Walsh Figure  Contents -  Orbital Energy Level Diagrams

Walsh Diagrams

This is a molecular orbital interpretation for the geometries of the ground and excited states of molecules. We would like to be able to predict whether the molecule XH₂ is linear or bent. The idea is to observe that the linear molecule HXH has a higher symmetry (D $_{\infty h}$) than the bent XH₂ molecule (C_2v). Molecular orbitals that do not interact in D $_{\infty h}$ because they are of different symmetry, may interact in C_2v because they then have the same symmetry. We use the following axes
 
     
 The symmetries of the symmetry orbitals in the two situations are
 

	D $_{\infty h}$	C2v
2sX	$\sigma_g$	a1
2pzX	$\pi_u$	a1
2pxX	$\pi_u$	b1
2pyX	$\sigma_u$	b2
1sA-1sB	$\sigma_u$	b2
1sA+1sB	$\sigma_g$	a1

 
The molecular orbitals of the linear molecule are in ascending order of energy $1\sigma_g,1\sigma_u,\pi_u,2\sigma_g^*,2\sigma_u^*$. On bending, one of the $\pi_u$ orbitals, 2p_zX, has the same symmetry as the $2\sigma_g^*$, c(1s_A+1s_B)-2s_X, orbital and interacts with it forming a strongly bonding orbital viz
     Thus the orbital energy ordering of the bent molecule is 1a₁,1b₂,2a₁,1b₁,3a₁^*,2b₂^*. If the molecule has more than 4 valence electrons its ground state will be bent, the degree of which will depend upon the occupancy of the 2a₁ orbital. Thus we have the following results for XH₂ molecules.
BeH₂....... $1\sigma_g^21\sigma_u^2$....... $^1\Sigma_g^+$ linear
BH₂........1a₁²1b₂²2a₁¹.........²A₁.....bent, 131^o
..............1a₁²1b₂²1b₁¹..........²B $_1\equiv\ ^2\Pi_u$ linear
These two states form a Renner-Teller pair, two separate states which become degenerate at linearity.
      CH₂........ 1a₁²1b₂²2a₁¹1b₁¹....³B₁......bent, 134^o, this state is lowest in energy because of the exchange integral (Hund's rule)
..............1a₁²1b₂²2a₁².......¹A₁.......bent, 102^o, 9kcal/mol higher in energy.
NH₂........1a₁²1b₂²2a₁²1b₁¹.....²B₁.....bent, 103^o
...............1a₁²1b₂²2a₁¹1b₁²....²A₁.....bent, 144^o
These are again two components of a Renner-Teller pair; they are degenerate $^2\Pi_u$ at linearity.    
 
 OH₂.........1a₁²1b₂²2a₁²1b₁²......¹A₁...bent, 105^o
The more the 2a₁ orbital is occupied, the more strongly the molecule is bent and the higher the bending vibrational frequency. Thus for H₂O we have the following configurations and bending vibrational frequency.
H₂O........¹A₁.......1a₁²1b₂²2a₁²1b₁²...1595cm^-1
H₂O⁺......²B₁......1a₁²1b₂²2a₁²1b₁¹....1431cm^-1
H₂O⁺......²A₁......1a₁²1b₂²2a₁¹1b₁²....873cm^-1
If an excited state has a very different geometry from the ground state, then the electronic transition will be broad as many vibrational energies of the excited state are accessible. Thus the in photoelectron spectrum a band corresponding to the removal of an electron from the nonbonding 1b₁ orbital is narrow, while one corresponding to the removal of an electron from the bonding 2a₁ orbital is broad, with many bending states of the ion being formed. ( a vibrational progression). This is demonstrated in the photoelectron spectrum of H₂O.
   
 The first band on the right corresponds to the removal of an electron from the least stable orbital 1b₁. It is fairly sharp, as the geometry of the molecule does not change much on ionisation. The middle band shows many bending vibrations, corresponding to the removal of an electron from the 2a₁ orbital which most effects the bending. The lefthand band corresponds to the removal of an electron from the 1b₂ bonding orbital, and shows a progression in the stretching vibration which is not very well resolved.

  
 Walsh Figure  Contents -  Orbital Energy Level Diagrams

1998-09-23