Lecture35Notes

Lecture35Notes - Chem 120A 04/21/06 Spring 2006 READING:...

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Chem 120A Molecular states: H 2 04/21/06 Spring 2006 Lecture 35 READING: Engel (QCS): Ch. 12-13, Ch. 15.2 McQuarrie and Simon: Ch. 9 Summary of H + 2 and MO labels In Lecture 34 we used a variational wavefunction in order to approximately solve the Schr¨odinger equation for the H + 2 molecular ion. Using a LCAO-MO wavefunction we set up the secular determinant then found the energies of the ground and first excited states of the molecule (Fig. 1). The corresponding wavefunctions are a symmetric, bonding σ g = 1 2 ( 1 s A + 1 s B ) orbital for the ground state and an antisymmetric, antibonding u = 1 2 ( 1 s A 1 s B ) orbital for the first excited state. We saw that the MOs are labeled according to their characteristic z -component of angular momentum and whether or not they possess symmetry with respect to inversion. The total angular momentum quantum number l is not a good quantum number for the MOs, but because a diatomic molecule possesses symmetry with respect to rotations about its bond axis, the z - component of the angular momentum is still well defined and | M L | is a good quantum number. We therefore label the MOs according to | L z | as for | L z | = 0, π for | L z | = 1, and δ for | L z | = 2. The value of | L z | also indicates the number of nodes that a MO has along the molecular axis, and so the labels give some indication of the energy of a MO. In addition to the Greek letters, we also add labels of g or u and + or . The g and u labels stand for gerade and ungerade respectively and they indicate whether or not a MO has inversion symmetry. The labels + and are added as a superscript. If a MO has symmetry with respect to reflection through a plane that includes the molecular bond axis it is given the label + . If it does not possess this reflection symmetry it is given the label . bonding antibonding R eq σ g σ u Figure 1: Energies of the molecular orbitals of H + 2 calculated using a variational wavefunction (Lecture 34.) We can also construct molecular orbital energy diagrams in order to understand bonding in diatomic molecules. Chem 120A, Spring 2006, Lecture 35 1
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The MO diagram for O 2 is shown in Fig. 2. Notice that in the MOs are numbered in addition to having labels. (Note, the numbering scheme is different for homonuclear and heteronuclear diatomics (Engel 13.6-13.7.) In addition, the labels g and u are not used for heteronuclear diatomics.) The electrons fill the MOs according to the Aufbau principle, following Hund’s rules, analogous to the filling of orbitals in multi-electron atoms. The electron configuration for the molecule can be written down by looking at the MO diagram. In the ex- ample of O 2 , the lowest energy configuration is ( 1 σ g ) 2 ( 1 u ) 2 ( 2 g ) 2 ( 2 u ) 2 ( 1 π u ) 2 ( 1 u ) 2 ( 1 g ) 1 ( 1 g ) 1 .We can also assign a label to the overall electron configuration of the molecule. Similar to the case of assigning
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Lecture35Notes - Chem 120A 04/21/06 Spring 2006 READING:...

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