# Atomic and Molecular Quantum Numbers

## What are quantum numbers?

We know from quantum mechanics that observable quantities are quantized. Each quantum number corresponds to an observable quantity or combination of quantities and describe the quantized states that characterize a system. That is, they are the set of numerical values that give acceptable solutions to the Schrodinger equation for that system. There is no single set of quantum numbers that completely describes every system; different types of systems are fully described by different quantum numbers.

## Quantum numbers in an atom

An electron in an atom is described completely by four quantum numbers: ${\displaystyle n}$, ${\displaystyle \ell }$, ${\displaystyle m}$, and ${\displaystyle s}$. (Fewer quantum numbers can be made to describe an electron if you choose carefully, since the spin-orbit interaction relates some of these numbers. However, this is the standard way.)

The principal quantum number ${\displaystyle n=1,2,...}$ describes the electron's energy level, or shell. As ${\displaystyle n}$ increases, the average distance of the electron from the proton increases.

The orbital quantum number ${\displaystyle \ell =0,1,2,...n-1}$, also called the azimuthal quantum number or the angular quantum number, describes the shape (or subshell) of the electron's shell. In chemistry, ${\displaystyle \ell =0}$ is called the s orbital, ${\displaystyle \ell =1}$ is called the p orbital, ${\displaystyle \ell =2}$ is called the d orbital, and ${\displaystyle \ell =3}$ is called the f orbital. ${\displaystyle \ell }$ corresponds to the orbital angular momentum of the electron as follows:

${\displaystyle L^{2}=\hbar ^{2}\ell (\ell +1)}$

The value of ${\displaystyle \ell }$ ranges from 0 to ${\displaystyle n-1}$ because the first p orbital appears in the second (${\displaystyle n=2}$) shell.

Figure 1: Outline of the periodic table of elements showing the orbital in which that element's valence electrons are found.

The magnetic quantum number ${\displaystyle m_{\ell }=-\ell ,1-\ell ,2-\ell ,...,0...,\ell -2,\ell -1,\ell }$ describes the specific orbital within the subshell. The s subshell contains just one orbital: ${\displaystyle m_{\ell }=0}$ is the only allowed value. The d subshell contains five orbitals, since ${\displaystyle m_{\ell }}$ can range from -2 to +2. ${\displaystyle m_{\ell }}$ corresponds to the projection of the electron's orbital angular momentum along a specified axis:

${\displaystyle L_{z}=\hbar m_{\ell }}$

The spin projection quantum number ${\displaystyle m_{s}=-s,1-s,2-s,...,s-2,s-1,s}$ describes the spin (intrinsic angular momentum) of the electron within an orbital. Here ${\displaystyle s}$ is the spin quantum number, an intrinsic property of a particle, and ${\displaystyle m_{s}}$ is related to the projection of the spin angular momentum ${\displaystyle S}$ along a specified axis:

${\displaystyle S_{z}=\hbar m_{s}}$

An electron has spin number ${\displaystyle s=\pm 1/2}$, so the allowed values of ${\displaystyle m_{s}}$ are -s and +s, or ${\displaystyle m_{s}=-1/2}$ or ${\displaystyle 1/2}$. Because electrons are fermions (i.e. they have a half-integer spin), they behave according to the Pauli exclusion principle, so an orbital cannot contain two electrons with the same spin.

Example: The valence electron of an aluminum atom is located in the 3p1 orbital, so it has quantum numbers ${\displaystyle n=3}$ (3rd electron shell), ${\displaystyle \ell =1}$ (p orbital subshell), ${\displaystyle m_{\ell }=-1,0,}$ or 1, and ${\displaystyle m_{s}=-1/2}$ or ${\displaystyle 1/2}$.

## Total Angular Momentum Numbers

The spin-orbit interaction is caused by electromagnetic interaction between the electron's spin and the magnetic field generated by the electron's orbit around the nucleus, shifting an electron's energy levels slightly. This can be thought of as a Zeeman effect due to the internal magnetic field of an atom. One consequence of the spin-orbit interaction is that the orbital angular momentum ${\displaystyle L}$ and the spin angular momentum ${\displaystyle S}$ are no longer independent, and a new set of quantum numbers should be used that describe the total angular momentum.

1) The principal quantum number as before.

2) The total angular momentum quantum number ${\displaystyle j=|\ell \pm s|}$ gives the total angular momentum as:

${\displaystyle J^{2}=\hbar ^{2}j(j+1)}$

3) The projection of ${\displaystyle J}$ along a specified axis ${\displaystyle m_{j}=-j,1-j,2-j,...,0,...,j-2,j-1,j}$ satisfies ${\displaystyle m_{j}=m_{\ell }+m_{s}}$ and ${\displaystyle |m_{\ell }+m_{s}\leq j|}$

4) The parity is positive for states that come from even ${\displaystyle \ell }$ and negative for states that come from odd ${\displaystyle \ell }$, such that ${\displaystyle P=(-1)^{\ell }}$.

## Quantum Numbers in Diatomic Molecules

Rotation:

A diatomic molecule may rotate about a direction perpendicular to the bond axis, giving rise to angular momentum ${\displaystyle N}$. However, a molecule also possesses electronic angular momentum ${\displaystyle L}$. These two momenta couple to make a total angular momentum ${\displaystyle J=N+L}$ which is conserved, though ${\displaystyle N}$ and ${\displaystyle L}$ are not themselves conserved since one type of angular momentum could be converted to the other. The notation gets a bit weird here: in the electron case capital letters denote actual angular momenta, i.e. the factors of ${\displaystyle \hbar }$ are in there, but in the case of rotating molecules a capital ${\displaystyle J}$ is used as a quantum number directly. That is, you'll often see transitions labeled ${\displaystyle J=1\rightarrow 0}$, meaning that a molecule has moved from the first excited rotational state to its ground state. The total angular momentum is given by ${\displaystyle J(J+1)h^{2}}$.

Since ${\displaystyle N}$ must be perpendicular to the bond axis to be nonzero, ${\displaystyle J_{z}=L_{z}}$ and ${\displaystyle L_{z}}$ is conserved, too, meaning that ${\displaystyle M_{L}}$ is is a good quantum number to use to describe the electronic contribution to the total angular momentum. Since the energy of the system is the same whether ${\displaystyle M_{L}}$ is positive or negative, ${\displaystyle \Lambda =|M_{L}|}$ is usually used instead. Of course, there may also be a small contribution from electron spins, which is similarly parameterized as ${\displaystyle \Sigma =|M_{s}|}$. The total is again conserved, with ${\displaystyle \Omega =\Sigma +\Lambda }$ describing the total angular momentum projected onto the bond axis.

Vibration:

A diatomic molecule vibrates approximately as a simple harmonic oscillator, and its vibration state can be described by a single quantum number ${\displaystyle \nu }$, where ${\displaystyle E=(\nu +1/2)h\omega }$ are the energy levels. Anharmonic terms proportional to ${\displaystyle \nu ^{2}}$ or higher order terms are possible but generally very small for diatomic molecules.

Rovibrational spectra:

The total internal energy of a molecule is the sum of its rotational and vibrational energy. The selection rule for vibration is ${\displaystyle \Delta \nu =\pm 1}$. In reality contributions from ${\displaystyle \Delta \nu =\pm 2}$, ${\displaystyle \Delta \nu =\pm 3}$, etc. also arise in a spectrum at much lower energies due to the anharmonic terms. For rotation, ${\displaystyle \Delta J=0,\pm 1}$ is the selection rule; however, in a linear molecule ${\displaystyle \Delta J=0}$ is only allowed if there is angular momentum perpendicular to the rotation axis, which is possible either if a molecular bending mode is excited or if there's an unpaired electron and ${\displaystyle \Omega \neq 0}$ (as in, for example, the molecules NO and OH).

A rovibrational spectrum is classified into three branches, P, Q, and R, which correspond to ${\displaystyle \Delta J=-1}$ (low energy), 0 (medium energy), and 1 (high energy), respectively.