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 == Electron energy levels == 

 The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. This not only includes one-electron systems such as the hydrogen atom, singly-ionized helium, doubly ionized lithium, but it includes positronium and Rydberg states of any atom where one electron is far away from everything else. It can be used for K-line X-ray transition calculations if other assumptions are added (see Moseley's law below). In high energy physics, 
 it can be used to calculate the masses of heavy quark mesons. 

 To calculate the orbits requires two assumptions: 

 1. Classical mechanics 
 :The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force. 
 :: 
 :where me is the mass, e is the charge of the electron and ke is Coulomb's constant. This determines the speed at any radius: 
 ::  

 : It also determines the total energy at any radius: 
 ::  

 :The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the potential energy, which is true for non circular orbits too by the virial theorem. 

 :For larger nuclei, the only change is that kee2 is everywhere replaced by Zkee2 where Z is the number of protons. For positronium, me is replaced by its reduced mass (). 

 2. Quantum rule 

 :The angular momentum  is an integer multiple of ħ: 
 :: 

 :Substituting the expression for the velocity gives an equation for r in terms of n: 
 :: 
 :so that the allowed orbit radius at any n is: 
 :: 

 :The smallest possible value of r is called the Bohr radius and is equal to: 
 ::  

 :The energy of the n-th level is determined by the radius: 
 :: 

 An electron in the lowest energy level of hydrogen () therefore has 13.6 eV less energy than a motionless electron infinitely far from the nucleus.  The next energy level ()  is −3.4 eV.  The third (n = 3) is −1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom. 

 The combination of natural constants in the energy formula is called the Rydberg energy (RE): 
 : 

 This expression is clarified by interpreting it in combinations which form more natural units: 
 :  is the rest mass energy of the electron (511 keV/c) 
 :  is the fine structure constant 
 :  

 Since this derivation is with the assumption that the nucleus is orbited by one electron, we can generalize this result by letting the nucleus have a charge q = Ze where Z is the atomic number. This will now give us energy levels for hydrogenic atoms, which can serve as a rough order-of-magnitude approximation of the actual energy levels. So, for nuclei with Z protons, the energy levels are (to a rough approximation): 
 : 

 The actual energy levels cannot be solved analytically for more than one electron (see n-body problem) because the electrons are not only affected by the nucleus but also interact with each other via the Coulomb Force. However, the analytic soltuion can be approximated using the Hartee-Fock method, which involves replacing ""Z"" with ""Z - b"" where b is constant representing [[elect