hsefunctional.F90 111 KB
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!--------------------------------------------------------------------------------
! Copyright (c) 2016 Peter Grünberg Institut, Forschungszentrum Jülich, Germany
! This file is part of FLEUR and available as free software under the conditions
! of the MIT license as expressed in the LICENSE file in more detail.
!--------------------------------------------------------------------------------

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! Provides all procedures needed to implement the HSE-exchange functional
! within a FLAPW framework
! Author: M. Schlipf 2009
MODULE m_hsefunctional
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  USE m_judft
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  IMPLICIT NONE
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#ifdef __PGI
    REAL,EXTERNAL ::erfc
#endif
  ! Constant omega of the HSE exchange functional
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   REAL, PARAMETER :: omega_HSE = 0.11
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  ! Constant for the maximum number of G points
  INTEGER, PARAMETER :: maxNoGPts = 50

  ! these arrays are calculated once and then reused
  LOGICAL, ALLOCATABLE   :: already_known(:)
  REAL, ALLOCATABLE      :: known_potential(:,:)
#ifdef CPP_INVERSION
  REAL, ALLOCATABLE      :: known_fourier_trafo(:,:,:)
#else
  COMPLEX, ALLOCATABLE   :: known_fourier_trafo(:,:,:)
#endif

  ! variables needed for fourier transformation
  PRIVATE already_known, known_potential, known_fourier_trafo

  ! functions needed internally
  PRIVATE calcYlm, calcSphBes, my_dot_product3x1, my_dot_product3x3, my_dot_product4x1, &
       my_dot_product4x4, my_sum3d, my_sum4d, gPtsSummation, approximateIntegral, generateIntegrals, &
       integrateY1ExpErfc, integrateY3ExpErfc, integrateY5ExpErfc, integrateY7ExpErfc, &
       integrateY9ExpErfc, calculateF, calculateG, calculateH

  INTERFACE my_sum
     MODULE PROCEDURE my_sum3d, my_sum4d
  END INTERFACE

  INTERFACE my_dot_product
     MODULE PROCEDURE my_dot_product4x1, my_dot_product4x4, my_dot_product3x1, my_dot_product3x3
  END INTERFACE

  INTERFACE gPtsSummation
     MODULE PROCEDURE crc_gPtsSummation, rrr_gPtsSummation
  END INTERFACE

CONTAINS

  ! Calculate the enhancement factor of the HSE03 functional
  ! References:
  ! [1] Heyd, Scuseria, Ernzerhof: Hybrid functionals based on a screened Coulomb potential,
  !     J. Chem. Phys. 118 (2003) 8207-8215
  ! [2] Heyd, Scuseria: Assessment and validation of a screened Coulomb hybrid density
  !     functional, J. Chem. Phys. 120 (2004) 7274-7280
  ! [3] Ernzerhof, Perdew: Generalized gradient approximation to the angle- and system-
  !     averaged exchange hole, J. Chem. Phys. 109 (1998) 3313-3319
  ! Input:  kF       - kF = (3 pi^2 rho) ** 1/3
  !         s_inp    - reduced gradient = |grad rho| / 2 kF rho
  ! Output: F_x      - HSE03 enhancement factor
  !         dFx_ds   - derivative of this factor with respect to s
  !         d2Fx_ds2 - second derivative with respect to s
  SUBROUTINE calculateEnhancementFactor(kF, s_inp, F_x, dFx_Ds, d2Fx_Ds2, dFx_dkF, d2Fx_dsdkF)

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    IMPLICIT NONE

    REAL, INTENT(IN)  :: kF, s_inp
    REAL, INTENT(OUT) :: F_x, dFx_ds, d2Fx_ds2
    REAL, INTENT(OUT) :: dFx_dkF, d2Fx_dsdkF

    ! Helper variables 
    REAL :: r1_kF, &                   ! 1 / kF
         omega_kF,omega_kF_Sqr, &      ! omega_HSE / kF, (omega/kF)^2
         s, s2, &                      ! reduced gradient s and s^2
         s_si2, r2s_si3, r6s_si4, &    ! quotient s_chg / s_inp^n (n = 2,3,4)
         F, G, H, &                    ! results of the functions F, G, and H defined in [3]
         Hs2, dHs2_ds, d2Hs2_ds2, &    ! H * s^2 and derivatives with respect to s
         Fs2, dFs2_ds, d2Fs2_ds2, &    ! F * s^2 and derivatives with respect to s
         dGs2_ds, d2Gs2_ds2, &         ! derivatives with respect to s of G s^2
         C_term, dCt_ds, d2Ct_ds2, &   ! C * (1 + F(s) s^2) and derivatives with respect to s
         E_term, dEt_ds, d2Et_ds2, &   ! E * (1 + G(s) s^2) and derivatives with respect to s
         D_term                        ! D + H s^2

    ! integrals of y^n Exp(-ay^2) Erfc(w/k y) for n = 1, 3, 5, 7, 9
    REAL :: intY1ExpErfc,intY3ExpErfc,intY5ExpErfc,intY7ExpErfc,intY9ExpErfc
    ! integrals of 2/sqrt(pi) y^n Exp(-arg*y^2) for n = 2, 4, 6, 8
    REAL :: r1_arg, intY2Gauss, intY4Gauss, intY6Gauss, intY8Gauss

    ! approximation of the not analytical part of the integral and derivatives
    REAL :: appInt, dAppInt_ds, d2AppInt_ds2
    REAL :: dAppInt_dkF, d2AppInt_dsdkF

    REAL, PARAMETER :: r8_9 = 8.0/9.0  ! 8/9
    REAL, PARAMETER :: &               ! Parameters of the exchange hole as defined in [3]
         B = -0.37170836, C = -0.077215461,&
         D =  0.57786348, E = -0.051955731

    REAL, PARAMETER :: &               ! Correction of the reduced gradient to ensure Lieb-Oxford bound [2]
         s_thresh = 8.3, s_max = 8.572844, s_chg = 18.796223
    LOGICAL :: correction              ! Has the value of s been corrected

    ! If a large value of s would violate the Lieb-Oxford bound, the value of s is reduced, 
    ! so that this condition is fullfilled
    correction = s_inp > s_thresh
    IF (correction) THEN
       s_si2 = s_chg / (s_inp * s_inp)
       s     = s_max - s_si2
    ELSE
       s = s_inp
    END IF

    ! Calculate different helper variables
    r1_kF        = 1.0 / kF
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    omega_kF     = omega_hse * r1_kF !was omega_VHSE()
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    omega_kF_Sqr = omega_kF * omega_kF

    ! calculate the functions H and F in [3] and its derivatives
    CALL calculateH(s, H, dHs2_ds, d2Hs2_ds2)
    CALL calculateF(s, H, dHs2_ds, d2Hs2_ds2, F, dFs2_ds, d2Fs2_ds2) 
    s2  = s * s
    Hs2 = H * s2
    Fs2 = F * s2
    CALL calculateG(s2, Fs2, dFs2_ds, d2Fs2_ds2, Hs2, dHs2_ds, d2Hs2_ds2,& !Input
                    G, dGs2_ds, d2Gs2_ds2)                                 !Output

    C_term   = C * (1 + s2 * F)
    dCt_ds   = C * dFs2_ds
    d2Ct_ds2 = C * d2Fs2_ds2
    E_term   = E * (1 + s2 * G)
    dEt_ds   = E * dGs2_ds
    d2Et_ds2 = E * d2Gs2_ds2
    D_term   = D + Hs2

    ! approximate the integral using an expansion of the error function (cf. [2])
    CALL approximateIntegral(omega_kF, Hs2, D_term, dHs2_ds, d2Hs2_ds2,& !Input
         appInt, dAppInt_ds, d2AppInt_ds2, dAppInt_dkF, d2AppInt_dsdkF)  !Output

    ! Calculate the integrals
    !
    ! inf
    !   /                2   2      /       \
    !  |     n  -(D+H(s)s ) y      | omega   |
    !  | dy y  e               Erfc| ----- y |
    !  |                           |   k     |
    ! /                             \   F   /
    !  0
    ! for n = 1, 3, 5, 7, and 9
    ! 
    intY1ExpErfc = integrateY1ExpErfc(omega_kf, omega_kf_Sqr, D_term)
    intY3ExpErfc = integrateY3ExpErfc(omega_kf, omega_kf_Sqr, D_term)
    intY5ExpErfc = integrateY5ExpErfc(omega_kf, omega_kf_Sqr, D_term)
    intY7ExpErfc = integrateY7ExpErfc(omega_kf, omega_kf_Sqr, D_term)
    intY9ExpErfc = integrateY9ExpErfc(omega_kf, omega_kf_Sqr, D_term)

    ! Calculate the integrals
    !
    !       inf
    !         /          /                      2    \
    !   2    |     n    |          2   2   omega   2  |
    ! -----  | dy y  Exp| -(D+H(s)s ) y  - ------ y   |
    !  ____  |          |                   k ^2      |
    ! V Pi  /            \                   F       /
    !        0
    ! for n = 2, 4, 6, 8
    !
    r1_arg     = 1.0 / (D_term + omega_kF_Sqr)
    intY2Gauss = 0.5 * SQRT(r1_arg) * r1_arg
    intY4Gauss = 1.5 * intY2Gauss   * r1_arg
    intY6Gauss = 2.5 * intY4Gauss   * r1_arg
    intY8Gauss = 3.5 * intY6Gauss   * r1_arg

    ! Calculate the integral 
    !  inf
    !    /                 /       \
    !   |                 | omega   |
    !   | dy y J(s,y) Erfc| ----- y |
    !   |                 |   k     |
    !  /                   \   F   /
    !   0
    ! where J(s, y) is the exchange hole defined in [3]    
    ! the exchange factor is proportional to this integral
    F_x = - r8_9 * (appInt + B * intY1ExpErfc + C_term * intY3ExpErfc + E_term * intY5ExpErfc)

    ! Calculate the derivatives with respect to s using that the derivatative of the integral
    ! yields higher orders of the same kind of integral intY1 -> -intY3 -> intY5 ... times
    ! the derivative of the exponent
    dFx_ds   = -r8_9 * (dAppInt_ds - (B * intY3ExpErfc + C_term * intY5ExpErfc + E_term * intY7ExpErfc) * dHs2_ds &
                                   + dCt_ds * intY3ExpErfc + dEt_ds * intY5ExpErfc)
    d2Fx_ds2 = -r8_9 * (d2AppInt_ds2 + (B * intY5ExpErfc + C_term * intY7ExpErfc + E_term * intY9ExpErfc) * dHs2_ds**2 &
                                     - (B * intY3ExpErfc + C_term * intY5ExpErfc + E_term * intY7ExpErfc) * d2Hs2_ds2 &
                                     - 2.0 * (dCt_ds * intY5ExpErfc + dEt_ds * intY7ExpErfc) * dHs2_ds &
                                     + d2Ct_ds2 * intY3ExpErfc + d2Et_ds2 * intY5ExpErfc)

    dFx_dkF    = -r8_9 * r1_kF * (omega_kF * (B * intY2Gauss + C_term * intY4Gauss + E_term * intY6Gauss) + dAppInt_dkF)
    d2Fx_dsdkF = -r8_9 * r1_kF * ( d2AppInt_dsdkF + omega_kF * ( dCt_ds * intY4Gauss + dEt_ds * intY6Gauss &
         - (B * intY4Gauss + C_term * intY6Gauss + E_term * intY8Gauss) * dHs2_ds ) )

    ! Correction to the derivatives, if s_inp > s_thresh
    IF (correction) THEN
      r2s_si3    = 2.0 * s_si2   / s_inp
      r6s_si4    = 3.0 * r2s_si3 / s_inp
      d2Fx_ds2   = d2Fx_ds2   * r2s_si3**2 - dFx_ds * r6s_si4
      d2Fx_dsdkF = d2Fx_dsdkF * r2s_si3
      dFx_ds     = dFx_ds     * r2s_si3
    END IF

  END SUBROUTINE calculateEnhancementFactor

  ! Approximation for the first part of the exchange hole integral
  ! Calculated using the algorithm described in [2].
  ! Additionally the first and second derivative of this approximation with
  ! respect to s are calculated
  ! Input:  omega_kF       - omega / kF 
  !         Hs2            - H s^2
  !         D_Hs2          - D + H s^2
  !         dHs2_ds        - d (H s^2) / ds 
  !         d2Hs2_ds2      - d^2 (H s^2) / ds^2
  ! Output: appInt         - approximation for one part of the exchange hole integral 
  !         dAppInt_ds     - first derivative with respect to s
  !         d2AppInt_ds2   - second derivative with respect to s
  !         dAppInt_dkF    - first derivative with respect to kF
  !         d2AppInt_dsdkF - mixed derivative with respect to s and kF
  SUBROUTINE approximateIntegral(omega_kF, Hs2, D_Hs2, dHs2_ds, d2Hs2_ds2,&
       appInt, dAppInt_ds, d2AppInt_ds2, dAppInt_dkF, d2AppInt_dsdkF)

    USE m_exponential_integral, ONLY: calculateExponentialIntegral, gauss_laguerre

    IMPLICIT NONE

    REAL, INTENT(IN)  :: omega_kF, Hs2, D_Hs2, dHs2_ds, d2Hs2_ds2
    REAL, INTENT(OUT) :: appInt, dAppInt_ds, d2AppInt_ds2
    REAL, INTENT(OUT) :: dAppInt_dkF, d2AppInt_dsdkF

    REAL    :: w2, bw, r2bw, bw_Hs2, bw_D_Hs2
    ! variables for temporary storage of the integrals, the prefactors and the dot_product
    REAL    :: integral(0:12), a_omegaI(0:8), aI_omegaI(0:8), dotpr, dotpr2
    INTEGER :: i

    REAL    :: r2w2, r4w2, r2w2_Hs2, r2w2_D_Hs2, arg, exp_e1, dAppInt_dh, d2AppInt_dh2

    ! parameters of the erfc fit as given in [2]
    REAL, PARAMETER :: A_2 = 0.5080572, scale = 1.125 / A_2,&
         a(1:8) = (/-1.128223946706117, 1.452736265762971, &
                    -1.243162299390327, 0.971824836115601, &
                    -0.568861079687373, 0.246880514820192, &
                    -0.065032363850763, 0.008401793031216 /),&
         b = 1.455915450052607, cutoff = 14.0

    ! Calculate helper variables
    w2       = omega_kF**2
    bw       = b * w2
    r2bw     = 2.0 * bw
    bw_Hs2   = bw + Hs2
    bw_D_Hs2 = bw + D_Hs2

    IF (bw_Hs2 < cutoff) THEN
       ! integrate the erfc approximation times the exchange hole
       CALL generateIntegrals(bw_Hs2, bw_D_Hs2, integral)

       ! combine the solutions of the integrals with the appropriate prefactors
       a_omegaI(0)    = 1.0
       a_omegaI(1:8)  = a * (/ (omega_kF**i,i=1,8) /)
       aI_omegaI      = (/ (i * a_omegaI(i),i=0,8) /)
       appInt         = DOT_PRODUCT(a_omegaI, integral(0: 8))
       dotpr          = DOT_PRODUCT(a_omegaI, integral(2:10))
       dAppInt_ds     = -dotpr * dHs2_ds 
       dAppInt_dkF    = r2bw * dotpr - DOT_PRODUCT(aI_omegaI, integral(0:8))
       dotpr2         = DOT_PRODUCT(a_omegaI, integral(4:12))
       d2AppInt_ds2   = dotpr2 * dHs2_ds**2 - dotpr * d2Hs2_ds2
       d2AppInt_dsdkF = -(r2bw * dotpr2 - DOT_PRODUCT(aI_omegaI, integral(2:10))) * dHs2_ds
    ELSE
       r2w2           = 2.0 * w2
       r4w2           = 4.0 * w2
       r2w2_Hs2       = r2w2 + Hs2 
       r2w2_D_Hs2     = r2w2 + D_Hs2
       arg            = scale * r2w2_Hs2
       exp_e1         = gauss_laguerre(arg)
       appInt         = A_2 * (LOG(r2w2_Hs2 / r2w2_D_Hs2) + exp_e1)
       dAppInt_dh     = -A_2 / r2w2_D_Hs2 + 1.125 * exp_e1
       d2AppInt_dh2   = 1.125 / r2w2_Hs2 - A_2 / r2w2_D_Hs2**2 - 1.265625 / A_2 * exp_e1
       dAppInt_ds     = dAppInt_dh * dHs2_ds
       dAppInt_dkF    = -dAppInt_dh * r4w2
       d2AppInt_ds2   = d2AppInt_dh2 * dHs2_ds**2 + dAppInt_dh * d2Hs2_ds2
       d2AppInt_dsdkF = d2AppInt_dh2 * dHs2_ds * r4w2
    END IF

  END SUBROUTINE approximateIntegral

  ! Generate the integrals for n = 0, 1, ..., 12
  !
  ! inf
  !   /        /        2                  \      /                        \
  !  |     n  |  A   -Dy           A        |    |    /   omega      2\   2 |
  !  | dy y   | --- e     - --------------  | Exp| - (  b ----- + H s  ) y  |
  !  |        |  y            /    4   2\   |    |    \     k         /     |
  ! /          \            y( 1 + - Ay  ) /      \          F             /
  !  0                        \    9    /
  !
  ! Input:  bw_Hs2   - b (omega/kF)^2 + H s^2
  !         bw_D_Hs2 - b (omega/kF)^2 + D + H s^2
  ! Output: integral - array with the calculated integrals
  ! To simplify the calculation use integral(n+2) = - d(integral(n))/d(b omega/kF)
  SUBROUTINE generateIntegrals(bw_Hs2, bw_D_Hs2, integral)
    USE m_exponential_integral, ONLY: calculateExponentialIntegral, gauss_laguerre, series_laguerre
    USE m_constants
    IMPLICIT NONE

    REAL, INTENT(IN)  :: bw_Hs2, bw_D_Hs2
    REAL, INTENT(OUT) :: integral(0:12)

    ! Helper variables
    REAL :: bw_Hs2_Sqr, bw_Hs2_Cub, sqrt_bw_Hs2,&
         bw_D_Hs2_Sqr, bw_D_Hs2_Cub, bw_D_Hs2_Tet, sqrt_bw_D_Hs2, &
         arg, sqrt_arg, r1_arg, e1_arg, exp_arg, exp_erfc,&
         term1, factor2, term2, arg_n, sum_term, add_term, half_i2_1
    INTEGER :: i

    ! A is an exchange hole parameter [3] and b the fit parameter for erfc [2]
    REAL, PARAMETER :: &
         A = 1.0161144, A_2 = A / 2.0, scale = 2.25 / A, sqrtA =  1.008025,& !sqrt(A) 
         b = 1.455915450052607

    ! Calculate many helper variables
    bw_Hs2_Sqr    = bw_Hs2 * bw_Hs2
    bw_Hs2_Cub    = bw_Hs2 * bw_Hs2_Sqr
    sqrt_bw_Hs2   = SQRT(bw_Hs2)
    bw_D_Hs2_Sqr  = bw_D_Hs2 * bw_D_Hs2
    bw_D_Hs2_Cub  = bw_D_Hs2 * bw_D_Hs2_Sqr
    bw_D_Hs2_Tet  = bw_D_Hs2_Sqr * bw_D_Hs2_Sqr
    sqrt_bw_D_Hs2 = SQRT(bw_D_Hs2)

    ! arg = 9/4 * (b omega/kF + H s^2) / A
    arg      = scale * bw_Hs2
    sqrt_arg = SQRT(arg)
    r1_arg   = 1.0 / arg

    ! calculate e^(arg), E1(arg), and erfc(sqrt(arg))
    exp_arg  = EXP(arg)
    exp_erfc = exp_arg * erfc(sqrt_arg)

    IF (arg > series_laguerre) THEN
       term2    = gauss_laguerre(arg)
    ELSE
       CALL calculateExponentialIntegral(arg, e1_arg)
       term2    = exp_arg * e1_arg
    END IF

    ! The n = 0 integral is 
    ! A/2 ( ln((b omega/kF + H s^2) / (b omega/kF + D + H s^2))
    !     + e^(arg) E1(arg) )
    integral(0) = A_2 * (LOG(bw_Hs2 / bw_D_Hs2) + term2)

    ! Calculate now all even n's by successive derivation
    ! The log(...) term gives term proportional to 1/(b omega/kF + D + H s^2)^i
    ! The e^(arg) E1(arg) term reproduces itself with a prefactor and 
    ! generates an additional 1/arg term which produces higher 1/arg^i terms
    ! when further deriviated
    term1       = A_2 / bw_D_Hs2
    factor2     = -1.125
    arg_n       = -1.0 / arg
    integral(2) = term1 + factor2 * term2

    DO i = 1, 5
       term1   = term1 / bw_D_Hs2 * REAL(i)
       factor2 = -factor2 * scale
       term2   = term2 + arg_n

       integral(2*(i+1)) = term1 + factor2 * term2

       arg_n = -arg_n * REAL(i) / arg
    END DO

    ! The n = 1 integral is
    ! A/2 ( sqrt(pi) / sqrt( b omega/kF + D + H s2 )
    ! - 3/4 sqrt(A) pi e^(arg) erfc(sqrt(arg))
    term1   = A_2 * SQRT(PI_const) / sqrt_bw_D_Hs2
    term2   = PI_const * exp_erfc
    factor2 = -0.75 * sqrtA

    integral(1) = term1 + factor2 * term2

    ! Calculate now all uneven n's by successive derivation
    ! The 1 / sqrt(...) term gives higher orders of 1 / (...)^((2i+1)/2)
    ! The e^(arg) erfc(sqrt(arg)) term reproduces itself with a prefactor 
    ! and generates an additional 1/sqrt(arg) term which produces higher
    ! 1/(arg)^((2i+1)/2) terms when further deriviated
    sum_term  = -1.125 * SQRT(pi_const) / sqrt_bw_Hs2
    add_term  = sum_term
    half_i2_1 = -0.5

    DO i = 3, 11, 2
       factor2     = -factor2 * scale
       term1       = -term1 * half_i2_1 / bw_D_Hs2
       integral(i) = term1 + term2 * factor2 + sum_term

       add_term    = -add_term * half_i2_1 / bw_Hs2
       sum_term    = -sum_term * scale + add_term
       half_i2_1   = half_i2_1 - 1.0
    ENDDO

  END SUBROUTINE generateIntegrals

  ! Calculate the integral
  !
  ! inf
  !   /                  /       \         /                 ________________ \
  !  |       -a y^2     | omega   |    1  |     omega   /   /     / omega \2   |
  !  | dy y e       Erfc| ----- y | = --- | 1 - -----  /   / a + (  -----  )   |
  !  |                  |   k     |   2 a |       k   /   V       \   k   /    |
  ! /                    \   F   /         \       F                   F      /
  !  0
  !
  ! Input:  omega_kF     - omega / kF
  !         omega_kF_Sqr - (omega / kF)^2
  !         a            - factor of the gaussian function
  ! Return: result of the integration
  REAL FUNCTION integrateY1ExpErfc(omega_kF, omega_kF_Sqr, a)
    IMPLICIT NONE

    REAL, INTENT(IN)  :: omega_kF, omega_kF_Sqr, a

    ! helper variable sqrt(a + (omega/kF)^2)
    REAL :: sqrt_term

    ! calculate helper variable
    sqrt_term = SQRT(a + omega_kF_Sqr)

    ! calculate the integral
    integrateY1ExpErfc = 0.5 * (1 - omega_kF / sqrt_term) / a

  END FUNCTION integrateY1ExpErfc

  ! Calculate the integral
  !
  ! inf
  !   /                   /       \           /                   ________________                   ________________3 \
  !  |     3  -a y^2     | omega   |     1   |       omega   /   /     / omega \2       omega   /   /     / omega \2    |
  !  | dy y  e       Erfc| ----- y | = ----- | 2 - 2 -----  /   / a + (  -----  )  - a  -----  /   / a + (  -----  )    |
  !  |                   |   k     |   4 a^2 |         k   /   V       \   k   /          k   /   V       \   k   /     |
  ! /                     \   F   /           \         F                   F              F                   F       /
  !  0
  !
  ! Input:  omega_kF     - omega / kF
  !         omega_kF_Sqr - (omega / kF)^2
  !         a            - factor of the gaussian function
  ! Return: result of the integration
  REAL FUNCTION integrateY3ExpErfc(omega_kF, omega_kF_Sqr, a)
    IMPLICIT NONE

    REAL, INTENT(IN)  :: omega_kF, omega_kF_Sqr, a

    ! helper variables
    REAL :: term, sqrt_term, sqrt_term_3

    ! calculate helper variables
    term        = a + omega_kF_Sqr
    sqrt_term   = SQRT(term)
    sqrt_term_3 = term * sqrt_term

    ! calculate the integral
    integrateY3ExpErfc = 0.25 * (2.0 - a * omega_kF / sqrt_term_3 - 2.0 * omega_kF / sqrt_term) / (a * a)

  END FUNCTION integrateY3ExpErfc

  ! Calculate the integral
  !
  ! inf                                     /                                   \
  !   /                   /       \        |                  /              2\  |
  !  |     5  -a y^2     | omega   |    1  |       omega     |      a     3 a  | |
  !  | dy y  e       Erfc| ----- y | = --- | 1 - ----------  | 1 + --- + ----- | |
  !  |                   |   k     |   a^3 |     k  sqrt(x)  |     2 x   8 x^2 | |
  ! /                     \   F   /        |      F           \               /  |
  !  0                                      \                                   /
  ! where
  !          /     \ 2
  !         | omega |
  ! x = a + | ----- |
  !         |   k   |
  !          \   F /
  !
  ! Input:  omega_kF     - omega / kF
  !         omega_kF_Sqr - (omega / kF)^2
  !         a            - factor of the gaussian function
  ! Return: result of the integration
  REAL FUNCTION integrateY5ExpErfc(omega_kF, omega_kF_Sqr, a)
    IMPLICIT NONE

    REAL, INTENT(IN)  :: omega_kF, omega_kF_Sqr, a

    ! helper variables
    REAL :: term, sqrt_term, a_Sqr

    ! calculate helper variables
    term        = a + omega_kF_Sqr
    sqrt_term   = SQRT(term)
    a_Sqr       = a * a

    ! calculate the integral
    integrateY5ExpErfc = (1.0 - (0.375 * a_Sqr / (term * term) + a / (2.0 * term) + 1.0) * omega_kF / sqrt_term) / (a_Sqr * a)

  END FUNCTION integrateY5ExpErfc

  ! Calculate the integral
  !
  ! inf                                     /                                            \
  !   /                   /       \        |                  /              2       3 \  |
  !  |     7  -a y^2     | omega   |    3  |       omega     |      a     3 a     5 a   | |
  !  | dy y  e       Erfc| ----- y | = --- | 1 - ----------  | 1 + --- + ----- + ------ | |
  !  |                   |   k     |   a^4 |     k  sqrt(x)  |     2 x   8 x^2   16 x^3 | |
  ! /                     \   F   /        |      F           \                        /  |
  !  0                                      \                                            /
  ! where
  !          /     \ 2
  !         | omega |
  ! x = a + | ----- |
  !         |   k   |
  !          \   F /
  !
  ! Input:  omega_kF     - omega / kF
  !         omega_kF_Sqr - (omega / kF)^2
  !         a            - factor of the gaussian function
  ! Return: result of the integration
  REAL FUNCTION integrateY7ExpErfc(omega_kF, omega_kF_Sqr, a)
    IMPLICIT NONE

    REAL, INTENT(IN)  :: omega_kF, omega_kF_Sqr, a

    ! helper variables
    REAL :: term2, term, sqrt_term, a_Sqr

    ! calculate helper variables
    term        = a + omega_kF_Sqr
    term2       = term * term
    sqrt_term   = SQRT(term)
    a_Sqr       = a * a

    ! calculate the integral
    integrateY7ExpErfc = 3.0  / (a_Sqr * a_Sqr) * (1.0 - omega_kF / sqrt_term * &
         (0.3125 * (a_Sqr * a) / (term2 * term) + 0.375 * a_Sqr / term2 + a / (2.0 * term) + 1.0))

  END FUNCTION integrateY7ExpErfc

  ! Calculate the integral
  !
  ! inf                                     /                                                      \
  !   /                   /       \        |                  /              2       3           \  |
  !  |     9  -a y^2     | omega   |    12 |       omega     |      a     3 a     5 a      35 a^4 | |
  !  | dy y  e       Erfc| ----- y | = --- | 1 - ----------  | 1 + --- + ----- + ------ + ------- | |
  !  |                   |   k     |   a^5 |     k  sqrt(x)  |     2 x   8 x^2   16 x^3   128 x^4 | |
  ! /                     \   F   /        |      F           \                                  /  |
  !  0                                      \                                                      /
  ! where
  !          /     \ 2
  !         | omega |
  ! x = a + | ----- |
  !         |   k   |
  !          \   F /
  !
  ! Input:  omega_kF     - omega / kF
  !         omega_kF_Sqr - (omega / kF)^2
  !         a            - factor of the gaussian function
  ! Return: result of the integration
  REAL FUNCTION integrateY9ExpErfc(omega_kF, omega_kF_Sqr, a)
    IMPLICIT NONE

    REAL, INTENT(IN)  :: omega_kF, omega_kF_Sqr, a

    ! helper variables
    REAL :: term2, term, sqrt_term, a_Sqr, a_Tet

    ! calculate helper variables
    term        = a + omega_kF_Sqr
    term2       = term * term
    sqrt_term   = SQRT(term)
    a_Sqr       = a * a
    a_Tet       = a_Sqr * a_Sqr

    ! calculate the integral
    integrateY9ExpErfc = 12.0 * (1.0 - (0.2734375 * a_Tet / (term2 * term2) + 0.3125 * (a_Sqr * a) / (term2 * term) &
         + 0.375 * a_Sqr / term2 + a / (2.0 * term) + 1.0) * omega_kF / sqrt_term) / (a_Tet * a)

  END FUNCTION integrateY9ExpErfc

  ! Calculate the function F(s) given in [3]
  ! Input:  s         - reduced gradient
  !         H         - value of the function H(s)
  !         dHs2_ds   - first derivative of H(s)s^2
  !         dsHs2_ds2 - second derivative of H(s)s^2
  ! Output: F         - value of the function F
  !         dFs2_ds   - first derivative of F(s)s^2
  !         d2Fs2_ds2 - second derivative of F(s)s^2
  SUBROUTINE calculateF(s, H, dHs2_ds, d2Hs2_ds2, F, dFs2_ds, d2Fs2_ds2)
    IMPLICIT NONE

    REAL, INTENT(IN)  :: s, H, dHs2_ds, d2Hs2_ds2
    REAL, INTENT(OUT) :: F, dFs2_ds, d2Fs2_ds2

    REAL, PARAMETER   :: slope = 6.4753871
    REAL, PARAMETER   :: shift = 0.4796583

    ! calculate the function F(s), d(s^2F(s))/ds, and d^2(s^2F(s))/ds^2
    F         = slope * H + shift
    dFs2_ds   = slope * dHs2_ds + 2.0 * s * shift
    d2Fs2_ds2 = slope * d2Hs2_ds2 + 2.0 * shift
  END SUBROUTINE calculateF

  ! Calculate the function G(s) given in [3]
  ! Input:  s2        - s^2 where s is the reduced gradient
  !         Fs2       - F(s) s^2
  !         dFs2_ds   - first derivative of F(s)s^2 with respect to s
  !         d2Fs2_ds2 - second derivative of F(s)s^2
  !         Hs2       - H(s) s^2
  !         dHs2_ds   - first derivative of H(s)s^2 with respect to s
  !         d2Hs2_ds2 - second derivative of H(s)s^2
  ! Output: G         - value of the function G
  !         dGs2_ds   - first derivative of G(s)s^2 with respect to s
  !         d2Gs2_ds2 - second derivative of G(s)s^2
  SUBROUTINE calculateG(s2, Fs2, dFs2_ds, d2Fs2_ds2, Hs2, dHs2_ds, d2Hs2_ds2, G, dGs2_ds, d2Gs2_ds2)
    IMPLICIT NONE

    REAL, INTENT(IN)  :: s2, Fs2, dFs2_ds, d2Fs2_ds2, Hs2, dHs2_ds, d2Hs2_ds2
    REAL, INTENT(OUT) :: G, dGs2_ds, d2Gs2_ds2

    ! helper variables
    REAL :: AHs2_1_2, AHs2_3_2, r1_Fs2, D_Hs2, D_Hs2Sqr, D_Hs2Cub, &
         D_Hs2_5_2, D_Hs2_7_2, D_Hs2_9_2, D_Hs2_11_2
    REAL :: part1, dpart1_dh, d2part1_dh2
    REAL :: arg1, arg2, exp_erfc,&
         part2, dpart2_dh, d2part2_dh2
    REAL :: alpha, Ebeta, r3Pi_4_alpha, beta_s2, Ebeta_s2,&
         dalpha_dh, d2alpha_dh2, dalpha_df, d2alpha_dfdh, dbeta_dh, d2beta_dh2

    ! parameters of the exchange hole given in [3]
    REAL, PARAMETER :: PI_75 = 2.356194490192344929, SQRT_PI = 1.77245385090551602729816748334,&
         A = 1.0161144, sqrtA = 1.008025, r9_4A = 2.25 / A,& !sqrt(A), 9/4A
         B = -0.37170836,&
         C = -0.077215461,&
         D =  0.57786348,&
         E = -0.051955731

    ! calculate the helper variables
    AHs2_1_2   = sqrtA * SQRT(Hs2)
    AHs2_3_2   = AHs2_1_2 * A * Hs2
    r1_Fs2     = 1.0 + Fs2
    D_Hs2      = D + Hs2
    D_Hs2Sqr   = D_Hs2 * D_Hs2
    D_Hs2Cub   = D_Hs2 * D_Hs2Sqr
    D_Hs2_5_2  = D_Hs2Sqr * SQRT(D_Hs2)
    D_Hs2_7_2  = D_Hs2_5_2 * D_Hs2
    D_Hs2_9_2  = D_Hs2_5_2 * D_Hs2Sqr
    D_Hs2_11_2 = D_Hs2_5_2 * D_Hs2Cub

    ! calculate first part of the term called 'a' in [3] eq. (A2) and its derivatives with respect to H(s)s^2 and F(s)s^2
    !
    !                          2      2           2 2          2 3
    !          __ 15E + 6C(1+Fs )(D+Hs ) + 4B(D+Hs )  + 8A(D+Hs ) 
    ! part1 = VPi ------------------------------------------------
    !                                    2  7/2
    !                        16 ( D + H s  )
    !
    part1 = SQRT_PI * (15.0*E + 6.0*C * r1_Fs2 * D_Hs2 + 4.0*B * D_Hs2Sqr + 8.0*A * D_Hs2Cub) / (16.0 * D_Hs2_7_2)
    !
    !                               2      2            2 2          2 3
    ! d part1     __ 105E + 30C(1+Fs )(D+Hs ) + 12B(D+Hs )  + 8A(D+Hs ) 
    ! ------- = -VPi ---------------------------------------------------
    ! d(Hs^2)                                  2  9/2
    !                              32 ( D + H s  )
    !
    dpart1_dh = -SQRT_PI * (105.0*E + 30.0*C * r1_Fs2 * D_Hs2 + 12.0*B * D_Hs2Sqr + 8.0*A * D_Hs2Cub) / (32.0 * D_Hs2_9_2)
    !
    !  2                             2      2            2 2           2 3
    ! d  part1    __ 945E + 210C(1+Fs )(D+Hs ) + 60B(D+Hs )  + 24A(D+Hs ) 
    ! -------- = VPi -----------------------------------------------------
    !        2                                   2  11/2
    ! d(Hs^2)                        64 ( D + H s  )
    !
    d2part1_dh2 = SQRT_PI * (945.0*E + 210.0*C * r1_Fs2 * D_Hs2 + 60.0*B * D_Hs2Sqr + 24.0*A * D_Hs2Cub) / (64.0 * D_Hs2_11_2)
    !
    ! d part1    __       3 C 
    ! ------- = VPi -----------------
    ! d(Fs^2)                  2  5/2
    !               8 ( D + H s  )
    !
    dalpha_df = SQRT_PI * 0.375 * C / D_Hs2_5_2
    !
    !      2
    !     d part1         __       15 C 
    ! --------------- = -VPi ------------------
    ! d(Fs^2) d(Hs^2)                    2  7/2
    !                        16 ( D + H s  )
    !
    d2alpha_dfdh = -2.5 * dalpha_df / D_Hs2

    ! calculate second part of the term called 'a' in [3] eq. (A2) and its derivatives
    !                                            ________
    !                       /     2 \       /   /     2   \
    !         3 Pi  ___    | 9 H s   |     |   / 9 H s     |
    ! part2 = ---- V A  Exp| ------- | Erfc|  /  -------   |
    !          4           |   4 A   |     | V     4 A     |
    !                       \       /       \             /
    !
    arg1     = r9_4A * Hs2
    arg2     = SQRT(arg1)
    exp_erfc = EXP(arg1) * erfc(arg2)
    part2    = PI_75 * sqrtA * exp_erfc
    !
    !                      /                                \
    ! d part2   3 Pi  ___ |  9                    3          |
    ! ------- = ---- V A  | --- exp_erfc - ----------------  |
    ! d(Hs^2)    4        | 4 A                          2   |
    !                      \               2 sqrt(Pi A Hs ) /
    !
    dpart2_dh = PI_75 * sqrtA * (r9_4A * exp_erfc - 1.5 / (SQRT_PI * AHs2_1_2) )
    !
    !  2                    /                                 2    \
    ! d  part2   3 Pi  ___ |   81                  6 A - 27 Hs      |
    ! -------- = ---- V A  | ------ exp_erfc + -------------------- |
    !        2    4        | 16 A^2                           2 3/2 |
    ! d(Hs^2)               \                  8 sqrt(Pi)(A Hs )   /
    !
    d2part2_dh2 = PI_75 * sqrtA * (r9_4A**2 * exp_erfc + (6.0*A - 27.0*Hs2) / (8.0 * SQRT_PI * AHs2_3_2) )

    ! calculate the 'a' and 'b' terms ([3] eq. (A2) and (A3) )
    !
    ! a = part1 - part2
    !
    !             __  2
    !         15 VPi s
    ! b = -----------------
    !                2  7/2
    !     16 (D + H s  )
    !
    alpha   = part1 - part2
    beta_s2 = 0.9375 * SQRT_PI / D_Hs2_7_2
    Ebeta   = E * beta_s2 * s2

    ! the derivatives of a with respect to Fs^2 and Hs^2
    !
    !    da      d part1   d part2
    ! -------- = ------- - -------
    ! d (Hs^2)   d(Hs^2)   d(Hs^2)
    !
    dalpha_dh = dpart1_dh - dpart2_dh
    !
    !    2        2          2
    !   d a      d part1    d part2
    ! -------- = -------- - --------
    !        2          2          2
    ! d(Hs^2)    d(Hs^2)    d(Hs^2)
    !
    d2alpha_dh2 = d2part1_dh2 - d2part2_dh2
    !
    !    da      d part1
    ! -------- = -------
    ! d (Fs^2)   d(Fs^2)
    !
    !        2               2
    !       d a             d part1
    ! --------------- = ---------------
    ! d(Fs^2) d(Hs^2)   d(Fs^2) d(Hs^2)
    !
    ! (this expressions are already defined earlier)
    !
    ! and the derivatives of b/s^2 with respect to Hs^2
    !                        __
    ! d(b/s^2)          105 VPi          7  b / s^2
    ! -------- = - ----------------- = - -  --------
    ! d (Hs^2)                2  9/2     2         2
    !              32 (D + H s  )           D + H s
    !
    dbeta_dh = -3.5 * beta_s2 / D_Hs2
    !
    !  2                     __
    ! d (b/s^2)         945 VPi          9  d(b/s^2)
    ! --------- = ------------------ = - -  --------
    !         2              2  11/2     2  d (Hs^2)
    ! d (Hs^2)    64 (D + H s  )
    !
    d2beta_dh2 = -4.5 * dbeta_dh / D_Hs2

    ! calculate the function G(s) and its derivatives
    ! 
    !       3Pi/4 + a
    ! G = - ---------
    !          E b
    !
    r3Pi_4_alpha = PI_75 + alpha
    Ebeta_s2     = E * beta_s2
    G            = - r3Pi_4_alpha / Ebeta
    !
    !             /  /3 Pi    \  d(b/s^2)   /           d a    \  d(Hs^2)       d a   d(Fs^2) 
    !            <  ( ---- + a ) -------   /  b/s^2  - -------  > -------  -  ------- ------- 
    ! d (Gs^2)    \  \ 4      /  d(Hs^2)  /            d(Hs^2) /    d s       d(Fs^2)   ds    
    ! -------- = ----------------------------------------------------------------------------
    !    ds                                     E b/s^2
    !
    dGs2_ds = ( (r3Pi_4_alpha * dbeta_dh / beta_s2 - dalpha_dh) * dHs2_ds - dalpha_df * dFs2_ds ) / Ebeta_s2
    !
    !                                                     /3Pi    \  /             2\           /           \
    !                                                    ( --- + a )( b  h" + b   h' ) + 2b  h'( f'a  + h'a  ) + ...
    !                                            2        \ 4     /  \ h       hh   /      h    \   f      h/
    !  2         -f" a  - h" a  - 2 f' h' a   - h' a   + -----------------------------------------------------------
    ! d (Gs^2)        f       h            fh       hh                                b/s^2
    ! -------- = ---------------------------------------------------------------------------------------------------
    !   ds^2                                               E b/s^2
    !
    !            /3Pi    \   2  2
    !           ( --- + a ) b  h'
    !            \ 4     /   h
    ! ... = - 2 -----------------
    !                  b/s^2
    !
    ! where the primes indicate derivative with respect to s and the subscripts derivatives with respect to the subscript
    ! and f and h are abbrevations for f = Fs^2 and h = Hs^2
    !
    d2Gs2_ds2 = (-d2Fs2_ds2 * dalpha_df - d2Hs2_ds2 * dalpha_dh &
         - 2.0 * dFs2_ds * dHs2_ds * d2alpha_dfdh - dHs2_ds**2 * d2alpha_dh2 &
                  + ( r3Pi_4_alpha * (dbeta_dh * d2Hs2_ds2 + d2beta_dh2 * dHs2_ds**2) &
                      + 2.0 * dbeta_dh * dHs2_ds * (dFs2_ds * dalpha_df + dHs2_ds * dalpha_dh) &
                      - 2.0 * r3Pi_4_alpha * (dbeta_dh*dHs2_ds)**2 / beta_s2 ) / beta_s2 ) / Ebeta_s2
  END SUBROUTINE calculateG

  ! Calculate the function H(s) given in [3]
  ! Input:  s         - reduced gradient
  ! Output: H         - value of the function H
  !         dHs2_ds   - first derivative d(s^2*H(s))/ds
  !         d2Hs2_ds2 - second derivative d^2(s^2H(s))/ds^2
  SUBROUTINE calculateH(s,H,dHs2_ds,d2Hs2_ds2)
    IMPLICIT NONE

    REAL, INTENT(IN)  :: s
    REAL, INTENT(OUT) :: H, dHs2_ds, d2Hs2_ds2

    ! helper variables
    REAL :: s2, s3, s4, s5, s6
    REAL :: numer, denom
    REAL :: dnum_ds, dden_ds
    REAL :: d2num_ds2, d2den_ds2

    ! parameters given in [3]
    REAL, PARAMETER :: &
         a1 = 0.00979681,&
         a2 = 0.0410834,&
         a3 = 0.187440,&
         a4 = 0.00120824,&
         a5 = 0.0347188

    ! calculate helper variables
    s2 = s  * s
    s3 = s2 * s
    s4 = s2 * s2
    s5 = s3 * s2
    s6 = s4 * s2

    ! calculate function H(s) with [3] eq. (A5)
    !
    !            2         4
    !        a  s   +  a  s
    !         1         2
    ! H = -------------------------
    !             4       5       6
    !     1 + a  s  + a  s  + a  s
    !          3       4       5
    !
    numer = a1 * s2 + a2 * s4
    denom = 1.0 + a3 * s4 + a4 * s5 + a5 * s6
    H     = numer / denom

    ! calculate the first derivative of s^2 H(s)
    !
    !     /    \
    !  d | f(x) |   f'(x) - f(x)g'(x) / g(x)
    ! -- | ---- | = ------------------------
    ! dx | g(x) |            g(x)
    !     \    /
    !
    numer   = numer * s2
    dnum_ds = 4.0 * s3 * a1 + 6.0 * s5 * a2
    dden_ds = 4.0 * s3 * a3 + 5.0 * s4 * a4 + 6.0 * s5 * a5
    dHs2_ds = ( dnum_ds - (dden_ds * numer)/denom ) / denom

    ! calculate the second derivative of s^2 H(s)
    !
    !                                                               2
    !                         2 f'(x)g'(x) + f(x)g"(x) - 2 f(x)g'(x) /g(x)
    !    2  /    \    f"(x) - --------------------------------------------
    !   d  | f(x) |                              g(x)
    ! ---- | ---- | = ----------------------------------------------------
    ! dx^2 | g(x) |                          g(x)
    !       \    /
    !
    d2num_ds2 = 12.0 * s2 * a1 + 30.0 * s4 * a2
    d2den_ds2 = 12.0 * s2 * a3 + 20.0 * s3 * a4 + 30.0 * s4 * a5
    d2Hs2_ds2 = ( d2num_ds2 - ( 2.0*dnum_ds*dden_ds + numer*d2den_ds2 - 2.0*numer*dden_ds**2/denom ) / denom ) / denom
  END SUBROUTINE calculateH

  ! Calculate several Fourier transformations
  ! a) the Fourier transformed potential
  !                                                 /          \
  !          /      | erf |       \     4 Pi       |   |k+G|^2  |
  ! V (k) = ( k + G | --- | k + G  ) = -------  Exp| - -------  |                             [1]
  !  G       \      |  r  |       /    |k+G|^2     |    4 w^2   |
  !                                                 \          /
  !
  ! b) muffin-tin basis function
  !                                                            R
  !                                     -L                      /
  !           /      |      \     4 Pi i         ^     -i G R  |     2
  ! MT (k) = ( k + G | M     ) = ---------- Y ( k+G ) e        | dr r  Phi(r) j ( |k+G| r )  [2]
  !   G,I     \      |  k,I /     Sqrt(Om)   LM                |               L
  !                                                           /
  !                                                            0
  !
  ! c) interstitial basis function
  !                                                      -----
  !          /      |      \                       4 Pi   \     -i(G - G_I)R_a  Sin(|G_I - G| R_a) - |G_I - G| R_a Cos(|G_I - G| R_a) 
  ! IN    = ( k + G | M     ) = kronecker(G,G ) - ------   )   e               ------------------------------------------------------- [3]
  !   G,I    \      |  k,I /                 I      Om    /                                          |G_I - G|^3
  !                                                      -----
  !                                                        a
  !                                                                            \_________________________ ___________________________/
  !                                                                                                      V
  !                                                                                                     I_a
  !
  ! In the code:
  ! V_G(k):  potential
  ! MT_G(k): muffintin
  ! IN_G:    interstitial
  ! I_a:     inter_atom
  !
  ! Input:
  ! rmsh        - array of radial mesh points
  ! rmt         - radius of the muffin tin
  ! dx          - logarithmic increment of radial mesh
  ! jri         - number of radial mesh points
  ! jmtd        - dimension of radial mesh points
  ! bk          - one(!) k-vector for which FT is calculated
  ! bmat        - reciprocal lattice vector for all directions
  ! vol         - volume of the unit cell
  ! ntype       - number of atom types
  ! neq         - number of symmetry-equivalent atoms of atom type i
  ! natd        - dimesion of atoms in unit cell
  ! taual       - vector of atom positions (internal coordinates)
  ! lcutm       - l cutoff for mixed basis
  ! maxlcutm    - maximum of all these l cutoffs
  ! nindxm      - number of radial functions of mixed basis
  ! maxindxm    - maximum of these numbers
  ! gptm        - reciprocal lattice vectors of the mixed basis (internal coord.)
  ! ngptm       - number of vectors (for treated k-vector)
  ! pgptm       - pointer to the appropriate g-vector (for treated k-vector)
  ! gptmd       - dimension of gptm
  ! basm        - mixed basis functions (mt + inter) for treated k-vector
  ! lexp        - cutoff of spherical harmonics expansion of plane wave
  ! noGPts      - no g-vectors used for Fourier trafo
  ! Output:
  ! potential   - Fourier transformation of the potential
  ! muffintin   - Fourier transformation of all MT functions
  ! interstital - Fourier transformation of all IR functions
  SUBROUTINE calculate_fourier_transform( &
      ! Input
      rmsh,rmt,dx,jri,jmtd,bk,&
      bmat,vol,ntype,neq,natd,taual,lcutm,maxlcutm,&
      nindxm,maxindxm,gptm,ngptm,pgptm,gptmd,&
      basm,noGPts,irank,&
      ! Output
      potential,muffintin,interstitial)

    USE m_constants
    USE m_olap,      ONLY : gptnorm
    USE m_util,      ONLY : sphbessel,pure_intgrf,intgrf_init,intgrf_out,NEGATIVE_EXPONENT_WARNING,NEGATIVE_EXPONENT_ERROR

    IMPLICIT NONE

    ! scalar input
    INTEGER, INTENT(IN)    :: natd,ntype,maxlcutm
    INTEGER, INTENT(IN)    :: jmtd,irank
    INTEGER, INTENT(IN)    :: maxindxm
    INTEGER, INTENT(IN)    :: gptmd, noGPts
    REAL, INTENT(IN)       :: vol

    ! array input
    INTEGER, INTENT(IN)    :: lcutm(ntype)
    INTEGER, INTENT(IN)    :: nindxm(0:maxlcutm,ntype),neq(ntype)
    INTEGER, INTENT(IN)    :: jri(ntype)
    INTEGER, INTENT(IN)    :: gptm(3,gptmd)
    INTEGER, INTENT(IN)    :: ngptm
    INTEGER, INTENT(IN)    :: pgptm(ngptm)

    REAL, INTENT(IN)       :: bk(3)
    REAL, INTENT(IN)       :: rmsh(jmtd,ntype),rmt(ntype),dx(ntype)
    REAL, INTENT(IN)       :: basm(jmtd,maxindxm,0:maxlcutm,ntype)
    REAL, INTENT(IN)       :: bmat(3,3)!,amat(3,3)
    REAL, INTENT(IN)       :: taual(3,natd)

    ! array output
    REAL,    INTENT(OUT)   :: potential(noGPts)                           ! Fourier transformed potential
    COMPLEX, INTENT(OUT)   :: muffintin(noGPts,maxindxm,&                 ! muffin-tin overlap integral
                                        (maxlcutm+1)**2,ntype,MAXVAL(neq))
    COMPLEX, INTENT(OUT)   :: interstitial(noGPts,gptmd)                  ! interstistial overlap intergral

    ! private scalars
    INTEGER                :: cg,cg2,ci,cl,cn,cr                          ! counter variables
    REAL                   ::  r2Pi, r4Pi, pi_omega2                   !  2*Pi, 4*Pi, Pi/omega^2
    REAL                   :: sVol, r4Pi_sVol, r4Pi_Vol                   ! sqrt(vol), 4*Pi/sqrt(Vol), 4*Pi/Vol
    REAL                   :: omega, r1_omega2, r1_4omega2
!     REAL,    PARAMETER     :: omega = omega_VHSE()                        ! omega of the HSE functional
!     REAL,    PARAMETER     :: r1_omega2  = 1.0 / omega**2                 ! 1/omega^2
!     REAL,    PARAMETER     :: r1_4omega2 = 0.25 * r1_omega2               ! 1/(4 omega^2)
    COMPLEX, PARAMETER     :: img = (0d0,1d0)                             ! i

    ! private arrays
    INTEGER                :: gPts(3,noGPts)                              ! g vectors (internal units)
    INTEGER                :: gPts_gptm(3,noGpts,gptmd)                   ! gPts - gptm
    INTEGER                :: natdPtr(ntype+1)                            ! pointer to all atoms of one type
    REAL,    ALLOCATABLE   :: gridf(:,:)                                  ! grid for radial integration
    REAL                   :: k_G(3,noGPts)                               ! k + G
    REAL                   :: AbsK_G(noGPts),AbsK_G2(noGPts)              ! abs(k+G), abs(k+G)^2
    REAL                   :: arg(noGPts)                                 ! abs(k+G)^2 / (4*omega^2)
    REAL                   :: sphbesK_Gr(noGPts,jmtd,0:maxlcutm,ntype)    ! spherical bessel function of abs(k+G)r
    TYPE(intgrf_out)       :: intgrMT(noGPts,maxindxm,0:maxlcutm,ntype)   ! integration in muffin-tin
    REAL                   :: abs_dg(noGpts,gptmd)                        ! abs(gPts - gptm)
    COMPLEX                :: imgl(0:maxlcutm)                            ! i^l
    COMPLEX                :: Ylm(noGPts,(maxlcutm+1)**2)                 ! spherical harmonics for k+G and all lm
    COMPLEX                :: expIGR(noGPts,ntype,MAXVAL(neq))            ! exp(-iGR) for all atom types
    COMPLEX                :: sumInter_atom(noGpts,gptmd)                 ! sum over inter-atom factors

    ! Calculate helper variables
    r2Pi       = 2.0 * pi_const
    r4Pi       = 4.0 * pi_const
    sVol       = SQRT(vol)
    r4Pi_sVol  = r4Pi / sVol
    r4Pi_Vol   = r4Pi / Vol
1040
    omega      = omega_hse!omega_VHSE()
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    r1_omega2  = 1.0 / omega**2
    r1_4omega2 = 0.25 * r1_omega2
    pi_omega2  = pi_const * r1_omega2

    ! calculate pointers for all atom-types to all atoms
    natdPtr(ntype+1) = natd
    DO cn=ntype,1,-1
       natdPtr(cn) = natdPtr(cn+1) - neq(cn)
    END DO

    ! Define imgl(l) = img**l
    imgl(0) = 1.0
    DO ci=1,maxlcutm
       imgl(ci) = imgl(ci-1) * img
    END DO

    ! generate grid for fast radial integration
    CALL intgrf_init(ntype,jmtd,jri,dx,rmsh,gridf)

    ! Set all arrays to 0
    gPts           = 0
    k_G            = 0
    AbsK_G         = 0
    arg            = 0
    sphbesK_Gr     = 0
    intgrMT%value  = 0
    intgrMT%ierror = 0
    Ylm            = 0
    expIGR         = 0
    gPts_gptm      = 0
    abs_dg         = 0
    sumInter_atom  = 0
    potential      = 0
    muffintin      = 0
    interstitial   = 0

! Calculate the muffin-tin basis function overlap integral
!                                                            R
!                                     -L                      /
!           /      |      \     4 Pi i         ^     -i G R  |     2
! MT (k) = ( k + G | M     ) = ---------- Y ( k+G ) e        | dr r  Phi(r) j ( |k+G| r )  [2]
!   G,I     \      |  k,I /     Sqrt(Om)   LM                |               L
!                                                           /
!                                                            0
    IF (ngptm < noGpts) STOP 'hsefunctional: error calculating Fourier coefficients, noGpts too large'

    gPts(:,:) = gptm(:,pgptm(1:noGPts))
1088
#ifndef __PGI 
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    gpoints: FORALL ( cg=1:noGPts )
      ntypesA: FORALL ( cn=1:ntype )
        ! Calculate the phase factor exp(-iGR) for all atoms
        FORALL ( ci=1:neq(cn) )
          expIGR(cg,cn,ci)        = EXP( -r2Pi * img * DOT_PRODUCT(taual(:,natdPtr(cn)+ci),gPts(:,cg)) )
          muffintin(cg,:,:,cn,ci) = r4Pi_sVol * expIGR(cg,cn,ci)
        END FORALL

        ! Multiplication with i^-L
        FORALL ( cl=0:lcutm(cn) )
          muffintin(cg,:,cl*cl+1:(cl+1)*(cl+1),cn,:) = muffintin(cg,:,cl*cl+1:(cl+1)*(cl+1),cn,:) / imgl(cl)
        END FORALL

      END FORALL ntypesA

      ! Calculate the k+G, abs(k+G), and abs(k+G)^2
      k_G(:,cg)   = MATMUL( gPts(:,cg)+bk(:), bmat(:,:) )
      AbsK_G2(cg) = SUM( k_G(:,cg)**2 )
      AbsK_G(cg)  = SQRT(AbsK_G2(cg))

      ! Spherical harmonics are calculated for all lm's and k+G's
      Ylm(cg,:) = calcYlm(k_G(:,cg),maxlcutm)

      ! Perform the integration in eq.[2] for all muffin-tins
      ntypesB: FORALL ( cn=1:ntype )

        ! Multiplication with the spherical harmonic
        FORALL ( cl=1:(lcutm(cn)+1)**2 )
          muffintin(cg,:,cl,cn,:) = muffintin(cg,:,cl,cn,:) * Ylm(cg,cl)
        END FORALL

        ! Calculate the spherical bessel function
        FORALL ( cr=1:jri(cn) )
          sphbesK_Gr(cg,cr,:,cn) = calcSphBes(AbsK_G(cg)*rmsh(cr,cn),maxlcutm)
        END FORALL
        ! integrate the function and multiplicate to the result
        FORALL ( cl=0:lcutm(cn) )
          FORALL ( ci=1:nindxm(cl,cn) )
            intgrMT(cg,ci,cl,cn) = pure_intgrf(rmsh(:,cn) * basm(:,ci,cl,cn) * &
              sphbesK_Gr(cg,:,cl,cn),jri,jmtd,rmsh,dx,ntype,cn,gridf)
            muffintin(cg,ci,cl*cl+1:(cl+1)*(cl+1),cn,:) = &
              muffintin(cg,ci,cl*cl+1:(cl+1)*(cl+1),cn,:) * intgrMT(cg,ci,cl,cn)%value
          END FORALL
        END FORALL

      END FORALL ntypesB

! calculate the overlap with the interstitial basis function
!                                                      -----
!          /      |      \                       4 Pi   \     -i(G - G_I)R_a  Sin(|G_I - G| R_a) - |G_I - G| R_a Cos(|G_I - G| R_a) 
! IN    = ( k + G | M     ) = kronecker(G,G ) - ------   )   e               ------------------------------------------------------- [3]
!   G,I    \      |  k,I /                 I      Om    /                                          |G_I - G|^3
!                                                      -----
!                                                        a
!                                                                            \_________________________ ___________________________/
!                                                                                                      V
!                                                                                                     I_a
      ! Calculate the difference of the G vectors and its absolute value
      FORALL ( cg2=1:gptmd )

        gPts_gptm(:,cg,cg2) = gPts(:,cg) - gptm(:,cg2)
        abs_dg(cg,cg2)      = gptnorm( gPts_gptm(:,cg,cg2),bmat )

      END FORALL

    END FORALL gpoints

    ! Check if any of the integrations failed and abort if one did
    IF ( ANY( intgrMT%ierror == NEGATIVE_EXPONENT_ERROR ) ) THEN
        IF ( irank == 0 ) WRITE(6,*) 'intgrf: Warning! Negative exponent x in extrapolation a+c*r**x'
    ELSEIF ( ANY( intgrMT%ierror == NEGATIVE_EXPONENT_WARNING ) ) THEN
      IF ( irank == 0 ) WRITE(6,*) 'intgrf: Negative exponent x in extrapolation a+c*r**x'
      STOP       'intgrf: Negative exponent x in extrapolation a+c*r**x'
    END IF

    ! Calculate the interstitial value with eq.[3] using the limit
    !           3
    !        R_a
    ! I_a = ------
    !         3
    ! if there's no difference of the two G vectors
    WHERE ( abs_dg == 0 )
      sumInter_atom = DOT_PRODUCT(rmt**3, neq) / 3.0
      interstitial  = 1.0 - r4Pi_Vol * sumInter_atom
    ELSEWHERE
      sumInter_atom = calculateSummation(abs_dg,gPts_gptm)
      interstitial  = -r4Pi_Vol * sumInter_atom
    END WHERE

! Calculate the Fourier transformed potential
!                                                 /          \
!          /      | erf |       \      4 Pi      |   |k+G|^2  |
! V (k) = ( k + G | --- | k + G' ) = -------  Exp| - -------  | kronecker(G,G')             [1]
!  G       \      |  r  |       /    |k+G|^2     |    4 w^2   |
!                                                 \          /
    WHERE ( absK_G.NE.0 )
      ! Calculate (k+G)^2 / 4*w^2
      arg = -r1_4omega2 * AbsK_G2

      ! The potential is calculated using eq.[1]
      potential = r4Pi / AbsK_G2 * EXP(arg)

! Calculate the Fourier transformed potential for the full(!) potential
!
!          /          | erfc |          \      Pi
! V (k) = ( k + G = 0 | ---- | k + G = 0 ) = -----
!  G       \          |   r  |          /     w^2
!
    ELSEWHERE
      ! the negative sign is added to compensate the sign when the
      ! contribution is added to the Coulomb matrix
      potential = -pi_omega2
    endwhere

    DEALLOCATE( gridf )
1205 1206 1207
#else
    call judft_error("hsefunctional not implemented for PGI")
#endif
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  CONTAINS

    ! Calculates the inter-atom parts and the summation over all atoms:
    ! -----
    !  \     -i(G - G_I)R_a  Sin(|G_I - G| R_a) - |G_I - G| R_a Cos(|G_I - G| R_a) 
    !   )   e               -------------------------------------------------------
    !  /                                          |G_I - G|^3
    ! -----
    !   a
    ! Input:  abs_dg    - absolute value |G_I - G|
    !         gPts_gptm - vector G - G_I
    PURE FUNCTION calculateSummation(abs_dg,gPts_gptm)
      IMPLICIT NONE
      INTEGER, INTENT(IN) :: gPts_gptm(3,noGPts,gptmd)
      REAL, INTENT(IN)    :: abs_dg(noGPts,gptmd)
      COMPLEX             :: calculateSummation(noGPts,gptmd)
      INTEGER             :: cn,ci                                     ! counter variables
      REAL                :: abs_dgR(noGPts,gptmd,ntype)               ! abs(gPts - gptm)*R (R: radius MT)
      REAL                :: inter_atom(noGPts,gptmd,ntype)            ! inter-atom interaction for interstitial
      COMPLEX             :: expIdGR(noGPts,gptmd,ntype,MAXVAL(neq))   ! exp(-i(gPts-gptm)R)
      COMPLEX             :: sumExpIdGR(noGPts,gptmd,ntype)            ! sum over atom of same type

      atoms: FORALL ( cn=1:ntype )
        !                                                         -i(G-G_I)R_a
        ! Calculate for all similar atoms (same type) the factor e
        ! use that the G vectors and atomic positions are given in internal units
        FORALL ( ci=1:neq(cn) )
          expIdGR(:,:,cn,ci) = EXP( -r2pi * img * my_dot_product( gPts_gptm(:,:,:),taual(:,natdPtr(cn)+ci) ) )
        END FORALL
        ! Add all factors for atoms of the same type
        sumExpIdGR(:,:,cn) = my_sum(expIdGR(:,:,cn,:))

        ! Calculate the inter-atom factor which is the same for all atoms of the same type
        abs_dgR(:,:,cn)    = abs_dg * rmt(cn)
        inter_atom(:,:,cn) = ( SIN(abs_dgR(:,:,cn)) - abs_dgR(:,:,cn) * COS(abs_dgR(:,:,cn)) ) / abs_dg**3
      END FORALL atoms

      ! Add the factors of all atoms together
      calculateSummation = my_dot_product(sumExpIdGR, inter_atom)
    END FUNCTION calculateSummation

  END SUBROUTINE calculate_fourier_transform

  ! Calculate several Fourier transformations
  ! a) the Fourier transformed potential
  !                                                 /          \
  !          /      | erf |       \     4 Pi       |   |k+G|^2  |
  ! V (k) = ( k + G | --- | k + G  ) = -------  Exp| - -------  |                             [1]
  !  G       \      |  r  |       /    |k+G|^2     |    4 w^2   |
  !                                                 \          /
  !
  ! b) muffin-tin basis function
  !                                                            R
  !                                     -L                      /
  !           /      |      \     4 Pi i         ^     -i G R  |     2
  ! MT (k) = ( k + G | M     ) = ---------- Y ( k+G ) e        | dr r  Phi(r) j ( |k+G| r )  [2]
  !   G,I     \      |  k,I /     Sqrt(Om)   LM                |               L
  !                                                           /
  !                                                            0
  !
  ! In the code:
  ! V_G(k):  potential
  ! MT_G(k): muffintin
  !
  ! Input:
  ! rmsh        - array of radial mesh points
  ! rmt         - radius of the muffin tin
  ! dx          - logarithmic increment of radial mesh
  ! jri         - number of radial mesh points
  ! jmtd        - dimension of radial mesh points
  ! bk          - one(!) k-vector for which FT is calculated
  ! bmat        - reciprocal lattice vector for all directions
  ! vol         - volume of the unit cell
  ! ntype       - number of atom types
  ! neq         - number of symmetry-equivalent atoms of atom type i
  ! natd        - dimesion of atoms in unit cell
  ! taual       - vector of atom positions (internal coordinates)
  ! lcutm       - l cutoff for mixed basis
  ! maxlcutm    - maximum of all these l cutoffs
  ! nindxm      - number of radial functions of mixed basis
  ! maxindxm    - maximum of these numbers
  ! gptm        - reciprocal lattice vectors of the mixed basis (internal coord.)
  ! ngptm       - number of vectors (for treated k-vector)
  ! pgptm       - pointer to the appropriate g-vector (for treated k-vector)
  ! gptmd       - dimension of gptm
  ! basm        - mixed basis functions (mt + inter) for treated k-vector
  ! lexp        - cutoff of spherical harmonics expansion of plane wave
  ! noGPts      - no g-vectors used for Fourier trafo
  ! Output:
  ! potential   - Fourier transformation of the potential
  ! muffintin   - Fourier transformation of all MT functions
  SUBROUTINE calculate_fourier_transform_once( &
      ! Input
      rmsh,rmt,dx,jri,jmtd,bk,ikpt,nkptf,           &
      bmat,vol,ntype,neq,natd,taual,lcutm,maxlcutm, &
      nindxm,maxindxm,gptm,ngptm,pgptm,gptmd,       &
      nbasp,basm,noGPts,invsat,invsatnr,irank,      &
      ! Output
      potential, fourier_trafo)

    USE m_constants
    USE m_util,      ONLY : sphbessel,pure_intgrf,intgrf_init,intgrf_out,NEGATIVE_EXPONENT_WARNING,NEGATIVE_EXPONENT_ERROR
    USE m_trafo,     ONLY : symmetrize

    IMPLICIT NONE

    ! scalar input
    INTEGER, INTENT(IN)    :: natd,ntype,maxlcutm
    INTEGER, INTENT(IN)    :: ikpt,nkptf,jmtd
    INTEGER, INTENT(IN)    :: maxindxm,irank
    INTEGER, INTENT(IN)    :: gptmd,noGPts
    REAL, INTENT(IN)       :: vol

    ! array input
    INTEGER, INTENT(IN)    :: lcutm(ntype)
    INTEGER, INTENT(IN)    :: nindxm(0:maxlcutm,ntype),neq(ntype)
    INTEGER, INTENT(IN)    :: jri(ntype)
    INTEGER, INTENT(IN)    :: gptm(3,gptmd)
    INTEGER, INTENT(IN)    :: ngptm,nbasp
    INTEGER, INTENT(IN)    :: pgptm(ngptm)
    INTEGER, INTENT(IN)    :: invsat(natd),invsatnr(natd)

    REAL, INTENT(IN)       :: bk(3)
    REAL, INTENT(IN)       :: rmsh(jmtd,ntype),rmt(ntype),dx(ntype)
    REAL, INTENT(IN)       :: basm(jmtd,maxindxm,0:maxlcutm,ntype)
    REAL, INTENT(IN)       :: bmat(3,3)
    REAL, INTENT(IN)       :: taual(3,natd)

    ! array output
    REAL,    INTENT(OUT)   :: potential(noGPts)                           ! Fourier transformed potential
#ifdef CPP_INVERSION
    REAL,    INTENT(OUT)   :: fourier_trafo(nbasp,noGPts) !muffintin_out(nbasp,noGPts)
#else
    COMPLEX, INTENT(OUT)   :: fourier_trafo(nbasp,noGPts) !muffintin_out(nbasp,noGPts)
#endif

    ! private scalars
    INTEGER                :: cg,cg2,ci,cl,cm,cn,cr                       ! counter variables
    REAL                   :: r2Pi, r4Pi, pi_omega2                   ! Pi, 2*Pi, 4*Pi, Pi/omega^2
    REAL                   :: sVol, r4Pi_sVol, r4Pi_Vol                   ! sqrt(vol), 4*Pi/sqrt(Vol), 4*Pi/Vol
    REAL                   :: omega, r1_omega2, r1_4omega2
!     REAL,    PARAMETER     :: omega = omega_VHSE()                        ! omega of the HSE functional
!     REAL,    PARAMETER     :: r1_omega2  = 1.0 / omega**2                 ! 1/omega^2
!     REAL,    PARAMETER     :: r1_4omega2 = 0.25 * r1_omega2               ! 1/(4 omega^2)
    COMPLEX, PARAMETER     :: img = (0d0,1d0)                             ! i

    ! private arrays
    INTEGER                :: gPts(3,noGPts)                              ! g vectors (internal units)
    INTEGER                :: natdPtr(ntype+1)                            ! pointer to all atoms of one type
    INTEGER                :: ptrType(nbasp),ptrEq(nbasp),&               ! pointer from ibasp to corresponding
                              ptrLM(nbasp),ptrN(nbasp)                    ! type, atom, l and m, and n
    REAL,    ALLOCATABLE   :: gridf(:,:)                                  ! grid for radial integration
    REAL                   :: k_G(3,noGPts)                               ! k + G
    REAL                   :: AbsK_G(noGPts),AbsK_G2(noGPts)              ! abs(k+G), abs(k+G)^2
    REAL                   :: arg(noGPts)                                 ! abs(k+G)^2 / (4*omega^2)
    REAL                   :: sphbesK_Gr(noGPts,jmtd,0:maxlcutm,ntype)    ! spherical bessel function of abs(k+G)r
    TYPE(intgrf_out)       :: intgrMT(noGPts,maxindxm,0:maxlcutm,ntype)   ! integration in muffin-tin
    COMPLEX                :: imgl(0:maxlcutm)                            ! i^l
    COMPLEX                :: Ylm(noGPts,(maxlcutm+1)**2)                 ! spherical harmonics for k+G and all lm
    COMPLEX                :: expIGR(noGPts,ntype,MAXVAL(neq))            ! exp(-iGR) for all atom types
    COMPLEX                :: muffintin(noGPts,maxindxm,&                 ! muffin-tin overlap integral
                                        (maxlcutm+1)**2,ntype,MAXVAL(neq))
#ifdef CPP_INVERSION
    COMPLEX                :: sym_muffintin(nbasp,noGPts)                 ! symmetrized muffin tin
#endif

    LOGICAL, SAVE          :: first_entry = .TRUE.                        ! allocate arrays in first entry

    ! allocate arrays in first entry reuse later
    IF ( first_entry ) THEN
      ALLOCATE ( already_known(nkptf),                       &            ! stores which elements are known
                 known_potential(maxNoGPts,nkptf),           &            ! stores the potential for all k-points
                 known_fourier_trafo(nbasp, maxNoGPts,nkptf) )            ! stores the fourier transform of the mixed basis
      ! initialization
      already_known       = .FALSE.
      known_potential     = 0.0
      known_fourier_trafo = 0.0
      ! unset flag as arrays are allocated
      first_entry         = .FALSE.
    ELSE
      ! check if size of arrays has changed and stop with error if they did
      IF ( SIZE(already_known) /= nkptf )         STOP 'hsefunctional: Array size changed!'
      IF ( SIZE(known_fourier_trafo,1) /= nbasp ) STOP 'hsefunctional: Array size changed!'
    END IF

    ! if the current k-point was not calculated yet
    IF ( .NOT.(already_known(ikpt)) ) THEN

      ! Calculate helper variables
      r2Pi       = 2.0 * pi_const
      r4Pi       = 4.0 * pi_const
      sVol       = SQRT(vol)
      r4Pi_sVol  = r4Pi / sVol
      r4Pi_Vol   = r4Pi / Vol
1402
      omega      = omega_hse!omega_VHSE()
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      r1_omega2  = 1.0 / omega**2
      r1_4omega2 = 0.25 * r1_omega2
      pi_omega2  = pi_const * r1_omega2

      ! calculate pointers for all atom-types to all atoms
      natdPtr(ntype+1) = natd
      DO cn=ntype,1,-1
        natdPtr(cn) = natdPtr(cn+1) - neq(cn)
      END DO

      ! Define imgl(l) = img**l
      imgl(0) = 1.0
      DO ci=1,maxlcutm
        imgl(ci) = imgl(ci-1) * img
      END DO

      ! generate grid for fast radial integration
      CALL intgrf_init(ntype,jmtd,jri,dx,rmsh,gridf)

      ! Set all arrays to 0
      gPts           = 0
      k_G            = 0
      AbsK_G         = 0
      arg            = 0
      sphbesK_Gr     = 0
      intgrMT%value  = 0
      intgrMT%ierror = 0
      Ylm            = 0
      expIGR         = 0
      potential      = 0
      muffintin      = 0

! Calculate the muffin-tin basis function overlap integral
!                                                            R
!                                     -L                      /
!           /      |      \     4 Pi i         ^     -i G R  |     2
! MT (k) = ( k + G | M     ) = ---------- Y ( k+G ) e        | dr r  Phi(r) j ( |k+G| r )  [2]
!   G,I     \      |  k,I /     Sqrt(Om)   LM                |               L
!                                                           /
!                                                            0
      IF (ngptm < noGpts) STOP 'hsefunctional: error calculating Fourier coefficients, noGpts too large'

      gPts(:,:) = gptm(:,pgptm(1:noGPts))

      gpoints: FORALL ( cg=1:noGPts )
        ntypesA: FORALL ( cn=1:ntype )

          ! Calculate the phase factor exp(-iGR) for all atoms
          FORALL ( ci=1:neq(cn) )
            expIGR(cg,cn,ci)        = EXP( -r2Pi * img * DOT_PRODUCT(taual(:,natdPtr(cn)+ci),gPts(:,cg)) )
            muffintin(cg,:,:,cn,ci) = r4Pi_sVol * expIGR(cg,cn,ci)
          END FORALL

          ! Multiplication with i^-L
          FORALL ( cl=0:lcutm(cn) )
            muffintin(cg,:,cl*cl+1:(cl+1)*(cl+1),cn,:) = muffintin(cg,:,cl*cl+1:(cl+1)*(cl+1),cn,:) / imgl(cl)
          END FORALL

        END FORALL ntypesA

        ! Calculate the k+G, abs(k+G), and abs(k+G)^2
        k_G(:,cg)   = MATMUL( gPts(:,cg)+bk(:), bmat(:,:) )
        AbsK_G2(cg) = SUM( k_G(:,cg)**2 )
        AbsK_G(cg)  = SQRT(AbsK_G2(cg))

        ! Spherical harmonics are calculated for all lm's and k+G's
        Ylm(cg,:) = calcYlm(k_G(:,cg),maxlcutm)

        ! Perform the integration in eq.[2] for all muffin-tins
        ntypesB: FORALL ( cn=1:ntype )

          ! Multiplication with the spherical harmonic
          FORALL ( cl=1:(lcutm(cn)+1)**2 )
            muffintin(cg,:,cl,cn,:) = muffintin(cg,:,cl,cn,:) * Ylm(cg,cl)
          END FORALL

          ! Calculate the spherical bessel function
          FORALL ( cr=1:jri(cn) )
            sphbesK_Gr(cg,cr,:,cn) = calcSphBes(AbsK_G(cg)*rmsh(cr,cn),maxlcutm)
          END FORALL
          ! integrate the function and multiplicate to the result
          FORALL ( cl=0:lcutm(cn) )
            FORALL ( ci=1:nindxm(cl,cn) )
              intgrMT(cg,ci,cl,cn) = pure_intgrf(rmsh(:,cn) * basm(:,ci,cl,cn) * &
                sphbesK_Gr(cg,:,cl,cn),jri,jmtd,rmsh,dx,ntype,cn,gridf)
              muffintin(cg,ci,cl*cl+1:(cl+1)*(cl+1),cn,:) = &
                muffintin(cg,ci,cl*cl+1:(cl+1)*(cl+1),cn,:) * intgrMT(cg,ci,cl,cn)%value
            END FORALL
          END FORALL

        END FORALL ntypesB

      END FORALL gpoints

      ! Check if any of the integrations failed and abort if one did
      IF ( ANY( intgrMT%ierror == NEGATIVE_EXPONENT_ERROR ) ) THEN
          IF ( irank == 0 ) WRITE(6,*) 'intgrf: Warning! Negative exponent x in extrapolation a+c*r**x'
      ELSEIF ( ANY( intgrMT%ierror == NEGATIVE_EXPONENT_WARNING ) ) THEN
        IF ( irank == 0 ) WRITE(6,*) 'intgrf: Negative exponent x in extrapolation a+c*r**x'
        STOP       'intgrf: Negative exponent x in extrapolation a+c*r**x'
      END IF

! Calculate the Fourier transformed potential
!                                                 /          \
!          /      | erf |       \      4 Pi      |   |k+G|^2  |
! V (k) = ( k + G | --- | k + G' ) = -------  Exp| - -------  | kronecker(G,G')             [1]
!  G       \      |  r  |       /    |k+G|^2     |    4 w^2   |
!                                                 \          /
      WHERE ( absK_G.NE.0 )
        ! Calculate (k+G)^2 / 4*w^2
        arg = -r1_4omega2 * AbsK_G2

        ! The potential is calculated using eq.[1]
        potential = r4Pi / AbsK_G2 * EXP(arg)

! Calculate the Fourier transformed potential for the full(!) potential
!
!          /          | erfc |          \      Pi
! V (k) = ( k + G = 0 | ---- | k + G = 0 ) = -----
!  G       \          |   r  |          /     w^2
!
      ELSEWHERE
        ! the negative sign is added to compensate the sign when the
        ! contribution is added to the Coulomb matrix
        potential = -pi_omega2
      endwhere

      DEALLOCATE( gridf )

      !
      ! Create pointer which correlate the position in the array with the
      ! appropriate indices of the MT mixed basis function
      !
      cg = 0
      DO cn=1,ntype
        DO ci=1,neq(cn)
          DO cl=0,lcutm(cn)
            DO cm=-cl,cl
              DO cr=1,nindxm(cl,cn)
                cg = cg + 1
                ptrType(cg) = cn
                ptrEq(cg)   = ci
                ptrLM(cg)   = (cl+1)*cl+cm+1
                ptrN(cg)    = cr
              END DO
            END DO
          END DO
        END DO
      END DO
      IF ( nbasp /= cg ) STOP 'hsefunctional: wrong array size: nbasp'

#ifdef CPP_INVERSION
      ! Symmetrize muffin tin fourier transform
      DO ci=1,nbasp
        sym_muffintin(ci,:noGPts) = muffintin(:,ptrN(ci),ptrLM(ci),ptrType(ci),ptrEq(ci))
      END DO
      DO cg=1,noGPts
        CALL symmetrize( sym_muffintin(:,cg),1,nbasp,2,.FALSE., &
                         ntype,ntype,neq,lcutm,maxlcutm,        &
                         nindxm,natd,invsat,invsatnr )
      END DO
      ! store the fourier transform of the muffin tin basis
      known_fourier_trafo(:,:,ikpt) = REAL( sym_muffintin )
#else
      ! store the fourier transform of the muffin tin basis
      DO ci=1,nbasp
        known_fourier_trafo(ci,:noGPts,ikpt) = CONJG( muffintin(:,ptrN(ci),ptrLM(ci),ptrType(ci),ptrEq(ci)) )
      END DO
#endif
      ! store the fourier transform of the potential
      known_potential(:noGPts,ikpt)   = potential
      ! set the flag so that the fourier transform is not calculated again
      already_known(ikpt)             = .TRUE.
      IF ( MINVAL( ABS(potential) ) > 1e-12 ) THEN
        WRITE(*,*) 'hsefunctional: Warning! Smallest potential bigger than numerical precision!', MINVAL( ABS(potential) )
        WRITE(*,*) 'Perhaps you should increase the number of g-Points used and recompile!'
      END IF

    END IF

    ! return the fourier transform
    potential     = known_potential(:noGPts,ikpt)
    fourier_trafo = known_fourier_trafo(:,:noGPts,ikpt)

  END SUBROUTINE calculate_fourier_transform_once

  ! Correct the pure Coulomb Matrix with by subtracting the long-range component
  !
  !  /     |      |      \     /     |       |      \     /     |     |      \
  ! ( M    | V    | M     ) = ( M    | V     | M     ) - ( M    | V   | M     )
  !  \ k,I |  HSE |  k,J /     \ k,I |  coul |  k,J /     \ k,I |  LR |  k,J /
  !
  ! The long-range component is given py the potential
  !
  !         erf( w r )
  ! V (r) = ----------
  !  LR         r
  !
  ! Input:
  ! rmsh     - array of radial mesh points
  ! rmt      - radius of the muffin tin
  ! dx       - logarithmic increment of radial mesh
  ! jri      - number of radial mesh points
  ! jmtd     - dimension of radial mesh points
  ! nkptf    - number of total k-points
  ! nkptd    - dimension of k-points
  ! nkpti    - number of irreducible k-points in window (in KPTS)
  ! bk       - k-vector for all irreduble k-points
  ! bmat     - reciprocal lattice vector for all directions
  ! vol      - volume of unit cell
  ! ntype    - number of atom types
  ! neq      - number of symmetry-equivalent atoms of atom type i
  ! natd     - dimension of atoms in unit cell
  ! taual    - vector of atom positions (internal coordinates)
  ! lcutm    - l cutoff for mixed basis
  ! maxlcutm - maximum of all these l cutoffs
  ! nindxm   - number of radial functions of mixed basis
  ! maxindxm - maximum of these numbers
  ! gptm     - reciprocal lattice vectors of the mixed basis (internal coord.)
  ! ngptm    - number of vectors
  ! pgptm    - pointer to the appropriate g-vector
  ! gptmd    - dimension of gptm
  ! basm     - radial mixed basis functions (mt + inter)
  ! lexp     - cutoff of spherical harmonics expansion of plane wave
  ! maxbasm  - maximum number of mixed basis functions
  ! nbasm    - number of mixed basis function
  ! invsat   - number of inversion-symmetric atom
  ! invsatnr - number of inversion-symmetric atom
  ! Inout:
  ! coulomb  - Coulomb matrix which is changed
  SUBROUTINE change_coulombmatrix(&
      ! Input
      rmsh,rmt,dx,jri,jmtd,nkptf,nkptd,nkpti,bk,      &
      bmat,vol,ntype,neq,natd,taual,lcutm,maxlcutm,   &
      nindxm,maxindxm,gptm,ngptm,pgptm,gptmd,         &
      basm,lexp,maxbasm,nbasm,invsat,invsatnr,irank,  &
      ! Input & output
      coulomb)

    USE m_trafo, ONLY: symmetrize
    USE m_wrapper, ONLY: packmat, unpackmat, diagonalize, inverse
    USE m_olap,  ONLY: olap_pw

    IMPLICIT NONE

    ! scalar input
    INTEGER, INTENT(IN)    :: natd,ntype,maxlcutm,lexp
    INTEGER, INTENT(IN)    :: jmtd,nkptf,nkptd,nkpti
    INTEGER, INTENT(IN)    :: maxindxm
    INTEGER, INTENT(IN)    :: gptmd,irank
    INTEGER, INTENT(IN)    :: maxbasm
    REAL, INTENT(IN)       :: vol

    ! array input
    INTEGER, INTENT(IN)    :: lcutm(ntype)
    INTEGER, INTENT(IN)    :: nindxm(0:maxlcutm,ntype),neq(ntype)
    INTEGER, INTENT(IN)    :: jri(ntype)
    INTEGER, INTENT(IN)    :: gptm(3,gptmd)
    INTEGER, INTENT(IN)    :: ngptm(nkptf)
    INTEGER, INTENT(IN)    :: pgptm(MAXVAL(ngptm),nkptf)
    INTEGER, INTENT(IN)    :: nbasm(nkptf)
    INTEGER, INTENT(IN)    :: invsat(natd),invsatnr(natd)

    REAL, INTENT(IN)       :: bk(3,nkptd)
    REAL, INTENT(IN)       :: rmsh(jmtd,ntype),rmt(ntype),dx(ntype)
    REAL, INTENT(IN)       :: basm(jmtd,maxindxm,0:maxlcutm,ntype)
    REAL, INTENT(IN)       :: bmat(3,3)
    REAL, INTENT(IN)       :: taual(3,natd)

    ! array inout
#ifdef CPP_INVERSION
    REAL, INTENT(INOUT)    :: coulomb(maxbasm*(maxbasm+1)/2,nkpti)
#else
    COMPLEX, INTENT(INOUT) :: coulomb(maxbasm*(maxbasm+1)/2,nkpti)
#endif

    ! private scalars
    INTEGER                :: ikpt,itype,ieq,il,im,iindxm,idum,n1,n2,ok    ! counters and other helper variables
    INTEGER                :: noGPts,nbasp                                 ! no used g-vectors, no MT functions

    ! private arrays
    INTEGER, ALLOCATABLE   :: ptrType(:),ptrEq(:),ptrL(:),ptrM(:),ptrN(:)  ! Pointer
    REAL                   :: potential(maxNoGPts)                         ! Fourier transformed potential
    COMPLEX                :: muffintin(maxNoGPts,maxindxm,&               ! muffin-tin overlap integral
                                        (maxlcutm+1)**2,ntype,MAXVAL(neq))
    COMPLEX                :: interstitial(maxNoGPts,gptmd)                ! interstistial overlap intergral
    COMPLEX, ALLOCATABLE   :: coulmat(:,:)                                 ! helper array to symmetrize coulomb

    ! Check size of arrays
    IF (nkpti > nkptd) STOP 'hsefunctional: missmatch in dimension of arrays'
    nbasp = maxbasm - MAXVAL(ngptm)
    IF ( ANY( nbasm - ngptm /= nbasp ) ) STOP 'hsefunctional: wrong assignment of nbasp'

    !
    ! Create pointer which correlate the position in the array with the
    ! appropriate indices of the MT mixed basis function
    !
    ALLOCATE( ptrType(nbasp), ptrEq(nbasp), ptrL(nbasp), ptrM(nbasp), ptrN(nbasp) )
    nbasp = 0
    DO itype=1,ntype
       DO ieq=1,neq(itype)
          DO il=0,lcutm(itype)
             DO im=-il,il
                DO iindxm=1,nindxm(il,itype)
                   nbasp = nbasp + 1
                   ptrType(nbasp) = itype
                   ptrEq(nbasp)   = ieq
                   ptrL(nbasp)    = il
                   ptrM(nbasp)    = im
                   ptrN(nbasp)    = iindxm
                END DO
             END DO
          END DO
       END DO
    END DO

    !
    ! Change the Coulomb matrix for all k-points
    !
    DO ikpt=1,nkpti
      ! use the same g-vectors as in the mixed basis
      ! adjust the limit of the array if necessary
      IF (ngptm(ikpt) < maxNoGPts) THEN
        noGPts = ngptm(ikpt)
      ELSE
        noGPts = maxNoGPts
      END IF

      !
      ! Calculate the Fourier transform of the mixed basis and the potential
      !
      CALL calculate_fourier_transform( &
           ! Input
           rmsh,rmt,dx,jri,jmtd,bk(:,ikpt),&
           bmat,vol,ntype,neq,natd,taual,lcutm,maxlcutm,&
           nindxm,maxindxm,gptm,ngptm(ikpt),pgptm(:,ikpt),gptmd,&
           basm,noGPts,irank,&
           ! Output
           potential,muffintin,interstitial)
      interstitial = CONJG(interstitial)

      ! Helper matrix for temporary storage of the attenuated Coulomb matrix
      ALLOCATE ( coulmat(nbasm(ikpt),nbasm(ikpt)) , stat=ok )
      IF ( ok /= 0 ) STOP 'hsefunctional: failure at matrix allocation'
      coulmat = 0
      !
      ! Calculate the difference of the Coulomb matrix by the attenuation
      !
      DO n1 = 1,nbasp
        DO n2 = 1,n1
          ! muffin tin - muffin tin contribution
          coulmat(n2,n1) = - gPtsSummation( noGpts,&
                                muffintin(:,ptrN(n2),(ptrL(n2)+1)*ptrL(n2)+ptrM(n2)+1,ptrType(n2),ptrEq(n2)),potential,&
                                CONJG(muffintin(:,ptrN(n1),(ptrL(n1)+1)*ptrL(n1)+ptrM(n1)+1,ptrType(n1),ptrEq(n1))) )
          coulmat(n1,n2) = CONJG(coulmat(n2,n1))
        END DO
      END DO
      DO n1 = nbasp+1,nbasm(ikpt)
        DO n2 = 1,n1
          IF ( n2 <= nbasp ) THEN
            ! muffin tin - interstitial contribution
            coulmat(n2,n1) = - gPtsSummation( noGPts,&
                                  muffintin(:,ptrN(n2),(ptrL(n2)+1)*ptrL(n2)+ptrM(n2)+1,ptrType(n2),ptrEq(n2)),&
                                  potential,CONJG(interstitial(:,pgptm(n1-nbasp,ikpt))) )
            coulmat(n1,n2) = CONJG(coulmat(n2,n1))
          ELSE
            ! interstitial - interstitial contribution
            coulmat(n2,n1) = - gPtsSummation( noGPts,&
                                  interstitial(:,pgptm(n2-nbasp,ikpt)),potential,&
                                  CONJG(interstitial(:,pgptm(n1-nbasp,ikpt))) )
            coulmat(n1,n2) = CONJG(coulmat(n2,n1))
          END IF
        END DO
      END DO

#ifdef CPP_INVERSION
      ! symmetrize matrix if system has inversion symmetry
      CALL symmetrize(coulmat,nbasm(ikpt),nbasm(ikpt),3,.FALSE., &
                      ntype,ntype,neq,lcutm,maxlcutm, &
                      nindxm,natd,invsat,invsatnr)
#endif
      ! add the changes to the Coulomb matrix
      coulomb(:nbasm(ikpt)*(nbasm(ikpt)+1)/2,ikpt) = packmat( coulmat ) + coulomb(:nbasm(ikpt)*(nbasm(ikpt)+1)/2,ikpt)
      DEALLOCATE ( coulmat )

    END DO

    DEALLOCATE( ptrType, ptrEq, ptrL, ptrM, ptrN )

  END SUBROUTINE change_coulombmatrix

  ! Correct the pure Coulomb Matrix with by subtracting the long-range component
  ! during execution time
  !
  !  /     |      |      \     /     |       |      \     /     |     |      \
  ! ( M    | V    | M     ) = ( M    | V     | M     ) - ( M    | V   | M     )
  !  \ k,I |  HSE |  k,J /     \ k,I |  coul |  k,J /     \ k,I |  LR |  k,J /
  !
  ! The long-range component is given py the potential
  !
  !         erf( w r )
  ! V (r) = ----------
  !  LR         r
  !
  ! Input:
  ! rmsh     - array of radial mesh points
  ! rmt      - radius of the muffin tin
  ! dx       - logarithmic increment of radial mesh
  ! jri      - number of radial mesh points
  ! jmtd     - dimension of radial mesh points
  ! bk       - k-vector for this k-point
  ! ikpt     - number of this k-point
  ! nkptf    - number of total k-points
  ! bmat     - reciprocal lattice vector for all directions
  ! vol      - volume of unit cell
  ! ntype    - number of atom types
  ! neq      - number of symmetry-equivalent atoms of atom type i
  ! natd     - dimension of atoms in unit cell
  ! taual    - vector of atom positions (internal coordinates)
  ! lcutm    - l cutoff for mixed basis
  ! maxlcutm - maximum of all these l cutoffs
  ! nindxm   - number of radial functions of mixed basis
  ! maxindxm - maximum of these numbers
  ! gptm     - reciprocal lattice vectors of the mixed basis (internal coord.)
  ! ngptm    - number of vectors
  ! pgptm    - pointer to the appropriate g-vector
  ! gptmd    - dimension of gptm
  ! basm     - radial mixed basis functions (mt + inter)
  ! nbasm    - number of mixed basis function
  ! nobd     - dimension of occupied bands
  ! nbands   - number of bands
  ! nsst     - size of indx
  ! indx     - pointer to bands
  ! invsat   - number of inversion-symmetric atom
  ! invsatnr - number of inversion-symmetric atom
  ! cprod    - scalar product of mixed basis and wavefunction product basis
  ! wl_iks   -
  ! n_q      -
  ! Return:
  ! Change of the Coulomb matrix
  FUNCTION dynamic_hse_adjustment( &
      rmsh,rmt,dx,jri,jmtd,bk,ikpt,nkptf,bmat,vol,&
      ntype,neq,natd,taual,lcutm,maxlcutm,nindxm,maxindxm,&
      gptm,ngptm,pgptm,gptmd,basm,nbasm,&
      nobd,nbands,nsst,ibando,psize,indx,invsat,invsatnr,irank,&
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      cprod_r,cprod_c,l_real,wl_iks,n_q)
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    USE m_trafo, ONLY: symmetrize
    USE m_olap,  ONLY: olap_pw, olap_pwp
    USE m_wrapper, ONLY: diagonalize, dotprod, matvec, packmat, inverse

    IMPLICIT NONE

    ! scalar input
    INTEGER, INTENT(IN)  :: natd,ntype,maxlcutm
    INTEGER, INTENT(IN)  :: jmtd,ikpt,nkptf
    INTEGER, INTENT(IN)  :: maxindxm
    INTEGER, INTENT(IN)  :: gptmd,irank
    INTEGER, INTENT(IN)  :: nbasm,nobd,nbands,ibando,psize
    INTEGER, INTENT(IN)  :: n_q
    REAL, INTENT(IN)     :: vol

    ! array input
    INTEGER, INTENT(IN)  :: lcutm(ntype)
    INTEGER, INTENT(IN)  :: nindxm(0:maxlcutm,ntype),neq(ntype)
    INTEGER, INTENT(IN)  :: jri(ntype)
    INTEGER, INTENT(IN)  :: gptm(3,gptmd)
    INTEGER, INTENT(IN)  :: ngptm
    INTEGER, INTENT(IN)  :: pgptm(ngptm)
    INTEGER, INTENT(IN)  :: nsst(nbands),indx(nbands,nbands)
    INTEGER, INTENT(IN)  :: invsat(natd),invsatnr(natd)
    REAL, INTENT(IN)     :: bk(3)
    REAL, INTENT(IN)     :: rmsh(jmtd,ntype),rmt(ntype),dx(ntype)
    REAL, INTENT(IN)     :: basm(jmtd,maxindxm,0:maxlcutm,ntype)
    REAL, INTENT(IN)     :: bmat(3,3)
    REAL, INTENT(IN)     :: taual(3,natd)
    REAL, INTENT(IN)     :: wl_iks(nobd)
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    REAL, INTENT(IN)     :: cprod_r(nbasm,psize,nbands)
    COMPLEX, INTENT(IN)  :: cprod_c(nbasm,psize,nbands)
    LOGICAL,INTENT(IN)   :: l_real
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    ! return type definition
    COMPLEX              :: dynamic_hse_adjustment(nbands,nbands)

    ! private scalars
    INTEGER              :: noGpts,nbasp
    INTEGER              :: itype,ieq,il,im,iindxm,ibasp
    INTEGER              :: igpt,iobd,iobd0,isst,iband1,iband2
    COMPLEX              :: cdum,gPtsSum

    ! private arrays
    INTEGER, ALLOCATABLE :: ptrType(:),ptrEq(:),ptrL(:),ptrM(:),ptrN(:) ! pointer to reference muffintin-array
    REAL                 :: potential(maxNoGPts)                        ! Fourier transformed potential
    COMPLEX              :: muffintin(maxNoGPts,maxindxm,&              ! muffin-tin overlap integral
                                        (maxlcutm+1)**2,ntype,MAXVAL(neq))
    COMPLEX              :: interstitial(maxNoGPts,gptmd)               ! interstistial overlap intergral
#ifdef CPP_INVERSION
    REAL                 :: fourier_trafo(nbasm-ngptm,maxNoGPts)        ! Fourier trafo of all mixed basis functions
#else
    COMPLEX              :: fourier_trafo(nbasm-ngptm,maxNoGPts)        ! Fourier trafo of all mixed basis functions
#endif

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    REAL                 :: cprod_fourier_trafo_r(maxNoGpts,psize,nbands)  ! Product of cprod and Fourier trafo
    COMPLEX              :: cprod_fourier_trafo_c(maxNoGpts,psize,nbands)  ! Product of cprod and Fourier trafo
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    ! Initialisation
    dynamic_hse_adjustment = 0.0
    potential              = 0.0
    fourier_trafo          = 0.0
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    cprod_fourier_trafo_r    = 0.0
    cprod_fourier_trafo_c    = 0.0
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    noGPts                 = MIN(ngptm,maxNoGPts)
    nbasp                  = nbasm - ngptm

    !
    ! Calculate the fourier transform of the mixed basis once
    ! If it was already calculated load the old results
    !
    CALL calculate_fourier_transform_once(          &
      rmsh,rmt,dx,jri,jmtd,bk,ikpt,nkptf,           &
      bmat,vol,ntype,neq,natd,taual,lcutm,maxlcutm, &
      nindxm,maxindxm,gptm,ngptm,pgptm,gptmd,       &
      nbasp,basm,noGPts,invsat,invsatnr,irank,      &
      potential,fourier_trafo)

    ! Calculate the Fourier transform of the 'normal' basis
    ! by summing over the mixed basis
    DO igpt = 1,noGPts
      DO iobd0 = 1,psize!ibando,min(ibando+psize,nobd)
        iobd = iobd0 + ibando - 1
        IF( iobd .GT. nobd ) CYCLE
        DO iband1 = 1,nbands
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