Single electron ions are any nucleus
surrounded by only one electron, examples are He+, Li2+,
Be3+, B4+, …. Energy levels of these ions are calculated
like in Hydrogen atom : \(e_n=-\frac{13.6 Z^2}{n^2}\)
Monday, October 12, 2026
Single electron ions
Hubble constant
Just a quick thought, in ideal and simplified condition, the expansion rate of universe is constant, hence Hubble constant is also constant over time. Therefore Hubble constant can't be used for estimating the age of the universe, because either we measure today or we measure 5 billion years from now, the Hubble constant still shows 1/(13.8 billion years).
Ionisation energies of Lithium atoms in ground state
I calculated the ionisation energies of Lithium atoms in ground state. Ionisation energy of an atom is the energy required to remove electron(s) from that neutral atom. Lithium has three electrons, hence it has three ionisation energies :
transition | ionisation energy (eV) | electron configuration | spectroscopy term symbol [1,2] |
Li ® Li+ | 5.39 [3,4] | 1s22s ® 1s2 | 2S1/2 ® 1S0 |
Li+ ® Li2+ | 75.64 [5] | 1s2 ® 1s | 1S0 ® 2S1/2 |
Li2+ ® Li3+ | 122.45 [6] | 1s ® none | 2S1/2 ® none |
A reminder, spectroscopy term symbol
is 2S+1LJ with S,
L, J are total spin, total angular momentum, total spin plus angular momentum.
I wanted to obtain these three numbers 5.39, 75.64, 122.45 via computational
simulation. The software package that I selected is Nwchem [7]. It
runs within Linux Ubuntu. So some minimum understanding of Linux operating
system is desirable here. And I ran it within Windows 11 operating system.
In quantum chemistry calculations, I
used density functional theory (DFT) [8] with 6-31G basis set [9] and b3lyp hybrid exchange
correlation functional [10]. Although other better methods can be
used.
Below are my steps of the calculations :
1 | installed WSL (Windows Subsystem
for Linux – by Microsoft) in Windows 11 | in Windows 11 command prompt, type wsl -- install |
2 | installed Linux Ubuntu in WSL |
|
3 | updated/upgraded the linux
operating system | sudo apt update sudo apt upgrade |
4 | installed parallel processing software
to take advantage of my multicore computer, e.g. Open MPI or MPICH (optional) | sudo apt install
openmpi-bin libopenmpi-dev or sudo apt install mpich
libmpich-dev |
5 | installed Nwchem | sudo apt install nwchem |
6 | created working directories | C:\Users\username\Documents\nwchem C:\Users\username\Documents\nwchem\perm C:\Users\username\Documents\nwchem\scratch |
7 | went into nwchem directory | cd /mnt/c/Users/username/Documents/nwchem |
8 | calculated the ground state energy
of Li (using 8 processors) electric charge = 0 electron configuration is 1s22s,
then S=1/2, multiplicity = 2S+1 = 2 | input file : Li3el.nw [11] mpirun –np 8 nwchem Li3el.nw
> Li3el.out output file : Li3el.out Total DFT energy = -7.490902307210 Hartree ELi = -7.491 ´ 27.211 = -203.838 eV |
9 | calculated the ground state energy
of Li+ (using 8 processors) electric charge = 1 electron configuration is 1s2,
then S=0, multiplicity = 2S+1 = 1 | input file : Li2el.nw [12] mpirun –np 8 nwchem Li2el.nw
> Li2el.out output file : Li2el.out Total DFT energy = -7.284534796564 Hartree ELi+ = -7.284 ´ 27.211 = -198.205 eV |
10 | calculated the ground state energy
of Li2+ (using 8 processors) electric charge = 2 electron configuration is 1s, then
S=1/2, multiplicity = 2S+1 = 2 | input file : Li1el.nw [13] mpirun –np 8 nwchem Li1el.nw
> Li1el.out output file : Li1el.out Total DFT energy = -4.446963826727 Hartree ELi2+ = -4.447 ´ 27.211 = -121.007 eV |
11 | calculated first ionisation energy | E1st = ELi+ - ELi = -198.205 + 203.838 = 5.633 eV |
12 | calculated second ionisation energy | E2nd = ELi2+ - ELi+ = -121.007 + 198.205 = 77.198 eV |
13 | calculated third ionisation energy | E3rd = 0 - ELi2+ = 121.007 eV |
[1] CN Banwell, Fundamentals of Molecular Spectroscopy, 3rd
ed., McGraw-Hill, 1983, p. 179.
[2] DJ Griffiths, Introduction
to Quantum Mechanics, Prentice Hall, 1995, p. 191.
[3] Wolfram Mathematica
ll =
SpectralLineData[EntityClass["AtomicLevel", {"Lithium",
1}]];
aa =
Length[ll] (* 182 *)
Table[{ll[[i]]["ElectronConfigurationString"],
ll[[i]]["TermSymbol"],ll[[i]]["Energy"]}, {i, 1, 182}] //
TableForm
browsed through the list, I picked
last data.
[4] BH Bransden & CJ Joachain, Physics of atoms and molecules, Longman
Scientific & Technical, 1983, p. 302.
[5] Wolfram Mathematica
ll =
SpectralLineData[EntityClass["AtomicLevel", {"Lithium",
2}]];
aa =
Length[ll] (* 109 *)
Table[{ll[[i]]["ElectronConfigurationString"],
ll[[i]]["TermSymbol"],ll[[i]]["Energy"]}, {i, 1, 109}] //
TableForm
browsed through the list, I picked
last data
[6] Wolfram Mathematica
ll =
SpectralLineData[EntityClass["AtomicLevel", {"Lithium",
3}]];
aa =
Length[ll] (* 149 *)
Table[{ll[[i]]["ElectronConfigurationString"],
ll[[i]]["TermSymbol"],ll[[i]]["Energy"]}, {i, 1, 149}] //
TableForm
browsed through the list, I picked
last data
[7] https://nwchemgit.github.io/index.html.
Among a range of selection of software packages, e.g. Gaussian (https://gaussian.com/),
Abinit (https://www.abinit.org/), VASP
(Vienna Ab-initio Simulation Package) (https://vasp.at/),
ADF (Amsterdam Density Functional) (https://www.scm.com/amsterdam-modeling-suite/adf/),
ORCA (https://www.faccts.de/orca/),
Quantum Espresso (https://www.quantum-espresso.org/),
etc.
[9] https://www.basissetexchange.org/
[10] https://mazziotti.uchicago.edu/qcdictionary/b_terms/b3lyp/
|
[11] Li3el.nw title "Li with 3
electrons" permanent_dir ./perm scratch_dir ./scratch charge 0 geometry
Li 0 0 0 end basis
* library 6-31G end dft
xc b3lyp
mult 2 end task dft |
[12] Li2el.nw title "Li with 2
electrons" permanent_dir ./perm scratch_dir ./scratch charge 1 geometry
Li 0 0 0 end basis
* library 6-31G end dft
xc b3lyp
mult 1 end task dft |
[13] Li1el.nw title "Li with 1
electrons" permanent_dir ./perm scratch_dir ./scratch charge 2 geometry
Li 0 0 0 end basis
* library 6-31G end dft
xc b3lyp
mult 2 end task dft |