CHEM 342. Spring 2002. PS#6 Answers PS#4 Answers PS#5 Questions PS#5 Answers.doc (Word 97)
Problem Set #5 Answers
Note: Relevant Chapters in Mortimer are Chapters 17 and 18. BornOppenheimer Approximation 1. What is the main assumption of the BornOppenheimer approximation? The BornOppenheimer 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 BornOppenheimer approximation, write the Hamiltonian for the H_{2} molecule (2 electrons). where r_{A1} is the distance between nucleus A and electron 1; r_{A2} is the distance between nucleus A and electron 2; r_{B1} is the distance between nucleus B and electron 1; r_{B2} is the distance between nucleus B and electron 2; r_{12} is the distance between electron 1 and electron 2; and R_{AB} is the distance between nucleus A and nucleus B (and remains constant by the BornOppenheimer approximation).
Variation Method; LCAOMO 3. Determine the energy for in terms of H_{aa}, H_{ab}, and S using the variation method. The secular determinant for this system is In this case, , giving or
Normalization; Hybrid Orbitals 4. Show that the sp^{2} 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^{+}, O^{2}, F^{}, Na^{+}, and Mg^{2+}? 6. Which of the following transitions are allowed in the normal electronic emission spectrum of an atom
For a singleelectron transition, and = any integer.
7. Write the electronic configurations for N_{2}, N_{2}^{+}, N_{2}^{}. 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 Li_{2}, 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 (a) The transition is allowed. (b) The transition is forbidden because . (c)
The transition is allowed.
11. The groundlevel 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.

