Basis The concept of a basis is critical for quantum chemistry. We can represent any wavefunction...

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:

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x − a( )n

{ }n=0

∞The set is a basis for any smooth function f(x).

Why? TAYLOR SERIES:

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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

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sin(x) =−1( )

nx 2n+1

2n +1( )!= x −

x 3

3!+x 5

5!−x 7

7!+L

n=0

∞

∑

Let us define

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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:

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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:

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ψn (x) =2

Lsinnπx

L

Using this basis, can we represent the “tent” function?

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Sn (x) = c iψ i(x)i=0

n

∑ = Iii=0

n

∑define

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Final example