RealQM: Ionization Energies for Atoms with 1 Valence Electron

We will now let RealQM compute the 1st ionization energy E in for atoms with 1 valence electron and compare with the following data from NIST Atomic Data base including electron configuration with number of electrons in a sequence of shells with increasing radii: 

• H        1                                          E = 0.5000 Hartree
• Li       2+1                                      E =  0.198
• Na      2+8+1                                  E = 0.188
• K       2+8+8+1                              E = 0.159 
• Cu     2+8+18+1                            E = 0.213
• Rb     2+8+18+8+1                        E = 0.154
• Ag     2+8+18+18+1                      E = 0.205
• Ce     2+8+18+18+8+1                  E = 0.143
• Au     2+8+18+32+18+1                E = 0.339
• Fr      2+8+18+32+18+8+1            E = 0.145 (est)

We see two remarkable features of E: 

1. Steady decrease H, Li, Na, K, Rb, Ce, Fr all with shell sequence ending with 1 or 8+1.
2. Higher energy for Cu, Ag and Au all ending with 18+1.

We now compare with RealQM recalling that electrons have non-overlapping supports meeting at a free boundary and so form a sequence of distinct spherical shells with a certain number of electrons in each shell with the electron wave functions meeting at a free boundary between shells with continuity and zero normal derivative. 

We can thus model an atom with one valence electron as a +1 kernel surrounded by one electron wave function with support outside some distance R to the kernel, with thus the void with distance smaller than R representing the net of the ion without the valence electron. We compute the following energies for different R in atomic units using this code (with print out for R = 2 below):

• R = 0            E = 0.500
• R = 0.4         E = 0.426
• R = 0.6         E = 0.368
• R = 0.8         E = 0.320
• R = 1.0         E = 0.280
• R = 1.2         E = 0.247
• R = 1.4         E = 0.219
• R = 1.6         E = 0.196
• R = 1.8         E = 0.177
• R = 2.0         E = 0.161
• R = 2.2         E = 0.149
• R = 2.4         E = 0.141

We see a steady decrease of E with increasing R in the code, which we can match to the above NIST data as follows:

• Li     R = 1.6
• Na    R = 1.8
• K      R = 2.0
• Rb    R = 2.2
• Ce    R = 2.4

This is to be expected since R increases with number of shells as long as the electron configuration ends with 8+1. 

We now turn to Cu, Ag and Au having larger E which matches to smaller R. The radius of Cu thus comes out as smaller than that of K, that of Ag smaller than Rb, and that of Au smaller than Ce. We can see this as an effect of the 18+1 ending of these atoms with electrons densely packed into an 18 shell.

Altogether RealQM matches with observation as concerns 1st ionization of atoms with one valence electron. The relative success can be attributed to the fact that in RealQM electrons only have support in one shell, to be compared with Standard Quantum Mechanics where electrons have presence in all shells.  

R=2.0 Electron in red/yellow around void in white.

I will complement with RealQM for full shell configurations giving information on relevant R for each atom.