quantum chemistry and...

Chem 4502 Prof. Doreen Leopold __________________________________ 12/23/2015 (Wednesday) Name (Optional) Quantum Chemistry and Spectroscopy

Final Exam (185 points, 37 questions, 22 pages)

• Students have 2 hours to do this exam (1:30 PM - 3:30 PM).

• It has 37 multiple choice questions worth 5 points each, for a total of 185 points.

Each question will be graded as 0 or 5 points; there is no partial credit.

There is just one correct answer for each question, and no penalty for incorrect answers.

If more than one answer is bubbled in, no credit will be given for that question.

• On the bubble sheets, please bubble in your name (last name first), X.500, student ID number, and

sign the back.

• You can take the question portion of the exam with you. The answers will be posted later today.

• There are 6 - 8 questions on the material potentially covered on midterm Exams 1, 2, 3, and 4,

and 8 questions on molecular spectroscopy (Chapter 13 and Homework 10).

• You may use a non-programmable, non-graphing calculator.

No notes are allowed.

The equation sheets (2 pages) are included at the end of the exam (and may be torn off).

• Final exam grades, a bar graph with statistics, and a description of the end-of-semester grading curve

will be posted by Wednesday, December 30.

PLEASE DO NOT TURN THE PAGE UNTIL IT IS ANNOUNCED THAT YOU MAY BEGIN.

2

Your Exam is Version A. Please check that your bubble sheet is premarked with the correct version.

Part I. (6 questions, 30 points) - Questions selected from among the following topics:

2-Slit Experiment, Dawn of Quantum Mechanics, Classical Wave Equation, Complex Numbers,

Schrödinger Equation, Operators, Eigenvalue Problems, Wave Functions

(Homeworks 1-3 and Problem Set 1).

1. Consider a 2-slit type of experiment performed with single electrons (all having the same

kinetic energy). Which statement(s) is/are true concerning the possibility of observing an

interference pattern on the detector? Choose the best single answer among A - F.

A. Many electrons must be in the apparatus at a given time to observe an interference pattern.

B. To observe an interference pattern, one must determine which slit each electron passes through.

C. To observe an interference pattern, both slits must be open at the same time.

D. An interference pattern cannot be observed using electrons (but it can be observed using photons).

E. An interference pattern is observed each time a single electron passes through the apparatus.

F. More than one of the above statements are true.

2. In a fictional universe, the value of Planck's constant is different than in ours.

There, a 1900's style photoelectric effect experiment on a certain metal gave the following

results. (Recall that the "stopping voltage", which is negative with respect to the metal, is

the voltage required to stop the photoelectrons.)

frequency of light magnitude of stopping voltage

5.5 x 1014 s-1 8.0 V

8.0 x 1014 s-1 15.0 V

Use the above data to solve for h in that universe (in the usual units of J•s).

A. 2.3 x 10-34

B. 6.6 x 10-34

C. 2.3 x 10-33

D. 3.0 x 10-33

E. 4.5 x 10-33

F. 9.0 x 10-33

G. 1.1 x 10-18

H. 1.5 x 10-14

I. 1.9 x 10-14

J. 2.8 x 10-14

3

3. Consider a particle of mass m (in kilograms) with de Broglie wavelength = 2a,

where a is a constant with units of meters.

Calculate the kinetic energy of the particle (in Joules), in terms of m, a and h.

A. 3 h / ( 2 a )

B. h / a

C. h / (2 a)

D. 4 h2 / a2

E. 9 h2 / (4 a2 )

F. h2 / a2

G. h2 / ( 2 a2 )

H. 9 h2 / (8 m a2 )

I. 4 h2 / (8 m a2 )

J. h2 / (8 m a2 )

4. What is the minimum uncertainty in the speed of an electron (in units of m/s) known to be

located within a 3.0 Å interval ? Choose the closest answer.

A. 2 x 108

B. 2 x 106

C. 2 x 103

D. 0.2

E. 2 x 10-2

F. 2 x 10-4

G. 2 x 10-9

H. 2 x 10-13

I. 2 x 10-17

J. 2 x 10-24

4

5. In the emission spectrum of the hydrogen atom, what is the longest wavelength line in the

Paschen series (which terminates at n = 3), in units of cm-1 ?

A. 5,300

B. 6,900

C. 9,100

D. 12,200

E. 24,400

F. 36,600

G. 61,000

H. 97,500

I. 109,700

J. 768,000

6. Wave numbers, (cm-1) = 1 / , are often used to describe radiation in the infrared (IR) region.

For IR radiation with = 1,500 cm-1, what is the photon energy (in Joules)?

A. 6.7 x 10-6

B. 5.2 x 10-9

C. 2.2 x 10-14

D. 8.3 x 10-16

E. 1.3 x 10-18

F. 3.0 x 10-20

G. 3.0 x 10-22

H. 4.6 x 10-24

I. 1.3 x 10-26

J. 4.6 x 10-28

5

Part II (8 questions, 40 points) - Questions selected from among the following topics:

Particle-in-a-Box, Probability and Statistics, Harmonic Oscillator, Postulates and Principles of

Quantum Mechanics (Homeworks 4 & 5 and Problem Set 2).

7. The one-dimensional particle-in-a-box model can be used to model electronic transitions of

butadiene (H2C=CH–CH=CH2), in which the and * molecular orbitals are nondegenerate.

According to this "free electron model", if the length of the molecule is 6.0 Å, what photon energy is

required to promote an electron from the HOMO to the * LUMO?

A. 3.3 x 10-20 J

B. 1.7 x 10-19 J

C. 3.4 x 10-19 J

D. 5.0 x 10-19 J

E. 8.4 x 10-19 J

F. 1.2 x 10-18 J

G. 1.5 x 10-18 J

H. 2.7 x 10-18 J

8. Assume that a two-dimensional Hamiltonian operator is separable into one-dimensional terms,

Ĥ(x,y) = Ĥ(x) + Ĥ(y) , where the one-dimensional solutions to the Schrödinger equation are

Ĥx (nx) = E(nx) (nx) and

Ĥy (ny) = E(ny) (ny).

Which of the following substitutions will allow us to solve the two-dimensional Schrödinger equation,

Ĥ(x,y) (nx, ny) = E(nx, ny) (nx, ny) ?

A. (nx, ny) = (nx) (ny) and E(nx, ny) = E(nx) E(ny)

B. (nx, ny) = (nx) (ny) and E(nx, ny) = E(nx) + E(ny)

C. (nx, ny) = (nx) + (ny) and E(nx, ny) = E(nx) E(ny)

D. (nx, ny) = (nx) + (ny) and E(nx, ny) = E(nx) + E(ny)

6

9. In view of the selection rules for vibrational spectroscopy, which of the following molecules

are not expected to absorb infrared light? Choose the best answer.

O2 , CO2 , H2O

A. O2 only

B. CO2 only

C. H2O only

D. O2 and CO2

E. O2 and H2O

F. H2O and CO2

G. All 3 molecules will absorb infrared light

H. None of these molecules will absorb infrared light

10. The vibrational frequency of 39K35Cl is 278 cm-1. Assuming the harmonic oscillator

approximation, calculate the force constant of the chemical bond in diatomic

potassium chloride, in units of N / m ( = kg / s2).

A. 9.3 x 10-24

B. 9.3 x 10-20

C. 1.6 x 10-12

D. 8.4 x 10-3

E. 9.2

F. 84

G. 130

H. 300

I. 320

J. 5.1 x 1028

7

11. For a system characterized by a wave function that is a linear combination of two or more

eigenfunctions of an operator, a postulate of quantum mechanics predicts the possible outcomes of a

single, ideal measurement of the observable property associated with that operator. According to this

postulate, this type of measurement will yield:

A. a value that is less than the eigenvalue associated with the lowest energy eigenfunction

B. a linear combination of two or more eigenvalues of this operator

C. the average value of two or more eigenvalues of this operator

D. one of the eigenvalues of this operator

E. the eigenvalue associated with the lowest energy eigenfunction

12. Which of the following statements is/are true concerning the harmonic oscillator model for the

vibrational motion of a diatomic molecule?

I. The potential energy operator is (-ℏ2/(2µ)) d 2/dx 2 (where µ is the reduced mass).

II. For a molecule in an eigenstate (with quantum number v) of the Hamiltonian operator,

the distance between the two nuclei is well-defined.

III. As quantum number v increases, the atoms are more likely to be found near the classical

turning points.

A. I only

B. II only

C. III only

D. I and II only

E. I and III only

F. II and III only

G. I, II and III are all true

H. None of these 3 statements (I, II, III) are true.

.

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13. Which of the following statements is/are true concerning tunneling?

I. Tunneling is predicted in both the harmonic oscillator and the particle-in-a-box models.

II. In the harmonic oscillator model, the probability of tunneling decreases as the vibrational

quantum number increases.

III. For the ground vibrational state, according to the harmonic oscillator approximation, the

tunneling probability for HBr is greater than for its deuterium isotope, DBr.

A. I only

B. II only

C. III only

D. I and II only

E. I and III only

F. II and III only



14. Consider a particle of mass m in a one-dimensional box with walls at x = -a and x = +a, in a

state characterized by the following wave function:

What is the energy of the particle in this state?

A. h2 / (32 m a2)

B. h2 / (8 m a2)

C. 9 h2 / (32 m a2)

D. h2 / (2 m a2)

E. 25 h2 / (32 m a2)

F. 9 h2 / (8 m a2)

G. 2 h2 / (m a2)

H. 8 h2 / (m a2)

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Part III (7 questions, 35 points) - Questions selected from among the following topics:

Rigid Rotator, Hydrogen Atom, Angular Momentum Vectors, Spherical Coordinates

(Homeworks 6 & 7 and Problem Set 3).

15. For the normalized wave function for the ground state of the hydrogen atom (1s), what is the

meaning of the following expression?

A. the average value of the kinetic energy

B. the average value of the potential energy

C. the average value of the total energy

D. the most probable value of the kinetic energy

E. the most probable value of the potential energy

F. the most probable value of the total energy

G. the probability to find the electron between r = 0 and r =

16. Below is a graph of the radial distribution function (proportional to r 2 R

2) for one of the

hydrogen atomic orbitals. Which orbital is it?

A. 1s

B. 2s

C. 2p

D. 3s

E. 3p

F. 3d

drrr

e r

r

ss

0

21

*1

0

2

41

4

5 0 10 15 20 r / a0

r 2 R

2

10

17. The angular momentum vector diagram shown below represents one of the spherical harmonics

having specific values of ℓ and m. What is the best description of the corresponding orbital?

A. s

B. px

C. py

D. pz

E. dz2

F. dyz

G. dxz

H. dxy

I. dx2-y2

18. The following is a probability density plot for one of the hydrogen atomic orbitals.

Which orbital is it?

A. 1s

B. 2s

C. 2p

D. 3s

E. 3p

F. 3d

G. 4s

y

z

2 ℏ

11

19. What is the value of r at the classical turning point for a hydrogen atom in n = 3?

(Hint: use one of the equations for the energy on the equation sheet.)

A. a0

B. 1.5 a0

C. 2 a0

D. 3 a0

E. 4 a0

F. 6 a0

G. 8 a0

H. 12 a0

I. 18 a0

J. 24 a0

20. According to the rigid rotator approximation, which of the following statement(s) is/are true for a

rotating gas phase molecule? Choose the best answer.

I. In the lowest energy state, the molecule has zero rotational kinetic energy.

II. The selection rule for the absorption or emission of light is: J = 0, 1, 2, 3, ...

III. To have a pure rotational spectrum, a molecule must have a permanent dipole moment.

A. I only

B. II only

C. III only

D. I and II only

E. I and III only

F. II and III only



12

21. The microwave spectrum of 39K127I consists of a series of lines spaced by

3600 MHz. The reduced mass of 39K127I is 4.95 x 10-26 kg.

What is this molecule's equilibrium bond length?

A. 3.1 x 10-7 m

B. 3.6 x 10-10 m

C. 3.1 x 10-10 m

D. 2.7 x 10-10 m

E. 2.2 x 10-10 m

F. 1.7 x 10-10 m

G. 1.2 x 10-10 m

H. 9.4 x 10-11 m

I. 4.7 x 10-14 m

J. 9.4 x 10-20 m

Part IV (8 questions, 40 points) - Questions selected from among the following topics:

Helium Atom, Variational Method; Multielectron Atoms; Determinants, Diatomic Molecules

(Homeworks 7 - 9 and Problem Set 4).

22. Slater determinants are used for writing atomic and molecular wave functions because they satisfy which of the following requirements? Choose the best answer.

A. The wave function is zero if any two electron labels are interchanged.

B. The wave function changes sign if any two electron labels are interchanged.

C. The wave function is zero if any two electrons occupy the same spin-orbital.

D. The wave function changes sign if any two electrons occupy the same spin-orbital.

E. both A and C

F. both A and D

G. both B and C

H. both B and D

13

23. For homonuclear diatomics like O2, combining one 2s and three 2p atomic orbitals on each atom

yields ________ bonding and ________ antibonding molecular orbitals.

A. 2 bonding and 2 antibonding

B. 3 bonding and 3 antibonding

C. 4 bonding and 4 antibonding

D. 8 bonding and 8 antibonding

E. 2 bonding and 0 antibonding

F. 4 bonding and 0 antibonding

G. 4 bonding and 2 antibonding

H. 6 bonding and 0 antibonding

I. 6 bonding and 2 antibonding

J. 6 bonding and 4 antibonding

24. For O2, which of the following statements is false regarding the excited 3u "B" state?

A. The transition from the ground state is allowed by the spectroscopic spin selection rule.

B. The bond order is higher in the B state than in the ground state.

C. The B state has a longer bond length and a lower vibrational frequency than the ground state.

D. The B state can be accessed from the ground state by the absorption of ultraviolet light.

E. More than one of the above statements (A-D) is false.

14

25. In the simplest LCAO-MO description of H2+, the probability for finding the electron in the

bonding molecular orbital can be calculated as:

Here, the last term (indicated by the bracket)

A. is due to constructive interference

B. is due to destructive interference

C. is due to electron-electron repulsion

D. is zero because the atomic orbitals are orthogonal

E. approaches zero as the internuclear separation decreases

F. more than one of the above statements (A - E) are true

26. The term symbols associated with the 2p3 electron configuration of the nitrogen atom are

2P, 2D, and 4S. According to Hund's Rules,

the lowest energy term is __________ and the highest energy term is __________ .

A. lowest 2P, highest 2D

B. lowest 2P, highest 4S

C. lowest 2D, highest 2P

D. lowest 2D, highest 4S

E. lowest 4S, highest 2P

F. lowest 4S, highest 2D

15

27. The variational principle involves the calculation of the quantity:

d

dHE

*

* ˆ

In this expression, in general,

is _______________________________________________ ;

H is _____________________________ ;

and the variational principle states that _________________________ .

A. is the true ground state wave function; H is the exact Hamiltonian operator;

states that E is less than the true ground state energy

B. is the true ground state wave function; H is an approximate Hamiltonian operator;


C. is a trial wave function; H is the exact Hamiltonian operator;


D. is a trial wave function; H is an approximate Hamiltonian operator;


E. is the true ground state wave function; H is the exact Hamiltonian operator;

states that E is not less than the true ground state energy

F. is the true ground state wave function; H is an approximate Hamiltonian operator;


G. is a trial wave function; H is the exact Hamiltonian operator;


H. is a trial wave function; H is an approximate Hamiltonian operator;


16

28. Which of the following statements is/are true concerning Hartree-Fock calculations?

I. The instantaneous correlations of the electrons' motions are neglected.

II. The total energy of the He atom is calculated to be about 1 eV more negative than the true energy.

III. The total energy of an atom is calculated as the sum of the energies of the occupied spin-orbitals.

A. I only

B. II only

C. III only

D. I and II

E. I and III

F. II and III

G. I, II and III

H. None of these statements (I, II, III) are true.

29. What is the highest occupied MO in the N2+ molecular ion?

A. u 2pz

B. g

C. g 2pz

D. u

E. u 2s

F. g 2s

17

Part V. (8 questions, 40 points)

Topics: Molecular Spectroscopy (Homework 10)

30. Which of the following statements is/are true regarding the harmonic oscillator (HO) as

compared with the Morse potential models for molecular vibrations?

A. In the HO model, the energy levels are equally spaced, but in the Morse potential model,

spacings between consecutive vibrational levels increase with increasing energy.

B. In the HO model, there is only a finite number of vibrational levels, but in the Morse potential

model, there is an infinite number.

C. In the HO model, the average amplitude of the vibration remains the same as the quantum number

v increases, but in the Morse potential model, the amplitude decreases with increasing v.

D. More than one of the above statements are true.

E. None of the above statements (A-C) are true.

31. For a gas phase molecule in a sealed container, which of the following choices correctly ranks

(smallest to largest) the typical energy spacings for various types of quantum states?

A. electronic < translational < vibrational < rotational

B. rotational < vibrational < translational < electronic

C. rotational < translational < vibrational < electronic

D. rotational < electronic < translational < vibrational

E. translational < vibrational < rotational < electronic

F. translational < rotational < vibrational < electronic

G. translational < electronic < rotational < vibrational

H. vibrational < translational < rotational < electronic

I. vibrational < rotational < translational < electronic

J. vibrational < rotational < electronic < translational

18

32. Consider a diatomic molecule with rotational constant B = 10 cm-1. At 300 K, what will be the

relative population of molecules in the J = 3 rotational state as compared with J = 0?

A. about 2 times more molecules in J = 3 than in J = 0

B. about 4 times more molecules in J = 3 than in J = 0

C. about 7 times more molecules in J = 3 than in J = 0

D. about 12 times more molecules in J = 3 than in J = 0

E. about 2 times more molecules in J = 0 than in J = 3

F. about 4 times more molecules in J = 0 than in J = 3

G. about 7 times more molecules in J = 0 than in J = 3

H. about 12 times more molecules in J = 0 than in J = 3

33. For benzene (C6H6), what are the numbers of translational, rotational, and vibrational degrees

of freedom?

A. 1 translational, 2 rotational and 12 vibrational

B. 1 translational, 3 rotational and 32 vibrational

C. 2 translational, 3 rotational and 31 vibrational

D. 3 translational, 2 rotational and 31 vibrational

E. 3 translational, 2 rotational and 36 vibrational

F. 3 translational, 3 rotational and 12 vibrational

G. 3 translational, 3 rotational and 30 vibrational

H. 12 translational, 12 rotational and 12 vibrational

19

34. In the rotational-vibrational spectrum of a gas phase diatomic molecule, the absorption lines are

spaced by about 20 cm-1. If the vibrational frequency is 2,000 cm-1, what is the "energy" (in cm-1) of

the transition from J = 3 in the lower vibrational state to J = 2 in the upper vibrational state?

A. 2120 cm-1

B. 2060 cm-1

C. 2000 cm-1

D. 1960 cm-1

E. 1940 cm-1

F. 1880 cm-1

G. 240 cm-1

H. 180 cm-1

I. 120 cm-1

J. 60 cm-1

20

35. In the infrared spectrum of carbon monoxide (12C16O), the fundamental line is observed at 2143

cm-1 and the first overtone occurs at 4260 cm-1. What are the values of the harmonic frequency (e)

and the anharmonicity constant (exe) for this molecule?

A. e = 2117 cm-1, exe = 13 cm-1

B. e = 2117 cm-1, exe = 26 cm-1

C. e = 2130 cm-1, exe = 13 cm-1

D. e = 2143 cm-1, exe = 26 cm-1

E. e = 2143 cm-1, exe = 2117 cm-1

F. e = 2156 cm-1, exe = 7.5 cm-1

G. e = 2156 cm-1, exe = 13 cm-1

H. e = 2169 cm-1, exe = 13 cm-1

I. e = 2169 cm-1, exe = 26 cm-1

J. e = 4260 cm-1, exe = 2117 cm-1

36. The visible emission spectrum of gas phase I2 following laser excitation at 514 nm shows a long

vibrational progression. Which of the following statements is/are true concerning this spectrum?

A. The ground and excited electronic states have similar equilibrium bond lengths.

B. Most of the emitted light has wavelengths shorter than 514 nm.

C. I2 relaxes to the zero point level of the excited electronic state before emitting visible light.

D. I2 relaxes to a wide range of vibrational levels in the ground electronic state.

E. More than one of the above statements are true.

37. In this course, we have applied the Schrodinger equation to various systems. For which system(s)

did we set the potential energy term equal to zero when setting up the Schrödinger equation? Choose

the best answer.

A. particle-in-a-box

B. harmonic oscillator

C. rigid rotator

D. hydrogen atom

E. helium atom

F. H2 molecule

G. two of the above

H. three of the above (among A - F)

21

Possibly Useful Equations, Conversions and Constants Chem 4502

c = 2.998 x 108 m/s k B = 0.695 cm-1 / K

h = 6.626 x 10−34 J·s = h/(2π) = 1.055 x 10−34 J·s

e = 1.602 x 10−19 C

1 eV = 1.602 x 10−19 J (corresponds to 8066 cm-1)

me = 9.109 x 10−31 kg mp = 1.673 x 10−27 kg amu = 1.661 x 10-27 kg

H atom

eix = cos x + i sin x

Classical wave equation ∂2u(x,t) / ∂x2 = (1/2) ∂2u(x,t) / ∂t2

Normal modes of a vibrating string of length : un(x,t) = An cos (ωnt + n) sin (n π x / )

Schrödinger equation: (−2 / (2m)) d

2Ψ / dx2 + V(x)Ψ(x) = EΨ(x)

Momentum operator: = -i ∂/∂x

--------------------------------------------------------------------------------------------------------------------- PIB ψn(x) = (2/a)½ sin (nπx / a) En = n2h2 / (8ma2) HO where a = (μk)½

/

Ev = (v+½) ho where o = (1 / (2π)) (k/μ)½ and μ = m1m2 / (m1+m2) (diatomic)

---------------------------------------------------------------------------------------------------------------------

Rotational motion: K = ½ Iω2 = L2 / 2I I = μr 2 L = Iω = mrv = v/r

Rigid rotator: EJ = ( 2/2I ) J(J+1) E (cm-1) = B J(J+1)

B (Hz) = h / ( 8π2I ) B (cm-1) = h / ( 8π2cI )

Hydrogen atom:

1s = (1/π)½ (1/ao)

3/2 e - r / ao m() = (1/(2)½) e im

Angular momentum: L2 = ħ2 (+1) Lz = m Lz = - i ħ ∂/∂

Spherical coordinates:

dV = r 2 sinθ dr dθ d

-a x2 / 2 ψ0(x) = (a / π)¼ e

En = - me e 4 1 = - 1 e2 = - 1 (13.6 eV) 8ε0

2h2 n2 n2 (4πε0) 2ao n2

ao = ε0 h2 = 0.529 Å

π me e2

ν ~ = 109,678 (1/n1

2 - 1/n22) cm-1

˜

˜

ˆ

Px

En = - me e

4 1 8ε0

2h2 n2 r = ε0 h

2 n2 π me e

2

ν ~ = 109,678 (1/n1

2 - 1/n22) cm-1

22

Boltzmann distribution: Nj / Ni = (gj / gi) e k = 0.695 cm-1

/ K

Morse potential: G(v) = (v+½) e - (v+½)2 exe De = e2 / (4 exe)

G(v) - G(0) = v e - v(v+1) exe

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quantum chemistry and...

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