-- 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")

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

-------------------------- 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()
