Conventions

Spherical Coordinates

Spherical coordinates \((r, \theta, \phi)\) are used extensively throughout the molecular, atomic, and angular grid modules. The radius \(r \in [0, \infty)\), azimuthal \(\theta \in [-\pi, \pi]\) and polar \(\phi \in [0, \pi)\) angles are defined from Cartesian coordinates as:

\[\begin{split} \begin{align} r &= \sqrt{x^2 + y^2 + z^2}\\ \theta &= \text{arctan2} \bigg(\frac{y}{x}\bigg)\\ \phi &= \arccos \bigg(\frac{z}{r}\bigg) \end{align} \end{split}\]

such that when the radius is zero \(r=0\), the angles are zero \(\theta, \phi = 0\).

Grid offers a utility function to convert to spherical coordinates:

import numpy as np
from grid.utils import convert_cart_to_sph

# Generate random set of points
cart_pts = np.random.uniform((-100, 100), size=(1000, 3))
# Convert to spherical coordinates
spher_pts = convert_cart_to_sph(cart_pts)
# Convert to spherical coordinates, centered at [1, 1, 1]
spher_pts = convert_cart_to_sph(cart_pts, center=np.array([1.0, 1.0, 1.0]))

The atomic grid class can also convert its points to spherical coordinates:

from grid.atomgrid import AtomGrid

atom_grid = AtomGrid(...)
spher_pts = atom_grid.convert_cartesian_to_spherical()

With the convention that when \(r=0\), the angles within \(\{(0, \theta_j, \phi_j)\}\) in that shell are obtained from an angular grid \(\{(\theta_j, \phi_j)\}\) with some degree.

Spherical Harmonics

The real spherical harmonics are defined using complex spherical harmonics:

\[\begin{split} Y_{lm}(\theta, \phi) = \begin{cases} i(Y^m_l(\theta, \phi) - (-1)^m Y_l^{-m}(\theta, \phi)) & \text{if } m < 0 \\ Y_l^0 & \text{if } m = 0 \\ (Y^{-|m|}_{l}(\theta, \phi) + (-1)^m Y_l^m(\theta, \phi)) & \text{if } m > 0 \end{cases}, \end{split}\]

where the degree \(l \in \mathbb{N}\), order \(m \in \{-l, \cdots, l\}\) and \(Y^m_l\) is the complex spherical harmonic. The Condon-Shortley phase is not included.

Alternatively, using the associated Legendre polynomials \(P_l^m\) (without the Condon-Shortley phase):

\[\begin{split} Y_{lm}(\theta, \phi) = \begin{cases} \sqrt{\frac{(2l + 1)(l - m)!}{4\pi(l + m)!}} \sqrt{2} \cos(m\theta) P_l^m(\cos(\phi)) & \text{if } m < 0 \\ \frac{1}{2}\sqrt{\frac{1}{\pi}} & \text{if } m = 0 \\ \sqrt{\frac{(2l + 1)(l - m)!}{4\pi(l + m)!}} \sqrt{2} \sin(m\theta) P_l^m(\cos(\phi)) & \text{if } m > 0 \end{cases} \end{split}\]

Grid offers the generate_real_spherical_harmonics function:

from grid.utils import generate_real_spherical_harmonics

spher_pts = np.array(...)
theta = spher_pts[:, 1]
phi = spher_pts[:, 2]
# Generate all degrees up to l=2
spherical_harmonics = generate_real_spherical_harmonics(2, theta, phi)

Ordering

The spherical harmonics are ordered by degree \(l\) in ascending order. For each degree \(l\), the orders \(m\) are in HORTON2 order:

\[m = [0, 1, -1, 2, -2, \cdots, l, -l].\]

Angular Grids

The angular grids integrate functions over the unit sphere. The quadrature weights are chosen so that the spherical harmonics are orthonormal:

\[ \int_{-\pi}^{\pi} \int_0^{\pi} Y_{l_1}^{m_1} Y_{l_2}^{m_2} \sin(\phi)\, d\theta\, d\phi = \delta_{l_1, l_2} \delta_{m_1, m_2} \]

The weights also sum to \(4\pi\).

Nested Grids

Angular grids of different degrees can be very close to one another. The following shows the mean with standard deviation and maximum distance between an angular grid of one degree and the next higher degree. The * indicates Lebedev-Laikov grids with negative weights.

Table 1 Lebedev-Laikov Grids

Degree	Number Points	Mean (std) Distance	Max Distance
3	6	0.0(0.0)	0.0
5	18	0.0(0.0)	0.0
7	26	0.14(0.15)	0.31
9	38	0.19(0.15)	0.31
11	50	0.15(0.15)	0.31
13*	74	0.11(0.082)	0.2
15	86	0.052(0.042)	0.12
17	110	0.065(0.047)	0.14
19	146	0.048(0.048)	0.14
21	170	0.035(0.027)	0.089
23	194	0.061(0.047)	0.16
25*	230	0.058(0.036)	0.12
27*	266	0.045(0.035)	0.13
29	302	0.055(0.036)	0.14
31	350	0.042(0.028)	0.1
35	434	0.054(0.021)	0.092
41	590	0.048(0.019)	0.082
47	770	0.043(0.017)	0.074
53	974	0.038(0.015)	0.065
59	1202	0.035(0.014)	0.06
65	1454	0.032(0.012)	0.054
71	1730	0.03(0.011)	0.05
77	2030	0.027(0.011)	0.047
83	2354	0.026(0.0099)	0.043
89	2702	0.024(0.0093)	0.041
95	3074	0.023(0.0088)	0.039
101	3470	0.021(0.0083)	0.036
107	3890	0.02(0.0078)	0.035
113	4334	0.019(0.0075)	0.033
119	4802	0.018(0.007)	0.031
125	5294	0.018(0.0066)	0.03

Table 2 Symmetric Spherical t-Design Grids

Degree	Number Points	Mean (std) Distance	Max Distance
1	2	0.0(0.0)	0.0
3	6	0.34(0.24)	0.55
5	12	0.22(0.13)	0.35
7	32	0.16(0.076)	0.26
9	48	0.14(0.071)	0.26
11	70	0.14(0.045)	0.23
13	94	0.12(0.047)	0.2
15	120	0.11(0.041)	0.19
17	156	0.088(0.038)	0.16
19	192	0.078(0.036)	0.14
21	234	0.077(0.033)	0.14
23	278	0.075(0.027)	0.13
25	328	0.066(0.029)	0.11
27	380	0.061(0.023)	0.11
29	438	0.057(0.024)	0.1
31	498	0.06(0.021)	0.1
33	564	0.047(0.02)	0.094
35	632	0.05(0.019)	0.085
37	706	0.048(0.018)	0.085
39	782	0.039(0.017)	0.074
41	864	0.045(0.017)	0.079
43	948	0.043(0.015)	0.074
45	1038	0.041(0.015)	0.071
47	1130	0.037(0.014)	0.068
49	1228	0.033(0.014)	0.064
51	1328	0.034(0.013)	0.062
53	1434	0.034(0.013)	0.061
55	1542	0.03(0.012)	0.054
57	1656	0.034(0.011)	0.057
59	1772	0.03(0.011)	0.054
61	1894	0.025(0.011)	0.05
63	2018	0.029(0.011)	0.052
65	2148	0.029(0.011)	0.052
67	2280	0.025(0.01)	0.05
69	2418	0.027(0.0095)	0.048
71	2558	0.027(0.0093)	0.048
73	2704	0.022(0.0095)	0.045
75	2852	0.024(0.0091)	0.044
77	3006	0.02(0.0085)	0.041
79	3162	0.023(0.0086)	0.041
81	3324	0.023(0.0084)	0.042
83	3488	0.02(0.0081)	0.04
85	3658	0.022(0.0083)	0.04
87	3830	0.022(0.0081)	0.039
89	4008	0.018(0.0076)	0.036
91	4188	0.02(0.0073)	0.036
93	4374	0.018(0.0075)	0.036
95	4562	0.02(0.0074)	0.036
97	4756	0.019(0.007)	0.034
99	4952	0.016(0.0067)	0.031
101	5154	0.018(0.0067)	0.034
103	5358	0.016(0.0064)	0.031
105	5568	0.018(0.0064)	0.031
107	5780	0.018(0.0063)	0.032
109	5998	0.015(0.0062)	0.03
111	6218	0.017(0.0062)	0.03
113	6444	0.017(0.0061)	0.029
115	6672	0.014(0.006)	0.027
117	6906	0.016(0.006)	0.029
119	7142	0.016(0.0056)	0.029
121	7384	0.016(0.0056)	0.028
123	7628	0.015(0.0056)	0.027
125	7878	0.013(0.0054)	0.027
127	8130	0.015(0.0056)	0.027
129	8388	0.014(0.0053)	0.027
131	8648	0.012(0.0051)	0.025
133	8914	0.014(0.0054)	0.025
135	9182	0.012(0.005)	0.025
137	9456	0.014(0.0052)	0.025
139	9732	0.014(0.0048)	0.024
141	10014	0.013(0.005)	0.025
143	10298	0.011(0.0047)	0.023
145	10588	0.013(0.0049)	0.023
147	10880	0.01(0.0047)	0.022
149	11178	0.013(0.0047)	0.023
151	11478	0.013(0.0045)	0.023
153	11784	0.01(0.0044)	0.022
155	12092	0.012(0.0045)	0.022
157	12406	0.01(0.0043)	0.021
159	12722	0.012(0.0044)	0.022
161	13044	0.012(0.0044)	0.021
163	13368	0.0096(0.0042)	0.02
165	13698	0.011(0.0043)	0.021
167	14030	0.011(0.0042)	0.02
169	14368	0.0092(0.004)	0.02
171	14708	0.011(0.004)	0.02
173	15054	0.011(0.004)	0.02
175	15402	0.0086(0.0039)	0.02
177	15756	0.01(0.004)	0.019
179	16112	0.0085(0.0039)	0.018
181	16474	0.01(0.0038)	0.019
183	16838	0.011(0.0037)	0.019
185	17208	0.01(0.0038)	0.018
187	17580	0.0087(0.0036)	0.018
189	17958	0.0081(0.0036)	0.017
191	18338	0.0098(0.0036)	0.018
193	18724	0.0099(0.0036)	0.018
195	19112	0.0098(0.0035)	0.018
197	19506	0.0078(0.0034)	0.017
199	19902	0.0076(0.0034)	0.017
201	20304	0.0095(0.0034)	0.017
203	20708	0.0094(0.0033)	0.017
205	21118	0.0075(0.0033)	0.016
207	21530	0.0091(0.0033)	0.017
209	21948	0.009(0.0033)	0.016
211	22368	0.0089(0.0033)	0.016
213	22794	0.0072(0.0032)	0.016
215	23222	0.009(0.0032)	0.016
217	23656	0.0071(0.0031)	0.016
219	24092	0.0087(0.0032)	0.016
221	24534	0.0069(0.0031)	0.015
223	24978	0.0086(0.0031)	0.016
225	25428	0.0084(0.003)	0.015
227	25880	0.0066(0.003)	0.015
229	26338	0.0083(0.003)	0.015
231	26798	0.0082(0.003)	0.015
233	27264	0.0065(0.003)	0.015
235	27732	0.0081(0.003)	0.015
237	28206	0.0064(0.0028)	0.015
239	28682	0.0079(0.0029)	0.015
241	29164	0.0062(0.0028)	0.014
243	29648	0.006(0.0028)	0.014
245	30138	0.0078(0.0028)	0.014
247	30630	0.0077(0.0028)	0.014
249	31128	0.006(0.0027)	0.013
251	31628	0.0076(0.0028)	0.014
253	32134	0.0076(0.0027)	0.014
255	32642	0.0075(0.0027)	0.014
257	33156	0.0059(0.0026)	0.013
259	33672	0.0074(0.0026)	0.013
261	34194	0.0058(0.0026)	0.013
263	34718	0.0071(0.0027)	0.013
265	35248	0.0055(0.0025)	0.013
267	35780	0.0071(0.0026)	0.013
269	36318	0.0071(0.0026)	0.013
271	36858	0.0054(0.0024)	0.012
273	37404	0.007(0.0025)	0.013
275	37952	0.007(0.0025)	0.012
277	38506	0.0054(0.0024)	0.012
279	39062	0.0068(0.0025)	0.012
281	39624	0.0052(0.0024)	0.012
283	40188	0.0068(0.0025)	0.012
285	40758	0.0068(0.0024)	0.012
287	41330	0.0066(0.0025)	0.012
289	41908	0.0051(0.0023)	0.011
291	42488	0.0065(0.0024)	0.012
293	43074	0.0051(0.0023)	0.011
295	43662	0.0065(0.0023)	0.012
297	44256	0.005(0.0023)	0.011
299	44852	0.0064(0.0023)	0.011
301	45454	0.0064(0.0023)	0.011
303	46058	0.0063(0.0023)	0.011
305	46668	0.0049(0.0023)	0.011
307	47280	0.0062(0.0023)	0.011
309	47898	0.0047(0.0022)	0.011
311	48518	0.0061(0.0022)	0.011
313	49144	0.0047(0.0022)	0.011
315	49772	0.0061(0.0022)	0.011
317	50406	0.0059(0.0022)	0.011
319	51042	0.0059(0.0022)	0.011
321	51684	0.0046(0.0021)	0.011
323	52328	0.0047(0.0021)	0.01