= Solution
Neglecting intersite <orbital overlap>, one normalized real set of <symmetry-adapted linear combinations> is
$$
\boxed{
\begin{aligned}
\chi_{A_1}&=(s_A+s_B+s_C)/\sqrt3,\\
\chi_{E,1}&=(2s_A-s_B-s_C)/\sqrt6,\\
\chi_{E,2}&=(s_B-s_C)/\sqrt2.
\end{aligned}}
$$
The coefficient vectors are <orthonormal>. The first has equal phase at all three sites; the second opposes site $A$ to $B,C$; the third has zero amplitude at $A$ and opposite phases at $B,C$. Any orthonormal rotation of the two $E$ functions is equally valid. With nonzero intersite <orbital overlap>, normalization must instead use the overlap <matrix>.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-1-ammonia-symmetry-orbitals.png]
{title=Original coefficient-and-phase sketches of the three hydrogen <symmetry orbitals> in <ammonia>}
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