Ex4: Atomic and Molecular Structure

Posted by:Prof. Lexx Posted on:May 27,2016

• More on the He atom.
• Chemical Bond
• The Born Oppenheimer Approximation
• The Hydrogen Molecule Ion
• Molecular Orbital Method
Items (1)-(6) should be submitted for marking
Five questions will be checked and graded.
Due: Submit in class on April 02.
3 points per day are taken for late return.
1. In a diatomic molecule A2 the mass of each atom is M. The potential energy
of the two atoms, as a function of the internuclear distance, is given by V (r) =
ae− r[1 − r], a > 0, > 0.
(i) Plot the potential energy V (r) as a function of r. What is the bond energy and
the average bond length? Express your results in terms of and a only.
(ii) Write an expression for the potential energy close to the equilibrium point,
assuming an harmonic potential. In this harmonic approximation, calculate the energy
required for exciting the molecule from its ground state to the first excited state.
2. (i) Write the Hamiltonian for a general AB molecule. The A atom has n electrons.
The B atom has p electrons. The atomic masses are different.
(ii)What is the Hamiltonian under the Born-Oppenheimer approximation? Explain.
3. Consider the following species: N+2 , N2, N−
2 and N2−
(i) Write the ground state configurations for these species.
(ii) Arrange them in terms of increasing bond energy and bond length on the basis
of their bond order.
CHM221-CHM225 Winter 2013
4. A helium atom is excited from the ground state to the state 2s14p1 by absorption
of ultraviolet light. Assume that the 2s electron moves in the unscreened Coulomb
field of the nucleus (Z = 2) and the 4p electron in the fully screened Coulomb
potential (Z = 1). Obtain the energy of this level (in eV) and the corresponding
wavelength (in °A) of the ultraviolet light required to effect this excitation.
5. Two identical noninteracting spin-1/2 particles of mass m are placed in a onedimensional
harmonic oscillator.
(a) Write the corresponding Hamiltonian. Define all terms.
(b) Determine the ground state and its energy. What is the total spin of the system?
(c) Determine the energy and eigenstates for the first excited state when the two
particles are in either a total spin-0 state or in a total spin-1 state.
In (b) and (c) use the general notation n to denote the eigenstates of the harmonic
oscillator (n is the quantum number associated with the harmonic oscillator solution).
No need to use explicit expressions.
6. In the circles in the following wavefunction, fill in the correct signs (+ or −) that
make the overall function an acceptable approximate wavefunction for the He atom
(r1 and r2 are the coordinates for electron ’1’ and ’2’). Is it a singlet or a triplet
(1, 2) = e−r1/ae−2r2/a (1) (2)

e−2r1/ae−r2/a (2) (1)

e−r1/ae−2r2/a (2) (1)

e−2r1/ae−r2/a (1) (2)
CHM221-CHM225 Winter 2013
Problems that MAY be discussed and/or solved by the TA:
1. 19.23 19.26 19.30 19.31 19.32
2. He atom. (i) Use first order perturbation theory and calculate the corrected ground
state energy of the He atom.
(ii) Again, using first-order perturbation theory, write a formal expression for the
energy correction to the first excited state of He atom (triplet or singlet)
3. A helium atom is excited from the ground state to the state 2s2 by absorption of
ultraviolet light. Obtain the energy of this level (in eV) and the corresponding
wavelength (in °A) of the ultraviolet light required to effect this excitation.
4. The potential energy surface between two nucleus in a molecule is often modeled
using the “Morse potential” defined as
V (R) = D1 − e− (R−R0)2
. (1)
Here R0 is a constant and R is a parameter, denoting the inter-nuclei distance.
(i) What is the equilibrium distance between the two nuclei under this potential?
(ii) What is the dissociation energy, ignoring the ground vibrational energy?
(iii) Expand V (R) around the equilibrium distance to the second order. Find the
vibrational frequency, at low energies.
(iv) What is the energy of the ground vibrational state, in the harmonic approximation?
5. The CN molecule can absorb light in the ∼ 1eV range. It can be shown that this
absorption is due to electronic transition, not vibrational. In contrast, the CN−
molecule does not absorb in that range. Explain this fact based on the molecules
molecular orbitals.
6. A specie which is paramagnetic has unpaired electrons meaning it can be affected by
an external magnetic field. O2 has two unpaired electrons in the doubly degenerate
g orbital.
Which of these ions would you expect to be paramagnetic, O+2 , O−
2 and O2−
2 ?
7. Display the MO energy level diagrams for H2, F2 and HF. Note, the valence shell
ionization energies, in eV, of H1S is -13.6; F2S=-40.2 and F2p = −18.6 .
CHM221-CHM225 Winter 2013
8. Consider the two molecules N2 and O2 .
(a) Write down their electronic configurations, i.e., (g1s)2(
(b) Draw Schematically (on the same plot), Born-Oppenheimer potential surfaces
for the ground electronic state of each molecule. Annotate the axes and explain
your considerations.
Note: There are 14 electrons in the N2 molecule and 16 electrons in O2.


Prof. Lexx

Overview I have a Bachelor of Arts, majoring in English and minoring in History. I have worked in various fields ranging from academic research to freelance translating to editing to customer support and data entry. These include editing my old university's newspaper as well as co-leading their creative writing team; serving as a junior member in the history department's research network; publishing music and film reviews for several different magazines and webzines; and translating papers and books for numerous researchers in various languages, including Russian, Spanish, and Romanian. I am also a skilled typist, with a rate of at least 70 words per minute, and have myself digitized dozens of books and essays for both private and commercial use. I have a very strong work ethic, and make sure to prioritize the task I am given so that it is completed as quickly as possible. I am organized and disciplined, ensuring a job done professionally and efficiently.

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