Orbital Distances


The original explanation of an electron’s orbital was proposed by Niels Bohr in 1913 using a model that is very similar to the Earth’s orbit around the Sun – hence the name orbital. The Earth is attracted to the Sun through the gravitational force but it remains in orbit due to its velocity – an equal centripetal force to counter the force of gravity. In the Bohr model, it was assumed that a proton attracts an electron similar to the Sun attracting the Earth. In this model, gravity is replaced by the electrostatic force.

The problem with the Bohr model is that it worked well for hydrogen and helium but not for atoms with multiple electrons. It also cannot account for the probabilistic nature of the electron cloud or the electron’s spin. In reality, the electron has a probable distance from the atomic nucleus and the orbital is the average distance it is likely to be.

Another issue is the difference between a proton and a positron that have the same charge. In the case of the positron, the electron is attracted to it and annihilates. In the case of a proton, it finds its orbital.  What is the difference between the proton and positron that would cause this to occur?

bohr model of electron's orbit

Bohr Model – Electron Orbital




An electron does not circle an atom like a planet circles the Sun. Instead, it is both pushed and pulled at the same time. Bohr originally assumed that the proton was an elementary particle with a single, positive charge. By 1968, the proton was found to be a composite particle consisting of smaller particles (three quarks). In the proton page, the newer discovery of the pentaquark (discovered 2015) is used as the model of the proton, which shows a composite particle of 4 quarks and 1 anti-quark. In this theory the pentaquark is believed to be 4 highly-energetic electrons and a positron at the center.

proton pentaquark model



The composite structure of the proton allows both an attractive force and a repelling force. The orbital electron is attracted to the proton at any angle by the electric force between the positron in the center and the electron. This force decreases with the square of the distance between the particles. The orbital electron is also repelled at the axis between tetrahedral quarks, referred to as an orbital force. Within the proton, this is the strong force. Beyond standing waves, it is a repelling force and it decreases rapidly at the cube of distance.

orbital attractive and repelling forces of atom animated


Orbital Distance

This model allows the position of the electron to be calculated with classical mechanics. The equation and its calculations are compared to measured results, shown in brief on this page and in more detail in their respective pages.

Using the tetrahedral structure of the quarks in the pentaquark model, there are six axes in which the axial force of gluons would repel the electron.  As the nucleus continues to spin, this changes the point where the sum of force is zero.  The electric force, which is the attractive force, remains constant.  But the repelling, orbital force is continually changing based on the spin of the proton.  The constant attractive force is why the photon ionization energies of electrons are constant and predictable yet the distance of the electron is not.  This is modeled below.
Electron Path in Proton causing probability cloud


This proposed structure of the proton not only leads to accurate distance and photon energy calculations but it explains the curious probability nature of the electron and the shape of atomic orbitals. The electron does not take a straight path to orbit the nucleus like a planet circling the Sun. Instead, it has the probability of being in a particular location around the the atomic nucleus such as below.

Probability Cloud


Orbital distances for elements with more than one orbital becomes a complex calculation of constructive wave interference with other electrons in the atom, in addition to the constructive and destructive waves from the nucleus.  The distances and angles of the other electrons are required to compute the final orbital distance of the affected electron.  Rules for establishing these angles and distances in equations are shown on the Equation page.

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


Orbital Distance Solution 

The Equation page shows the steps to calculate an orbital by hand, which is helpful for hydrogen and helium. Beyond helium, a series of equations needs to be solved simultaneously. This section highlights the simplified method to use Mathcad as a software tool to solve the equations. To simplify the complex Mathcad solution, the equation was simplified to only the orbital radii as the unknowns.  The Bohr radius was removed from F2 but is added back after the solution of radius (r) to get to meters.  Thus, the Mathcad solution provides a ratio of distance to the Bohr radius.

1) Sum of forces of extra electrons using shorthand notation.  Now, another electron is added as F3 but this force scales similarly for any additional electrons, so the same steps are repeated for F4,F5, etc.

Sum of forces equation

2) F1 is the attractive electric force based on the number of protons (Z) and one electron.   F2 is the axial, repelling force where Q1 and Q2 are the number of same-spin protons in alignment.  To simplify the equation, they are set to equal and now become Q2.  F3 is an electron (ey) that will affect the electron considered for its orbital, from the distance r3.  F3 can be repeated for other electrons with forces at other radii.

electron orbitals derivation

3) Simplify above and expand r3 from distance rule.

Mathcad electron orbitals equation

4) Temporarily remove Bohr radius (re/⍺2) from above equation.  This makes the solution easier but the result is now a ratio of the Bohr radius.  This value (re/⍺2) needs to be re-added to convert to orbital distances from a ratio of the Bohr radius to the actual orbital distance in meters.

Mathcad electron orbitals equation without Bohr radius

5) The equation expands to the right with more electrons at same distance where j is the total number of orbitals. x is the axis between the affected electron and the nucleus.  y will expand for each electron and the angle needs to be determined.  Electron angle rules were established to estimate these angles.

Orbital Distance Solution (function of Bohr Radius)


When using the Orbital Distance Solution equation, each electron affects all others and so all distances need to be solved simultaneously.  Equations were arranged to be solved with Mathcad to generate the orbital distances for each electron in an atom (illustrated below).  Solving for rx, this solution provides the radius in terms of a ratio to the Bohr radius. The details of the methodology and its calculations can be found in the Atomic Orbitals paper.




Proof of the energy wave explanation for atomic orbital distances is the calculations of:



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


Equation: Orbital Distance equation (from above)

  • Z = 1 (proton)
  • Q = 1 (same-spin proton in alignment)
  • j = 0 (no additional forces acting on the single electron)

Hydrogen 1s Calculation Bohr ratio

Result: rx = 1

The solution is a ratio relative to the Bohr radius. To arrive at distance in meters, multiple the Bohr radius:

Hydrogen 1s Calculation Distance

Result: 5.2918E-11 m
Comments: 0.000% difference from the CODATA value of the Bohr radius



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


Equation: Orbital Distance equation (from above)

  • Z = 2 (2 protons)
  • Q = 1 (same-spin proton in alignment)
  • j = 1 (one additional force due to another electron)
  • ey1 = 1 (one additional electron acting as a force on the electron being calculated)

Helium Distance Calculation Bohr Radius

Result: rx = 4/7 (0.571)

The solution is a ratio relative to the Bohr radius. To arrive at distance in meters, multiple the Bohr radius:

Helium Distance Calculation

Result: 3.02E-11 m
Comments: The orbital distance for helium 1s is calculated to be 30.2 pm (in picometers).  This is compared with an estimated radius of 31 pm from experiments.



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