# Temperature

The effect of temperature can be added by calculating excited states and expectation values of excited states. The temperature dependent expectation value is then created using Boltzmann statistics.

A small example:

Temperature.Quanty
-- Sofar we calculated eigenstates and expectation values (or spectra) of these
-- eigenstates. At 0 K one would measure the expectation value of the lowest eigenstate
-- at finite temperature one would measure an average over several states weighted by
-- Boltzmann statistics. In this example we calculate the temperature dependent
-- x-ray absorption spectra of NiO. (Ni L23 edge 2p to 3d) within the ligand-field
-- theory approximation

-- The first part is an exact copy of example 41

Verbosity(0)

-- 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)

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

-- We calculate the x-ray absorption spectra for z, right circular and left circular polarized light for the 3 lowest eigen-states. (9 spectra in total)

XASSpectra = CreateSpectra(XASHamiltonian, {TXASz, TXASr, TXASl}, psiList, {{"Emin",-15}, {"Emax",25}, {"NE",2000}, {"Gamma",0.1}})

-- and put some additional energy broadening on it (0.4 Gaussian and energy dependent lorenzian)
XASSpectra.Broaden(0.4, {{-3.7, 0.45}, {-2.2, 0.65}, { 0.0, 0.65}, { 1.0, 2.00}, { 6  , 2.00}, { 8  , 0.80}, {13.2, 0.80}, {14.0, 0.90}, {16.0, 0.90}, {17.0, 2.00}})

-- and now we start to do things different in order to include temperature averaging
-- We have calculated three states. The energy of these states is:
Enlist = {}
for i=1,#psiList do
Enlist[i] = psiList[i] * Hamiltonian * psiList[i]
end
-- In order to calculate occupation numbers we need to calculate E^(-Energy/(kb T))
-- If the ground-state has an energy of -3.5 eV and we calculate this pre-factor at a temperature
-- of 10 Kelvin we get E^(3.5/(10*8.6*10^(-5))) = 8.3 * 10^(1763) a number so large it does
-- not fit in the computer memory. The solution is simple, we need to make sure the lowest
-- energy is zero.
for i=2,#psiList do
Enlist[i] = Enlist[i] - Enlist[1]
end
Enlist[1]=0

-- Besides the energy I would like to look at the magnetic moment, both spin and angular part
-- so also here we calculate the expectation values in a list
Szlist = {}
Lzlist = {}
for i=1,#psiList do
Szlist[i] = psiList[i] * OppSz_3d * psiList[i]
Lzlist[i] = psiList[i] * OppLz_3d * psiList[i]
end

-- We can now calcualte the temperature dependent expectation values:
p={}
print("Temperature, Total Energy,  Magnetic Moment, Sz,            Lz")
for T=1,1000,10 do
Z=0
SzT=0
LzT=0
ET=0
for i=1,#psiList do
p[i] = exp(-Enlist[i]/(EnergyUnits.Kelvin.value * T))
Z = Z + p[i]
SzT = SzT + p[i] * Szlist[i]
LzT = LzT + p[i] * Lzlist[i]
ET  = ET  + p[i] * Enlist[i]
end
SzT = SzT / Z
LzT = LzT / Z
MzT = -LzT - 2*SzT

io.write(string.format("%8i ",T))
io.write(string.format("%14.8f ",ET))
io.write(string.format("%14.8f   ",MzT))
io.write(string.format("%14.8f ",SzT))
io.write(string.format("%14.8f\n",LzT))
end

-- Note that in order to see the magnetic phase transition you need to make Hz in the Hamiltonian
-- temperature dependent. This can be done using self consistent loops. (The mathematica version
-- has an example of this, will make it here at some point as well)

-- The first column shows you how much energy you need to add to the system in order to heat
-- it. (specific heat) Be aware though that most of the specific heat is due to phonons, not
-- included in this calculation. It does capture nicely the electronic contribution to the
-- specific heat. (Including the change at cross overs for excited states (important in rare-
-- earth systems) and Lambda peaks for phase transitions

-- Now the temperature dependent XAS:

-- The object XASSpectra contains 9 spectra. For 3 different polarizations and 3 different states
-- What we need to do is to sum them according to Boltzmann statistics.

T = 100
Z=0
for i=1,#psiList do
p[i] = exp(-Enlist[i]/(EnergyUnits.Kelvin.value * T))
Z = Z + p[i]
end
-- we now create the Bolzmann summ for z, right and left polarized absorption (at T=100 Kelvin)
XASSpectraT100 = Spectra.Sum(XASSpectra,{p[1]/Z,p[2]/Z,p[3]/Z, 0,0,0, 0,0,0},{0,0,0, p[1]/Z,p[2]/Z,p[3]/Z, 0,0,0},{0,0,0, 0,0,0, p[1]/Z,p[2]/Z,p[3]/Z})

-- and the same at 1000 Kelvin
T = 1000
Z=0
for i=1,#psiList do
p[i] = exp(-Enlist[i]/(EnergyUnits.Kelvin.value * T))
Z = Z + p[i]
end
-- we now create the Bolzmann summ for z, right and left polarized absorption (at T=1000 Kelvin) (note again that we still have a large magnetization here as the exchange-field is not temperature dependent)
XASSpectraT1000 = Spectra.Sum(XASSpectra,{p[1]/Z,p[2]/Z,p[3]/Z, 0,0,0, 0,0,0},{0,0,0, p[1]/Z,p[2]/Z,p[3]/Z, 0,0,0},{0,0,0, 0,0,0, p[1]/Z,p[2]/Z,p[3]/Z})

-- We can print the spectra to file
XASSpectraT100.Print({{"file","XASSpectraT100.dat"}})
XASSpectraT1000.Print({{"file","XASSpectraT1000.dat"}})

gnuplotInput = [[
set autoscale
set xtic auto
set ytic auto
set style line  1 lt 1 lw 1 lc rgb "#0000FF"
set style line  2 lt 1 lw 1 lc rgb "#FF0000"
set style line  3 lt 1 lw 1 lc rgb "#00FF00"

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

set out 'XASSpecT.ps'
set size 1.0, 0.6
set terminal postscript portrait enhanced color  "Times" 12

energyshift=857.6
intensityscale=48
set xrange [847:877]

plot "XASSpectraT100.dat"  using ($1+energyshift):((-$3-$5-$7) * intensityscale) title 'isotropic theory T=100K' with lines ls 1,\
"XASSpectraT1000.dat" using ($1+energyshift):((-$3-$5-$7) * intensityscale) title 'isotropic theory T=1000K' with lines ls 2,\
"NiO_Experiment/XAS_L23_PRB_57_11623_1998" using 1:2 title 'isotropic experiment' with lines ls 3,\
"XASSpectraT100.dat"  using ($1+energyshift):(($5-$7) * intensityscale) title 'XMCD theory T=100K' with lines ls 1,\ "XASSpectraT1000.dat" using ($1+energyshift):(($5-$7) * intensityscale) title 'XMCD theory T=1000K' with lines ls 2

]]

-- write the gnuplot script to a file
file = io.open("XASSpecT.gnuplot", "w")
file:write(gnuplotInput)
file:close()

-- and finally call gnuplot to execute the script
os.execute("gnuplot XASSpecT.gnuplot")
-- as I like pdf to view and eps to include in the manuel I transform the format
os.execute(" ps2pdf XASSpecT.ps ; ps2eps XASSpecT.ps ;  mv XASSpecT.eps temp.eps ; eps2eps temp.eps XASSpecT.eps ; rm temp.eps")

The output is:

Temperature.out
  #    <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
Temperature, Total Energy,  Magnetic Moment, Sz,            Lz
1     0.00000000     2.09639244      -0.90751752    -0.28135740
11     0.00000000     2.09639244      -0.90751752    -0.28135740
21     0.00000000     2.09639244      -0.90751752    -0.28135740
31     0.00000000     2.09639244      -0.90751752    -0.28135740
41     0.00000000     2.09639244      -0.90751752    -0.28135740
51     0.00000000     2.09639244      -0.90751752    -0.28135740
61     0.00000000     2.09639244      -0.90751752    -0.28135740
71     0.00000000     2.09639240      -0.90751750    -0.28135739
81     0.00000002     2.09639208      -0.90751736    -0.28135735
91     0.00000011     2.09639041      -0.90751664    -0.28135713
101     0.00000042     2.09638445      -0.90751406    -0.28135633
111     0.00000128     2.09636784      -0.90750687    -0.28135410
121     0.00000327     2.09632960      -0.90749031    -0.28134898
131     0.00000724     2.09625331      -0.90745727    -0.28133877
141     0.00001431     2.09611725      -0.90739835    -0.28132054
151     0.00002587     2.09589513      -0.90730217    -0.28129080
161     0.00004345     2.09555736      -0.90715590    -0.28124556
171     0.00006869     2.09507255      -0.90694596    -0.28118063
181     0.00010326     2.09440895      -0.90665860    -0.28109175
191     0.00014878     2.09353581      -0.90628050    -0.28097481
201     0.00020678     2.09242436      -0.90579920    -0.28082596
211     0.00027865     2.09104862      -0.90520345    -0.28064171
221     0.00036565     2.08938584      -0.90448341    -0.28041902
231     0.00046884     2.08741683      -0.90363076    -0.28015531
241     0.00058915     2.08512592      -0.90263871    -0.27984850
251     0.00072732     2.08250098      -0.90150202    -0.27949695
261     0.00088395     2.07953318      -0.90021685    -0.27909949
271     0.00105950     2.07621676      -0.89878071    -0.27865534
281     0.00125430     2.07254875      -0.89719232    -0.27816410
291     0.00146857     2.06852868      -0.89545148    -0.27762573
301     0.00170243     2.06415828      -0.89355892    -0.27704043
311     0.00195590     2.05944119      -0.89151624    -0.27640872
321     0.00222896     2.05438273      -0.88932572    -0.27573129
331     0.00252149     2.04898959      -0.88699027    -0.27500905
341     0.00283333     2.04326969      -0.88451332    -0.27424305
351     0.00316429     2.03723192      -0.88189871    -0.27343450
361     0.00351411     2.03088598      -0.87915065    -0.27258468
371     0.00388252     2.02424225      -0.87627363    -0.27169500
381     0.00426920     2.01731161      -0.87327236    -0.27076690
391     0.00467384     2.01010537      -0.87015173    -0.26980191
401     0.00509608     2.00263511      -0.86691677    -0.26880157
411     0.00553555     1.99491262      -0.86357258    -0.26776747
421     0.00599187     1.98694984      -0.86012432    -0.26670120
431     0.00646466     1.97875875      -0.85657719    -0.26560437
441     0.00695351     1.97035131      -0.85293636    -0.26447859
451     0.00745802     1.96173945      -0.84920700    -0.26332544
461     0.00797777     1.95293497      -0.84539423    -0.26214651
471     0.00851235     1.94394955      -0.84150309    -0.26094337
481     0.00906134     1.93479468      -0.83753856    -0.25971755
491     0.00962431     1.92548164      -0.83350554    -0.25847056
501     0.01020084     1.91602149      -0.82940880    -0.25720389
511     0.01079052     1.90642502      -0.82525301    -0.25591899
521     0.01139292     1.89670276      -0.82104276    -0.25461725
531     0.01200762     1.88686496      -0.81678245    -0.25330005
541     0.01263421     1.87692156      -0.81247641    -0.25196873
551     0.01327227     1.86688221      -0.80812881    -0.25062458
561     0.01392140     1.85675621      -0.80374369    -0.24926884
571     0.01458119     1.84655259      -0.79932494    -0.24790272
581     0.01525123     1.83628003      -0.79487633    -0.24652738
591     0.01593114     1.82594688      -0.79040147    -0.24514394
601     0.01662053     1.81556118      -0.78590385    -0.24375348
611     0.01731900     1.80513065      -0.78138681    -0.24235703
621     0.01802619     1.79466269      -0.77685356    -0.24095557
631     0.01874172     1.78416436      -0.77230714    -0.23955007
641     0.01946523     1.77364244      -0.76775051    -0.23814142
651     0.02019636     1.76310338      -0.76318645    -0.23673048
661     0.02093476     1.75255334      -0.75861763    -0.23531809
671     0.02168008     1.74199819      -0.75404658    -0.23390503
681     0.02243199     1.73144349      -0.74947573    -0.23249203
691     0.02319017     1.72089454      -0.74490736    -0.23107982
701     0.02395428     1.71035636      -0.74034365    -0.22966906
711     0.02472402     1.69983370      -0.73578666    -0.22826038
721     0.02549907     1.68933105      -0.73123833    -0.22685440
731     0.02627915     1.67885267      -0.72670050    -0.22545167
741     0.02706395     1.66840255      -0.72217491    -0.22405274
751     0.02785320     1.65798447      -0.71766318    -0.22265810
761     0.02864662     1.64760198      -0.71316687    -0.22126824
771     0.02944393     1.63725840      -0.70868740    -0.21988360
781     0.03024489     1.62695686      -0.70422614    -0.21850459
791     0.03104922     1.61670029      -0.69978434    -0.21713161
801     0.03185669     1.60649141      -0.69536319    -0.21576503
811     0.03266706     1.59633277      -0.69096380    -0.21440517
821     0.03348008     1.58622675      -0.68658719    -0.21305237
831     0.03429555     1.57617555      -0.68223432    -0.21170691
841     0.03511323     1.56618121      -0.67790606    -0.21036908
851     0.03593291     1.55624561      -0.67360325    -0.20903911
861     0.03675440     1.54637050      -0.66932663    -0.20771725
871     0.03757748     1.53655749      -0.66507689    -0.20640371
881     0.03840196     1.52680804      -0.66085468    -0.20509868
891     0.03922767     1.51712348      -0.65666057    -0.20380234
901     0.04005441     1.50750505      -0.65249509    -0.20251486
911     0.04088201     1.49795385      -0.64835873    -0.20123639
921     0.04171030     1.48847088      -0.64425191    -0.19996706
931     0.04253913     1.47905702      -0.64017502    -0.19870698
941     0.04336832     1.46971308      -0.63612841    -0.19745627
951     0.04419774     1.46043977      -0.63211237    -0.19621502
961     0.04502722     1.45123768      -0.62812719    -0.19498331
971     0.04585664     1.44210737      -0.62417308    -0.19376121
981     0.04668585     1.43304928      -0.62025025    -0.19254878
991     0.04751472     1.42406379      -0.61635886    -0.19134607