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
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- Fall '19
- dr. ahmed