The power spectrum is a very impoverished representation of the original
function
q
, because all
p
l
coefficients are rotationally invariant
independently
,
i.e. different
l
channels are decoupled. This representation, although rotationally
invariant, is, in turn, severely incomplete.
The incompleteness of the power spectrum can be demonstrated by the fol-
lowing example. Assuming a function
f
in the form
f
ð
^
r
Þ ¼
X
l
1
m
¼±
l
1
a
m
Y
l
1
m
ð
^
r
Þ þ
X
l
2
m
¼±
l
2
b
m
Y
l
2
m
ð
^
r
Þ
;
ð
2
:
39
Þ
its power spectrum elements are
p
l
1
¼ j
a
j
2
and
p
l
1
¼ j
b
j
2
. Thus only the
length
of the
vectors
a
and
b
are constrained by the power spectrum, their relative orientation is
lost, i.e. the information content of channels
l
1
and
l
2
becomes decoupled. Figure
2.2
shows two different angular functions,
f
1
=
Y
22
+
Y
2–2
+
Y
33
+
Y
3–3
and
f
2
=
Y
21
+
Y
2–1
+
Y
32
+
Y
3–2
that have the same power spectrum
p
2
=
2 and
p
3
=
2.
2.3.3 Bispectrum
We will now generalise the concept of the power spectrum in order to obtain a
more complete
set of invariants via
the coupling of the
different
angular
2.3
Rotationally Invariant Features
13

momentum channels [
13
]. Let us consider the direct product
c
l
1
µ
c
l
2
;
which
transforms under a rotation as
c
l
1
µ
c
l
2
!
D
ð
l
l
Þ
µ
D
ð
l
2
Þ
µ
¶
c
l
1
µ
c
l
2
ð
Þ
:
ð
2
:
40
Þ
It follows from the representation theory of groups that the direct product of two
irreducible representations can be decomposed into direct sum of irreducible
representations of the same group. In case of the SO(3) group, the direct product of
two Wigner-matrices can be decomposed into a direct sum of Wigner-matrices in
the form
D
ð
l
l
Þ
µ
D
ð
l
2
Þ
¼
C
l
1
;
l
2
± ²
y
a
l
1
þ
l
2
l
¼j
l
1
±
l
2
j
D
ð
l
Þ
"
#
C
l
1
;
l
2
;
ð
2
:
41
Þ
where
C
l
1
;
l
2
denote the Clebsch–Gordan coefficients. The matrices of Clebsch–
Gordan coefficients are themselves unitary, hence the vector
C
l
1
;
l
2
c
l
1
µ
c
l
2
ð
Þ
transforms as
C
l
1
;
l
2
c
l
1
µ
c
l
2
ð
Þ !
a
l
1
þ
l
2
l
¼j
l
1
±
l
2
j
D
ð
l
Þ
"
#
C
l
1
;
l
2
c
l
1
µ
c
l
2
ð
Þ
:
ð
2
:
42
Þ
We define
g
l
1
;
l
2
;
l
as
a
l
1
þ
l
2
l
¼j
l
1
±
l
2
j
g
l
1
;
l
2
;
l
¶
C
l
1
;
l
2
c
l
1
µ
c
l
2
ð
Þ
;
ð
2
:
43
Þ
i.e. the
g
l
1
;
l
2
;
l
is that part of the RHS which transforms under rotation as
g
l
1
;
l
2
;
l
!
D
ð
l
Þ
g
l
1
;
l
2
;
l
:
ð
2
:
44
Þ
Analogously to the power spectrum, the bispectrum components or cubic invari-
ants, can be written as
Fig. 2.2
Two different
angular functions that share
the same power spectrum
coefficients
14
2
Representation of Atomic Environments

b
l
1
;
l
2
;
l
¼
c
y
l
g
l
1
;
l
2
;
l
;
ð
2
:
45
Þ
which are invariant to rotations:
b
l
1
;
l
2
;
l
¼
c
y
l
g
l
1
;
l
2
;
l
!
c
l
D
ð
l
Þ
µ
¶
y
D
ð
l
Þ
g
l
1
;
l
2
;
l
¼
c
y
l
g
l
1
;
l
2
;
l
ð
2
:
46
Þ
Kondor showed that the bispectrum of the SO(3) space is not complete, i.e. the
bispectrum does not determine uniquely the original function. This is a deficiency
due to the fact that the unit sphere,
S
2
is a homogeneous space. However, he states

#### You've reached the end of your free preview.

Want to read all 96 pages?

- Fall '19
- dr. ahmed