exatomic.algorithms.basis module

Basis Function Manipulation

Functions for managing and manipulating basis set data. Many of the ordering schemes used in computational codes can be generated programmatically with the right numerical function. This is preferred to an explicit parsing and storage of a given basis set ordering scheme.

exatomic.algorithms.basis.gen_enum_cartesian(lmax)

Cartesian powers in the order expected for overlap computation up to order lmax.

Parameters:

lmax (int) – highest order angular momentum quantum number

exatomic.algorithms.basis.spherical_harmonics(lmax)

Symbolic real spherical harmonics up to order lmax.

sh = spherical_harmonics(6)  # Inclusive, so computes up to l = 6
sh[3][-3]                    # symbolic f-phi angular function

Parameters:

lmax (int) – highest order angular momentum quantum number

exatomic.algorithms.basis.solid_harmonics(lmax, scaled=False)

Symbolic real solid harmonics up to order lmax.

sh = solid_harmonics(6)  # Inclusive, so computes up to l = 6
sh[3][-3]                # symbolic f-phi angular function

Parameters:

• lmax (int) – highest order angular momentum quantum number

• scaled (bool) – if scaled, includes factor of 1 / (2 * np.pi ** 0.5)

exatomic.algorithms.basis.car2sph(sh, cart, orderedp=True)

Cartesian to spherical transform matrices.

sh = solid_harmonics(8)        # symbolic solid harmonics
cart = gen_enum_cartesian(8)   # cartesian powers in a defined order
c2s = car2sph(sh, cart)        # dictionary of {l: transform_matrix}

Parameters:

• sh (OrderedDict) – symbolic solid harmonics

• cart (OrderedDict) – cartesian powers in a defined order

• orderedp (bool) – order l=1 as [‘x’, ‘y’, ‘z’], not [-1, 0, 1] (default True)

Returns:

c2s (OrderedDict) – cartesian to spherical transform matrices

exatomic.algorithms.basis.diff_expr(expr, cart='x', order=1)

Compute the nth order derivative symbolically with respect to cart.

Parameters:

• expr (symbolic) – sympy or symengine expression

• cart (str) – ‘x’, ‘y’, or ‘z’

• order (int) – order of differentiation

Returns:

expr (symbolic) – The symbolic derivative

exatomic.algorithms.basis.evaluate_expr(expr, xs, ys, zs, arr=None, alpha=None)

Evaluate symbolic expression on a numerical grid.

Parameters:

• expr (symbolic) – sympy or symengine expression

• xs (np.ndarray) – 1D-array of x values

• ys (np.ndarray) – 1D-array of y values

• zs (np.ndarray) – 1D-array of z values

• arr (np.ndarray) – additional 1D-array to multiply expression by

• alpha (float) – multiply expression by gaussian with exponent alpha

Note

See exatomic.algorithms.orbital_util.numerical_grid_from_field_params() for grid construction details.

class exatomic.algorithms.basis.BasisFunctions(uni, frame=0, cartp=True)

Bases: object

Composition wrapper class that leverages symbolic expressions using symengine and numexpr, using values extracted from the numerical Shell jitclasses, to evaluate basis functions on a numerical grid.

Parameters:

• uni (exatomic.core.universe.Universe) – a universe with basis set

• frame (int) – frame corresponding to basis set (default 0)

• cartp (bool) – forces p function ordering as (x, y, z) not (-1, 0, 1)

integrals()

Compute the overlap matrix using primitive cartesian integrals.

enum_shell(shl)

Return a generator over angular momentum degrees of freedom.

Parameters:

shl (exatomic.algorithms.numerical.Shell) – basis set shell

evaluate(xs=None, ys=None, zs=None, irrep=None, verbose=False)

Evaluate basis functions on a numerical grid.

Parameters:

• xs (np.ndarray) – 1D-array of x values

• ys (np.ndarray) – 1D-array of y values

• zs (np.ndarray) – 1D-array of z values

• verbose (bool) – print code pathway

• irrep (int,OrderedDict) – irrep or {irrep: [vectors] for irrep in irreps}

Note

Default behavior returns symbolic expressions if xs is None. See exatomic.algorithms.orbital_util.numerical_grid_from_field_params() for grid construction details.

evaluate_diff(xs, ys, zs, cart='x', verbose=False)

Evaluate basis function derivatives on a numerical grid.

Parameters:

• xs (np.ndarray) – 1D-array of x values

• ys (np.ndarray) – 1D-array of y values

• zs (np.ndarray) – 1D-array of z values

• cart (str) – derivative with respect to cart (in [‘x’, ‘y’, ‘z’])

• verbose (bool) – print code pathway

Note

See numerical_grid_from_field_params() for grid construction details.

exatomic.algorithms.basis.compute_uncontracted_basis_set_order(uni)