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Low qr form of Pseudoscalar multipole operator: \( P_K = S^5_K = t^K(q)(i\gamma^0\gamma^5)\) NOTE: If K=0, omega should be (ea-eb); for K=1 should be q = alpha*omega!

#include <EM_multipole_lowqr.hpp>

Inheritance diagram for DiracOperator::S5k_lowq:

Public Member Functions
	S5k_lowq (const Grid &gr, int K, double omega, double alpha=PhysConst::alpha)

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

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

void	updateFrequency (const double omega) override final
	NOTE: If K=0, omega should be (ea-eb); for K=1 should be q = alpha*omega!

Public Member Functions inherited from DiracOperator::EM_multipole
const SphericalBessel::JL_table *	jl () const
	Returns the precomputed Bessel table pointer (may be nullptr).

std::string	name () const override
	Returns a human-readable label, e.g. "T^E_1", "T^M5_2", "t_1", "P_1".

double	angularF (const int ka, const int kb) const override
	Angular factor linking the radial integral to the reduced matrix element: \( \langle a \\| h \\| b \rangle = F(\kappa_a,\kappa_b) \cdot \int \! dr \).

void	updateRank (int new_K) override
	Updates the tensor rank and adjusts parity accordingly.

std::unique_ptr< TensorOperator >	clone () const override
	Creates a fully independent copy of this operator at its current (rank, frequency) state via the MultipoleOperator factory.

	EM_multipole (const EM_multipole &)=default

EM_multipole &	operator= (const EM_multipole &)=default

	EM_multipole (EM_multipole &&)=default

EM_multipole &	operator= (EM_multipole &&)=default

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).

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>

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

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

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

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

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

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.

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::EM_multipole
	EM_multipole (int rank_k, Parity pi, double constant, const std::vector< double > &vec, Realness RorI, bool freq_dep, const Grid grid, char type, char comp, bool low_q, const SphericalBessel::JL_table jl=nullptr, char form='V')
	Initialise the EM_multipole base layer.

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::EM_multipole
const Grid *	m_grid {nullptr}

double	m_omega
	Current frequency; cached by each derived updateFrequency().

char	m_type {}

char	m_comp {}

bool	m_low_q {}

char	m_form {}

const SphericalBessel::JL_table *	m_jl {nullptr}

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

◆ radial_rhs()

DiracSpinor DiracOperator::S5k_lowq::radial_rhs	(	const int	kappa_a,
		const DiracSpinor &	Fb
	)		const

inlinefinaloverridevirtual

Computes the right-hand spinor dF_b for the radial integral.

Returns \( \delta F_b \) such that the radial integral satisfies:

\[ R_{ab} = F_a \cdot \delta F_b = \int_0^\infty \left(f_a\,\delta f_b + g_a\,\delta g_b\right)\,{\rm d}r \]

The default implementation constructs \( \delta F_b \) using the stored radial function \( v(r) \) and the angular coefficients:

\[ \delta F_b(r) = c\,v(r) \begin{pmatrix} C_{ff}\,f_b(r) + C_{fg}\,g_b(r) \\ C_{gf}\,f_b(r) + C_{gg}\,g_b(r) \end{pmatrix} \]

This is used by reduced_rhs() to build \( \langle a \| \hat{h} \| b \rangle \) as a spinor-valued quantity, enabling perturbation theory and TDHF. Override this for operators whose radial structure cannot be expressed in this standard form.

Parameters

kappa_a	Relativistic quantum number \( \kappa_a \) of the bra state (needed to evaluate the angular coefficients).
Fb	Ket DiracSpinor \( F_b \) .

Returns: DiracSpinor \( \delta F_b \) .

Warning: If this is overridden, radialIntegral() should also be overridden consistently (and vice versa), so that reducedME() and reduced_rhs() remain consistent.

Reimplemented from DiracOperator::TensorOperator.

◆ radialIntegral()

double DiracOperator::S5k_lowq::radialIntegral	(	const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)		const

inlinefinaloverridevirtual

Radial integral R_ab, defined by RME = angularF(a,b) * radialIntegral(a,b).

Returns the radial part \( R_{ab} \) of the reduced matrix element:

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

where \( A_{ab} \) is angularF().

The default implementation evaluates \( R_{ab} = F_a \cdot \delta F_b \) , using the default radial structure:

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

Override this for operators that do not fit this standard form. If radial_rhs() is also overridden, both must remain mutually consistent.

Warning: If radial_rhs() is overridden but radialIntegral() is not (or vice versa), reducedME() and reduced_rhs() will give inconsistent results.

Reimplemented from DiracOperator::TensorOperator.

◆ updateFrequency()

void DiracOperator::S5k_lowq::updateFrequency ( const double omega )

inlinefinaloverridevirtual

NOTE: If K=0, omega should be (ea-eb); for K=1 should be q = alpha*omega!

Reimplemented from DiracOperator::TensorOperator.

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

DiracOperator/Operators/EM_multipole_lowqr.hpp

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Public Member Functions

Additional Inherited Members

Member Function Documentation

◆ radial_rhs()

◆ radialIntegral()

◆ updateFrequency()