How Can the First Ionization Energy of Helium be Accurately Calculated Using Basic Quantum Mechanics?

How Can the First Ionization Energy of Helium be Accurately Calculated Using Basic Quantum Mechanics?


There is [a book which summarizes the fairly accurate calculation of the First Ionization Energy of Helium](https://www.lulu.com/shop/stephen-elliott/magnetics-and-radially-symmetric-helium-bh/ebook/product-18npv8vz.html?q=stephen%20elliott%20helium&page=1&pageSize=4), but it would be nice to see if there is a more direct answer than the one calculated, that might have better accuracy even.


The experimental ionization energies for Hydrogen and Helium are given at this link: [Experimental Values of Ionization Energies](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Ionization_Energy/Ionization_Energies) and in [Table 3 in the linked Chemistry book here](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Ionization_Energy):


Table 3: Ionization Energies (kJ/mol)





| Element | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| H | 1312 |  |  |  |  |  |  |  |
| He | 2372 | 5250 |  |  |  |  |  |  |
| Li | 520 | 7297 | 11810 |  |  |  |  |  |
| Be | 899 | 1757 | 14845 | 21000 |  |  |  |  |
| B | 800 | 2426 | 3659 | 25020 | 32820 |  |  |  |
| C | 1086 | 2352 | 4619 | 6221 | 37820 | 47260 |  |  |
| N | 1402 | 2855 | 4576 | 7473 | 9442 | 53250 | 64340 |  |
| O | 1314 | 3388 | 5296 | 7467 | 10987 | 13320 | 71320 | 84070 |
| F | 1680 | 3375 | 6045 | 8408 | 11020 | 15160 | 17860 | 92010 |
| Ne | 2080 | 3963 | 6130 | 9361 | 12180 | 15240 |  |  |
| Na | 496 | 4563 | 6913 | 9541 | 13350 | 16600 | 20113 | 25666 |
| Mg | 737 | 1450 | 7731 | 10545 | 13627 | 17995 | 21700 | 25662 |



Are these the most currently-accepted experimentally-confirmed values?


Now how can these energies be calculated using radially-symmetric basic Quantum Mechanics Theory accurately?


I have made [fairly accurate Helium ionization calculations at this link](https://www.lulu.com/shop/stephen-elliott/magnetics-and-radially-symmetric-helium-bh/ebook/product-18npv8vz.html?q=stephen%20elliott%20helium&page=1&pageSize=4). What are other accurate approaches that use just the radially-symmetric Schrödinger wave equation, with a few small simplifications?




There is no need to reach the experimental results by basic methods, the wave function corresponding to $He$ atom has no analytical expression. The numerical methods used do not often bare a significant scientific meaning. Trying to reach this experimental result is more or less a computational challenge. The expansion of the ground-state function in a minimal basis-set leads to an accurate description depending on a right choice of the basis functions for this expansion. We are more interested in the suitable basis functions for reducing the computational cost.



> 
> The current version of the book that I downloaded, my calculated ionization energy for Helium was 22.1433 eV (2136.6417 kJ/Mol ) conversion at unitsconverters.com/en/Ev/Particle-To-Kj/Mol/Utu-6180-6179 . Then the accuracy is (2372−2136.6417)/2372∗100= +9.9% accurate
> 
> 
> 


Assuming that your second ionization energy is given by the analytical formula of hydrogenic atoms $E\_2 = \frac{Z^2}{2n^2}$ a.u. , your total energy, related to the first ionization energy $E\_1=22.14$ eV will be $E\_{tot} = -(E\_1+E\_2) = -76.54$ eV. This ground state energy is not accurate at all for a numerical method. [A variational method](https://en.wikipedia.org/wiki/Helium_atom) applied considering $V\_{ee}$ as a perturbation gives an analytical solution of $E\_{tot} = - 77.7$ eV more accurate that your numerical result.


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