What are the rules for constructing the wave functions for the hybrid orbitals?

The rules for constructing the wave functions for the hybrid orbitals are given below:

1. Firstly we have to take the linear combination of wave function of the respective atomic orbitals for constructing the wave function of the hybrid orbitals. Hence the wave function of the ith hybrid orbital formed from s and p atomic orbitals is:
Ψ _i = a_i ф_s + b_i ф_px + c_iф_py + d_i ф_pz

Where ф _s, ф _px, ф _py, ф _pz represents the atomic orbital wave functions which constitute the orthonormal set.

The wave function ф_n and ф_m are said to form an orthonormal set if:

The coefficients of the wave functions a_i,b_i, c_i, d_i can be calculated by using the following three generations:

(i) Each wave function is normalized, i.e.
a²_i + b²_i + c²_i + d²_i = 1

(ii) Each hybrid orbital in the set of hybrid orbitals is orthogonal to the other hybrid orbitals, i.e.
a_ia_j + b _ib_j + c_i c_j+d_id_j = 0

(iii) The squares of the coefficients of component wave functions which is summed over all the hybrid orbitals which participate, equal unity i.e.

∑ a²_i = 1

2.To obtain the equivalent hybrid orbitals, we have to specify a Cartesian co-ordinate system and along one co-ordinate axis place one of the ith hybrid orbitals. Then place as many as hybrid orbitals possible in the plane containing this axis and the other co-ordinate axes.

3.As the s orbital is spherical which is symmetric, each equivalent hybrid orbital of a set contains 1/ n ^1/2 of the s orbital which is distributed among n orbitals.

Therefore the coefficient of ф _s in each orbitals is 1/ n ^1/2