Understanding Hund's Second Rule: What is a Singlet or Triplet?

[McLaren Rulez]

Can anyone explain the second rule, because the Wikipedia page is not very clear?

Hund's zeroth rule - Ignore all inner shells and focus on the outermost shell.
Hund's first rule - Put the electrons such that they maximize spin, $s$.

So far so good. Hund's second rule appears to be simply that the we should maximize $l$ (after rule 1). That is, if I have say $3p^2$, then the two electrons should be in $l=1$ and $l=0$. Is there anything more to it than that?

https://en.wikipedia.org/wiki/Hund's_rules talks about singlets and triplets and that the second rule is never used until Ti. I don't understand either of these statements. What is the meaning of singlets and triplets in this context? Also, in my example above, we used Hund's second rule for a 3p orbital so why is Ti the lowest element where this rule is used? Thank you.

 

[DrClaude]

So far so good. Hund's second rule appears to be simply that the we should maximize $l$ (after rule 1). That is, if I have say $3p^2$, then the two electrons should be in $l=1$ and $l=0$. Is there anything more to it than that?

You are mixing up $l$ and $m_l$. If you have 3p², then you have two electrons with $l=1$. When you add up the two orbital angular momenta with $l=1$, you can get $L=2, 1, 0$, so the second rule would tell you that the ground state would be $L=2$. (This is not actually the case for 3p² because of the first rule, see below.)

https://en.wikipedia.org/wiki/Hund's_rules talks about singlets and triplets and that the second rule is never used until Ti. I don't understand either of these statements. What is the meaning of singlets and triplets in this context? Also, in my example above, we used Hund's second rule for a 3p orbital so why is Ti the lowest element where this rule is used? Thank you.

A term with $S=0$ is called is singlet (because $2S+1=1$) while a term with $S=1$ is called a triplet (because $2S+1=3$). The first rule tells you that the triplet will be lowest in energy. In the case of 3p², the possible terms are ³P, ¹D, and ¹S. Therefore, by the first rule, the ground state is ³P. Since there are no other triplet terms, you don't need the second rule. It is only when you have two d electrons that multiplicity alone will not tell you which is the ground state.

 

[McLaren Rulez]

A term with $S=0$ is called is singlet (because $2S+1=1$) while a term with $S=1$ is called a triplet (because $2S+1=3$). The first rule tells you that the triplet will be lowest in energy. In the case of 3p², the possible terms are ³P, ¹D, and ¹S. Therefore, by the first rule, the ground state is ³P. Since there are no other triplet terms, you don't need the second rule. It is only when you have two d electrons that multiplicity alone will not tell you which is the ground state.

Thank you for your reply. Can I check what exactly ³P, ¹D, and ¹S notation means? Sorry, if this is an obvious question but I'm guessing the 3 and the 1 refer to the triplet and the singlet but what are the P, D, and S?

 

[DrClaude]

Spectroscopic notation: $^{2S+1}L_J$, with $S$ the total spin, $L$ the total orbital angular momentum, and $J$ the total angular momentum. $L$ is expressed as a capital letter, similar to single-electron orbitals. In what I wrote, I omitted the possible values of $J$ (you need the third Hund rule to find which $J$ for the ground state).

 

[McLaren Rulez]

Spectroscopic notation: $^{2S+1}L_J$, with $S$ the total spin, $L$ the total orbital angular momentum, and $J$ the total angular momentum. $L$ is expressed as a capital letter, similar to single-electron orbitals.

Thank you. Sorry to ask more questions but how did you know that $^3P, ^1D$, and $^1S$ were the available options? In other words, why is it that the way spin adds up also determine $L$? For example, why can I not have $S=1$ and $L=0$ which is the state $^3S$ in spectroscopic notation?

 

[DrClaude]

Thank you. Sorry to ask more questions but how did you know that $^3P, ^1D$, and $^1S$ were the available options? In other words, why is it that the way spin adds up also determine $L$? For example, why can I not have $S=1$ and $L=0$ which is the state $^3S$ in spectroscopic notation?

There is no ³S because of the Pauli exclusion principle. The only way to figure this out is to write all possible microstates (combinations of $m_l$ and $m_s$), and then see which terms they lead to.

Have a look at https://www.academia.edu/28235099/Term_symbols_for_multi-electron_atoms for more information on how to do that,

 

[McLaren Rulez]

There is no ³S because of the Pauli exclusion principle. The only way to figure this out is to write all possible microstates (combinations of $m_l$ and $m_s$), and then see which terms they lead to.

Have a look at https://www.academia.edu/28235099/Term_symbols_for_multi-electron_atoms for more information on how to do that,

Thank you for the detailed document. I think I mostly get it except for how the $L$ value is known from a given $M_L$. If I look at the microstates, for example, the fifth one in your table $M_L = 0$ but the $L$ value of this can be either $0, 1$ or $2$. How is it that it is labelled as $^3P$ and not $^3S$?

 

[DrClaude]

Thank you for the detailed document. I think I mostly get it except for how the $L$ value is known from a given $M_L$. If I look at the microstates, for example, the fifth one in your table $M_L = 0$ but the $L$ value of this can be either $0, 1$ or $2$. How is it that it is labelled as $^3P$ and not $^3S$?

$^3$P is made up of $M_L = 1, 0, -1$ with $M_S = 1, 0, -1$, you need all those microstates to form the term. Once you have used the microstate with $M_L=0$ and $M_S=1$ to build up $^3$P, there is no such microstate left to make a $^3$S. That's why have have to form the terms starting from the highest value of $L$.

 

[McLaren Rulez]

Ah I see! Thank you a million DrClaude. You've really helped me finally get it!