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atom_operators.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 Calculate the atomic operator matrices
!> \author jgh
!> \date 03.03.2008
!> \version 1.0
!>
! **************************************************************************************************
MODULE atom_operators
USE ai_onecenter, ONLY: &
sg_coulomb, sg_erf, sg_erfc, sg_exchange, sg_gpot, sg_kinetic, sg_kinnuc, sg_nuclear, &
sg_overlap, sg_proj_ol, sto_kinetic, sto_nuclear, sto_overlap
USE atom_types, ONLY: &
atom_basis_gridrep, atom_basis_type, atom_compare_grids, atom_integrals, &
atom_potential_type, atom_state, cgto_basis, ecp_pseudo, gth_pseudo, gto_basis, lmat, &
no_pseudo, num_basis, release_atom_basis, sgp_pseudo, sto_basis, upf_pseudo
USE atom_utils, ONLY: &
atom_solve, contract2, contract2add, contract4, coulomb_potential_numeric, integrate_grid, &
numpot_matrix, slater_density, wigner_slater_functional
USE dkh_main, ONLY: dkh_atom_transformation
USE input_constants, ONLY: &
barrier_conf, do_dkh0_atom, do_dkh1_atom, do_dkh2_atom, do_dkh3_atom, do_nonrel_atom, &
do_sczoramp_atom, do_zoramp_atom, poly_conf
USE kinds, ONLY: dp
USE mathconstants, ONLY: gamma1,&
sqrt2
USE mathlib, ONLY: jacobi
USE periodic_table, ONLY: ptable
USE physcon, ONLY: c_light_au
USE qs_grid_atom, ONLY: grid_atom_type
#include "./base/base_uses.f90"
IMPLICIT NONE
PRIVATE
CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'atom_operators'
PUBLIC :: atom_int_setup, atom_ppint_setup, atom_int_release, atom_ppint_release
PUBLIC :: atom_relint_setup, atom_relint_release, atom_basis_projection_overlap
PUBLIC :: calculate_model_potential
CONTAINS
! **************************************************************************************************
!> \brief Set up atomic integrals.
!> \param integrals atomic integrals
!> \param basis atomic basis set
!> \param potential pseudo-potential
!> \param eri_coulomb setup one-centre Coulomb Electron Repulsion Integrals
!> \param eri_exchange setup one-centre exchange Electron Repulsion Integrals
!> \param all_nu compute integrals for all even integer parameters [0 .. 2*l]
!> REDUNDANT, AS THIS SUBROUTINE IS NEVER INVOKED WITH all_nu = .TRUE.
! **************************************************************************************************
SUBROUTINE atom_int_setup(integrals, basis, potential, &
eri_coulomb, eri_exchange, all_nu)
TYPE(atom_integrals), INTENT(INOUT) :: integrals
TYPE(atom_basis_type), INTENT(INOUT) :: basis
TYPE(atom_potential_type), INTENT(IN), OPTIONAL :: potential
LOGICAL, INTENT(IN), OPTIONAL :: eri_coulomb, eri_exchange, all_nu
CHARACTER(len=*), PARAMETER :: routineN = 'atom_int_setup'
INTEGER :: handle, i, ii, info, ipiv(1000), l, l1, &
l2, ll, lwork, m, m1, m2, mm1, mm2, n, &
n1, n2, nn1, nn2, nu, nx
REAL(KIND=dp) :: om, rc, ron, sc, x
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: cpot, w, work
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: omat, vmat
REAL(KIND=dp), DIMENSION(:, :), POINTER :: eri
CALL timeset(routineN, handle)
IF (integrals%status == 0) THEN
n = MAXVAL(basis%nbas)
integrals%n = basis%nbas
IF (PRESENT(eri_coulomb)) THEN
integrals%eri_coulomb = eri_coulomb
ELSE
integrals%eri_coulomb = .FALSE.
END IF
IF (PRESENT(eri_exchange)) THEN
integrals%eri_exchange = eri_exchange
ELSE
integrals%eri_exchange = .FALSE.
END IF
IF (PRESENT(all_nu)) THEN
integrals%all_nu = all_nu
ELSE
integrals%all_nu = .FALSE.
END IF
NULLIFY (integrals%ovlp, integrals%kin, integrals%core, integrals%conf)
DO ll = 1, SIZE(integrals%ceri)
NULLIFY (integrals%ceri(ll)%int, integrals%eeri(ll)%int)
END DO
ALLOCATE (integrals%ovlp(n, n, 0:lmat))
integrals%ovlp = 0._dp
ALLOCATE (integrals%kin(n, n, 0:lmat))
integrals%kin = 0._dp
integrals%status = 1
IF (PRESENT(potential)) THEN
IF (potential%confinement) THEN
ALLOCATE (integrals%conf(n, n, 0:lmat))
integrals%conf = 0._dp
m = basis%grid%nr
ALLOCATE (cpot(1:m))
IF (potential%conf_type == poly_conf) THEN
rc = potential%rcon
sc = potential%scon
cpot(1:m) = (basis%grid%rad(1:m)/rc)**sc
ELSEIF (potential%conf_type == barrier_conf) THEN
om = potential%rcon
ron = potential%scon
rc = ron + om
DO i = 1, m
IF (basis%grid%rad(i) < ron) THEN
cpot(i) = 0.0_dp
ELSEIF (basis%grid%rad(i) < rc) THEN
x = (basis%grid%rad(i) - ron)/om
x = 1._dp - x
cpot(i) = -6._dp*x**5 + 15._dp*x**4 - 10._dp*x**3 + 1._dp
x = (rc - basis%grid%rad(i))**2/om/(basis%grid%rad(i) - ron)
cpot(i) = cpot(i)*x
ELSE
cpot(i) = 1.0_dp
END IF
END DO
ELSE
CPABORT("")
END IF
CALL numpot_matrix(integrals%conf, cpot, basis, 0)
DEALLOCATE (cpot)
END IF
END IF
SELECT CASE (basis%basis_type)
CASE DEFAULT
CPABORT("")
CASE (GTO_BASIS)
DO l = 0, lmat
n = integrals%n(l)
CALL sg_overlap(integrals%ovlp(1:n, 1:n, l), l, basis%am(1:n, l), basis%am(1:n, l))
CALL sg_kinetic(integrals%kin(1:n, 1:n, l), l, basis%am(1:n, l), basis%am(1:n, l))
END DO
IF (integrals%eri_coulomb) THEN
ll = 0
DO l1 = 0, lmat
n1 = integrals%n(l1)
nn1 = (n1*(n1 + 1))/2
DO l2 = 0, l1
n2 = integrals%n(l2)
nn2 = (n2*(n2 + 1))/2
IF (integrals%all_nu) THEN
nx = MIN(2*l1, 2*l2)
ELSE
nx = 0
END IF
DO nu = 0, nx, 2
ll = ll + 1
CPASSERT(ll <= SIZE(integrals%ceri))
ALLOCATE (integrals%ceri(ll)%int(nn1, nn2))
integrals%ceri(ll)%int = 0._dp
eri => integrals%ceri(ll)%int
CALL sg_coulomb(eri, nu, basis%am(1:n1, l1), l1, basis%am(1:n2, l2), l2)
END DO
END DO
END DO
END IF
IF (integrals%eri_exchange) THEN
ll = 0
DO l1 = 0, lmat
n1 = integrals%n(l1)
nn1 = (n1*(n1 + 1))/2
DO l2 = 0, l1
n2 = integrals%n(l2)
nn2 = (n2*(n2 + 1))/2
DO nu = ABS(l1 - l2), l1 + l2, 2
ll = ll + 1
CPASSERT(ll <= SIZE(integrals%eeri))
ALLOCATE (integrals%eeri(ll)%int(nn1, nn2))
integrals%eeri(ll)%int = 0._dp
eri => integrals%eeri(ll)%int
CALL sg_exchange(eri, nu, basis%am(1:n1, l1), l1, basis%am(1:n2, l2), l2)
END DO
END DO
END DO
END IF
CASE (CGTO_BASIS)
DO l = 0, lmat
n = integrals%n(l)
m = basis%nprim(l)
IF (n > 0 .AND. m > 0) THEN
ALLOCATE (omat(m, m))
CALL sg_overlap(omat(1:m, 1:m), l, basis%am(1:m, l), basis%am(1:m, l))
CALL contract2(integrals%ovlp(1:n, 1:n, l), omat(1:m, 1:m), basis%cm(1:m, 1:n, l))
CALL sg_kinetic(omat(1:m, 1:m), l, basis%am(1:m, l), basis%am(1:m, l))
CALL contract2(integrals%kin(1:n, 1:n, l), omat(1:m, 1:m), basis%cm(1:m, 1:n, l))
DEALLOCATE (omat)
END IF
END DO
IF (integrals%eri_coulomb) THEN
ll = 0
DO l1 = 0, lmat
n1 = integrals%n(l1)
nn1 = (n1*(n1 + 1))/2
m1 = basis%nprim(l1)
mm1 = (m1*(m1 + 1))/2
DO l2 = 0, l1
n2 = integrals%n(l2)
nn2 = (n2*(n2 + 1))/2
m2 = basis%nprim(l2)
mm2 = (m2*(m2 + 1))/2
IF (integrals%all_nu) THEN
nx = MIN(2*l1, 2*l2)
ELSE
nx = 0
END IF
DO nu = 0, nx, 2
ll = ll + 1
CPASSERT(ll <= SIZE(integrals%ceri))
ALLOCATE (integrals%ceri(ll)%int(nn1, nn2))
integrals%ceri(ll)%int = 0._dp
ALLOCATE (omat(mm1, mm2))
eri => integrals%ceri(ll)%int
CALL sg_coulomb(omat, nu, basis%am(1:m1, l1), l1, basis%am(1:m2, l2), l2)
CALL contract4(eri, omat, basis%cm(1:m1, 1:n1, l1), basis%cm(1:m2, 1:n2, l2))
DEALLOCATE (omat)
END DO
END DO
END DO
END IF
IF (integrals%eri_exchange) THEN
ll = 0
DO l1 = 0, lmat
n1 = integrals%n(l1)
nn1 = (n1*(n1 + 1))/2
m1 = basis%nprim(l1)
mm1 = (m1*(m1 + 1))/2
DO l2 = 0, l1
n2 = integrals%n(l2)
nn2 = (n2*(n2 + 1))/2
m2 = basis%nprim(l2)
mm2 = (m2*(m2 + 1))/2
DO nu = ABS(l1 - l2), l1 + l2, 2
ll = ll + 1
CPASSERT(ll <= SIZE(integrals%eeri))
ALLOCATE (integrals%eeri(ll)%int(nn1, nn2))
integrals%eeri(ll)%int = 0._dp
ALLOCATE (omat(mm1, mm2))
eri => integrals%eeri(ll)%int
CALL sg_exchange(omat, nu, basis%am(1:m1, l1), l1, basis%am(1:m2, l2), l2)
CALL contract4(eri, omat, basis%cm(1:m1, 1:n1, l1), basis%cm(1:m2, 1:n2, l2))
DEALLOCATE (omat)
END DO
END DO
END DO
END IF
CASE (STO_BASIS)
DO l = 0, lmat
n = integrals%n(l)
CALL sto_overlap(integrals%ovlp(1:n, 1:n, l), basis%ns(1:n, l), basis%as(1:n, l), &
basis%ns(1:n, l), basis%as(1:n, l))
CALL sto_kinetic(integrals%kin(1:n, 1:n, l), l, basis%ns(1:n, l), basis%as(1:n, l), &
basis%ns(1:n, l), basis%as(1:n, l))
END DO
CPASSERT(.NOT. integrals%eri_coulomb)
CPASSERT(.NOT. integrals%eri_exchange)
CASE (NUM_BASIS)
CPABORT("")
END SELECT
! setup transformation matrix to get an orthogonal basis, remove linear dependencies
NULLIFY (integrals%utrans, integrals%uptrans)
n = MAXVAL(basis%nbas)
ALLOCATE (integrals%utrans(n, n, 0:lmat), integrals%uptrans(n, n, 0:lmat))
integrals%utrans = 0._dp
integrals%uptrans = 0._dp
integrals%nne = integrals%n
lwork = 10*n
ALLOCATE (omat(n, n), vmat(n, n), w(n), work(lwork))
DO l = 0, lmat
n = integrals%n(l)
IF (n > 0) THEN
omat(1:n, 1:n) = integrals%ovlp(1:n, 1:n, l)
CALL jacobi(omat(1:n, 1:n), w(1:n), vmat(1:n, 1:n))
omat(1:n, 1:n) = vmat(1:n, 1:n)
ii = 0
DO i = 1, n
IF (w(i) > basis%eps_eig) THEN
ii = ii + 1
integrals%utrans(1:n, ii, l) = omat(1:n, i)/SQRT(w(i))
END IF
END DO
integrals%nne(l) = ii
IF (ii > 0) THEN
omat(1:ii, 1:ii) = MATMUL(TRANSPOSE(integrals%utrans(1:n, 1:ii, l)), integrals%utrans(1:n, 1:ii, l))
DO i = 1, ii
integrals%uptrans(i, i, l) = 1._dp
END DO
CALL dgesv(ii, ii, omat(1:ii, 1:ii), ii, ipiv, integrals%uptrans(1:ii, 1:ii, l), ii, info)
CPASSERT(info == 0)
END IF
END IF
END DO
DEALLOCATE (omat, vmat, w, work)
END IF
CALL timestop(handle)
END SUBROUTINE atom_int_setup
! **************************************************************************************************
!> \brief ...
!> \param integrals ...
!> \param basis ...
!> \param potential ...
! **************************************************************************************************
SUBROUTINE atom_ppint_setup(integrals, basis, potential)
TYPE(atom_integrals), INTENT(INOUT) :: integrals
TYPE(atom_basis_type), INTENT(INOUT) :: basis
TYPE(atom_potential_type), INTENT(IN) :: potential
CHARACTER(len=*), PARAMETER :: routineN = 'atom_ppint_setup'
INTEGER :: handle, i, ii, j, k, l, m, n
REAL(KIND=dp) :: al, alpha, rc
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: cpot, xmat
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: omat, spmat
REAL(KIND=dp), DIMENSION(:), POINTER :: rad
CALL timeset(routineN, handle)
IF (integrals%ppstat == 0) THEN
n = MAXVAL(basis%nbas)
integrals%n = basis%nbas
NULLIFY (integrals%core, integrals%hnl)
ALLOCATE (integrals%hnl(n, n, 0:lmat))
integrals%hnl = 0._dp
ALLOCATE (integrals%core(n, n, 0:lmat))
integrals%core = 0._dp
ALLOCATE (integrals%clsd(n, n, 0:lmat))
integrals%clsd = 0._dp
integrals%ppstat = 1
SELECT CASE (basis%basis_type)
CASE DEFAULT
CPABORT("")
CASE (GTO_BASIS)
SELECT CASE (potential%ppot_type)
CASE (no_pseudo, ecp_pseudo)
DO l = 0, lmat
n = integrals%n(l)
CALL sg_nuclear(integrals%core(1:n, 1:n, l), l, basis%am(1:n, l), basis%am(1:n, l))
END DO
CASE (gth_pseudo)
alpha = 1._dp/potential%gth_pot%rc/SQRT(2._dp)
DO l = 0, lmat
n = integrals%n(l)
ALLOCATE (omat(n, n), spmat(n, 5))
omat = 0._dp
CALL sg_erf(omat(1:n, 1:n), l, alpha, basis%am(1:n, l), basis%am(1:n, l))
integrals%core(1:n, 1:n, l) = -potential%gth_pot%zion*omat(1:n, 1:n)
DO i = 1, potential%gth_pot%ncl
omat = 0._dp
CALL sg_gpot(omat(1:n, 1:n), i - 1, potential%gth_pot%rc, l, basis%am(1:n, l), basis%am(1:n, l))
integrals%core(1:n, 1:n, l) = integrals%core(1:n, 1:n, l) + &
potential%gth_pot%cl(i)*omat(1:n, 1:n)
END DO
IF (potential%gth_pot%lpotextended) THEN
DO k = 1, potential%gth_pot%nexp_lpot
DO i = 1, potential%gth_pot%nct_lpot(k)
omat = 0._dp
CALL sg_gpot(omat(1:n, 1:n), i - 1, potential%gth_pot%alpha_lpot(k), l, &
basis%am(1:n, l), basis%am(1:n, l))
integrals%core(1:n, 1:n, l) = integrals%core(1:n, 1:n, l) + &
potential%gth_pot%cval_lpot(i, k)*omat(1:n, 1:n)
END DO
END DO
END IF
IF (potential%gth_pot%lsdpot) THEN
DO k = 1, potential%gth_pot%nexp_lsd
DO i = 1, potential%gth_pot%nct_lsd(k)
omat = 0._dp
CALL sg_gpot(omat(1:n, 1:n), i - 1, potential%gth_pot%alpha_lsd(k), l, &
basis%am(1:n, l), basis%am(1:n, l))
integrals%clsd(1:n, 1:n, l) = integrals%clsd(1:n, 1:n, l) + &
potential%gth_pot%cval_lsd(i, k)*omat(1:n, 1:n)
END DO
END DO
END IF
spmat = 0._dp
m = potential%gth_pot%nl(l)
DO i = 1, m
CALL sg_proj_ol(spmat(1:n, i), l, basis%am(1:n, l), i - 1, potential%gth_pot%rcnl(l))
END DO
integrals%hnl(1:n, 1:n, l) = MATMUL(spmat(1:n, 1:m), &
MATMUL(potential%gth_pot%hnl(1:m, 1:m, l), TRANSPOSE(spmat(1:n, 1:m))))
DEALLOCATE (omat, spmat)
END DO
CASE (upf_pseudo)
CALL upfint_setup(integrals, basis, potential)
CASE (sgp_pseudo)
CALL sgpint_setup(integrals, basis, potential)
CASE DEFAULT
CPABORT("")
END SELECT
CASE (CGTO_BASIS)
SELECT CASE (potential%ppot_type)
CASE (no_pseudo, ecp_pseudo)
DO l = 0, lmat
n = integrals%n(l)
m = basis%nprim(l)
ALLOCATE (omat(m, m))
CALL sg_nuclear(omat(1:m, 1:m), l, basis%am(1:m, l), basis%am(1:m, l))
CALL contract2(integrals%core(1:n, 1:n, l), omat(1:m, 1:m), basis%cm(1:m, 1:n, l))
DEALLOCATE (omat)
END DO
CASE (gth_pseudo)
alpha = 1._dp/potential%gth_pot%rc/SQRT(2._dp)
DO l = 0, lmat
n = integrals%n(l)
m = basis%nprim(l)
IF (n > 0 .AND. m > 0) THEN
ALLOCATE (omat(m, m), spmat(n, 5), xmat(m))
omat = 0._dp
CALL sg_erf(omat(1:m, 1:m), l, alpha, basis%am(1:m, l), basis%am(1:m, l))
omat(1:m, 1:m) = -potential%gth_pot%zion*omat(1:m, 1:m)
CALL contract2(integrals%core(1:n, 1:n, l), omat(1:m, 1:m), basis%cm(1:m, 1:n, l))
DO i = 1, potential%gth_pot%ncl
omat = 0._dp
CALL sg_gpot(omat(1:m, 1:m), i - 1, potential%gth_pot%rc, l, basis%am(1:m, l), basis%am(1:m, l))
omat(1:m, 1:m) = potential%gth_pot%cl(i)*omat(1:m, 1:m)
CALL contract2add(integrals%core(1:n, 1:n, l), omat(1:m, 1:m), basis%cm(1:m, 1:n, l))
END DO
IF (potential%gth_pot%lpotextended) THEN
DO k = 1, potential%gth_pot%nexp_lpot
DO i = 1, potential%gth_pot%nct_lpot(k)
omat = 0._dp
CALL sg_gpot(omat(1:m, 1:m), i - 1, potential%gth_pot%alpha_lpot(k), l, &
basis%am(1:m, l), basis%am(1:m, l))
omat(1:m, 1:m) = potential%gth_pot%cval_lpot(i, k)*omat(1:m, 1:m)
CALL contract2add(integrals%core(1:n, 1:n, l), omat(1:m, 1:m), basis%cm(1:m, 1:n, l))
END DO
END DO
END IF
IF (potential%gth_pot%lsdpot) THEN
DO k = 1, potential%gth_pot%nexp_lsd
DO i = 1, potential%gth_pot%nct_lsd(k)
omat = 0._dp
CALL sg_gpot(omat(1:m, 1:m), i - 1, potential%gth_pot%alpha_lsd(k), l, &
basis%am(1:m, l), basis%am(1:m, l))
omat(1:m, 1:m) = potential%gth_pot%cval_lsd(i, k)*omat(1:m, 1:m)
CALL contract2add(integrals%clsd(1:n, 1:n, l), omat(1:m, 1:m), basis%cm(1:m, 1:n, l))
END DO
END DO
END IF
spmat = 0._dp
k = potential%gth_pot%nl(l)
DO i = 1, k
CALL sg_proj_ol(xmat(1:m), l, basis%am(1:m, l), i - 1, potential%gth_pot%rcnl(l))
spmat(1:n, i) = MATMUL(TRANSPOSE(basis%cm(1:m, 1:n, l)), xmat(1:m))
END DO
IF (k > 0) THEN
integrals%hnl(1:n, 1:n, l) = MATMUL(spmat(1:n, 1:k), &
MATMUL(potential%gth_pot%hnl(1:k, 1:k, l), &
TRANSPOSE(spmat(1:n, 1:k))))
END IF
DEALLOCATE (omat, spmat, xmat)
END IF
END DO
CASE (upf_pseudo)
CALL upfint_setup(integrals, basis, potential)
CASE (sgp_pseudo)
CALL sgpint_setup(integrals, basis, potential)
CASE DEFAULT
CPABORT("")
END SELECT
CASE (STO_BASIS)
SELECT CASE (potential%ppot_type)
CASE (no_pseudo, ecp_pseudo)
DO l = 0, lmat
n = integrals%n(l)
CALL sto_nuclear(integrals%core(1:n, 1:n, l), basis%ns(1:n, l), basis%as(1:n, l), &
basis%ns(1:n, l), basis%as(1:n, l))
END DO
CASE (gth_pseudo)
rad => basis%grid%rad
m = basis%grid%nr
ALLOCATE (cpot(1:m))
rc = potential%gth_pot%rc
alpha = 1._dp/rc/SQRT(2._dp)
! local pseudopotential, we use erf = 1/r - erfc
integrals%core = 0._dp
DO i = 1, m
cpot(i) = potential%gth_pot%zion*erfc(alpha*rad(i))/rad(i)
END DO
DO i = 1, potential%gth_pot%ncl
ii = 2*(i - 1)
cpot(1:m) = cpot(1:m) + potential%gth_pot%cl(i)*(rad/rc)**ii*EXP(-0.5_dp*(rad/rc)**2)
END DO
IF (potential%gth_pot%lpotextended) THEN
DO k = 1, potential%gth_pot%nexp_lpot
al = potential%gth_pot%alpha_lpot(k)
DO i = 1, potential%gth_pot%nct_lpot(k)
ii = 2*(i - 1)
cpot(1:m) = cpot(1:m) + potential%gth_pot%cval_lpot(i, k)*(rad/al)**ii*EXP(-0.5_dp*(rad/al)**2)
END DO
END DO
END IF
CALL numpot_matrix(integrals%core, cpot, basis, 0)
DO l = 0, lmat
n = integrals%n(l)
ALLOCATE (omat(n, n))
omat = 0._dp
CALL sto_nuclear(omat(1:n, 1:n), basis%ns(1:n, l), basis%as(1:n, l), &
basis%ns(1:n, l), basis%as(1:n, l))
integrals%core(1:n, 1:n, l) = integrals%core(1:n, 1:n, l) - potential%gth_pot%zion*omat(1:n, 1:n)
DEALLOCATE (omat)
END DO
IF (potential%gth_pot%lsdpot) THEN
cpot = 0._dp
DO k = 1, potential%gth_pot%nexp_lsd
al = potential%gth_pot%alpha_lsd(k)
DO i = 1, potential%gth_pot%nct_lsd(k)
ii = 2*(i - 1)
cpot(:) = cpot + potential%gth_pot%cval_lsd(i, k)*(rad/al)**ii*EXP(-0.5_dp*(rad/al)**2)
END DO
END DO
CALL numpot_matrix(integrals%clsd, cpot, basis, 0)
END IF
DO l = 0, lmat
n = integrals%n(l)
! non local pseudopotential
ALLOCATE (spmat(n, 5))
spmat = 0._dp
k = potential%gth_pot%nl(l)
DO i = 1, k
rc = potential%gth_pot%rcnl(l)
cpot(:) = sqrt2/SQRT(gamma1(l + 2*i - 1))*rad**(l + 2*i - 2)*EXP(-0.5_dp*(rad/rc)**2)/rc**(l + 2*i - 0.5_dp)
DO j = 1, basis%nbas(l)
spmat(j, i) = integrate_grid(cpot, basis%bf(:, j, l), basis%grid)
END DO
END DO
integrals%hnl(1:n, 1:n, l) = MATMUL(spmat(1:n, 1:k), &
MATMUL(potential%gth_pot%hnl(1:k, 1:k, l), &
TRANSPOSE(spmat(1:n, 1:k))))
DEALLOCATE (spmat)
END DO
DEALLOCATE (cpot)
CASE (upf_pseudo)
CALL upfint_setup(integrals, basis, potential)
CASE (sgp_pseudo)
CALL sgpint_setup(integrals, basis, potential)
CASE DEFAULT
CPABORT("")
END SELECT
CASE (NUM_BASIS)
CPABORT("")
END SELECT
! add ecp_pseudo using numerical representation of basis
IF (potential%ppot_type == ecp_pseudo) THEN
! scale 1/r potential
integrals%core = -potential%ecp_pot%zion*integrals%core
! local potential
m = basis%grid%nr
rad => basis%grid%rad
ALLOCATE (cpot(1:m))
cpot = 0._dp
DO k = 1, potential%ecp_pot%nloc
n = potential%ecp_pot%nrloc(k)
alpha = potential%ecp_pot%bloc(k)
cpot(:) = cpot + potential%ecp_pot%aloc(k)*rad**(n - 2)*EXP(-alpha*rad**2)
END DO
CALL numpot_matrix(integrals%core, cpot, basis, 0)
! non local pseudopotential
DO l = 0, MIN(potential%ecp_pot%lmax, lmat)
cpot = 0._dp
DO k = 1, potential%ecp_pot%npot(l)
n = potential%ecp_pot%nrpot(k, l)
alpha = potential%ecp_pot%bpot(k, l)
cpot(:) = cpot + potential%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)
integrals%hnl(i, j, l) = integrate_grid(cpot, basis%bf(:, i, l), basis%bf(:, j, l), basis%grid)
integrals%hnl(j, i, l) = integrals%hnl(i, j, l)
END DO
END DO
END DO
DEALLOCATE (cpot)
END IF
END IF
CALL timestop(handle)
END SUBROUTINE atom_ppint_setup
! **************************************************************************************************
!> \brief ...
!> \param integrals ...
!> \param basis ...
!> \param potential ...
! **************************************************************************************************
SUBROUTINE upfint_setup(integrals, basis, potential)
TYPE(atom_integrals), INTENT(INOUT) :: integrals
TYPE(atom_basis_type), INTENT(INOUT) :: basis
TYPE(atom_potential_type), INTENT(IN) :: potential
CHARACTER(len=4) :: ptype
INTEGER :: i, j, k1, k2, la, lb, m, n
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, potential%upf_pot%r, potential%upf_pot%rab)
! local pseudopotential
integrals%core = 0._dp
CALL numpot_matrix(integrals%core, potential%upf_pot%vlocal, gbasis, 0)
ptype = ADJUSTL(TRIM(potential%upf_pot%pseudo_type))
IF (ptype(1:2) == "NC" .OR. ptype(1:2) == "US") THEN
! non local pseudopotential
n = MAXVAL(integrals%n(:))
m = potential%upf_pot%number_of_proj
ALLOCATE (spmat(n, m))
spmat = 0.0_dp
DO i = 1, m
la = potential%upf_pot%lbeta(i)
DO j = 1, gbasis%nbas(la)
spmat(j, i) = integrate_grid(potential%upf_pot%beta(:, i), gbasis%bf(:, j, la), gbasis%grid)
END DO
END DO
DO i = 1, m
la = potential%upf_pot%lbeta(i)
DO j = 1, m
lb = potential%upf_pot%lbeta(j)
IF (la == lb) THEN
DO k1 = 1, gbasis%nbas(la)
DO k2 = 1, gbasis%nbas(la)
integrals%hnl(k1, k2, la) = integrals%hnl(k1, k2, la) + &
spmat(k1, i)*potential%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, potential%upf_pot%l_max
IF (la == potential%upf_pot%l_local) CYCLE
m = SIZE(potential%upf_pot%vsemi(:, la + 1))
ALLOCATE (spot(m))
spot(:) = potential%upf_pot%vsemi(:, la + 1) - potential%upf_pot%vlocal(:)
n = basis%nbas(la)
DO i = 1, n
DO j = i, n
integrals%core(i, j, la) = integrals%core(i, j, la) + &
integrate_grid(spot(:), &
gbasis%bf(:, i, la), gbasis%bf(:, j, la), gbasis%grid)
integrals%core(j, i, la) = integrals%core(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 upfint_setup
! **************************************************************************************************
!> \brief ...
!> \param integrals ...
!> \param basis ...
!> \param potential ...
! **************************************************************************************************
SUBROUTINE sgpint_setup(integrals, basis, potential)
TYPE(atom_integrals), INTENT(INOUT) :: integrals
TYPE(atom_basis_type), INTENT(INOUT) :: basis
TYPE(atom_potential_type), INTENT(IN) :: potential
INTEGER :: i, ia, j, l, m, n, na
REAL(KIND=dp) :: a, c, rc, 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
integrals%core = 0._dp
ALLOCATE (cpot(m))
cpot = 0.0_dp
zval = potential%sgp_pot%zion
DO i = 1, m
rc = rad(i)/potential%sgp_pot%ac_local/SQRT(2.0_dp)
cpot(i) = cpot(i) - zval/rad(i)*erf(rc)
END DO
DO i = 1, potential%sgp_pot%n_local
cpot(:) = cpot(:) + potential%sgp_pot%c_local(i)*EXP(-potential%sgp_pot%a_local(i)*rad(:)**2)
END DO
CALL numpot_matrix(integrals%core, cpot, basis, 0)
DEALLOCATE (cpot)
! nonlocal pseudopotential
integrals%hnl = 0.0_dp
IF (potential%sgp_pot%has_nonlocal) THEN
ALLOCATE (pgauss(1:m))
n = potential%sgp_pot%n_nonlocal
!
DO l = 0, potential%sgp_pot%lmax
CPASSERT(l <= UBOUND(basis%nbas, 1))
IF (.NOT. potential%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 = potential%sgp_pot%a_nonlocal(j)
c = potential%sgp_pot%c_nonlocal(j, i, l)
pgauss(:) = pgauss(:) + c*EXP(-a*rad(:)**2)*rad(:)**l
END DO
DO ia = 1, na
qmat(ia, i) = SUM(basis%bf(:, ia, l)*pgauss(:)*basis%grid%wr(:))
END DO
END DO
DO i = 1, na
DO j = i, na
DO ia = 1, n
integrals%hnl(i, j, l) = integrals%hnl(i, j, l) &
+ qmat(i, ia)*qmat(j, ia)*potential%sgp_pot%h_nonlocal(ia, l)
END DO
integrals%hnl(j, i, l) = integrals%hnl(i, j, l)
END DO
END DO
DEALLOCATE (qmat)
END DO
DEALLOCATE (pgauss)
END IF
END SUBROUTINE sgpint_setup
! **************************************************************************************************
!> \brief ...
!> \param integrals ...
!> \param basis ...
!> \param reltyp ...
!> \param zcore ...
!> \param alpha ...
! **************************************************************************************************
SUBROUTINE atom_relint_setup(integrals, basis, reltyp, zcore, alpha)
TYPE(atom_integrals), INTENT(INOUT) :: integrals
TYPE(atom_basis_type), INTENT(INOUT) :: basis
INTEGER, INTENT(IN) :: reltyp
REAL(dp), INTENT(IN) :: zcore
REAL(dp), INTENT(IN), OPTIONAL :: alpha
CHARACTER(len=*), PARAMETER :: routineN = 'atom_relint_setup'
INTEGER :: dkhorder, handle, i, k1, k2, l, m, n, nl
REAL(dp) :: ascal
REAL(dp), ALLOCATABLE, DIMENSION(:) :: cpot
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: modpot
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :) :: ener, sps
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:, :, :) :: hmat, pvp, sp, tp, vp, wfn
CALL timeset(routineN, handle)
SELECT CASE (reltyp)
CASE DEFAULT
CPABORT("")
CASE (do_nonrel_atom, do_zoramp_atom, do_sczoramp_atom)
dkhorder = -1
CASE (do_dkh0_atom)
dkhorder = 0
CASE (do_dkh1_atom)
dkhorder = 1
CASE (do_dkh2_atom)
dkhorder = 2
CASE (do_dkh3_atom)
dkhorder = 3
END SELECT
SELECT CASE (reltyp)
CASE DEFAULT
CPABORT("")
CASE (do_nonrel_atom)
! nothing to do
NULLIFY (integrals%tzora, integrals%hdkh)
CASE (do_zoramp_atom, do_sczoramp_atom)
NULLIFY (integrals%hdkh)
IF (integrals%zorastat == 0) THEN
n = MAXVAL(basis%nbas)
ALLOCATE (integrals%tzora(n, n, 0:lmat))
integrals%tzora = 0._dp
m = basis%grid%nr
ALLOCATE (modpot(1:m), cpot(1:m))
CALL calculate_model_potential(modpot, basis%grid, zcore)
! Zora potential
cpot(1:m) = modpot(1:m)/(4._dp*c_light_au*c_light_au - 2._dp*modpot(1:m))
cpot(1:m) = cpot(1:m)/basis%grid%rad2(1:m)
CALL numpot_matrix(integrals%tzora, cpot, basis, 0)
DO l = 0, lmat
nl = basis%nbas(l)
integrals%tzora(1:nl, 1:nl, l) = REAL(l*(l + 1), dp)*integrals%tzora(1:nl, 1:nl, l)
END DO
cpot(1:m) = cpot(1:m)*basis%grid%rad2(1:m)
CALL numpot_matrix(integrals%tzora, cpot, basis, 2)
!
! scaled ZORA
IF (reltyp == do_sczoramp_atom) THEN
ALLOCATE (hmat(n, n, 0:lmat), wfn(n, n, 0:lmat), ener(n, 0:lmat), pvp(n, n, 0:lmat), sps(n, n))
hmat(:, :, :) = integrals%kin + integrals%tzora
! model potential
CALL numpot_matrix(hmat, modpot, basis, 0)
! eigenvalues and eigenvectors
CALL atom_solve(hmat, integrals%utrans, wfn, ener, basis%nbas, integrals%nne, lmat)
! relativistic kinetic energy
cpot(1:m) = c_light_au*c_light_au/(2._dp*c_light_au*c_light_au - modpot(1:m))**2
cpot(1:m) = cpot(1:m)/basis%grid%rad2(1:m)
pvp = 0.0_dp
CALL numpot_matrix(pvp, cpot, basis, 0)
DO l = 0, lmat
nl = basis%nbas(l)
pvp(1:nl, 1:nl, l) = REAL(l*(l + 1), dp)*pvp(1:nl, 1:nl, l)
END DO
cpot(1:m) = cpot(1:m)*basis%grid%rad2(1:m)
CALL numpot_matrix(pvp, cpot, basis, 2)
! calculate psi*pvp*psi and the scaled orbital energies
! actually, we directly calculate the energy difference
DO l = 0, lmat
nl = basis%nbas(l)
DO i = 1, integrals%nne(l)
IF (ener(i, l) < 0._dp) THEN
ascal = SUM(wfn(1:nl, i, l)*MATMUL(pvp(1:nl, 1:nl, l), wfn(1:nl, i, l)))
ener(i, l) = ener(i, l)*ascal/(1.0_dp + ascal)
ELSE
ener(i, l) = 0.0_dp
END IF
END DO
END DO
! correction term is calculated as a projector
hmat = 0.0_dp
DO l = 0, lmat
nl = basis%nbas(l)
DO i = 1, integrals%nne(l)
DO k1 = 1, nl
DO k2 = 1, nl
hmat(k1, k2, l) = hmat(k1, k2, l) + ener(i, l)*wfn(k1, i, l)*wfn(k2, i, l)
END DO
END DO
END DO
! transform with overlap matrix
sps(1:nl, 1:nl) = MATMUL(integrals%ovlp(1:nl, 1:nl, l), &
MATMUL(hmat(1:nl, 1:nl, l), integrals%ovlp(1:nl, 1:nl, l)))
! add scaling correction to tzora
integrals%tzora(1:nl, 1:nl, l) = integrals%tzora(1:nl, 1:nl, l) - sps(1:nl, 1:nl)
END DO
DEALLOCATE (hmat, wfn, ener, pvp, sps)
END IF
!
DEALLOCATE (modpot, cpot)
integrals%zorastat = 1
END IF
CASE (do_dkh0_atom, do_dkh1_atom, do_dkh2_atom, do_dkh3_atom)
NULLIFY (integrals%tzora)
IF (integrals%dkhstat == 0) THEN
n = MAXVAL(basis%nbas)
ALLOCATE (integrals%hdkh(n, n, 0:lmat))
integrals%hdkh = 0._dp
m = MAXVAL(basis%nprim)
ALLOCATE (tp(m, m, 0:lmat), sp(m, m, 0:lmat), vp(m, m, 0:lmat), pvp(m, m, 0:lmat))
tp = 0._dp
sp = 0._dp
vp = 0._dp
pvp = 0._dp
SELECT CASE (basis%basis_type)
CASE DEFAULT
CPABORT("")
CASE (GTO_BASIS, CGTO_BASIS)
DO l = 0, lmat
m = basis%nprim(l)
IF (m > 0) THEN
CALL sg_kinetic(tp(1:m, 1:m, l), l, basis%am(1:m, l), basis%am(1:m, l))
CALL sg_overlap(sp(1:m, 1:m, l), l, basis%am(1:m, l), basis%am(1:m, l))
IF (PRESENT(alpha)) THEN
CALL sg_erfc(vp(1:m, 1:m, l), l, alpha, basis%am(1:m, l), basis%am(1:m, l))
ELSE
CALL sg_nuclear(vp(1:m, 1:m, l), l, basis%am(1:m, l), basis%am(1:m, l))
END IF
CALL sg_kinnuc(pvp(1:m, 1:m, l), l, basis%am(1:m, l), basis%am(1:m, l))
vp(1:m, 1:m, l) = -zcore*vp(1:m, 1:m, l)
pvp(1:m, 1:m, l) = -zcore*pvp(1:m, 1:m, l)
END IF
END DO
CASE (STO_BASIS)
CPABORT("")
CASE (NUM_BASIS)
CPABORT("")
END SELECT
CALL dkh_integrals(integrals, basis, dkhorder, sp, tp, vp, pvp)
integrals%dkhstat = 1
DEALLOCATE (tp, sp, vp, pvp)
ELSE
CPASSERT(ASSOCIATED(integrals%hdkh))
END IF
END SELECT
CALL timestop(handle)
END SUBROUTINE atom_relint_setup
! **************************************************************************************************
!> \brief ...
!> \param integrals ...
!> \param basis ...
!> \param order ...
!> \param sp ...
!> \param tp ...
!> \param vp ...
!> \param pvp ...
! **************************************************************************************************
SUBROUTINE dkh_integrals(integrals, basis, order, sp, tp, vp, pvp)
TYPE(atom_integrals), INTENT(INOUT) :: integrals
TYPE(atom_basis_type), INTENT(INOUT) :: basis
INTEGER, INTENT(IN) :: order
REAL(dp), DIMENSION(:, :, 0:) :: sp, tp, vp, pvp
INTEGER :: l, m, n
REAL(dp), DIMENSION(:, :, :), POINTER :: hdkh
CPASSERT(order >= 0)
hdkh => integrals%hdkh