405 lines
14 KiB
Fortran
405 lines
14 KiB
Fortran
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!=======================================================================
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!
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MODULE GAMMA_ASYMPT
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!
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USE ACCURACY_REAL
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!
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! It contains the following functions/subroutines:
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!
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! * FUNCTION GAMMA_0_3D(RS,T)
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!
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! * FUNCTION GAMMA_I_3D(RS,T)
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!
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! * FUNCTION GAMMA_0_2D(RS,T)
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!
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! * FUNCTION GAMMA_I_2D(RS,T)
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!
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! * FUNCTION G0_INF_2D(X,RS,T)
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!
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!
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CONTAINS
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!=======================================================================
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!
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FUNCTION GAMMA_0_3D(RS,T)
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!
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! This function computes the coefficient gamma0 so that:
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!
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! lim (q --> 0) G(q) = gamma_0 (q / k_F)^2
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!
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!
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! References: (1) S. Ichimaru, Rev. Mod. Phys. 54, 1017-1059 (1982)
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! (3) K. Utsumi and S. Ichimaru, Phys. Rev. B 22, 1522-1533 (1980)
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!
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! Input parameters:
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!
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : temperature (SI)
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!
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! Author : D. Sébilleau
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!
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! Last modified : 2 Dec 2020
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!
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!
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USE LF_VALUES, ONLY : GQ_TYPE,G0_TYPE
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USE SF_VALUES, ONLY : SQ_TYPE
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USE ENERGIES, ONLY : EC_TYPE
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!
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USE REAL_NUMBERS, ONLY : TWO,FOUR,HALF,FOURTH
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USE PI_ETC, ONLY : PI
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USE FERMI_SI, ONLY : KF_SI
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USE UTILITIES_1, ONLY : ALFA
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USE CORRELATION_ENERGIES, ONLY : DERIVE_EC_3D
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USE SCREENING_VEC, ONLY : THOMAS_FERMI_VECTOR
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USE SPECIFIC_INT_2
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!
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IMPLICIT NONE
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!
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INTEGER :: IN_MODE,NMAX
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!
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REAL (WP), INTENT(IN) :: RS,T
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REAL (WP) :: GAMMA_0_3D
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!
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REAL (WP) :: ALPHA
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REAL (WP) :: RS2,RS3
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REAL (WP) :: D_EC_1,D_EC_2
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REAL (WP) :: KS,X_TF
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REAL (WP) :: X_MAX,IN
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!
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IF(G0_TYPE == 'EC') THEN !
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!
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ALPHA = ALFA('3D') !
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RS2 = RS * RS !
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RS3 = RS2 * RS !
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!
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! Computing the correlation energy derivatives
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!
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CALL DERIVE_EC_3D(EC_TYPE,1,5,RS,T,D_EC_1,D_EC_2) !
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!
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GAMMA_0_3D = FOURTH - (PI * ALPHA / 24.0E0_WP) * & ! ref. (1) eq. (3.30a)
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(RS3 * D_EC_2 - TWO * RS2 * D_EC_1) !
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!
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ELSE IF(G0_TYPE == 'SQ') THEN !
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!
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IN_MODE = 2 !
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NMAX = 1000 ! number of integration points
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X_MAX = 50.0E0_WP !
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!
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! Computing Thomas-Fermi screening vector
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!
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CALL THOMAS_FERMI_VECTOR('3D',KS) !
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X_TF = KS / KF_SI ! q_{TF} / k_F
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!
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! Computing the integral
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!
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CALL INT_SQM1(NMAX,X_MAX,IN_MODE,RS,T,X_TF,0,SQ_TYPE, & !
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GQ_TYPE,IN)
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!
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GAMMA_0_3D = - HALF * IN ! ref. (3) eq. (5.5)
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!
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END IF !
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!
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END FUNCTION GAMMA_0_3D
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!
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!=======================================================================
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!
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FUNCTION GAMMA_I_3D(RS,T)
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!
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! This function computes the coefficient gamma_i so that:
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!
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! lim (q --> 0) I(q) = gamma_i (q / k_F)^2 --> 3D
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!
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!
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! References: (1) S. Ichimaru, Rev. Mod. Phys. 54, 1017-1059 (1982)
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! (2) N. Iwamoto, Phys. Rev. A 30, 3289-3304 (1984)
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! (3) K. Utsumi and S. Ichimaru, Phys. Rev. B 22, 1522-1533 (1980)
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!
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! Input parameters:
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!
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : temperature (SI)
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!
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! Author : D. Sébilleau
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!
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! Last modified : 2 Dec 2020
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!
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!
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USE LF_VALUES, ONLY : GQ_TYPE,GI_TYPE
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USE SF_VALUES, ONLY : SQ_TYPE
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USE ENERGIES, ONLY : EC_TYPE
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!
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USE REAL_NUMBERS, ONLY : TWO,FOUR,TEN,FIFTH
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USE PI_ETC, ONLY : PI
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USE FERMI_SI, ONLY : KF_SI
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USE UTILITIES_1, ONLY : ALFA
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USE CORRELATION_ENERGIES, ONLY : DERIVE_EC_3D,EC_3D
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USE SCREENING_VEC, ONLY : THOMAS_FERMI_VECTOR
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USE SPECIFIC_INT_2
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!
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IMPLICIT NONE
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!
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INTEGER :: IN_MODE,NMAX
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!
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REAL (WP), INTENT(IN) :: RS,T
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REAL (WP) :: GAMMA_I_3D
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!
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REAL (WP) :: ALPHA
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REAL (WP) :: RS2
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REAL (WP) :: EC,D_EC_1,D_EC_2
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REAL (WP) :: KS,X_TF
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REAL (WP) :: X_MAX,IN
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!
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IF(GI_TYPE == 'EC') THEN !
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!
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ALPHA = ALFA('3D') !
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RS2 = RS * RS !
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!
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! Computing the correlation energy derivatives
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!
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EC = EC_3D(EC_TYPE,1,RS,T) !
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CALL DERIVE_EC_3D(EC_TYPE,1,5,RS,T,D_EC_1,D_EC_2) !
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!
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GAMMA_I_3D = 0.15E0_WP - (PI * ALPHA / TEN) * & ! ref. (1) eq. (3.30b)
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(RS2 * D_EC_1 + TWO* RS * EC) !
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!
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ELSE IF(GI_TYPE == 'SQ') THEN !
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!
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IN_MODE = 1 !
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NMAX = 1000 ! number of integration points
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X_MAX = 50.0E0_WP !
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!
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! Computing Thomas-Fermi screening vector
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!
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CALL THOMAS_FERMI_VECTOR('3D',KS) !
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X_TF = KS / KF_SI ! q_{TF} / k_F
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!
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! Computing the integral
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!
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CALL INT_SQM1(NMAX,X_MAX,IN_MODE,RS,T,X_TF,0,SQ_TYPE, & !
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GQ_TYPE,IN)
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!
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GAMMA_I_3D = - FIFTH * IN ! ref. (3) eq. (5.6)
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!
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END IF !
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!
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END FUNCTION GAMMA_I_3D
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!
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!=======================================================================
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!
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FUNCTION GAMMA_0_2D(RS,T)
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!
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! This function computes the coefficient gamma0 so that:
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!
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! lim (q --> 0) G(q) = gamma_i (q / k_F) --> 2D
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!
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!
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! References: (2) N. Iwamoto, Phys. Rev. A 30, 3289-3304 (1984)
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! (3) K. Utsumi and S. Ichimaru, Phys. Rev. B 22, 1522-1533 (1980)
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!
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! Input parameters:
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!
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : temperature (SI)
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!
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! Author : D. Sébilleau
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!
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! Last modified : 3 Dec 2020
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!
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!
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USE LF_VALUES, ONLY : GQ_TYPE,G0_TYPE
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USE SF_VALUES, ONLY : SQ_TYPE
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USE ENERGIES, ONLY : EC_TYPE
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!
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USE REAL_NUMBERS, ONLY : ONE,HALF,EIGHTH
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USE PI_ETC, ONLY : PI_INV
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USE FERMI_SI, ONLY : KF_SI
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USE UTILITIES_1, ONLY : ALFA
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USE CORRELATION_ENERGIES, ONLY : DERIVE_EC_2D
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USE ENERGIES, ONLY : EC_TYPE
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USE SCREENING_VEC, ONLY : THOMAS_FERMI_VECTOR
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USE SPECIFIC_INT_2
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!
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IMPLICIT NONE
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!
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INTEGER :: IN_MODE,NMAX
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!
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REAL (WP), INTENT(IN) :: RS,T
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REAL (WP) :: GAMMA_0_2D
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!
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REAL (WP) :: ALPHA,RS2,RS3
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REAL (WP) :: D_EC_1,D_EC_2
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REAL (WP) :: KS,X_TF
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REAL (WP) :: X_MAX,IN
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!
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IF(G0_TYPE == 'EC') THEN !
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!
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ALPHA = ALFA('2D') !
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RS2 = RS * RS !
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RS3 = RS2 * RS !
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!
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! Computing the correlation energy derivatives
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!
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CALL DERIVE_EC_2D(EC_TYPE,1,RS,T,D_EC_1,D_EC_2) !
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!
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GAMMA_0_2D = PI_INV + EIGHTH * ALPHA * ( & !
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RS2 * D_EC_1 - RS3 * D_EC_2 & ! ref. (1) eq. (3.6c)
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) !
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!
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ELSE IF(G0_TYPE == 'SQ') THEN !
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!
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IN_MODE = 2 !
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NMAX = 1000 ! number of integration points
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X_MAX = 50.0E0_WP !
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!
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! Computing Thomas-Fermi screening vector
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!
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CALL THOMAS_FERMI_VECTOR('2D',KS) !
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X_TF = KS / KF_SI ! q_{TF} / k_F
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!
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! Computing the integral
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!
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CALL INT_SQM1(NMAX,X_MAX,IN_MODE,RS,T,X_TF,0,SQ_TYPE, & !
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GQ_TYPE,IN)
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!
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GAMMA_0_2D = - HALF * IN ! ref. (3) eq. (5.5)
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!
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END IF !
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!
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END FUNCTION GAMMA_0_2D
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!
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!=======================================================================
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!
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FUNCTION GAMMA_I_2D(RS,T)
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!
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! This function computes the coefficient gamma0 so that:
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!{\bf 33}
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! lim (q --> 0) I(q) = gamma_i (q / k_F) --> 2D
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!
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!
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! References: (2) N. Iwamoto, Phys. Rev. A 30, 3289-3304 (1984)
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! (3) K. Utsumi and S. Ichimaru, Phys. Rev. B 22, 1522-1533 (1980)
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!
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! Input parameters:
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!
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : temperature (SI)
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!
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! Author : D. Sébilleau
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!
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! Last modified : 3 Dec 2020
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!
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!
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USE LF_VALUES, ONLY : GQ_TYPE,GI_TYPE
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USE SF_VALUES, ONLY : SQ_TYPE
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USE ENERGIES, ONLY : EC_TYPE
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!
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USE REAL_NUMBERS, ONLY : TWO,FIVE,SIX,FIFTH
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USE PI_ETC, ONLY : PI_INV
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USE FERMI_SI, ONLY : KF_SI
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USE UTILITIES_1, ONLY : ALFA
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USE CORRELATION_ENERGIES, ONLY : DERIVE_EC_2D,EC_2D
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USE ENERGIES, ONLY : EC_TYPE
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USE SCREENING_VEC, ONLY : THOMAS_FERMI_VECTOR
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USE SPECIFIC_INT_2
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!
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IMPLICIT NONE
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!
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INTEGER :: IN_MODE,NMAX
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!
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REAL (WP), INTENT(IN) :: RS,T
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REAL (WP) :: GAMMA_I_2D
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!
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REAL (WP) :: ALPHA,RS2
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REAL (WP) :: EC,D_EC_1,D_EC_2
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REAL (WP) :: KS,X_TF
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REAL (WP) :: X_MAX,IN
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!
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IF(GI_TYPE == 'EC') THEN !
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!
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ALPHA = ALFA('3D') !
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RS2 = RS * RS !
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!
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! Computing the correlation energy derivatives
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!
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EC = EC_2D(EC_TYPE,RS,T) !
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CALL DERIVE_EC_2D(EC_TYPE,1,RS,T,D_EC_1,D_EC_2) !
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!
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GAMMA_I_2D = FIVE * PI_INV / SIX - & !
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(FIVE * ALPHA / 16.0E0_WP) * & !
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(RS2 * D_EC_1 + TWO * RS * EC) ! ref. (2) eq. (D9c)
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!
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ELSE IF(GI_TYPE == 'SQ') THEN !
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!
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IN_MODE = 1 !
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NMAX = 1000 ! number of integration points
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X_MAX = 50.0E0_WP !
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!
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! Computing Thomas-Fermi screening vector
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!
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CALL THOMAS_FERMI_VECTOR('3D',KS) !
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X_TF = KS / KF_SI ! q_{TF} / k_F
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!
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! Computing the integral
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!
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CALL INT_SQM1(NMAX,X_MAX,IN_MODE,RS,T,X_TF,0,SQ_TYPE, & !
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GQ_TYPE,IN)
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!
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GAMMA_I_2D = - FIFTH * IN ! ref. (3) eq. (5.6)
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!
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END IF !
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!
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END FUNCTION GAMMA_I_2D
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!
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!=======================================================================
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!
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FUNCTION G0_INF_2D(X,RS,T)
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!
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! This function computes G(0,infinity), the value of the dynamic
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! local-field correction for q --> 0 and omega = infinity, for 2D systems
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!
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! References: (1) B. Tanatar, Phys. Lett. A 158, 153-157 (1991)
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!
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! Input parameters:
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!
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! * X : dimensionless factor --> X = q / (2 * k_F)
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : temperature (SI)
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!
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! Author : D. Sébilleau
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!
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! Last modified : 2 Dec 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ONE,SEVEN,EIGHT,HALF,THIRD
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USE SQUARE_ROOTS, ONLY : SQR2
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USE PI_ETC, ONLY : PI_INV
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USE CORRELATION_ENERGIES, ONLY : DERIVE_EC_2D,EC_2D
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USE ENERGIES, ONLY : EC_TYPE
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: X,RS,T
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REAL (WP) :: G0_INF_2D
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!
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REAL (WP) :: Y
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REAL (WP) :: A
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REAL (WP) :: EC,D_EC_1,D_EC_2
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!
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Y = X + X ! Y = q / k_F
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!
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A = ONE / SQR2 !
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!
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! Computing the correlation energy derivatives
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!
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EC = EC_2D(EC_TYPE,RS,T) !
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CALL DERIVE_EC_2D(EC_TYPE,1,RS,T,D_EC_1,D_EC_2) !
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!
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G0_INF_2D = Y *(0.20E0_WP * HALF * THIRD * PI_INV + & !
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SEVEN * A * RS * EC / EIGHT + & ! ref. (1) eq. (7)
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19.0E0_WP * A * RS * D_EC_1 / 16.0E0_WP) !
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!
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END FUNCTION G0_INF_2D
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!
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END MODULE GAMMA_ASYMPT
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