Molecular Orbital Theory: 2s/3s & 2p Orthogonality Questions

In summary: The radial part of 2p has a node, as it is proportional to r. So when you integrate it with the radial part of 2s, you get zero because of the node. In summary, molecular orbital theory states that the orbitals have zero overlap, meaning the integral of two orbitals is zero. This is due to the fact that the orbitals are eigenfunctions of the Hamiltonian and form an orthonormal basis. The orthogonality of 2s and 3s orbitals is achieved through a node, as seen in the case of hydrogen. Similarly, the orthogonality of 2p and 2s orbitals is also achieved through a node in the radial part of the orbital
  • #1
Master J
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I have a few questions on molecular orbital theory which I hope you guys can help me settle!

So I understand orthogonality meaning that the molecular orbitals have zero overlap, due to the Pauli Exclusion Principle.

How do a 2s and 3s molecular orbital achieve orthogonality? Is it due to a node? Does the 3s electron density penetrate the inner 2s at all?

And how do 2p and 2s orbitals achieve orthogonality?
 
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  • #2
Master J said:
So I understand orthogonality meaning that the molecular orbitals have zero overlap, due to the Pauli Exclusion Principle.

It doesn't mean zero overlap, but rather that for two orbitals, [tex]\psi_1, \psi_2[/tex] the integral [tex]\int_{-\infty}^{\infty}\psi_1^\ast\psi_2 dx = 0[/tex]
So they can overlap as much as they want, as long as the overall integral becomes zero.

This isn't due to the exclusion principle, but due to the fact that the orbitals are eigenfunctions of the Hamiltonian and form an orthonormal basis of a Hilbert space.

How do a 2s and 3s molecular orbital achieve orthogonality? Is it due to a node? Does the 3s electron density penetrate the inner 2s at all?

2s and 3s are atomic orbitals. But just look at the hydrogen case (just to simplify, I'll take 1s and 2s):
[tex]\psi_{1s} = e^{-r}\quad\psi_{2s}=(1-\frac{r}{2})e^{-r/2}[/tex]

Obviously the 2s orbital has a node, it must change sign at r=2 given the (1-r/2). Integrate [tex]\int_0^{\infty}r^2\psi_1\psi_2 dr[/tex] and see what you get.
(the r^2 comes in because you're integrating the radial wave function spherically)

And how do 2p and 2s orbitals achieve orthogonality?

The same way.
 
  • #3


Molecular orbital theory explains the behavior of electrons in a molecule by describing the formation of molecular orbitals from the atomic orbitals of the constituent atoms. The concept of orthogonality in molecular orbitals refers to the lack of overlap between different molecular orbitals, which is essential for the stability and properties of the molecule.

To answer your first question, the 2s and 3s molecular orbitals achieve orthogonality because they have different spatial distributions. The 3s orbital has a larger size and extends further from the nucleus compared to the 2s orbital. This results in a lack of overlap between the two orbitals, making them orthogonal.

In terms of the electron density, the 3s orbital does penetrate the inner 2s orbital to a certain extent, but this does not affect the orthogonality of the two orbitals. The important factor is the spatial distribution of the orbitals rather than the electron density.

For the second question, the 2p and 2s orbitals achieve orthogonality due to their different orientations in space. The 2p orbitals are oriented along the x, y, and z axes, while the 2s orbital is spherically symmetric. This difference in orientation leads to a lack of overlap between the orbitals, making them orthogonal.

I hope this helps to clarify your questions on molecular orbital theory and orthogonality. Keep exploring and asking questions to deepen your understanding of this important concept in chemistry.
 

1. What is Molecular Orbital Theory?

Molecular Orbital Theory (MOT) is a theoretical approach in chemistry that describes the behavior of electrons in a molecule. It is based on the concept of molecular orbitals, which are mathematical functions that represent the probability of finding an electron in a particular region of space within a molecule.

2. What is the significance of 2s/3s & 2p orthogonality in Molecular Orbital Theory?

2s/3s & 2p orthogonality refers to the idea that the 2s, 3s, and 2p orbitals of an atom are mutually perpendicular to each other. In other words, they do not overlap or interact with each other. This is important in MOT because it allows for the proper combination of atomic orbitals to form molecular orbitals.

3. How does 2s/3s & 2p orthogonality affect the energy levels of molecular orbitals?

The 2s/3s & 2p orthogonality plays a crucial role in determining the energy levels of molecular orbitals. In MOT, the combination of atomic orbitals leads to the formation of bonding and anti-bonding molecular orbitals. The energy levels of these molecular orbitals are determined by the degree of overlap and interaction between the atomic orbitals. The 2s/3s & 2p orthogonality ensures that only the appropriate orbitals overlap and contribute to the formation of molecular orbitals, resulting in distinct energy levels.

4. What are the implications of violating 2s/3s & 2p orthogonality in Molecular Orbital Theory?

If 2s/3s & 2p orthogonality is violated, it can lead to incorrect predictions in MOT. This is because the incorrect combination of atomic orbitals can result in energy levels that do not accurately represent the actual energy levels of the molecule. This can affect the overall stability and reactivity of the molecule, leading to inaccurate experimental results.

5. How is 2s/3s & 2p orthogonality determined in Molecular Orbital Theory?

2s/3s & 2p orthogonality is determined through mathematical calculations and theoretical models. The degree of overlap and interaction between atomic orbitals can be predicted using various mathematical equations and principles, such as the LCAO (Linear Combination of Atomic Orbitals) method. Experimental data, such as spectroscopic measurements, can also be used to validate the predictions made by MOT calculations.

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