Atomic Structure Theory

82 3 Self-Consistent Fields
I(nala) =
∞
0
dr
1
2
dPnala
dr
2
+ la(la +1)
2r2 P 2 Z
nala −
r P 2
nala . (3.52)
We will need this term later in this section.
Next, we examine the direct Coulomb matrix element gabab. To evaluate
this quantity, we make use of the decomposition of 1/r12 given in (3.15).
Further, we use the well-known identity
Pl(cos θ) =
l
(−1) m C l −m(ˆr1)C l m(ˆr2) , (3.53)
m=−l
to express the Legendre polynomial of cos θ, whereθis the angle between the
two vectors r1 and r2, in terms of the angular coordinates of the two vectors
in an arbitrary coordinate system. Here, as in Chapter 1, the quantities Cl m(ˆr)
are tensor operators, defined in terms of spherical harmonics by:
C l
4π
m(ˆr) =
2l +1 Ylm(ˆr) .
With the aid of the above decomposition, we find:
gabab =
∞ l
(−1) m
l=0 m=−l
∞
0
dr2P 2 nblb (r2)
∞
0
dr1P 2 nala (r1)
l r< r l+1
>
dΩ1Y ∗
lama (ˆr1)C l −m(ˆr1)Ylama (ˆr1)
dΩ2Y ∗
lbmb (ˆr2)C l m(ˆr2)Ylbmb (ˆr2). (3.54)
The angular integrals can be expressed in terms of reduced matrix elements
of the tensor operator C l m using the Wigner-Eckart theorem. We find
gabab =
∞
l=0
lama
✻
− l0
lama
lbmb
✻
− l0
lbmb
〈la||C l ||la〉〈lb||C l ||lb〉 Rl(nala,nblb,nala,nblb) ,
(3.55)
where
∞
∞
l r< Rl(a, b, c, d) = dr1Pa(r1)Pc(r1) dr2Pb(r2)Pd(r2)
0
0
r l+1
. (3.56)
>
These integrals of products of four radial orbitals are called Slater integrals.
The Slater integrals can be written in terms of multipole potentials. We define
the potentials vl(a, b, r) by
∞
l r< vl(a, b, r1) = dr2Pa(r2)Pb(r2)
0
r l+1
. (3.57)
>