Atomic Orbitals and Their Quantum Numbers

In order to understand the coordination polyhedra encountered in inorganic chemistry, it is useful to understand the properties of the atomic orbitals on the central atom, which can hybridize to form the observed coordination polyhedra. In this connection atomic orbitals correspond to the one-particle wave functions T, obtained iS spherical harmonicshy solution of the following second-order differential equation in which the potential energy V is spherically symmetric  

the spherical harmonics 4 obtained by solving equation (8) are functions of either the three spatial coordinates x, y, and z or the corresponding spherical polar coordinates r, 0, and ( ) definedby the equations 

Furthermore, a set of linearly independent wave functions can be found such that 4 can be factored into the following product  

All atomic orbitals are orthogonal, i.e., the overlap between different atomic orbitals is zero. 

The extent of the s orbital (with / = 0) is independent of direction, i.e., the s orbital is spherically symmetrical and its wave function T sis independent of 0 and (().. In view of this and in view of the uniqueness of the ns orbital for a given value of n, there is no polynomial designation for the s orbital. 

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