grid.molgrid module

Molecular grid class.

class MolGrid(atnums, atgrids, aim_weights, store=False)

Bases: Grid

Molecular grid class for integration of three-dimensional functions.

Molecular grid is defined here to be a weighted average of \(M\) atomic grids (see AtomGrid). This is defined by a atom in molecule weights (or nuclear weight functions) \(w_n(r)\) for each center n such that \(\sum^M_n w_n(r) = 1\) for all points \(r\in\mathbb{R}^3\). [1]

References

[1]

Becke, Axel D. “A multicenter numerical integration scheme for polyatomic molecules.” The Journal of chemical physics 88.4 (1988): 2547-2553.

__init__(atnums, atgrids, aim_weights, store=False)

Initialize class.

Parameters:

• atnums (np.ndarray(M, 3)) – Atomic number of \(M\) atoms in molecule.

• atgrids (list[AtomGrid]) – List of atomic grids of size \(M\) for each atom in molecule.

• aim_weights (Callable or np.ndarray(sum^M_n N_n,)) – Atoms in molecule weights \({ {w_n(r_k)}_k^{N_i}}_n^{M}\), where \(N_i\) is the number of points in the ith atomic grid.

• store (bool) – If true, then the atomic grids atgrids are stored as attribute atgrids.

property aim_weights

Get the atom in molecules/nuclear weight function of the molecular grid.

Returns:

List of atomic weights where \(K = \sum_n N_i\) and \(N_i\) is the number of points in the ith atomic grid.

Return type:

ndarray(K,)

property atcoords

Center/Atomic coordinates.

Type:

ndarray(M, 3)

property atgrids

List of atomic grids for each center. Returns None if store false.

Type:

List[AtomGrid]

property atweights

Get the atomic grid weights for all points, i.e. without atom in molecule weights.

Returns:

List of weights of atomic grid for each point,where \(K = \sum_n N_i\) and \(N_i\) is the number of points in the ith atomic grid.

Return type:

ndarray(K,)

classmethod from_preset(atnums, atcoords, preset, rgrid=None, aim_weights=None, rotate=37, store=False)

Construct molecular grid wih preset parameters.

Parameters:

• atnums (np.ndarray(M,)) – Array of atomic numbers.

• atcoords (np.ndarray(M, 3)) – Atomic coordinates of atoms.

• preset ((str, list[str], dict[int: str])) – Preset grid accuracy scheme. If string is provided, then preset is used for all atoms, either it is specified by a list, or a dictionary whose keys are from atnums. These predefined grid specify the radial sectors and their corresponding number of Lebedev grid points. Supported preset options include: ‘coarse’, ‘medium’, ‘fine’, ‘veryfine’, ‘ultrafine’, and ‘insane’. Other options include the “standard grids”: ‘sg_0’, ‘sg_1’, ‘sg_2’, and ‘sg_3’, and the Ochsenfeld grids: ‘g1’, ‘g2’, ‘g3’, ‘g4’, ‘g5’, ‘g6’, and ‘g7’, with higher number indicating greater accuracy but denser grid. See Notes for more information.

• rgrid ((OneDGrid, list[OneDGrid], dict[int: OneDGrid], None), optional) – One dimensional radial grid. If of type OneDGrid then this radial grid is used for all atoms. If a list is provided, then ith grid correspond to the ith atom. If dictionary is provided, then the keys correspond to the atnums[i] attribute. If None, a default radial grid (PowerRTransform of UniformInteger grid) is constructed based on the given atomic numbers.

• aim_weights (Callable or np.ndarray(K,), optional) – Atoms in molecule weights. If None, then aim_weights is Becke weights with order=3.

• rotate (bool or int, optional) – Flag to set auto rotation for atomic grid, if given int, the number will be used as a seed to generate rantom matrix.

• store (bool, optional) – Store atomic grid as a class attribute.

Notes

• The standard and Ochsenfeld presets were not designed with symmetric spherical t-design in mind.

• The “standard grids” [2] “SG-0” and “SG-1” are designed for large molecules with LDA (GGA) functionals, whereas “SG-2” and “SG-3” are designed for Meta-GGA functionals and B95/Minnesota functionals, respectively.

• The Ochsenfeld pruned grids [3] are obtained based on the paper.

References

[2]

Y. Shao, et al. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys. 113, 184-215 (2015)

[3]

Laqua, H., Kussmann, J., & Ochsenfeld, C. (2018). An improved molecular partitioning scheme for numerical quadratures in density functional theory. The Journal of Chemical Physics, 149(20).

classmethod from_pruned(atnums, atcoords, radius, r_sectors, d_sectors=50, *, s_sectors=None, rgrid=None, aim_weights=None, rotate=37, store=False)

Initialize a MolGrid instance with pruned method from AtomGrid.

Parameters:

• atnums (ndarray(M,)) – List of atomic numbers for each atom.

• atcoords (np.ndarray(M, 3)) – Three-dimensional Cartesian coordinates for each atoms in atomic units.

• radius (float, List[float]) – The atomic radius to be multiplied with r_sectors in atomic units (to make the radial sectors atom specific). If float, then the same atomic radius is used for all atoms, otherwise a list with \(M\) elements is used, where \(M\) is the number of atoms in the molecule. If list, then the ith element is used for the ith atom.

• r_sectors (list of List[float]) – List of sequences of the boundary radius (in atomic units) specifying sectors of the pruned radial grid of \(M\) atoms. For the first atom, the first sector is (0, radius*r_sectors[0][0]), then (radius*r_sectors[0][0], radius*r_sectors[0][1]), and so on. See AtomGrid.from_pruned for more information.

• d_sectors (int or list of List[int], optional) – List of sequences of the angular degrees for radial sectors of \(M\) atoms. If a number is given, then the same number of degrees is used for all sectors of all atoms.

• s_sectors (int or list of List[int] or None, optional, keyword-only) – List of sequences of angular sizes for each radial sector of of \(M\) atoms. If both d_sectors and s_sectors are given, s_sectors is used.

• rgrid (OneDGrid or List[OneDGrid] or Dict[int: OneDGrid], optional) – One dimensional grid for the radial component. If a list is provided,then ith grid correspond to the ith atom. If dictionary is provided, then the keys are correspond to the atnums[i] attribute. If None, then using atomic numbers it will generate a default radial grid (PowerRTransform of UniformInteger grid).

• aim_weights (Callable or np.ndarray(sum^M_n N_n,), optional) – Atoms in molecule/nuclear weights \({ {w_n(r_k)}_k^{N_i}}_n^{M}\), where \(N_i\) is the number of points in the ith atomic grid. If None, then aim_weights is Becke weights with order=3.

• rotate (bool or int , optional) – Flag to set auto rotation for atomic grid, if given int, the number will be used as a seed to generate random matrix.

• store (bool, optional) – Flag to store each original atomic grid information.

Returns:

Molecular grid class for integration.

Return type:

MolGrid

classmethod from_size(atnums, atcoords, size, rgrid=None, aim_weights=None, rotate=37, store=False)

Initialize a MolGrid instance with Horton Style input.

Parameters:

• atnums (np.ndarray(M, 3)) – Atomic number of \(M\) atoms in molecule.

• atcoords (np.ndarray(N, 3)) – Cartesian coordinates for each atoms

• size (int) – Number of points on each shell of angular grid.

• rgrid (OneDGrid or None, optional) – One-dimensional grid to construct the atomic grid. If none, then default radial grid is generated based on atomic numbers.

• aim_weights (Callable or np.ndarray(K,), optional) – Atoms in molecule weights. If None, then aim_weights is Becke weights with order=3.

• rotate (bool or int , optional) – Flag to set auto rotation for atomic grid, if given int, the number will be used as a seed to generate rantom matrix.

• store (bool, optional) – Flag to store each original atomic grid information

Returns:

MolGrid instance with specified grid property

Return type:

MolGrid

get_atomic_grid(index)

Get atomic grid corresponding to the given atomic index.

Parameters:

index (int) – index of atom starting from 0 to \(N-1\) where \(N\) is the number of atoms in molecule.

Returns:

If store=True, the AtomGrid instance used is returned. If store=False, the LocalGrid containing points and weights of AtomGrid is returned.

Return type:

AtomGrid or LocalGrid

get_localgrid(center, radius)

Create a grid containing points within the given radius of center.

Parameters:

• center (float or np.array(M,)) – Cartesian coordinates of the center of the local grid.

• radius (float) – Radius of sphere around the center. When equal to np.inf, the local grid coincides with the whole grid, which can be useful for debugging.

Returns:

Instance of LocalGrid.

Return type:

LocalGrid

property indices

Get the indices of the molecular grid.

Returns:

List of indices \([i_0, i_1, \cdots]\) that whose indices range [i_k, i_{k+1}] specify which points in points correspond to kth atomic grid.

Return type:

ndarray(M + 1,)

integrate(*value_arrays)

Integrate over the whole grid for given multiple value arrays.

Product of all value_arrays will be computed element-wise then integrated on the grid with its weights:

\[\int w(x) \prod_i f_i(x) dx.\]

Parameters:

*value_arrays (np.ndarray(N, )) – One or multiple value array to integrate.

Returns:

The calculated integral over given integrand or function

Return type:

float

interpolate(func_vals)

Return function that interpolates (and its derivatives) from function values.

Consider a real-valued function \(f(r, \theta, \phi) = \sum_A f_A(r, \theta, \phi)\) written as a sum of atomic centers \(f_A(r, \theta, \phi)\). Each of these functions can be further decomposed based on the atom grid centered at \(A\):

\[f_A(r, \theta, \phi) = \sum_l \sum_{m=-l}^l \sum_i \rho^A_{ilm}(r) Y_{lm}(\theta, \phi)\]

A cubic spline is used to interpolate the radial functions \(\sum_i \rho^A_{ilm}(r)\) based on the atomic grid centered at \(A\). This is then multipled by the corresponding spherical harmonics at all \((\theta_j, \phi_j)\) angles and summed to obtain approximation to \(f_A\). This is then further summed over all centers to get \(f\).

Parameters:

func_vals (ndarray(sum_i N_i,)) – The function values evaluated on all \(N_i\) points on the \(i\)th atomic grid.

Returns:

Callable function that interpolates the function and its derivative provided. The function takes the following attributes:

pointsndarray(N, 3)

Cartesian coordinates of \(N\) points to evaluate the splines on.

derivint, optional

If deriv is zero, then only returns function values. If it is one, then returns the first derivative of the interpolated function with respect to either Cartesian or spherical coordinates. Only higher-order derivatives (deriv=2,3) are supported for the derivatives wrt to radial components.

deriv_sphericalbool

If True, then returns the derivatives with respect to spherical coordinates \((r, \theta, \phi)\). Default False.

only_radial_derivbool

If true, then the derivative wrt to radius \(r\) is returned.

This function returns the following.

ndarray(M,…):

The interpolated function values or its derivatives with respect to Cartesian \((x,y,z)\) or if deriv_spherical then \((r, \theta, \phi)\) or if only_radial_derivs then derivative wrt to \(r\) is only returned.

Return type:

Callable[[ndarray(M, 3), int] -> ndarray(M)]

moments(orders, centers, func_vals, type_mom='cartesian', return_orders=False)

Compute the multipole moment integral of a function over centers.

The Cartesian type moments are:

\[m_{n_x, n_y, n_z} = \int (x - X_c)^{n_x} (y - Y_c)^{n_y} (z - Z_c)^{n_z} f(r) dr,\]

where \(\textbf{R}_c = (X_c, Y_c, Z_c)\) is the center of the moment, \(f(r)\) is the function, and \((n_x, n_y, n_z)\) are the Cartesian orders.

The spherical/pure moments with \((l, m)\) parameter are:

\[m_{lm} = \int | \textbf{r} - \textbf{R}_c|^l S_l^m(\theta, \phi) f(\textbf{r}) d\textbf{r},\]

where \(S_l^m\) is a regular, real solid harmonic.

The radial moments with \(n\) parameter are:

\[m_n = \int | \textbf{r} - \textbf{R}_c|^{n} f(\textbf{r}) d\textbf{r}\]

The radial combined with spherical/pure moments \((n, l, m)\) are:

\[m_{nlm} = \int | \textbf{r} - \textbf{R}_c|^{n+1} S_l^m(\theta, \phi) f(\textbf{r}) d\textbf{r}\]

Parameters:

• orders (int) – Generates all orders with Horton order depending on the type of the multipole moment type_mom.

• centers (ndarray(M, 3)) – The centers \(\textbf{R}_c\) of the moments to compute from.

• func_vals (ndarray(N,)) – The function \(f\) is evaluated at all \(N\) points on the integration grid.

• type_mom (str) – The type of multipole moments: “cartesian”, “pure”, “radial” and “pure-radial”.

• return_orders (bool) – If true, it will also return a list of size \(L\) of the orders corresponding to each integral/row of the output.

Returns:

Computes the moment integral of the function at the mth center for all orders. If return_orders is true, then this also returns a list that describes what each row/order is, e.g. for Cartesian, [(0, 0, 0), (1, 0, 0) ,…].

Return type:

ndarray(L, M), or (ndarray(L, M), list)

property points

Positions of the grid points.

Type:

np.ndarray(N,) or np.ndarray(N, M)

save(filename)

Save molecular grid attributes as a npz file.

Parameters:

filename (str) – The path/name of the .npz file.

property size

the total number of points on the grid.

Type:

int

property weights

the weights of each grid point.

Type:

np.ndarray(N,)