**Secular equations** Using the variationial theorem we can find eigenvalues and eigenvectors of the Hamiltonian. We can obtain them through the so called secular equations which are derived as follows:

E = <Φ|H|Φ> / <Φ|Φ>

where:

Φ is the trial wave function

H is the Hamiltonian operator

E is the average energy of the system

A trial wave function Φ is chosen to be a linear combination of linear independent known functions satisfying the same boundary conditions as the exact wave function:

Φ = ∑

_{i}a

_{i}ψ

_{i} Note that this is not the same as expanding Φ in the basis formed by eigenfunctions of H. In the expansion above <ψ

_{i}|ψ

_{j}> is probably not 0, the functions are linear independent but they do not form an orthonormal basis.

E = <Φ|H|Φ> / <Φ|Φ>

= <∑

_{i}a

_{i}ψ

_{i}|H|∑

_{j}a

_{j}ψ

_{j}> / <∑

_{i}a

_{i}ψ

_{i}|∑

_{j}a

_{j}ψ

_{j}>

= ∑

_{i}∑

_{j}a

^{*}_{i}a

_{j}<ψ

_{i}|H|ψ

_{j}> / ∑

_{i}∑

_{j}a

^{*}_{i}a

_{j}<ψ

_{i}|ψ

_{j}>

= ∑

_{ij}a

^{*}_{i}a

_{j}H

_{ij} / ∑

_{ij}a

^{*}_{i}a

_{j}S

_{ij} We arrive at E = = ∑

_{ij}a

^{*}_{i}a

_{j}H

_{ij} / ∑

_{ij}a

^{*}_{i}a

_{j}S

_{ij} where:

H

_{ij} = <ψ

_{i}|H|ψ

_{j}> ; resonance integral

H

_{ii} = <ψ

_{i}|H|ψ

_{i}> ; corresponds to the energy of a single electron occupying this function i; essentially it is the ionisation potential of the AO (atomic orbital) in the environment of the surrounding molecule.

S

_{ij} = <ψ

_{i}|ψ

_{j}> ; overlap integral

ψ

_{i} ; AO = atomic orbital i

The secular equations are derived from this expression. We are looking for a minimum of the energy (see the variation principle) so we seek:

δE/δa

_{k} = 0 for all k

This leads to N equations with N unknowns a

_{i} From linear algebra we know that there is a non-trivial solution (the trivial solutions are all coefficients a

_{i}=0) if the determinant formed from the coefficients of the unknowns is zero:

|H

_{ki} - ES

_{ki}| = 0

This is the secular equation. It gives N roots E, N energies E where each value of E

_{j} gives a different set of coefficients a

_{ij}. These coefficients will define an optimal wave function Φ

_{j} within the given basis set:

Φ

_{j} = ∑

_{i=1..N}a

_{ij}ψ

_{i} See also:

Variational theorem

http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1203365153