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bse_properties.F
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!--------------------------------------------------------------------------------------------------!
! 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 Routines for computing excitonic properties, e.g. exciton diameter, from the BSE
!> \par History
!> 10.2024 created [Maximilian Graml]
! **************************************************************************************************
MODULE bse_properties
USE bse_util, ONLY: fm_general_add_bse,&
print_bse_nto_cubes,&
reshuffle_eigvec,&
trace_exciton_descr
USE cp_files, ONLY: close_file,&
open_file
USE cp_fm_basic_linalg, ONLY: cp_fm_scale_and_add,&
cp_fm_trace
USE cp_fm_diag, ONLY: cp_fm_svd
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_submatrix,&
cp_fm_release,&
cp_fm_set_all,&
cp_fm_to_fm,&
cp_fm_to_fm_submat,&
cp_fm_to_fm_submat_general,&
cp_fm_type
USE cp_log_handling, ONLY: cp_get_default_logger,&
cp_logger_get_default_unit_nr,&
cp_logger_type
USE input_section_types, ONLY: section_vals_get_subs_vals,&
section_vals_type,&
section_vals_val_get
USE kinds, ONLY: dp
USE mathconstants, ONLY: pi
USE mp2_types, ONLY: mp2_type
USE parallel_gemm_api, ONLY: parallel_gemm
USE physcon, ONLY: c_light_au,&
evolt
USE qs_environment_types, ONLY: get_qs_env,&
qs_environment_type
USE qs_mo_types, ONLY: allocate_mo_set,&
deallocate_mo_set,&
init_mo_set,&
mo_set_type
#include "./base/base_uses.f90"
IMPLICIT NONE
PRIVATE
CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'bse_properties'
PUBLIC :: exciton_descr_type
PUBLIC :: get_exciton_descriptors, get_oscillator_strengths, compute_absorption_spectrum, &
calculate_NTOs
! TYPE definitions for exciton wavefunction descriptors
TYPE exciton_descr_type
REAL(KIND=dp), DIMENSION(3) :: r_e = 0.0_dp, &
r_h = 0.0_dp, &
r_e_sq = 0.0_dp, &
r_h_sq = 0.0_dp, &
r_e_h = 0.0_dp, &
r_e_shift = 0.0_dp, &
r_h_shift = 0.0_dp, &
cov_e_h = 0.0_dp
REAL(KIND=dp) :: sigma_e = 0.0_dp, &
sigma_h = 0.0_dp, &
cov_e_h_sum = 0.0_dp, &
corr_e_h = 0.0_dp, &
diff_r_abs = 0.0_dp, &
diff_r_sqr = 0.0_dp, &
norm_XpY = 0.0_dp
LOGICAL :: flag_TDA = .FALSE.
END TYPE exciton_descr_type
CONTAINS
! **************************************************************************************************
!> \brief Compute and return BSE dipoles d_r^n = sqrt(2) sum_ia < ψ_i | r | ψ_a > ( X_ia^n + Y_ia^n )
!> and oscillator strengths f^n = 2/3 * Ω^n sum_r∈(x,y,z) ( d_r^n )^2
!> Prelim Ref.: Eqs. (23), (24)
!> in J. Chem. Phys. 152, 044105 (2020); https://doi.org/10.1063/1.5123290
!> \param fm_eigvec_X ...
!> \param Exc_ens ...
!> \param fm_dipole_ai_trunc ...
!> \param trans_mom_bse BSE dipole vectors in real space per excitation level
!> \param oscill_str Oscillator strength per excitation level
!> \param polarizability_residues Residues of polarizability ("tensorial oscillator strength")
!> per excitation level
!> \param mp2_env ...
!> \param homo_red ...
!> \param virtual_red ...
!> \param unit_nr ...
!> \param fm_eigvec_Y ...
! **************************************************************************************************
SUBROUTINE get_oscillator_strengths(fm_eigvec_X, Exc_ens, fm_dipole_ai_trunc, &
trans_mom_bse, oscill_str, polarizability_residues, &
mp2_env, homo_red, virtual_red, unit_nr, &
fm_eigvec_Y)
TYPE(cp_fm_type), INTENT(IN) :: fm_eigvec_X
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:), &
INTENT(IN) :: Exc_ens
TYPE(cp_fm_type), DIMENSION(3) :: fm_dipole_ai_trunc
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :, :), &
INTENT(OUT) :: trans_mom_bse
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:), &
INTENT(OUT) :: oscill_str
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :, :), &
INTENT(OUT) :: polarizability_residues
TYPE(mp2_type), INTENT(IN) :: mp2_env
INTEGER, INTENT(IN) :: homo_red, virtual_red, unit_nr
TYPE(cp_fm_type), INTENT(IN), OPTIONAL :: fm_eigvec_Y
CHARACTER(LEN=*), PARAMETER :: routineN = 'get_oscillator_strengths'
INTEGER :: handle, idir, jdir, n
TYPE(cp_fm_struct_type), POINTER :: fm_struct_dipole_MO_trunc_reordered, &
fm_struct_trans_mom_bse
TYPE(cp_fm_type) :: fm_eigvec_XYsum
TYPE(cp_fm_type), DIMENSION(3) :: fm_dipole_MO_trunc_reordered, &
fm_dipole_per_dir, fm_trans_mom_bse
CALL timeset(routineN, handle)
CALL cp_fm_struct_create(fm_struct_dipole_MO_trunc_reordered, fm_eigvec_X%matrix_struct%para_env, &
fm_eigvec_X%matrix_struct%context, 1, homo_red*virtual_red)
CALL cp_fm_struct_create(fm_struct_trans_mom_bse, fm_eigvec_X%matrix_struct%para_env, &
fm_eigvec_X%matrix_struct%context, 1, homo_red*virtual_red)
! Include excitonic amplitudes in dipoles, i.e. obtain "BSE dipoles":
! \vec{D}_n = sqrt(2) * sum_{i,a} \vec{D}_ai (X_{ai}^{(n)} + Y_{ai}^{(n)})
! Reorder dipoles in order to execute the sum over i and a by parallel gemm
DO idir = 1, 3
CALL cp_fm_create(fm_dipole_MO_trunc_reordered(idir), matrix_struct=fm_struct_dipole_MO_trunc_reordered, &
name="dipoles_mo_reordered")
CALL cp_fm_set_all(fm_dipole_MO_trunc_reordered(idir), 0.0_dp)
CALL fm_general_add_bse(fm_dipole_MO_trunc_reordered(idir), fm_dipole_ai_trunc(idir), 1.0_dp, &
1, 1, &
1, virtual_red, &
unit_nr, (/2, 4, 3, 1/), mp2_env)
CALL cp_fm_release(fm_dipole_per_dir(idir))
END DO
DO idir = 1, 3
CALL cp_fm_create(fm_trans_mom_bse(idir), matrix_struct=fm_struct_trans_mom_bse, &
name="excitonic_dipoles")
CALL cp_fm_set_all(fm_trans_mom_bse(idir), 0.0_dp)
END DO
! If TDA is invoked, Y is not present as it is simply 0
CALL cp_fm_create(fm_eigvec_XYsum, matrix_struct=fm_eigvec_X%matrix_struct, name="excit_amplitude_sum")
CALL cp_fm_set_all(fm_eigvec_XYsum, 0.0_dp)
CALL cp_fm_to_fm(fm_eigvec_X, fm_eigvec_XYsum)
IF (PRESENT(fm_eigvec_Y)) THEN
CALL cp_fm_scale_and_add(1.0_dp, fm_eigvec_XYsum, 1.0_dp, fm_eigvec_Y)
END IF
DO idir = 1, 3
CALL parallel_gemm('N', 'N', 1, homo_red*virtual_red, homo_red*virtual_red, SQRT(2.0_dp), &
fm_dipole_MO_trunc_reordered(idir), fm_eigvec_XYsum, 0.0_dp, fm_trans_mom_bse(idir))
END DO
! Get oscillator strengths themselves
ALLOCATE (oscill_str(homo_red*virtual_red))
ALLOCATE (trans_mom_bse(3, 1, homo_red*virtual_red))
ALLOCATE (polarizability_residues(3, 3, homo_red*virtual_red))
trans_mom_bse(:, :, :) = 0.0_dp
! Sum over all directions
DO idir = 1, 3
CALL cp_fm_get_submatrix(fm_trans_mom_bse(idir), trans_mom_bse(idir, :, :))
END DO
DO n = 1, homo_red*virtual_red
DO idir = 1, 3
DO jdir = 1, 3
polarizability_residues(idir, jdir, n) = 2.0_dp*Exc_ens(n)*trans_mom_bse(idir, 1, n)*trans_mom_bse(jdir, 1, n)
END DO
END DO
oscill_str(n) = 2.0_dp/3.0_dp*Exc_ens(n)*SUM(ABS(trans_mom_bse(:, :, n))**2)
END DO
CALL cp_fm_struct_release(fm_struct_dipole_MO_trunc_reordered)
CALL cp_fm_struct_release(fm_struct_trans_mom_bse)
DO idir = 1, 3
CALL cp_fm_release(fm_dipole_MO_trunc_reordered(idir))
CALL cp_fm_release(fm_trans_mom_bse(idir))
CALL cp_fm_release(fm_dipole_ai_trunc(idir))
END DO
CALL cp_fm_release(fm_eigvec_XYsum)
CALL timestop(handle)
END SUBROUTINE get_oscillator_strengths
! **************************************************************************************************
!> \brief Computes and returns absorption spectrum for the frequency range and broadening
!> provided by the user.
!> Prelim Ref.: C. Ullrich, Time-Dependent Density-Functional Theory: Concepts and Applications
!> (Oxford University Press, Oxford, 2012), Eq. 7.51
!> \param oscill_str ...
!> \param polarizability_residues ...
!> \param Exc_ens ...
!> \param info_approximation ...
!> \param unit_nr ...
!> \param mp2_env ...
! **************************************************************************************************
SUBROUTINE compute_absorption_spectrum(oscill_str, polarizability_residues, Exc_ens, &
info_approximation, unit_nr, mp2_env)
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:), &
INTENT(IN) :: oscill_str
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :, :), &
INTENT(IN) :: polarizability_residues
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:), &
INTENT(IN) :: Exc_ens
CHARACTER(LEN=10) :: info_approximation
INTEGER, INTENT(IN) :: unit_nr
TYPE(mp2_type), INTENT(INOUT) :: mp2_env
CHARACTER(LEN=*), PARAMETER :: routineN = 'compute_absorption_spectrum'
CHARACTER(LEN=10) :: eta_str, width_eta_format_str
CHARACTER(LEN=40) :: file_name_crosssection, &
file_name_spectrum
INTEGER :: handle, i, idir, j, jdir, k, num_steps, &
unit_nr_file, width_eta
REAL(KIND=dp) :: eta, freq_end, freq_start, freq_step, &
omega
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: abs_cross_section, abs_spectrum
REAL(KIND=dp), DIMENSION(:), POINTER :: eta_list
TYPE(cp_logger_type), POINTER :: logger
CALL timeset(routineN, handle)
freq_step = mp2_env%bse%bse_spectrum_freq_step_size
freq_start = mp2_env%bse%bse_spectrum_freq_start
freq_end = mp2_env%bse%bse_spectrum_freq_end
eta_list => mp2_env%bse%bse_eta_spectrum_list
! Calculate number of steps to fit given frequency range
num_steps = NINT((freq_end - freq_start)/freq_step) + 1
DO k = 1, SIZE(eta_list)
eta = eta_list(k)
! Some magic to get a nice formatting of the eta value in filenames
width_eta = MAX(1, INT(LOG10(eta)) + 1) + 4
WRITE (width_eta_format_str, "(A2,I0,A3)") '(F', width_eta, '.3)'
WRITE (eta_str, width_eta_format_str) eta*evolt
! Filename itself
file_name_spectrum = 'BSE'//TRIM(ADJUSTL(info_approximation))//'eta='//TRIM(eta_str)//'.spectrum'
file_name_crosssection = 'BSE'//TRIM(ADJUSTL(info_approximation))//'eta='//TRIM(eta_str)//'.crosssection'
! First column is frequency in eV, second column is imaginary part of the trace of the polarizability
! The following 9 columns are the entries of the polarizability tensor
ALLOCATE (abs_spectrum(num_steps, 11))
abs_spectrum(:, :) = 0.0_dp
! Also calculate and print the photoabsorption cross section tensor
! σ_{µ µ'}(ω) = 4πω Im{α_{µ µ'}(ω)} / c
ALLOCATE (abs_cross_section(num_steps, 11))
abs_cross_section(:, :) = 0.0_dp
! Calculate the imaginary part of the mean dipole polarizability α_{avg}(ω)
! which is given by (cf. C. Ullrichs Book on TDDFT, Eq. 7.51)
! We introduce an additional - due to his convention for charge vs particle density, see also:
! Computer Physics Communications, 208:149–161, November 2016
! https://doi.org/10.1016/j.cpc.2016.06.019
! α_{avg}(ω) = - \sum_{n=1}^{N_exc} \frac{f_n}{(ω+iη)² - (Ω^n)²}
! and then the imaginary part is (in the limit η -> 0)
! Im[α_{avg}(ω)] = - \sum_{n=1}^{N_exc} f_n * η / ((ω - Ω^n)² + η²)
! where f_n are the oscillator strengths and E_exc the excitation energies
! For the full polarizability tensor, we have
! α_{µ µ'}(ω) = - \sum_n [2 Ω^n d^n_µ d^n_µ'] / [(ω+iη)^2- (Ω^n)^2]
! = - \sum_n "polarizability_residues" / [(ω+iη)^2- (Ω^n)^2]
DO i = 1, num_steps
omega = freq_start + (i - 1)*freq_step
abs_spectrum(i, 1) = omega
DO j = 1, SIZE(oscill_str)
abs_spectrum(i, 2) = abs_spectrum(i, 2) - oscill_str(j)* &
AIMAG(1/((omega + CMPLX(0.0, eta, kind=dp))**2 - Exc_ens(j)**2))
DO idir = 1, 3
DO jdir = 1, 3
! Factor 2 from formula for tensor is already in the polarizability_residues
! to follow the same convention as the oscillator strengths
abs_spectrum(i, 2 + (idir - 1)*3 + jdir) = abs_spectrum(i, 2 + (idir - 1)*3 + jdir) &
- polarizability_residues(idir, jdir, j)* &
AIMAG(1/((omega + CMPLX(0.0, eta, kind=dp))**2 - Exc_ens(j)**2))
END DO
END DO
END DO
END DO
! Extract cross section σ from polarizability tensor
DO i = 1, num_steps
omega = abs_spectrum(i, 1)
abs_cross_section(i, 1) = omega
abs_cross_section(i, 2:) = 4.0_dp*pi*abs_spectrum(i, 2:)*omega/c_light_au
END DO
!For debug runs: Export an entry of the two tensors to allow regtests on spectra
IF (mp2_env%bse%bse_debug_print) THEN
IF (unit_nr > 0) THEN
WRITE (unit_nr, '(T2,A10,T13,A,T65,F16.4)') 'BSE|DEBUG|', &
'Averaged dynamical dipole polarizability at 8.2 eV:', &
abs_spectrum(83, 2)
WRITE (unit_nr, '(T2,A10,T13,A,T65,F16.4)') 'BSE|DEBUG|', &
'Averaged photoabsorption cross section at 8.2 eV:', &
abs_cross_section(83, 2)
END IF
END IF
! Print it to file
logger => cp_get_default_logger()
IF (logger%para_env%is_source()) THEN
unit_nr_file = cp_logger_get_default_unit_nr()
ELSE
unit_nr_file = -1
END IF
IF (unit_nr_file > 0) THEN
CALL open_file(file_name_crosssection, unit_number=unit_nr_file, &
file_status="UNKNOWN", file_action="WRITE")
WRITE (unit_nr_file, '(A,A6)') "# Photoabsorption cross section σ_{µ µ'}(ω) = -4πω/c * Im[ \sum_n "// &
"[2 Ω^n d_µ^n d_µ'^n] / [(ω+iη)²- (Ω^n)²] ] from Bethe Salpeter equation for method ", &
TRIM(ADJUSTL(info_approximation))
WRITE (unit_nr_file, '(A20,1X,10(2X,A20,1X))') "# Frequency (eV)", "σ_{avg}(ω)", "σ_xx(ω)", &
"σ_xy(ω)", "σ_xz(ω)", "σ_yx(ω)", "σ_yy(ω)", "σ_yz(ω)", "σ_zx(ω)", &
"σ_zy(ω)", "σ_zz(ω)"
DO i = 1, num_steps
WRITE (unit_nr_file, '(11(F20.8,1X))') abs_cross_section(i, 1)*evolt, abs_cross_section(i, 2:11)
END DO
CALL close_file(unit_nr_file)
END IF
DEALLOCATE (abs_cross_section)
IF (unit_nr_file > 0) THEN
CALL open_file(file_name_spectrum, unit_number=unit_nr_file, &
file_status="UNKNOWN", file_action="WRITE")
WRITE (unit_nr_file, '(A,A6)') "# Imaginary part of polarizability α_{µ µ'}(ω) = -\sum_n "// &
"[2 Ω^n d_µ^n d_µ'^n] / [(ω+iη)²- (Ω^n)²] from Bethe Salpeter equation for method ", &
TRIM(ADJUSTL(info_approximation))
WRITE (unit_nr_file, '(A20,1X,10(2X,A20,1X))') "# Frequency (eV)", "Im{α_{avg}(ω)}", "Im{α_xx(ω)}", &
"Im{α_xy(ω)}", "Im{α_xz(ω)}", "Im{α_yx(ω)}", "Im{α_yy(ω)}", "Im{α_yz(ω)}", "Im{α_zx(ω)}", &
"Im{α_zy(ω)}", "Im{α_zz(ω)}"
DO i = 1, num_steps
WRITE (unit_nr_file, '(11(F20.8,1X))') abs_spectrum(i, 1)*evolt, abs_spectrum(i, 2:11)
END DO
CALL close_file(unit_nr_file)
END IF
DEALLOCATE (abs_spectrum)
END DO
IF (unit_nr > 0) THEN
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A,A)') &
'BSE|', "Printed optical absorption spectrum to local files, e.g. ", file_name_spectrum
WRITE (unit_nr, '(T2,A4,T7,A,A)') &
'BSE|', "as well as photoabsorption cross section to, e.g. ", file_name_crosssection
WRITE (unit_nr, '(T2,A4,T7,A52)') &
'BSE|', "using the Eq. (7.51) from C. Ullrichs Book on TDDFT:"
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T10,A73)') &
'BSE|', "Im{α_{avg}(ω)} = -Im{\sum_{n=1}^{N_exc} \frac{f_n}{(ω+iη)² - (Ω^n)²}}"
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A)') &
'BSE|', "or for the full polarizability tensor:"
WRITE (unit_nr, '(T2,A4,T10,A)') &
'BSE|', "α_{µ µ'}(ω) = -\sum_n [2 Ω^n d_µ^n d_µ'^n] / [(ω+iη)²- (Ω^n)²]"
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A)') &
'BSE|', "as well as Eq. (7.48):"
WRITE (unit_nr, '(T2,A4,T10,A)') &
'BSE|', "σ_{µ µ'}(ω) = 4πω Im{α_{µ µ'}(ω)} / c"
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A)') &
'BSE|', "with transition moments d_µ^n, oscillator strengths f_n,"
WRITE (unit_nr, '(T2,A4,T7,A)') &
'BSE|', "excitation energies Ω^n and the speed of light c."
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A)') &
'BSE|', "Please note that we adopt an additional minus sign for both quantities,"
WRITE (unit_nr, '(T2,A4,T7,A)') &
'BSE|', "due to the convention for charge vs particle density as done in MolGW:"
WRITE (unit_nr, '(T2,A4,T7,A)') &
'BSE|', "https://doi.org/10.1016/j.cpc.2016.06.019."
WRITE (unit_nr, '(T2,A4)') 'BSE|'
END IF
CALL timestop(handle)
END SUBROUTINE
! **************************************************************************************************
!> \brief ...
!> \param fm_X ...
!> \param fm_Y ...
!> \param mo_coeff ...
!> \param homo ...
!> \param virtual ...
!> \param info_approximation ...
!> \param oscill_str ...
!> \param qs_env ...
!> \param unit_nr ...
!> \param mp2_env ...
! **************************************************************************************************
SUBROUTINE calculate_NTOs(fm_X, fm_Y, &
mo_coeff, homo, virtual, &
info_approximation, &
oscill_str, &
qs_env, unit_nr, mp2_env)
TYPE(cp_fm_type), INTENT(IN) :: fm_X
TYPE(cp_fm_type), INTENT(IN), OPTIONAL :: fm_Y
TYPE(cp_fm_type), DIMENSION(:), INTENT(IN) :: mo_coeff
INTEGER, INTENT(IN) :: homo, virtual
CHARACTER(LEN=10) :: info_approximation
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: oscill_str
TYPE(qs_environment_type), POINTER :: qs_env
INTEGER, INTENT(IN) :: unit_nr
TYPE(mp2_type), INTENT(INOUT) :: mp2_env
CHARACTER(LEN=*), PARAMETER :: routineN = 'calculate_NTOs'
CHARACTER(LEN=20), DIMENSION(2) :: nto_name
INTEGER :: handle, homo_irred, i, i_nto, info_svd, &
j, n_exc, n_nto, nao_full, nao_trunc
INTEGER, DIMENSION(:), POINTER :: stride
LOGICAL :: append_cube, cube_file
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: eigval_svd_squ
REAL(KIND=dp), DIMENSION(:), POINTER :: eigval_svd
TYPE(cp_fm_struct_type), POINTER :: fm_struct_m, fm_struct_mo_coeff, &
fm_struct_nto_holes, &
fm_struct_nto_particles, &
fm_struct_nto_set
TYPE(cp_fm_type) :: fm_eigvl, fm_eigvr_t, fm_m, fm_mo_coeff, fm_nto_coeff_holes, &
fm_nto_coeff_particles, fm_nto_set, fm_X_ia, fm_Y_ai
TYPE(mo_set_type), ALLOCATABLE, DIMENSION(:) :: nto_set
TYPE(section_vals_type), POINTER :: bse_section, input, nto_section
CALL timeset(routineN, handle)
CALL get_qs_env(qs_env=qs_env, &
input=input)
bse_section => section_vals_get_subs_vals(input, "DFT%XC%WF_CORRELATION%RI_RPA%GW%BSE")
nao_full = qs_env%mos(1)%nao
nao_trunc = homo + virtual
! This is not influenced by the BSE cutoff
homo_irred = qs_env%mos(1)%homo
! M will have a block structure and is quadratic in homo+virtual, i.e.
! occ virt
! | 0 X_i,a | occ = homo
! M = | Y_a,i 0 | virt = virtual
!
! X and Y are here not the eigenvectors X_ia,n - instead we fix n and reshape the combined ia index
! Notice the index structure of the lower block, i.e. X is transposed
CALL cp_fm_struct_create(fm_struct_m, &
fm_X%matrix_struct%para_env, fm_X%matrix_struct%context, &
nao_trunc, nao_trunc)
CALL cp_fm_struct_create(fm_struct_mo_coeff, &
fm_X%matrix_struct%para_env, fm_X%matrix_struct%context, &
nao_full, nao_trunc)
CALL cp_fm_struct_create(fm_struct_nto_holes, &
fm_X%matrix_struct%para_env, fm_X%matrix_struct%context, &
nao_full, nao_trunc)
CALL cp_fm_struct_create(fm_struct_nto_particles, &
fm_X%matrix_struct%para_env, fm_X%matrix_struct%context, &
nao_full, nao_trunc)
CALL cp_fm_create(fm_mo_coeff, matrix_struct=fm_struct_mo_coeff, &
name="mo_coeff")
! Here, we take care of possible cutoffs
! Simply truncating the matrix causes problems with the print function
! Therefore, we keep the dimension, but set the coefficients of truncated indices to 0
CALL cp_fm_to_fm_submat_general(mo_coeff(1), fm_mo_coeff, &
nao_full, nao_trunc, &
1, homo_irred - homo + 1, &
1, 1, &
mo_coeff(1)%matrix_struct%context)
! Print some information about the NTOs
IF (unit_nr > 0) THEN
WRITE (unit_nr, '(T2,A4,T7,A)') 'BSE|', &
'The Natural Transition Orbital (NTO) pairs φ_I(r_e) and χ_I(r_h) for a fixed'
WRITE (unit_nr, '(T2,A4,T7,A)') 'BSE|', &
'excitation index n are obtained by singular value decomposition of T'
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T15,A)') 'BSE|', &
' = (0 X)'
IF (PRESENT(fm_Y)) THEN
WRITE (unit_nr, '(T2,A4,T15,A)') 'BSE|', &
'T = (Y^T 0)'
ELSE
WRITE (unit_nr, '(T2,A4,T15,A)') 'BSE|', &
'T = (0 0)'
END IF
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T15,A)') 'BSE|', &
'T = U Λ V^T'
WRITE (unit_nr, '(T2,A4,T15,A)') 'BSE|', &
'φ_I(r_e) = \sum_p V_pI ψ_p(r_e)'
WRITE (unit_nr, '(T2,A4,T15,A)') 'BSE|', &
'χ_I(r_h) = \sum_p U_pI ψ_p(r_e)'
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A)') 'BSE|', &
'where we have introduced'
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A,T20,A)') &
'BSE|', "ψ_p:", "occupied and virtual molecular orbitals,"
WRITE (unit_nr, '(T2,A4,T7,A,T20,A)') &
'BSE|', "φ_I(r_e):", "NTO state for the electron,"
WRITE (unit_nr, '(T2,A4,T7,A,T20,A)') &
'BSE|', "χ_I(r_h):", "NTO state for the hole,"
WRITE (unit_nr, '(T2,A4,T7,A,T20,A)') &
'BSE|', "Λ:", "diagonal matrix of NTO weights λ_I,"
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A)') 'BSE|', &
"The NTOs are calculated with the following settings:"
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T7,A,T71,I10)') 'BSE|', 'Number of excitations, for which NTOs are computed', &
mp2_env%bse%num_print_exc_ntos
IF (mp2_env%bse%eps_nto_osc_str > 0.0_dp) THEN
WRITE (unit_nr, '(T2,A4,T7,A,T71,F10.3)') 'BSE|', 'Threshold for oscillator strength f^n', &
mp2_env%bse%eps_nto_osc_str
ELSE
WRITE (unit_nr, '(T2,A4,T7,A,T71,A10)') 'BSE|', 'Threshold for oscillator strength f^n', &
ADJUSTL("---")
END IF
WRITE (unit_nr, '(T2,A4,T7,A,T72,F10.3)') 'BSE|', 'Threshold for NTO weights (λ_I)^2', &
mp2_env%bse%eps_nto_eigval
END IF
! Write the header of NTO info table
IF (unit_nr > 0) THEN
WRITE (unit_nr, '(T2,A4)') 'BSE|'
IF (.NOT. PRESENT(fm_Y)) THEN
WRITE (unit_nr, '(T2,A4,T7,A)') 'BSE|', &
'NTOs from solving the BSE within the TDA:'
ELSE
WRITE (unit_nr, '(T2,A4,T7,A)') 'BSE|', &
'NTOs from solving the BSE without the TDA:'
END IF
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T8,A12,T22,A8,T33,A14,T62,A)') 'BSE|', &
'Excitation n', "TDA/ABBA", "Index of NTO I", 'NTO weights (λ_I)^2'
END IF
DO j = 1, mp2_env%bse%num_print_exc_ntos
n_exc = mp2_env%bse%bse_nto_state_list_final(j)
! Takes care of unallocated oscill_str array in case of Triplet
IF (mp2_env%bse%eps_nto_osc_str > 0.0_dp) THEN
! Check actual values
IF (oscill_str(n_exc) < mp2_env%bse%eps_nto_osc_str) THEN
! Print skipped levels to table
IF (unit_nr > 0) THEN
WRITE (unit_nr, '(T2,A4)') 'BSE|'
WRITE (unit_nr, '(T2,A4,T8,I12,T24,A6,T42,A39)') 'BSE|', &
n_exc, info_approximation, "Skipped (Oscillator strength too small)"
END IF
CYCLE
END IF
END IF
CALL cp_fm_create(fm_m, matrix_struct=fm_struct_m, &
name="single_part_trans_dm")
CALL cp_fm_set_all(fm_m, 0.0_dp)
CALL cp_fm_create(fm_nto_coeff_holes, matrix_struct=fm_struct_nto_holes, &
name="nto_coeffs_holes")
CALL cp_fm_set_all(fm_nto_coeff_holes, 0.0_dp)
CALL cp_fm_create(fm_nto_coeff_particles, matrix_struct=fm_struct_nto_particles, &
name="nto_coeffs_particles")
CALL cp_fm_set_all(fm_nto_coeff_particles, 0.0_dp)
! Reshuffle from X_ia,n_exc to X_i,a
CALL reshuffle_eigvec(fm_X, fm_X_ia, homo, virtual, n_exc, &
.FALSE., unit_nr, mp2_env)
! Copy X to upper block in M, i.e. starting from column homo+1
CALL cp_fm_to_fm_submat(fm_X_ia, fm_m, &
homo, virtual, &
1, 1, &
1, homo + 1)
CALL cp_fm_release(fm_X_ia)
! Copy Y if present
IF (PRESENT(fm_Y)) THEN
! Reshuffle from Y_ia,n_exc to Y_a,i
CALL reshuffle_eigvec(fm_Y, fm_Y_ai, homo, virtual, n_exc, &
.TRUE., unit_nr, mp2_env)
! Copy Y^T to lower block in M, i.e. starting from row homo+1
CALL cp_fm_to_fm_submat(fm_Y_ai, fm_m, &
virtual, homo, &
1, 1, &
homo + 1, 1)
CALL cp_fm_release(fm_Y_ai)
END IF
! Now we compute the SVD of M_{occ+virt,occ+virt}, which yields
! M = U * Lambda * V^T
! Initialize matrices and arrays to store left/right eigenvectors and singular values
CALL cp_fm_create(matrix=fm_eigvl, &
matrix_struct=fm_m%matrix_struct, &
name="LEFT_SINGULAR_MATRIX")
CALL cp_fm_set_all(fm_eigvl, alpha=0.0_dp)
CALL cp_fm_create(matrix=fm_eigvr_t, &
matrix_struct=fm_m%matrix_struct, &
name="RIGHT_SINGULAR_MATRIX")
CALL cp_fm_set_all(fm_eigvr_t, alpha=0.0_dp)
ALLOCATE (eigval_svd(nao_trunc))
eigval_svd(:) = 0.0_dp
info_svd = 0
CALL cp_fm_svd(fm_m, fm_eigvl, fm_eigvr_t, eigval_svd, info_svd)
IF (info_svd /= 0) THEN
IF (unit_nr > 0) THEN
CALL cp_warn(__LOCATION__, &
"SVD for computation of NTOs not successful. "// &
"Skipping print of NTOs.")
IF (info_svd > 0) THEN
CALL cp_warn(__LOCATION__, &
"PDGESVD detected heterogeneity. "// &
"Decreasing number of MPI ranks might solve this issue.")
END IF
END IF
! Release matrices to avoid memory leaks
CALL cp_fm_release(fm_m)
CALL cp_fm_release(fm_nto_coeff_holes)
CALL cp_fm_release(fm_nto_coeff_particles)
ELSE
! Rescale singular values as done in Martin2003 (10.1063/1.1558471)
ALLOCATE (eigval_svd_squ(nao_trunc))
eigval_svd_squ(:) = eigval_svd(:)**2
! Sanity check for TDA: In case of TDA, the sum should be \sum_ia |X_ia|^2 = 1
IF (.NOT. PRESENT(fm_Y)) THEN
IF (ABS(SUM(eigval_svd_squ) - 1) >= 1e-5) THEN
CPWARN("Sum of NTO coefficients deviates from 1!")
END IF
END IF
! Create NTO coefficients for later print to grid via TDDFT routine
! Apply U = fm_eigvl to MO coeffs, which yields hole states
CALL parallel_gemm("N", "N", nao_full, nao_trunc, nao_trunc, 1.0_dp, fm_mo_coeff, fm_eigvl, 0.0_dp, &
fm_nto_coeff_holes)
! Apply V^T = fm_eigvr_t to MO coeffs, which yields particle states
CALL parallel_gemm("N", "T", nao_full, nao_trunc, nao_trunc, 1.0_dp, fm_mo_coeff, fm_eigvr_t, 0.0_dp, &
fm_nto_coeff_particles)
!Release intermediary work matrices
CALL cp_fm_release(fm_m)
CALL cp_fm_release(fm_eigvl)
CALL cp_fm_release(fm_eigvr_t)
! Transfer NTO coefficients to sets
nto_name(1) = 'Hole_coord'
nto_name(2) = 'Particle_coord'
ALLOCATE (nto_set(2))
! Extract number of significant NTOs
n_nto = 0
DO i_nto = 1, nao_trunc
IF (eigval_svd_squ(i_nto) > mp2_env%bse%eps_nto_eigval) THEN
n_nto = n_nto + 1
ELSE
! Since svd orders in descending order, we can exit the loop if smaller
EXIT
END IF
END DO
IF (unit_nr > 0) THEN
WRITE (unit_nr, '(T2,A4)') 'BSE|'
DO i_nto = 1, n_nto
WRITE (unit_nr, '(T2,A4,T8,I12,T24,A6,T41,I6,T71,F10.5)') 'BSE|', &
n_exc, info_approximation, i_nto, eigval_svd_squ(i_nto)
END DO
END IF
CALL cp_fm_struct_create(fm_struct_nto_set, template_fmstruct=fm_struct_nto_holes, &
ncol_global=n_nto)
CALL cp_fm_create(fm_nto_set, fm_struct_nto_set)
DO i = 1, 2
CALL allocate_mo_set(nto_set(i), nao_trunc, n_nto, 0, 0.0_dp, 2.0_dp, 0.0_dp)
CALL init_mo_set(nto_set(i), fm_ref=fm_nto_set, name=nto_name(i))
END DO
CALL cp_fm_release(fm_nto_set)
CALL cp_fm_struct_release(fm_struct_nto_set)
! Fill NTO sets
CALL cp_fm_to_fm(fm_nto_coeff_holes, nto_set(1)%mo_coeff, ncol=n_nto)
CALL cp_fm_to_fm(fm_nto_coeff_particles, nto_set(2)%mo_coeff, ncol=n_nto)
! Cube files
nto_section => section_vals_get_subs_vals(bse_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_bse_nto_cubes(qs_env, nto_set, n_exc, info_approximation, &
stride, append_cube, nto_section)
END IF
CALL cp_fm_release(fm_nto_coeff_holes)
CALL cp_fm_release(fm_nto_coeff_particles)
DEALLOCATE (eigval_svd)
DEALLOCATE (eigval_svd_squ)
DO i = 1, 2
CALL deallocate_mo_set(nto_set(i))
END DO
DEALLOCATE (nto_set)
END IF
END DO
CALL cp_fm_release(fm_mo_coeff)
CALL cp_fm_struct_release(fm_struct_m)
CALL cp_fm_struct_release(fm_struct_nto_holes)
CALL cp_fm_struct_release(fm_struct_nto_particles)
CALL cp_fm_struct_release(fm_struct_mo_coeff)
CALL timestop(handle)
END SUBROUTINE calculate_NTOs
! **************************************************************************************************
!> \brief ...
!> \param exc_descr Allocated and initialized on exit
!> \param fm_X_ia ...
!> \param fm_multipole_ij_trunc ...
!> \param fm_multipole_ab_trunc ...
!> \param fm_multipole_ai_trunc ...
!> \param i_exc ...
!> \param homo ...
!> \param virtual ...
!> \param fm_Y_ia ...
! **************************************************************************************************
SUBROUTINE get_exciton_descriptors(exc_descr, fm_X_ia, &
fm_multipole_ij_trunc, fm_multipole_ab_trunc, &
fm_multipole_ai_trunc, &
i_exc, homo, virtual, &
fm_Y_ia)
TYPE(exciton_descr_type), ALLOCATABLE, &
DIMENSION(:) :: exc_descr
TYPE(cp_fm_type), INTENT(IN) :: fm_X_ia
TYPE(cp_fm_type), ALLOCATABLE, DIMENSION(:), &
INTENT(IN) :: fm_multipole_ij_trunc, &
fm_multipole_ab_trunc, &
fm_multipole_ai_trunc
INTEGER, INTENT(IN) :: i_exc, homo, virtual
TYPE(cp_fm_type), INTENT(IN), OPTIONAL :: fm_Y_ia
CHARACTER(LEN=*), PARAMETER :: routineN = 'get_exciton_descriptors'
INTEGER :: handle, i_dir
INTEGER, DIMENSION(3) :: mask_quadrupole
LOGICAL :: flag_TDA
REAL(KIND=dp) :: norm_X, norm_XpY, norm_Y
REAL(KIND=dp), DIMENSION(3) :: r_e_h_XX, r_e_h_XY, r_e_h_YX, r_e_h_YY, &
r_e_sq_X, r_e_sq_Y, r_e_X, r_e_Y, &
r_h_sq_X, r_h_sq_Y, r_h_X, r_h_Y
TYPE(cp_fm_struct_type), POINTER :: fm_struct_ab, fm_struct_ia
TYPE(cp_fm_type) :: fm_work_ba, fm_work_ia, fm_work_ia_2
CALL timeset(routineN, handle)
IF (PRESENT(fm_Y_ia)) THEN
flag_TDA = .FALSE.
ELSE
flag_TDA = .TRUE.
END IF
! translates 1,2,3 to diagonal entries of quadrupoles xx, yy, zz
! Ordering in quadrupole moments is x, y, z, xx, xy, xz, yy, yz, zz
mask_quadrupole = (/4, 7, 9/)
CALL cp_fm_struct_create(fm_struct_ia, &
context=fm_X_ia%matrix_struct%context, nrow_global=homo, ncol_global=virtual)
CALL cp_fm_struct_create(fm_struct_ab, &
context=fm_X_ia%matrix_struct%context, nrow_global=virtual, ncol_global=virtual)
r_e_X(:) = 0.0_dp
r_e_Y(:) = 0.0_dp
r_h_X(:) = 0.0_dp
r_h_Y(:) = 0.0_dp
r_e_sq_X(:) = 0.0_dp
r_h_sq_X(:) = 0.0_dp
r_e_sq_Y(:) = 0.0_dp
r_h_sq_Y(:) = 0.0_dp
r_e_h_XX(:) = 0.0_dp
r_e_h_XY(:) = 0.0_dp
r_e_h_YX(:) = 0.0_dp
r_e_h_YY(:) = 0.0_dp
norm_X = 0.0_dp
norm_Y = 0.0_dp
norm_XpY = 0.0_dp
! Initialize values of exciton descriptors
exc_descr(i_exc)%r_e(:) = 0.0_dp
exc_descr(i_exc)%r_h(:) = 0.0_dp
exc_descr(i_exc)%r_e_sq(:) = 0.0_dp
exc_descr(i_exc)%r_h_sq(:) = 0.0_dp
exc_descr(i_exc)%r_e_h(:) = 0.0_dp
exc_descr(i_exc)%flag_TDA = flag_TDA
exc_descr(i_exc)%norm_XpY = 0.0_dp
! Norm of X
CALL cp_fm_trace(fm_X_ia, fm_X_ia, norm_X)
norm_XpY = norm_X
! Norm of Y
IF (.NOT. flag_TDA) THEN
CALL cp_fm_trace(fm_Y_ia, fm_Y_ia, norm_Y)
norm_XpY = norm_XpY + norm_Y
END IF
exc_descr(i_exc)%norm_XpY = norm_XpY
! <r_h>_X = Tr[ X^T µ_ij X + Y µ_ab Y^T ] = X_ai µ_ij X_ja + Y_ia µ_ab Y_bi
DO i_dir = 1, 3
! <r_h>_X = X_ai µ_ij X_ja + ...
CALL trace_exciton_descr(fm_X_ia, fm_multipole_ij_trunc(i_dir), fm_X_ia, r_h_X(i_dir))
r_h_X(i_dir) = r_h_X(i_dir)/norm_XpY
IF (.NOT. flag_TDA) THEN
! <r_h>_X = ... + Y_ia µ_ab Y_bi
CALL trace_exciton_descr(fm_Y_ia, fm_Y_ia, fm_multipole_ab_trunc(i_dir), r_h_Y(i_dir))
r_h_Y(i_dir) = r_h_Y(i_dir)/norm_XpY
END IF
END DO
exc_descr(i_exc)%r_h(:) = r_h_X(:) + r_h_Y(:)
! <r_e>_X = Tr[ X µ_ab X^T + Y^T µ_ij Y ] = X_ia µ_ab X_bi + Y_ai µ_ij Y_ja
DO i_dir = 1, 3
! <r_e>_X = work_ib X_bi + ... = X_ib^T work_ib + ...
CALL trace_exciton_descr(fm_X_ia, fm_X_ia, fm_multipole_ab_trunc(i_dir), r_e_X(i_dir))
r_e_X(i_dir) = r_e_X(i_dir)/norm_XpY
IF (.NOT. flag_TDA) THEN
! <r_e>_X = ... + Y_ai µ_ij Y_ja
CALL trace_exciton_descr(fm_Y_ia, fm_multipole_ij_trunc(i_dir), fm_Y_ia, r_e_Y(i_dir))
r_e_Y(i_dir) = r_e_Y(i_dir)/norm_XpY
END IF
END DO
exc_descr(i_exc)%r_e(:) = r_e_X(:) + r_e_Y(:)
! <r_h^2>_X = Tr[ X^T M_ij X + Y M_ab Y^T ] = X_ai M_ij X_ja + Y_ia M_ab Y_bi
DO i_dir = 1, 3
! <r_h^2>_X = X_ai M_ij X_ja + ...
CALL trace_exciton_descr(fm_X_ia, fm_multipole_ij_trunc(mask_quadrupole(i_dir)), &
fm_X_ia, r_h_sq_X(i_dir))
r_h_sq_X(i_dir) = r_h_sq_X(i_dir)/norm_XpY
IF (.NOT. flag_TDA) THEN
! <r_h^2>_X = ... + Y_ia M_ab Y_bi
CALL trace_exciton_descr(fm_Y_ia, fm_Y_ia, &
fm_multipole_ab_trunc(mask_quadrupole(i_dir)), r_h_sq_Y(i_dir))
r_h_sq_Y(i_dir) = r_h_sq_Y(i_dir)/norm_XpY
END IF
END DO
exc_descr(i_exc)%r_h_sq(:) = r_h_sq_X(:) + r_h_sq_Y(:)
! <r_e^2>_X = Tr[ X M_ab X^T + Y^T M_ij Y ] = X_ia M_ab X_bi + Y_ai M_ij Y_ja
DO i_dir = 1, 3
! <r_e^2>_X = work_ib X_bi + ... = X_ib^T work_ib + ...
CALL trace_exciton_descr(fm_X_ia, fm_X_ia, &
fm_multipole_ab_trunc(mask_quadrupole(i_dir)), r_e_sq_X(i_dir))
r_e_sq_X(i_dir) = r_e_sq_X(i_dir)/norm_XpY
IF (.NOT. flag_TDA) THEN
! <r_e^2>_X = ... + Y_ai M_ij Y_ja
CALL trace_exciton_descr(fm_Y_ia, fm_multipole_ij_trunc(mask_quadrupole(i_dir)), &
fm_Y_ia, r_e_sq_Y(i_dir))
r_e_sq_Y(i_dir) = r_e_sq_Y(i_dir)/norm_XpY
END IF
END DO
exc_descr(i_exc)%r_e_sq(:) = r_e_sq_X(:) + r_e_sq_Y(:)
! <r_h r_e>_X
! = Tr[ X^T µ_ij X µ_ab + Y^T µ_ij Y µ_ab + X µ_ai Y µ_ai + Y µ_ai X µ_ai]
! = X_bj µ_ji X_ia µ_ab + Y_bj µ_ji Y_ia µ_ab + X_ia µ_aj Y_jb µ_bi + Y_ia µ_aj X_jb µ_bi
CALL cp_fm_create(fm_work_ia, fm_struct_ia)
CALL cp_fm_create(fm_work_ia_2, fm_struct_ia)
CALL cp_fm_create(fm_work_ba, fm_struct_ab)
DO i_dir = 1, 3
! First term - X^T µ_ij X µ_ab
CALL cp_fm_set_all(fm_work_ia, 0.0_dp)
CALL cp_fm_set_all(fm_work_ia_2, 0.0_dp)
! work_ib = X_ia µ_ab
CALL parallel_gemm("N", "N", homo, virtual, virtual, 1.0_dp, &
fm_X_ia, fm_multipole_ab_trunc(i_dir), 0.0_dp, fm_work_ia)
! work_ja_2 = µ_ji work_ia
CALL parallel_gemm("N", "N", homo, virtual, homo, 1.0_dp, &
fm_multipole_ij_trunc(i_dir), fm_work_ia, 0.0_dp, fm_work_ia_2)
! <r_h r_e>_X = work_ia_2 X_bj + ... = X^T work_ia_2 + ...
CALL cp_fm_trace(fm_X_ia, fm_work_ia_2, r_e_h_XX(i_dir))
r_e_h_XX(i_dir) = r_e_h_XX(i_dir)/norm_XpY
IF (.NOT. flag_TDA) THEN
! Second term - Y^T µ_ij Y µ_ab
CALL cp_fm_set_all(fm_work_ia, 0.0_dp)
CALL cp_fm_set_all(fm_work_ia_2, 0.0_dp)
! work_ib = Y_ia µ_ab
CALL parallel_gemm("N", "N", homo, virtual, virtual, 1.0_dp, &
fm_Y_ia, fm_multipole_ab_trunc(i_dir), 0.0_dp, fm_work_ia)
! work_ja_2 = µ_ji work_ia
CALL parallel_gemm("N", "N", homo, virtual, homo, 1.0_dp, &
fm_multipole_ij_trunc(i_dir), fm_work_ia, 0.0_dp, fm_work_ia_2)
! <r_h r_e>_X = work_ia_2 Y_bj + ... = Y^T work_ia_2 + ...
CALL cp_fm_trace(fm_Y_ia, fm_work_ia_2, r_e_h_YY(i_dir))
r_e_h_YY(i_dir) = r_e_h_YY(i_dir)/norm_XpY
! Third term - X µ_ai Y µ_ai = X_ia µ_aj Y_jb µ_bi
! Reshuffle for usage of trace (where first argument is transposed)
! = µ_aj Y_jb µ_bi X_ia =
! \__________/
! fm_work_ai
! fm_work_ai = µ_aj Y_jb µ_bi
! fm_work_ia = µ_ib Y_bj µ_ja
! \_____/
! fm_work_ba
CALL cp_fm_set_all(fm_work_ba, 0.0_dp)
CALL cp_fm_set_all(fm_work_ia, 0.0_dp)
! fm_work_ba = Y_bj µ_ja
CALL parallel_gemm("T", "T", virtual, virtual, homo, 1.0_dp, &
fm_Y_ia, fm_multipole_ai_trunc(i_dir), 0.0_dp, fm_work_ba)
! fm_work_ia = µ_ib fm_work_ba
CALL parallel_gemm("T", "N", homo, virtual, virtual, 1.0_dp, &
fm_multipole_ai_trunc(i_dir), fm_work_ba, 0.0_dp, fm_work_ia)
! <r_h r_e>_X = ... + X_ia µ_aj Y_jb µ_bi
CALL cp_fm_trace(fm_work_ia, fm_X_ia, r_e_h_XY(i_dir))
r_e_h_XY(i_dir) = r_e_h_XY(i_dir)/norm_XpY
! Fourth term - Y µ_ai X µ_ai = Y_ia µ_aj X_jb µ_bi
! Reshuffle for usage of trace (where first argument is transposed)
! = µ_aj X_jb µ_bi Y_ia =
! \__________/
! fm_work_ai
! fm_work_ai = µ_aj X_jb µ_bi
! fm_work_ia = µ_ib X_bj µ_ja
! \_____/
! fm_work_ba
CALL cp_fm_set_all(fm_work_ba, 0.0_dp)
CALL cp_fm_set_all(fm_work_ia, 0.0_dp)
! fm_work_ba = Y_bj µ_ja
CALL parallel_gemm("T", "T", virtual, virtual, homo, 1.0_dp, &
fm_X_ia, fm_multipole_ai_trunc(i_dir), 0.0_dp, fm_work_ba)
! fm_work_ia = µ_ib fm_work_ba
CALL parallel_gemm("T", "N", homo, virtual, virtual, 1.0_dp, &
fm_multipole_ai_trunc(i_dir), fm_work_ba, 0.0_dp, fm_work_ia)
! <r_h r_e>_X = ... + X_ia µ_aj Y_jb µ_bi
CALL cp_fm_trace(fm_work_ia, fm_Y_ia, r_e_h_YX(i_dir))
r_e_h_YX(i_dir) = r_e_h_YX(i_dir)/norm_XpY
END IF
END DO
exc_descr(i_exc)%r_e_h(:) = r_e_h_XX(:) + r_e_h_XY(:) + r_e_h_YX(:) + r_e_h_YY(:)
CALL cp_fm_release(fm_work_ia)
CALL cp_fm_release(fm_work_ia_2)
CALL cp_fm_release(fm_work_ba)
! diff_r_abs = |<r_h>_X - <r_e>_X|
exc_descr(i_exc)%diff_r_abs = SQRT(SUM((exc_descr(i_exc)%r_h(:) - exc_descr(i_exc)%r_e(:))**2))
! σ_e = sqrt( <r_e^2>_X - <r_e>_X^2 )
exc_descr(i_exc)%sigma_e = SQRT(SUM(exc_descr(i_exc)%r_e_sq(:)) - SUM(exc_descr(i_exc)%r_e(:)**2))
! σ_h = sqrt( <r_h^2>_X - <r_h>_X^2 )
exc_descr(i_exc)%sigma_h = SQRT(SUM(exc_descr(i_exc)%r_h_sq(:)) - SUM(exc_descr(i_exc)%r_h(:)**2))
! COV(r_e, r_h) = < r_e r_h >_X - < r_e >_X < r_h >_X
exc_descr(i_exc)%cov_e_h_sum = 0.0_dp
exc_descr(i_exc)%cov_e_h(:) = 0.0_dp
DO i_dir = 1, 3
exc_descr(i_exc)%cov_e_h(i_dir) = exc_descr(i_exc)%r_e_h(i_dir) &
- exc_descr(i_exc)%r_e(i_dir)*exc_descr(i_exc)%r_h(i_dir)
exc_descr(i_exc)%cov_e_h_sum = exc_descr(i_exc)%cov_e_h_sum + &
exc_descr(i_exc)%r_e_h(i_dir) - exc_descr(i_exc)%r_e(i_dir)*exc_descr(i_exc)%r_h(i_dir)
END DO
! root-mean-square e-h separation
exc_descr(i_exc)%diff_r_sqr = SQRT(exc_descr(i_exc)%diff_r_abs**2 + &
exc_descr(i_exc)%sigma_e**2 + exc_descr(i_exc)%sigma_h**2 &
- 2*exc_descr(i_exc)%cov_e_h_sum)
! e-h-correlation coefficient R_eh = COV(r_e, r_h) / ( σ_e σ_h )
exc_descr(i_exc)%corr_e_h = exc_descr(i_exc)%cov_e_h_sum/(exc_descr(i_exc)%sigma_e*exc_descr(i_exc)%sigma_h)
! Expectation values of r_e and r_h
exc_descr(i_exc)%r_e_shift(:) = exc_descr(i_exc)%r_e(:)
exc_descr(i_exc)%r_h_shift(:) = exc_descr(i_exc)%r_h(:)
CALL cp_fm_struct_release(fm_struct_ia)