Computational Physics - Department of Physics

14.5 Variational Monte Carlo for atoms 469

1

R

∂

∂r

(r^2

∂R

∂r

)+

2 mrke^2

̄h^2

+

2 mr^2

̄h^2

E=Cr, (14.14)

whereCrandCφare constants. The angle-dependent differential equations result in the so-
called spherical harmonic functions as solutions, with quantum numberslandml. These
functions are given by

Ylml(θ,φ) =P(θ)F(φ) =

√

( 2 l+ 1 )(l−ml)!

4 π(l+ml)!

Plml(cos(θ))exp(imlφ),

withPlmlbeing the associated Legendre polynomials. They can be rewritten as

Ylml(θ,φ) =sin|ml|(θ)×(polynom(cosθ))exp(imlφ),

with the following selected examples

Y 00 =

√

1

4 π,

forl=ml= 0 ,

Y 10 =

√

3

4 π

cos(θ),

forl= 1 ogml= 0 ,

Y 1 ± 1 =∓ 1

√

3

8 π

sin(θ)exp(±iφ),

forl= 1 ogml=± 1 , and

Y 20 =

√

5

16 π

(3 cos^2 (θ)− 1 )

forl= 2 ogml= 0. The quantum numberslandml represent the orbital momentum and
projection of the orbital momentum, respectively and take the valuesl≥ 0 ,l= 0 , 1 , 2 ,...and
ml=−l,−l+ 1 ,...,l− 1 ,l. The spherical harmonics forl≤ 3 are listed in Table 14.1.

Spherical Harmonics

ml\l 0 1 2 3

+3 −^18 (^35 π)^1 /^2 sin^3 θe+^3 iφ

+2^14 (^152 π)^1 /^2 sin^2 θe+^2 iφ^14 (^1052 π)^1 /^2 cosθsin^2 θe+^2 iφ

+1 −^12 ( 23 π)^1 /^2 sinθe+iφ−^12 (^152 π)^1 /^2 cosθsinθe+iφ−^18 (^212 π)^1 /^2 (5 cos^2 θ− 1 )sinθe+iφ

(^02) π^11 / 2 12 (π^3 )^1 /^2 cosθ^14 (π^5 )^1 /^2 (3 cos^2 θ− 1 )^14 (π^7 )^1 /^2 ( 2 −5 sin^2 θ)cosθ
-1 +^12 ( 23 π)^1 /^2 sinθe−iφ+^12 (^152 π)^1 /^2 cosθsinθe−iφ+^18 (^212 π)^1 /^2 (5 cos^2 θ− 1 )sinθe−iφ
-2^14 (^152 π)^1 /^2 sin^2 θe−^2 iφ^14 (^1052 π)^1 /^2 cosθsin^2 θe−^2 iφ
-3 +^18 (^35 π)^1 /^2 sin^3 θe−^3 iφ
Table 14.1Spherical harmonicsYlmlfor the lowestlandmlvalues.
We focus now on the radial equation, which can be rewritten as