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/**********************************************************************
XC_PBE.c:
XC_PBE.c is a subroutine to calculate the exchange-correlation
potential developed by Perdew, Burke and Ernzerhof within
generalized gradient approximation.
This routine was written by T.Ozaki, based on the original fortran
code provided by the SIESTA group through their website.
Thanks to them.
Ref: J.P.Perdew, K.Burke & M.Ernzerhof, PRL 77, 3865 (1996)
Log of XC_PBE.c:
22/Nov/2001 Released by T.Ozaki
***********************************************************************/
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include "openmx_common.h"
#define FOUTHD 4.0/3.0
#define HALF 0.50
#define THD 1.0/3.0
#define THRHLF 1.50
#define THRHLF 1.50
#define TWO 2.0
#define TWOTHD 2.0/3.0
#define beta 0.0667250
#define gamma (1.0 - log(TWO))/(PI*PI)
#define mu beta*PI*PI/3.0
#define kappa 0.8040
void XC_PBE(double dens[2], double GDENS[3][2], double Exc[2],
double DEXDD[2], double DECDD[2],
double DEXDGD[3][2], double DECDGD[3][2])
{
int IS,IX;
double Ec_unif[1],Vc_unif[2];
double dt,rs,zeta;
double den_min,gd_min,phi,t,ks,kF,f1,f2,f3,f4;
double A,H,Fc,Fx;
double GDMT,GDT[3];
double DRSDD,DKFDD,DKSDD,DZDD[2],DPDZ;
double DECUDD,DPDD,DTDD,DF1DD,DF2DD,DF3DD,DF4DD,DADD;
double DHDD,DFCDD[2],DTDGD,DF3DGD,DF4DGD,DHDGD,DFCDGD[3][2];
double DS[2],GDMS,KFS,s,f,DFDD,DFXDD[2],Vx_unif[2],Ex_unif[1];
double GDS,DSDGD,DSDD,DF1DGD,DFDGD,DFXDGD[3][2];
double D[2],GD[3][2],GDM[2];
/****************************************************
Lower bounds of density and its gradient
to avoid divisions by zero
****************************************************/
den_min = 1.0e-10;
gd_min = 1.0e-10;
/****************************************************
Translate density and its gradient to new variables
****************************************************/
dens[0] = largest(0.5*den_min,dens[0]);
dens[1] = largest(0.5*den_min,dens[1]);
D[0] = dens[0];
D[1] = dens[1];
dt = largest(den_min,dens[0] + dens[1]);
for (IX=0; IX<=2; IX++){
GD[IX][0] = GDENS[IX][0];
GD[IX][1] = GDENS[IX][1];
GDT[IX] = GDENS[IX][0] + GDENS[IX][1];
}
GDM[0] = sqrt(GD[0][0]*GD[0][0] + GD[1][0]*GD[1][0] + GD[2][0]*GD[2][0]);
GDM[1] = sqrt(GD[0][1]*GD[0][1] + GD[1][1]*GD[1][1] + GD[2][1]*GD[2][1]);
GDMT = sqrt(GDT[0]*GDT[0] + GDT[1]*GDT[1] + GDT[2]*GDT[2]);
GDMT = largest(gd_min, GDMT);
/****************************************************
Local correlation energy and potential
****************************************************/
XC_PW91C(dens, Ec_unif, Vc_unif);
/****************************************************
Total correlation energy
****************************************************/
rs = pow(3.0/(4.0*PI*dt),THD);
kF = pow(3.0*PI*PI*dt,THD);
ks = sqrt(4.0*kF/PI);
zeta = (dens[0] - dens[1])/dt;
zeta = largest( -1.0 + den_min,zeta);
zeta = smallest( 1.0 - den_min,zeta);
phi = 0.50*(pow(1.0 + zeta,TWOTHD)
+ pow(1.0 - zeta,TWOTHD));
t = GDMT/(2.0*phi*ks*dt);
f1 = Ec_unif[0]/(gamma*pow(phi,3.0));
f2 = exp(-f1);
A = beta/gamma/(f2 - 1.0);
f3 = t*t + A*t*t*t*t;
f4 = beta/gamma * f3/(1.0 + A*f3);
H = gamma*pow(phi,3.0)*log(1.0 + f4);
Fc = Ec_unif[0] + H;
/****************************************************
Correlation energy derivatives
****************************************************/
DRSDD = -(THD*rs/dt);
DKFDD = THD*kF/dt;
DKSDD = HALF*ks*DKFDD/kF;
DZDD[0] = 1.0/dt - zeta/dt;
DZDD[1] = -(1.0/dt) - zeta/dt;
DPDZ = HALF*TWOTHD*(1.0/pow(1.0 + zeta,THD) - 1.0/pow(1.0 - zeta,THD));
for (IS=0; IS<=1; IS++){
DECUDD = (Vc_unif[IS] - Ec_unif[0])/dt;
DPDD = DPDZ*DZDD[IS];
DTDD = (-t)*(DPDD/phi + DKSDD/ks + 1.0/dt);
DF1DD = f1*(DECUDD/Ec_unif[0] - 3.0*DPDD/phi);
DF2DD = (-f2)*DF1DD;
DADD = (-A)*DF2DD/(f2 - 1.0);
DF3DD = (2.0*t + 4.0*A*t*t*t) * DTDD + DADD*t*t*t*t;
DF4DD = f4*(DF3DD/f3 - (DADD*f3+A*DF3DD)/(1.0 + A*f3));
DHDD = 3.0*H*DPDD/phi;
DHDD = DHDD + gamma*phi*phi*phi*DF4DD/(1.0 + f4);
DFCDD[IS] = Vc_unif[IS] + H + dt * DHDD;
for (IX=0; IX<=2; IX++){
DTDGD = (t/GDMT)*GDT[IX]/GDMT;
DF3DGD = DTDGD*(2.0*t + 4.0*A*t*t*t);
DF4DGD = f4*DF3DGD*(1.0/f3 - A/(1.0 + A*f3));
DHDGD = gamma*phi*phi*phi*DF4DGD/(1.0 + f4);
DFCDGD[IX][IS] = dt*DHDGD;
}
}
/****************************************************
Exchange energy and potential
****************************************************/
Fx = 0.0;
for (IS=0; IS<=1; IS++){
DS[IS] = largest(den_min, 2.0*D[IS]);
GDMS = largest(gd_min, 2.0*GDM[IS]);
KFS = pow(3.0*PI*PI*DS[IS],THD);
s = GDMS/(2.0*KFS*DS[IS]);
f1 = 1.0 + mu*s*s/kappa;
f = 1.0 + kappa - kappa/f1;
/****************************************************
Note nspin=1 in call to XC_EX
****************************************************/
XC_EX(1, DS[IS], DS, Ex_unif, Vx_unif);
Fx = Fx + DS[IS]*Ex_unif[0]*f;
DKFDD = THD * KFS/DS[IS];
DSDD = s*(-(DKFDD/KFS) - 1.0/DS[IS]);
DF1DD = 2.0*(f1 - 1.0)*DSDD/s;
DFDD = kappa*DF1DD/(f1*f1);
DFXDD[IS] = Vx_unif[0]*f + DS[IS]*Ex_unif[0]*DFDD;
for (IX=0; IX<=2; IX++){
GDS = 2.0*GD[IX][IS];
DSDGD = (s/GDMS)*GDS/GDMS;
DF1DGD = 2.0*mu*s*DSDGD/kappa;
DFDGD = kappa*DF1DGD/(f1*f1);
DFXDGD[IX][IS] = DS[IS]*Ex_unif[0]*DFDGD;
}
}
Fx = HALF*Fx/dt;
/****************************************************
Set output arguments
****************************************************/
Exc[0] = Fx;
Exc[1] = Fc;
for (IS=0; IS<=1; IS++){
DEXDD[IS] = DFXDD[IS];
DECDD[IS] = DFCDD[IS];
for (IX=0; IX<=2; IX++){
DEXDGD[IX][IS] = DFXDGD[IX][IS];
DECDGD[IX][IS] = DFCDGD[IX][IS];
}
}
}
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