LCAO Theory: Evaluating the Coefficients

The two ways of calculating values:

Differentiation of the Energy equation
The Matrix Method

Differentiation of the Energy equation

The objective is to find an expression for the orbital weighting coefficients, such that their choice of value minimises the energy. The molecular orbital energy can be derived by differentation of the energy expression: first with respect to `c_1` and then to `c_2`- each differentiation provides an expression that we set to zero (ie., minimum), we then solve the set of equations simultaneously.

We begin with the expression derived for the energy of a molecular orbital, constructed from two atomic orbitals,

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

Minimisation with respect to c₁

First we will differentiate with respect to `c_1`,

`del/(delc_1) E(c_1^2+2c_1c_2S+c_2^2)=del/(delc_1)(c_1^2alpha_1+2c_1c_2beta+c_2^2alpha_2)`

Left hand side

We will first evaluate the left hand side. We note that since both E and the bracketed term depend on `c_1`, we use the product rule,

`del/(delc_1) E(c_1^2+2c_1c_2S+c_2^2)= Edel/(delc_1)(c_1^2+2c_1c_2S+c_2^2)+(c_1^2+2c_1c_2S+c_2^2)del/(delc_1)E`

Since differentiation is a linear operation, we exploit the distributive property and apply the differentiation to each term individually.

`del/(delc_1) E(c_1^2+2c_1c_2S+c_2^2)= E(del/(delc_1)c_1^2+2del/(delc_1)c_1c_2S+del/(delc_1)c_2^2)+(c_1^2+2c_1c_2S+c_2^2)del/(delc_1)E`

`=E(2c_1+2c_2S)+(c_1^2+2c_1c_2S+c_2^2)del/(delc_1)E`

Right hand side

We now perform the same differentiation operation to the right hand side. Again exploiting the distributive property of differentiation, we expand the brackets, moving the differentiation operation inside,

`del/(delc_1)(c_1^2alpha_1+2c_1c_2beta+c_2^2alpha_2)=(del/(delc_1)c_1^2alpha_1+2del/(delc_1)c_1c_2beta+del/(delc_1)c_2^2alpha_2)`

`=(2c_1alpha_1+2c_2beta)`

Equating the two sides

Having found the derivatives of both lhs and rhs, we can equate the two:

`E(2c_1+2c_2S)+(c_1^2+2c_1c_2S+c_2^2)(delE)/(delc_1)=(2c_1alpha_1+2c_2beta)`

...as per the method, we are aiming to minimise the energy, and hence we set `(delE)/(delc_1)=0`. The result is one of a pair of secular equations - so called because they are also the characteristic equation (or secular determinant) of the system of equations when treated as a matrix, see below. Whilst we have not employed matrices in this derivation, they afford equivalent solutions. The equation becomes,

`E(2c_1+2c_2S)=2c_1alpha_1+2c_2beta`

We expand the brackets and factorise for the coefficients, c₁ and c₂

`2c_1E+2c_2ES=2c_1alpha_1+2c_2beta rarr 0=c_1(2alpha_1-2E)+c_2(2beta-2ES)`

Minimisation with respect to c₂

We now perform the same differentiation operation, but this time with respect to `c_2`,

`del/(delc_2) E(c_1^2+2c_1c_2S+c_2^2)=del/(delc_2)(c_1^2alpha_1+2c_1c_2beta+c_2^2alpha_2)`

Left hand side

Firstly, the left hand side. Again using the product rule,

`del/(delc_2) E(c_1^2+2c_1c_2S+c_2^2)= Edel/(delc_2)(c_1^2+2c_1c_2S+c_2^2)+(c_1^2+2c_1c_2S+c_2^2)del/(delc_2)E`

Moving the differentiation operator inside the brackets,

`=E(del/(delc_2)c_1^2+2del/(delc_2)c_1c_2S+del/(delc_2)c_2^2)+(c_1^2+2c_1c_2S+c_2^2)del/(delc_2)E`

...and obtain

`=E(2c_1S+2c_2)+(c_1^2+2c_1c_2S+c_2^2)del/(delc_2)E`

Right hand side

Now the right hand side, with respect to `c_2`

`del/(delc_2)(c_1^2alpha_1+2c_1c_2beta+c_2^2alpha_2)=(del/(delc_2)c_1^2alpha_1+2del/(delc_2)c_1c_2beta+del/(delc_2)c_2^2alpha_2)`

`=(2c_1beta+2c_2alpha_2)`

Equating the two sides

`E(2c_1S+2c_2)+(c_1^2+2c_1c_2S+c_2^2)del/(delc_2)E=(2c_1beta+2c_2alpha_2)`

We again set the energy derivative to zero, `(delE)/(delc_2)=0`, and arrive at the second of the pair of secular equations,

`E(2c_1S+2c_2)=2c_1beta+2c_2alpha_2`

We expand the brackets and factorise for the coefficients, c₁ and c₂

`2c_1ES+2c_2E=2c_1beta+2c_2alpha_2 rarr 0=c_1beta-2c_1ES+2c_2alpha_2-2c_2E`

Hence,

`c_1(2beta-2ES)+c_2(2alpha_2-2E)=0`

Simultaneous Solution to the Secular Equations

Having derived the two secular equations, we now solve them simultaneously.

`c_1(2beta-2ES)+c_2(2alpha_2-2E)=0`

`c_1(2alpha_1-2E)+c_2(2beta-2ES)=0`

Matrix method

this is not finished

Variational principle to obtain values for the weighting coefficients,

`[[alpha_1-E,beta-ES],[beta-ES,alpha_2-E]]=0`

We evaluate the secular determinant and obtain two solutions,

`(alpha_1-E)c_1+(beta-ES)c_2=0`
and
`(beta-ES)c_1+(alpha_2-E)c_2=0`