MBPT::Goldstone

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Second-order correlation potential, Sigma, from Goldstone diagrams.

Forms the second-order correlation potential for a valence state, as a non-local matrix on a radial sub-grid (every stride-th point of the full grid), by direct summation over a finite basis of internal core (hole) and excited states. Direct and exchange diagrams, with optional effective screening (fk) and hole-particle (etak) factors, and the two-body Breit correction.

Used by CorrelationPotential: for all diagrams (Goldstone method), or for the exchange and Breit diagrams only (Feynman method).

#include <Goldstone.hpp>

Public Member Functions
	Goldstone (const std::vector< DiracSpinor > &basis, const std::vector< DiracSpinor > &core, std::size_t i0, std::size_t stride, std::size_t size, int n_min_core=1, bool include_G=false, const HF::Breit *Br=nullptr)
	Splits basis into internal holes and excited states; forms Yk table.
GMatrix	Sigma_direct (int kappa_v, double en_v, const std::vector< double > &fk={}, const std::vector< double > &etak={}, int n_max_core=99) const
	Calculate Direct part of correlation potential.
GMatrix	Sigma_exchange (int kappa_v, double en_v, const std::vector< double > &fk={}, bool fk_both_lines=false) const
	Calculate Exchange part of correlation potential.
GMatrix	Sigma_both (int kappa_v, double en_v, const std::vector< double > &fk={}, const std::vector< double > &etak={}, int n_max_core=99, bool fk_both_lines=false) const
	Calculate both parts (direct + exchange) of correlation potential.
GMatrix	dSigma_Breit2 (int kappa_v, double en_v, const std::vector< double > &fk={}, const std::vector< double > &etak={}, int n_max_core=99, int max_n_breit=-1) const
	Calculates 2-body Breit correction to correlation potential.
const std::pair< Basis, Basis > &	basis () const
	Internal {holes, excited} states (holes: core states with n >= n_min)
const Coulomb::YkTable &	Yeh () const
	Yk table for each excited-hole pair.
std::size_t	stride () const
	Sub-grid stride: every stride-th point of the full grid.
int	n_min () const
	Minimum n of core states included as internal holes.
int	lmax () const
	Maximum l of the internal excited states.

Constructor & Destructor Documentation

Goldstone()

MBPT::Goldstone::Goldstone	(	const std::vector< DiracSpinor > &	basis,
		const std::vector< DiracSpinor > &	core,
		std::size_t	i0,
		std::size_t	stride,
		std::size_t	size,
		int	n_min_core = 1,
		bool	include_G = false,
		const HF::Breit *	Br = nullptr
	)

Splits basis into internal holes and excited states; forms Yk table.

Basis states found in core with n >= n_min_core are the internal holes; basis states not in core are the excited states. Calculates the Yk integrals for each excited-hole pair.

Parameters

basis	Basis for internal lines (core and excited states); must not be empty
core	Core states: used to split basis into holes and excited states
i0	Full-grid index of the first sub-grid point
stride	Sub-grid stride: every stride-th point of the full grid
size	Number of sub-grid points
n_min_core	Minimum n of core states included as internal holes (lower core states are frozen)
include_G	Include small-component (fg, gf, gg) parts of Sigma
Br	Breit operator, for dSigma_Breit2 (copied; nullptr: none)

Member Function Documentation

Sigma_direct()

GMatrix MBPT::Goldstone::Sigma_direct	(	int	kappa_v,
		double	en_v,
		const std::vector< double > &	fk = {},
		const std::vector< double > &	etak = {},
		int	n_max_core = 99
	)		const

Calculate Direct part of correlation potential.

Direct diagrams, summed over internal holes and excited states, at energy en_v. Each multipolarity k is scaled by fk[k] * etak[k].

Parameters

kappa_v	Dirac quantum number of the valence state
en_v	Energy at which Sigma is formed
fk	Effective screening factors, for each multipole k (missing: 1)
etak	Effective hole-particle factors, for each k (missing: 1)
n_max_core	Maximum n of internal hole states (<= 0: all)

Returns

Direct part of Sigma, on the sub-grid.

Sigma_exchange()

GMatrix MBPT::Goldstone::Sigma_exchange	(	int	kappa_v,
		double	en_v,
		const std::vector< double > &	fk = {},
		bool	fk_both_lines = false
	)		const

Calculate Exchange part of correlation potential.

fk are effective screening factors, applied to the multipole k of the outer Coulomb line; with fk_both_lines, also to the multipole of the inner line (inside \( P^k \)), i.e., \( f_k f_l \).

Returns

Exchange part of Sigma, on the sub-grid.

Sigma_both()

GMatrix MBPT::Goldstone::Sigma_both	(	int	kappa_v,
		double	en_v,
		const std::vector< double > &	fk = {},
		const std::vector< double > &	etak = {},
		int	n_max_core = 99,
		bool	fk_both_lines = false
	)		const

Calculate both parts (direct + exchange) of correlation potential.

Direct and exchange diagrams formed together, in a single sum. Parameters as for Sigma_direct() and Sigma_exchange(); here, n_max_core applies to both the direct and exchange diagrams.

Returns

Direct plus exchange parts of Sigma, on the sub-grid.

dSigma_Breit2()

GMatrix MBPT::Goldstone::dSigma_Breit2	(	int	kappa_v,
		double	en_v,
		const std::vector< double > &	fk = {},
		const std::vector< double > &	etak = {},
		int	n_max_core = 99,
		int	max_n_breit = -1
	)		const

Calculates 2-body Breit correction to correlation potential.

Second-order diagrams with one Coulomb line replaced by the two-body Breit interaction (factor of 2: either line), including direct and exchange parts. Each multipolarity k is scaled by fk[k]; etak[k] scales the direct part only.

Parameters

kappa_v	Dirac quantum number of the valence state
en_v	Energy at which Sigma is formed
fk	Effective screening factors, for each multipole k
etak	Effective hole-particle factors, for each k (direct part only)
n_max_core	Maximum n of internal hole states (<= 0: all)
max_n_breit	Maximum n of internal excited states (<= 0: all)

Returns

Breit correction to Sigma, on the sub-grid.

Note

Must have Breit: returns zero if no Breit operator was given to the constructor.

basis()

const std::pair< Basis, Basis > & MBPT::Goldstone::basis ( ) const

inline

Internal {holes, excited} states (holes: core states with n >= n_min)

Yeh()

const Coulomb::YkTable & MBPT::Goldstone::Yeh ( ) const

inline

Yk table for each excited-hole pair.

stride()

std::size_t MBPT::Goldstone::stride ( ) const

inline

Sub-grid stride: every stride-th point of the full grid.

n_min()

int MBPT::Goldstone::n_min ( ) const

inline

Minimum n of core states included as internal holes.

lmax()

int MBPT::Goldstone::lmax ( ) const

inline

Maximum l of the internal excited states.

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

• MBPT/Goldstone.hpp
• MBPT/Goldstone.cpp