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Correlation potential, Sigma, for valence states.

Core-valence correlations, stored as a non-local matrix for each kappa, on a radial sub-grid (every stride-th point of the full grid, between r0 and rmax). Formed at a given energy (see formSigma). Read from file if it exists, so it need only be formed once (see write).

Direct diagrams are formed using either the Goldstone (basis sum) or Feynman (Green's function) technique; the Feynman technique allows all-order inclusion of screening and the hole-particle interaction. Exchange diagrams always use Goldstone, with optional effective screening factors fk.

The action of Sigma on a state (SigmaFv) includes:

lambda scaling, \( \Sigma \to \lambda\Sigma \) (see scale_Sigma)
ladder correction, \( \Sigma_L \), if a ladder file was given
large-r tail, \( a_{\rm eff}/r^4 \), beyond the sub-grid, if tail_extrapolation is set (experimental; see fit_tail_coefficient)

Optionally also forms the energy derivative, \( \partial\Sigma/\partial\varepsilon \) (see dSigmaFv).

Note: Not thread safe: formSigma modifies internal state.

#include <CorrelationPotential.hpp>

Public Member Functions
	CorrelationPotential (const std::string &fname, const HF::HartreeFock *vHF, const std::vector< DiracSpinor > &basis, double r0, double rmax, std::size_t stride, int n_min_core, SigmaMethod method, bool include_g=false, bool include_Breit_b2=false, int n_max_breit=0, const FeynmanOptions &Foptions={}, bool calculate_fk=true, const std::vector< double > &fk={}, const std::vector< double > &etak={}, const std::string &ladder_file="", bool form_derivative=false, bool fk_both_lines=false, bool tail_extrapolation=false)
	Reads Sigma from file if it exists; otherwise sets up to form it.

void	formSigma (int kappa, double ev, int n, const DiracSpinor *Fv=nullptr)
	Forms and stores Sigma for given kappa, at energy `ev`.

bool	empty () const
	True if no Sigma matrices are stored.

const GMatrix *	getSigma (int kappa, int n=0) const
	Pointer to the stored Sigma matrix for given kappa (and n).

double	getLambda (int kappa, int n=0) const
	Scaling factor, lambda, for given kappa (and n); 1 if not present.

void	clear ()
	Removes all stored Sigma matrices (not ladder or dSigma/dE)

DiracSpinor	SigmaFv (const DiracSpinor &Fv) const
	Returns Sigma\|Fv>, including lambda scaling, ladder, and tail.

DiracSpinor	operator() (const DiracSpinor &Fv) const
	Same as SigmaFv.

void	print_de (const std::vector< DiracSpinor > &valence)
	Prints MBPT(2) energy correction for each valence state.

void	print_tail (const std::vector< DiracSpinor > &valence) const
	Prints a_eff for each valence state with a fitted large-r tail.

bool	has_derivative () const
	True if any dSigma/dE matrices are present (see form_derivative)

const SigmaData *	get_derivative (int kappa, int n=0) const
	Pointer to dSigma/dE data for given kappa (and n); nullptr if not present.

DiracSpinor	dSigmaFv (const DiracSpinor &Fv) const
	Returns lambda * dSigma/dE \|Fv>.

void	scale_Sigma (const std::vector< double > &lambdas)
	Stores scaling factors, lambda, for each stored Sigma.

void	scale_Sigma (double lambda, int kappa, int n=0)
	Stores scaling factor, lambda, for Sigma with given kappa and n.

void	print_scaling () const
	Prints the scaling factors to screen.

void	print_info () const
	Prints kappa, energy, lambda, and if dSigma/dE exists, for each Sigma.

std::string	method_string () const
	Short string identifying the method (main differences only).

std::string	lambda_string () const
	Fitting (scaling) factors as a string, "kappa=value,".

std::vector< double >	average_fk (int l_max=2) const
	Average of the stored fk screening factors, over l up to `l_max`.

void	print_subGrid () const
	Prints the sub-grid parameters to screen.

void	write (const std::string &fname)
	Writes Sigma matrices to file `fname`.

const SigmaData *	get (int kappa, int n=0) const
	Pointer to the stored Sigma data for given kappa (and n).

CorrelationPotential &	operator= (const CorrelationPotential &)=default

	CorrelationPotential (const CorrelationPotential &)=default

Constructor & Destructor Documentation

◆ CorrelationPotential()

MBPT::CorrelationPotential::CorrelationPotential	(	const std::string &	fname,
		const HF::HartreeFock *	vHF,
		const std::vector< DiracSpinor > &	basis,
		double	r0,
		double	rmax,
		std::size_t	stride,
		int	n_min_core,
		SigmaMethod	method,
		bool	include_g = `false`,
		bool	include_Breit_b2 = `false`,
		int	n_max_breit = `0`,
		const FeynmanOptions &	Foptions = `{}`,
		bool	calculate_fk = `true`,
		const std::vector< double > &	fk = `{}`,
		const std::vector< double > &	etak = `{}`,
		const std::string &	ladder_file = `""`,
		bool	form_derivative = `false`,
		bool	fk_both_lines = `false`,
		bool	tail_extrapolation = `false`
	)

Reads Sigma from file if it exists; otherwise sets up to form it.

If fname exists, Sigma is read from it. Otherwise, sets up the Goldstone/Feynman method; Sigma itself is only formed by formSigma. The ladder file, if given, is read in either case.

Parameters

fname	Sigma file (read if it exists; see write)
vHF	Hartree-Fock (core, grid, Breit). Pointer is stored: must outlive this object
basis	Basis for the Goldstone sums (and print_de)
r0	Sub-grid first point (aB)
rmax	Sub-grid last point (aB)
stride	Sub-grid stride: every stride-th point of the full grid
n_min_core	Minimum n of core states included in polarisation (lower core states are frozen)
method	Goldstone or Feynman, for direct diagrams
include_g	Include small-component (fg, gf, gg) parts of Sigma
include_Breit_b2	Include two-body Breit (B2) correction (only if `vHF` includes Breit)
n_max_breit	Maximum n of excited states in the B2 correction (0: all)
Foptions	Options for the Feynman method (screening, hole-particle, etc.)
calculate_fk	Calculate fk screening factors for exchange, for each state (Feynman, with screening); else `fk` used
fk	Effective screening factors, for each multipole k
etak	Effective hole-particle factors, for each k (Goldstone only)
ladder_file	File with ladder correction, Sigma_L (empty: none)
form_derivative	Also form dSigma/dE (see dSigmaFv)
fk_both_lines	Apply fk to both Coulomb lines of the exchange diagrams (else outer only)
tail_extrapolation	Extrapolate Sigma beyond the sub-grid as a_eff/r^4 (experimental; see fit_tail_coefficient)

Note: If read from file, the sub-grid and include_g are taken from the file, not the inputs.

Member Function Documentation

◆ formSigma()

void MBPT::CorrelationPotential::formSigma	(	int	kappa,
		double	ev,
		int	n,
		const DiracSpinor *	Fv = `nullptr`
	)

Forms and stores Sigma for given kappa, at energy ev.

If Sigma already exists for kappa and n (any n if n <= 0), it is not re-formed; only the tail fit and dSigma/dE are added if required and missing (e.g., Sigma read from file).

Parameters

kappa	Dirac quantum number
ev	Energy at which Sigma is formed
n	Principal quantum number: allows distinct Sigma for each n of the same kappa
Fv	State Sigma is formed for (optional): used to calculate fk screening factors, fit the large-r tail, and print the energy shift. Must have kappa = `kappa`.

Warning: Not thread safe.

◆ empty()

bool MBPT::CorrelationPotential::empty ( ) const

inline

True if no Sigma matrices are stored.

◆ getSigma()

const GMatrix * MBPT::CorrelationPotential::getSigma	(	int	kappa,
		int	n = `0`
	)		const

Pointer to the stored Sigma matrix for given kappa (and n).

See get() for the choice of n.

Returns: Pointer to the Sigma matrix; nullptr if not present.

Note: The raw matrix: does not include lambda, the ladder part, or the large-r tail (use SigmaFv for those).

◆ getLambda()

double MBPT::CorrelationPotential::getLambda	(	int	kappa,
		int	n = `0`
	)		const

Scaling factor, lambda, for given kappa (and n); 1 if not present.

◆ clear()

void MBPT::CorrelationPotential::clear ( )

inline

Removes all stored Sigma matrices (not ladder or dSigma/dE)

◆ SigmaFv()

DiracSpinor MBPT::CorrelationPotential::SigmaFv ( const DiracSpinor & Fv ) const

Returns Sigma|Fv>, including lambda scaling, ladder, and tail.

\( \lambda (\Sigma + \Sigma_L) \ket{F_v} \): scaled by lambda, and includes the ladder correction \( \Sigma_L \) if present. Includes the large-r tail if it was fitted (see tail_extrapolation option).

Returns: Sigma|Fv>; |0> if neither Sigma nor Sigma_L exists for kappa_v.

◆ operator()()

DiracSpinor MBPT::CorrelationPotential::operator() ( const DiracSpinor & Fv ) const

inline

Same as SigmaFv.

◆ print_de()

void MBPT::CorrelationPotential::print_de ( const std::vector< DiracSpinor > & valence )

Prints MBPT(2) energy correction for each valence state.

Direct, exchange, and total, and \( \langle v|\Sigma|v \rangle \) using the stored Sigma matrix.

◆ print_tail()

void MBPT::CorrelationPotential::print_tail ( const std::vector< DiracSpinor > & valence ) const

Prints a_eff for each valence state with a fitted large-r tail.

Also prints \( \langle v|\Sigma|v \rangle \) without (S0) and with (S) the tail. Prints nothing if tail_extrapolation is off.

◆ has_derivative()

bool MBPT::CorrelationPotential::has_derivative ( ) const

inline

True if any dSigma/dE matrices are present (see form_derivative)

◆ get_derivative()

const SigmaData * MBPT::CorrelationPotential::get_derivative	(	int	kappa,
		int	n = `0`
	)		const

Pointer to dSigma/dE data for given kappa (and n); nullptr if not present.

◆ dSigmaFv()

DiracSpinor MBPT::CorrelationPotential::dSigmaFv ( const DiracSpinor & Fv ) const

Returns lambda * dSigma/dE |Fv>.

\( \lambda \, (\partial\Sigma/\partial\varepsilon) \ket{F_v} \). The energy derivative of Sigma, formed by finite difference alongside Sigma itself (option form_derivative), so it corresponds to the actual method used (Goldstone/Feynman, screening, etc.). Scaled by the same lambda as the base Sigma.

Returns: lambda * dSigma/dE |Fv>; |0> if no derivative exists for this kappa.

Note: The ladder correction (Sigma_L) has no energy derivative: it is included in SigmaFv() but not here.

◆ scale_Sigma() [1/2]

void MBPT::CorrelationPotential::scale_Sigma ( const std::vector< double > & lambdas )

Stores scaling factors, lambda, for each stored Sigma.

\( \Sigma \to \lambda\Sigma \). One factor for each stored Sigma, in the order they were formed.

◆ scale_Sigma() [2/2]

void MBPT::CorrelationPotential::scale_Sigma	(	double	lambda,
		int	kappa,
		int	n = `0`
	)

Stores scaling factor, lambda, for Sigma with given kappa and n.

\( \Sigma \to \lambda\Sigma \). If n <= 0, for all Sigma with given kappa.

◆ print_scaling()

void MBPT::CorrelationPotential::print_scaling ( ) const

Prints the scaling factors to screen.

◆ print_info()

void MBPT::CorrelationPotential::print_info ( ) const

Prints kappa, energy, lambda, and if dSigma/dE exists, for each Sigma.

◆ method_string()

std::string MBPT::CorrelationPotential::method_string ( ) const

Short string identifying the method (main differences only).

E.g., "Feynman, all-order" or "Goldstone". Intended for file-cache keys.

◆ lambda_string()

std::string MBPT::CorrelationPotential::lambda_string ( ) const

Fitting (scaling) factors as a string, "kappa=value,".

Rounded to 4 dp; empty if none scaled. For file-cache keys (e.g., the ci-file hash).

◆ average_fk()

std::vector< double > MBPT::CorrelationPotential::average_fk ( int l_max = 2 ) const

Average of the stored fk screening factors, over l up to l_max.

Averaged over the lowest Sigma of each l. For re-use of the Sigma_1 screening in Sigma_2 (CI), where there is one effective fk per Coulomb line but no unique state: the low-l factors are the relevant ones for the CI valence space (higher-l factors can differ significantly, e.g., f/g at k = 0, but such states rarely contribute). Weighted by l, not kappa: the fine-structure pair of each l is averaged first, then the mean is taken over the ls (so s counts the same as p, d, ...). Truncated to the shortest stored list.

Parameters

l_max Maximum l included in the average (CI uses 2: s, p, d).

Returns: Averaged fk, for each k; empty if none stored.

◆ print_subGrid()

void MBPT::CorrelationPotential::print_subGrid ( ) const

Prints the sub-grid parameters to screen.

◆ write()

void MBPT::CorrelationPotential::write ( const std::string & fname )

inline

Writes Sigma matrices to file fname.

Includes the sub-grid, dSigma/dE, and fk. Not written: lambda, a_eff (re-fitted in formSigma), or the ladder correction.

◆ get()

const SigmaData * MBPT::CorrelationPotential::get	(	int	kappa,
		int	n = `0`
	)		const

Pointer to the stored Sigma data for given kappa (and n).

If n <= 0, the first stored Sigma with given kappa. Otherwise, the one with given n; if none, the nearest lower n, else the first with given kappa.

Returns: Pointer to Sigma data (matrix, energy formed at, lambda, etc.); nullptr if not present.

Note: Sigma matrix is raw (no lambda, ladder, or tail; see SigmaFv).

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

MBPT/CorrelationPotential.hpp
MBPT/CorrelationPotential.cpp

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Public Member Functions

Constructor & Destructor Documentation

◆ CorrelationPotential()

Member Function Documentation

◆ formSigma()

◆ empty()

◆ getSigma()

◆ getLambda()

◆ clear()

◆ SigmaFv()

◆ operator()()

◆ print_de()

◆ print_tail()

◆ has_derivative()

◆ get_derivative()

◆ dSigmaFv()

◆ scale_Sigma() [1/2]

◆ scale_Sigma() [2/2]

◆ print_scaling()

◆ print_info()

◆ method_string()

◆ lambda_string()

◆ average_fk()

◆ print_subGrid()

◆ write()

◆ get()