LCAO Theory: Constructing Molecular Orbitals

We begin by stating the molecular orbital is a linear combination of atomic orbitals,

`Psi=c_1phi_1+c_2phi_2+...=sum_i (c_iphi_i)`

The energy of the molecular orbital is obtained as the eigenvalue of the Hamiltonian equation,

`hatHPsi=EPsi`

`Psi^**hatHPsi=Psi^**EPsi`

`int Psi^**hatHPsi d tau=EintPsi^**Psid tau`

`E=(int Psi^**hatHPsi d tau)/(intPsi^**Psid tau)`

Apply the variational principle to obtain the weighting coefficients, and orbital energy.

For a heteronuclear diatomic

`E=(int Psi^**hatHPsi d tau)/(intPsi^**Psid tau)=(int (c_1phi_1^**+c_2phi_2^**)hatH(c_1phi_1+c_2phi_2) d tau)/(int(c_1phi_1^**+c_2phi_2^**)(c_1phi_1+c_2phi_2)d tau)`

Expanding the brackets,

`E=(int (c_1^2phi_1^**hatHphi_1+c_1c_2phi_1^**hatHphi_2+c_2c_1phi_2^**hatHphi_1+c_2^2phi_2^**hatHphi_2) d tau)/(int(c_1phi_1^**c_1phi_1+c_2phi_2^**c_1phi_1+c_1phi_1^**c_2phi_2+c_2phi_2^**c_2phi_2)d tau)`

Then moving the integrals inside,

`E=(c_1^2int phi_1^**hatHphi_1d tau+c_1c_2int phi_1^**hatHphi_2d tau+c_1c_2int phi_2^**hatHphi_1d tau+c_2^2int phi_2^**hatHphi_2 d tau)/(c_1^2int phi_1^**phi_1d tau+c_1c_2int phi_2^**phi_1d tau+c_1c_2int phi_1^**phi_2d tau+c_2^2int phi_2^**phi_2d tau)`

At this point it is worth defining and explaining the various integrals:

Integral	Definition
`alpha_1=int phi_1^**hatHphi_1d tau`	Coulomb integral - the (negative) energy of the electron residing completely in one particular atomic orbital
`beta_12=int phi_1^**hatHphi_2d tau`	Resonance integral - the energy (positive for destruction, negative for construction) arising from the interaction between the two orbitals
`S=int phi_1phi_2d tau=int phi_2phi_1d tau`	Overlap integral - represents the extent of the interaction (between 0 and 1, and always significantly smaller than 1)
`1=int phi_1^**phi_1d tau`	probability over all space is definite and since wavefunctions are normalised, this equals one (for more information see wavefunction normalisation)

Thus,

`E=(c_1^2alpha_1+c_1c_2beta_(12)+c_1c_2beta_(21)+c_2^2alpha_2)/(c_1^2+2c_1c_2S+c_2^2)`

if `beta_(21)=beta_(12)` then

`E=(c_1^2alpha_1+2c_1c_2beta+c_2^2alpha_2)/(c_1^2+2c_1c_2S+c_2^2)`

We rearrange and expand the brackets,

`E(c_1^2+2c_1c_2S+c_2^2)=(c_1^2alpha_1+2c_1c_2beta+c_2^2alpha_2)`

To find the energy, we have to minimise `(delE)/(delc_1)` and `(delE)/(delc_2)`, (click here for method)

`E(c_1^2+2c_1c_2S+c_2^2)=(c_1^2alpha_1+2c_1c_2beta+c_2^2alpha_2)`

Having investigated the quantitative formulation of molecular orbital theory, it might be of interest to see a qualitative picture of how we go about constructing molecular orbitals from their atomic orbitals. Of course, it is possible to determine in this quantitative sense which combination of orbitals will give rise to bonding, anti-bonding and non-bonding molecular orbitals, however through the application of group-theory and symmetry, we can more rapidly establish which orbital combinations are of interest to chemistry.

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