241 lines
6.9 KiB
Fortran
241 lines
6.9 KiB
Fortran
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!---------------------------------------------------------------------
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!
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MODULE BASIC_FUNCTIONS
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!
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! This module contains basic functions of general use:
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!
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! * Lorentzian
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! * Gaussian
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! * Numerical Dirac Delta
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! * Slater-type and Gaussian-type orbitals
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!
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! Modules used: ACCURACY_REAL
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! PI_ETC
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! SQUARE_ROOTS
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! REAL_NUMBERS
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! FACTORIALS
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!
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USE ACCURACY_REAL
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USE PI_ETC
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USE SQUARE_ROOTS
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USE REAL_NUMBERS
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USE FACTORIALS
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!
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CONTAINS
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!
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!---------------------------------------------------------------------
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!
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FUNCTION LORENTZIAN(X,X0,GAMMA)
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!
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! This function computes the Lorentzian function L(x) normalized to unity:
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!
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! L(x) = 1/pi * gamma / ([x - x0]^2 + gamma^2)
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!
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! Input parameters:
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!
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! X : value at which the function is computed
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! X0 : center paramameter
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! GAMMA : half width at half maximum
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!
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! Output parameter:
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!
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! LORENTZIAN : value of the function at X
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Aug 2020
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!
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: X,X0,GAMMA
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REAL (WP) :: LORENTZIAN
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!
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LORENTZIAN = PI_INV * GAMMA / ( (X-X0)*(X-X0) + GAMMA*GAMMA )!
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!
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END FUNCTION LORENTZIAN
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!
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!---------------------------------------------------------------------
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!
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FUNCTION GAUSSIAN(X,X0,SIGMA)
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!
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! This function computes the Gaussian function G(x) normalized to unity:
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!
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! G(x) = (1/sigma sqrt(2 pi)) * exp[ -1/2 * (x-x0)^2 / sigma^2 ]
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!
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!
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! Input parameters:
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!
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! X : value at which the function is computed
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! X0 : expected value
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! SIGMA : variance
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!
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! Output parameter:
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!
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! GAUSSIAN : value of the function at X
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!
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! Note: the half width at half maximum is given by sigma * sqrt[2 Ln(2)]
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Aug 2020
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: X,X0,SIGMA
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REAL (WP) :: GAUSSIAN
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!
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REAL (WP) :: EXP
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!
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GAUSSIAN = PI_INV * EXP(-HALF * (X - X0) * (X - X0) / & !
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(SIGMA * SIGMA) ) / (SIGMA * SQR2) !
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!
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END FUNCTION GAUSSIAN
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!
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!---------------------------------------------------------------------
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!
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FUNCTION DELTA(X,I_D,EPSI)
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!
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! This function computes the numerical Dirac Delta function
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!
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!
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! Input parameters:
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!
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! X : value at which the function is computed
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! I_D : type of numerical approximation used
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!
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! = 1 : lim [ eps -> 0 ] 1/pi eps/(x^2 + eps^2)
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! = 2 : lim [ eps -> 0 ] 1/(pi*x) sin(x/eps)
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! = 3 : lim [ eps -> 0 ] 1/2 eps |x|^{eps-1}
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! = 4 : lim [ eps -> 0 ] 1/(2*sqrt(pi eps}) e^{- x^2/(4 eps)}
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! = 5 : lim [ eps -> 0 ] 1/(2 eps) e^{- |x|/eps)
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! = 6 : lim [ eps -> 0 ] 1/eps Ai(x/eps)
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! = 7 : lim [ eps -> 0 ] 1/eps J_{1/eps}([x+1]/eps)
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! = 8 : lim [ eps -> 0 ] | 1/eps e^{- x^2/eps} Ln(2x/eps)
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! EPSI : small eps value
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!
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! Output parameter:
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!
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! DELTA : value of the function at X
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Aug 2020
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: X
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REAL (WP) :: DELTA
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REAL (WP) :: EPSI
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!
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REAL (WP) :: SIN,ABS,EXP,SQRT
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!
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INTEGER, INTENT(IN) :: I_D
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!
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IF(I_D == 1) THEN !
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DELTA = PI_INV * EPSI / ( X * X + EPSI * EPSI) !
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ELSE IF(I_D == 2) THEN !
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DELTA = PI_INV * SIN( X / EPSI) / X !
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ELSE IF(I_D == 3) THEN !
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DELTA = HALF * EPSI * ABS(X)**(EPSI-1) !
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ELSE IF(I_D == 4) THEN !
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DELTA = HALF * EXP( - FOURTH * X * X / EPSI) / & !
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(SQR_PI * SQRT(EPSI)) !
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ELSE IF(I_D == 5) THEN !
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DELTA = HALF * EXP(- ABS(X) / EPSI) /EPSI !
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END IF
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!
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END FUNCTION DELTA
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!
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!---------------------------------------------------------------------
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!
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FUNCTION STO(N,ALPHA,R)
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!
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! This function computes Slater-type orbitals normalized to unity::
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!
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! STO = A * r^{n-1} * e^{- zeta r}
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!
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!
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! Input parameters:
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!
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! N : principal quantum number
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! ALPHA : function parameter
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! R : value at which the function is computed
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!
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! Output parameter:
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!
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! STO : value of the function at R
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Aug 2020
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: ALPHA,R
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REAL (WP) :: STO
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REAL (WP) :: A,TWOZ
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!
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REAL (WP) :: SQRT,EXP
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!
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INTEGER, INTENT(IN) :: N
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!
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TWOZ = TWO * ALPHA !
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A = TWOZ**N * SQRT(TWOZ / FAC(2*N)) !
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STO = A * R**(N-1) * EXP(-ALPHA * R) ! normalisation constant
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!
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END FUNCTION STO
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!
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!---------------------------------------------------------------------
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!
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FUNCTION GTO(N,ALPHA,R)
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!
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! This function computes Gaussian-type orbitals normalized to unity::
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!
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! GTO = A * r^{n-1} * e^{- zeta r^2}
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!
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!
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! Input parameters:
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!
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! N : principal quantum number
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! ALPHA : function parameter
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! R : value at which the function is computed
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!
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! Output parameter:
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!
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! GTO : value of the function at R
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Aug 2020
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: ALPHA,R
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REAL (WP) :: GTO
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REAL (WP) :: A,TWOZ,NN
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REAL (WP) :: GAMMA
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!
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REAL (WP) :: SQRT,EXP
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!
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INTEGER, INTENT(IN) :: N
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!
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TWOZ = TWO * ALPHA !
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NN = N + HALF !
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GAMMA = SQR_PI * FAC(2*N) / (FOUR**N * FAC(N)) !
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A = SQRT(TWO * TWOZ**NN / GAMMA) ! normalisation constant
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GTO = A * R**(N-1) * EXP(-ALPHA * R * R) !
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!
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END FUNCTION GTO
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!
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END MODULE BASIC_FUNCTIONS
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