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

[tex]\|u\|=\sqrt{\langle u,u\rangle}[/tex]

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

[tex]\langle x_i,x_j\rangle=\delta_{ij}[/tex]

where [itex]\delta_{ij}[/itex] 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.

 

[Entropia]

The orthonormality of a basis set refers to the mathematical property that the basis functions are both orthogonal and normalized. Orthogonal means that the basis functions are perpendicular to each other, while normalized means that the magnitude of each basis function is equal to 1.

In practical terms, this means that the basis functions in a set are independent from each other and do not overlap. This is important in quantum chemistry, as it allows for simplification of calculations and more accurate results.

In the context of Mulliken Population Analysis, the orthonormality of the basis set allows for the simplification of integrals, making it easier to determine the distribution of electrons in a molecule. This is because the orthonormality property ensures that the basis functions do not interfere with each other, allowing for a more accurate representation of the electron density.

Overall, the orthonormality of a basis set is a crucial aspect in quantum chemistry, as it allows for more efficient and accurate calculations of molecular properties.