DETERMINATION OF WAVE FUNCTION OF sp² & sp³ hybrid orbital

INTRODUCTION:- In Hybridization , there are three types of hybrid orbital – sp, sp², sp³ .Subscripts 2, 3 indicates that the shapes of hybrid orbitals are different from those of the parent orbitals. e.g.:- s orbital is spherical & p orbital is dumb-bell shaped and thus, the shape of the bigger lobe of sp hybrid orbital which results from the mixing of one s & one p orbital, is different. It has the characteristics of both the mixing orbitals, and thus has an oval shape. The bigger lobes of sp² hybrid orbitals which result from the combination of one s & two p orbitals are pear shaped because of the greater contribution of two dumb-bell shaped p orbitals. In the case of sp³ hybrid orbitals which result from the mixing of one s & three p orbitals, the contribution of three p orbitals predominates over the contribution of one s orbital.



RULES FOR CONSRUCTING WAVE
FUNCTIONS FOR HYBRID ORBITALS
*1:- The Wave functions of hybrid orbitals are constructed by taking into consideration the linear combination of the wave function of the appropriate AOs. Thus, wave function of the ith hybrid orbital formed from s & p AOs is
Ψi = a i фs +bi фpx+ci фpy+di фpz
Where фs ,фpx ,фpy ,ф pz are the atomic orbital wave functions which constitute an orthogonal set .Two wave functions фm & фn
Are said to form a orthogonal set if
∫ фn фn dґ = 1
The coefficients of wave functions ai , bi ,ci ,di can be calculated by three conditions:-
• Each wave function of a hybrid orbital is normalized , that is,
∫ Ψi2 dґ = 1.It can be shown that for such a normalized wave function
ai2 +bi2 +ci2 +di2 = 1
• Each hybrid orbital in the set of hybrid orbitals is orthogonal to the other hybrid orbitals, i.e. ∫Ψ iΨjdΓ =0
From this relation, it can be shown that
aiaj +bibj +cicj +didj =0
• The squares of the coefficients of component wave-functions to the hybrid orbital wave functions over all the hybrid orbitals wherein they participate, equal unity, i.e.,
ai2 =1
This is because pure orbitals must be completely utilized in formation of hybrid orbitals.
*2:- Since the s orbital is spherically symmetric, each equivalent hybrid orbital of a set contains 1∕√n of the s orbital distributed in n orbitals , i.e. the coefficient of ф in each hybrid orbital is
1∕√n


DETERMINATION OF WAVE FUNCTIONS FOR THE sp HYBRID ORBITALS:-
Since one of the sp hybrid orbitals is to be placed along the +z axis, we shall invoke the hybridization of s, p ,p orbitals are in the XZ plane.The Wave functions for the three sp hybrid orbitals are:-