Calculations – Ionization Energies

Calculations – Ionization Energies (H to Ca)

Hydrogen to Calcium

The energy of a photon that is absorbed or emitted by any atom can be calculated using the classical constant form or the wave constant form of the Transverse Energy Equation, along with 1) the electron’s distance and 2) constructive/destructive wave interference (amplitude factor) on an electron transitioning orbitals in an atom.  These two variables are found in the tables on the following pages: orbital distance calculations and amplitude factor calculations.

More than 200 calculations are compared to measured results of photon energies required for ionization.  An explanation is provided in the Atomic Orbitals paper for the variation of calculations versus measured results. Ionization energies were calculated in Mj per mole.


Ionization Energy of the First (Outermost) Electron – Neutral Element

Ionization Energy of Neutral Elements


Ionization Energy of 1s1 Electron – Ionized Element

Ionization Energy of 1s Electron - Ionized Atom


Ionization Energy of 1s2 Electron – Ionized Element

Ionization Energy of 1s2 Electron - Ionized Atom


Ionization Energy of 2s and 2p Electrons – Ionized Element

The charts for the 2s and 2p electron ionization energies are summarized quickly below, but the details can be found in the Atomic Orbitals paper.

Ionization energy of 2nd orbital - Ionized Atoms



Example Calculation

Boron – 2p 

An example is the photon energy required to ionize an electron in the 2p subshell of boron. In this case, both distance and amplitude must be known.

Step 1) The 2p orbital distance (r0) comes from the distance table.  It is 1.41Note that radius (1.41) is converted to meters by multiplying by the Bohr radius (found in wave constant terms) – more details on this in the Atomic Orbitals paper.  

Step 2) The amplitude factor for Boron 2p (δB2p) comes from the amplitude factor table.  Neutral boron has Z=5 protons and e=5 electrons, and rx is found from the distance table used above (1.41). The calculation is shown again as:

Step 3) The variables are now known for the Transverse Energy Equation (using wave constant form):

Equation: Transverse Energy equation

  • r0 = a0(1.41) = 7.4614E-11 m (from above)
  • r = infinity (leaves the atom in ionization)
  • δB2p = 0.8546 (from above)

After converting from joules to megajoules per mole using Avogadro’s number (NA) and dividing by 106, the calculated result is -0.80 Mj/mol.  

-0.80 megajoules per mole (Mj/mol)
Comments: This matches the measured result for Boron which is -.80 Mj/mol.

Source Data: All graphs and tables shown here for orbitals distances can be found in the downloadable spreadsheet.  The solutions for orbitals were first generated with Mathcad, available on the same page, and then placed into the spreadsheet. Further information on the derivation of the equations and how to replicate them is in the Atomic Orbitals paper.