Connection of term symbols with specific microstates for atomic carbon
To add on to what Feodoran has mentioned, you might be interested in looking at the Clebsch–Gordan coefficients, which are essentially the coefficients of the microstates in the term symbols. You can look these up in e.g. the appendices of Atkins Molecular Quantum Mechanics, 5th ed. Note that each term symbol is a collection of states, so a $D$ term (for example) which has $L  = 2$ would comprise five states: one each with $M_L = -2, -1, 0, +1, +2$. These five states can in general be denoted as $|L, M_L\rangle$.

$L$, which is a total angular momentum, is obtained by a coupling of two individual angular momenta. For two electrons in p-orbitals, these individual angular momenta are $l_1 = 1$ (with $m_{l1} = -1, 0, +1$) and $l_2 = 1$ (with $m_{l2} = -1, 0, +1$). The microstates in the table specify both values of $m_{l1}$ and $m_{l2}$, so they are simultaneous eigenstates of $m_{l1}$ and $m_{l2}$, i.e. they can be denoted by $|m_{l1},m_{l2}\rangle$.

Now, in general each coupled representation $|L, M_L\rangle$ will be a linear combination of the microstates, or uncoupled representations $|m_{l1},m_{l2}\rangle$. It is in general not possible to construct a state which is simultaneously an eigenstate of $L, M_L, m_{l1}$, and $m_{l2}$, because the relevant operators do not commute (refer to Atkins or any other QM text on coupling of angular momentum). There are examples of one-to-one correspondence between the coupled and uncoupled representations – for example, microstate 13 perfectly corresponds to $|L = 2, M_L = +2\rangle$, but it is not general, so:

$$ |L, M_L\rangle = \sum_{m_{l1},m_{l2}} c^{L, M_L}_{m_{l1},m_{l2}} |m_{l1},m_{l2}\rangle$$

The Clebsch–Gordan coefficients are simply the $c^{L, M_L}_{m_{l1},m_{l2}}$ in the expansion above. Usually the notation is slightly different – so instead of $(L, M_L, m_{l1}, m_{l2})$, it is usually $(j, m_j, m_{j1}, m_{j2})$ – but the concept is entirely the same.

Using the coefficients for $j_1 = j_2 = 1$ in Atkins (which is relevant to the $\mathrm{p^2}$ configuration of atomic carbon), we find that the $^1S$ term is given by

$$|L = 0, M_L = 0\rangle = \underbrace{\frac{1}{\sqrt 3} |m_{l1}= 1, m_{l2} = -1\rangle + \frac{1}{\sqrt 3} |m_{l1}= -1, m_{l2} = 1\rangle}_{\text{a combination of microstates 8 and 11}} - \underbrace{\frac{1}{\sqrt 3} |m_{l1}= 0, m_{l2} = 0\rangle}_{\text{microstate 14}}$$

It's slightly more confusing because there is a spin component as well, which the coefficients do not cover, and this is why I've given a handwavy "combination of microstates 8 and 11". This "combination" is needed to generate the singlet spin wavefunction $2^{-1/2}(|\alpha\beta\rangle - |\beta\alpha\rangle)$.



As an aside, the familiar singlet and triplet spin wavefunctions

$$\underbrace{|\alpha\alpha\rangle, |\beta\beta\rangle, \frac{1}{\sqrt 2}(|\alpha\beta\rangle + |\beta\alpha\rangle)}_{\text{triplet spin wavefunctions}}, \underbrace{\frac{1}{\sqrt 2}(|\alpha\beta\rangle + |\beta\alpha\rangle)}_{\text{singlet spin wavefunction}}$$

are also coupled representations. These four wavefunctions are, respectively,

$$|S=1, M_S=+1\rangle, |S=1, M_S=-1\rangle, |S=1, M_S=0\rangle, |S=0, M_S=0\rangle$$

(Different notation again, but it is the same concept.) The uncoupled representations are the states in which each electron has been assigned a spin, i.e. $|\alpha\alpha\rangle, |\alpha\beta\rangle, |\beta\alpha\rangle, |\beta\beta\rangle$. For example:

$$|m_{s1} = +1/2, m_{s2} = +1/2\rangle = |\alpha\alpha\rangle$$

and so we can see a simpler example of how the coefficients link the two representations:

$$\begin{align}
|S=1, M_S=+1\rangle &amp;= |m_{s1} = +1/2, m_{s2} = -1/2 \rangle \\
|S=1, M_S=-1\rangle &amp;= |m_{s1} = -1/2, m_{s2} = -1/2 \rangle \\
|S=1, M_S=0\rangle &amp;= \frac{1}{\sqrt 2}|m_{s1} = +1/2, m_{s2} = -1/2 \rangle + \frac{1}{\sqrt 2}|m_{s1} = -1/2, m_{s2} = +1/2 \rangle  \\
|S=0, M_S=0\rangle &amp;= \frac{1}{\sqrt 2}|m_{s1} = +1/2, m_{s2} = -1/2 \rangle - \frac{1}{\sqrt 2}|m_{s1} = -1/2, m_{s2} = +1/2 \rangle \\
\end{align}$$