Bond angles around a tetravalent central atom

There are several practical algorithms for building molecular geometries based on bond lengths, bond angles, and torsional angles. There seem to be few discussions of the effect changing one angle has on the remaining bond angles depending upon local symmetry. For example, in methane, CH$_{4}$, the H-C-H bond angles are all tetrahedral, i.e., $\alpha$ = 109.4712206... deg. If one considers CH$_{3}$F, a molecule with C$_{3v}$ symmetry, how are the H-C-F bond angles related to the H-C-H bond angles? This study derives the bond angle relationships for a 4-bonded central atom such as a saturated C atom. For a 4-bonded central atom (6 bond angles) the possible local point group symmetries are T$_{d}$(0), D$_{2d}$(1), C$_{3v}$(1), C$_{2v}$(1), D$_{2}$(2), C$_{2}$(3), C$_{s}$(3), and C$_{1}$(4). The numbers in parentheses are the degrees of freedom, i.e., the number of angles which can be assigned arbitrary values with the remaining angles fixed by symmetry. Analytical formulas relating the bond angles for each of the eight possible symmetries are derived. Also, formulas have been derived for the five possible symmetries of a planar 4-bonded atom, D$_{4h}$(0), D$_{2h}$(1), C$_{2v}$(pendant, 1), C$_{2v}$(trapezoid, 2), and C$_{s}$(3); the three possible structures of planar 3-bonded atoms, (D$_{3h}$(0), C$_{2v}$ (1), and C$_{s}$(2); the three possible symmetries of pyramidal 3-bonded atoms, C$_{3v}$(1), C$_{s}$(2), and C$_{1}$ (3); and the trivial case of 2-bonded atoms, D$_{{\infty}h}$(0) and C$_{2v}$(bent 1). There are also six distinct 4-bonded central atom structures with all the bonds directed into a hemisphere, C$_{4v}$(1), C$_{2v}$(2), C$_{2}$(3), C$_{s}$(trapezoid 3), C$_{s}$(pendant 3), and C$_{1}$(4), geometries rarely seen in molecules.Ope