VII.B.8. (XII.E.1.)

Notes on converting between angles in tetravalent centers

C_3V molecules

In a C3v molecule such as CH3F, the angles FCH and HCH are related by symmetry. The angles a (FCH) and b (HCH) are related using the following formulas: Please note that the latter solution is not unique, and that solving for angle a may return 180° - a

C_2V pentatomic molecules

In a C2V molecule such as CH2F2 the angle HCF can be related to the two angles HCH and FCF by symmetry. The angle e (HCF) is related to the angles c (HCH) and d (FCF) by: Please note that the latter solution is not unique, and that solving for angle e may return 180° - e

C_s tetravalent molecules

We use fluorochloromethane as an example. We know: angle c (H-C-H) (there is only one) angle d (F-C-Cl) (there is only one) angle e (H-C-Cl) (there are two equivilant) We want angle f (F-C-H) (there are two equivilant)

Arbitrary tetravalent center

In a arbitrary tetravalent center their are six angles. We define these angles as: a is angle H-C-Cl b is angle F-C-Br c is angle H-C-Br d is angle F-C-Cl e is angle Cl-C-Br f is angle H-C-F Assume we want angle f as a function of the other five angles. We can work with the cosines of the angles: ca = cos(a), cb = cos(b),etc. Define p and q: p = sqrt( (ca2 + cc2 + ce2 - 2*ca*cc*ce - 1) * ((ca2 + cb2 + cd2 - 2*ca*cb*cd - 1) ) q = ca*(cb*ce + cc*cd) - cb*cc - cd*ce The two solutions for cos(f) are: cf = ( p + q)/(ca2 - 1) cf = (-p + q)/(ca2 - 1)

Dihedral from three angles

Given the three angles: a is angle F-C-Cl b is angle H-C-Cl c is angle H-C-F We want the dihedral angle d (the angle between the two planes defined by F-C-Cl and H-C-Cl).