# Statistical Thermodynamics and Rate Theories/Electronic energy

Atoms and molecules can hold different energy levels that correspond to different distributions in how the electron density is distributed around them. These are counted as electronic degrees of freedom.

For example, the simplest atom is the hydrogen atom, with 1 proton and 1 electron. The attraction between the proton and electron in the hydrogen atom is governed by Coulomb's Law,

 principal quantum number n=1,2,3,... angular momentum quantum number $l=0,1,...,n-1$ magnetic quantum number $m_{l}=-l,-l+1,...,0,l-1,l$ spin quantum number $m_{s}=+{\frac {1}{2}},-{\frac {1}{2}}$ ${\mathcal {V}}(r)=-{\frac {e^{2}}{4\pi \epsilon _{0}r}}$ Where $e$ is the elementary charge of an electron, $r$ is the distance between the proton and electron in the hydrogen atom, and $\epsilon _{o}$ is the vacuum permittivity constant. When the Schrödinger equation is solved for this potential, four quantum numbers result, n, l, ml, and ms. This means that the particles can only take on certain discrete values of energy. The electronic energy of the hydrogen atom depends only on the principal quantum number $n$ via the following equation,

$\epsilon _{n}=-{\frac {m_{e}e^{4}}{32\pi ^{2}\epsilon _{o}^{2}\hbar ^{2}}}{\frac {1}{n^{2}}}$ Where $m_{e}$ is the mass of an electron. This means that all sets of angular momentum, l, and magnetic quantum numbers, ml, and ms, with the same value of n are degenerate, as they do not appear in this solution to the Schrödinger equation. This quantization of allowed energies give rise to the electronic energy levels of the hydrogen atom. This equation can also be used to calculate the energy levels of hydrogen like species such as He+. An atomic factor, $Z^{2}$ , would be added in the numerator to account for the extra protons in the atom, which would pull the electrons in more than the hydrogen atom, and thereby increase the energy.

## Example

Calculation of the energy for the excitation of O2 from the triplet to singlet electronic state.

Singlet oxygen comes from the electronic excitation of triple oxygen, the energy is experimentally 0.98 electron volts per mole of electrons. The figure below illustrates the different states of oxygen. 1Δg and 1Σg+ represents singlet oxygen as it has a spin state equal to one, the triplet degenerate state is represented by 3Σg-, the transition is represented by 1Δg3Σg-