 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2023/paper-1/a1/c/solution

Denote the two complex spatial atomic orbitals by

ψ_{+}

and

ψ_{-}

, retaining the same spin state. With the phases specified by the supplied spherical harmonics, choose

ψ_{x} = \frac{ψ _{-} - ψ _{+}}{2}, ψ_{y} = \frac{i ( ψ _{-} + ψ _{+} )}{2} .

The two original atomic orbitals are orthonormal, so the squared coefficients add to one and these combinations are normalized and mutually orthogonal. Explicitly,

ψ_{x} = \frac{1}{32 π} x e^{- r /2}, ψ_{y} = \frac{1}{32 π} y e^{- r /2},

which are real spatial wavefunctions. An overall sign on either orbital has no physical consequence.

These are the familiar $2 p_{x}$ and $2 p_{y}$ p orbitals. Each has two lobes of opposite wavefunction sign, with an atomic-orbital nodal plane through the atomic nucleus:

x = 0

for

2 p_{x}

and

y = 0

for

2 p_{y}

. The sign is a phase convention, not a positive or negative electric charge; the probability density is nonnegative on both lobes.

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