Basis

1 / 13

Basis

Télécharger la présentation

Basis

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

1. Basis The concept of a basis is critical for quantum chemistry. We can represent any wavefunction using the basis of our choice. The basis we choose is normally the 1-electron AO’s (hydrogen atom orbitals) since we know these exactly.

2. Simple example of an infinite dimensional basis: The set is a basis for any smooth function f(x). Why? TAYLOR SERIES: Let us define because we can’t go up to ∞ on the computer. Example: take a=0 and f(x)=sin(x). Taylor says

3. Taylor says

4. What does this have to do with quantum mechanics? The link is through Sturm-Liouville theory. This theory guarantees certain properties of the solutions of a class of differential equations which includes the time-indep. Schroedinger equation. It says the set of solutions for a particular Hamiltonian forms an orthonormal basis, and also that the energy values can be ordered: (we will use this ordering property to prove the variational theorem) Since HF is concerned with 1-electron wavefunctions, we can take as our basis the eigenvectors of the hydrogen atom Hamiltonian, since we know these exactly and we are guaranteed that they, indeed, form a basis for any 1-electron wavefunction.

5. A simple example: let us take our basis to be the eigenvectors of the particle-in-a-box Hamiltonian: Using this basis, can we represent the “tent” function? define

6. zoom up on the peak…

7. zoom up even more…

8. Final example