The Molecular Orbital Wavefunction  Contents -  The Molecular Orbital Approximation

Molecular Orbitals

For H₂, the molecular orbital is

$$\sigma_g({\bf r}) N\{{\rm 1s}_A(r_A)+{\rm 1s}_B(r_B)\}$$ = (16)

= C(e^-r_A+e^-r_B) (17)

 N,C are normalisation constants. The coordinates are
   
 and the molecular orbital looks like
   
 if it is evaluated along the molecular axis. This $\sigma_g$ molecular orbital is a linear combination of atomic orbitals. (MO-LCAO).
For the water molecule H₂O, we can construct a molecular orbital which is a combination of 2p_zO and 1s functions on each H

$$\phi({\bf r})=c_1{\rm 2p}_{zO}+c_2({\rm 1s}_A+{\rm 1s}_B)$$ (18)

 c₁ and c₂ are mixing coefficients and take numerical values. This orbital may be sketched:
     
 Another molecular orbital for H₂O can be constructed from the 2p_yO atomic orbital and the 1s functions

$$\phi({\bf r})=c_3{\rm 2p}_{yO}+c_4({\rm 1s}_A-{\rm 1s}_B)$$ (19)

 and it is sketched
   
 
 The above are both bonding orbitals, corresponding to a build up of charge between the O and H atoms. In general we write

$$\phi_i \sum_{\alpha}c_{\alpha i}\eta_{\alpha}$$

$$c_{1i}\eta_1+c_{2i}\eta_2+...c_{mi}\eta_m$$ = (21)

 where $\eta_{\alpha}$ are the atomic orbitals and $\phi_i$ are the molecular orbitals. In practice today we do not use hydrogen-like functions e^-r because they are too difficult to work with, but instead we use gaussian functions e^-ar2.
Thus for H₂

$$\sigma_g({\bf r})=C(e^{-ar_A^2}+e^{-ar_B^2})$$ (22)

    
 The Molecular Orbital Wavefunction  Contents -  The Molecular Orbital Approximation

1998-09-23