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
• Cs     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, Cs, 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
• Cs    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 Cs. 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.

Ionization: Helium to Neon…

Ionization energies (observed) for the 2nd row of the periodic table starting with Helium and ending with Neon follows the following pattern (energy in kJ/mol with 1 Hartree = 2625 kJ/mol):

RealQM gives the following energies (with shell configuration given and list values corresponding to the above graph):

• He 2             (code)  = -2.90       (list -2.903)
• He+ 1           (code)  = -2.00      (list -2.00)
• Li 2+1          (code)  = -7.47      (list -7.48)
• Li+ 2            (code)  = -7.17      (list -7.28)
• Be 2+2          (code) = -14.6       (list -14.5)
• Be+  2+1       (code) = -14.0       (list -14.2)
• B  2+2+1       (code) =  -24.4        (list -24.5)
• B+  2+2         (code) =  -24.2      (list -24.2)
• C 2+4            (code)  = -37.7      (list -37.7)
• C 2+2+2        (code)  = -37.8       (list -37.7)
• C+ 2+2+1     (code)  = -37.3       (list -37.3)
• O 2+4+2       (code)  = -74.9       (list -74.8)
• O+ 2+4+1     (code) = -74.4        (list -74.3)
• N 2+2+3       (code) = -54.4       (list -54.4)
• N+ 2+4         (code)  = -53.7      (list -53.9)
• F 2+4+3        (code) = -99.6       (list -99.4)
• F+ 2+4+2      (code) = -98.8       (list -98.7)
• Ne 2+4+4      (code) = -128.6     (list -128.5)
• Ne+ 2+4+3    (code) = -127.5     (list -127.7)
• Na 2+4+4+1  (code) = -162     (list -162)

Fairly good agreement on 50^3 mesh with list values in the graph capturing the big ionization energies for He and Ne and steady increase from Li with kernel charge/pull, modulo the kinks Be-B and N-O. In RealQM terms the decrease Be-B can be connected to the 1 electron valence of B compared to 2 for Be, and the 2 electron valence of O compared to the 3 electron valence of N. Continue on next row starting with Na on 100^3 mesh…

Ionization: Carbon1+ and Carbon2+

Let us now test RealQM on ionization where one or more electrons are ripped off an atom at some energy expense, starting with the example of Carbon with 6 electrons. 

We have seen that a shell configuration of 2+4 (2 electrons in 1st shell and 4 electrons in 2nd shell) gives a total energy of -37.7 Hartree in correspondence with list value (code). This configuration matches CH4 (code)

On the other hand, the configuration 2+2+2 has about the same energy (code), while the electrons in the 3rd shell are less tightly bound to the kernel than the ones in the 2nd shell of the 2+4 configuration, and so would require less energy to be ripped off. 

We thus let RealQM compute the energy of Carbon1+ with one electron removed in a 2+2+1 configuration to get a total energy of -37.3 Hartree in agreement with list value (code), with thus an ionization energy of 0.4 Hartree. 

We continue with Carbon2+ with configuration 2+2 and get -36.4 with ionization energy 0.9 again in agreement with list value (code).  

Note that to compute ionization energy by subtracting total energies of atom and ion, requires three correct decimal places and so is a bit delicate, because of the required mesh size cut-off of the singular kernel potential. 

We see that both 2+4 and 2+2+2 configurations for Carbon have about the same total energy and so may both be possible, while as concerns ionization the 2+2+2 configuration in agreement with list value seems to preferred, because the ionization to 2+3 requires much bigger energy (code).

Next objective is to test if RealQM can make sense of the following pattern for 1st ionisation energies: