TABLE OF CONTENTS
ABINIT/m_xcpositron [ Modules ]
NAME
m_xcpositron
FUNCTION
Compute electron-positron correlation potentials and energy density.
COPYRIGHT
Copyright (C) 1998-2024 ABINIT group (GJ,MT) This file is distributed under the terms of the GNU General Public License, see ~abinit/COPYING or http://www.gnu.org/copyleft/gpl.txt .
SOURCE
16 #if defined HAVE_CONFIG_H 17 #include "config.h" 18 #endif 19 20 #include "abi_common.h" 21 22 module m_xcpositron 23 24 use defs_basis 25 use m_errors 26 use m_abicore 27 28 use m_numeric_tools, only : invcb 29 30 implicit none 31 32 private
ABINIT/xcpositron [ Functions ]
NAME
xcpositron
FUNCTION
Compute electron-positron correlation potentials and energy density. Used electron-positron correlation functional is controlled by ixcpositron argument. Returns Fxc, Vxc_pos, Vxc_el from input rhor_pos and rhor_el for positron and electrons.
INPUTS
grhoe2(ngr)=square of the gradient of electronic density rhoe (needed for GGA)
ixcpositron=type of electron-positron correlation functional:
1: LDA zero positron density limit parametrized by Arponen & Pajanne
and provided by Boronski & Nieminen [1,2]
11: LDA zero positron density limit parametrized by Arponen & Pajanne
and fitted by Sterne & Kaiser [1,3]
2: LDA electron-positron correlation
provided by Puska, Seitsonen, and Nieminen [1,4]
3: GGA zero positron density limit parametrized by Arponen & Pajanne
and provided by Boronski & Nieminen [1,2,5]
31: GGA zero positron density limit parametrized by Arponen & Pajanne
and fitted by Sterne & Kaiser [1,3,5]
See references below
ngr=size of grho2 array (0 if LDA, npt if GGA)
npt=number of real space points on which density is provided
posdensity0_limit=True if we are in the zero positron density limit
rhoer(npt)=electron density (bohr^-3)
rhopr(npt)=positron density (bohr^-3)
OUTPUT
fnxc(npt)=correlation energy per unit volume fxc vxce(npt)=correlation potential for electron dfxc/drhoe (hartree) vxcp(npt)=correlation potential for positron dfxc/drhop (hartree) vxcegr(ngr)= 1/|gradRhoe| dfxc/d|gradRhoe| (empty if LDA, i.e. ngr=0) Optional outputs: dvxce(npt)=partial second derivatives of the xc energy wr to the electronic density dvxce(:)=dVxce/dRhoe dvxcp(npt)=partial second derivatives of the xc energy wr to the positronic density dvxcp(:)=dVxcp/drhop
NOTES
References for electron-positron correlation functionals:
[1] J. Arponen and E. Pajanne, Ann. Phys. (N.Y.) 121, 343 (1979) [[cite:Arponen1979a]].
[2] E. Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986) [[cite:Boronski1986]].
[3] P.A. Sterne and J.H. Kaiser, Phys. Rev. B 43, 13892 (1991) [[cite:Sterne1991]].
[4] M.J. Puska, A.P. Seitsonen and R.M. Nieminen, Phys. Rev. B 52, 10947 (1994) [[cite:Puska1994]].
[5] B. Barbiellini, M.J. Puska, T. Torsti and R.M.Nieminen, Phys. Rev. B 51, 7341 (1995) [[cite:Barbiellini1995]]
SOURCE
91 subroutine xcpositron(fnxc,grhoe2,ixcpositron,ngr,npt,posdensity0_limit,rhoer,rhopr,vxce,vxcegr,vxcp,& 92 & dvxce,dvxcp) ! optional arguments 93 94 !Arguments ------------------------------------ 95 !scalars 96 integer,intent(in) :: ixcpositron,ngr,npt 97 logical,intent(in) :: posdensity0_limit 98 !arrays 99 real(dp),intent(in) :: grhoe2(ngr),rhoer(npt),rhopr(npt) 100 real(dp),intent(out) :: fnxc(npt),vxce(npt),vxcegr(ngr),vxcp(npt) 101 real(dp),intent(out),optional :: dvxce(npt),dvxcp(npt) 102 103 !Local variables------------------------------- 104 !scalars 105 integer,parameter :: idebug=0 106 integer :: ipt 107 logical :: gga,need_dvxce,need_dvxcp 108 real(dp),parameter :: alpha_gga=0.22_dp 109 real(dp),parameter :: ap_a1=-1.56_dp,ap_b1=0.051_dp,ap_c1=-0.081_dp,ap_d1=1.14_dp 110 real(dp),parameter :: ap_a2=-0.92305_dp,ap_b2=-0.05459_dp 111 real(dp),parameter :: ap_a3=-0.6298_dp,ap_b3=-13.15111_dp,ap_c3=2.8655_dp 112 real(dp),parameter :: ap_a4=-179856.2768_dp,ap_b4=186.4207_dp,ap_c4=-0.524_dp 113 real(dp),parameter :: ap_psn_limit=0.7_dp 114 real(dp),parameter :: ap_psn_1=0.9_dp*ap_psn_limit,ap_psn_2=1.1_dp*ap_psn_limit 115 real(dp),parameter :: fpi3=third*four_pi 116 real(dp),parameter :: psn_aa=69.7029_dp,psn_ba=-107.4927_dp,psn_bb=141.8458_dp 117 real(dp),parameter :: psn_ca=23.7182_dp,psn_cb=-33.6472_dp ,psn_cc=5.21152_dp 118 real(dp),parameter :: sk_a=-1.56_dp,sk_b=0.1324_dp,sk_c=-4.092_dp,sk_d=51.96_dp,sk_e=0.7207_dp 119 real(dp),parameter :: rsfac=0.6203504908994000_dp 120 real(dp) :: arse,brse,crse,darse,dbrse,dcrse,d2arse,d2brse,d2crse 121 real(dp) :: d2eps,deps,dexc,dexcdg,dexc_p,d2expgga,dexpgga,d2invrs,dinvrs,d2kf,dkf,d2nqtf2,dnqtf2 122 real(dp) :: drse,drsp,d2exc,d2exc_p,d2rse,d2rsp,d2sqr,dsqr 123 real(dp) :: eexp,eps,exc,exc_p,expgga,invf,dinvf,d2invf,invrhoe,invrhop,invrs,invsqr 124 real(dp) :: kf,logrs,nqtf2,opr2,ratio_ap,ratio_psn,rhoe,rhop,rse,rsp,sqr 125 character(len=500) :: msg 126 !arrays 127 real(dp),allocatable :: rsepts(:),rsppts(:) 128 129 ! ************************************************************************* 130 131 gga=(ngr==npt) 132 need_dvxce=present(dvxce) 133 need_dvxcp=present(dvxcp) 134 135 if (gga.and.ixcpositron==2) then 136 msg = 'xcpositron: GGA not yet implemented for ixcpositron=2 !' 137 ABI_ERROR(msg) 138 end if 139 if (posdensity0_limit.and.ixcpositron==2) then 140 msg = 'xcpositron: ixcpositron=2 cannot be treated in the zero positron density limit !' 141 ABI_ERROR(msg) 142 end if 143 if (abs(ixcpositron)/=1.and.ixcpositron/=11.and.ixcpositron/=2.and.ixcpositron/=3.and.ixcpositron/=31) then 144 msg = 'xcpositron: unknown electron-positron correlation !' 145 ABI_ERROR(msg) 146 end if 147 148 !Compute density radii for rhor_el, rhor_pos 149 ABI_MALLOC(rsepts,(npt)) 150 call invcb(rhoer(:),rsepts,npt) 151 rsepts(:)=rsfac*rsepts(:) 152 if (ixcpositron==2) then 153 ABI_MALLOC(rsppts,(npt)) 154 call invcb(rhopr(:),rsppts,npt) 155 rsppts(:)=rsfac*rsppts(:) 156 end if 157 158 !Loop over grid points 159 !---------------------------------------------------- 160 do ipt=1,npt 161 162 rhoe=rhoer(ipt) 163 rhop=rhopr(ipt) 164 exc=zero;dexc=zero;d2exc=zero;dexcdg=zero 165 166 rse=rsepts(ipt) 167 invrhoe=one/rhoe 168 drse=-third*rse*invrhoe 169 if (need_dvxce) d2rse= four/nine*rse*invrhoe**2 170 171 ! Arponen & Pajane parametrization for electron 172 if (ixcpositron/=11.and.ixcpositron/=31) then 173 if (rse<0.302_dp) then 174 invrs=one/rse;invsqr=sqrt(invrs);logrs=log(rse) 175 exc =ap_a1*invsqr+(ap_b1*logrs+ap_c1)*logrs+ap_d1 176 dexc=drse*invrs*(-half*ap_a1*invsqr+two*ap_b1*logrs+ap_c1) 177 if (need_dvxce) d2exc=(d2rse/drse-drse*invrs)*dexc+drse**2*invrs**2*(quarter*ap_a1*invsqr+two*ap_b1) 178 else if (rse>=0.302_dp.and.rse<=0.56_dp) then 179 invrs=one/rse 180 exc =ap_a2+ap_b2*invrs**2 181 dexc=-drse*ap_b2*two*invrs**3 182 if (need_dvxce) d2exc=d2rse/drse*dexc+six*drse**2*ap_b2*invrs**4 183 else if (rse>0.56_dp.and.rse<=8.0_dp) then 184 invrs=one/(rse+2.5_dp) 185 dinvrs=-drse*invrs**2 186 ! jmb : d2rse initialized only if need_dvxce = .True. 187 if (need_dvxce) d2invrs=-d2rse*invrs**2-two*invrs*drse**2 188 exc =ap_a3+ap_b3*invrs**2+ap_c3*invrs 189 dexc=two*ap_b3*invrs*dinvrs+ap_c3*dinvrs 190 if (need_dvxce) d2exc=two*ap_b3*dinvrs**2+(two*ap_b3*invrs+ap_c3)*d2invrs 191 else 192 exc =ap_a4*rhoe**2+ap_b4*rhoe+ap_c4 193 dexc =two*ap_a4*rhoe+ap_b4 194 if (need_dvxce) d2exc=two*ap_a4 195 end if 196 197 ! Sterne & Kaiser parametrization for electron 198 else 199 eexp=exp(-(rse+sk_c)**2/sk_d) 200 opr2=(one+rse**2) 201 arse=atan(rse) 202 exc = sk_a/sqrt(arse)+sk_b*eexp+sk_e 203 dexc= -(two*sk_b*eexp*(sk_c+rse)/sk_d + sk_a/(two*opr2*sqrt(arse)**3))*drse 204 if (need_dvxce) d2exc=-(two*sk_b*eexp*(sk_c+rse)/sk_d + sk_a/(two*opr2*arse**1.5_dp))*d2rse & 205 & +(two*sk_b*eexp*(two*sk_c**2-sk_d+four*sk_c*rse+two*rse**2)/sk_d**2 & 206 & +sk_a*(three+four*rse*arse)/(four*opr2**2*sqrt(arse)**5))*drse**2 207 end if 208 209 ! Puska, Seitsonen and Nieminen parametrization for positron 210 if (ixcpositron==2.and.rse>=ap_psn_1) then 211 rsp=rsppts(ipt) 212 invrhop=one/rhop 213 drsp=-third*rsp*invrhop 214 if (need_dvxcp) d2rsp= four/nine*rsp*invrhop**2 215 exc_p=zero;dexc_p=zero;d2exc_p=zero 216 if (rsp<0.302_dp) then 217 invrs=one/rsp;invsqr=sqrt(invrs);logrs=log(rsp) 218 exc_p =ap_a1*invsqr+(ap_b1*logrs+ap_c1)*logrs+ap_d1 219 dexc_p=drsp*invrs*(-half*ap_a1*invsqr+two*ap_b1*logrs+ap_c1) 220 if (need_dvxcp) d2exc_p=(d2rsp/drsp-drsp*invrs)*dexc_p+drsp**2*invrs**2*(quarter*ap_a1*invsqr+two*ap_b1) 221 else if (rsp>=0.302_dp.and.rsp<=0.56_dp) then 222 invrs=one/rsp 223 exc_p =ap_a2+ap_b2*invrs**2 224 dexc_p=-drsp*ap_b2*two*invrs**3 225 if (need_dvxcp) d2exc_p=d2rsp/drsp*dexc_p+six*drsp**2*ap_b2*invrs**4 226 else if (rsp>0.56_dp.and.rsp<=8.0_dp) then 227 invrs=one/(rsp+2.5_dp) 228 dinvrs=-drsp*invrs**2 229 ! jmb : d2rsp initialized only if need_dvxcp = .True.* 230 if (need_dvxcp) d2invrs=-d2rsp*invrs**2-two*invrs*drsp**2 231 exc_p =ap_a3+ap_b3*invrs**2+ap_c3*invrs 232 dexc_p=two*ap_b3*invrs*dinvrs+ap_c3*dinvrs 233 if (need_dvxcp) d2exc_p=two*ap_b3*dinvrs**2+(two*ap_b3*invrs+ap_c3)*d2invrs 234 else 235 exc_p =ap_a4*rhop**2+ap_b4*rhop+ap_c4 236 dexc_p =two*ap_a4*rhop+ap_b4 237 if (need_dvxcp) d2exc_p=two*ap_a4 238 end if 239 end if 240 241 ! GGA correction 242 if (gga) then 243 kf=(three*pi*pi*rhoe)**third 244 nqtf2=(rhoe*sqrt(four*kf/pi))**2 245 eps=grhoe2(ipt)/nqtf2 246 if (eps<zero) then 247 ABI_ERROR('xcpositron: problem, negative GGA espilon !') 248 end if 249 expgga=exp(-alpha_gga*eps*third) 250 251 dkf=pi*pi/(sqrt(three*pi*pi*rhoe)**third) 252 d2kf=-two*pi*pi*pi*pi*(three*pi*pi*rhoe)**(-5.0_dp/3.0_dp) 253 sqr=sqrt(four*kf/pi) 254 dsqr=(four*dkf/pi)/(two*sqr) 255 d2sqr=two/(pi*sqr*dkf)*(d2kf*sqr-dsqr*dkf) 256 nqtf2=(rhoe*sqr)**two 257 dnqtf2=two*(sqr+rhoe*dsqr)*rhoe*sqr 258 d2nqtf2=two*(rhoe*sqr*(two*dsqr+rhoe*d2sqr) & 259 & +sqr*(sqr+rhoe*dsqr) & 260 & +rhoe*(sqr+rhoe*dsqr) ) 261 deps=-grhoe2(ipt)*dnqtf2/(nqtf2**two) 262 d2eps=-grhoe2(ipt)/(nqtf2*nqtf2*dnqtf2)*(d2nqtf2*nqtf2*nqtf2-two*nqtf2*dnqtf2*dnqtf2) 263 dexpgga=-alpha_gga*third*deps*expgga 264 d2expgga=-alpha_gga*third*(d2eps*expgga+deps*dexpgga) 265 266 exc = exc *expgga 267 dexc=(dexc*expgga+exc*dexpgga) 268 if (need_dvxce) d2exc=d2exc*expgga+two*dexc*dexpgga+exc*d2expgga 269 if (abs(grhoe2(ipt))<1.e24_dp) dexcdg=-exc*alpha_gga*two_thirds/nqtf2 270 end if 271 272 ! Computation of XC energy, potentials and kernels 273 ! Zero positron density limit 274 if (ixcpositron/=2.or.rse<ap_psn_1) then 275 fnxc(ipt)=rhop*exc 276 vxce(ipt)=rhop*dexc 277 vxcp(ipt)=exc 278 if (need_dvxce) dvxce(ipt)=rhop*d2exc 279 if (need_dvxcp) dvxcp(ipt)=zero 280 if (gga) vxcegr(ipt)=rhop*dexcdg 281 else 282 ! Puska, Seitsonen and Nieminen functional 283 arse=psn_aa+psn_ba*rse+psn_ca*rse**2 284 brse=psn_ba+psn_bb*rse+psn_cb*rse**2 285 crse=psn_ca+psn_cb*rse+psn_cc*rse**2 286 darse=(psn_ba+two*psn_ca*rse)*drse 287 dbrse=(psn_bb+two*psn_cb*rse)*drse 288 dcrse=(psn_cb+two*psn_cc*rse)*drse 289 invf=arse+brse*rsp+crse*rsp**2+invrhop/exc+invrhoe/exc_p 290 fnxc(ipt)=one/invf 291 dinvf=darse+dbrse*rsp+dcrse*rsp**2-invrhop*dexc/exc**2-invrhoe**2/exc_p 292 vxce(ipt)=-dinvf/invf**2 293 if (need_dvxce) then 294 d2arse=darse*d2rse/drse+two*psn_ca*drse**2 295 d2brse=dbrse*d2rse/drse+two*psn_cb*drse**2 296 d2crse=dcrse*d2rse/drse+two*psn_cc*drse**2 297 d2invf=d2arse+d2brse*rsp+d2crse*rsp**2 & 298 & +invrhop*(two*dexc**2/exc-d2exc)/exc**2+two*invrhoe**3/exc_p 299 dvxce(ipt)=(two*dinvf**2/invf-d2invf)/invf**2 300 end if 301 dinvf=(brse+two*crse*rsp)*drsp-invrhop**2/exc-invrhoe*dexc_p/exc_p**2 302 vxcp(ipt)=-dinvf/invf**2 303 if (need_dvxcp) then 304 d2invf=two*crse*drsp+(brse+two*crse*rsp)*d2rsp & 305 & +two*invrhop**3/exc+invrhoe*(two*dexc_p**2/exc_p-d2exc_p)/exc_p**2 306 dvxcp(ipt)=(two*dinvf**2/invf-d2invf)/invf**2 307 end if 308 ! For small rse, use pure Arponen/Pajanne functional 309 ! Around the limit (rse=0.7, see PSN paper), switch smoothly from PSN to AP 310 if (rse>=ap_psn_1.and.rse<=ap_psn_2) then 311 ratio_psn=(rse-ap_psn_1)/(ap_psn_2-ap_psn_1);ratio_ap=one-ratio_psn 312 fnxc(ipt)=ratio_psn*fnxc(ipt)+ratio_ap*rhop*exc 313 vxce(ipt)=ratio_psn*vxce(ipt)+ratio_ap*rhop*dexc 314 vxcp(ipt)=ratio_psn*vxcp(ipt)+ratio_ap*exc 315 if (need_dvxce) dvxce(ipt)=ratio_psn*dvxce(ipt)+ratio_ap*rhop*d2exc 316 if (need_dvxcp) dvxcp(ipt)=ratio_psn*dvxcp(ipt) 317 end if 318 end if 319 320 ! Debug statements: use polynomial functionals 321 if (idebug>0) then 322 if (idebug==4) then ! order 4 323 fnxc(ipt)=tol3*((rhop**4+rhoe**4)/12._dp+(rhop**3*rhoe+rhop*rhoe**3)/3._dp+rhop**2*rhoe**2) 324 vxce(ipt)=tol3*((rhop**3*rhoe+rhop*rhoe**3)/3._dp+rhop**2*rhoe+rhop*rhoe**2) 325 vxcp(ipt)=tol3*((rhop**3*rhoe+rhop*rhoe**3)/3._dp+rhop**2*rhoe+rhop*rhoe**2) 326 if (need_dvxce) dvxce(ipt)=tol3*(rhop**3/3._dp+rhop*rhoe**2+rhop**2+two*rhop*rhoe) 327 if (need_dvxcp) dvxcp(ipt)=tol3*(rhoe**3/3._dp+rhoe*rhop**2+rhoe**2+two*rhop*rhoe) 328 end if 329 if (idebug==3) then ! order 3 330 fnxc(ipt)=tol3*((rhop**3+rhoe**3)*third+rhop**2*rhoe+rhop*rhoe**2) 331 vxce(ipt)=tol3*(rhop+rhoe)**2 332 vxcp(ipt)=tol3*(rhop+rhoe)**2 333 if (need_dvxce) dvxce(ipt)=tol3*two*rhoe 334 if (need_dvxcp) dvxcp(ipt)=tol3*two*rhop 335 end if 336 if (idebug==2) then ! order 2 337 fnxc(ipt)=tol3*(rhop+rhoe)**2 338 vxce(ipt)=tol3*two*(rhop+rhoe) 339 vxcp(ipt)=tol3*two*(rhop+rhoe) 340 if (need_dvxce) dvxce(ipt)=tol3*two 341 if (need_dvxcp) dvxcp(ipt)=tol3*two 342 end if 343 if (idebug==1) then ! order 1 344 fnxc(ipt)=tol3*(rhop+rhoe) 345 vxce(ipt)=tol3 346 vxcp(ipt)=tol3 347 if (need_dvxce) dvxce(ipt)=zero 348 if (need_dvxcp) dvxcp(ipt)=zero 349 end if 350 end if 351 352 end do ! ipt 353 354 ABI_FREE(rsepts) 355 if (ixcpositron==2) then 356 ABI_FREE(rsppts) 357 end if 358 359 !Convert everything in Hartree units 360 fnxc(:)=half*fnxc(:) 361 vxce(:)=half*vxce(:) 362 vxcp(:)=half*vxcp(:) 363 if (need_dvxce) dvxce(:)=half*dvxce(:) 364 if (need_dvxcp) dvxcp(:)=half*dvxcp(:) 365 if (gga) vxcegr(:)=half*vxcegr(:) 366 367 end subroutine xcpositron