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atom_sgp.F
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atom_sgp.F
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!--------------------------------------------------------------------------------------------------!
! CP2K: A general program to perform molecular dynamics simulations !
! Copyright 2000-2023 CP2K developers group <https://cp2k.org> !
! !
! SPDX-License-Identifier: GPL-2.0-or-later !
!--------------------------------------------------------------------------------------------------!
! **************************************************************************************************
MODULE atom_sgp
USE ai_onecenter, ONLY: sg_overlap
USE atom_types, ONLY: &
atom_basis_gridrep, atom_basis_type, atom_ecppot_type, atom_p_type, atom_type, &
create_opmat, init_atom_basis_default_pp, opmat_type, release_atom_basis, release_opmat
USE atom_upf, ONLY: atom_upfpot_type
USE atom_utils, ONLY: integrate_grid,&
numpot_matrix
USE input_constants, ONLY: ecp_pseudo,&
gaussian,&
gth_pseudo,&
no_pseudo,&
upf_pseudo
USE input_section_types, ONLY: section_vals_get,&
section_vals_type
USE kahan_sum, ONLY: accurate_dot_product
USE kinds, ONLY: dp
USE mathconstants, ONLY: dfac,&
fourpi,&
rootpi,&
sqrt2
USE mathlib, ONLY: diamat_all,&
get_pseudo_inverse_diag
USE powell, ONLY: opt_state_type,&
powell_optimize
#include "./base/base_uses.f90"
IMPLICIT NONE
TYPE atom_sgp_potential_type
LOGICAL :: has_nonlocal
INTEGER :: n_nonlocal
INTEGER :: lmax
LOGICAL, DIMENSION(0:5) :: is_nonlocal = .FALSE.
REAL(KIND=dp), DIMENSION(:), POINTER :: a_nonlocal => Null()
REAL(KIND=dp), DIMENSION(:, :), POINTER :: h_nonlocal => Null()
REAL(KIND=dp), DIMENSION(:, :, :), POINTER :: c_nonlocal => Null()
LOGICAL :: has_local
INTEGER :: n_local
REAL(KIND=dp) :: zval
REAL(KIND=dp) :: ac_local
REAL(KIND=dp), DIMENSION(:), POINTER :: a_local => Null()
REAL(KIND=dp), DIMENSION(:), POINTER :: c_local => Null()
LOGICAL :: has_nlcc
INTEGER :: n_nlcc
REAL(KIND=dp), DIMENSION(:), POINTER :: a_nlcc => Null()
REAL(KIND=dp), DIMENSION(:), POINTER :: c_nlcc => Null()
END TYPE
PRIVATE
PUBLIC :: atom_sgp_potential_type, atom_sgp_release
PUBLIC :: atom_sgp_construction, sgp_construction
CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'atom_sgp'
! **************************************************************************************************
CONTAINS
! **************************************************************************************************
!> \brief ...
!> \param sgp_pot ...
!> \param ecp_pot ...
!> \param upf_pot ...
!> \param error ...
! **************************************************************************************************
SUBROUTINE sgp_construction(sgp_pot, ecp_pot, upf_pot, error)
TYPE(atom_sgp_potential_type) :: sgp_pot
TYPE(atom_ecppot_type), OPTIONAL :: ecp_pot
TYPE(atom_upfpot_type), OPTIONAL :: upf_pot
REAL(KIND=dp), DIMENSION(3) :: error
INTEGER :: i, n
LOGICAL :: is_ecp, is_upf
REAL(KIND=dp) :: errcc, rcut
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: cgauss, cutpots, cutpotu
TYPE(atom_basis_type) :: basis
TYPE(opmat_type), POINTER :: core, hnl, score, shnl
! define basis
CALL init_atom_basis_default_pp(basis)
is_ecp = .FALSE.
IF (PRESENT(ecp_pot)) is_ecp = .TRUE.
is_upf = .FALSE.
IF (PRESENT(upf_pot)) is_upf = .TRUE.
CPASSERT(.NOT. (is_ecp .AND. is_upf))
! upf has often very small grids, use a smooth cutoff function
IF (is_upf) THEN
n = SIZE(upf_pot%r)
ALLOCATE (cutpotu(n))
rcut = MAXVAL(upf_pot%r)
CALL cutpot(cutpotu, upf_pot%r, rcut, 2.5_dp)
n = basis%grid%nr
ALLOCATE (cutpots(n))
CALL cutpot(cutpots, basis%grid%rad, rcut, 2.5_dp)
ELSE
n = basis%grid%nr
ALLOCATE (cutpots(n))
cutpots = 1.0_dp
END IF
! generate the transformed potentials
IF (is_ecp) THEN
CALL ecp_sgp_constr(ecp_pot, sgp_pot, basis)
ELSEIF (is_upf) THEN
CALL upf_sgp_constr(upf_pot, sgp_pot, basis)
ELSE
CPABORT("")
END IF
NULLIFY (core, hnl)
CALL create_opmat(core, basis%nbas)
CALL create_opmat(hnl, basis%nbas, 5)
NULLIFY (score, shnl)
CALL create_opmat(score, basis%nbas)
CALL create_opmat(shnl, basis%nbas, 5)
!
IF (is_ecp) THEN
CALL ecpints(hnl%op, basis, ecp_pot)
ELSEIF (is_upf) THEN
CALL upfints(core%op, hnl%op, basis, upf_pot, cutpotu, sgp_pot%ac_local)
ELSE
CPABORT("")
END IF
!
CALL sgpints(score%op, shnl%op, basis, sgp_pot, cutpots)
!
error = 0.0_dp
IF (sgp_pot%has_local) THEN
n = MIN(3, UBOUND(core%op, 3))
error(1) = MAXVAL(ABS(core%op(:, :, 0:n) - score%op(:, :, 0:n)))
END IF
IF (sgp_pot%has_nonlocal) THEN
n = MIN(3, UBOUND(hnl%op, 3))
error(2) = MAXVAL(ABS(hnl%op(:, :, 0:n) - shnl%op(:, :, 0:n)))
END IF
IF (sgp_pot%has_nlcc) THEN
IF (is_upf) THEN
n = SIZE(upf_pot%r)
ALLOCATE (cgauss(n))
cgauss = 0.0_dp
DO i = 1, sgp_pot%n_nlcc
cgauss(:) = cgauss(:) + sgp_pot%c_nlcc(i)*EXP(-sgp_pot%a_nlcc(i)*upf_pot%r(:)**2)
END DO
errcc = SUM((cgauss(:) - upf_pot%rho_nlcc(:))**2*upf_pot%r(:)**2*upf_pot%rab(:))
errcc = SQRT(errcc/REAL(n, KIND=dp))
DEALLOCATE (cgauss)
ELSE
CPABORT("")
END IF
error(3) = errcc
END IF
!
IF (is_upf) THEN
DEALLOCATE (cutpotu)
DEALLOCATE (cutpots)
ELSE
DEALLOCATE (cutpots)
END IF
!
CALL release_opmat(score)
CALL release_opmat(shnl)
CALL release_opmat(core)
CALL release_opmat(hnl)
CALL release_atom_basis(basis)
END SUBROUTINE sgp_construction
! **************************************************************************************************
!> \brief ...
!> \param atom_info ...
!> \param input_section ...
!> \param iw ...
! **************************************************************************************************
SUBROUTINE atom_sgp_construction(atom_info, input_section, iw)
TYPE(atom_p_type), DIMENSION(:, :), POINTER :: atom_info
TYPE(section_vals_type), POINTER :: input_section
INTEGER, INTENT(IN) :: iw
INTEGER :: i, n, ppot_type
LOGICAL :: do_transform, explicit, is_ecp, is_upf
REAL(KIND=dp) :: errcc, rcut
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: cgauss, cutpots, cutpotu
TYPE(atom_ecppot_type), POINTER :: ecp_pot
TYPE(atom_sgp_potential_type) :: sgp_pot
TYPE(atom_type), POINTER :: atom_ref
TYPE(atom_upfpot_type), POINTER :: upf_pot
TYPE(opmat_type), POINTER :: core, hnl, score, shnl
CALL section_vals_get(input_section, explicit=explicit)
IF (.NOT. explicit) RETURN
IF (iw > 0) WRITE (iw, '(/," ",79("*"),/,T24,A,/," ",79("*"))') "SEPARABLE GAUSSIAN PSEUDOPOTENTIAL"
atom_ref => atom_info(1, 1)%atom
ppot_type = atom_ref%potential%ppot_type
SELECT CASE (ppot_type)
CASE (gth_pseudo)
IF (iw > 0) WRITE (iw, '(" GTH Pseudopotential is already in SGP form. ")')
do_transform = .FALSE.
CASE (ecp_pseudo)
do_transform = .TRUE.
is_ecp = .TRUE.
is_upf = .FALSE.
ecp_pot => atom_ref%potential%ecp_pot
CASE (upf_pseudo)
do_transform = .TRUE.
is_ecp = .FALSE.
is_upf = .TRUE.
upf_pot => atom_ref%potential%upf_pot
CASE (no_pseudo)
IF (iw > 0) WRITE (iw, '(" No Pseudopotential available for transformation. ")')
do_transform = .FALSE.
CASE DEFAULT
CPABORT("")
END SELECT
! generate the transformed potentials
IF (do_transform) THEN
IF (is_ecp) THEN
CALL ecp_sgp_constr(ecp_pot, sgp_pot, atom_ref%basis)
ELSEIF (is_upf) THEN
CALL upf_sgp_constr(upf_pot, sgp_pot, atom_ref%basis)
ELSE
CPABORT("")
END IF
END IF
! Check the result
IF (do_transform) THEN
NULLIFY (core, hnl)
CALL create_opmat(core, atom_ref%basis%nbas)
CALL create_opmat(hnl, atom_ref%basis%nbas, 5)
NULLIFY (score, shnl)
CALL create_opmat(score, atom_ref%basis%nbas)
CALL create_opmat(shnl, atom_ref%basis%nbas, 5)
!
! upf has often very small grids, use a smooth cutoff function
IF (is_upf) THEN
n = SIZE(upf_pot%r)
ALLOCATE (cutpotu(n))
rcut = MAXVAL(upf_pot%r)
CALL cutpot(cutpotu, upf_pot%r, rcut, 2.5_dp)
n = atom_ref%basis%grid%nr
ALLOCATE (cutpots(n))
CALL cutpot(cutpots, atom_ref%basis%grid%rad, rcut, 2.5_dp)
ELSE
n = atom_ref%basis%grid%nr
ALLOCATE (cutpots(n))
cutpots = 1.0_dp
END IF
!
IF (is_ecp) THEN
CALL ecpints(hnl%op, atom_ref%basis, ecp_pot)
ELSEIF (is_upf) THEN
CALL upfints(core%op, hnl%op, atom_ref%basis, upf_pot, cutpotu, sgp_pot%ac_local)
ELSE
CPABORT("")
END IF
!
CALL sgpints(score%op, shnl%op, atom_ref%basis, sgp_pot, cutpots)
!
IF (sgp_pot%has_local) THEN
n = MIN(3, UBOUND(core%op, 3))
errcc = MAXVAL(ABS(core%op(:, :, 0:n) - score%op(:, :, 0:n)))
IF (iw > 0) THEN
WRITE (iw, '(" Local part of pseudopotential")')
WRITE (iw, '(" Number of basis functions ",T77,i4)') sgp_pot%n_local
WRITE (iw, '(" Max. abs. error of matrix elements ",T65,f16.8)') errcc
END IF
END IF
IF (sgp_pot%has_nonlocal) THEN
errcc = MAXVAL(ABS(hnl%op - shnl%op))
IF (iw > 0) THEN
WRITE (iw, '(" Nonlocal part of pseudopotential")')
WRITE (iw, '(" Max. l-quantum number",T77,i4)') sgp_pot%lmax
WRITE (iw, '(" Number of projector basis functions ",T77,i4)') sgp_pot%n_nonlocal
WRITE (iw, '(" Max. abs. error of matrix elements ",T69,f12.8)') errcc
END IF
END IF
IF (sgp_pot%has_nlcc) THEN
IF (is_upf) THEN
n = SIZE(upf_pot%r)
ALLOCATE (cgauss(n))
cgauss = 0.0_dp
DO i = 1, sgp_pot%n_nlcc
cgauss(:) = cgauss(:) + sgp_pot%c_nlcc(i)*EXP(-sgp_pot%a_nlcc(i)*upf_pot%r(:)**2)
END DO
errcc = SUM((cgauss(:) - upf_pot%rho_nlcc(:))**2*upf_pot%r(:)**2*upf_pot%rab(:))
errcc = SQRT(errcc/REAL(n, KIND=dp))
DEALLOCATE (cgauss)
ELSE
CPABORT("")
END IF
IF (iw > 0) THEN
WRITE (iw, '(" Non-linear core correction: core density")')
WRITE (iw, '(" Number of basis functions ",T77,i4)') sgp_pot%n_nlcc
WRITE (iw, '(" RMS error of core density ",T69,f12.8)') errcc
END IF
END IF
!
IF (is_upf) THEN
DEALLOCATE (cutpotu)
DEALLOCATE (cutpots)
ELSE
DEALLOCATE (cutpots)
END IF
!
CALL release_opmat(score)
CALL release_opmat(shnl)
CALL release_opmat(core)
CALL release_opmat(hnl)
END IF
CALL atom_sgp_release(sgp_pot)
IF (iw > 0) WRITE (iw, '(" ",79("*"))')
END SUBROUTINE atom_sgp_construction
! **************************************************************************************************
!> \brief ...
!> \param ecp_pot ...
!> \param sgp_pot ...
!> \param basis ...
! **************************************************************************************************
SUBROUTINE ecp_sgp_constr(ecp_pot, sgp_pot, basis)
TYPE(atom_ecppot_type) :: ecp_pot
TYPE(atom_sgp_potential_type) :: sgp_pot
TYPE(atom_basis_type) :: basis
INTEGER :: i, ia, ir, j, k, l, n, na, nl, nr
REAL(KIND=dp) :: alpha, eee, ei
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: al, cl, cpot, pgauss
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: cmat, qmat, score, sinv, smat, tmat
REAL(KIND=dp), DIMENSION(:), POINTER :: rad
sgp_pot%has_nlcc = .FALSE.
sgp_pot%n_nlcc = 0
sgp_pot%has_local = .FALSE.
sgp_pot%n_local = 0
! transform semilocal potential into a separable local form
sgp_pot%has_nonlocal = .TRUE.
!
nl = 12
!
sgp_pot%n_nonlocal = nl
sgp_pot%lmax = ecp_pot%lmax
ALLOCATE (sgp_pot%a_nonlocal(nl))
ALLOCATE (sgp_pot%h_nonlocal(nl, 0:ecp_pot%lmax))
ALLOCATE (sgp_pot%c_nonlocal(nl, nl, 0:ecp_pot%lmax))
!
ALLOCATE (al(nl), cl(nl))
ALLOCATE (smat(nl, nl), sinv(nl, nl))
ALLOCATE (tmat(nl, nl), cmat(nl, nl))
al = 0.0_dp
DO ir = 1, nl
al(ir) = 80.0_dp*0.60_dp**(ir - 1)
END DO
!
sgp_pot%a_nonlocal(1:nl) = al(1:nl)
!
nr = basis%grid%nr
rad => basis%grid%rad
ALLOCATE (cpot(nr), pgauss(nr))
DO l = 0, ecp_pot%lmax
na = basis%nbas(l)
ALLOCATE (score(na, na), qmat(na, nl))
cpot = 0._dp
DO k = 1, ecp_pot%npot(l)
n = ecp_pot%nrpot(k, l)
alpha = ecp_pot%bpot(k, l)
cpot(:) = cpot + ecp_pot%apot(k, l)*rad**(n - 2)*EXP(-alpha*rad**2)
END DO
DO i = 1, na
DO j = i, na
score(i, j) = integrate_grid(cpot, basis%bf(:, i, l), basis%bf(:, j, l), basis%grid)
score(j, i) = score(i, j)
END DO
END DO
! overlap basis with projectors
DO i = 1, nl
pgauss(:) = EXP(-al(i)*rad(:)**2)*rad(:)**l
eee = rootpi/(2._dp**(l + 2)*dfac(2*l + 1))/(2._dp*al(i))**(l + 1.5_dp)
pgauss(:) = pgauss(:)/SQRT(eee)
DO ia = 1, na
qmat(ia, i) = SUM(basis%bf(:, ia, l)*pgauss(:)*basis%grid%wr(:))
END DO
END DO
! tmat = qmat * score * qmat
tmat(1:nl, 1:nl) = MATMUL(TRANSPOSE(qmat(1:na, 1:nl)), MATMUL(score(1:na, 1:na), qmat(1:na, 1:nl)))
smat(1:nl, 1:nl) = MATMUL(TRANSPOSE(qmat(1:na, 1:nl)), qmat(1:na, 1:nl))
CALL get_pseudo_inverse_diag(smat(1:nl, 1:nl), sinv(1:nl, 1:nl), 1.e-10_dp)
cmat(1:nl, 1:nl) = MATMUL(sinv(1:nl, 1:nl), MATMUL(tmat(1:nl, 1:nl), sinv(1:nl, 1:nl)))
cmat(1:nl, 1:nl) = (cmat(1:nl, 1:nl) + TRANSPOSE(cmat(1:nl, 1:nl)))*0.5_dp
CALL diamat_all(cmat(1:nl, 1:nl), cl(1:nl), .TRUE.)
!
! get back unnormalized Gaussians
DO i = 1, nl
ei = rootpi/(2._dp**(l + 2)*dfac(2*l + 1))/(2._dp*al(i))**(l + 1.5_dp)
cmat(i, 1:nl) = cmat(i, 1:nl)/SQRT(ei)
END DO
sgp_pot%h_nonlocal(1:nl, l) = cl(1:nl)
sgp_pot%c_nonlocal(1:nl, 1:nl, l) = cmat(1:nl, 1:nl)
sgp_pot%is_nonlocal(l) = .TRUE.
!
DEALLOCATE (score, qmat)
END DO
DEALLOCATE (cpot, pgauss)
DEALLOCATE (al, cl, smat, sinv, tmat, cmat)
END SUBROUTINE ecp_sgp_constr
! **************************************************************************************************
!> \brief ...
!> \param upf_pot ...
!> \param sgp_pot ...
!> \param basis ...
! **************************************************************************************************
SUBROUTINE upf_sgp_constr(upf_pot, sgp_pot, basis)
TYPE(atom_upfpot_type) :: upf_pot
TYPE(atom_sgp_potential_type) :: sgp_pot
TYPE(atom_basis_type) :: basis
CHARACTER(len=4) :: ptype
INTEGER :: ia, ib, ipa, ipb, ir, la, lb, na, nl, &
np, nr
LOGICAL :: nl_trans
REAL(KIND=dp) :: cpa, cpb, eee, ei, errcc, errloc, rc, &
x(2), zval
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: al, ccharge, cgauss, cl, pgauss, pproa, &
pprob, tv, vgauss, vloc, ww
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: cmat, qmat, score, sinv, smat, tmat
TYPE(atom_basis_type) :: gbasis
TYPE(opt_state_type) :: ostate
IF (upf_pot%is_ultrasoft .OR. upf_pot%is_paw .OR. upf_pot%is_coulomb) THEN
sgp_pot%has_nonlocal = .FALSE.
sgp_pot%n_nonlocal = 0
sgp_pot%has_local = .FALSE.
sgp_pot%n_local = 0
sgp_pot%has_nlcc = .FALSE.
sgp_pot%n_nlcc = 0
RETURN
END IF
! radial grid
nr = SIZE(upf_pot%r)
! weights for integration
ALLOCATE (ww(nr))
ww(:) = upf_pot%r(:)**2*upf_pot%rab(:)
! start with local potential
sgp_pot%has_local = .TRUE.
! fit local potential to Gaussian form
ALLOCATE (vloc(nr), vgauss(nr))
! smearing of core charge
zval = upf_pot%zion
! Try to find an optimal Gaussian charge distribution
CALL erffit(sgp_pot%ac_local, upf_pot%vlocal, upf_pot%r, zval)
sgp_pot%zval = zval
DO ir = 1, nr
IF (upf_pot%r(ir) < 1.e-12_dp) THEN
vgauss(ir) = -SQRT(2.0_dp)*zval/rootpi/sgp_pot%ac_local
ELSE
rc = upf_pot%r(ir)/sgp_pot%ac_local/SQRT(2.0_dp)
vgauss(ir) = -zval/upf_pot%r(ir)*erf(rc)
END IF
END DO
vloc(:) = upf_pot%vlocal(:) - vgauss(:)
!
CALL atom_basis_gridrep(basis, gbasis, upf_pot%r, upf_pot%rab)
!
nl = 12
ALLOCATE (al(nl), cl(nl))
ostate%nf = 0
ostate%nvar = 2
x(1) = 1.00_dp !starting point of geometric series
x(2) = 1.20_dp !factor of series
ostate%rhoend = 1.e-12_dp
ostate%rhobeg = 5.e-2_dp
ostate%maxfun = 1000
ostate%iprint = 1
ostate%unit = -1
ostate%state = 0
DO
IF (ostate%state == 2) THEN
DO ir = 1, nl
al(ir) = x(1)*x(2)**(ir - 1)
END DO
CALL pplocal_error(nl, al, cl, vloc, vgauss, gbasis, upf_pot%r, ww, 1, ostate%f)
END IF
IF (ostate%state == -1) EXIT
CALL powell_optimize(ostate%nvar, x, ostate)
END DO
ostate%state = 8
CALL powell_optimize(ostate%nvar, x, ostate)
DO ir = 1, nl
al(ir) = x(1)*x(2)**(ir - 1)
END DO
CALL pplocal_error(nl, al, cl, vloc, vgauss, gbasis, upf_pot%r, ww, 1, errloc)
!
ALLOCATE (sgp_pot%a_local(nl), sgp_pot%c_local(nl))
sgp_pot%n_local = nl
sgp_pot%a_local(1:nl) = al(1:nl)
sgp_pot%c_local(1:nl) = cl(1:nl)
DEALLOCATE (vloc, vgauss)
DEALLOCATE (al, cl)
CALL release_atom_basis(gbasis)
!
ptype = ADJUSTL(TRIM(upf_pot%pseudo_type))
IF (ptype(1:2) == "NC" .OR. ptype(1:2) == "US") THEN
nl_trans = .FALSE.
ELSE IF (ptype(1:2) == "SL") THEN
nl_trans = .TRUE.
ELSE
CPABORT("Pseudopotential type: ["//ADJUSTL(TRIM(ptype))//"] not known")
END IF
! purely local pseudopotentials
IF (upf_pot%l_max < 0) THEN
sgp_pot%n_nonlocal = 0
sgp_pot%lmax = -1
sgp_pot%has_nonlocal = .FALSE.
ELSE
! Non-local pseudopotential in Gaussian form
IF (nl_trans) THEN
sgp_pot%has_nonlocal = .TRUE.
! semi local pseudopotential
! fit to nonlocal form
! get basis representation on UPF grid
nl = 8
!
sgp_pot%n_nonlocal = nl
sgp_pot%lmax = upf_pot%l_max
ALLOCATE (sgp_pot%a_nonlocal(nl))
ALLOCATE (sgp_pot%h_nonlocal(nl, 0:upf_pot%l_max))
ALLOCATE (sgp_pot%c_nonlocal(nl, nl, 0:upf_pot%l_max))
!
ALLOCATE (al(nl), cl(nl))
ALLOCATE (smat(nl, nl), sinv(nl, nl))
ALLOCATE (tmat(nl, nl), cmat(nl, nl))
al = 0.0_dp
DO ir = 1, nl
al(ir) = 10.0_dp*0.60_dp**(ir - 1)
END DO
!
sgp_pot%a_nonlocal(1:nl) = al(1:nl)
!
CALL atom_basis_gridrep(basis, gbasis, upf_pot%r, upf_pot%rab)
ALLOCATE (pgauss(nr), vloc(nr))
DO la = 0, upf_pot%l_max
IF (la == upf_pot%l_local) CYCLE
sgp_pot%is_nonlocal(la) = .TRUE.
na = gbasis%nbas(la)
ALLOCATE (score(na, na), qmat(na, nl))
! Reference matrix
vloc(:) = upf_pot%vsemi(:, la + 1) - upf_pot%vlocal(:)
DO ia = 1, na
DO ib = ia, na
score(ia, ib) = SUM(vloc(:)*gbasis%bf(:, ia, la)*gbasis%bf(:, ib, la)*ww(:))
score(ib, ia) = score(ia, ib)
END DO
END DO
! overlap basis with projectors
DO ir = 1, nl
pgauss(:) = EXP(-al(ir)*upf_pot%r(:)**2)*upf_pot%r(:)**la
eee = rootpi/(2._dp**(la + 2)*dfac(2*la + 1))/(2._dp*al(ir))**(la + 1.5_dp)
pgauss(:) = pgauss(:)/SQRT(eee)
DO ia = 1, na
qmat(ia, ir) = SUM(gbasis%bf(:, ia, la)*pgauss(:)*ww)
END DO
END DO
! tmat = qmat * score * qmat
tmat(1:nl, 1:nl) = MATMUL(TRANSPOSE(qmat(1:na, 1:nl)), MATMUL(score(1:na, 1:na), qmat(1:na, 1:nl)))
smat(1:nl, 1:nl) = MATMUL(TRANSPOSE(qmat(1:na, 1:nl)), qmat(1:na, 1:nl))
CALL get_pseudo_inverse_diag(smat(1:nl, 1:nl), sinv(1:nl, 1:nl), 1.e-10_dp)
cmat(1:nl, 1:nl) = MATMUL(sinv(1:nl, 1:nl), MATMUL(tmat(1:nl, 1:nl), sinv(1:nl, 1:nl)))
cmat(1:nl, 1:nl) = (cmat(1:nl, 1:nl) + TRANSPOSE(cmat(1:nl, 1:nl)))*0.5_dp
CALL diamat_all(cmat(1:nl, 1:nl), cl(1:nl), .TRUE.)
!
! get back unnormalized Gaussians
DO ir = 1, nl
ei = rootpi/(2._dp**(la + 2)*dfac(2*la + 1))/(2._dp*al(ir))**(la + 1.5_dp)
cmat(ir, 1:nl) = cmat(ir, 1:nl)/SQRT(ei)
END DO
sgp_pot%h_nonlocal(1:nl, la) = cl(1:nl)
sgp_pot%c_nonlocal(1:nl, 1:nl, la) = cmat(1:nl, 1:nl)
sgp_pot%is_nonlocal(la) = .TRUE.
DEALLOCATE (score, qmat)
END DO
! SQRT(4PI)
sgp_pot%c_nonlocal = sgp_pot%c_nonlocal/SQRT(fourpi)
CALL release_atom_basis(gbasis)
DEALLOCATE (pgauss, vloc)
DEALLOCATE (al, cl, smat, sinv, tmat, cmat)
ELSE
sgp_pot%has_nonlocal = .TRUE.
! non local pseudopotential
ALLOCATE (pproa(nr), pprob(nr), pgauss(nr))
np = upf_pot%number_of_proj
nl = 8
ALLOCATE (al(nl), cl(nl))
ALLOCATE (smat(nl, nl), sinv(nl, nl))
ALLOCATE (tmat(nl, nl), cmat(nl, nl))
al = 0.0_dp
cl = 0.0_dp
DO ir = 1, nl
al(ir) = 10.0_dp*0.60_dp**(ir - 1)
END DO
!
sgp_pot%lmax = MAXVAL(upf_pot%lbeta(:))
sgp_pot%n_nonlocal = nl
ALLOCATE (sgp_pot%a_nonlocal(nl))
ALLOCATE (sgp_pot%h_nonlocal(nl, 0:sgp_pot%lmax))
ALLOCATE (sgp_pot%c_nonlocal(nl, nl, 0:sgp_pot%lmax))
!
sgp_pot%a_nonlocal(1:nl) = al(1:nl)
!
CALL atom_basis_gridrep(basis, gbasis, upf_pot%r, upf_pot%rab)
DO la = 0, sgp_pot%lmax
sgp_pot%is_nonlocal(la) = .TRUE.
na = gbasis%nbas(la)
ALLOCATE (score(na, na), qmat(na, nl))
! Reference matrix
score = 0.0_dp
DO ipa = 1, np
lb = upf_pot%lbeta(ipa)
IF (la /= lb) CYCLE
pproa(:) = upf_pot%beta(:, ipa)
DO ipb = 1, np
lb = upf_pot%lbeta(ipb)
IF (la /= lb) CYCLE
pprob(:) = upf_pot%beta(:, ipb)
eee = upf_pot%dion(ipa, ipb)
DO ia = 1, na
cpa = SUM(pproa(:)*gbasis%bf(:, ia, la)*ww(:))
DO ib = ia, na
cpb = SUM(pprob(:)*gbasis%bf(:, ib, la)*ww(:))
score(ia, ib) = score(ia, ib) + cpa*eee*cpb
score(ib, ia) = score(ia, ib)
END DO
END DO
END DO
END DO
! overlap basis with projectors
DO ir = 1, nl
pgauss(:) = EXP(-al(ir)*upf_pot%r(:)**2)*upf_pot%r(:)**la
eee = rootpi/(2._dp**(la + 2)*dfac(2*la + 1))/(2._dp*al(ir))**(la + 1.5_dp)
pgauss(:) = pgauss(:)/SQRT(eee)
DO ia = 1, na
qmat(ia, ir) = SUM(gbasis%bf(:, ia, la)*pgauss(:)*ww)
END DO
END DO
! tmat = qmat * score * qmat
tmat(1:nl, 1:nl) = MATMUL(TRANSPOSE(qmat(1:na, 1:nl)), MATMUL(score(1:na, 1:na), qmat(1:na, 1:nl)))
smat(1:nl, 1:nl) = MATMUL(TRANSPOSE(qmat(1:na, 1:nl)), qmat(1:na, 1:nl))
CALL get_pseudo_inverse_diag(smat(1:nl, 1:nl), sinv(1:nl, 1:nl), 1.e-10_dp)
cmat(1:nl, 1:nl) = MATMUL(sinv(1:nl, 1:nl), MATMUL(tmat(1:nl, 1:nl), sinv(1:nl, 1:nl)))
cmat(1:nl, 1:nl) = (cmat(1:nl, 1:nl) + TRANSPOSE(cmat(1:nl, 1:nl)))*0.5_dp
CALL diamat_all(cmat(1:nl, 1:nl), cl(1:nl), .TRUE.)
!
! get back unnormalized Gaussians
DO ir = 1, nl
ei = rootpi/(2._dp**(la + 2)*dfac(2*la + 1))/(2._dp*al(ir))**(la + 1.5_dp)
cmat(ir, 1:nl) = cmat(ir, 1:nl)/SQRT(ei)
END DO
sgp_pot%h_nonlocal(1:nl, la) = cl(1:nl)
sgp_pot%c_nonlocal(1:nl, 1:nl, la) = cmat(1:nl, 1:nl)
sgp_pot%is_nonlocal(la) = .TRUE.
DEALLOCATE (score, qmat)
END DO
! SQRT(4PI)
sgp_pot%c_nonlocal = sgp_pot%c_nonlocal/SQRT(fourpi)
CALL release_atom_basis(gbasis)
DEALLOCATE (pgauss, pproa, pprob)
DEALLOCATE (al, cl, smat, sinv, tmat, cmat)
END IF
END IF
IF (upf_pot%core_correction) THEN
sgp_pot%has_nlcc = .TRUE.
ELSE
sgp_pot%has_nlcc = .FALSE.
sgp_pot%n_nlcc = 0
END IF
! fit core charge to Gaussian form
IF (sgp_pot%has_nlcc) THEN
ALLOCATE (ccharge(nr), cgauss(nr))
ccharge(:) = upf_pot%rho_nlcc(:)
nl = 8
ALLOCATE (al(nl), cl(nl), tv(nl))
ALLOCATE (smat(nl, nl), sinv(nl, nl))
al = 0.0_dp
cl = 0.0_dp
DO ir = 1, nl
al(ir) = 10.0_dp*0.6_dp**(ir - 1)
END DO
! calculate integrals
smat = 0.0_dp
sinv = 0.0_dp
tv = 0.0_dp
CALL sg_overlap(smat(1:nl, 1:nl), 0, al(1:nl), al(1:nl))
DO ir = 1, nl
cgauss(:) = EXP(-al(ir)*upf_pot%r(:)**2)
tv(ir) = SUM(cgauss*ccharge*ww)
END DO
CALL get_pseudo_inverse_diag(smat(1:nl, 1:nl), sinv(1:nl, 1:nl), 1.e-10_dp)
cl(1:nl) = MATMUL(sinv(1:nl, 1:nl), tv(1:nl))
cgauss = 0.0_dp
DO ir = 1, nl
cgauss(:) = cgauss(:) + cl(ir)*EXP(-al(ir)*upf_pot%r(:)**2)
END DO
errcc = SUM((cgauss - ccharge)**2*ww)
ALLOCATE (sgp_pot%a_local(nl), sgp_pot%c_local(nl))
sgp_pot%n_nlcc = nl
sgp_pot%a_nlcc(1:nl) = al(1:nl)
sgp_pot%c_nlcc(1:nl) = cl(1:nl)
DEALLOCATE (ccharge, cgauss)
DEALLOCATE (al, cl, tv, smat, sinv)
END IF
DEALLOCATE (ww)
END SUBROUTINE upf_sgp_constr
! **************************************************************************************************
!> \brief ...
!> \param sgp_pot ...
! **************************************************************************************************
SUBROUTINE atom_sgp_release(sgp_pot)
TYPE(atom_sgp_potential_type) :: sgp_pot
IF (ASSOCIATED(sgp_pot%a_nonlocal)) DEALLOCATE (sgp_pot%a_nonlocal)
IF (ASSOCIATED(sgp_pot%h_nonlocal)) DEALLOCATE (sgp_pot%h_nonlocal)
IF (ASSOCIATED(sgp_pot%c_nonlocal)) DEALLOCATE (sgp_pot%c_nonlocal)
IF (ASSOCIATED(sgp_pot%a_local)) DEALLOCATE (sgp_pot%a_local)
IF (ASSOCIATED(sgp_pot%c_local)) DEALLOCATE (sgp_pot%c_local)
IF (ASSOCIATED(sgp_pot%a_nlcc)) DEALLOCATE (sgp_pot%a_nlcc)
IF (ASSOCIATED(sgp_pot%c_nlcc)) DEALLOCATE (sgp_pot%c_nlcc)
END SUBROUTINE atom_sgp_release
! **************************************************************************************************
!> \brief ...
!> \param core ...
!> \param hnl ...
!> \param basis ...
!> \param upf_pot ...
!> \param cutpotu ...
!> \param ac_local ...
! **************************************************************************************************
SUBROUTINE upfints(core, hnl, basis, upf_pot, cutpotu, ac_local)
REAL(KIND=dp), DIMENSION(:, :, 0:) :: core, hnl
TYPE(atom_basis_type), INTENT(INOUT) :: basis
TYPE(atom_upfpot_type) :: upf_pot
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: cutpotu
REAL(KIND=dp), INTENT(IN) :: ac_local
CHARACTER(len=4) :: ptype
INTEGER :: i, j, k1, k2, la, lb, m, n
REAL(KIND=dp) :: rc, zval
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: spot
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: spmat
TYPE(atom_basis_type) :: gbasis
! get basis representation on UPF grid
CALL atom_basis_gridrep(basis, gbasis, upf_pot%r, upf_pot%rab)
! local pseudopotential
core = 0._dp
n = SIZE(upf_pot%r)
ALLOCATE (spot(n))
spot(:) = upf_pot%vlocal(:)
zval = upf_pot%zion
DO i = 1, n
IF (upf_pot%r(i) < 1.e-12_dp) THEN
spot(i) = spot(i) + sqrt2*zval/rootpi/ac_local
ELSE
rc = upf_pot%r(i)/ac_local/sqrt2
spot(i) = spot(i) + zval/upf_pot%r(i)*erf(rc)
END IF
END DO
spot(:) = spot(:)*cutpotu(:)
CALL numpot_matrix(core, spot, gbasis, 0)
DEALLOCATE (spot)
hnl = 0._dp
ptype = ADJUSTL(TRIM(upf_pot%pseudo_type))
IF (ptype(1:2) == "NC" .OR. ptype(1:2) == "US") THEN
! non local pseudopotential
n = MAXVAL(gbasis%nbas(:))
m = upf_pot%number_of_proj
ALLOCATE (spmat(n, m))
spmat = 0.0_dp
DO i = 1, m
la = upf_pot%lbeta(i)
DO j = 1, gbasis%nbas(la)
spmat(j, i) = integrate_grid(upf_pot%beta(:, i), gbasis%bf(:, j, la), gbasis%grid)
END DO
END DO
DO i = 1, m
la = upf_pot%lbeta(i)
DO j = 1, m
lb = upf_pot%lbeta(j)
IF (la == lb) THEN
DO k1 = 1, gbasis%nbas(la)
DO k2 = 1, gbasis%nbas(la)
hnl(k1, k2, la) = hnl(k1, k2, la) + spmat(k1, i)*upf_pot%dion(i, j)*spmat(k2, j)
END DO
END DO
END IF
END DO
END DO
DEALLOCATE (spmat)
ELSE IF (ptype(1:2) == "SL") THEN
! semi local pseudopotential
DO la = 0, upf_pot%l_max
IF (la == upf_pot%l_local) CYCLE
m = SIZE(upf_pot%vsemi(:, la + 1))
ALLOCATE (spot(m))
spot(:) = upf_pot%vsemi(:, la + 1) - upf_pot%vlocal(:)
spot(:) = spot(:)*cutpotu(:)
n = basis%nbas(la)
DO i = 1, n
DO j = i, n
hnl(i, j, la) = hnl(i, j, la) + &
integrate_grid(spot(:), &
gbasis%bf(:, i, la), gbasis%bf(:, j, la), gbasis%grid)
hnl(j, i, la) = hnl(i, j, la)
END DO
END DO
DEALLOCATE (spot)
END DO
ELSE
CPABORT("Pseudopotential type: ["//ADJUSTL(TRIM(ptype))//"] not known")
END IF
! release basis representation on UPF grid
CALL release_atom_basis(gbasis)
END SUBROUTINE upfints
! **************************************************************************************************
!> \brief ...
!> \param hnl ...
!> \param basis ...
!> \param ecp_pot ...
! **************************************************************************************************
SUBROUTINE ecpints(hnl, basis, ecp_pot)
REAL(KIND=dp), DIMENSION(:, :, 0:) :: hnl
TYPE(atom_basis_type), INTENT(INOUT) :: basis
TYPE(atom_ecppot_type) :: ecp_pot
INTEGER :: i, j, k, l, m, n
REAL(KIND=dp) :: alpha
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: cpot
REAL(KIND=dp), DIMENSION(:), POINTER :: rad
rad => basis%grid%rad
m = basis%grid%nr
ALLOCATE (cpot(1:m))
! non local pseudopotential
hnl = 0.0_dp
DO l = 0, ecp_pot%lmax
cpot = 0._dp
DO k = 1, ecp_pot%npot(l)
n = ecp_pot%nrpot(k, l)
alpha = ecp_pot%bpot(k, l)
cpot(:) = cpot(:) + ecp_pot%apot(k, l)*rad(:)**(n - 2)*EXP(-alpha*rad(:)**2)
END DO
DO i = 1, basis%nbas(l)
DO j = i, basis%nbas(l)
hnl(i, j, l) = integrate_grid(cpot, basis%bf(:, i, l), basis%bf(:, j, l), basis%grid)
hnl(j, i, l) = hnl(i, j, l)
END DO
END DO
END DO
DEALLOCATE (cpot)
END SUBROUTINE ecpints
! **************************************************************************************************
!> \brief ...
!> \param core ...
!> \param hnl ...
!> \param basis ...
!> \param sgp_pot ...
!> \param cutpots ...
! **************************************************************************************************
SUBROUTINE sgpints(core, hnl, basis, sgp_pot, cutpots)
REAL(KIND=dp), DIMENSION(:, :, 0:) :: core, hnl
TYPE(atom_basis_type), INTENT(INOUT) :: basis
TYPE(atom_sgp_potential_type) :: sgp_pot
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: cutpots
INTEGER :: i, ia, j, l, m, n, na
REAL(KIND=dp) :: a, c, zval
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: cpot, pgauss
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: qmat
REAL(KIND=dp), DIMENSION(:), POINTER :: rad
rad => basis%grid%rad
m = basis%grid%nr
! local pseudopotential
ALLOCATE (cpot(m))
IF (sgp_pot%has_local) THEN
zval = sgp_pot%zval
core = 0._dp
cpot = 0.0_dp
DO i = 1, sgp_pot%n_local
cpot(:) = cpot(:) + sgp_pot%c_local(i)*EXP(-sgp_pot%a_local(i)*rad(:)**2)
END DO
cpot(:) = cpot(:)*cutpots(:)
CALL numpot_matrix(core, cpot, basis, 0)
END IF
DEALLOCATE (cpot)
! nonlocal pseudopotential
IF (sgp_pot%has_nonlocal) THEN
hnl = 0.0_dp
ALLOCATE (pgauss(1:m))
n = sgp_pot%n_nonlocal
!
DO l = 0, sgp_pot%lmax
CPASSERT(l <= UBOUND(basis%nbas, 1))
IF (.NOT. sgp_pot%is_nonlocal(l)) CYCLE
! overlap (a|p)
na = basis%nbas(l)
ALLOCATE (qmat(na, n))
DO i = 1, n
pgauss(:) = 0.0_dp
DO j = 1, n
a = sgp_pot%a_nonlocal(j)
c = sgp_pot%c_nonlocal(j, i, l)
pgauss(:) = pgauss(:) + c*EXP(-a*rad(:)**2)*rad(:)**l
END DO
pgauss(:) = pgauss(:)*cutpots(:)
DO ia = 1, na
qmat(ia, i) = SUM(basis%bf(:, ia, l)*pgauss(:)*basis%grid%wr(:))
END DO
END DO
qmat = SQRT(fourpi)*qmat
DO i = 1, na
DO j = i, na
DO ia = 1, n
hnl(i, j, l) = hnl(i, j, l) + qmat(i, ia)*qmat(j, ia)*sgp_pot%h_nonlocal(ia, l)
END DO
hnl(j, i, l) = hnl(i, j, l)
END DO
END DO
DEALLOCATE (qmat)
END DO
DEALLOCATE (pgauss)
END IF
END SUBROUTINE sgpints
! **************************************************************************************************
!> \brief ...
!> \param ac ...
!> \param vlocal ...
!> \param r ...
!> \param z ...
! **************************************************************************************************
SUBROUTINE erffit(ac, vlocal, r, z)
REAL(KIND=dp), INTENT(OUT) :: ac
REAL(KIND=dp), DIMENSION(:), INTENT(IN) :: vlocal, r
REAL(KIND=dp), INTENT(IN) :: z
REAL(KIND=dp), PARAMETER :: rcut = 1.4_dp
INTEGER :: i, j, m, m1
REAL(KIND=dp) :: a1, a2, an, e2, en, rc
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: epot, rval, vpot