gamma.F

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!--------------------------------------------------------------------------------------------------!
!   CP2K: A general program to perform molecular dynamics simulations                              !
!   Copyright 2000-2024 CP2K developers group <https://cp2k.org>                                   !
!                                                                                                  !
!   SPDX-License-Identifier: GPL-2.0-or-later                                                      !
!--------------------------------------------------------------------------------------------------!
 
! **************************************************************************************************
!> \brief Calculation of the incomplete Gamma function F_n(t) for multi-center
!>      integrals over Cartesian Gaussian functions.
!> \par History
!>      - restructured and cleaned (24.05.2004,MK)
!> \author Matthias Krack (07.01.1999)
! **************************************************************************************************
MODULE gamma
 
   USE kinds,                           ONLY: dp
   USE mathconstants,                   ONLY: ifac,&
                                              pi
#include "../base/base_uses.f90"
 
   IMPLICIT NONE
 
   PRIVATE
 
! *** Global parameters ***
 
   CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'gamma'
   REAL(KIND=dp), PARAMETER  :: teps = 1.0e-13_dp
 
! *** Maximum n value of the tabulated F_n(t) function values ***
 
   INTEGER, SAVE :: current_nmax = -1
 
! *** F_n(t) table ***
 
   REAL(KIND=dp), DIMENSION(:, :), ALLOCATABLE, SAVE :: ftable
 
! *** Public subroutines ***
 
   PUBLIC :: deallocate_md_ftable, &
             fgamma_0, &
             fgamma_ref, &
             init_md_ftable
 
CONTAINS
 
! **************************************************************************************************
!> \brief   Build a table of F_n(t) values in the range tmin <= t <= tmax
!>          with a stepsize of tdelta up to n equal to nmax.
!> \param nmax ...
!> \param tmin ...
!> \param tmax ...
!> \param tdelta ...
!> \date    11.01.1999
!> \par Parameters
!>       - nmax  : Maximum n value of F_n(t).
!>       - tdelta: Difference between two consecutive t abcissas (step size).
!>       - tmax  : Maximum t value.
!>       - tmin  : Minimum t value.
!> \author  MK
!> \version 1.0
! **************************************************************************************************
   SUBROUTINE create_md_ftable(nmax, tmin, tmax, tdelta)
 
      INTEGER, INTENT(IN)                                :: nmax
      REAL(KIND=dp), INTENT(IN)                          :: tmin, tmax, tdelta
 
      INTEGER                                            :: itab, itabmax, itabmin, n
      REAL(KIND=dp)                                      :: t
 
      IF (current_nmax > -1) THEN
         CALL cp_abort(__location__, &
                       "An incomplete Gamma function table is already "// &
                       "allocated. Use the init routine for an update")
      END IF
 
      IF (nmax < 0) THEN
         CALL cp_abort(__location__, &
                       "A negative n value for the initialization of the "// &
                       "incomplete Gamma function is invalid")
      END IF
 
!   *** Check arguments ***
 
      IF ((tmax < 0.0_dp) .OR. &
          (tmin < 0.0_dp) .OR. &
          (tdelta <= 0.0_dp) .OR. &
          (tmin > tmax)) THEN
         cpabort("Invalid arguments")
      END IF
 
      n = nmax + 6
 
      itabmin = floor(tmin/tdelta)
      itabmax = ceiling((tmax - tmin)/tdelta)
 
      ALLOCATE (ftable(0:n, itabmin:itabmax))
      ftable = 0.0_dp
 
!   *** Fill table ***
 
      DO itab = itabmin, itabmax
         t = real(itab, dp)*tdelta
         ftable(0:n, itab) = fgamma_ref(n, t)
      END DO
 
!   *** Save initialization status ***
 
      current_nmax = nmax
 
   END SUBROUTINE create_md_ftable
 
! **************************************************************************************************
!> \brief   Deallocate the table of F_n(t) values.
!> \date    24.05.2004
!> \author  MK
!> \version 1.0
! **************************************************************************************************
   SUBROUTINE deallocate_md_ftable()
 
      IF (current_nmax > -1) THEN
 
         DEALLOCATE (ftable)
 
         current_nmax = -1
 
      END IF
 
   END SUBROUTINE deallocate_md_ftable
 
! **************************************************************************************************
!> \brief   Calculation of the incomplete Gamma function F(t) for multicenter
!>          integrals over Gaussian functions. f returns a vector with all
!>          F_n(t) values for 0 <= n <= nmax.
!> \param nmax ...
!> \param t ...
!> \param f ...
!> \date    08.01.1999,
!> \par History
!>          09.06.1999, MK : Changed from a FUNCTION to a SUBROUTINE
!> \par Literature
!>       L. E. McMurchie, E. R. Davidson, J. Comp. Phys. 26, 218 (1978)
!> \par Parameters
!>       - f   : The incomplete Gamma function F_n(t).
!>       - nmax: Maximum n value of F_n(t).
!>       - t   : Argument of the incomplete Gamma function.
!>       - kmax: Maximum number of iterations.
!>       - expt: Exponential term in the upward recursion of F_n(t).
!> \author  MK
!> \version 1.0
! **************************************************************************************************
   SUBROUTINE fgamma_0(nmax, t, f)
 
      INTEGER, INTENT(IN)                                :: nmax
      REAL(kind=dp), INTENT(IN)                          :: t
      REAL(kind=dp), DIMENSION(0:nmax), INTENT(OUT)      :: f
 
      INTEGER                                            :: itab, k, n
      REAL(kind=dp)                                      :: expt, g, tdelta, tmp, ttab
 
!   *** Calculate F(t) ***
 
      IF (t < teps) THEN
 
!     *** Special cases: t = 0 ***
 
         DO n = 0, nmax
            f(n) = 1.0_dp/real(2*n + 1, dp)
         END DO
 
      ELSE IF (t <= 12.0_dp) THEN
 
!     *** 0 < t < 12 -> Taylor expansion ***
 
         tdelta = 0.1_dp
 
!     *** Pretabulation of the F_n(t) function ***
!     *** for the Taylor series expansion      ***
 
         IF (nmax > current_nmax) THEN
            CALL init_md_ftable(nmax)
         END IF
 
         itab = nint(t/tdelta)
         ttab = real(itab, dp)*tdelta
 
         f(nmax) = ftable(nmax, itab)
 
         tmp = 1.0_dp
         DO k = 1, 6
            tmp = tmp*(ttab - t)
            f(nmax) = f(nmax) + ftable(nmax + k, itab)*tmp*ifac(k)
         END DO
 
         expt = exp(-t)
 
!     *** Use the downward recursion relation to ***
!     *** generate the remaining F_n(t) values   ***
 
         DO n = nmax - 1, 0, -1
            f(n) = (2.0_dp*t*f(n + 1) + expt)/real(2*n + 1, dp)
         END DO
 
      ELSE
 
!     *** t > 12 ***
         tmp = 1.0_dp/t ! reciprocal value
 
         IF (t <= 15.0_dp) THEN
 
!       *** 12 < t <= 15 -> Four term polynom expansion ***
 
            g = 0.4999489092_dp - 0.2473631686_dp*tmp + &
                0.321180909_dp*tmp**2 - 0.3811559346_dp*tmp**3
            f(0) = 0.5_dp*sqrt(pi*tmp) - g*exp(-t)*tmp
 
         ELSE IF (t <= 18.0_dp) THEN
 
!       *** 15 < t <= 18 -> Three term polynom expansion ***
 
            g = 0.4998436875_dp - 0.24249438_dp*tmp + 0.24642845_dp*tmp**2
            f(0) = 0.5_dp*sqrt(pi*tmp) - g*exp(-t)*tmp
 
         ELSE IF (t <= 24.0_dp) THEN
 
!       *** 18 < t <= 24 -> Two term polynom expansion ***
 
            g = 0.499093162_dp - 0.2152832_dp*tmp
            f(0) = 0.5_dp*sqrt(pi*tmp) - g*exp(-t)*tmp
 
         ELSE IF (t <= 30.0_dp) THEN
 
!       *** 24 < t <= 30 -> One term polynom expansion ***
 
            g = 0.49_dp
            f(0) = 0.5_dp*sqrt(pi*tmp) - g*exp(-t)*tmp
 
         ELSE
 
!       *** t > 30 -> Asymptotic formula ***
 
            f(0) = 0.5_dp*sqrt(pi*tmp)
 
         END IF
 
         IF (t > real(2*nmax + 36, dp)) THEN
            expt = 0.0_dp
         ELSE
            expt = exp(-t)
         END IF
 
!     *** Use the upward recursion relation to ***
!     *** generate the remaining F_n(t) values ***
 
         DO n = 1, nmax
            f(n) = (0.5_dp*tmp)*(real(2*n - 1, dp)*f(n - 1) - expt)
         END DO
 
      END IF
 
   END SUBROUTINE fgamma_0
 
! **************************************************************************************************
!> \brief   Calculation of the incomplete Gamma function F(t) for multicenter
!>          integrals over Gaussian functions. f returns a vector with all
!>          F_n(t) values for 0 <= n <= nmax.
!> \param nmax ...
!> \param t ...
!> \param f ...
!> \date    08.01.1999
!> \par Literature
!>       L. E. McMurchie, E. R. Davidson, J. Comp. Phys. 26, 218 (1978)
!> \par Parameters
!>       - f   : The incomplete Gamma function F_n(t).
!>       - nmax: Maximum n value of F_n(t).
!>       - t   : Argument of the incomplete Gamma function.
!> \author  MK
!> \version 1.0
! **************************************************************************************************
   SUBROUTINE fgamma_1(nmax, t, f)
 
      INTEGER, INTENT(IN)                                :: nmax
      REAL(kind=dp), DIMENSION(:), INTENT(IN)            :: t
      REAL(kind=dp), DIMENSION(SIZE(t, 1), 0:nmax), &
         INTENT(OUT)                                     :: f
 
      INTEGER                                            :: i, itab, k, n
      REAL(kind=dp)                                      :: expt, g, tdelta, tmp, ttab
 
      DO i = 1, SIZE(t, 1)
 
!     *** Calculate F(t) ***
 
         IF (t(i) < teps) THEN
 
!       *** Special cases: t = 0 ***
 
            DO n = 0, nmax
               f(i, n) = 1.0_dp/real(2*n + 1, dp)
            END DO
 
         ELSE IF (t(i) <= 12.0_dp) THEN
 
!       *** 0 < t < 12 -> Taylor expansion ***
 
            tdelta = 0.1_dp
 
!       *** Pretabulation of the F_n(t) function ***
!       *** for the Taylor series expansion      ***
 
            IF (nmax > current_nmax) THEN
               CALL init_md_ftable(nmax)
            END IF
 
            itab = nint(t(i)/tdelta)
            ttab = real(itab, dp)*tdelta
 
            f(i, nmax) = ftable(nmax, itab)
 
            tmp = 1.0_dp
            DO k = 1, 6
               tmp = tmp*(ttab - t(i))
               f(i, nmax) = f(i, nmax) + ftable(nmax + k, itab)*tmp*ifac(k)
            END DO
 
            expt = exp(-t(i))
 
!       *** Use the downward recursion relation to ***
!       *** generate the remaining F_n(t) values   ***
 
            DO n = nmax - 1, 0, -1
               f(i, n) = (2.0_dp*t(i)*f(i, n + 1) + expt)/real(2*n + 1, dp)
            END DO
 
         ELSE
 
!       *** t > 12 ***
 
            IF (t(i) <= 15.0_dp) THEN
 
!         *** 12 < t <= 15 -> Four term polynom expansion ***
 
               g = 0.4999489092_dp - 0.2473631686_dp/t(i) + &
                   0.321180909_dp/t(i)**2 - 0.3811559346_dp/t(i)**3
               f(i, 0) = 0.5_dp*sqrt(pi/t(i)) - g*exp(-t(i))/t(i)
 
            ELSE IF (t(i) <= 18.0_dp) THEN
 
!         *** 15 < t <= 18 -> Three term polynom expansion ***
 
               g = 0.4998436875_dp - 0.24249438_dp/t(i) + 0.24642845_dp/t(i)**2
               f(i, 0) = 0.5_dp*sqrt(pi/t(i)) - g*exp(-t(i))/t(i)
 
            ELSE IF (t(i) <= 24.0_dp) THEN
 
!         *** 18 < t <= 24 -> Two term polynom expansion ***
 
               g = 0.499093162_dp - 0.2152832_dp/t(i)
               f(i, 0) = 0.5_dp*sqrt(pi/t(i)) - g*exp(-t(i))/t(i)
 
            ELSE IF (t(i) <= 30.0_dp) THEN
 
!         *** 24 < t <= 30 -> One term polynom expansion ***
 
               g = 0.49_dp
               f(i, 0) = 0.5_dp*sqrt(pi/t(i)) - g*exp(-t(i))/t(i)
 
            ELSE
 
!         *** t > 30 -> Asymptotic formula ***
 
               f(i, 0) = 0.5_dp*sqrt(pi/t(i))
 
            END IF
 
            IF (t(i) > real(2*nmax + 36, dp)) THEN
               expt = 0.0_dp
            ELSE
               expt = exp(-t(i))
            END IF
 
!       *** Use the upward recursion relation to ***
!       *** generate the remaining F_n(t) values ***
 
            DO n = 1, nmax
               f(i, n) = 0.5_dp*(real(2*n - 1, dp)*f(i, n - 1) - expt)/t(i)
            END DO
 
         END IF
 
      END DO
 
   END SUBROUTINE fgamma_1
 
! **************************************************************************************************
!> \brief   Calculation of the incomplete Gamma function F_n(t) using a
!>          spherical Bessel function expansion. fgamma_ref returns a
!>          vector with all F_n(t) values for 0 <= n <= nmax.
!>          For t values greater than 50 an asymptotic formula is used.
!>          This function is expected to return accurate F_n(t) values
!>          for any combination of n and t, but the calculation is slow
!>          and therefore the function may only be used for a pretabulation
!>          of F_n(t) values or for reference calculations.
!> \param nmax ...
!> \param t ...
!> \return ...
!> \date    07.01.1999
!> \par Literature
!>        F. E. Harris, Int. J. Quant. Chem. 23, 1469 (1983)
!> \par Parameters
!>       - expt   : Exponential term in the downward recursion of F_n(t).
!>       - factor : Prefactor of the Bessel function expansion.
!>       - nmax   : Maximum n value of F_n(t).
!>       - p      : Product of the Bessel function quotients.
!>       - r      : Quotients of the Bessel functions.
!>       - sumterm: One term in the sum over products of Bessel functions.
!>       - t      : Argument of the incomplete Gamma function.
!> \author  MK
!> \version 1.0
! **************************************************************************************************
   FUNCTION fgamma_ref(nmax, t) RESULT(f)
 
      INTEGER, INTENT(IN)                                :: nmax
      REAL(kind=dp), INTENT(IN)                          :: t
      REAL(kind=dp), DIMENSION(0:nmax)                   :: f
 
      INTEGER, PARAMETER                                 :: kmax = 50
      REAL(kind=dp), PARAMETER                           :: eps = epsilon(0.0_dp)
 
      INTEGER                                            :: j, k, n
      REAL(kind=dp)                                      :: expt, factor, p, sumterm, sumtot, term
      REAL(kind=dp), DIMENSION(kmax+10)                  :: r
 
!   ------------------------------------------------------------------
!   *** Initialization ***
 
      f(:) = 0.0_dp
 
      IF (t < teps) THEN
 
!     *** Special case: t = 0 => analytic expression ***
 
         DO n = 0, nmax
            f(n) = 1.0_dp/real(2*n + 1, dp)
         END DO
 
      ELSE IF (t <= 50.0_dp) THEN
 
!     *** Initialize ratios of Bessel functions ***
 
         r(kmax + 10) = 0.0_dp
 
         DO j = kmax + 9, 1, -1
            r(j) = -t/(real(4*j + 2, dp) - t*r(j + 1))
         END DO
 
         factor = 2.0_dp*sinh(0.5_dp*t)*exp(-0.5_dp*t)/t
 
         DO n = 0, nmax
 
!       *** Initialize iteration ***
 
            sumtot = factor/real(2*n + 1, dp)
            term = 1.0_dp
 
!       *** Begin the summation and recursion ***
 
            DO k = 1, kmax
 
               term = term*real(2*n - 2*k + 1, dp)/real(2*n + 2*k + 1, dp)
 
!         *** Product of Bessel function quotients ***
 
               p = 1.0_dp
 
               DO j = 1, k
                  p = p*r(j)
               END DO
 
               sumterm = factor*term*p*real(2*k + 1, dp)/real(2*n + 1, dp)
 
               IF (abs(sumterm) < eps) THEN
 
!           *** Iteration converged ***
 
                  EXIT
 
               ELSE IF (k == kmax) THEN
 
!           *** No convergence with kmax iterations ***
 
                  cpabort("Maximum number of iterations reached")
 
               ELSE
 
!           *** Add the current term to the sum and continue the iteration ***
 
                  sumtot = sumtot + sumterm
 
               END IF
 
            END DO
 
            f(n) = sumtot
 
         END DO
 
      ELSE
 
!     *** Use asymptotic formula for t > 50 ***
 
         f(0) = 0.5_dp*sqrt(pi/t)
 
!     *** Use the upward recursion relation to ***
!     *** generate the remaining F_n(t) values ***
 
         expt = exp(-t)
 
         DO n = 1, nmax
            f(n) = 0.5_dp*(real(2*n - 1, dp)*f(n - 1) - expt)/t
         END DO
 
      END IF
 
   END FUNCTION fgamma_ref
 
! **************************************************************************************************
!> \brief   Initialize a table of F_n(t) values in the range 0 <= t <= 12 with
!>            a stepsize of 0.1 up to n equal to nmax for the Taylor series
!>            expansion used by McMurchie-Davidson (MD).
!> \param nmax ...
!> \date    10.06.1999
!> \par Parameters
!>       - nmax   : Maximum n value of F_n(t).
!> \author  MK
!> \version 1.0
! **************************************************************************************************
   SUBROUTINE init_md_ftable(nmax)
      INTEGER, INTENT(IN)                                :: nmax
 
      IF (nmax < 0) THEN
         CALL cp_abort(__location__, &
                       "A negative n value for the initialization of the "// &
                       "incomplete Gamma function is invalid")
      END IF
 
!   *** Check, if the current initialization is sufficient ***
 
      IF (nmax > current_nmax) THEN
 
         CALL deallocate_md_ftable()
 
!     *** Pretabulation of the F_n(t) function ***
!     *** for the Taylor series expansion      ***
 
         CALL create_md_ftable(nmax, 0.0_dp, 12.0_dp, 0.1_dp)
 
      END IF
 
   END SUBROUTINE init_md_ftable
 
END MODULE gamma