2019-07-09 17:26:36 +02:00
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subroutine lsdsr(rs,z,mu,excsr,vxcsrup,vxcsrdown)
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ccc Hartree atomic units used
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ccc for given density parameter 'rs', spin polarization 'z'
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ccc and cutoff parameter 'mu'
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ccc gives the complementary short-range exchange-correlation
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ccc energy (i.e., xc energy of jellium minus xc energy of long-range
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ccc interacting electron gas) => 'excsr'
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ccc and the corresponding exchange-correlation potentials for
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ccc spin-up and spin-down electrons => 'vxcsrup','vxcsrdown'
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ccc from Paziani, Moroni, Gori-Giorgi, and Bachelet, cond-mat/0601353
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2019-07-15 12:07:54 +02:00
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2019-07-09 17:26:36 +02:00
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implicit none
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double precision rs,z,mu,excsr,vxcsrup,vxcsrdown
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double precision eclr,exlr,ec,ecd,ecz,ex
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double precision vclrup,vclrdown,vxlrup,vxlrdown
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double precision vxup,vxdown,vcup,vcdown
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double precision pi,alpha,cf
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pi=dacos(-1.d0)
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alpha=(4.d0/9.d0/pi)**(1.d0/3.d0)
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cf=1.d0/alpha
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ex=-3.d0*cf/rs/8.d0/pi*((1.d0+z)**(4.d0/3.d0)+
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$ (1.d0-z)**(4.d0/3.d0))
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2019-07-09 23:09:32 +02:00
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ex = 0d0
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2019-07-09 17:26:36 +02:00
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vxup=-(1.d0+z)**(1.d0/3.d0)*(3.d0/2.d0/pi)**(2.d0/3.d0)/rs
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vxdown=-(1.d0-z)**(1.d0/3.d0)*(3.d0/2.d0/pi)**(2.d0/3.d0)/rs
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vxup = 0d0
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vxdown = 0d0
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call ecPW(rs,z,ec,ecd,ecz)
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vcup=ec-rs/3.d0*ecd-(z-1.d0)*ecz
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vcdown=ec-rs/3.d0*ecd-(z+1.d0)*ecz
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call exchangelr(rs,z,mu,exlr)
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exlr = 0d0
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call vexchangelr(rs,z,mu,vxlrup,vxlrdown)
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vxlrup = 0d0
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vxlrdown = 0d0
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2019-07-09 23:09:32 +02:00
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2019-07-09 17:26:36 +02:00
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call ecorrlr(rs,z,mu,eclr)
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call vcorrlr(rs,z,mu,vclrup,vclrdown)
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2019-07-09 23:09:32 +02:00
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2019-07-09 17:26:36 +02:00
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excsr=ex+ec-(exlr+eclr)
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vxcsrup=vxup+vcup-(vxlrup+vclrup)
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vxcsrdown=vxdown+vcdown-(vxlrdown+vclrdown)
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return
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end
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subroutine ecorrlr(rs,z,mu,eclr)
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ccc Hartree atomic units used
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ccc for given density parameter rs, spin polarization z
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ccc and cutoff parameter mu
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ccc gives the correlation energy of the LR gas
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ccc => eclr
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implicit none
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double precision rs,z,mu,eclr,ec,ecd,ecz
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double precision pi,alpha,cf,phi
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double precision g0,dpol,d2anti,d3anti,Qrpa
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double precision coe2,coe3,coe4,coe5
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double precision a1,a2,a3,a4,b0
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double precision q1a,q2a,q3a,t1a,t2a,t3a,adib
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pi=dacos(-1.d0)
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alpha=(4.d0/9.d0/pi)**(1.d0/3.d0)
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cf=1.d0/alpha
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phi=((1.d0+z)**(2.d0/3.d0)+(1.d0-z)**(2.d0/3.d0))/2.d0
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cc parameters from the fit
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adib = 0.784949d0
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q1a = -0.388d0
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q2a = 0.676d0
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q3a = 0.547d0
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t1a = -4.95d0
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t2a = 1.d0
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t3a = 0.31d0
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b0=adib*rs
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d2anti=(q1a*rs+q2a*rs**2)*exp(-abs(q3a)*rs)/rs**2
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d3anti=(t1a*rs+t2a*rs**2)*exp(-abs(t3a)*rs)/rs**3
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coe2=-3.d0/8.d0/rs**3*(1.d0-z**2)*(g0(rs)-0.5d0)
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coe3=-(1.d0-z**2)*g0(rs)/(sqrt(2.d0*pi)*rs**3)
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if(abs(z).eq.1.d0) then
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coe4=-9.d0/64.d0/rs**3*(dpol(rs)
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$ -cf**2*2**(5.d0/3.d0)/5.d0/rs**2)
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coe5=-9.d0/40.d0/(sqrt(2.d0*pi)*rs**3)*dpol(rs)
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else
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coe4=-9.d0/64.d0/rs**3*(((1.d0+z)/2.d0)**2*
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$ dpol(rs*(2/(1.d0+z))**(1.d0/3.d0))+((1.d0-z)/2.d0)**2
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$ *dpol(rs*(2.d0/(1.d0-z))**(1.d0/3.d0))+
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$ (1.-z**2)*d2anti-cf**2/10.d0*((1.d0+z)**(8.d0/3.d0)
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$ +(1.-z)**(8.d0/3.d0))/rs**2)
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coe5=-9.d0/40.d0/(sqrt(2.d0*pi)*rs**3)*(((1.d0+z)/2.d0)**2
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$ *dpol(rs*(2.d0/(1.d0+z))**(1.d0/3.d0))+((1.d0-z)/2.d0)**2
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$ *dpol(rs*(2.d0/(1.d0-z))**(1.d0/3.d0))+(1.d0-z**2)*
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$ d3anti)
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endif
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call ecPW(rs,z,ec,ecd,ecz)
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a1=4.d0*b0**6*coe3+b0**8*coe5
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a2=4.d0*b0**6*coe2+b0**8*coe4+6.d0*b0**4*ec
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a3=b0**8*coe3
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a4=b0**6*(b0**2*coe2+4.d0*ec)
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eclr=(phi**3*Qrpa(mu*sqrt(rs)/phi)+a1*mu**3+a2*mu**4+a3*mu**5+
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$ a4*mu**6+b0**8*mu**8*ec)/((1.d0+b0**2*mu**2)**4)
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return
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end
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subroutine vcorrlr(rs,z,mu,vclrup,vclrdown)
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ccc Hartree atomic units used
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ccc for given density parameter rs, spin polarization z
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ccc and cutoff mu it gives the correlation LSD potential for LR interaction
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ccc => vclrup (spin-up electrons), vclrdown (spin-down electrons)
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implicit none
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double precision rs,z,mu,eclr,eclrrs,eclrz,vclrup,vclrdown
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double precision ec,ecd,ecz
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double precision pi,alpha,cf,phi
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double precision g0,dpol,d2anti,d3anti,Qrpa
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double precision g0d,dpold,d2antid,d3antid,Qrpad,x
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double precision coe2,coe3,coe4,coe5
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double precision coe2rs,coe3rs,coe4rs,coe5rs
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double precision coe2z,coe3z,coe4z,coe5z
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double precision a1,a2,a3,a4,a5,b0,a1rs,a2rs,a3rs,a4rs,a5rs,
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$ b0rs,a1z,a2z,a3z,a4z,a5z,b0z
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double precision q1a,q2a,q3a,t1a,t2a,t3a,adib
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pi=dacos(-1.d0)
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alpha=(4.d0/9.d0/pi)**(1.d0/3.d0)
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cf=1.d0/alpha
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phi=((1.d0+z)**(2.d0/3.d0)+(1.d0-z)**(2.d0/3.d0))/2.d0
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cc parameters from the fit
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adib = 0.784949d0
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q1a = -0.388d0
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q2a = 0.676d0
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q3a = 0.547d0
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t1a = -4.95d0
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t2a = 1.d0
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t3a = 0.31d0
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b0=adib*rs
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d2anti=(q1a+q2a*rs)*exp(-q3a*rs)/rs
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d3anti=(t1a+t2a*rs)*exp(-t3a*rs)/rs**2
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d2antid=-((q1a + q1a*q3a*rs + q2a*q3a*rs**2)/
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- rs**2)*exp(-q3a*rs)
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d3antid=-((rs*t2a*(1 + rs*t3a) + t1a*(2 + rs*t3a))/
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- rs**3)*exp(-rs*t3a)
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coe2=-3.d0/8.d0/rs**3*(1.d0-z**2)*(g0(rs)-0.5d0)
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coe2rs=-3.d0/8.d0/rs**3*(1.d0-z**2)*g0d(rs)+
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$ 9.d0/8.d0/rs**4*(1.d0-z**2)*(g0(rs)-0.5d0)
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coe2z=-3.d0/8.d0/rs**3*(-2.d0*z)*(g0(rs)-0.5d0)
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coe3=-(1.d0-z**2)*g0(rs)/(sqrt(2.d0*pi)*rs**3)
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coe3rs=-(1.d0-z**2)*g0d(rs)/(sqrt(2.d0*pi)*rs**3)+
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$ 3.d0*(1.d0-z**2)*g0(rs)/(sqrt(2.d0*pi)*rs**4)
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coe3z=2.d0*z*g0(rs)/(sqrt(2.d0*pi)*rs**3)
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if(abs(z).eq.1.d0) then
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coe4=-9.d0/64.d0/rs**3*(dpol(rs)
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$ -cf**2*2**(5.d0/3.d0)/5.d0/rs**2)
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coe4rs=-3.d0/rs*coe4-9.d0/64.d0/rs**3*(dpold(rs)
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$ +2.d0*cf**2*2**(5.d0/3.d0)/5.d0/rs**3)
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coe4z=-9.d0/64.d0/rs**3*(dpol(rs)-rs/6.d0*dpold(rs)-2.d0*d2anti
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$ -4.d0/15.d0/rs**2*cf**2*2.d0**(5.d0/3.d0))*z
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coe5=-9.d0/40.d0/(sqrt(2.d0*pi)*rs**3)*dpol(rs)
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coe5rs=-3.d0/rs*coe5-9.d0/40.d0/(sqrt(2.d0*pi)*rs**3)*dpold(rs)
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coe5z=-9.d0/40.d0/(sqrt(2.d0*pi)*rs**3)*(dpol(rs)-rs/6.d0*
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$ dpold(rs)-2.d0*d3anti)*z
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else
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coe4=-9.d0/64.d0/rs**3*(((1.d0+z)/2.d0)**2*
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$ dpol(rs*(2/(1.d0+z))**(1.d0/3.d0))+((1.d0-z)/2.d0)**2
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$ *dpol(rs*(2.d0/(1.d0-z))**(1.d0/3.d0))+
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$ (1.-z**2)*d2anti-cf**2/10.d0*((1.d0+z)**(8.d0/3.d0)
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$ +(1.-z)**(8.d0/3.d0))/rs**2)
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coe4rs=-3.d0/rs*coe4-9.d0/64.d0/rs**3*(
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$ ((1.d0+z)/2.d0)**(5.d0/3.d0)*dpold(rs*(2/(1.d0+z))**
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$ (1.d0/3.d0))+((1.d0-z)/2.d0)**(5.d0/3.d0)*
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$ dpold(rs*(2/(1.d0-z))**(1.d0/3.d0))+(1.d0-z**2)*
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$ d2antid+cf**2/5.d0*((1.d0+z)**(8.d0/3.d0)
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$ +(1.d0-z)**(8.d0/3.d0))/rs**3)
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coe4z=-9.d0/64.d0/rs**3*(1.d0/2.d0*(1.d0+z)*
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$ dpol(rs*(2/(1.d0+z))**(1.d0/3.d0))-1.d0/2.d0*(1.d0-z)*
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$ dpol(rs*(2/(1.d0-z))**(1.d0/3.d0))-rs/6.d0*
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$ ((1.d0+z)/2.d0)**(2.d0/3.d0)*dpold(rs*(2/(1.d0+z))
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$ **(1.d0/3.d0))+rs/6.d0*((1.d0-z)/2.d0)**(2.d0/3.d0)
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$ *dpold(rs*(2/(1.d0-z))**(1.d0/3.d0))-2.d0*z*d2anti-
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$ 4.d0/15.d0/rs**2*cf**2*((1.d0+z)**(5.d0/3.d0)-
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$ (1.d0-z)**(5.d0/3.d0)))
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coe5=-9.d0/40.d0/(sqrt(2.d0*pi)*rs**3)*(((1.d0+z)/2.d0)**2
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$ *dpol(rs*(2.d0/(1.d0+z))**(1.d0/3.d0))+((1.d0-z)/2.d0)**2
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$ *dpol(rs*(2.d0/(1.d0-z))**(1.d0/3.d0))+(1.d0-z**2)*
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$ d3anti)
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coe5rs=-3.d0/rs*coe5-9.d0/(40.d0*sqrt(2.d0*pi)*rs**3)*(
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$ ((1.d0+z)/2.d0)**(5.d0/3.d0)*dpold(rs*(2/(1.d0+z))**
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$ (1.d0/3.d0))+((1.d0-z)/2.d0)**(5.d0/3.d0)*
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$ dpold(rs*(2/(1.d0-z))**(1.d0/3.d0))+(1.d0-z**2)*
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$ d3antid)
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coe5z=-9.d0/40.d0/(sqrt(2.d0*pi)*rs**3)*(1.d0/2.d0*(1.d0+z)*
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$ dpol(rs*(2/(1.d0+z))**(1.d0/3.d0))-1.d0/2.d0*(1.d0-z)*
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$ dpol(rs*(2/(1.d0-z))**(1.d0/3.d0))-rs/6.d0*
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$ ((1.d0+z)/2.d0)**(2.d0/3.d0)*dpold(rs*(2/(1.d0+z))
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$ **(1.d0/3.d0))+rs/6.d0*((1.d0-z)/2.d0)**(2.d0/3.d0)
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$ *dpold(rs*(2/(1.d0-z))**(1.d0/3.d0))-2.d0*z*d3anti)
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endif
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call ecPW(rs,z,ec,ecd,ecz)
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a1=4.d0*b0**6*coe3+b0**8*coe5
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a1rs=24.d0*adib*b0**5*coe3+4.d0*b0**6*coe3rs+8.d0*adib*b0**7*
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$ coe5+b0**8*coe5rs
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a1z=4.d0*b0**6*coe3z+b0**8*coe5z
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a2=4.d0*b0**6*coe2+b0**8*coe4+6.d0*b0**4*ec
|
|
|
|
|
a2rs=24.d0*adib*b0**5*coe2+4.d0*b0**6*coe2rs+8.d0*adib*b0**7*
|
|
|
|
|
$ coe4+b0**8*coe4rs+24.d0*adib*b0**3*ec+6.d0*b0**4*ecd
|
|
|
|
|
a2z=4.d0*b0**6*coe2z+b0**8*coe4z+6.d0*b0**4*ecz
|
|
|
|
|
|
|
|
|
|
a3=b0**8*coe3
|
|
|
|
|
a3rs=8.d0*adib*b0**7*coe3+b0**8*coe3rs
|
|
|
|
|
a3z=b0**8*coe3z
|
|
|
|
|
|
|
|
|
|
a4=b0**6*(b0**2*coe2+4.d0*ec)
|
|
|
|
|
a4rs=8.d0*adib*b0**7*coe2+b0**8*coe2rs+24.d0*adib*b0**5*ec+
|
|
|
|
|
$ 4.d0*b0**6*ecd
|
|
|
|
|
a4z=b0**6*(b0**2*coe2z+4.d0*ecz)
|
|
|
|
|
|
|
|
|
|
a5=b0**8*ec
|
|
|
|
|
a5rs=8.d0*adib*b0**7*ec+b0**8*ecd
|
|
|
|
|
a5z=b0**8*ecz
|
|
|
|
|
|
|
|
|
|
x=mu*sqrt(rs)/phi
|
|
|
|
|
|
|
|
|
|
eclr=(phi**3*Qrpa(x)+a1*mu**3+a2*mu**4+a3*mu**5+
|
|
|
|
|
$ a4*mu**6+a5*mu**8)/((1.d0+b0**2*mu**2)**4)
|
|
|
|
|
|
|
|
|
|
eclrrs=-4.d0/(1.d0+b0**2*mu**2)*2.d0*adib*b0*mu**2*eclr+
|
|
|
|
|
$ 1.d0/((1.d0+b0**2*mu**2)**4)*(phi**2*mu/(2.d0*sqrt(rs))
|
|
|
|
|
$ *Qrpad(x)+
|
|
|
|
|
$ a1rs*mu**3+a2rs*mu**4+a3rs*mu**5+a4rs*mu**6+a5rs*mu**8)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
if(z.eq.1.d0) then
|
|
|
|
|
vclrup=eclr-rs/3.d0*eclrrs
|
|
|
|
|
vclrdown=0.d0
|
|
|
|
|
elseif(z.eq.-1.d0) then
|
|
|
|
|
vclrup=0.d0
|
|
|
|
|
vclrdown=eclr-rs/3.d0*eclrrs
|
|
|
|
|
else
|
|
|
|
|
|
|
|
|
|
eclrz=(phi**2*((1.d0+z)**(-1.d0/3.d0)-(1.d0-z)**(-1.d0/3.d0))
|
|
|
|
|
$ *Qrpa(x)-phi*Qrpad(x)*mu*sqrt(rs)*((1.d0+z)**(-1.d0/3.d0)
|
|
|
|
|
$ -(1.d0-z)**(-1.d0/3.d0))/3.d0+
|
|
|
|
|
$ a1z*mu**3+a2z*mu**4+a3z*mu**5+
|
|
|
|
|
$ a4z*mu**6+a5z*mu**8)/((1.d0+b0**2*mu**2)**4)
|
|
|
|
|
|
|
|
|
|
vclrup=eclr-rs/3.d0*eclrrs-(z-1.d0)*eclrz
|
|
|
|
|
vclrdown=eclr-rs/3.d0*eclrrs-(z+1.d0)*eclrz
|
|
|
|
|
endif
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
double precision function g0(x)
|
|
|
|
|
ccc on-top pair-distribution function
|
|
|
|
|
ccc Gori-Giorgi and Perdew, PRB 64, 155102 (2001)
|
|
|
|
|
ccc x -> rs
|
|
|
|
|
implicit none
|
|
|
|
|
double precision C0f,D0f,E0f,F0f,x
|
|
|
|
|
C0f = 0.0819306d0
|
|
|
|
|
D0f = 0.752411d0
|
|
|
|
|
E0f = -0.0127713d0
|
|
|
|
|
F0f = 0.00185898d0
|
|
|
|
|
g0=(1.d0-(0.7317d0-D0f)*x+C0f*x**2+E0f*x**3+
|
|
|
|
|
$ F0f*x**4)*exp(-abs(D0f)*x)/2.d0
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
double precision function g0d(rs)
|
|
|
|
|
ccc derivative of on-top pair-distribution function
|
|
|
|
|
ccc Gori-Giorgi and Perdew, PRB 64, 155102 (2001)
|
|
|
|
|
implicit none
|
|
|
|
|
double precision Bg0,Cg0,Dg0,Eg0,Fg0,rs
|
|
|
|
|
Cg0 = 0.0819306d0
|
|
|
|
|
Fg0 = 0.752411d0
|
|
|
|
|
Dg0 = -0.0127713d0
|
|
|
|
|
Eg0 = 0.00185898d0
|
|
|
|
|
Bg0 =0.7317d0-Fg0
|
|
|
|
|
g0d=(-Bg0+2*Cg0*rs+3*Dg0*rs**2+4*Eg0*rs**3)/2.d0*exp(-Fg0*rs)
|
|
|
|
|
- - (Fg0*(1 - Bg0*rs + Cg0*rs**2 + Dg0*rs**3 + Eg0*rs**4))/
|
|
|
|
|
- 2.d0*exp(-Fg0*rs)
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
double precision function dpol(rs)
|
|
|
|
|
implicit none
|
|
|
|
|
double precision cf,pi,rs,p2p,p3p
|
|
|
|
|
pi=dacos(-1.d0)
|
|
|
|
|
cf=(9.d0*pi/4.d0)**(1.d0/3.d0)
|
|
|
|
|
p2p = 0.04d0
|
|
|
|
|
p3p = 0.4319d0
|
|
|
|
|
dpol=2.d0**(5.d0/3.d0)/5.d0*cf**2/rs**2*(1.d0+(p3p-0.454555d0)*rs)
|
|
|
|
|
$ /(1.d0+p3p*rs+p2p*rs**2)
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
double precision function dpold(rs)
|
|
|
|
|
implicit none
|
|
|
|
|
double precision cf,pi,rs,p2p,p3p
|
|
|
|
|
pi=dacos(-1.d0)
|
|
|
|
|
cf=(9.d0*pi/4.d0)**(1.d0/3.d0)
|
|
|
|
|
p2p = 0.04d0
|
|
|
|
|
p3p = 0.4319d0
|
|
|
|
|
dpold=2.d0**(5.d0/3.d0)/5.d0*cf**2*
|
|
|
|
|
- (-2. + (0.454555 - 4.*p3p)*rs +
|
|
|
|
|
- (-4.*p2p +
|
|
|
|
|
- (0.90911 - 2.*p3p)*p3p)*rs**2
|
|
|
|
|
- + p2p*(1.363665 - 3.*p3p)*
|
|
|
|
|
- rs**3)/
|
|
|
|
|
- (rs**3*(1. + p3p*rs + p2p*rs**2)**2)
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
double precision function Qrpa(x)
|
|
|
|
|
implicit none
|
|
|
|
|
double precision pi,a2,b2,c2,d2,x,Acoul
|
|
|
|
|
pi=dacos(-1.d0)
|
|
|
|
|
Acoul=2.d0*(log(2.d0)-1.d0)/pi**2
|
|
|
|
|
a2 = 5.84605d0
|
|
|
|
|
c2 = 3.91744d0
|
|
|
|
|
d2 = 3.44851d0
|
|
|
|
|
b2=d2-3.d0/(2.d0*pi*Acoul)*(4.d0/(9.d0*pi))**(1.d0/3.d0)
|
|
|
|
|
Qrpa=Acoul*log((1.d0+a2*x+b2*x**2+c2*x**3)/(1.d0+a2*x+d2*x**2))
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
double precision function Qrpad(x)
|
|
|
|
|
implicit none
|
|
|
|
|
double precision pi,a2,b2,c2,d2,x,Acoul
|
|
|
|
|
pi=dacos(-1.d0)
|
|
|
|
|
Acoul=2.d0*(log(2.d0)-1.d0)/pi**2
|
|
|
|
|
a2 = 5.84605d0
|
|
|
|
|
c2 = 3.91744d0
|
|
|
|
|
d2 = 3.44851d0
|
|
|
|
|
b2=d2-3.d0/(2.d0*pi*Acoul)*(4.d0/(9.d0*pi))**(1.d0/3.d0)
|
|
|
|
|
Qrpad=Acoul*((x*(b2*(2.d0 + a2*x) +
|
|
|
|
|
- c2*x*(3.d0 + 2.d0*a2*x) +
|
|
|
|
|
- d2*(-2.d0 - a2*x + c2*x**3)))/
|
|
|
|
|
- ((1.d0 + a2*x + d2*x**2)*
|
|
|
|
|
- (1.d0 + a2*x + b2*x**2 + c2*x**3)))
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
subroutine exchangelr(rs,z,mu,exlr)
|
|
|
|
|
ccc Hartree atomic units used
|
|
|
|
|
ccc for given density parameter rs, spin polarization z
|
|
|
|
|
ccc and cutoff mu it gives the exchange energy of the LR gas
|
|
|
|
|
ccc => exlr
|
|
|
|
|
implicit none
|
|
|
|
|
double precision rs,z,mu,exlr
|
|
|
|
|
double precision pi,alpha,fx,y
|
|
|
|
|
pi=dacos(-1.d0)
|
|
|
|
|
alpha=(4.d0/9.d0/pi)**(1.d0/3.d0)
|
|
|
|
|
if(abs(z).eq.1.d0) then
|
|
|
|
|
y=mu*alpha*rs/2.d0/2.d0**(1.d0/3.d0)
|
|
|
|
|
fx=-((-3*y + 4*y**3 +(2*y - 4*y**3)*exp(-1./(4.*y**2)) +
|
|
|
|
|
$ sqrt(pi)*erf(1/(2.*y)))/pi)
|
|
|
|
|
exlr=mu*fx
|
|
|
|
|
else
|
|
|
|
|
y=mu*alpha*rs/2.d0/(1.+z)**(1.d0/3.d0)
|
|
|
|
|
fx=-((-3*y + 4*y**3 +(2*y - 4*y**3)*exp(-1./(4.*y**2)) +
|
|
|
|
|
$ sqrt(pi)*erf(1/(2.*y)))/pi)
|
|
|
|
|
exlr=(1.d0+z)*mu*fx/2.d0
|
|
|
|
|
y=mu*alpha*rs/2.d0/(1.-z)**(1.d0/3.d0)
|
|
|
|
|
fx=-((-3*y + 4*y**3 +(2*y - 4*y**3)*exp(-1./(4.*y**2)) +
|
|
|
|
|
$ sqrt(pi)*erf(1/(2.*y)))/pi)
|
|
|
|
|
exlr=exlr+(1.d0-z)*mu*fx/2.d0
|
|
|
|
|
endif
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
subroutine vexchangelr(rs,z,mu,vxlrup,vxlrdown)
|
|
|
|
|
ccc Hartree atomic units used
|
|
|
|
|
ccc for given density parameter rs, spin polarization z
|
|
|
|
|
ccc and cutoff mu it gives the exchange LSD potential for LR interaction
|
|
|
|
|
ccc => vxlrup (spin-up electrons), vxlrdown (spin-down electrons)
|
|
|
|
|
implicit none
|
|
|
|
|
double precision rs,z,mu,vxlrup,vxlrdown
|
|
|
|
|
double precision pi,alpha,fx,fx1,y,exlr,derrs,derz
|
|
|
|
|
pi=dacos(-1.d0)
|
|
|
|
|
alpha=(4.d0/9.d0/pi)**(1.d0/3.d0)
|
|
|
|
|
if(z.eq.1.d0) then
|
|
|
|
|
vxlrup=(rs*alpha*mu**2)/
|
|
|
|
|
- (2**(1.d0/3.d0)*pi) - (rs*alpha*mu**2)/(2**(1.d0/3.d0)*pi)*
|
|
|
|
|
- exp(-2**(2.d0/3.d0)/(rs**2*alpha**2*mu**2)) -
|
|
|
|
|
- (mu*erf(2**(1.d0/3.d0)/(rs*alpha*mu)))/sqrt(Pi)
|
|
|
|
|
vxlrdown=0.d0
|
|
|
|
|
elseif(z.eq.-1.d0) then
|
|
|
|
|
vxlrdown=(rs*alpha*mu**2)/
|
|
|
|
|
- (2**(1.d0/3.d0)*pi) - (rs*alpha*mu**2)/(2**(1.d0/3.d0)*pi)*
|
|
|
|
|
- exp(-2**(2.d0/3.d0)/(rs**2*alpha**2*mu**2)) -
|
|
|
|
|
- (mu*erf(2**(1.d0/3.d0)/(rs*alpha*mu)))/sqrt(Pi)
|
|
|
|
|
vxlrup=0.d0
|
|
|
|
|
else
|
|
|
|
|
y=mu*alpha*rs/2.d0/(1.+z)**(1.d0/3.d0)
|
|
|
|
|
fx=-((-3*y + 4*y**3 +(2*y - 4*y**3)*exp(-1./(4.*y**2)) +
|
|
|
|
|
$ sqrt(pi)*erf(1/(2.*y)))/pi)
|
|
|
|
|
fx1=(3.d0*(1 + (-4.d0 + 4.d0*exp(-1.d0/(4.d0*y**2)))*y**2))/pi
|
|
|
|
|
derrs=1.d0/4.d0*(1.d0+z)**(2.d0/3.d0)*mu**2*alpha*fx1
|
|
|
|
|
derz=1.d0/2.d0*mu*fx-1.d0/6.d0*fx1*mu*y
|
|
|
|
|
vxlrup=rs/3.d0*derrs+(z-1.d0)*derz
|
|
|
|
|
vxlrdown=rs/3.d0*derrs+(z+1.d0)*derz
|
|
|
|
|
|
|
|
|
|
y=mu*alpha*rs/2.d0/(1.-z)**(1.d0/3.d0)
|
|
|
|
|
fx=-((-3*y + 4*y**3 +(2*y - 4*y**3)*exp(-1./(4.*y**2)) +
|
|
|
|
|
$ sqrt(pi)*erf(1/(2.*y)))/pi)
|
|
|
|
|
fx1=(3.d0*(1 + (-4.d0 + 4.d0*exp(-1.d0/(4.d0*y**2)))*y**2))/pi
|
|
|
|
|
derrs=1.d0/4.d0*(1.d0-z)**(2.d0/3.d0)*mu**2*alpha*fx1
|
|
|
|
|
derz=-1.d0/2.d0*mu*fx+1.d0/6.d0*fx1*mu*y
|
|
|
|
|
vxlrup=vxlrup+rs/3.d0*derrs+(z-1.d0)*derz
|
|
|
|
|
vxlrdown=vxlrdown+rs/3.d0*derrs+(z+1.d0)*derz
|
|
|
|
|
|
|
|
|
|
call exchangelr(rs,z,mu,exlr)
|
|
|
|
|
vxlrup=exlr-vxlrup
|
|
|
|
|
vxlrdown=exlr-vxlrdown
|
|
|
|
|
endif
|
|
|
|
|
return
|
|
|
|
|
end
|
|
|
|
|
|
2019-07-09 23:09:32 +02:00
|
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
|
|
|
c correlation energy and its derivative w.r.t. rs and z at mu=infinity
|
|
|
|
|
c Perdew & Wang PRB 45, 13244 (1992)
|
|
|
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
|
|
|
subroutine ecPW(x,y,ec,ecd,ecz)
|
|
|
|
|
c in Hartree; ec=ec(rs,zeta)
|
|
|
|
|
c x -> rs; y -> zeta
|
|
|
|
|
ccc ecd is d/drs ec
|
|
|
|
|
ccc ecz is d/dz ec
|
|
|
|
|
implicit none
|
|
|
|
|
double precision pi,f02,ff,x,y,ec,ecd,ec0,ec0d,ec1,ec1d,
|
|
|
|
|
$ aaa,G,Gd,alfac,alfacd,ecz
|
|
|
|
|
pi=dacos(-1.d0)
|
|
|
|
|
|
|
|
|
|
f02=4.d0/(9.d0*(2.d0**(1.d0/3.d0)-1.d0))
|
|
|
|
|
|
|
|
|
|
ff=((1.d0+y)**(4.d0/3.d0)+(1.d0-y)**(4.d0/3.d0)-
|
|
|
|
|
$ 2.d0)/(2.d0**(4.d0/3.d0)-2.d0)
|
|
|
|
|
|
|
|
|
|
aaa=(1.d0-log(2.d0))/pi**2
|
|
|
|
|
call GPW(x,aaa,0.21370d0,7.5957d0,3.5876d0,
|
|
|
|
|
$ 1.6382d0,0.49294d0,G,Gd)
|
|
|
|
|
ec0=G
|
|
|
|
|
ec0d=Gd
|
|
|
|
|
|
|
|
|
|
aaa=aaa/2.d0
|
|
|
|
|
call GPW(x,aaa,0.20548d0,14.1189d0,6.1977d0,
|
|
|
|
|
$ 3.3662d0,0.62517d0,G,Gd)
|
|
|
|
|
ec1=G
|
|
|
|
|
ec1d=Gd
|
|
|
|
|
call GPW(x,0.016887d0,0.11125d0,10.357d0,3.6231d0,
|
|
|
|
|
$ 0.88026d0,0.49671d0,G,Gd)
|
|
|
|
|
alfac=-G
|
|
|
|
|
alfacd=-Gd
|
|
|
|
|
|
|
|
|
|
ec=ec0+alfac*ff/f02*(1.d0-y**4)+(ec1-ec0)*ff*y**4
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ecd=ec0d+alfacd*ff/f02*(1.d0-y**4)+(ec1d-ec0d)*
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$ ff*y**4
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ecz=alfac*(-4.d0*y**3)*ff/f02+alfac*(1.d0-y**4)/f02*
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$ 4.d0/3.d0*((1.d0+y)**(1.d0/3.d0)-(1.d0-y)**(1.d0/3.d0))/
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$ (2.d0**(4.d0/3.d0)-2.d0)+(ec1-ec0)*(4.d0*y**3*ff+
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$ 4.d0/3.d0*((1.d0+y)**(1.d0/3.d0)-(1.d0-y)**(1.d0/3.d0))/
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$ (2.d0**(4.d0/3.d0)-2.d0)*y**4)
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return
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end
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subroutine GPW(x,Ac,alfa1,beta1,beta2,beta3,beta4,G,Gd)
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ccc Gd is d/drs G
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implicit none
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double precision G,Gd,Ac,alfa1,beta1,beta2,beta3,beta4,x
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G=-2.d0*Ac*(1.d0+alfa1*x)*dlog(1.d0+1.d0/(2.d0*
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$ Ac*(beta1*x**0.5d0+
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$ beta2*x+beta3*x**1.5d0+beta4*x**2)))
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Gd=(1.d0+alfa1*x)*(beta2+beta1/(2.d0*sqrt(x))+3.d0*beta3*
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$ sqrt(x)/2.d0+2.d0*beta4*x)/((beta1*sqrt(x)+beta2*x+
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$ beta3*x**(3.d0/2.d0)+beta4*x**2)**2*(1.d0+1.d0/
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$ (2.d0*Ac*(beta1*sqrt(x)+beta2*x+beta3*x**(3.d0/2.d0)+
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$ beta4*x**2))))-2.d0*Ac*alfa1*dlog(1.d0+1.d0/(2.d0*Ac*
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$ (beta1*sqrt(x)+beta2*x+beta3*x**(3.d0/2.d0)+
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$ beta4*x**2)))
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return
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end
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ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
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ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
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