Question

While studying basis sets, I came across Gaussian type orbitals (GTOs), which are used as atomic basis sets. I reviewed the polar form of these orbitals, which are very similar to hydrogen atom orbitals. I then looked at the Cartesian form of the same orbitals and tried to derive the polar form from the Cartesian form using simple polar substitutions.

xyz=rcos(θ)sin(ϕ);=rsin(θ)sin(ϕ);=rcos(θ)

To my surprise, I was unable to get the same polar expression. Both the polar and Cartesian forms of the GTO are attached for reference. Would someone who has worked on this derivation be willing to share it? It would be very helpful.

Cartesian form: Gℓ,m,α(r) = Nℓ,m x^ℓₓ y^ℓ_y z^ ℓ_z e⁻αr² polar form: Gℓ,m,α(r) = Nℓ,mrℓe⁻^αʳ²Yℓ,m(θ,ϕ)

Answer 1

To derive the polar form of Gaussian type orbitals (GTOs) from the Cartesian form, you need to substitute the Cartesian variables (x, y, z) with the polar variables (r, θ, ϕ). The polar expression for GTO is given by Gℓ,m,α(r) = Nℓ,mrℓe⁻ˣʳ²Yℓ,m(θ,ϕ), where Nℓ,m is a normalization constant, r is the radial distance, θ is the polar angle, ϕ is the azimuthal angle, and Yℓ,m(θ,ϕ) is the spherical harmonic function.

Step-by-step explanation:

The polar and Cartesian forms of Gaussian type orbitals (GTOs) are related to each other through a mathematical transformation called polar substitution. To derive the polar form from the Cartesian form, you need to substitute the Cartesian variables (x, y, z) with the polar variables (r, θ, ϕ). The polar expression for GTO is given by Gℓ,m,α(r) = Nℓ,mrℓe⁻ˣʳ²Yℓ,m(θ,ϕ), where Nℓ,m is a normalization constant, r is the radial distance, θ is the polar angle, ϕ is the azimuthal angle, and Yℓ,m(θ,ϕ) is the spherical harmonic function.