# Mathematical relation between bond angle and fraction of s or p character

In Concise Inorganic Chemistry by J.D.Lee (Adapted by Sudarsan Guha, Fourth Edition), on page 75, under the topic "Effect of Electronegativity - When the surrounding atom is same with different central atom having lone pair" it is given:

The effect of electronegativity as postulated in VSEPR theory explains the order of the angle for the above molecules (Hydrides of groups 15 and 16) but cannot rationalize very small angles (~$$90^\circ$$) in $$\ce{PH3,AsH3,SbH3}$$ and $$\ce{H2S,H2Se,H2Te}$$ with respect to $$\ce{NH3}$$ and $$\ce{H2O}$$ respectively.

To explain this, Drago suggested an empirical rule$$^{[1]}$$ which is compatible with the energitics of the hybridisation. It states that if the central atom is in the third row or below in the periodic table, the lone pair will occupy a stereochemically inactive s orbital$$^{[2]}$$, and the bonding will be through p orbitals, and bond angles will be nearly $$90^\circ$$ if the electronegativity of the surrounding atom is less than or equal to $$2.5$$.

The above rule is based upon the relation between hybridisation and bond angle for two or more equivalent s-p hybrid orbitals, where the fraction of s character (S) or fraction of p character (P) is given by the relationship:

$$\cos \theta = \frac{S}{S-1}=\frac{P-1}{P}$$

for $$\theta \in (90^\circ,180^\circ)$$

The following are links to previous questions asked on Chemistry Stack Exchange regarding concepts in this quoted text.

[1] : Drago's Rule : What is Drago's rule? Does it really exist?

[2] : Stereochemically (in)active s orbital : What is a stereochemically active or inactive s orbital?

This formula,

$$\cos \theta = \frac{S}{S-1}=\frac{P-1}{P}$$

relates the bond angle $$\theta$$ and the fraction of s/p characters. I tried to plug in some standard values.

For example, On substituting the expression with S=0.25 or P=0.75, I get $$\frac{S}{S-1}=-0.33...$$ and $$\frac{P-1}{P}=-0.33...$$. I know they both give the same as in sp$$^n$$ hybridisation where only s and p orbitals are involved, the sum of fractions of the s character and p character i.e., S+P=1, and I was able to interconvert the expressions $$\frac{S}{S-1}$$ and $$\frac{P-1}{P}$$ with simple substitution - S=1-P or P=1-S.

On finding the cosine inverse of -0.33...,

$$\cos ^{-1} (-0.33...)=109.471^{\circ}$$

which is exactly equal to the tetrahedral angle ($$109.471^{\circ}$$). From this, we can obtain the bond angle in sp3 hybridisation. And similarly, we can do it for sp, and sp2 hybridisation which I've skipped here.

It is also possible to the reverse with this formula, i.e., given the bond angle we can compute the fraction of s or p character. For example, for $$\ce{AsH3}$$, the H-As-H angle is $$91.8 ^{\circ}$$, and from the calculation, it can be shown that each As-H bond consists of almost 97% p character and 3% s character.

My doubt is how did they arrive at the given formula? Is there any derivation* for this or is that simply an experimental result? Will this work for all values of S and P (under the constraint S+P=1)? What is the logic behind this formula?