Definition and use of spherical harmonics in Siesta
A shortcut:
l = 1, M :
−1
−y
0
z
1
l = 2, M :
−x
−2
xy
−1
0
−yz
3z 2 −r2
1
2
−xz
x2 −y 2
Where and how defined:
spher_harm.f
C
C
C
C
C
C
C
C
C
C
C
C
module spher_harm
...
subroutine rlylm( LMAX, R, RLY, GRLY )
...
FINDS REAL SPHERICAL HARMONICS MULTIPLIED BY R**L: RLY=R**L*YLM,
AND THEIR GRADIENTS GRLY, AT POINT R:
YLM = C * PLM( COS(THETA) ) * SIN(M*PHI)
FOR
M < 0
YLM = C * PLM( COS(THETA) ) * COS(M*PHI)
FOR
M >= 0
WITH (THETA,PHI) THE POLAR ANGLES OF R, C A POSITIVE NORMALIZATION
CONSTANT AND PLM ASSOCIATED LEGENDRE POLYNOMIALS.
THE ORDER OF THE Y’S IS THAT IMPLICIT IN THE NESTED LOOPS
DO L=0,LMAX
DO M=-L,L
WITH A UNIFIED INDEX ILM=1,...,LMAX**2 INCREASING BY ONE UNIT IN THE
INNER LOOP.
WRITTEN BY J.M.SOLER. AUG/96
Complex harmonics:
s
| lmi = Clm Plm (cos ϑ) eimϕ =
Pl (u) =
2l + 1 (l − |m|)!
d|m| Pl (cos ϑ) imϕ
· (−1)m sin|m| ϑ
e
;
4π (l + |m|)!
d cos|m| ϑ
(−1)l dl
(1 − u2 )l
2l l! dul
⇒ Legendre polynomials:
P0 = 0 ;
1
P2 = (3u2 − 1) ;
2
P1 = u ;
1
P3 = (5x3 − 3x) . . .
2
Real harmonics | lM i and their relation to complex ones | lmi:
| lM i =
| lmi =
1
1
lM̄ = √
√ [ | lmi + (−1)m | l, −mi ] ;
[ | lmi − (−1)m | l, −mi ]
2
i 2
1
1
√ [ | lM i + i| l, M̄ i ] ;
| l, −mi = √ [ | lM i − i| l, M̄ i ] .
2
2
1
1
|00i = √
4π
r
r
3
3 z
|10i =
cos ϑ →
;
4π
4π rr
i
1 h
3
| 1 1 i = √ | 1 I i + i | 1 Ī i = −
sin ϑ eiϕ
8π
2
i r 3
1 h
| 1 −1 i = √ −| 1 I i + i | 1 Ī i =
sin ϑ e−iϕ
8π
2
r
i
1 h
3
| 1 I i = √ | 1 1 i − | 1 −1 i = −
sin ϑ cos ϕ
4π
2
r
i
1 h
3
√
| 1 Ī i =
| 1 1 i + | 1 −1 i = −
sin ϑ sin ϕ
4π
i 2
r
→
→
3 x
;
4π r
r
3 y
−
;
4π r
−
5
5 3z 2 −r2
|20i =
(3 cos2 ϑ − 1) →
;
16π
16π
r2
r
i
1 h
15
| 2 1 i = √ | 2 I i + i | 2 Ī i = −
sin ϑ cos ϑ eiϕ
8π
2
i r 15
1 h
| 2 −1 i = √ −| 2 I i + i | 2 Ī i =
sin ϑ cos ϑ e−iϕ
8π
2
r
r
i
1 h
15
15 2xz
| 2 I i = √ | 2 1 i − |2 −1 i = −
sin 2ϑ cos ϕ → −
;
16π
16π
r2
2
r
r
i
15
15 2yz
1 h
| 2 Ī i = √ | 2 1 i + |2 −1 i = −
sin 2ϑ sin ϕ → −
16π
16π r2
i 2
r
i
15
1 h
| 2 2 i = √ | 2 II i + i | 2 ĪI i =
sin2 ϑ ei2ϕ
32π
2
i r 15
1 h
√
| 2 −2 i =
| 2 II i − i | 2 ĪI i =
sin2 ϑ e−i2ϕ
32π
2
r
i r 15
1 h
15 x2 − y 2
2
| 2 II i = √ | 2 2 i + |2 −2 i =
sin ϑ cos 2ϕ →
;
16π
16π r2
2
r
i r 15
i h
15 2xy
2
| 2 ĪI i = √ | 2 2 i − |2 −2 i =
sin ϑ sin 2ϕ →
16π
16π r2
2
r
r
2