Problem Set #5 Answers
Note:Relevant Chapters in Mortimer are Chapters 17 and 18.
1. What is the main assumption of the Born-Oppenheimer approximation?
The Born-Oppenheimer approximation assumes that the nuclei are stationary, and electron motion can be treated separately. Fixed bond distances and bond angles are assumed, and a Hamiltonian operator is written for electronic motion only (Mortimer, p.648).
2. Using the Born-Oppenheimer approximation, write the Hamiltonian for the H2 molecule (2 electrons).
where rA1 is the distance between nucleus A and electron 1; rA2 is the distance between nucleus A and electron 2; rB1 is the distance between nucleus B and electron 1; rB2 is the distance between nucleus B and electron 2; r12 is the distance between electron 1 and electron 2; and RAB is the distance between nucleus A and nucleus B (and remains constant by the Born-Oppenheimer approximation).
Variation Method; LCAO-MO
3. Determine the energy for in terms of Haa, Hab, and S using the variation method.
The secular determinant for this system is
In this case, , giving
Normalization; Hybrid Orbitals
4. Show that the sp2 hybrid orbital is normalized if the s and p orbitals are also normalized.
Note: We used ; ; and for normalized, orthogonal s and p orbitals.
Electronic Configuration for Atoms & Molecules; Pauli Exclusion Principle
5. What are the electron configurations for H-, Li+, O2-, F-, Na+, and Mg2+?
6. Which of the following transitions are allowed in the normal electronic emission spectrum of an atom
For a single-electron transition, and = any integer.
7. Write the electronic configurations for N2, N2+, N2-.
The configurations are
Term Symbols for Diatomics; Electronic Transitions
8. Determine the complete term symbol for each of the following electronic configurations
for which and
for which . Therefore and the state is . Also, the function is ungerade because it is a product of three ungerade functions. The term symbol is therefore .
9. Write the electronic configuration for Li2, and predict the term symbol for the ground level.
The electronic configuration is . For this configuration, . Since , we know that and S = 0, which results in the symbol . Because the wave function for a molecular orbital does not change sign upon reflection across the xz plane, the + superscript is used. The parity can be found by multiplying the parities of the orbitals being used, according to the laws of odd (u) and even (g) multiplication:.
Therefore, , and the complete term symbol is .
10. Which of the following electronic transitions are allowed?
The selection rules for electronic transitions in diatomic molecules are
The transition is allowed.
(b) The transition is forbidden because .
The transition is allowed.
11. The ground-level term for a heteronuclear diatomic molecule is . Write the term symbols of the electronic transitions allowed for this molecule.
According to the transition rules (see the previous problem), the transition must be ; the superscript must be 3; and or . The allowed transitions are and . Note that the notations g and u are not used for heteronuclear diatomic molecules.