Trying to understand lowest configurations of carbon My study group is debating about which are the three lowest configurations of carbon. I've been arguing that the electron has to jump to the 3s level for the configuration to be different. Others have suggested that the two valence electrons just have to change their $m$ and $s$ numbers on the 2p level. We are using Morrison's Modern Physics and having trouble settling this issue within the text. We are aware of Hund's rule, so some of the problem is about exactly what is meant by "configuration." We want to understand this problem and do the work ourselves, but we are installing doubt in one another. Can someone clarify "configuration" and maybe suggest the general approach appropriate here?
 A: By 'electron configuration' can be understood the way an atom's electrons are arranged in atomic orbitals, in accordance with Pauli's Exclusion Principle, the Aufbau Principle and Hund's Rule, of the lowest possible total energy (known as the Ground State).
For carbon (Z=6), six electron have to be placed in the correct atomic orbitals.
The first 2 occupy the lowest energy atomic orbital possible, that is 1s, so we have $1s^2$ for the first term.
For the remaining four electrons, the next two lowest available atomic orbitals are 2s and 2p and following the above rules that gives us $2s^2$ and $2p^2$. Bearing in mind that to satisfy Hund's Rule the latter two 2p electrons are divided over one $p_x$ and one $p_y$ sub-orbital, each with one electron of the same spin quantum number ($m_s=-\frac{1}{2} \text{or} +\frac{1}{2}$).
Overall we can write the electron configuration of carbon as:
$1s^2 2s^2 2p^2$ or with some added detail $1s^22s^22p_x^12p_y^1$ and because $[He]=1s^2$, carbon's electron configuration (ground state) can be written as:
$[C]=[He]2s^22p^2$ = $[He]2s^22p_x^12p_y^1$.
The first excited state of carbon $C^*$, and the one that explains the existence of $C(+4)$ chemical compounds, is $[He]2s^12p_x^12p_y^12p_z^1$ where all three lone 2p electrons have the same $m_s$ value.
A: Electron configuration means the distribution of electrons in atomic (or molecular) orbitals. Here you seem to have an idea dealing with angular and spin momenta.
Hund's rule can tell us what the lowest electron configuration is and firstly originated from the way to choose a state of lowest interaction between total momentum and entire atomic environment within given electron configuration.
By Hund's rule, lowest electron configuration of carbon atom is $[He]2s^22p^2$. If we go further, $[He]2s^22p_i^12p_j^1$ where $i,j=x,y,z$ and both spins need to be same.
Why? That's the way to lower the interaction between total momentum and any other factors. Atomic Hamiltonian has several terms; kinetic energy of electrons + repulsion between electrons + attraction between electrons and protons + (spin-orbit interaction) + (etc). Here spin-orbit interaction is the one of main ideas of the interaction between total angular momentum and other factors. You might be confused if you try to understand this using $\sum m$ and $\sum s$ merely, so don't be worry about how to explain the interaction numerically at this time.
If you and your group want to study the structure of atomic Hamiltonian strictly, any other configurations that deviate from Hund's-ground-state will be next excited states, since they produce more powerful momenta-quantum mechanical factors interaction that you cannot figure out yet.
Since 3 $p$ orbitals are equivalent and generally change of interaction between total momentum and atomic environment is smaller than change of kinetic energy/repulsion potential of electrons, by changing states of electrons,
$[He]2s^22p_i^12p_j^1$ (two $p$ electrons have opposite spin)
$[He]2s^22p_i^2$
$[He]2s^12p_i^12p_j^12p_k^1$ (since $2s$ and $2p$ have different energy with more than one electron)
are next three lowest configurations and unfortunately we cannot determine the order of them here.
Well, if you are not familiar with the very structure of atomic Hamiltonian, we can just concentrate on the energy difference between atomic orbitals and spin-spin interaction. At this time,
$[He]2s^12p_i^12p_j^12p_k^1$
$[He]2p_i^22p_j^12p_k^1$
$[He]2p_i^22p_j^2$
are next three lowest configurations and they are written in (low energy -> high energy) order.
