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.

Born-Oppenheimer Approximation

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

or

 

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

    1. 2s to 1s
    2. 2p to 1s
    3. 3d to 2p

For a single-electron transition, and = any integer.

    1. not allowed,
    2. allowed,
    3. allowed,

 

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

 

  1. , the single unpaired electron gives a doublet state because S = 1/2 so that . The symbol is
  2. , the term symbol is
  3. , the electrons must be paired. Therefore, S = 0 and the state is a singlet. The product of two ungerade functions is gerade. The term symbol is
  4. , the electron could be in either the or the state. Therefore, there is a degenerate pair of states with and the term symbol is .
  5. , there are two possible 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

(a)

The transition is allowed.

(b) The transition is forbidden because .

(c)

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.