Differences

This shows you the differences between two versions of the page.

Link to this comparison view

documentation:tutorials:nio_ligand_field:energy_level_diagram [2016/10/10 09:41] (current)
Line 1: Line 1:
 +{{indexmenu_n>​3}}
 +====== 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:
 +<code Quanty 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"​)
 +</​code>​
 +###
 +
 +As in example 4 Quanty returns a nice plot. Note that one can add labeling. For this have a look at example 4.
 +| {{:​documentation:​tutorials:​nio_ligand_field:​energyleveldiagram.png?​nolink |}} |
 +^Energy level diagram showing the energies of the different multiplets as a function of $10Dq$ ^
 +
 +
 +###
 +The script does not write to standard output.
 +###
 +
 +
 +===== Table of contents =====
 +{{indexmenu>​.#​1|msort}}
Print/export