CHEM 4300 Take-Home Final Exam
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CHEM 4300 Take-Home Final Exam
1 Multiple Choice (20 points total, 1 point each). Check, mark or indicate the answer(s).
1.1 The eigenvalues of a general matrix cannot be complex. True False
1.2 The Hamiltonian in Cartesian coordinates is 
, and the wavefunction is 
, then the energy is:
For the next two questions: The states of the two-step well potential have energies and wavefunction amplitudes shown at right.
1.3 The states of energy greater than 1 Hartree are:
unbound continuum bound discrete
1.4 The tunneling splitting is the difference in energy between which levels
top of well and bottom average of two lowest and top of well
highest an 2nd highest 2nd lowest and lowest
1.5 Choose the polynomial-related portion of the wavefunction solutions for the harmonic oscillator, rigid rotor, and the radial part of the hydrogen atom, respectively.
Legendre, Laguerre, Hermite Hermite, Laguerre, Legendre Hermite, Legendre, Laguerre
1.6 The blackbody spectrum of an object at 865 K has a maximum closest to which wavelength
3.4 µm 630 µm 3400 µm 0.63 µm
1.7 In what wavelength regime is the radiation maximally emitted by the walls of a room at 298 K?
Microwave Infrared Visible Ultraviolet X-ray
1.8 Write the letter of the model problem energy levels which go with the H-atom.
______
1.9 The ionizing transitions shown in the photoelectron spectrum of CO(g) diatomic molecule (at right) arise from different _____.
vibrations total electronic energy
orbitals spin-orbit states rotations
1.10 Choose the p*-molecular orbital of a diatomic molecule.
1.11 The ground state of the O+ atomic cation is 2P3/2. True False
1.12 The term symbol for the ground state of C atom is 3P0 3P1 3P2 1D2 1S0
1.13 The CH4 molecule is an asymmetric top. True False
1.14 There are 4 kinds of integrals that result when one makes a variational treatment of the energy of the H2+ ion using a basis set of two 1s orbitals - one on each nucleus. Which one is most responsible for the large well in the electronic potential energy about the equilibrium bond distance?
Coulomb (J) Exchange (K) One Electron H-Like Overlap
1.15 Choose the region in the drawing which represents a semiconductor in solid state band theory (see Section 4.4.3 in textbook).
left middle right
1.16 Choose the scaling parameter for two dimensional growth (see Section 4.4.5 in textbook).
1.17 Which of the different, two electron, He electronic states have the largest value of electron-electron repulsion?
singlet 1s12p1 triplet 1s12p1 singlet 1s2 triplet 2s14p1
1.18 The bond order of the beryllium dimer, Be2 is 1. True False
1.19 The definition of temperature in statistical mechanics is a derivative of energy with respect to entropy at constant volume and number of particles.
True False
1.20 In the canonical ensemble, the average energy can be related to a derivative of the natural log of the partition function with respect to temperature (constant N and V).
True False
2 Brief Questions (20 points total, 1 point each) Answer briefly and mathematically if possible.
2.1 When potential is independent of time, the total wavefunction can be written as a product of spatial and temporal parts as 
. Determine the period in seconds of a 2s electron in an H-atom using the approximate energy (3 sig. figs) of 
in atomic units.
2.2 The MATLAB code to find the eigenvalues of a matrix is
H=[1 1 0; 0 1 1;1 0 1]
[C E1]=eigs(H); % find the eigenvectors (C) and eigenvalues (E)
Type in the elements of the matrix H= 
being addressed by the first line and report the eigenvalues _____ ______ _____.
2.3 Write the Secular Equation for the matrices 
and 
.
2.4 What is the result of this operation 
?
14 
2.5 Finish the reaction expression corresponding for the total electronic energy of the HF diatomic molecule
2.6 The uncertainty principle is 
where 
. If we choose the position operator to be the same as in classical physics, i.e. 
, what is the corresponding quantum mechanical operator for the z-component of momentum? Be mathematical if possible.
2.7 What is the excess kinetic energy (in eV) of an electron ejected from a Nickel metal surface, with a work function of 5.01 eV, by a laser operating at 205 nm? __________
2.8 Write an expression for the commutator of two operators: 
2.9 What is the value of 
for H atom in atomic units using the general formula 
?
2.10 CH3CH2CH2OH has ______ vibrational degrees of freedom (fundamental vibrations).
2.11 What matrix is diagonalized for rotational constants? _____________________________
2.12 What matrix is diagonalized for vibrational fundamentals? ___________________________
2.13 Write out the internal electronic Hamiltonian for H2+ (i.e. within the Born-Oppernheimer approximation) in atomic units based on the diagram at right
2.14 In the canonical ensemble, write an expression for the average pressure in terms of the partition function Q.
2.15 What is the difference between the Canonical Ensemble and the Grand Canonical Ensemble (see Fig. 5.8 in text)?
2.16 A molecule with N atoms has 3N degrees of freedom. What are the other two types of degrees of freedom besides vibrations? ______________________ ____________________________
2.17 Sketch/draw the amplitude vs radial distance for the
radial distribution function of a 3s orbital. Label you axes.
The next three questions correspond to the homonuclear diatomic energy diagram at right.
2.18 What is the highest occupied molecular orbital (HOMO) of the O2 diatomic molecule?
pu 2p p*g 2p sg 2p s*u 2p sg 2s
2.19 What is the bond order of the C2 and O2 diatomic molecules, respectively?
2, 2 1 ,1 2 ,2.5 2.5, 2 1, 2
2.20 How many unpaired electrons have the O2- and F2+ diatomic ions, respectively?
1, 2 2, 2 1, 1 2, 3 3, 2
3 HO Raising and Lowering Operators (10 points). A diatomic molecule with effective mass of m is modeled as a harmonic oscillator with 
where 
and 
. The raising and lowering operators are defined in terms of the position (displacement from equilibrium) and momentum operators as 
and 
where 
. The diatomic displacement from equilibrium is x=
where 
is the diatomic molecule’s bond distance and 
is the equilibrium bond distance.
3.1 (2 points) Write an expression for the operator 
in terms of 
and 
3.2 (2 points) Recalling that 
is displacement from equilibrium, explain briefly in words why 
in terms of raising and lowering operators, Kronecker Delta functions, and your answer to 4.1.
3.3 (5 points) Evaluate the root-mean-square displacement, 
, using raising and lowering operators. Show all work for credit.
3.4 (1 point) In light of the previous answer, does a harmonic oscillator move vibrationally in the ground state? Yes No
4 H Atom. (10 points). The wavefunction of the 1s ground state of the H atom is
in spherical coordinates (
).
4.1 (1 point) What is the value of 
for H atom? _________
For the next two questions: The H atom orbitals in the figure at right all have the same energy and comprise a single shell, i.e. share a common principal quantum number.
4.2 (1 point) What is n, the principal quantum number? _______
4.3 (1 point) What are the 
,
quantum numbers of the 3rd and 4th orbitals from the left?
4.4 (2 points) Write a specific integral expression for calculating the expectation value of radial distance for a 1s electron in the H-atom using 
, 
, 
, and a spherical coordinate expression for the volume element.
4.5 (5 points) Evaluate the above expression in atomic units for a 1s electron in the H-atom. Recall that 
and show your work.
5 Diatomic Matrix Approach (10 points).
The drawing at right integrates the electronic, vibrational, and rotation aspects of the energy of a diatomic molecule.
5.1 (1 point) Which of A, B, or C labels the electronic part of the problem?
5.2 (1 point) Which of A, B, or C labels the rotational part of the problem?
Approach the H2+ diatomic molecule using a minimal basis set of a 1sA orbital on an H atom (labelled A), a 1sB orbital on an H atom (labelled B), and a bond distance R between A and B. Using the Variational Matrix Approach, the Hamiltonian is 2x2 and one solves the secular equation, 
.
5.2 (2 points) Write out the internal electronic Hamiltonian for H2+ (i.e. within the Born-Oppernheimer approximation) in atomic units based on the diagram above
There are certain integrals which are important in this 2x2 Hamiltonian. The diagonal integrals of the overlap matrix are 
and the off diagonal integrals of the overlap matrix are 
The diagonal Hamiltonian matrix elements are defined as 
and the off-diagonal Hamiltonian matrix elements are defined as 
.
5.3 (1 point) Give the value of 
______
5.4 (4 points) Find the eigenvalues of the secular equation, 
, in terms of 
, 
, 
, and 
. [Hint: analytically, this is one where both terms are squared, so you don't need to use the quadratic equation.]
5.5 (1 point) If 
, 
Hartree, and 
Hartree, find the energy of the ground state of H2+ in atomic units using 
.
6 Polyatomic Rotations (10 points). The moment of inertia tensor of the H2O water molecule ground state with 
and 
is
where the units are mass(distance)2, more specifically daltons(angstroms)2, i.e. 
.
6.1 (1 point) Give a term or phrase for the eigenvalues of I.
6.2 (5 points) Numerically, find the eigenvalues (with Matlab) which are in daltons(angstroms)2, i.e. 
. Show your work including the Matlab program and results.
6.3 (3 points) Determine the rotational constants in cm-1 which go with these eigenvalues. [A rotation constant in cm-1 is 
.]
6.4 (1 point) Choose the type of rotor of H2O ground state.
spherical prolate oblate asymmetric linear
7 Polyatomic Vibrations (5 points). A computational study of the CH3F(g) molecule at the MP2/6-31G(d) level of theory found the fundamental vibrational frequencies for CH3F are 
cm-1 (singly degenerate), 
cm-1 (doubly degenerate), 
cm-1 (singly degenerate), 
1566 cm-1 (doubly degenerate), 
3115 cm-1 (singly degenerate), and 
3215 cm-1 (doubly degenerate).
7.1 (1 point) What is the number of CH3F fundamental vibrations? _____
7.2 (2 points) If the vibrational state is specified by (
, then the ground state is 
). What is the energy in cm-1 of the 
) excited vibrational state relative to the ground vibrational state?
7.3 (2 points) What is the harmonic estimate of the vibrational zero point energy in cm-1? Show work.
8 Polyatomic Electronic Excited States (5 points). WebMO has its own menu option to calculate electronically excited states if you don’t need to optimize the geometry of the electronically excited states.
Optimize the ground electronic state geometry of the H2C=CHF molecule at the HF/6-311+G(2d,p) level of theory. [This basis set is a standard option in WebMO called “Accurate”]
8.1 (1 point) Report the optimized total electronic energy and the C-C bond distance.
8.2 (1 point) Capture an image of the highest occupied molecular orbital and place it in the image box at right. Give the number of this orbital for later reference.
Calculate the transition energies to electronically excited singlet states from the ground state geometry at the HF/6-311+G(2d,p) level of theory using the WebMO option “Excited States and UV-Vis” with the ground state geometry. [This performs a CIS (configuration interaction singles) theory calculation of the ten lowest energy excited states.] Go to the “Raw Output” to answer the following questions.
8.3 (1 point) Report the vertical transition energy in eV and the oscillator strength to the lowest excited energy level (“Excited State 1). Report the four most dominant orbital contributions for the transition to the lowest excited energy level (“Excited State 1).
8.4 (1 point) Consider the reported orbital contribution of the transition to Excited State 1. Identify the ground state orbital from which all four of the orbital contributions originate. Which of the ground state orbitals makes the strongest contribution to upper state of this transition? Explain how you know.
8.5 (1 point) Report the excited state # which has the strongest transition from the ground state geometry. Give the vertical transition energy in eV and the oscillator strength of this transition.
|
Species |
Electronic Energy (Hartrees) |
|
C2H4 singlet |
-78.2850278 |
|
H doublet |
-0.4982329 |
|
C2H3 doublet |
-77.6040842 |
|
C2H3F singlet |
-177.3034147 |
|
F doublet |
-99.4872711 |
9 Molecules and Computation, H2C=CH2 and H2C=CHF (10 points). A set of ab initio
MP2/6-31G(d) calculations were performed on the C2H4 and C2H3F singlet, gas phase molecules, as well as their dissociation fragments yielding the optimized total electronic energies of species tabulated at right.
9.1 (1 point) Write a reaction which corresponds to the total electronic energy of C2H4.
9.2 (3 points) Write a reaction which corresponds to the breaking of a C-H bond in C2H4 and calculate the MP2/6-31G(d) value of the C-H bond strength in atomic units using the table at top right.
9.3 (3 points) Write a reaction which corresponds to the breaking of a C-F bond in C2H3F and calculate the MP2/6-31G(d) value of the C-F bond strength in atomic units using the table at top right.
9.4 (1 point) Circle the strongest of the bonds considered above. C-H C-F
9.5 (2 points) Define the electron correlation energy, EEC, in terms of the Hartree-Fock energy, EHF.
10 Statistical Mechanics of Diatomic Molecule (20 points total). The Matlab programs used for the last homework may prove useful is this problem.
10.1 (2 points) Give an expression for the harmonic vibrational partition function as a function of temperature for a diatomic molecule using a sum over the states.
10.2 (2 points) Give an expression for the rigid rotor partition function as a function of temperature for a diatomic molecule using a sum over the states.
10.3 (2 points) Given a high temperature expression for the rigid rotor partition for a diatomic molecule using the rotational constant in cm-1.
10.4 (2 points) What is the value of the high temperature rigid rotor partition function for the HF diatomic at 500 K? Show work.
10.5 (6 points) What fraction of the HF population is in the vibration rotation state 
at 500 K? Show work.
10.6 (6 points) What fraction of the HF population is in the vibration rotation state 
at 500 K? Show work.
2022-12-09