Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781285969770
Author: Ball
Publisher: Cengage
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Chapter 18, Problem 18.33E
Interpretation Introduction

Interpretation:

The expected symmetry pairings of the nuclear and rotational wavefunctions is to be predicted.

Concept introduction:

The symmetry pairings of the nuclear and rotational wavefunctions is given as follows:

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym,Jeven     degen:(I+1)(2I+1)     degen:(2J+1)

Expert Solution & Answer
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Answer to Problem 18.33E

The expected symmetry pairings of the nuclear and rotational wavefunctions are,

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:5

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:9

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:13

Explanation of Solution

It is given that the diatomic oxygen has an antisymmetric ground electronic state. The oxygen nuclei are bosons (I=0).

The symmetry pairings of the nuclear and rotational wavefunctions is given as follows:

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym,Jeven     degen:(I+1)(2I+1)     degen:(2J+1)…(1)

Substitute (I=0) in equation (1).

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym,Jeven       degen:1            degen:(2J+1)…(2)

For even values of J, the system will be symmetric.

Substitute J=2 in equation (2).

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1           degen:(2(2)+1):5

Substitute J=4 in equation (2).

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1           degen:(2(4)+1):9

Substitute J=6 in equation (2).

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1           degen:(2(6)+1):13

Hence, the expected symmetry pairings of the nuclear and rotational wavefunctions are,

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:5

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:9

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:13

Conclusion

The expected symmetry pairings of the nuclear and rotational wavefunctions are,

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:5

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:9

Ψtot=    Ψnuc         ×     Ψrotsym      sym               sym       degen:1             degen:13

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