# Using Hund's rule, find the fundamental term of an atom

• fluidistic
In summary: Hund's zeroth rule is what allows the other shells to contribute to the spin. Without it, each electron would have 1/2 spin only.
fluidistic
Gold Member

## Homework Statement

Using Hund's rule, find the fundamental term of an atom whose last incomplete subshell contains 3 electrons d. Do the same with 4 electrons p.

## Homework Equations

The 3 Hund's rules.

## The Attempt at a Solution

By fundamental term I'm guessing they mean the ground state term.
Despite knowing Hund's rules, I don't know how I can start the problem without having the term symbols of the atom(s).
I somehow made an attempt, for the 3 electrons d.
Since the subshell is d, it means each electrons will have the quantum numbers $l=2$. So that $L=6$. Also, since there are 3 electrons, $S=3/2$. Thus J goes from $L-S=9/2$ to $L+S=15/2$.
That d shell is half filled so that J will equal to $9/2$ for the ground state.
L could in principle for 6 (but in practice no?!), this would make my answer as $^{16}H_{9/2}$. Which is -I'm 100% sure-, totally wrong.
What I don't understand is that L seems to go from 0 to 6 as if one adds the $m_l$ 's of the 3 electrons instead of the $3 l$'s.
Any help is appreciated.

By Hund's zeroth rule, all the other shells are filled, so they contribute nothing to spin or angular momentum.

There are 5 d orbitals and 3 p orbitals. Hund's first rule is that they have the maximum spin allowed for these shells following the exclusion principle. Hence, for d3, s=3/2, and for p4, the spin is 1/2+1/2+1/2-1/2 = 1.

Hund's second rule is that L is the maximum for given S in each shell. d's separate orbitals have L = 2, 1, 0, -1, -2, while p's are 1, 0, -1. Using what we found in the last step, d3 has L = 2+1+0 = 3, and P4 has 1+0+-1+1 = 1.

Hund's third rule is that J = L+S for more-than-half filled shells, and J=|L-S| for less-than-half filled shells. (notice that both equations give the same answer for exactly half-filled shells). Hence for d3, J = |3/2-3|=3/2, and for p4, J=1+1=2.

In the "Hieroglyphic" notation, these would be:

d3 = 4F3/2
p4 = 3P2Edit: in retrospect, it seems like the source of your confusion is that there are actually 4 Hund's rules! The zeroth rule is crucial and allows you to neglect everything besides the outermost shells. Also, remember the exclusion principle: there can only be two electrons per orbital, and if there are two, their spins must be antiparallel.

Edit 2: corrections are in bold. Now it should make sense.

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Jolb said:
By Hund's zeroth rule, all the other shells are filled, so they contribute nothing to spin or angular momentum.

There are 5 d orbitals and 3 p orbitals. Hund's first rule is that they have the maximum spin allowed for these shells following the exclusion principle. Hence, for d3, s=3/2, and for p4, the spin is 1/2+1/2+1/2-1/2 = 1.

Hund's second rule is that L is the maximum for given S in each shell. d's separate orbitals have L = 2, 1, 0, -1, -2, while p's are 1, 0, -1. Using what we found in the last step, d3 has L = 2+1+0 = 3, and P4 has 1+0+-1+1 = 1.

Hund's third rule is that J = L+S for less-than-half filled shells, and J=|L-S| for more-than-half filled shells. Hence for d3, J = |3/2-3|=3/2, and for p4, J=1+1=2.

In the "Hieroglyphic" notation, these would be:

d3 = 4F3/2
p4 = 3P2

Edit: in retrospect, it seems like the source of your confusion is that there are actually 4 Hund's rules! The zeroth rule is crucial and allows you to neglect everything besides the outermost shells. Also, remember the exclusion principle: there can only be two electrons per orbital, and if there are two, their spins must be antiparallel.
Thank you, I was not expecting a full and detailed answer (in fact the rules of the forum forbids this!) but I have nothing else to say than a big thank you, that helped me to understand a lot. I'm lacking books on this topic.
I've carefully been through your sentences and I've a small doubt.
Why did you consider that for the d³ electrons the shell was more than half filled and for the p⁴ electrons it was less than half filled? Except this point, I understood everything you made, thanks to your comments.

I think trying to answer your question without giving the full answer would be much more trouble--the best way I could possibly explain this would be by working an example for you, and this is actually a much better example problem than I could think of easily. Plus, a good example would include both a more-than-half-filled example and a less-than-half-filled example, like this one. Hund's rules are one of the most heuristic things you learn in QM--I personally have never heard a satisfactory theoretical discussion of them. So please forgive me, Gods of PF.Oops, I think I made a typo above. I had "more" and "less" reversed.
Anyway, p shells have 3 orbitals. Two electrons can inhabit each orbital. Thus there can be at most 6 electrons in a P orbital. 4>6/2. So p4 is MORE than half filled, so you use J=L+S

I'll go back and find my typo so the post makes sense.

Last edited:
Jolb said:
I think trying to answer your question without giving the full answer would be much more trouble--the best way I could possibly explain this would be by working an example for you, and this is actually a much better example problem than I could think of easily. Plus, a good example would include both a more-than-half-filled example and a less-than-half-filled example, like this one. Hund's rules are one of the most heuristic things you learn in QM--I personally have never heard a satisfactory theoretical discussion of them. So please forgive me, Gods of PF.

Oops, I think I made a typo above. I had "more" and "less" reversed.
Anyway, p shells have 3 orbitals. Two electrons can inhabit each orbital. Thus there can be at most 6 electrons in a P orbital. 4>6/2. So p4 is MORE than half filled, so you use J=L+S

I'll go back and find my typo so the post makes sense.
Thank you for everything. You left everything as clear as possible to me.

## 1. What is Hund's rule and how does it apply to finding the fundamental term of an atom?

Hund's rule is a principle in quantum mechanics that describes the way electrons are distributed in an atom's orbitals. It states that electrons will occupy separate orbitals with the same energy, and only pair up when all orbitals with that energy level are occupied.

## 2. Why is it important to use Hund's rule when finding the fundamental term of an atom?

Using Hund's rule allows us to determine the most stable electron configuration for an atom, which is important in understanding its chemical and physical properties. It also helps us to predict the reactivity and bonding of atoms.

## 3. How do you apply Hund's rule to find the fundamental term of an atom?

To find the fundamental term of an atom, we first need to determine the number of valence electrons it has. Then, we follow Hund's rule by placing one electron in each orbital with the same energy before pairing them up in orbitals with opposite spins. The fundamental term is determined by the number of unpaired electrons.

## 4. Can Hund's rule be applied to all atoms?

Yes, Hund's rule can be applied to all atoms as it is a fundamental principle in quantum mechanics that governs the behavior of electrons in atoms. However, it is more commonly used for atoms with multiple electron orbitals.

## 5. What other factors should be considered when using Hund's rule to find the fundamental term of an atom?

Aside from the number of valence electrons, the atomic number and electron configuration of an atom should also be taken into account when applying Hund's rule. These factors can affect the energy levels and stability of the atom's electron configuration, and ultimately determine the fundamental term.

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