Solving Hydrogen Molecule Basis Functions

In summary, the conversation discussed the formulation of 6 basis functions using four possible single particle states with two electrons in a molecule of hydrogen, taking into consideration the Pauli exclusion principle. The basis functions included both spin triplet and spin singlet states, with the labels A and B representing the two different atoms. The possibility of AA or BB pairs was also addressed, with the conclusion that they are not included in the basis functions.
  • #1
rasko
6
0
A molecule of hydrogen. We assume that the only possible states of
its two electrons (indistinguishable) are [tex] |A\uparrow\rangle,
|A\downarrow\rangle, |B\uparrow\rangle[/tex] and [tex]|B\downarrow\rangle[/tex]
(A=1s-orbital at atom A, B=1s-orbital at atom B).
Formulate the 6 basis functions using the four possible single
particle states above. (Don't forget the Pauli-principle!)


Here is my solution:

Total spin S=1 or 0.
[tex]|1,1\rangle=|A\uparrow\rangle|B\uparrow\rangle[/tex],
[tex]|1,0\rangle=\frac{1}{\sqrt{2}}(|A\uparrow\rangle|B\downarrow\rangle+|A\downarrow\rangle|B\uparrow\rangle)[/tex],
[tex]|1,-1\rangle=|A\downarrow\rangle|B\downarrow\rangle[/tex],
[tex]|0,0\rangle=\frac{1}{\sqrt{2}}(|A\uparrow\rangle|B\downarrow\rangle-|A\downarrow\rangle|B\uparrow\rangle)[/tex].

Are they basis functions? There should be 6 but I got only 4.
 
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  • #2
You've written down spin triplet and spin singlet states, and those would be all you had if your single particle states were [itex]\left|\uparrow\right>[/itex] and [itex]\left|\downarrow\right>[/itex], but your states aren't labelled by just the spin - you have labels A and B too. However, all of the states you have written are pairs A and B. What about AA or BB pairs?
 
  • #3
Hi, Mute. I think there should be no AA or BB pairs. Because at the same time A or B has only one state.
 
  • #4
Why do you think that? It's not given anywhere in the problem statement that each atom can only have one electron.

Remember that the Pauli exclusion principle only rules out two indist. particles in the *exact* same state...
 
  • #5
2Tesla and Mute, thanks. I understand now.

|0,0>=|A up>|A down>
|0,0>=|B up>|B down>

Thanks u for ur tips.
 

What is the purpose of solving hydrogen molecule basis functions?

The purpose of solving hydrogen molecule basis functions is to accurately describe the electronic structure and behavior of hydrogen molecules. This is important for understanding chemical reactions, predicting properties of new molecules, and designing new materials.

How is solving hydrogen molecule basis functions done?

Solving hydrogen molecule basis functions involves using mathematical techniques, such as linear algebra and quantum mechanics, to determine the wavefunction of the molecule. This wavefunction describes the distribution of electrons in the molecule and can be used to calculate various properties.

What are the challenges of solving hydrogen molecule basis functions?

One of the main challenges of solving hydrogen molecule basis functions is the large number of calculations required. Since the wavefunction depends on the position of each electron, it is necessary to calculate the energy and position of each electron at every point in space. This can be computationally intensive and require advanced algorithms.

What are the benefits of solving hydrogen molecule basis functions?

By accurately solving hydrogen molecule basis functions, scientists can gain a better understanding of the electronic structure and behavior of hydrogen molecules. This can lead to advancements in fields such as chemistry, materials science, and energy production. Additionally, the techniques used to solve hydrogen molecule basis functions can be applied to other molecules and systems.

Are there any limitations to solving hydrogen molecule basis functions?

One limitation of solving hydrogen molecule basis functions is that it assumes the nuclei are stationary and do not interact with each other. This is known as the Born-Oppenheimer approximation. However, in some cases, this assumption may not hold and can lead to inaccuracies in the results. Additionally, solving hydrogen molecule basis functions can become more complex when considering larger molecules with multiple atoms.

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