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In A Primer on Quantum Numbers and Spectroscopic Notation Contents, the concept of a term and level is described as

a) The level is the set of 2J+1 states with specific values of L, S, and J. The difference in the energy between two levels gives the wavelength or frequency of an atomic transition.

b) The term is the set of levels characterized by a specific S and L.

1) Is there a better way to understand the subtle difference among the concept of a level, a state, and a term. A example would be helpful.

2) A side question is what are the German equivalents of term, state and level. What do they call it in atomic spectroscopy literature? Thanks.

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The figure below shows the situation between configuration for a $p^2$ configuration, terms, levels and states. The word 'state' tends to be used colloquially to mean any of Term, Level or State.

The Configuration such as $(1s)^2$ or ...$(2p)^2$ etc. tell us which orbitals are occupied.

These are split with electrostatic (Coulomb) coupling to form Terms.

The Term Symbol informs about the angular momentum. The number of levels in each Term is $(2J+1)$, the number of States is $(2L+1)(2S+1)$ .

The term symbol is

$$\large ^{2S+1}L_J$$

where $S$ is the total spin and $L$ the orbital angular momentum. $J$ represents the angular momentum as quantum numbers $(J=L+S)$ and is determined by adding individual spin numbers according to Glebsch-Gordon series, i.e. if $q_1,\,q_2$ are angular momentum quantum numbers then the series is all terms (separated by unit steps) with values $q_1+q_2 , q_1+q_2-1,\cdots \to |q_1-q_2|$. (There may be only be term).

The Terms are split with a magnetic interaction such as spin-orbit coupling, i.e coupling of spin angular momentum with that of the orbital. It is not present for S orbitals because here $L=0$.

Levels are distinguished by total angular momentum $J$, and are $2J+1$ degenerate.

An external magnetic field will remove all remaining degeneracy to form States. These are distinguished by $m_J$ and there are $2J+1$ of them.

term level states

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  • $\begingroup$ Excellent. Could you share the reference for this figure? So in a rigorous sense, when we talk about electronic transitions in atomic spectra, the transitions are in levels. And transition among states can be seen in Zeeman effect. $\endgroup$
    – AChem
    Aug 30, 2019 at 12:14
  • $\begingroup$ I used the figure in lectures years ago, and I don't recall where it came from so perhaps drew it. And yes Zeeman effect (or Stark) splits levels to make states so this spectroscopy identifies states. I think this was how these energy levels were discovered in the first place; anomalous Zeeman effect and all that history. $\endgroup$
    – porphyrin
    Aug 31, 2019 at 15:02
  • $\begingroup$ The concept of terms is really old, and the origin is most likely from the series concept, where each member of series is called a term. $\endgroup$
    – AChem
    Aug 31, 2019 at 15:30
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To answer the side question about German language terminology, I'll pick up terms from @porphyrin's answer.

But first of all, term = Term (same as in maths)

The word 'state' tends to be used colloquially to mean any of Term, Level or State.

All those colloquial states are Zustände (pl., sg. Zustand) in German or you can use Term, Niveau (level) and Zustand (state). I'd say that in German, Zustand is an Oberbegriff (umbrella term) that includes specific types of Zustände such as Terme (Zustand with fixed L and S) and Niveaus (Zustand with fixed L, S, and J).

angular momentum = Drehimpuls (drehen = to turn, Impuls = momentum)
But Eigendrehimpuls ("intrinsic angular momentum") is usually just Spin also in German.
Thus, the Anzahl der Zustände (2S + 1) is the Spinmultiplizität.

spin-orbit-coupling = Spin-Bahn-Kopplung or Spin-Bahn-Wechselwirkung

degenerate = entartet

See e.g. The German wiki page https://de.wikipedia.org/wiki/Spin-Bahn-Kopplung#LS-Kopplung_bei_mehreren_Elektronen for all those terms in their native environment.

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