I'm studying term symbols. I can derive them mechanically from the electron configuration using the complete table of microstates.
Let's consider a typical example - an atomic carbon with the electron configuration $1s^22s^22p^2$.
It has states ${}^3P_0, {}^3P_1, {}^3P_2, {}^1D_2, {}^1S_0$.
But now, how can I derive the symmetries for each of these states? I need to compute their energies in some specialized software (Molpro), but I don't see the exact connection between the atomic term symbol and its symmetry.
Question
Let's work with the $D_{2h}$ point group.
The ${}^1S_0$ state is supposed to be totally symmetric, i.e. to have $A_g$ irreducible representation.
But how can I determine it without previously knowing?
I know, thank to this answer, that ${}^1S_0$ wavefunction should be a linear combination of 3 microstates:
\begin{align} |L = 0, M_L = 0\rangle & = \frac{1}{\sqrt 3} \left( |m_{l1}= 1, m_{l2} = -1\rangle \right. \\ & \qquad \qquad \left. + |m_{l1}= -1, m_{l2} = 1\rangle \right. \\ & \qquad \qquad \left. - |m_{l1}= 0, m_{l2} = 0\rangle \right) \end{align}
But I see no way to use it to determine the ${}^1S_0$ symmetry in $D_{2h}$ or any other point group...
The second example are ${}^3P_0, {}^3P_1, {}^3P_2$ states - they're supposed to have $B_{1g}, B_{2g}, B_{3g}$ irreducible representations. But again - how can I derive them just from the electron configuration and the term symbol?
I understand basics of molecular symmetry and I can draw the molecular orbital diagrams for ground states. My confusion lies mostly in the situation, where I can theoretically describe the electron configuration using direct product of irreducible representation of single orbitals (like it's described in this answer), but I don't know what to do, when the state (described by an atomic term symbol) is a linear combination of multiple such configurations (microstates).
I'd be very grateful for an explanation using basic terms and maybe a derivation step-by-step example or pictures, as I'm a real beginner and the topic seems to be quite confusing so far.
