qs_tddfpt2_properties.F

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!--------------------------------------------------------------------------------------------------!
!   CP2K: A general program to perform molecular dynamics simulations                              !
!   Copyright 2000-2024 CP2K developers group <https://cp2k.org>                                   !
!                                                                                                  !
!   SPDX-License-Identifier: GPL-2.0-or-later                                                      !
!--------------------------------------------------------------------------------------------------!
 
MODULE qs_tddfpt2_properties
   USE atomic_kind_types,               ONLY: atomic_kind_type
   USE bibliography,                    ONLY: martin2003,&
                                              cite_reference
   USE cell_types,                      ONLY: cell_type
   USE cp_blacs_env,                    ONLY: cp_blacs_env_type
   USE cp_cfm_basic_linalg,             ONLY: cp_cfm_solve
   USE cp_cfm_types,                    ONLY: cp_cfm_create,&
                                              cp_cfm_release,&
                                              cp_cfm_set_all,&
                                              cp_cfm_to_fm,&
                                              cp_cfm_type,&
                                              cp_fm_to_cfm
   USE cp_control_types,                ONLY: dft_control_type,&
                                              tddfpt2_control_type
   USE cp_dbcsr_operations,             ONLY: copy_dbcsr_to_fm,&
                                              copy_fm_to_dbcsr,&
                                              cp_dbcsr_sm_fm_multiply,&
                                              dbcsr_allocate_matrix_set,&
                                              dbcsr_deallocate_matrix_set
   USE cp_fm_basic_linalg,              ONLY: cp_fm_scale,&
                                              cp_fm_scale_and_add,&
                                              cp_fm_trace
   USE cp_fm_diag,                      ONLY: choose_eigv_solver,&
                                              cp_fm_geeig
   USE cp_fm_struct,                    ONLY: cp_fm_struct_create,&
                                              cp_fm_struct_release,&
                                              cp_fm_struct_type
   USE cp_fm_types,                     ONLY: cp_fm_create,&
                                              cp_fm_get_info,&
                                              cp_fm_release,&
                                              cp_fm_set_all,&
                                              cp_fm_to_fm,&
                                              cp_fm_type,&
                                              cp_fm_vectorsnorm
   USE cp_log_handling,                 ONLY: cp_get_default_logger,&
                                              cp_logger_get_default_io_unit,&
                                              cp_logger_type
   USE cp_output_handling,              ONLY: cp_p_file,&
                                              cp_print_key_finished_output,&
                                              cp_print_key_should_output,&
                                              cp_print_key_unit_nr
   USE cp_realspace_grid_cube,          ONLY: cp_pw_to_cube
   USE dbcsr_api,                       ONLY: &
        dbcsr_copy, dbcsr_get_block_p, dbcsr_get_info, dbcsr_init_p, dbcsr_iterator_blocks_left, &
        dbcsr_iterator_next_block, dbcsr_iterator_start, dbcsr_iterator_stop, dbcsr_iterator_type, &
        dbcsr_p_type, dbcsr_set, dbcsr_type
   USE input_constants,                 ONLY: tddfpt_dipole_berry,&
                                              tddfpt_dipole_length,&
                                              tddfpt_dipole_velocity
   USE input_section_types,             ONLY: section_vals_get_subs_vals,&
                                              section_vals_type,&
                                              section_vals_val_get
   USE kahan_sum,                       ONLY: accurate_dot_product
   USE kinds,                           ONLY: default_path_length,&
                                              dp,&
                                              int_8
   USE mathconstants,                   ONLY: twopi,&
                                              z_one,&
                                              z_zero
   USE message_passing,                 ONLY: mp_comm_type,&
                                              mp_para_env_type,&
                                              mp_request_type
   USE molden_utils,                    ONLY: write_mos_molden
   USE moments_utils,                   ONLY: get_reference_point
   USE parallel_gemm_api,               ONLY: parallel_gemm
   USE particle_list_types,             ONLY: particle_list_type
   USE particle_types,                  ONLY: particle_type
   USE physcon,                         ONLY: evolt
   USE pw_env_types,                    ONLY: pw_env_get,&
                                              pw_env_type
   USE pw_poisson_types,                ONLY: pw_poisson_type
   USE pw_pool_types,                   ONLY: pw_pool_p_type,&
                                              pw_pool_type
   USE pw_types,                        ONLY: pw_c1d_gs_type,&
                                              pw_r3d_rs_type
   USE qs_collocate_density,            ONLY: calculate_wavefunction
   USE qs_environment_types,            ONLY: get_qs_env,&
                                              qs_environment_type
   USE qs_kind_types,                   ONLY: qs_kind_type
   USE qs_ks_types,                     ONLY: qs_ks_env_type
   USE qs_mo_types,                     ONLY: allocate_mo_set,&
                                              deallocate_mo_set,&
                                              get_mo_set,&
                                              init_mo_set,&
                                              mo_set_type,&
                                              set_mo_set
   USE qs_moments,                      ONLY: build_berry_moment_matrix
   USE qs_neighbor_list_types,          ONLY: neighbor_list_set_p_type
   USE qs_operators_ao,                 ONLY: rrc_xyz_ao
   USE qs_overlap,                      ONLY: build_overlap_matrix
   USE qs_subsys_types,                 ONLY: qs_subsys_get,&
                                              qs_subsys_type
   USE qs_tddfpt2_types,                ONLY: tddfpt_ground_state_mos
   USE string_utilities,                ONLY: integer_to_string
   USE util,                            ONLY: sort
#include "./base/base_uses.f90"
 
   IMPLICIT NONE
 
   PRIVATE
 
   CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'qs_tddfpt2_properties'
 
   ! number of first derivative components (3: d/dx, d/dy, d/dz)
   INTEGER, PARAMETER, PRIVATE          :: nderivs = 3
   INTEGER, PARAMETER, PRIVATE          :: maxspins = 2
 
   PUBLIC :: tddfpt_dipole_operator, tddfpt_print_summary, tddfpt_print_excitation_analysis, &
             tddfpt_print_nto_analysis
 
! **************************************************************************************************
 
CONTAINS
 
! **************************************************************************************************
!> \brief Compute the action of the dipole operator on the ground state wave function.
!> \param dipole_op_mos_occ  2-D array [x,y,z ; spin] of matrices where to put the computed quantity
!>                           (allocated and initialised on exit)
!> \param tddfpt_control     TDDFPT control parameters
!> \param gs_mos             molecular orbitals optimised for the ground state
!> \param qs_env             Quickstep environment
!> \par History
!>    * 05.2016 created as 'tddfpt_print_summary' [Sergey Chulkov]
!>    * 06.2018 dipole operator based on the Berry-phase formula [Sergey Chulkov]
!>    * 08.2018 splited of from 'tddfpt_print_summary' and merged with code from 'tddfpt'
!>              [Sergey Chulkov]
!> \note \parblock
!>       Adapted version of the subroutine find_contributions() which was originally created
!>       by Thomas Chassaing on 02.2005.
!>
!>       The relation between dipole integrals in velocity and length forms are the following:
!>       \f[<\psi_i|\nabla|\psi_a> = <\psi_i|\vec{r}|\hat{H}\psi_a> - <\hat{H}\psi_i|\vec{r}|\psi_a>
!>                                 = (\epsilon_a - \epsilon_i) <\psi_i|\vec{r}|\psi_a> .\f],
!>       due to the commutation identity:
!>       \f[\vec{r}\hat{H} - \hat{H}\vec{r} = [\vec{r},\hat{H}] = [\vec{r},-1/2 \nabla^2] = \nabla\f] .
!>       \endparblock
! **************************************************************************************************
   SUBROUTINE tddfpt_dipole_operator(dipole_op_mos_occ, tddfpt_control, gs_mos, qs_env)
      TYPE(cp_fm_type), ALLOCATABLE, DIMENSION(:, :), &
         INTENT(inout)                                   :: dipole_op_mos_occ
      TYPE(tddfpt2_control_type), POINTER                :: tddfpt_control
      TYPE(tddfpt_ground_state_mos), DIMENSION(:), &
         INTENT(in)                                      :: gs_mos
      TYPE(qs_environment_type), POINTER                 :: qs_env
 
      CHARACTER(LEN=*), PARAMETER :: routinen = 'tddfpt_dipole_operator'
 
      INTEGER                                            :: handle, i_cos_sin, icol, ideriv, irow, &
                                                            ispin, jderiv, nao, ncols_local, &
                                                            ndim_periodic, nrows_local, nspins
      INTEGER, DIMENSION(:), POINTER                     :: col_indices, row_indices
      INTEGER, DIMENSION(maxspins)                       :: nmo_occ, nmo_virt
      REAL(kind=dp)                                      :: eval_occ
      REAL(kind=dp), CONTIGUOUS, DIMENSION(:, :), &
         POINTER                                         :: local_data_ediff, local_data_wfm
      REAL(kind=dp), DIMENSION(3)                        :: kvec, reference_point
      TYPE(cell_type), POINTER                           :: cell
      TYPE(cp_blacs_env_type), POINTER                   :: blacs_env
      TYPE(cp_cfm_type), ALLOCATABLE, DIMENSION(:)       :: gamma_00, gamma_inv_00
      TYPE(cp_fm_struct_type), POINTER                   :: fm_struct
      TYPE(cp_fm_type)                                   :: ediff_inv, rrc_mos_occ, wfm_ao_ao, &
                                                            wfm_mo_virt_mo_occ
      TYPE(cp_fm_type), ALLOCATABLE, DIMENSION(:)        :: s_mos_virt
      TYPE(cp_fm_type), ALLOCATABLE, DIMENSION(:, :)     :: dberry_mos_occ, gamma_real_imag, opvec
      TYPE(dbcsr_p_type), DIMENSION(:), POINTER          :: berry_cossin_xyz, matrix_s, rrc_xyz, scrm
      TYPE(dft_control_type), POINTER                    :: dft_control
      TYPE(neighbor_list_set_p_type), DIMENSION(:), &
         POINTER                                         :: sab_orb
      TYPE(pw_env_type), POINTER                         :: pw_env
      TYPE(pw_poisson_type), POINTER                     :: poisson_env
      TYPE(qs_ks_env_type), POINTER                      :: ks_env
 
      CALL timeset(routinen, handle)
 
      NULLIFY (blacs_env, cell, matrix_s, pw_env)
      CALL get_qs_env(qs_env, blacs_env=blacs_env, cell=cell, matrix_s=matrix_s, pw_env=pw_env)
 
      nspins = SIZE(gs_mos)
      CALL dbcsr_get_info(matrix_s(1)%matrix, nfullrows_total=nao)
      DO ispin = 1, nspins
         nmo_occ(ispin) = SIZE(gs_mos(ispin)%evals_occ)
         nmo_virt(ispin) = SIZE(gs_mos(ispin)%evals_virt)
      END DO
 
      ! +++ allocate dipole operator matrices (must be deallocated elsewhere)
      ALLOCATE (dipole_op_mos_occ(nderivs, nspins))
      DO ispin = 1, nspins
         CALL cp_fm_get_info(gs_mos(ispin)%mos_occ, matrix_struct=fm_struct)
 
         DO ideriv = 1, nderivs
            CALL cp_fm_create(dipole_op_mos_occ(ideriv, ispin), fm_struct)
         END DO
      END DO
 
      ! +++ allocate work matrices
      ALLOCATE (s_mos_virt(nspins))
      DO ispin = 1, nspins
         CALL cp_fm_get_info(gs_mos(ispin)%mos_virt, matrix_struct=fm_struct)
         CALL cp_fm_create(s_mos_virt(ispin), fm_struct)
         CALL cp_dbcsr_sm_fm_multiply(matrix_s(1)%matrix, &
                                      gs_mos(ispin)%mos_virt, &
                                      s_mos_virt(ispin), &
                                      ncol=nmo_virt(ispin), alpha=1.0_dp, beta=0.0_dp)
      END DO
 
      ! check that the chosen dipole operator is consistent with the periodic boundary conditions used
      CALL pw_env_get(pw_env, poisson_env=poisson_env)
      ndim_periodic = count(poisson_env%parameters%periodic == 1)
 
      ! select default for dipole form
      IF (tddfpt_control%dipole_form == 0) THEN
         CALL get_qs_env(qs_env, dft_control=dft_control)
         IF (dft_control%qs_control%xtb) THEN
            IF (ndim_periodic == 0) THEN
               tddfpt_control%dipole_form = tddfpt_dipole_length
            ELSE
               tddfpt_control%dipole_form = tddfpt_dipole_berry
            END IF
         ELSE
            tddfpt_control%dipole_form = tddfpt_dipole_velocity
         END IF
      END IF
 
      SELECT CASE (tddfpt_control%dipole_form)
      CASE (tddfpt_dipole_berry)
         IF (ndim_periodic /= 3) THEN
            CALL cp_warn(__location__, &
                         "Fully periodic Poisson solver (PERIODIC xyz) is needed "// &
                         "for oscillator strengths based on the Berry phase formula")
         END IF
 
         NULLIFY (berry_cossin_xyz)
         ! index: 1 = Re[exp(-i * G_t * t)],
         !        2 = Im[exp(-i * G_t * t)];
         ! t = x,y,z
         CALL dbcsr_allocate_matrix_set(berry_cossin_xyz, 2)
 
         DO i_cos_sin = 1, 2
            CALL dbcsr_init_p(berry_cossin_xyz(i_cos_sin)%matrix)
            CALL dbcsr_copy(berry_cossin_xyz(i_cos_sin)%matrix, matrix_s(1)%matrix)
         END DO
 
         ! +++ allocate berry-phase-related work matrices
         ALLOCATE (gamma_00(nspins), gamma_inv_00(nspins), gamma_real_imag(2, nspins), opvec(2, nspins))
         ALLOCATE (dberry_mos_occ(nderivs, nspins))
         DO ispin = 1, nspins
            NULLIFY (fm_struct)
            CALL cp_fm_struct_create(fm_struct, nrow_global=nmo_occ(ispin), &
                                     ncol_global=nmo_occ(ispin), context=blacs_env)
 
            CALL cp_cfm_create(gamma_00(ispin), fm_struct)
            CALL cp_cfm_create(gamma_inv_00(ispin), fm_struct)
 
            DO i_cos_sin = 1, 2
               CALL cp_fm_create(gamma_real_imag(i_cos_sin, ispin), fm_struct)
            END DO
            CALL cp_fm_struct_release(fm_struct)
 
            ! G_real C_0, G_imag C_0
            CALL cp_fm_get_info(gs_mos(ispin)%mos_occ, matrix_struct=fm_struct)
            DO i_cos_sin = 1, 2
               CALL cp_fm_create(opvec(i_cos_sin, ispin), fm_struct)
            END DO
 
            ! dBerry * C_0
            DO ideriv = 1, nderivs
               CALL cp_fm_create(dberry_mos_occ(ideriv, ispin), fm_struct)
               CALL cp_fm_set_all(dberry_mos_occ(ideriv, ispin), 0.0_dp)
            END DO
         END DO
 
         DO ideriv = 1, nderivs
            kvec(:) = twopi*cell%h_inv(ideriv, :)
            DO i_cos_sin = 1, 2
               CALL dbcsr_set(berry_cossin_xyz(i_cos_sin)%matrix, 0.0_dp)
            END DO
            CALL build_berry_moment_matrix(qs_env, berry_cossin_xyz(1)%matrix, &
                                           berry_cossin_xyz(2)%matrix, kvec)
 
            DO ispin = 1, nspins
               ! i_cos_sin = 1: cos (real) component; opvec(1) = gamma_real C_0
               ! i_cos_sin = 2: sin (imaginary) component; opvec(2) = gamma_imag C_0
               DO i_cos_sin = 1, 2
                  CALL cp_dbcsr_sm_fm_multiply(berry_cossin_xyz(i_cos_sin)%matrix, &
                                               gs_mos(ispin)%mos_occ, &
                                               opvec(i_cos_sin, ispin), &
                                               ncol=nmo_occ(ispin), alpha=1.0_dp, beta=0.0_dp)
               END DO
 
               CALL parallel_gemm('T', 'N', nmo_occ(ispin), nmo_occ(ispin), nao, &
                                  1.0_dp, gs_mos(ispin)%mos_occ, opvec(1, ispin), &
                                  0.0_dp, gamma_real_imag(1, ispin))
 
               CALL parallel_gemm('T', 'N', nmo_occ(ispin), nmo_occ(ispin), nao, &
                                  -1.0_dp, gs_mos(ispin)%mos_occ, opvec(2, ispin), &
                                  0.0_dp, gamma_real_imag(2, ispin))
 
               CALL cp_fm_to_cfm(msourcer=gamma_real_imag(1, ispin), &
                                 msourcei=gamma_real_imag(2, ispin), &
                                 mtarget=gamma_00(ispin))
 
               ! gamma_inv_00 = Q = [C_0^T (gamma_real - i gamma_imag) C_0] ^ {-1}
               CALL cp_cfm_set_all(gamma_inv_00(ispin), z_zero, z_one)
               CALL cp_cfm_solve(gamma_00(ispin), gamma_inv_00(ispin))
 
               CALL cp_cfm_to_fm(msource=gamma_inv_00(ispin), &
                                 mtargetr=gamma_real_imag(1, ispin), &
                                 mtargeti=gamma_real_imag(2, ispin))
 
               ! dBerry_mos_occ is identical to dBerry_psi0 from qs_linres_op % polar_operators()
               CALL parallel_gemm("N", "N", nao, nmo_occ(ispin), nmo_occ(ispin), &
                                  1.0_dp, opvec(1, ispin), gamma_real_imag(2, ispin), &
                                  0.0_dp, dipole_op_mos_occ(1, ispin))
               CALL parallel_gemm("N", "N", nao, nmo_occ(ispin), nmo_occ(ispin), &
                                  -1.0_dp, opvec(2, ispin), gamma_real_imag(1, ispin), &
                                  1.0_dp, dipole_op_mos_occ(1, ispin))
 
               DO jderiv = 1, nderivs
                  CALL cp_fm_scale_and_add(1.0_dp, dberry_mos_occ(jderiv, ispin), &
                                           cell%hmat(jderiv, ideriv), dipole_op_mos_occ(1, ispin))
               END DO
            END DO
         END DO
 
         ! --- release berry-phase-related work matrices
         CALL cp_fm_release(opvec)
         CALL cp_fm_release(gamma_real_imag)
         DO ispin = nspins, 1, -1
            CALL cp_cfm_release(gamma_inv_00(ispin))
            CALL cp_cfm_release(gamma_00(ispin))
         END DO
         DEALLOCATE (gamma_00, gamma_inv_00)
         CALL dbcsr_deallocate_matrix_set(berry_cossin_xyz)
 
         NULLIFY (fm_struct)
         CALL cp_fm_struct_create(fm_struct, nrow_global=nao, ncol_global=nao, context=blacs_env)
         CALL cp_fm_create(wfm_ao_ao, fm_struct)
         CALL cp_fm_struct_release(fm_struct)
 
         ! trans_dipole = 2|e|/|G_mu| * Tr Imag(evects^T * (gamma_real - i gamma_imag) * C_0 * gamma_inv_00) +
         !                2|e|/|G_mu| * Tr Imag(C_0^T * (gamma_real - i gamma_imag) * evects * gamma_inv_00) ,
         !
         ! Taking into account the symmetry of the matrices 'gamma_real' and 'gamma_imag' and the fact
         ! that the response wave-function is a real-valued function, the above expression can be simplified as
         ! trans_dipole = 4|e|/|G_mu| * Tr Imag(evects^T * (gamma_real - i gamma_imag) * C_0 * gamma_inv_00)
         !
         ! 1/|G_mu| = |lattice_vector_mu| / (2*pi) .
         DO ispin = 1, nspins
            ! wfm_ao_ao = S * mos_virt * mos_virt^T
            CALL parallel_gemm('N', 'T', nao, nao, nmo_virt(ispin), &
                               1.0_dp/twopi, s_mos_virt(ispin), gs_mos(ispin)%mos_virt, &
                               0.0_dp, wfm_ao_ao)
 
            DO ideriv = 1, nderivs
               CALL parallel_gemm('N', 'N', nao, nmo_occ(ispin), nao, &
                                  1.0_dp, wfm_ao_ao, dberry_mos_occ(ideriv, ispin), &
                                  0.0_dp, dipole_op_mos_occ(ideriv, ispin))
            END DO
         END DO
 
         CALL cp_fm_release(wfm_ao_ao)
         CALL cp_fm_release(dberry_mos_occ)
 
      CASE (tddfpt_dipole_length)
         IF (ndim_periodic /= 0) THEN
            CALL cp_warn(__location__, &
                         "Non-periodic Poisson solver (PERIODIC none) is needed "// &
                         "for oscillator strengths based on the length operator")
         END IF
 
         ! compute components of the dipole operator in the length form
         NULLIFY (rrc_xyz)
         CALL dbcsr_allocate_matrix_set(rrc_xyz, nderivs)
 
         DO ideriv = 1, nderivs
            CALL dbcsr_init_p(rrc_xyz(ideriv)%matrix)
            CALL dbcsr_copy(rrc_xyz(ideriv)%matrix, matrix_s(1)%matrix)
         END DO
 
         CALL get_reference_point(reference_point, qs_env=qs_env, &
                                  reference=tddfpt_control%dipole_reference, &
                                  ref_point=tddfpt_control%dipole_ref_point)
 
         CALL rrc_xyz_ao(op=rrc_xyz, qs_env=qs_env, rc=reference_point, order=1, &
                         minimum_image=.false., soft=.false.)
 
         NULLIFY (fm_struct)
         CALL cp_fm_struct_create(fm_struct, nrow_global=nao, ncol_global=nao, context=blacs_env)
         CALL cp_fm_create(wfm_ao_ao, fm_struct)
         CALL cp_fm_struct_release(fm_struct)
 
         DO ispin = 1, nspins
            CALL cp_fm_get_info(gs_mos(ispin)%mos_occ, matrix_struct=fm_struct)
            CALL cp_fm_create(rrc_mos_occ, fm_struct)
 
            ! wfm_ao_ao = S * mos_virt * mos_virt^T
            CALL parallel_gemm('N', 'T', nao, nao, nmo_virt(ispin), &
                               1.0_dp, s_mos_virt(ispin), gs_mos(ispin)%mos_virt, &
                               0.0_dp, wfm_ao_ao)
 
            DO ideriv = 1, nderivs
               CALL cp_dbcsr_sm_fm_multiply(rrc_xyz(ideriv)%matrix, &
                                            gs_mos(ispin)%mos_occ, &
                                            rrc_mos_occ, &
                                            ncol=nmo_occ(ispin), alpha=1.0_dp, beta=0.0_dp)
 
               CALL parallel_gemm('N', 'N', nao, nmo_occ(ispin), nao, &
                                  1.0_dp, wfm_ao_ao, rrc_mos_occ, &
                                  0.0_dp, dipole_op_mos_occ(ideriv, ispin))
            END DO
 
            CALL cp_fm_release(rrc_mos_occ)
         END DO
 
         CALL cp_fm_release(wfm_ao_ao)
         CALL dbcsr_deallocate_matrix_set(rrc_xyz)
 
      CASE (tddfpt_dipole_velocity)
         ! generate overlap derivatives
         CALL get_qs_env(qs_env, ks_env=ks_env, sab_orb=sab_orb)
         NULLIFY (scrm)
         CALL build_overlap_matrix(ks_env, matrix_s=scrm, nderivative=1, &
                                   basis_type_a="ORB", basis_type_b="ORB", &
                                   sab_nl=sab_orb)
 
         DO ispin = 1, nspins
            NULLIFY (fm_struct)
            CALL cp_fm_struct_create(fm_struct, nrow_global=nmo_virt(ispin), &
                                     ncol_global=nmo_occ(ispin), context=blacs_env)
            CALL cp_fm_create(ediff_inv, fm_struct)
            CALL cp_fm_create(wfm_mo_virt_mo_occ, fm_struct)
            CALL cp_fm_struct_release(fm_struct)
 
            CALL cp_fm_get_info(ediff_inv, nrow_local=nrows_local, ncol_local=ncols_local, &
                                row_indices=row_indices, col_indices=col_indices, local_data=local_data_ediff)
            CALL cp_fm_get_info(wfm_mo_virt_mo_occ, local_data=local_data_wfm)
 
!$OMP       PARALLEL DO DEFAULT(NONE), &
!$OMP                PRIVATE(eval_occ, icol, irow), &
!$OMP                SHARED(col_indices, gs_mos, ispin, local_data_ediff, ncols_local, nrows_local, row_indices)
            DO icol = 1, ncols_local
               ! E_occ_i ; imo_occ = col_indices(icol)
               eval_occ = gs_mos(ispin)%evals_occ(col_indices(icol))
 
               DO irow = 1, nrows_local
                  ! ediff_inv_weights(a, i) = 1.0 / (E_virt_a - E_occ_i)
                  ! imo_virt = row_indices(irow)
                  local_data_ediff(irow, icol) = 1.0_dp/(gs_mos(ispin)%evals_virt(row_indices(irow)) - eval_occ)
               END DO
            END DO
!$OMP       END PARALLEL DO
 
            DO ideriv = 1, nderivs
               CALL cp_dbcsr_sm_fm_multiply(scrm(ideriv + 1)%matrix, &
                                            gs_mos(ispin)%mos_occ, &
                                            dipole_op_mos_occ(ideriv, ispin), &
                                            ncol=nmo_occ(ispin), alpha=1.0_dp, beta=0.0_dp)
 
               CALL parallel_gemm('T', 'N', nmo_virt(ispin), nmo_occ(ispin), nao, &
                                  1.0_dp, gs_mos(ispin)%mos_virt, dipole_op_mos_occ(ideriv, ispin), &
                                  0.0_dp, wfm_mo_virt_mo_occ)
 
               ! in-place element-wise (Schur) product;
               ! avoid allocation of a temporary [nmo_virt x nmo_occ] matrix which is needed
               ! for cp_fm_schur_product() subroutine call
 
!$OMP          PARALLEL DO DEFAULT(NONE), &
!$OMP                   PRIVATE(icol, irow), &
!$OMP                   SHARED(ispin, local_data_ediff, local_data_wfm, ncols_local, nrows_local)
               DO icol = 1, ncols_local
                  DO irow = 1, nrows_local
                     local_data_wfm(irow, icol) = local_data_wfm(irow, icol)*local_data_ediff(irow, icol)
                  END DO
               END DO
!$OMP          END PARALLEL DO
 
               CALL parallel_gemm('N', 'N', nao, nmo_occ(ispin), nmo_virt(ispin), &
                                  1.0_dp, s_mos_virt(ispin), wfm_mo_virt_mo_occ, &
                                  0.0_dp, dipole_op_mos_occ(ideriv, ispin))
            END DO
 
            CALL cp_fm_release(wfm_mo_virt_mo_occ)
            CALL cp_fm_release(ediff_inv)
         END DO
         CALL dbcsr_deallocate_matrix_set(scrm)
 
      CASE DEFAULT
         cpabort("Unimplemented form of the dipole operator")
      END SELECT
 
      ! --- release work matrices
      CALL cp_fm_release(s_mos_virt)
 
      CALL timestop(handle)
   END SUBROUTINE tddfpt_dipole_operator
 
! **************************************************************************************************
!> \brief Print final TDDFPT excitation energies and oscillator strengths.
!> \param log_unit           output unit
!> \param evects             TDDFPT trial vectors (SIZE(evects,1) -- number of spins;
!>                           SIZE(evects,2) -- number of excited states to print)
!> \param evals              TDDFPT eigenvalues
!> \param ostrength          TDDFPT oscillator strength
!> \param mult               multiplicity
!> \param dipole_op_mos_occ  action of the dipole operator on the ground state wave function
!>                           [x,y,z ; spin]
!> \param dipole_form ...
!> \par History
!>    * 05.2016 created [Sergey Chulkov]
!>    * 06.2016 transition dipole moments and oscillator strengths [Sergey Chulkov]
!>    * 07.2016 spin-unpolarised electron density [Sergey Chulkov]
!>    * 08.2018 compute 'dipole_op_mos_occ' in a separate subroutine [Sergey Chulkov]
!> \note \parblock
!>       Adapted version of the subroutine find_contributions() which was originally created
!>       by Thomas Chassaing on 02.2005.
!>
!>       Transition dipole moment along direction 'd' is computed as following:
!>       \f[ t_d(spin) = Tr[evects^T dipole\_op\_mos\_occ(d, spin)] .\f]
!>       \endparblock
! **************************************************************************************************
   SUBROUTINE tddfpt_print_summary(log_unit, evects, evals, ostrength, mult, &
                                   dipole_op_mos_occ, dipole_form)
      INTEGER, INTENT(in)                                :: log_unit
      TYPE(cp_fm_type), DIMENSION(:, :), INTENT(in)      :: evects
      REAL(kind=dp), DIMENSION(:), INTENT(in)            :: evals
      REAL(kind=dp), DIMENSION(:), INTENT(inout)         :: ostrength
      INTEGER, INTENT(in)                                :: mult
      TYPE(cp_fm_type), DIMENSION(:, :), INTENT(in)      :: dipole_op_mos_occ
      INTEGER, INTENT(in)                                :: dipole_form
 
      CHARACTER(LEN=*), PARAMETER :: routinen = 'tddfpt_print_summary'
 
      CHARACTER(len=1)                                   :: lsd_str
      CHARACTER(len=20)                                  :: mult_str
      INTEGER                                            :: handle, ideriv, ispin, istate, nspins, &
                                                            nstates
      REAL(kind=dp)                                      :: osc_strength
      REAL(kind=dp), ALLOCATABLE, DIMENSION(:, :, :)     :: trans_dipoles
 
      CALL timeset(routinen, handle)
 
      nspins = SIZE(evects, 1)
      nstates = SIZE(evects, 2)
 
      IF (nspins > 1) THEN
         lsd_str = 'U'
      ELSE
         lsd_str = 'R'
      END IF
 
      ! *** summary header ***
      IF (log_unit > 0) THEN
         CALL integer_to_string(mult, mult_str)
         WRITE (log_unit, '(/,1X,A1,A,1X,A)') lsd_str, "-TDDFPT states of multiplicity", trim(mult_str)
         SELECT CASE (dipole_form)
         CASE (tddfpt_dipole_berry)
            WRITE (log_unit, '(1X,A,/)') "Transition dipoles calculated using Berry operator formulation"
         CASE (tddfpt_dipole_length)
            WRITE (log_unit, '(1X,A,/)') "Transition dipoles calculated using length formulation"
         CASE (tddfpt_dipole_velocity)
            WRITE (log_unit, '(1X,A,/)') "Transition dipoles calculated using velocity formulation"
         CASE DEFAULT
            cpabort("Unimplemented form of the dipole operator")
         END SELECT
 
         WRITE (log_unit, '(T10,A,T19,A,T37,A,T69,A)') "State", "Excitation", &
            "Transition dipole (a.u.)", "Oscillator"
         WRITE (log_unit, '(T10,A,T19,A,T37,A,T49,A,T61,A,T67,A)') "number", "energy (eV)", &
            "x", "y", "z", "strength (a.u.)"
         WRITE (log_unit, '(T10,72("-"))')
      END IF
 
      ! transition dipole moment
      ALLOCATE (trans_dipoles(nstates, nderivs, nspins))
      trans_dipoles(:, :, :) = 0.0_dp
 
      ! nspins == 1 .AND. mult == 3 : spin-flip transitions are forbidden due to symmetry reasons
      IF (nspins > 1 .OR. mult == 1) THEN
         DO ispin = 1, nspins
            DO ideriv = 1, nderivs
               CALL cp_fm_trace(evects(ispin, :), dipole_op_mos_occ(ideriv, ispin), &
                                trans_dipoles(:, ideriv, ispin))
            END DO
         END DO
 
         IF (nspins == 1) THEN
            trans_dipoles(:, :, 1) = sqrt(2.0_dp)*trans_dipoles(:, :, 1)
         ELSE
            trans_dipoles(:, :, 1) = sqrt(trans_dipoles(:, :, 1)**2 + trans_dipoles(:, :, 2)**2)
         END IF
      END IF
 
      ! *** summary information ***
      DO istate = 1, nstates
         osc_strength = 2.0_dp/3.0_dp*evals(istate)* &
                        accurate_dot_product(trans_dipoles(istate, :, 1), trans_dipoles(istate, :, 1))
         ostrength(istate) = osc_strength
         IF (log_unit > 0) THEN
            WRITE (log_unit, '(1X,A,T9,I7,T19,F11.5,T31,3(1X,ES11.4E2),T69,ES12.5E2)') &
               "TDDFPT|", istate, evals(istate)*evolt, trans_dipoles(istate, 1:nderivs, 1), osc_strength
         END IF
      END DO
 
      ! punch a checksum for the regs
      IF (log_unit > 0) THEN
         WRITE (log_unit, '(/,T2,A,E14.6)') 'TDDFPT : CheckSum  =', sqrt(sum(evals**2))
      END IF
 
      DEALLOCATE (trans_dipoles)
 
      CALL timestop(handle)
   END SUBROUTINE tddfpt_print_summary
 
! **************************************************************************************************
!> \brief Print excitation analysis.
!> \param log_unit           output unit
!> \param evects             TDDFPT trial vectors (SIZE(evects,1) -- number of spins;
!>                           SIZE(evects,2) -- number of excited states to print)
!> \param evals              TDDFPT eigenvalues
!> \param gs_mos             molecular orbitals optimised for the ground state
!> \param matrix_s           overlap matrix
!> \param min_amplitude      the smallest excitation amplitude to print
!> \par History
!>    * 05.2016 created as 'tddfpt_print_summary' [Sergey Chulkov]
!>    * 08.2018 splited of from 'tddfpt_print_summary' [Sergey Chulkov]
! **************************************************************************************************
   SUBROUTINE tddfpt_print_excitation_analysis(log_unit, evects, evals, gs_mos, matrix_s, min_amplitude)
      INTEGER, INTENT(in)                                :: log_unit
      TYPE(cp_fm_type), DIMENSION(:, :), INTENT(in)      :: evects
      REAL(kind=dp), DIMENSION(:), INTENT(in)            :: evals
      TYPE(tddfpt_ground_state_mos), DIMENSION(:), &
         INTENT(in)                                      :: gs_mos
      TYPE(dbcsr_type), POINTER                          :: matrix_s
      REAL(kind=dp), INTENT(in)                          :: min_amplitude
 
      CHARACTER(LEN=*), PARAMETER :: routinen = 'tddfpt_print_excitation_analysis'
 
      CHARACTER(len=5)                                   :: spin_label
      INTEGER                                            :: handle, icol, iproc, irow, ispin, &
                                                            istate, nao, ncols_local, nrows_local, &
                                                            nspins, nstates, state_spin
      INTEGER(kind=int_8)                                :: iexc, imo_occ, imo_virt, ind, nexcs, &
                                                            nexcs_local, nexcs_max_local, &
                                                            nmo_virt_occ, nmo_virt_occ_alpha
      INTEGER(kind=int_8), ALLOCATABLE, DIMENSION(:)     :: inds_local, inds_recv, nexcs_recv
      INTEGER(kind=int_8), DIMENSION(1)                  :: nexcs_send
      INTEGER(kind=int_8), DIMENSION(maxspins)           :: nmo_occ8, nmo_virt8
      INTEGER, ALLOCATABLE, DIMENSION(:)                 :: inds
      INTEGER, DIMENSION(:), POINTER                     :: col_indices, row_indices
      INTEGER, DIMENSION(maxspins)                       :: nmo_occ, nmo_virt
      LOGICAL                                            :: do_exc_analysis
      REAL(kind=dp), ALLOCATABLE, DIMENSION(:)           :: weights_local, weights_neg_abs_recv, &
                                                            weights_recv
      REAL(kind=dp), CONTIGUOUS, DIMENSION(:, :), &
         POINTER                                         :: local_data
      TYPE(cp_blacs_env_type), POINTER                   :: blacs_env
      TYPE(cp_fm_struct_type), POINTER                   :: fm_struct
      TYPE(cp_fm_type), ALLOCATABLE, DIMENSION(:)        :: s_mos_virt, weights_fm
      TYPE(mp_para_env_type), POINTER                    :: para_env
      TYPE(mp_request_type)                              :: send_handler, send_handler2
      TYPE(mp_request_type), ALLOCATABLE, DIMENSION(:)   :: recv_handlers, recv_handlers2
 
      CALL timeset(routinen, handle)
 
      nspins = SIZE(evects, 1)
      nstates = SIZE(evects, 2)
      do_exc_analysis = min_amplitude < 1.0_dp
 
      CALL cp_fm_get_info(gs_mos(1)%mos_occ, context=blacs_env, para_env=para_env)
      CALL dbcsr_get_info(matrix_s, nfullrows_total=nao)
 
      DO ispin = 1, nspins
         nmo_occ(ispin) = SIZE(gs_mos(ispin)%evals_occ)
         nmo_virt(ispin) = SIZE(gs_mos(ispin)%evals_virt)
         nmo_occ8(ispin) = SIZE(gs_mos(ispin)%evals_occ, kind=int_8)
         nmo_virt8(ispin) = SIZE(gs_mos(ispin)%evals_virt, kind=int_8)
      END DO
 
      ! *** excitation analysis ***
      IF (do_exc_analysis) THEN
         cpassert(log_unit <= 0 .OR. para_env%is_source())
         nmo_virt_occ_alpha = int(nmo_virt(1), int_8)*int(nmo_occ(1), int_8)
 
         IF (log_unit > 0) THEN
            WRITE (log_unit, "(1X,A)") "", &
               "-------------------------------------------------------------------------------", &
               "-                            Excitation analysis                              -", &
               "-------------------------------------------------------------------------------"
            WRITE (log_unit, '(8X,A,T27,A,T49,A,T69,A)') "State", "Occupied", "Virtual", "Excitation"
            WRITE (log_unit, '(8X,A,T28,A,T49,A,T69,A)') "number", "orbital", "orbital", "amplitude"
            WRITE (log_unit, '(1X,79("-"))')
 
            IF (nspins == 1) THEN
               state_spin = 1
               spin_label = '     '
            END IF
         END IF
 
         ALLOCATE (s_mos_virt(nspins), weights_fm(nspins))
         DO ispin = 1, nspins
            CALL cp_fm_get_info(gs_mos(ispin)%mos_virt, matrix_struct=fm_struct)
            CALL cp_fm_create(s_mos_virt(ispin), fm_struct)
            CALL cp_dbcsr_sm_fm_multiply(matrix_s, &
                                         gs_mos(ispin)%mos_virt, &
                                         s_mos_virt(ispin), &
                                         ncol=nmo_virt(ispin), alpha=1.0_dp, beta=0.0_dp)
 
            NULLIFY (fm_struct)
            CALL cp_fm_struct_create(fm_struct, nrow_global=nmo_virt(ispin), ncol_global=nmo_occ(ispin), &
                                     context=blacs_env)
            CALL cp_fm_create(weights_fm(ispin), fm_struct)
            CALL cp_fm_struct_release(fm_struct)
         END DO
 
         nexcs_max_local = 0
         DO ispin = 1, nspins
            CALL cp_fm_get_info(weights_fm(ispin), nrow_local=nrows_local, ncol_local=ncols_local)
            nexcs_max_local = nexcs_max_local + int(nrows_local, int_8)*int(ncols_local, int_8)
         END DO
 
         ALLOCATE (weights_local(nexcs_max_local), inds_local(nexcs_max_local))
 
         DO istate = 1, nstates
            nexcs_local = 0
            nmo_virt_occ = 0
 
            ! analyse matrix elements locally and transfer only significant
            ! excitations to the master node for subsequent ordering
            DO ispin = 1, nspins
               ! compute excitation amplitudes
               CALL parallel_gemm('T', 'N', nmo_virt(ispin), nmo_occ(ispin), nao, 1.0_dp, s_mos_virt(ispin), &
                                  evects(ispin, istate), 0.0_dp, weights_fm(ispin))
 
               CALL cp_fm_get_info(weights_fm(ispin), nrow_local=nrows_local, ncol_local=ncols_local, &
                                   row_indices=row_indices, col_indices=col_indices, local_data=local_data)
 
               ! locate single excitations with significant amplitudes (>= min_amplitude)
               DO icol = 1, ncols_local
                  DO irow = 1, nrows_local
                     IF (abs(local_data(irow, icol)) >= min_amplitude) THEN
                        ! number of non-negligible excitations
                        nexcs_local = nexcs_local + 1
                        ! excitation amplitude
                        weights_local(nexcs_local) = local_data(irow, icol)
                        ! index of single excitation (ivirt, iocc, ispin) in compressed form
                        inds_local(nexcs_local) = nmo_virt_occ + int(row_indices(irow), int_8) + &
                                                  int(col_indices(icol) - 1, int_8)*nmo_virt8(ispin)
                     END IF
                  END DO
               END DO
 
               nmo_virt_occ = nmo_virt_occ + nmo_virt8(ispin)*nmo_occ8(ispin)
            END DO
 
            IF (para_env%is_source()) THEN
               ! master node
               ALLOCATE (nexcs_recv(para_env%num_pe), recv_handlers(para_env%num_pe), recv_handlers2(para_env%num_pe))
 
               ! collect number of non-negligible excitations from other nodes
               DO iproc = 1, para_env%num_pe
                  IF (iproc - 1 /= para_env%mepos) THEN
                     CALL para_env%irecv(nexcs_recv(iproc:iproc), iproc - 1, recv_handlers(iproc), 0)
                  ELSE
                     nexcs_recv(iproc) = nexcs_local
                  END IF
               END DO
 
               DO iproc = 1, para_env%num_pe
                  IF (iproc - 1 /= para_env%mepos) &
                     CALL recv_handlers(iproc)%wait()
               END DO
 
               ! compute total number of non-negligible excitations
               nexcs = 0
               DO iproc = 1, para_env%num_pe
                  nexcs = nexcs + nexcs_recv(iproc)
               END DO
 
               ! receive indices and amplitudes of selected excitations
               ALLOCATE (weights_recv(nexcs), weights_neg_abs_recv(nexcs))
               ALLOCATE (inds_recv(nexcs), inds(nexcs))
 
               nmo_virt_occ = 0
               DO iproc = 1, para_env%num_pe
                  IF (nexcs_recv(iproc) > 0) THEN
                     IF (iproc - 1 /= para_env%mepos) THEN
                        ! excitation amplitudes
                        CALL para_env%irecv(weights_recv(nmo_virt_occ + 1:nmo_virt_occ + nexcs_recv(iproc)), &
                                            iproc - 1, recv_handlers(iproc), 1)
                        ! compressed indices
                        CALL para_env%irecv(inds_recv(nmo_virt_occ + 1:nmo_virt_occ + nexcs_recv(iproc)), &
                                            iproc - 1, recv_handlers2(iproc), 2)
                     ELSE
                        ! data on master node
                        weights_recv(nmo_virt_occ + 1:nmo_virt_occ + nexcs_recv(iproc)) = weights_local(1:nexcs_recv(iproc))
                        inds_recv(nmo_virt_occ + 1:nmo_virt_occ + nexcs_recv(iproc)) = inds_local(1:nexcs_recv(iproc))
                     END IF
 
                     nmo_virt_occ = nmo_virt_occ + nexcs_recv(iproc)
                  END IF
               END DO
 
               DO iproc = 1, para_env%num_pe
                  IF (iproc - 1 /= para_env%mepos .AND. nexcs_recv(iproc) > 0) THEN
                     CALL recv_handlers(iproc)%wait()
                     CALL recv_handlers2(iproc)%wait()
                  END IF
               END DO
 
               DEALLOCATE (nexcs_recv, recv_handlers, recv_handlers2)
            ELSE
               ! working node: send the number of selected excited states to the master node
               nexcs_send(1) = nexcs_local
               CALL para_env%isend(nexcs_send, para_env%source, send_handler, 0)
               CALL send_handler%wait()
 
               IF (nexcs_local > 0) THEN
                  ! send excitation amplitudes
                  CALL para_env%isend(weights_local(1:nexcs_local), para_env%source, send_handler, 1)
                  ! send compressed indices
                  CALL para_env%isend(inds_local(1:nexcs_local), para_env%source, send_handler2, 2)
 
                  CALL send_handler%wait()
                  CALL send_handler2%wait()
               END IF
            END IF
 
            ! sort non-negligible excitations on the master node according to their amplitudes,
            ! uncompress indices and print summary information
            IF (para_env%is_source() .AND. log_unit > 0) THEN
               weights_neg_abs_recv(:) = -abs(weights_recv)
               CALL sort(weights_neg_abs_recv, int(nexcs), inds)
 
               WRITE (log_unit, '(T7,I8,F10.5,A)') istate, evals(istate)*evolt, " eV"
 
               DO iexc = 1, nexcs
                  ind = inds_recv(inds(iexc)) - 1
                  IF (nspins > 1) THEN
                     IF (ind < nmo_virt_occ_alpha) THEN
                        state_spin = 1
                        spin_label = '(alp)'
                     ELSE
                        state_spin = 2
                        ind = ind - nmo_virt_occ_alpha
                        spin_label = '(bet)'
                     END IF
                  END IF
 
                  imo_occ = ind/nmo_virt8(state_spin) + 1
                  imo_virt = mod(ind, nmo_virt8(state_spin)) + 1
 
                  WRITE (log_unit, '(T27,I8,1X,A5,T48,I8,1X,A5,T70,F9.6)') imo_occ, spin_label, &
                     nmo_occ8(state_spin) + imo_virt, spin_label, weights_recv(inds(iexc))
               END DO
            END IF
 
            ! deallocate temporary arrays
            IF (para_env%is_source()) &
               DEALLOCATE (weights_recv, weights_neg_abs_recv, inds_recv, inds)
         END DO
 
         DEALLOCATE (weights_local, inds_local)
         IF (log_unit > 0) THEN
            WRITE (log_unit, "(1X,A)") &
               "-------------------------------------------------------------------------------"
         END IF
      END IF
 
      CALL cp_fm_release(weights_fm)
      CALL cp_fm_release(s_mos_virt)
 
      CALL timestop(handle)
 
   END SUBROUTINE tddfpt_print_excitation_analysis
 
! **************************************************************************************************
!> \brief Print natural transition orbital analysis.
!> \param qs_env             Information on Kinds and Particles
!> \param evects             TDDFPT trial vectors (SIZE(evects,1) -- number of spins;
!>                           SIZE(evects,2) -- number of excited states to print)
!> \param evals              TDDFPT eigenvalues
!> \param ostrength ...
!> \param gs_mos             molecular orbitals optimised for the ground state
!> \param matrix_s           overlap matrix
!> \param print_section      ...
!> \par History
!>    * 06.2019 created [JGH]
! **************************************************************************************************
   SUBROUTINE tddfpt_print_nto_analysis(qs_env, evects, evals, ostrength, gs_mos, matrix_s, print_section)
      TYPE(qs_environment_type), POINTER                 :: qs_env
      TYPE(cp_fm_type), DIMENSION(:, :), INTENT(in)      :: evects
      REAL(kind=dp), DIMENSION(:), INTENT(in)            :: evals, ostrength
      TYPE(tddfpt_ground_state_mos), DIMENSION(:), &
         INTENT(in)                                      :: gs_mos
      TYPE(dbcsr_type), POINTER                          :: matrix_s
      TYPE(section_vals_type), POINTER                   :: print_section
 
      CHARACTER(LEN=*), PARAMETER :: routinen = 'tddfpt_print_nto_analysis'
      INTEGER, PARAMETER                                 :: ntomax = 10
 
      CHARACTER(LEN=20), DIMENSION(2)                    :: nto_name
      INTEGER                                            :: handle, i, ia, icg, iounit, ispin, &
                                                            istate, j, nao, nlist, nmax, nmo, &
                                                            nnto, nspins, nstates
      INTEGER, DIMENSION(2)                              :: iv
      INTEGER, DIMENSION(2, ntomax)                      :: ia_index
      INTEGER, DIMENSION(:), POINTER                     :: slist, stride
      LOGICAL                                            :: append_cube, cube_file, explicit
      REAL(kind=dp)                                      :: os_threshold, sume, threshold
      REAL(kind=dp), ALLOCATABLE, DIMENSION(:)           :: eigvals
      REAL(kind=dp), ALLOCATABLE, DIMENSION(:, :)        :: eigenvalues
      REAL(kind=dp), DIMENSION(ntomax)                   :: ia_eval
      TYPE(cp_fm_struct_type), POINTER                   :: fm_mo_struct, fm_struct
      TYPE(cp_fm_type)                                   :: sev, smat, tmat, wmat, work, wvec
      TYPE(cp_fm_type), ALLOCATABLE, DIMENSION(:)        :: teig
      TYPE(cp_logger_type), POINTER                      :: logger
      TYPE(mo_set_type), ALLOCATABLE, DIMENSION(:)       :: nto_set
      TYPE(particle_type), DIMENSION(:), POINTER         :: particle_set
      TYPE(qs_kind_type), DIMENSION(:), POINTER          :: qs_kind_set
      TYPE(section_vals_type), POINTER                   :: molden_section, nto_section
 
      CALL timeset(routinen, handle)
 
      logger => cp_get_default_logger()
      iounit = cp_logger_get_default_io_unit(logger)
 
      IF (btest(cp_print_key_should_output(logger%iter_info, print_section, &
                                           "NTO_ANALYSIS"), cp_p_file)) THEN
 
         CALL cite_reference(martin2003)
 
         CALL section_vals_val_get(print_section, "NTO_ANALYSIS%THRESHOLD", r_val=threshold)
         CALL section_vals_val_get(print_section, "NTO_ANALYSIS%INTENSITY_THRESHOLD", r_val=os_threshold)
         CALL section_vals_val_get(print_section, "NTO_ANALYSIS%STATE_LIST", explicit=explicit)
 
         IF (explicit) THEN
            CALL section_vals_val_get(print_section, "NTO_ANALYSIS%STATE_LIST", i_vals=slist)
            nlist = SIZE(slist)
         ELSE
            nlist = 0
         END IF
 
         IF (iounit > 0) THEN
            WRITE (iounit, "(1X,A)") "", &
               "-------------------------------------------------------------------------------", &
               "-                            Natural Orbital analysis                         -", &
               "-------------------------------------------------------------------------------"
         END IF
 
         nspins = SIZE(evects, 1)
         nstates = SIZE(evects, 2)
         CALL dbcsr_get_info(matrix_s, nfullrows_total=nao)
 
         DO istate = 1, nstates
            IF (os_threshold > ostrength(istate)) THEN
               IF (iounit > 0) THEN
                  WRITE (iounit, "(1X,A,I6)") "  Skipping state ", istate
               END IF
               cycle
            END IF
            IF (nlist > 0) THEN
               IF (.NOT. any(slist == istate)) THEN
                  IF (iounit > 0) THEN
                     WRITE (iounit, "(1X,A,I6)") "  Skipping state ", istate
                  END IF
                  cycle
               END IF
            END IF
            IF (iounit > 0) THEN
               WRITE (iounit, "(1X,A,I6,T30,F10.5,A)") "  STATE NR. ", istate, evals(istate)*evolt, " eV"
            END IF
            nmax = 0
            DO ispin = 1, nspins
               CALL cp_fm_get_info(evects(ispin, istate), matrix_struct=fm_struct, ncol_global=nmo)
               nmax = max(nmax, nmo)
            END DO
            ALLOCATE (eigenvalues(nmax, nspins))
            eigenvalues = 0.0_dp
            ! SET 1: Hole states
            ! SET 2: Particle states
            nto_name(1) = 'Hole_states'
            nto_name(2) = 'Particle_states'
            ALLOCATE (nto_set(2))
            DO i = 1, 2
               CALL allocate_mo_set(nto_set(i), nao, ntomax, 0, 0.0_dp, 1.0_dp, 0.0_dp)
               CALL cp_fm_get_info(evects(1, istate), matrix_struct=fm_struct)
               CALL cp_fm_struct_create(fmstruct=fm_mo_struct, template_fmstruct=fm_struct, &
                                        ncol_global=ntomax)
               CALL cp_fm_create(tmat, fm_mo_struct)
               CALL init_mo_set(nto_set(i), fm_ref=tmat, name=nto_name(i))
               CALL cp_fm_release(tmat)
               CALL cp_fm_struct_release(fm_mo_struct)
            END DO
            !
            ALLOCATE (teig(nspins))
            ! hole states
            ! Diagonalize X(T)*S*X
            DO ispin = 1, nspins
               associate(ev => evects(ispin, istate))
                  CALL cp_fm_get_info(ev, matrix_struct=fm_struct, ncol_global=nmo)
                  CALL cp_fm_create(sev, fm_struct)
                  CALL cp_fm_struct_create(fmstruct=fm_mo_struct, template_fmstruct=fm_struct, &
                                           nrow_global=nmo, ncol_global=nmo)
                  CALL cp_fm_create(tmat, fm_mo_struct)
                  CALL cp_fm_create(teig(ispin), fm_mo_struct)
                  CALL cp_dbcsr_sm_fm_multiply(matrix_s, ev, sev, ncol=nmo, alpha=1.0_dp, beta=0.0_dp)
                  CALL parallel_gemm('T', 'N', nmo, nmo, nao, 1.0_dp, ev, sev, 0.0_dp, tmat)
               END associate
 
               CALL choose_eigv_solver(tmat, teig(ispin), eigenvalues(1:nmo, ispin))
 
               CALL cp_fm_struct_release(fm_mo_struct)
               CALL cp_fm_release(tmat)
               CALL cp_fm_release(sev)
            END DO
            ! find major determinants i->a
            ia_index = 0
            sume = 0.0_dp
            nnto = 0
            DO i = 1, ntomax
               iv = maxloc(eigenvalues)
               ia_eval(i) = eigenvalues(iv(1), iv(2))
               ia_index(1:2, i) = iv(1:2)
               sume = sume + ia_eval(i)
               eigenvalues(iv(1), iv(2)) = 0.0_dp
               nnto = nnto + 1
               IF (sume > threshold) EXIT
            END DO
            ! store hole states
            CALL set_mo_set(nto_set(1), nmo=nnto)
            DO i = 1, nnto
               ia = ia_index(1, i)
               ispin = ia_index(2, i)
               CALL cp_fm_get_info(gs_mos(ispin)%mos_occ, ncol_global=nmo)
               CALL cp_fm_get_info(teig(ispin), matrix_struct=fm_struct)
               CALL cp_fm_struct_create(fmstruct=fm_mo_struct, template_fmstruct=fm_struct, &
                                        nrow_global=nmo, ncol_global=1)
               CALL cp_fm_create(tmat, fm_mo_struct)
               CALL cp_fm_struct_release(fm_mo_struct)
               CALL cp_fm_get_info(gs_mos(1)%mos_occ, matrix_struct=fm_struct)
               CALL cp_fm_struct_create(fmstruct=fm_mo_struct, template_fmstruct=fm_struct, &
                                        ncol_global=1)
               CALL cp_fm_create(wvec, fm_mo_struct)
               CALL cp_fm_struct_release(fm_mo_struct)
               CALL cp_fm_to_fm(teig(ispin), tmat, 1, ia, 1)
               CALL parallel_gemm('N', 'N', nao, 1, nmo, 1.0_dp, gs_mos(ispin)%mos_occ, &
                                  tmat, 0.0_dp, wvec)
               CALL cp_fm_to_fm(wvec, nto_set(1)%mo_coeff, 1, 1, i)
               CALL cp_fm_release(wvec)
               CALL cp_fm_release(tmat)
            END DO
            ! particle states
            ! Solve generalized eigenvalue equation:  (S*X)*(S*X)(T)*v = lambda*S*v
            CALL set_mo_set(nto_set(2), nmo=nnto)
            DO ispin = 1, nspins
               associate(ev => evects(ispin, istate))
                  CALL cp_fm_get_info(ev, matrix_struct=fm_struct, nrow_global=nao, ncol_global=nmo)
                  ALLOCATE (eigvals(nao))
                  eigvals = 0.0_dp
                  CALL cp_fm_create(sev, fm_struct)
                  CALL cp_dbcsr_sm_fm_multiply(matrix_s, ev, sev, ncol=nmo, alpha=1.0_dp, beta=0.0_dp)
               END associate
               CALL cp_fm_struct_create(fmstruct=fm_mo_struct, template_fmstruct=fm_struct, &
                                        nrow_global=nao, ncol_global=nao)
               CALL cp_fm_create(tmat, fm_mo_struct)
               CALL cp_fm_create(smat, fm_mo_struct)
               CALL cp_fm_create(wmat, fm_mo_struct)
               CALL cp_fm_create(work, fm_mo_struct)
               CALL cp_fm_struct_release(fm_mo_struct)
               CALL copy_dbcsr_to_fm(matrix_s, smat)
               CALL parallel_gemm('N', 'T', nao, nao, nmo, 1.0_dp, sev, sev, 0.0_dp, tmat)
               CALL cp_fm_geeig(tmat, smat, wmat, eigvals, work)
               DO i = 1, nnto
                  IF (ispin == ia_index(2, i)) THEN
                     icg = 0
                     DO j = 1, nao
                        IF (abs(eigvals(j) - ia_eval(i)) < 1.e-6_dp) THEN
                           icg = j
                           EXIT
                        END IF
                     END DO
                     IF (icg == 0) THEN
                        CALL cp_warn(__location__, &
                                     "Could not locate particle state associated with hole state.")
                     ELSE
                        CALL cp_fm_to_fm(wmat, nto_set(2)%mo_coeff, 1, icg, i)
                     END IF
                  END IF
               END DO
               DEALLOCATE (eigvals)
               CALL cp_fm_release(sev)
               CALL cp_fm_release(tmat)
               CALL cp_fm_release(smat)
               CALL cp_fm_release(wmat)
               CALL cp_fm_release(work)
            END DO
            ! print
            IF (iounit > 0) THEN
               sume = 0.0_dp
               DO i = 1, nnto
                  sume = sume + ia_eval(i)
                  WRITE (iounit, "(T6,A,i2,T30,A,i1,T42,A,F8.5,T63,A,F8.5)") &
                     "Particle-Hole state:", i, " Spin:", ia_index(2, i), &
                     "Eigenvalue:", ia_eval(i), " Sum Eigv:", sume
               END DO
            END IF
            ! Cube and Molden files
            nto_section => section_vals_get_subs_vals(print_section, "NTO_ANALYSIS")
            CALL section_vals_val_get(nto_section, "CUBE_FILES", l_val=cube_file)
            CALL section_vals_val_get(nto_section, "STRIDE", i_vals=stride)
            CALL section_vals_val_get(nto_section, "APPEND", l_val=append_cube)
            IF (cube_file) THEN
               CALL print_nto_cubes(qs_env, nto_set, istate, stride, append_cube, nto_section)
            END IF
            CALL get_qs_env(qs_env, qs_kind_set=qs_kind_set, particle_set=particle_set)
            molden_section => section_vals_get_subs_vals(print_section, "MOS_MOLDEN")
            CALL write_mos_molden(nto_set, qs_kind_set, particle_set, molden_section)
            !
            DEALLOCATE (eigenvalues)
            CALL cp_fm_release(teig)
            !
            DO i = 1, 2
               CALL deallocate_mo_set(nto_set(i))
            END DO
            DEALLOCATE (nto_set)
         END DO
 
         IF (iounit > 0) THEN
            WRITE (iounit, "(1X,A)") &
               "-------------------------------------------------------------------------------"
         END IF
 
      END IF
 
      CALL timestop(handle)
 
   END SUBROUTINE tddfpt_print_nto_analysis
 
! **************************************************************************************************
!> \brief ...
!> \param vin ...
!> \param vout ...
!> \param mos_occ ...
!> \param matrix_s ...
! **************************************************************************************************
   SUBROUTINE project_vector(vin, vout, mos_occ, matrix_s)
      TYPE(dbcsr_type)                                   :: vin, vout
      TYPE(cp_fm_type), INTENT(IN)                       :: mos_occ
      TYPE(dbcsr_type), POINTER                          :: matrix_s
 
      CHARACTER(LEN=*), PARAMETER                        :: routinen = 'project_vector'
 
      INTEGER                                            :: handle, nao, nmo
      REAL(kind=dp)                                      :: norm(1)
      TYPE(cp_fm_struct_type), POINTER                   :: fm_struct, fm_vec_struct
      TYPE(cp_fm_type)                                   :: csvec, svec, vec
 
      CALL timeset(routinen, handle)
 
      CALL cp_fm_get_info(mos_occ, matrix_struct=fm_struct, nrow_global=nao, ncol_global=nmo)
      CALL cp_fm_struct_create(fmstruct=fm_vec_struct, template_fmstruct=fm_struct, &
                               nrow_global=nao, ncol_global=1)
      CALL cp_fm_create(vec, fm_vec_struct)
      CALL cp_fm_create(svec, fm_vec_struct)
      CALL cp_fm_struct_release(fm_vec_struct)
      CALL cp_fm_struct_create(fmstruct=fm_vec_struct, template_fmstruct=fm_struct, &
                               nrow_global=nmo, ncol_global=1)
      CALL cp_fm_create(csvec, fm_vec_struct)
      CALL cp_fm_struct_release(fm_vec_struct)
 
      CALL copy_dbcsr_to_fm(vin, vec)
      CALL cp_dbcsr_sm_fm_multiply(matrix_s, vec, svec, ncol=1, alpha=1.0_dp, beta=0.0_dp)
      CALL parallel_gemm('T', 'N', nmo, 1, nao, 1.0_dp, mos_occ, svec, 0.0_dp, csvec)
      CALL parallel_gemm('N', 'N', nao, 1, nmo, -1.0_dp, mos_occ, csvec, 1.0_dp, vec)
      CALL cp_fm_vectorsnorm(vec, norm)
      cpassert(norm(1) > 1.e-14_dp)
      norm(1) = sqrt(1._dp/norm(1))
      CALL cp_fm_scale(norm(1), vec)
      CALL copy_fm_to_dbcsr(vec, vout, keep_sparsity=.false.)
 
      CALL cp_fm_release(csvec)
      CALL cp_fm_release(svec)
      CALL cp_fm_release(vec)
 
      CALL timestop(handle)
 
   END SUBROUTINE project_vector
 
! **************************************************************************************************
!> \brief ...
!> \param va ...
!> \param vb ...
!> \param res ...
! **************************************************************************************************
   SUBROUTINE vec_product(va, vb, res)
      TYPE(dbcsr_type)                                   :: va, vb
      REAL(kind=dp), INTENT(OUT)                         :: res
 
      CHARACTER(LEN=*), PARAMETER                        :: routinen = 'vec_product'
 
      INTEGER                                            :: blk, group_handle, handle, icol, irow
      LOGICAL                                            :: found
      REAL(kind=dp), DIMENSION(:, :), POINTER            :: vba, vbb
      TYPE(dbcsr_iterator_type)                          :: iter
      TYPE(mp_comm_type)                                 :: group
 
      CALL timeset(routinen, handle)
 
      res = 0.0_dp
 
      CALL dbcsr_get_info(va, group=group_handle)
      CALL group%set_handle(group_handle)
      CALL dbcsr_iterator_start(iter, va)
      DO WHILE (dbcsr_iterator_blocks_left(iter))
         CALL dbcsr_iterator_next_block(iter, irow, icol, vba, blk)
         CALL dbcsr_get_block_p(vb, row=irow, col=icol, block=vbb, found=found)
         res = res + sum(vba*vbb)
         cpassert(found)
      END DO
      CALL dbcsr_iterator_stop(iter)
      CALL group%sum(res)
 
      CALL timestop(handle)
 
   END SUBROUTINE vec_product
 
! **************************************************************************************************
!> \brief ...
!> \param qs_env ...
!> \param mos ...
!> \param istate ...
!> \param stride ...
!> \param append_cube ...
!> \param print_section ...
! **************************************************************************************************
   SUBROUTINE print_nto_cubes(qs_env, mos, istate, stride, append_cube, print_section)
 
      TYPE(qs_environment_type), POINTER                 :: qs_env
      TYPE(mo_set_type), DIMENSION(:), INTENT(IN)        :: mos
      INTEGER, INTENT(IN)                                :: istate
      INTEGER, DIMENSION(:), POINTER                     :: stride
      LOGICAL, INTENT(IN)                                :: append_cube
      TYPE(section_vals_type), POINTER                   :: print_section
 
      CHARACTER(LEN=default_path_length)                 :: filename, my_pos_cube, title
      INTEGER                                            :: i, iset, nmo, unit_nr
      LOGICAL                                            :: mpi_io
      TYPE(atomic_kind_type), DIMENSION(:), POINTER      :: atomic_kind_set
      TYPE(cell_type), POINTER                           :: cell
      TYPE(cp_fm_type), POINTER                          :: mo_coeff
      TYPE(cp_logger_type), POINTER                      :: logger
      TYPE(dft_control_type), POINTER                    :: dft_control
      TYPE(particle_list_type), POINTER                  :: particles
      TYPE(particle_type), DIMENSION(:), POINTER         :: particle_set
      TYPE(pw_c1d_gs_type)                               :: wf_g
      TYPE(pw_env_type), POINTER                         :: pw_env
      TYPE(pw_pool_p_type), DIMENSION(:), POINTER        :: pw_pools
      TYPE(pw_pool_type), POINTER                        :: auxbas_pw_pool
      TYPE(pw_r3d_rs_type)                               :: wf_r
      TYPE(qs_kind_type), DIMENSION(:), POINTER          :: qs_kind_set
      TYPE(qs_subsys_type), POINTER                      :: subsys
 
      logger => cp_get_default_logger()
 
      CALL get_qs_env(qs_env=qs_env, dft_control=dft_control, pw_env=pw_env)
      CALL pw_env_get(pw_env, auxbas_pw_pool=auxbas_pw_pool, pw_pools=pw_pools)
      CALL auxbas_pw_pool%create_pw(wf_r)
      CALL auxbas_pw_pool%create_pw(wf_g)
 
      CALL get_qs_env(qs_env, subsys=subsys)
      CALL qs_subsys_get(subsys, particles=particles)
 
      my_pos_cube = "REWIND"
      IF (append_cube) THEN
         my_pos_cube = "APPEND"
      END IF
 
      CALL get_qs_env(qs_env=qs_env, &
                      atomic_kind_set=atomic_kind_set, &
                      qs_kind_set=qs_kind_set, &
                      cell=cell, &
                      particle_set=particle_set)
 
      DO iset = 1, 2
         CALL get_mo_set(mo_set=mos(iset), mo_coeff=mo_coeff, nmo=nmo)
         DO i = 1, nmo
            CALL calculate_wavefunction(mo_coeff, i, wf_r, wf_g, atomic_kind_set, qs_kind_set, &
                                        cell, dft_control, particle_set, pw_env)
            IF (iset == 1) THEN
               WRITE (filename, '(a4,I3.3,I2.2,a11)') "NTO_STATE", istate, i, "_Hole_State"
            ELSEIF (iset == 2) THEN
               WRITE (filename, '(a4,I3.3,I2.2,a15)') "NTO_STATE", istate, i, "_Particle_State"
            END IF
            mpi_io = .true.
            unit_nr = cp_print_key_unit_nr(logger, print_section, '', extension=".cube", &
                                           middle_name=trim(filename), file_position=my_pos_cube, &
                                           log_filename=.false., ignore_should_output=.true., mpi_io=mpi_io)
            IF (iset == 1) THEN
               WRITE (title, *) "Natural Transition Orbital Hole State", i
            ELSEIF (iset == 2) THEN
               WRITE (title, *) "Natural Transition Orbital Particle State", i
            END IF
            CALL cp_pw_to_cube(wf_r, unit_nr, title, particles=particles, stride=stride, mpi_io=mpi_io)
            CALL cp_print_key_finished_output(unit_nr, logger, print_section, '', &
                                              ignore_should_output=.true., mpi_io=mpi_io)
         END DO
      END DO
 
      CALL auxbas_pw_pool%give_back_pw(wf_g)
      CALL auxbas_pw_pool%give_back_pw(wf_r)
 
   END SUBROUTINE print_nto_cubes
 
END MODULE qs_tddfpt2_properties