Energies of Helium Isoelectronic Series Using Perimetric Coordinates
Energies of Helium Isoelectronic Series Using Perimetric Coordinates
The most accurate computations on the ground state of the helium atom and its isoelectronic series followed from the work of Pekeris [1]. For an Sstate, the wavefunction depends on just three coordinates, say , and , which can be represented as the sides of a planar triangle. The perimetric coordinates , , have the advantage that they automatically satisfy the triangle inequalities and each independently varies from 0 to ∞. Pekeris's original computation made use of an expansion in perimetric coordinates containing 1058 terms, leading to the essentially exact nonrelativistic groundstate energy =2.903724375 hartrees. In this Demonstration, we introduce the use of perimetric coordinates in computations on the twoelectron isoelectronic series , , , …, , corresponding to in the Hamiltonian
r
1
r
2
r
12
u=++
r
1
r
2
r
12
v=+
r
1
r
2
r
12
w=+
r
1
r
2
r
12
E
0

H
He
+
Li
8+
Ne
Z=1,2,…,10
H=++
1
2
2
∇
1
2
∇
2
Z
r
1
Z
r
2
1
r
12
The wavefunctions considered by Pekeris were expansions in the form
ψ(u,v,w)=(u)(u)(u)
ξuηvξw
e
∑
nmk
f
n
f
m
f
k
We consider a much more modest version with
ψ(u,v,w)=(1+γ(u+v))
αZ(u+v)βZw
e
where , , can be chosen such as to minimize the variational integral
α
β
γ
ℰ=∫ψHψdτ∫dτ
2
ψ
The optimized results can be obtained directly by checking "show optimized values."
The corresponding ionization energies of each atom is also shown, given by , where is the groundstate energy of the singleelectron ion.
IP=(ℰ2)
2
Z
/2
2
Z
The atomic energies and ionization energies are represented on bar graphs, with the exact nonrelativistic values written in for reference. The variationally determined energy must, of necessity, be higher (less negative) than the exact value.