DiracODE::Internal::DiracDerivative

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Derivative matrix for the radial Dirac equation, dF/du = D(u)*F(u) + S(u).

Implements AdamsMoulton::DerivativeMatrix<std::size_t, double>, using the grid index i as the argument type (T = std::size_t). The ODE is solved in terms of the grid parameter u (where r = r(u)), so all matrix elements include a dr/du Jacobian factor.

The radial Dirac equation for a central potential v(r) is:

\[ \frac{d}{du}\begin{pmatrix}f\\g\end{pmatrix} = \frac{dr}{du} \begin{pmatrix} -\kappa/r + \alpha H_\text{mag} & \alpha(\varepsilon - v) + 2mc \\ \alpha(v - \varepsilon) & \kappa/r - \alpha H_\text{mag} \end{pmatrix} \begin{pmatrix}f\\g\end{pmatrix} + S \]

where \( c = 1/\alpha \) is the speed of light in atomic units. The optional inhomogeneous source S encodes the exchange interaction:

\[ S_f = -\alpha \, [V_x F_a]_g \, \frac{dr}{du}, \quad S_g = +\alpha \, [V_x F_a]_f \, \frac{dr}{du}. \]

Note

Non-copyable; the constructor stores raw pointers to the grid and potential arrays, which must outlive this object.

#include <BoundState.hpp>

Inheritance diagram for DiracODE::Internal::DiracDerivative:

Public Member Functions
	DiracDerivative (const Grid &in_grid, const std::vector< double > &in_v, const int in_k, const double in_en, const double in_alpha, const std::vector< double > &V_off_diag={}, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const iFa0=nullptr, double zion=1, double in_mass=1.0)
	Constructs the Dirac derivative matrix for a given orbital and potential.
double	a (std::size_t i) const final
	D matrix elements (see DiracDerivative for definitions); index i is grid point.
double	b (std::size_t i) const final
double	c (std::size_t i) const final
double	d (std::size_t i) const final
double	Sf (std::size_t i) const final
	Inhomogeneous source terms from exchange potential VxFa.
double	Sg (std::size_t i) const final
	DiracDerivative (const DiracDerivative &)=delete
void	operator= (const DiracDerivative &)=delete

Public Attributes
const Grid *const	pgr
const std::vector< double > *const	v
const std::vector< double > *const	Hmag
const DiracSpinor *const	VxFa
const DiracSpinor *const	Fa0
const double	zion = 1.0
const int	k
const double	en
const double	alpha
const double	cc
double	mass

Constructor & Destructor Documentation

DiracDerivative()

DiracODE::Internal::DiracDerivative::DiracDerivative	(	const Grid &	in_grid,
		const std::vector< double > &	in_v,
		const int	in_k,
		const double	in_en,
		const double	in_alpha,
		const std::vector< double > &	V_off_diag = {},
		const DiracSpinor *const	VxFa = nullptr,
		const DiracSpinor *const	iFa0 = nullptr,
		double	zion = 1,
		double	in_mass = 1.0
	)

Constructs the Dirac derivative matrix for a given orbital and potential.

Parameters

in_grid	Radial grid.
in_v	Local potential v(r).
in_k	Orbital kappa quantum number.
in_en	Orbital energy.
in_alpha	Fine-structure constant (alpha).
V_off_diag	Optional off-diagonal (magnetic) potential H_mag(r). If empty, treated as zero.
VxFa	Optional exchange potential VxFa. If nullptr, ignored.
iFa0	Optional inhomogeneous source spinor. If nullptr, ignored.
zion	Effective ionic charge (used for boundary conditions).
in_mass	Effective particle mass in atomic units (default 1).

Member Function Documentation

a()

double DiracODE::Internal::DiracDerivative::a ( std::size_t  i ) const

finalvirtual

D matrix elements (see DiracDerivative for definitions); index i is grid point.

Implements AdamsMoulton::DerivativeMatrix< std::size_t, double >.

b()

double DiracODE::Internal::DiracDerivative::b ( std::size_t  i ) const

finalvirtual

Implements AdamsMoulton::DerivativeMatrix< std::size_t, double >.

c()

double DiracODE::Internal::DiracDerivative::c ( std::size_t  i ) const

finalvirtual

Implements AdamsMoulton::DerivativeMatrix< std::size_t, double >.

d()

double DiracODE::Internal::DiracDerivative::d ( std::size_t  i ) const

finalvirtual

Implements AdamsMoulton::DerivativeMatrix< std::size_t, double >.

Sf()

double DiracODE::Internal::DiracDerivative::Sf ( std::size_t  i ) const

finalvirtual

Inhomogeneous source terms from exchange potential VxFa.

Reimplemented from AdamsMoulton::DerivativeMatrix< std::size_t, double >.

Sg()

double DiracODE::Internal::DiracDerivative::Sg ( std::size_t  i ) const

finalvirtual

Reimplemented from AdamsMoulton::DerivativeMatrix< std::size_t, double >.

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The documentation for this struct was generated from the following files:

• DiracODE/BoundState.hpp
• DiracODE/BoundState.cpp