Why Isn't the Ground State Electronic Structure of the Lithium Atom 1s3?
The purpose of this tutorial is to point out that if all that mattered in the determination of atomic structure was
energy minimization, the electronic structure of lithium would be 1s3, rather than 1s22s1.
To deal with this issue we choose the following scaled hydrogenic orbitals for the lithium atom's electrons:
Ψ1s α3
πexp α− r⋅
()
⋅=Ψ2s α3
32 π⋅ 2αr⋅−
()
⋅exp α− r⋅
2
⎛
⎜
⎝
⎞
⎟
⎠
⋅=
Using this basis set we find the following expressions (in terms of the variational parameter α) for the
expectation values for the various contributions to the electronic energy of the lithium atom.
Nuclear charge: Z3:=
Kinetic energy integrals: T1s α
()
α2
2
:= T2s α
()
α2
8
:=
VN1s α
()
Z−α⋅:= VN2s α
()
Z
4
−α⋅:=
Electron-nucleus potential energy integrals:
Electron-electron potential energy integrals: V1s1s α
()
5
8α⋅:= V1s2s α
()
17
81 α⋅:=
We now calculate the ground-state energy of lithium assuming it has the 1s22s1 electronic configuration. The
total electronic energy consists of nine contributions: three kinetic energy terms, three electron-nucleus
potential energy terms, and three electron-electron potential energy contriubtions.
ELi α
()
2T
1s α
()
⋅2V
N1s α
()
⋅+ V1s1s α
()
+T2s α
()
+VN2s α
()
+2V
1s2s α
()
⋅+:=
Minimization of the energy with respect to the variational parameter, α, yields the following result:
ααELi α
()
d
d0=solve α,
float 4,2.536→:= ELi α
()
7.2333−=
Compared to the experimental ground-state energy -7.478 Eh (the negative of the successive ionization
energies of the lithium atom) this result is in error by 3.3%. This result is satisfactory, indicating that the
theoretical model has some merit. We could do better, of course, but it would cost something in terms of
computational effort and simplicity of the model.
Now we calculate the energy of the hypothetical 1s3 electronic configuration for lithium using the same basis
functions. Again, the total electronic energy consists of nine contributions: three kinetic energy terms,
three electron-nucleus potential energy terms, and three electron-electron potential energy
contriubtions.
ELi α
()
3T
1s α
()
⋅3V
N1s α
()
⋅+ 3V
1s1s α
()
⋅+:=
