Calculations – Atoms

Calculations – Orbital Distances

This web site provides the framework for the calculation of the electron’s position in the atom, and its associated energy levels, using only classical mechanics.  It removes the need to have a separate set of quantum rules and equations for the electron’s behavior.  The classical calculation for hydrogen matches the Bohr radius with 0.000% difference at 5.29177 picometers.

The classical explanation of the electron’s position in an atomic orbit is that it is being pushed and pulled at the same time by both a spherical, attractive and an axial, repulsive force.  The Bohr model assumes that there is only an attractive charge in the atom’s nucleus similar to the gravitational pull of the Sun.

The chart below is the results of the largest orbital distance calculated using methods on this site and compared to the measured or estimated results for the first 20 neutral elements (results are in picometers).  Below, on this page, are more than 450 orbital distances that have been calculated using the same method.  The method is described in the page on orbital distances.

Atomic Radii - Calculated with Classical Mechanics Equations


Measuring atomic orbitals for all elements is difficult due to the small size of the atom and the probability of the electron, so some variation in the graphs are expected. Therefore, a second method was also chosen to test the orbital distance calculations using ionization energies of electrons which is more accurate. These results were shown in the Photon calculations section, as orbital distance is a required variable to accurately calculate ionization energies.



Calculations – Orbital Distance Tables

Hydrogen to Calcium

Orbital distances are calculated using the aforementioned methods and using Mathcad to simultaneously solve a series of equations for the point where the sum of forces are zero on the affected electron (the Mathcad files can be found here). These tables summarize the orbital distances for neutral atoms and for ionized atoms containing one to ten electrons, for each of the orbitals (1s, 2s, 2p, 3s, 4p and 4s). Calculations are provided from hydrogen (H) to calcium (Ca).

Ionized atoms are calculated in a similar method using the Mathcad solutions, but changing the number of protons (Z) in the solution.  For example, Ca18+ is calcium with 2 electrons. This is the same electron configuration as helium, so the helium Mathcad solution is used, but the Z value is changed to Z=20 instead of Z=2.


Neutral Atoms

The results are a ratio of the Bohr radius. E.g. Hydrogen 1s orbital distance is 1.00 * 5.29177 x 10 11 meters, or 52.92 pm.

Neutral Atoms - Orbital Distances - Ratio of Bohr Radius


Ionized Atoms – 1 to 6 Electrons

Ionized Atoms 1 to 6 Electrons - Orbital Distances - Ratio of Bohr Radius


Ionized Atoms – 7 to 12 Electrons

Ionized Atoms 7 to 12 Electrons - Orbital Distances - Ratio of Bohr Radius



Calculations – Amplitude Factors

Hydrogen to Calcium

The amplitude factor is a measurement of constructive or destructive wave interference.  When one or more particles are located in two groups at a single distance (r), the rule for amplitude factor is simple.  The waves are added or subtracted based on the positive or negative charge of the particle where a single proton-electron combination is one.  In this configuration, the amplitude factor is Z-e+1 where Z is the number of protons and e is the number of electrons.

Beyond the 1s orbital, it becomes more complicated as electrons have varying distances, yet they affect each other. However, the amplitude factor resembles the shape and structure of the orbitals.  There is a pattern for s subshells and p subshells, and further, the p subshell is split into two parts based on the spin of the proton 2p[1-3] (spin up) versus 2p[4-6] (spin down).  The following are the equations for determining the amplitude factors.

Amplitude Factor Equations

Amplitude Factor Equations


Neutral Atoms

Using the above amplitude factor equations, the following table was produced. The last row in the table (1s*) is a special Amplitude Factor Equation – 1s Orbital that is used when the distance is not calculated (and the Bohr radius is used instead).

Amplitude Factor Table - Neutral Elements


Ionized Atoms

Using the amplitude factor equations, ionized atoms can be determined by modifying the number of protons (Z) and electrons (e).  These tabes for all of the ionized atoms are not inserted here, but they can be found in the spreadsheet with all of the calculations from this web site.



Example Calculations

Orbital Distances – 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.


Example#1) Hydrogen

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


Example #2) Helium

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



Orbital Distances – 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 section on orbital distances where each orbital was calculated for hydrogen to calcium, including ionized versions of these elements.


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.