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						  September 20, 1996 
Dear Colleague,


Please find below the routines for our new PBE functional, and  a
uuencoded  gzipped  postscipt  file of the accompanying paper (to
unpack, first uudecode, then gunzip).

The PBE is intended to replace PW91 and our earlier GGA's for Exc.
For most purposes, it should yield energies very close to those of
PW91.  However, the derivation and analytic form have been simplified,
and some spurious wiggles in the xc potential have been eliminated.

Individuals receiving the first round of this e-mail will also get
a hard copy of the PBE manuscript by regular mail.  This manuscript
has just been accepted by Phys. Rev. Lett.  Best regards,
 
John, Kieron, and Matthias.  
-----------------------------------------------------------------
c PBE alpha2.1:
c Perdew-Burke-Ernzerhof generalized gradient approximation to the
c density functional for exchange-correlation energy of a many-electron
c system.
c  --------------------------------------------------------------------
c |WARNING!  PBE is a simplification of PW91, which yields almost      |
c |identical numerical results with simpler formulas from a simpler    |
c |derivation.  If you should find significant DIFFERENCES between     |
c |PBE and PW91 results, please consult kieron@merlin.phy.tulane.edu   |
c |or perdew@mailhost.tcs.tulane.edu.  Thank you.                      |
c  --------------------------------------------------------------------
c Note: Neglects small grad (zeta) contributions to the correlation
c energy.
c
c Programs implement functional in PBE paper, July 1996 version.
c 
c----------------------------------------------------------------------
c Main program testing PBE subroutines for exchange-correlation energy
c and potentials, by application to unequal exponential
c up- and down-spin densities, showing that the functional derivative
c (exchange-correlation potential) correctly generates the energy change
c due to a small change in the density.
c Kieron Burke, July 2, 1996.
C Atomic units are used, so all energies are in hartrees and all 
c distances in bohrs.  
c 1 hartree=27.2116eV=627.5kcal/mol; 1bohr=0.529E-10m.
c The output should be:
c Fup Fdn Zup Zdn             Exc           CHNG1          CHNG
c 1.0  .0 1.0  .5  -.311916530229   .000000000000   .0000000000
c 1.0  .2 1.0  .5  -.336377065446  -.053880102756  -.0538804290
c 1.0  .4 1.0  .5  -.369084886012  -.120463328976  -.1204642921
c 1.0  .6 1.0  .5  -.406088525151  -.193370595518  -.1933723422
c 1.0  .8 1.0  .5  -.446305936853  -.271252139632  -.2712547575
c 1.0 1.0 1.0  .5  -.489150144888  -.353405855349  -.3534094042
c 1.0 1.0  .5  .5  -.341059977353  -.316599687356  -.3166037653
c 1.0 1.0 1.0 1.0  -.653407740519  -.309758886707  -.3097606837
c 1.0 1.0 1.5 1.5  -.962039224827  -.307820467953  -.3078216918
c 1.0 1.0 2.0 2.0 -1.269410948459  -.307021487395  -.3070225637
c----------------------------------------------------------------------
c----------------------------------------------------------------------
      IMPLICIT REAL*8 (A-H,O-Z)
      parameter(thrd=1.d0/3.d0,thrd2=2.d0*thrd)
      pi=4.d0*datan(1.d0)
      CONF=(3.D0*PI*pi)**THRD
      CONRS=(3.D0/(4.D0*PI))**THRD
      write(6,*)'Fup Fdn Zup Zdn             Exc'
     1,'           CHNG1          CHNG'
c----------------------------------------------------------------------
c----------------------------------------------------------------------
C BEGIN THE LOOP THAT SELECTS A TRIAL DENSITY
c spin-densities are of the form
c          rho(r)=f*(Z**3/pi)*dexp(-2*Z*r)
c delzdn=small change in zdn to test potentials
c jdens=counter for which density
      DO JDENS = 1,10
        FUP=1.D0
        FDN=0.2D0*(JDENS-1)
        ZUP=1.D0
        ZDN=0.5D0
        IF(JDENS.GT.6)then
	  FDN=1.D0
          ZUP=0.5D0+0.5D0*(JDENS-7)
          ZDN=ZUP
	endif
        DELZDN=1D-5
c----------------------------------------------------------------------
c----------------------------------------------------------------------
C BEGIN THE LOOP THAT INCREMENTS THE DENSITY DIFFERENTIALLY
c kdif=1=>density as above
c kdif=2=>Zdn changed by DELZDN
        DO KDIF=1,2
          IF(KDIF.EQ.2)ZDN=ZDN+DELZDN
c----------------------------------------------------------------------
c----------------------------------------------------------------------
C BEGIN THE RADIAL LOOP
c sumexc=integrated exchange-correlation energy 
c chng1=integrated xc energy change, based on vxc
c nr=number of points in radial loop
c rf=final value of r in integrals
c dr=change in r
c wt=weight of r in trapezoidal rule
c dup=up density
c agrup=|grad up|
c delgrup=(grad up).(grad |grad up|) 
c uplap=grad^2 up=Laplacian of up
c dn,agrdn,delgrdn,dnlap=corresponding down quantities
c d=up+dn
c agrad=|grad rho|
c delgrad=(grad rho).(grad |grad rho|) 
          sumexc=0.0D0
          CHNG1=0.0D0
	  nr=10000
	  rf=20.d0
	  dr=rf/real(nr)
          DO I=1,nr
            R=I*dr
            WT=4.d0*PI*R*R*dr
            DUP=FUP*(ZUP**3/PI)*DEXP(-2.D0*ZUP*R)
            DDN=FDN*(ZDN**3/PI)*DEXP(-2.D0*ZDN*R)
            ZDNNU=ZDN+DELZDN
            DELDDN=FDN*(ZDNNU**3/PI)*DEXP(-2.D0*ZDNNU*R)-DDN
	    agrup=2.d0*zup*dup
	    delgrup=8.d0*(zup**3)*dup*dup
	    uplap=4.d0*zup*dup*(zup-1.d0/r)
	    agrdn=2.d0*zdn*ddn
	    delgrdn=8.d0*(zdn**3)*ddn*ddn
	    dnlap=4.d0*zdn*ddn*(zdn-1.d0/r)
            D=DUP+DDN
            agrad=2.d0*(ZUP*DUP+ZDN*DDN)
	    delgrad=4.d0*agrad*(ZUP**2*DUP+ZDN**2*DDN)
            call easypbe(dup,agrup,delgrup,uplap,ddn,agrdn,delgrdn,
     1           dnlap,agrad,delgrad,1,1,
     1           exlsd,vxuplsd,vxdnlsd,eclsd,vcuplsd,vcdnlsd,
     1           expw91,vxuppw91,vxdnpw91,ecpw91,vcuppw91,vcdnpw91,
     1           expbe,vxuppbe,vxdnpbe,ecpbe,vcuppbe,vcdnpbe)
	    sumexc=sumexc+d*(expbe+ecpbe)*wt
            CHNG1=CHNG1+(vxdnpbe+vcdnpbe)*DELDDN*WT/DELZDN
	  enddo
          IF(KDIF.EQ.1)then
	    sumEXCO=sumEXC
	  endif
        enddo
c----------------------------------------------------------------------
c----------------------------------------------------------------------
C  CHNG: DIRECT XC ENERGY INCREMENT
C  IF THE FUNCTIONAL DERIVATIVE IS CORRECT, THEN CHNG1=CHNG
        CHNG=(sumEXC-sumEXCO)/DELZDN
        PRINT 200,FUP,FDN,ZUP,ZDN,sumEXC,CHNG1,chng
      enddo
      STOP
  200 FORMAT(4f4.1,2f16.12,f14.10)
      END
c----------------------------------------------------------------------
c######################################################################
c----------------------------------------------------------------------
      subroutine easypbe(up,agrup,delgrup,uplap,dn,agrdn,delgrdn,dnlap,
     1           agr,delgr,lcor,lpot,
     1           exlsd,vxuplsd,vxdnlsd,eclsd,vcuplsd,vcdnlsd,
     1           expw91,vxuppw91,vxdnpw91,ecpw91,vcuppw91,vcdnpw91,
     1           expbe,vxuppbe,vxdnpbe,ecpbe,vcuppbe,vcdnpbe)
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c EASYPBE is a driver for the PBE subroutines, using simple inputs
c K. Burke, May 14, 1996.
c inputs: up=up density
c	: agrup=|grad up|
c	: delgrup=(grad up).(grad |grad up|) 
c	: uplap=grad^2 up=Laplacian of up
c	: dn,agrdn,delgrdn,dnlap=corresponding down quantities
c	: agr=|grad rho|
c	: delgr=(grad rho).(grad |grad rho|) 
c	: lcor=flag to do correlation(=0=>don't)
c	: lpot=flag to do potential(=0=>don't)
c outputs: exlsd=LSD exchange energy density, so that
c		ExLSD=int d^3r rho(r) exlsd(r)
c	 : vxuplsd=up LSD exchange potential
c	 : vxdnlsd=down LSD exchange potential
c        : exclsd=LSD exchange-correlation energy density
c	 : vxcuplsd=up LSD exchange-correlation potential
c	 : vxcdnlsd=down LSD exchange-correlation potential
c        : expw91,vxuppw91,vxdnpw91,ecpw91,etc.=PW91 quantities
c        : expbe,vxuppbe,vxdnpbe,ecpbe,etc.=PBE quantities
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c needed constants:
c pi32=3 pi**2
c alpha=(9pi/4)**thrd
      implicit real*8(a-h,o-z)
      parameter(thrd=1.d0/3.d0,thrd2=2.d0*thrd)
      parameter(pi32=29.608813203268075856503472999628d0)
      parameter(pi=3.1415926535897932384626433832795d0)
      parameter(alpha=1.91915829267751300662482032624669d0)
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c PBE exchange
c use  Ex[up,dn]=0.5*(Ex[2*up]+Ex[2*dn]) (i.e., exact spin-scaling)
c do up exchange
c fk=local Fermi wavevector for 2*up=(3 pi^2 (2up))^(1/3) 
c s=dimensionless density gradient=|grad rho|/ (2*fk*rho)_(rho=2*up)
c u=delgrad/(rho^2*(2*fk)**3)_(rho=2*up)
c v=Laplacian/(rho*(2*fk)**2)_(rho=2*up)
      rho2=2.d0*up
      if(rho2.gt.1d-18)then
        fk=(pi32*rho2)**thrd
        s=2.d0*agrup/(2.d0*fk*rho2)
        u=4.d0*delgrup/(rho2*rho2*(2.d0*fk)**3)
        v=2.d0*uplap/(rho2*(2.d0*fk)**2)
        call exchpbe(rho2,s,u,v,0,lpot,exuplsd,vxuplsd)
        call exchpw91(rho2,s,u,v,exuppw91,vxuppw91)
        call exchpbe(rho2,s,u,v,1,lpot,exuppbe,vxuppbe)
      else
	exuplsd=0.d0
	vxuplsd=0.d0
	exuppw91=0.d0
	vxuppw91=0.d0
	exuppbe=0.d0
	vxuppbe=0.d0
      endif
c repeat for down
      rho2=2.d0*dn
      if(rho2.gt.1d-18)then
        fk=(pi32*rho2)**thrd
        s=2.d0*agrdn/(2.d0*fk*rho2)
        u=4.d0*delgrdn/(rho2*rho2*(2.d0*fk)**3)
        v=2.d0*dnlap/(rho2*(2.d0*fk)**2)
        call exchpbe(rho2,s,u,v,0,lpot,exdnlsd,vxdnlsd)
        call exchpw91(rho2,s,u,v,exdnpw91,vxdnpw91)
        call exchpbe(rho2,s,u,v,1,lpot,exdnpbe,vxdnpbe)
      else
	exdnlsd=0.d0
	vxdnlsd=0.d0
	exdnpw91=0.d0
	vxdnpw91=0.d0
	exdnpbe=0.d0
	vxdnpbe=0.d0
      endif
10    continue 
c construct total density and contribution to ex
      rho=up+dn
      exlsd=(exuplsd*up+exdnlsd*dn)/rho
      expw91=(exuppw91*up+exdnpw91*dn)/rho
      expbe=(exuppbe*up+exdnpbe*dn)/rho
      if(lcor.eq.0)return
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c Now do correlation
c zet=(up-dn)/rho
c g=phi(zeta)
c rs=(3/(4pi*rho))^(1/3)=local Seitz radius=alpha/fk
c sk=Ks=Thomas-Fermi screening wavevector=sqrt(4fk/pi)
c twoksg=2*Ks*phi
c t=correlation dimensionless gradient=|grad rho|/(2*Ks*phi*rho)
c uu=delgrad/(rho^2*twoksg^3)
c rholap=Laplacian
c vv=Laplacian/(rho*twoksg^2)
c ww=(|grad up|^2-|grad dn|^2-zet*|grad rho|^2)/(rho*twoksg)^2
c ec=lsd correlation energy
c vcup=lsd up correlation potential
c vcdn=lsd down correlation potential
c h=gradient correction to correlation energy
c dvcup=gradient correction to up correlation potential
c dvcdn=gradient correction to down correlation potential
      if(rho.lt.1.d-18)return
      zet=(up-dn)/rho
      g=((1.d0+zet)**thrd2+(1.d0-zet)**thrd2)/2.d0
      fk=(pi32*rho)**thrd
      rs=alpha/fk
      sk=sqrt(4.d0*fk/pi)
      twoksg=2.d0*sk*g
      t=agr/(twoksg*rho)
      uu=delgr/(rho*rho*twoksg**3)
      rholap=uplap+dnlap
      vv=rholap/(rho*twoksg**2)
      ww=(agrup**2-agrdn**2-zet*agr**2)/(rho*rho*twoksg**2)
      call CORPBE(RS,ZET,T,UU,VV,WW,1,lpot,ec,vcup,vcdn,
     1                  H,DVCUP,DVCDN)
      eclsd=ec
      ecpbe=ec+h
      vcuplsd=vcup
      vcdnlsd=vcdn
      vcuppbe=vcup+dvcup
      vcdnpbe=vcdn+dvcdn
      call CORLSD(RS,ZET,EC,VCUP,VCDN,ECRS,ECZET,ALFC)
      call CORPW91(RS,ZET,G,EC,ECRS,ECZET,T,UU,VV,WW,H,DVCUP,DVCDN)
      ecpw91=ec+h
      vcuppw91=vcup+dvcup
      vcdnpw91=vcdn+dvcdn
      return
      end
c----------------------------------------------------------------------
c######################################################################
c----------------------------------------------------------------------
      SUBROUTINE EXCHPBE(rho,S,U,V,lgga,lpot,EX,VX)
c----------------------------------------------------------------------
C  PBE EXCHANGE FOR A SPIN-UNPOLARIZED ELECTRONIC SYSTEM
c  K Burke's modification of PW91 codes, May 14, 1996
c  Modified again by K. Burke, June 29, 1996, with simpler Fx(s)
c----------------------------------------------------------------------
c----------------------------------------------------------------------
C  INPUT rho : DENSITY
C  INPUT S:  ABS(GRAD rho)/(2*KF*rho), where kf=(3 pi^2 rho)^(1/3)
C  INPUT U:  (GRAD rho)*GRAD(ABS(GRAD rho))/(rho**2 * (2*KF)**3)
C  INPUT V: (LAPLACIAN rho)/(rho*(2*KF)**2)
c   (for U,V, see PW86(24))
c  input lgga:  (=0=>don't put in gradient corrections, just LDA)
c  input lpot:  (=0=>don't get potential and don't need U and V)
C  OUTPUT:  EXCHANGE ENERGY PER ELECTRON (EX) AND POTENTIAL (VX)
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c References:
c [a]J.P.~Perdew, K.~Burke, and M.~Ernzerhof, submiited to PRL, May96
c [b]J.P. Perdew and Y. Wang, Phys. Rev.  B {\bf 33},  8800  (1986);
c     {\bf 40},  3399  (1989) (E).
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c Formulas:
c   	e_x[unif]=ax*rho^(4/3)  [LDA]
c ax = -0.75*(3/pi)^(1/3)
c	e_x[PBE]=e_x[unif]*FxPBE(s)
c	FxPBE(s)=1+uk-uk/(1+ul*s*s)                 [a](13)
c uk, ul defined after [a](13) 
c----------------------------------------------------------------------
c----------------------------------------------------------------------
      IMPLICIT REAL*8 (A-H,O-Z)
      parameter(thrd=1.d0/3.d0,thrd4=4.d0/3.d0)
      parameter(pi=3.14159265358979323846264338327950d0)
      parameter(ax=-0.738558766382022405884230032680836d0)
      parameter(um=0.2195149727645171d0,uk=0.8040d0,ul=um/uk)
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c construct LDA exchange energy density
      exunif = AX*rho**THRD
      if(lgga.eq.0)then
	ex=exunif
        vx=ex*thrd4
	return
      endif
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c construct PBE enhancement factor
      S2 = S*S
      P0=1.d0+ul*S2
      FxPBE = 1d0+uk-uk/P0
      EX = exunif*FxPBE
      if(lpot.eq.0)return
c----------------------------------------------------------------------
c----------------------------------------------------------------------
C  ENERGY DONE. NOW THE POTENTIAL:
c  find first and second derivatives of Fx w.r.t s.
c  Fs=(1/s)*d FxPBE/ ds
c  Fss=d Fs/ds
      Fs=2.d0*uk*ul/(P0*P0)
      Fss=-4.d0*ul*S*Fs/P0
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c calculate potential from [b](24) 
      VX = exunif*(THRD4*FxPBE-(U-THRD4*S2*s)*FSS-V*FS)
      RETURN
      END
c----------------------------------------------------------------------
c######################################################################
c----------------------------------------------------------------------
      SUBROUTINE CORPBE(RS,ZET,T,UU,VV,WW,lgga,lpot,ec,vcup,vcdn,
     1                  H,DVCUP,DVCDN)
c----------------------------------------------------------------------
c  Official PBE correlation code. K. Burke, May 14, 1996.
C  INPUT: RS=SEITZ RADIUS=(3/4pi rho)^(1/3)
C       : ZET=RELATIVE SPIN POLARIZATION = (rhoup-rhodn)/rho
C       : t=ABS(GRAD rho)/(rho*2.*KS*G)  -- only needed for PBE
C       : UU=(GRAD rho)*GRAD(ABS(GRAD rho))/(rho**2 * (2*KS*G)**3)
C       : VV=(LAPLACIAN rho)/(rho * (2*KS*G)**2)
C       : WW=(GRAD rho)*(GRAD ZET)/(rho * (2*KS*G)**2
c       :  UU,VV,WW, only needed for PBE potential
c       : lgga=flag to do gga (0=>LSD only)
c       : lpot=flag to do potential (0=>energy only)
c  output: ec=lsd correlation energy from [a]
c        : vcup=lsd up correlation potential
c        : vcdn=lsd dn correlation potential
c        : h=NONLOCAL PART OF CORRELATION ENERGY PER ELECTRON
c        : dvcup=nonlocal correction to vcup
c        : dvcdn=nonlocal correction to vcdn
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c References:
c [a] J.P.~Perdew, K.~Burke, and M.~Ernzerhof, 
c     {\sl Generalized gradient approximation made simple}, sub.
c     to Phys. Rev.Lett. May 1996.
c [b] J. P. Perdew, K. Burke, and Y. Wang, {\sl Real-space cutoff
c     construction of a generalized gradient approximation:  The PW91
c     density functional}, submitted to Phys. Rev. B, Feb. 1996.
c [c] J. P. Perdew and Y. Wang, Phys. Rev. B {\bf 45}, 13244 (1992).
c----------------------------------------------------------------------
c----------------------------------------------------------------------
      IMPLICIT REAL*8 (A-H,O-Z)
c thrd*=various multiples of 1/3
c numbers for use in LSD energy spin-interpolation formula, [c](9).
c      GAM= 2^(4/3)-2
c      FZZ=f''(0)= 8/(9*GAM)
c numbers for construction of PBE
c      gamma=(1-log(2))/pi^2
c      bet=coefficient in gradient expansion for correlation, [a](4).
c      eta=small number to stop d phi/ dzeta from blowing up at 
c          |zeta|=1.
      parameter(thrd=1.d0/3.d0,thrdm=-thrd,thrd2=2.d0*thrd)
      parameter(sixthm=thrdm/2.d0)
      parameter(thrd4=4.d0*thrd)
      parameter(GAM=0.5198420997897463295344212145565d0)
      parameter(fzz=8.d0/(9.d0*GAM))
      parameter(gamma=0.03109069086965489503494086371273d0)
      parameter(bet=0.06672455060314922d0,delt=bet/gamma)
      parameter(eta=1.d-12)
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c find LSD energy contributions, using [c](10) and Table I[c].
c EU=unpolarized LSD correlation energy
c EURS=dEU/drs
c EP=fully polarized LSD correlation energy
c EPRS=dEP/drs
c ALFM=-spin stiffness, [c](3).
c ALFRSM=-dalpha/drs
c F=spin-scaling factor from [c](9).
c construct ec, using [c](8)
      rtrs=dsqrt(rs)
      CALL gcor2(0.0310907D0,0.21370D0,7.5957D0,3.5876D0,1.6382D0,
     1    0.49294D0,rtrs,EU,EURS)
      CALL gcor2(0.01554535D0,0.20548D0,14.1189D0,6.1977D0,3.3662D0,
     1    0.62517D0,rtRS,EP,EPRS)
      CALL gcor2(0.0168869D0,0.11125D0,10.357D0,3.6231D0,0.88026D0,
     1    0.49671D0,rtRS,ALFM,ALFRSM)
      ALFC = -ALFM
      Z4 = ZET**4
      F=((1.D0+ZET)**THRD4+(1.D0-ZET)**THRD4-2.D0)/GAM
      EC = EU*(1.D0-F*Z4)+EP*F*Z4-ALFM*F*(1.D0-Z4)/FZZ
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c LSD potential from [c](A1)
c ECRS = dEc/drs [c](A2)
c ECZET=dEc/dzeta [c](A3)
c FZ = dF/dzeta [c](A4)
      ECRS = EURS*(1.D0-F*Z4)+EPRS*F*Z4-ALFRSM*F*(1.D0-Z4)/FZZ
      FZ = THRD4*((1.D0+ZET)**THRD-(1.D0-ZET)**THRD)/GAM
      ECZET = 4.D0*(ZET**3)*F*(EP-EU+ALFM/FZZ)+FZ*(Z4*EP-Z4*EU
     1        -(1.D0-Z4)*ALFM/FZZ)
      COMM = EC -RS*ECRS/3.D0-ZET*ECZET
      VCUP = COMM + ECZET
      VCDN = COMM - ECZET
      if(lgga.eq.0)return
c----------------------------------------------------------------------
c----------------------------------------------------------------------
c PBE correlation energy
c G=phi(zeta), given after [a](3)
c DELT=bet/gamma
c B=A of [a](8)
      G=((1.d0+ZET)**thrd2+(1.d0-ZET)**thrd2)/2.d0
      G3 = G**3
      PON=-EC/(G3*gamma)
      B = DELT/(DEXP(PON)-1.D0)
      B2 = B*B
      T2 = T*T
      T4 = T2*T2
      RS2 = RS*RS
      RS3 = RS2*RS
      Q4 = 1.D0+B*T2
      Q5 = 1.D0+B*T2+B2*T4
      H = G3*(BET/DELT)*DLOG(1.D0+DELT*Q4*T2/Q5)
      if(lpot.eq.0)return
c----------------------------------------------------------------------
c----------------------------------------------------------------------
C ENERGY DONE. NOW THE POTENTIAL, using appendix E of [b].
      G4 = G3*G
      T6 = T4*T2
      RSTHRD = RS/3.D0
      GZ=(((1.d0+zet)**2+eta)**sixthm-
     1((1.d0-zet)**2+eta)**sixthm)/3.d0
      FAC = DELT/B+1.D0
      BG = -3.D0*B2*EC*FAC/(BET*G4)
      BEC = B2*FAC/(BET*G3)
      Q8 = Q5*Q5+DELT*Q4*Q5*T2
      Q9 = 1.D0+2.D0*B*T2
      hB = -BET*G3*B*T6*(2.D0+B*T2)/Q8
      hRS = -RSTHRD*hB*BEC*ECRS
      FACT0 = 2.D0*DELT-6.D0*B
      FACT1 = Q5*Q9+Q4*Q9*Q9
      hBT = 2.D0*BET*G3*T4*((Q4*Q5*FACT0-DELT*FACT1)/Q8)/Q8
      hRST = RSTHRD*T2*hBT*BEC*ECRS
      hZ = 3.D0*GZ*h/G + hB*(BG*GZ+BEC*ECZET)
      hT = 2.d0*BET*G3*Q9/Q8
      hZT = 3.D0*GZ*hT/G+hBT*(BG*GZ+BEC*ECZET)
      FACT2 = Q4*Q5+B*T2*(Q4*Q9+Q5)
      FACT3 = 2.D0*B*Q5*Q9+DELT*FACT2
      hTT = 4.D0*BET*G3*T*(2.D0*B/Q8-(Q9*FACT3/Q8)/Q8)
      COMM = H+HRS+HRST+T2*HT/6.D0+7.D0*T2*T*HTT/6.D0
      PREF = HZ-GZ*T2*HT/G
      FACT5 = GZ*(2.D0*HT+T*HTT)/G
      COMM = COMM-PREF*ZET-UU*HTT-VV*HT-WW*(HZT-FACT5)
      DVCUP = COMM + PREF
      DVCDN = COMM - PREF
      RETURN
      END
c----------------------------------------------------------------------
c######################################################################
c----------------------------------------------------------------------
      SUBROUTINE GCOR2(A,A1,B1,B2,B3,B4,rtrs,GG,GGRS)
c slimmed down version of GCOR used in PW91 routines, to interpolate
c LSD correlation energy, as given by (10) of
c J. P. Perdew and Y. Wang, Phys. Rev. B {\bf 45}, 13244 (1992).
c K. Burke, May 11, 1996.
      IMPLICIT REAL*8 (A-H,O-Z)
      Q0 = -2.D0*A*(1.D0+A1*rtrs*rtrs)
      Q1 = 2.D0*A*rtrs*(B1+rtrs*(B2+rtrs*(B3+B4*rtrs)))
      Q2 = DLOG(1.D0+1.D0/Q1)
      GG = Q0*Q2
      Q3 = A*(B1/rtrs+2.D0*B2+rtrs*(3.D0*B3+4.D0*B4*rtrs))
      GGRS = -2.D0*A*A1*Q2-Q0*Q3/(Q1*(1.d0+Q1))
      RETURN
      END
c----------------------------------------------------------------------
c######################################################################
c----------------------------------------------------------------------
      SUBROUTINE EXCHPW91(D,S,U,V,EX,VX)
C  GGA91 EXCHANGE FOR A SPIN-UNPOLARIZED ELECTRONIC SYSTEM
C  INPUT D : DENSITY
C  INPUT S:  ABS(GRAD D)/(2*KF*D)
C  INPUT U:  (GRAD D)*GRAD(ABS(GRAD D))/(D**2 * (2*KF)**3)
C  INPUT V: (LAPLACIAN D)/(D*(2*KF)**2)
C  OUTPUT:  EXCHANGE ENERGY PER ELECTRON (EX) AND POTENTIAL (VX)
      IMPLICIT REAL*8 (A-H,O-Z)
      parameter(a1=0.19645D0,a2=0.27430D0,a3=0.15084D0,a4=100.d0)
      parameter(ax=-0.7385588D0,a=7.7956D0,b1=0.004d0)
      parameter(thrd=0.333333333333D0,thrd4=1.33333333333D0)
c for Becke exchange, set a3=b1=0
      FAC = AX*D**THRD
      S2 = S*S
      S3 = S2*S
      S4 = S2*S2
      P0 = 1.D0/DSQRT(1.D0+A*A*S2)
      P1 = DLOG(A*S+1.D0/P0)
      P2 = DEXP(-A4*S2)
      P3 = 1.D0/(1.D0+A1*S*P1+B1*S4)
      P4 = 1.D0+A1*S*P1+(A2-A3*P2)*S2
      F = P3*P4
      EX = FAC*F
C  LOCAL EXCHANGE OPTION
C     EX = FAC
C  ENERGY DONE. NOW THE POTENTIAL:
      P5 = B1*S2-(A2-A3*P2)
      P6 = A1*S*(P1+A*S*P0)
      P7 = 2.D0*(A2-A3*P2)+2.D0*A3*A4*S2*P2-4.D0*B1*S2*F
      FS = P3*(P3*P5*P6+P7)
      P8 = 2.D0*S*(B1-A3*A4*P2)
      P9 = A1*P1+A*A1*S*P0*(3.D0-A*A*S2*P0*P0)
      P10 = 4.D0*A3*A4*S*P2*(2.D0-A4*S2)-8.D0*B1*S*F-4.D0*B1*S3*FS
      P11 = -P3*P3*(A1*P1+A*A1*S*P0+4.D0*B1*S3)
      FSS = P3*P3*(P5*P9+P6*P8)+2.D0*P3*P5*P6*P11+P3*P10+P7*P11
      VX = FAC*(THRD4*F-(U-THRD4*S3)*FSS-V*FS)
C  LOCAL EXCHANGE OPTION:
C     VX = FAC*THRD4
      RETURN
      END
c----------------------------------------------------------------------
c######################################################################
c----------------------------------------------------------------------
      SUBROUTINE CORLSD(RS,ZET,EC,VCUP,VCDN,ECRS,ECZET,ALFC)
C  UNIFORM-GAS CORRELATION OF PERDEW AND WANG 1991
C  INPUT: SEITZ RADIUS (RS), RELATIVE SPIN POLARIZATION (ZET)
C  OUTPUT: CORRELATION ENERGY PER ELECTRON (EC), UP- AND DOWN-SPIN
C     POTENTIALS (VCUP,VCDN), DERIVATIVES OF EC WRT RS (ECRS) & ZET (ECZET)
C  OUTPUT: CORRELATION CONTRIBUTION (ALFC) TO THE SPIN STIFFNESS
      IMPLICIT REAL*8 (A-H,O-Z)
      parameter(gam=0.5198421D0,fzz=1.709921D0)
      parameter(thrd=0.333333333333D0,thrd4=1.333333333333D0)
      F = ((1.D0+ZET)**THRD4+(1.D0-ZET)**THRD4-2.D0)/GAM
      CALL GCOR(0.0310907D0,0.21370D0,7.5957D0,3.5876D0,1.6382D0,
     1    0.49294D0,1.00D0,RS,EU,EURS)
      CALL GCOR(0.01554535D0,0.20548D0,14.1189D0,6.1977D0,3.3662D0,
     1    0.62517D0,1.00D0,RS,EP,EPRS)
      CALL GCOR(0.0168869D0,0.11125D0,10.357D0,3.6231D0,0.88026D0,
     1    0.49671D0,1.00D0,RS,ALFM,ALFRSM)
C  ALFM IS MINUS THE SPIN STIFFNESS ALFC
      ALFC = -ALFM
      Z4 = ZET**4
      EC = EU*(1.D0-F*Z4)+EP*F*Z4-ALFM*F*(1.D0-Z4)/FZZ
C  ENERGY DONE. NOW THE POTENTIAL:
      ECRS = EURS*(1.D0-F*Z4)+EPRS*F*Z4-ALFRSM*F*(1.D0-Z4)/FZZ
      FZ = THRD4*((1.D0+ZET)**THRD-(1.D0-ZET)**THRD)/GAM
      ECZET = 4.D0*(ZET**3)*F*(EP-EU+ALFM/FZZ)+FZ*(Z4*EP-Z4*EU
     1        -(1.D0-Z4)*ALFM/FZZ)
      COMM = EC -RS*ECRS/3.D0-ZET*ECZET
      VCUP = COMM + ECZET
      VCDN = COMM - ECZET
      RETURN
      END
c----------------------------------------------------------------------
c######################################################################
c----------------------------------------------------------------------
      SUBROUTINE GCOR(A,A1,B1,B2,B3,B4,P,RS,GG,GGRS)
C  CALLED BY SUBROUTINE CORLSD
      IMPLICIT REAL*8 (A-H,O-Z)
      P1 = P + 1.D0
      Q0 = -2.D0*A*(1.D0+A1*RS)
      RS12 = DSQRT(RS)
      RS32 = RS12**3
      RSP = RS**P
      Q1 = 2.D0*A*(B1*RS12+B2*RS+B3*RS32+B4*RS*RSP)
      Q2 = DLOG(1.D0+1.D0/Q1)
      GG = Q0*Q2
      Q3 = A*(B1/RS12+2.D0*B2+3.D0*B3*RS12+2.D0*B4*P1*RSP)
      GGRS = -2.D0*A*A1*Q2-Q0*Q3/(Q1**2+Q1)
      RETURN
      END
c----------------------------------------------------------------------
c######################################################################
c----------------------------------------------------------------------
      SUBROUTINE CORpw91(RS,ZET,G,EC,ECRS,ECZET,T,UU,VV,WW,H,
     1                   DVCUP,DVCDN)
C  pw91 CORRELATION, modified by K. Burke to put all arguments 
c  as variables in calling statement, rather than in common block
c  May, 1996.
C  INPUT RS: SEITZ RADIUS
C  INPUT ZET: RELATIVE SPIN POLARIZATION
C  INPUT T: ABS(GRAD D)/(D*2.*KS*G)
C  INPUT UU: (GRAD D)*GRAD(ABS(GRAD D))/(D**2 * (2*KS*G)**3)
C  INPUT VV: (LAPLACIAN D)/(D * (2*KS*G)**2)
C  INPUT WW:  (GRAD D)*(GRAD ZET)/(D * (2*KS*G)**2
C  OUTPUT H: NONLOCAL PART OF CORRELATION ENERGY PER ELECTRON
C  OUTPUT DVCUP,DVCDN:  NONLOCAL PARTS OF CORRELATION POTENTIALS
      IMPLICIT REAL*8 (A-H,O-Z)
      parameter(xnu=15.75592D0,cc0=0.004235D0,cx=-0.001667212D0)
      parameter(alf=0.09D0)
      parameter(c1=0.002568D0,c2=0.023266D0,c3=7.389D-6,c4=8.723D0)
      parameter(c5=0.472D0,c6=7.389D-2,a4=100.D0)
      parameter(thrdm=-0.333333333333D0,thrd2=0.666666666667D0)
      BET = XNU*CC0
      DELT = 2.D0*ALF/BET
      G3 = G**3
      G4 = G3*G
      PON = -DELT*EC/(G3*BET)
      B = DELT/(DEXP(PON)-1.D0)
      B2 = B*B
      T2 = T*T
      T4 = T2*T2
      T6 = T4*T2
      RS2 = RS*RS
      RS3 = RS2*RS
      Q4 = 1.D0+B*T2
      Q5 = 1.D0+B*T2+B2*T4
      Q6 = C1+C2*RS+C3*RS2
      Q7 = 1.D0+C4*RS+C5*RS2+C6*RS3
      CC = -CX + Q6/Q7
      R0 = 0.663436444d0*rs
      R1 = A4*R0*G4
      COEFF = CC-CC0-3.D0*CX/7.D0
      R2 = XNU*COEFF*G3
      R3 = DEXP(-R1*T2)
      H0 = G3*(BET/DELT)*DLOG(1.D0+DELT*Q4*T2/Q5)
      H1 = R3*R2*T2
      H = H0+H1
C  LOCAL CORRELATION OPTION:
C     H = 0.0D0
C  ENERGY DONE. NOW THE POTENTIAL:
      CCRS = (C2+2.*C3*RS)/Q7 - Q6*(C4+2.*C5*RS+3.*C6*RS2)/Q7**2
      RSTHRD = RS/3.D0
      R4 = RSTHRD*CCRS/COEFF
      GZ = ((1.D0+ZET)**THRDM - (1.D0-ZET)**THRDM)/3.D0
      FAC = DELT/B+1.D0
      BG = -3.D0*B2*EC*FAC/(BET*G4)
      BEC = B2*FAC/(BET*G3)
      Q8 = Q5*Q5+DELT*Q4*Q5*T2
      Q9 = 1.D0+2.D0*B*T2
      H0B = -BET*G3*B*T6*(2.D0+B*T2)/Q8
      H0RS = -RSTHRD*H0B*BEC*ECRS
      FACT0 = 2.D0*DELT-6.D0*B
      FACT1 = Q5*Q9+Q4*Q9*Q9
      H0BT = 2.D0*BET*G3*T4*((Q4*Q5*FACT0-DELT*FACT1)/Q8)/Q8
      H0RST = RSTHRD*T2*H0BT*BEC*ECRS
      H0Z = 3.D0*GZ*H0/G + H0B*(BG*GZ+BEC*ECZET)
      H0T = 2.*BET*G3*Q9/Q8
      H0ZT = 3.D0*GZ*H0T/G+H0BT*(BG*GZ+BEC*ECZET)
      FACT2 = Q4*Q5+B*T2*(Q4*Q9+Q5)
      FACT3 = 2.D0*B*Q5*Q9+DELT*FACT2
      H0TT = 4.D0*BET*G3*T*(2.D0*B/Q8-(Q9*FACT3/Q8)/Q8)
      H1RS = R3*R2*T2*(-R4+R1*T2/3.D0)
      FACT4 = 2.D0-R1*T2
      H1RST = R3*R2*T2*(2.D0*R4*(1.D0-R1*T2)-THRD2*R1*T2*FACT4)
      H1Z = GZ*R3*R2*T2*(3.D0-4.D0*R1*T2)/G
      H1T = 2.D0*R3*R2*(1.D0-R1*T2)
      H1ZT = 2.D0*GZ*R3*R2*(3.D0-11.D0*R1*T2+4.D0*R1*R1*T4)/G
      H1TT = 4.D0*R3*R2*R1*T*(-2.D0+R1*T2)
      HRS = H0RS+H1RS
      HRST = H0RST+H1RST
      HT = H0T+H1T
      HTT = H0TT+H1TT
      HZ = H0Z+H1Z
      HZT = H0ZT+H1ZT
      COMM = H+HRS+HRST+T2*HT/6.D0+7.D0*T2*T*HTT/6.D0
      PREF = HZ-GZ*T2*HT/G
      FACT5 = GZ*(2.D0*HT+T*HTT)/G
      COMM = COMM-PREF*ZET-UU*HTT-VV*HT-WW*(HZT-FACT5)
      DVCUP = COMM + PREF
      DVCDN = COMM - PREF
C  LOCAL CORRELATION OPTION:
C     DVCUP = 0.0D0
C     DVCDN = 0.0D0
      RETURN
      END
c----------------------------------------------------------------------
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