Verbosity(0)
 
-- here we calculate the 1s 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 1s 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)
 
NF=22
NB=0
IndexDn_1s={ 0}
IndexUp_1s={ 1}
IndexDn_3d={ 2, 4, 6, 8,10}
IndexUp_3d={ 3, 5, 7, 9,11}
IndexDn_Ld={12,14,16,18,20}
IndexUp_Ld={13,15,17,19,21}
 
-- 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_1s = NewOperator("Number", NF, IndexUp_1s, IndexUp_1s, {1})
OppNDn_1s = NewOperator("Number", NF, IndexDn_1s, IndexDn_1s, {1})
OppN_1s = OppNUp_1s + OppNDn_1s
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
OppUsdF0 = NewOperator("U", NF, IndexUp_1s, IndexDn_1s, IndexUp_3d, IndexDn_3d, {1}, {0})
OppUsdG2 = NewOperator("U", NF, IndexUp_1s, IndexDn_1s, IndexUp_3d, IndexDn_3d, {0}, {1})

-- next we define the Quadrupole operator. The quadrupole operator is given as epsilon.r * k.r
-- with epsilon the polarization vector of the light, r the unit position vector and
-- k the momentum vector of the light
-- We can expand the position vector on (renormalized) spherical harmonics and use
-- the crystal-field operator to create the quadrupole operator. 
-- Note that a polarisation in the x direction and momentum in the y direction gives the 
-- same operator as for the case where the polarisation is in the y direction and the 
-- momentum is in the x direction

-- There are 5 possible quadrupole operators and all other polarisations and k directions
-- can be made from linear combinations
-- yz "polarisation"
Akm = {{2,-1,I*sqrt(1/2)},{2, 1,I*sqrt(1/2)}}
TXASyz = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_1s, IndexDn_1s, Akm)
-- xz "polarisation"
Akm = {{2,-1,  sqrt(1/2)},{2, 1, -sqrt(1/2)}}
TXASxz = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_1s, IndexDn_1s, Akm)
-- xy "polarisation"
Akm = {{2,-2,I*sqrt(1/2)},{2, 2,-I*sqrt(1/2)}}
TXASxy = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_1s, IndexDn_1s, Akm)
-- 3z-r^2 "polarisation"
Akm = {{2,0,1}}
TXASz2 = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_1s, IndexDn_1s, Akm)
-- x^2-y^2 "polarisation"
Akm = {{2,-2,  sqrt(1/2)},{2, 2,  sqrt(1/2)}}
TXASx2y2 = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_1s, IndexDn_1s, Akm)

-- once all operators are defined we can set some parameter values.

-- 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 1s^1 L^10 d^n+1 configuration has an energy 0
-- The 1s^1 L^9  d^n+2 configuration has an energy Delta + Udd - Usd
-- The 1s^1 L^8  d^n+3 configuration has an energy 2*Delta + 3*Udd - 2*Usd
--
-- If we relate this to the onsite energy of the s and d orbitals we find
-- 2 es + 10 eL +  n    ed + n(n-1)     Udd/2 + 2 n     Usd == 0
-- 2 es +  9 eL + (n+1) ed + (n+1)n     Udd/2 + 2 (n+1) Usd == Delta
-- 2 es +  8 eL + (n+2) ed + (n+1)(n+2) Udd/2 + 2 (n+2) Usd == 2*Delta+Udd
-- 1 es + 10 eL + (n+1) ed + (n+1)(n)   Udd/2 + 1 (n+1) Usd == 0
-- 1 es +  9 eL + (n+2) ed + (n+2)(n+1) Udd/2 + 1 (n+2) Usd == Delta+Udd-Usd
-- 1 es +  8 eL + (n+3) ed + (n+3)(n+2) Udd/2 + 1 (n+3) Usd == 2*Delta+3*Udd-2*Usd
-- 6 equations with 3 unknowns, but with interdependence yield:
-- esfinal = (10*Delta + (1+nd)*(nd*Udd/2-(10+nd)*Usd)) / (12+nd)
-- edfinal = (10*Delta - nd*(23+nd)*Udd/2-44*Usd) / (12+nd)
-- eLfinal = ((1+nd)*(nd*Udd/2+2*Usd)-(2+nd)*Delta) / (12+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
Usd     =  1.0
Delta   =  4.7
-- parameters obtained from DFT (PRB 85, 165113 (2012))
F2dd    = 11.14 
F4dd    =  6.87
G2sd    =  0.057
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)
 
esfinal = (10*Delta + (1+nd)*(nd*Udd/2-(10+nd)*Usd)) / (12+nd)
edfinal = (10*Delta - nd*(23+nd)*Udd/2-44*Usd) / (12+nd)
eLfinal = ((1+nd)*(nd*Udd/2+2*Usd)-(2+nd)*Delta) / (12+nd)
 
F0dd    = Udd + (F2dd+F4dd) * 2/63
F0sd    =  Usd + G2sd*1/10
 
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 + esfinal * OppN_1s + F0sd * OppUsdF0 + G2sd * OppUsdG2 
 
-- 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, {"00 1111111111 0000000000",8,8}, {"11 0000000000 1111111111",12,12}}
 
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.
XASSpectra = CreateSpectra(XASHamiltonian, {TXASyz,TXASxz,TXASxy,TXASz2,TXASx2y2}, psiList, {{"Emin",-5}, {"Emax",10}, {"NE",3000}, {"Gamma",0.1}})
-- XASSpectra will contain 15 spectra in the end. We have 3 functions in psiList and 5 
-- polarizations. We furthermore define the energy range to be from -5 to +10
-- using 3000 energy points (delta E = 10 meV) and an internal lifetime during the calculation
-- of 100 meV (Full width Half maximum)

-- we can use several operations on a spectrum object. For one we can sum several spectra
-- the function Spectra.Sum sums the spectra with the prefactors given in the list afterwards
-- sum the spectra of the ground-state for different polarizations in order to get the
-- isotropic spectrum
XASIsoSpectra = Spectra.Sum(XASSpectra,{1,0,0, 1,0,0, 1,0,0, 1,0,0, 1,0,0})
-- broaden the spectra, note that a method changes the spectrum object, here we broaden by
-- a Gaussian of FWHM 0.4 and Lorenzian of FWHM 1.0
XASIsoSpectra.Broaden(0.4,1.0)

-- in order to plot the spectra we write them to file in ASCII format
-- note that the calculated object, i.e. <psi | T^dag 1/(w-H+IG/2) T | psi>
-- is complex. Minus the imaginary part is the absorption.
XASSpectra.Print({{"file","XASSpec.dat"}})
XASIsoSpectra.Print({{"file","XASIsoSpec.dat"}})

-- in order to plot we use gnuplot. You can use any program, but gnuplot is nice as it can
-- be called from within Quanty. We first define the gnuplot script as a string inside Quanty
gnuplotInput = [[
set autoscale 
set xtic auto 
set ytic auto 
set style line  1 lt 1 lw 1 lc rgb "#000000"
set style line  2 lt 1 lw 1 lc rgb "#FF0000"
set style line  3 lt 1 lw 3 lc rgb "#000000"

set xlabel "E (eV)" font "Times,12"
set ylabel "Intensity (arb. units)" font "Times,8"

set out 'XASSpec.ps'
set size 1.0, 1.0
set terminal postscript landscape enhanced color  "Times" 8

set multiplot layout 5, 3

plot "XASSpec.dat"  u 1:(-$3 ) title '  yz-polarized Sz=-1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$5 ) title '  yz-polarized Sz= 0' with lines ls  1
plot "XASSpec.dat"  u 1:(-$7 ) title '  yz-polarized Sz= 1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$9 ) title '  xz-polarized Sz=-1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$11) title '  xz-polarized Sz= 0' with lines ls  1
plot "XASSpec.dat"  u 1:(-$13) title '  xz-polarized Sz= 1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$15) title '  xy-polarized Sz=-1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$17) title '  xy-polarized Sz= 0' with lines ls  1
plot "XASSpec.dat"  u 1:(-$19) title '  xy-polarized Sz= 1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$21) title 'x2y2-polarized Sz=-1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$23) title 'x2y2-polarized Sz= 0' with lines ls  1
plot "XASSpec.dat"  u 1:(-$25) title 'x2y2-polarized Sz= 1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$27) title '  z2-polarized Sz=-1' with lines ls  1
plot "XASSpec.dat"  u 1:(-$29) title '  z2-polarized Sz= 0' with lines ls  1
plot "XASSpec.dat"  u 1:(-$31) title '  z2-polarized Sz= 1' with lines ls  1

unset multiplot

energyshift=8331
intensityscale=64

plot "XASSpec.dat" using ($1+energyshift):((-$3-$9-$15-$21-$27) * intensityscale) title 'isotropic theory' with lines ls 1


set size 1.0, 0.6
intensityscale=48
set out 'XASIsoSpec.ps'

plot "XASIsoSpec.dat" using ($1+energyshift):((-$3) * intensityscale) title 'isotropic theory' with lines ls 3

]]

-- next we save this string to a file
file = io.open("XASSpec.gnuplot", "w")
file:write(gnuplotInput)
file:close()

-- and finally call gnuplot to execute the script
os.execute("gnuplot XASSpec.gnuplot")
-- as I like pdf to view I transform the format
os.execute(" ps2pdf XASSpec.ps ; ps2pdf XASIsoSpec.ps")