# How can I find the ground state?

Jun 27, 2014

The ground state is the state that is occupied by the most part of the atoms of the same element at room temperature, because it is lower in energy.

It can be experimentally recognized from the absorption spectra of the atomized element in the gas phase: there is a series of lines, in the spectrum, whose frequency reaches a maximum value. That value corresponds to the minimum energy that you should furnish to extract the less bounded electron from the atom, that is the first "ionization energy", and that energy is the same of the GROUND STATE.

In general, to that state correspond many different microstates having exactly the same energy, but different occupation and spin of the valence electrons in the partially filled orbitals.
The most stable electronic configuration DOES NOT correspond to the ground state you are asking about (these are two completely different concepts), because there are usually several different states, with different energies, corresponding to the same electronic configuration , but ONLY ONE is the ground state, the one with the lowest energy.

For example, a simple six electron atom as the carbon atom has the most stable electronic configuration represented by 1s² 2s² 2p².
The two 2p electrons can be placed in 15 different combinations in the three available 2p orbitals ($2 {p}_{x}$, $2 {p}_{y}$, $2 {p}_{z}$ corresponding to orbital magnetic momentum -1, 0, +1) and with electron spins up (↑ = +½) or down (↓ = -½). 9 microstates constitute the same state (and energy) called ³P, in which the total magnetic momentum can assume three different values (0, 1, 2). We'll see that these three states in the triplet don't have exactly the same energy and only one is the * true ground state * . 5 microstates correspond to a so called "singlet" state ¹D in which the total electron spin is zero, whereas the total magnetic momentum is 2. One microstate corresponds to the ¹S singlet state in which the spin, the orbital and total magnetic momentum are zero.

Well, of the five states (a triple plus two singlets) ³P, ¹D and ¹S, of unionized carbon, belonging to the same electronic configuration 1s² 2s² 2p², one of the ³P states (they have very similar, but not equal energies) is the most stable (has the very lowest energy).

Among the different states corresponding to the same most stable electronic configuration, the true ground state can be determined following the three Hund's rules.

1st Hund's rule of maximum molteplicity predicts that one of the three ³P states in the triplet should be the GROUND STATE because ³P corresponds to the state with the "highest multiplicity", i.e. the one having more different values of electron spin (the superscript ³).
The second Hund's rule applies to those states that have the same multiplicity. This rule states that in these cases the ground state is the one with the highest total spin (electronic + orbital), that increases from S = 0, P =1, D=2, F = 3, G=4... But this is not the case of carbon, with only one triplet, which "wins" against the two singlets.
There is also a third Hund's rule, that applies to carbon. This rule says that in a multiplet if the partially filled orbitals are half-filled or less than that, (as in carbon which has two 2p electrons by a maximum occupation of six 2p electrons in the filled 2p subshell) the lowest energy state in a multiplet is the one with the minumum total spin, that is 0. If the unfilled subshell has more than half of the maximum number of electrons (e.g. oxygen, $2 {s}^{2}$$2 {p}^{4}$) the third Hund's rule applies conversely, thus the ground state will be the multiplet term with the maximum total spin.

I know this stuff is somewhat complicated, but this is a matter of facts, and you should prefer the facts to the hearsays, if you are interested to science.

Most of chemistry teachers just know & teach the electronic configuration, and a SINGLE and WRONG Hund's rule, whereby a carbon atom has two paired electrons and a spin constantly equal to +½ + ½ = 1. This is simply FALSE. The maximum magnetic momentum of the ³P states is 2 and derives from the coupling of electron spin and orbital momentum in nine different combinatins (microstates) of which three have total momentum 0, three have total momentum 1 and three have total momentum 2.

You can find an easy accessable source to all this topic from here:
http://physics.nist.gov/PhysRefData/Handbook/periodictable.htm

Choose carbon from the periodic table, and you learn soon that:
C I Ground State 1s²2s²2p²; ³P_0 (electronic configuration plus ground state with capital letter.) the "0" lowerscript indicates the term of the triplet with minimum total angular momentum as the ground state.

Now, if you choose "Energy level" for the unionized atom (correspond to "I" in the symbol "C I") you will find a mess of excited states

The five lower energy states will correspond to the five states (a triplet and two singlets) with the most stable electronic configuration, 1s² 2s² 2p². The value of J = 0 indicates the true ground, or zero energy, state.

You can also find a map of all the energy levels (states) of unionized carbon, or Grotrian diagram, from this link, putting C i in the "Spectrum" entry: http://physics.nist.gov/PhysRefData/ASD/lines_form.html

I believe that deep understanding of a single ground state, of such an important element as carbon, even if you can't understand everything, is enormously more important than to be able to apply some easy mechanical rule to determine the stablest electronic configuration (n+l rule) that you can find in every general chemistry book.