= Solution
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 block
$$
H_{A_1}=\begin{pmatrix}\alpha&3\beta\\3\beta&\alpha\end{pmatrix}.
$$
For 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 is
$$
H_{T_2}=\begin{pmatrix}\alpha&\beta\\\beta&\alpha\end{pmatrix}.
$$
Their <eigenvalues> give the <cube nearest-neighbour Hückel model>:
$$
\boxed{E=\alpha+3\beta\ (a_1),\quad\alpha+\beta\ (t_2),\quad
\alpha-\beta\ (t_2),\quad\alpha-3\beta\ (a_1).}
$$
For the bonding convention $\beta<0$ these are in increasing energy order, with <statistical weights> $1,3,3,1$. The eight valence <electrons> occupy \b[$(a_1)^2(t_2)^6$], 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.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-1-cubic-cluster-spectrum.png]
{title=Original <cube nearest-neighbour Hückel model> energy levels and eight-electron occupancy}
Back to article page
