E''= 1/12 [(1)(3)(2)+(2)(0)(-1)+(3)(1)(0)+(1)(3)(-2)+(2)(0)(1)+(3)(1)(0)] = 0 E ''
Thus, LGO's symmetry labels are a1' + e'
b. How to determine the shapes of the LGOS?
-LGO's symmetry labels are a1' + e'. There are 3 LGOs that can be made out of these symmetry labels,one LGO is from a1' and two LGOs are from e' due to doubly degenerate. In order to determine the shape of each LGO, we would use the wavefunctions.
-Three hydrogens in BH3 are assigned with Ψ1, Ψ2, Ψ3. Now lets look at how each Ψ is affected by the symmetry operations of the D3h and their results are completed in the following table:
D3h
E
C3
C23
C2
C2’
C2’’
σh
S3
S23
σv
σv’
σv’’
Ψ1
Ψ1
Ψ2
Ψ3
Ψ1
Ψ3
Ψ2
Ψ1
Ψ2
Ψ3
Ψ1
Ψ3
Ψ2
a1’
1
1
1
1
1
1
1
1
1
1
1
1
LGO1
Ψ1
Ψ2
Ψ3
Ψ1
Ψ3
Ψ2
Ψ1
Ψ2
Ψ3
Ψ1
Ψ3
Ψ2
Ψ (a1') = 4Ψ1+4Ψ2+4Ψ3
= 4(Ψ1+Ψ2+Ψ3)
Ψ(a1')= 1/√3 (Ψ1+Ψ2+Ψ3)
-The shape of the LGO1 is
D3h
E
C3
C23
C2
C2’
C2’’
σh
S3
S23
σv
σv’
σv’’
Ψ1
Ψ1
Ψ2
Ψ3
Ψ1
Ψ3
Ψ2
Ψ1
Ψ2
Ψ3
Ψ1
Ψ3
Ψ2
e’
2
-1
-1
0
0
0
2
-1
-1
0
0
0
LGO1
2Ψ1
-Ψ2
-Ψ3
0
0
0
2Ψ1
-Ψ2
-Ψ3
0
0
0
Ψ(e') = 4 (Ψ1)-2 (Ψ2)-2 (Ψ3)
= 2[ 2(Ψ1)-Ψ2-Ψ3]
Ψ(e') = 1/ √6 (2 Ψ1-Ψ2-Ψ3)
-The shape of the LGO2 is
-Noticed that in the LGO2, we have 1 nodal plane which is the horizonal line between the positive charge and negative charge. Therefore, the LGO3 (doubly degenerate with e') would also 1 nodal plane and its wavefunction would be Ψ(e') = 1/√2 (Ψ2-Ψ3). The shape of the LGO3 is
MO diagram
-As we can see in this diagram, the energy level of 3 LGOs are higher than the 2s orbital and below the 2 p orbital dued to the electronegativy of both Boron and Hydrogen. Hydrogen has higher electronegativity than boron, therefore hydrogen would have lower energy level in the MO diagram.
-In addition, B has 3 electrons in the valence electrons and 3 hydrogens have total 3 electrons. Therefore, the total number of electrons filled in orbitals are 6. With all of the informations above about symmetry labels of B atom and the 3 LGOs, we now construct the MO diagram of BH3. Noticed that, the bonding formation only happens to atoms that have the same symmetry labels. 2s orbital and LGO(1) would contribute 1 electron to give 2 spin pairs electrons at the a1' energy level. 2px and 2py orbitals would bond to the LGO(2) and LGO(3), which give 2 spin pairs electrons at the e' energy level.
FIGURE 3: MO diagram for the formation of BH3
-there is a detailed explaination that performed in this video
References
Pople, John A.. Approximate Molecular Orbital Theory (Advanced Chemistry). New York: Mcgraw-Hill (Tx), 1970. Print., Phy Sci Engr Library QD461. P66
Noebe, Guillermo (EDT)/, Ronald D. (EDT)/ Abel, and Phillip B. (EDT) Bozzolo. Applied Computational Materials Modeling. Berlin and Heidelberg: Springer Verlag, 2007. Print. Phy Sci Engr Library TA 404.23.A66
Housecroft, Catherine, and Alan G. Sharpe. Housecroft Inorganic Chemistry 3e (3rd Edition). 3 ed. Alexandria, VA: Prentice Hall, 2008. 128