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 Incoming links: Tetrahedral s-orbital symmetry combinations

Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A5 / b / iii / Solution 2026-10-20

Write

L_{i}

for the lithium

2 s

orbital at site

i

and

N_{i}

for the sodium

3 s

orbital. Reading the printed axes, the lithium coordinates have signs

1 = (-, +, +)

,

2 = (+, -, +)

,

3 = (+, +, -)

and

4 = (-, -, -)

; the sodium sites are

5 = (+, +, +)

,

6 = (-, -, +)

,

7 = (-, +, -)

and

8 = (+, -, -)

. The normalized tetrahedral s-orbital symmetry combinations are

A_{1} T_{2 x} T_{2 y} T_{2 z} Li (L_{1} + L_{2} + L_{3} + L_{4}) /2 (- L_{1} + L_{2} + L_{3} - L_{4}) /2 (L_{1} - L_{2} + L_{3} - L_{4}) /2 (L_{1} + L_{2} - L_{3} - L_{4}) /2 Na (N_{5} + N_{6} + N_{7} + N_{8}) /2 (N_{5} - N_{6} - N_{7} + N_{8}) /2 (N_{5} - N_{6} + N_{7} - N_{8}) /2 (N_{5} + N_{6} - N_{7} - N_{8}) /2

Each

T_{2}

row takes its signs from the corresponding coordinate, so the three functions transform together like

(x, y, z)

. These coefficient vectors are orthonormal when intersite orbital overlap is neglected. A common sign reversal of any row is physically immaterial.

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