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Copy pathCoulomb_Interaction.c
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Coulomb_Interaction.c
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/**********************************************************************
Coulomb_Interaction.c:
Coulomb_Interaction.c is a subroutine to calculate
Coulomb interaction tensor for general LDA+U scheme.
by Siheon Ryee.
***********************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
#include "openmx_common.h"
#include "Inputtools.h"
#include "mpi.h"
/* Coulomb interaction tensor */
static double FAC(int num);
static int MIN(int a, int b);
static int MAX(int a, int b);
static double Wigner3j(int l1, int l2, int l3, int m1, int m2, int m3);
static double Gaunt_SR(int l, int k, int m1, int m2);
static dcomplex StoR(int l, int m1, int m2);
static double Coulomb_Matrix(int num, double F0, double JH, int l, int a, int b, int c, int d);
/*****************************/
/* Yukawa potential */
static double Bessel_j(int l2max, int k_in, double x);
static double Bessel_h(int k_in, double x);
static double Integrate_Bessel(int mode, int N, double start, double end, int isp, int l, int mul, int k, double Ndi, double lambda);
static double Integrate_Bessel0(int mode, int N, double start, double end, int isp, int l, int mul, int k, double Ndi);
/********************/
void Coulomb_Interaction(){
int Npoints,Nmesh,N_iter,i,j,l,mul,mm;
double Ndi,Nend,lambda_lower,lambda_upper,lambda_test;
int dum_Nmul,dum_l,dum_mul;
double Uval,Jval,delta_U,J_temp;
int NZUJ;
int a,b,c,d;
int numprocs,myid;
MPI_Comm_size(mpi_comm_level1,&numprocs);
MPI_Comm_rank(mpi_comm_level1,&myid);
/* Calculate Yukawa potential */
if(Yukawa_on==1){
if (myid==Host_ID && 0<level_stdout){
printf("\n");
printf("*******************************************************\n"); fflush(stdout);
printf(" Calculating Thomas-Fermi screening length \n" ); fflush(stdout);
printf("*******************************************************\n\n"); fflush(stdout);
}
/* Allocating arrays */
Npoints = 1600; /* Arbitrarily large value */
Bessel_j_Array0 = (double**)malloc(sizeof(double*)*(Nmul+1));
for(i=0; i<=Nmul; i++){
Bessel_j_Array0[i] = (double*)malloc(sizeof(double)*Npoints);
for(mm=0; mm<Npoints; mm++){
Bessel_j_Array0[i][mm] = 0.0;
}
}
Bessel_h_Array0 = (double**)malloc(sizeof(double*)*(Nmul+1));
for(i=0; i<=Nmul; i++){
Bessel_h_Array0[i] = (double*)malloc(sizeof(double)*Npoints);
for(mm=0; mm<Npoints; mm++){
Bessel_h_Array0[i][mm] = 0.0;
}
}
dum_Nmul=1;
for(i=0; i<SpeciesNum; i++){
for(dum_l=0; dum_l<=Spe_MaxL_Basis[i]; dum_l++){
for(dum_mul=0; dum_mul<Spe_Num_Basis[i][dum_l]; dum_mul++){
Uval = Hub_U_Basis[i][dum_l][dum_mul];
Jval = Hund_J_Basis[i][dum_l][dum_mul];
if (Uval > 1.0e-5 || Jval > 1.0e-5){
U[dum_Nmul] = Uval;
B_spe[dum_Nmul] = i;
B_l[dum_Nmul] = dum_l;
B_mul[dum_Nmul] = dum_mul;
B_cut[dum_Nmul] = Spe_Atom_Cut1[i];
dum_Nmul++;
}
}
}
}
/* End allocating arrays */
for(i=1; i<=Nmul; i++){
/* parameters for calculating Slater integrals */
/* unit of lambda: 1/a.u. */
/* All quantities are in atomic unit */
lambda_lower = 0.1;
if (U[i] >= 1.5) lambda_upper = 10.0;
else if (U[i] >= 0.5) lambda_upper = 20.0;
else lambda_upper = 40.0;
delta_U = 0.0005/eV2Hartree; /* criterion for convergence */
Ndi=0.01;
Nend=B_cut[i]+1.0;
Nmesh=(int)(Nend/Ndi);
N_iter=200; /* maximum # of iteration */
/***********************************************/
/* Calculating unscreened Slater integrals */
/* for(mm=0; mm<Nmesh; mm++){
Bessel_j_Array0[i][mm] = Integrate_Bessel0(0,i,0.0,(double)(mm)*Ndi,B_spe[i],B_l[i],B_mul[i],0,Ndi);
Bessel_h_Array0[i][mm] = Integrate_Bessel0(1,i,(double)(mm)*Ndi,Nend,B_spe[i],B_l[i],B_mul[i],0,Ndi);
}
Slater_F0[i] = Integrate_Bessel0(2,i,0.0,(Nend-1.0),B_spe[i],B_l[i],B_mul[i],0,Ndi);
for(mm=0; mm<Nmesh; mm++){
Bessel_j_Array0[i][mm] = Integrate_Bessel0(0,i,0.0,(double)(mm)*Ndi,B_spe[i],B_l[i],B_mul[i],2,Ndi);
Bessel_h_Array0[i][mm] = Integrate_Bessel0(1,i,(double)(mm)*Ndi,Nend,B_spe[i],B_l[i],B_mul[i],2,Ndi);
}
Slater_F2[i] = Integrate_Bessel0(2,i,0.0,(Nend-1.0),B_spe[i],B_l[i],B_mul[i],2,Ndi);
for(mm=0; mm<Nmesh; mm++){
Bessel_j_Array0[i][mm] = Integrate_Bessel0(0,i,0.0,(double)(mm)*Ndi,B_spe[i],B_l[i],B_mul[i],4,Ndi);
Bessel_h_Array0[i][mm] = Integrate_Bessel0(1,i,(double)(mm)*Ndi,Nend,B_spe[i],B_l[i],B_mul[i],4,Ndi);
}
Slater_F4[i] = Integrate_Bessel0(2,i,0.0,(Nend-1.0),B_spe[i],B_l[i],B_mul[i],4,Ndi);
if(B_l[i]==3){
for(mm=0; mm<Nmesh; mm++){
Bessel_j_Array0[i][mm] = Integrate_Bessel0(0,i,0.0,(double)(mm)*Ndi,B_spe[i],B_l[i],B_mul[i],6,Ndi);
Bessel_h_Array0[i][mm] = Integrate_Bessel0(1,i,(double)(mm)*Ndi,Nend,B_spe[i],B_l[i],B_mul[i],6,Ndi);
}
Slater_F6[i] = Integrate_Bessel0(2,i,0.0,(Nend-1.0),B_spe[i],B_l[i],B_mul[i],6,Ndi);
}
printf("F0= %f, F2= %f, F4= %f\n", Slater_F0[i], Slater_F2[i], Slater_F4[i]); */
/* Finding Thomas-Fermi screening length of Hubbard U(=F0) */
for(j=1; j<=N_iter; j++){
lambda_test = 0.5*(lambda_lower+lambda_upper);
for(mm=0; mm<Nmesh; mm++){
Bessel_j_Array0[i][mm] = Integrate_Bessel(0,i,0.0,(double)(mm)*Ndi,B_spe[i],B_l[i],B_mul[i],0,Ndi,lambda_test);
Bessel_h_Array0[i][mm] = Integrate_Bessel(1,i,(double)(mm)*Ndi,Nend,B_spe[i],B_l[i],B_mul[i],0,Ndi,lambda_test);
}
Slater_F0[i] = Integrate_Bessel(2,i,0.0,(Nend-1.0),B_spe[i],B_l[i],B_mul[i],0,Ndi,lambda_test);
if(Slater_F0[i] <= U[i]+delta_U && Slater_F0[i] >= U[i]-delta_U){
lambda[i]=lambda_test;
break;
}
if((Slater_F0[i]-U[i])>0.0) lambda_lower = lambda_test;
else lambda_upper = lambda_test;
if(j==N_iter){
if(myid==Host_ID && 0<level_stdout){
printf("!!!!!!!!!!!!!!!!!!!!! WARNING !!!!!!!!!!!!!!!!!!!!!!! \n"); fflush(stdout);
printf(" Slater integrals are not correctly calculated.\n"); fflush(stdout);
printf("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! \n"); fflush(stdout);
printf("\n"); fflush(stdout);
}
}
} /* N_iter */
/* Calculating F2, F4, and F6 by using Yukawa potential */
for(mm=0; mm<Nmesh; mm++){
Bessel_j_Array0[i][mm] = Integrate_Bessel(0,i,0.0,(double)(mm)*Ndi,B_spe[i],B_l[i],B_mul[i],2,Ndi,lambda[i]);
Bessel_h_Array0[i][mm] = Integrate_Bessel(1,i,(double)(mm)*Ndi,Nend,B_spe[i],B_l[i],B_mul[i],2,Ndi,lambda[i]);
}
Slater_F2[i] = Integrate_Bessel(2,i,0.0,(Nend-1.0),B_spe[i],B_l[i],B_mul[i],2,Ndi,lambda[i]);
for(mm=0; mm<Nmesh; mm++){
Bessel_j_Array0[i][mm] = Integrate_Bessel(0,i,0.0,(double)(mm)*Ndi,B_spe[i],B_l[i],B_mul[i],4,Ndi,lambda[i]);
Bessel_h_Array0[i][mm] = Integrate_Bessel(1,i,(double)(mm)*Ndi,Nend,B_spe[i],B_l[i],B_mul[i],4,Ndi,lambda[i]);
}
Slater_F4[i] = Integrate_Bessel(2,i,0.0,(Nend-1.0),B_spe[i],B_l[i],B_mul[i],4,Ndi,lambda[i]);
if(B_l[i]==3){
for(mm=0; mm<Nmesh; mm++){
Bessel_j_Array0[i][mm] = Integrate_Bessel(0,i,0.0,(double)(mm)*Ndi,B_spe[i],B_l[i],B_mul[i],6,Ndi,lambda[i]);
Bessel_h_Array0[i][mm] = Integrate_Bessel(1,i,(double)(mm)*Ndi,Nend,B_spe[i],B_l[i],B_mul[i],6,Ndi,lambda[i]);
}
Slater_F6[i] = Integrate_Bessel(2,i,0.0,(Nend-1.0),B_spe[i],B_l[i],B_mul[i],6,Ndi,lambda[i]);
}
if(B_l[i]==1){
J_temp = (Slater_F2[i])/5.0;
}
if(B_l[i]==2){
J_temp = (Slater_F2[i] + Slater_F4[i])/14.0;
}
if(B_l[i]==3){
J_temp = (286.0*Slater_F2[i] + 195.0*Slater_F4[i] + 250.0*Slater_F6[i])/6435.0;
}
J[i] = J_temp;
if (myid==Host_ID && level_stdout>0){
printf("<species: %s, angular momentum= %d, multiplicity number= %d>\n", SpeName[B_spe[i]], B_l[i], B_mul[i]);
printf(" TF-screening-length lambda= %f 1/au\n", lambda[i]);
printf(" Hubbard U= %f eV\n", U[i]*eV2Hartree);
printf(" Hund J= %f eV\n", J[i]*eV2Hartree);
printf(" Slater F0= %f eV\n", Slater_F0[i]*eV2Hartree);
printf(" Slater F2= %f eV\n", Slater_F2[i]*eV2Hartree);
printf(" Slater F4= %f eV\n", Slater_F4[i]*eV2Hartree);
if(B_l[i]==3){
printf(" Slater F6= %f eV\n", Slater_F6[i]*eV2Hartree);
}
if(B_l[i]==2){
printf(" F4/F2= %f\n", Slater_F4[i]/Slater_F2[i]);
}
if(B_l[i]==3){
printf(" F4/F2= %f\n", Slater_F4[i]/Slater_F2[i]);
printf(" F6/F4= %f\n", Slater_F6[i]/Slater_F4[i]);
}
printf("\n");
}
} /* Nmul */
/* Freeing arrays */
for(i=0; i<=Nmul; i++){
free(Bessel_j_Array0[i]);
free(Bessel_h_Array0[i]);
}
} /* Yukawa_on==1 */
/**********************************************
Storing values to Coulomb interaction array
(only when Hub_Type=2 is used)
**********************************************/
if(Yukawa_on != 1){ /* U and J from input */
/* Storing U,J values to the array */
dum_Nmul=1;
for(i=0; i<SpeciesNum; i++){
for(dum_l=0; dum_l<=Spe_MaxL_Basis[i]; dum_l++){
for(dum_mul=0; dum_mul<Spe_Num_Basis[i][dum_l]; dum_mul++){
Uval = Hub_U_Basis[i][dum_l][dum_mul];
Jval = Hund_J_Basis[i][dum_l][dum_mul];
NZUJ = Nonzero_UJ[i][dum_l][dum_mul];
if((Uval > 1.0e-5 || Jval > 1.0e-5) && (NZUJ>0)){
for(a=0; a<(2*dum_l+1); a++){
for(b=0; b<(2*dum_l+1); b++){
for(c=0; c<(2*dum_l+1); c++){
for(d=0; d<(2*dum_l+1); d++){
Coulomb_Array[NZUJ][a][b][c][d]
= Coulomb_Matrix(dum_Nmul,Uval,Jval,dum_l,a,b,c,d);
}
}
}
}
dum_Nmul++;
}
}
}
}
/* for(a=0; a<(2*2+1); a++){
for(b=0; b<(2*2+1); b++){
printf("a=%d b=%d, U=%f\n", a,b,Coulomb_Array[0][a][b][a][b]*eV2Hartree);
}
}*/
}
/*************************************************************
Allocating Hubbard U & Hund J values from Slater integrals
*************************************************************/
if(Yukawa_on==1){ /* U from input and J from Yukawa */
/* initialize J values */
for (i=0; i<SpeciesNum; i++){
for (l=0; l<=Spe_MaxL_Basis[i]; l++){
for (mul=0; mul<Spe_Num_Basis[i][l]; mul++){
Hund_J_Basis[i][l][mul]=0.0;
}
}
}
/* Storing U,J values to the array */
dum_Nmul=1;
for(i=0; i<SpeciesNum; i++){
for(dum_l=0; dum_l<=Spe_MaxL_Basis[i]; dum_l++){
for(dum_mul=0; dum_mul<Spe_Num_Basis[i][dum_l]; dum_mul++){
Uval = Hub_U_Basis[i][dum_l][dum_mul];
NZUJ = Nonzero_UJ[i][dum_l][dum_mul];
if ((Uval > 1.0e-5) && (NZUJ>0)){
Hund_J_Basis[i][dum_l][dum_mul] = J[dum_Nmul];
Jval = J[dum_Nmul];
for(a=0; a<(2*dum_l+1); a++){
for(b=0; b<(2*dum_l+1); b++){
for(c=0; c<(2*dum_l+1); c++){
for(d=0; d<(2*dum_l+1); d++){
Coulomb_Array[NZUJ][a][b][c][d]
= Coulomb_Matrix(dum_Nmul,Uval,Jval,dum_l,a,b,c,d);
}
}
}
}
dum_Nmul++;
}
}
}
}
}
/****************************************************
Generating AMF array
(only when around-mean-field double-counting is used)
****************************************************/
if(Hub_U_switch==1 && Hub_Type==2 && (dc_Type==2 || dc_Type==4)){
AMF_Array = (double*****)malloc(sizeof(double****)*(Nmul+1));
for(i=0; i<=Nmul; i++){
AMF_Array[i] = (double****)malloc(sizeof(double***)*4);
for(a=0; a<4; a++){
AMF_Array[i][a] = (double***)malloc(sizeof(double**)*2);
for(b=0; b<2; b++){
AMF_Array[i][a][b] = (double**)malloc(sizeof(double*)*7);
for(c=0; c<7; c++){
AMF_Array[i][a][b][c] = (double*)malloc(sizeof(double)*7);
for(d=0; d<7; d++){
AMF_Array[i][a][b][c][d] = 0.0;
}
}
}
}
}
}
} /* Yukawa_potential */
/******************************************************
Integrate r^2*R*R*j form "start" to "end"
R: Radial function of atomic orbital
with angular moment "l" and multiplicity "mul"
j: Spherical Bessel function
******************************************************/
/******************************************************
Integrate r^2*R*R*h form "start" to "end"
R: Radial function of atomic orbital
with angular moment "l" and multiplicity "mul"
h: Spherical Hankel function of the first kind
******************************************************/
/*static double Integrate_Bessel0(int mode, int N, double start, double end, int isp, int l, int mul, int k, double Ndi){
double i,di,f_i1,f_i2,f_i3,sum_f;
double j0,j0_di,j0_2di;
double h0,h0_di,h0_2di;
int step;
di = 0.001;
sum_f = 0.0;
switch (mode){
case 0:
if(end>start){
for(i=start; i<=end; i+=2.0*di){
j0 = pow(i,(double)k);
j0_di = pow(i+di,(double)k);
j0_2di = pow(i+2.0*di,(double)k);
f_i1 = i*i*RadialF(isp, l, mul, i)*RadialF(isp, l, mul, i) * j0;
f_i2 = (i+di)*(i+di)*RadialF(isp, l, mul, i+di)*RadialF(isp, l, mul, i+di) * j0_di;
f_i3 = (i+2.0*di)*(i+2.0*di)*RadialF(isp, l, mul, i+2.0*di)*RadialF(isp, l, mul, i+2.0*di) * j0_2di;
if (i > end-2.0*di) break;
sum_f += (di/3.0)*(f_i1+4.0*f_i2+f_i3);
}
}
else{
sum_f = 0.0;
}
printf("Bessel = %f\n", sum_f);
break;
case 1:
if(end>start){
for(i=start; i<=end; i+=2.0*di){
h0 = 1.0/(i*pow(i,(double)k));
h0_di = 1.0/((i+di)*pow(i+di,(double)k));
h0_2di = 1.0/((i+2.0*di)*pow(i+2.0*di,(double)k));
f_i1 = i*i*RadialF(isp, l, mul, i)*RadialF(isp, l, mul, i) * h0;
f_i2 = (i+di)*(i+di)*RadialF(isp, l, mul, i+di)*RadialF(isp, l, mul, i+di) * h0_di;
f_i3 = (i+2.0*di)*(i+2.0*di)*RadialF(isp, l, mul, i+2.0*di)*RadialF(isp, l, mul, i+2.0*di) * h0_2di;
if (i > end-2.0*di) break;
sum_f += (di/3.0)*(f_i1+4.0*f_i2+f_i3);
}
}
else{
sum_f = 0.0;
}
printf("Hankel = %f\n", sum_f);
break;
step = 0;
case 2:
for(i=start; i<=end; i+=2.0*Ndi){
j0 = pow(i,(double)k);
j0_di = pow(i+Ndi,(double)k);
j0_2di = pow(i+2.0*Ndi,(double)k);
h0 = 1.0/(i*pow(i,(double)k));
h0_di = 1.0/((i+Ndi)*pow(i+Ndi,(double)k));
h0_2di = 1.0/((i+2.0*Ndi)*pow(i+2.0*Ndi,(double)k));
f_i1 = i*i*RadialF(isp,l,mul,i)*RadialF(isp,l,mul,i)
*(Bessel_j_Array0[N][step]*h0
+Bessel_h_Array0[N][step]*j0);
f_i2 = (i+Ndi)*(i+Ndi)*RadialF(isp,l,mul,i+Ndi)*RadialF(isp,l,mul,i+Ndi)
*(Bessel_j_Array0[N][step+1]*h0_di
+Bessel_h_Array0[N][step+1]*j0_di);
f_i3 = (i+2.0*Ndi)*(i+2.0*Ndi)*RadialF(isp,l,mul,i+2.0*Ndi)*RadialF(isp,l,mul,i+2.0*Ndi)
*(Bessel_j_Array0[N][step+2]*h0_2di
+Bessel_h_Array0[N][step+2]*j0_2di);
sum_f += (Ndi/3.0)*(f_i1+4.0*f_i2+f_i3);
step+=2;
}
printf("Total = %f\n", sum_f);
break;
}
return sum_f;
}
*/
static double Integrate_Bessel(int mode, int N, double start, double end, int isp, int l, int mul, int k, double Ndi, double lambda){
double i,di,f_i1,f_i2,f_i3,sum_f;
int step;
di = 0.001;
sum_f = 0.0;
switch (mode){
case 0: /* Spherical Bessel function */
if(end>start){
for(i=start; i<=end; i+=2.0*di){
/* Integration using Simpson's rule */
f_i1 = i*i*RadialF(isp, l, mul, i)*RadialF(isp, l, mul, i) * Bessel_j(2*l,k,lambda*i);
f_i2 = (i+di)*(i+di)*RadialF(isp, l, mul, i+di)*RadialF(isp, l, mul, i+di) * Bessel_j(2*l,k,lambda*(i+di));
f_i3 = (i+2.0*di)*(i+2.0*di)*RadialF(isp, l, mul, i+2.0*di)*RadialF(isp, l, mul, i+2.0*di) * Bessel_j(2*l,k,lambda*(i+2.0*di));
if (i > end-2.0*di) break;
sum_f += (di/3.0)*(f_i1+4.0*f_i2+f_i3);
}
}
else{
sum_f = 0.0;
}
break;
case 1: /* Spherical Hankel function */
if(end>start){
for(i=start; i<=end; i+=2.0*di){
/* Integration using Simpson's rule */
f_i1 = i*i*RadialF(isp, l, mul, i)*RadialF(isp, l, mul, i) * Bessel_h(k,lambda*i);
f_i2 = (i+di)*(i+di)*RadialF(isp, l, mul, i+di)*RadialF(isp, l, mul, i+di) * Bessel_h(k,lambda*(i+di));
f_i3 = (i+2.0*di)*(i+2.0*di)*RadialF(isp, l, mul, i+2.0*di)*RadialF(isp, l, mul, i+2.0*di) * Bessel_h(k,lambda*(i+2.0*di));
if (i > end-2.0*di) break;
sum_f += (di/3.0)*(f_i1+4.0*f_i2+f_i3);
}
}
else{
sum_f = 0.0;
}
break;
step = 0;
case 2: /* Calculating Slater integral */
for(i=start; i<=end; i+=2.0*Ndi){
/* Integration using Simpson's rule */
f_i1 = i*i*RadialF(isp,l,mul,i)*RadialF(isp,l,mul,i)
*(Bessel_j_Array0[N][step]*Bessel_h(k,lambda*i)
+Bessel_h_Array0[N][step]*Bessel_j(2*l,k,lambda*i));
f_i2 = (i+Ndi)*(i+Ndi)*RadialF(isp,l,mul,i+Ndi)*RadialF(isp,l,mul,i+Ndi)
*(Bessel_j_Array0[N][step+1]*Bessel_h(k,lambda*(i+Ndi))
+Bessel_h_Array0[N][step+1]*Bessel_j(2*l,k,lambda*(i+Ndi)));
f_i3 = (i+2.0*Ndi)*(i+2.0*Ndi)*RadialF(isp,l,mul,i+2.0*Ndi)*RadialF(isp,l,mul,i+2.0*Ndi)
*(Bessel_j_Array0[N][step+2]*Bessel_h(k,lambda*(i+2.0*Ndi))
+Bessel_h_Array0[N][step+2]*Bessel_j(2*l,k,lambda*(i+2.0*Ndi)));
sum_f += (Ndi/3.0)*(f_i1+4.0*f_i2+f_i3);
step+=2;
}
sum_f = sum_f*lambda*(((double)(2*k+1))*pow(-1.0,(double)(k+1)));
break;
}
return sum_f;
}
/****************************************************
Bessel_h and Bessel_j are based on the
subroutine of Elk code created by F. Bultmark,
F. Cricchio, L. Nordstrom and J. K. Dewhurst.
Computes variations of the spherical Bessel function
$b_l(x)=i^lh^{(1)}_l(ix)$, for real argument $x$ and
$l=0,1,\ldots,l_{\rm max}$. The recursion relation
$$ j_{l+1}(x)=\frac{2l+1}{x}j_l(x)+j_{l-1}(x) $$
is used upwards. For starting values there are
$$ b_0(x)=-\frac{e^{-x}}{x};\qquad b_1(x)=b_0(x)\left\{1+\frac{1}{x}
\right\}. $$
For $x\ll 1$ the asymtotic forms
$$ b_l(x)\approx\frac{-(2l-1)!!}{(-x)^{l+1}} $$
are used.
****************************************************/
static double Bessel_h(int k_in, double x){
int l;
double xi,xmin,t1,t2;
double b0,b1,b_dum;
double f;
if (x==0.0){
x = 1.0e-9;
}
xmin = 1.0e-8;
xi = 1.0/x;
b0 = -xi*exp(-x);
b1 = b0*(1.0+xi);
if(x <= xmin){
b0 = -xi;
t1 = -1.0;
t2 = xi;
f = b0;
if(k_in > 0){
for(l=1; l<=k_in; l++){
t1 = t1*((double)(2*l-1));
t2 = t2*xi;
b_dum = t2*t1;
}
f = b_dum;
}
}
if(x > xmin){
if (k_in==0){
f = b0;
}
else if (k_in==1){
f = b1;
}
else{
b_dum = 0.0;
for(l=2; l<=k_in; l++){
b_dum = ((double)(2*l-1))*b1*xi + b0;
b0 = b1;
b1 = b_dum;
}
f = b_dum;
}
}
return f;
}
/***********************************************************************
Computes variations of the spherical Bessel function, $a_l(x)=i^lj_l(ix)$,
for real argument $x$ and $l=0,1,\ldots,l_{\rm max}$. The recursion relation
$$ j_{l+1}(x)=\frac{2l+1}{x}j_l(x)+j_{l-1}(x) $$
is used upwards. For starting values there are
$$ a_0(x)=\frac{\sinh(x)}{x};\qquad a_1(x)=\frac{a_0(x)-\cosh(x)}{x} $$.
For $x\ll 1$ the asymtotic forms
$$ a_l(x)\approx\frac{(-x)^l}{(2l+1)!!} $$
are used.
***********************************************************************/
static double Bessel_j(int l2max, int k_in, double x){
int l;
double xi,xmin,t1,t2;
double a0,a1,a_dum;
double f;
if(l2max==0){
xmin = 1.0e-6;
}
else if(l2max==2){
xmin = 1.0e-4;
}
else if(l2max==4){
xmin = 1.0e-2;
}
else{
xmin = 1.0;
}
xi = 1.0/x;
a0 = xi*sinh(x);
a1 = xi*(a0-cosh(x));
if(x <= xmin){
a0 = 1.0;
t1 = 1.0;
t2 = 1.0;
f = a0;
if(k_in > 0){
for(l=1; l<=k_in; l++){
t1 = t1/((double)(2*l+1));
t2 = -t2*x;
a_dum = t2*t1;
}
f = a_dum;
}
}
if(x > xmin){
if (k_in==0){
f = a0;
}
else if (k_in==1){
f = a1;
}
else{
a_dum = 0.0;
for(l=2; l<=k_in; l++){
a_dum = ((double)(2*l-1))*a1*xi + a0;
a0 = a1;
a1 = a_dum;
}
f = a_dum;
}
}
return f;
}
/**********************************************************************
Calculating Coulomb interaction matrix in real harmonics basis.
U(a,b,c,d) = <a, b | Vee | c, d>
= Sum_over_m1,m2,m3,m4[S_a,m1 * S_b,m2 *
{sum_over_k=0,2,4,..[a_k(m1,m2,m3,m4)*F[k]}*
S^*_c,m3 * S^*_d,m4
***********************************************************************/
static double Coulomb_Matrix(int num, double F0, double JH, int l, int a, int b, int c, int d){
double Slater_F[7];
double Coulomb_Sph;
dcomplex Coulomb_Real, TRANS;
int i,j,m,n,kk;
if(Yukawa_on==1){
switch (l){
case 1:
Slater_F[0] = F0;
Slater_F[2] = Slater_F2[num];
break;
case 2:
Slater_F[0] = F0;
Slater_F[2] = Slater_F2[num];
Slater_F[4] = Slater_F4[num];
break;
case 3:
Slater_F[0] = F0;
Slater_F[2] = Slater_F2[num];
Slater_F[4] = Slater_F4[num];
Slater_F[6] = Slater_F6[num];
break;
}
}
else{
switch (l){
case 1:
Slater_F[0] = F0;
Slater_F[2] = 5.0*JH;
break;
case 2:
Slater_F[0] = F0;
Slater_F[2] = (14.0/(1.0 + slater_ratio))*JH;
Slater_F[4] = slater_ratio*Slater_F[2];
break;
case 3:
Slater_F[0] = F0;
Slater_F[2] = (6435.0/(286.0 + 195.0*0.668 + 250.0*0.494))*JH;
Slater_F[4] = 0.668*Slater_F[2];
Slater_F[6] = 0.494*Slater_F[2];
break;
}
}
Coulomb_Real = Complex(0.0,0.0);
for (i=0; i<(2*l+1); i++){
for (j=0; j<(2*l+1); j++){
for (m=0; m<(2*l+1); m++){
for (n=0; n<(2*l+1); n++){
TRANS = Cmul(Cmul(StoR(l,a,i),StoR(l,b,j)),Cmul(Conjg(StoR(l,c,m)),Conjg(StoR(l,d,n))));
Coulomb_Sph = 0.0;
for (kk=0; kk<=2*l; kk+=2){
if ((i+j) == (m+n)){
Coulomb_Sph += Gaunt_SR(l,kk,l-i,l-m)*Gaunt_SR(l,kk,l-n,l-j)*Slater_F[kk];
}
else{
Coulomb_Sph += 0.0;
}
}
Coulomb_Real.r += Cmul(TRANS, Complex(Coulomb_Sph,0.0)).r;
}
}
}
}
return Coulomb_Real.r;
}
/*********************************************************************
Transformation matrix from spherical to real harmonics,
S_m1,m2 (0 <= (m1, m2) < 2l+1).
Real | Spherical
y_l0 = Y^l_0
y_lm = 1/sqrt(2)*[Y^l_-m + (-1)^m*Y^l_m]
y_l(-m) = i/sqrt(2)*[Y^l_-m - (-1)^m*Y^l_m]
Real | Spherical
px S |11>
py <-- |10>
pz |1-1>
-----------------------------
dz^2 |22>
dx^2-y^2 S |21>
dxy <-- |20>
dzx |2-1>
dyz |2-2>
-----------------------------
fz^3 |33>
fxz^2 |32>
fyz^2 S |31>
fz(x^2-y^2) <-- |30>
fxyz |3-1>
fx^3-3xy^2 |3-2>
f3yx^2-y^3 |3-3>
**********************************************************************/
static dcomplex StoR(int l, int m1, int m2){
dcomplex result, S2R;
result = Complex(0.0,0.0);
switch(l){
case 1:
if (m1 == 0){
if (m2 != 1){
result = Complex((double)(m2-l), 0.0);
}
else{
result = Complex(0.0,0.0);
}
}
else if (m1 == 1){
if (m2 != 1){
result = Complex(0.0,1.0);
}
else{
result = Complex(0.0,0.0);
}
}
else{
if (m1 ==2 && m2 == 1){
result = Complex(sqrt(2.0),0.0);
}
else{
result = Complex(0.0,0.0);
}
}
break;
case 2:
if (m1 == 0){
if (m2 == 2){
result = Complex(sqrt(2.0),0.0);
}
else{
result = Complex(0.0,0.0);
}
}
else if (m1 == 1){
if (m2 == 0 || m2 == 4){
result = Complex(1.0,0.0);
}
else{
result = Complex(0.0,0.0);
}
}
else if (m1 == 2){
if (m2 == 0 || m2 == 4){
result = Complex(0.0,((double)(m2-l)/2.0));
}
else{
result = Complex(0.0,0.0);
}
}
else if (m1 == 3){
if (m2 == 1 || m2 == 3){
result = Complex((double)(m2-l),0.0);
}
else{
result = Complex(0.0,0.0);
}
}
else{
if (m1 == 4 && (m2 == 1 || m2 == 3)){
result = Complex(0.0,1.0);
}
else{
result = Complex(0.0,0.0);
}
}
break;
case 3:
if (m1 == 0 && m2 == 3){
result = Complex(sqrt(2.0),0.0);
}
else if (m1 == 1){
if (m2 == 2 || m2 == 4){
result = Complex((double)(m2-l),0.0);
}
}
else if (m1 == 2){
if (m2 == 2 || m2 == 4){
result = Complex(0.0,1.0);
}
}
else if (m1 == 3){
if (m2 == 1 || m2 == 5){
result = Complex(1.0,0.0);
}
}
else if (m1 == 4){
if (m2 == 1 || m2 == 5){
result = Complex(0.0,((double)(m2-l)/2.0));
}
}
else if (m1 == 5){
if (m2 == 0 || m2 == 6){
result = Complex((double)(m2-l)/3.0,0.0);
}
}
else if (m1 == 6){
if (m2 == 0 || m2 == 6){
result = Complex(0.0,1.0);
}
}
else{
result = Complex(0.0,0.0);
}
break;
}
S2R = Cmul(result, Complex(1.0/sqrt(2.0),0.0));
return S2R;
}
/*******************************************************************************
Gaunt coeffcient calculator.
c_k(l,m1;l,m3) = sqrt(4*pi/(2k+1)) * integral(Y^*_l,m1 * Y_k,m-m' * Y_l,m2)
********************************************************************************/
static double Gaunt_SR(int l, int k, int m1, int m2){
double ck;
ck = pow(-1.0, -(double)m1)*sqrt((2.0*(double)l+1.0)*(2.0*(double)l+1.0))*
Wigner3j(l,k,l,0,0,0)*Wigner3j(l,k,l,-m1,m1-m2,m2);
return ck;
}
/**************************************************************
Wigner3-j symbol calculator using Racah formula.
Algorithm is based on the python code by David Terr.
/ l1 l2 l3 \
| |
\ m1 m2 m3 /
***************************************************************/
static double Wigner3j(int l1, int l2, int l3, int m1, int m2, int m3){
double t_coeff, tabc;
int tc1, tc2, tc3, tc4;
int a1, a2, b1, b2, c1, c2;
int count;
int x1, x2, x3, x4, x5;
double x;
int t1, t2, t3, t4, t5;
int tmin, tmax;
tc1 = l1 + l2 - l3;
tc2 = l1 - l2 + l3;
tc3 = -l1 + l2 + l3;
tc4 = l1 + l2 + l3 + 1;
t_coeff = FAC(tc1)*FAC(tc2)*FAC(tc3)/FAC(tc4);
a1 = (l1 + m1);
a2 = (l1 - m1);
b1 = (l2 + m2);
b2 = (l2 - m2);
c1 = (l3 + m3);
c2 = (l3 - m3);
tabc = pow(-1.0,(double)(l1-l2-m3))*sqrt(t_coeff*FAC(a1)*FAC(a2)*FAC(b1)*FAC(b2)*FAC(c1)*FAC(c2));
t1 = l2 - m1 - l3;