Solid harm prefact

[arthur-lin1027]

The inner product obtaining our expansion coefficients erroneously contained the solid harmonic prefactor. I added code to divide through by the prefactor. This causes the expansion coefficients to actually converge to the underlying atomic gaussian, which is necessary.

I currently cleaning up an example notebook that shows that the expansion of an isotropic gaussian does indeed converge.

However, an anisotropic gaussian does not yet converge... I have some suspicions.

[curiosity54]

hey, i had a rather quick look, but was the purpose here to modify the density expansion coefficients? I am afraid that adding the additional 1/sqrt{2l+1} affects how higher order features are constructed down the line could you perhaps give me an overview of how the density expansion is now normalized?

[arthur-lin1027]

hey, i had a rather quick look, but was the purpose here to modify the density expansion coefficients? I am afraid that adding the additional 1/sqrt{2l+1} affects how higher order features are constructed down the line could you perhaps give me an overview of how the density expansion is now normalized?

Hi @curiosity54 ,

I made these changes under the assumption that a linear combination of expansion coefficients and basis functions should recreate the underlying gaussian. I identified that this 1/sqrt(2l+1) factor was erroneously included and needed to be divided out in order to recreate the underlying gaussian.

[arthur-lin1027]

To reiterate one more time, I'm trying to unify equations 15 and 16 within the AniSOAP paper, but right below equation 16, we make an incorrect statement saying that

with $R_l^m(r) = r^l * Y_l^m(\hat{r})$

This is incorrect, because actually $R_l^m(r) = \sqrt{\frac{4\pi}{2l+1}}r^l Y_l^m(\hat{r})$. Hence I need to divide out this factor, because in the code, we are calculating $R_l^m$, but we actually want $r^l Y_l^m(\hat{r})$ (eqn 15 in the paper)

[curiosity54]

okay, thank you! i was a little unsure of the prefactor being there in the definition of real-valued spherical harmonics but all good if this makes the anisoap and isotropic gaussian expansion consistent. (ps. incidentally, this made me realize there's a typo in Eq 17, l' must be equal to l)

[rosecers]

That’s fine, only a suggestion

On Apr 30, 2024, at 6:10 PM, arthur-lin1027 @.***> wrote:

@arthur-lin1027 commented on this pull request.

In anisoap/representations/ellipsoidal_density_projection.py https://github.com/cersonsky-lab/AniSOAP/pull/38#discussion_r1585643426:

• solid_harm_prefact = np.sqrt((4 np.pi) / (np.arange(lmax + 1) 2 + 1))
• scaled_sph_to_cart = []
• for l in range(lmax + 1):
• scaled_sph_to_cart.append(sph_to_cart[l] / solid_harm_prefact[l]) cannot actually make this change, because sph_to_cart is a list of 5-dimensional arrays that change depending on l. np.divide cannot understand how to unify the dimensions so I just use a for loop, plus it's more explicit here.

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