Valence-Bond Theory  Contents -  The Molecular Orbital Wavefunction

The Equations for the Orbitals

The orbitals are the solution of a Schrodinger-like equation

$$F\phi_i=\epsilon_i\phi_i$$ (31)

 where $\epsilon_i$ is the orbital energy, F is the hamiltonian (in the molecular orbital approximation). If we substitute the MO-LCAO expression we get

$$(F-\epsilon_i)\sum c_{ri}\eta_r=0$$ (32)

 Multiply on the left $\eta_s$ and integrate to give

$$\sum_r^M\langle\eta_s\vert F-\epsilon_i\vert\eta_r\rangle c_{ri}=0$$ (33)

 where we have introduced the notation

$$\langle\eta_s\vert F\vert\eta_r\rangle=\int\eta_sF\eta_rdv=F_{sr}=F_{rs}$$ (34)

$$\langle\eta_s\eta_r\rangle=\int\eta_s\eta_rdv=S_{sr}=S_{rs}$$ (35)

 We can therefore write the equations

$$\sum_r^M(F_{sr}-\epsilon_i S_{sr})c_{ri}=0$$ (36)

 or

$${\bf (F-\epsilon_{\rm i} S) c_{\rm i}=0}$$ (37)

 These are of course the secular equations and they have been written in an eigenvector notation.    
 Valence-Bond Theory  Contents -  The Molecular Orbital Wavefunction

1998-09-23