: Question 1-6 in the attached files

CHM221-CHM225 Winter 2013

Ex4: Atomic and Molecular Structure
Covering:
• 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−βR [1 − βR], A > 0, β > 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 β 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 en-ergy 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 N22−
(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.





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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 one-dimensional 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 φN to denote the eigenstates of the harmonic oscillator (N is the quantum number associated with the harmonic oscillator solu-tion). 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 state?


Ψ(1, 2) = E−R1/AE−2R2/Aα(1)β(2) E−2R1/AE−R2/Aα(2)β(1) E−R1/AE−2R2/Aα(2)β(1)
E−2R1/AE−R2/Aα(1)β(2)

















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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 asV (R) = D 1 − E−α(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 approxi-mation?

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 1πG∗ orbital.
Which of these ions would you expect to be paramagnetic, O+2, O−2 and O22− ?
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 .

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CHM221-CHM225 Winter 2013

8. Consider the two molecules N2 and O2 .
(a) Write down their electronic configurations, i.e., (σG 1S)2(σU∗1S)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.
















































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