Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A5 / b

The $z$ axis is a $C_2$ axis through the centers of opposite cube faces: a half turn permutes atoms within each species. A body diagonal through sites 1 and 8 is a $C_3$ axis: a third turn fixes those two opposite vertices and cycles the other three vertices of each tetrahedral sublattice. These are symmetry operations of the tetrahedral symmetry group $T_d$; a quarter turn about $z$ would interchange the two chemical species and is not a symmetry of the specified cluster.

For the tetrahedral symmetry classes $(E,8C_3,3C_2,6S_4,6{\sigma }_d)$, each set of four $s$ atomic orbitals hasThe counts are the fixed sites of each symmetry operation: a threefold rotation fixes one vertex of each tetrahedron, a twofold or improper fourfold operation fixes none, and a diagonal mirror fixes two. Character orthogonality reduces each set toThe dimensions $1+3=4$ provide a check. Thus the complete eight-orbital model contains two copies of each of $A_1$ and $T_2$, which can mix across the two species.

Write $L_i$ for the lithium $2s$ orbital at site $i$ and $N_i$ for the sodium $3s$ 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 areEach $T_2$ 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.

Use the stipulated common Hückel Coulomb parameter $\alpha$, an equal Hückel resonance integral $\beta$ along each cube edge, and zero coupling elsewhere; take an orthonormal site basis. Each lithium vertex has three sodium neighbors. The totally symmetric combinations therefore have the blockFor a coordinate-like $T_2$ combination, the sum of the three neighboring sodium coordinate signs equals the lithium coordinate sign. Thus each of the three equivalent blocks isTheir eigenvalues give the cube nearest-neighbour Hückel model:For the bonding convention $\beta <0$ these are in increasing energy order, with statistical weights $1,3,3,1$. The eight valence electrons occupy $(a_1{)}^2(t_2{)}^6$, filling the four bonding spatial molecular orbitals. Their total one-electron energy is $8\alpha +12\beta$. This is a deliberately simplified model: unequal atomic energies or unequal overlaps require a generalized calculation.