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Direct correlation potential diagrams via the Feynman technique.

Constructs Hartree-Fock Green's functions, the polarisation operator, and the direct Sigma diagrams, as matrices on a radial sub-grid. The direct Sigma is formed by integrating, along Im(w) at fixed Re(w) = omre, the product of the Green's function \( G(\varepsilon_v + \omega) \) and the Coulomb-polarisation-Coulomb operator, \( Q\Pi Q(\omega) \).

Allows all-order inclusion of:

screening of the Coulomb interaction: \( Q\Pi Q \to Q\Pi X Q \), with \( X = [1 + iQ\Pi]^{-1} \)
the hole-particle interaction, in the polarisation operator

Exchange diagrams are not formed here (see Goldstone).

#include <Feynman.hpp>

Public Member Functions
	Feynman (const HF::HartreeFock *vHF, std::size_t i0, std::size_t stride, std::size_t size, const FeynmanOptions &options, int n_min_core, bool include_G, bool verbose=true, const std::string &ident="", bool form_qpq=true)
	Sets up the Feynman method on given sub-grid, and forms QPiQ.

bool	screening () const
	True if Coulomb screening is included.

bool	hole_particle () const
	True if hole-particle interaction is included.

std::size_t	stride () const
	Returns stride used for sub-grid.

int	n_min () const
	Minimum n of core states in the polarisation loop.

int	max_k () const
	Maximum multipolarity, k.

double	omre () const
	Real part of frequency used for integration.

double	w0 () const
	First non-zero point of the frequency grid along Im(w)

double	wratio () const
	Ratio of the logarithmic frequency grid along Im(w)

int	lmax () const
	Maximum l for internal lines.

ComplexGMatrix	green (int kappa, std::complex< double > en, GreenStates states=GreenStates::both) const
	Green's function for given kappa, at complex energy `en`.

ComplexGMatrix	green_complex_dirac (int kappa, std::complex< double > en) const
	Green's function, solving the Dirac equation directly at complex en.

ComplexGMatrix	green_dyson (int kappa, std::complex< double > en) const
	Green's function by the Dyson (resolvent) method.

ComplexGMatrix	green_basis (int kappa, std::complex< double > en, const std::vector< DiracSpinor > &basis) const
	Green's function for given kappa, as a sum over an explicit basis.

std::vector< ComplexRMatrix >	polarisation_each_k (std::complex< double > omega, bool hole_particle) const
	Polarisation operator pi^k(w), for each k = 0..max_k separately.

std::vector< std::vector< ComplexRMatrix > >	polarisation_wk () const
	Polarisation operator pi^k(w) on the frequency grid, for each k.

void	form_qpiq (const std::string &ident="")
	Forms QPiQ along the frequency grid (required by Sigma_direct).

void	form_qpiq (const std::vector< std::vector< ComplexRMatrix > > &pi_wk)
	Forms QPiQ (with screening, if set) from given polarisation op.

bool	has_qpiq () const
	True once QPiQ has been formed.

bool	read_qpiq (const std::string &ident)
	Reads QPiQ from the disk cache for `ident` (file prefix).

bool	write_qpiq (const std::string &ident)
	Writes QPiQ to the disk cache for ident; false if not written.

GMatrix	Sigma_direct (int kappa_v, double en_v, std::optional< int > k={}) const
	Direct part of the correlation potential, for kappa_v at en_v.

std::vector< GMatrix >	Sigma_direct_each_k (int kappa_v, double en_v) const
	Direct part of correlation potential, for each multipole k.

const ComplexRMatrix &	get_qk (int k) const
	Returns (reference to) radial Coulomb matrix, q^k.

const GMatrix &	get_Vx_kappa (int kappa) const
	Returns (ref to) radial exchange matrix Vx_kappa. Nb: includes dri*drj.

const ComplexGMatrix &	get_Pcore (int kappa) const
	Returns (ref to) summed core projection operator, for given kappa.

Feynman &	operator= (const Feynman &)=default

	Feynman (const Feynman &)=default

Constructor & Destructor Documentation

◆ Feynman()

MBPT::Feynman::Feynman	(	const HF::HartreeFock *	vHF,
		std::size_t	i0,
		std::size_t	stride,
		std::size_t	size,
		const FeynmanOptions &	options,
		int	n_min_core,
		bool	include_G,
		bool	verbose = `true`,
		const std::string &	ident = `""`,
		bool	form_qpq = `true`
	)

Sets up the Feynman method on given sub-grid, and forms Q*Pi*Q.

Forms the radial Coulomb (q^k), core projection, and Hartree-Fock exchange matrices, and the frequency quadrature along Im(w). Q*Pi*Q is the expensive step: only needed for Sigma_direct (the Green's functions etc. do not need it), and can be skipped (form_qpq = false) and formed later with form_qpiq().

Parameters

vHF	Hartree-Fock (core, potential, grid). Pointer is stored: must outlive this object
i0	Sub-grid: index of first point on the full grid
stride	Sub-grid stride: every stride-th point of full grid
size	Number of sub-grid points
options	Screening, hole-particle, internal lines, frequency grid, Green's function method (see FeynmanOptions)
n_min_core	Minimum n of core states in the polarisation loop
include_G	Include small (g) components throughout. Always on if `vHF` includes Breit
verbose	Print summary of options
ident	File prefix for the QPiQ disk cache ("" or "false": no cache)
form_qpq	Form QPiQ on construction

Member Function Documentation

◆ screening()

bool MBPT::Feynman::screening ( ) const

inline

True if Coulomb screening is included.

◆ hole_particle()

bool MBPT::Feynman::hole_particle ( ) const

inline

True if hole-particle interaction is included.

◆ stride()

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

inline

Returns stride used for sub-grid.

◆ n_min()

int MBPT::Feynman::n_min ( ) const

inline

Minimum n of core states in the polarisation loop.

◆ max_k()

int MBPT::Feynman::max_k ( ) const

inline

Maximum multipolarity, k.

◆ omre()

double MBPT::Feynman::omre ( ) const

inline

Real part of frequency used for integration.

◆ w0()

double MBPT::Feynman::w0 ( ) const

inline

First non-zero point of the frequency grid along Im(w)

◆ wratio()

double MBPT::Feynman::wratio ( ) const

inline

Ratio of the logarithmic frequency grid along Im(w)

◆ lmax()

int MBPT::Feynman::lmax ( ) const

inline

Maximum l for internal lines.

◆ green()

ComplexGMatrix MBPT::Feynman::green	(	int	kappa,
		std::complex< double >	en,
		GreenStates	states = `GreenStates::both`
	)		const

Green's function for given kappa, at complex energy en.

For states = both, the full Hartree-Fock Green's function. With complex_green set (see FeynmanOptions): solved directly at complex energy for small Im(en), and by the Dyson method at large Im(en) (where it is consistent with the discrete radial sums). Otherwise, Dyson method throughout.

Parameters

kappa	Dirac quantum number
en	Complex energy
states	Which states to include: both : core and excited core : core only, sum over core states a of \|a>

◆ green_complex_dirac()

ComplexGMatrix MBPT::Feynman::green_complex_dirac	(	int	kappa,
		std::complex< double >	en
	)		const

Green's function, solving the Dirac equation directly at complex en.

cf. green(), which mixes this with the Dyson method.

◆ green_dyson()

ComplexGMatrix MBPT::Feynman::green_dyson	(	int	kappa,
		std::complex< double >	en
	)		const

Green's function by the Dyson (resolvent) method.

Solves at Re(en), then shifts to complex energy. cf. green(), which mixes the two methods.

◆ green_basis()

ComplexGMatrix MBPT::Feynman::green_basis	(	int	kappa,
		std::complex< double >	en,
		const std::vector< DiracSpinor > &	basis
	)		const

Green's function for given kappa, as a sum over an explicit basis.

\( G = \sum_n \ket{n}\bra{n}/(\varepsilon - \varepsilon_n) \), over states in basis with given kappa.

◆ polarisation_each_k()

std::vector< ComplexRMatrix > MBPT::Feynman::polarisation_each_k	(	std::complex< double >	omega,
		bool	hole_particle
	)		const

Polarisation operator pi^k(w), for each k = 0..max_k separately.

Not summed over k:

\[ \Pi^k(\omega) = i \sum_{a,n} \frac{(C^k_{an})^2}{2k+1} \, F_a^\dagger \left[ G^{\rm ex}_n(\varepsilon_a - \omega) + G^{\rm ex}_n(\varepsilon_a + \omega) \right] F_a, \]

with core states a (n >= n_min) and excited-state Green's functions for each internal kappa_n. Green's functions are computed once and re-used across all k.

Parameters

omega	Complex frequency
hole_particle	Include the hole-particle interaction (independent of the option set on construction)

◆ polarisation_wk()

std::vector< std::vector< ComplexRMatrix > > MBPT::Feynman::polarisation_wk ( ) const

Polarisation operator pi^k(w) on the frequency grid, for each k.

This is the expensive stage of forming Q*Pi*Q. Nothing is stored: pass the result to form_qpiq(pi_wk) to form Q*Pi*Q (for this object, or for another that differs only in screening). Depends on the hole-particle option, not on screening.

Returns: pi^k(w) at every point of the frequency grid, indexed [iw][k].

◆ form_qpiq() [1/2]

void MBPT::Feynman::form_qpiq ( const std::string & ident = "" )

Forms Q*Pi*Q along the frequency grid (required by Sigma_direct).

With ident, reads from the disk cache if a matching file exists, else forms and writes it (see read_qpiq/write_qpiq).

◆ form_qpiq() [2/2]

void MBPT::Feynman::form_qpiq ( const std::vector< std::vector< ComplexRMatrix > > & pi_wk )

Forms Q*Pi*Q (with screening, if set) from given polarisation op.

pi_wk is from polarisation_wk(), and may be shared between Feynman objects that differ only in screening. With screening, \( Q\Pi Q \to Q\Pi X Q \), with \( X = [1 + iQ\Pi]^{-1} \).

Warning: pi_wk must be on the same sub-grid and frequency grid (checked only by assert).

◆ has_qpiq()

bool MBPT::Feynman::has_qpiq ( ) const

inline

True once Q*Pi*Q has been formed.

◆ read_qpiq()

bool MBPT::Feynman::read_qpiq ( const std::string & ident )

Reads Q*Pi*Q from the disk cache for ident (file prefix).

Returns: False if no matching file (or ident is "" or "false").

◆ write_qpiq()

bool MBPT::Feynman::write_qpiq ( const std::string & ident )

Writes Q*Pi*Q to the disk cache for ident; false if not written.

◆ Sigma_direct()

GMatrix MBPT::Feynman::Sigma_direct	(	int	kappa_v,
		double	en_v,
		std::optional< int >	k = `{}`
	)		const

Direct part of the correlation potential, for kappa_v at en_v.

Summed over multipoles k (unless k given) and internal kappa. See class description.

Parameters

kappa_v	Valence Dirac quantum number
en_v	Energy at which Sigma is formed
k	If given, only this multipole (zero matrix if out of range)

Warning: Q*Pi*Q must be formed first (see form_qpiq); checked only by assert.

◆ Sigma_direct_each_k()

std::vector< GMatrix > MBPT::Feynman::Sigma_direct_each_k	(	int	kappa_v,
		double	en_v
	)		const

Direct part of correlation potential, for each multipole k.

Not summed: Sigma_d = sum of these. Same cost as a single Sigma_direct call, since the Green's functions are shared by all k.

Warning: Q*Pi*Q must be formed first (see form_qpiq).

◆ get_qk()

const ComplexRMatrix & MBPT::Feynman::get_qk ( int k ) const

inline

Returns (reference to) radial Coulomb matrix, q^k.

\( q^k_{ij} = (r_<^k / r_>^{k+1}) \, dr_j \): includes the right-hand integration measure (drj) only.

◆ get_Vx_kappa()

const GMatrix & MBPT::Feynman::get_Vx_kappa ( int kappa ) const

inline

Returns (ref to) radial exchange matrix Vx_kappa. Nb: includes dri*drj.

◆ get_Pcore()

const ComplexGMatrix & MBPT::Feynman::get_Pcore ( int kappa ) const

inline

Returns (ref to) summed core projection operator, for given kappa.

\( P = \sum_a \ket{a}\bra{a} \). No integration measure.

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

MBPT/Feynman.hpp
MBPT/Feynman.cpp

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Public Member Functions

Constructor & Destructor Documentation

◆ Feynman()

Member Function Documentation

◆ screening()

◆ hole_particle()

◆ stride()

◆ n_min()

◆ max_k()

◆ omre()

◆ w0()

◆ wratio()

◆ lmax()

◆ green()

◆ green_complex_dirac()

◆ green_dyson()

◆ green_basis()

◆ polarisation_each_k()

◆ polarisation_wk()

◆ form_qpiq() [1/2]

◆ form_qpiq() [2/2]

◆ has_qpiq()

◆ read_qpiq()

◆ write_qpiq()

◆ Sigma_direct()

◆ Sigma_direct_each_k()

◆ get_qk()

◆ get_Vx_kappa()

◆ get_Pcore()