Data Types
type	ion_interaction_t

Functions/Subroutines
subroutine, public	ion_interaction_init (this, namespace, space, natoms)

subroutine, public	ion_interaction_init_parallelization (this, natoms, mc)

subroutine, public	ion_interaction_end (this)

subroutine, public	ion_interaction_calculate (this, space, latt, atom, natoms, pos, lsize, energy, force, energy_components, force_components)
	Top level routine for computing electrostatic energies and forces between ions. More...

real(real64) function	jellium_slab_energy_periodic (space, atom, lsize)
	Electrostatic energy of a periodic jellium slab. More...

real(real64) function	jellium_self_energy_finite (dist, latt, atom, lsize)
	Electrostatic self-interaction for jellium instances, with orthogonal cells. More...

subroutine	ion_interaction_finite (dist, space, atom, pos, lsize, energy, force)
	Electrostatic Ewald energy and forces for finite systems. More...

subroutine	ion_interaction_periodic (this, space, latt, atom, natoms, pos, energy, force, energy_components, force_components)
	Total Ewald electrostatic energy and forces, for 1D, 2D and 3D systems. More...

subroutine	ewald_short (dist, space, latt, atom, pos, alpha, ereal, force)
	Short range component of the Ewald electrostatic energy and force. More...

subroutine	ewald_self_interaction (dist, atom, alpha, eself, charge)
	@ brief Ewald self-interaction energy More...

subroutine	ewald_long_3d (this, space, latt, atom, natoms, pos, efourier, force, charge)
	Computes the long-range part of the 3D Ewald summation. More...

subroutine	ewald_long_2d (this, space, latt, atom, natoms, pos, efourier, force)
	In-Chul Yeh and Max L. Berkowitz, J. Chem. Phys. 111, 3155 (1999). More...

subroutine	pseudopotential_correction_3d (dist, latt, atom, charge, epseudo)
	G=0 component of Ewald energy arising from the pseudopotentials, for 3D systems. More...

subroutine, public	ion_interaction_stress (this, space, latt, atom, natoms, pos, stress_ii)
	Computes the contribution to the stress tensor the ion-ion energy. More...

subroutine	ion_interaction_stress_short (this, space, latt, atom, natoms, pos, stress_short)
	Computes the short-range contribution to the stress tensor the ion-ion energy. More...

subroutine	ewald_3d_stress (this, space, latt, atom, natoms, pos, stress_Ewald)
	Computes the contribution to the stress tensor from the 3D Ewald sum. More...

subroutine	ewald_2d_stress (this, space, latt, atom, natoms, pos, stress_Ewald)
	Computes the contribution to the stress tensor from the 2D Ewald sum. More...

real(real64) function	screening_function_2d (alpha, z_ij, gg_abs, erfc)
	Auxiliary function for the Ewald 2D stress. More...

subroutine, public	ion_interaction_test (space, latt, atom, natoms, pos, lsize, namespace, mc)

Variables
integer, parameter	ion_component_real = 1

integer, parameter	ion_component_self = 2

integer, parameter	ion_component_fourier = 3

integer, parameter	ion_num_components = 3

Function/Subroutine Documentation

◆ ion_interaction_init()

subroutine, public ion_interaction_oct_m::ion_interaction_init	(	type(ion_interaction_t), intent(out)	this,
		type(namespace_t), intent(in)	namespace,
		class(space_t), intent(in)	space,
		integer, intent(in)	natoms
	)

Definition at line 162 of file ion_interaction.F90.

◆ ion_interaction_init_parallelization()

subroutine, public ion_interaction_oct_m::ion_interaction_init_parallelization	(	type(ion_interaction_t), intent(inout)	this,
		integer, intent(in)	natoms,
		type(multicomm_t), intent(in)	mc
	)

Definition at line 194 of file ion_interaction.F90.

◆ ion_interaction_end()

subroutine, public ion_interaction_oct_m::ion_interaction_end ( type(ion_interaction_t), intent(inout) this )

Definition at line 212 of file ion_interaction.F90.

◆ ion_interaction_calculate()

subroutine, public ion_interaction_oct_m::ion_interaction_calculate	(	type(ion_interaction_t), intent(inout)	this,
		class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(1:space%dim,1:natoms), intent(in)	pos,
		real(real64), dimension(:), intent(in)	lsize,
		real(real64), intent(out)	energy,
		real(real64), dimension(:, :), intent(out)	force,
		real(real64), dimension(:), intent(out), optional	energy_components,
		real(real64), dimension(:, :, :), intent(out), optional	force_components
	)

Top level routine for computing electrostatic energies and forces between ions.

For additional details about this routine, see http:

Parameters

[in,out]	this	Ion interaction instance
[in]	space	System dimensions and boundary conditions
[in]	latt	Crystal lattice
[in]	atom	Atoms
[in]	natoms	Number of atoms == size(atoms)
[in]	pos	Atomic positions
[in]	lsize	Box half-lengths
[out]	energy	Total ion-ion electrostatic energy
[out]	force	Total force on each ion
[out]	energy_components	Energy contributions
[out]	force_components	Force contributions

Definition at line 228 of file ion_interaction.F90.

◆ jellium_slab_energy_periodic()

real(real64) function ion_interaction_oct_m::jellium_slab_energy_periodic	(	class(space_t), intent(in)	space,
		type(atom_t), dimension(:), intent(in)	atom,
		real(real64), dimension(:), intent(in)	lsize
	)

private

Electrostatic energy of a periodic jellium slab.

From the inner potential been \( v(z) = 4\pi \rho_0 ( z^2/2 + L^2/8)\) we get the energy as \(U = 1/2 \int \rho(r) v(r) d^3r = \pi / 3 \rho^2 A L^3\)

Parameters

[in]	space	System dimensions and boundary conditions
[in]	atom	Atoms
[in]	lsize	Box half-lengths

Returns: Energy of periodic jellium slab

Definition at line 282 of file ion_interaction.F90.

◆ jellium_self_energy_finite()

real(real64) function ion_interaction_oct_m::jellium_self_energy_finite	(	type(distributed_t), intent(in)	dist,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		real(real64), dimension(:), intent(in)	lsize
	)

private

Electrostatic self-interaction for jellium instances, with orthogonal cells.

If no atoms are of species SPECIES_JELLIUM or SPECIES_JELLIUM_SLAB, the returned energy is zero.

Note: One could make the function more readable using pack:
jellium_indices = pack([(i, i=1,natoms)], species == species_jellium)

jellium_slab_indices = pack([(i, i=1,natoms)], species == species_jellium_slab)

but it means looping over natoms an additional two times with no parallelism. @endnote

Parameters

[in]	dist	Atom MPI distribution
[in]	latt	Crystal lattice
[in]	atom	Atoms
[in]	lsize	Box half-lengths

Returns: Jellium contribution to finite electrostatic energy

Definition at line 318 of file ion_interaction.F90.

◆ ion_interaction_finite()

subroutine ion_interaction_oct_m::ion_interaction_finite	(	type(distributed_t), intent(in)	dist,
		class(space_t), intent(in)	space,
		type(atom_t), dimension(:), intent(in)	atom,
		real(real64), dimension(:,:), intent(in)	pos,
		real(real64), dimension(:), intent(in)	lsize,
		real(real64), intent(out)	energy,
		real(real64), dimension(:, :), intent(out)	force
	)

private

Electrostatic Ewald energy and forces for finite systems.

Parameters

[in]	dist	Atom distribution
[in]	space	System dimensions
[in]	atom	Atoms
[in]	pos	Ion positions
[in]	lsize	Box half-lengths
[out]	energy	Total electrostatic energy
[out]	force	Forces

Definition at line 361 of file ion_interaction.F90.

◆ ion_interaction_periodic()

subroutine ion_interaction_oct_m::ion_interaction_periodic	(	type(ion_interaction_t), intent(in)	this,
		class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(1:space%dim,1:natoms), intent(in)	pos,
		real(real64), intent(out)	energy,
		real(real64), dimension(:, :), intent(out)	force,
		real(real64), dimension(:), intent(out), optional	energy_components,
		real(real64), dimension(:, :, :), intent(out), optional	force_components
	)

private

Total Ewald electrostatic energy and forces, for 1D, 2D and 3D systems.

Parameters

[in]	this	Ion interaction instance
[in]	space	System dimensions and boundary conditions
[in]	latt	Crystal lattice
[in]	atom	Atoms
[in]	natoms	Number of atoms in system
[in]	pos	Atomic positions
[out]	energy	Total ion-ion electrostatic energy
[out]	force	Total force on each ion
[out]	energy_components	Ion-ion energy contributions
[out]	force_components	Force contributions

Definition at line 412 of file ion_interaction.F90.

◆ ewald_short()

subroutine ion_interaction_oct_m::ewald_short	(	type(distributed_t), intent(in)	dist,
		class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		real(real64), dimension(:, :), intent(in)	pos,
		real(real64), intent(in)	alpha,
		real(real64), intent(out)	ereal,
		real(real64), dimension(:, :), intent(inout)	force
	)

private

Short range component of the Ewald electrostatic energy and force.

Computes the energy:

\[ E_{\text{short}} = \frac{1}{2} \sum_{i \neq j}^{N_{atom}} \sum_{\mathbf{T}} \frac{Z_i Z_j \operatorname{erfc}\left(\sqrt{\alpha} |\mathbf{r}_{i j} + \mathbf{T}|\right)}{|\mathbf{r}_{i j} + \mathbf{T}|}, \]

and the force:

\[ \mathbf{f}_j = \sum_i \sum_{\mathbf{T}} -Z_i Z_j (\mathbf{r}_{ij} + \mathbf{T}) \frac{\operatorname{erfc}}{|\mathbf{r}_{ij} + \mathbf{T}|} + \frac{2\alpha}{\sqrt{\pi}}\frac{e^{-\alpha |\mathbf{r}_{ij} + \mathbf{T}|^2}}{|\mathbf{r}_{ij} + \mathbf{T}|^2} \]

Note on the Implementation

The middle loop over atoms only iterates over the upper triangle, resulting in a large load imbalance for MPI, however it should still be an improvement w.r.t. iterating over all atoms. OMP cannot be used to collapse non-rectangular loops.

One could alternatively collapse the loops over the whole matrix and distribute a single index, so each process would get its own set of (ia, ja, icopy). The individual indices could be recovered algebraically.

Parameters

[in]	dist	Atomic distribution
[in]	space	System dims and boundary conditions
[in]	latt	Lattice
[in]	atom	Atom properties
[in]	pos	Ion positions
[in]	alpha	Broadening parameter, controlling range separation
[out]	ereal	Realspace contribution to Ewald energy
[in,out]	force	Forces.

Definition at line 500 of file ion_interaction.F90.

◆ ewald_self_interaction()

subroutine ion_interaction_oct_m::ewald_self_interaction	(	type(distributed_t), intent(in)	dist,
		type(atom_t), dimension(:), intent(in)	atom,
		real(real64), intent(in)	alpha,
		real(real64), intent(out)	eself,
		real(real64), intent(out)	charge
	)

private

@ brief Ewald self-interaction energy

The force exerted on an atom by itself is by definition zero. Also returns the total ion charge.

Parameters

[in]	dist	Atomic distribution
[in]	atom	Atom properties
[in]	alpha	Broadening parameter, controlling range separation
[out]	eself	Self-interaction energy
[out]	charge	Total ionic charge

Definition at line 582 of file ion_interaction.F90.

◆ ewald_long_3d()

subroutine ion_interaction_oct_m::ewald_long_3d	(	type(ion_interaction_t), intent(in)	this,
		class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(:,:), intent(in)	pos,
		real(real64), intent(inout)	efourier,
		real(real64), dimension(:, :), intent(inout)	force,
		real(real64), intent(in)	charge
	)

private

Computes the long-range part of the 3D Ewald summation.

Parameters

[in]	this	Ion interaction instance
[in]	space	System dimensions and boundary conditions
[in]	latt	Crystal lattice
[in]	atom	Atoms
[in]	natoms	Number of atoms in system
[in]	pos	(spacedim, natoms)
[in,out]	efourier	Long-range part of total ion-ion energy
[in,out]	force	(spacedim, natoms)
[in]	charge	Total ionic charge

Definition at line 610 of file ion_interaction.F90.

◆ ewald_long_2d()

subroutine ion_interaction_oct_m::ewald_long_2d	(	type(ion_interaction_t), intent(in)	this,
		class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(1:space%dim,1:natoms), intent(in)	pos,
		real(real64), intent(inout)	efourier,
		real(real64), dimension(:, :), intent(inout)	force
	)

private

In-Chul Yeh and Max L. Berkowitz, J. Chem. Phys. 111, 3155 (1999).

Parameters

[in,out] force (spacedim, natoms)

Definition at line 696 of file ion_interaction.F90.

◆ pseudopotential_correction_3d()

subroutine ion_interaction_oct_m::pseudopotential_correction_3d	(	type(distributed_t), intent(in)	dist,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		real(real64)	charge,
		real(real64), intent(out)	epseudo
	)

private

G=0 component of Ewald energy arising from the pseudopotentials, for 3D systems.

See J. Ihm, A. Zunger, M.L. Cohen, J. Phys. C 12, 4409 (1979) This is the term \alpha_1 or Eq. 12.22 in the book of R. Martin.

Parameters

[in]	dist	Atom MPI distribution
[in]	latt	Lattice
[in]	atom	Atoms
[out]	epseudo	Ewald energy correction
	charge	Total ionic charge

Definition at line 901 of file ion_interaction.F90.

◆ ion_interaction_stress()

subroutine, public ion_interaction_oct_m::ion_interaction_stress	(	type(ion_interaction_t), intent(inout)	this,
		class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(:,:), intent(in)	pos,
		real(real64), dimension(space%dim, space%dim), intent(out)	stress_ii
	)

Computes the contribution to the stress tensor the ion-ion energy.

Definition at line 929 of file ion_interaction.F90.

◆ ion_interaction_stress_short()

subroutine ion_interaction_oct_m::ion_interaction_stress_short	(	type(ion_interaction_t), intent(inout)	this,
		class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(1:space%dim,1:natoms), intent(in)	pos,
		real(real64), dimension(1:space%dim, 1:space%dim), intent(out)	stress_short
	)

private

Computes the short-range contribution to the stress tensor the ion-ion energy.

The formula that is implemented here corresponds to the short-range part of the expression B1 of Nielsen and Martin, Stresses in semiconductors: Ab initio calculations on Si, Ge, and GaAs PRB 32, 3792 (1985).

\[ \sigma_{\alpha\beta}^{\rm Ewald-SR} = \frac{\sqrt{\epsilon}}{2\Omega} \sum_{\mathbf{\tau\sigma T}} Z_\tau Z_{\sigma} H(\sqrt{\epsilon}D) \frac{D_\alpha D_{\beta}}{D^2}\Bigg|_{\mathbf{D=x_{\sigma}-x_{\tau}+T\neq0}}\,, \]

where the function \(H(x)\) is

\[ H(x) = \frac{\partial[{\rm erfc}(x)]}{\partial x} - \frac{{\rm erfc}(x)}{x} \]

Definition at line 982 of file ion_interaction.F90.

◆ ewald_3d_stress()

subroutine ion_interaction_oct_m::ewald_3d_stress	(	type(ion_interaction_t), intent(inout)	this,
		class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(1:space%dim,1:natoms), intent(in)	pos,
		real(real64), dimension(3, 3), intent(out)	stress_Ewald
	)

private

Computes the contribution to the stress tensor from the 3D Ewald sum.

The formula that is implemented here correspond to the long-range part of the expression B1 of Nielsen and Martin, Stresses in semiconductors: Ab initio calculations on Si, Ge, and GaAs PRB 32, 3792 (1985).

\[ \sigma_{\alpha\beta}^{\rm Ewald-LR} = \frac{\pi}{2\Omega^2\epsilon} \sum_{\mathbf{G\neq0}} \frac{e^{-G^2/4\epsilon}}{G^2/4\epsilon}\left|\sum_\tau Z_\tau e^{i\mathbf{G}\cdot\mathbf{x}_\tau}\right|^2 \Big(2\frac{G_\alpha G_\beta}{G^2}(G^2/4\epsilon + 1) - \delta_{\alpha\beta}\Big) \nonumber\ + \frac{\pi}{2\Omega^2\epsilon}\left(\sum_{\tau} Z_\tau\right)^2\delta_{\alpha\beta}\,, \]

Definition at line 1067 of file ion_interaction.F90.

◆ ewald_2d_stress()

subroutine ion_interaction_oct_m::ewald_2d_stress	(	type(ion_interaction_t), intent(inout)	this,
		type(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(1:space%dim,1:natoms), intent(in)	pos,
		real(real64), dimension(3, 3), intent(out)	stress_Ewald
	)

private

Computes the contribution to the stress tensor from the 2D Ewald sum.

The formula that is implemented here is derived from the corresponding energy

\[ \frac{\partial \gamma_{\rm Ewald, LR-2D}}{\partial \epsilon_{\alpha\beta}}\Bigg|_{\vec{\epsilon}=\vec{I}} = - \delta_{\alpha\beta} \gamma_{\rm Ewald, LR-2D} + \frac{\pi}{4A\alpha}\sum_{ij}Z_i Z_{j}\sum_{G\neq 0}\cos(\vec{G}\cdot\vec{r}_{ij})\frac{G_\alpha G_\beta}{G^2}\nonumber\ \Bigg( e^{G z_{ij}}\Bigg[ 2\alpha\Big(\frac{1}{G}-z_{ij}){\rm erfc}(\alpha z_{ij} + \frac{G}{2\alpha}) -{\rm erfc`}(\alpha z_{ij} + \frac{G}{2\alpha})\Bigg]\nonumber\ + e^{-G z_{ij}}\Bigg[ 2\alpha\Big(\frac{1}{G}+z_{ij}){\rm erfc}(-\alpha z_{ij} + \frac{G}{2\alpha}) +{\rm erfc`}(-\alpha z_{ij} + \frac{G}{2\alpha})\Bigg] \Bigg)\,. \]

Definition at line 1185 of file ion_interaction.F90.

◆ screening_function_2d()

real(real64) function ion_interaction_oct_m::screening_function_2d	(	real(real64), intent(in)	alpha,
		real(real64), intent(in)	z_ij,
		real(real64), intent(in)	gg_abs,
		real(real64), intent(out)	erfc
	)

private

Auxiliary function for the Ewald 2D stress.

Definition at line 1328 of file ion_interaction.F90.

◆ ion_interaction_test()

subroutine, public ion_interaction_oct_m::ion_interaction_test	(	class(space_t), intent(in)	space,
		type(lattice_vectors_t), intent(in)	latt,
		type(atom_t), dimension(:), intent(in)	atom,
		integer, intent(in)	natoms,
		real(real64), dimension(1:space%dim,1:natoms), intent(in)	pos,
		real(real64), dimension(:), intent(in)	lsize,
		type(namespace_t), intent(in)	namespace,
		type(multicomm_t), intent(in)	mc
	)

Definition at line 1345 of file ion_interaction.F90.

Variable Documentation

◆ ion_component_real

integer, parameter ion_interaction_oct_m::ion_component_real = 1

private

Definition at line 154 of file ion_interaction.F90.

◆ ion_component_self

integer, parameter ion_interaction_oct_m::ion_component_self = 2

private

Definition at line 154 of file ion_interaction.F90.

◆ ion_component_fourier

integer, parameter ion_interaction_oct_m::ion_component_fourier = 3

private

Definition at line 154 of file ion_interaction.F90.

◆ ion_num_components

integer, parameter ion_interaction_oct_m::ion_num_components = 3

private

Definition at line 154 of file ion_interaction.F90.

Data Types

Functions/Subroutines

Variables

Function/Subroutine Documentation

◆ ion_interaction_init()

◆ ion_interaction_init_parallelization()

◆ ion_interaction_end()

◆ ion_interaction_calculate()

◆ jellium_slab_energy_periodic()

◆ jellium_self_energy_finite()

◆ ion_interaction_finite()

◆ ion_interaction_periodic()

◆ ewald_short()

Note on the Implementation

◆ ewald_self_interaction()

◆ ewald_long_3d()

◆ ewald_long_2d()

◆ pseudopotential_correction_3d()

◆ ion_interaction_stress()

◆ ion_interaction_stress_short()

◆ ewald_3d_stress()

◆ ewald_2d_stress()

◆ screening_function_2d()

◆ ion_interaction_test()

Variable Documentation

◆ ion_component_real

◆ ion_component_self

◆ ion_component_fourier

◆ ion_num_components