Basis set orthonormality property

[filippo]

What does orthonormality of a basis set i.e. {χ_i} stand for? I am reading the Mulliken Population Analysis and there are integrals that are simplified by this property of basis sets and I can't quite catch what is it.

 

[Fredrik]

Do you know what an inner product (also called scalar product) is? Two vectors u and v are said to be orthogonal if their inner product is zero: <u,v>=0. The quantity

$$\|u\|=\sqrt{\langle u,u\rangle}$$

is called the norm of u. A basis is said to be orthonormal if all its members have norm 1 and are orthogonal to each other, i.e. if

$$\langle x_i,x_j\rangle=\delta_{ij}$$

where $\delta_{ij}$ is the Kronecker delta (=1 when i=j, and =0 otherwise).

Intuitively, you can think of this as meaning that the basis vectors all have length 1 and are perpendicular to each other, but we're really talking about generalizations of those concepts.