=

2228ml h n E n = ()ν

h v E v 2/1+=

μ

π

k v 21=

)1(822+=

J J I

h E J π

2

21

n

E -

=(atomic unit)

()()1*=?μμαμαd ()()1*=?μμβμβd ()()()()0**==??μμαμβμμβμαd d

αα21?=z s

ββ2

1

?-=z s φ

??

-= i l z ?

())2sin(41

21sin 2

ax a

x ax dx -=?

θcos 2222ab b a c ++=

)2cos(81

)2sin(44sin 322

bx b

bx b x x dx bx x --=? 22

222221c

l b k a h d ++= ij R

j i g R R δχχ=∑)()(

()R R g a a R

χχ∑Γ=

)(1*

Relative atomic mass: Ag-108 O-16

1. A system is in one of three states described by the wavefunctions 1ψ,2ψ and

3ψ. The wavefunctions are linear combinations of the orthonormal

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

16/57b E =. Which wavefunction correctly describes the state of the system? a) 32112141411φφφψ++=

; b) 32124

11

4121φφφψ++=; c) 32134

1

41121φφφψ++=

Briefly 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 )tan()(ax N x =ψ for ∞≤≤x 0; (3 points) b) Chemist B proposes )exp()(2/1ax Nx x -=ψ for ∞≤≤x 0; (3 points) c) Chemist C proposes )sin()(ax N x =ψ for ∞≤≤x 0 （3 points ） Which chemist is mostly likely to be correct? Justify your answer.

3. Consider the wavefunction ()βαψC N +=, where α and β are spin

functions and N and C are constants. a) Normalize ψ; (4 points)

b) Find Z s for the wavefunction ψ. (4 points)

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 Schr?dinger equation can be written:

ψψφE ma =??-22222 and the eigenfunctions are ()()φπ

φψim exp 21=

a). What are the eigenvalues of this equation? (3 points)

b). When the benzene molecular ion C 6H 6+ is placed in a magnetic field of strength B oriented in the z direction, a perturbation potential z l B V ?γ= is added to the Hamiltonian of each electron ( z l ? is the orbital angular momentum operator, γ is a constant). Evaluate the perturbation energy 'E for this effect. （4 points ）

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

()[]

2221exp r r +-=αψ

in this function, α 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 αα2

11411trial -=

E What is the optimal value of α ? 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 s 1 functions: ()[]21'exp r r Z +=ψ 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.

a) The operator, F

?, that acts on ψ to generate the exponential of ψ: ψψe F

=? （3 points ） b) The operator, F

?, that acts on ψ by integrating it over the range from 0 to x : ?

=x

dx x F 0)(?ψψ. （3 points ）

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

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

()()F 2H 1H F 21z p c s c +=σ

Letting, ()H 11s f = and ()F 22z p f =, and taking a bunch of integral we obtain

the following matrix elements: eV 6.1311-=H , eV 4.1722-=H , eV 0.52112-==H H

12211==S S , 1.02112==S S

Note, that 11H represents the energy of the s 1 electron of a hydrogen atom,

relative to a free electron and 22H represents the energy of the p 2 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 22? secular equation and solve it for the two orbital energies 1E and 2E . （4 points ）

b) Substitute the orbital energy back into the secular equations, and find the ratio of the coefficients 1c and 2c . （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:

9. Consider the H 2O molecule and the basis of the valence orbitals H s 1(A), H s 1(B),

O s 2 and the three O p 2-orbitals. a) Which point group is H 2O? （2 points ）

b) Taking two s 1 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 H 2O using projection operators; Are these orbitals orthogonal? (5 points)

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

f) Determine the resulting representations for the product of 2121B B A A ??? in the point group of H 2O.(2 points)

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) 2s ; b)5p ; c)11p s ; d)1d (9 points)

11. Solid silver exists as a face-centered cubic crystal with A

a 09.4 . a) 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 4d , 5d , 6d , 7d high and low spin configurations in (a)

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