# Energy level diagram

In order to do temperature averaging it is important to understand the number of excited states that are important. One can learn a lot by looking at the energy level diagram. Here we plot one for Ni$^{2+}$.

The input file is:

Energy_level_diagram.Quanty
Verbosity(0)

-- In order to understand the physics / chemistry of a system it is often good
-- to make energy level diagrams. i.e. plot the eigen-state energy as a function
-- of some parameter one varies.

-- Here we create the energy level diagram of all 190 states of NiO in the ligand
-- field approximation as a function of the Ni onsite crystal-field strenght

NF=20
NB=0
IndexDn_3d={ 0, 2, 4, 6, 8}
IndexUp_3d={ 1, 3, 5, 7, 9}
IndexDn_Ld={10,12,14,16,18}
IndexUp_Ld={11,13,15,17,19}

-- 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_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)

-- 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+U
--
-- If we relate this to the onsite energy of the p 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)
-- ep = nd*((1+nd)*U/2-Delta)/(10+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)
U       =  7.3
Delta   =  4.7
-- parameters obtained from DFT (PRB 85, 165113 (2012))
F2dd    = 11.142
F4dd    =  6.874
tenDq   =  0.56
tenDqL  =  1.44
Veg     =  2.06
Vt2g    =  1.21
zeta_3d =  0.081
Bz      =  0.000001

ed      = (10*Delta-nd*(19+nd)*U/2)/(10+nd)
eL      = nd*((1+nd)*U/2-Delta)/(10+nd)
F0dd    = U+(F2dd+F4dd)*2/63

Hamiltonian0 =  F0dd*OppF0_3d + F2dd*OppF2_3d + F4dd*OppF4_3d + zeta_3d*Oppldots_3d + Bz*(2*OppSz_3d + OppLz_3d)
+ tenDqL*OpptenDq_Ld + Veg * OppVeg + Vt2g * OppVt2g
+ ed * OppN_3d + eL * OppN_Ld

Npsi=190
StartRestrictions = {NF, NB, {"1111111111 0000000000",8,8}, {"0000000000 1111111111",10,10}}
psiList = Eigensystem(Hamiltonian0, StartRestrictions, Npsi)

file = assert( io.open("EnergyLevelDiagram", "w"))

for i=0, 30 do
tenDq = 0.1*i
file:write(string.format("%14.7E ",tenDq))
Hamiltonian=Hamiltonian0 + tenDq * OpptenDq_3d
Eigensystem(Hamiltonian, psiList)
for key,value in pairs(psiList) do
energy = value * Hamiltonian * value
file:write(string.format("%14.7E ",energy))
end
file:write("\n")
end

file:close()

gnuplotInput = [[
set autoscale
set xtic auto
set ytic auto
set style line  1 lt 1 lw 1 lc rgb "#000000"

set xlabel "10Dq (eV)" font "Times,12"
set ylabel "Energy (eV)" font "Times,12"

set out 'EnergyLevelDiagram.ps'
set size 1.0, 1.0
set terminal postscript portrait enhanced color  "Times" 8

plot for [i=2:191] "EnergyLevelDiagram" using 1:i notitle with lines ls  1
]]

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

-- call gnuplot to execute the script
os.execute("gnuplot EnergyLevelDiagram.gnuplot")
-- change the postscript file to pdf or eps
os.execute("ps2pdf EnergyLevelDiagram.ps ; ps2eps EnergyLevelDiagram.ps ; mv EnergyLevelDiagram.eps temp.eps ; eps2eps temp.eps EnergyLevelDiagram.eps ; rm temp.eps")

As in example 4 Quanty returns a nice plot. Note that one can add labeling. For this have a look at example 4.

Energy level diagram showing the energies of the different multiplets as a function of $10Dq$ The script does not write to standard output.

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