pyextal.jaxMultipole.radial¶

Fourier-Bessel transforms of Slater-type radial functions.

Every radial integral the model needs is one of

\[\langle j_l \rangle_n(k; \alpha) = \frac{\alpha^{n+3}}{(n+2)!} \int_0^\infty r^{n+2} e^{-\alpha r} j_l(kr)\, dr\]

– the transform of a unit-normalised Slater density alpha^(n+3)/(n+2)! r^n e^(-alpha r). The spherical core and valence factors are sums of <j_0> over pairs of basis functions (jaxMultipole.bank.pair_table); the deformation terms are a single <j_l> each.

Stable closed form¶

The integral is a Gauss hypergeometric function,

\[\langle j_l \rangle_n = C_{n,l}\, u^l\, {}_2F_1\!\left(\tfrac{n+l+3}{2}, \tfrac{n+l+4}{2}; l+\tfrac32; -u^2\right), \qquad u = k/\alpha, \quad C_{n,l} = \frac{(n+l+2)!}{(2l+1)!!\,(n+2)!},\]

and a Pfaff transformation turns it into (1+u^2)^(-p) times a terminating polynomial in t = u^2/(1+u^2), which lies in [0, 1):

n - l even: p = (n+l+4)/2, degree (n-l)/2, c_q = ((l-n)/2)_q ((n+l+4)/2)_q / ((l+3/2)_q q!);
n - l odd: p = (n+l+3)/2, degree (n-l+1)/2, c_q = ((n+l+3)/2)_q ((l-n-1)/2)_q / ((l+3/2)_q q!).

Both need n >= l - 1. The coefficients are exact rationals, the polynomial is evaluated by Horner’s rule, and nothing cancels at small k – unlike the textbook route through m!/(alpha - ik)^(m+1) and the trigonometric expansion of j_l, which for n = l = 4 is already 41 times off at k = 0.05 in float64 and useless in float32. It is exact at k = 0 (1 for l = 0, 0 otherwise) and smooth in alpha, so derivatives in kappa are clean. For example <j_0>_2 = (1 - u^2)/(1 + u^2)^4.

Units: k and alpha must share a length unit. The banks give exponents in bohr^-1, so the model passes k = 4 pi s * BOHR.

Functions¶

`fb_coefficients`(n, l)	Exact constants of `sto_fb()`.
`sto_fb`(k, alpha, n, l)	`<j_l>_n(k; alpha)` for a unit-normalised Slater density.
`k_bohr`(s)	`4 pi s` in bohr^-1, from `s = sin(theta)/lambda` in Angstrom^-1.
`spherical_f`(k, table[, kappa])	Scattering factor of a spherical shell, one electron's worth.
`deformation_fb`(k, l, n_l, zeta_l[, kappa_p])	`<j_l>` of a deformation radial function `R_l ~ r^n_l e^(-kappa' zeta_l r)`.

Module Contents¶

pyextal.jaxMultipole.radial.fb_coefficients(n, l)¶

Exact constants of sto_fb().

Parameters:

n (int) – Power of r in the Slater density.
l (int) – Order of the Bessel function.

Returns:

(C, p, coeffs) with C the prefactor, p the integer power of 1/(1+u^2) and coeffs the polynomial in t, lowest order first, all as floats.

Return type:

tuple

Raises:

ValueError – If n < l - 1, where the integral diverges at r = 0.

pyextal.jaxMultipole.radial.sto_fb(k, alpha, n, l)¶

<j_l>_n(k; alpha) for a unit-normalised Slater density.

Parameters:

k (array-like) – Scattering vector magnitude 4 pi s, any shape.
alpha (array-like) – Slater exponent(s), broadcasting against k.
n (int) – Power of r.
l (int) – Bessel order.

Returns:

The transform, broadcast shape, dtype of k * alpha.

Return type:

jnp.ndarray

pyextal.jaxMultipole.radial.k_bohr(s)¶: 4 pi s in bohr^-1, from s = sin(theta)/lambda in Angstrom^-1.

pyextal.jaxMultipole.radial.spherical_f(k, table, kappa=1.0)¶

Scattering factor of a spherical shell, one electron’s worth.

Parameters:

k (array-like) – 4 pi s in bohr^-1.
table (dict) – N -> (w, alpha) from jaxMultipole.bank.pair_table.
kappa (float | array) – Expansion parameter; the density kappa^3 rho(kappa r) scatters as f(k/kappa).

Returns:

f(k), shape of k; zero for an empty table.

Return type:

jnp.ndarray

pyextal.jaxMultipole.radial.deformation_fb(k, l, n_l, zeta_l, kappa_p=1.0)¶

<j_l> of a deformation radial function R_l ~ r^n_l e^(-kappa' zeta_l r).

Parameters:

k (array-like) – 4 pi s in bohr^-1.
l (int) – Multipole order.
n_l (int) – Power of r; n_l >= l is required for a density regular at the nucleus (the transform itself only needs n_l >= l - 1).
zeta_l (float) – Exponent, bohr^-1.
kappa_p (float | array) – kappa’.

Returns:

Same shape as k.

Return type:

jnp.ndarray