# Quantum mechanics/Atomic spectra
## Combination of two orbital angular momenta (different orbitals)

In classical physics, the total angular momentum is the (vector) sum of the angular momenta of the particles present in the system; the resulting total depends on the orientation of the components. In quantum mechanics, if a system has two particles with angular momentum L$_1$ and L$_2$ and angular momentum quantum numbers l$_1$ and l$_2$, we can define a total angular momentum L=L$_1$+L$_2$. L behaves as any angular momentum and the eigenvalues of ${\hat {\mathbf {L} }}^{2}$ are ħ²L(L+1), i.e.
${\hat {\mathbf {L} }}^{2}\psi =\hbar ^{2}L(L+1)\psi$ (Eq. 1)
L is the total angular momentum quantum number and can take values |l$_1$ - l$_2$|, |l$_1$ - l$_2$|+1, ..., l$_1$ + l$_2$.
Example 1. Consider an atom with one electron in orbital 2p (l$_1$=1) and one electron in orbital 3d (l$_2$=2). The total angular momentum quantum number for this system can be L=1,2,3. States with different total angular momentum have different energy so that there are (at least) 3 different states deriving from a configuration 2p3d.
Example 2. Consider the excited He atom with configuration 1s2p. The allowed quantum number for this configuration is just L=1.
The component along the z axis of the angular momentum is ${\hat {L}}_{z}$ and its eigenvalues can take the values ħm$_L$ with m$_L$ = -L, -L+1, ..., L.
Wavefunctions with L=0, 1, 2, 3,... are indicated with the capital letters S, P, D, F,...

## Combination of two spin angular momenta (different orbitals)

In a system with two electrons in different orbitals, you can define that total spin as the sum of the individual spins: ${\hat {\mathbf {S} }}={\hat {\mathbf {S} }}_{1}+{\hat {\mathbf {S} }}_{2}$. The total spin behaves as any angular momentum, and the eigenvalues of ${\hat {\mathbf {S} }}^{2}$ are ħ²S(S+1), i.e.
${\hat {\mathbf {S} }}^{2}\psi =\hbar ^{2}S(S+1)\psi$ (Eq. 2)
S is the total spin quantum number and takes the value between |s$_1$ - s$_2$| and s$_1$ + s$_2$, i.e. between 0 and 1 in the case of 2 electrons, because the individual spin quantum numbers are each equal to ½.
The component along the z axis of the total spin is ${\hat {S}}_{z}$ and its eigenvalues can take the values ħm$_S$ with m$_S$ = -S, -S+1, ..., S. There are in total 2S+1 eigenvalues. 2S+1 is called spin multiplicity. States with multiplicity 1, 2, 3, 4,... are called singlet, doublet, triple, quartet, etc.
In the two examples in the section above, the total spin can take the value of 0 and 1 and so the configuration gives rise to a singlet and a triplet. The energy of a wavefunction depends on the spin multiplicity (or S).

## Spectroscopic term

The electronic configuration is not enough to describe the wavefunction, because for each configuration there are different ways to combine the orbital and spin angular momenta of the electrons (different combinations have different energies). The labels of a wavefunction which indicate the total angular momentum (S, P, D,...) and the spin multiplicity (singlet, doublet, triplet) are called spectroscopic terms.
The spectroscopic terms originating from the configuration 2p3d of the example 1 above are $^1$P, $^1$D, $^1$F, $^3$P, $^3$D, $^3$F.
The spectroscopic terms originating from the configuration 1s2p of the excited helium atom are $^1$P and $^3$P.
You can neglect the electrons in filled sub-shells, so that the spectroscopic term of the ground state of lithium 1s$^2$2s is $^2$S. The spectroscopic terms of the excited configuration of beryllium 1s$^2$2s2p are $^1$P and $^3$P.

## Electrons in equivalent orbitals

In sections 1 and 2 we considered electrons in different orbitals. If they are in the same orbital the calculation of the allowed spectroscopic terms is slightly more laborious. Consider atomic carbon with 2 electrons in 2p. You have to arrange the electrons in all possible ways allowed by the Pauli principle and then identify by inspection the spectroscopic terms on the basis of the values of m$_S$ and m$_L$ The spectroscopic terms of atomic carbon are $^1$D, $^3$P, $^1$S.

## Hund's empirical rule

It is useful to know what is the lowest energy spectroscopic term originating from a given configuration. Hund's rule states that:
* The level with the largest multiplicity has the lowest energy.
* For a given multiplicity, the term with the largest L has the lowest energy.
The ground state of atomic carbon is $^3$P (which has the largest multiplicity). This is why in elementary textbooks, electrons in partially filled shells have parallel spin.

## The last complication: spin-orbit coupling

Everything said so far implied that the spin does not appear in the Hamiltonian and its only effect is to change the structure of the spatial wavefunction. It is true that this is the main role of spin, but there is a very small term in the Hamiltonian that depends on the spin and this is called the spin-orbit coupling (which will not be written explicitly). Spin orbit coupling is important only for heavy atoms or when very accurate energy levels are needed (spectroscopy). The effect of spin orbit coupling is to split the spectroscopic terms in energy levels. The number of energy levels originating from a spectroscopic term is given by the possible values of the total angular momentum J = L + S (angular plus spin) with eigenfunction
${\hat {\mathbf {J} }}^{2}\psi =\hbar ^{2}J(J+1)\psi$
The quantum number J can take the values |L - S|, |L - S|+1, ..., L + S.
The spectroscopic term $^3$P (L=1, S=1) can have J=0,1,2 and therefore 3 finely spaced energy levels indicated as $^3$P$_0$, $^3$P$_1$, $^3$P$_2$.

## Summary

Lessons 12-13 give an example of how quantum mechanics is used when the problem is too difficult to be solved exactly. You have to proceed by successive approximations, each time more refined. The aim is to understand the experiment (in this case, how many energy levels are present in an atom) and not to predict the exact value of the energy levels.
* The first level of approximation is the independent electron approximation. Each electron is in an orbital and the energy is determined only by the orbital occupation or electronic configuration.
* The second level of approximation is considering that the energy is different if the total orbital angular momentum and spin angular momentum are different. We differentiate the energy in spectroscopic terms depending on the values of L and S.
* The most accurate level takes into account the spin orbit coupling. The energy levels are identified by the total angular momentum J.
Exercise
Find the spectroscopic terms originating from the following configurations (two electrons in two non-equivalent orbitals):
1s2s
1s3s
2p3p
2s3d
Find the spectroscopic term originating from the ground states configuration of the sodium atom and the boron atom.
Find the spin-orbit energy levels of hydrogen with electron in 2p and 3d.
Solutions
$^3$S, $^1$S
$^3$S, $^1$S
$^3$D, $^3$P, $^3$S, $^1$D, $^1$P, $^1$S
$^3$D, $^1$D
Na: [1s$^2$2s$^2$2p$^6$]3s → $^2$S; B: [1s$^2$2s$^2$]2p$^1$ → $^2$P
2p configuration corresponds to the term $^2$P (L=1, S=½). The energy level corresponds to J=1-½, 1+½ i.e. $^2$P$_{1/2}$ and $^2$P$_{3/2}$. For the 3d configuration the term is $^2$D and the energy levels are $^2$D$_{3/2}$ and $^2$D$_{5/2}$ :
Next: Lesson 13 - Introduction to molecular orbital theory