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Magnetic dipole operator: <a||M1||b>

\[ <a||M1||b> = R (k_a + k_b) <-k_a||C^1||k_b> \]

\[R = \frac{-3}{\alpha^2\omega} \int (f_ag_b+g_af_b) j_1(kr) \, dr\]

\( k = \omega/c = \omega*\alpha \). Negative sign (and alpha) puts into units |mu_B|. For k<<1 (static case): j1(kr) -> (r*k)/3,

\[R = \frac{-1}{\alpha} \int (f_ag_b+g_af_b) r \, dr\]

Probably, alpha should always by actual alpha, just units, not relativistic?

Be careful with this operator - compare with: U. I. Safronova, M. S. Safronova, and W. R. Johnson, Phys. Rev. A 95, 042507 (2017).

#include <M1.hpp>

Inheritance diagram for DiracOperator::M1:

Public Member Functions
	M1 (const Grid &gr, const double alpha, const double omega=0.0)

M1 &	operator= (const M1 &)=delete

	M1 (const M1 &)=default

std::string	name () const override final
	Returns "name" of operator (e.g., 'E1')

std::string	units () const override final
	Returns units of operator as a string (usually au, may be MHz, etc.)

double	angularF (const int ka, const int kb) const override final
	Angular factor A_ab linking the radial integral to the RME.

double	angularCff (int, int) const override final
	Angular coefficient C_ff for the f_a*f_b term of the radial integral.

double	angularCgg (int, int) const override final
	Angular coefficient C_gg for the g_a*g_b term of the radial integral.

double	angularCfg (int, int) const override final
	Angular coefficient C_fg for the f_a*g_b term of the radial integral.

double	angularCgf (int, int) const override final
	Angular coefficient C_gf for the g_a*f_b term of the radial integral.

void	updateFrequency (const double omega) override final
	Updates the operator for a new frequency omega.

Public Member Functions inherited from DiracOperator::TensorOperator
	TensorOperator (const TensorOperator &)=default

TensorOperator &	operator= (const TensorOperator &)=default

	TensorOperator (TensorOperator &&)=default

TensorOperator &	operator= (TensorOperator &&)=default

bool	freqDependantQ () const
	Returns true if the operator is frequency-dependent (requires updateFrequency() calls).

bool	isZero (int ka, int kb) const
	Returns true if <a\|h\|b> = 0 by rank/parity selection rules.

bool	isZero (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Overload taking DiracSpinors; forwards to isZero(ka, kb).

bool	selectrion_rule (int twoJA, int piA, int twoJB, int piB) const
	Returns true if the matrix element is non-zero by angular momentum and parity selection rules (arguments are 2j and pi as integers).

virtual void	updateRank (int)

const std::vector< double > &	getv () const
	Returns a const ref to the stored vector v.

double	getc () const
	Returns the "overall" constant c.

bool	imaginaryQ () const
	returns true if operator is imaginary (has imag MEs)

int	rank () const
	Rank k of operator.

int	parity () const
	returns parity, as integer (+1 or -1)

int	symm_sign (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	returns relative sign between <a\|\|x\|\|b> and <b\|\|x\|\|a>

double	angularCxy (uint8_t x, uint8_t y, int kappa_a, int kappa_b) const
	Dispatches to angularCff/fg/gf/gg based on component indices x, y.

virtual std::unique_ptr< TensorOperator >	clone () const
	Returns a polymorphic copy of the operator at its current state, or nullptr if cloning is not supported by the derived class.

virtual DiracSpinor	radial_rhs (const int kappa_a, const DiracSpinor &Fb) const
	Computes the right-hand spinor dF_b for the radial integral.

virtual double	radialIntegral (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Radial integral R_ab, defined by RME = angularF(a,b) * radialIntegral(a,b).

double	rme3js (int twoja, int twojb, int two_mb=1, int two_q=0) const
	3j-symbol factor linking the full ME to the RME.

double	rme3js (const DiracSpinor &Fa, const DiracSpinor &Fb, int two_mb=1, int two_q=0) const
	Overload of rme3js taking DiracSpinors.

DiracSpinor	reduced_rhs (const int ka, const DiracSpinor &Fb) const
	Returns angularF(ka,kb) * radial_rhs(ka,Fb); spinor-valued RME action on Fb, used in perturbation theory/TDHF.

DiracSpinor	reduced_lhs (const int ka, const DiracSpinor &Fb) const
	As reduced_rhs but for the conjugate direction; Fb * reduced_lhs(ka, Fb) = <b\|\|h\|\|a>.

double	reducedME (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Returns the reduced matrix element <a\|\|h\|\|b> = A_ab * R_ab.

double	fullME (const DiracSpinor &Fa, const DiracSpinor &Fb, std::optional< int > two_ma=std::nullopt, std::optional< int > two_mb=std::nullopt, std::optional< int > two_q=std::nullopt) const
	Returns "full" matrix element, for optional (ma, mb, q) [taken as int 2*]. If not specified, returns z-component (q=0), with ma=mb=min(ja,jb)

double	matel_factor (MatrixElementType type, int twoJa, int twoJb) const
	Returns the factor to convert a reduced ME to a different form (Reduced, Stretched, or HFConstant); see MatrixElementType.

double	matel_factor (MatrixElementType type, const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Overload of matel_factor taking DiracSpinors.

Additional Inherited Members
Protected Member Functions inherited from DiracOperator::TensorOperator
	TensorOperator (int rank_k, Parity pi, double constant=1.0, const std::vector< double > &vec={}, Realness RorI=Realness::real, bool freq_dep=false)
	Constructs a specific tensor operator. Called by derived classes.

Protected Attributes inherited from DiracOperator::TensorOperator
int	m_rank

Parity	m_parity

Realness	m_Realness

bool	m_freqDependantQ {false}

double	m_constant

std::vector< double >	m_vec

Member Function Documentation

◆ name()

std::string DiracOperator::M1::name ( ) const

inlinefinaloverridevirtual

Returns "name" of operator (e.g., 'E1')

Reimplemented from DiracOperator::TensorOperator.

◆ units()

std::string DiracOperator::M1::units ( ) const

inlinefinaloverridevirtual

Returns units of operator as a string (usually au, may be MHz, etc.)

Reimplemented from DiracOperator::TensorOperator.

◆ angularF()

double DiracOperator::M1::angularF	(	const int	,
		const int
	)		const

inlinefinaloverridevirtual

Angular factor A_ab linking the radial integral to the RME.

All derived operators must implement this. It gives the purely angular part of the reduced matrix element:

\[ \langle a \| \hat{h} \| b \rangle \equiv A_{ab} \cdot R_{ab} \]

where \( R_{ab} \) is returned by radialIntegral(). For most operators, \( A_{ab} \) is a product of Clebsch-Gordan / 3j coefficients and depends only on \( \kappa_a, \kappa_b \) (and the rank \( k \) and parity \( \pi \) of the operator).

Note: This is a pure virtual function – every derived operator must provide an implementation.

Implements DiracOperator::TensorOperator.

◆ angularCff()

double DiracOperator::M1::angularCff	(	int	kappa_a,
		int	kappa_b
	)		const

inlinefinaloverridevirtual

Angular coefficient C_ff for the f_a*f_b term of the radial integral.

The default radial integral is structured as:

\[ R_{ab} = c\int_0^\infty v(r)\left( C_{ff}\,f_a f_b + C_{fg}\,f_a g_b + C_{gf}\,g_a f_b + C_{gg}\,g_a g_b \right)\,{\rm d}r \]

These coefficients are often constants, but may depend on \( \kappa_a, \kappa_b \) for operators with angular-momentum-dependent coupling between large and small components (e.g., spin-dependent operators). Override in derived classes as needed.

Parameters

kappa_a	kappa \( \kappa_a \) for left-hand-side (bra)
kappa_b	kappa \( \kappa_b \) for right-hand-side (ket)

Note: Only relevant when using the default radial_rhs()/radialIntegral(). If those are overridden, these are not called.

Reimplemented from DiracOperator::TensorOperator.

◆ angularCgg()

double DiracOperator::M1::angularCgg	(	int	,
		int
	)		const

inlinefinaloverridevirtual

Angular coefficient C_gg for the g_a*g_b term of the radial integral.

Reimplemented from DiracOperator::TensorOperator.

◆ angularCfg()

double DiracOperator::M1::angularCfg	(	int	,
		int
	)		const

inlinefinaloverridevirtual

Angular coefficient C_fg for the f_a*g_b term of the radial integral.

Reimplemented from DiracOperator::TensorOperator.

◆ angularCgf()

double DiracOperator::M1::angularCgf	(	int	,
		int
	)		const

inlinefinaloverridevirtual

Angular coefficient C_gf for the g_a*f_b term of the radial integral.

Reimplemented from DiracOperator::TensorOperator.

◆ updateFrequency()

void DiracOperator::M1::updateFrequency ( const double )

inlinefinaloverridevirtual

Updates the operator for a new frequency omega.

Must be overridden by any frequency-dependent operator (i.e., where freqDependantQ() returns true). Called before computing matrix elements whenever the frequency changes.

The base class implementation aborts – if a frequency-dependent operator is constructed but this function is not overridden, it will abort at runtime when called.

Parameters

omega Frequency in atomic units.

Warning: Must be implemented in any derived class that sets freq_dep=true. Calling this on a non-frequency-dependent operator is a logic error.

Reimplemented from DiracOperator::TensorOperator.

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

DiracOperator/Operators/M1.hpp

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Public Member Functions

Additional Inherited Members

Member Function Documentation

◆ name()

◆ units()

◆ angularF()

◆ angularCff()

◆ angularCgg()

◆ angularCfg()

◆ angularCgf()

◆ updateFrequency()