ExternalField::TDHFbasis

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Similar to TDHF, but uses a basis expansion to include core-polarisation (RPA) corrections to matrix elements of an external field.

Like the TDHF method, but uses a basis to expand the dPsi corrections. Solves

\[ (H + \delta V - \epsilon \pm \omega)\,\delta\psi = -(h + \delta V)\psi \]

by expanding \(\delta\psi\) over the basis:

\[ \delta\psi = \sum_n \frac{|n\rangle\langle n|(h + \delta V)|\psi\rangle}{\epsilon - \epsilon_n \pm \omega} \]

#include <TDHFbasis.hpp>

Inheritance diagram for ExternalField::TDHFbasis:

Public Member Functions
	TDHFbasis (const DiracOperator::TensorOperator *const h, const HF::HartreeFock *const hf, const std::vector< DiracSpinor > &basis)
	Constructs TDHFbasis for operator h using the provided basis.
virtual void	solve_core (const double omega, int max_its=100, const bool print=true) override final
	Solves TDHF equations self-consistently for core electrons at frequency omega.
virtual Method	method () const override final
	Returns RPA method.
DiracSpinor	form_dPsi (const DiracSpinor &Fv, const double omega, dPsiType XorY, const int kappa_beta, const std::vector< DiracSpinor > &spectrum, StateType st=StateType::ket, bool incl_dV=true) const
	Forms dF_v for valence state Fv (including core pol.): single kappa.
std::vector< DiracSpinor >	form_dPsis (const DiracSpinor &Fv, const double omega, dPsiType XorY, const std::vector< DiracSpinor > &spectrum, StateType st=StateType::ket, bool incl_dV=true) const
	Forms the perturbed wavefunctions dF_v for a valence state for all allowed kappa channels.
TDHFbasis &	operator= (const TDHFbasis &)=delete
	TDHFbasis (const TDHFbasis &)=default
Public Member Functions inherited from ExternalField::TDHF
	TDHF (const DiracOperator::TensorOperator *const h, const HF::HartreeFock *const hf)
	Constructs TDHF for operator h.
virtual void	clear () override final
	Clears the internal state back to pre solve_core()
double	dV (const DiracSpinor &Fa, const DiracSpinor &Fb, bool conj) const
	Returns reduced matrix element \(\langle a||\delta V||b\rangle\), or the conjugate \(\langle a||\delta V^\dagger||b\rangle\) if conj=true.
virtual double	dV (const DiracSpinor &Fa, const DiracSpinor &Fb) const override final
	Calculate reduced matrix element <n||dV||m>
DiracSpinor	dV_rhs (int kappa_n, const DiracSpinor &Fm, bool conj=false) const override
	Calculates reduced right-hand-side, projected onto kappa: [dV|phi_m]_kappa.
const std::vector< DiracSpinor > &	get_dPsis (const DiracSpinor &Fc, dPsiType XorY) const
	Returns const ref to dPsi orbitals for given core orbital Fc.
const DiracSpinor &	get_dPsi_x (const DiracSpinor &Fc, dPsiType XorY, const int kappa_x) const
	Returns const ref to dPsi orbital of given kappa.
DiracSpinor	solve_dPsi (const DiracSpinor &Fv, const double omega, dPsiType XorY, const int kappa_beta, const MBPT::CorrelationPotential *const Sigma=nullptr, StateType st=StateType::ket, bool incl_dV=true) const
	Forms \(\delta\psi_v\) for valence state Fv (including core pol.): single kappa.
std::vector< DiracSpinor >	solve_dPsis (const DiracSpinor &Fv, const double omega, dPsiType XorY, const MBPT::CorrelationPotential *const Sigma=nullptr, StateType st=StateType::ket, bool incl_dV=true) const
	Forms \(\delta\psi_v\) for all kappa channels; see solve_dPsi.
TDHF &	operator= (const TDHF &)=delete
	TDHF (const TDHF &)=default
Public Member Functions inherited from ExternalField::CorePolarisation
double	last_eps () const
	Returns eps (convergance) of last solve_core run.
double	last_its () const
	Returns its (# of iterations) of last solve_core run.
double	last_omega () const
	Returns omega (frequency) of last solve_core run.
int	rank () const
int	parity () const
bool	imagQ () const
double &	eps_target ()
	Convergance target.
double	eps_target () const
	Convergance target.
double	eta () const
	Damping factor; 0 means no damping. Must have 0 <= eta < 1.
void	set_eta (double eta)
	Set/update damping factor; 0 means no damping. Must have 0 <= eta < 1.
CorePolarisation &	operator= (const CorePolarisation &)=delete
	CorePolarisation (const CorePolarisation &)=default

Additional Inherited Members
Protected Member Functions inherited from ExternalField::CorePolarisation
	CorePolarisation (const DiracOperator::TensorOperator *const h)
Protected Attributes inherited from ExternalField::TDHF
std::vector< std::vector< DiracSpinor > >	m_X {}
std::vector< std::vector< DiracSpinor > >	m_Y {}
std::vector< std::vector< DiracSpinor > >	m_hFcore {}
const HF::HartreeFock *const	p_hf
const std::vector< DiracSpinor >	m_core
const double	m_alpha
const HF::Breit *const	p_VBr
Protected Attributes inherited from ExternalField::CorePolarisation
const DiracOperator::TensorOperator *	m_h
double	m_core_eps {1.0}
int	m_core_its {0}
double	m_core_omega {0.0}
int	m_rank
int	m_pi
bool	m_imag
double	m_eta {0.4}
double	m_eps {1.0e-10}

Constructor & Destructor Documentation

TDHFbasis()

ExternalField::TDHFbasis::TDHFbasis	(	const DiracOperator::TensorOperator *const	h,
		const HF::HartreeFock *const	hf,
		const std::vector< DiracSpinor > &	basis
	)

Constructs TDHFbasis for operator h using the provided basis.

Parameters

h	External field operator.
hf	HF::HartreeFock object defining the core.
basis	Single-particle basis used to expand dPsi. The entire basis is used; it is the caller's responsibility to ensure all required states are present.

Member Function Documentation

solve_core()

void ExternalField::TDHFbasis::solve_core ( const double  omega, int  max_its = 100, const bool  print = true  )

finaloverridevirtual

Solves TDHF equations self-consistently for core electrons at frequency omega.

Will iterate up to a maximum of max_its. Set max_its=1 to get first-order correction [note: no damping is used for first iteration]. If print=true, will write progress to screen.

Reimplemented from ExternalField::TDHF.

method()

virtual Method ExternalField::TDHFbasis::method ( ) const

inlinefinaloverridevirtual

Returns RPA method.

Reimplemented from ExternalField::TDHF.

form_dPsi()

DiracSpinor ExternalField::TDHFbasis::form_dPsi	(	const DiracSpinor &	Fv,
		const double	omega,
		dPsiType	XorY,
		const int	kappa_beta,
		const std::vector< DiracSpinor > &	spectrum,
		StateType	st = StateType::ket,
		bool	incl_dV = true
	)		const

Forms dF_v for valence state Fv (including core pol.): single kappa.

Solves

\[ (H + \Sigma - \epsilon - \omega)X = -(h + \delta V - \delta\epsilon)\psi \]

or

\[ (H + \Sigma - \epsilon + \omega)Y = -(h^\dagger + \delta V^\dagger - \delta\epsilon)\Psi \]

Returns \( \chi_\beta \) for given kappa_beta, where

\[ X_{j,m} = (-1)^{j_\beta-m}tjs(j,k,j;-m,0,m)\chi_j \]

Parameters

Fv	Valence state \(\psi_v\).
omega	Perturbation frequency \(\omega\).
XorY	Selects X or Y solution; see dPsiType.
kappa_beta	Kappa quantum number of the target channel.
spectrum	Single-particle spectrum used to expand the solution.
st	Bra or ket convention; see StateType.
incl_dV	Include the induced potential \(\delta V\) if true.

Note

To include correlations, use a basis with correlations.

To exclude excitations to occupied states, remove them from the basis.

form_dPsis()

std::vector< DiracSpinor > ExternalField::TDHFbasis::form_dPsis	(	const DiracSpinor &	Fv,
		const double	omega,
		dPsiType	XorY,
		const std::vector< DiracSpinor > &	spectrum,
		StateType	st = StateType::ket,
		bool	incl_dV = true
	)		const

Forms the perturbed wavefunctions dF_v for a valence state for all allowed kappa channels.

Solves the inhomogeneous equation for the perturbed wavefunction \(\delta\psi_v\) of the valence state \(\psi_v\) at frequency \(\omega\). The solutions \([\delta\psi_v]_\kappa\) are returned for all allowed \(\kappa\) values (see form_dPsi).

The calculation uses the supplied single-particle spectrum and may include the induced potential \(\delta V\) if incl_dV is true.

Parameters

Fv	Valence state \(\psi_v\).
omega	Perturbation frequency \(\omega\).
XorY	Selects the X or Y solution; see dPsiType.
spectrum	Single-particle spectrum used to construct the solution.
st	Bra or ket convention; see StateType.
incl_dV	Include the induced potential \(\delta V\) if true.

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

• ExternalField/TDHFbasis.hpp
• ExternalField/TDHFbasis.cpp