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—documentation:tutorials:nio_ligand_field:xas_l23_as_conductivity_tensor [2016/10/10 09:41] (current) – external edit 127.0.0.1
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 +{{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:
 +\begin{equation}
 +I(\omega,\epsilon) = -\mathrm{Im}[\epsilon^* \cdot \sigma(\omega) \cdot \epsilon],
 +\end{equation}
 +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. 
 +<code Quanty XAS_tensor.Quanty>
 +-- 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()
 +</code>
 +###
 +
 +###
 +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:
 +<file Quanty_Output XAS_tensor.out>
 +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
 +</file>
 +###
 +
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort}}

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