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documentation:tutorials:nio_crystal_field:xas_l23_as_conductivity_tensor [2016/10/08 21:27] – created Maurits W. Haverkortdocumentation:tutorials:nio_crystal_field:xas_l23_as_conductivity_tensor [2018/03/21 10:19] (current) – Stefano Agrestini
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 +{{indexmenu_n>12}}
 +====== 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>
 +-- This tutorial calculates the 2p to 3d x-ray absorption spectra of Ni in NiO using
 +-- crystal field theory
 +
 +-- 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
 +
 +-- Within crystal-field theory the solid is approximated by a single atom in an effective 
 +-- potential. Although an extremely crude approximation it is useful for some cases.
 +-- For correlated transition metal insulators it captures the right symmetry of the 
 +-- localized open d-shell. It is useful to determine magnetic g-factors, energies of d-d
 +-- excitations or core level x-ray absorption. (2p to 3d excitations L23 edges) 
 +
 +-- One should notice that the effective crystal-field potential is an affective potential
 +-- it is there to mimic the interaction with neighboring ligand atoms. In real materials
 +-- there do not exist such large electro static potentials.
 +
 +-- In order to do crystal-field theory for NiO we need to define a Ni d-shell.
 +-- A d-shell has 10 elements and we label again the even spin-orbitals to be spin down
 +-- and the odd spin-orbitals to be spin up. In order to calculate 2p to 3d excitations we
 +-- also need a Ni 2p shell. We thus have a total of 10+6=16 fermions, 6 Ni-2p and 10 Ni-3d
 +-- spin-orbitals
 +
 +NF=16
 +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}
 +
 +-- We define several operators acting on the Ni -3d shell
 +
 +OppSx   =NewOperator("Sx"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppSy   =NewOperator("Sy"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppSz   =NewOperator("Sz"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppSsqr =NewOperator("Ssqr" ,NF, IndexUp_3d, IndexDn_3d)
 +OppSplus=NewOperator("Splus",NF, IndexUp_3d, IndexDn_3d)
 +OppSmin =NewOperator("Smin" ,NF, IndexUp_3d, IndexDn_3d)
 +
 +OppLx   =NewOperator("Lx"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppLy   =NewOperator("Ly"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppLz   =NewOperator("Lz"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppLsqr =NewOperator("Lsqr" ,NF, IndexUp_3d, IndexDn_3d)
 +OppLplus=NewOperator("Lplus",NF, IndexUp_3d, IndexDn_3d)
 +OppLmin =NewOperator("Lmin" ,NF, IndexUp_3d, IndexDn_3d)
 +
 +OppJx   =NewOperator("Jx"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppJy   =NewOperator("Jy"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppJz   =NewOperator("Jz"   ,NF, IndexUp_3d, IndexDn_3d)
 +OppJsqr =NewOperator("Jsqr" ,NF, IndexUp_3d, IndexDn_3d)
 +OppJplus=NewOperator("Jplus",NF, IndexUp_3d, IndexDn_3d)
 +OppJmin =NewOperator("Jmin" ,NF, IndexUp_3d, IndexDn_3d)
 +
 +Oppldots=NewOperator("ldots",NF, IndexUp_3d, IndexDn_3d)
 +
 +-- The Coulomb interaction
 +
 +OppF0 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {1,0,0})
 +OppF2 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {0,1,0})
 +OppF4 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {0,0,1})
 +
 +-- The crystal-field operator
 +
 +Akm = PotentialExpandedOnClm("Oh",2,{0.6,-0.4})
 +OpptenDq = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
 +
 +-- Count the number of eg and t2g electrons
 +
 +Akm = PotentialExpandedOnClm("Oh",2,{1,0})
 +OppNeg = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
 +Akm = PotentialExpandedOnClm("Oh",2,{0,1})
 +OppNt2g = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
 +
 +-- new for core level spectroscopy are operators that define the interaction acting on the
 +-- Ni-2p shell. There is actually only one of these interactions, which is the Ni-2p
 +-- spin-orbit interaction
 +
 +Oppcldots= NewOperator("ldots", NF, IndexUp_2p, IndexDn_2p)
 +
 +-- we also need to define the Coulomb interaction between the Ni 2p- and Ni 3d-shell
 +-- Again the interaction (e^2/(|r_i-r_j|)) is expanded on spherical harmonics. For the interaction
 +-- between two shells we need to consider two cases. For the direct interaction a 2p electron
 +-- scatters of a 3d electron into a 2p and 3d electron. The radial integrals involve
 +-- the square of a 2p radial wave function at coordinate 1 and the square of a 3d radial
 +-- wave function at coordinate 2. The transfer of angular momentum can either be 0 or 2.
 +-- These processes are called direct and the resulting Slater integrals are F[0] and F[2].
 +-- The second proces involves a 2p electron scattering of a 3d electron into the 3d shell
 +-- and at the same time the 3d electron scattering into a 2p shell. These exchange processes
 +-- involve radial integrals over the product of a 2p and 3d radial wave function. The transfer
 +-- of angular momentum in this case can be 1 or 3 and the Slater integrals are called G1 and G3.
 +
 +-- In Quanty you can enter these processes by labeling 4 indices for the orbitals, once
 +-- the 2p shell with spin up, 2p shell with spin down, 3d shell with spin up and 3d shell with
 +-- spin down. Followed by the direct Slater integrals (F0 and F2) and the exchange Slater 
 +-- integrals (G1 and G3)
 +
 +-- Here we define the operators separately and later sum them with appropriate prefactors
 +
 +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})
 +
 +-- next we define the dipole operator. The dipole operator is given as epsilon.r
 +-- with epsilon the polarization vector of the light and r the unit position vector
 +-- We can expand the position vector on (renormalized) spherical harmonics and use
 +-- the crystal-field operator to create the dipole operator. 
 +
 +-- x polarized light is defined as x = Cos[phi]Sin[theta] = sqrt(1/2) ( C_1^{(-1)} - C_1^{(1)})
 +Akm = {{1,-1,sqrt(1/2)},{1, 1,-sqrt(1/2)}}
 +TXASx = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
 +-- y polarized light is defined as y = Sin[phi]Sin[theta] = sqrt(1/2) I ( C_1^{(-1)} + C_1^{(1)})
 +Akm = {{1,-1,sqrt(1/2)*I},{1, 1,sqrt(1/2)*I}}
 +TXASy = Chop(NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm))
 +-- z polarized light is defined as z = Cos[theta] = C_1^{(0)}
 +Akm = {{1,0,1}}
 +TXASz = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
 +
 +
 +-- once all operators are defined we can set some parameter values.
 +
 +-- the value of U drops out of a crystal-field calculation as the total number of electrons
 +-- is always the same
 +U       =  0.000 
 +-- F2 and F4 are often referred to in the literature as J_{Hund}. They represent the energy
 +-- differences between different multiplets. Numerical values can be found in the back of
 +-- my PhD. thesis for example. http://arxiv.org/abs/cond-mat/0505214 
 +F2dd    = 11.142 
 +F4dd    =  6.874
 +-- F0 is not the same as U, although they are related. Unimportant in crystal-field theory
 +-- the difference between U and F0 is so important that I do include it here. (U=0 so F0 is not)
 +F0dd    = U+(F2dd+F4dd)*2/63
 +-- in crystal field theory U drops out of the equation, also true for the interaction between the 
 +-- Ni 2p and Ni 3d electrons
 +Upd     =  0.000 
 +-- The Slater integrals between the 2p and 3d shell, again the numerical values can be found
 +-- in the back of my PhD. thesis. (http://arxiv.org/abs/cond-mat/0505214)
 +F2pd    =  6.667
 +G1pd    =  4.922
 +G3pd    =  2.796
 +-- F0 is not the same as U, although they are related. Unimportant in crystal-field theory
 +-- the difference between U and F0 is so important that I do include it here. (U=0 so F0 is not)
 +F0pd    =  Upd + G1pd*1/15 + G3pd*3/70
 +-- tenDq in NiO is 1.1 eV as can be seen in optics or using IXS to measure d-d excitations
 +tenDq   =  1.100
 +-- the Ni 3d spin-orbit is small but finite
 +zeta_3d =  0.081
 +-- the Ni 2p spin-orbit is very large and should not be scaled as theory is quite accurate here
 +zeta_2p = 11.498
 +-- we can add a small magnetic field, just to get nice expectation values. (units in eV... )
 +Bz      = 0.000001
 +-- In mean field theory the neighboring Ni sites give an effective potential acting on the
 +-- spin only when magnetically ordered. This exchange field in NiO is 6 J with J=27 meV.
 +Hex     = 6*0.027 -- see Europhys. Lett., 32 259 (1995) [ and Phys. Rev. B 82, 094403 (2010) ]
 +
 +-- the total Hamiltonian is the sum of the different operators multiplied with their prefactor
 +Hamiltonian = Chop(F0dd*OppF0 + F2dd*OppF2 + F4dd*OppF4 + tenDq*OpptenDq + zeta_3d*Oppldots + Bz*(2*OppSz+OppLz) + Hex*sqrt(1/6)*(OppSx+OppSy+2*OppSz))
 +
 +-- We normally do not include core-valence interactions between filed and partially filled 
 +-- shells as it drops out anyhow. For the XAS we thus need to define a "different" 
 +-- (more complete) Hamiltonian.
 +XASHamiltonian = Hamiltonian + zeta_2p * Oppcldots + F0pd * OppUpdF0 + F2pd * OppUpdF2 + G1pd * OppUpdG1 + G3pd * OppUpdG3
 +
 +-- We saw in the previous example that NiO has a ground-state doublet with occupation 
 +-- t2g^6 eg^2 and S=1 (S^2=S(S+1)=2). The next state is 1.1 eV higher in energy and thus
 +-- unimportant for the ground-state upto temperatures of 10 000 Kelvin. We thus restrict 
 +-- the calculation to the lowest 3 eigenstates.
 +Npsi=3
 +-- in order to make sure we have a filling of 8
 +-- electrons we need to define some restrictions
 +-- We need to restrict the occupation of the Ni-2p shell to 6 for the ground state and for
 +-- the Ni 3d-shell to 8.
 +StartRestrictions = {NF, NB, {"111111 0000000000",6,6}, {"000000 1111111111",8,8}}
 +
 +-- And calculate the lowest 3 eigenfunctions
 +psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi)
 +
 +-- In order to get some information on these eigenstates it is good to plot expectation values
 +-- We first define a list of all the operators we would like to calculate the expectation value of
 +oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSx, OppLx, OppSy, OppLy, OppSz, OppLz, Oppldots, OppF2, OppF4, OppNeg, OppNt2g};
 +
 +-- next we loop over all operators and all states and print the expectation value
 +print(" <E>    <S^2>  <L^2>  <J^2>  <S_x>  <L_x>  <S_y>  <L_y>  <S_z>  <L_z>  <l.s>  <F[2]> <F[4]> <Neg>  <Nt2g>");
 +for i = 1,#psiList do
 +  for j = 1,#oppList do
 +    expectationvalue = Chop(psiList[i]*oppList[j]*psiList[i])
 +    io.write(string.format("%6.3f ",Complex.Re(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_crystal_field:sigmatensor_method1.png?nolink |}}|
 +^ XAS spectra ($2p$ to $3d$) in form of a conductivity tensor ($\sigma$). For a particular polarization $\varepsilon$ the measured spectrum is $-\mathrm{Im}[\epsilon^* \cdot \sigma(\omega) \cdot \epsilon]$ ^
 +
 +###
 +The resulting spectra are for method 2 are:
 +###
 +|{{:documentation:tutorials:nio_crystal_field:sigmatensor_method2.png?nolink |}}|
 +^ XAS spectra ($2p$ to $3d$) in form of a conductivity tensor ($\sigma$). For a particular polarization $\varepsilon$ the measured spectrum is $-\mathrm{Im}[\epsilon^* \cdot \sigma(\omega) \cdot \epsilon]$ ^
 + 
 +###
 +The difference is:
 +###
 +|{{ :documentation:tutorials:nio_crystal_field:xassigma_diff.png?nolink |}}|
 +^ Difference between calculation with method 1 and method 2 (should be zero) ^
 +
 +###
 +The output of the script is:
 +<file Quanty_Output XAS_tensor.out>
 +text produced as output
 +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  2.335271462951932E+00
 +Start of BlockOperatorPsiSerialRestricted
 +Start of BlockGroundState. Converge 3 states to an energy with relative variance smaller than  1.490116119384766E-06
 +
 +Start of BlockOperatorPsiSerial
 + <E>    <S^2>  <L^2>  <J^2>  <S_x>  <L_x>  <S_y>  <L_y>  <S_z>  <L_z>  <l.s>  <F[2]> <F[4]> <Neg>  <Nt2g>
 +-2.882  1.999 12.000 15.052 -0.406 -0.117 -0.406 -0.117 -0.813 -0.234 -0.306 -1.020 -0.878  2.009  5.991 
 +-2.721  1.999 12.000 15.142 -0.001 -0.000 -0.001 -0.000 -0.002 -0.001 -0.330 -1.020 -0.878  2.011  5.989 
 +-2.560  1.999 12.000 15.211  0.405  0.116  0.405  0.116  0.810  0.233 -0.349 -1.020 -0.878  2.012  5.988 
 +Start of LanczosTriDiagonalizeMC
 +Start of LanczosTriDiagonalizeMC
 +Start of LanczosTriDiagonalizeMC
 +Start of LanczosTriDiagonalizeMC
 +Start of LanczosTriDiagonalizeMC
 +Start of LanczosTriDiagonalizeMC
 +Start of LanczosTriDiagonalizeMC
 +Start of LanczosTriDiagonalizeMC
 +Start of LanczosTriDiagonalizeMC
 +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 LanczosBlockTriDiagonalizeMC
 +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
 +      0:00:04 |            1 |       0 | Mehtod1
 +      0:00:16 |            2 |       0 | Broaden
 +      0:00:01 |            1 |       0 | Mehtod2
 +###
 +
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort}}

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