basis the concept of a basis is critical for quantum chemistry. we can represent any wavefunction...
TRANSCRIPT
BasisThe 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:
€
x − a( )n
{ }n=0
∞The set is a basis for any smooth function f(x).
Why? TAYLOR SERIES:
€
f (x) =f (n )(a)
n!n=0
∞
∑ x − a( )n= cn x − a( )
n
n=0
∞
∑
Example: take a=0 and f(x)=sin(x).
Taylor says
€
sin(x) =−1( )
nx 2n+1
2n +1( )!= x −
x 3
3!+x 5
5!−x 7
7!+L
n=0
∞
∑
Let us define
€
Sn = c i x − a( )i
i=0
n
∑ because we can’t go up to ∞ on the computer.
Taylor says
€
sin(x) =−1( )
nx 2n+1
2n +1( )!= x −
x 3
3!+x 5
5!−x 7
7!+L
n=0
∞
∑
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:
€
E0 < E1 < E2 <L < En <L(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:
€
ψn (x) =2
Lsinnπx
L
Using this basis, can we represent the “tent” function?
€
Sn (x) = c iψ i(x)i=0
n
∑ = Iii=0
n
∑define
zoom up on the peak…
zoom up even more…
Final example