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atom_utils.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 Some basic routines for atomic calculations
!> \author jgh
!> \date 01.04.2008
!> \version 1.0
!>
! **************************************************************************************************
MODULE atom_utils
USE ai_onecenter, ONLY: sg_overlap,&
sto_overlap
USE ai_overlap, ONLY: overlap_ab_s,&
overlap_ab_sp
USE ao_util, ONLY: exp_radius
USE atom_types, ONLY: &
CGTO_BASIS, GTO_BASIS, NUM_BASIS, STO_BASIS, atom_basis_type, atom_gthpot_type, &
atom_hfx_type, atom_potential_type, atom_state, atom_type, ecp_pseudo, eri, gth_pseudo, &
lmat, no_pseudo, sgp_pseudo, upf_pseudo
USE basis_set_types, ONLY: srules
USE cp_files, ONLY: close_file,&
get_unit_number,&
open_file
USE input_constants, ONLY: do_rhf_atom,&
do_rks_atom,&
do_rohf_atom,&
do_uhf_atom,&
do_uks_atom
USE kahan_sum, ONLY: accurate_dot_product
USE kinds, ONLY: default_string_length,&
dp
USE mathconstants, ONLY: dfac,&
fac,&
fourpi,&
maxfac,&
pi,&
rootpi
USE mathlib, ONLY: binomial_gen,&
invmat_symm
USE orbital_pointers, ONLY: deallocate_orbital_pointers,&
init_orbital_pointers
USE orbital_transformation_matrices, ONLY: deallocate_spherical_harmonics,&
init_spherical_harmonics
USE periodic_table, ONLY: nelem,&
ptable
USE physcon, ONLY: bohr
USE qs_grid_atom, ONLY: grid_atom_type
USE splines, ONLY: spline3ders
USE string_utilities, ONLY: uppercase
#include "./base/base_uses.f90"
IMPLICIT NONE
PRIVATE
CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'atom_utils'
PUBLIC :: atom_condnumber, atom_completeness, atom_basis_condnum
PUBLIC :: atom_set_occupation, get_maxl_occ, get_maxn_occ
PUBLIC :: atom_denmat, atom_density, atom_core_density
PUBLIC :: integrate_grid, atom_trace, atom_solve
PUBLIC :: coulomb_potential_numeric, coulomb_potential_analytic
PUBLIC :: exchange_numeric, exchange_semi_analytic
PUBLIC :: numpot_matrix, ceri_contract, eeri_contract, err_matrix
PUBLIC :: slater_density, wigner_slater_functional, atom_local_potential
PUBLIC :: atom_orbital_charge, atom_orbital_nodes, atom_consistent_method
PUBLIC :: atom_orbital_max, atom_wfnr0, get_rho0
PUBLIC :: contract2, contract4, contract2add
! ZMP added public subroutines
PUBLIC :: atom_read_external_density
PUBLIC :: atom_read_external_vxc
PUBLIC :: atom_read_zmp_restart
PUBLIC :: atom_write_zmp_restart
!-----------------------------------------------------------------------------!
INTERFACE integrate_grid
MODULE PROCEDURE integrate_grid_function1, &
integrate_grid_function2, &
integrate_grid_function3
END INTERFACE
! **************************************************************************************************
CONTAINS
! **************************************************************************************************
!> \brief Set occupation of atomic orbitals.
!> \param ostring list of electronic states
!> \param occupation ...
!> \param wfnocc ...
!> \param multiplicity ...
!> \par History
!> * 11.2009 unrestricted KS and HF methods [Juerg Hutter]
!> * 08.2008 created [Juerg Hutter]
! **************************************************************************************************
SUBROUTINE atom_set_occupation(ostring, occupation, wfnocc, multiplicity)
CHARACTER(LEN=default_string_length), &
DIMENSION(:), POINTER :: ostring
REAL(Kind=dp), DIMENSION(0:lmat, 10) :: occupation, wfnocc
INTEGER, INTENT(OUT), OPTIONAL :: multiplicity
CHARACTER(len=2) :: elem
CHARACTER(LEN=default_string_length) :: pstring
INTEGER :: i, i1, i2, ielem, is, jd, jf, jp, js, k, &
l, mult, n, no
REAL(Kind=dp) :: e0, el, oo
occupation = 0._dp
CPASSERT(ASSOCIATED(ostring))
CPASSERT(SIZE(ostring) > 0)
no = SIZE(ostring)
is = 1
! look for multiplicity
mult = -1 !not specified
IF (is <= no) THEN
IF (INDEX(ostring(is), "(") /= 0) THEN
i1 = INDEX(ostring(is), "(")
i2 = INDEX(ostring(is), ")")
CPASSERT((i2 - i1 - 1 > 0) .AND. (i2 - i1 - 1 < 3))
elem = ostring(is) (i1 + 1:i2 - 1)
IF (INDEX(elem, "HS") /= 0) THEN
mult = -2 !High spin
ELSE IF (INDEX(elem, "LS") /= 0) THEN
mult = -3 !Low spin
ELSE
READ (elem, *) mult
END IF
is = is + 1
END IF
END IF
IF (is <= no) THEN
IF (INDEX(ostring(is), "CORE") /= 0) is = is + 1 !Pseudopotential detected
END IF
IF (is <= no) THEN
IF (INDEX(ostring(is), "none") /= 0) is = is + 1 !no electrons, used with CORE
END IF
IF (is <= no) THEN
IF (INDEX(ostring(is), "[") /= 0) THEN
! core occupation from element [XX]
i1 = INDEX(ostring(is), "[")
i2 = INDEX(ostring(is), "]")
CPASSERT((i2 - i1 - 1 > 0) .AND. (i2 - i1 - 1 < 3))
elem = ostring(is) (i1 + 1:i2 - 1)
ielem = 0
DO k = 1, nelem
IF (elem == ptable(k)%symbol) THEN
ielem = k
EXIT
END IF
END DO
CPASSERT(ielem /= 0)
DO l = 0, MIN(lmat, UBOUND(ptable(ielem)%e_conv, 1))
el = 2._dp*(2._dp*REAL(l, dp) + 1._dp)
e0 = ptable(ielem)%e_conv(l)
DO k = 1, 10
occupation(l, k) = MIN(el, e0)
e0 = e0 - el
IF (e0 <= 0._dp) EXIT
END DO
END DO
is = is + 1
END IF
END IF
DO i = is, no
pstring = ostring(i)
CALL uppercase(pstring)
js = INDEX(pstring, "S")
jp = INDEX(pstring, "P")
jd = INDEX(pstring, "D")
jf = INDEX(pstring, "F")
CPASSERT(js + jp + jd + jf > 0)
IF (js > 0) THEN
CPASSERT(jp + jd + jf == 0)
READ (pstring(1:js - 1), *) n
READ (pstring(js + 1:), *) oo
CPASSERT(n > 0)
CPASSERT(oo >= 0._dp)
CPASSERT(occupation(0, n) == 0)
occupation(0, n) = oo
END IF
IF (jp > 0) THEN
CPASSERT(js + jd + jf == 0)
READ (pstring(1:jp - 1), *) n
READ (pstring(jp + 1:), *) oo
CPASSERT(n > 1)
CPASSERT(oo >= 0._dp)
CPASSERT(occupation(1, n - 1) == 0)
occupation(1, n - 1) = oo
END IF
IF (jd > 0) THEN
CPASSERT(js + jp + jf == 0)
READ (pstring(1:jd - 1), *) n
READ (pstring(jd + 1:), *) oo
CPASSERT(n > 2)
CPASSERT(oo >= 0._dp)
CPASSERT(occupation(2, n - 2) == 0)
occupation(2, n - 2) = oo
END IF
IF (jf > 0) THEN
CPASSERT(js + jp + jd == 0)
READ (pstring(1:jf - 1), *) n
READ (pstring(jf + 1:), *) oo
CPASSERT(n > 3)
CPASSERT(oo >= 0._dp)
CPASSERT(occupation(3, n - 3) == 0)
occupation(3, n - 3) = oo
END IF
END DO
wfnocc = 0._dp
DO l = 0, lmat
k = 0
DO i = 1, 10
IF (occupation(l, i) /= 0._dp) THEN
k = k + 1
wfnocc(l, k) = occupation(l, i)
END IF
END DO
END DO
!Check for consistency with multiplicity
IF (mult /= -1) THEN
! count open shells
js = 0
DO l = 0, lmat
k = 2*(2*l + 1)
DO i = 1, 10
IF (wfnocc(l, i) /= 0._dp .AND. wfnocc(l, i) /= REAL(k, dp)) THEN
js = js + 1
i1 = l
i2 = i
END IF
END DO
END DO
IF (js == 0 .AND. mult == -2) mult = 1
IF (js == 0 .AND. mult == -3) mult = 1
IF (js == 0) THEN
CPASSERT(mult == 1)
END IF
IF (js == 1) THEN
l = i1
i = i2
k = NINT(wfnocc(l, i))
IF (k > (2*l + 1)) k = 2*(2*l + 1) - k
IF (mult == -2) mult = k + 1
IF (mult == -3) mult = MOD(k, 2) + 1
CPASSERT(MOD(k + 1 - mult, 2) == 0)
END IF
IF (js > 1 .AND. mult /= -2) THEN
CPASSERT(mult == -2)
END IF
END IF
IF (PRESENT(multiplicity)) multiplicity = mult
END SUBROUTINE atom_set_occupation
! **************************************************************************************************
!> \brief Return the maximum orbital quantum number of occupied orbitals.
!> \param occupation ...
!> \return ...
!> \par History
!> * 08.2008 created [Juerg Hutter]
! **************************************************************************************************
PURE FUNCTION get_maxl_occ(occupation) RESULT(maxl)
REAL(Kind=dp), DIMENSION(0:lmat, 10), INTENT(IN) :: occupation
INTEGER :: maxl
INTEGER :: l
maxl = 0
DO l = 0, lmat
IF (SUM(occupation(l, :)) /= 0._dp) maxl = l
END DO
END FUNCTION get_maxl_occ
! **************************************************************************************************
!> \brief Return the maximum principal quantum number of occupied orbitals.
!> \param occupation ...
!> \return ...
!> \par History
!> * 08.2008 created [Juerg Hutter]
! **************************************************************************************************
PURE FUNCTION get_maxn_occ(occupation) RESULT(maxn)
REAL(Kind=dp), DIMENSION(0:lmat, 10), INTENT(IN) :: occupation
INTEGER, DIMENSION(0:lmat) :: maxn
INTEGER :: k, l
maxn = 0
DO l = 0, lmat
DO k = 1, 10
IF (occupation(l, k) /= 0._dp) maxn(l) = maxn(l) + 1
END DO
END DO
END FUNCTION get_maxn_occ
! **************************************************************************************************
!> \brief Calculate a density matrix using atomic orbitals.
!> \param pmat electron density matrix
!> \param wfn atomic wavefunctions
!> \param nbas number of basis functions
!> \param occ occupation numbers
!> \param maxl maximum angular momentum to consider
!> \param maxn maximum principal quantum number for each angular momentum
!> \par History
!> * 08.2008 created [Juerg Hutter]
! **************************************************************************************************
PURE SUBROUTINE atom_denmat(pmat, wfn, nbas, occ, maxl, maxn)
REAL(KIND=dp), DIMENSION(:, :, 0:), INTENT(INOUT) :: pmat
REAL(KIND=dp), DIMENSION(:, :, 0:), INTENT(IN) :: wfn
INTEGER, DIMENSION(0:lmat), INTENT(IN) :: nbas
REAL(KIND=dp), DIMENSION(0:, :), INTENT(IN) :: occ
INTEGER, INTENT(IN) :: maxl
INTEGER, DIMENSION(0:lmat), INTENT(IN) :: maxn
INTEGER :: i, j, k, l, n
pmat = 0._dp
n = SIZE(wfn, 2)
DO l = 0, maxl
DO i = 1, MIN(n, maxn(l))
DO k = 1, nbas(l)
DO j = 1, nbas(l)
pmat(j, k, l) = pmat(j, k, l) + occ(l, i)*wfn(j, i, l)*wfn(k, i, l)
END DO
END DO
END DO
END DO
END SUBROUTINE atom_denmat
! **************************************************************************************************
!> \brief Map the electron density on an atomic radial grid.
!> \param density computed electron density
!> \param pmat electron density matrix
!> \param basis atomic basis set
!> \param maxl maximum angular momentum to consider
!> \param typ type of the matrix to map:
!> RHO -- density matrix;
!> DER -- first derivatives of the electron density;
!> KIN -- kinetic energy density;
!> LAP -- second derivatives of the electron density.
!> \param rr abscissa on the radial grid (required for typ == 'KIN')
!> \par History
!> * 08.2008 created [Juerg Hutter]
! **************************************************************************************************
SUBROUTINE atom_density(density, pmat, basis, maxl, typ, rr)
REAL(KIND=dp), DIMENSION(:), INTENT(OUT) :: density
REAL(KIND=dp), DIMENSION(:, :, 0:), INTENT(IN) :: pmat
TYPE(atom_basis_type), INTENT(IN) :: basis
INTEGER, INTENT(IN) :: maxl
CHARACTER(LEN=*), OPTIONAL :: typ
REAL(KIND=dp), DIMENSION(:), INTENT(IN), OPTIONAL :: rr
CHARACTER(LEN=3) :: my_typ
INTEGER :: i, j, l, n
REAL(KIND=dp) :: ff
my_typ = "RHO"
IF (PRESENT(typ)) my_typ = typ(1:3)
IF (my_typ == "KIN") THEN
CPASSERT(PRESENT(rr))
END IF
density = 0._dp
DO l = 0, maxl
n = basis%nbas(l)
DO i = 1, n
DO j = i, n
ff = pmat(i, j, l)
IF (i /= j) ff = 2._dp*pmat(i, j, l)
IF (my_typ == "RHO") THEN
density(:) = density(:) + ff*basis%bf(:, i, l)*basis%bf(:, j, l)
ELSE IF (my_typ == "DER") THEN
density(:) = density(:) + ff*basis%dbf(:, i, l)*basis%bf(:, j, l) &
+ ff*basis%bf(:, i, l)*basis%dbf(:, j, l)
ELSE IF (my_typ == "KIN") THEN
density(:) = density(:) + 0.5_dp*ff*( &
basis%dbf(:, i, l)*basis%dbf(:, j, l) + &
REAL(l*(l + 1), dp)*basis%bf(:, i, l)*basis%bf(:, j, l)/rr(:))
ELSE IF (my_typ == "LAP") THEN
density(:) = density(:) + ff*basis%ddbf(:, i, l)*basis%bf(:, j, l) &
+ ff*basis%bf(:, i, l)*basis%ddbf(:, j, l) &
+ 2._dp*ff*basis%dbf(:, i, l)*basis%bf(:, j, l)/rr(:) &
+ 2._dp*ff*basis%bf(:, i, l)*basis%dbf(:, j, l)/rr(:)
ELSE
CPABORT("")
END IF
END DO
END DO
END DO
! this factor from the product of two spherical harmonics
density = density/fourpi
END SUBROUTINE atom_density
! **************************************************************************************************
!> \brief ZMP subroutine to write external restart file.
!> \param atom information about the atomic kind
!> \date 07.10.2013
!> \author D. Varsano [[email protected]]
!> \version 1.0
! **************************************************************************************************
SUBROUTINE atom_write_zmp_restart(atom)
TYPE(atom_type), INTENT(IN) :: atom
INTEGER :: extunit, i, k, l, n
extunit = get_unit_number()
CALL open_file(file_name=atom%zmp_restart_file, file_status="UNKNOWN", &
file_form="FORMATTED", file_action="WRITE", &
unit_number=extunit)
n = SIZE(atom%orbitals%wfn, 2)
WRITE (extunit, *) atom%basis%nbas
DO l = 0, atom%state%maxl_occ
DO i = 1, MIN(n, atom%state%maxn_occ(l))
DO k = 1, atom%basis%nbas(l)
WRITE (extunit, *) atom%orbitals%wfn(k, i, l)
END DO
END DO
END DO
CALL close_file(unit_number=extunit)
END SUBROUTINE atom_write_zmp_restart
! **************************************************************************************************
!> \brief ZMP subroutine to read external restart file.
!> \param atom information about the atomic kind
!> \param doguess flag that indicates that the restart file has not been read,
!> so the initial guess is required
!> \param iw output file unit
!> \date 07.10.2013
!> \author D. Varsano [[email protected]]
!> \version 1.0
! **************************************************************************************************
SUBROUTINE atom_read_zmp_restart(atom, doguess, iw)
TYPE(atom_type), INTENT(INOUT) :: atom
LOGICAL, INTENT(INOUT) :: doguess
INTEGER, INTENT(IN) :: iw
INTEGER :: er, extunit, i, k, l, maxl, n
INTEGER, DIMENSION(0:lmat) :: maxn, nbas
INQUIRE (file=atom%zmp_restart_file, exist=atom%doread)
IF (atom%doread) THEN
WRITE (iw, fmt="(' ZMP | Restart file found')")
extunit = get_unit_number()
CALL open_file(file_name=atom%zmp_restart_file, file_status="OLD", &
file_form="FORMATTED", file_action="READ", &
unit_number=extunit)
READ (extunit, *, IOSTAT=er) nbas
IF (er .NE. 0) THEN
WRITE (iw, fmt="(' ZMP | ERROR! Restart file unreadable')")
WRITE (iw, fmt="(' ZMP | ERROR! Starting ZMP calculation form initial atomic guess')")
doguess = .TRUE.
atom%doread = .FALSE.
ELSE IF (nbas(1) .NE. atom%basis%nbas(1)) THEN
WRITE (iw, fmt="(' ZMP | ERROR! Restart file contains a different basis set')")
WRITE (iw, fmt="(' ZMP | ERROR! Starting ZMP calculation form initial atomic guess')")
doguess = .TRUE.
atom%doread = .FALSE.
ELSE
nbas = atom%basis%nbas
maxl = atom%state%maxl_occ
maxn = atom%state%maxn_occ
n = SIZE(atom%orbitals%wfn, 2)
DO l = 0, atom%state%maxl_occ
DO i = 1, MIN(n, atom%state%maxn_occ(l))
DO k = 1, atom%basis%nbas(l)
READ (extunit, *) atom%orbitals%wfn(k, i, l)
END DO
END DO
END DO
doguess = .FALSE.
END IF
CALL close_file(unit_number=extunit)
ELSE
WRITE (iw, fmt="(' ZMP | WARNING! Restart file not found')")
WRITE (iw, fmt="(' ZMP | WARNING! Starting ZMP calculation form initial atomic guess')")
END IF
END SUBROUTINE atom_read_zmp_restart
! **************************************************************************************************
!> \brief ZMP subroutine to read external density from linear grid of density matrix.
!> \param density external density
!> \param atom information about the atomic kind
!> \param iw output file unit
!> \date 07.10.2013
!> \author D. Varsano [[email protected]]
!> \version 1.0
! **************************************************************************************************
SUBROUTINE atom_read_external_density(density, atom, iw)
REAL(KIND=dp), DIMENSION(:), INTENT(OUT) :: density
TYPE(atom_type), INTENT(INOUT) :: atom
INTEGER, INTENT(IN) :: iw
CHARACTER(LEN=default_string_length) :: filename
INTEGER :: extunit, ir, j, k, l, maxl_occ, maxnbas, &
nbas, nr
LOGICAL :: ldm
REAL(KIND=dp) :: rr
REAL(KIND=dp), ALLOCATABLE :: pmatread(:, :, :)
filename = atom%ext_file
ldm = atom%dm
extunit = get_unit_number()
CALL open_file(file_name=filename, file_status="OLD", &
file_form="FORMATTED", file_action="READ", &
unit_number=extunit)
IF (.NOT. ldm) THEN
READ (extunit, *) nr
IF (nr .NE. atom%basis%grid%nr) THEN
IF (iw > 0) WRITE (iw, fmt="(' ZMP | ERROR! External grid dimension ',I4,' differs from internal grid ',I4)") &
nr, atom%basis%grid%nr
IF (iw > 0) WRITE (iw, fmt="(' ZMP | ERROR! Stopping ZMP calculation')")
CPABORT("")
END IF
DO ir = 1, nr
READ (extunit, *) rr, density(ir)
IF (ABS(rr - atom%basis%grid%rad(ir)) .GT. atom%zmpgrid_tol) THEN
IF (iw > 0) WRITE (iw, fmt="(' ZMP | ERROR! Grid points do not coincide: ')")
IF (iw > 0) WRITE (iw, fmt='(" ZMP |",T20,"R_out[bohr]",T36,"R_in[bohr]",T61,"R_diff[bohr]")')
IF (iw > 0) WRITE (iw, fmt='(" ZMP |",T14,E24.15,T39,E24.15,T64,E24.15)') &
rr, atom%basis%grid%rad(ir), ABS(rr - atom%basis%grid%rad(ir))
CPABORT("")
END IF
END DO
CALL close_file(unit_number=extunit)
ELSE
READ (extunit, *) maxl_occ
maxnbas = MAXVAL(atom%basis%nbas)
ALLOCATE (pmatread(maxnbas, maxnbas, 0:maxl_occ))
pmatread = 0.0
DO l = 0, maxl_occ
nbas = atom%basis%nbas(l)
READ (extunit, *) ! Read empty line
DO k = 1, nbas
READ (extunit, *) (pmatread(j, k, l), j=1, k)
DO j = 1, k
pmatread(k, j, l) = pmatread(j, k, l)
END DO
END DO
END DO
CALL close_file(unit_number=extunit)
CALL atom_density(density, pmatread, atom%basis, maxl_occ, typ="RHO")
extunit = get_unit_number()
CALL open_file(file_name="rho_target.dat", file_status="UNKNOWN", &
file_form="FORMATTED", file_action="WRITE", unit_number=extunit)
IF (iw > 0) WRITE (iw, fmt="(' ZMP | Writing target density from density matrix')")
WRITE (extunit, fmt='("# Target density built from density matrix : ",A20)') filename
WRITE (extunit, fmt='("#",T10,"R[bohr]",T36,"Rho[au]")')
nr = atom%basis%grid%nr
DO ir = 1, nr
WRITE (extunit, fmt='(T1,E24.15,T26,E24.15)') &
atom%basis%grid%rad(ir), density(ir)
END DO
DEALLOCATE (pmatread)
CALL close_file(unit_number=extunit)
END IF
END SUBROUTINE atom_read_external_density
! **************************************************************************************************
!> \brief ZMP subroutine to read external v_xc in the atomic code.
!> \param vxc external exchange-correlation potential
!> \param atom information about the atomic kind
!> \param iw output file unit
!> \author D. Varsano [[email protected]]
! **************************************************************************************************
SUBROUTINE atom_read_external_vxc(vxc, atom, iw)
REAL(KIND=dp), DIMENSION(:), INTENT(OUT) :: vxc
TYPE(atom_type), INTENT(INOUT) :: atom
INTEGER, INTENT(IN) :: iw
CHARACTER(LEN=default_string_length) :: adum, filename
INTEGER :: extunit, ir, nr
REAL(KIND=dp) :: rr
filename = atom%ext_vxc_file
extunit = get_unit_number()
CALL open_file(file_name=filename, file_status="OLD", &
file_form="FORMATTED", file_action="READ", &
unit_number=extunit)
READ (extunit, *)
READ (extunit, *) adum, nr
IF (nr .NE. atom%basis%grid%nr) THEN
IF (iw > 0) WRITE (iw, fmt="(' ZMP | ERROR! External grid dimension ',I4,' differs from internal grid ',I4)") &
nr, atom%basis%grid%nr
IF (iw > 0) WRITE (iw, fmt="(' ZMP | ERROR! Stopping ZMP calculation')")
CPABORT("")
END IF
DO ir = 1, nr
READ (extunit, *) rr, vxc(ir)
IF (ABS(rr - atom%basis%grid%rad(ir)) .GT. atom%zmpvxcgrid_tol) THEN
IF (iw > 0) WRITE (iw, fmt="(' ZMP | ERROR! Grid points do not coincide: ')")
IF (iw > 0) WRITE (iw, fmt='(" ZMP |",T20,"R_out[bohr]",T36,"R_in[bohr]",T61,"R_diff[bohr]")')
IF (iw > 0) WRITE (iw, fmt='(" ZMP |",T14,E24.15,T39,E24.15,T64,E24.15)') &
rr, atom%basis%grid%rad(ir), ABS(rr - atom%basis%grid%rad(ir))
CPABORT("")
END IF
END DO
END SUBROUTINE atom_read_external_vxc
! **************************************************************************************************
!> \brief ...
!> \param charge ...
!> \param wfn ...
!> \param rcov ...
!> \param l ...
!> \param basis ...
! **************************************************************************************************
PURE SUBROUTINE atom_orbital_charge(charge, wfn, rcov, l, basis)
REAL(KIND=dp), INTENT(OUT) :: charge
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: wfn
REAL(KIND=dp), INTENT(IN) :: rcov
INTEGER, INTENT(IN) :: l
TYPE(atom_basis_type), INTENT(IN) :: basis
INTEGER :: i, j, m, n
REAL(KIND=dp) :: ff
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: den
charge = 0._dp
m = SIZE(basis%bf, 1)
ALLOCATE (den(m))
n = basis%nbas(l)
den = 0._dp
DO i = 1, n
DO j = 1, n
ff = wfn(i)*wfn(j)
den(1:m) = den(1:m) + ff*basis%bf(1:m, i, l)*basis%bf(1:m, j, l)
END DO
END DO
DO i = 1, m
IF (basis%grid%rad(i) > rcov) den(i) = 0._dp
END DO
charge = SUM(den(1:m)*basis%grid%wr(1:m))
DEALLOCATE (den)
END SUBROUTINE atom_orbital_charge
! **************************************************************************************************
!> \brief ...
!> \param corden ...
!> \param potential ...
!> \param typ ...
!> \param rr ...
!> \par History
!> * 01.2017 rewritten [Juerg Hutter]
!> * 03.2010 extension of GTH pseudo-potential definition [Juerg Hutter]
! **************************************************************************************************
SUBROUTINE atom_core_density(corden, potential, typ, rr)
REAL(KIND=dp), DIMENSION(:), INTENT(INOUT) :: corden
TYPE(atom_potential_type), INTENT(IN) :: potential
CHARACTER(LEN=*), OPTIONAL :: typ
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: rr
CHARACTER(LEN=3) :: my_typ
INTEGER :: i, j, m, n
LOGICAL :: reverse
REAL(KIND=dp) :: a, a2, cval, fb
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: fe, rc, rhoc, rval
my_typ = "RHO"
IF (PRESENT(typ)) my_typ = typ(1:3)
SELECT CASE (potential%ppot_type)
CASE (no_pseudo, ecp_pseudo)
! we do nothing
CASE (gth_pseudo)
IF (potential%gth_pot%nlcc) THEN
m = SIZE(corden)
ALLOCATE (fe(m), rc(m))
n = potential%gth_pot%nexp_nlcc
DO i = 1, n
a = potential%gth_pot%alpha_nlcc(i)
a2 = a*a
! note that all terms are computed with rc, not rr
rc(:) = rr(:)/a
fe(:) = EXP(-0.5_dp*rc(:)*rc(:))
DO j = 1, potential%gth_pot%nct_nlcc(i)
cval = potential%gth_pot%cval_nlcc(j, i)
IF (my_typ == "RHO") THEN
corden(:) = corden(:) + fe(:)*rc**(2*j - 2)*cval
ELSE IF (my_typ == "DER") THEN
corden(:) = corden(:) - fe(:)*rc**(2*j - 1)*cval/a
IF (j > 1) THEN
corden(:) = corden(:) + REAL(2*j - 2, dp)*fe(:)*rc**(2*j - 3)*cval/a
END IF
ELSE IF (my_typ == "LAP") THEN
fb = 2._dp*cval/a
corden(:) = corden(:) - fb*fe(:)/rr(:)*rc**(2*j - 1)
corden(:) = corden(:) + fe(:)*rc**(2*j)*cval/a2
IF (j > 1) THEN
corden(:) = corden(:) + fb*REAL(2*j - 2, dp)*fe(:)/rr(:)*rc**(2*j - 3)
corden(:) = corden(:) + REAL((2*j - 2)*(2*j - 3), dp)*fe(:)*rc**(2*j - 4)*cval/a2
corden(:) = corden(:) - REAL(2*j - 2, dp)*fe(:)*rc**(2*j - 2)*cval/a2
END IF
ELSE
CPABORT("")
END IF
END DO
END DO
DEALLOCATE (fe, rc)
END IF
CASE (upf_pseudo)
IF (potential%upf_pot%core_correction) THEN
m = SIZE(corden)
n = potential%upf_pot%mesh_size
reverse = .FALSE.
IF (rr(1) > rr(m)) reverse = .TRUE.
ALLOCATE (rhoc(m), rval(m))
IF (reverse) THEN
DO i = 1, m
rval(i) = rr(m - i + 1)
END DO
ELSE
rval(1:m) = rr(1:m)
END IF
IF (my_typ == "RHO") THEN
CALL spline3ders(potential%upf_pot%r(1:n), potential%upf_pot%rho_nlcc(1:n), rval(1:m), &
ynew=rhoc(1:m))
ELSE IF (my_typ == "DER") THEN
CALL spline3ders(potential%upf_pot%r(1:n), potential%upf_pot%rho_nlcc(1:n), rval(1:m), &
dynew=rhoc(1:m))
ELSE IF (my_typ == "LAP") THEN
CALL spline3ders(potential%upf_pot%r(1:n), potential%upf_pot%rho_nlcc(1:n), rval(1:m), &
d2ynew=rhoc(1:m))
ELSE
CPABORT("")
END IF
IF (reverse) THEN
DO i = 1, m
rval(i) = rr(m - i + 1)
corden(i) = corden(i) + rhoc(m - i + 1)
END DO
ELSE
corden(1:m) = corden(1:m) + rhoc(1:m)
END IF
DEALLOCATE (rhoc, rval)
END IF
CASE (sgp_pseudo)
IF (potential%sgp_pot%has_nlcc) THEN
CPABORT("not implemented")
END IF
CASE DEFAULT
CPABORT("Unknown PP type")
END SELECT
END SUBROUTINE atom_core_density
! **************************************************************************************************
!> \brief ...
!> \param locpot ...
!> \param gthpot ...
!> \param rr ...
! **************************************************************************************************
PURE SUBROUTINE atom_local_potential(locpot, gthpot, rr)
REAL(KIND=dp), DIMENSION(:), INTENT(INOUT) :: locpot
TYPE(atom_gthpot_type), INTENT(IN) :: gthpot
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: rr
INTEGER :: i, j, m, n
REAL(KIND=dp) :: a
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: fe, rc
m = SIZE(locpot)
ALLOCATE (fe(m), rc(m))
rc(:) = rr(:)/gthpot%rc
DO i = 1, m
locpot(i) = -gthpot%zion*erf(rc(i)/SQRT(2._dp))/rr(i)
END DO
n = gthpot%ncl
fe(:) = EXP(-0.5_dp*rc(:)*rc(:))
DO i = 1, n
locpot(:) = locpot(:) + fe(:)*rc**(2*i - 2)*gthpot%cl(i)
END DO
IF (gthpot%lpotextended) THEN
DO j = 1, gthpot%nexp_lpot
a = gthpot%alpha_lpot(j)
rc(:) = rr(:)/a
fe(:) = EXP(-0.5_dp*rc(:)*rc(:))
n = gthpot%nct_lpot(j)
DO i = 1, n
locpot(:) = locpot(:) + fe(:)*rc**(2*i - 2)*gthpot%cval_lpot(i, j)
END DO
END DO
END IF
DEALLOCATE (fe, rc)
END SUBROUTINE atom_local_potential
! **************************************************************************************************
!> \brief ...
!> \param rmax ...
!> \param wfn ...
!> \param rcov ...
!> \param l ...
!> \param basis ...
! **************************************************************************************************
PURE SUBROUTINE atom_orbital_max(rmax, wfn, rcov, l, basis)
REAL(KIND=dp), INTENT(OUT) :: rmax
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: wfn
REAL(KIND=dp), INTENT(IN) :: rcov
INTEGER, INTENT(IN) :: l
TYPE(atom_basis_type), INTENT(IN) :: basis
INTEGER :: i, m, n
REAL(KIND=dp) :: ff
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: dorb
m = SIZE(basis%bf, 1)
ALLOCATE (dorb(m))
n = basis%nbas(l)
dorb = 0._dp
DO i = 1, n
ff = wfn(i)
dorb(1:m) = dorb(1:m) + ff*basis%dbf(1:m, i, l)
END DO
rmax = -1._dp
DO i = 1, m - 1
IF (basis%grid%rad(i) < 2*rcov) THEN
IF (dorb(i)*dorb(i + 1) < 0._dp) THEN
rmax = MAX(rmax, basis%grid%rad(i))
END IF
END IF
END DO
DEALLOCATE (dorb)
END SUBROUTINE atom_orbital_max
! **************************************************************************************************
!> \brief ...
!> \param node ...
!> \param wfn ...
!> \param rcov ...
!> \param l ...
!> \param basis ...
! **************************************************************************************************
PURE SUBROUTINE atom_orbital_nodes(node, wfn, rcov, l, basis)
INTEGER, INTENT(OUT) :: node
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: wfn
REAL(KIND=dp), INTENT(IN) :: rcov
INTEGER, INTENT(IN) :: l
TYPE(atom_basis_type), INTENT(IN) :: basis
INTEGER :: i, m, n
REAL(KIND=dp) :: ff
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: orb
node = 0
m = SIZE(basis%bf, 1)
ALLOCATE (orb(m))
n = basis%nbas(l)
orb = 0._dp
DO i = 1, n
ff = wfn(i)
orb(1:m) = orb(1:m) + ff*basis%bf(1:m, i, l)
END DO
DO i = 1, m - 1
IF (basis%grid%rad(i) < rcov) THEN
IF (orb(i)*orb(i + 1) < 0._dp) node = node + 1
END IF
END DO
DEALLOCATE (orb)
END SUBROUTINE atom_orbital_nodes
! **************************************************************************************************
!> \brief ...
!> \param value ...
!> \param wfn ...
!> \param basis ...
! **************************************************************************************************
PURE SUBROUTINE atom_wfnr0(value, wfn, basis)
REAL(KIND=dp), INTENT(OUT) :: value
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: wfn
TYPE(atom_basis_type), INTENT(IN) :: basis
INTEGER :: i, m, n
value = 0._dp
m = MAXVAL(MINLOC(basis%grid%rad))
n = basis%nbas(0)
DO i = 1, n
value = value + wfn(i)*basis%bf(m, i, 0)
END DO
END SUBROUTINE atom_wfnr0
! **************************************************************************************************
!> \brief Solve the generalised eigenproblem for every angular momentum.
!> \param hmat Hamiltonian matrix
!> \param umat transformation matrix which reduces the overlap matrix to its unitary form
!> \param orb atomic wavefunctions
!> \param ener atomic orbital energies
!> \param nb number of contracted basis functions
!> \param nv number of linear-independent contracted basis functions
!> \param maxl maximum angular momentum to consider
!> \par History
!> * 08.2008 created [Juerg Hutter]
! **************************************************************************************************
SUBROUTINE atom_solve(hmat, umat, orb, ener, nb, nv, maxl)
REAL(KIND=dp), DIMENSION(:, :, 0:), INTENT(IN) :: hmat, umat
REAL(KIND=dp), DIMENSION(:, :, 0:), INTENT(INOUT) :: orb
REAL(KIND=dp), DIMENSION(:, 0:), INTENT(INOUT) :: ener
INTEGER, DIMENSION(0:), INTENT(IN) :: nb, nv
INTEGER, INTENT(IN) :: maxl
INTEGER :: info, l, lwork, m, n
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: w, work
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: a
CPASSERT(ALL(nb >= nv))
orb = 0._dp
DO l = 0, maxl
n = nb(l)
m = nv(l)
IF (n > 0 .AND. m > 0) THEN
lwork = 10*m
ALLOCATE (a(n, n), w(n), work(lwork))
a(1:m, 1:m) = MATMUL(TRANSPOSE(umat(1:n, 1:m, l)), MATMUL(hmat(1:n, 1:n, l), umat(1:n, 1:m, l)))
CALL dsyev("V", "U", m, a(1:m, 1:m), m, w(1:m), work, lwork, info)
a(1:n, 1:m) = MATMUL(umat(1:n, 1:m, l), a(1:m, 1:m))
m = MIN(m, SIZE(orb, 2))
orb(1:n, 1:m, l) = a(1:n, 1:m)
ener(1:m, l) = w(1:m)
DEALLOCATE (a, w, work)
END IF
END DO
END SUBROUTINE atom_solve
! **************************************************************************************************
!> \brief THIS FUNCTION IS NO LONGER IN USE.
!> \param fun ...
!> \param deps ...
!> \return ...
! **************************************************************************************************
FUNCTION prune_grid(fun, deps) RESULT(nc)
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: fun
REAL(KIND=dp), INTENT(IN), OPTIONAL :: deps
INTEGER :: nc
INTEGER :: i, nr
REAL(KIND=dp) :: meps
meps = 1.e-14_dp
IF (PRESENT(deps)) meps = deps
nr = SIZE(fun)
nc = 0
DO i = nr, 1, -1
IF (ABS(fun(i)) > meps) THEN
nc = i
EXIT
END IF
END DO