grid.rtransform module

Transformation from 1D intervals [a, b] to other 1D intervals [c, d].

class BaseTransform

Bases: ABC

Abstract class for transformation.

__init__()
property codomain

Transformation codomain.

Type

tuple

abstract deriv(x)

Abstract method for 1st derivative of transformation.

abstract deriv2(x)

Abstract method for the second derivative of transformation.

abstract deriv3(x)

Abstract method for the third derivative of transformation.

property domain

Transformation domain.

Type

tuple

abstract inverse(r)

Abstract method for inverse transformation.

abstract transform(x)

Abstract method for transformation.

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class BeckeRTransform(rmin, R, trim_inf=True)

Bases: BaseTransform

Becke Transformation.

The Becke transformation transforms from \([-1, 1]\) to \([r_{min}, \infty)\) according to

\[r(x) = R \frac{1 + x}{1 - x} + r_{min}.\]

The inverse transformation is given by

\[x(r) = \frac{r - r_{min} - R} {r - r_{min} + R}.\]

References

1

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

property R

the scale factor for the transformed array.

Type

float

__init__(rmin, R, trim_inf=True)

Construct Becke transform, \([-1, 1]\) to :math`[r_{min}, infty)`.

Parameters
  • rmin (float) – The minimum coordinate \(r_{min}\) in the transformed interval \([r_{min}, \infty)\).

  • R (float) – The scale factor used in the transformation.

  • trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of Becke transformation.

\[\frac{dr_i}{dx_i} = 2R \frac{1}{(1-x)^2}\]
Parameters

x (np.array(N,)) – One-dimensional array in the domain \([-1, 1]\).

Returns

First derivative of Becke transformation at each point.

Return type

np.ndarray(N,)

deriv2(x)

Compute the second derivative of Becke transformation.

\[\frac{d^2r}{dx^2} = 4R \frac{1}{1-x^3}\]
Parameters

x (np.array(N,)) – One-dimensional array in the domain \([-1, 1]\).

Returns

Second derivative of Becke transformation at each point.

Return type

np.ndarray(N,)

deriv3(x)

Compute the third derivative of Becke transformation.

\[\frac{d^3r}{dx^3} = 12R \frac{1}{1 - x^4}\]
Parameters

x (np.array(N,)) – One-dimensional array in the domain \([-1, 1]\).

Returns

Third derivative of Becke transformation at each point.

Return type

np.ndarray(N,)

property domain

Transformation domain.

Type

tuple

static find_parameter(array, rmin, radius)

Compute R such that half of the points in \([r_{min}, \infty)\) are within radius.

Parameters
  • array (np.ndarray(N,)) – One-dimensional array in the domain \([-1, 1]\).

  • rmin (float) – Minimum value for transformed array.

  • radius (float) – Atomic radius of interest.

Returns

The optimal value of scale factor R.

Return type

float

inverse(r)

Transform \([r_{mi}n, \infty)\) back to original \([-1, 1]\).

\[x_i = \frac{r_i - r_{min} - R} {r_i - r_{min} + R}\]
Parameters

r (np.ndarray(N,)) – One-dimensional array in the codomain \([r_{min}, \infty)\).

Returns

One dimensional array in \([-1, 1]\).

Return type

np.ndarray(N,)

property rmin

the minimum value for the transformed array.

Type

float

transform(x)

Transform from \([-1, 1]\) to \([r_{min}, \infty)\).

\[r_i = R \frac{1 + x_i}{1 - x_i} + r_{min}\]
Parameters

x (np.ndarray(N,)) – One-dimensional array in the domain \([-1, 1]\).

Returns

Transformed array located between \([r_min, \infty)\).

Return type

np.ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class ExpRTransform(rmin, rmax, b=None)

Bases: BaseTransform

Exponential transform from \([0, \infty)\) to \([r_{min}, r_{max}]\).

This transformation is given by

\[r(x) = r_{min} e^{x \log\bigg(\frac{r_{max}}{r_{min} / b} \bigg)},\]
where \(b\) maps to rmax. If None, then the \(b\) is taken to be the maximum

from the first grid that is being transformed. This transformation always maps zero to \(r_{min}\).

The inverse transformation is given by

\[x(r) = \frac{\log\big(\frac{r}{r_{min}} \big) b}{\log(\frac{r_{max}}{r_{min}})}\]
__init__(rmin, rmax, b=None)

Initialize exp transform instance.

Parameters
  • rmin (float) – Minimum value for transformed points.

  • rmax (float) – Maximum value for transformed points.

  • b (float) – Maximum \(b\) of a prespecified radial grid \([0, b]\) such that \(b\) maps to rmax. If None, then the maximum is taken and stored from the grid that is transformed initially.

property b

Parameter \(b\) that maps/transforms to \(r_{max}\).

Type

float

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of exponential transform.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation.

Returns

First derivative of transformation at x.

Return type

ndarray(N,)

deriv2(x)

Compute the second derivative of exponential transform.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation.

Returns

Second derivative of transformation at x.

Return type

ndarray(N,)

deriv3(x)

Compute the third derivative of exponential transform.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation.

Returns

Third derivative of transformation at x.

Return type

ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Compute the inverse of exponential transform.

\[x(r_i) = \frac{\log\big(\frac{r}{r_{min}} \big) b}{\log(\frac{r_{max}}{r_{min}})},\]

where \(b\) is a prespecified parameter

Parameters

r (ndarray(N,)) – One-dimensional array in the domain of the transformation.

Returns

Inverse transformation at r.

Return type

ndarray(N,)

property rmax

the value of rmax.

Type

float

property rmin

the value of rmin.

Type

float

set_maximum_parameter_b(x)

Sets up the parameter b from taken the maximum over x.

transform(x)

Perform exponential transform.

\[r = r_{min} e^{x \log\bigg(\frac{r_{max}}{r_{min} / b} \bigg)},\]

where \(b\) is a prespecified parameter that maps to \(r_{max}\).

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation.

Returns

The transformation of x in the co-domain of the transformation.

Return type

ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class HandyModRTransform(rmin, rmax, m, trim_inf=True)

Bases: BaseTransform

Modified Handy Transformation class from \([-1, 1]\) to \([r_{min}, r_{max}]\).

This transformation is given by

\[r(x) = \frac{(1+x)^m (r_{max} - r_{min})} { 2^m (1 - 2^m + r_{max} - r_{min}) - (1 + x)^m (r_{max} - r_{min} - 2^m )} + r_{min},\]

where \(m > 0\).

The inverse transformation is given by

\[x(r) = 2 \sqrt[m]{ \frac{(r - r_{min})(r_{max} - r_{min} - 2^m + 1)} {(r - r_{min})(r_{max} - r_{min} - 2^m) + r_{max} - r_{min}} } - 1.\]
__init__(rmin, rmax, m, trim_inf=True)

Construct a modified Handy transform from \([-1, 1]\) to \([r_{min}, r_{max}]\).

Parameters
  • rmin (float) – The minimum coordinate for transformed radial array.

  • rmax (float) – The maximum coordinate for transformed radial array.

  • m (integer m > 0) – Free parameter, m must be > 0.

  • trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of modified Handy transformation.

\[\frac{dr}{dx} = -\frac{ 2^m m (r_{max}-r_{min})(2^m-r_{max}+r_{min}-1)(1+x)^{m-1}} {\left( 2^m (2^m-1-r_{max}+r_{min})-(2^m-r_{max} + r_{min})(1 + x)^m\right)^2}\]
Parameters

x (ndarrray(N,)) – One dimensional array in \([-1,1]\).

Returns

The first derivative of Handy transformation at each point.

Return type

ndarrray(N,)

deriv2(x)

Compute the second derivative of modified Handy transformation.

Parameters

x (ndarrray(N,)) – One dimensional array in \([-1,1]\).

Returns

The second derivative of Handy transformation at each point.

Return type

ndarrray(N,)

deriv3(x)

Compute the third derivative of modified Handy transformation.

Parameters

x (ndarrray(N,)) – One dimensional array in \([-1,1]\).

Returns

The third derivative of Handy transformation at each point.

Return type

ndarrray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Inverse transform from \([r_{min},r_{max}]\) to \([-1,1]\).

\[x_i = 2 \sqrt[m]{ \frac{(r_i - r_{min})(r_{max} - r_{min} - 2^m + 1)} {(r_i - r_{min})(r_{max} - r_{min} - 2^m) + r_{max} - r_{min}} } - 1\]
Parameters

r (ndarrray(N,)) – One dimensional array in \([r_{min},\infty)\).

Returns

The original one dimensional array in \([-1,1]\).

Return type

ndarrray(N,)

property m

Free and extra parameter, m must be > 0.

Type

integer

property rmax

The maximum value for the transformed radial array.

Type

float

property rmin

The minimum value for the transformed radial array.

Type

float

transform(x)

Transform given array \([-1,1]\) to radial array \([r_{min},r_{max}]\).

\[r_i = \frac{(1+x_i)^m (r_{max} - r_{min})} { 2^m (1 - 2^m + r_{max} - r_{min}) - (1 + x_i)^m (r_{max} - r_{min} - 2^m )} + r_{min}\]
Parameters

x (ndarray(N,)) – One dimensional array in \([-1,1]\).

Returns

One dimensional array in \([r_{min},r_{max}]\).

Return type

ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class HandyRTransform(rmin, R, m, trim_inf=True)

Bases: BaseTransform

Handy Transformation class from \([-1, 1]\) to \([r_{min}, \infty)\).

This transformation is given by

\[r(x) = R \left( \frac{1+x}{1-x} \right)^m + r_{min}.\]

The inverse transformations is given by

\[x(r) = \frac{\sqrt[m]{r-r_{min}} - \sqrt[m]{R}} {\sqrt[m]{r-r_{min}} + \sqrt[m]{R}}.\]
property R

The scale factor for the transformed radial array.

Type

float

__init__(rmin, R, m, trim_inf=True)

Construct Handy transformation.

Parameters
  • rmin (float) – The minimum coordinate for transformed radial array.

  • R (float) – The scale factor for transformed radial array.

  • m (integer m > 0) – Free parameter, m must be > 0.

  • trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of Handy transformation.

\[\frac{dr}{dx} = 2mR \frac{(1+x)^{m-1}}{(1-x)^{m+1}}\]
Parameters

x (ndarray(N,)) – One dimensional array with values between \([-1,1]\).

Returns

The first derivative of Handy transformation at each point.

Return type

ndarray(N,)

deriv2(x)

Compute the second derivative of Handy transformation.

\[\frac{d^2r}{dx^2} = 4mR (m + x) \frac{(1+x)^{m-2}}{(1-x)^{m+2}}\]
Parameters

x (ndarray(N,)) – One dimensional array in \([-1,1]\).

Returns

The second derivative of Handy transformation at each point.

Return type

ndarray(N,)

deriv3(x)

Compute the third derivative of Handy transformation.

\[\frac{d^3r}{dx^3} = 4mR ( 1 + 6 m x + 2 m^2 + 3 x^2) \frac{(1+x)^{m-3}}{(1-x)^{m+3}}\]
Parameters

array (np.ndarray(N,)) – One dimensional array in \([-1,1]\).

Returns

The third derivative of Handy transformation at each point.

Return type

np.ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Inverse transform from \([r_{min},\infty)\) to \([-1,1]\).

\[x_i = \frac{\sqrt[m]{r_i-r_{min}} - \sqrt[m]{R}} {\sqrt[m]{r_i-r_{min}} + \sqrt[m]{R}}\]
Parameters

r (np.ndarray(N,)) – One dimensional array in \([r_{min},\infty)\).

Returns

One-dimensional array in \([-1,1]\).

Return type

np.ndarray(N,)

property m

Free and extra parameter, m must be > 0.

Type

integer

property rmin

The minimum value for the transformed radial array.

Type

float

transform(x)

Transform from \([-1,1]\) to \([r_{min},\infty)\).

\[r_i = R \left( \frac{1+x_i}{1-x_i} \right)^m + r_{min}\]
Parameters

x (np.ndarray(N,)) – One dimensional array in \([-1,1]\).

Returns

One dimensional array in \([r_{min},\infty)\).

Return type

np.ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class HyperbolicRTransform(a, b)

Bases: BaseTransform

Hyperbolic transform from :math`[0, infty)` to \([0, \infty)\).

The transformation is given by

\[r(x) = \frac{a x}{(1 - bx)},\]

where \(b ( N - 1) \geq 1\), and \(N\) is the number of points in x.

The inverse transformation is given by

\[x(r) = \frac{r}{a + br}\]
__init__(a, b)

Hyperbolic transform class.

Parameters
  • a (float) – parameter a to determine hyperbolic function

  • b (float) – parameter b to determine hyperbolic function

property a

value of parameter a.

Type

float

property b

value of parameter b.

Type

float

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of hyperbolic transformation.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

First derivative of transformation at x.

Return type

ndarray(N,)

deriv2(x)

Compute the second derivative of hyperbolic transformation.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

Second derivative of transformation at x.

Return type

ndarray(N,)

deriv3(x)

Compute the third derivative of hyperbolic transformation.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

Third derivative of transformation at x.

Return type

ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Compute the inverse of hyperbolic transformation.

\[x(r) = \frac{r}{a + br}\]
Parameters

r (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

Inverse transformation at r.

Return type

ndarray(N,)

transform(x)

Perform hyperbolic transformation.

\[r_i = \frac{a x_i}{(1 - bx_i)},\]

where \(b ( N - 1) \geq 1\), and \(N\) is the number of points in x.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

The transformation of x to the co-domain of the transformation.

Return type

ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class IdentityRTransform

Bases: BaseTransform

Identity Transform class.

The identity transform class trivially transforms from \([0, \infty)\) to \([0, \infty)\) given by

\[r(x) = x.\]

The inverse transformation is given by

\[x(r) = r.\]
__init__()
property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of identity transform.

Parameters

x (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).

Returns

First derivative of identity transformation at x.

Return type

ndarray(N,)

deriv2(x)

Compute the second derivative of identity transform.

Parameters

x (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).

Returns

Second derivative of identity transformation at x.

Return type

ndarray(N,)

deriv3(x)

Compute the third derivative of identity transform.

Parameters

x (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).

Returns

Third derivative of identity transformation at x.

Return type

np.ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Compute the inverse of identity transform.

Parameters

r (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).

Returns

Inverse transformation of the identity transformation at x.

Return type

np.ndarray(N,)

transform(x)

Perform given array into itself.

\[r_i = x_i\]
Parameters

x (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).

Returns

Identity transformation at each point x.

Return type

ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class InverseRTransform(transform)

Bases: BaseTransform

Inverse transformation class for any general transformation.

__init__(transform)

Construct InverseRTransform instance.

Parameters

transform (BaseTransform) – One-dimension transformation instance.

property codomain

Transformation codomain.

Type

tuple

deriv(r)

Compute the first derivative of inverse transformation.

\[\frac{dx}{dr} = (\frac{dr}{dx})^{-1}\]
Parameters

r (np.array(N,)) – One-dimensional array in the co-domain r of the original transformation.

Returns

First derivative of the inverse transformation at each point r.

Return type

np.ndarray(N,)

deriv2(r)

Compute the second derivative of inverse transformation.

\[\frac{d^2 x}{dr^2} = - \frac{d^2 r}{dx^2} (\frac{dx}{dr})^3\]
Parameters

r (np.array(N,)) – One-dimensional array in the co-domain r of the original transformation.

Returns

Second derivative of the inverse transformation at each point r.

Return type

np.ndarray(N,)

deriv3(r)

Compute the third derivative of inverse transformation.

\[\frac{d^3 x}{dr^3} = -\frac{d^3 r}{dx^3} (\frac{dx}{dr})^4 + 3 (\frac{d^2 r}{dx^2})^2 (\frac{dx}{dr})^5\]
Parameters

r (np.array(N,)) – One-dimensional array in the co-domain r of the original transformation.

Returns

Third derivative of inverse transformation at each point r.

Return type

np.ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(x)

Transform array to the co-domain of the provided transformation.

This transformation is equivalent to the transformation function of the original transformation (i.e. OriginTF.transform).

Parameters

x (np.ndarray) – One-dimension array in the domain x of the original transformation.

Returns

One-dimensional array in the co-domain r of the original transformation.

Return type

np.ndarray

transform(r)

Transform array back to the original, domain of the provided transformation.

This transformation is equivalent to the inverse function of the original transformation (i.e. OriginTF.inverse).

Parameters

r (np.ndarray(N,)) – One-dimensional array in the co-domain r of the original transformation.

Returns

Original one-dimensional array in the domain x of the original transformation.

Return type

np.ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class KnowlesRTransform(rmin, R, k, trim_inf=True)

Bases: BaseTransform

Knowles Transformation from \([-1, 1]\) to \([r_{min}, \infty)\).

The transformation is given by

\[r(x) = r_{min} - R \log \left( 1 - 2^{-k} (x + 1)^k \right),\]

where \(k > 0\) and \(R\) is the scaling parameter.

The inverse transformation is given by

\[x(r) = 2 \sqrt[k]{1-\exp \left( -\frac{r-r_{min}}{R}\right)}-1\]
property R

The scale factor for the transformed radial array.

Type

float

__init__(rmin, R, k, trim_inf=True)

Construct Knowles transformation class.

Parameters
  • rmin (float) – The minimum coordinate for transformed radial array.

  • R (float) – The scale factor for transformed radial array.

  • k (integer k > 0) – Free parameter, k must be > 0.

  • trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of Knowles transformation.

\[\frac{dr}{dx} = kR \frac{(1+x_i)^{k-1}}{2^k-(1+x_i)^k}\]
Parameters

x (ndarray(N,)) – One dimensional array with values between \([-1,1]\).

Returns

The first derivative of Knowles transformation at each point.

Return type

ndarray(N,)

deriv2(x)

Compute the second derivative of Knowles transformation.

\[\frac{d^2r}{dx^2} = kR \frac{(1+x_i)^{k-2} \left(2^k(k-1) + (1+x_i)^k \right)}{\left( 2^k-(1+x_i)^k\right)^2}\]
Parameters

x (ndarray(N,)) – One dimensional array in \([-1,1]\).

Returns

The second derivative of Knowles transformation at each point.

Return type

ndarray(N,)

deriv3(x)

Compute the third derivative of Knowles transformation.

\[\frac{d^3r}{dx^3} = kR \frac{(1+x_i)^{k-3} \left( 4^k (k-1) (k-2) + 2^k (k-1)(k+4)(1+x_i)^k +2(1+x_i)^k \right)}{\left( 2^k - (1+x_i)^k \right)^3}\]
Parameters

x (ndarray(N,)) – One dimensional array with values between \([-1,1]\).

Returns

The third derivative of Knowles transformation at each point.

Return type

ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Inverse of transformation from \([r_{min},\infty)\) to \([-1,1]\).

\[x_i = 2 \sqrt[k]{1-\exp \left( -\frac{r_i-r_{min}}{R}\right)}-1\]
Parameters

r (ndarray(N,)) – One-dimensional array in \([r_{min},\infty)\).

Returns

The inverse transformation in \([-1,1]\).

Return type

ndarray(N,)

property k

Free and extra parameter, k must be > 0.

Type

float

property rmin

The minimum value for the transformed radial array.

Type

float

transform(x)

Transform from \([-1,1]\) to \([r_{min},\infty)\).

\[r_i = r_{min} - R \log \left( 1 - 2^{-k} (x_i + 1)^k \right)\]
Parameters

x (np.ndarray(N,)) – One dimensional array in \([-1,1]\).

Returns

One dimensional array in \([r_{min},\infty)\).

Return type

np.ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class LinearFiniteRTransform(rmin, rmax)

Bases: BaseTransform

Linear finite transformation from \([-1, 1]\) to \([r_{min}, r_{max}]\).

The Linear transformation from finite interval \([-1, 1]\) to finite interval \([r_{min}, r_{max}]\) is given by

\[r(x) = \frac{r_{max} - r_{min}}{2} (1 + x) + r_{min}.\]

The inverse transformation is given by

\[x(r) = \frac{2 r - (r_{max} + r_{min})}{r_{max} - r_{min}}\]
__init__(rmin, rmax)

Construct linear transformation instance.

Parameters
  • rmin (float) – Minimum value for transformed interval.

  • rmax (float) – Maximum value for transformed interval.

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the 1st order derivative.

\[\frac{dr}{dx} = \frac{r_{max} - r_{min}}{2}\]
Parameters

x (ndarray) – One-dimensional array in the domain \([-1, 1]\).

Returns

First order derivative at given points.

Return type

float or np.ndarray

deriv2(x)

Compute the second order derivative.

\[\frac{d^2 r}{dx^2} = 0\]
Parameters

x (ndarray) – One-dimensional array in the domain \([-1, 1]\).

Returns

Second order derivative at given points.

Return type

float or np.ndarray

deriv3(x)

Compute the third order derivative.

\[\frac{d^2 r}{dx^2} = 0\]
Parameters

x (ndarray) – One-dimensional array in the domain \([-1, 1]\).

Returns

Third order derivative at given points.

Return type

ndarray

property domain

Transformation domain.

Type

tuple

inverse(r)

Compute the inverse of the transformation.

\[x_i = \frac{2 r_i - (r_{max} + r_{min})}{r_{max} - r_{min}}\]
Parameters

r (ndarray) – One-dimensional array in the co-domain \([r_{min}, r_{max}]\).

Returns

One-dimensional array in the domain \([-1, 1]\).

Return type

ndarray

transform(x)

Transform from interval \([-1, 1]\) to \([r_{min}, r_{max}]\).

\[r_i = \frac{r_{max} - r_{min}}{2} (1 + x_i) + r_{min}.\]

This transformation maps \(r_i(-1) = r_{min}\) and \(r_i(1) = r_{max}\).

Parameters

x (ndarray) – One-dimensional array in the domain \([-1, 1]\).

Returns

Transformed points between \([r_{min}, r_{max}]\)

Return type

float or np.ndarray

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class LinearInfiniteRTransform(rmin, rmax, b=None)

Bases: BaseTransform

Linear transform from interval \([0, \infty)\) to \([r_{min}, r_{max})\).

This transformation linearly maps the infinite interval \([0, \infty)\) to a finite interval \([r_{min}, r_{max}]\) given by

\[r(x) = \frac{(r_{max} - r_{min})}{b} x + r_{min},\]

where \(r(b) = r_{max}\). If None, then the \(b\) is taken to be the maximum from the first grid that is being transformed. This transformation always maps zero to \(r_{min}\).

The original goal is to transform the UniformGrid, equally-spaced integers from 0 to N-1, to \([r_{min}, r_{max}]\).

The inverse is given by

\[x(r) = (r - r_{min}) \frac{\max_i (r_i)}{r_{max} - r_{min}}\]
__init__(rmin, rmax, b=None)

Initialize linear transform class.

Parameters
  • rmin (float) – Define the lower end of the linear transform

  • rmax (float) – Define the upper end of the linear transform

  • b (float) – Maximum \(b\) of a prespecified radial grid \([0, b]\) such that \(b\) maps to rmax. If None, then the maximum is taken and stored from the grid that is transformed initially.

property b

Parameter such that \(r(b) = r_{max}\).

Type

float

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of linear transformation.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain \([0, \infty)\).

Returns

First derivative of transformation at x.

Return type

ndarray(N,)

deriv2(x)

Compute the second derivative of linear transformation.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain \([0, \infty)\).

Returns

Second derivative of transformation at x.

Return type

ndarray(N,)

deriv3(x)

Compute the third derivative of linear transformation.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain \([0, \infty)\).

Returns

Third derivative of transformation at x.

Return type

ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Compute the inverse of linear transformation.

\[x_i = (r - r_{min}) / frac{b}{r_{max} - r_{min}}\]
Parameters

r (ndarray(N,)) – One-dimensional array in the domain \([r_{min}, r_{max}]\).

Returns

Inverse of transformation from coordinate \(r\) to \(x\).

Return type

ndarray(N,)

property rmax

rmax value of the tf.

Type

float

property rmin

rmin value of the tf.

Type

float

set_maximum_parameter_b(x)

Sets up the parameter b from taken the maximum over some grid x.

transform(x)

Transform from interval \([0, \infty)\) to \([r_{min}, r_{max}]\).

\[r_i = \frac{(r_{max} - r_{min})}{\max_i x_i} x_i + r_{min},\]

where \(N\) is the number of points. The goal is to transform equally-spaced integers from 0 to N-1, to \([r_{min}, r_{max}]\).

Parameters

x (ndarray(N,)) – One-dimensional array in the domain \([0, \infty)\).

Returns

Transformed points between \([r_{min}, r_{max}]\).

Return type

ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class MultiExpRTransform(rmin, R, trim_inf=True)

Bases: BaseTransform

MultiExp Transformation class from \([-1,1]\) to \([r_{min}, \infty)\).

The transformation is given by

\[r(x) = -R \log \left( \frac{x + 1}{2} \right) + r_{min}\]

The inverse of this transformation is given by

\[x(r) = 2 \exp \left( \frac{-(r - r_{min})}{R} \right) - 1\]

References

2

Gill, Peter MW, and Siu-Hung Chien. “Radial quadrature for multiexponential integrands.” Journal of computational chemistry 24.6 (2003): 732-740.

property R

The scale factor for the transformed radial array.

Type

float

__init__(rmin, R, trim_inf=True)

Construct MultiExp transform from \([-1,1]\) to \([r_{min}, \infty)\).

Parameters
  • rmin (float) – The minimum coordinate for transformed radial array.

  • R (float) – The scale factor for transformed radial array.

  • trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute the first derivative of MultiExp transformation.

\[\frac{dr}{dx} = -\frac{R}{1+x}\]
Parameters

x (ndarray(N,)) – One dimensional in \([-1, 1]\).

Returns

The first derivative of MultiExp transformation at each point.

Return type

ndarray(N,)

deriv2(x)

Compute the second derivative of MultiExp transformation.

\[\frac{d^2r}{dx^2} = \frac{R}{(1 + x)^2}\]
Parameters

x (ndarray(N,)) – One dimensional in \([-1,1]\).

Returns

The second derivative of MultiExp transformation at each point.

Return type

ndarray(N,)

deriv3(x)

Compute the third derivative of MultiExp transformation.

\[\frac{d^3r}{dx^3} = -\frac{2R}{(1 + x)^3}\]
Parameters

x (ndarray(N,)) – One dimensional array in \([-1,1]\).

Returns

The third derivative of MultiExp transformation at each point.

Return type

ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Inverse of transform from \([r_{min},\infty)\) to \([-1,1]\).

\[x_i = 2 \exp \left( \frac{-(r_i - r_{min})}{R} \right) - 1\]
Parameters

r (np.ndarray(N,)) – One-dimensional array in \([r_{min}, \infty)\).

Returns

The inverse of transformation in \([-1, 1]\).

Return type

np.ndarray(N,)

property rmin

The minimum value for the transformed radial array.

Type

float

transform(x)

Transform from [-1,1] to \([r_{min},\infty)\).

\[r_i = -R \log \left( \frac{x_i + 1}{2} \right) + r_{min}\]
Parameters

x (ndarray(N,)) – One dimensional array with values between \([-1,1]\).

Returns

Transformed array located between \([r_{min},\infty)\).

Return type

ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid

class PowerRTransform(rmin, rmax, b=None)

Bases: BaseTransform

Power transform class from \([0, \infty)\) to \([r_{min}, r_{max}]\).

This transformations is given by

\[r(x) = r_{min} (x + 1)^{\frac{\log(r_{max} - \log(r_{min}}{\log(b + 1)}},\]

such that \(r(b) = r_{max}\).

The inverse of the transformation is given by

\[x(r) = \frac{r}{r_{min}}^{\frac{\log(N)}{\log(r_{max}) - \log(r_{min})}} - 1.\]
__init__(rmin, rmax, b=None)

Initialize power transform instance.

Parameters
  • rmin (float) – Minimum value for transformed points

  • rmax (float) – Maximum value for transformed points

  • b (float) – The parameter b that maps to \(r_{max}\).

property b

Parameter \(b\) that maps/transforms to \(r_{max}\).

Type

float

property codomain

Transformation codomain.

Type

tuple

deriv(x)

Compute first derivative of power transform.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

First derivative of transformation at x.

Return type

ndarray(N,)

deriv2(x)

Compute second derivative of power transform.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

Second derivative of transformation at x.

Return type

ndarray(N,)

deriv3(x)

Compute third derivative of power transform.

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

Third derivative of transformation at x.

Return type

ndarray(N,)

property domain

Transformation domain.

Type

tuple

inverse(r)

Compute the inverse of power transform.

\[x(r) = \frac{r}{r_{min}}^{\frac{\log(b + 1)}{\log(r_{max}) - \log(r_{min})}} - 1\]

such that \(r(b) = r_{max}\).

Parameters

r (ndarray(N,)) – One-dimensional array in the co-domain of the transformation.

Returns

Inverse of transformation at r.

Return type

ndarray(N,)

property rmax

the value of rmax.

Type

float

property rmin

the value of rmin.

Type

float

set_maximum_parameter_b(x)

Sets up the parameter b from taken the maximum over x.

transform(x)

Perform power transform.

\[r = r_{min} (x + 1)^{\frac{\log(r_{max} - \log(r_{min}}{b + 1}},\]

such that \(r(b) = r_{max}\).

Parameters

x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).

Returns

The transformation of x to the co-domain of the transformation.

Return type

ndarray(N,)

transform_1d_grid(oned_grid)

Generate a new integral grid by transforming the provided grid.

\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
Parameters

oned_grid (OneDGrid) – An instance of one-dimensional grid.

Returns

Transformed one-dimensional grid spanning a different domain.

Return type

OneDGrid