# Conventions

## Spherical Coordinates

Spherical coordinates $$(r, \theta, \phi)$$ is used extensively throughout the molecular, atomic, and angular grid modules. Particularly, the radius $$r \in [0, \infty)$$, azumuthal $$\theta \in [-\pi, \pi]$$ 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 &= arc\cos \bigg(\frac{z}{r}\bigg) \end{align}\end{split}

such that when the radius is zero $$r=0$$, then 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 be used to convert its points to spherical coordinates

from grid.atomgrid import AtomGrid

# Construct atomic grid object
atom_grid = AtomGrid(...)

# Convert atomic grid points centered at the nucleus to spherical coordinates
spher_pts = atom_grid.convert_cartesian_to_spherical()


With the convention that when $$r=0$$, then the angles within $$\{(0, \theta_j, \phi_j)\}$$ in that shell is obtained from a 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, it can be written using the associated Legendre polynomials $$P_l^m$$ (without the Conway phase):

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

Grid offers generate_real_spherical_harmonics function to generate the real spherical harmonics:

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 first ordered by the degree $$l$$ in ascending order.

For each degree $$l$$, the orders $$m$$ are in HORTON2 order defined as:

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

## Angular Grids

The angular grids is responsible for integrating functions over the unit-sphere. The quadrature weights are specifically chosen so that the spherical harmonics are orthonormal:

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

Further, the quadrature weights are all chosen that the weights sum up 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 consequent angular grid with higher degree. The * indicates Lebedev-Laikov grids with negative weights.

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

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