nitrogen.basis¶

Basis set functions including discrete-variable representations (DVRs) and finite-basis representations (FBRs). The main objects are the GriddedBasis class and its sub-classes GenericDVR and NDBasis. See Discrete-variable representation (DVR) bases for a tutorial.

General gridded bases
`GriddedBasis`	General quadrature grid basis.
`ConcatenatedBasis`	Direct sum of `GriddedBasis`
Discrete-variable representation bases
`GenericDVR`	Parent DVR basis class.
`SimpleDVR`	Simple one-dimensional DVRs.
`Contracted`	Contracted DVR.
FBR quadrature bases
`NDBasis`	Parent class for \(n\)-d quadrature bases
`SinCosBasis`	A sine-cosine (real Fourier) basis.
`LegendreLMCosBasis`	Associated Legendre polynomials.
`RealSphericalHBasis`	Real spherical harmonics.
`Real2DHOBasis`	Two-dimensional harmonic oscillator.
`RadialHOBasis`	Radial HO basis in \(d\) dimensions.

class nitrogen.basis.ConcatenatedBasis(bases)¶

Bases: nitrogen.basis.genericbasis.GriddedBasis

A concatenated set of GriddedBasis basis sets. It is assumed these have compatible grid methods.

Create a GriddedBasis

Parameters

gridpts ((nd, ng) array_like) – The ng grid points for each of the nd coordinates.
nb (int) – The number of basis functions.
wgtfun (DFun, optional) – The integration weight function

class nitrogen.basis.Contracted(U, prim_dvr)¶

Bases: nitrogen.basis.dvr.GenericDVR

A contracted basis DVR class (usually for PO-DVRs).

prim_dvr¶

The primitive basis

Type: GenericDVR

W¶

The transformation matrix from the contracted DVR to the primitive DVR

Type: ndarray

Parameters

U ((n,m) ndarray) – A unitary operator defining the contracted basis. n is the size of the primitive basis. m is the size of the contracted basis.
prim_dvr (GenericDVR) – The primitive DVR

class nitrogen.basis.GenericDVR(grid, D, D2)¶

Bases: nitrogen.basis.genericbasis.GriddedBasis

A super-class for generic 1D DVRs.

num¶

The number of DVR grid points

Type: int

grid¶

The DVR grid points

Type: ndarray

D¶

First derivative operator.

Type: ndarray

D2¶

Second derivative operator.

Type: ndarray

Create a GriddedBasis

Parameters

gridpts ((nd, ng) array_like) – The ng grid points for each of the nd coordinates.
nb (int) – The number of basis functions.
wgtfun (DFun, optional) – The integration weight function

matchfun(f)¶

Calculate DVR coefficients to match a function at DVR grid points.

Parameters: f (function) – The function to be matched.
Returns: coeff – A (num,1) array with the DVR basis function coefficients.
Return type: ndarray

wfs(q)¶

Evaluate DVR basis wavefunctions.

Parameters: q (ndarray) – A 1-D array of coordinate values.
Returns: wfs – An array of shape (q.size, num) containing the values of the DVR wavefunctions evaluated at coordinate values q.
Return type: ndarray

class nitrogen.basis.GriddedBasis(gridpts, nb, wgtfun=None)¶

Bases: object

A generic super-class for gridded basis sets, including DVRs and FBRs equipped with quadrature grids.

nb¶

The number of basis functions

Type: int

nd¶

The number of grid coordinates

Type: int

ng¶

The number of grid points

Type: int

gridpts¶

The grid points

Type: (nd,ng) ndarray

wgtfun¶

An integration weight function. If None, this is unity.

Type: DFun

Create a GriddedBasis

Parameters

gridpts ((nd, ng) array_like) – The ng grid points for each of the nd coordinates.
nb (int) – The number of basis functions.
wgtfun (DFun, optional) – The integration weight function

getDQuadOp(var)¶: calculate the explicit basis-to-quadrature derivative grid transformation matrix

getQuadOp()¶: calculate the explicit basis-to-quadrature grid transformation matrix

class nitrogen.basis.LegendreLMCosBasis(m, lmax, Nq=None)¶

Bases: nitrogen.basis.ndbasis.NDBasis

Associated Legendre polynomials with cosine argument, \(F_\ell^m(\theta) \propto P_\ell^m(\cos\theta)\). See LegendreLMCos.

Quadrature is performed with a Gauss-Legendre grid.

These functions are eigenfunctions of the differential operator

\[-\frac{\partial^2}{\partial \theta^2} - \cot \theta \frac{\partial}{\partial \theta} + \frac{m^2}{\sin^2\theta}\]

with eigenvalue \(\ell(\ell+1)\).

m¶

The associated Legendre order.

Type: int

l¶

A list of l-indices of the basis functions.

Type: ndarray

See also

nitrogen.special.LegendreLMCos: DFun sub-class for associated Legendre functions.

Create a LegendreLMCosBasis.

Parameters

m (int) – The associated Legendre order.
lmax (int) – The maximum l index.
Nq (int, optional) – The number of quadrature points. The default is 2*lmax + 1.

class nitrogen.basis.NDBasis(basisfun, wgtfun, qgrid, wgt)¶

Bases: nitrogen.basis.genericbasis.GriddedBasis

A generic multi-dimensional finite basis representation supporting quadrature integration/transformation. NDBasis objects define a set of \(N_b\) basis functions, \(\phi_i(\vec{x})\), \(i = 0,\ldots,N_b - 1\), where \(\vec{x}\) is an \(n_d\)-dimensional coordinate vector. Matrix elements with these functions are defined with respect to a weighted integral,

\[\langle \phi_i \vert \phi_j \rangle = \int d\vec{x}\,\Omega(\vec{x}) \phi_i(\vec{x}) \phi_j(\vec{x}),\]

where \(\Omega(\vec{x})\) is the weight function (NDBasis.wgtfun).

These integrals can be approximated with a quadrature over \(N_q\) (possibly scattered) grid points \(\vec{x}_k\),

\[\int d\vec{x}\, \Omega(\vec{x}) f(\vec{x}) \approx \sum_{k=0}^{N_q-1} w_k f(\vec{x}_k),\]

where \(w_k\) are the quadrature weights (NDBasis.wgt).

nd¶

The number of dimensions (i.e. coordinates)

Type: int

Nb¶

The number of basis functions

Type: int

Nq¶

The number of quadrature points

Type: int

qgrid¶

The quadrature grid.

Type: (nd,`Nq`) ndarray

wgt¶

The quadrature weights.

Type: (Nq,) ndarray

bas¶

The basis functions evaluated on the quadrature grid.

Type: (Nb,`Nq`) ndarray

basisfun¶

An Nb-output-valued DFun of the basis functions with nd input variables.

Type: DFun

wgtfun¶

The weight function associated with matrix element integrals of the basis.

Type: DFun

Initialize a generic NDBasis.

Parameters

basisfun (DFun) – The basis function DFun.
wgtfun (DFun or None) – The integration weight function. If None, this is assumed to be unity.
qgrid ((nd, Nq) ndarray) – The quadrature points.
wgt ((Nq,) ndarray) – The quadrature weights.

fbrToQuad(v, axis=0)¶

Transform an axis from the FBR to the quadrature grid representation.

Parameters: v ((...,`Nb`,...) ndarray) – An array with the axis index spanned by this basis.
Returns: w – The transformed array.
Return type: (…, Nq, …) ndarray

quadD(x, var, axis=0)¶

Apply the quadrature representation derivative operator.

Parameters

x ((...,`Nq`,...) ndarray) – The quadrature representation array.
var (int) – The coordinate of the derivative.
axis (int, optional) – The quadrature axis. The default is 0.

Returns

y – The result.

Return type

(…,`Nq`,…) ndarray

quadDH(x, var, axis=0)¶

Apply the quadrature representation D^dagger operator.

Parameters

x ((...,`Nq`,...) ndarray) – The quadrature representation array.
var (int) – The coordinate of the derivative.
axis (int, optional) – The quadrature axis. The default is 0.

Returns

y – The result.

Return type

(…,`Nq`,…) ndarray

quadToFbr(w, axis=0)¶

Transform an axis from the quadrature representation to the FBR.

Parameters: w ((...,`Nq`,...) ndarray) – An array with the axis index spanned by this quadrature.
Returns: v – The transformed array.
Return type: (…, Nb, …) ndarray

class nitrogen.basis.RadialHOBasis(vmax, rmax, ell, d=2, Nr=None)¶

Bases: nitrogen.basis.ndbasis.NDBasis

A radial basis for a harmonic oscillator in \(d\) dimensions.

Create a RadialHO basis.

Parameters

vmax (int) – The number of basis functions is vmax - ell + 1. This is not the conventional vibrational quantum number.
rmax (float) – The radial extent of the basis.
ell (scalar) – The generalized angular momentum quantum number.
d (int) – The dimension, \(d\).
Nr (int, optional) – The number of quadrature points over \(r\). The default is vmax + 3.

class nitrogen.basis.Real2DHOBasis(vmax, rmax, ell=None, Nr=None, Nphi=None, angle='rad')¶

Bases: nitrogen.basis.ndbasis.NDBasis

Real 2-D isotropic harmonic oscillator basis functions in cylindrical coordinates. \(\chi_n^{\ell}(r,\phi) = R_n^{\ell}(r)f_{\ell}(\phi)\). See Real2DHO.

Quadrature is performed with a direct product of a Gauss-Laguerre-type grid over \(r\) and a uniform Fourier grid over \(\phi\).

The integration weight function is \(\Omega(r,\phi) = r\).

v¶

The \(v\) quantum numbers, where \(v = 2n + \vert \ell \vert\).

Type: ndarray

ell¶

The \(\ell\) quantum numbers.

Type: ndarray

n¶

The \(n\) Laguerre degree.

Type: ndarray

rmax¶

The radial extent of the basis.

Type: float

alpha¶

The radial scaling parameter, \(\alpha\), corresponding to radial extent R.

Type: float

angle¶

The angular units.

Type: {‘rad’, ‘deg’}

See also

nitrogen.special.Real2DHO: DFun sub-class real 2-D harmonic oscillator wavefunctions.
nitrogen.special.RadialHO: DFun sub-class for d-dimensional radial harmonic oscillator wavefunctions.
nitrogen.special.SinCosDFun: DFun sub-class for sine-cosine basis.

Create a Real2DHO basis.

Parameters

vmax (int) – The maximum vibrational quantum number \(v\), in the conventional sum-of-modes sense.
rmax (float) – The radial extent of the basis.
ell (scalar or 1-D array_like, optional) – The angular momentum quantum number. If scalar, then all \(\ell\) with \(|\ell| \leq\) abs(ell) will be included. If array_like, then ell lists all (signed) \(\ell\) values to be included. A value of None is equivalent to ell = vmax. The default is None.
Nr (int, optional) – The number of quadrature points over \(r\). The default is vmax + 1.
Nphi (int, optional) – The number of quadrature points over \(\phi\). The default is \(2(\ell_{max} + 1)\).
angle ({'rad', 'deg'}) – The angular unit. The default is ‘rad’.

class nitrogen.basis.RealSphericalHBasis(m, lmax, Ntheta=None, Nphi=None)¶

Bases: nitrogen.basis.ndbasis.NDBasis

Real spherical harmonics, \(\Phi_\ell^m(\theta,\phi) = F_\ell^m(\theta)f_m(\phi)\). See RealSphericalH.

The integration weight function is \(\Omega(\theta,\phi) = \sin(\theta)\). Quadrature is performed with a direct product of a Gauss-Legendre grid over \(\theta\) and a uniform Fourier grid over \(\phi\).

These functions are eigenfunctions of the differential operator

\[-\frac{\partial^2}{\partial \theta^2} - \cot \theta \frac{\partial}{\partial \theta} - \frac{\partial_\phi^2}{\sin^2\theta}\]

with eigenvalue \(\ell(\ell+1)\).

m¶

The projection quantum number \(m\).

Type: ndarray

l¶

The azimuthal quantum number \(\ell\).

Type: ndarray

See also

nitrogen.special.RealSphericalH: DFun sub-class real spherical harmonics.
nitrogen.special.LegendreLMCos: DFun sub-class for associated Legendre polynomials.
nitrogen.special.SinCosDFun: DFun sub-class for sine-cosine basis.

Create a RealSphericalH basis.

Parameters

m (int or array_like) – The associated Legendre order(s).
lmax (int) – The maximum \(\ell\) quantum number.
Ntheta (int, optional) – The number of quadrature points over \(\theta\). The default is 2*`lmax` + 1.
Nphi (int, optional) – The number of quadrature points over \(\phi\). The default is 2(mmax+1).

class nitrogen.basis.SimpleDVR(start=0, stop=1, num=10, basis='sinc')¶

Bases: nitrogen.basis.dvr.GenericDVR

Standard 1D DVRs.

start¶

The DVR grid starting value.

Type: float

stop¶

The DVR grid stopping value.

Type: float

basis¶

The DVR basis type.

Type: {‘sinc’,’ho’,’fourier’, ‘legendre’}

Create a DVR object.

Parameters

start (float, optional) – The DVR grid starting value. The default is 0.
stop (float, optional) – The DVR grid stopping value. The default is 1.
num (int, optional) – The number of DVR grid points. The default is 10.
basis ({'sinc','ho','fourier','legendre'}, optional) – The DVR basis type. The default is ‘sinc’. See Notes for details.

Notes

Each basis option constructs a one-dimensional DVR over a grid defined by start, stop, and num. All basis types have a simple \(dq\) volume element. Descriptions for each follow:

basis value	Description
`'sinc'`	\(\text{sinc}\) basis
`'ho'`	Harmonic oscillator DVR.
`'fourier'`	Fourier DVR (a periodic version of `'sinc'`).
`'legendre'`	Legendre polynomial DVR.

class nitrogen.basis.SinCosBasis(m=10, Nq=None, angle='rad')¶

Bases: nitrogen.basis.ndbasis.NDBasis

A 1-D, real sine/cosine basis set,

\[\begin{split}f_m(\phi) &= 1/\sqrt{\pi} \sin(|m|\phi) &\ldots m < 0\\ &= 1/\sqrt{2\pi} &\ldots m = 0\\ &= 1/\sqrt{\pi} \cos(m \phi) &\ldots m > 0\end{split}\]

m¶

The m quantum number of each basis function.

Type: ndarray

angle¶

The angular unit.

Type: {‘rad’, ‘deg’}

See also

nitrogen.special.SinCosDFun: DFun sub-class for sine-cosine basis set.

A real sine-cosine Fourier basis.

Parameters

m (int or 1-D array_like of int, optional) – If scalar, the \(2|m|+1\) basis functions with index \(\leq |m|\) will be included. If array_like, then m lists all m-indices to be included. The default is m = 10.
Nq (int, optional) – The number of quadrature points. The default is 2*max(abs(m)) + 3
angle ({'rad', 'deg'}) – The angular unit. The default is ‘rad’.

class nitrogen.basis.StructuredBasis(bases, structure)¶

Bases: nitrogen.basis.ndbasis.NDBasis

Multi-dimensional bases that have a structured product form, such as spherical harmonics, D-matrices, etc, and direct product quadrature grids.

Initialize a generic StructuredBasis.

Parameters

bases (list of NDBasis objects) – The basis function factors
structure ((Nb, nfs) array_like) – The product structure of each basis function in terms of the indices of each factor basis in bases.

Notes

Currently only 1-dimensional factor bases are supported. This may be expanded in the future.

nitrogen.basis.basisShape(bases)¶

Determine the direct-product shape of a set of GriddedBasis and/or scalars.

Parameters

bases (list of GriddedBasis and scalar) – The direct-product factors. Scalars become singleton dimensions.

Returns

shape (tuple) – The basis shape.
size (int) – The total size.

nitrogen.basis.basisVar(bases)¶

Return the list of active coordinates (i.e. non-scalar entries) represented by a list of GriddedBasis or scalars

Parameters: bases (list of GriddedBasis and scalar) – The direct-product factors. Scalars become singleton dimensions.
Returns: var – The ordered list of active coordinates.
Return type: list

nitrogen.basis.coordAxis(bases)¶

Determine the direct-product axis to which each coordinate belongs

Parameters: bases (list of GriddedBasis and scalar) – The direct-product factors. Scalars become singleton dimensions.
Returns: axis_of_coord – The direct-product axis of each coordinate.
Return type: list

nitrogen.basis.coordSubcoord(bases)¶

Determine the intra-basis coordinate index of each coordinate represented by the complete basis. Scalar singletons will just take 0.

Parameters: bases (list of GriddedBasis and scalar) – The direct-product factors. Scalars become singleton dimensions.
Returns: axis_of_coord – The intra-basis coordinate sub-index of each coordinate.
Return type: list

nitrogen.basis.sameGrid(A, B)¶

Check whether GriddedBases A and B have the same grid. This also handles scalar input, in which case simple equality is checked.

A, B : GriddedBasis or scalar

Returns: True if equal grids, otherwise False.
Return type: bool

nitrogen.basis.ops¶

DVR grid operations

nitrogen.basis.ops.opDD_grid(x, f, h, G, Dlist)¶

Evaluate the matrix-vector operation for \(\sum_{k,\ell} f \partial_k^\dagger G^{kl} h \partial_\ell f\)

Parameters

x (ndarray) – Input grid
f (ndarray) – Weighting functions evaluated over grid.
h (ndarray) – Weighting functions evaluated over grid.
G (ndarray) – Inverse metric grids in packed storage for active coordinates only.
Dlist (list of ndarrays) – List of the D operator for each dimension. An entry of None will be skipped.

Returns

y – The result grid

Return type

ndarray

nitrogen.basis.ops.opD_grid(x, f, Dlist)¶

Evaluate the matrix-vector operation for \(\sum_{k} -\partial_k^\dagger f_k + f_k \partial_k\)

Parameters

x (ndarray) – Input grid
f (ndarray) – Weighting functions evaluated over grid for each active vibrational index k.
Dlist (list of ndarrays) – List of the D operator for each dimension. An entry of None for inactive coordinates will be skipped.

Returns

y – The result grid

Return type

ndarray

nitrogen.basis.ops.opO(x, O, axis)¶

Apply matrix operator O to a single axis

Parameters

x (ndarray) – Input grid
O (ndarray) – An (m,n) matrix
axis (int) – The axis O acts on. x.shape[axis] must equal n.

Returns

y – The result array, with y.shape[axis] equal to m.

Return type

ndarray

nitrogen.basis.ops.opO_coeff(x, Olist, c=None)¶

Evaluate the matrix-vector operation for \(\sum_{k} c_k \hat{O}_k\)

Parameters

x (ndarray) – Input grid
Olist (list of ndarrays) – List of the matrix operator for each dimension. An entry of None for inactive coordinates will be skipped.
c (array_like, optional) – Coefficients for each term in sum, including inactive operators (whose values are ignored). If None, this is ignored.

Returns

y – The result grid

Return type

ndarray

nitrogen.basis.ops.opTensorO(x, Os)¶

Apply a sequence of matrix operators to each axis.

Parameters

x (ndarray) – Input grid
Os (list of ndarray) – The operator matrix for each axis. An entry of None is the same as identity.

Returns

y – The result array.

Return type

ndarray

nitrogen.basis¶

nitrogen.basis.ops¶

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