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Learn how basis sets form the foundation for wavefunction representation in quantum chemistry, with examples related to Taylor series and Sturm-Liouville theory. Understand the significance of choosing proper basis functions for accurate calculations.
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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.
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
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.
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
