# Differences

This shows you the differences between two versions of the page.

 — documentation:tutorials:nio_ligand_field:xas_l23_as_conductivity_tensor [2016/10/10 09:41] (current) 2016/10/09 15:48 Maurits W. Haverkort created 2016/10/09 15:48 Maurits W. Haverkort created Line 1: Line 1: + {{indexmenu_n>​13}} + ====== XAS $L_{2,3}$ as conductivity tensor ====== + ### + Absorption spectra are polarization dependent. In principle one can choose an infinite different number of polarizations. Calculating for each different experimental geometry (or polarization) a new spectrum is cumbersome and not needed. The material properties are given by the conductivity tensor. For dipole transitions a 3 by 3 matrix. The absorption spectra for a given experiment are then found by the relation: + + I(\omega,​\epsilon) = -\mathrm{Im}[\epsilon^* \cdot \sigma(\omega) \cdot \epsilon], + + with $\epsilon$ the polarization vector, $\omega$ the photon energy, $\sigma(\omega)$ the energy dependent conductivity tensor, and $I$ the measured intensity. Quanty can calculate the conductivity tensor. This is an extra option given to the function CreateSpectra (\{"​Tensor",​true\}). + ### + + ### + The example below calculates the conductivity tensor at the Ni $L_{2,3}$ edge. We show two different methods. The first calculates 9 spectra and by linear combining them retrieves the tensor. Method two uses a Block algorithm. ​ + ​ + -- here we calculate the 2p to 3d x-ray absorption of NiO within the Ligand-field theory + -- approximation. The first part of the script is very much the same as calculating + -- the ground-state with the addition that we now also need a 2p core shell in the basis + + -- from the previous example we know that within NiO there are 3 states close to each other + -- and then there is an energy gap of about 1 eV. We thus only need to consider the 3 + -- lowest states (Npsi=3 later on) + + -- the spectra are represented as a 3 by 3 tensor, the conductivity tensor. We show two + -- different methods to calculate this tensor, once creating 9 spectra with different + -- polarizations,​ once using the option Tensor in the CreateSpectra function + + NF=26 + NB=0 + IndexDn_2p={ 0, 2, 4} + IndexUp_2p={ 1, 3, 5} + IndexDn_3d={ 6, 8,10,12,14} + IndexUp_3d={ 7, 9,11,13,15} + IndexDn_Ld={16,​18,​20,​22,​24} + IndexUp_Ld={17,​19,​21,​23,​25} + + -- angular momentum operators on the d-shell + + OppSx_3d ​  ​=NewOperator("​Sx" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppSy_3d ​  ​=NewOperator("​Sy" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppSz_3d ​  ​=NewOperator("​Sz" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppSsqr_3d =NewOperator("​Ssqr"​ ,NF, IndexUp_3d, IndexDn_3d) + OppSplus_3d=NewOperator("​Splus",​NF,​ IndexUp_3d, IndexDn_3d) + OppSmin_3d =NewOperator("​Smin"​ ,NF, IndexUp_3d, IndexDn_3d) + + OppLx_3d ​  ​=NewOperator("​Lx" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppLy_3d ​  ​=NewOperator("​Ly" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppLz_3d ​  ​=NewOperator("​Lz" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppLsqr_3d =NewOperator("​Lsqr"​ ,NF, IndexUp_3d, IndexDn_3d) + OppLplus_3d=NewOperator("​Lplus",​NF,​ IndexUp_3d, IndexDn_3d) + OppLmin_3d =NewOperator("​Lmin"​ ,NF, IndexUp_3d, IndexDn_3d) + + OppJx_3d ​  ​=NewOperator("​Jx" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppJy_3d ​  ​=NewOperator("​Jy" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppJz_3d ​  ​=NewOperator("​Jz" ​  ,NF, IndexUp_3d, IndexDn_3d) + OppJsqr_3d =NewOperator("​Jsqr"​ ,NF, IndexUp_3d, IndexDn_3d) + OppJplus_3d=NewOperator("​Jplus",​NF,​ IndexUp_3d, IndexDn_3d) + OppJmin_3d =NewOperator("​Jmin"​ ,NF, IndexUp_3d, IndexDn_3d) + + Oppldots_3d=NewOperator("​ldots",​NF,​ IndexUp_3d, IndexDn_3d) + + -- Angular momentum operators on the Ligand shell + + OppSx_Ld ​  ​=NewOperator("​Sx" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppSy_Ld ​  ​=NewOperator("​Sy" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppSz_Ld ​  ​=NewOperator("​Sz" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppSsqr_Ld =NewOperator("​Ssqr"​ ,NF, IndexUp_Ld, IndexDn_Ld) + OppSplus_Ld=NewOperator("​Splus",​NF,​ IndexUp_Ld, IndexDn_Ld) + OppSmin_Ld =NewOperator("​Smin"​ ,NF, IndexUp_Ld, IndexDn_Ld) + + OppLx_Ld ​  ​=NewOperator("​Lx" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppLy_Ld ​  ​=NewOperator("​Ly" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppLz_Ld ​  ​=NewOperator("​Lz" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppLsqr_Ld =NewOperator("​Lsqr"​ ,NF, IndexUp_Ld, IndexDn_Ld) + OppLplus_Ld=NewOperator("​Lplus",​NF,​ IndexUp_Ld, IndexDn_Ld) + OppLmin_Ld =NewOperator("​Lmin"​ ,NF, IndexUp_Ld, IndexDn_Ld) + + OppJx_Ld ​  ​=NewOperator("​Jx" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppJy_Ld ​  ​=NewOperator("​Jy" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppJz_Ld ​  ​=NewOperator("​Jz" ​  ,NF, IndexUp_Ld, IndexDn_Ld) + OppJsqr_Ld =NewOperator("​Jsqr"​ ,NF, IndexUp_Ld, IndexDn_Ld) + OppJplus_Ld=NewOperator("​Jplus",​NF,​ IndexUp_Ld, IndexDn_Ld) + OppJmin_Ld =NewOperator("​Jmin"​ ,NF, IndexUp_Ld, IndexDn_Ld) + + -- total angular momentum + OppSx = OppSx_3d + OppSx_Ld + OppSy = OppSy_3d + OppSy_Ld + OppSz = OppSz_3d + OppSz_Ld + OppSsqr = OppSx * OppSx + OppSy * OppSy + OppSz * OppSz + OppLx = OppLx_3d + OppLx_Ld + OppLy = OppLy_3d + OppLy_Ld + OppLz = OppLz_3d + OppLz_Ld + OppLsqr = OppLx * OppLx + OppLy * OppLy + OppLz * OppLz + OppJx = OppJx_3d + OppJx_Ld + OppJy = OppJy_3d + OppJy_Ld + OppJz = OppJz_3d + OppJz_Ld + OppJsqr = OppJx * OppJx + OppJy * OppJy + OppJz * OppJz + + -- define the coulomb operator + -- we here define the part depending on F0 seperately from the part depending on F2 + -- when summing we can put in the numerical values of the slater integrals + + OppF0_3d =NewOperator("​U",​ NF, IndexUp_3d, IndexDn_3d, {1,0,0}) + OppF2_3d =NewOperator("​U",​ NF, IndexUp_3d, IndexDn_3d, {0,1,0}) + OppF4_3d =NewOperator("​U",​ NF, IndexUp_3d, IndexDn_3d, {0,0,1}) + + -- define onsite energies - crystal field + -- Akm = {{k1,​m1,​Akm1},​{k2,​m2,​Akm2},​ ... } + + Akm = PotentialExpandedOnClm("​Oh",​ 2, {0.6,-0.4}) + OpptenDq_3d = NewOperator("​CF",​ NF, IndexUp_3d, IndexDn_3d, Akm) + OpptenDq_Ld = NewOperator("​CF",​ NF, IndexUp_Ld, IndexDn_Ld, Akm) + + Akm = PotentialExpandedOnClm("​Oh",​ 2, {1,0}) + OppNeg_3d = NewOperator("​CF",​ NF, IndexUp_3d, IndexDn_3d, Akm) + OppNeg_Ld = NewOperator("​CF",​ NF, IndexUp_Ld, IndexDn_Ld, Akm) + Akm = PotentialExpandedOnClm("​Oh",​ 2, {0,1}) + OppNt2g_3d = NewOperator("​CF",​ NF, IndexUp_3d, IndexDn_3d, Akm) + OppNt2g_Ld = NewOperator("​CF",​ NF, IndexUp_Ld, IndexDn_Ld, Akm) + + OppNUp_2p = NewOperator("​Number",​ NF, IndexUp_2p, IndexUp_2p, {1,1,1}) + OppNDn_2p = NewOperator("​Number",​ NF, IndexDn_2p, IndexDn_2p, {1,1,1}) + OppN_2p = OppNUp_2p + OppNDn_2p + OppNUp_3d = NewOperator("​Number",​ NF, IndexUp_3d, IndexUp_3d, {1,​1,​1,​1,​1}) + OppNDn_3d = NewOperator("​Number",​ NF, IndexDn_3d, IndexDn_3d, {1,​1,​1,​1,​1}) + OppN_3d = OppNUp_3d + OppNDn_3d + OppNUp_Ld = NewOperator("​Number",​ NF, IndexUp_Ld, IndexUp_Ld, {1,​1,​1,​1,​1}) + OppNDn_Ld = NewOperator("​Number",​ NF, IndexDn_Ld, IndexDn_Ld, {1,​1,​1,​1,​1}) + OppN_Ld = OppNUp_Ld + OppNDn_Ld + + -- define L-d interaction + + Akm = PotentialExpandedOnClm("​Oh",​ 2, {1,0}) + OppVeg ​ = NewOperator("​CF",​ NF, IndexUp_3d, IndexDn_3d, IndexUp_Ld, IndexDn_Ld,​Akm) +  NewOperator("​CF",​ NF, IndexUp_Ld, IndexDn_Ld, IndexUp_3d, IndexDn_3d, Akm) + Akm = PotentialExpandedOnClm("​Oh",​ 2, {0,1}) + OppVt2g = NewOperator("​CF",​ NF, IndexUp_3d, IndexDn_3d, IndexUp_Ld, IndexDn_Ld,​Akm) +  NewOperator("​CF",​ NF, IndexUp_Ld, IndexDn_Ld, IndexUp_3d, IndexDn_3d, Akm) + + -- core valence interaction + + Oppcldots= NewOperator("​ldots",​ NF, IndexUp_2p, IndexDn_2p) + OppUpdF0 = NewOperator("​U",​ NF, IndexUp_2p, IndexDn_2p, IndexUp_3d, IndexDn_3d, {1,0}, {0,0}) + OppUpdF2 = NewOperator("​U",​ NF, IndexUp_2p, IndexDn_2p, IndexUp_3d, IndexDn_3d, {0,1}, {0,0}) + OppUpdG1 = NewOperator("​U",​ NF, IndexUp_2p, IndexDn_2p, IndexUp_3d, IndexDn_3d, {0,0}, {1,0}) + OppUpdG3 = NewOperator("​U",​ NF, IndexUp_2p, IndexDn_2p, IndexUp_3d, IndexDn_3d, {0,0}, {0,1}) + + -- dipole transition + + t=math.sqrt(1/​2) + + Akm = {{1,​-1,​t},​{1,​ 1,-t}} + TXASx = NewOperator("​CF",​ NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm) + Akm = {{1,​-1,​t*I},​{1,​ 1,t*I}} + TXASy = NewOperator("​CF",​ NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm) + Akm = {{1,0,1}} + TXASz = NewOperator("​CF",​ NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm) + + TXASr = t*(TXASx - I * TXASy) + TXASl =-t*(TXASx + I * TXASy) + + -- 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 + Hz      =  0.120 + + 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) + Hz * OppSz_3d + 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)+ Hz * OppSz_3d + 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 1111111111 0000000000",​8,​8},​ {"​111111 0000000000 1111111111",​16,​16}} + + psiList = Eigensystem(Hamiltonian,​ StartRestrictions,​ Npsi) + oppList={Hamiltonian,​ OppSsqr, OppLsqr, OppJsqr, 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(" ​ #    <​E> ​     <​S^2> ​   <​L^2> ​   <​J^2> ​   <​S_z^3d>​ <​L_z^3d>​ <​l.s> ​   <​F[2]> ​  <​F[4]> ​  <​Neg^3d>​ <​Nt2g^3d><​Neg^Ld>​ <​Nt2g^Ld><​N^3d>"​);​ + 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 + + -- calculating the spectra is simple and single line once all operators and wave-functions + -- are defined. + + --------------------------- Method 1 ----------------------------- + -- in order to create the tensor we define 9 spectra using operators that are combinations + -- of x, y and z polarized light + + TXASypz ​ =  sqrt(1/​2)*(TXASy + TXASz) ​ + TXASzpx ​ =  sqrt(1/​2)*(TXASz + TXASx) ​ + TXASxpy ​ =  sqrt(1/​2)*(TXASx + TXASy) ​ + TXASypiz =  sqrt(1/​2)*(TXASy + I * TXASz) ​ + TXASzpix =  sqrt(1/​2)*(TXASz + I * TXASx) ​ + TXASxpiy =  sqrt(1/​2)*(TXASx + I * TXASy) ​ + + TimeStart("​Mehtod1"​) + XASSpectra = CreateSpectra(XASHamiltonian,​ {TXASx,​TXASy,​TXASz,​TXASypz,​TXASzpx,​TXASxpy,​TXASypiz,​TXASzpix,​TXASxpiy},​ psiList[1], {{"​Emin",​-10},​ {"​Emax",​20},​ {"​NE",​3000},​ {"​Gamma",​0.1}}) + TimeEnd("​Mehtod1"​) + + -- Broaden these 9 spectra + TimeStart("​Broaden"​) + XASSpectra.Broaden(0.4,​ {{-3.7, 0.45}, {-2.2, 0.65}, { 0.0, 0.65}, { 8  , 0.80}, {13.2, 0.80}, {14.0, 1.075}, {16.0, 1.075}}) + TimeEnd("​Broaden"​) + + -- linear combine them into a tensor (note that the order here is given by the list of operators in the CreateSpectra function + + XASSigma_method1 = Spectra.Sum(XASSpectra,​{1,​0,​0, ​              ​0,​0,​0,​ 0, 0,0}, {-(1-I)/​2,​-(1-I)/​2,​0,​ 0,0,1, 0,0,-I}, {-(1+I)/​2,​0,​-(1+I)/​2,​ 0,1,0, 0,I,0 } + ,​{-(1+I)/​2,​-(1+I)/​2,​0,​ 0,0,1, 0, 0,I}, {0,1,0, 0,0,0, 0,​0,​0}, ​               {0,​-(1-I)/​2,​-(1-I)/​2,​ 1,0,0, -I,0,0} + ,​{-(1-I)/​2,​0,​-(1-I)/​2,​ 0,1,0, 0,-I,0}, {0,​-(1+I)/​2,​-(1+I)/​2,​ 1,0,0, I,0, 0}, {0,0,1, 0,0,0, 0,0,0}) + + XASSigma_method1.Print({{"​file","​XASSigma_method1.dat"​}}) + + -- prepare the gnuplot output for Sigma + gnuplotInput = [[ + set autoscale ​  # scale axes automatically + set xtic auto   # set xtics automatically + set ytic auto   # set ytics automatically + set style line  1 lt 1 lw 2 lc 1 + set style line  2 lt 1 lw 2 lc 3 + + set xlabel "E (eV)" font "​Times,​10"​ + set ylabel "​Intensity (arb. units)"​ font "​Times,​10"​ + + set yrange [-0.3:0.3] + + set out '​SigmaTensor_method1.ps'​ + set size 1.0, 1.0 + set terminal postscript portrait enhanced color  "​Times"​ 8 + + set multiplot layout 6, 3 + + plot "​XASSigma_method1.dat"​ u 1:2  title '​Re[{/​Symbol s}_{xx}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:3  title '​Im[{/​Symbol s}_{xx}]'​ with lines ls 2 + plot "​XASSigma_method1.dat"​ u 1:4  title '​Re[{/​Symbol s}_{xy}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:5  title '​Im[{/​Symbol s}_{xy}]'​ with lines ls 2 + plot "​XASSigma_method1.dat"​ u 1:6  title '​Re[{/​Symbol s}_{xz}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:7  title '​Im[{/​Symbol s}_{xz}]'​ with lines ls 2 + plot "​XASSigma_method1.dat"​ u 1:8  title '​Re[{/​Symbol s}_{yx}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:9  title '​Im[{/​Symbol s}_{yx}]'​ with lines ls 2 + plot "​XASSigma_method1.dat"​ u 1:10 title '​Re[{/​Symbol s}_{yy}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:11 title '​Im[{/​Symbol s}_{yy}]'​ with lines ls 2 + plot "​XASSigma_method1.dat"​ u 1:12 title '​Re[{/​Symbol s}_{yz}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:13 title '​Im[{/​Symbol s}_{yz}]'​ with lines ls 2 + plot "​XASSigma_method1.dat"​ u 1:14 title '​Re[{/​Symbol s}_{zx}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:15 title '​Im[{/​Symbol s}_{zx}]'​ with lines ls 2 + plot "​XASSigma_method1.dat"​ u 1:16 title '​Re[{/​Symbol s}_{zy}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:17 title '​Im[{/​Symbol s}_{zy}]'​ with lines ls 2 + plot "​XASSigma_method1.dat"​ u 1:18 title '​Re[{/​Symbol s}_{zz}]'​ with lines ls 1,\ + "​XASSigma_method1.dat"​ u 1:19 title '​Im[{/​Symbol s}_{zz}]'​ with lines ls 2 + + unset multiplot + ]] + + print("​Prepare gnuplot-file for Sigma"​) + + -- write the gnuplot script to a file + file = io.open("​SigmaTensor_method1.gnuplot",​ "​w"​) + file:​write(gnuplotInput) + file:​close() + + print(""​) + print("​Execute the gnuplot to produce plots and convert the output into a pdf-file"​) + + -- call gnuplot to execute the script + os.execute("​gnuplot SigmaTensor_method1.gnuplot ; ps2pdf SigmaTensor_method1.ps ; ps2eps SigmaTensor_method1.ps ;  mv SigmaTensor_method1.eps temp.eps ; eps2eps temp.eps SigmaTensor_method1.eps ; rm temp.eps"​) + + --------------------------- Method 2 ----------------------------- + + + TimeStart("​Mehtod2"​) + XASSigma_method2,​ SigmaTri = CreateSpectra(XASHamiltonian,​ {TXASx,​TXASy,​TXASz},​ psiList[1], {{"​Emin",​-10},​ {"​Emax",​20},​ {"​NE",​3000},​ {"​Gamma",​0.1},​ {"​Tensor",​true}}) + TimeEnd("​Mehtod2"​) + + -- Broaden these 9 spectra + TimeStart("​Broaden"​) + XASSigma_method2.Broaden(0.4,​ {{-3.7, 0.45}, {-2.2, 0.65}, { 0.0, 0.65}, { 8  , 0.80}, {13.2, 0.80}, {14.0, 1.075}, {16.0, 1.075}}) + TimeEnd("​Broaden"​) + + XASSigma_method2.Print({{"​file","​XASSigma_method2.dat"​}}) + + -- prepare the gnuplot output for Sigma + gnuplotInput = [[ + set autoscale ​  # scale axes automatically + set xtic auto   # set xtics automatically + set ytic auto   # set ytics automatically + set style line  1 lt 1 lw 2 lc 1 + set style line  2 lt 1 lw 2 lc 3 + + set xlabel "E (eV)" font "​Times,​10"​ + set ylabel "​Intensity (arb. units)"​ font "​Times,​10"​ + + set yrange [-0.3:0.3] + + set out '​SigmaTensor_method2.ps'​ + set size 1.0, 1.0 + set terminal postscript portrait enhanced color  "​Times"​ 8 + + set multiplot layout 6, 3 + + plot "​XASSigma_method2.dat"​ u 1:2  title '​Re[{/​Symbol s}_{xx}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:3  title '​Im[{/​Symbol s}_{xx}]'​ with lines ls 2 + plot "​XASSigma_method2.dat"​ u 1:4  title '​Re[{/​Symbol s}_{xy}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:5  title '​Im[{/​Symbol s}_{xy}]'​ with lines ls 2 + plot "​XASSigma_method2.dat"​ u 1:6  title '​Re[{/​Symbol s}_{xz}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:7  title '​Im[{/​Symbol s}_{xz}]'​ with lines ls 2 + plot "​XASSigma_method2.dat"​ u 1:8  title '​Re[{/​Symbol s}_{yx}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:9  title '​Im[{/​Symbol s}_{yx}]'​ with lines ls 2 + plot "​XASSigma_method2.dat"​ u 1:10 title '​Re[{/​Symbol s}_{yy}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:11 title '​Im[{/​Symbol s}_{yy}]'​ with lines ls 2 + plot "​XASSigma_method2.dat"​ u 1:12 title '​Re[{/​Symbol s}_{yz}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:13 title '​Im[{/​Symbol s}_{yz}]'​ with lines ls 2 + plot "​XASSigma_method2.dat"​ u 1:14 title '​Re[{/​Symbol s}_{zx}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:15 title '​Im[{/​Symbol s}_{zx}]'​ with lines ls 2 + plot "​XASSigma_method2.dat"​ u 1:16 title '​Re[{/​Symbol s}_{zy}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:17 title '​Im[{/​Symbol s}_{zy}]'​ with lines ls 2 + plot "​XASSigma_method2.dat"​ u 1:18 title '​Re[{/​Symbol s}_{zz}]'​ with lines ls 1,\ + "​XASSigma_method2.dat"​ u 1:19 title '​Im[{/​Symbol s}_{zz}]'​ with lines ls 2 + + unset multiplot + ]] + + print("​Prepare gnuplot-file for Sigma"​) + + -- write the gnuplot script to a file + file = io.open("​SigmaTensor_method2.gnuplot",​ "​w"​) + file:​write(gnuplotInput) + file:​close() + + print(""​) + print("​Execute the gnuplot to produce plots and convert the output into a pdf-file"​) + + -- call gnuplot to execute the script + os.execute("​gnuplot SigmaTensor_method2.gnuplot ; ps2pdf SigmaTensor_method2.ps ; ps2eps SigmaTensor_method2.ps ;  mv SigmaTensor_method2.eps temp.eps ; eps2eps temp.eps SigmaTensor_method2.eps ; rm temp.eps"​) + + -------------------------- difference ------------------------ + + XASSigma_diff = XASSigma_method2 - XASSigma_method1 + + + XASSigma_diff.Print({{"​file","​XASSigma_diff.dat"​}}) + + -- prepare the gnuplot output for Sigma + gnuplotInput = [[ + set autoscale ​  # scale axes automatically + set xtic auto   # set xtics automatically + set ytic auto   # set ytics automatically + set style line  1 lt 1 lw 2 lc 1 + set style line  2 lt 1 lw 2 lc 3 + + set xlabel "E (eV)" font "​Times,​10"​ + set ylabel "​Intensity (arb. units * 1000 000 000)" font "​Times,​10"​ + + set yrange [-0.3:0.3] + + scale = 1000000000 + + set out '​XASSigma_diff.ps'​ + set size 1.0, 1.0 + set terminal postscript portrait enhanced color  "​Times"​ 8 + + set multiplot layout 6, 3 + + plot "​XASSigma_diff.dat"​ u 1:​($2*scale) ​ title '​Re[{/​Symbol s}_{xx}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($3*scale) ​ title '​Im[{/​Symbol s}_{xx}]'​ with lines ls 2 + plot "​XASSigma_diff.dat"​ u 1:​($4*scale) ​ title '​Re[{/​Symbol s}_{xy}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($5*scale) ​ title '​Im[{/​Symbol s}_{xy}]'​ with lines ls 2 + plot "​XASSigma_diff.dat"​ u 1:​($6*scale) ​ title '​Re[{/​Symbol s}_{xz}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($7*scale) ​ title '​Im[{/​Symbol s}_{xz}]'​ with lines ls 2 + plot "​XASSigma_diff.dat"​ u 1:​($8*scale) ​ title '​Re[{/​Symbol s}_{yx}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($9*scale) ​ title '​Im[{/​Symbol s}_{yx}]'​ with lines ls 2 + plot "​XASSigma_diff.dat"​ u 1:​($10*scale) title '​Re[{/​Symbol s}_{yy}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($11*scale) title '​Im[{/​Symbol s}_{yy}]'​ with lines ls 2 + plot "​XASSigma_diff.dat"​ u 1:​($12*scale) title '​Re[{/​Symbol s}_{yz}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($13*scale) title '​Im[{/​Symbol s}_{yz}]'​ with lines ls 2 + plot "​XASSigma_diff.dat"​ u 1:​($14*scale) title '​Re[{/​Symbol s}_{zx}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($15*scale) title '​Im[{/​Symbol s}_{zx}]'​ with lines ls 2 + plot "​XASSigma_diff.dat"​ u 1:​($16*scale) title '​Re[{/​Symbol s}_{zy}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($17*scale) title '​Im[{/​Symbol s}_{zy}]'​ with lines ls 2 + plot "​XASSigma_diff.dat"​ u 1:​($18*scale) title '​Re[{/​Symbol s}_{zz}]'​ with lines ls 1,\ + "​XASSigma_diff.dat"​ u 1:​($19*scale) title '​Im[{/​Symbol s}_{zz}]'​ with lines ls 2 + + unset multiplot + ]] + + print("​Prepare gnuplot-file for Sigma"​) + + -- write the gnuplot script to a file + file = io.open("​XASSigma_diff.gnuplot",​ "​w"​) + file:​write(gnuplotInput) + file:​close() + + print(""​) + print("​Execute the gnuplot to produce plots and convert the output into a pdf-file"​) + + -- call gnuplot to execute the script + os.execute("​gnuplot XASSigma_diff.gnuplot ; ps2pdf XASSigma_diff.ps ; ps2eps XASSigma_diff.ps ;  mv XASSigma_diff.eps temp.eps ; eps2eps temp.eps XASSigma_diff.eps ; rm temp.eps"​) + + + ---------------- overview of timing ------------------- + TimePrint() + ​ + ### + + ### + The resulting spectra are for method 1 are: + | {{:​documentation:​tutorials:​nio_ligand_field:​sigmatensor_method1.png?​nolink |}} | + ^ $2p$ to $3d$ excitations for all possible polarizations represented as a conductivity tensor. Calculated using 9 spectra of different rotated polarization and linear combining. ^ + ### + + ### + The resulting spectra are for method 2 are: + | {{:​documentation:​tutorials:​nio_ligand_field:​sigmatensor_method2.png?​nolink |}} | + ^ $2p$ to $3d$ excitations for all possible polarizations represented as a conductivity tensor. Calculated using a block Lanczos method. ^ + ### + + ### + The difference is: + | {{:​documentation:​tutorials:​nio_ligand_field:​xassigma_diff.png?​nolink |}} | + ^ Difference between the conductivity tensor calculated using method 1 or 2 ^ + ### + + ### + The output of the script is: + ​ + Start of BlockGroundState. Converge 3 states to an energy with relative variance smaller than  1.490116119384766E-06 + + Start of BlockOperatorPsiSerialRestricted + Outer loop   1, Number of Determinants: ​       45        45 last variance ​ 1.159112471523181E+00 + Start of BlockOperatorPsiSerialRestricted + Start of BlockGroundState. Converge 3 states to an energy with relative variance smaller than  1.490116119384766E-06 + + Start of BlockOperatorPsiSerial + Outer loop   1, Number of Determinants: ​       32       100 last variance ​ 8.359599622001442E+00 + Start of BlockOperatorPsiSerial + Outer loop   2, Number of Determinants: ​      ​100 ​      138 last variance ​ 1.315915512292445E+00 + Start of BlockOperatorPsiSerial + #    <​E> ​     <​S^2> ​   <​L^2> ​   <​J^2> ​   <​S_z^3d>​ <​L_z^3d>​ <​l.s> ​   <​F[2]> ​  <​F[4]> ​  <​Neg^3d>​ <​Nt2g^3d><​Neg^Ld>​ <​Nt2g^Ld><​N^3d>​ + 1   ​-3.503 ​   1.999   ​12.000 ​  ​15.095 ​  ​-0.908 ​  ​-0.281 ​  ​-0.305 ​  ​-1.042 ​  ​-0.924 ​   2.186    5.990    3.825    6.000    8.175 + 2   ​-3.395 ​   1.999   ​12.000 ​  ​15.160 ​  ​-0.004 ​  ​-0.002 ​  ​-0.322 ​  ​-1.043 ​  ​-0.925 ​   2.189    5.988    3.823    6.000    8.178 + 3   ​-3.286 ​   1.999   ​12.000 ​  ​15.211 ​   0.903    0.278   ​-0.336 ​  ​-1.043 ​  ​-0.925 ​   2.193    5.987    3.820    6.000    8.180 + Start of LanczosTriDiagonalizeRR + Start of LanczosTriDiagonalizeRC + Start of LanczosTriDiagonalizeRR + Start of LanczosTriDiagonalizeRC + Start of LanczosTriDiagonalizeRR + Start of LanczosTriDiagonalizeRC + Start of LanczosTriDiagonalizeRC + Start of LanczosTriDiagonalizeRC + Start of LanczosTriDiagonalizeRC + Spectra printed to file: XASSigma_method1.dat + Prepare gnuplot-file for Sigma + + Execute the gnuplot to produce plots and convert the output into a pdf-file + Start of LanczosBlockTriDiagonalize + Start of LanczosBlockTriDiagonalizeRC + Spectra printed to file: XASSigma_method2.dat + Prepare gnuplot-file for Sigma + + Execute the gnuplot to produce plots and convert the output into a pdf-file + Spectra printed to file: XASSigma_diff.dat + Prepare gnuplot-file for Sigma + + Execute the gnuplot to produce plots and convert the output into a pdf-file + Timing results + ​Total_time | NumberOfRuns | Running | Name + 0:00:05 |            1 |       0 | Mehtod1 + 0:00:17 |            2 |       0 | Broaden + 0:00:02 |            1 |       0 | Mehtod2 + ​ + ### + + + ===== Table of contents ===== + {{indexmenu>​.#​1|msort}}