The four quantum numbers of interest are n (principal quantum number), l (angular momentum), m_l (magnetic), and m_s (spin).
A generic 4d_(z^2) orbital has n = 4 and l = 2. n = 4 specifies the energy level, and l specifies the orbital's shape. s -> l = 0, p -> l = 1, etc. Thus, its m_l varies as 0, pm1, pm2, and the orbital has projections above the plane and below the plane.
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Depending on how full the orbital is, m_s varies. If it happens to be a 4d^1 configuration, for example, then one of five orbitals are filled (d_(x^2-y^2), d_(z^2), d_(xy), d_(xz), d_(yz)) with one electron. In that case, the electron is, by default, spin pm1/2. Thus, m_s = pm1/2.
In this case, it would give a term symbol of ""^(2)D_("1/2"), ""^(2)D_("3/2"), and ""^(2)D_("5/2"). The notation is:
""^(2S+1) L_("J")
where J = L+S.
(The most stable one would be the ""^(2)D_("1/2") state, according to Hund's rules for less-than-half-filled orbitals with the same S and the same L.)
Here, the spin multiplicity is 2S+1 = 2("1/2")+1 = 2, and the total angular momentum J = L + S = |m_l| + |m_s|
= 0 + "1/2", 1 + "1/2", and 2 + "1/2" = "1/2", "3/2", and "5/2".
(2 - "1/2" = 1 + "1/2", and 1 - "1/2" = 0 + "1/2", which are duplicates, while by the selection rules, DeltaL = 0, pm1, DeltaS = 0, and DeltaJ =0, pm1 )
