-- This example calculates the fluorescence yield spectra for NiO (L23M45, i.e. 2p to -- 3d excitation and decay from 3s to 2p) within the ligand field theory approximation -- We use the definitions of all operators and basis orbitals as defined in the file -- 44_50_include and can afterwards directly continue by creating the Hamiltonian -- and calculating the spectra dofile("Include.Quanty") -- The parameters and scheme needed is the same as the one used for XAS -- We follow the energy definitions as introduced in the group of G.A. Sawatzky (Groningen) -- J. Zaanen, G.A. Sawatzky, and J.W. Allen PRL 55, 418 (1985) -- for parameters of specific materials see -- A.E. Bockquet et al. PRB 55, 1161 (1996) -- After some initial discussion the energies U and Delta refer to the center of a configuration -- The L^10 d^n configuration has an energy 0 -- The L^9 d^n+1 configuration has an energy Delta -- The L^8 d^n+2 configuration has an energy 2*Delta+Udd -- -- If we relate this to the onsite energy of the L and d orbitals we find -- 10 eL + n ed + n(n-1) U/2 == 0 -- 9 eL + (n+1) ed + (n+1)n U/2 == Delta -- 8 eL + (n+2) ed + (n+1)(n+2) U/2 == 2*Delta+U -- 3 equations with 2 unknowns, but with interdependence yield: -- ed = (10*Delta-nd*(19+nd)*U/2)/(10+nd) -- eL = nd*((1+nd)*Udd/2-Delta)/(10+nd) -- -- For the final state we/they defined -- The 2p^5 L^10 d^n+1 configuration has an energy 0 -- The 2p^5 L^9 d^n+2 configuration has an energy Delta + Udd - Upd -- The 2p^5 L^8 d^n+3 configuration has an energy 2*Delta + 3*Udd - 2*Upd -- -- If we relate this to the onsite energy of the p and d orbitals we find -- 6 ep + 10 eL + n ed + n(n-1) Udd/2 + 6 n Upd == 0 -- 6 ep + 9 eL + (n+1) ed + (n+1)n Udd/2 + 6 (n+1) Upd == Delta -- 6 ep + 8 eL + (n+2) ed + (n+1)(n+2) Udd/2 + 6 (n+2) Upd == 2*Delta+Udd -- 5 ep + 10 eL + (n+1) ed + (n+1)(n) Udd/2 + 5 (n+1) Upd == 0 -- 5 ep + 9 eL + (n+2) ed + (n+2)(n+1) Udd/2 + 5 (n+2) Upd == Delta+Udd-Upd -- 5 ep + 8 eL + (n+3) ed + (n+3)(n+2) Udd/2 + 5 (n+3) Upd == 2*Delta+3*Udd-2*Upd -- 6 equations with 3 unknowns, but with interdependence yield: -- epfinal = (10*Delta + (1+nd)*(nd*Udd/2-(10+nd)*Upd) / (16+nd) -- edfinal = (10*Delta - nd*(31+nd)*Udd/2-90*Upd) / (16+nd) -- eLfinal = ((1+nd)*(nd*Udd/2+6*Upd)-(6+nd)*Delta) / (16+nd) -- -- -- -- note that ed-ep = Delta - nd * U and not Delta -- note furthermore that ep and ed here are defined for the onsite energy if the system had -- locally nd electrons in the d-shell. In DFT or Hartree Fock the d occupation is in the end not -- nd and thus the onsite energy of the Kohn-Sham orbitals is not equal to ep and ed in model -- calculations. -- -- note furthermore that ep and eL actually should be different for most systems. We happily ignore this fact -- -- We normally take U and Delta as experimentally determined parameters -- number of electrons (formal valence) nd = 8 -- parameters from experiment (core level PES) Udd = 7.3 Upd = 8.5 Delta = 4.7 -- parameters obtained from DFT (PRB 85, 165113 (2012)) F2dd = 11.14 F4dd = 6.87 F2pd = 6.67 G1pd = 4.92 G3pd = 2.80 tenDq = 0.56 tenDqL = 1.44 Veg = 2.06 Vt2g = 1.21 zeta_3d = 0.081 zeta_2p = 11.51 Bz = 0.000001 H112 = 0 ed = (10*Delta-nd*(19+nd)*Udd/2)/(10+nd) eL = nd*((1+nd)*Udd/2-Delta)/(10+nd) epfinal = (10*Delta + (1+nd)*(nd*Udd/2-(10+nd)*Upd)) / (16+nd) edfinal = (10*Delta - nd*(31+nd)*Udd/2-90*Upd) / (16+nd) eLfinal = ((1+nd)*(nd*Udd/2+6*Upd) - (6+nd)*Delta) / (16+nd) F0dd = Udd + (F2dd+F4dd) * 2/63 F0pd = Upd + (1/15)*G1pd + (3/70)*G3pd Hamiltonian = F0dd*OppF0_3d + F2dd*OppF2_3d + F4dd*OppF4_3d + zeta_3d*Oppldots_3d + Bz*(2*OppSz_3d + OppLz_3d) + H112 * (OppSx_3d+OppSy_3d+2*OppSz_3d)/sqrt(6) + tenDq*OpptenDq_3d + tenDqL*OpptenDq_Ld + Veg * OppVeg + Vt2g * OppVt2g + ed * OppN_3d + eL * OppN_Ld XASHamiltonian = F0dd*OppF0_3d + F2dd*OppF2_3d + F4dd*OppF4_3d + zeta_3d*Oppldots_3d + Bz*(2*OppSz_3d + OppLz_3d)+ H112 * (OppSx_3d+OppSy_3d+2*OppSz_3d)/sqrt(6) + tenDq*OpptenDq_3d + tenDqL*OpptenDq_Ld + Veg * OppVeg + Vt2g * OppVt2g + edfinal * OppN_3d + eLfinal * OppN_Ld + epfinal * OppN_2p + zeta_2p * Oppcldots + F0pd * OppUpdF0 + F2pd * OppUpdF2 + G1pd * OppUpdG1 + G3pd * OppUpdG3 -- we now can create the lowest Npsi eigenstates: Npsi=3 -- in order to make sure we have a filling of 8 electrons we need to define some restrictions StartRestrictions = {NF, NB, {"000000 00 1111111111 0000000000",8,8}, {"111111 11 0000000000 1111111111",18,18}} psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi) oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSx_3d, OppLx_3d, OppSy_3d, OppLy_3d, OppSz_3d, OppLz_3d, Oppldots_3d, OppF2_3d, OppF4_3d, OppNeg_3d, OppNt2g_3d, OppNeg_Ld, OppNt2g_Ld, OppN_3d} -- print of some expectation values print(" # "); for i = 1,#psiList do io.write(string.format("%3i ",i)) for j = 1,#oppList do expectationvalue = Chop(psiList[i]*oppList[j]*psiList[i]) io.write(string.format("%8.3f ",expectationvalue)) end io.write("\n") end -- here we calculate the x-ray absorption spectra, not the main task of this file, but nice to do so we can compare XASSpectra = CreateSpectra(XASHamiltonian, {T2p3dz, T2p3dr, T2p3dl}, psiList, {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}}); XASSpectra.Print({{"file","FYL23M1_XAS.dat"}}); -- and we calculate the FY spectra FYSpectra = CreateFluorescenceYield(XASHamiltonian, {T2p3dz, T2p3dr, T2p3dl}, {T3s2px, T3s2py, T3s2pz}, psiList, {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}}) FYSpectra.Print({{"file","FYL23M1_Spec.dat"}}) -- in order to plot both the XAS and FY spectra we can define a gnuplot script gnuplotInput = [[ set autoscale set xtic auto set ytic auto set style line 1 lt 1 lw 1 lc rgb "#000000" set style line 2 lt 1 lw 1 lc rgb "#FF0000" set style line 3 lt 1 lw 1 lc rgb "#00FF00" set style line 4 lt 1 lw 1 lc rgb "#0000FF" set xlabel "E (eV)" font "Times,10" set ylabel "Intensity (arb. units)" font "Times,10" set out 'FYL23M1.ps' set terminal postscript portrait enhanced color "Times" 8 size 7.5,6 set yrange [0:0.6] set size 1,1 set multiplot layout 3, 3 plot "FYL23M1_XAS.dat" u 1:(-$3 ) title 'z-polarized Sz=-1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$2) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$4) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$6) title 'FY - z out' with lines ls 4 plot "FYL23M1_XAS.dat" u 1:(-$5 ) title 'z-polarized Sz= 0' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$8) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$10) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$12) title 'FY - z out' with lines ls 4 plot "FYL23M1_XAS.dat" u 1:(-$7 ) title 'z-polarized Sz= 1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$14) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$16) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$18) title 'FY - z out' with lines ls 4 plot "FYL23M1_XAS.dat" u 1:(-$9 ) title 'r-polarized Sz=-1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$20) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$22) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$24) title 'FY - z out' with lines ls 4 plot "FYL23M1_XAS.dat" u 1:(-$11) title 'r-polarized Sz= 0' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$26) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$28) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$30) title 'FY - z out' with lines ls 4 plot "FYL23M1_XAS.dat" u 1:(-$13) title 'r-polarized Sz= 1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$32) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$34) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$36) title 'FY - z out' with lines ls 4 plot "FYL23M1_XAS.dat" u 1:(-$15) title 'l-polarized Sz=-1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$38) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$40) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$42) title 'FY - z out' with lines ls 4 plot "FYL23M1_XAS.dat" u 1:(-$17) title 'l-polarized Sz= 0' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$44) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$46) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$48) title 'FY - z out' with lines ls 4 plot "FYL23M1_XAS.dat" u 1:(-$19) title 'l-polarized Sz= 1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ "FYL23M1_Spec.dat" u 1:(4*$50) title 'FY - x out' with lines ls 2,\ "FYL23M1_Spec.dat" u 1:(4*$52) title 'FY - y out' with lines ls 3,\ "FYL23M1_Spec.dat" u 1:(4*$54) title 'FY - z out' with lines ls 4 unset multiplot ]] -- write the gnuplot script to a file file = io.open("FYL23M1.gnuplot", "w") file:write(gnuplotInput) file:close() -- call gnuplot to execute the script os.execute("gnuplot FYL23M1.gnuplot") -- and change the ps to pdf and eps