Calculations – Orbital Distances

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


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 Calculations

Hydrogen and helium can be calculated manually due to one radius required to solve for 1s, described here.  Beyond helium, a solver for simultaneous equations must be used to solve for multiple distances.  This section describes the method to use a solver, such as MathCad, to solve for the remaining elements that are placed into the tables.


Lithium to Calcium

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.

The equations in Mathcad are available to download for the first 20 elements, but here, some of the solutions are annotated to describe the pattern for the equations and how the solutions are built and used to all the elements.


Lithium – Example Calculation

The equations become more complex beginning with lithium (Z=3) because it begins a new orbital (2s).  Therefore, a second equation is required to simultaneously solve the 1s and 2s orbital distances.  Each new orbital requires a new equation and appends more repulsive electrons to each equation being solved.  These explanations are annotated along with the Mathcad solution below.

lithium mathcad solution for orbital distances

Lithium: Mathcad solution of 1s and 2s orbital distances (ratio of Bohr radius)

The solution provides the 1s and 2s orbital distances as a ratio of the Bohr radius as 0.397 and 3.272 respectively.  In picometers, these distances are 21 pm and 173 pm.


Boron – Example Calculation

Boron is the next example, as it now begins the transition to the 2p orbital.  Since this is a third distance to calculate, a third equation is added and each equation expands to the right to include the effect of the electron at the 2p orbital distance.  Also, this is the first time that the electron angles for the p orbital (θp) needs to be considered.  Again, annotations below explain the changes at boron to construct the equations that yield the orbital distances.

boron mathcad solution

Boron: Mathcad solution of 1s, 2s and 2p orbital distances (ratio of Bohr radius)

The solution provides the 1s, 2s and 2p orbital distances as a ratio of the Bohr radius as 0.226, 1.643 and 1.41 respectively.  In picometers, these distances are 11.9 pm, 86.9 and 74.6 pm.


Argon – Example Calculation

The pattern continues for equations and electron angles.  Argon, with 18 protons, now includes five orbital distances for 1s, 2s, 2p, 3s and 3p.  Mathcad now simultaneously solves 5 unknowns (orbital radii) using 5 equations.  The electron angles have been color coded below because there is a distinct pattern.

 Argon orbital equations


Argon: Mathcad solution of 1s, 2s, 2p, 3s and 3p orbital distances (ratio of Bohr radius)


The complete set of Mathcad solutions for all neutral elements from hydrogen to calcium are available to download in Mathcad file format.  The ionized elements from hydrogen to calcium were also calculated and put into the tables in the downloadable spreadsheet, although their Mathcad solutions were not provided.  Ionized elements simply need to change the Z variable in the Mathcad solution to obtain the ionized element distance.  For example, Li1+ uses the configuration and Mathcad solution for He since they both have two electrons.  However, the Z value is modified to be Z=3, which is lithium.



Further information on the derivation of the equations and how to replicate them is in the Atomic Orbitals paper.