= Solution
For E, each BH supplies two skeletal <electrons> and each <phosphorus> vertex supplies three after retaining a <lone pair>. Hence $10(2)+2(3)=26$ <electrons>, or thirteen pairs, matching a twelve-vertex <closo cluster>. \b[Place the two adjacent P vertices on a <regular icosahedron>.] The remaining ten <boron> vertices fall into four symmetry classes: two sites adjacent to both P vertices, four adjacent to exactly one, two adjacent to neither but opposite the common-neighbour pair, and two opposite the P pair. The populations are \b[$2:4:2:2$], so four <boron> resonances are expected. Interchanging the identical <phosphorus> vertices is allowed when determining chemical equivalence.
For F, a ten-vertex <bicapped square antiprism> offers a consistent closo model. There are eight BH vertices, one B vertex bearing the $\mathrm{NEt_3}$ donor instead of H, and one P vertex. Their skeletal contributions are respectively $16$, $3$ and $3$, giving \b[22 <electrons>, eleven pairs]. The donor-bearing B contribution follows from B plus the donated <lone pair> minus the exoskeletal B–N bond, $3+2-2=3$.
One explicit arrangement consistent with six <boron> signals puts P at one axial cap and the $\mathrm{NEt_3}$ substituent on one B vertex of the nearer square. The remaining <mirror plane> through that B, the opposite vertex of its square, and the cap axis leaves three B sites individually fixed: the donor-bearing B, its opposite square vertex and the other axial cap. The other six B sites form three exchanged pairs. Thus the nine <borons> give \b[three singleton environments and three two-site environments], six signals in all. The original compound has a crystallographically characterized positional assignment; the spectrum alone asks for a compatible structure and does not establish a unique positional isomer. A <single-crystal X-ray diffraction> determination or assigned heteronuclear correlations would distinguish alternatives.<Synthesis of polyhedral phosphaboranes: crystal structure of 6-(C2H5)3N·2-PB9H8 by John L. Little and others>[1989, pp.1087–1091, synthesis of diphosphaborane and crystallographic characterization of the amine-containing side product]{ref}
\Image[/past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1-cluster-skeletons.png]
{title=Original skeletal models and boron-site equivalence classes}
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