Walsh Diagrams  Contents -  To Determine Symmetry Orbitals

Orbital Energy Level Diagrams

The idea here is that symmetry orbitals of the same symmetry mix and push one another apart.

This is understood by considering a two level system. We have two orbitals, $\phi_1,\phi_2$ with energies $\epsilon_1,\epsilon_2$, $\epsilon_2>\epsilon_1$, of the same symmetry which interact through a perturbation. We solve the secular equations to obtain the new orbitals $\psi=a_1\phi_1+a_2\phi_2$. The secular equations are

$\left(\begin{array}{cc}F_{11} - \epsilon & F_{12} \\F_{21} & F_{22} - \epsilon \\\end{array}\right)$ $\left(\begin{array}{c}a_1 \\a_2 \\\end{array}\right)$ = $\left(\begin{array}{c}0 \\0 \\\end{array}\right)$ The interaction is responsible for $F_{12}=\Delta$ not being zero. We assume the interaction does not change the diagonal elements, thus $F_{11}=\epsilon_1,F_{22}=\epsilon_2$. The solutions of the secular equations are

$$\epsilon_{\pm}=\frac{1}{2}(\epsilon_1+\epsilon_2)\pm\frac{1}{2}((\epsilon_1-\epsilon_2)^2+4\Delta^2)^{\frac{1}{2}}$$ (50)

 If, as usual, the perturbation is much smaller than the energy difference, then we obtain

$$\epsilon_{\pm}=\frac{1}{2}(\epsilon_1+\epsilon_2)\pm\frac{1}{2}((\epsilon_1-\epsilon_2)(1+\frac{2\Delta^2}{(\epsilon_2-\epsilon_1)^2})$$ (51)

 or

$$\epsilon_+=\epsilon_1-\frac{\Delta^2}{\epsilon_2-\epsilon_1}$$ (52)

$$\epsilon_-=\epsilon_2+\frac{\Delta^2}{\epsilon_2-\epsilon_1}$$ (53)

 Thus the lower level is pushed down, and the upper level pushed up viz
 
     This two level picture is important for understanding HOMO LUMO interactions of molecules (the HOMO is the highest occupied molecular orbital of molecule A, and the LUMO is the lowest unoccupied orbital of molecule B; these orbitals are obtained by solving the secular equations of each molecule. Now the molecules interact and the two level picture gives
 
     
  The magnitude of the interaction will depend upon the magnitude of F₁₂, which itself is determined by their overlap and relative size. The larger coefficients in the molecular orbitals indicate where most of the electrons are (eg on the O in a carbonyl HOMO and on the C in a carbonyl LUMO)

In H₂O the A₁, B₂ combinations of 1s_A,1s_B will mix with 2p_zO,2p_yO. To construct the diagram we use the fact that the 2s atomic orbital is lower in energy than the 2p orbitals (why?), and that the 1s H orbitals are highest in energy. Thus we get the following orbital energy diagram
 

    Using an aufbau process to fill the orbitals with electrons, we therefore predict that the ground state electronic configuration is 2a₁²1b₂²3a₁²1b₁², where we have used the small letter convention, and remembered that the inner electrons occupy 1a₁.
The diagram for NH₃ may similarly be constructed:
     
 The ground state configuration is 2a₁²1e⁴3a₁².
The diagram for SF₆ without and with the inclusion of 3d orbitals is shown
 

 
 
Without the 3d orbitals, the 1a_1g and 1t_1u are bonding, and the 1e_g combination of the 2p_F orbitals is nonbonding. The addition of the 3d_S orbitals lowers the 1e_g orbital energy. Group theory cannot say whether the effect is small or large; in this case the effect is significant and it is important to include `d basis functions' on S to correctly describe the bonding situation.

  
 Walsh Diagrams  Contents -  To Determine Symmetry Orbitals
1998-09-23