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documentation:tutorials:nio_ligand_field:xas_l23_as_conductivity_tensor [2016/10/10 09:41] (current)
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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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