As far as I recall, the following steps need to be taken:

1)you neglect all core orbitals

2)you determine the irreducible representation (irrep) of each incompletely occubied valence orbital (here: 2p, although it's nasty for atoms because their groups have infinite elements)

3)you calculate the direct product of all valence orbital electron irreps which should provide a reducible representation of your state

4)you deconvolute the reducible representation into irreps (http://www-schuetz.chemie.uni-regensburg.de/files/grouptheory/theochem_grouptheory_part4.pdf, page 20)

5)you now know the symmetries, but you still have to assign them to the term symbols.

Jürgen Weippert Thank you! So, let's have a nitrogen atom, for example, with the electron configuration 1s^22s^2p^3.

The only open-shell electrons are in 2p^3. Let's have 4S_3/2 state, i.e. every electron spin-up in a separate orbital. How will we determine its symmetry, if there are in fact 4 states included (we can see https://rh.jhu.edu/courses/3411a/driver/node73.html)?

Let's work with D2h point group.

Using your approach I see, that px, py and pz orbitals are represented by B3u, B2u and B1u irreps, respectively.Their direct product is Au - that seems correct, but I don't understand two things here - do we need to compute explicitly 4 states to get an average from them? Or are they included somehow "implicitly"?

Also, how would I proceed with another term, let's say 2P? Now we have multiple options of electron configuration (i.e. microstates). There will be always one full orbital, one empty and one half-filled. How can I properly derive the symmetry here?

For the issues having to do with chemistry (in particular for effects of crystal fields, which break the continuous rotational symmetry of central fields associated with closed shells in atomic physics), consult [1]. For various vector coupling methods (LS, jj, etc.), consult [2,3,4] (consult in particular Chapter 6 in [3] and Chapter 9 in [4]). In the case of two angular momenta, the relevant expansion coefficients are the well-known Clebsch-Gordan coefficients. For more angular momenta, Racah's fractional parentage method is used, for which the relevant expansion coefficients are the 6-j, 9-j, etc., symbols.

[1] DM Bishop, Group Theory and Chemistry (Dover, New York, 1993).

[2] AR Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, 1957).

[3] IG Kaplan, Symmetry of Many-Electron Systems (Academic Press, New York, 1975).

[4] Z Rudzikas, Theoretical Atomic Spectroscopy (Cambridge University Press, 1997).