Question

Physical Chemistry Questions:

3. Write down the simplest appropriate LCAO (linear combinations of atomic orbitals) variational wavefunction and give the explicit expressions for the atomic orbitals used, i.e., how they depend on the coordinates.

4. Write down the variational formula for the energy and give explicit expressions for the integrals that have to be computed. Don’t compute all the integrals, as it is too time consuming, but point out which are easy to evaluate and which are equivalent. Give an interpretation of the terms (e.g., “coulomb integral”)

Answer 1

LCAO does not give exact solutions to the one-electron Schrödinger equation, only approximate solutions.

The Variational Principle

For a particular wavefunction, the estimate (expectation value) of the orbital energy (E - previously referred to as ? ) is given by:

The variational principle states that the value of E given by the above equation is always greater than the true energy for the exact solution, from which it follows that the best approximate solution (i.e. the best values for the coefficients in the LCAO construction) can be obtained by minimising the value of the energy, E , given by this equation.