bfit.model

Models used for fitting.

Module Contents

Classes

AtomicGaussianDensity	Gaussian density model for modeling the electronic density of a single atom.
MolecularGaussianDensity	Molecular Atom-Centered Gaussian Density Model.

class bfit.model.AtomicGaussianDensity(points, center=None, num_s=1, num_p=0, normalize=False)

Gaussian density model for modeling the electronic density of a single atom.

Atomic Gaussian density is a linear combination of Gaussian functions of S-type and p-type functions:

\[f(x) := \sum_i c_i e^{-\alpha_i |x - c|^2} + \sum_j d_j |x - c|^2 e^{-\beta_j |x - c|^2}\]

where \(c_i, d_i\) are the coefficients of s-type and p-type Gaussian functions, \(\alpha_i, \beta_j\) are teh exponents of the s-type and p-type Gaussian functions, \(c\) is the center of all Gaussian functions. \(x\) is the real coordinates, can be multi-dimensional.

Construct class representing atomic density modeled as Gaussian functions.

Parameters

• points (ndarray, (N, D)) – Grid points where N is the number of points and D is the number of dimensions.

• center (ndarray (D,), optional) – The D-dimensional coordinates of the single center. If None, then the center is the origin of all zeros.

• num_s (int, optional) – Number of s-type Gaussian basis functions.

• num_p (int, optional) – Number of p-type Gaussian basis functions.

• normalize (bool, optional) – Whether to normalize Gaussian basis functions.

property points(self)

Return the grid points.

property radii(self)

Return the distance of grid points from center of Gaussian(s).

property num_s(self)

Return the number of s-type Gaussian basis functions.

property num_p(self)

Return the number of p-type Gaussian basis functions.

property nbasis(self)

Return the total number of Gaussian basis functions.

property natoms(self)

Return the number of basis functions centers.

property prefactor(self)

Obtain list of exponents for the prefactors.

change_numb_s_and_numb_p(self, new_s, new_p)

Change the number of s-type and p-type Gaussians.

Parameters

• new_s (int) – New number of s-type Gaussians.

• new_p (int) – New number of p-type Gaussians.

evaluate(self, coeffs, expons, deriv=False)

Compute linear combination of Gaussian basis & its derivatives on the grid points.

\[f(x):= \sum_i c_i e^{-\alpha_i |x - c|^2} + \sum_j d_j |x - c|^2 e^{-\beta_j |x - c|^2}\]

where \(c_i, d_i\) are the coefficients of s-type and p-type Gaussian functions, \(\alpha_i, \beta_j\) are teh exponents of the s-type and p-type Gaussian functions, \(c\) is the center of all Gaussian functions. \(x\) is the real coordinates, can be multi-dimensional.

Parameters

• coeffs (ndarray(nbasis,)) – The coefficients \(c_i\) of num_s s-type Gaussian basis functions followed by the coefficients \(d_j\) of num_p p-type Gaussian basis functions.

• expons (ndarray(nbasis,)) – The exponents \(\alpha_i\) of num_s s-type Gaussian basis functions followed by the exponents \(\beta_j\) of num_p p-type Gaussian basis functions.

• deriv (bool, optional) – Whether to compute derivative of Gaussian basis functions w.r.t. coefficients & exponents.

Returns

• g (ndarray, (N,)) – The linear combination of Gaussian basis functions evaluated on the grid points.

• dg (ndarray, (N, 2 * nbasis)) – The derivative of a linear combination of Gaussian basis functions w.r.t. coefficients & exponents, respectively, evaluated on the grid points. Only returned if deriv=True.

class bfit.model.MolecularGaussianDensity(points, coords, basis, normalize=False)

Molecular Atom-Centered Gaussian Density Model.

The Molecular Gaussian Density model is based on multiple centers each associated with a Gaussian density model (s or p-type) of any dimension.

\[f(x) := \sum_j \bigg[ \sum_{i =1}^{M^s_j} c_{ji} e^{-\alpha_{ji} |x - m_j|^2} + \sum_{i=1}^{M_j^p}d_{ji} |x - m_j|^2 e^{-\beta_{ji} |x - m_j|^2} \bigg]\]

where \(c_{ji}, d_{ji}\) are the ith coefficients of s-type and p-type functions of the jth center, \(\alpha_{ji}, \beta_{ji}\) are the ith exponents of S-type and P-type functions of the jth center, \(M_j^s, M_j^p\) is the total number of s-type or p-type Gaussians functions of the jth center respectively, \(m_j\) is the coordinate of the jth center, and \(x\) is the real coordinates of the point. It can be of any dimension.

Construct the MolecularGaussianDensity class.

Parameters

• points (ndarray, (N, D)) – The grid points, where N is the number of grid points and D is the dimension.

• coords (ndarray, (M, D)) – The atomic coordinates (M centers) on which Gaussian basis are centered.

• basis (ndarray, (M, 2)) – The number of S-type & P-type Gaussian basis functions placed on each center.

• normalize (bool, optional) – Whether to normalize Gaussian basis functions.

property points(self)

Get grid points.

property nbasis(self)

Get the total number of Gaussian basis functions.

property radii(self)

Get the distance of grid points from center of each basis function.

property natoms(self)

Get number of basis functions centers.

property prefactor(self)

Get the pre-factor of Gaussian basis functions to make it normalized.

Only used if attribute normalize is true.

assign_basis_to_center(self, index)

Assign the Gaussian basis function to the atomic center.

Parameters

index (int) – The index of Gaussian basis function.

Returns

index – The index of atomic center.

Return type

int

evaluate(self, coeffs, expons, deriv=False)

Compute linear combination of Gaussian basis & its derivatives on the grid points.

The Molecular Gaussian is defined to be:

\[f(x) := \sum_j \bigg[ \sum_{i =1}^{M^s_j} c_{ji} e^{-\alpha_{ji} |x - m_j|^2} + \sum_{i=1}^{M_j^p}d_{ji} |x - m_j|^2 e^{-\beta_{ji} |x - m_j|^2} \bigg]\]

where \(c_{ji}, d_{ji}\) are the ith coefficients of s-type and p-type functions of the jth center, \(\alpha_{ji}, \beta_{ji}\) are the ith exponents of S-type and P-type functions of the jth center, \(M_j^s, M_j^p\) is the total number of s-type or p-type Gaussians functions of the jth center respectively, \(m_j\) is the coordinate of the jth center, and \(x\) is the real coordinates of the point. It can be of any dimension.

Parameters

• coeffs (ndarray, (nbasis,)) – The coefficients of num_s s-type Gaussian basis functions followed by the coefficients of num_p p-type Gaussian basis functions for an atom, then repeat for the next atom.

• expons (ndarray, (nbasis,)) – The exponents of num_s s-type Gaussian basis functions followed by the exponents of num_p p-type Gaussian basis functions for an atom, then repeat for the next atom.

• deriv (bool, optional) – Whether to compute derivative of Gaussian basis functions w.r.t. coefficients & exponents.

Returns

• g (ndarray, (N,)) – The linear combination of Gaussian basis functions evaluated on the grid points.

• dg (ndarray, (N, nbasis)) – The derivative of linear combination of Gaussian basis functions w.r.t. coefficients & exponents, respectively, evaluated on the grid points. Only returned if deriv=True.