Ewald sums to represent integrals in direct and reciprocal lattice. More...

Functions/Subroutines
subroutine, public	eri_mme_2c_get_rads (la_max, lb_max, zeta, zetb, a_mm, g_min, r_min, sum_precision, g_rad, r_rad)
	Get summation radii for 2c integrals.

subroutine, public	eri_mme_3c_get_rads (la_max, lb_max, lc_max, zeta, zetb, zetc, a_mm, g_min, r_min, sum_precision, g_rads_1, r_rads_2, r_rads_3)
	Get summation radii for 3c integrals.

subroutine, public	eri_mme_2c_get_bounds (hmat, h_inv, vol, is_ortho, g_min, r_min, la_max, lb_max, zeta, zetb, a_mm, sum_precision, n_sum_1d, n_sum_3d, g_bounds, g_rad, r_bounds, r_rad)
	Get summation bounds for 2c integrals.

subroutine, public	eri_mme_3c_get_bounds (hmat, h_inv, vol, is_ortho, g_min, r_min, la_max, lb_max, lc_max, zeta, zetb, zetc, a_mm, sum_precision, n_sum_1d, n_sum_3d, g_bounds_1, g_rads_1, r_bounds_2, r_rads_2, r_bounds_3, r_rads_3)
	Get summation bounds for 3c integrals.

pure subroutine, public	pgf_sum_2c_gspace_1d (s_g, r, alpha, inv_lgth, g_c)
	Compute Ewald-like sum for 2-center ERIs in G space in 1 dimension S_G(l, alpha) = (-i)^linv_lgthsum_G( C(l, alpha, G) exp(iGR) ), with C(l, alpha, r) = r^l exp(-alphar^2), dG = inv_lgthtwopi and G = -G_bounddG, (-G_bound + 1)dG, ..., G_bound*dG for all l < = l_max.

pure subroutine, public	pgf_sum_2c_gspace_3d (s_g, l_max, r, alpha, h_inv, g_c, g_rad, vol, potential, pot_par)
	As pgf_sum_2c_gspace_1d but 3d sum required for non-orthorhombic cells.

pure subroutine, public	pgf_sum_2c_rspace_1d (s_r, r, alpha, lgth, r_c)
	Compute Ewald-like sum for 2-center ERIs in R space in 1 dimension S_R(l, alpha) = SQRT(alpha/pi) sum_R'( H(l, alpha, R-R') ), with H(l, alpha, R) = (-d/dR)^l exp(-alphaR^2), dR = lgth and R' = -R_mindR, (-R_min + 1)dR, ..., R_maxdR, for all l < = l_max.

pure subroutine, public	pgf_sum_2c_rspace_3d (s_r, l_max, r, alpha, hmat, h_inv, r_c, r_rad)
	As pgf_sum_2c_rspace_1d but 3d sum required for non-orthorhombic cells.

pure subroutine, public	pgf_sum_2c_gspace_1d_deltal (s_g, alpha, inv_lgth, g_min, g_c, delta_l, prefactor)
	Compute 1d sum S_G(l, alpha) = inv_lgthsum_G( C(l, alpha, delta_l, G) ) with C(l, alpha, delta_l, G) = prefactor\|G\|^(l-delta_l) exp(-alphaG^2) if G not equal 0 C(l = 0, alpha, delta_l, 0) = 1, C(l>0, alpha, delta_l, 0) = 0 dG = inv_lgthtwopi and G = -G_bounddG, (-G_bound + 1)dG, ..., G_bound*dG for all l < = l_max.

subroutine, public	pgf_sum_3c_1d (s_g, ra, rb, rc, zeta, zetb, zetc, a_mm, lgth, r_bounds_3)
	Compute Ewald-like sum for 3-center integrals in 1 dimension S_G(l, m, n, alpha, beta, gamma) = i^(l+m+n)(-1)^(l+m)inv_lgth^2* sum_G sum_G'( exp(i G R1) C(l,alpha,G) C(m,beta,G'-G) C(n,gamma,G') exp(i G' R2) ) for all l < = l_max, m <= m_max, n <= n_max. a_mm is the minimax exponent. alpha = 1/(4 zeta), beta = 1/(4 zetb), gamma = 1/(4 zetc) + a_mm R1 = RB-RA; R2 = RC-RB Note on method / order arguments: Three equivalent methods (Poisson summation) to compute this sum over Cartesian Gaussians C or Hermite Gaussians H and reciprocal lattice vectors G or direct lattice vectors R:

subroutine, public	pgf_sum_3c_3d (s_g, la_max, lb_max, lc_max, ra, rb, rc, zeta, zetb, zetc, a_mm, hmat, h_inv, vol, g_bounds_1, r_bounds_2, r_bounds_3, g_rads_1, r_rads_2, r_rads_3, method, method_out, order)
	As pgf_sum_3c_1d but 3d sum required for non-orthorhombic cells.

pure real(kind=dp) function, dimension(3), public	ellipsoid_bounds (s_rad, s_to_e)
	Compute bounding box for ellipsoid. This is needed in order to find summation bounds for sphere for sums over non-orthogonal lattice vectors.

pure elemental subroutine, public	get_l (lco, l, lx, ly, lz)
	...

Detailed Description

Ewald sums to represent integrals in direct and reciprocal lattice.

History: 2015 09 created

Author: Patrick Seewald

Function/Subroutine Documentation

◆ eri_mme_2c_get_rads()

subroutine, public eri_mme_lattice_summation::eri_mme_2c_get_rads	(	integer, intent(in)	la_max,
		integer, intent(in)	lb_max,
		real(kind=dp), intent(in)	zeta,
		real(kind=dp), intent(in)	zetb,
		real(kind=dp), intent(in)	a_mm,
		real(kind=dp), intent(in)	g_min,
		real(kind=dp), intent(in)	r_min,
		real(kind=dp), intent(in)	sum_precision,
		real(kind=dp), intent(out), optional	g_rad,
		real(kind=dp), intent(out), optional	r_rad
	)

Get summation radii for 2c integrals.

Parameters

la_max	...
lb_max	...
zeta	...
zetb	...
a_mm	...
G_min	...
R_min	...
sum_precision	...
G_rad	...
R_rad	...

Definition at line 72 of file eri_mme_lattice_summation.F.

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◆ eri_mme_3c_get_rads()

subroutine, public eri_mme_lattice_summation::eri_mme_3c_get_rads	(	integer, intent(in)	la_max,
		integer, intent(in)	lb_max,
		integer, intent(in)	lc_max,
		real(kind=dp), intent(in)	zeta,
		real(kind=dp), intent(in)	zetb,
		real(kind=dp), intent(in)	zetc,
		real(kind=dp), intent(in)	a_mm,
		real(kind=dp), intent(in), optional	g_min,
		real(kind=dp), intent(in), optional	r_min,
		real(kind=dp), intent(in)	sum_precision,
		real(kind=dp), dimension(3), intent(out), optional	g_rads_1,
		real(kind=dp), dimension(3), intent(out), optional	r_rads_2,
		real(kind=dp), dimension(2), intent(out), optional	r_rads_3
	)

Get summation radii for 3c integrals.

Parameters

la_max	...
lb_max	...
lc_max	...
zeta	...
zetb	...
zetc	...
a_mm	...
G_min	...
R_min	...
sum_precision	...
G_rads_1	...
R_rads_2	...
R_rads_3	...

Definition at line 109 of file eri_mme_lattice_summation.F.

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◆ eri_mme_2c_get_bounds()

subroutine, public eri_mme_lattice_summation::eri_mme_2c_get_bounds	(	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,
		logical, intent(in)	is_ortho,
		real(kind=dp), intent(in)	g_min,
		real(kind=dp), intent(in)	r_min,
		integer, intent(in)	la_max,
		integer, intent(in)	lb_max,
		real(kind=dp), intent(in)	zeta,
		real(kind=dp), intent(in)	zetb,
		real(kind=dp), intent(in)	a_mm,
		real(kind=dp), intent(in)	sum_precision,
		integer(kind=int_8), dimension(2, 3), intent(out)	n_sum_1d,
		integer(kind=int_8), dimension(2), intent(out)	n_sum_3d,
		real(kind=dp), dimension(3), intent(out)	g_bounds,
		real(kind=dp), intent(out)	g_rad,
		real(kind=dp), dimension(3), intent(out)	r_bounds,
		real(kind=dp), intent(out)	r_rad
	)

Get summation bounds for 2c integrals.

Parameters

hmat	...
h_inv	...
vol	...
is_ortho	...
G_min	...
R_min	...
la_max	...
lb_max	...
zeta	...
zetb	...
a_mm	...
sum_precision	...
n_sum_1d	...
n_sum_3d	...
G_bounds	...
G_rad	...
R_bounds	...
R_rad	...

Definition at line 174 of file eri_mme_lattice_summation.F.

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◆ eri_mme_3c_get_bounds()

subroutine, public eri_mme_lattice_summation::eri_mme_3c_get_bounds	(	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,
		logical, intent(in)	is_ortho,
		real(kind=dp), intent(in)	g_min,
		real(kind=dp), intent(in)	r_min,
		integer, intent(in)	la_max,
		integer, intent(in)	lb_max,
		integer, intent(in)	lc_max,
		real(kind=dp), intent(in)	zeta,
		real(kind=dp), intent(in)	zetb,
		real(kind=dp), intent(in)	zetc,
		real(kind=dp), intent(in)	a_mm,
		real(kind=dp), intent(in)	sum_precision,
		integer(kind=int_8), dimension(3, 3), intent(out)	n_sum_1d,
		integer(kind=int_8), dimension(3), intent(out)	n_sum_3d,
		real(kind=dp), dimension(3, 3)	g_bounds_1,
		real(kind=dp), dimension(3), intent(out)	g_rads_1,
		real(kind=dp), dimension(3, 3)	r_bounds_2,
		real(kind=dp), dimension(3), intent(out)	r_rads_2,
		real(kind=dp), dimension(2, 3)	r_bounds_3,
		real(kind=dp), dimension(2), intent(out)	r_rads_3
	)

Get summation bounds for 3c integrals.

Parameters

hmat	...
h_inv	...
vol	...
is_ortho	...
G_min	...
R_min	...
la_max	...
lb_max	...
lc_max	...
zeta	...
zetb	...
zetc	...
a_mm	...
sum_precision	...
n_sum_1d	...
n_sum_3d	...
G_bounds_1	...
G_rads_1	...
R_bounds_2	...
R_rads_2	...
R_bounds_3	...
R_rads_3	...

Definition at line 241 of file eri_mme_lattice_summation.F.

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◆ pgf_sum_2c_gspace_1d()

pure subroutine, public eri_mme_lattice_summation::pgf_sum_2c_gspace_1d	(	real(kind=dp), dimension(0:), intent(out)	s_g,
		real(kind=dp), intent(in)	r,
		real(kind=dp), intent(in)	alpha,
		real(kind=dp), intent(in)	inv_lgth,
		real(kind=dp), intent(in)	g_c
	)

Compute Ewald-like sum for 2-center ERIs in G space in 1 dimension S_G(l, alpha) = (-i)^l*inv_lgth*sum_G( C(l, alpha, G) exp(iGR) ), with C(l, alpha, r) = r^l exp(-alpha*r^2), dG = inv_lgth*twopi and G = -G_bound*dG, (-G_bound + 1)*dG, ..., G_bound*dG for all l < = l_max.

Parameters

S_G	...
R	...
alpha	...
inv_lgth	...
G_c	...

Note: S_G is real.

Definition at line 358 of file eri_mme_lattice_summation.F.

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◆ pgf_sum_2c_gspace_3d()

pure subroutine, public eri_mme_lattice_summation::pgf_sum_2c_gspace_3d	(	real(kind=dp), dimension(:), intent(out)	s_g,
		integer, intent(in)	l_max,
		real(kind=dp), dimension(3), intent(in)	r,
		real(kind=dp), intent(in)	alpha,
		real(kind=dp), dimension(3, 3), intent(in)	h_inv,
		real(kind=dp), dimension(3), intent(in)	g_c,
		real(kind=dp), intent(in)	g_rad,
		real(kind=dp), intent(in)	vol,
		integer, intent(in), optional	potential,
		real(kind=dp), intent(in), optional	pot_par
	)

As pgf_sum_2c_gspace_1d but 3d sum required for non-orthorhombic cells.

Parameters

S_G	...
l_max	...
R	...
alpha	...
h_inv	...
G_c	...
G_rad	...
vol	...
coulomb	...

Note: MMME Method is not very efficient for non-orthorhombic cells

Definition at line 413 of file eri_mme_lattice_summation.F.

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◆ pgf_sum_2c_rspace_1d()

pure subroutine, public eri_mme_lattice_summation::pgf_sum_2c_rspace_1d	(	real(kind=dp), dimension(0:), intent(out)	s_r,
		real(kind=dp), intent(in)	r,
		real(kind=dp), intent(in)	alpha,
		real(kind=dp), intent(in)	lgth,
		real(kind=dp), intent(in)	r_c
	)

Compute Ewald-like sum for 2-center ERIs in R space in 1 dimension S_R(l, alpha) = SQRT(alpha/pi) sum_R'( H(l, alpha, R-R') ), with H(l, alpha, R) = (-d/dR)^l exp(-alpha*R^2), dR = lgth and R' = -R_min*dR, (-R_min + 1)*dR, ..., R_max*dR, for all l < = l_max.

Parameters

S_R	...
R	...
alpha	...
lgth	...
R_c	...

Note: result is equivalent to pgf_sum_2c_gspace_1d with S_R(l, alpha) = S_G(l, 1/(4*alpha))

Definition at line 513 of file eri_mme_lattice_summation.F.

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◆ pgf_sum_2c_rspace_3d()

pure subroutine, public eri_mme_lattice_summation::pgf_sum_2c_rspace_3d	(	real(kind=dp), dimension(:), intent(out)	s_r,
		integer, intent(in)	l_max,
		real(kind=dp), dimension(3), intent(in)	r,
		real(kind=dp), intent(in)	alpha,
		real(kind=dp), dimension(3, 3), intent(in)	hmat,
		real(kind=dp), dimension(3, 3), intent(in)	h_inv,
		real(kind=dp), dimension(3), intent(in)	r_c,
		real(kind=dp), intent(in)	r_rad
	)

As pgf_sum_2c_rspace_1d but 3d sum required for non-orthorhombic cells.

Parameters

S_R	...
l_max	...
R	...
alpha	...
hmat	...
h_inv	...
R_c	...
R_rad	...

Note: MMME Method is not very efficient for non-orthorhombic cells

Definition at line 569 of file eri_mme_lattice_summation.F.

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◆ pgf_sum_2c_gspace_1d_deltal()

pure subroutine, public eri_mme_lattice_summation::pgf_sum_2c_gspace_1d_deltal	(	real(kind=dp), dimension(0:), intent(out)	s_g,
		real(kind=dp), intent(in)	alpha,
		real(kind=dp), intent(in)	inv_lgth,
		integer, intent(in)	g_min,
		integer, intent(in)	g_c,
		real(kind=dp), intent(in)	delta_l,
		real(kind=dp), intent(in)	prefactor
	)

Compute 1d sum S_G(l, alpha) = inv_lgth*sum_G( C(l, alpha, delta_l, G) ) with C(l, alpha, delta_l, G) = prefactor*|G|^(l-delta_l) exp(-alpha*G^2) if G not equal 0 C(l = 0, alpha, delta_l, 0) = 1, C(l>0, alpha, delta_l, 0) = 0 dG = inv_lgth*twopi and G = -G_bound*dG, (-G_bound + 1)*dG, ..., G_bound*dG for all l < = l_max.

Parameters

S_G	...
alpha	...
inv_lgth	...
G_min	...
G_c	...
delta_l	...
prefactor	...

Note: needed for cutoff error estimate

Definition at line 655 of file eri_mme_lattice_summation.F.

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◆ pgf_sum_3c_1d()

subroutine, public eri_mme_lattice_summation::pgf_sum_3c_1d	(	real(kind=dp), dimension(0:, 0:, 0:), intent(out)	s_g,
		real(kind=dp), intent(in)	ra,
		real(kind=dp), intent(in)	rb,
		real(kind=dp), intent(in)	rc,
		real(kind=dp), intent(in)	zeta,
		real(kind=dp), intent(in)	zetb,
		real(kind=dp), intent(in)	zetc,
		real(kind=dp), intent(in)	a_mm,
		real(kind=dp), intent(in)	lgth,
		real(kind=dp), dimension(2), intent(in)	r_bounds_3
	)

Compute Ewald-like sum for 3-center integrals in 1 dimension S_G(l, m, n, alpha, beta, gamma) = i^(l+m+n)*(-1)^(l+m)*inv_lgth^2* sum_G sum_G'( exp(i G R1) C(l,alpha,G) C(m,beta,G'-G) C(n,gamma,G') exp(i G' R2) ) for all l < = l_max, m <= m_max, n <= n_max. a_mm is the minimax exponent. alpha = 1/(4 zeta), beta = 1/(4 zetb), gamma = 1/(4 zetc) + a_mm R1 = RB-RA; R2 = RC-RB Note on method / order arguments: Three equivalent methods (Poisson summation) to compute this sum over Cartesian Gaussians C or Hermite Gaussians H and reciprocal lattice vectors G or direct lattice vectors R:

method 1: sum_G sum_G' C(G) C(G,G') C(G')
method 2: sum_G sum_R C(G) C(R)

method 3: sum_R sum_R' H(R, R') The order parameter selects the Gaussian functions over which the sum is performed method 1: order = 1, 2, 3 method 2: order = 1, 2, 3 method 3: order = 1 If method and order are not present, the method / order that converges fastest is automatically chosen.

Parameters

S_G	...
RA	...
RB	...
RC	...
zeta	...
zetb	...
zetc	...
a_mm	...
lgth	...
G_bounds_1	...
R_bounds_2	...
R_bounds_3	...
method	...
method_out	...
order	...

Definition at line 735 of file eri_mme_lattice_summation.F.

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◆ pgf_sum_3c_3d()

subroutine, public eri_mme_lattice_summation::pgf_sum_3c_3d	(	real(kind=dp), dimension(:, :, :), intent(out)	s_g,
		integer, intent(in)	la_max,
		integer, intent(in)	lb_max,
		integer, intent(in)	lc_max,
		real(kind=dp), dimension(3), intent(in)	ra,
		real(kind=dp), dimension(3), intent(in)	rb,
		real(kind=dp), dimension(3), intent(in)	rc,
		real(kind=dp), intent(in)	zeta,
		real(kind=dp), intent(in)	zetb,
		real(kind=dp), intent(in)	zetc,
		real(kind=dp), intent(in)	a_mm,
		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), dimension(3, 3), intent(in)	g_bounds_1,
		real(kind=dp), dimension(3, 3), intent(in)	r_bounds_2,
		real(kind=dp), dimension(2, 3), intent(in)	r_bounds_3,
		real(kind=dp), dimension(3), intent(in)	g_rads_1,
		real(kind=dp), dimension(3), intent(in)	r_rads_2,
		real(kind=dp), dimension(2), intent(in)	r_rads_3,
		integer, intent(in)	method,
		integer, intent(out), optional	method_out,
		integer, intent(in), optional	order
	)

As pgf_sum_3c_1d but 3d sum required for non-orthorhombic cells.

Parameters

S_G	...
la_max	...
lb_max	...
lc_max	...
RA	...
RB	...
RC	...
zeta	...
zetb	...
zetc	...
a_mm	...
hmat	...
h_inv	...
vol	...
G_bounds_1	...
R_bounds_2	...
R_bounds_3	...
G_rads_1	...
R_rads_2	...
R_rads_3	...
method	...
method_out	...
order	...

Definition at line 120267 of file eri_mme_lattice_summation.F.

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◆ ellipsoid_bounds()

pure real(kind=dp) function, dimension(3), public eri_mme_lattice_summation::ellipsoid_bounds	(	real(kind=dp), intent(in)	s_rad,
		real(kind=dp), dimension(3, 3), intent(in)	s_to_e
	)

Compute bounding box for ellipsoid. This is needed in order to find summation bounds for sphere for sums over non-orthogonal lattice vectors.

Parameters

s_rad	sphere radius
s_to_e	sphere to ellipsoid trafo

Returns: ...

Definition at line 121049 of file eri_mme_lattice_summation.F.

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◆ get_l()

pure elemental subroutine, public eri_mme_lattice_summation::get_l	(	integer, intent(in)	lco,
		integer, intent(out)	l,
		integer, intent(out), optional	lx,
		integer, intent(out), optional	ly,
		integer, intent(out), optional	lz
	)

...

Parameters

lco	...
l	...
lx	...
ly	...
lz	...

Definition at line 121092 of file eri_mme_lattice_summation.F.

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Functions/Subroutines

Detailed Description

Function/Subroutine Documentation

◆ eri_mme_2c_get_rads()

◆ eri_mme_3c_get_rads()

◆ eri_mme_2c_get_bounds()

◆ eri_mme_3c_get_bounds()

◆ pgf_sum_2c_gspace_1d()

◆ pgf_sum_2c_gspace_3d()

◆ pgf_sum_2c_rspace_1d()

◆ pgf_sum_2c_rspace_3d()

◆ pgf_sum_2c_gspace_1d_deltal()

◆ pgf_sum_3c_1d()

◆ pgf_sum_3c_3d()

◆ ellipsoid_bounds()

◆ get_l()