Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A2 / a / i

Solution

The Pauli exclusion principle requires the complete electron wavefunction to change sign on exchange of any two electrons, including both position and spin labels. Let

a = 1 s

and

b = 2 s

be normalized, orthogonal spatial atomic orbitals, and let

α, β

denote spin up and down.

The symmetric and antisymmetric spatial functions are

Φ_{\pm} = \frac{a ( 1 ) b ( 2 ) \pm b ( 1 ) a ( 2 )}{2} .

Ψ_{S = 0} = Φ_{+} \frac{α ( 1 ) β ( 2 ) - β ( 1 ) α ( 2 )}{2} .

The three triplet states share the antisymmetric spatial factor:

Ψ_{1, + 1} Ψ_{1, 0} Ψ_{1, - 1} = Φ_{-} α (1) α (2), = Φ_{-} \frac{α ( 1 ) β ( 2 ) + β ( 1 ) α ( 2 )}{2}, = Φ_{-} β (1) β (2) .

Every factor and complete product is normalized. Here the subscripts are

(S, M_{S})

1 s 2 s

in each case.

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