# Derivation of the orbital analysis equation and its physical significance

## Summary:

unable to prove/derive theoretically the relation between bond angle and s and p character of a hybridized orbital.
$$cos\theta = \frac {s}{1-s} = \frac{p-1}{p}$$
in this equation ##\theta## is the bond angle and ##s## and ##p## are the fractional s-character of the orbital and p-character of the orbital.
This is equation is used rigorously in showing that the s-character of the axial orbitals in a ##sp^3d## hybridized orbital is 0 and hence it is also used to prove bents rule which states that the most electronegative element in a ##sp^3d## hybridized orbital takes the axial positions of the trigonal bipyramid that is formed. But I am unable to find a formal proof/derivation of this equation on the internet or in my textbook.
Is this relation between the bond angle and the s/p character very obvious that it does not need a proof/derivation?
(also I am assuming there might be an experimental proof for this formula but I am not looking for that rather a theoretical proof of this equation will be nice)

Borek
Mentor
Can't say this relation makes sense, for s>0.5 cos(θ) would be larger than 1.

Can't say this relation makes sense, for s>0.5 cos(θ) would be larger than 1.
I am so sorry I made a typo it is actually $$cos\theta = \frac{s}{s-1}$$

Borek
Mentor
Still fails, now for s>0.5.

Still fails, now for s>0.5.
I thought it was clear no hybridized orbital can have an s character more than 50%(0.5 fractional) as sp hybridized orbital has the least number of orbitals(2) combining and has 50% s-character in both orbitals. Hence the equation does not fail.
also, this very fact was used in Dragos rule...

Borek
Mentor
OK, for "standard", non fractional hybridizations it will hold.

Apparently I know a bit too much to properly understand the problem as defined here.

OK, for "standard", non fractional hybridizations it will hold.

Apparently I know a bit too much to properly understand the problem as defined here.
maybe this particular equation isn't very popular(because it isn't very general?)

Borek
Mentor
I don't remember being taught it. Doesn't mean much.

On the second thought I could get s and p reversed. My bad.