# Atomic orbital orthogonality

## Main Question or Discussion Point

I am wondering how two orbitals of same n values can be orthogonal, for example how are a 2s and 3s orbital orthogonal?
What I understand is a property of orthogonality is the product of the two wave functions integrate to zero over all space. I tried to look at this graphically and categorize overlapping regions as either positive or negative products and then cancel out positive and negative regions to yield zero, but what I am having trouble is that the 3s orbital is larger than the 2s orbital, so how can they possible integrate to zero?
If someone could also explain the significance/implications of all orbitals being orthogonal that would be helpful too! I do not understand the importance of orthogonality in orbitals!
Thank you!

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Quantum Defect
Homework Helper
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I am wondering how two orbitals of same n values can be orthogonal, for example how are a 2s and 3s orbital orthogonal?
What I understand is a property of orthogonality is the product of the two wave functions integrate to zero over all space. I tried to look at this graphically and categorize overlapping regions as either positive or negative products and then cancel out positive and negative regions to yield zero, but what I am having trouble is that the 3s orbital is larger than the 2s orbital, so how can they possible integrate to zero?
If someone could also explain the significance/implications of all orbitals being orthogonal that would be helpful too! I do not understand the importance of orthogonality in orbitals!
Thank you!
There are radial nodes that you usually cannot see in the higher n (n>=2) representations, as typically drawn. 2s has one radial node, 3s has two radial nodes. On either side of the radial node there is a change in sign for the wave function.

When you do the integral for any of the products you will get positive regions cancelling the negative regions.

There are radial nodes that you usually cannot see in the higher n (n>=2) representations, as typically drawn. 2s has one radial node, 3s has two radial nodes. On either side of the radial node there is a change in sign for the wave function.

When you do the integral for any of the products you will get positive regions cancelling the negative regions.
Ok thank you. I may be heading in the wrong direction... But how do the product of a 1s x 3s and then product of a 2s x 3s both integrate to zero when 1s and 2s orbitals are not equal to each other?

Quantum Defect
Homework Helper
Gold Member
Ok thank you. I may be heading in the wrong direction... But how do the product of a 1s x 3s and then product of a 2s x 3s both integrate to zero when 1s and 2s orbitals are not equal to each other?
You might look into some Linear Algebra, Differential Equations textbooks. For the H-atom wave functions, you will have an infinite set of wavefunctions, that are solutions to the Schroedinger Eq. Each l=0 wavefunction will be "orthogonal" to the other, as long as n_1<> n_2.

This is kind of like the way that the unit vectors in 3D space are all orthogonal to one another. x_hat dot y_hat = 0, x_hat dot z_hat = 0, y_hat dot z_hat =0.

The solutions to the Schoredinger Eq for H are like these basis vectors in 3D space, except the space is infinite.