The Pauli Exclusion Principle — Question 3

Question 3

Nonorthogonal orbitals change the normalization of an antisymmetric pair

Two normalized spin orbitals a and b have real overlap $S=\langle a|b\rangle=3/5$. Use $\Psi=C[a(1)b(2)-b(1)a(2)]$ with C positive.

1. Find the normalization constant.

2. Discuss the limit $S\to1$ and why the formula cannot normalize two identical orbitals.

Question 3 - Solution

Nonorthogonal orbitals change the normalization of an antisymmetric pair

(a) Method and calculation

$$\|\Psi\|^2=2C^2(1-|S|^2).$$$$\boxed{C=[2(1-9/25)]^{-1/2}=5/(4\sqrt2)}.$$

(b) Interpretation and further calculation

$$S\to1:\quad\|a(1)b(2)-b(1)a(2)\|^2\to0.$$$$b=a:\quad\Psi\equiv0\Rightarrow\boxed{\text{no normalized duplicate state}}.$$

Physical check

A diverging normalization factor at coincident orbitals does not rescue the exactly zero determinant. Distinct limiting paths may approach different second orbitals after orthogonalization.