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LCAO Wave Functions

Key Foundations of MO Theory

Eigenfunctions and Eigenvalues


  • An operator is a set of instructions that acts on an adjacent function.
  • For a given operator, n function is called an eigenfunction if the operator returns the original function times a constant.
  • An eigenvalue is the constant which is returned when a given operator acts on an eigenfunction:
  • e.g. (operator)(eigenfunction)=(eigenvalue)(eigenfunction)
\[ \hat{H}\Psi=E\Psi \]

Variational Principle

The variational principle states that for any guess of the wavefunction \(\Phi\), the calculated value of the hamiltonian operator \(H\), will never result in lower energy than the exact ground state energy of the system

  • \(\Phi=\) the MO wavefunction
  • \(E_0=\) the exact ground state energy of the system
\[ \frac{\int\Phi^*H\Phi dr}{\int\Phi^8\Phi dr}\geq E_0 \]

The best perfect wavefunction will be the one which is equal to \(E_0\)

Molecular Orbitals from LCAOs

The one-electron molecular orbital \(\phi\) is the liinear combination of the atomic orbitals (basis functions) \(\varphi\), weighted by some coefficient \(a\)

\[ \phi=\sum\limits_{i=1}^N{a_i\varphi_i} \]
  • \(\Phi\) is the many-electron wave function, formed of the slater determinants of the occupied \(\phi\).
  • To pick the basis set we start with atomic basis functions \(\varphi\) (s, p, d, f functions) since we know that they are going to be centred on the atoms and we sum them together.
  • We use the variational principle to iteratively optimise these steps


Hartree Product is the sum of one-electron molecular orbitals

\[ \sum\limits_{i=1}^N{\phi_i} \]

Minimising the basis set

Calculating the energy

For a one-electron orbital \(\Phi\) we evaluate the equation:

\[ E=\frac{\int\bigg(\sum\limits_i{a_i^*\varphi_i^*}\bigg)H\bigg(\sum\limits_j{a_j\varphi_j}\bigg)d\bf{r}}{\int\bigg(\sum\limits_i{a_i^*\varphi_i^*}\bigg)\bigg(\sum\limits_j{a_j\varphi_j}\bigg)d\bf{r}}=\frac{\sum\limits_{ij}a_i^*a_j\int\varphi_i^* H\varphi_jd\bf{r}}{\sum\limits_{ij}a_i^*a_j\int\varphi_i^* \varphi_jd\bf{r}} \]

\(i\) runs over basis functions

\(j\) runs over MOs

Where we sum the basis functions and coefficients and their complex conjugates and calculate the expectation value for the hamiltonian (top) and divide it by the integral over all space of the product.

  • The integral \(\int\varphi_i^* H\varphi_jd\bf{r}\) is denoted as \(H_{ij}\) and is called the “resonance integral”
  • The integral \(\int\varphi_i^*\varphi_jd\bf{r}\) is denoted as \(S_{ij}\) and is called the “overlap integral”

Minimisaiton condition

To minimise the energy we are trying to obtain:

\[ \frac{\partial E}{\partial a_k}=0\hspace{2cm} \text{for all }k \]

That is that all the partial derivatives of all the \(a\)‘s must equal zero - They’re all at a stationary point

If we take all the partial derivative of the energy calculation, we get \(N\) linear equations with the form

Where we are now treating \(a_i\) as the variable and \((H_{ki}-ES_{ki})\) as the coefficient term. If we break that down further, we can evaluate \(H_{ki}\) and \(S_{ki}\), but we need to determine the energy that results in the equation \(=0\).

\[ \sum\limits_{i=1}^N{a_i(H_{ki}-ES_{ki})}=0\hspace{2cm} \text{for all }k \]

The Secular Determinant

Since we need to evaluate all of the coefficients as one, we can use the secular equation as:

\[ \begin{vmatrix} H_{11}-ES_{11} & H_{12}-ES_{12} & \cdots & H_{1N}-ES_{1N} \\ H_{21}-ES_{21} & H_{22}-ES_{22} & \cdots & H_{2N}-ES_{2N} \\ \vdots & \vdots & \ddots & \vdots\\ H_{N1}-ES_{N1} & H_{N2}-ES_{N2} & \cdots & H_{NN}-ES_{NN} \end{vmatrix}=0 \]

Fundamental algebra tells us however that this determinant is going to be a polynomial of order \(N\) (\(E^N\)) but also that there will be \(N\) possible values of \(E\) that will give us a satisfactory answer (some of these may be degenerate or complex…)

The steps to calculate the MO:

  1. Select a set of \(N\) basis functions
  2. Determine all (\(N(N-1)/2\)) values of both \(H_{ij}\) and \(S_{ij}\)
    • While the determinant grows at a rate \(N^2\), because of the symmetry of the matrix, there are fewer solutions.
  3. From all the values of \(H_{ij}\) and \(S_{ij}\), form the secular determinant and determine the values of \(E_{j}\) that satisfy the secular equation.
  4. For each \(E_j\), solve the set of linear equations \(\bigg({a_i(H_{ki}-ES_{ki})}\bigg)\) to determine the basis set coefficients \(a_{ij}\) for that MO