The decisive feature is the word heteronuclear, and it turns on the Coulomb integrals.

The two atoms are different, so their Coulomb integrals must differ. In a heteronuclear molecule A-B the atoms have different electronegativities and therefore different atomic orbital energies, written \(\alpha_A\) and \(\alpha_B\). Any expression using a single \(\alpha\) and squaring the term \((\alpha - E)^{2}\) describes a homonuclear molecule, where the two atoms are identical. That immediately eliminates the two options written that way.

The secular determinant is

\(\begin{vmatrix} \alpha_A - E & \beta \\ \beta & \alpha_B - E \end{vmatrix} = 0\),

which expands to \((\alpha_A - E)(\alpha_B - E) - \beta^{2} = 0\).

The overlap integral is neglected here. In the general treatment the off-diagonal elements are \(\beta - ES\), because the two atomic orbitals are not orthogonal. But the standard Hückel (zero-overlap) approximation sets \(S = 0\), reducing those elements to a bare \(\beta\). This is the usual convention in which secular determinants are quoted, and it is the form required here.

Solving the resulting quadratic gives two roots — one below both \(\alpha_A\) and \(\alpha_B\) (bonding) and one above both (antibonding).

The consequence of \(\alpha_A \ne \alpha_B\) is chemically important: the molecular orbitals are unequally weighted. The bonding MO resembles the orbital of the more electronegative atom and the antibonding MO resembles the other, and that unequal mixing is precisely what produces bond polarity. The larger the gap between \(\alpha_A\) and \(\alpha_B\), the more ionic the bond and the weaker the covalent interaction.

Hence the secular determinant is (α_A - E)(α_B - E) - β² = 0.