sirius::Matching_coefficients Class Reference

#include <matching_coefficients.hpp>

Public Member Functions
	Matching_coefficients (Unit_cell const &unit_cell__, fft::Gvec const &gkvec__)
	Constructor. More...
template<bool conjugate, typename T , typename = std::enable_if_t<!std::is_scalar<T>::value>>
void	generate (Atom const &atom__, sddk::mdarray< T, 2 > &alm__) const
	Generate plane-wave matching coefficients for the radial solutions of a given atom. More...
auto const &	gkvec () const

Private Member Functions
template<int N, bool conjugate, typename T , typename = std::enable_if_t<!std::is_scalar<T>::value>>
void	generate (int ngk, std::vector< std::complex< double > > const &phase_factors__, int iat, int l, int lm, int nu, r3::matrix< double > const &A, T *alm) const
	Generate matching coefficients for a specific \( \ell \) and order. More...

Private Attributes
Unit_cell const &	unit_cell_
	Description of the unit cell. More...
fft::Gvec const &	gkvec_
	Description of the G+k vectors. More...
std::vector< double >	gkvec_len_
sddk::mdarray< std::complex< double >, 2 >	gkvec_ylm_
	Spherical harmonics Ylm(theta, phi) of the G+k vectors. More...
sddk::mdarray< std::complex< double >, 4 >	alm_b_
	Precomputed values for the linear equations for matching coefficients. More...

Detailed Description

The following matching conditions must be fulfilled:

\[ \frac{\partial^j}{\partial r^j} \sum_{L \nu} A_{L \nu}^{\bf k}({\bf G})u_{\ell \nu}(r) Y_{L}(\hat {\bf r}) \bigg|_{R^{MT}} = \frac{\partial^j}{\partial r^j} \frac{4 \pi}{\sqrt \Omega} e^{i{\bf (G+k)\tau}} \sum_{L}i^{\ell} j_{\ell}(|{\bf G+k}|r) Y_{L}^{*}(\widehat {\bf G+k}) Y_{L}(\hat {\bf r}) \bigg|_{R^{MT}} \]

where \( L = \{ \ell, m \} \). Dropping sum over L we arrive to the following system of linear equations:

\[ \sum_{\nu} \frac{\partial^j u_{\ell \nu}(r)}{\partial r^j} \bigg|_{R^{MT}} A_{L \nu}^{\bf k}({\bf G}) = \frac{4 \pi}{\sqrt \Omega} e^{i{\bf (G+k)\tau}} i^{\ell} \frac{\partial^j j_{\ell}(|{\bf G+k}|r)}{\partial r^j} \bigg|_{R^{MT}} Y_{L}^{*}(\widehat {\bf G+k}) \]

The matching coefficients are then equal to:

\[ A_{L \nu}^{\bf k}({\bf G}) = \sum_{j} \bigg[ \frac{\partial^j u_{\ell \nu}(r)}{\partial r^j} \bigg|_{R^{MT}} \bigg]_{\nu j}^{-1} \frac{\partial^j j_{\ell}(|{\bf G+k}|r)}{\partial r^j} \bigg|_{R^{MT}} \frac{4 \pi}{\sqrt \Omega} i^{\ell} e^{i{\bf (G+k)\tau}} Y_{L}^{*}(\widehat {\bf G+k}) \]

Definition at line 53 of file matching_coefficients.hpp.

Constructor & Destructor Documentation

Matching_coefficients()

sirius::Matching_coefficients::Matching_coefficients ( Unit_cell const &  unit_cell__, fft::Gvec const &  gkvec__  )

inline

Constructor.

Definition at line 112 of file matching_coefficients.hpp.

Member Function Documentation

generate() [1/2]

template<int N, bool conjugate, typename T , typename = std::enable_if_t<!std::is_scalar<T>::value>>

void sirius::Matching_coefficients::generate ( int  ngk, std::vector< std::complex< double > > const &  phase_factors__, int  iat, int  l, int  lm, int  nu, r3::matrix< double > const &  A, T *  alm  ) const

inlineprivate

Generate matching coefficients for a specific \( \ell \) and order.

Parameters

[in]	ngk	Number of G+k vectors.
[in]	phase_factors	Phase factors of G+k vectors.
[in]	iat	Index of atom type.
[in]	l	Orbital quantum nuber.
[in]	lm	Composite l,m index.
[in]	nu	Order of radial function \( u_{\ell \nu}(r) \) for which coefficients are generated.
[in]	A	Inverse matrix of radial derivatives.
[out]	alm	Pointer to alm coefficients.

Definition at line 81 of file matching_coefficients.hpp.

generate() [2/2]

template<bool conjugate, typename T , typename = std::enable_if_t<!std::is_scalar<T>::value>>

void sirius::Matching_coefficients::generate ( Atom const &  atom__, sddk::mdarray< T, 2 > &  alm__  ) const

inline

Generate plane-wave matching coefficients for the radial solutions of a given atom.

Parameters

[in]	atom	Atom, for which matching coefficients are generated.
[out]	alm	Array of matching coefficients with dimension indices \( ({\bf G+k}, \xi) \).

Definition at line 193 of file matching_coefficients.hpp.

gkvec()

auto const & sirius::Matching_coefficients::gkvec ( ) const

inline

Definition at line 296 of file matching_coefficients.hpp.

Member Data Documentation

unit_cell_

Unit_cell const& sirius::Matching_coefficients::unit_cell_

private

Description of the unit cell.

Definition at line 57 of file matching_coefficients.hpp.

gkvec_

fft::Gvec const& sirius::Matching_coefficients::gkvec_

private

Description of the G+k vectors.

Definition at line 60 of file matching_coefficients.hpp.

gkvec_len_

std::vector<double> sirius::Matching_coefficients::gkvec_len_

private

Definition at line 62 of file matching_coefficients.hpp.

gkvec_ylm_

sddk::mdarray<std::complex<double>, 2> sirius::Matching_coefficients::gkvec_ylm_

private

Spherical harmonics Ylm(theta, phi) of the G+k vectors.

Definition at line 65 of file matching_coefficients.hpp.

alm_b_

sddk::mdarray<std::complex<double>, 4> sirius::Matching_coefficients::alm_b_

private

Precomputed values for the linear equations for matching coefficients.

Definition at line 68 of file matching_coefficients.hpp.

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The documentation for this class was generated from the following file:

• matching_coefficients.hpp