Hund's Rule for Determining Term Symbol Energy Order

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Hund's rules help determine the energy order of term symbols based on total angular momentum J, stating that for less than half-filled subshells, the lowest J-value corresponds to the lowest energy, while for more than half-filled subshells, the highest J-value does. The application of these rules is straightforward for configurations involving a single orbital, such as p^5 or d^3. However, complications arise with configurations involving multiple orbitals, like p^5d^1 or d^9f^1, as it becomes unclear which orbital to apply the half-filled condition to. It is noted that Hund's rules apply only to equivalent electrons, making the d^9 configuration more than half-filled, but the application to d^9f^1 is less certain. Overall, the discussion highlights the challenges in applying Hund's rules to complex electron configurations.
Mr_Allod
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Homework Statement
Use Hund's rules to determine the energy order of the term symbols of a ##d^9f^1## electron configuration.
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Hello there, for the above question I have no issue finding the term symbols but I am a little unsure about employing Hund's rules to the electron configuration, particularly those referring to the energies based on the total angular momentum J. These state:

- In a less than ##\frac12##-filled subshell Lowest J-value is Lowest energy
- In a more than ##\frac12##-filled subshell Highest J-value is Lowest energy

For configurations where only one orbital is involved such as ##p^5,d^3## etc. this is easy to apply. But what about configurations where two orbitals are involved such as ##p^5d^1## or ##d^9f^1##? In these cases to which orbital do we apply the less/more than ##\frac 12##-filled condition?
 
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Hund's rules apply only to equivalent electrons. It would work for d9, but applying it to d9f1 is iffy.

In any case, considering that this is the problem you were given, the equivalent electrons being the d9, I would consider the subshell as more than half-filled.
 
DrClaude said:
Hund's rules apply only to equivalent electrons. It would work for d9, but applying it to d9f1 is iffy.

In any case, considering that this is the problem you were given, the equivalent electrons being the d9, I would consider the subshell as more than half-filled.
Thank you for the answer, I've had a very hard time finding a straight answer to this question anywhere so it's nice to have something to go on.
 

Thread 'Help needed with Elliptic Integrals'

I want to find the solution to the integral ##\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}## I can see that ##\frac{d^2u}{d\theta^2} = A +Bu+Cu^2## is a Weierstrass elliptic function, which can be generated from ##\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3## (A = 0, B=-1, C=3) So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck. TerryW
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