!--------------------------------------------------------------------------------------------------!
!   CP2K: A general program to perform molecular dynamics simulations                              !
!   Copyright 2000-2025 CP2K developers group <https://cp2k.org>                                   !
!                                                                                                  !
!   SPDX-License-Identifier: GPL-2.0-or-later                                                      !
!--------------------------------------------------------------------------------------------------!

! **************************************************************************************************
!> \brief The implicit (generalized) Poisson solver
!> \par History
!>   06.2014 created [Hossein Bani-Hashemian]
!>   11.2015 - dealt with missing grid points of periodic grids while performing dct;
!>           - revised solver for Neumann and mixed boundary setups.
!> \author Hossein Bani-Hashemian
! **************************************************************************************************
MODULE ps_implicit_methods
   USE bibliography,                    ONLY: BaniHashemian2016,&
                                              cite_reference
   USE cp_log_handling,                 ONLY: cp_get_default_logger,&
                                              cp_logger_get_default_unit_nr,&
                                              cp_logger_type
   USE dct,                             ONLY: &
        dct_type, dct_type_init, neumannX, neumannXY, neumannXYZ, neumannXZ, neumannY, neumannYZ, &
        neumannZ, pw_expand, pw_shrink
   USE dielectric_methods,              ONLY: derive_fft,&
                                              dielectric_create
   USE dielectric_types,                ONLY: dielectric_type
   USE dirichlet_bc_methods,            ONLY: dirichlet_boundary_region_setup
   USE dirichlet_bc_types,              ONLY: dbc_tile_release
   USE kahan_sum,                       ONLY: accurate_sum
   USE kinds,                           ONLY: dp,&
                                              int_8
   USE mathconstants,                   ONLY: fourpi,&
                                              pi
   USE ps_implicit_types,               ONLY: MIXED_BC,&
                                              MIXED_PERIODIC_BC,&
                                              NEUMANN_BC,&
                                              PERIODIC_BC,&
                                              ps_implicit_type
   USE pw_grid_types,                   ONLY: pw_grid_type
   USE pw_methods,                      ONLY: pw_axpy,&
                                              pw_copy,&
                                              pw_integral_ab,&
                                              pw_scale,&
                                              pw_transfer,&
                                              pw_zero
   USE pw_poisson_types,                ONLY: greens_fn_type,&
                                              pw_poisson_parameter_type,&
                                              pw_poisson_type
   USE pw_pool_types,                   ONLY: pw_pool_create,&
                                              pw_pool_release,&
                                              pw_pool_type
   USE pw_types,                        ONLY: pw_c1d_gs_type,&
                                              pw_r3d_rs_type
#include "../base/base_uses.f90"

   IMPLICIT NONE
   PRIVATE
   CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'ps_implicit_methods'

   PUBLIC ps_implicit_create, &
      implicit_poisson_solver_periodic, &
      implicit_poisson_solver_neumann, &
      implicit_poisson_solver_mixed_periodic, &
      implicit_poisson_solver_mixed

   INTERFACE ps_implicit_compute_ehartree
      MODULE PROCEDURE compute_ehartree_periodic_bc, &
         compute_ehartree_mixed_bc
   END INTERFACE ps_implicit_compute_ehartree

   REAL(dp), PRIVATE, PARAMETER         :: large_error = 1.0E4_dp

CONTAINS

! **************************************************************************************************
!> \brief  Creates implicit Poisson solver environment
!> \param pw_pool pool of pw grid
!> \param poisson_params poisson_env parameters
!> \param dct_pw_grid discrete cosine transform (extended) grid
!> \param green green function for FFT based inverse Laplacian
!> \param ps_implicit_env implicit env to be created
!> \par History
!>       06.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_create(pw_pool, poisson_params, dct_pw_grid, green, ps_implicit_env)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(pw_poisson_parameter_type), INTENT(INOUT)     :: poisson_params
      TYPE(pw_grid_type), INTENT(IN), POINTER            :: dct_pw_grid
      TYPE(greens_fn_type), INTENT(IN), POINTER          :: green
      TYPE(ps_implicit_type), INTENT(INOUT), POINTER     :: ps_implicit_env

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_create'

      INTEGER                                            :: boundary_condition, handle, j, &
                                                            n_contacts, neumann_directions
      TYPE(pw_pool_type), POINTER                        :: pw_pool_xpndd

      CALL timeset(routineN, handle)

      CALL cite_reference(BaniHashemian2016)

      IF (.NOT. ASSOCIATED(ps_implicit_env)) THEN
         ALLOCATE (ps_implicit_env)

         ps_implicit_env%do_dbc_cube = poisson_params%dbc_params%do_dbc_cube
         boundary_condition = poisson_params%ps_implicit_params%boundary_condition
         neumann_directions = poisson_params%ps_implicit_params%neumann_directions

! create dielectric
         NULLIFY (ps_implicit_env%dielectric)
         SELECT CASE (boundary_condition)
         CASE (PERIODIC_BC, MIXED_PERIODIC_BC)
            CALL dielectric_create(ps_implicit_env%dielectric, pw_pool, poisson_params%dielectric_params)
         CASE (NEUMANN_BC, MIXED_BC)
            CALL pw_pool_create(pw_pool_xpndd, pw_grid=dct_pw_grid)
            CALL dielectric_create(ps_implicit_env%dielectric, pw_pool_xpndd, poisson_params%dielectric_params)
            CALL pw_pool_release(pw_pool_xpndd)
         END SELECT

! initial guess
         NULLIFY (ps_implicit_env%initial_guess)

! v_eps
         NULLIFY (ps_implicit_env%v_eps)
         ALLOCATE (ps_implicit_env%v_eps)
         CALL pw_pool%create_pw(ps_implicit_env%v_eps)
         CALL pw_zero(ps_implicit_env%v_eps)

! constraint charge
         NULLIFY (ps_implicit_env%cstr_charge)
         SELECT CASE (boundary_condition)
         CASE (MIXED_PERIODIC_BC)
            ALLOCATE (ps_implicit_env%cstr_charge)
            CALL pw_pool%create_pw(ps_implicit_env%cstr_charge)
            CALL pw_zero(ps_implicit_env%cstr_charge)
         CASE (MIXED_BC)
            CALL pw_pool_create(pw_pool_xpndd, pw_grid=dct_pw_grid)
            ALLOCATE (ps_implicit_env%cstr_charge)
            CALL pw_pool_xpndd%create_pw(ps_implicit_env%cstr_charge)
            CALL pw_zero(ps_implicit_env%cstr_charge)
            CALL pw_pool_release(pw_pool_xpndd)
         END SELECT

! initialize energies
         ps_implicit_env%ehartree = 0.0_dp
         ps_implicit_env%electric_enthalpy = 0.0_dp
! times called
         ps_implicit_env%times_called = 0

! dct env
         IF (boundary_condition .EQ. MIXED_BC .OR. boundary_condition .EQ. NEUMANN_BC) THEN
            CALL dct_type_init(pw_pool%pw_grid, neumann_directions, ps_implicit_env%dct_env)
         END IF

! prepare dirichlet bc
         CALL dirichlet_boundary_region_setup(pw_pool, poisson_params, ps_implicit_env%contacts)
         CALL ps_implicit_prepare_blocks(pw_pool, dct_pw_grid, green, poisson_params, ps_implicit_env)
         ! release tiles if they are not supposed to be written into cube files
         IF ((boundary_condition .EQ. MIXED_PERIODIC_BC .OR. boundary_condition .EQ. MIXED_BC) .AND. &
             (.NOT. poisson_params%dbc_params%do_dbc_cube)) THEN
            n_contacts = SIZE(ps_implicit_env%contacts)
            DO j = 1, n_contacts
               CALL dbc_tile_release(ps_implicit_env%contacts(j)%dirichlet_bc, pw_pool)
            END DO
         END IF

      END IF

      CALL timestop(handle)

   END SUBROUTINE ps_implicit_create

! **************************************************************************************************
!> \brief  implicit Poisson solver for periodic boundary conditions
!> \param poisson_env poisson environment
!> \param density electron density
!> \param v_new electrostatic potential
!> \param ehartree Hartree energy
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE implicit_poisson_solver_periodic(poisson_env, density, v_new, ehartree)

      TYPE(pw_poisson_type), INTENT(IN)                  :: poisson_env
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: density
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: v_new
      REAL(dp), INTENT(OUT), OPTIONAL                    :: ehartree

      CHARACTER(LEN=*), PARAMETER :: routineN = 'implicit_poisson_solver_periodic'

      INTEGER                                            :: handle, iter, max_iter, outp_unit, &
                                                            times_called
      LOGICAL                                            :: reached_max_iter, reached_tol, &
                                                            use_zero_initial_guess
      REAL(dp)                                           :: nabs_error, omega, pres_error, tol
      TYPE(dielectric_type), POINTER                     :: dielectric
      TYPE(greens_fn_type), POINTER                      :: green
      TYPE(ps_implicit_type), POINTER                    :: ps_implicit_env
      TYPE(pw_pool_type), POINTER                        :: pw_pool
      TYPE(pw_r3d_rs_type)                               :: g, PxQAinvxres, QAinvxres, res_new, &
                                                            res_old, v_old

      CALL timeset(routineN, handle)

      pw_pool => poisson_env%pw_pools(poisson_env%pw_level)%pool
      dielectric => poisson_env%implicit_env%dielectric
      green => poisson_env%green_fft
      ps_implicit_env => poisson_env%implicit_env

      tol = poisson_env%parameters%ps_implicit_params%tol
      omega = poisson_env%parameters%ps_implicit_params%omega
      max_iter = poisson_env%parameters%ps_implicit_params%max_iter
      use_zero_initial_guess = poisson_env%parameters%ps_implicit_params%zero_initial_guess
      times_called = ps_implicit_env%times_called

! check if this is the first scf iteration
      IF (times_called .EQ. 0) CALL ps_implicit_initial_guess_create(ps_implicit_env, pw_pool)

      CALL pw_pool%create_pw(g)
      CALL pw_pool%create_pw(v_old)
      CALL pw_pool%create_pw(res_old)
      CALL pw_pool%create_pw(res_new)
      CALL pw_pool%create_pw(QAinvxres)
      CALL pw_pool%create_pw(PxQAinvxres)

      IF (use_zero_initial_guess) THEN
         CALL pw_zero(v_old)
      ELSE
         CALL pw_copy(ps_implicit_env%initial_guess, v_old)
      END IF

      g%array = fourpi*density%array/dielectric%eps%array

! res_old = g - \Delta(v_old) - P(v_old)
      CALL apply_poisson_operator_fft(pw_pool, green, dielectric, v_old, res_old)
      CALL pw_scale(res_old, -1.0_dp)
      CALL pw_axpy(g, res_old)

! evaluate \Delta^-1(res_old)
      CALL apply_inv_laplace_operator_fft(pw_pool, green, res_old, QAinvxres)

      iter = 1
      DO

! v_new = v_old + \omega * QAinvxres_old
         CALL pw_scale(QAinvxres, omega)
         CALL pw_copy(QAinvxres, v_new)
         CALL pw_axpy(v_old, v_new)

! res_new = res_old - \omega * ( \Delta(QAinvxres_old) + P(QAinvxres_old) )
!         = (1 - \omega) * res_old - \omega * PxQAinvxres
         CALL apply_P_operator(pw_pool, dielectric, QAinvxres, PxQAinvxres)
         CALL pw_copy(PxQAinvxres, res_new)
         CALL pw_scale(res_new, -1.0_dp)
         CALL pw_axpy(res_old, res_new, 1.0_dp - omega)

! compute the error
         CALL ps_implicit_compute_error_fft(pw_pool, green, res_new, v_old, v_new, QAinvxres, &
                                            pres_error, nabs_error)
! output
         CALL ps_implicit_output(iter, pres_error, nabs_error, outp_unit)
         IF (PRESENT(ehartree)) THEN
            CALL ps_implicit_compute_ehartree(density, v_new, ehartree)
            CALL ps_implicit_report_ehartree(ps_implicit_env, outp_unit, ehartree)
            ps_implicit_env%ehartree = ehartree
         ELSE
            IF (outp_unit .GT. 0) WRITE (outp_unit, '(A1,/)')
         END IF

         iter = iter + 1
         reached_max_iter = iter .GT. max_iter
         reached_tol = pres_error .LE. tol
         IF (pres_error .GT. large_error) &
            CPABORT("Poisson solver did not converge.")
         ps_implicit_env%times_called = ps_implicit_env%times_called + 1
         IF (reached_max_iter .OR. reached_tol) EXIT

! v_old = v_new, res_old = res_new
         CALL pw_copy(v_new, v_old)
         CALL pw_copy(res_new, res_old)

      END DO
      CALL ps_implicit_print_convergence_msg(iter, max_iter, outp_unit)

      IF ((times_called .NE. 0) .AND. (.NOT. use_zero_initial_guess)) &
         CALL pw_copy(v_new, ps_implicit_env%initial_guess)

      IF (PRESENT(ehartree)) ehartree = ps_implicit_env%ehartree
! compute the extra contribution to the Hamiltonian due to the presence of dielectric
      BLOCK
         TYPE(pw_r3d_rs_type) :: v_eps
         v_eps%pw_grid => ps_implicit_env%v_eps%pw_grid
         v_eps%array => ps_implicit_env%v_eps%array
         CALL ps_implicit_compute_veps(pw_pool, dielectric, v_new, v_eps)
      END BLOCK

      CALL pw_pool%give_back_pw(g)
      CALL pw_pool%give_back_pw(v_old)
      CALL pw_pool%give_back_pw(res_old)
      CALL pw_pool%give_back_pw(res_new)
      CALL pw_pool%give_back_pw(QAinvxres)
      CALL pw_pool%give_back_pw(PxQAinvxres)

      CALL timestop(handle)

   END SUBROUTINE implicit_poisson_solver_periodic

! **************************************************************************************************
!> \brief  implicit Poisson solver: zero-average solution of the Poisson equation
!>         subject to homogeneous Neumann boundary conditions
!> \param poisson_env poisson environment
!> \param density electron density
!> \param v_new electrostatic potential
!> \param ehartree Hartree energy
!> \par History
!>       02.2015 created [Hossein Bani-Hashemian]
!>       11.2015 revised [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE implicit_poisson_solver_neumann(poisson_env, density, v_new, ehartree)

      TYPE(pw_poisson_type), INTENT(IN)                  :: poisson_env
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: density
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: v_new
      REAL(dp), INTENT(OUT), OPTIONAL                    :: ehartree

      CHARACTER(LEN=*), PARAMETER :: routineN = 'implicit_poisson_solver_neumann'

      INTEGER                                            :: handle, iter, max_iter, &
                                                            neumann_directions, outp_unit, &
                                                            times_called
      LOGICAL                                            :: reached_max_iter, reached_tol, &
                                                            use_zero_initial_guess
      REAL(dp)                                           :: nabs_error, omega, pres_error, tol, &
                                                            vol_scfac
      TYPE(dct_type), POINTER                            :: dct_env
      TYPE(dielectric_type), POINTER                     :: dielectric
      TYPE(greens_fn_type), POINTER                      :: green
      TYPE(ps_implicit_type), POINTER                    :: ps_implicit_env
      TYPE(pw_pool_type), POINTER                        :: pw_pool, pw_pool_xpndd
      TYPE(pw_r3d_rs_type)                               :: density_xpndd, g, PxQAinvxres, &
                                                            QAinvxres, res_new, res_old, &
                                                            v_eps_xpndd, v_new_xpndd, v_old

      CALL timeset(routineN, handle)

      pw_pool => poisson_env%pw_pools(poisson_env%pw_level)%pool
      dielectric => poisson_env%implicit_env%dielectric
      green => poisson_env%green_fft
      ps_implicit_env => poisson_env%implicit_env
      dct_env => ps_implicit_env%dct_env

      tol = poisson_env%parameters%ps_implicit_params%tol
      omega = poisson_env%parameters%ps_implicit_params%omega
      max_iter = poisson_env%parameters%ps_implicit_params%max_iter
      use_zero_initial_guess = poisson_env%parameters%ps_implicit_params%zero_initial_guess
      neumann_directions = poisson_env%parameters%ps_implicit_params%neumann_directions
      times_called = ps_implicit_env%times_called

      SELECT CASE (neumann_directions)
      CASE (neumannXYZ)
         vol_scfac = 8.0_dp
      CASE (neumannXY, neumannXZ, neumannYZ)
         vol_scfac = 4.0_dp
      CASE (neumannX, neumannY, neumannZ)
         vol_scfac = 2.0_dp
      END SELECT

      CALL pw_pool_create(pw_pool_xpndd, pw_grid=poisson_env%dct_pw_grid)

! check if this is the first scf iteration
      IF (times_called .EQ. 0) CALL ps_implicit_initial_guess_create(ps_implicit_env, pw_pool_xpndd)

      CALL pw_pool_xpndd%create_pw(g)
      CALL pw_pool_xpndd%create_pw(v_old)
      CALL pw_pool_xpndd%create_pw(res_old)
      CALL pw_pool_xpndd%create_pw(res_new)
      CALL pw_pool_xpndd%create_pw(QAinvxres)
      CALL pw_pool_xpndd%create_pw(PxQAinvxres)
      CALL pw_pool_xpndd%create_pw(density_xpndd)
      CALL pw_pool_xpndd%create_pw(v_new_xpndd)
      CALL pw_pool_xpndd%create_pw(v_eps_xpndd)

      IF (use_zero_initial_guess) THEN
         CALL pw_zero(v_old)
      ELSE
         CALL pw_copy(ps_implicit_env%initial_guess, v_old)
      END IF

      CALL pw_expand(neumann_directions, &
                     dct_env%recv_msgs_bnds, dct_env%dests_expand, dct_env%srcs_expand, &
                     dct_env%flipg_stat, dct_env%bounds_shftd, density, density_xpndd)
      CALL pw_expand(neumann_directions, &
                     dct_env%recv_msgs_bnds, dct_env%dests_expand, dct_env%srcs_expand, &
                     dct_env%flipg_stat, dct_env%bounds_shftd, v_new, v_new_xpndd)

      g%array = fourpi*density_xpndd%array/dielectric%eps%array

! res_old = g - \Delta(v_old) - P(v_old)
      CALL apply_poisson_operator_dct(pw_pool_xpndd, green, dielectric, v_old, res_old)
      CALL pw_scale(res_old, -1.0_dp)
      CALL pw_axpy(g, res_old)

! evaluate \Delta^-1(res_old)
      CALL apply_inv_laplace_operator_dct(pw_pool_xpndd, green, res_old, QAinvxres)

      iter = 1
      DO

! v_new = v_old + \omega * QAinvxres_old
         CALL pw_scale(QAinvxres, omega)
         CALL pw_copy(QAinvxres, v_new_xpndd)
         CALL pw_axpy(v_old, v_new_xpndd)

! res_new = res_old - \omega * ( \Delta(QAinvxres_old) + P(QAinvxres_old) )
!         = (1 - \omega) * res_old - \omega * PxQAinvxres
         CALL apply_P_operator(pw_pool_xpndd, dielectric, QAinvxres, PxQAinvxres)
         CALL pw_copy(PxQAinvxres, res_new)
         CALL pw_scale(res_new, -1.0_dp)
         CALL pw_axpy(res_old, res_new, 1.0_dp - omega)

! compute the error
         CALL ps_implicit_compute_error_dct(pw_pool_xpndd, green, res_new, v_old, v_new_xpndd, QAinvxres, &
                                            pres_error, nabs_error)
! output
         CALL ps_implicit_output(iter, pres_error, nabs_error, outp_unit)
         IF (PRESENT(ehartree)) THEN
            CALL ps_implicit_compute_ehartree(density_xpndd, v_new_xpndd, ehartree)
            CALL ps_implicit_report_ehartree(ps_implicit_env, outp_unit, ehartree/vol_scfac)
            ps_implicit_env%ehartree = ehartree/vol_scfac
         ELSE
            IF (outp_unit .GT. 0) WRITE (outp_unit, '(A1,/)')
         END IF

         iter = iter + 1
         reached_max_iter = iter .GT. max_iter
         reached_tol = pres_error .LE. tol
         IF (pres_error .GT. large_error) &
            CPABORT("Poisson solver did not converge.")
         ps_implicit_env%times_called = ps_implicit_env%times_called + 1
         IF (reached_max_iter .OR. reached_tol) EXIT

! v_old = v_new, res_old = res_new
         CALL pw_copy(v_new_xpndd, v_old)
         CALL pw_copy(res_new, res_old)

      END DO
      CALL ps_implicit_print_convergence_msg(iter, max_iter, outp_unit)

      CALL pw_shrink(neumann_directions, dct_env%dests_shrink, dct_env%srcs_shrink, &
                     dct_env%bounds_local_shftd, v_new_xpndd, v_new)

      IF ((times_called .NE. 0) .AND. (.NOT. use_zero_initial_guess)) &
         CALL pw_copy(v_new_xpndd, ps_implicit_env%initial_guess)

      IF (PRESENT(ehartree)) ehartree = ps_implicit_env%ehartree
! compute the extra contribution to the Hamiltonian due to the presence of dielectric
! veps has to be computed for the expanded data and then shrunk otherwise we loose accuracy
      CALL ps_implicit_compute_veps(pw_pool_xpndd, dielectric, v_new_xpndd, v_eps_xpndd)
      BLOCK
         TYPE(pw_r3d_rs_type) :: v_eps
         v_eps%pw_grid => ps_implicit_env%v_eps%pw_grid
         v_eps%array => ps_implicit_env%v_eps%array
         CALL pw_shrink(neumann_directions, dct_env%dests_shrink, dct_env%srcs_shrink, &
                        dct_env%bounds_local_shftd, v_eps_xpndd, v_eps)
      END BLOCK

      CALL pw_pool_xpndd%give_back_pw(g)
      CALL pw_pool_xpndd%give_back_pw(v_old)
      CALL pw_pool_xpndd%give_back_pw(res_old)
      CALL pw_pool_xpndd%give_back_pw(res_new)
      CALL pw_pool_xpndd%give_back_pw(QAinvxres)
      CALL pw_pool_xpndd%give_back_pw(PxQAinvxres)
      CALL pw_pool_xpndd%give_back_pw(density_xpndd)
      CALL pw_pool_xpndd%give_back_pw(v_new_xpndd)
      CALL pw_pool_xpndd%give_back_pw(v_eps_xpndd)
      CALL pw_pool_release(pw_pool_xpndd)

      CALL timestop(handle)

   END SUBROUTINE implicit_poisson_solver_neumann

! **************************************************************************************************
!> \brief  implicit Poisson solver for mixed-periodic boundary conditions (periodic + Dirichlet)
!> \param poisson_env poisson environment
!> \param density electron density
!> \param v_new electrostatic potential
!> \param electric_enthalpy electric enthalpy
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE implicit_poisson_solver_mixed_periodic(poisson_env, density, v_new, electric_enthalpy)

      TYPE(pw_poisson_type), INTENT(IN)                  :: poisson_env
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: density
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: v_new
      REAL(dp), INTENT(OUT), OPTIONAL                    :: electric_enthalpy

      CHARACTER(LEN=*), PARAMETER :: routineN = 'implicit_poisson_solver_mixed_periodic'

      INTEGER :: data_size, handle, iter, j, lb1, lb2, lb3, max_iter, n_contacts, n_tiles_tot, ng, &
         ngpts_local, nt, nt_tot, outp_unit, times_called, ub1, ub2, ub3
      INTEGER(KIND=int_8)                                :: ngpts
      INTEGER, DIMENSION(2, 3)                           :: bounds_local
      INTEGER, DIMENSION(3)                              :: npts_local
      LOGICAL                                            :: reached_max_iter, reached_tol, &
                                                            use_zero_initial_guess
      REAL(dp)                                           :: Axvbar_avg, ehartree, eta, g_avg, &
                                                            gminusAxvbar_avg, nabs_error, omega, &
                                                            pres_error, tol
      REAL(dp), ALLOCATABLE, DIMENSION(:) :: Btxlambda_new, Btxlambda_old, Bxv_bar, Bxv_new, &
         lambda0, lambda_new, lambda_newNeta, lambda_old, QSxlambda, v_bar1D, v_D, v_new1D, w
      REAL(dp), ALLOCATABLE, DIMENSION(:, :)             :: B, Bt, QS, Rinv
      REAL(dp), ALLOCATABLE, DIMENSION(:, :, :)          :: Btxlambda_new3D, Btxlambda_old3D
      TYPE(dielectric_type), POINTER                     :: dielectric
      TYPE(greens_fn_type), POINTER                      :: green
      TYPE(ps_implicit_type), POINTER                    :: ps_implicit_env
      TYPE(pw_grid_type), POINTER                        :: pw_grid
      TYPE(pw_pool_type), POINTER                        :: pw_pool
      TYPE(pw_r3d_rs_type)                               :: Axvbar, g, PxQAinvxres, QAinvxres, &
                                                            res_new, res_old, v_old

      CALL timeset(routineN, handle)

      pw_pool => poisson_env%pw_pools(poisson_env%pw_level)%pool
      pw_grid => pw_pool%pw_grid
      dielectric => poisson_env%implicit_env%dielectric
      green => poisson_env%green_fft
      ps_implicit_env => poisson_env%implicit_env

      ngpts_local = pw_grid%ngpts_local
      ngpts = pw_grid%ngpts
      npts_local = pw_grid%npts_local
      bounds_local = pw_grid%bounds_local
      tol = poisson_env%parameters%ps_implicit_params%tol
      omega = poisson_env%parameters%ps_implicit_params%omega
      max_iter = poisson_env%parameters%ps_implicit_params%max_iter
      use_zero_initial_guess = poisson_env%parameters%ps_implicit_params%zero_initial_guess
      times_called = ps_implicit_env%times_called

      n_contacts = SIZE(ps_implicit_env%contacts)
      n_tiles_tot = 0
      DO j = 1, n_contacts
         n_tiles_tot = n_tiles_tot + ps_implicit_env%contacts(j)%dirichlet_bc%n_tiles
      END DO

      IF (pw_grid%para%blocked) THEN
         data_size = PRODUCT(npts_local)
      ELSE IF (pw_grid%para%ray_distribution) THEN
         data_size = ngpts_local
      ELSE ! parallel run with np = 1
         data_size = PRODUCT(npts_local)
      END IF

! check if this is the first scf iteration
      IF (times_called .EQ. 0) CALL ps_implicit_initial_guess_create(ps_implicit_env, pw_pool)

      ALLOCATE (B(n_tiles_tot, data_size))
      ALLOCATE (Bt(data_size, n_tiles_tot))
      ALLOCATE (QS(n_tiles_tot, n_tiles_tot))
      ALLOCATE (Rinv(n_tiles_tot + 1, n_tiles_tot + 1))

      B(:, :) = ps_implicit_env%B
      Bt(:, :) = ps_implicit_env%Bt
      QS(:, :) = ps_implicit_env%QS
      Rinv(:, :) = ps_implicit_env%Rinv
      CALL get_voltage(poisson_env%parameters%dbc_params%time, ps_implicit_env%v_D, ps_implicit_env%osc_frac, &
                       ps_implicit_env%frequency, ps_implicit_env%phase, v_D)

      lb1 = bounds_local(1, 1); ub1 = bounds_local(2, 1)
      lb2 = bounds_local(1, 2); ub2 = bounds_local(2, 2)
      lb3 = bounds_local(1, 3); ub3 = bounds_local(2, 3)

      ALLOCATE (lambda0(n_tiles_tot), lambda_old(n_tiles_tot), lambda_new(n_tiles_tot))
      ALLOCATE (Btxlambda_old(data_size), Btxlambda_new(data_size))
      ALLOCATE (Btxlambda_old3D(lb1:ub1, lb2:ub2, lb3:ub3), Btxlambda_new3D(lb1:ub1, lb2:ub2, lb3:ub3))
      ALLOCATE (QSxlambda(n_tiles_tot))
      ALLOCATE (w(n_tiles_tot + 1))
      ALLOCATE (lambda_newNeta(n_tiles_tot + 1))
      ALLOCATE (v_bar1D(data_size))
      ALLOCATE (Bxv_bar(n_tiles_tot))

      ALLOCATE (v_new1D(data_size))
      ALLOCATE (Bxv_new(n_tiles_tot))

      CALL pw_pool%create_pw(g)
      CALL pw_pool%create_pw(v_old)
      CALL pw_pool%create_pw(res_old)
      CALL pw_pool%create_pw(res_new)
      CALL pw_pool%create_pw(QAinvxres)
      CALL pw_pool%create_pw(PxQAinvxres)
      CALL pw_pool%create_pw(Axvbar)

      IF (use_zero_initial_guess) THEN
         CALL pw_zero(v_old)
         lambda0 = 0.0_dp
      ELSE
         CALL pw_copy(ps_implicit_env%initial_guess, v_old)
         lambda0(:) = ps_implicit_env%initial_lambda
      END IF

      g%array = fourpi*density%array/dielectric%eps%array
      g_avg = accurate_sum(g%array)/ngpts

      lambda_old(:) = lambda0

! res_old = g - \Delta(v_old) - P(v_old) - B^t * \lambda_old
      CALL apply_poisson_operator_fft(pw_pool, green, dielectric, v_old, res_old)
      CALL pw_scale(res_old, -1.0_dp)
      CALL pw_axpy(g, res_old)
      IF (data_size .NE. 0) THEN
         CALL DGEMV('N', data_size, n_tiles_tot, 1.0_dp, Bt, data_size, lambda_old, 1, 0.0_dp, Btxlambda_old, 1)
      END IF
      CALL convert_1dto3d(ps_implicit_env%idx_1dto3d, Btxlambda_old, Btxlambda_old3D)
      res_old%array = res_old%array - Btxlambda_old3D

! evaluate \Delta^-1(res_old)
      CALL apply_inv_laplace_operator_fft(pw_pool, green, res_old, QAinvxres)

      iter = 1
      DO

! v_new (v_bar) = v_old + \omega * QAinvxres_old
         CALL pw_scale(QAinvxres, omega)
         CALL pw_copy(QAinvxres, v_new)
         CALL pw_axpy(v_old, v_new)

! evaluate 1^t * (g - \Delta(\bar{v}) - P(\bar{v}))
!        = 1^t * (g - P(\bar{v}))
         CALL apply_P_operator(pw_pool, dielectric, v_new, Axvbar)
         Axvbar_avg = accurate_sum(Axvbar%array)/ngpts
         gminusAxvbar_avg = g_avg - Axvbar_avg
         CALL pw_grid%para%group%sum(gminusAxvbar_avg)

! evaluate Q_S * \lambda + v_D - B * \bar{v}
         CALL DGEMV('N', n_tiles_tot, n_tiles_tot, 1.0_dp, QS, n_tiles_tot, lambda_old, 1, 0.0_dp, QSxlambda, 1)
         v_bar1D(ps_implicit_env%idx_1dto3d) = RESHAPE(v_new%array, (/data_size/))
         CALL DGEMV('N', n_tiles_tot, data_size, 1.0_dp, B, n_tiles_tot, v_bar1D, 1, 0.0_dp, Bxv_bar, 1)
         CALL pw_grid%para%group%sum(Bxv_bar)
! solve R [\lambda; \eta] = [Q_S * \lambda + v_D - B * \bar{v}; 1^t * (g - \Delta(\bar{v}) - P(\bar{v}))]
         w = 0.0_dp
         w(:) = (/QSxlambda + v_D - Bxv_bar, gminusAxvbar_avg/)
         CALL DGEMV('N', n_tiles_tot + 1, n_tiles_tot + 1, 1.0_dp, Rinv, n_tiles_tot + 1, w, 1, 0.0_dp, lambda_newNeta, 1)
         lambda_new(:) = lambda_newNeta(1:n_tiles_tot)
         eta = lambda_newNeta(n_tiles_tot + 1)

! v_new = v_bar + 1 * \eta
         v_new%array = v_new%array + eta/ngpts

! evaluate B^t * \lambda_new
         IF (data_size .NE. 0) THEN
            CALL DGEMV('N', data_size, n_tiles_tot, 1.0_dp, Bt, data_size, lambda_new, 1, 0.0_dp, Btxlambda_new, 1)
         END IF
         CALL convert_1dto3d(ps_implicit_env%idx_1dto3d, Btxlambda_new, Btxlambda_new3D)

! res_new = res_old - \omega * ( \Delta(QAinvxres_old) + P(QAinvxres_old) ) - B^t * ( \lambda_new - \lambda_old )
!         = (1 - \omega) * res_old - \omega * P(QAinvxres_old) - B^t * ( \lambda_new - \lambda_old )
         CALL pw_zero(res_new)
         CALL apply_P_operator(pw_pool, dielectric, QAinvxres, PxQAinvxres)
         CALL pw_axpy(PxQAinvxres, res_new, -1.0_dp)
         CALL pw_axpy(res_old, res_new, 1.0_dp - omega)
         res_new%array = res_new%array + Btxlambda_old3D - Btxlambda_new3D

! compute the error
         CALL ps_implicit_compute_error_fft(pw_pool, green, res_new, v_old, v_new, QAinvxres, &
                                            pres_error, nabs_error)
! output
         CALL ps_implicit_output(iter, pres_error, nabs_error, outp_unit)
         IF (PRESENT(electric_enthalpy)) THEN
            CALL ps_implicit_compute_ehartree(dielectric, density, Btxlambda_new3D, v_new, ehartree, electric_enthalpy)
            CALL ps_implicit_report_ehartree(ps_implicit_env, outp_unit, ehartree)
            ps_implicit_env%ehartree = ehartree
            ps_implicit_env%electric_enthalpy = electric_enthalpy
         ELSE
            IF (outp_unit .GT. 0) WRITE (outp_unit, '(A1,/)')
         END IF

! verbose output
         IF (poisson_env%parameters%dbc_params%verbose_output) THEN
            v_new1D(ps_implicit_env%idx_1dto3d) = RESHAPE(v_new%array, (/data_size/))
            CALL DGEMV('N', n_tiles_tot, data_size, 1.0_dp, B, n_tiles_tot, v_new1D, 1, 0.0_dp, Bxv_new, 1)
            CALL pw_grid%para%group%sum(Bxv_new)
            IF (outp_unit .GT. 0) THEN
               WRITE (outp_unit, '(T3,A,A)') "======== verbose ", REPEAT('=', 61)
               WRITE (outp_unit, '(T20,A)') "Drgn       tile      vhartree      lambda "
               WRITE (outp_unit, '(T19,A)') REPEAT('-', 46)
               nt_tot = 1
               DO ng = 1, n_contacts
                  DO nt = 1, ps_implicit_env%contacts(ng)%dirichlet_bc%n_tiles
                     WRITE (outp_unit, '(T17,I6,5X,I6,3X,E13.4,E13.4)') ng, nt, Bxv_new(nt_tot), lambda_new(nt_tot)
                     nt_tot = nt_tot + 1
                  END DO
               END DO
               WRITE (outp_unit, '(T3,A)') REPEAT('=', 78)
            END IF
         END IF

! check the convergence
         iter = iter + 1
         reached_max_iter = iter .GT. max_iter
         reached_tol = pres_error .LE. tol
         ps_implicit_env%times_called = ps_implicit_env%times_called + 1
         IF (pres_error .GT. large_error) &
            CPABORT("Poisson solver did not converge.")
         IF (reached_max_iter .OR. reached_tol) EXIT

! update
         CALL pw_copy(v_new, v_old)
         lambda_old(:) = lambda_new
         CALL pw_copy(res_new, res_old)
         Btxlambda_old3D(:, :, :) = Btxlambda_new3D

      END DO
      CALL ps_implicit_print_convergence_msg(iter, max_iter, outp_unit)

      IF ((times_called .NE. 0) .AND. (.NOT. use_zero_initial_guess)) THEN
         CALL pw_copy(v_new, ps_implicit_env%initial_guess)
         ps_implicit_env%initial_lambda(:) = lambda_new
      END IF

      ps_implicit_env%cstr_charge%array = Btxlambda_new3D
      IF (PRESENT(electric_enthalpy)) electric_enthalpy = ps_implicit_env%electric_enthalpy
! compute the extra contribution to the Hamiltonian due to the presence of dielectric
      BLOCK
         TYPE(pw_r3d_rs_type) :: tmp
         tmp%pw_grid => ps_implicit_env%v_eps%pw_grid
         tmp%array => ps_implicit_env%v_eps%array
         CALL ps_implicit_compute_veps(pw_pool, dielectric, v_new, tmp)
      END BLOCK

      CALL pw_pool%give_back_pw(g)
      CALL pw_pool%give_back_pw(v_old)
      CALL pw_pool%give_back_pw(res_old)
      CALL pw_pool%give_back_pw(res_new)
      CALL pw_pool%give_back_pw(QAinvxres)
      CALL pw_pool%give_back_pw(PxQAinvxres)
      CALL pw_pool%give_back_pw(Axvbar)

      CALL timestop(handle)

   END SUBROUTINE implicit_poisson_solver_mixed_periodic

! **************************************************************************************************
!> \brief  implicit Poisson solver for mixed boundary conditions (Neumann + Dirichlet)
!> \param poisson_env poisson environment
!> \param density electron density
!> \param v_new electrostatic potential
!> \param electric_enthalpy electric enthalpy
!> \par History
!>       10.2014 created [Hossein Bani-Hashemian]
!>       11.2015 revised [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE implicit_poisson_solver_mixed(poisson_env, density, v_new, electric_enthalpy)

      TYPE(pw_poisson_type), INTENT(IN)                  :: poisson_env
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: density
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: v_new
      REAL(dp), INTENT(OUT), OPTIONAL                    :: electric_enthalpy

      CHARACTER(LEN=*), PARAMETER :: routineN = 'implicit_poisson_solver_mixed'

      INTEGER :: data_size, handle, iter, j, lb1, lb2, lb3, max_iter, n_contacts, n_tiles_tot, &
         neumann_directions, ng, ngpts_local, nt, nt_tot, outp_unit, times_called, ub1, ub2, ub3
      INTEGER(KIND=int_8)                                :: ngpts
      INTEGER, DIMENSION(2, 3)                           :: bounds_local
      INTEGER, DIMENSION(3)                              :: npts_local
      LOGICAL                                            :: reached_max_iter, reached_tol, &
                                                            use_zero_initial_guess
      REAL(dp)                                           :: Axvbar_avg, ehartree, eta, g_avg, &
                                                            gminusAxvbar_avg, nabs_error, omega, &
                                                            pres_error, tol, vol_scfac
      REAL(dp), ALLOCATABLE, DIMENSION(:) :: Btxlambda_new, Btxlambda_old, Bxv_bar, Bxv_new, &
         lambda0, lambda_new, lambda_newNeta, lambda_old, QSxlambda, v_bar1D, v_D, v_new1D, w
      REAL(dp), ALLOCATABLE, DIMENSION(:, :)             :: B, Bt, QS, Rinv
      REAL(dp), ALLOCATABLE, DIMENSION(:, :, :)          :: Btxlambda_new3D, Btxlambda_old3D
      TYPE(dct_type), POINTER                            :: dct_env
      TYPE(dielectric_type), POINTER                     :: dielectric
      TYPE(greens_fn_type), POINTER                      :: green
      TYPE(ps_implicit_type), POINTER                    :: ps_implicit_env
      TYPE(pw_grid_type), POINTER                        :: dct_pw_grid, pw_grid
      TYPE(pw_pool_type), POINTER                        :: pw_pool, pw_pool_xpndd
      TYPE(pw_r3d_rs_type)                               :: Axvbar, density_xpndd, g, PxQAinvxres, &
                                                            QAinvxres, res_new, res_old, &
                                                            v_eps_xpndd, v_new_xpndd, v_old

      CALL timeset(routineN, handle)

      pw_pool => poisson_env%pw_pools(poisson_env%pw_level)%pool
      pw_grid => pw_pool%pw_grid
      dielectric => poisson_env%implicit_env%dielectric
      green => poisson_env%green_fft
      ps_implicit_env => poisson_env%implicit_env
      dct_env => ps_implicit_env%dct_env

      dct_pw_grid => poisson_env%dct_pw_grid
      ngpts_local = dct_pw_grid%ngpts_local
      ngpts = dct_pw_grid%ngpts
      npts_local = dct_pw_grid%npts_local
      bounds_local = dct_pw_grid%bounds_local
      tol = poisson_env%parameters%ps_implicit_params%tol
      omega = poisson_env%parameters%ps_implicit_params%omega
      max_iter = poisson_env%parameters%ps_implicit_params%max_iter
      use_zero_initial_guess = poisson_env%parameters%ps_implicit_params%zero_initial_guess
      neumann_directions = poisson_env%parameters%ps_implicit_params%neumann_directions
      times_called = ps_implicit_env%times_called

      SELECT CASE (neumann_directions)
      CASE (neumannXYZ)
         vol_scfac = 8.0_dp
      CASE (neumannXY, neumannXZ, neumannYZ)
         vol_scfac = 4.0_dp
      CASE (neumannX, neumannY, neumannZ)
         vol_scfac = 2.0_dp
      END SELECT

      n_contacts = SIZE(ps_implicit_env%contacts)
      n_tiles_tot = 0
      DO j = 1, n_contacts
         n_tiles_tot = n_tiles_tot + ps_implicit_env%contacts(j)%dirichlet_bc%n_tiles
      END DO

      IF (dct_pw_grid%para%blocked) THEN
         data_size = PRODUCT(npts_local)
      ELSE IF (dct_pw_grid%para%ray_distribution) THEN
         data_size = ngpts_local
      ELSE ! parallel run with np = 1
         data_size = PRODUCT(npts_local)
      END IF

      CALL pw_pool_create(pw_pool_xpndd, pw_grid=dct_pw_grid)

! check if this is the first scf iteration
      IF (times_called .EQ. 0) CALL ps_implicit_initial_guess_create(ps_implicit_env, pw_pool_xpndd)

      ALLOCATE (B(n_tiles_tot, data_size))
      ALLOCATE (Bt(data_size, n_tiles_tot))
      ALLOCATE (QS(n_tiles_tot, n_tiles_tot))
      ALLOCATE (Rinv(n_tiles_tot + 1, n_tiles_tot + 1))

      B(:, :) = ps_implicit_env%B
      Bt(:, :) = ps_implicit_env%Bt
      QS(:, :) = ps_implicit_env%QS
      Rinv(:, :) = ps_implicit_env%Rinv
      CALL get_voltage(poisson_env%parameters%dbc_params%time, ps_implicit_env%v_D, ps_implicit_env%osc_frac, &
                       ps_implicit_env%frequency, ps_implicit_env%phase, v_D)

      lb1 = bounds_local(1, 1); ub1 = bounds_local(2, 1)
      lb2 = bounds_local(1, 2); ub2 = bounds_local(2, 2)
      lb3 = bounds_local(1, 3); ub3 = bounds_local(2, 3)

      ALLOCATE (lambda0(n_tiles_tot), lambda_old(n_tiles_tot), lambda_new(n_tiles_tot))
      ALLOCATE (Btxlambda_old(data_size), Btxlambda_new(data_size))
      ALLOCATE (Btxlambda_old3D(lb1:ub1, lb2:ub2, lb3:ub3), Btxlambda_new3D(lb1:ub1, lb2:ub2, lb3:ub3))
      ALLOCATE (QSxlambda(n_tiles_tot))
      ALLOCATE (w(n_tiles_tot + 1))
      ALLOCATE (lambda_newNeta(n_tiles_tot + 1))
      ALLOCATE (v_bar1D(data_size))
      ALLOCATE (Bxv_bar(n_tiles_tot))

      ALLOCATE (v_new1D(data_size))
      ALLOCATE (Bxv_new(n_tiles_tot))

      CALL pw_pool_xpndd%create_pw(g)
      CALL pw_pool_xpndd%create_pw(v_old)
      CALL pw_pool_xpndd%create_pw(res_old)
      CALL pw_pool_xpndd%create_pw(res_new)
      CALL pw_pool_xpndd%create_pw(QAinvxres)
      CALL pw_pool_xpndd%create_pw(PxQAinvxres)
      CALL pw_pool_xpndd%create_pw(Axvbar)
      CALL pw_pool_xpndd%create_pw(density_xpndd)
      CALL pw_pool_xpndd%create_pw(v_new_xpndd)
      CALL pw_pool_xpndd%create_pw(v_eps_xpndd)

      IF (use_zero_initial_guess) THEN
         CALL pw_zero(v_old)
         lambda0 = 0.0_dp
      ELSE
         CALL pw_copy(ps_implicit_env%initial_guess, v_old)
         lambda0(:) = ps_implicit_env%initial_lambda
      END IF

      CALL pw_expand(neumann_directions, &
                     dct_env%recv_msgs_bnds, dct_env%dests_expand, dct_env%srcs_expand, &
                     dct_env%flipg_stat, dct_env%bounds_shftd, density, density_xpndd)
      CALL pw_expand(neumann_directions, &
                     dct_env%recv_msgs_bnds, dct_env%dests_expand, dct_env%srcs_expand, &
                     dct_env%flipg_stat, dct_env%bounds_shftd, v_new, v_new_xpndd)

      g%array = fourpi*density_xpndd%array/dielectric%eps%array
      g_avg = accurate_sum(g%array)/ngpts

      lambda_old(:) = lambda0

! res_old = g - \Delta(v_old) - P(v_old) - B^t * \lambda_old
      CALL apply_poisson_operator_dct(pw_pool_xpndd, green, dielectric, v_old, res_old)
      CALL pw_scale(res_old, -1.0_dp)
      CALL pw_axpy(g, res_old)
      IF (data_size .NE. 0) THEN
         CALL DGEMV('N', data_size, n_tiles_tot, 1.0_dp, Bt, data_size, lambda_old, 1, 0.0_dp, Btxlambda_old, 1)
      END IF
      CALL convert_1dto3d(ps_implicit_env%idx_1dto3d, Btxlambda_old, Btxlambda_old3D)
      res_old%array = res_old%array - Btxlambda_old3D

! evaluate \Delta^-1(res_old)
      CALL apply_inv_laplace_operator_dct(pw_pool_xpndd, green, res_old, QAinvxres)

      iter = 1
      DO

! v_new (v_bar) = v_old + \omega * QAinvxres_old
         CALL pw_scale(QAinvxres, omega)
         CALL pw_copy(QAinvxres, v_new_xpndd)
         CALL pw_axpy(v_old, v_new_xpndd)

! evaluate 1^t * (g - \Delta(\bar{v}) - P(\bar{v}))
!        = 1^t * (g - P(\bar{v}))
         CALL apply_P_operator(pw_pool_xpndd, dielectric, v_new_xpndd, Axvbar)
         Axvbar_avg = accurate_sum(Axvbar%array)/ngpts
         gminusAxvbar_avg = g_avg - Axvbar_avg
         CALL dct_pw_grid%para%group%sum(gminusAxvbar_avg)

! evaluate Q_S * \lambda + v_D - B * \bar{v}
         CALL DGEMV('N', n_tiles_tot, n_tiles_tot, 1.0_dp, QS, n_tiles_tot, lambda_old, 1, 0.0_dp, QSxlambda, 1)
         v_bar1D(ps_implicit_env%idx_1dto3d) = RESHAPE(v_new_xpndd%array, (/data_size/))
         CALL DGEMV('N', n_tiles_tot, data_size, 1.0_dp, B, n_tiles_tot, v_bar1D, 1, 0.0_dp, Bxv_bar, 1)
         CALL dct_pw_grid%para%group%sum(Bxv_bar)
! solve R [\lambda; \eta] = [Q_S * \lambda + v_D - B * \bar{v}; 1^t * (g - \Delta(\bar{v}) - P(\bar{v}))]
         w = 0.0_dp
         w(:) = (/QSxlambda + v_D - Bxv_bar, gminusAxvbar_avg/)
         CALL DGEMV('N', n_tiles_tot + 1, n_tiles_tot + 1, 1.0_dp, Rinv, n_tiles_tot + 1, w, 1, 0.0_dp, lambda_newNeta, 1)
         lambda_new(:) = lambda_newNeta(1:n_tiles_tot)
         eta = lambda_newNeta(n_tiles_tot + 1)

! v_new = v_bar + 1 * \eta
         v_new_xpndd%array = v_new_xpndd%array + eta/ngpts

! evaluate B^t * \lambda_new
         IF (data_size .NE. 0) THEN
            CALL DGEMV('N', data_size, n_tiles_tot, 1.0_dp, Bt, data_size, lambda_new, 1, 0.0_dp, Btxlambda_new, 1)
         END IF
         CALL convert_1dto3d(ps_implicit_env%idx_1dto3d, Btxlambda_new, Btxlambda_new3D)

! res_new = res_old - \omega * ( \Delta(QAinvxres_old) + P(QAinvxres_old) ) - B^t * ( \lambda_new - \lambda_old )
!         = (1 - \omega) * res_old - \omega * P(QAinvxres_old) - B^t * ( \lambda_new - \lambda_old )
         CALL pw_zero(res_new)
         CALL apply_P_operator(pw_pool_xpndd, dielectric, QAinvxres, PxQAinvxres)
         CALL pw_axpy(PxQAinvxres, res_new, -1.0_dp)
         CALL pw_axpy(res_old, res_new, 1.0_dp - omega)
         res_new%array = res_new%array - Btxlambda_new3D + Btxlambda_old3D

! compute the error
         CALL ps_implicit_compute_error_dct(pw_pool_xpndd, green, res_new, v_old, v_new_xpndd, QAinvxres, &
                                            pres_error, nabs_error)
! output
         CALL ps_implicit_output(iter, pres_error, nabs_error, outp_unit)
         IF (PRESENT(electric_enthalpy)) THEN
            CALL ps_implicit_compute_ehartree(dielectric, density_xpndd, Btxlambda_new3D, v_new_xpndd, &
                                              ehartree, electric_enthalpy)
            CALL ps_implicit_report_ehartree(ps_implicit_env, outp_unit, ehartree/vol_scfac)
            ps_implicit_env%ehartree = ehartree/vol_scfac
            ps_implicit_env%electric_enthalpy = electric_enthalpy/vol_scfac
         ELSE
            IF (outp_unit .GT. 0) WRITE (outp_unit, '(A1,/)')
         END IF

! verbose output
         IF (poisson_env%parameters%dbc_params%verbose_output) THEN
            v_new1D(ps_implicit_env%idx_1dto3d) = RESHAPE(v_new_xpndd%array, (/data_size/))
            CALL DGEMV('N', n_tiles_tot, data_size, 1.0_dp, B, n_tiles_tot, v_new1D, 1, 0.0_dp, Bxv_new, 1)
            CALL pw_grid%para%group%sum(Bxv_new)
            IF (outp_unit .GT. 0) THEN
               WRITE (outp_unit, '(T3,A)') "======== verbose "//REPEAT('=', 61)
               WRITE (outp_unit, '(T20,A)') "Drgn       tile      vhartree      lambda "
               WRITE (outp_unit, '(T19,A)') REPEAT('-', 46)
               nt_tot = 1
               DO ng = 1, n_contacts
                  DO nt = 1, ps_implicit_env%contacts(ng)%dirichlet_bc%n_tiles
                     WRITE (outp_unit, '(T17,I6,5X,I6,3X,E13.4,E13.4)') ng, nt, Bxv_new(nt_tot), lambda_new(nt_tot)
                     nt_tot = nt_tot + 1
                  END DO
               END DO
               WRITE (outp_unit, '(T3,A)') REPEAT('=', 78)
            END IF
         END IF

! check the convergence
         iter = iter + 1
         reached_max_iter = iter .GT. max_iter
         reached_tol = pres_error .LE. tol
         ps_implicit_env%times_called = ps_implicit_env%times_called + 1
         IF (pres_error .GT. large_error) &
            CPABORT("Poisson solver did not converge.")
         IF (reached_max_iter .OR. reached_tol) EXIT

! update
         CALL pw_copy(v_new_xpndd, v_old)
         lambda_old(:) = lambda_new
         CALL pw_copy(res_new, res_old)
         Btxlambda_old3D(:, :, :) = Btxlambda_new3D

      END DO
      CALL ps_implicit_print_convergence_msg(iter, max_iter, outp_unit)

      CALL pw_shrink(neumann_directions, dct_env%dests_shrink, dct_env%srcs_shrink, &
                     dct_env%bounds_local_shftd, v_new_xpndd, v_new)

      IF ((times_called .NE. 0) .AND. (.NOT. use_zero_initial_guess)) THEN
         CALL pw_copy(v_new_xpndd, ps_implicit_env%initial_guess)
         ps_implicit_env%initial_lambda(:) = lambda_new
      END IF

      ps_implicit_env%cstr_charge%array = Btxlambda_new3D
      IF (PRESENT(electric_enthalpy)) electric_enthalpy = ps_implicit_env%electric_enthalpy
! compute the extra contribution to the Hamiltonian due to the presence of dielectric
      CALL ps_implicit_compute_veps(pw_pool_xpndd, dielectric, v_new_xpndd, v_eps_xpndd)
      CALL pw_shrink(neumann_directions, dct_env%dests_shrink, dct_env%srcs_shrink, &
                     dct_env%bounds_local_shftd, v_eps_xpndd, ps_implicit_env%v_eps)

      CALL pw_pool_xpndd%give_back_pw(g)
      CALL pw_pool_xpndd%give_back_pw(v_old)
      CALL pw_pool_xpndd%give_back_pw(res_old)
      CALL pw_pool_xpndd%give_back_pw(res_new)
      CALL pw_pool_xpndd%give_back_pw(QAinvxres)
      CALL pw_pool_xpndd%give_back_pw(PxQAinvxres)
      CALL pw_pool_xpndd%give_back_pw(Axvbar)
      CALL pw_pool_xpndd%give_back_pw(density_xpndd)
      CALL pw_pool_xpndd%give_back_pw(v_new_xpndd)
      CALL pw_pool_xpndd%give_back_pw(v_eps_xpndd)
      CALL pw_pool_release(pw_pool_xpndd)

      CALL timestop(handle)

   END SUBROUTINE implicit_poisson_solver_mixed

! **************************************************************************************************
!> \brief  allocates and zeroises initial guess for implicit (iterative) Poisson solver
!> \param ps_implicit_env the implicit env containing the initial guess
!> \param pw_pool pool of pw grid
!> \par History
!>       06.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_initial_guess_create(ps_implicit_env, pw_pool)

      TYPE(ps_implicit_type), INTENT(INOUT), POINTER     :: ps_implicit_env
      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_initial_guess_create'

      INTEGER                                            :: handle, n_tiles_tot

      CALL timeset(routineN, handle)

      n_tiles_tot = SIZE(ps_implicit_env%v_D)
      NULLIFY (ps_implicit_env%initial_guess)
      ALLOCATE (ps_implicit_env%initial_guess)
      CALL pw_pool%create_pw(ps_implicit_env%initial_guess)
      CALL pw_zero(ps_implicit_env%initial_guess)
      ALLOCATE (ps_implicit_env%initial_lambda(n_tiles_tot))
      ps_implicit_env%initial_lambda = 0.0_dp

      CALL timestop(handle)

   END SUBROUTINE ps_implicit_initial_guess_create

! **************************************************************************************************
!> \brief  prepare blocks B, Bt, QS, R^-1, v_D
!> \param pw_pool_orig original pw grid
!> \param dct_pw_grid DCT (extended) grid
!> \param green green functions for FFT based inverse Laplacian
!> \param poisson_params paramaters of the poisson_env
!> \param ps_implicit_env the implicit_env that stores the blocks
!> \par History
!>       10.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_prepare_blocks(pw_pool_orig, dct_pw_grid, green, &
                                         poisson_params, ps_implicit_env)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool_orig
      TYPE(pw_grid_type), INTENT(IN), POINTER            :: dct_pw_grid
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(pw_poisson_parameter_type), INTENT(IN)        :: poisson_params
      TYPE(ps_implicit_type), INTENT(INOUT), POINTER     :: ps_implicit_env

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_prepare_blocks'

      INTEGER :: data_size, handle, i, indx1, indx2, info, j, k, l, lb1, lb2, lb3, n_contacts, &
         n_tiles, n_tiles_tot, neumann_directions, ngpts_local, ub1, ub2, ub3, unit_nr
      INTEGER(KIND=int_8)                                :: ngpts
      INTEGER, ALLOCATABLE, DIMENSION(:)                 :: ipiv
      INTEGER, DIMENSION(2, 3)                           :: bounds, bounds_local
      INTEGER, DIMENSION(3)                              :: npts, npts_local
      LOGICAL                                            :: done_preparing
      REAL(dp)                                           :: tile_volume, vol_scfac
      REAL(dp), ALLOCATABLE, DIMENSION(:)                :: Bxunit_vec, test_vec, work_arr
      REAL(dp), ALLOCATABLE, DIMENSION(:, :)             :: QAinvxBt, R
      TYPE(cp_logger_type), POINTER                      :: logger
      TYPE(dct_type), POINTER                            :: dct_env
      TYPE(pw_grid_type), POINTER                        :: pw_grid_orig
      TYPE(pw_pool_type), POINTER                        :: pw_pool_xpndd
      TYPE(pw_r3d_rs_type)                               :: pw_in, pw_out

      CALL timeset(routineN, handle)

      pw_grid_orig => pw_pool_orig%pw_grid

      logger => cp_get_default_logger()
      IF (logger%para_env%is_source()) THEN
         unit_nr = cp_logger_get_default_unit_nr(logger, local=.TRUE.)
      ELSE
         unit_nr = -1
      END IF

      SELECT CASE (poisson_params%ps_implicit_params%boundary_condition)
      CASE (MIXED_BC)

         ngpts_local = dct_pw_grid%ngpts_local
         ngpts = dct_pw_grid%ngpts
         npts_local = dct_pw_grid%npts_local
         npts = dct_pw_grid%npts
         bounds_local = dct_pw_grid%bounds_local
         bounds = dct_pw_grid%bounds
         dct_env => ps_implicit_env%dct_env

         neumann_directions = poisson_params%ps_implicit_params%neumann_directions

         SELECT CASE (neumann_directions)
         CASE (neumannXYZ)
            vol_scfac = 8.0_dp
         CASE (neumannXY, neumannXZ, neumannYZ)
            vol_scfac = 4.0_dp
         CASE (neumannX, neumannY, neumannZ)
            vol_scfac = 2.0_dp
         END SELECT

! evaluate indices for converting 3D arrays into 1D arrays
         lb1 = bounds_local(1, 1); ub1 = bounds_local(2, 1)
         lb2 = bounds_local(1, 2); ub2 = bounds_local(2, 2)
         lb3 = bounds_local(1, 3); ub3 = bounds_local(2, 3)

         IF (dct_pw_grid%para%blocked) THEN
            data_size = PRODUCT(npts_local)
         ELSE IF (dct_pw_grid%para%ray_distribution) THEN
            data_size = ngpts_local
         ELSE ! parallel run with np = 1
            data_size = PRODUCT(npts_local)
         END IF

         ALLOCATE (ps_implicit_env%idx_1dto3d(data_size))
         l = 1
         ! Suppress OpenMP (at least the Intel compiler has an issue here)
         ! An automatic OpenMP parallelization of this loop might be tricky
         ! because of the l incrementation
!$OMP PARALLEL IF(.FALSE.)
!$OMP DO
         DO k = lb3, ub3
            DO j = lb2, ub2
               DO i = lb1, ub1
                  ps_implicit_env%idx_1dto3d(l) = (i - lb1 + 1) + &
                                                  (j - lb2)*npts_local(1) + &
                                                  (k - lb3)*npts_local(1)*npts_local(2)
                  l = l + 1
               END DO
            END DO
         END DO
!$OMP END DO
!$OMP END PARALLEL

         n_contacts = SIZE(ps_implicit_env%contacts)
         n_tiles_tot = 0
         DO j = 1, n_contacts
            n_tiles_tot = n_tiles_tot + ps_implicit_env%contacts(j)%dirichlet_bc%n_tiles
         END DO

         ALLOCATE (ps_implicit_env%B(n_tiles_tot, data_size))
         ALLOCATE (ps_implicit_env%Bt(data_size, n_tiles_tot))
         ALLOCATE (ps_implicit_env%QS(n_tiles_tot, n_tiles_tot))
         ALLOCATE (ps_implicit_env%Rinv(n_tiles_tot + 1, n_tiles_tot + 1))
         ALLOCATE (ps_implicit_env%v_D(n_tiles_tot))
         ALLOCATE (ps_implicit_env%osc_frac(n_tiles_tot))
         ALLOCATE (ps_implicit_env%frequency(n_tiles_tot))
         ALLOCATE (ps_implicit_env%phase(n_tiles_tot))

         ALLOCATE (QAinvxBt(data_size, n_tiles_tot))
         ALLOCATE (Bxunit_vec(n_tiles_tot))
         ALLOCATE (test_vec(n_tiles_tot))
         ALLOCATE (R(n_tiles_tot + 1, n_tiles_tot + 1))
         ALLOCATE (work_arr(n_tiles_tot + 1), ipiv(n_tiles_tot + 1)) ! LAPACK work and ipiv arrays

! prepare pw_pool for evaluating inverse Laplacian of tile_pw's using DCT
         CALL pw_pool_create(pw_pool_xpndd, pw_grid=dct_pw_grid)

! set up B, B^t, (\Delta^-1)*B^t
         indx1 = 1
         DO j = 1, n_contacts
            n_tiles = ps_implicit_env%contacts(j)%dirichlet_bc%n_tiles
            indx2 = indx1 + n_tiles - 1
            DO i = 1, n_tiles

               CALL pw_pool_xpndd%create_pw(pw_in)
               CALL pw_expand(neumann_directions, &
                              dct_env%recv_msgs_bnds, dct_env%dests_expand, dct_env%srcs_expand, &
                              dct_env%flipg_stat, dct_env%bounds_shftd, &
                              ps_implicit_env%contacts(j)%dirichlet_bc%tiles(i)%tile%tile_pw, pw_in)

               tile_volume = ps_implicit_env%contacts(j)%dirichlet_bc%tiles(i)%tile%volume
               CALL pw_scale(pw_in, 1.0_dp/(vol_scfac*tile_volume)) ! normalize tile_pw
               ps_implicit_env%Bt(ps_implicit_env%idx_1dto3d, indx1 + i - 1) = RESHAPE(pw_in%array, (/data_size/))

               CALL pw_pool_xpndd%create_pw(pw_out)
               CALL apply_inv_laplace_operator_dct(pw_pool_xpndd, green, pw_in, pw_out)
               QAinvxBt(ps_implicit_env%idx_1dto3d, indx1 + i - 1) = RESHAPE(pw_out%array, (/data_size/))
               ! the electrostatic potential has opposite sign by internal convention
               ps_implicit_env%v_D(indx1 + i - 1) = -1.0_dp*ps_implicit_env%contacts(j)%dirichlet_bc%v_D
               ps_implicit_env%osc_frac(indx1 + i - 1) = ps_implicit_env%contacts(j)%dirichlet_bc%osc_frac
               ps_implicit_env%frequency(indx1 + i - 1) = ps_implicit_env%contacts(j)%dirichlet_bc%frequency
               ps_implicit_env%phase(indx1 + i - 1) = ps_implicit_env%contacts(j)%dirichlet_bc%phase

               CALL pw_pool_xpndd%give_back_pw(pw_in)
               CALL pw_pool_xpndd%give_back_pw(pw_out)
            END DO
            indx1 = indx2 + 1
         END DO
         ps_implicit_env%B(:, :) = TRANSPOSE(ps_implicit_env%Bt)

! evaluate QS = - B*(\Delta^-1)*B^t
         IF (data_size .NE. 0) THEN
            CALL DGEMM('N', 'N', n_tiles_tot, n_tiles_tot, data_size, &
                       -1.0_dp, ps_implicit_env%B, n_tiles_tot, QAinvxBt, &
                       data_size, 0.0_dp, ps_implicit_env%QS, n_tiles_tot)
         END IF
         CALL pw_grid_orig%para%group%sum(ps_implicit_env%QS)

! evaluate B*1
         Bxunit_vec(:) = SUM(ps_implicit_env%B, 2)/ngpts
         CALL pw_grid_orig%para%group%sum(Bxunit_vec)
! set up R = [QS B*1; (B*1)^t 0]
         R = 0.0_dp
         R(1:n_tiles_tot, 1:n_tiles_tot) = ps_implicit_env%QS
         R(1:n_tiles_tot, n_tiles_tot + 1) = Bxunit_vec
         R(n_tiles_tot + 1, 1:n_tiles_tot) = Bxunit_vec
! evaluate R^(-1)
         ps_implicit_env%Rinv(:, :) = R
         CALL DGETRF(n_tiles_tot + 1, n_tiles_tot + 1, ps_implicit_env%Rinv, n_tiles_tot + 1, ipiv, info)
         IF (info .NE. 0) &
            CALL cp_abort(__LOCATION__, &
                          "R is (nearly) singular! Either two Dirichlet constraints are identical or "// &
                          "you need to reduce the number of tiles.")
         CALL DGETRI(n_tiles_tot + 1, ps_implicit_env%Rinv, n_tiles_tot + 1, ipiv, work_arr, n_tiles_tot + 1, info)
         IF (info .NE. 0) &
            CPABORT("Inversion of R failed!")

         DEALLOCATE (QAinvxBt, Bxunit_vec, R, work_arr, ipiv)
         CALL pw_pool_release(pw_pool_xpndd)

         done_preparing = .TRUE.
         CALL pw_grid_orig%para%group%sum(done_preparing)
         IF ((unit_nr .GT. 0) .AND. done_preparing) THEN
            WRITE (unit_nr, "(T3,A,/,T3,A,/,A)") "POISSON| ... Done. ", REPEAT('-', 78)
         END IF

      CASE (MIXED_PERIODIC_BC)

         ngpts_local = pw_grid_orig%ngpts_local
         ngpts = pw_grid_orig%ngpts
         npts_local = pw_grid_orig%npts_local
         npts = pw_grid_orig%npts
         bounds_local = pw_grid_orig%bounds_local
         bounds = pw_grid_orig%bounds
         dct_env => ps_implicit_env%dct_env

! evaluate indices for converting 3D arrays into 1D arrays
         lb1 = bounds_local(1, 1); ub1 = bounds_local(2, 1)
         lb2 = bounds_local(1, 2); ub2 = bounds_local(2, 2)
         lb3 = bounds_local(1, 3); ub3 = bounds_local(2, 3)

         IF (pw_grid_orig%para%blocked) THEN
            data_size = PRODUCT(npts_local)
         ELSE IF (pw_grid_orig%para%ray_distribution) THEN
            data_size = ngpts_local
         ELSE ! parallel run with np = 1
            data_size = PRODUCT(npts_local)
         END IF

         ALLOCATE (ps_implicit_env%idx_1dto3d(data_size))
         l = 1
         ! Suppress OpenMP (at least the Intel compiler has an issue here)
         ! An automatic OpenMP parallelization of this loop might be tricky
         ! because of the l incrementation
!$OMP PARALLEL IF(.FALSE.)
!$OMP DO
         DO k = lb3, ub3
            DO j = lb2, ub2
               DO i = lb1, ub1
                  ps_implicit_env%idx_1dto3d(l) = (i - lb1 + 1) + &
                                                  (j - lb2)*npts_local(1) + &
                                                  (k - lb3)*npts_local(1)*npts_local(2)
                  l = l + 1
               END DO
            END DO
         END DO
!$OMP END DO
!$OMP END PARALLEL

         n_contacts = SIZE(ps_implicit_env%contacts)
         n_tiles_tot = 0
         DO j = 1, n_contacts
            n_tiles_tot = n_tiles_tot + ps_implicit_env%contacts(j)%dirichlet_bc%n_tiles
         END DO

         ALLOCATE (ps_implicit_env%B(n_tiles_tot, data_size))
         ALLOCATE (ps_implicit_env%Bt(data_size, n_tiles_tot))
         ALLOCATE (ps_implicit_env%QS(n_tiles_tot, n_tiles_tot))
         ALLOCATE (ps_implicit_env%Rinv(n_tiles_tot + 1, n_tiles_tot + 1))
         ALLOCATE (ps_implicit_env%v_D(n_tiles_tot))
         ALLOCATE (ps_implicit_env%osc_frac(n_tiles_tot))
         ALLOCATE (ps_implicit_env%frequency(n_tiles_tot))
         ALLOCATE (ps_implicit_env%phase(n_tiles_tot))

         ALLOCATE (QAinvxBt(data_size, n_tiles_tot))
         ALLOCATE (Bxunit_vec(n_tiles_tot))
         ALLOCATE (test_vec(n_tiles_tot))
         ALLOCATE (R(n_tiles_tot + 1, n_tiles_tot + 1))
         ALLOCATE (work_arr(n_tiles_tot + 1), ipiv(n_tiles_tot + 1))

! set up B, B^t, (\Delta^-1)*B^t
         indx1 = 1
         DO j = 1, n_contacts
            n_tiles = ps_implicit_env%contacts(j)%dirichlet_bc%n_tiles
            indx2 = indx1 + n_tiles - 1
            DO i = 1, n_tiles
               CALL pw_pool_orig%create_pw(pw_in)
               CALL pw_copy(ps_implicit_env%contacts(j)%dirichlet_bc%tiles(i)%tile%tile_pw, pw_in)

               tile_volume = ps_implicit_env%contacts(j)%dirichlet_bc%tiles(i)%tile%volume
               CALL pw_scale(pw_in, 1.0_dp/tile_volume) ! normalize tile_pw
               ps_implicit_env%Bt(ps_implicit_env%idx_1dto3d, indx1 + i - 1) = RESHAPE(pw_in%array, (/data_size/))

               CALL pw_pool_orig%create_pw(pw_out)
               CALL apply_inv_laplace_operator_fft(pw_pool_orig, green, pw_in, pw_out)
               QAinvxBt(ps_implicit_env%idx_1dto3d, indx1 + i - 1) = RESHAPE(pw_out%array, (/data_size/))
               ! the electrostatic potential has opposite sign by internal convention
               ps_implicit_env%v_D(indx1 + i - 1) = -1.0_dp*ps_implicit_env%contacts(j)%dirichlet_bc%v_D
               ps_implicit_env%osc_frac(indx1 + i - 1) = ps_implicit_env%contacts(j)%dirichlet_bc%osc_frac
               ps_implicit_env%frequency(indx1 + i - 1) = ps_implicit_env%contacts(j)%dirichlet_bc%frequency
               ps_implicit_env%phase(indx1 + i - 1) = ps_implicit_env%contacts(j)%dirichlet_bc%phase

               CALL pw_pool_orig%give_back_pw(pw_in)
               CALL pw_pool_orig%give_back_pw(pw_out)
            END DO
            indx1 = indx2 + 1
         END DO
         ps_implicit_env%B(:, :) = TRANSPOSE(ps_implicit_env%Bt)

! evaluate QS = - B*(\Delta^-1)*B^t
         IF (data_size .NE. 0) THEN
            CALL DGEMM('N', 'N', n_tiles_tot, n_tiles_tot, data_size, &
                       -1.0_dp, ps_implicit_env%B, n_tiles_tot, QAinvxBt, &
                       data_size, 0.0_dp, ps_implicit_env%QS, n_tiles_tot)
         END IF
         CALL pw_grid_orig%para%group%sum(ps_implicit_env%QS)

! evaluate B*1
         Bxunit_vec(:) = SUM(ps_implicit_env%B, 2)/ngpts
         CALL pw_grid_orig%para%group%sum(Bxunit_vec)
! set up R = [QS B*1; (B*1)^t 0]
         R = 0.0_dp
         R(1:n_tiles_tot, 1:n_tiles_tot) = ps_implicit_env%QS
         R(1:n_tiles_tot, n_tiles_tot + 1) = Bxunit_vec
         R(n_tiles_tot + 1, 1:n_tiles_tot) = Bxunit_vec
! evaluate R^(-1)
         ps_implicit_env%Rinv(:, :) = R
         CALL DGETRF(n_tiles_tot + 1, n_tiles_tot + 1, ps_implicit_env%Rinv, n_tiles_tot + 1, ipiv, info)
         IF (info .NE. 0) &
            CALL cp_abort(__LOCATION__, &
                          "R is (nearly) singular! Either two Dirichlet constraints are identical or "// &
                          "you need to reduce the number of tiles.")
         CALL DGETRI(n_tiles_tot + 1, ps_implicit_env%Rinv, n_tiles_tot + 1, ipiv, work_arr, n_tiles_tot + 1, info)
         IF (info .NE. 0) &
            CPABORT("Inversion of R failed!")

         DEALLOCATE (QAinvxBt, Bxunit_vec, R, work_arr, ipiv)

         done_preparing = .TRUE.
         CALL pw_grid_orig%para%group%sum(done_preparing)
         IF ((unit_nr .GT. 0) .AND. done_preparing) THEN
            WRITE (unit_nr, "(T3,A,/,T3,A,/,A)") "POISSON| ... Done. ", REPEAT('-', 78)
         END IF

      CASE (PERIODIC_BC, NEUMANN_BC)

         ALLOCATE (ps_implicit_env%idx_1dto3d(1))
         ALLOCATE (ps_implicit_env%B(1, 1))
         ALLOCATE (ps_implicit_env%Bt(1, 1))
         ALLOCATE (ps_implicit_env%QS(1, 1))
         ALLOCATE (ps_implicit_env%Rinv(1, 1))
         ALLOCATE (ps_implicit_env%v_D(1))
         ALLOCATE (ps_implicit_env%osc_frac(1))
         ALLOCATE (ps_implicit_env%frequency(1))
         ALLOCATE (ps_implicit_env%phase(1))

         ps_implicit_env%idx_1dto3d = 1
         ps_implicit_env%B = 0.0_dp
         ps_implicit_env%Bt = 0.0_dp
         ps_implicit_env%QS = 0.0_dp
         ps_implicit_env%Rinv = 0.0_dp
         ps_implicit_env%v_D = 0.0_dp

      CASE DEFAULT
         CALL cp_abort(__LOCATION__, &
                       "Please specify the type of boundary conditions using the "// &
                       "input file keyword BOUNDARY_CONDITIONS.")
      END SELECT

      CALL timestop(handle)

   END SUBROUTINE ps_implicit_prepare_blocks

! **************************************************************************************************
!> \brief   Evaluates the action of the operator P on a given matrix v, defined
!>          as:   P(v) := - \nabla_r(\ln(\eps)) \cdot \nabla_r(v)
!> \param pw_pool pool of pw grid
!> \param dielectric dielectric_type containing eps
!> \param v input matrix
!> \param Pxv action of the operator P on v
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE apply_P_operator(pw_pool, dielectric, v, Pxv)

      TYPE(pw_pool_type), POINTER                        :: pw_pool
      TYPE(dielectric_type), INTENT(IN), POINTER         :: dielectric
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: v
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: Pxv

      CHARACTER(LEN=*), PARAMETER                        :: routineN = 'apply_P_operator'

      INTEGER                                            :: handle, i
      TYPE(pw_r3d_rs_type), DIMENSION(3)                 :: dv

      CALL timeset(routineN, handle)

      DO i = 1, 3
         CALL pw_pool%create_pw(dv(i))
      END DO

      CALL derive_fft(v, dv, pw_pool)
      ASSOCIATE (dln_eps => dielectric%dln_eps)
         Pxv%array = -(dv(1)%array*dln_eps(1)%array + &
                       dv(2)%array*dln_eps(2)%array + &
                       dv(3)%array*dln_eps(3)%array)
      END ASSOCIATE

      DO i = 1, 3
         CALL pw_pool%give_back_pw(dv(i))
      END DO

      CALL timestop(handle)

   END SUBROUTINE apply_P_operator

! **************************************************************************************************
!> \brief  Evaluates the action of the inverse of the Laplace operator on a given 3d matrix
!> \param pw_pool pool of pw grid
!> \param green green functions for FFT based inverse Laplacian
!> \param pw_in pw_in (density)
!> \param pw_out pw_out (potential)
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE apply_inv_laplace_operator_fft(pw_pool, green, pw_in, pw_out)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: pw_in
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: pw_out

      CHARACTER(LEN=*), PARAMETER :: routineN = 'apply_inv_laplace_operator_fft'

      INTEGER                                            :: handle, ig, ng
      REAL(dp)                                           :: prefactor
      TYPE(pw_c1d_gs_type)                               :: pw_in_gs
      TYPE(pw_grid_type), POINTER                        :: pw_grid

      CALL timeset(routineN, handle)

! here I divide by fourpi to cancel out the prefactor fourpi in influence_fn
      prefactor = 1.0_dp/fourpi

      pw_grid => pw_pool%pw_grid
      ng = SIZE(pw_grid%gsq)

      CALL pw_pool%create_pw(pw_in_gs)

      CALL pw_transfer(pw_in, pw_in_gs)
      DO ig = 1, ng
         pw_in_gs%array(ig) = prefactor*pw_in_gs%array(ig)*green%influence_fn%array(ig)
      END DO
      CALL pw_transfer(pw_in_gs, pw_out)

      CALL pw_pool%give_back_pw(pw_in_gs)

      CALL timestop(handle)

   END SUBROUTINE apply_inv_laplace_operator_fft

! **************************************************************************************************
!> \brief  Evaluates the action of the inverse of the Laplace operator on a given
!>         3d matrix using DCT-I
!> \param pw_pool pool of pw grid
!> \param green the greens_fn_type data holding a valid dct_influence_fn
!> \param pw_in pw_in (density)
!> \param pw_out pw_out (potential)
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!>       11.2015 revised [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE apply_inv_laplace_operator_dct(pw_pool, green, pw_in, pw_out)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: pw_in
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: pw_out

      CHARACTER(LEN=*), PARAMETER :: routineN = 'apply_inv_laplace_operator_dct'

      INTEGER                                            :: handle, ig, ng
      REAL(dp)                                           :: prefactor
      TYPE(pw_c1d_gs_type)                               :: pw_in_gs
      TYPE(pw_grid_type), POINTER                        :: pw_grid

      CALL timeset(routineN, handle)

! here I divide by fourpi to cancel out the prefactor fourpi in influence_fn
      prefactor = 1.0_dp/fourpi

      pw_grid => pw_pool%pw_grid
      ng = SIZE(pw_grid%gsq)

      CALL pw_pool%create_pw(pw_in_gs)

      CALL pw_transfer(pw_in, pw_in_gs)
      DO ig = 1, ng
         pw_in_gs%array(ig) = prefactor*pw_in_gs%array(ig)*green%dct_influence_fn%array(ig)
      END DO
      CALL pw_transfer(pw_in_gs, pw_out)

      CALL pw_pool%give_back_pw(pw_in_gs)

      CALL timestop(handle)

   END SUBROUTINE apply_inv_laplace_operator_dct

! **************************************************************************************************
!> \brief  Evaluates the action of the Laplace operator on a given 3d matrix
!> \param pw_pool pool of pw grid
!> \param green green functions for FFT based inverse Laplacian
!> \param pw_in pw_in (potential)
!> \param pw_out pw_out (density)
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE apply_laplace_operator_fft(pw_pool, green, pw_in, pw_out)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: pw_in
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: pw_out

      CHARACTER(LEN=*), PARAMETER :: routineN = 'apply_laplace_operator_fft'

      INTEGER                                            :: g0_index, handle, ig, ng
      LOGICAL                                            :: have_g0
      REAL(dp)                                           :: prefactor
      TYPE(pw_c1d_gs_type)                               :: pw_in_gs
      TYPE(pw_grid_type), POINTER                        :: pw_grid

      CALL timeset(routineN, handle)

! here I multiply by fourpi to cancel out the prefactor fourpi in influence_fn
      prefactor = fourpi

      pw_grid => pw_pool%pw_grid
      ng = SIZE(pw_in%pw_grid%gsq)
      have_g0 = green%influence_fn%pw_grid%have_g0

      CALL pw_pool%create_pw(pw_in_gs)

      CALL pw_transfer(pw_in, pw_in_gs)

      IF (have_g0) THEN
         g0_index = green%influence_fn%pw_grid%first_gne0 - 1
         pw_in_gs%array(g0_index) = 0.0_dp
      END IF
      DO ig = green%influence_fn%pw_grid%first_gne0, ng
         pw_in_gs%array(ig) = prefactor*(pw_in_gs%array(ig)/green%influence_fn%array(ig))
      END DO

      CALL pw_transfer(pw_in_gs, pw_out)

      CALL pw_pool%give_back_pw(pw_in_gs)

      CALL timestop(handle)

   END SUBROUTINE apply_laplace_operator_fft

! **************************************************************************************************
!> \brief  Evaluates the action of the Laplace operator on a given 3d matrix using DCT-I
!> \param pw_pool pool of pw grid
!> \param green the greens_fn_type data holding a valid dct_influence_fn
!> \param pw_in pw_in (potential)
!> \param pw_out pw_out (density)
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!>       11.2015 revised [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE apply_laplace_operator_dct(pw_pool, green, pw_in, pw_out)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: pw_in
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: pw_out

      CHARACTER(LEN=*), PARAMETER :: routineN = 'apply_laplace_operator_dct'

      INTEGER                                            :: g0_index, handle, ig, ng
      LOGICAL                                            :: have_g0
      REAL(dp)                                           :: prefactor
      TYPE(pw_c1d_gs_type)                               :: pw_in_gs
      TYPE(pw_grid_type), POINTER                        :: pw_grid

      CALL timeset(routineN, handle)

! here I multiply by fourpi to cancel out the prefactor fourpi in influence_fn
      prefactor = fourpi

      pw_grid => pw_pool%pw_grid
      ng = SIZE(pw_in%pw_grid%gsq)
      have_g0 = green%dct_influence_fn%pw_grid%have_g0

      CALL pw_pool%create_pw(pw_in_gs)

      CALL pw_transfer(pw_in, pw_in_gs)

      IF (have_g0) THEN
         g0_index = green%dct_influence_fn%pw_grid%first_gne0 - 1
         pw_in_gs%array(g0_index) = 0.0_dp
      END IF
      DO ig = green%dct_influence_fn%pw_grid%first_gne0, ng
         pw_in_gs%array(ig) = prefactor*(pw_in_gs%array(ig)/green%dct_influence_fn%array(ig))
      END DO

      CALL pw_transfer(pw_in_gs, pw_out)

      CALL pw_pool%give_back_pw(pw_in_gs)

      CALL timestop(handle)

   END SUBROUTINE apply_laplace_operator_dct

! **************************************************************************************************
!> \brief  Evaluates the action of the generalized Poisson operator on a given 3d matrix.
!> \param pw_pool pool of pw grid
!> \param green green functions for FFT based inverse Laplacian
!> \param dielectric dielectric environment
!> \param v potential
!> \param density density
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE apply_poisson_operator_fft(pw_pool, green, dielectric, v, density)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(dielectric_type), INTENT(IN), POINTER         :: dielectric
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: v
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: density

      CHARACTER(LEN=*), PARAMETER :: routineN = 'apply_poisson_operator_fft'

      INTEGER                                            :: handle
      TYPE(pw_r3d_rs_type)                               :: Pxv

      CALL timeset(routineN, handle)

      CALL pw_pool%create_pw(Pxv)

      CALL apply_P_operator(pw_pool, dielectric, v, Pxv)
      CALL apply_laplace_operator_fft(pw_pool, green, v, density)
      CALL pw_axpy(Pxv, density)

      CALL pw_pool%give_back_pw(Pxv)

      CALL timestop(handle)

   END SUBROUTINE apply_poisson_operator_fft

! **************************************************************************************************
!> \brief  Evaluates the action of the generalized Poisson operator on a given
!>         3d matrix using DCT-I.
!> \param pw_pool pool of pw grid
!> \param green the greens_fn_type data holding a valid dct_influence_fn
!> \param dielectric dielectric environment
!> \param v potential
!> \param density density
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!>       11.2015 revised [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE apply_poisson_operator_dct(pw_pool, green, dielectric, v, density)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(dielectric_type), INTENT(IN), POINTER         :: dielectric
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: v
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: density

      CHARACTER(LEN=*), PARAMETER :: routineN = 'apply_poisson_operator_dct'

      INTEGER                                            :: handle
      TYPE(pw_r3d_rs_type)                               :: Pxv

      CALL timeset(routineN, handle)

      CALL pw_pool%create_pw(Pxv)

      CALL apply_P_operator(pw_pool, dielectric, v, Pxv)
      CALL apply_laplace_operator_dct(pw_pool, green, v, density)
      CALL pw_axpy(Pxv, density)

      CALL pw_pool%give_back_pw(Pxv)

      CALL timestop(handle)

   END SUBROUTINE apply_poisson_operator_dct

! **************************************************************************************************
!> \brief Computes the extra contribution (v_eps)
!>        v_eps = - \frac{1}{8*\pi} * |\nabla_r(v)|^2 * \frac{d \eps}{d \rho}
!>  to the functional derivative of the Hartree energy wrt the density, being
!>  attributed to the dependency of the dielectric constant to the charge density.
!> [see V. M. Sanchez, M. Sued, and D. A. Scherlis, J. Chem. Phys. 131, 174108 (2009)]
!> \param pw_pool pool of the original plane-wave grid
!> \param dielectric dielectric environment
!> \param v Hartree potential
!> \param v_eps v_eps
!> \par History
!>       08.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_compute_veps(pw_pool, dielectric, v, v_eps)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(dielectric_type), INTENT(IN), POINTER         :: dielectric
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: v
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: v_eps

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_compute_veps'

      INTEGER                                            :: handle, i
      REAL(dp)                                           :: eightpi
      TYPE(pw_r3d_rs_type)                               :: dv2
      TYPE(pw_r3d_rs_type), DIMENSION(3)                 :: dv

      CALL timeset(routineN, handle)

      eightpi = 2*fourpi

      CALL pw_pool%create_pw(dv2)
      DO i = 1, 3
         CALL pw_pool%create_pw(dv(i))
      END DO

      CALL derive_fft(v, dv, pw_pool)

! evaluate |\nabla_r(v)|^2
      dv2%array = dv(1)%array**2 + dv(2)%array**2 + dv(3)%array**2

      v_eps%array = -(1.0_dp/eightpi)*(dv2%array*dielectric%deps_drho%array)

      CALL pw_pool%give_back_pw(dv2)
      DO i = 1, 3
         CALL pw_pool%give_back_pw(dv(i))
      END DO

      CALL timestop(handle)

   END SUBROUTINE ps_implicit_compute_veps

! **************************************************************************************************
!> \brief Computes the Hartree energy
!> \param density electronic density
!> \param v Hartree potential
!> \param ehartree Hartree energy
!> \par History
!>       06.2015 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE compute_ehartree_periodic_bc(density, v, ehartree)

      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: density, v
      REAL(dp), INTENT(OUT)                              :: ehartree

      CHARACTER(LEN=*), PARAMETER :: routineN = 'compute_ehartree_periodic_bc'

      INTEGER                                            :: handle

      CALL timeset(routineN, handle)

! E_H = \frac{1}{2} * \int \rho * v dr
      ehartree = 0.5_dp*pw_integral_ab(density, v)

      CALL timestop(handle)

   END SUBROUTINE compute_ehartree_periodic_bc

! **************************************************************************************************
!> \brief Computes the Hartree energy
!> \param dielectric dielectric environment
!> \param density electronic density
!> \param Btxlambda B^t * \lambda (\lambda is the vector of Lagrange multipliers
!>                  and B^t is the transpose of the boundary operator
!> \param v Hartree potential
!> \param ehartree Hartree energy
!> \param electric_enthalpy electric enthalpy
!> \par History
!>       06.2015 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE compute_ehartree_mixed_bc(dielectric, density, Btxlambda, v, ehartree, electric_enthalpy)

      TYPE(dielectric_type), INTENT(IN), POINTER         :: dielectric
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: density
      REAL(dp), ALLOCATABLE, DIMENSION(:, :, :), &
         INTENT(IN)                                      :: Btxlambda
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: v
      REAL(dp), INTENT(OUT)                              :: ehartree, electric_enthalpy

      CHARACTER(LEN=*), PARAMETER :: routineN = 'compute_ehartree_mixed_bc'

      INTEGER                                            :: handle
      REAL(dp)                                           :: dvol, ehartree_rho, ehartree_rho_cstr
      TYPE(pw_grid_type), POINTER                        :: pw_grid

      CALL timeset(routineN, handle)

      pw_grid => v%pw_grid

      dvol = pw_grid%dvol

! E_H = \frac{1}{2} * \int \rho * v dr + \frac{1}{8 \pi} * \int Btxlambda * v dr
! the sign of the second term depends on the sign chosen for the Lagrange multiplier
! term in the variational form
      ehartree_rho = accurate_sum(density%array*v%array)
      ehartree_rho_cstr = accurate_sum(dielectric%eps%array*Btxlambda*v%array/fourpi)
      ehartree_rho = 0.5_dp*ehartree_rho*dvol
      ehartree_rho_cstr = 0.5_dp*ehartree_rho_cstr*dvol
      CALL pw_grid%para%group%sum(ehartree_rho)
      CALL pw_grid%para%group%sum(ehartree_rho_cstr)
      electric_enthalpy = ehartree_rho + ehartree_rho_cstr
      ehartree = ehartree_rho - ehartree_rho_cstr

      CALL timestop(handle)

   END SUBROUTINE compute_ehartree_mixed_bc

! **************************************************************************************************
!> \brief  Computes the (normalized) preconditioned residual norm error and the
!>         normalized absolute error
!> \param pw_pool pool of the original plane-wave grid
!> \param green greens functions for FFT based inverse Laplacian
!> \param res_new residual
!> \param v_old old v
!> \param v_new new v
!> \param QAinvxres_new Delta^-1(res_new)
!> \param pres_error preconditioned residual norm error
!> \param nabs_error normalized absolute error
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_compute_error_fft(pw_pool, green, res_new, v_old, v_new, &
                                            QAinvxres_new, pres_error, nabs_error)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: res_new, v_old, v_new
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: QAinvxres_new
      REAL(dp), INTENT(OUT)                              :: pres_error, nabs_error

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_compute_error_fft'

      INTEGER                                            :: handle
      REAL(dp)                                           :: vol

      CALL timeset(routineN, handle)

      vol = pw_pool%pw_grid%vol

! evaluate \Delta^-1(res) = \Delta^-1 (g - \Delta(v_new) - P(v_new) + Bt \lambda)
      CALL apply_inv_laplace_operator_fft(pw_pool, green, res_new, QAinvxres_new)
! (normalized) preconditioned residual norm error :
      pres_error = accurate_sum(QAinvxres_new%array(:, :, :)**2)
      CALL pw_pool%pw_grid%para%group%sum(pres_error)
      pres_error = SQRT(pres_error)/vol

! normalized absolute error :
! nabs_error := \frac{\| v_old - v_new \|}{volume}
      nabs_error = accurate_sum(ABS(v_old%array - v_new%array)**2)
      CALL pw_pool%pw_grid%para%group%sum(nabs_error)
      nabs_error = SQRT(nabs_error)/vol

      CALL timestop(handle)

   END SUBROUTINE ps_implicit_compute_error_fft

! **************************************************************************************************
!> \brief  Computes the (normalized) preconditioned residual norm error and the
!>         normalized absolute error
!> \param pw_pool pool of the original plane-wave grid
!> \param green the greens_fn_type data holding a valid dct_influence_fn
!> \param res_new residual
!> \param v_old old v
!> \param v_new new v
!> \param QAinvxres_new Delta^-1(res_new)
!> \param pres_error preconditioned residual norm error
!> \param nabs_error normalized absolute error
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!>       11.2015 revised [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_compute_error_dct(pw_pool, green, res_new, v_old, v_new, &
                                            QAinvxres_new, pres_error, nabs_error)

      TYPE(pw_pool_type), INTENT(IN), POINTER            :: pw_pool
      TYPE(greens_fn_type), INTENT(IN)                   :: green
      TYPE(pw_r3d_rs_type), INTENT(IN)                   :: res_new, v_old, v_new
      TYPE(pw_r3d_rs_type), INTENT(INOUT)                :: QAinvxres_new
      REAL(dp), INTENT(OUT)                              :: pres_error, nabs_error

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_compute_error_dct'

      INTEGER                                            :: handle
      REAL(dp)                                           :: vol

      CALL timeset(routineN, handle)

      vol = pw_pool%pw_grid%vol

! evaluate \Delta^-1(res) = \Delta^-1 (g - \Delta(v_new) - P(v_new) + Bt \lambda)
      CALL apply_inv_laplace_operator_dct(pw_pool, green, res_new, QAinvxres_new)
! (normalized) preconditioned residual norm error :
      pres_error = accurate_sum(QAinvxres_new%array(:, :, :)**2)
      CALL pw_pool%pw_grid%para%group%sum(pres_error)
      pres_error = SQRT(pres_error)/vol

! normalized absolute error :
! nabs_error := \frac{\| v_old - v_new \|}{volume}
      nabs_error = accurate_sum(ABS(v_old%array - v_new%array)**2)
      CALL pw_pool%pw_grid%para%group%sum(nabs_error)
      nabs_error = SQRT(nabs_error)/vol

      CALL timestop(handle)

   END SUBROUTINE ps_implicit_compute_error_dct

! **************************************************************************************************
!> \brief  output of the implicit (iterative) Poisson solver
!> \param iter current iteration
!> \param pres_error preconditioned residual norm error
!> \param nabs_error normalized absolute error
!> \param outp_unit output unit
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_output(iter, pres_error, nabs_error, outp_unit)

      INTEGER, INTENT(IN)                                :: iter
      REAL(dp), INTENT(IN)                               :: pres_error, nabs_error
      INTEGER, INTENT(OUT)                               :: outp_unit

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_output'
      INTEGER, PARAMETER                                 :: low_print_level = 1

      INTEGER                                            :: handle
      TYPE(cp_logger_type), POINTER                      :: logger

      CALL timeset(routineN, handle)

      logger => cp_get_default_logger()
      IF (logger%para_env%is_source()) THEN
         outp_unit = cp_logger_get_default_unit_nr(logger, local=.TRUE.)
      ELSE
         outp_unit = -1
      END IF

      IF (logger%iter_info%print_level .GT. low_print_level) THEN
         IF ((outp_unit .GT. 0) .AND. (iter .EQ. 1)) THEN
            WRITE (outp_unit, '(T3,A)') &
               "POISSON|   iter        pres error      nabs error        E_hartree    delta E"
         END IF

         IF (outp_unit .GT. 0) THEN
            WRITE (outp_unit, '(T3,A,I6,5X,E13.4,3X,E13.4)', ADVANCE='NO') &
               "POISSON| ", iter, pres_error, nabs_error
         END IF
      END IF

      CALL timestop(handle)

   END SUBROUTINE ps_implicit_output

! **************************************************************************************************
!> \brief  reports the Hartree energy in every iteration
!> \param ps_implicit_env the implicit poisson solver environment
!> \param outp_unit output unit
!> \param ehartree Hartree energy
!> \par History
!>       07.2014 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_report_ehartree(ps_implicit_env, outp_unit, ehartree)

      TYPE(ps_implicit_type)                             :: ps_implicit_env
      INTEGER, INTENT(IN)                                :: outp_unit
      REAL(dp), INTENT(IN)                               :: ehartree

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_report_ehartree'
      INTEGER, PARAMETER                                 :: low_print_level = 1

      INTEGER                                            :: handle
      TYPE(cp_logger_type), POINTER                      :: logger

      logger => cp_get_default_logger()
      CALL timeset(routineN, handle)
      IF (logger%iter_info%print_level .GT. low_print_level) THEN
         IF (outp_unit .GT. 0) WRITE (outp_unit, '(F19.10,E10.2)') &
            ehartree, ehartree - ps_implicit_env%ehartree
      END IF
      CALL timestop(handle)

   END SUBROUTINE ps_implicit_report_ehartree

! **************************************************************************************************
!> \brief  reports the final number of iteration
!> \param iter the iteration number after exiting the main loop
!> \param max_iter maximum number of iterations
!> \param outp_unit output unit
!> \par History
!>       02.2016 created [Hossein Bani-Hashemian]
!> \author Mohammad Hossein Bani-Hashemian
! **************************************************************************************************
   SUBROUTINE ps_implicit_print_convergence_msg(iter, max_iter, outp_unit)

      INTEGER, INTENT(IN)                                :: iter, max_iter, outp_unit

      CHARACTER(LEN=*), PARAMETER :: routineN = 'ps_implicit_print_convergence_msg'

      CHARACTER(LEN=12)                                  :: msg
      INTEGER                                            :: handle, last_iter

      CALL timeset(routineN, handle)

      last_iter = iter - 1

      IF (outp_unit .GT. 0) THEN
         IF (last_iter .EQ. max_iter) THEN
            WRITE (outp_unit, '(T3,A)') &
               "POISSON| No convergence achieved within the maximum number of iterations."
         END IF
         IF (last_iter .LT. max_iter) THEN
            IF (last_iter .EQ. 1) THEN
               msg = " iteration."
            ELSE
               msg = " iterations."
            END IF
            WRITE (outp_unit, '(T3,A,I0,A)') &
               "POISSON| Poisson solver converged in ", last_iter, msg
         END IF
      END IF
      CALL timestop(handle)

   END SUBROUTINE ps_implicit_print_convergence_msg

! **************************************************************************************************
!> \brief  converts a 1D array to a 3D array (contiguous layout)
!> \param idx_1dto3d mapping of indices
!> \param arr1d input 1D array
!> \param arr3d input 3D array
! **************************************************************************************************
   SUBROUTINE convert_1dto3d(idx_1dto3d, arr1d, arr3d)

      INTEGER, DIMENSION(:), INTENT(IN)                  :: idx_1dto3d
      REAL(dp), DIMENSION(:), INTENT(IN)                 :: arr1d
      REAL(dp), ALLOCATABLE, DIMENSION(:, :, :), &
         INTENT(INOUT)                                   :: arr3d

      CHARACTER(LEN=*), PARAMETER                        :: routineN = 'convert_1dto3d'

      INTEGER                                            :: handle, i, j, k, l, lb1, lb2, lb3, &
                                                            npts1, npts2, npts3, ub1, ub2, ub3

      CALL timeset(routineN, handle)

      lb1 = LBOUND(arr3d, 1); ub1 = UBOUND(arr3d, 1)
      lb2 = LBOUND(arr3d, 2); ub2 = UBOUND(arr3d, 2)
      lb3 = LBOUND(arr3d, 3); ub3 = UBOUND(arr3d, 3)

      npts1 = ub1 - lb1 + 1
      npts2 = ub2 - lb2 + 1
      npts3 = ub3 - lb3 + 1

      DO l = 1, SIZE(idx_1dto3d)
         k = ((idx_1dto3d(l) - 1)/(npts1*npts2)) + lb3
         j = ((idx_1dto3d(l) - 1) - (k - lb3)*npts1*npts2)/npts1 + lb2
         i = idx_1dto3d(l) - ((j - lb2)*npts1 + (k - lb3)*npts1*npts2) + lb1 - 1
         arr3d(i, j, k) = arr1d(l)
      END DO

      CALL timestop(handle)

   END SUBROUTINE convert_1dto3d

! **************************************************************************************************
!> \brief Returns the voltage of a tile. In case an alternating field is used, the oltage is a function of time
!> \param time ...
!> \param v_D ...
!> \param osc_frac ...
!> \param frequency ...
!> \param phase ...
!> \param v_D_new ...
! **************************************************************************************************
   SUBROUTINE get_voltage(time, v_D, osc_frac, frequency, phase, v_D_new)

      REAL(dp), INTENT(IN)                               :: time
      REAL(dp), DIMENSION(:), INTENT(IN)                 :: v_D, osc_frac, frequency, phase
      REAL(dp), ALLOCATABLE, DIMENSION(:), INTENT(OUT)   :: v_D_new

      CHARACTER(LEN=*), PARAMETER                        :: routineN = 'get_voltage'

      INTEGER                                            :: handle, i

      CALL timeset(routineN, handle)

      ALLOCATE (v_D_new(SIZE(v_D)))

      DO i = 1, SIZE(v_D)
         v_D_new(i) = v_D(i)*(1 - osc_frac(i)) + &
                      v_D(i)*osc_frac(i)*COS(2*pi*time*frequency(i) + phase(i))
      END DO

      CALL timestop(handle)

   END SUBROUTINE get_voltage

END MODULE ps_implicit_methods
