Finite element calculations for the helium atom∗ 1 Introduction

wave function as: ψ ± l = �l k=0 u k±
l (r1, r2, θ)D k±
l . Substitute ψ ± l into (2.5). Since D k±
l
independent solutions of (2.6), in view of (2.9), we have
L(u k±
l ) +
� l(l + 1) + k 2 (ctg 2 θ − 1)
r 2 1
+ k2
r2 2 sin 2 �
u
θ
k±
l
− Cl,k−1(k − 1)(1 − δk0)ctgθ
r2 u
1
(k−1)±
l
+ Cl,k−1(1 − δk0)(1 + δk1)
r 2 1
∂u (k−1)±
l
∂θ
+ 2(k + 1)2 (1 − δkl)ctgθ
r2 u
1
(k+1)±
l
+ 2(k + 1)(1 − δkl)
r 2 1
∂u (k+1)±
l
∂θ
= Eu k±
l , (2.10)
where δij = 1, if i = j; δij = 0, if i �= j.
Since D 0−
l = 0, we set u0− l = 0. For any given l ≥ 0, (2.10) are two independent
systems of equations. The unknown functions of one are u 0+
l , u 1+
l , · · · , u l+
l satisfying l + 1
equations(the index k varies from 0 to l in (2.10)). Those of another are u 1−
l , u 2−
l , · · · , u l−
l
satisfying l equations(k = 1, 2, · · · , l). They are used for different stationary state problems.
3 Weak forms of energy equations
By the discussion in section 2, the magnetic number m = 0 means that the z-component
Mz of the angular momentum M is zero, so ∂ψ/∂φ ′ = 0 by (2.7). If l = 0 (S-state), (2.6)
has only constant solutions, so V +
0 consists of all square integrable functions depending
on r1, r2, θ. If l = 1 (P -state), all independent solutions of (2.6) are
6
are
cos θ ′ , sin θ ′ cos φ, sin θ ′ sin φ, (3.1)
so V +
1 = span{cos θ ′ , sin θ ′ cos φ} and V −
1 = span{sin θ ′ sin φ}. If l = 2 (D-state), all
independent solutions of (2.6) are
so V +
2
3 cos 2 θ ′ − 1, sin 2 θ ′ sin 2φ, sin 2 θ ′ cos 2φ, sin 2θ ′ sin φ, sin 2θ ′ cos φ, (3.2)
= span{sin2 θ ′ sin 2φ,
sin 2θ ′ cos φ}. Since each of the five spaces is invariant under the Hamiltonian H, any
eigenfunction ψp of P states must be in V +
1 or V −
1 . Similar results hold for S and D
= span{3 cos 2 θ ′ − 1, sin 2θ ′ cos φ, sin 2 θ ′ cos 2φ} and V −
2
states. Now we show that the wave functions ψp and ψd of 1s2p and 1s3d states belong
to V +
1 and V +
2 respectively.
If we assume one electron is in a s-state and neglect the interaction between two
electrons, the angular part of any wave function is a linear combination of the following
functions:
const. (S states), cos θ1, cos θ2 (P states), 3 cos2 θ1 − 1
,
2
3 cos2 θ2 − 1
2
(D states). (3.3)