vp.special.boys

The Boys functions F_0(T) .. F_nmax(T), as an array of nmax + 1 values, with F_n(T) = int_0^1 t^(2n) exp(-T t^2) dt. Orders 0 to 16, T >= 0.

Part of the distribution vipyr; it runs from a checkout of the repository. Planned on PyPI as pip install vipyr.

Signature

vp.special.boys(n_max, T)

Parameters

n_max : int, required

Range: >= 0, <= 16.

T : float, required

Range: >= 0.

Returns

A frozen BoysResult. Iterating it gives the fields below in order; it also has params, tests, log and check().

F : ndarray[float64]

Shape n_max + 1.

Raises

vipyr.ParameterError

A ValueError: an argument outside its range, choices, rank or shape, or one of a group given without the rest; its .param names the argument.

TypeError

An argument of the wrong type, or an unknown keyword.

vipyr.VivBuildError

An ImportError: no C compiler, or the program's C does not build.

Examples

import vipyr as vp
r = vp.special.boys(n_max=8, T=2.5)
print(r.F)
assert r.check()

[0.54629197 0.09284139 0.03928784 0.02287084 0.01560217 0.01166691
 0.0092502  0.00763353 0.00648359]

Each example ran as written when these pages were generated, and the output shown is what it printed.

Checked against

Reference

mpmath at 50 digits: F_n(T) = T^-(n+1/2) gammainc(n+1/2, 0, T) / 2, and F_n(0) = 1/(2n+1)

Compared by

relative error, worst over every order

Tolerance

rel 5 × 10⁻¹⁵

Why that tolerance

lib/hf/boys measures 1.7 × 10⁻¹⁵ against its own mpmath table on [0, 200]; 5 × 10⁻¹⁵ allows for arguments off that table

Credits

• Checked against PySCF 2.5.0 (Apache-2.0), which runs as an external oracle; no PySCF code is copied. The Gaussian basis sets come from the Basis Set Exchange.

Source

• Manifest: kernels/k1/k1.export.toml
• Viv: kernels/k1/special.viv
• Viv: lib/hf/boys.viv
• The defs it reaches and the calls between them: map of the code