How come that Esinglet < Etriplet and Hund’s rule are both correct?

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In summary, The conversation discusses the singlet and triplet energies of two electrons, with one person mentioning that according to Hund's rule, the ground spin state should be symmetric. However, another person points out that this may not be the case for atoms with more than two electrons, as the proof for (1) only applies to two electrons and may not be accurate for larger numbers. They also discuss the calculation of energy over symmetric and anti-symmetric functions to support this argument.
  • #1
Alex83
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Hi all, I will be most grateful if someone could help me with this:

1) It can be shown that for two electrons the singlet energy is lower
then the triplet energy.

2) According to Hund’s rule the ground spin state is symmetric.

Why (1) is not correct for atoms where Hund’s rule can be used?

Maybe the proof of (1) is wrong for more then two electrons but I can’t figure why.

With thanks,
Alex.
 
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  • #2
Of course. You get a singlet and a triplet when adding two spins only. Once there are more than 2 (3 or more), things are longer to calculate. Sides, an electron in an atomic shell has orbital angular momentum, too.

Daniel.
 
  • #3
The proof of (1) is for general symmetric and anti symmetric functions, it goes like that:

1) One can minimize the energy over all the symmetric functions and call the result Es and fs(r), then minimize over all the anti symmetric functions and call it Et and ft(r).

2) Prove that |ft(r)| yields the same mean energy as ft(r), Et.

3) |ft(r)| is obviously symmetric and so Et>Es.

I think that for more then two electrons (2) is wrong but I don’t know why.
 
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1. Why is the energy of a singlet state lower than that of a triplet state?

In quantum mechanics, the energy of a state is determined by the arrangement of electrons within that state. In a singlet state, the electrons have opposite spin, resulting in a lower energy due to their more stable configuration. In a triplet state, the electrons have the same spin, resulting in a higher energy due to their less stable configuration.

2. How does Hund's rule relate to the energy of states in an atom?

Hund's rule states that electrons will occupy empty orbitals of the same energy before pairing up in orbitals. This leads to a lower energy overall for the atom since electrons are in their most stable arrangement. This rule applies to both singlet and triplet states, contributing to the lower energy of singlet states.

3. Can you explain the concept of spin in relation to singlet and triplet states?

Electron spin is an intrinsic property of particles and can have two values: up or down. In singlet states, the electrons have opposite spins, resulting in a net spin of zero. In triplet states, the electrons have the same spin, resulting in a net spin of one. Since states with a lower net spin are more stable, singlet states have a lower energy than triplet states.

4. Are there any exceptions to the rule that singlet states have lower energy than triplet states?

Yes, there are some cases where the energy of a triplet state may be lower than that of a singlet state. This can occur when there is a strong electron-electron repulsion, causing the paired electrons in a singlet state to have a higher energy than the unpaired electrons in a triplet state.

5. How does Hund's rule and the energy of singlet and triplet states impact the chemical and physical properties of atoms?

The energy differences between singlet and triplet states can affect the reactivity and stability of atoms. Atoms with a higher net spin state may be more reactive due to their higher energy, while atoms with a lower net spin state may be more stable. Additionally, Hund's rule can impact the arrangement of electrons in molecular orbitals, influencing the properties of molecules such as polarity and bond strength.

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