Assumptions and Examples

Assumptions – Atomic Orbitals

Orbital distances are calculated using the statics rule from classical mechanics that an object will remain at rest when the sum of the forces is zero. This requires an assumption that the proton has an attractive and repelling force, as described by the pentaquark structure of the proton in the overview. In addition, each and every electron in the atom affects one another. Additional rules were created for the angles and distances of these electrons. They are described graphically below.


Electron Angles

Because of tetrahedral alignment from the quarks in the proton, most angles are at 0° or 60° in relation to the proton.  As orbitals become more complex, they are computed as an average of these angles.  The angle is in relation to the proton because the sum of forces that is calculated in the equations is the linear direction between the atom’s nucleus and the affected electron.

Electron Angles

Electron angles in relation to the proton


The s orbitals have a common angle:

s orbital angle


The p orbitals are a mix of these two angles*:

p orbital angle


There are a few exceptions that have this angle**:

exceptions angle


* The angles are averaged across the entire solution
** When sodium and magnesium begin building the 3s orbital, they have this angle


Electron Distances

The axis between the nucleus and the electron being measured is the line where the forces will be calculated (where the sum of the forces is zero).  The attractive force (F1) and proton’s repulsive force (F2) are on this axis.  Each additional electron in the atom has a repulsive force that may be on a difference axis and its distance is computed based on its orbital and the electron angle from above.

Electron distance rule

Electron distances in relation to the affected electron (being calculated for orbital distance)



Examples – Orbital Distances of Hydrogen and Helium

Using the classical mechanics method discussed in the overview section, orbital distances were calculated for each electron in each orbital for elements from hydrogen to calcium, including their ionized elements. These calculated orbital distances were validated by two methods:

  1. Comparing calculated distances against the known distance of the largest orbital radius.
  2. Comparing calculated distances against ionization energies of all orbitals, using the Transverse Energy Equation, which requires electron distance.

These were placed in the Atomic Orbitals papers and summarized on this site. Here, examples will be provided to manually calculate the 1s orbitals, then equations built to use computer programs (e.g. Mathcad) to solve more complex orbitals with multiple electrons at various distances.



A single proton and electron at the ground state is known as the Bohr radius.  It has two forces: an attractive and repelling force.

Hydrogen- The attractive force and repulsive force of an electron in orbit

Hydrogen: The attractive force and repulsive force of an electron in orbit


1) Shorthand notation for electron energy (Ee) and electron classical radius (re) in the Force Equation.  These values match the CODATA values and were derived in Particle Energy and Interaction.  Although they can be represented in wave constants, for simplified equations and readability they are expressed as single constants.  Shorthand notation is used in these equations because they will become complex with multiple electrons and it maintains readability in the next section.

electron energy and radius in wave constants


2) The Force Equation for the attractive, Coulomb force using shorthand notation.

Force 1 - hydrogen


3) The Force Equation for the repelling, axial force using shorthand notation.

Force 2 - hydrogen


4) Set (1) equal to (2). Ee and re always appear in every force and will cancel.  For shorthand, they will be removed from all future derivations beyond hydrogen.

Eq 3 - hydrogen


5) Shorthand. Z is the number of protons. For hydrogen it is one (1). Q1 and Qare also one (1) for ground state hydrogen.

sum of forces zero - hydrogen


6) Solve for r and the solution is 5.2918 x 10-11 m (52.9 pm).  This is the exact value of the Bohr radius.

bohr radius derived



Helium adds a second electron which is also in the 1s orbital, placed in the position as shown below.

Helium now with two protons and one extra electron

Helium: Similar to hydrogen, but now with two protons and one extra electron


1) A new repelling, Coulomb force (F3) from the additional electron at a distance of 2r will be calculated. Q1 and Q2 are one because one electron (at distance 2r) affects the one electron being calculated.  Using shorthand notation (without Ee and re which cancel).

Force 3 - helium


2) The attractive force (F1) is set equal to the two repelling forces (F2 and F3).  This is where the sum of forces is zero and the electron rests.  F1 now has two protons (Z=2).  F2 is the same repelling axial force of 1 proton.  Q1 and Q2 are 1 due to one proton in alignment.  Using shorthand.

helium - sum of forces is zero


3) Solve for r.  Helium 1s orbital distance is calculated to be 3.02 x 10-11 m (30.2 pm).  This is compared with an estimated radius of 31 pm from experiments.

helium 1s orbital distance



Examples – Orbital Distances of Lithium to Calcium

Beyond the 1s orbital, there are electrons at various unknown distances that affect the electron that will be ionized.  Each of these electrons much be considered for the total resultant force.

Complex interaction of forces on an affected electron. The distances and angles of each particle are required.


An equation can be arranged for each of the unknown distances.  To simultaneously solve these distances and equations, Mathcad solver was used. The method for creating these equations is explained in the next section on orbital distances where each orbital was calculated for hydrogen to calcium, including ionized versions of these elements.



The Mathcad solutions are available to download here (.mcdx file).  These examples, and more, are found in the Atomic Orbitals paper.