-- 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 ",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("Method1")
XASSpectra = CreateSpectra(XASHamiltonian, {TXASx,TXASy,TXASz,TXASypz,TXASzpx,TXASxpy,TXASypiz,TXASzpix,TXASxpiy}, psiList[1], {{"Emin",-10}, {"Emax",20}, {"NE",3000}, {"Gamma",0.1}})
TimeEnd("Method1")

-- 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")

--------------------------- Method 2 -----------------------------


TimeStart("Method2")
XASSigma_method2, SigmaTri = CreateSpectra(XASHamiltonian, {TXASx,TXASy,TXASz}, psiList[1], {{"Emin",-10}, {"Emax",20}, {"NE",3000}, {"Gamma",0.1}, {"Tensor",true}})
TimeEnd("Method2")

-- 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")

-------------------------- 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")


---------------- overview of timing -------------------
TimePrint()