Methods aiming for error estimate and automatic cutoff calibration. integrals. More...

Functions/Subroutines
subroutine, public	calibrate_cutoff (hmat, h_inv, G_min, vol, zet_min, l_mm, zet_max, l_max_zet, n_minimax, cutoff_l, cutoff_r, tol, delta, cutoff, err_mm, err_c, C_mm, para_env, print_calib, unit_nr)
	Find optimal cutoff minimizing errors due to minimax approximation and due to finite cutoff using bisection on the difference of the errors. More...

subroutine, public	cutoff_minimax_error (cutoff, hmat, h_inv, vol, G_min, zet_min, l_mm, zet_max, l_max_zet, n_minimax, minimax_aw, err_mm, err_ctff, C_mm, para_env)
	Compute upper bounds for the errors of 2-center ERI's (P\|P) due to minimax approximation and due to finite cutoff, where P is a normalized Hermite Gaussian. More...

subroutine, public	minimax_error (cutoff, hmat, vol, G_min, zet_min, l_mm, n_minimax, minimax_aw, err_mm, delta_mm, potential, pot_par)
	Minimax error, simple analytical formula Note minimax error may blow up for small exponents. This is also observed numerically, but in this case, error estimate is no upper bound. More...

subroutine, public	cutoff_error (cutoff, h_inv, G_min, zet_max, l_max_zet, n_minimax, minimax_aw, err_ctff, C_mm, para_env)
	Cutoff error, estimating G > G_c part of Ewald sum by using C/3 * 1/(Gx^2Gy^2Gz^2)^1/3 as an upper bound for 1/G^2 (AM-GM inequality) and its minimax approximation (factor C). More...

Detailed Description

Methods aiming for error estimate and automatic cutoff calibration. integrals.

History: 2015 09 created

Author: Patrick Seewald

Function/Subroutine Documentation

◆ calibrate_cutoff()

subroutine, public eri_mme_error_control::calibrate_cutoff	(	real(kind=dp), dimension(3, 3), intent(in)	hmat,
		real(kind=dp), dimension(3, 3), intent(in)	h_inv,
		real(kind=dp), intent(in)	G_min,
		real(kind=dp)	vol,
		real(kind=dp), intent(in)	zet_min,
		integer, intent(in)	l_mm,
		real(kind=dp), intent(in)	zet_max,
		integer, intent(in)	l_max_zet,
		integer, intent(in)	n_minimax,
		real(kind=dp), intent(in)	cutoff_l,
		real(kind=dp), intent(in)	cutoff_r,
		real(kind=dp), intent(in)	tol,
		real(kind=dp), intent(in)	delta,
		real(kind=dp), intent(out)	cutoff,
		real(kind=dp), intent(out)	err_mm,
		real(kind=dp), intent(out)	err_c,
		real(kind=dp), intent(out)	C_mm,
		type(mp_para_env_type), intent(in)	para_env,
		logical, intent(in)	print_calib,
		integer, intent(in)	unit_nr
	)

Find optimal cutoff minimizing errors due to minimax approximation and due to finite cutoff using bisection on the difference of the errors.

Parameters

hmat	...
h_inv	...
G_min	...
vol	...
zet_min	Minimum exponent
l_mm	Total ang. mom. quantum number
zet_max	Max. exponents to estimate cutoff error
l_max_zet	Max. total ang. mom. quantum numbers to estimate cutoff error
n_minimax	Number of terms in minimax approximation
cutoff_l	Initial guess of lower bound for cutoff
cutoff_r	Initial guess of upper bound for cutoff
tol	Tolerance (cutoff precision)
delta	to modify initial guess interval
cutoff	Best cutoff
err_mm	Minimax error
err_c	Cutoff error
C_mm	Scaling constant to generalize AM-GM upper bound estimate to minimax approx.
para_env	...
print_calib	...
unit_nr	...

Definition at line 63 of file eri_mme_error_control.F.

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◆ cutoff_minimax_error()

subroutine, public eri_mme_error_control::cutoff_minimax_error	(	real(kind=dp), intent(in)	cutoff,
		real(kind=dp), dimension(3, 3), intent(in)	hmat,
		real(kind=dp), dimension(3, 3), intent(in)	h_inv,
		real(kind=dp), intent(in)	vol,
		real(kind=dp), intent(in)	G_min,
		real(kind=dp), intent(in)	zet_min,
		integer, intent(in)	l_mm,
		real(kind=dp), intent(in)	zet_max,
		integer, intent(in)	l_max_zet,
		integer, intent(in)	n_minimax,
		real(kind=dp), dimension(:), intent(out)	minimax_aw,
		real(kind=dp), intent(out)	err_mm,
		real(kind=dp), intent(out)	err_ctff,
		real(kind=dp), intent(out)	C_mm,
		type(mp_para_env_type), intent(in)	para_env
	)

Compute upper bounds for the errors of 2-center ERI's (P|P) due to minimax approximation and due to finite cutoff, where P is a normalized Hermite Gaussian.

Parameters

cutoff	...
hmat	...
h_inv	...
vol	...
G_min	...
zet_min	Exponent of P to estimate minimax error
l_mm	total ang. mom. quantum number of P to estimate minimax error
zet_max	Max. exponents of P to estimate cutoff error
l_max_zet	Max. total ang. mom. quantum numbers of P to estimate cutoff error
n_minimax	Number of terms in minimax approximation
minimax_aw	Minimax coefficients
err_mm	Minimax error
err_ctff	Cutoff error
C_mm	Scaling constant to generalize AM-GM upper bound estimate to minimax approx.
para_env	...

Definition at line 202 of file eri_mme_error_control.F.

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◆ minimax_error()

subroutine, public eri_mme_error_control::minimax_error	(	real(kind=dp), intent(in)	cutoff,
		real(kind=dp), dimension(3, 3), intent(in)	hmat,
		real(kind=dp), intent(in)	vol,
		real(kind=dp), intent(in)	G_min,
		real(kind=dp), intent(in)	zet_min,
		integer, intent(in)	l_mm,
		integer, intent(in)	n_minimax,
		real(kind=dp), dimension(:), intent(out)	minimax_aw,
		real(kind=dp), intent(out)	err_mm,
		real(kind=dp), intent(out)	delta_mm,
		integer, intent(in), optional	potential,
		real(kind=dp), intent(in), optional	pot_par
	)

Minimax error, simple analytical formula Note minimax error may blow up for small exponents. This is also observed numerically, but in this case, error estimate is no upper bound.

Parameters

cutoff	...
hmat	...
vol	...
G_min	...
zet_min	Exponent of P to estimate minimax error
l_mm	total ang. mom. quantum number of P to estimate minimax error
n_minimax	Number of terms in minimax approximation
minimax_aw	Minimax coefficients
err_mm	Minimax error
delta_mm	...
potential	...
pot_par	...

Definition at line 240 of file eri_mme_error_control.F.

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◆ cutoff_error()

subroutine, public eri_mme_error_control::cutoff_error	(	real(kind=dp), intent(in)	cutoff,
		real(kind=dp), dimension(3, 3), intent(in)	h_inv,
		real(kind=dp), intent(in)	G_min,
		real(kind=dp), intent(in)	zet_max,
		integer, intent(in)	l_max_zet,
		integer, intent(in)	n_minimax,
		real(kind=dp), dimension(:), intent(inout)	minimax_aw,
		real(kind=dp), intent(out)	err_ctff,
		real(kind=dp), intent(out)	C_mm,
		type(mp_para_env_type), intent(in)	para_env
	)

Cutoff error, estimating G > G_c part of Ewald sum by using C/3 * 1/(Gx^2*Gy^2*Gz^2)^1/3 as an upper bound for 1/G^2 (AM-GM inequality) and its minimax approximation (factor C).

Note: usually, minimax approx. falls off faster than 1/G**2, so C should be approximately 1. The error is calculated for all l up to l_max and golden section search algorithm is applied to find the exponent that maximizes cutoff error.

Parameters

cutoff	...
h_inv	...
G_min	...
zet_max	Max. exponents of P to estimate cutoff error
l_max_zet	Max. total ang. mom. quantum numbers of P to estimate cutoff error
n_minimax	Number of terms in minimax approximation
minimax_aw	Minimax coefficients
err_ctff	Cutoff error
C_mm	Scaling constant to generalize AM-GM upper bound estimate to minimax approx.
para_env	...

Definition at line 285 of file eri_mme_error_control.F.

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Functions/Subroutines

Detailed Description

Function/Subroutine Documentation

◆ calibrate_cutoff()

◆ cutoff_minimax_error()

◆ minimax_error()

◆ cutoff_error()