# Atomic orbital orthogonality

• fsci
In summary, the property of orthogonality in orbitals means that the product of two wave functions integrates to zero over all space. This is significant because it allows for the cancellation of positive and negative regions, resulting in a value of zero. Even though the 3s orbital is larger than the 2s orbital, they can still integrate to zero due to the presence of radial nodes in higher n representations. This concept is similar to the orthogonal unit vectors in 3D space. The solutions to the Schrodinger equation for H behave like basis vectors in a infinite space.

#### fsci

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!

fsci said:
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.

Quantum Defect said:
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?

fsci said:
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.

## What is atomic orbital orthogonality?

Atomic orbital orthogonality is a concept in quantum mechanics that refers to the idea that the wave functions of different atomic orbitals are perpendicular to each other. This means that they do not overlap and have no shared regions, resulting in zero probability of finding an electron in both orbitals simultaneously.

## Why is atomic orbital orthogonality important?

Atomic orbital orthogonality is important because it allows us to treat each atomic orbital as a separate entity and make accurate predictions about the behavior of electrons in atoms. It also helps in simplifying complex calculations in quantum mechanics.

## How is atomic orbital orthogonality determined?

Atomic orbital orthogonality is determined by solving the Schrodinger equation for each orbital and then using mathematical techniques to determine the overlap between the wave functions. If the overlap is zero, then the orbitals are considered to be orthogonal.

## What happens when two atomic orbitals are not orthogonal?

When two atomic orbitals are not orthogonal, they can overlap and create regions of non-zero probability for finding an electron in both orbitals simultaneously. This can affect the accuracy of calculations and predictions in quantum mechanics.

## Can atomic orbitals be partially orthogonal?

No, atomic orbitals cannot be partially orthogonal. They are either completely orthogonal or not orthogonal at all. Partial orthogonality would mean that there is some overlap between the orbitals, which is not possible according to the principles of quantum mechanics.